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--- abstract: 'So far, privacy models follow two paradigms. The first paradigm, termed *inferential privacy* in this paper, focuses on the risk due to statistical inference of sensitive information about a target record from other records in the database. The second paradigm, known as *differential privacy*, focuses on the risk to an individual when included in, versus when not included in, the database. The contribution of this paper consists of two parts. The first part presents a critical analysis on differential privacy with two results: (i) the differential privacy mechanism does not provide inferential privacy, (ii) the impossibility result about achieving Dalenius’s privacy goal [@Dwork06] is based on an adversary simulated by a Turing machine, but a *human* adversary may behave differently; consequently, the practical implication of the impossibility result remains unclear. The second part of this work is devoted to a solution addressing three major drawbacks in previous approaches to inferential privacy: lack of flexibility for handling variable sensitivity, poor utility, and vulnerability to auxiliary information.' author: - | Ke Wang, Peng Wang\ [email protected],[email protected]\ Simon Fraser University - | Ada Waichee Fu\ [email protected]\ Chinese University of Hong Kong - | Raywong Chi-Wing Wong\ [email protected]\ Hong Kong University of Science and Technology title: 'Inferential or Differential: Privacy Laws Dictate' --- Introduction ============ There has been a significant interest in the analysis of data sets whose individual records are too sensitive to expose directly. Examples include medical records, financial data, insurance data, web query logs, user rating data for recommender systems, personal data from social networks, etc. Data of this kind provide rich information for data analysis in a variety of important applications, but access to such data may pose a significant risk to individual privacy, as illustrated in the following example. \[example1\] A hospital maintains an online database for answering count queries on medical data like the table $T$ in Table \[tab:example1\]. $T$ contains three columns, Gender, Zipcode, and Disease, where Disease is a sensitive attribute. Suppose that an adversary tries to infer the disease of an individual Alice, with the background knowledge that Alice, a female living in the area with Zipcode 61434, has a record in $T$. The adversary issues the following two queries $Q_1$ and $Q_2$: $Q_1$: SELECT COUNT(\*) FROM T WHERE Gender=F AND Zipcode=61434 $Q_2$: SELECT COUNT(\*) FROM T WHERE Gender=F AND Zipcode=61434 AND Disease=HIV Each query returns the number of participants (records) who match the description in the WHERE clause. Suppose that the answers for $Q_1$ and $Q_2$ are $x$ and $y$, respectively. The adversary then estimates that Alice has HIV with probability $y/x$, and if $y/x$ and $x$ are “sufficiently large", there will be a privacy breach. Gender Zipcode Disease -------- --------- ------------- M 54321 Brain Tumor M 54322 Indigestion F 61234 Cancer F 61434 HIV ... ... ... : A table $T$[]{data-label="tab:example1"} Inferential vs Differential --------------------------- In the above example, the adversary infers that the rule $$(Gender=F \wedge Zipcode=61434) \rightarrow (Disease=HIV)$$ holds with the probability $y/x$ and that Alice has HIV with the probability $y/x$, assuming that the (diseases of) records follow some underlying probability distribution. This type of reasoning, which learns information about *one record* from the statistics of *other records*, is found in many advanced applications such as recommender systems, prediction models, viral marketing, social tagging, and social networks. The same technique could be misused to infer sensitive information about an individual like in the above example. According to the *Privacy Act* of Canada, publishing the above query answers would breach Alice’s privacy because they disclose Alice’s disease with a high accuracy. In this paper, *inferential privacy* refers to the requirement of limiting the statistical inference of sensitive information about a target record from other records in the database. See [@Adam89] for a list of works in this field. One recent breakthrough in the study of privacy preservation is *differential privacy* [@Dwork06][@springerlink:10.1007/11681878.14]. In an “impossibility result", the authors of [@Dwork06][@springerlink:10.1007/11681878.14] showed that it is impossible to achieve Dalenius’s absolute privacy goal for statistical databases: anything that can be learned about a respondent from the statistical database should be learnable without access to the database. Instead of limiting what can be learnt about one record from other records, the differential privacy mechanism hides the presence or absence of a participant in the database, by producing noisy query answers such that the distribution of query answers changes very little when the database differs in any *single* record. The following definition is from [@Blum08]. A randomized function $K$ gives $\varepsilon$-differential privacy if for all data sets $T$ and $T'$ differing on at most one record, for all queries $Q$, and for all outputs $x$, $Pr[K(T,Q) = x] \leq exp(\varepsilon) Pr[K(T',Q) = x]$. With a small $\varepsilon$, the presence or absence of an individual is hidden because $T$ and $T'$ are almost equally likely to be the underlying database that produces the final output of the query. Some frequently cited claims of the differential privacy mechanism are that it provides privacy without any assumptions about the data and that it protects against arbitrary background information. But there is no free lunch in data privacy, as pointed out by Kifer and Machanavajjhala recently [@Kifer2011]. Their study shows that assumptions about the data and the adversaries are required if hiding the *evidence* of participation, instead of the presence/absence of records in the database, is the privacy goal, which they argue should be a major privacy definition. Contributions ------------- The contribution of this paper consists of two parts. In the first part, we argue that differential privacy is insufficient because it does not provide inferential privacy. We present two specific results: - (Section 2.1) Using a *differential inference theorem*, we show that the noisy query answers returned by the differential privacy mechanism may derive an inference probability that is arbitrarily close to the inference probability obtained from the noise-free query answers. This study suggests that providing inferential privacy remains a meaningful research problem, despite the protection of differential privacy. - (Section 2.2) While the impossibility result in [@Dwork06] is based on an adversary simulated by a Turing machine, a *human* adversary may behave differently when evaluating the sensitivity of information. We use the Terry Gross example, which is a key motivation of differential privacy, to explain this point. This study suggests that the practical implication of the impossibility result remains unclear. Given that inferential privacy remains relevant, the second part of this work is devoted to stronger solutions for inferential privacy. Previous approaches suffer from three major limitations. Firstly, most solutions are unable to handle sensitive values that have skewed distribution and varied sensitivity. For example, with the Occupation attribute in the Census data (Section 7) having the minimum and maximum frequency of 0.18% and 7.5%, the maximum $\ell$-diversity [@MKG+006] that can be provided is $13$-diversity because of the eligibility requirement $1/\ell \geq 7.5\%$ [@XT06b]. Therefore, it is impossible to protect the infrequent items at the tail of the distribution or more sensitive items by a larger $\ell$-diversity, say $50$-diversity, which is more than 10 times the prior 0.18%. Secondly, even if it is possible to achieve such $\ell$-diversity, enforcing $\ell$-diversity with a large $\ell$ across *all* sensitive values leads to a large information loss. Finally, previous solutions are vulnerable to additional auxiliary information [@TaoCorr08][@Kifer:2009:APD:1559845.1559861][@Li09]. We address these issues in three steps. - (Section 3) To address the first two limitations in the above, we consider a sensitive attribute with domain values $x_1,\cdots, x_m$ such that each $x_i$ has a different sensitivity, thus, a tolerance $f'_i$ on inference probability. We consider a bucketization problem in which buckets of *different* sizes can be formed to accommodate different requirements $f'_i$. The goal is to find a collection of buckets for a given set of records so that a notion of information loss related to bucket size is minimized and the privacy constraint $f'_i$ of all $x_i$’s is satisfied. - (Sections 4, 5, and 7) We present an efficient algorithm for the case of two distinct bucket sizes (but many buckets) with *guaranteed optimality*, and a heuristic algorithm for the general case. The empirical study on real life data sets shows that both solutions are good approximations of optimal solutions in the general case and better deal with a sensitive attribute of skewed distribution and varied sensitivity. - (Section 6) We adapt our solutions to guard against two previously identified strong attacks, corruption attack [@TaoCorr08] and negative association attack [@Kifer:2009:APD:1559845.1559861][@Li09] (see more details in Section 6). Related Work ------------ Limiting statistical disclosure has been a topic extensively studied in the field of statistical databases, see [@Adam89] for a list of works. This problem was recently examined in the context of privacy preserving data publishing and some representative privacy models include $\rho_1$-$\rho_2$ privacy [@Evfimievski:2003:LPB:773153.773174], $\ell$-diversity principle [@MKG+006], and $t$-closeness[@N07]. All of these works assume uniform sensitivity across all sensitive values. One exception is the personalized privacy in [@XT06a] where a record owner can specify his/her privacy threshold. Another exception is [@LW10] where each sensitive value may have a different privacy setting. To achieve the privacy goal, these works require a taxonomy of domain values to generalize the attributes, thus, cannot be applied if such taxonomy is not available. The study in [@XT06b] shows that generalized attributes are not useful for count queries on raw values. Dealing with auxiliary information is a hard problem in data privacy [@TaoCorr08][@Kifer:2009:APD:1559845.1559861][@Li09], and so far there is little satisfactory solution. There have been a great deal of works in differential privacy since the pioneer work [@springerlink:10.1007/11681878.14][@Dwork06]. This includes, among others, contingency table releases [@Barak:2007:PAC:1265530.1265569], estimating the degree distribution of social networks [@Hay09], histogram queries [@Hay10] and the number of permissible queries [@XT08]. These works are concerned with applications of differential privacy in various scenarios. Unlike previous works, the authors of [@Kifer2011] argue that hiding the evidence of participation, instead of the presence/absence of records in the database, should be a major privacy definition, and this privacy goal cannot be achieved with making assumptions about the data and the adversaries. Analyzing Differential Privacy {#differential} ============================== This section presents a critical analysis on the differential privacy mechanism. In Section \[independent\] we show that the differential privacy mechanism allows violation of inferential privacy. In Section \[Gross\] we argue that a human adversary may behave differently from some assumptions made in the impossibility result of [@Dwork06], thus, the practical implication of the impossibility result remains unclear. On Violating Inferential Privacy {#independent} -------------------------------- One popularized claim of the differential privacy mechanism is that it protects an individual’s information even if an attacker knows about all other individuals in the data. We quote the original discussion from [@Blum:2005:PPS:1065167.1065184] (pp 3): > “If there is information about a row that can be learned from other rows, this information is not truly under the control of that row. Even if the row in question were to sequester itself away in a high mountaintop cave, information about the row that can be gained from the analysis of other rows is still available to an adversary. It is for this reason that we focus our attention on those inferences that can be made about rows without the help of others." In other words, the differential privacy framework does not consider violation to inferential privacy and the reason is that it is not under the control of the target row. Two points need clarification. Firstly, a user submits her sensitive data to an organization because she trusts that the organization will do everything possible to protect her sensitive information; indeed, the data publisher has full control in how to release the data or query answers in order to protect individual privacy. Secondly, learning information about one record from other records could pose a risk to an individual if the learnt information is accurate about the individual. This type of learning assumes that records follow some underlying probability distribution, which is widely adapted by prediction models in many real applications. Under this assumption, suppose $Q_1$ and $Q_2$ in Example \[example1\] have the answers $x=100$ and $y=99$, even if Alice’s record is removed from the database, it is still valid to infer that Alice has HIV with a high probability. Next, we show that even if the differential privacy mechanism adds noises to the answers for queries $Q_1$ and $Q_2$, Alice’s disease can still be inferred using the noisy answers. Let $x$ and $y$ be the true answers to $Q_1$ and $Q_2$. We assume that $x$ and $y$ are non-zero. The differential privacy mechanism will return the noisy answers $X=x+\xi_1$ and $Y=y+\xi_2$ for $Q_1$ and $Q_2$, after adding noises $\xi_1$ and $\xi_2$. Consider the most used Laplace distribution $Lap(b)=\frac{1}{2b} exp(-|\xi|/b)$ for the noise $\xi$, where $b$ is the scale factor. The mean $E[\xi]$ is zero and the variance $var[\xi]$ is $2b^2$. The next theorem is due to [@Dwork06]. [@Dwork06] \[th:countquery\] For a count query $Q$, the mechanism $K$ that adds independently generated noise $\xi$ with distribution $Lap(1/\varepsilon)$ to the output enjoys $\varepsilon$-differential privacy. The next theorem shows that $Y/X$ is a good approximation of $y/x$. \[variance\] Given two queries $Q_1$ and $Q_2$ as above, let $x$ and $y$ be the true answers and let $X$ and $Y$ be the answers returned by the $\varepsilon$-differential privacy mechanism. $E[\frac{Y}{X}]=\frac{y}{x}(1+\frac{2b^2}{x^2})$ and $var[\frac{Y}{X}]=\frac{2b^2}{x^2}(1+(\frac{y}{x})^2)$, where $b=1/\varepsilon$. Using the Taylor expansion technique [@Johnson80] [@Stuart98], the mean $E[\frac{Y}{X}]$ and variance $var[\frac{Y}{X}]$ of $Y/X$ can be approximated as follows: $$E[\frac{Y}{X}]\simeq \frac{E[Y]}{E[X]}+\frac{cov[X,Y]}{E[X]^2} + \frac{var[X]E[Y]}{E[X]^3}$$ $$var[\frac{Y}{X}]\simeq \frac{var[Y]}{E[X]^2} - \frac{2E[Y]}{E[X]^3} cov[X,Y] + \frac{E[Y]^2}{E[X]^4} var[X] \label{var}$$ $E[X]$ and $E[Y]$ are equal to the true answers $x$ and $y$ of $Q_1$ and $Q_2$. $var[X]$ and $var[Y]$ are $2b^2$ for $Lap(b)$. $cov[X,Y]=cov[x+\xi_1,y+\xi_2]=cov[\xi_1,\xi_2]$. Since $\xi_1$ and $\xi_2$ are unrelated, $cov[\xi_1,\xi_2]=0$. Simplifying the above equations, we get $E[\frac{Y}{X}]$ and $var[\frac{Y}{X}]$ as required. The next corollary follows from the fact that $\frac{y}{x}\leq 1$ and $b$ is a constant for a given $\varepsilon$-differential privacy mechanism $K$. \[c1\] Let $X,Y$ be defined as in Theorem \[variance\]. As the query size $x$ for $Q_1$ increases, $E[\frac{Y}{X}]$ gets arbitrarily close to $\frac{y}{x}$ and $var[\frac{Y}{X}]$ gets arbitrarily close to zero. Corollary \[c1\] suggests that $Y/X$, where $Y$ and $X$ are the noisy query answers returned by the differential privacy mechanism, can be a good estimate of the inference probability $y/x$ for a large query answer $x$. For example, for $\varepsilon=0.1$ and $x=100$, $\frac{2b^2}{x^2}=0.02$, and following Theorem \[variance\], $E[\frac{Y}{X}]$ is 1.02 times $\frac{y}{x}$; if $x=1000$, $E[\frac{Y}{X}]$ is 1.0002 times $\frac{y}{x}$. If $y/x$ is high, inferential privacy is violated. Note that $var[\frac{Y}{X}]$ is small in these cases. On The Impossibility Results {#Gross} ---------------------------- A key motivation behind differential privacy is the impossibility result about the Dalenius’s privacy goal [@Dwork06]. Intuitively, it says that for any privacy mechanism and any distribution satisfying certain conditions, there is always some particular piece of auxiliary information, $z$, so that $z$ alone is useless to an adversary who tries to win, while $z$ in combination with access to the data through the privacy mechanism permits the adversary to win with probability arbitrarily close to 1. The proof assumes an adversary simulated by a Turing machine. We argue that a *human* adversary, who also considers the “semantics" when evaluating the usefulness of information, may behave differently. Let us explain this point by the Terry Gross example that was originally used to capture the intuition of the impossibility result in [@Dwork08]. In the Terry Gross example, the exact height is considered private, thus, useful to an adversary, whereas the auxiliary information of being two inches shorter than an unknown average is considered not private, thus, not useful. Under this assumption, accessing the statistical database, which returns the average height, is to blame for disclosing Terry Gross’s privacy. Mathematically, knowing the exact height is a remarkable progress from knowing two inches shorter than an unknown average. However, to a *human* adversary, the information about how an individual *deviates from the statistics* already discloses the sensitive information, regardless of what the statistics is. For example, once knowing that someone took the HIV check-up ten times more frequently than an unknown average, his/her privacy is already leaked. Here, a human adversary is able to interpret “deviation" as a sensitive notion based on “life experiences", even though mathematically deviation does not derive the exact height. It is unclear whether such a human adversary can be simulated by a Turing machine. In practice, a realistic privacy definition does allow disclosure of sensitive information in a controlled manner and there are scenarios where it is possible to protect inferential privacy while retaining a reasonable level of data utility. For example, the study in [@FWY05] shows that the anonymized data is useful for training a classifier because the training does not depend on detailed personal information. Another scenario is when the utility metric is different from the adversary’s target. Suppose that the attribute $Disease$ is sensitive and the response attribute $R$ (to a medicine) is not. Learning the following rules does not violate privacy $(Disease=x_1) \rightarrow (R=Positive)$\ $(Disease=x_2) \rightarrow (R=Positive)$ in that a positive response does not indicate a specific disease with certainty. However, these rules are useful for a researcher to exclude the diseases $x_1$ and $x_2$ in the absence of a positive response. Even for a sensitive attribute like $Disease$, the varied sensitivity of domain values (such as Flu and HIV) could be leveraged to retain more utility for less sensitive values while ensuring strong protection for highly sensitive items. In the rest of the paper, we present an approach of leveraging such varied sensitivity to address some drawbacks in previous approaches to inferential privacy.
--- abstract: 'Motivated by the linear time algorithm that locates the eigenvalues of a cograph $G$ [@JTT2016], we investigate the multiplicity of eigenvalue $\lambda$ for $ \lambda \neq 0,-1.$ For cographs with [*balanced cotrees*]{} we determine explicitly the highest value for the multiplicity. The energy of a graph is defined as the sum of absolute values of the eigenvalues. A graph $G$ on $n$ vertices is said to be borderenergetic if its energy equals the energy of the complete graph $K_n.$ We present families of non-cospectral and borderenergetic cographs.' --- [**Multiplicity of eigenvalues of cographs**]{} [**Luiz Emilio Allem$^a,$ Fernando Tura$^{b,}$ [^1]** ]{} *$^a$ Instituto de Matemática, UFRGS, Porto Alegre, RS, 91509-900, Brazil\ [email protected]* *$^b$ Departamento de Matemática, UFSM, Santa Maria, RS, 97105-900, Brazil\ [email protected]* \#1 =0.30in Introduction {#intro} ============ We recall that the *spectrum* of a graph $G$ is the multiset of the eigenvalues of its adjacency matrix. The main goal of this paper is to discuss the multiplicity of eigenvalues of cographs. Cographs is an important class of graphs for its many applications. They have several alternative characterizations, for example, a cograph is graph which contains no path of length four as an induced subgraph [@Stewart] and because of this they are often simply called $P_4$ [*free*]{} graph in the literature. In particular it well known that any cograph has a canonical tree representation, called the [*cotree*]{} [@BSS2011]. The cotree will be relevant to this paper and will be described later. Our original motivation for considering cographs is to study the distribution of eigenvalues of graphs. It is known, for example, that any interval of the real line contains some eigenvalues of graphs, since, more generally, any root of a real-rooted monic polynomial with integer coefficients occurs as an eigenvalue of some tree [@Sal2015]. On the other hand it was proved (see [@Moha]) that no cograph has eingenvalues in the interval $(-1,0)$, a surprising result. In this paper, we turn to study the multiplicities of eigenvalues of cographs. In [@JTT2013] it was proved that all eigenvalues of threshold graphs (a subclass of cographs), except $-1$ and $0$ are simple. This motivates us to investigate further the multiplicities of cograph eigenvalues. Since the multiplicities of the eigenvalues $-1$ and $0$ are known [@BSS2011] we deal with eigenvalues that are different from 0 and -1. The multiplicities of graph eigenvalues are extensively studied by several authors. Bell [*et al.*]{} [@Bell] determined upper bound for the multiplicities of graphs. Later, Rowlinson in [@Row] studied the multiplicities of eigenvalues in trees. Recently, Bu [*et al.*]{} [@Bu] studied the multiplicities in graphs attaching one pendent path, generalizing some known results for trees and unicyclic graphs [@Row]. Different from the star complement technique used in the works above, our technique is based on an algorithm called [*Diagonalization*]{}, presented in [@JTT2016]. The Diagonalization finds, in $O(n)$ time, the number of eigenvalues of a cograph, by operating directly on the cotree of the cograph. The algorithm and the technique will be explained in the next section. We study cographs whose cotree is *balanced* (see definition in Section \[multi\]) and determine the multiplicity of some eigenvalues and an upper bound for the multiplicity of other eigenvalues. As an application of these results, we study the *energy* of families of cographs. Recall that if $G$ is a graph having eigenvalues $\lambda_1, \ldots, \lambda_n,$ its energy, denoted $E(G)$ is defined [@Gutman2015; @Gutman2012] as $ \sum_{i=1}^n | \lambda_i |.$ There are many results on energy and its applications in several areas, including in chemistry see [@Gutman2012] for more details and the references therein. It is well known that the complete graph $K_n$ has $E(K_n) = 2n-2$ and it is a natural and important research problem to determine graphs that have the same energy of the complete graph $K_n.$ A graph $G$ on $n$ vertices is said to be *borderenergetic* if its energy equals the energy of the complete graph $K_n.$ Some recent results on borderenergetic graphs are the following. In [@Gutman2015], it was shown that there exists borderenergetic graphs on order $n$ for each integer $n\geq 7,$ and all borderenergetic graphs with $7,8,$ and $9$ vertices were determined. In [@JTT2015] it was considered the classes of borderenergetic threshold graphs. For each $n\geq 3,$ it was determined $n-1$ threshold graphs on $n^2$ vertices, pairwise non-cospectral and equienergetic to the complete graph $K_{n^2}.$ Recently, Hou and Tao [@Hou], showed that for each $n\geq 2$ and $p\geq 1$ $(p \geq 2$ if $n=2),$ there are $n-1$ threshold graphs on $pn^2$ vertices, pairwise non-cospectral and equienergetic with the complete graph $K_{pn^2},$ generalizing the results in [@JTT2015]. In this paper, we continue this investigation in the class of cographs. More precisely, we determine two infinite families of cographs that are borderenergetic. Here is an outline of the remainder of this paper. In Section 2, we mention the representation of cographs by a cotree and explain the Diagonalization algorithm. In Section 3, we determine explicitly the multiplicity $m(\lambda)$ for some classes of cographs, except $0,-1$ and an upper bound for the remaining eigenvalues. In Section 4, as application, we present two families of integral non-cospectral and borderenergetic cographs. Notation and Preliminaries {#pre} ========================== Let $G= (V,E)$ be an undirected graph with vertex set $V$ and edge set $E,$ without loops or multiple edges. We denote the [*open neighborhood*]{} of $v,$ by $$N(v)=\{w|\{v,w\}\in E\}$$ and its [*closed neighborhood*]{} by $$N[v] = N(v) \cup \{v\}.$$ The [*adjacency matrix*]{} of $G$, denoted by $A=[a_{ij}]$, is a matrix whose rows and columns are indexed by the vertices of $G$, and is defined to have entries $$a_{ij} = \left\{ \begin{array}{rl} 1 &\mbox{ if $v_{i}v_{j}\in E$} \\ 0 &\mbox{ otherwise} \end{array} \right.$$ A value $\lambda$ is an [*eigenvalue*]{} of $G$ if $\det(A - \lambda I_n ) = 0$, and since $A$ is real and symmetric, its eigenvalues are real numbers. We denote $m(\lambda)$ the multiplicity of the eigenvalue $\lambda$ of $A.$ Cotrees ------- A cograph has been rediscovered independently by several authors since the 1960’s. Corneil, Lerchs and Burlingham [@Stewart] define cographs recursively by the following rules: 1. a graph on a single vertex is a cograph, 2. a finite union of cographs is a cograph, 3. a finite join of cographs is a cograph. In this note, we focus on representing the recursive construction of a cograph using its cotree, that we describe below. A cotree $T_G$ of a cograph $G$ is a rooted tree in which any interior vertex $w$ is either of $\cup$ type (corresponding to disjoint union) or $\otimes$ type (corresponding to join). The terminal vertices (leaves) are typeless and represent the vertices of the cograph $G.$ We say that the [*depth*]{} of the cotree is the number of edges of the longest path from the root to a leaf. To build a cotree for a connected cograph, we simply place a $\otimes$ at the tree’s root, placing $\cup$ on interior vertices with odd depth, and placing $\otimes$ on interior vertices with even depth. To build a cotree for a disconnected cograph, we place $\cup$ at the root, and place $\otimes$ at odd depths, and $\cup$ at even depths. All interior vertices have at least two children. In [@BSS2011] this structure is called [*minimal cotree*]{}, but throughout this paper we call it simply a cotree. Figure \[cotree\] shows a cograph and its cotree with depth equals to 4. \[scale=0.65,auto=left,every node/.style=[circle]{}\] i/in [1/,2/,3/,4/,5/,6/,7/,8/,9/]{}[ (i) at ([360/9 \* (i- 1)+90]{}:3) ;]{} in [3,5,6,7,9]{}[ in [1,2,...,]{} () – ();]{} in [8]{}[ in [1,2,3,4,]{} () – () ;]{} \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (p) at (-2,3) ; (o) at (3,4) ; (n) at (2,4) ; (q) at (1,4) ; (m) at (2,5) [$\otimes$]{}; (l) at (3,5) ; (k) at (3,6) ; (j) at (2,6) [$\cup$]{}; (h) at (2,7) [$\otimes$]{}; (g) at (0.85,6) [$\cup$]{}; (f) at (0.9,5) ; (a) at (0,5) [$\otimes$]{}; (b) at (-1,4) [$\cup$]{}; (c) at (0,4) ; (d) at (-1,3) ; (a) edge node\[left\] (b) (a) edge node\[below\] (c) (b) edge node\[left\] (d) (f) edge node\[right\](g) (g) edge node\[left\](a) (h) edge node\[right\](j) (h) edge node\[left\](g) (h) edge node\[left\](k) (j) edge node\[right\](l) (j) edge node\[below\](m) (m) edge node\[below\](n) (m) edge node\[right\](o) (b) edge node\[left\] (p) (m) edge node\[left\] (q); Two vertices $u$ and $v$ are [*duplicates*]{} if $N(u) = N(v)$ and [*coduplicates*]{} if $N[u]=N[v].$ In a cograph, any collection of mutually coduplicates (resp. duplicates) vertices, e.g. with the same neighbors and adjacent (resp. not adjacent) have a common parent of type $ \otimes$ (resp. $\cup$). In Figure \[cotree\], for example, we have that $v_{1}$ and $v_{2}$ are duplicates because $N(v_{1})=N(v_{2})$, while $v_5, v_6$ and $v_7$ are coduplicates. In fact, a recursive characterization of cographs in terms of the vertex duplication and co-duplication operations is given in [@Moha]. Diagonalization {#sub:diag} --------------- An algorithm for constructing a [*diagonal*]{} matrix congruent to $A + {x}I$, where $A$ is the adjacency matrix of a cograph, and ${x}$ is an arbitrary scalar, using $O(n)$ time and space was developed in [@JTT2016]. This algorithm will be the main tool of this article and, hence, we will make a brief review of the method. For more information, see [@JTT2016]. The algorithm’s input is the cotree $T_G$ and $x.$ Each leaf $v_i, i =1, \ldots,n$ has a value $d_i$ that represents the diagonal element of $A+xI.$ It initializes all entries $d_i$ with $x.$ Even though the operations represent rows and columns operations on the matrix $A+xI$, the algorithm is performed on the cotree itself and matrix is never actually used. In each iteration of the procedure, a pair $\{v_k, v_l\} $ of duplicates or coduplicates vertices with maximum depth is selected. Then the pair is processed, that is, assignments are given to $d_k$ and $d_l,$ such that either one or both rows (columns), corresponding to this vertices, are diagonalized. When a $k$ row(column) corresponding to vertex $v_k$ has been diagonalized then $v_k$ is removed from the cotree $T_G,$ it means that $d_k$ has a permanent final value. Then the algorithm moves to the cotree $T_G -v_k.$ The algorithm is shown in Figure \[algo\]. Now, we will present a few results from [@JTT2016] that will be used throughout the note. The following theorem is based on Sylvester’s Law of Inertia. [@JTT2016] \[mainB\] Let $D=[d_1,d_2,\ldots,d_n]$ be the diagonal returned by the diagonalization algorithm $(T_G,-x),$ and assume $D$ has $k_{+}$ positive values, $k_0$ zeros and $k_{-}$ negative values. The number of eigenvalues of $G$ that are greater than ${x}$ is exactly $k_{+}$. The number of eigenvalues of $G$ that are less than ${x}$ is exactly $k_{-}$. The multiplicity of ${x}$ is $k_{0}$. The following two lemmas show that, under certain conditions, we can control the assignments made at each iteration. [@JTT2016] \[lem1\] If $v_1, \ldots, v_m$ have parent $w= \otimes,$ each with diagonal value $y \neq1,$ then the algorithm performs $m-1$ iterations of [[**subcase 1a**]{}]{}  assigning, during iteration $j:$ $$d_k \leftarrow \frac{j+1}{j}(y-1)$$ $$d_l \leftarrow \frac{y+j}{j+1}$$ [@JTT2016] \[lem2\] If $v_1, \ldots, v_m$ have parent $w= \cup,$ each with diagonal value $y\neq 0,$ then the algorithm performs $m-1$ iterations of [[**subcase 2a**]{}]{}  assigning, during iteration $j:$ $$d_k \leftarrow \frac{(j+1)}{j}y$$ $$d_l \leftarrow \frac{y}{j+1}$$ The next three lemmas show that if we start an iteration with some known value then we can control the exit values. [@JTT2016] \[lem3\] If $\{v_k, v_l \}$ is a pair of coduplicate vertices processed by Diagonalization with assignments $0 \leq d_k, d_l < 1,$ then $d_k$ becomes permanently negative, and $d_l$ is assigned a value in $(0,1).$ \[lem4\] If $\{v_k, v_l \}$ is a pair of duplicate vertices processed by Diagonalization with the assignments $0 < d_k, d_l \leq 1,$ then $d_k$ becomes permanently positive, and $d_l$ is assigned a value in $(0,1).$ [**Proof:**]{} We notice that the algorithm executes [[**subcase 2a**]{}]{} , meaning that $d_k=\alpha + \beta >0$ and $d_l= \alpha\beta/(\alpha+\beta).$ The fact that $d_l>0$ is obvious. To see that $d_l < 1$, we observe that if $\alpha = \beta=1$, then $d_k =1/2$. If either (but not both) $\alpha$ or $\beta = 1$, then it is clear that $d_l =\alpha /(\alpha+1) < 1$. Now if $0 < \alpha ,\beta < 1$, then $d_l < 1$ follows from Lemma 3 of [@Chang08]. \[lem5\] During the execution of Diagonalize $(T_G, x)$ with $x \in (0,1),$ all diagonal values of vertices remaining on the cotree are in $(0,1).$ Furthermore, if $d_k$ corresponds to a permanent value of a removed vertex on $T_G -v_k,$ then $d_k \neq 0.$ [**Proof:**]{} Let $G$ be a cograph and $T_G$ its cotree. Initially all vertices on $T_G$ are in $(0,1).$ Suppose after $m$ iterations of Diagonalize all diagonal values of the cotree are in $(0,1)$ and no zero is assigned. Now consider iteration $m+1$ with a pair $\{ v_k, v_l\}$ and parent $w.$ If $w= \otimes$ then Lemma \[lem3\] guarantees the vertex $d_l$ remaining on the cotree is assigned a value in $(0,1)$ and the vertex $d_k$ is assigned a permanently negative value. If $w= \cup$ then Lemma \[lem4\] guarantees the vertex $d_l$ remaining on the cotree is assigned a value in $(0,1)$ and the vertex $d_k$ is assigned a permanently positive value, completing the proof. The next result follows from Lemma \[lem5\]. No cograph $G$ has eigenvalue in the interval $(-1,0).$ \[scale=0.8,auto=left,every node/.style=[circle,scale=0.7]{}\] \(o) at (2.3,4) ; (n) at (1.5,4) ; (q) at (0.75,4) ; (v) at (3,4) ; (v2) at (3.5,4) ; (z) at (4.2,4) ; (z2) at (4.8,4) ; (y3) at (5.4,4) ; (y4) at (6,4) ; (m) at (1.7,5) [$\otimes$]{}; (l) at (3,5) [$\otimes$]{}; (y) at (4.6,5) [$\otimes$]{}; (y2) at (5.4,5) [$\otimes$]{}; (k) at (4.6,6) [$\cup$]{}; (j) at (2,6) [$\cup$]{}; (h) at (2,7) [$\otimes$]{}; (g) at (0,6) [$\cup$]{}; (f) at (0.6,5) [$\otimes$]{}; (a) at (-1,5) [$\otimes$]{}; (b) at (-1.4,4) ; (c) at (-0.75,4) ; (e) at (0,4) ; \(a) edge node\[left\] (b) (a) edge node\[below\] (c) \(f) edge node\[below\] (e) (k) edge node\[below\] (y) (k) edge node\[below\] (y2) (y) edge node\[below\] (z) (y) edge node\[below\] (z2) (y2) edge node\[below\] (y3) (y2) edge node\[below\] (y4) (l) edge node\[below\] (v) (l) edge node\[below\] (v2) (f) edge node\[right\](g) (g) edge node\[left\](a) (f) edge node\[left\](q) (h) edge node\[right\](j) (h) edge node\[left\](g) (h) edge node\[left\](k) (j) edge node\[right\](l) (j) edge node\[below\](m) (m) edge node\[below\](n) (m) edge node\[right\](o) ; On the multiplicities of eigenvalues in balanced cotrees {#multi} ======================================================== In this section we study the eigenvalues of cographs that have balanced cotrees. We say that a cograph $G$ has a *balanced cotree* $T_G$ with depth $r$ if every interior vertex with depth $i$ in $T_G$ has the same number of interior vertices and the same number of leaves as direct successors, for $i \in \{1, \ldots,r-1\}$. We will use the notation $T_G ( a_1, \ldots, a_r | b_1, \ldots, b_r)$ to represent a balanced cotree of a cograph $G,$ where the root of $T_G$ has exactly $a_1$ immediate interior vertices and $b_1$ leaves. An interior vertex successor of the root has exactly $a_2$ immediate interior vertices and $b_2$ leaves, and so on. Thus, we will assume that $a_1, \ldots, a_{r-1}$ are positive integers and $a_{r}=0$. Additionally, we assume that $b_1, \ldots, b_{r-1}$ are non negative integer values and $b_{r}\geq 2$. Figure \[fig3\] shows the balanced cotree $T_G (3,2,0| 0,0,2).$ Regular balanced cotrees ------------------------ Here we will study eigenvalues of (regular) cographs $G$ that have balanced cotrees of the type $T_G (a_1, \ldots,a_{r-1}, 0| 0, \ldots, 0,b_r)$, whose order is $n =a_{1} a_2 \ldots a_{r-1} b_r$. We show in Figure \[Fig1\] a representation of general regular balanced cotree with odd $r$, meaning that level $r-1$ has vertices of type $\otimes$. \[scale=.8,auto=left,every node/.style=[circle,scale=0.7]{}\] \(a) at (3,8) [$\otimes$]{}; (b) at (2.5,7) [$\cup$]{}; (c) at (3.5,7) [$\cup$]{}; (2.7,7) – (3.2,7); (a) edge node\[left\] (b); (a) edge node\[left\] (c); \(d) at (3,6) [$\otimes$]{}; (e) at (4,6) [$\otimes$]{}; (3.2,6) – (3.8,6); (c) edge node\[left\] (d); (c) edge node\[left\] (e); (3,5.2) – (3,5.8); \(f) at (3,5) [$\otimes$]{}; (g) at (2.5,4) [$\cup$]{}; (h) at (3.5,4) [$\cup$]{}; (2.7,4) – (3.2,4); (f) edge node\[left\] (g); (f) edge node\[left\] (h); \(i) at (3,3) [$\otimes$]{}; (j) at (4,3) [$\otimes$]{}; (3.2,3) – (3.8,3); (h) edge node\[left\] (i); (h) edge node\[left\] (j); \(k) at (2.5,2) ; (l) at (3.5,2) ; (2.7,2) – (3.3,2); (i) edge node\[left\] (k); (i) edge node\[left\] (l); The next two theorems are known results and can be found, for example in [@BSS2011; @JTT2016]. \[Tre1\] Let G be a cograph with cotree $T_{G}$ having $\otimes$-nodes $\{w_{1},\ldots,w_{m}\}$, where $w_{i}$ has $t_{i} \geq 1$ terminal children. Then $m(-1) =\sum_{i=1}^{m}(t_{i}-1)$. \[Tre2\] Let G be a cograph with cotree $T_{G}$ having $\cup$-nodes $\{w_{1},\ldots,w_{m}\}$, where $w_{i}$ has $t_{i} \geq 1$ terminal children. If $G$ has $j\geq 0$ isolated vertices then $m(0)= j+\sum_{i=1}^{m}(t_{i}-1)$. Using the above results we can easily prove the next corollary. \[Cor1\] Let $G$ be a cograph with balanced cotree $T_G (a_1, \ldots,a_{r-1}, 0| 0, \ldots, 0,b_r)$ of order $n =a_{1} a_2 \ldots a_{r-1} b_r.$ 1. If $r$ is odd then $G$ has the eigenvalue $-1$ with multiplicity $a_1 a_2 \ldots a_{r-1}(b_{r} -1)$. 2. If $r$ is even then $G$ has the eigenvalue $0$ with multiplicity $a_1 a_2 \ldots a_{r-1}(b_{r} -1)$. \[Cor2\] Let $G$ be a balanced cotree $T_G (a_1, \ldots,a_{r-1}, 0| 0, \ldots,0, b_r)$ of a cograph $G$ of order $n =a_1 a_2 \ldots a_{r-1} b_r$. If $r$ is odd (even) then, counting multiplicities, the number of eigenvalues of $G$ other than $-1$ (0) is equal to $$a_1 a_2 \ldots a_{r-1}$$ [**Proof:**]{} Suppose $r$ is odd. Since $n=a_1 a_2 \ldots a_{r-1} b_r$ and $G$ has $a_1 a_2 \ldots a_{r-1}(b_r -1)$ coduplicates vertices, it follows that the number of eigenvalues that are distinct from $-1$ is equal to $n-a_1 a_2 \ldots a_{r-1}(b_r -1)=a_1 a_2 \ldots a_{r-1}.$ The case $r$ even is similar. \[lem6\] Let $G$ be a cograph with balanced cotree $T_G (a_1, \ldots,a_{r-1}, 0| 0, \ldots, 0,b_r)$ of order $n =a_{1} a_2 \ldots a_{r-1} b_r.$ 1. If $r$ is odd then $G$ has the eigenvalue $b_r -1$ with multiplicity $a_1 a_2 \ldots (a_{r-1} -1)$. 2. If $r$ is even then $G$ has the eigenvalue $-b_r $ with multiplicity $a_1 a_2 \ldots (a_{r-1} -1)$. \[scale=,5,auto=left,every node/.style=[circle,scale=0.7]{}\] \(f) at (3,5) [$\otimes$]{}; (g) at (2.5,4) [$\cup$]{}; (h) at (3.5,4) [$\cup$]{}; (2.7,4) – (3.2,4); (f) edge node\[left\] (g); (f) edge node\[left\] (h); \(i) at (3,3) [$\otimes$]{}; (j) at (4,3) [$\otimes$]{}; (3.2,3) – (3.8,3); (h) edge node\[left\] (i); (h) edge node\[left\] (j); \(k) at (3,2) ; (l) at (4,2) ; \(i) edge node\[left\] (k); (j) edge node\[left\] (l); (3.2,2) – (3.8,2); () at (3,1) ; () at (4,1) ; (3.2,1) – (3.8,1); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(f) at (3,5) [$\otimes$]{}; (g) at (2.5,4) [$\cup$]{}; (h) at (3.5,4) [$\cup$]{}; (2.7,4) – (3.2,4); (f) edge node\[left\] (g); (f) edge node\[left\] (h); \(i) at (3,3) ; (j) at (4,3) ; (3.2,3) – (3.8,3); (h) edge node\[left\] (i); (h) edge node\[left\] (j); () at (3,1) ; () at (4,1) ; (3.2,1) – (3.8,1); [**Proof:**]{} We assume that $r$ is odd. The case even is similar. Consider $ x= -(b_r -1)$ and execute the algorithm Diagonalization with input $(T_G, x).$ By Theorem \[mainB\] we have to prove that the algorithm creates $a_1 a_2 \ldots (a_{r-1} -1)$ null permanent values. Since $G$ has coduplicate vertices, see Figure \[Fig1\], and $ x\neq 1,$ we apply Lemma \[lem1\] and after $b_r -1$ iterations for each $\otimes$ vertice at level $r-1$ , the remaining vertices on the cotree receive $$d_l \leftarrow \frac{ -(b_r -1) + b_r -1}{b_r -1 +1} =0,$$ and the removed vertices receive $$d_k \leftarrow -\frac{j+1}{j} b_r<0,\mbox{ for } j = 1, \ldots, b_r -1.$$ This is illustrated on the left of Figure \[figure11\]. Now the leaves at level $r$ move up to the $\cup$ vertices as on the right of Figure \[figure11\] and we process them. Notice that we have duplicate leaves with null value. Then the algorithm performs [[**subcase 2b**]{}]{}  at the leaves in each vertex $\cup$ and it creates $a_1 a_2 \ldots (a_{r-1}-1)$ permanent zeros in the removed vertices. The remaining vertices keep the value zero, as shown on the left of Figure \[figure12\]. So $m(b_r -1) \geq a_1 a_2 \ldots (a_{r-1} -1).$ Now we show that no more permanent zeros are created. The zero value vertices now move up to the next $\bigotimes$ level. Notice that, see right of Figure \[figure12\], we have coduplicate vertices in the remaining tree with assignments equal to $0$. Using Lemma \[lem3\] once and then Lemma \[lem5\], we know that no null value will be generated and it proves that $m(b_r -1) = a_1 a_2 \ldots (a_{r-1} -1).$ \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(f) at (3,5) [$\otimes$]{}; (g) at (2.5,4) [$\cup$]{}; (h) at (3.5,4) [$\cup$]{}; (2.7,4) – (3.2,4); (f) edge node\[left\] (g); (f) edge node\[left\] (h); \(i) at (2.5,3) ; (j) at (3.5,3) ; (3.2,3) – (3.8,3); (g) edge node\[left\] (i); (h) edge node\[left\] (j); () at (3,1) ; () at (4,1) ; (3.2,1) – (3.8,1); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(f) at (3,5) [$\otimes$]{}; (g) at (2.5,4) ; (h) at (3.5,4) ; (2.7,4) – (3.2,4); (f) edge node\[left\] (g); (f) edge node\[left\] (h); In the next theorem we present a bound for the eigenvalues of regular balanced cotrees. \[main3\] Let $G$ be a cograph with balanced cotree $T_G (a_1, \ldots,a_{r-1}, 0| 0, \ldots, 0,b_r)$ order $n =a_{1} a_2 \ldots a_{r-1} b_r.$ 1. If $r$ is odd and $\lambda\neq -1$, $b_{r}-1$ then $m(\lambda)\leq a_{1}\cdots a_{r-2}$; 2. If $r$ is even and $\lambda\neq 0$, $-b_{r}$ then $m(\lambda)\leq a_{1}\cdots a_{r-2}$. [**Proof:**]{} Suppose that $r$ is odd. Then $m(\lambda) \leq n - m(-1) - m(b_r-1)=a_1 a_2 \ldots a_{r-1} b_r - a_1 a_2 \ldots a_{r-1}(b_{r} -1) - a_1 a_2 \ldots (a_{r-1} -1)= a_{1}\cdots a_{r-2}$. Non-regular balanced cotrees ---------------------------- We now define two types of cotrees depending on whether its depth $r$ is even or odd. Let $T_G (a_1, \ldots, a_{r-1},0| b_1, b_2,\ldots, b_r)$ be a balanced cotree defined as follows: If $r$ is even then, for $1 \leq i \leq r-1$, $\left\{ \begin{array}{lr} b_{i} =0 & \mbox{if $i$ is odd };\\ b_{i}\geq b_{r} & \mbox{if $i$ is even.} \end{array} \right.$ If $r$ is odd then, for $1 \leq i \leq r-1$, $\left\{ \begin{array}{lr} b_{i} =0 &\mbox{if $i$ is even}; \\ b_{i}\geq b_{r} &\mbox{if $i$ is odd.} \end{array} \right.$ \[lem7\] Let $G$ be a cograph with balanced cotree $T_G (a_1, \ldots,a_{r-1}, 0| b_1, b_2, \ldots,b_r)$ defined above. 1. If $r$ is odd then $G$ has the eigenvalue $b_r -1$ with multiplicity $a_1 a_2 \ldots (a_{r-1} -1)$. 2. If $r$ is even then $G$ has the eigenvalue $-b_r $ with multiplicity $a_1 a_2 \ldots (a_{r-1} -1)$. [**Proof:**]{} We assume that $r$ is even. The case odd is similar. The illustration of the initial configuration is given on the left of Figure \[figure19\]. Consider $ x= -b_r $ and execute the algorithm Diagonalization with input $(T_G, x).$ By Theorem \[mainB\] we have to prove that the algorithm creates at least $a_1 a_2 \ldots (a_{r-1} -1)$ permanent null values. Applying Lemma \[lem2\] at each vertex $\cup$ at level $r-1$, the following assignments are made $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}b_{r}>0, & j=1,\ldots,b_{r}-1; \\ d_{l} & \leftarrow & \frac{b_{r}}{b_{r}-1+1}=1. & \end{array}$$ The removed leaves have a permanent positive value and the remaining vertices have value $1$, as illustrated on the right of Figure \[figure19\]. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (1) at (3,3) [$\cup$]{}; (2) at (2,2) [$\otimes$]{}; (3) at (3,2) [$\otimes$]{}; (2.2,2) – (2.8,2); (4) at (3.5,2) ; (5) at (4.5,2) ; (3.7,2) – (4.3,2); \(6) at (2.5,1) [$\cup$]{}; (7) at (3.5,1) [$\cup$]{}; (2.7,1) – (3.3,1); \(8) at (2,0) ; (9) at (2.8,0) ; (2.2,0) – (2.6,0); \(10) at (3.2,0) ; (11) at (4,0) ; (3.4,0) – (3.8,0); \(1) edge node\[left\] (2); (1) edge node\[left\] (3); (1) edge node\[left\] (4); (1) edge node\[left\] (5); (3) edge node\[left\] (6); (3) edge node\[left\] (7); (6) edge node\[left\] (8); (6) edge node\[left\] (9); (7) edge node\[left\] (10); (7) edge node\[left\] (11); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (1) at (3,3) [$\cup$]{}; (2) at (2,2) [$\otimes$]{}; (3) at (3,2) [$\otimes$]{}; (2.2,2) – (2.8,2); (4) at (3.5,2) ; (5) at (4.5,2) ; (3.7,2) – (4.3,2); \(6) at (2.5,1) [$\cup$]{}; (7) at (3.5,1) [$\cup$]{}; (2.7,1) – (3.3,1); \(8) at (2.5,0) ; \(10) at (3.5,0) ; \(1) edge node\[left\] (2); (1) edge node\[left\] (3); (1) edge node\[left\] (4); (1) edge node\[left\] (5); (3) edge node\[left\] (6); (3) edge node\[left\] (7); (6) edge node\[left\] (8); (7) edge node\[left\] (10); Now the vertices remaining (with value 1) are moved up and become leaves of a $\bigotimes$ vertex, as seeing on the left of Figure \[figure21\]. We perform subcase 1b and then the $a_{1}\cdots a_{r-2}(a_{r-1}-1)$ removed vertices receive the value $0$ and the remaining leaves receive $1$ as shown on the right of Figure \[figure21\], and so $m(-b_{r})\geq a_1 a_2 \ldots (a_{r-1} -1)$. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(1) at (3,3) [$\cup$]{}; (2) at (2,2) [$\otimes$]{}; (3) at (3,2) [$\otimes$]{}; (2.2,2) – (2.8,2); (4) at (3.5,2) ; (5) at (4.5,2) ; (3.7,2) – (4.3,2); \(6) at (2.5,1) ; (7) at (3.5,1) ; (2.7,1) – (3.3,1); \(1) edge node\[left\] (2); (1) edge node\[left\] (3); (1) edge node\[left\] (4); (1) edge node\[left\] (5); (3) edge node\[left\] (6); (3) edge node\[left\] (7); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\]s \(1) at (3,3) [$\cup$]{}; (2) at (2,2) [$\otimes$]{}; (3) at (3,2) [$\otimes$]{}; (2.2,2) – (2.8,2); (4) at (3.5,2) ; (5) at (4.5,2) ; (3.7,2) – (4.3,2); (6) at (2,1) ; (7) at (3,1) ; \(1) edge node\[left\] (2); (1) edge node\[left\] (3); (1) edge node\[left\] (4); (1) edge node\[left\] (5); (2) edge node\[left\] (6); (3) edge node\[left\] (7); Now the vertices with value $1$ move to level $r-2$ as shown on the left of Figure \[figure23\]. At each vertex $\cup$ at level $r-3$ we start processing the vertices with value $1$, and by Lemma \[lem2\]: $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}1>0, & j=1,\ldots,a_{r-2}-1; \\ d_{l} & \leftarrow & \frac{1}{a_{r-2}-1+1}=\frac{1}{a_{r-2}}. & \end{array}$$ Then we process the vertices with value $b_{r}$ using Lemma \[lem2\]: $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}b_{r}>0, & j=1,\ldots,b_{r-2}-1; \\ d_{l} & \leftarrow & \frac{b_{r}}{b_{r-2}-1+1}=\frac{b_{r}}{b_{r-2}} & \end{array}$$ The right of Figure \[figure23\] represents the last iteration in each vertex $\cup$ at level $r-2$. Notice that each remaining leaf has a value in $(0,1]$ and using the same argument as in Lemma \[lem6\], we can prove that no more zeros are assigned and the remaining vertices on the cotree are in $(0,1)$, proving that $m(-b_r)=a_1 a_2 \ldots (a_{r-1} -1)$. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (1) at (3,3) [$\cup$]{}; (2) at (2,2) ; (3) at (3,2) ; (2.2,2) – (2.8,2); (4) at (3.5,2) ; (5) at (4.5,2) ; (3.7,2) – (4.3,2); (1) edge node\[left\] (2); (1) edge node\[left\] (3); (1) edge node\[left\] (4); (1) edge node\[left\] (5); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (1) at (3,3) [$\cup$]{}; (2) at (2.5,2) ; (3) at (3.5,2) ; (1) edge node\[left\] (2); (1) edge node\[left\] (3); Borderenergetic Cographs ======================== In this section we present some families of non-cospectral and borderenergetic cographs. Consider the cograph $G=K_{a} \otimes (a-1)(b-1)K_{b}$, of order $n=a+b(a-1)(b-1)$. We observe that $G$ has the balanced cotree $T_{G}(1, (a-1)(b-1), 0| a,0,b)$, represented in Figure \[Figu5\]. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] (a) at (2,2) [$\otimes$]{}; (b) at (2,1) ; (f) at (3.5,1) [$\cup$]{}; (c) at (0.75,1) ; (d) at (3,-1) ; (e) at (2,-1) ; (i) at (4,-1) ; (j) at (5,-1) ; \(g) at (2.5,0) [$\otimes$]{}; (h) at (4.5,0) [$\otimes$]{}; (a) edge node\[left\] (b) (a) edge node\[below\] (c) (a) edge node\[below\] (f) (h) edge node\[below\] (i) (h) edge node\[below\] (j) (g) edge node\[below\] (d) (g) edge node\[below\] (e) (f) edge node\[left\] (g) (f) edge node\[left\] (h); \[spec1\] Let $G=K_{a} \otimes (a-1)(b-1)K_{b}$ be the cograph $G$ of Figure \[Figu5\] of order $n=a+b(a-1)(b-1)$, for fixed values $a\geq b\geq 2$. The spectrum of $G$ is $$-(a-1)(b-1); -1; b-1; ab-1$$ with multiplicity $$1; (a-1)[(b-1)^{2}+1]; (a-1)(b-1)-1; 1,$$ respectively. [**Proof:**]{} Using Theorem \[Tre1\] with $m=(a-1)(b-1)+1$, $t_{1}=\cdots=t_{m-1}=b$ and $t_{m}=a$. We compute the multiplicity of $-1$: $$m(-1)=\sum_{i=1}^{m}(t_{i}-1)=(m-1)(b-1)+(a-1)=(a-1)[(b-1)^{2}+1].$$ Since $T_G(1,(a-1)(b-1),0|a,0,b)$ has a non regular balanced cotree, by Lemma \[lem7\], $b-1$ is an eigenvalue with multiplicity $(a-1)(b-1)-1.$ Now, we will prove, by Theorem \[mainB\], that $ab-1$ is an eigenvalue of multiplicity $1$ by showing that the algorithm D Diagonalize with input $(T_{G},-ab+1)$, creates a single zero in the $T_G$. We initialize the leaves with value $-ab+1\neq 1$. Then we can use Lemma \[lem1\], and for each $\otimes$ vertex, we have that $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}(-ab), & j=1,\ldots,b-1; \\ d_{l} & \leftarrow & -a+1& \end{array}$$ where $d_{k}$ represents the removed leaves and $d_{l}$ the remaining ones. The left of Figure \[figure30\] represents the cotree yet to be processed. Now, the leaves at depth 3 move up to the $\cup$ vertices at depth 2, as on the right of Figure \[figure30\]. \[scale=.9,auto=left,every node/.style=[circle,scale=0.9]{}\] \(a) at (2,2) [$\otimes$]{}; (b) at (2,1) ; (f) at (3.5,1) [$\cup$]{}; (c) at (0.75,1) ; \(d) at (2.5,-1) ; (i) at (4,-1) ; (j) at (4.5,-1) ; (g) at (2.5,0) [$\otimes$]{}; (h) at (4.5,0) [$\otimes$]{}; (a) edge node\[left\] (b) (a) edge node\[below\] (c) (a) edge node\[below\] (f) (h) edge node\[below\] (j) (g) edge node\[below\] (d) (f) edge node\[left\] (g) (f) edge node\[left\] (h); \[scale=.9,auto=left,every node/.style=[circle,scale=0.9]{}\] \(a) at (2,2) [$\otimes$]{}; (b) at (2,1) ; (f) at (3.5,1) [$\cup$]{}; (c) at (0.75,1) ; \(g) at (2.5,0) [$\otimes$]{}; (h) at (4.5,0) [$\otimes$]{}; (a) edge node\[left\] (b) (a) edge node\[below\] (c) (a) edge node\[below\] (f) (f) edge node\[left\] (g) (f) edge node\[left\] (h); In the next step we use Lemma \[lem2\] because the duplicate vertices at depth 2 have assignments equal to $-a+1\neq0$. We obtain $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}(-a+1), & j=1,\ldots,(a-1)(b-1)-1; \\ d_{l} & \leftarrow & \frac{-1}{b-1}.& \end{array}$$ As the left of Figure \[figure32\] shows, the remaining leaf at depth 2 moves up to depth 1, as on the right of Figure \[figure32\]. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(a) at (2,2) [$\otimes$]{}; (b) at (2,1) ; (f) at (3.5,1) [$\cup$]{}; (c) at (0.75,1) ; \(g) at (3.5,0) [$\otimes$]{}; \(a) edge node\[left\] (b) (a) edge node\[below\] (c) (a) edge node\[below\] (f) \(f) edge node\[left\] (g); \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(a) at (2,2) [$\otimes$]{}; (b) at (2,1) ; (f) at (3.5,1) [$\cup$]{}; (c) at (0.75,1) ; \(a) edge node\[left\] (b) (a) edge node\[below\] (c) (a) edge node\[below\] (f); At depth one, there are $a+1$ coduplicate vertices. $a$ with value $-ab+1$ and one with value $\frac{ -1}{b-1}$ as the right of Figure \[figure32\]. The algorithm processes, by Lemma \[lem1\], the leaves with value $-ab+1$ first and it generates the following assignments $$\begin{array}{cccc} d_{k} & \leftarrow & \frac{(j+1)}{j}(-ab), & j=1,\ldots,(a-1); \\ d_{l} & \leftarrow & -b+1.& \end{array}$$ The last step of the algorithm is to process the two remaining vertices whose values are $\alpha = -b+1$ and $\beta = \frac{-1}{b-1}.$ Since $\alpha, \beta<0$ then the algorithm performs [[**subcase 1a**]{}]{}  and assigns $$d_k \leftarrow \alpha + \beta -2 = \frac{-b^{2}}{b-1}\mbox{ and } d_l \leftarrow \frac{\alpha \beta -1}{\alpha +\beta -2} =0,$$ creating a negative value and a zero for the last two diagonal entries, so $m(ab-1)=1$. Using the fact that sum of eigenvalues must be zero, we obtain the remaining eigenvalue $-(a-1)(b-1)$ of $G$, proving the result. The following theorem follows directly from Lemma \[spec1\] and summarizes the results for the family of cographs represented in Figure \[Figu5\]. Let $G=K_{a} \otimes (a-1)(b-1)K_{b}$ be the cograph of order $n=a+b(a-1)(b-1)$, for fixed values $a\geq b\geq 2$ represented in Figure \[Figu5\]. Then $G$ is an integral cograph, non-cospectral and borderenergetic to $K_{n}$. [**Proof:**]{} It is well known that the Spec$(K_{n})=\{(-1)^{n-1},(n-1)^{1}\}$ and and, hence, $E(K_n) = 2(n-1)$. Using Lemma \[spec1\] we can compute the energy of $G$ as follows $$E(G)=(a-1)(b-1)+(1)(a-1)[(b-1)^{2}+1]+(b-1)[(a-1)(b-1)-1]+(ab-1)=2(n-1).$$ Consider now the cograph $G=(p+1)K_{2}\otimes(p+1)K_{2}$, of order $n=4p+4$, whose regular balanced cotree $T_{G}(2,p+1,0|0,0,2)$ is represented in Figure \[Figu6\]. \[scale=1,auto=left,every node/.style=[circle,scale=0.9]{}\] \(a) at (2,2) [$\otimes$]{}; (f) at (4,1) [$\cup$]{}; (c) at (0.5,1) [$\cup$]{}; (d) at (4,-1) ; (e) at (3,-1) ; (i) at (4.5,-1) ; (j) at (5.5,-1) ; \(m) at (-0.5,-1) ; (n) at (0.5,-1) ; (o) at (1,-1) ; (p) at (2,-1) ; \(k) at (0,0) [$\otimes$]{}; (b) at (1.5,0) [$\otimes$]{}; (g) at (3.5,0) [$\otimes$]{}; (h) at (5,0) [$\otimes$]{}; (c) edge node\[below\] (b) (c) edge node\[below\] (k) (b) edge node\[below\] (p) (b) edge node\[below\] (o) (k) edge node\[below\] (n) (k) edge node\[below\] (m) (a) edge node\[below\] (c) (a) edge node\[below\] (f) (h) edge node\[below\] (i) (h) edge node\[below\] (j) (g) edge node\[below\] (d) (g) edge node\[below\] (e) (f) edge node\[left\] (g) (f) edge node\[left\] (h); \[lem11\] Let $G=(p+1)K_{2}\otimes(p+1)K_{2}$ be a cograph of order $n=4p+4$, for a fixed value $p\geq 1$. Then the spectrum of $G$ is $$-(2p+1); -1; 1; 2p+3$$ with multiplicity $$1; 2(p+1); 2p; 1,$$ respectively. [**Proof:**]{} Notice that, using Theorem \[Tre1\], we can consider that $m=2(p+1)$, $t_{1}=\cdots=t_{m}=2$ . Then the multiplicity of $-1$ is $$\sum_{i=1}^{m}(t_{i}-1)=2(p+1).$$ Noticing that $T_{G}(2,p+1,0|0,0,2)$ is a regular balanced cotree, we can apply Lemma \[lem6\] to obtain that $m(1)=2p$. To obtain that $m(-(2p+1))=1$ we just execute the algorithm diagonalize with input $(T_{G},2p+1)$ and observe that it creates a single zero on the $T_G$. The eigenvalue $2p+3$ is determined by the fact that the eigenvalues must sum zero. Let $G=(p+1)K_{2}\otimes(p+1)K_{2}$ be the cograph of order $n=4p+4$, represented in Figure \[Figu6\], for a fixed value $p\geq 1$. Then $G$ is integral, non-cospectral and borderenergetic to $K_{n}$. [**Proof:**]{} Using Lemma \[lem11\] we have that $E(G)=8p+6$ and $E(K_{n})=2(n-1)=8p+6$. And the result follows. [99]{} F.K. Bell, P. 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--- abstract: 'We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similarly to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state dependent diffusion for arbitrary values of the stochastic prescription parameter.' author: - 'Miguel V. Moreno' - 'Daniel G. Barci' - Zochil González Arenas title: 'State dependent diffusion in a bistable potential: conditional probabilities and escape rates' --- Introduction ============ The physics of thermal or noise activation over a barrier has a long history. Nowadays, it is an important research topic due to the wide range of applications in several areas of science, such as physics, chemistry and biology as well [@ActivatedBarrierBook]. The simplest model to study this problem is a classical particle in a bistable potential, $U(x)$, whose dynamics is driven by an overdamped Langevin equation with additive white noise. In this context, an important physical quantity is the rate at which the particle escape out of a minimum of the potential. The seminal work of Kramers [@Kramers1940] stated the very simple formula $$r_{\rm add}= \frac{\sqrt{\omega_{\rm min}|\omega_{\rm max}|} }{\pi} e^{-\frac{\Delta U}{\sigma^2}} \label{eq:KramersEscapeRate}$$ where $r_{\rm add}$ is the escape rate, $\Delta U=U(x_{\rm max})-U(x_{\rm min})$ is the height of the potential barrier, $\sigma^2$ is the noise intensity and $\omega_{\rm min}=U''(x_{\rm min})$ and $\omega_{\rm max}=U''(x_{\rm max})$ are the local curvatures of the potential at its minimum ($x_{\rm min}$) and its maximum ($x_{\rm max}$), respectively (primes mean derivative with respect to $x$). We use the notation $r_{\rm add}$ to emphasize that this expression for the escape rate was computed assuming an [*additive noise*]{} stochastic differential equation. Equation (\[eq:KramersEscapeRate\]) is valid in the weak noise or high barrier approximation $\sigma^2\ll \Delta U$. Since this well-established result was defined, a lot of work has been done in order to compute more accurate expressions suitable to be applied to more realistic situations. The generalization of Eq. (\[eq:KramersEscapeRate\]) to multidimensional systems was (and still is) a big challenge [@Hangii1990]. Moreover, generalizations to different types of noise probability distributions have been also considered [@BrayMcKane1989; @McKane1-1990; @McKane2-1990; @McKane3-1990; @Jung2005; @Goulding2007]. On the other hand, there is an increasing interest for multiplicative noise stochastic systems. Some examples of multiplicative noise dynamics are given by the diffusion of particles near a wall [@Lancon2001; @Lancon2002; @Lubensky2007; @Volpe2010; @Volpe2011], micromagnetic dynamics [@GarciaPalacios1998; @Aron2014; @Arenas2018] and non-equilibrium transitions into absorbing states [@Hinrichsen2000]. There are two particular stochastic phenomena in which multiplicative noise plays an important role: noise-induced phase transitions [@Parrondo1994; @CastroWio1995; @Sancho2003; @Goldenfeld2015; @BarciMiguelZochil2016] and stochastic resonance [@Benzi1981; @Parisi1983; @Wio2007; @Wio2002]. In the last case, the escape rate is at the stem of the physical description of the observed phenomenology. One of the main questions that we address in this paper is how the Kramers escape rate of Eq. (\[eq:KramersEscapeRate\]) is modified when the dynamics is driven by a general multiplicative noise, modeled by a diffusion function $g(x)$. This topic have been rarely treated in the past and there is some controversy in the literature [@Jin2005; @NingLi2006; @Zheng2011; @FengGuo2011; @Byrne2014; @Rosas2016]. In particular, we study the dependence of the escape rate on the stochastic prescription, necessary to correctly define the multiplicative noise Langevin equation. This point is particularly relevant in order to compare analytic results with numerical simulations. Our main result is $$\begin{aligned} r_{\rm mult}=g^2(x_{\rm max}) \frac{ \sqrt{\tilde\omega_{\rm min}|\tilde \omega_{\rm max}| }}{\pi}\; e^{-\frac{\Delta U_{\rm eq}}{\sigma^2}}\; . \label{eq:mainresult} \end{aligned}$$ We used the notation $r_{\rm mult}$ to denote the escape rate in the [*multiplicative noise*]{} case. In general, we observe that the Arrhenius form of the Kramers result still remains. Another similarity with Eq. (\[eq:KramersEscapeRate\]) is that the escape rate does not depend on details, either of the potential or of the diffusion function. Instead, it only depends on the local properties of these functions at the maximum and minima of the potential. On the other hand, there are important differences between both results. Firstly, the original potential $U(x)$ has been replaced by the equilibrium potential $U_{\rm eq}(x)$, obtained from the solution of the asymptotic stationary Fokker-Planck equation (Eq. (\[eq:Ueq\])). The barrier height is, in this case, $ \Delta U_{\rm eq}=U_{\rm eq}(x_{\rm max}) - U_{\rm eq}(x_{\rm min})$. On the other hand, local curvatures have been renormalized by the diffusion function as $\tilde\omega_{\rm min}=g^2(x_{\rm min}) U''_{\rm eq}(x_{\rm min})$ and $\tilde\omega_{\rm max}=g^2(x_{\rm max}) U''_{\rm eq}(x_{\rm max})$. Finally, there is an overall factor $g^2(x_{\rm max})$ coming from a careful treatment of fluctuations. In the rest of the paper, we better describe the model and the technique used to compute Eq. (\[eq:mainresult\]), and discuss the result in more detail. Multiplicative stochastic processes can be study with different theoretical approaches. For numerical simulations [@Sivak2013], the Langevin approach seems to be more adequate. The Fokker-Planck equation is perhaps more appropriate to develop analytic calculations, specially in the long time stationary limit. In this context, techniques such as mean fields, perturbation theory and even renormalization group are also available [@Goldenfeld]. On the other hand, the path integral formulation of stochastic processes is the more natural technique to compute correlation and response functions [@WioBook2013]. Important progress has been recently reached in the path integral representation of multiplicative noise processes [@AronLeticia2010; @arenas2010; @arenas2012; @Arenas2012-2; @Miguel2015; @ArBaCuZoGus2016], despite the fact that this topic has been studied for a long time [@Janssen-RG]. The escape rate is just one ingredient of a more general problem that is the computation of conditional probabilities. Equilibrium properties, such as detailed balance, can be cast in terms of the conditional probability and its time reversal. Time reversal transformations, detailed-balance relations, as well as microscopic reversibility in multiplicative processes were studied in detail in Ref. [@Arenas2012-2]. More recently, we have presented a useful path integral technique to compute weak noise expansions [@Miguel2019]. The integration over fluctuations in the multiplicative case is not trivial. The reason is that the diffusion function produces an integration measure that resembles a curved time axis [@Zinn-Justin]. We have provided a local time reparametrization in order to integrate fluctuations [@Miguel2019]. In this paper, we compute the conditional probability of finding a particle in a well at large times $t/2$, provided it was in the same or the other well at $-t/2$. In the weak noise approximation, saddle points provide a set of diluted instanton and anti-instanton solutions. The diluted instanton gas approximation was first introduced in the context of quantum mechanics to compute the tunneling probability across a potential barrier [@Coleman1979]. In the context of an additive stochastic process, it was developed with great detail in Refs. [@Caroli1979; @Caroli1981]. From a technical point o view, we generalize the calculation of Ref. [@Caroli1981] to the multiplicative noise case, using the time reparametrization techniques introduced in Ref. [@Miguel2019]. The paper is organized as follows. In the next section, we present the equilibrium properties of a particle in a double-well potential under state dependent diffusion. In section \[sec:pathintegral\], we briefly review the path integral representation of a conditional probability in a multiplicative process and we show, in section \[sec:Weaknoise\], how to integrate fluctuations. We develop the dilute instanton gas approximation in section \[sec:instanton\], where we compute conditional probabilities and the escape rate. Finally, we discuss our results in section \[sec:discussion\]. We lead to the Appendix \[Ap:Zeromode\] some details of the calculation. Equilibrium properties of a particle in a double-well potential under state dependent diffusion {#sec:model} =============================================================================================== In this section, we describe the equilibrium properties of a model consisting of a single particle in a double-well potential coupled with a thermal bath with state dependent diffusion. We consider a conservative one dimensional system described by a potential energy $U(x)=U(-x)$ with a double minima structure. The thermal bath is characterized by the diffusion function $g(x)=g(-x)$. The reflection symmetry $x\to -x$ is not essential and most of our results do not depend on it. However, to keep the discussion as simple as possible, we focus in the symmetric model, leading the details of a more general asymmetric situation to a future presentation. In order to reach thermodynamic equilibrium at long times, the drift force $f(x)$ should be related with the classical potential $U(x)$ through a generalized Einstein relation [@arenas2012; @Arenas2012-2] $$f(x)=-\frac{1}{2} g^2(x) \frac{dU(x)}{dx}\;. \label{eq:Einstein}$$ In this way, the overdamped dynamics is driven by the Langevin equation $$\frac{dx}{dt} =-\frac{1}{2} g^2(x) \frac{dU(x)}{dx} + g(x) \eta(t), \label{eq:Langevin2}$$ where $\eta(t)$ obeys a Gaussian white noise distribution with $$\left\langle \eta(t)\right\rangle = 0 \;\;\mbox{,}\;\;\; \left\langle \eta(t)\eta(t')\right\rangle = \sigma^2 \delta(t-t')\; , \label{eq:whitenoise}$$ in which $\sigma^2$ measures the noise intensity. This equation is understood in the [*generalized Stratonovich*]{} [@Hanggi1978] prescription (also known as $\alpha-$prescription [@Janssen-RG]). The asymptotic long time equilibrium probability distribution is given by [@Arenas2012-2] $$P_{\rm eq}(x)={\cal N}\; e^{-\frac{1}{\sigma^2}U_{\rm eq}(x)}, \label{eq:Peq}$$ where ${\cal N}$ is a normalization constant and the equilibrium potential $$U_{\rm eq}(x)= U(x)+2(1-\alpha)\sigma^2\ln g(x)\; . \label{eq:Ueq}$$ The parameter $0\le \alpha\leq 1$ labels the particular stochastic prescription used to discretized the Langevin equation. For instance, $\alpha=0$ corresponds with Itô interpretation while $\alpha=1/2$ corresponds with the Stratonovich one. In this way, the equilibrium potential is not the bare classical potential, but it is corrected by the diffusion function $g(x)$. On the other hand, the case $\alpha=1$ corresponds with Hänggi-Klimontovich interpretation [@Hanggi1982; @Klimontovich]. This is the only prescription which leads to the Boltzmann distribution $U_{\rm eq}(x)=U(x)$. For this reason, this convention is sometimes called “thermal prescription”. Furthermore, this prescription is also known as anti-Itô and can be considered as the time reversal conjugated to the Itô prescription [@Arenas2012-2; @Miguel2015]. Although the techniques and results of this paper do not depend on details, either of $U(x)$ or of $g(x)$, it is convenient, just to visualize the equilibrium potential $U_{\rm eq}(x)$, to consider a very simple model. Let us take, for instance, $$U(x)=-\frac{1}{2} x^2 +\frac{1}{4} x^4 \; , \label{eq:U}$$ with the diffusion function $$g(x)=1+x^2. \label{eq:g}$$ The bare potential $U(x)$ has two degenerated minima at $x_{\rm min}=\pm 1$ and a local maximum at $x_{\rm max}=0$. The contribution of the multiplicative noise for the equilibrium potential is quite interesting. In the weak noise limit, the global two-minima structure remains the same. However, the minima are displaced to $$\begin{aligned} x_{\rm min}&=\pm (1-4\sigma^2(1-\alpha))^{1/4}\nonumber \\ & \sim \pm 1\mp\sigma^2(1-\alpha)+ O(\sigma^4)\; .\end{aligned}$$ For $\sigma^2\ge 1/4(1-\alpha)$, both minima melt in a single one, deeply changing the global structure of the potential. This dependence on the noise intensity resembles a second order phase transition, where the critical noise is given by $$\sigma_c=\frac{1}{2}\frac{1}{\sqrt{1-\alpha}}\;.$$ Interestingly, the critical noise depends on the stochastic prescription. For $\alpha\to 1$, $\sigma_c\to \infty$, meaning that, in the anti-Itô prescription, the double-well structure is preserved for all values of the noise. In Figure \[fig:Ueq\], we depict the equilibrium potential $U_{\rm eq}(x)$ given by Eq. (\[eq:Ueq\]) for the simple model specified by Eqs. (\[eq:U\]) and (\[eq:g\]), for different values of the parameters $\sigma$ and $\alpha$. In Figure \[fig:Ueq\]-(a), we show the equilibrium potential for $\sigma=0.45$ and different values of the stochastic prescription $\alpha=0,1/2,1$. We see that, for $\alpha=1$, $U_{\rm eq}=U$ and the minima are fixed at $x_{\rm min}=\pm 1$. However, in the Stratonovich and Itô prescriptions, the minima are displaced towards the origin. In Figure \[fig:Ueq\]-(b), the three curves are computed in the Itô prescription with different values of the noise $\sigma=1/5, 2/5, 2/3$. In this case, the minima approach zero when the noise grows and, for the value $\sigma=2/3>\sigma_c= 1/2$, the equilibrium potential has only one global minimum at $x_{\rm min}=0$. Conditional probabilities: path integral representation {#sec:pathintegral} ======================================================== We are interested in computing the conditional probability $P(x_f, t_f|x_i, t_i)$ of finding the system in the state $x_f$ at time $t_f$, provided the system was in the state $x_i$ at a previous time $t_i$. It is useful to express this quantity using a path integral representation [@Miguel2019]. It can be written as $$P(x_f, t_f|x_i, t_i)= e^{-\frac{\Delta U_{\rm eq}}{2\sigma^2}} K(x_f, t_f|x_i, t_i) \label{eq:PK}$$ where $\Delta U_{\rm eq}=U_{\rm eq}(x_f)-U_{\rm eq}(x_i)$ and the *propagator* $K(x_f, t_f|x_i, t_i)$ is given by $$K(x_f, t_f|x_i, t_i)=\int [{\cal D}x]\; e^{-\frac{1}{\sigma^2}\int_{t_i}^{t_f} dt \; L(x,\dot x)} \; . \label{eq:Propagator}$$ Here, the functional integration measure is $$[{\cal D}x]={\cal D}x\;{\det}^{-1} g=\lim\limits_{\substack{{N\to\infty} \\ {\Delta t\to 0}}} \prod_{n=0}^N \frac{dx_n}{\sqrt{\Delta t \;g^2(\frac{x_n+x_{n+1}}{2})}} \label{eq:Dx}$$ where $x_0=x_i$ and $x_N=x_f$. The Lagrangian can be written in the form, $$L= \frac{1}{2}\left(\frac{1}{g^2(x)}\right) \dot x^2+V(x) \; , \label{eq:L}$$ where $$V(x) = \frac{g^2}{2}\left[\left(\frac{U'_{\rm eq}}{2}\right)^2 - \sigma^2\left(\frac{U''_{\rm eq}}{2}+\frac{g'}{g} U'_{\rm eq} \right) \right] + \frac{\sigma^4}{4}\left(g g'\right)' . \label{eq:V}$$ The primes mean derivative with respect to $x$. Equation (\[eq:Propagator\]), with the Lagrangian defined by Eq. (\[eq:L\]), correctly describes the dynamics of the Langevin Eq. (\[eq:Langevin2\]) for arbitrary values of the parameter $0\le\alpha\le 1$ [@Miguel2019]. It is important to note that all the information about the stochastic prescription is codified in the structure of the equilibrium potential $U_{\rm eq}(x)$, contained in the definition of the potential $V(x)$, Eq. (\[eq:V\]). In this particular representation, the path integral measure given by Eq. (\[eq:Dx\]) is discretized symmetrically, allowing us to use normal calculus rules in the manipulation of the path integral (for more details on the subtleties of stochastic calculus in the path integral formulation, please see Ref. [@Arenas2012-2] and references therein). An interesting observation is that Eq. (\[eq:Propagator\]) coincides with the propagator of a quantum particle with position-dependent mass $m(x)=1/g^2(x)$ moving in a potential $V(x)$, written in the imaginary time path integral formalism $t\to -it$. The noise $\sigma^2$ plays the role of $\hbar$ in the quantum theory. At a classical level, the Lagrangian, Eq. (\[eq:L\]), represents a particle with variable mass moving in a potential $-V(x)$. The structure of the potential $-V(x)$ (Eq. (\[eq:V\])) is much more complex than $U(x)$ or even $U_{\rm eq}(x)$. In Fig. \[fig:potentialV\], we plotted the potential $-V(x)$ for the simple model displayed by Eq. (\[eq:U\]). All the curves have been plotted in the Itô prescription $\alpha=0$. The dashed lines corresponds to the additive noise case $g(x)=1$, while the continuous lines represent the potential in the multiplicative noise case, with $g(x)=1+x^2$. In Fig. \[fig:potentialV\]-(a) we fixed $\sigma=0.1$, while in Fig. \[fig:potentialV\]-(b), $\sigma=0.01$. The first observation is that $-V(x)$ has three maxima and two minima. The location of both non-zero maxima roughly coincides with the minima of the potential $U(x)$. The difference is of the order of $\sigma^2$. The main effect of the diffusion function is to increase the curvature at each maxima with a factor proportional to $g^2(x_{\rm max})>1$. An important feature that will be relevant to compute conditional probabilities is that the difference between the height of the peaks are of the order of $\sigma^2$. Thus, in a weak noise regime, the difference between the three maxima tends to disappear. In the extreme limit of $\sigma\to 0$, the potential $-V(x)$ has three degenerate maxima. This fact is clearly shown in Fig. \[fig:potentialV\]-(b). It is timely to note that the structure of $-V(x)$ is quite different from a similar calculus of the tunneling probability amplitude of a quantum particle [@Coleman1979]. In that case, the relevant potential is $-U(x)$, which has only two maxima. The appearance of a quasi-degenerate maximum at $x=0$ is proper of a classical stochastic process, even additive as well as multiplicative. Fluctuations and time reparametrization {#sec:Weaknoise} ======================================= The usual weak noise expansion consists in evaluating the path integral of Eq. (\[eq:Propagator\]) in the saddle-point approximation plus Gaussian fluctuations. Generally, multiplicative noise induces an integration measure that depends on the diffusion function $g(x)$. In Ref. [@Miguel2019], we have shown how to overcome this problem by means of a time reparametrization. In this section, we briefly review this technique since we will use it to compute conditional probabilities. The classical equation of motion is $$\frac{d^2 x}{dt^2}=g^2 V' +\frac{g'}{g} \dot x^2 \;. \label{eq:SadlePoint}$$ Despite the fact that this is a complicated nonlinear equation, using time translation symmetry, a first integral can be built up. We have $$\dot x_{cl}^2=2 g^2_{cl}\left(V_{cl}+H\right) \, . \label{eq:vsquare}$$ Here, $x_{cl}(t)$ is a solution of Eq. (\[eq:SadlePoint\]). The notation $x_{cl}$ stands for classical solution, resembling in some sense a semiclassical calculation in quantum mechanics. $H$ is an arbitrary constant, $g_{cl}=g(x_{cl}(t))$ and $V_{cl}=V(x_{cl}(t))$. Then, the solution of Eq. (\[eq:SadlePoint\]) can be expressed by a quadrature, $$t-t_0= \int_0^{x_{cl}} \frac{ds}{\sqrt{2 V_{\rm eff}(s)}}\; , \label{eq:firstIntegral}$$ where we have defined an effective potential, $$V_{\rm eff}(x)= g^2(x)\left[V(x)+H\right]. \label{eq:Veff}$$ These expressions have two arbitrary constants, $t_0$ and $H$, that should be determined by means of the boundary conditions $x_{cl}(t_i)=x_i$ and $x_{cl}(t_f)=x_f$. Thus, Eqs. (\[eq:firstIntegral\]) and (\[eq:Veff\]) implicitly define $x_{cl}(t)$, used as a starting point of the weak noise approximation. Let us assume, for the moment, that, given initial and final conditions, the classical solution $x_{cl}$ is unique. Then, we consider fluctuations around it $$x(t)=x_{cl}(t)+\delta x(t) \ , \label{eq:xfluct}$$ with boundary conditions $\delta x(t_i)=\delta x(t_f)=0$. Replacing Eq. (\[eq:xfluct\]) into Eq. (\[eq:Propagator\]) and keeping up to second-order terms in the fluctuations, we find for the propagator $$\begin{aligned} K(x_f, t_f|& x_i, t_i) = \label{eq:PropagatorF} \\ & ~~ e^{-\frac{1}{\sigma^2}S_{cl}} \int [{\cal D}\delta x]\; e^{-\frac{1}{2}\int dtdt'\; \delta x(t) O(t,t') \delta x(t') } \; , \nonumber\end{aligned}$$ where the classical action $S_{cl}$ is $$S_{cl}=\int_{t_i}^{t_f} dt \; L(x_{cl}(t),\dot x_{cl}(t)) \label{eq:Scl}$$ and the fluctuation kernel, $$\begin{aligned} O(t,t')&=-\frac{d~}{dt}\!\left( \frac{1}{g^2_{cl}}\frac{d\delta(t-t')}{dt}\right)\!+\!\left(\frac{1}{g^2_{cl}}V'_{\rm eff}(x_{cl})\right)'\!\!\delta(t-t'). \nonumber \\ & \label{eq:kernel}\end{aligned}$$ In Eq. (\[eq:PropagatorF\]), the functional integration measure is $$[{\cal D}\delta x]=\lim\limits_{\substack{{N\to\infty} \\ {\Delta t\to 0}}} \prod_{n=0}^N \frac{d\delta x_n}{\sqrt{\Delta t \;g^2(\frac{x_{cl}(t_n)+x_{cl}(t_{n+1})}{2})}}. \label{eq:Ddeltax}$$ Due to the time dependence of $g_{cl}=g(x_{cl}(t))$, the fluctuation kernel $O(t,t')$ is not trivial. On the other hand, the integration measure, Eq. (\[eq:Ddeltax\]), depends on the diffusion function $g(x(t))$. As a consequence, although the exponent in Eq. (\[eq:PropagatorF\]) is quadratic, the evaluation of the functional integral is cumbersome. In this case, to compute the fluctuation integral, we make a time reparametrization. For concreteness, we introduce a new time variable $\tau$ by means of $$\tau=\int_0^t g^2(x_{cl}(t')) dt'\; . \label{eq:reparametrization}$$ This is a nontrivial *local* scale transformation, weighted by the diffusion function evaluated at the classical solution $x_{cl}(t)$. Performing this time reparametrization, the fluctuation kernel transforms as $O(t,t')\to \Sigma(\tau,\tau')$ and takes the simpler form $$\Sigma(\tau,\tau')=\left[-\frac{d^2~~}{d\tau^2}+W[x_{cl}]\right]\delta(\tau-\tau') \label{eq:Sigma}$$ where $$W(x_{cl})= \frac{1}{g^2_{cl}}\left(\frac{1}{g^2_{cl}}V'_{\rm eff}(x_{cl})\right)'\; . \label{eq:W}$$ More important, after discretizing the reparametrized time axes $\tau$, the functional integration measure, Eq. (\[eq:Ddeltax\]) becomes $$[{\cal D}\delta x]=\lim\limits_{\substack{{N\to\infty} \\ {\Delta \tau\to 0}}} \prod_{n=0}^N \frac{d\delta x_n}{\sqrt{\Delta \tau}} \; , \label{eq:Ddeltaxtau}$$ in which the function $g(x_{cl})$ has been absorbed in the reparametrization. Thus, in the new time variable $\tau$, the functional integral over fluctuations can be formally evaluated, obtaining for the propagator $$K(x_f, t_f|x_i, t_i)=\left( \det\Sigma(\tau_i,\tau_f)\right)^{-1/2} e^{-\frac{1}{\sigma^2}S_{cl}(t_i,t_f)} \ , \label{eq:PropagatorWeaknoise}$$ where the relation between $(\tau_i,\tau_f)$ and $(t_i,t_f)$ is given through Eq. (\[eq:reparametrization\]). Equation (\[eq:PropagatorWeaknoise\]) is formally similar to the weak noise expansion in the additive noise case. However, in this case, the determinant is written in terms of a rescaled time parameter $\tau$. Thus, in order to compute a prefactor, we need to reparametrized the time variable, compute the determinant and, at the end, go back to the original time. In Ref. [@Miguel2019] we have successfully used this technique to compute conditional probabilities of an harmonic oscillator in a multiplicative noise environment. Here, we will use it to compute conditional probabilities in a double-well set-up. Probability of remaining in a well {#sec:instanton} ================================== In order to compute conditional probabilities, let us consider a potential $-V(x)$ with the general structure displayed in Figure \[fig:potentialV\]. We will consider that the potential has local maxima at $x=\pm a$ and $x=0$, while it has two minima, at $x=\pm x_p$. The difference $|V(a)-V(0)|\sim O(\sigma^2)$, in such a way that the three maxima are degenerated in the limit $\sigma\to 0$. As we have mention, the maxima at $x=\pm a$, roughly coincide with the minima of the bare potential $U(x)$. The difference is of order $\sigma^2$. We want to compute the probability of remaining in a minimum of $U(x)$, after some time $t$. Let us compute, for instance, the probability of remaining in the state $x=-a$, [*i.e.*]{}, the probability of finding the particle in the state $x=-a$ at a time $t/2$, provided it was in the same point, at a time $-t/2$. As the initial and final states coincide, $\Delta U_{\rm eq}=0$ and, from Eq. (\[eq:PK\]), we see that this conditional probability coincides with the propagator, $P \left(-a, t/2 | -a,-t/2\right)=K \left(-a,t/2| -a,-t/2\right)$. So, we are interested in the function $K(-a, -t/2~|-a, t/2)$ for very long times, $t\to\infty$. The main point is that for long times, there are a huge number of solutions (or approximate solutions) of the saddle-point equation which need to be considered in order to compute the path integral in the weak noise approximation. A trivial solution of Eq. (\[eq:SadlePoint\]) with initial and final conditions $x_{cl}(-t/2)=x_{cl}(t/2)=-a$ is $x_{cl}=-a$. In this case, the multiplicative noise has a trivial effect. Since $x_{cl}$ does not depend on time, the diffusion function $g_{cl}$ is a simple constant that renormalizes the noise intensity $\sigma$. Then, the contribution of this solution to $K(-a,t/2|-a,-t/2)$ can be easily computed obtaining, $$K^{(0)}(-a,t/2|-a,-t/2)= \left(\frac{g^2_aU''_{\rm eq}(a)}{2\pi\sigma^2}\right)^{1/2} \; , \label{eq:K0}$$ where $g_a=g(a)$. We are using the superscript $(0)$ to indicate the contribution of the constant solution to the propagator. Instantons/Anti-Instantons -------------------------- In the case of potentials with two degenerate maxima, there are topological time-dependent solutions of the equation of motion with finite action that interpolate between both maxima. These solutions are called instantons or anti-instantons and should be taken into account to compute the propagator. For very large time intervals, well separated superposition of instantons and anti-instantons will also contribute to the path integral in a nontrivial way. The technique of summation over these configurations, usually called instanton/anti-instanton diluted gas approximation, was developed by several authors to compute tunneling apmplitudes in quantum mechanics [@Coleman1979; @Brezin1977; @Bogomonly1980]. In stochastic processes, the technique was applied to the case of additive white noise in Ref. [@Caroli1981], in which the problem of a diffusion in a bistable potential was addressed. Some years later, the same technique was successfully applied to color noise processes [@BrayMcKane1989; @McKane1-1990; @McKane2-1990; @McKane3-1990]. Here, we will apply it to the multiplicative noise case. In the rest of this section we will closely follow the calculation of Ref. [@Caroli1981], emphasizing those steps that are proper of multiplicative noise. In addition to the constant solution, there are other time-dependent trajectories which begin and end at $x=-a$ for very long time intervals that will contribute to the propagator. In our case, the maximum at $x=0$ is quasi-degenerated with $x=\pm a$. For this reason, we expect that trajectories which begin at $x=-a$, go to approximately $x=0$ and then return to the original point, will also have an important weight in the functional integral. This type of trajectories are not exact solutions of the classical equation of motion, then, there will be a linear term in the fluctuations expansion. However, this term will be $O(\sigma^2)$ since, in the limit $\sigma\to 0$, it should disappear. We denote by $K^{(1)}\left(-a,t/2|-a,-t/2\right)$, the contribution of the trajectory $-a\to 0\to -a$ to the propagator. To compute it, we first rewrite the Lagrangian, Eq. (\[eq:L\]), in the following way $$L= \frac{1}{2}\left(\frac{1}{g^2(x)}\right) \dot x^2+V^{(0)}(x)+ \delta V(x) \ , \label{eq:Lmodified}$$ where we have defined the quantity $$\delta V(x)= V(x)-V^{(0)}(x)=\left\{ \begin{array}{ccl} 0, && x<-x_{p} \\ V_{0}-V_{a}, && x>-x_{p} \label{eq:deltaV} \end{array}\;. \right.$$ In the last expression, $-x_p$ is the position of the minimum of the potential $-V(x)$, $V_{a}=V(a)=V(-a)$ and $V_{0}=V(0)$. The specific form of $\delta V(x)$, as well as the specific value $x_p$ are not important. The final results will not depend on such details. Thus, the first two terms of Eq. (\[eq:Lmodified\]) describe the dynamics of a particle in a potential $-V^{(0)}$ with truly degenerated maxima, while $\delta V(x)\sim O(\sigma^2)$. Let us compute asymptotic solutions of the classical equation of motion for the potential $-V^{(0)}$. We define the “instanton”, $x_I(t)$, as the solution with initial and final conditions $x_{cl}(-t/2)=-a$ and $x_{cl}(t/2)=0$, for very large values of $t$. From Eq. (\[eq:firstIntegral\]), we have $$t-t_0= \int^{x_{I}}_{-x_p}\frac{dx}{\sqrt{2g^2(x)(V^{(0)}(x)-V_{a})}} \ , \label{eq:instanton}$$ where we fixed the conditions $x_I(t_0)=-x_p$ and $H=V_a$. These parameters guarantee the above mentioned initial and final conditions. We see, from Eq. (\[eq:instanton\]), that the integral is dominated by the region in which $V^{(0)}(x)-V_{a}\to 0$. It happens for $x\to 0 >-x_p$ or $x\to -a< -x_p$. Thus, to compute the integral we can expand $V^{(0)}(x)$ around $x=0$ and $x=-a$ to second order in powers of $x$ and $x+a$, respectively. Thus, in the harmonic approximation we have $$\begin{aligned} V^{(0)}_{h}(x)&= \left\{ \begin{array}{lcl} V_a + \frac{1}{2} V''_{0} x^2, && x>-x_p \\ \\ V_a + \frac{1}{2} V''_{a} (x+a)^2, && x<-x_p \end{array} \label{eq:harmonic} \right. \;.\end{aligned}$$ Using this approximation, we obtain for the instanton solution $$\begin{aligned} x_{I}(t) &\underset{{t\ll t_0}}{\sim} -a+(-x_{p}+a)\; e^{g_a(V''_{a})^{1/2}(t-t_0-\Delta_{ap})} \; , \label{eq:instanton-a} \\ x_{I}(t)&\underset{{t\gg t_0}}{\sim}\;- x_{p} \;e^{-g_0(V''_{0})^{1/2}(t-t_0-\Delta_{0p})}\;, \label{eq:instanton-b}\end{aligned}$$ where we have introduced the finite constants $$\begin{aligned} \Delta(x_i,&x_j) = \label{eq:inst44} \\ & \int_{x_i}^{x_j} \frac{dx}{\sqrt{2}} \left[\frac{1}{g(x)\sqrt{V^{(0)} - V_{a}}} - \frac{1}{g(x_i)\sqrt{V^{(0)}_{h} - V_{a}}}\right], \nonumber \end{aligned}$$ in such a way that, in Eq. (\[eq:instanton-b\]), $\Delta_{0p}=\Delta(0,x_p)$ and $\Delta_{ap}=\Delta(a,x_p)$. The instanton/anti-instanton pair of trajectories, corresponding with the path $-a\to 0\to -a$, can be written as $$x_{_{IA}}(t,t_0,t_1)= \left\{ \begin{array}{lcl} x_{_I}(t-t_0), && t<\frac{t_{0}+t_1}{2} \\ \\ x_{_I}(t_1-t), && t>\frac{t_{0}+t_1}{2} \label{eq:I-A} \end{array}\;, \right.$$ where $x_I(t)$ is given by Eqs. (\[eq:instanton-a\]) and (\[eq:instanton-b\]). A typical instanton/anti-instanton trajectory is shown in Figure \[fig:I-A\]. The classical action is computed by replacing Eq. (\[eq:I-A\]) into Eq. (\[eq:Lmodified\]) and integrating in time between $t_i=-t/2$ and $t_f=t/2$. We find $$\begin{aligned} S_{IA}(t,t_0,&t_1) = (V_{0}-V_{a})(t_{1}-t_{0}) + V_{a}t \label{eq:SIA} \\ &- \frac{x^{2}_p(V''_{0})^{1/2}}{g_0} e^{g_0(V''_{0})^{1/2}(t_0-t_1+2\Delta_{0p})} \nonumber \\ &+U_{\rm eq}(0) - U_{\rm eq}(a)+ \sigma^2\ln\left|\frac{U''_{\rm eq}(a)\;g^{2}_a\;(x_p+a)}{U''_{\rm eq}(0)\;g_{0}^2 \;x_p}\right| \nonumber \\ &+ \frac{\sigma^2}{2}\left[g_{a}^2 U''_{\rm eq}(a)\Delta_{pa} + g_{0}^2 U''_{\rm eq}(0)\Delta_{0p}\right] ,\nonumber \end{aligned}$$ where we have used the notation $S_{IA}=S_{cl}[x_{IA}]$, [*i.e.*]{}, the classical action computed at the instanton/anti-instanton configuration of Eq. (\[eq:I-A\]). The next step is to compute fluctuations around the instanton/anti-instanton solution. After the time reparametrization given by Eq. (\[eq:reparametrization\]), we are lead to the computation of the determinant $\det\hat{\Sigma}(\tau_f,\tau_i)$, where the operator $\hat\Sigma$ is given by Eq. (\[eq:Sigma\]), evaluated at $x_{cl}=x_{IA}(\tau)$. Due to time translation invariance, the determinant has zero modes. Similarly to the original computation of instanton fluctuations [@Coleman1979], we need to properly take into account translation modes, identifying translation fluctuations with the integration over the collective variables $t_0$ and $t_1$. We obtain (see Appendix \[Ap:Zeromode\]), $$\begin{aligned} &K^{(1)}\left(-a,\frac{t}{2}\Big\vert -a,-\frac{t}{2}\right) = {\cal N}\int^{t/2}_{-t/2} dt_0 \int^{t/2}_{t_0} dt_1 \;\times \label{eq:K01t0t1} \\ & g_a^2 \sqrt{S_I} g_0^2 \sqrt{S_A}\; \left[{\det}'{\hat{\Sigma}\left(\tau_f,\tau_i\right)}\right]^{-1/2} e^{-\frac{1}{\sigma^2}S_{_{IA}}(t,t_{0},t_{1})} \nonumber \end{aligned}$$ were $S_I=S_{cl}[x_I]$, $S_A=S_{cl}[x_A]$ and the prime in the determinant indicates that it should be evaluated excluding the zero modes. We use the notation $K^{(1)}$ to indicate the contribution of the path $-a\to 0\to -a$ to the propagator. This result is similar to the additive noise case [@Caroli1981]. The main difference is that the determinant is computed in a reparametrized time and the integration over collective variables $t_0$ and $t_1$ are renormalized by the diffusion function. The advantage of the reparametrized time is that the operator $\hat \Sigma$ has the simpler form of Eq. (\[eq:Sigma\]) and can be computed using the Gelfand-Yaglom theorem [@Dunne2008]. At the end of the calculation, we go back to the original time axes. Following tedious but usual procedures, we finally find $$K^{(1)} \left(-a,\frac{t}{2}\Big\vert -a,-\frac{t}{2}\right)= -g_{0}^{2} t \; K^{(0)} \;\Gamma \ , \label{eq:K1}$$ where $K^{(0)}$ is the contribution of the constant solution, given by Eq. (\[eq:K0\]), and $$\Gamma = \frac{\left(g^2_aU''_{\rm eq}(a)g^2_0|U''_{\rm eq}(0)|\right)^{1/2}}{2\pi } \exp\left\{-\frac{U_{\rm eq}(0) - U_{\rm eq}(a)}{\sigma^2}\right\}\;. \label{eq:Gamma}$$ We see that the contribution of an instanton/anti-instanton configuration to the propagator at long times, is a linear function of time. The structure of the coefficient $\Gamma$ is very interesting. All the information about the stochastic calculus is hidden in the definition of the equilibrium potential, $U_{\rm eq}$. On the other hand, it does not depend on the details of $U_{\rm eq}(x)$, but instead, it depends on the barrier height, $U_{\rm eq}(a)-U_{\rm eq}(0)$, and on the curvature at each maxima, $U''_{\rm eq}(0)$ and $U''_{\rm eq}(a)$. These properties are quite similar with the additive noise case, except for the fact that the original potential $U(x)$ is replaced by the equilibrium potential $U_{\rm eq}$ in the multiplicative case. Moreover, a remarkable feature of multiplicative noise, is that the diffusion function $g(x)$ renormalizes the curvature at each maxima. In this way, $K^{(1)}$ does not depend on the details of $g(x)$, but only on its value at the maxima, $g(0)$ and $g(a)$. Due to the structure of the potential $-V(x)$, there are other trajectories which contribute in a nontrivial way to the propagator; for instance, trajectories that begin in $x=-a$, go to $x=a$ passing through $x=0$, and return to $x=-a$. This kind of trajectories contains two instantons and two anti-instantons as shown in Figure \[fig:2IA\]. The contribution of these trajectories to the propagator can be computed following the same steps of the computation of the single instanton/anti-instanton case. We find, in this case, $$\begin{aligned} &K^{(2)} \left(-a,\frac{t}{2}\Big\vert -a,-\frac{t}{2}\right)= \frac{ (g_0^2 t)^2}{2!} \; K^{(0)}\; \Gamma^2 \ . \label{eq:K2}\end{aligned}$$ Thus, trajectories of the type $-a\to a \to -a$, produce a quadratic time contribution, the coefficient is simply $\Gamma^2$, where $\Gamma$ is given by Eq. (\[eq:Gamma\]). Kramers escape rate and time reversal transformation ---------------------------------------------------- To compute the conditional probability of remaining in a minimum after some time $t$, we need to sum up all the trajectories that begin and ends at $x=-a$ and which contribute to the propagator in a nontrivial way. Having in mind that $\Delta U_{\rm eq}=0$, this probability coincides with the propagator, $P \left(-a, t/2 | -a,-t/2\right)=K \left(-a,t/2| -a,-t/2\right)$. As described above, there are essentially three contributions to these paths: a constant one, $K^{(0)}$, given by Eq. (\[eq:K0\]), a linear term $K^{(1)}$ given by Eq. (\[eq:K1\]), corresponding to trajectories $-a\to 0 \to -a$ or, by symmetry, to $a\to 0 \to a$, and, finally, a quadratic term $K^{(2)}$ given by Eq. (\[eq:K2\]), related to the path $-a \to a \to -a$. Consider, for instance, a general trajectory containing $\ell_1$ paths of the type $-a\to 0 \to -a$ and $\ell_2$ paths of the type $a\to 0 \to a$, related with the linear function $K^{(1)}$. In addition, we allow $m$ paths of the type $-a\to a\to -a$, related with $K^{(2)}$. Then, this particular trajectory will contribute to the propagator with a term $$\begin{aligned} K^{(\ell_1,\ell_2,m)}&\left(-a,\frac{t}{2} \Big\vert -a,-\frac{t}{2}\right) = \nonumber \\ & \hspace{1cm} K^{(0)} \frac{(-g_0^2 t)^{\ell_1+\ell_2+2m}}{(\ell_1+\ell_2+2m)!} \Gamma^{\ell_1+\ell_2+2m} \ .\end{aligned}$$ By carefully counting the number of different paths which contribute to each trajectory labeled by $(\ell_1,\ell_2,m)$ and summing up, we finally arrive at the expression for the conditional probability, $$\begin{aligned} P\left(-a,\frac{t}{2}\Big\vert -a,-\frac{t}{2}\right)=\frac{1}{2}K^{(0)} \times \big(1+e^{-t/\tau_k}\big)\; . \label{eq:P-a-a}\end{aligned}$$ On the other hand, by using the same formalism, we easily find the expression for the conditional probability of finding the system in the state $x = a$ at time $t/2$, provided it was in the state $x = -a$ at a previous time $-t/2$, $$\begin{aligned} P\left(a,\frac{t}{2}\Big\vert -a,-\frac{t}{2}\right)=\frac{1}{2}K^{(0)} \times \big(1-e^{-t/\tau_k}\big)\; . \label{eq:P+a-a}\end{aligned}$$ In Eqs. (\[eq:P-a-a\]) and (\[eq:P+a-a\]), the inverse time parameter $\tau_k^{-1}$, which is equivalent to the Kramers escape rate, is given by $\tau_k^{-1}=r_{\rm mult}=2 g_0^2\ \Gamma$. Using Eq. (\[eq:Gamma\]), it is explicitly written as $$\begin{aligned} r_{\rm mult}=g_0^2 \frac{ \sqrt{g_a^2 U''_{\rm eq}(a) g_0^2 |U''_{\rm eq}(0)|}}{\pi}\; e^{-\frac{\Delta U_{\rm eq}}{\sigma^2}} \ , \label{eq:rmult}\end{aligned}$$ with $ \Delta U_{\rm eq}=U_{\rm eq}(0) - U_{\rm eq}(a)$. This is one of the main results of our paper. Comparing Eq. (\[eq:rmult\]) with the classical result of Eq. (\[eq:KramersEscapeRate\]), we clearly see the effect of the multiplicative noise. On one hand, the role of the original potential $U(x)$ is now played by the equilibrium potential $U_{\rm eq}(x)$ given by Eq. (\[eq:Ueq\]). This potential depends not only on the diffusion function $g(x)$ and the noise, but also on the stochastic prescription $\alpha$ that defines the original Langevin equation. On the other hand, the diffusion function $g(x)$ renormalizes the curvature of the equilibrium potential, in such a way that $\omega_{\rm min}=g_a^2 U''_{\rm eq}(a)$ and $\omega_{\rm max}=g_0^2 U''_{\rm eq}(0)$. There is also a global scaling factor given by $g^2(0)$. In order to gain more insight on this result, let us compare the Kramers escape rate with the expression of $r_{\rm mult}$. Using Eq. (\[eq:rmult\]), both quantities can be compared, since $r_{\rm add}=lim_{g\to 1} r_{\rm mult}$. We obtain, in the weak noise approximation, $$\frac{r_{\rm mult}}{r_{\rm add}}=|g_0|^{1+2\alpha} |g_a|^{3-2\alpha} \left(1+O(\sigma^2)\right) \; . \label{eq:raddmult}$$ It can be noticed that the relation between both escape rates does not depend on details of $g(x)$, but on its value at each maxima of $-V(x)$, $x=\pm a$ and $x=0$. As expected, Eq. (\[eq:raddmult\]) depends on the stochastic prescription parameter $\alpha$. For instance, in the case of the Stratonovich prescription, $\alpha=1/2$, $r_{\rm mult}/r_{\rm add}=g_0^2 g_a^2$. In this case, $g_0$ and $g_a$ have the same weight. On the other hand, in the Itô interpretation $\alpha=0$, $r_{\rm mult}/r_{\rm add}=g_0 g_a^3$ while in the thermal prescription, $\alpha=1$, $r_{\rm mult}/r_{\rm add}=g_0^3 g_a$. Indeed, Eq. (\[eq:raddmult\]) is invariant under the transformation $$\begin{aligned} \alpha & \longleftrightarrow 1-\alpha \\ 0 & \longleftrightarrow a \end{aligned}$$ which is nothing but a time reversal transformation [@Arenas2012-2]. The simplest way to understand this symmetry is by noting that the instanton solution $x_I(t)$ interpolates between the states $x=-a$ and $x=0$. The time reversal solution $x_I(-t)$ makes the inverse trajectory, [*i.e.*]{}, connecting $x=0$ with $x=a$. However, if the forward time process evolves with the $\alpha$ prescription, the backward evolution takes place with the $1-\alpha$ prescription. In this sense, one process is the time reversal conjugate of the other one. For this reason, the thermal prescription $\alpha=1$ is also called the anti-Itô interpretation. In fact, the only time reversal invariant prescription is the Stratonovich one, $\alpha=1/2$. For details on the time reversal transformation in multiplicative noise dynamics, please see Refs. [@arenas2012; @Arenas2012-2; @Miguel2015]. Let us finally mention that the escape rate in the multiplicative case may be greater or lower than in the additive case, depending essentially on the values of $g(0)$ and $g(a)$. Moreover, if the diffusion function $g(x)$ locally approaches zero at either $x=a$ or $x=0$, the escape rate goes to zero. This effect can be understood from the fact that the effective curvature, $\omega_{\rm min}=g_a^2 U''_{\rm eq}(a)$ or $\omega_{\rm max}=g_0^2U''_{\rm eq}(0)$ approaches zero when $g_a\to 0$ or $g_0\to 0$, respectively. Of course, our approximation $t \gg \tau_k$ is no longer valid in this limit. Summary and conclusions {#sec:discussion} ======================= We have considered the problem of a particle in a symmetric double-well potential $U(x)$, with a dynamics driven by an overdamped multiplicative Langevin equation characterized by a symmetric diffusion function $g(x)=g(-x)$. The stochastic differential equation was defined in the [*generalized Stratonovich*]{} prescription, parametrized by a continuum parameter $0\le\alpha\le 1$. This prescription contains the usual stochastic interpretations for particular values of the parameter $\alpha$. We have provided a path integral technique to compute conditional probabilities in the weak noise approximation. It was introduced a local time reparametrization which allows to exactly integrate fluctuations. Conditional probabilities were computed for long time intervals by generalizing the instanton/anti-instanton diluted gas approximation, already developed for the additive noise case [@Caroli1981]. From these probabilities, the escape rate was computed in the same approximation and the result was compared with the Kramers escape rate for additive noise dynamics. The main result of the paper is given by Eq. (\[eq:rmult\]). We found that the general structure of the escape rate keeps the Arrhenius form of the Kramers result. The main corrections are twofold. First, the equilibrium potential $U_{\rm eq}(x)$ of Eq. (\[eq:Ueq\]) plays the role of the bare potential $U(x)$. The potential $U_{\rm eq}(x)$ is the solution of the static Fokker-Plank equation and is generally different from $U(x)$ in the multiplicative noise case. Indeed, the only stochastic prescription in which $U_{\rm eq}(x)=U(x)$ is the anti-Itô prescription $\alpha=1$. On the other hand, the prefactor is modified by a renormalization of the curvatures produced by the diffusion function. Moreover, there is a global scale factor $g^2(0)$ that has its origin in the time reparametrization necessary to correctly compute fluctuations. In the weak noise limit, we found a simple relation between the Kramer escape rates computed with additive and multiplicative noise, given by Eq. (\[eq:raddmult\]). The obvious consistency check is that $r_{\rm mult}/r_{\rm add}=1$ in the limit $g(x)\to 1$. In addition, we observe that $g(0)$ and $g(a)$ enter with different weights depending on the prescription parameter $\alpha$. These weights are consistent with a time reversal transformation, that relates a stochastic process in the $\alpha$ prescription with its time reversal conjugate $1-\alpha$. Indeed, the Stratonovich convention $\alpha=1/2$ is the only one with time reversal invariance and, in this case, both maxima enter with the same weight. Although we have presented results for a system with full reflection symmetry $x\to -x$, the methods developed in this paper are completely general. We hope to communicate results for a more general non-symmetric case in the near future. Moreover, having analytic expressions for the conditional probability we can face the problem of stochastic resonance in multiplicative noise processes in a more solid bases. Zero modes in the multiplicative case {#Ap:Zeromode} ===================================== The relation of zero modes of the fluctuation operator and translation invariance is very well known in quantum mechanics [@Coleman1979], as well as in additive noise stochastic dynamics [@Caroli1981]. In this appendix, we focus on the effect produced by the diffusion function $g(x)$ in a multiplicative noise stochastic system. In order to compute fluctuations we perform a local time reparametrization given by Eq. (\[eq:reparametrization\]). We are lead to the computation of the integral $$I_F=\int [{\cal D}\delta x] \; e^{-\frac{1}{2}\int d\tau \delta x(\tau) \left(-\frac{d^2~}{d\tau^2}+W[x_{cl}]\right)\delta x(\tau)}\ ,$$ where $W$ is given by Eq. (\[eq:W\]). To compute it, we expand fluctuations in eigenfunctions of the fluctuation operator $\delta x(\tau)=\sum_n c_n \psi_n(\tau)$, where $$\left(-\frac{d^2~}{d\tau^2}+W[x_{cl}]\right)\psi_n(\tau)=\lambda_n\psi_n(\tau) \label{app:eigenfunction}$$ and the orthogonal eigenfunctions are normalized as $\int d\tau \psi_n(\tau)\psi_m(\tau)=\delta_{n,m}$. Then, $$I_F=\int \left(\Pi_k dc_k\right) \exp\left(-\frac{1}{2}\sum_n \lambda_n c_n^2\right) \label{app:IFc}$$ Consider, for simplicity, the instanton solution written in the reparametrized time $x_{cl}(\tau)=x_I(\tau)$, that interpolates between $x=-a$ for $\tau\to -\infty$ and $x=0$ for $\tau\to \infty$. Since the action is invariant under time translation, the instanton configuration $x_I(\tau-\tau_0)$ is a solution of the equation of motion for any value of $\tau_0$. This fact produces a zero eigenvalue of the fluctuation operator. In fact, it is simple to show that the function $$\psi_0(\tau)= A \; g^2(x_I(\tau)) \frac{d x_I}{d\tau}\; ,$$ with $A$ a normalization constant, satisfies Eq. (\[app:eigenfunction\]) with $\lambda_0=0$. On the other hand, the condition $\int d\tau \psi_0^2(\tau)=1$ fixes the arbitrary constant $A$ in the following way $$A^{-2}=\int_{-\infty}^{+\infty} d\tau g^4(x_I(\tau)) \left(\frac{d x_I}{d\tau}\right)^2 \; .$$ In the thin wall approximation, $(dx_I/d\tau)^2$ is a strongly localized function around $\tau\sim \tau_0$. Then, with a good accuracy, we can make the approximation $$\begin{aligned} A^{-2}&\sim g^4_a \int_{-\infty}^{+\infty} d\tau \left(\frac{d x_I}{d\tau}\right)^2 \sim g^4_a S_I \ ,\end{aligned}$$ where $S_I$ is the classical action evaluated at the instanton solution. Thus, in this approximation, accurate for high barriers, we find, $A=1/g^2_a \sqrt{S_I}$. Then, the zero mode is written as $$\psi_0(\tau)= \frac{1}{g_a^2\sqrt{S_I}} \; g^2(x_I(\tau)) \frac{d x_I}{d\tau}\; . \label{app:psi0}$$ The existence of this zero mode makes the integration over $dc_0$ in Eq. (\[app:IFc\]) divergent. Thus, in order to cure this problem we need to correctly take into account translation invariance. Let us expand fluctuations in the following way, $$\delta x(\tau)= c_0 \psi_0(\tau-\tau_0)+\sum_{k=1}^\infty c_k \psi_k(\tau-\tau_0)$$ where $\psi_k$ are eigenvectors with eigenvalues $\lambda_k\neq 0$ and $\psi_0$ is given by Eq. (\[app:psi0\]). Computing the variation of fluctuations under time translation, we have that $$d\delta x(\tau)= \frac{d x_I}{d\tau} d\tau_0\ . \label{app:dxt0}$$ On the other hand, a variation in the zero mode reads $$d\delta x(\tau)= \frac{1}{g_a^2\sqrt{S_I}} \; g^2(x_I(\tau)) \frac{d x_I}{d\tau} dc_0 \ . \label{app:dxc0}$$ Comparing Eqs. (\[app:dxt0\]) and (\[app:dxc0\]), and using the reparametrization identity $d\tau/dt=g^2(x_I)$, we immediately find $$dc_0=g_a^2 \sqrt{S_I} dt_0 \ .$$ In this way, $$\begin{aligned} I_F&=\int g_a^2 \sqrt{S_I} dt_0\left( \Pi_{n\neq 0} \lambda_n^{-1/2}(\tau_0)\right)\nonumber \\ &=\int dt_0 \;g_a^2\sqrt{S_I}\; \left( {\det}'\left(-\frac{d^2~}{d\tau^2}+W[x_{cl}]\right)\right)^{-1/2}\end{aligned}$$ where the prime means that the determinant should be computed without the zero mode. In this way, the usual interpretation of the zero mode as an integration in the collective variable $dt_0$ is still valid in the multiplicative case. However, the constant of proportionality is renormalized by the diffusion function $g_a^2$, computed at the minimum of the potential. The same reasoning applies to the anti-instanton solutions. However, in this case the variation is proportional to $g_0^2 \sqrt{S_A} dt_1$, where $g_0^2$ is evaluated at the maximum of the potential and $S_A$ is the classical action evaluated at the anti-instanton solution. This analysis leads to Eq. (\[eq:K01t0t1\]) for $K^{(1)}$. The Brazilian agencies, [*Fundação de Amparo à Pesquisa do Rio de Janeiro*]{} (FAPERJ), [*Conselho Nacional de Desenvolvimento Científico e Tecnológico*]{} (CNPq) and [*Coordenação de Aperfeiçoamento de Pessoal de Nível Superior*]{} (CAPES) - Finance Code 001, are acknowledged for partial financial support. MVM is partially supported by a Post-Doctoral fellowship by CAPES. [54]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**](https://books.google.com.br/books?id=xAa3CgAAQBAJ), Advanced Database Research and Development Series (, , ) [****,  ()](\doibase https://doi.org/10.1016/S0031-8914(40)90098-2) [****,  ()](\doibase 10.1103/RevModPhys.62.251) [****,  ()](\doibase 10.1103/PhysRevLett.62.493) [****,  ()](\doibase 10.1103/PhysRevA.41.644) [****,  ()](\doibase 10.1103/PhysRevA.41.657) [****, ()](\doibase 10.1103/PhysRevA.42.1982) [****,  ()](\doibase 10.1088/1367-2630/7/1/017) [****,  ()](\doibase 10.1103/PhysRevE.76.031128) [****,  ()](http://stacks.iop.org/0295-5075/54/i=1/a=028) [****,  ()](http://www.sciencedirect.com/science/article/pii/S0378437101005106) [****,  ()](\doibase 10.1103/PhysRevE.76.011123) [****,  ()](\doibase 10.1103/PhysRevLett.104.170602) [****,  ()](\doibase 10.1103/PhysRevE.83.041113) [****,  ()](\doibase 10.1103/PhysRevB.58.14937) [****,  ()](http://stacks.iop.org/1742-5468/2014/i=9/a=P09008) [****,  ()](\doibase https://doi.org/10.1016/j.physa.2018.06.126) [****,  ()](\doibase 10.1080/00018730050198152),  [****,  ()](\doibase 10.1103/PhysRevLett.73.3395) [****, ()](\doibase 10.1103/PhysRevLett.75.1691) [****,  ()](\doibase 10.1103/PhysRevE.67.046110) [****,  ()](\doibase 10.1103/PhysRevLett.115.158101) [****,  ()](http://stacks.iop.org/0295-5075/113/i=1/a=10009) [****,  ()](http://stacks.iop.org/0305-4470/14/i=11/a=006) [****,  ()](\doibase 10.1137/0143037),  [****,  ()](\doibase 10.1140/epjst/e2007-00173-0) [****,  ()](\doibase http://dx.doi.org/10.1016/S0378-4371(02)00493-4),  [****, ()](\doibase https://doi.org/10.1016/j.chaos.2005.02.026) [****, ()](\doibase 10.1088/0256-307x/23/12/016) [****,  ()](\doibase 10.1371/journal.pone.0017104) [****,  ()](\doibase 10.3938/jkps.58.1567) “,” in [**](\doibase 10.1002/9781118949702.ch7) (, ) Chap. , pp. ,  [****,  ()](\doibase 10.1103/PhysRevE.94.012101) [****,  ()](\doibase 10.1103/PhysRevX.3.011007) @noop [**]{} (, , ) [**](https://books.google.com.br/books?id= vKStkQEACAAJ) (, ) [****,  ()](http://stacks.iop.org/1742-5468/2010/i=11/a=P11018) [****, ()](\doibase 10.1103/PhysRevE.81.051113) [****, ()](\doibase 10.1103/PhysRevE.85.041122) [****,  ()](http://stacks.iop.org/1742-5468/2012/i=12/a=P12005) [****,  ()](\doibase 10.1103/PhysRevE.91.042103) [****,  ()](http://stacks.iop.org/1742-5468/2016/i=5/a=053207) @noop [**]{} (, , ) [****,  ()](\doibase 10.1103/PhysRevE.99.032125) @noop [**]{} (, , ) “,” in [**](\doibase 10.1007/978-1-4684-0991-8_16),  (, , ) pp.  [****, ()](\doibase 10.1007/BF01009609) [****, ()](\doibase 10.1007/BF01106788) [****,  ()](\doibase 10.5169/seals-114941) @noop [****,  ()]{} [****,  ()](\doibase 10.1070/PU1994v037n08ABEH000038) [****,  ()](\doibase 10.1103/PhysRevD.16.408) [****,  ()](\doibase https://doi.org/10.1016/0370-2693(80)91014-X) [****,  ()](http://stacks.iop.org/1751-8121/41/i=30/a=304006)
--- abstract: | In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D’Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called *periodic* $\sigma$-harmonic sense preserving homeomorphisms of $\mathbb C$ onto itself. author: - | Giovanni Alessandrini\ Dipartimento di Matematica e Informatica, Università di Trieste\ Via Valerio 12/b, 34100 Trieste, Italia, e-mail: `[email protected]`\ \ Vincenzo Nesi\ Dipartimento di Matematica, La Sapienza, Università di Roma,\ P. le A. Moro 2, 00185 Roma, Italia, e-mail: `[email protected]` title: | Beltrami operators,\ non–symmetric elliptic equations\ and quantitative Jacobian bounds --- [*2000 AMS Mathematics Classification Numbers*: 30C62, 35J55]{} [*Keywords*: Beltrami operators, quasiconformal mappings]{} Introduction. ============= In order to explain the results of this paper and their motivations, it is necessary to introduce a number of topics, and to illustrate their mutual relationships. These topics are Beltrami operators and their associated concept of $G$-convergence, non-symmetric elliptic operators in divergence form and $H$-convergence, $\sigma$-harmonic mappings. The $G$-convergence of Beltrami operators and the $K>3$ conjecture. ------------------------------------------------------------------- Recently Iwaniec et al. [@B0] and Bojarski et al. [@B], introduced a notion of $G$-convergence for Beltrami operators, aimed at generalizing to this context the well-known theory of $G$-convergence initiated by Spagnolo [@spa] and De Giorgi [@despa]. Let us recall their definitions and the main conjecture in [@B]. Let $\Omega$ be a bounded, simply connected open subset of $\mathbb R^2$, and, as usual, let us identify points $x=(x_1,x_2)\in\mathbb R^2$ with points $z\in \mathbb C$ through the relation $z=x_1+ix_2$ . Let $\nu$ and $\mu$ be two complex valued measurable functions defined on $\Omega$ and satisfying, for some $K\geq 1$, the following ellipticity condition $$\label{ellQC} |\mu|+|\nu|\leq \frac{K-1}{K+1} \ .$$ Consider the following first order non homogeneous Beltrami equation $$\label{1stordernh} \begin{array}{ll} f_{\overline{z}}-\mu f_z -\nu \overline{f_z}= g \ . \end{array}$$ Given a sequence of pairs of Beltrami coefficients $(\mu_j, \nu_j)$ and an extra pair $(\mu,\nu)$ all satisfying , for a fixed $K\geq 1$, one denotes by $\mathcal B_j$, $\mathcal B$ the differential operators defined as follows $$\label{boj} \mathcal B_j := \frac{\partial}{ \partial \overline{z}}-\mu_j \frac{\partial}{ \partial z} -\nu_j \overline{\frac{\partial}{ \partial z}} \ ,$$ $$\label{bo} \mathcal B := \frac{\partial}{ \partial \overline{z}}-\mu \frac{\partial}{ \partial z} -\nu \overline{\frac{\partial}{ \partial z}} \ ,$$ so that (\[1stordernh\]) can be rewritten as $$\mathcal B f= g \ .$$ The authors in [@B0] introduce the following definition, and prove Theorem \[main-B\] below. The sequence of differential operators $ \mathcal B_j$ is said to $G$-converge to $ \mathcal B$ if, for any sequence $f_j\in W^{1,2}(\Omega;\mathbb C)$ which converges weakly to $f\in W^{1,2}(\Omega;\mathbb C)$, and such that $\mathcal B_j f_j$ converges strongly in $L^2(\Omega;\mathbb C)$, one has $$\lim_{j\to +\infty} \mathcal B_j f_j =\mathcal B f$$ strongly in $L^2(\Omega;\mathbb C)$. \[main-B\] For any $K\in [1,3]$, the family of Beltrami operators defined by (\[bo\]) and satisfying (\[ellQC\]) is $G$-compact. In order to explain our new main results and to put the previous one into context, let us begin by explaining the main point in the proof of Theorem \[main-B\]. As previously outlined one of the main results in [@B0] is a compactness result obtained under an assumption of *small ellipticity*, that is, $K\leq 3$ in . The key to this result relies on the following issue. Let $\Omega$ be a bounded, open and convex set. Let $(\mu,\nu)$ be a Beltrami pair satisfying (\[ellQC\]) and let $\Phi$ and $\Psi$ be the solutions to $$\label{pair1storder} \left\{ \begin{array}{lr} \Phi_{\bar{z}}=\mu \Phi_z +\nu \overline{\Phi_z}\ , & \hbox{in $\Omega$}\ ,\\ \mathfrak{Re} \Phi = x_1 \ , & \hbox{on $\partial \Omega$}\ , \\ \Psi_{\bar{z}}=\mu \Psi_z +\nu \overline{\Psi_z}\ ,& \hbox{in $\Omega$}\ ,\\ \mathfrak{Re} \Psi = x_2\ , & \hbox{on $\partial \Omega$} \ , \end{array} \right.$$ where the boundary conditions are understood in the sense of $W^{1,2}(\Omega)$ traces. The pair $(\Phi,\Psi)$ is called a primary pair. In [@B] the authors formulate the following conjecture. \[conj1\] Let $(\mu,\nu)$ be complex valued measurable coefficients satisfying (\[ellQC\]). Then the pair of quasiconformal mappings $\Phi$ and $\Psi$ defined by (\[pair1storder\]) satisfies the following pointwise inequality: $$\label{det>0C} \begin{array}{llll} \mathfrak{Im} (\Phi_z \overline{\Psi_z})>0&\hbox{almost everywhere}&\hbox{in}&\Omega. \end{array}$$ In [@B0; @B] it is proven that, if Conjecture \[conj1\] holds, then Theorem \[main\] follows. As a consequence of our results we prove that holds and therefore we obtain the following result. \[main\] For any $K\in [1,+\infty)$, the family of Beltrami operators defined by (\[bo\]) and satisfying (\[ellQC\]) is $G$-compact. Very recently, Bojarski [@bo2] has proved a result which also implies Theorem \[main\] but does not solve Conjecture . More precisely he has proven that given $\Omega$ and a Beltrami pair $(\mu,\nu)$ satisfying there exists a [*primary pair*]{} $(\Phi,\Psi)$ so that $\Phi$ and $\Psi$ are quasiconformal mappings of the complex plane onto itself satisfying the Beltrami equations with coefficients $\mu$ and $\nu$ and satisfy . Bojarski’s primary pair is obtained by requiring the so-called [*hydrodynamical normalization*]{}, that is, by looking for a globally homeomorphic solution of $\mathbb C$ onto itself obtained as follows. First extend $(\mu,\nu)$ to be zero in the complement of $\Omega$. Then look for a solution of the new Beltrami equation defined on $\mathbb C$. Such a solution will be holomorphic near infinity. Then normalize the behaviour at infinity of such function. By the seminal work of Bojarski (see the references of [@bo2]), it is known that one obtains a quasiconformal mapping of $\mathbb C$ onto itself. This beautiful construction however does not set the question of whether the Dirichlet data in will provide us with a primary pair. We prove that this is the case in Theorem \[conj2\]. In fact we provide a large class of Dirichlet boundary data achieving the desired task. We use the combination of Theorem \[Kneser\] and Theorem \[t3.1\]. See Corollary \[pr-sol\]. Second order equations in divergence form, ellipticity and $H$-convergence. --------------------------------------------------------------------------- It is well known that Beltrami equations with complex dilatations $\nu$ and $\mu$ give rise in a very natural way to second order elliptic operators whose coefficient matrices $\sigma$ depend in an explicit way upon $\nu$ and $\mu$ and conversely. A brief review will be offered in the following subsection. The authors in [@B0; @B] use the notion of $G$-convergence for Beltrami operators also to induce a concept of $G$-convergence for second order non-symmetric operators in divergence form (see Definition 2 in [@B]) and to treat the $G$-convergence of second order non-divergence equations (see [@B0]). We shall not enter such issues in this note, however we observe that it is also instructive to recall the notion $H$-convergence introduced by Murat and Tartar for possibly non-symmetric, elliptic operators in divergence form. An easily accessible reference is [@mt]. The original work dates back to 1977 (see the quoted reference for more details). \[def.H\] Consider a bounded, open, simply connected set $\Omega\subset \mathbb R^2$. Given positive constants $\alpha$ and $\beta$, we say that a measurable function $\sigma$, defined on $\Omega$ with values into the space of $2\times2$ matrices, belongs to the class $\mathcal M (\alpha,\beta,\Omega)$ if one has $$\label{ellTM} \begin{array}{ccrllll} \sigma(z) \xi\cdot \xi&\geq& \alpha |\xi|^2&,&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $z\in\Omega$ ,}\\ \sigma^{-1}(z) \xi \cdot \xi &\geq & \beta^{-1} |\xi|^2&,&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $z\in\Omega$ \ .} \end{array}$$ It is obvious that, for $\lambda=\alpha$ and for some $M > 0$, such bounds are equivalent to the usual ellipticity bounds for second order elliptic operators, see for instance [@gt Chapter 8] $$\label{ellGT} \begin{array}{ccrllll} \sigma(z) \xi\cdot \xi&\geq& \lambda |\xi|^2,&&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $z\in\Omega$ ,}\\ \sum\limits_{i,j=1}^2|\sigma_{ij}(z)|^2& \leq & M \ ,& & \hbox{for a.e. $z\in\Omega$ .} \end{array}$$ Yet another notion, originally used for the $H$-convergence is the following. A matrix $\sigma$ with measurable entries belongs to $M(\lambda,\Lambda,\Omega)$ if $$\label{ellTMold} \begin{array}{cclllll} \sigma(z) \xi\cdot \xi&\geq& \lambda |\xi|^2\ ,&&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $z\in\Omega$ ,}\\ |\sigma(z) \xi |& \leq & \Lambda|\xi|\ ,&&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $z\in\Omega$ .} \end{array}$$ However, different ways of bounding sets of matrices $\sigma$ may or may not give rise to compact classes with respect to convergences of weak type. To explain this let us recall the notion of $H$-convergence [@mt]. We say that a sequence of elliptic matrices $\sigma_{j} \in \mathcal M (\alpha,\beta,\Omega)$ $H$-converges to $\sigma_0 \in \mathcal M (\alpha,\beta,\Omega)$ if for any $f\in H^{-1}(\Omega)$ the weak solution $u_j$ to $$\label{2ndorder} \begin{array}{llll} -{\rm div}(\sigma_{j} \nabla u_{j} )=f \ ,& \hbox{in $\Omega$}\ ,& u_{j}\in W^{1,2}_0(\Omega)\ , \end{array}$$ satisfies the following properties $$\label{weakconv} \left\{ \begin{array}{llll} u_{j} \rightharpoonup u_0\ , &\hbox{weakly in $W^{1,2}(\Omega)$}\ ,\\ \sigma_{j} \nabla u_{j}\rightharpoonup \sigma_0 \nabla u_0 \ , &\hbox{weakly in $L^{2}(\Omega)$ ,} \end{array} \right .$$ where $u_0$ denotes the weak solution to $$\label{2ndorder0} \begin{array}{llll} -{\rm div}(\sigma \nabla u_{0} )=f \ , & \hbox{in $\Omega$ ,}& u_{0}\in W^{1,2}_0(\Omega) \ . \end{array}$$ One of the main results in this theory is compactness. Given any sequence $\{\sigma_{j}\}\subset \mathcal M (\alpha,\beta,\Omega)$ there exists a subsequence which $H$-converges to some element of ${\mathcal M} (\alpha,\beta,\Omega)$. It is worth noting here that the compactness does indeed depend on the specific character of the ellipticity bounds given by Murat and Tartar. For instance, it is known that the set of matrices in $M(\lambda,\Lambda,\Omega)$, that is the set constrained by , is *not* compact for $H$-convergence. Murat and Tartar proved that a sequence of matrices in $M(\lambda,\Lambda,\Omega)$ admits (up to subsequence) an $H$-limit in the class $M\left(\lambda,\frac{\Lambda^2}{\lambda},\Omega\right)$. An explicit example given by Marcellini in [@mar] shows that there exist a sequence $\{\sigma_j\}\subset M(\lambda,\Lambda,\Omega)$ such that its $H$-limit $\sigma_0$ is constant (with respect to position) and satisfies $$\inf_{|\xi|=1}\sigma_0\xi\cdot\xi=\lambda \ , \sup_{|\xi|=1}|\sigma_0\xi| =(\Lambda^2/\lambda) \ .$$ Let us also recall that the approach of Murat and Tartar has been later extended to larger classes of operators (under the name of $G$-convergence) by Dal Maso, Chiadò-Piat and Defranceschi [@dmcpdf]. Beltrami equations, second order equations in divergence form and ellipticity. ------------------------------------------------------------------------------ Let us recall now the basic algebraic relationship between second order elliptic equations in divergence form and linear first order systems. Given $\sigma \in {\mathcal M}(\alpha, \beta,\Omega)$, let $u\in W^{1,2}_{\rm loc}(\Omega)$ be a weak solution to $$\label{e2.2} \begin{array}{ll} {\rm div}(\sigma \nabla u )=0 & \hbox{in $\Omega$}\ . \end{array}$$ Then there exists $\tilde u \in W^{1,2}_{\rm loc}(\Omega)$, called the *stream function* of $u$, such that one has $$\label{e2.3} \begin{array}{lllll} \nabla \tilde u= J \sigma \nabla u &\hbox{in}&\Omega&\ ,& J:=\left( \begin{array}{ccc} 0&-1\\ 1&0 \end{array} \right)\ . \end{array}$$ Setting $$\label{e2.4} F=u+i \tilde u$$ one has $F=u+i \tilde u \in W^{1,2}_{\rm loc}(\Omega;\mathbb R^2)$ and one writes, in complex notations, $$\label{1storder} \begin{array}{ll} F_{\bar{z}}=\mu F_z +\nu \bar{F_z}\ , & \hbox{in $\Omega$}\ , \end{array}$$ where, the so called complex dilatations $\mu , \nu$ are given by $$\label{SNU} \begin{array}{llll} \mu=\frac{\sigma_{22}-\sigma_{11}-i(\sigma_{12}+\sigma_{21})}{1+{\rm Tr\,}\sigma +\det \sigma}& \ ,&\nu =\frac{1-\det \sigma +i(\sigma_{12}-\sigma_{21})}{1+{\rm Tr\,}\sigma +\det \sigma}\ , \end{array}$$ and satisfy for some $K\geq 1$ only depending on $\alpha, \beta$, or in other words $F$ is a quasiregular mapping. In this paper we are interested in the opposite route, as well. Given measurable complex valued functions $\mu$ and $\nu$ satisfying , consider the matrix $\sigma$ defined as follows $$\label{S} \sigma:= \left( \begin{array}{lll} \frac{|1-\mu|^2-|\nu|^2}{|1+\nu|^2-|\mu|^2}&\frac{2 \mathfrak{Im} (\nu -\mu)}{|1+\nu|^2-|\mu|^2} \\ \\ \frac{-2 \mathfrak{Im} (\nu +\mu)}{|1+\nu|^2-|\mu|^2}& \frac{|1+\mu|^2-|\nu|^2}{|1+\nu|^2-|\mu|^2} \end{array} \right) \ ,$$ which is obtained just by inverting the algebraic system . One can check [@aron] that if holds for some for given $K\geq 1$, then there exists $\alpha,\beta > 0 $ such that holds for $\sigma$ as defined in . In short, ellipticity in the Beltrami sense implies ellipticity in the Murat & Tartar sense. The exact relationship between $K$ and $(\alpha,\beta)$ will not play a crucial role here. However, we shall prove the following. \[ellEG\] Let $(\mu, \nu)$ satisfy the ellipticity condition , let $\sigma$ be defined via . Then $\sigma$ satisfies with $$\label{alfabeta(K)} \begin{array}{lll} \alpha= \frac{1}{K}&\hbox{and}&\beta =K\ . \end{array}$$ Conversely assume that $\sigma\in \mathcal M (\lambda,\frac{1}{\lambda},\Omega)$ for some $\lambda \in (0,1]$ and let $(\mu, \nu)$ be defined by . Then $(\mu, \nu)$ satisfy the ellipticity condition with $K$ defined as follows $$\label{Koptimal} K=\frac{1+\sqrt{1- \lambda^2}}{\lambda} \ .$$ See Section \[dimo\] for a proof, which also shows the optimality of these bounds. Quasiconformal solutions to . ----------------------------- A question that is crucial in the mere formulation of Conjecture \[conj1\] is the following. *Is it possible to prescribe a Dirichlet boundary data $g$ on the real part of $F$ as defined in so that the solution to with that boundary data is globally one-to-one?* Or, equivalently, for $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$, consider the Dirichlet problem $$\label{2nddir} \left\{ \begin{array}{ll} {\rm div}(\sigma \nabla u )=0 \ ,& \hbox{in $\Omega$} \ ,\\ u=g\ , & \hbox{on $\partial \Omega$}\ . \end{array} \right.$$ *Under which condition on $g$ the mapping $F= u + i \tilde u$ is one-to-one?* We recall that solutions to the Beltrami equation are $K$-quasiregular mapping, therefore the question can be rephrased as requiring a boundary data which give rise to a global quasiconformal solution. Such issues turned out to be very important in applications of very different character [@ADV; @aron; @ln; @anII; @laszlo] and were addressed already in past years. The relevant notion in this context is *unimodality*. Assume that $\partial \Omega$ is a simple closed curve. We say that a continuous, real valued function $g$ on $\partial \Omega$ is *unimodal* if $\partial \Omega$ can be split into two simple arcs on which $g$ is separately monotone (increasing on one arc and decreasing on the other, once the orientation on $\partial \Omega$ is fixed). We shall also say that $g$ is *strictly unimodal* if it is strictly monotone on the same arcs. We shall prove the following. \[Thm:onetoone\]Let $F\in W^{1,2}_{loc}(\Omega,\mathbb{C})$ be a solution to such that $u=\mathfrak{Re} F \in C(\overline{\Omega})$. If $g=u|_{\partial \Omega}$ is unimodal then $F$ is one-to-one in $\Omega$. The above statement summarizes a circle of reasonings which, in the last two decades, has been repeatedly used in various contexts [@AannFI; @ADV; @aron; @an]. See in particular [@aron Proposition 3.7], where indeed an interior Hölder bound for $F^{-1}$ is obtained. A sketch of a proof is given, for the convenience of the reader in Section \[dimo\]. The first result in this direction we are aware of is due to Leonetti and Nesi [@ln Theorem 5]. Indeed they proved a stronger statement. *If $g$ is strictly unimodal and $F \in C(\overline{\Omega};\mathbb{C})$ then $F$ is one-to-one in $\overline{\Omega}$*. In fact, in [@ln] there are two additional assumptions, that $\Omega$ is a disk, and that $\sigma$ is symmetric, that is, in other words, $\mathfrak{Im}\, \nu = 0$. However, such assumptions are indeed immaterial, in fact we can always reduce to the case that $\Omega$ is a disk by a conformal mapping, and if $F$ solves then, as is well-known, it also solves a similar equation with $\nu=0$ and $\mu$ replaced by $$\tilde{\mu} = \mu + \frac{\overline{F_z}}{F_z}\nu \ .$$ Later, a result of the same sort was proven also in [@B Theorem 6.1]. In this case the assumptions are that $F\in W^{1,2}(\Omega,\mathbb{C})$ and that $g=\mathfrak{Re} F_0$ where $F_0$ is a given quasiconformal mapping whose one-to-one image is a convex domain. It is worth noticing that this last set of hypotheses clearly *implies* both $F \in C(\overline{\Omega};\mathbb{C})$ and the unimodality of $g$. $\sigma$-harmonic mappings. --------------------------- Now we review several known results about the so-called $\sigma$-harmonic mappings. We close this subsection by reformulating Conjecture \[conj1\] in the language of $\sigma$-harmonic mappings and stating Theorem \[conj2\] which proves Conjecture \[conj1\]. Possibly because of a slightly different language, several results which were published before [@B; @B0] may have escaped the authors’ attention. We review here those of more immediate relevance for Conjecture \[conj1\] and postpone a few of them to the following Sections. In order to rephrase what is already known it is convenient to use the following notation. We fix $\sigma\in {\mathcal M}(\alpha,\beta,\Omega)$ and we denote by $U=(u_1,u_2)$ the $W^{1,2}(\Omega,\mathbb R^2)$ solution to $$\label{pair2ndorder} \left\{ \begin{array}{lc} {\rm div}(\sigma \nabla u_1)=0 \ ,& \hbox{in $\Omega$}\ ,\\ u_1=x_1\ ,& \hbox{on $\partial \Omega$}\ ,\\ {\rm div}(\sigma \nabla u_2 )=0 \ ,& \hbox{in $\Omega$}\ ,\\ u_2=x_2\ , & \hbox{on $\partial \Omega$} \ .\\ \end{array} \right.$$ Finally we define the stream functions associated to $u_1$ and $u_2$ to be $\tilde u_1$ and $\tilde u_2$ respectively. Using these notations and recalling (\[pair1storder\]), we have the identities $$\label{translation} \begin{array}{lll} \Phi \equiv u_1 +i \tilde u_1&\ ,& \Psi \equiv u_2 + i \tilde u_2\ . \end{array}$$ Alessandrini and Nesi use the terms $\sigma$-harmonic functions and $\sigma$-harmonic mapping for $u_1, u_2$ and $U$ respectively. With this language, one can compute $$\label{equival} \mathfrak{Im} (\Phi_z \overline{\Psi_z})= (1+{\rm Tr\,}\sigma +\det \sigma)\det DU \ .$$ Note also that implies $${\rm Tr\,}\sigma \geq 2\alpha \ , \frac{{\rm Tr\,}\sigma}{\det \sigma} \geq 2\beta^{-1} \ ,$$ and hence $$\label{tracedet} (1+{\rm Tr\,}\sigma +\det \sigma) > 0 \ .$$ The interest of these calculations shall be evident after the following Theorem and Remark. \[conj2\] Let $\sigma\in {\mathcal M}(K^{-1},K,\Omega)$. If $\Omega$ is convex, then the $\sigma$-harmonic mapping $U$ defined by (\[pair2ndorder\]) satisfies $$\label{det>0} \begin{array}{llll} \det DU>0&\hbox{almost everywhere}&\hbox{in}&\Omega \ . \end{array}$$ \[metathm\] It is a straightforward matter to conclude that, by and , Theorem \[conj2\] proves Conjecture \[conj1\] and, consequently, Theorem \[main\]. A proof of Theorem \[conj2\] will be given in Section \[jacobian\]. The first result towards Theorem \[conj2\] was proven by Bauman, Marini and Nesi [@bmn]. They proved the assertion under the assumption that $\sigma$ is symmetric and of class $C^{\alpha}$. A further advance was obtained by Alessandrini and Nesi [@an] under the assumption that $\sigma$ is symmetric with measurable entries. The two papers follow a common scheme, first one proves that under suitable conditions on the boundary data (which are indeed satisfied for the problem when $\Omega$ is convex) the mapping $U$ is one-to-one. Here the guiding light is a conjecture by Radò [@r], which was first proved by H.Kneser [@k] and later, independently, by Choquet [@c], in the case when $U$ is harmonic. See Theorem \[Kneser\] below, for further details. Second, one proves that if $U$ is locally injective, and sense preserving, then $\det DU> 0$ almost everywhere. In this case the paradigmatic result, in the harmonic setting, is due to H. Lewy [@le]. Actually, in the harmonic case, and in the case $\sigma\in C^{\alpha}$, one obtains that $\det DU$ is strictly positive, uniformly on compact subsets. In the case when $\sigma$ has measurable entries, such uniform bound cannot hold true. Instead, in [@an] it is proven that for any subset $D$ compactly contained in $\Omega$ one has $$\label{BMO} \log (\det DU) \in \textrm{BMO}(D)$$ which, as is well-known implies that there exist $C\ , \epsilon > 0$ such that in any square $Q\subset \Omega$ one has $$\label{negint} \left(\frac{1}{|Q|} \int_Q (\det DU)^{\epsilon}dx\right) \left(\frac{1}{|Q|} \int_Q (\det DU)^{-\epsilon}dx\right) \leq C \$$ which clearly implies Theorem \[conj2\]. Therefore, when $\sigma$ is symmetric, the tools to prove Conjecture \[conj1\] were already available. Later Bojarski, D’Onofrio, Iwaniec and Sbordone addressed the more general question in the case when $\sigma$ is not necessarily symmetric. They proved Conjecture \[conj1\] in two cases. First when the coefficients are Hölder continuous so extending the results by Bauman et al. to the non-symmetric case. Second they proved the result when $K\leq 3$ so extending the result of Alessandrini and Nesi to the non-symmetric case in that regime. In the next two Sections we shall show that the procedure outlined above for the symmetric case and developed by the authors in [@an] also apply to the non-symmetric case. In fact these proofs already appeared in 2003 as a part of the *Laurea Thesis* of Natascia Fumolo [@nf], an undergraduate student of the first author. In this paper we present a much shorter version by outlining the very few slight changes needed to adapt the arguments in [@an]. On the other hand, some more delicate issues concerning the precise ellipticity constants, like in Proposition \[ellEG\] are treated in a more efficient way here. In Section \[basic\] below, we summarize some of the results obtained in [@an] which extend to the non-symmetric case in a straightforward fashion. Section \[jacobian\] contains the core results of this paper, the main result being Theorem \[t3.1\]. From the standpoint of primary pairs the main implication is Corollary \[pr-sol\]. In Section \[periodic\] we discuss consequences and improvements to Theorem \[t3.1\] in the case of periodic conductivities $\sigma$, which is relevant in the context of homogenization and also in connection to issues concerning the *rigidity* of gradient fields where quasiconvex hulls are defined either by using affine or periodic boundary conditions. We refer to [@laszlo], [@acn], [@achn], [@a] for more details. The main result here is Theorem \[tA\], which provides a novel, stronger, quantitative formulation of the non-vanishing of the Jacobian determinant, in terms of Muckenhoupt weights. Section \[dimo\] contains proofs of some auxiliary results. The final Section \[app\] collects further developments, remarks and connections with various relevant areas and applications. In §\[area\] we extend some area formulas first discussed in [@anL]. In §\[correct\] we lay a bridge towards the theory of *correctors* in homogenization. Finally §\[AAstala\] develops an application of the Theorem by Astala [@astala], generalizing results in [@ln] and [@anL]. Preliminaries. {#basic} ============== In this Section, $\Omega$ is a simply connected open subset of $\mathbb R^2$ and, for applications which will be discussed in Section \[periodic\], we also admit here that $\Omega$ be unbounded, possibly the whole $\mathbb R^2$. We consider matrix valued functions $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$ as defined in (\[ellTM\]). \[not1\] Let $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$ and let $U=(u_1,u_2)\in W^{1,2}_{\rm loc}(\Omega,\mathbb R^2)$ be $\sigma$-harmonic. We denote by $\tilde{U}:=(\tilde u_1,\tilde u_2)$ the *vectorial stream function* associated to $U$. Moreover, for any given non zero constant vector $\xi$ we set $f=U\cdot \xi +i\, \tilde{U}\cdot \xi$. Let $\Omega\subseteq \mathbb R^2$ be simply connected and open. Let $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$ and let $U=(u_1,u_2)\in W^{1,2}_{\rm loc}(\Omega,\mathbb R^2)$ be $\sigma$-harmonic. If for every non zero $\xi$, $f$ is univalent, then $U$ is univalent. The proof is identical to the proof of Proposition 1 in [@an]. In the latter symmetry of $\sigma$ was assumed but never used. Details can be found in [@nf]. \[t2.2\] Let $\Omega\subseteq \mathbb R^2$ be a simply connected and open set. Let $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$ and let $U=(u_1,u_2)\in W^{1,2}_{\rm loc}(\Omega,\mathbb R^2)$ be $\sigma$-harmonic. Adopt the Notation \[not1\]. We have that the following properties are equivalent: $$\begin{array}{lll} (i)&\hbox{$f$ is locally one-to-one for every non zero vector $\xi$}\ ,\\ (ii)&\hbox{$U$ is locally one-to-one for every non zero vector $\xi$}\ ,\\ (iii)&\hbox{$\tilde{U}$ is locally one-to-one for every non zero vector $\xi$}\ . \end{array}$$ Also in this case, the proof is identical to the proof of Theorem 3 in [@an], since symmetry of $\sigma$ was assumed but never used. In fact, additional equivalent conditions to $(i)-(iii)$ were stated in [@an], which involve the notion of *geometrical critical point*, we omit them here for the sake of simplicity. Details can be found in [@nf]. \[Kneser\] Let $\Omega$ be a bounded open set whose boundary is a simple closed curve and let $\sigma\in {\mathcal M}(\alpha,\beta,\Omega)$. Let $\phi=(\phi_1,\phi_2):\partial \Omega\to \mathbb R^2$ be a sense preserving homeomorphism of $\partial \Omega$ onto a simple closed curve $\Gamma$ which is the boundary of a *convex* domain $D$. Let $U\in W^{1,2}_{\rm loc}(\Omega;\mathbb R^2)\cap C^0(\overline{\Omega};\mathbb R^2)$ be the $\sigma$-harmonic mapping with components $u_1$ and $u_2$ solving $$\left\{ \begin{array}{lccc} {\rm div}(\sigma(x) \nabla u_i(x) )=0 \ ,& \hbox{in $\Omega$}&i=1,2\ ,\\ u_i=\phi_i\ ,& \hbox{on $\partial \Omega$}&i=1,2\ .\\ \end{array} \right.$$ Then $$\hbox{ $U$ is a sense preserving homeomorphism of $\overline{\Omega}$ onto $\overline{D}$}\ .$$ Again, the proof is identical to the proof of Theorem 4 in [@an], and details can be found in [@nf]. Theorem \[Kneser\] generalizes to the measurable, non-symmetric, context the celebrated result of H. Kneser [@k] who solved a problem raised by Radò [@r]. Jacobian of a $\sigma$-harmonic mapping: the bound. {#jacobian} =================================================== The main subject of this Section is the proof of Theorem \[conj2\]. We will preliminarily proof a much more general result, namely Theorem \[t3.1\]. We recall that, given an open set $D\subset \mathbb R^2$, $\phi \in L^1_{\rm loc} (D)$ belongs to ${\rm BMO} (D)$ if $$\| \phi \| _{*} = \sup_{Q\subset D} \left( \frac{1}{\mid Q\mid} \int_Q \mid \phi - \phi_Q\mid \right) <\infty$$ where $Q$ is any square in $D$ and $\phi_Q= \frac{1}{\mid Q\mid} \int_Q \phi$. Recall also that the normed space $({\rm BMO} (D), \|\cdot\|_*)$ is in fact a Banach space. The main object of this Section is the following. \[t3.1\] Let $\Omega$ be an open subset of $\mathbb R^2$, let $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$ and let $U \in W^{1,2}_{\rm loc}(\Omega,\mathbb R^2)$ be a $\sigma$-harmonic mapping which is locally one-to-one and sense preserving. For every $D\subset \subset \Omega$ we have $$\label{e3.1} \begin{array}{ll} \log (\det DU) \in {\rm BMO} (D)&. \end{array}$$ \[pr-sol\] Let $(\mu,\nu)$ be a Beltrami pair satisfying (\[ellQC\]) and let $\Phi$ and $\Psi$ be the solutions to $$\left\{ \begin{array}{lr} \Phi_{\bar{z}}=\mu \Phi_z +\nu \overline{\Phi_z}\ , & \hbox{in $\Omega$}\ ,\\ \mathfrak{Re} \Phi = \phi_1 \ , & \hbox{on $\partial \Omega$}\ , \\ \Psi_{\bar{z}}=\mu \Psi_z +\nu \overline{\Psi_z}\ ,& \hbox{in $\Omega$}\ ,\\ \mathfrak{Re} \Psi = \phi_2\ , & \hbox{on $\partial \Omega$} \ , \end{array} \right.$$ where $\phi=(\phi_1,\phi_2)$, as in Theorem \[Kneser\], defines the convex set $D$. Then $\Phi$ and $\Psi$ are quasiconformal mappings defined on $\Omega$ which satisfy the inequality $$\begin{array}{llll} \mathfrak{Im} (\Phi_z \overline{\Psi_z})>0&\hbox{almost everywhere}&\hbox{in}&\Omega. \end{array}$$ The proof of Theorem \[t3.1\] needs some preparation. It will be presented at the end of this Section. This part requires slightly more extended changes with respect to the work in [@an]. For this reason more details will be given. We recall below two fundamental results, Theorems \[t3.2\] and \[t3.3\], which will be needed for a proof of Theorem \[t3.1\]. \[t3.2\] Let $f$ be a quasiregular mapping on the open set $D\subset \mathbb R^2$, then for every $D^{\prime} \subset \subset D$ $$\log (\det Df) \in {\rm BMO} (D^{\prime}) \ .$$ [**Proof.**]{} See [@re Theorem 1, Remark 2]. $\Box$ \[t3.3\] Let $f:D\to G$ be a quasiconformal mapping, $D,G\subset \mathbb R^2$. For every $D^{\prime} \subset \subset D$, there exists $C>0$ such that $$\begin{array}{llllll} \|v \circ f \|_* \leq C \| v \|_*\ , &{\rm for}&{\rm every}&v\in {\rm BMO}(f(D^{\prime})) &. \end{array}$$ [**Proof.**]{} See [@re Theorem 4] and also [@jones p. 58].$\Box$ The next Theorem requires the notion of adjoint equation for a nondivergence elliptic operator. Let $G \subset \mathbb R^2$ be an open set. Let $a\in {\mathcal M}(\alpha,\beta,G)$. Set $$\begin{array}{ll} L = \sum\limits_{i,j=1}^2 a_{ij}\frac{\partial^2}{\partial x_i \partial x_j}&\ . \end{array}$$ We say that $v\in L^1_{\rm loc} (G)$ is a weak solution of the adjoint equation $$\label{e3.6} \begin{array}{lll} L^* v = 0 \ ,&{\rm in} &G \ , \end{array} %\eqno (3.6)$$ if $$\begin{array}{lllll} \int_G v L u =0\ ,&{\rm for}&{\rm every}& u\in W_0^{2,2}(G)&. \end{array}$$ We remark that, usually, the ellipticity bounds for $a$ are expressed in the form (\[ellGT\]), rather than (\[ellTM\]), but this plays no role here. \[t3.4\] For every $w\in L^2_{\rm loc}(G)$, $w\geq 0$, which is a weak solution of the adjoint equation (\[e3.6\]) we have $$\label{e3.7} \left(\frac{1}{\mid Q \mid} \int_Q w^2\right)^{\frac{1}{2}} \leq C\left( \frac{1}{\mid Q \mid} \int_Q w\right) %\eqno (3.7)$$ for every square $Q$ such that $ 2 Q \subset G$. Here $C>0$ only depends on the ellipticity constants $\alpha$ and $\beta$. [**Proof.**]{} This Theorem is a slight adaptation between [@bau Theorem 3.3] and [@fs Theorem 2.1]. A proof is readily obtained by following the arguments in [@fs]. The only additional ingredient which is needed here, is the observation that, with no need of any smoothness assumption on the coefficients of $L$, for the special case when the dimension is two (which is of interest here), for any ball $B \subset G$ and any $f \in L^2(B)$ there exists and it is unique, the strong solution $$u \in W^{2,2}(B)\cap W_0^{1,2}(B)$$ to the Dirichlet problem $$\left \{ \begin{array}{lll} Lu = f \ , & {\rm in}& B\ , \\ u=0 \ , &{\rm on} &\partial B\ , \end{array} \right.$$ see [@ta Theorem 3]. $\Box$ **Proof of Theorem \[t3.1\]. Preparation.** Let $U= (u_1,u_2)$ satisfy the hypotheses of Theorem \[t3.1\] and let $$\label{e3.8} f = u_1+ i \tilde{u}_1$$ be the quasiregular mapping introduced in Notation \[not1\] with $\xi=(1,0)$. In view of Theorem \[t2.2\], for every $z\in \Omega$, we can find a neighborhood $D$ of $z$, $D\subset \subset \Omega$ such that $U|_D$ and $f|_D$ (i.e. the restrictions of $U$ and $f$ to $D$) are univalent. Therefore, for the proof of Theorem \[t3.1\], it suffices to show that (\[e3.1\]) holds for any sufficiently small $D \subset \subset \Omega$, such that $U|_D$ and $f|_D$ are univalent. We set $$G = f|_D(D)$$ and $V:G\to \mathbb R^2$ given by $$\label{e3.9} V= U|_D \circ (f|_D)^{-1} %\eqno(3.9)$$ where, by definition $(f|_D)^{-1}:G\to D$. From now on, with a slight abuse of notation, we will drop the subscripts denoting restrictions to $D$. We have $DU = (DV \circ f) Df$, and hence $$\label{e3.10} \log (\det DU) = \log (\det DV) \circ f + \log (\det Df)~~. %\eqno (3.10)$$ In view of Theorems \[t3.2\] and \[t3.3\], the thesis will be proven as soon as we show that $\log (\det DV)$ belongs to on compact subsets of $G$. The advantage in replacing $U$ by $V$, lies in the observation that, in contrast with $\det DU$, $\det DV$ satisfies an equation of the type (\[e3.6\]) for a suitable choice of the operator $L^*$. In fact, letting $v_1$ and $\tilde{v}_1$ be the first component of $V$ and its stream function respectively, we can compute $$v_1(z) = u_1 \circ f^{-1}(z) = u_1 \circ (u_1+ i \tilde{u}_1)^{-1}(z) = x_1\ ,$$ $$\label{e3.11} \tilde{v}_1(z) = \tilde{u}_1 \circ f^{-1}(z) = \tilde{u}_1 \circ (u_1+ i \tilde{u}_1)^{-1}(z) = x_2\ . %\eqno (3.11)$$ Moreover, by definition, $$\label{e3.12} \nabla \tilde{v}_1 = J \tau \nabla v_1 \ , %\eqno (3.12)$$ where $$\label{e.13} \tau = T_f \sigma=\frac{Df \sigma Df^T}{\det Df} \circ f^{-1}\ . %\eqno (3.13)$$ Hence, using (\[e3.11\]) and (\[e3.12\]) $$\left( \begin{array}{ll} 0\\ 1 \end{array}\right) = \left(\begin{array}{ll} 0&-1\\ 1&0 \end{array} \right) \left( \begin{array}{ll} \tau_{11}&\tau_{12}\\ \tau_{12}&\tau_{22} \end{array} \right) \left( \begin{array}{ll} 1\\ 0 \end{array} \right)\ ,$$ that is $$\label{e3.14} \tau= \left( \begin{array}{ll} 1&b\\ 0&c \end{array} \right) %\eqno (3.14)$$ where, by construction, $$\label{cb} \begin{array}{lll} c= &\det \tau &= \det (\sigma \circ f^{-1})\in L^{\infty}(G)\ ,\\ b= &\tau_{12}&=(\sigma_{12}-\sigma_{21})\circ f^{-1}\in L^{\infty}(G)\ . \end{array}$$ For a given $\sigma$, let us denote $$\begin{array}{lll} \alpha_{\sigma}= \hbox{ $ \mathrm{ess}\inf\limits_{z\in \Omega} \left\{ \sigma(z)\xi\cdot \xi \right. $ such that $ \left.\ \xi\in\mathbb{R}^2, \ |\xi|=1 \right\} $}\ , \\ \frac{1}{\beta_{\sigma}}= \hbox{$\mathrm{ess}\inf\limits_{z\in \Omega}\left\{(\sigma(z))^{-1}\xi\cdot \xi \right.$ such that $ \left. \ \xi\in\mathbb{R}^2,\ |\xi|=1\right\}$}\ , \end{array}$$ that is, $\alpha_{\sigma}, \beta_{\sigma}$ are the best ellipticity constants $\alpha, \beta$ for which $\sigma \in \mathcal M (\alpha,\beta,\Omega)$ holds. We restrict our attention to the case when $\alpha_{\sigma}={\beta_{\sigma}}^{-1}:=K^{-1}$. A calculation that we omit shows that, if $\alpha_{\tau} , \beta_{\tau}$ are defined accordingly for $\tau$ in $G$, we have $$\begin{array}{lll} \label{e3.15} \alpha_{\tau}= {\rm ess}\inf\limits_{z\in G} \left\{\frac{c(z)+1 -\sqrt{(c(z)-1)^2+b(z)^2}}{2}\ \right\}\ ,\\ \frac{1}{\beta_{\tau}} = {\rm ess} \inf\limits_{z\in G} \left\{\frac{c(z)+1 -\sqrt{(c(z)-1)^2+b(z)^2}}{2 c(z)}\ \right\}\ . \end{array}$$ That is $\tau$ is elliptic in the sense of (\[ellTM\]) and a calculation shows that, in fact, one can take $$\label{alfabeta(tau)} \alpha_{\tau}=\frac{1}{\beta_{\tau}} =1-\sqrt{1-\frac{1}{K^2}} \ .$$ See Section \[dimo\] for a proof. Furthermore, by and (\[e3.11\]), $$\label{integrability} \det DV = \frac{\partial v_2}{\partial x_2} \in L^2(G)\ .$$ Consequently, $v_2$ satisfies $$\hbox{$ \frac{\partial }{\partial x_1} \left(\frac{\partial v_2}{\partial x_1}+b \frac{\partial v_2}{\partial x_2}\right)+\frac{\partial}{\partial x_2} \left(c\frac{\partial v_2}{\partial x_2}\right)= 0$ \ weakly in $G$}\ .$$ Differentiating the equation above with respect to $x_2$, we see that $w=\det DV$ is a distributional solution of $$\begin{array}{lllll} \frac{\partial^2 }{\partial x_1^2} w+ \frac{\partial^2 }{\partial x_1 \partial x_2}(b w)+ \frac{\partial^2 }{\partial x_2^2} (c w) =0 \ , &{\rm in}& G \ , \end{array}$$ that is, it is a distributional solution to the adjoint equation $$\label{e3.16} \begin{array}{lllll} L^* w =0\ ,&{\rm in}& G \end{array} %\eqno (3.16)$$ where $$L= \frac{\partial^2}{\partial x_1^2} + b \frac{\partial^2}{\partial x_1 \partial x_2}+c \frac{\partial^2}{\partial x_2^2}\ .$$ On use of (\[e3.16\]) and (\[e3.15\]) we may now apply Theorem \[t3.4\]. We summarize the resulting statement below. \[prop3.5\] For every square $Q$ such that $2Q\subset G$, we have $$\label{reverse} \left( \frac{1}{\mid Q\mid} \int_Q (\det DV)^2 \right) ^{\frac{1}{2}} \leq C \left(\frac{1}{\mid Q\mid} \int_Q \det DV\right),$$ where $C>0$ only depends on $\alpha$ and $\beta$. **Proof of Theorem \[t3.1\]. Conclusion.** A well known characterization of in terms of the reverse Hölder inequality (see, for instance, [@gcrdf Theorem 2.11 and Corollary 2.18] ), shows that Proposition \[prop3.5\] implies $\log (\det DV) \in \textrm{BMO}(G^{\prime})$ for every $G^{\prime} \subset \subset G$. Thus, possibly after replacing $D$ with $D^{\prime} = f^{-1} (G^{\prime})$, we have, by (\[e3.10\]) and Theorems \[t3.2\] and \[t3.3\] that $\log (\det DU) \in \textrm{BMO}(D)$.$\Box$ **Proof of Theorem \[conj2\].** Apply Theorem \[Kneser\] with $\phi_1=x_1 \ , \phi_2=x_2$ and $D=\Omega$, which, by assumption, is convex. Then use Theorem \[t3.1\] . $\Box$ We recall now that, in view of Remark \[metathm\], the proof of Theorem \[conj2\] concludes also the proof of Conjecture \[conj1\] and of Theorem \[main\]. The proof of Corollary \[pr-sol\] is also immediate. The periodic case. {#periodic} ================== In the homogenization theory, operators with periodic coefficients play an important role. We refer to the wide literature on the subject, see for instance [@blp] and [@m]. We want to remark here that our result has two interesting consequences in that particular setting. We set $Q=(0,1)\times (0,1)$ and we shall deal with functions which are $1$-periodic with respect to each of its variables $x$ and $y$, which we will call $Q$-[*periodic*]{}, or for short, [*periodic*]{}. For a given $2 \times 2$ matrix $A$, we write $U\in W^{1,2}_{\sharp,A}(Q;\mathbb R^2)$ for the space of zero average (on $Q$) vector fields $U$ such that $U-Ax\in W^{1,2}_{\sharp}(Q;\mathbb R^2)$, where $W^{1,2}_{\sharp}(Q;\mathbb R^2)$ denotes the completion of $Q$-periodic function with respect to the $W^{1,2}$ norm (see [@d] for more details). We are especially interested in boundary conditions of periodic type because of their central role in homogenization and in particular in the so-called $G$-closure problems. In fact, our starting point for this investigation has its origin in such type of applications. Given a $2 \times 2$ matrix $ A$, we denote by $U^A=(u_1^A,u_2^A)$ a solution (unique because of our normalization) of $$\label{e1.4} \left\{ \begin{array}{lll} {\rm div} (\sigma \nabla u_1^A )= 0\ , \hbox{ in $ \mathbb R^2$\ ,} \\ {\rm div} (\sigma \nabla u_2^A )= 0\ , \hbox{ in $ \mathbb R^2$\ ,} \\ U^A \in W^{1,2}_{\sharp,A} (\mathbb R^2,\mathbb R^2)\ . \end{array} \right.$$ The auxiliary problem (\[e1.4\]) is usually called the cell problem. Solutions to (\[e1.4\]) will be called, with a slight abuse of language, ${\it periodic} \ \sigma$-harmonic mappings. In the sequel, $\alpha, \beta>0$ and $\sigma \in {\mathcal M}(\alpha,\beta,\mathbb R^2)$ and $Q$-periodic are given. \[tA\] Let $A$ be a non singular $2 \times 2$ matrix and let $U^A$ be a solution to (\[e1.4\]). Then we have $$\label{hom} \hbox{$U^A$ is a homeomorphism of $\mathbb R^2$ onto itself}.$$ Moreover there exists positive constants $C,\delta$ only depending on $\alpha$ and $\beta$ such that, for every square $P\subset \mathbb R^2$ and any measurable set $E\subset P$ we have $$\label{quantitative} \int_E \frac{\det DU^A}{\det A}\geq C\left(\frac{|E|}{|P|}\right)^\delta \int_P \frac{\det DU^A}{\det A}\ .$$ Here, and in the sequel, integration is meant with respect to two-dimensional Lebesgue measure. It is worth observing that, when $P=Q$, the unit square, and $E\subset Q$, we obtain $$\label{global} \frac{|U^A(E)|}{|\det A|}\geq C |E|^\delta \ .$$ Which also trivially implies $$\label{qualiitative} \hbox{$\frac{\det DU^A}{\det A} >0 $ almost everywhere in $\mathbb R^2\ .$}$$ In fact, for any $\sigma$-harmonic homeomorphism $U$ the area formula $$\label{areaf2} |U(E)|=\int_E |\det DU|$$ holds, see [@anL Proposition 4.2], for a proof in the symmetric case, which however applies equally well to the present context. See also the discussion in the Section \[app\] below. It is anticipated that quantitative Jacobian bounds, like the one obtained in , are useful to prove new bounds for effective conductivity i.e. for classes of $H$-limits. See [@n] and [@acn]. In particular [@n Theorem 3.4] gives an explicit improved bound in terms of the constants $C$ and $\delta$ appearing in . Note the relevance of in [@n Definition 3.7] (thanks to the preceding discussion about the role of the boundary conditions in Section 2 of that paper). However, all such developments would require a careful derivation of bounds for $C$ and $\delta$ and are beyond the scope of this note. Before beginning the proof Theorem \[tA\], let us recall some basic facts about Muckenhoupt weights. A non negative measurable function $w=w(z)$ with $z\in \mathbb C$ is an $A_{\infty}$-weight if \(i) there exist constants $C,\delta>0$ such that for every square $P$ and every measurable set $E\subset P$ we have $$\label{5.7} \frac{\int_E w }{\int_P w }\leq C\left(\frac{|E|}{|P|}\right)^\delta.$$ Thus, as is well-known, the $A_{\infty}$ condition is a property of absolute continuity, *uniform at all scales*, of the measure $w\textrm{d}x$ with respect to Lebesgue measure $\textrm{d}x$. The following characterizations of $A_{\infty}$ are also well-known, see for instance [@cf Lemma 5]. \[Ainfequiv\] Condition (i) above is equivalent to (ii) and (iii) below. \(ii) There exist constants $N,\theta>0$ such that for every square $P$ $$\label{5.8} \left(\frac{1}{|P|} \int_P w^{1+\theta} \right)^{\frac{1}{1+\theta}} \leq N\left(\frac{1}{|P|} \int_P w \right).$$ \(iii) There exists constants $M,\eta>0$ such that for every square $P$ and every measurable set $E\subset P$, we have $$\label{5.9} \frac{\int_E w }{\int_P w }\geq M\left(\frac{|E|}{|P|}\right)^\eta.$$ We observe that the quantitative relationships among the pairs of constants $(C,\delta), (N,\theta)$ and $(M,\eta)$ appearing in the equivalent characterizations of $A_{\infty}$ can be constructively evaluated, see Vessella [@vess]. We shall also make use of the following observation. \[sigmatilde\] Let $\sigma\in {\mathcal M}(\alpha,\beta,\Omega)$ and let $u$ be $\sigma$-harmonic in $\Omega$. Then, up to a multiplicative scaling, we have that $u$ is also $\tilde{\sigma}$-harmonic with $$\label{eq:sigmatilde} \tilde{\sigma} = \sqrt{\frac{\beta}{\alpha}}\sigma \in {\mathcal M}\left(\sqrt{\frac{\alpha}{\beta}},\sqrt{\frac{\beta}{\alpha}},\Omega\right) \ .$$ Thus in the proof below, we may assume, without loss of generality, $\sigma\in {\mathcal M}(K^{-1},K,\Omega)$ with $K=\sqrt{\beta / \alpha}$. **Proof of Theorem \[tA\].** It suffices to treat the case when $A$ is the identity matrix $I$ because $U^A= A U^{I}$. From now on, for simplicity, we omit the superscript ${I}$. The proof of (\[hom\]) follows with no substantial changes the one in [@an Theorem 1]. The proof of (\[quantitative\]) consist of showing that $\det DU$ is a Muckenhoupt weight. We observe that the arguments of Theorem \[t3.1\] tell us that $(\det DU)^\epsilon$ is a Muckenhoupt weight for some sufficiently small $\epsilon>0$. Here we improve the result and show that this is true also for $\epsilon=1$. By Remark \[sigmatilde\], we may assume $\sigma\in {\mathcal M}(K^{-1},K,\Omega)$ with $K=\sqrt{\beta / \alpha}$. Using the notation of Section \[jacobian\], we have $U=V \circ f$ where $f$ now is a $K$-quasiconformal homeomorphism of $\mathbb C$ onto itself. Moreover $V$ satisfies (\[reverse\]) for all squares in $\mathbb C$. Recall also that $V$ is a $\tau$-harmonic homeomorphism of $\mathbb C$ onto itself with $\tau$ given by (\[e3.14\]), hence we also have that area formulas of the type (\[areaf2\]) also apply to $V$, and obviously to $f$ because of its quasiconformality. By (\[reverse\]) we deduce that $\det DV$ is an $A_{\infty}$-weight, and for suitable $M,\eta>0$ only depending on $K$, we have $$\label{5.10} \int_F \det DV \geq M \left(\frac{|F|}{|P|}\right)^{\eta} \int_P \det DV$$ for any square $P$ and any measurable set $F\subset P$. Since $f$ is $K$-quasiconformal, we have that $f$ satisfies the following condition, which can be viewed as one of the many manifestations of the bounded distortion property of quasiconformal mappings. *There exist $q\in (0,1)$ depending on $K$ only such that for every square $P\subset \mathbb C$, there exists a square $P^\prime \subset \mathbb C$ such that* $$q P^\prime \subset f(P) \subset P^\prime\ .$$ Here, if $l$ is the length of the side of $P^\prime$, we denote by $q P^\prime$ the square concentric to $P^\prime$ with side $q \cdot l$. We refer to [@lv Proof of Theorem 9.1] for a proof. Therefore, we have $f(E)\subset f(P)\subset P^\prime$ and hence $$\label{5.12} |U(E)| =|V(f(E))|\geq M \left( \frac{|f(E)|}{|P^\prime|} \right)^\eta |V(P^\prime)|\ .$$ Obviously, $$\hbox{$|V(P^\prime)|\geq |V(f(P))|$ and $|P^\prime| =\frac{1}{q^2} |q\,P^\prime| \leq \frac{1}{q^2}|f(P)|$}\ .$$ Therefore $$\label{5.13} |U(E)| \geq Q \,q^{2\eta} \left(\frac{|f(E)|}{|f(P)|}\right)^\eta |U(P)|\ .$$ By Gehring’s Theorem [@geh], we have that $\det Df$ satisfies a reverse Hölder inequality of the form $(ii)$ in Lemma \[Ainfequiv\], with constants only depending on $K$. By $(iii)$ in Lemma \[Ainfequiv\], there exists $L,\rho>0$ only depending on $K$ such that $$\label{5.14} \frac{|f(E)|}{|f(P)|} \geq L \left(\frac{|E|}{|P|}\right)^\rho$$ and finally, by and $$|U(E)| \geq Q ( q^2\, L)^\eta \left(\frac{|E|}{|P|}\right)^{\eta\,\rho} |U(B)|\ .$$ Thus follows.$\Box$ The $A_{\infty}$-property of the Jacobian determinant, obtained in Theorem \[tA\] for the periodic setting, is indeed an improvement of the bound obtained previously and which applies to the wider context of locally injective $\sigma$-harmonic mappings. Local versions of a bound like could be obtained as well for locally injective $\sigma$-harmonic mappings, however it is expected that a quantitative evaluation of the constants might be more involved in this case. Miscellaneous proofs. {#dimo} ===================== **Proof of Theorem \[Thm:onetoone\] (Sketch).** By the well-known Stoïlow representation, see for instance [@lv Chapter VI], there exists a quasiconformal mapping $\chi: \mathbb{C}\rightarrow \mathbb{C}$ such that $F$ factorizes as $F=H\circ\chi$ with $H$ holomorphic in $\chi(\Omega)$. Thus, up to the change of variable $\chi$, one can assume w.l.o.g. $\mu=\nu=0$. Then $u$ is harmonic and $\tilde{u}$ is its harmonic conjugate. Being $g$ unimodal, $u$ has no critical point inside $\Omega$ [@AannFI; @amsiam], moreover, by the maximum principle, for every $t\in(\min g , \max g)$ the level set $\{u>t\}$ is connected and the level line $\{u=t\}$ in $\Omega$ is a simple open arc. On $\{u=t\}$, $\tilde{u}$ has nonzero tangential derivative, hence it is strictly monotone there. Consequently, $F$ is one-to-one on $\Omega$.$\Box$ [**Proof of Proposition \[ellEG\].**]{} The proof of this Proposition is a calculus matter regarding matrices $\sigma$ and complex numbers $\mu, \nu$ linked by the relations , or equivalently . The dependence on the space variables $z=x_1+ix_2$ plays no role at this point, and thus we can neglect it. The inequalities can be viewed as lower bounds on the eigenvalues of the symmetric matrices $\frac{\sigma + \sigma^T}{2}$ and $\frac{\sigma^{-1} + (\sigma^{-1})^T}{2}$. In terms of $\mu , \nu$, the lower eigenvalues of such matrices are given by $$\frac{(1-|\mu|)^2- |\nu|^2}{|1+\nu|^2-|\mu|^2} \ , \ \frac{(1-|\mu|)^2- |\nu|^2}{|1-\nu|^2-|\mu|^2} \ ,$$ respectively. By computing the minima of such expressions as $\mu,\nu \in \mathbb{C}$ satisfy we obtain . It is worth noticing that such minima are achieved when $\nu=|\nu|$ in the first case, and when $\nu=-|\nu|$ in the second case. In either case, the corresponding $\sigma$ turns out to be symmetric. Viceversa, if we constrain $\mu,\nu$ to satisfy both limitations $$\label{constralfabeta} \frac{(1-|\mu|)^2- |\nu|^2}{|1+\nu|^2-|\mu|^2}\geq\lambda \ , \ \frac{(1-|\mu|)^2- |\nu|^2}{|1-\nu|^2-|\mu|^2} \geq\lambda\ ,$$ then the maximum of $|\mu|+|\nu|$ turns out to be $\sqrt{\frac{1-\lambda}{1+\lambda}}$ and follows. Note that in this case the maximum is achieved with $\mu , \nu$ satisfying $ \mu=0$ and $\mathfrak{Re}\nu =0$ which means $$\label{extr-matrices} \sigma=\left( \begin{array}{ccc} a&b\\ -b&a \end{array} \right)\hbox{ with } \ a=\lambda \ , \ b = \pm \sqrt{1-\lambda^2} \ .$$ Let us also recall the well-known fact that, if we a-priori assume $\sigma$ symmetric, then, under the constraints , the maximum of $|\mu|+|\nu|$ becomes $\frac{1-\lambda}{1+\lambda}$, that is $K=\frac{1}{\lambda}$. $\Box$ **Proof of .** As in the Proof of Proposition \[ellEG\], we can neglect the dependence on the space variables $z=x_1+ix_2$. The task here is to evaluate the minimum eigenvalue of the symmetric part of the matrices $\tau$ and of $\tau^{-1}$. It suffices to consider the case $\det \sigma\leq1$. Indeed, up to replacing $\sigma$ with $\sigma^{-1}$ we can always reduce to this case. Set $D=\det \sigma$, $T={\rm\,Tr} \sigma$ and $H=(\sigma_{12}-\sigma_{21})^2$. Elementary computations lead us to minimize the functions $$\label{6a} F(D,H)=\frac{D+1-\sqrt{(D-1)^2+H}}{2} \ ,$$ $$\label{6b} G(F,H)=\frac{F(D,H)}{D}\ ,$$ subject to the constraints $$\label{6c} \frac{T-\sqrt{T^2+H-4D}}{2}\geq \frac{1}{K}\ ,$$ $$\label{6d} \frac{T-\sqrt{T^2+H-4D}}{2D}\geq \frac{1}{K}\ .$$ Note that, being $D\leq 1$, we have that (\[6c\]) is always satisfied if (\[6d\]) holds and also that $G(D,H)\geq F(D,H)$ with equality when $D=1$. Thus we are reduced to compute $$\min\{F(D,H) \, \left|\right. 0\leq D \leq 1 \ , H,T\geq 0 \ ,\hbox{(\ref{6c}) holds}\}=1+\sqrt{1-\frac{1}{K^2}}\ .$$ The minimum is achieved when $$\label{minpoint} \begin{array}{cccc} T=\frac{2}{K}, &D=1,&\hbox{ and} &H=1-\frac{1}{K^2} \end{array}$$ which implies that $\sigma$ has the form (\[extr-matrices\]) with $\lambda = 1/K$. This proves that $\alpha_\tau$ as defined in satisfies . Consequently, by and we also obtain $ \beta_\tau=\frac{1}{\alpha_\tau}$, proving . $\Box$ Further results and connections. {#app} ================================ Area formulas for $\sigma$-harmonic mappings. {#area} --------------------------------------------- One of the original motivations to the study of Theorem \[Thm:onetoone\] came from homogenization and in particular the study of bounds for *effective conductivity*, that is, $H$-limits. So let $\sigma\in{\mathcal M} (\alpha,\beta,\mathbb R^2)$ be $Q$-periodic ($Q= (0,1)\times(0,1) $). By its associated $H$-limit we mean the constant matrix $\sigma_{\rm eff}$ also called the effective conductivity defined as the $H$-limit of $\sigma^{\epsilon}(z):=\sigma(\frac{z}{\epsilon})$ which, as is well-known, it is defined via cell problems as follows. For any vector $\xi\in \mathbb R^2$, one has $$\label{vp} \sigma_{\rm eff}\xi \cdot \xi=\min \left\{ \int_Q \sigma \nabla u \cdot \nabla u \,\left| \right. u-\xi\cdot x\in W^{1,2}_{\sharp}(Q;\mathbb R)\right\}.$$ Let $u^\xi$ be the minimizer of (\[vp\]) and let $\tilde u^\xi$ be its stream function. Using the notation of Section \[periodic\], we have $u^\xi = U^I \cdot \xi$. Set $f^\xi=u^\xi + i \tilde{ u}^{\xi}$. Notice that this quasiconformal mapping coincides with the one introduced in Notation \[not1\] when $U=U^I$. Here we use the superscript $\xi$ just in order to emphasize this dependence. For any nonzero vector $\xi\in \mathbb R^2$ one has $$\sigma_{\rm eff}\xi \cdot \xi=|f^\xi(Q)|.$$ [**Proof.**]{} We refer to [@anL Proposition 4.1]. Again in that context $\sigma$ was assumed to be symmetric but the hypotheses was not used. $\Box$ The previous result transforms the problem of the calculation of the effective conductivity into a geometrical one, finding the area of the set $f^\xi(Q)$. Next result has already been invoked in Section \[jacobian\]. \[area2\] Let $\Omega$ be a bounded, open, simply connected set. Let $\sigma\in {\mathcal M}(\alpha,\beta,\Omega)$ and let $U\in W^{1,2}(\Omega;\mathbb R^2)$ be a univalent $\sigma$-harmonic mapping onto an open set $D$. For any measurable set $E\subset \Omega$ and any function $\phi\in L^1(D;\mathbb R)$ one has $$\int_E \phi(U(x)) |\det DU(x)| \textrm{d}x = \int_{U(E)} \phi(y) \textrm{d}y\ .$$ [**Proof.**]{} We refer to [@anL Proposition 4.2]. Again in that context $\sigma$ was assumed to be symmetric but the hypotheses was not used. $\Box$ Correctors and $H$-convergence. {#correct} ------------------------------- In order to explain the meaning of our results in the context of $H$-convergence we need to recall the notion of correctors. It is convenient to use the operator ${\rm Div}$ which acts as the usual ${\rm div}$ operator on the rows of $2\times 2$ matrices. Let $\sigma_{\epsilon}$ be a sequence in ${\mathcal M} (\alpha,\beta,\Omega)$ which is $H$-converging to $\sigma_0$. Set $P^{\epsilon}=DU^{\epsilon}$ where, for $\omega$ open with $\omega\subset \subset \Omega$, one has that $U^{\epsilon}$ satisfies the following properties $$\left\{ \begin{array}{lrcc} U^{\epsilon}\in W^{1,2}(\omega;\mathbb R^2)\ ,\\ U^{\epsilon} \rightharpoonup {\rm Id}\ , &&& \hbox{ weakly in $W^{1,2}(\omega;\mathbb R^{2 \times 2}) $}\ ,\\ -{\rm Div} ( DU^{\epsilon} (\sigma_{\epsilon})^T) &\to &-{\rm Div} (\sigma_{0}^T)\ , &\hbox{ strongly in $ W^{-1,2}(\omega;\mathbb R^2)$\ .} \end{array} \right.$$ Then $P^\epsilon$ is called a corrector associated with $(\sigma_{\epsilon},\sigma_0)$. For the main properties of the correctors we refer to [@mt]. Let us just recall here that they exist and that, for a given sequence, $\sigma_{\epsilon}$ which is $H$-converging to $\sigma_0$, the difference between two such correctors converges strongly to zero in $L^2_{\rm loc}(\Omega;\mathbb R^{2\times 2})$. Our interest in this context is given by the following result. \[cor-prop\] Let $\sigma_{\epsilon}$ be a sequence in $\mathcal{M} (\alpha,\beta,\Omega)$ which is $H$-converging to $\sigma_{0}$. Set $U^{\epsilon}=(u^{\epsilon}_1,u^{\epsilon}_2)\in H^1(\Omega;\mathbb R^2)$ to be the unique solution to $$\label{correctors} \left\{ \begin{array}{lll} {\rm Div} (D U^{\epsilon} \sigma_{\epsilon}^T) ={\rm Div} (\sigma_{0} ^T)\ , &\hbox{ in $\Omega$}\ ,\\ (u_1^{\epsilon},u_2^{\epsilon})= (x_1,x_2)\ , &\hbox{ on $\partial \Omega$\ .} \end{array} \right.$$ Then $P^{\epsilon}=DU^{\epsilon}$ is a corrector associated with $(\sigma_{\epsilon},\sigma_0)$. Proposition \[cor-prop\] has a particularly simple interpretation in our language when $\sigma_{0}$ does not depend on position. In this case (which is of fundamental importance in the so called $G$-closure problems), is nothing else than a reformulation of the boundary value problem , or equivalently of , with $\sigma =\sigma_{\epsilon}$ and Proposition \[cor-prop\] says that the corrector can be identified, up to an $L^2$ strong remainder as the Jacobian matrix of an appropriate $\sigma$-harmonic mapping. Exponent of higher integrability. {#AAstala} --------------------------------- As a concluding remark, we observe a straightforward corollary to Proposition \[ellEG\] which we state as a Theorem for the reader’s convenience. \[astala\] Let $\sigma \in {\mathcal M} (\alpha,\beta,\Omega)$ and let $u\in W^{1,2}_{\rm loc}(\Omega)$ be a $\sigma$-harmonic function. Set $$\label{kappaastala} K= \sqrt{\frac{\beta}{\alpha}}+ \sqrt{\frac{\beta-\alpha}{\alpha}}\ .$$ Then $u\in W^{1,p}_{\rm loc}(\Omega)$ for any $$p\in \left.\left[2,\frac{2K}{K-1}\right. \right)\ .$$ **Proof.** As we noted already in Remark \[sigmatilde\], $u$ is also $\tilde{\sigma}$-harmonic with $\tilde{\sigma}$ given by , which belongs to ${\mathcal M} (\lambda,\lambda^{-1},\Omega)$ and $\lambda = \sqrt{\alpha/\beta}$. By Proposition \[ellEG\], $f=u+i \tilde{u}$ is $K$-quasiregular with $K$ given by . Then one applies the celebrated Astala’s Theorem [@astala].$\Box$ Let us emphasize here that the only, possibly new, observation is of algebraic nature. In the case when $\sigma$ is symmetric the [*algebraically*]{} optimal bound is known as was pointed out in [@ln] and [@anL] and achieved for some $\sigma$’s. Astala states explicitly in his paper fundamental paper [@astala] that the exact exponent for the $\sigma$-harmonic function seems to depend in a non obvious and complicated way on the entries of $\sigma$. Our calculation seems to set the algebraically optimal bound in the most general case of non-symmetric $\sigma$. Optimality, in the sense of the existence of a $\sigma$ showing that the exponent of higher integrability cannot be improved, in the context of non symmetric $\sigma$’s seems to be an open problem. Indeed, by the optimality conditions , the extremal $\sigma$ cannot be symmetric almost everywhere. Therefore it appears that the putative example must be of a new type. **Acknowledgements.** The research of the first author was supported in part by MiUR, PRIN no. 2006014115 . The research of the second author was supported in part by MiUR, PRIN no. 2006017833. [99]{} Albin, N. 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--- author: - | [Peter A. van der Helm]{}\ [*Laboratory of Experimental Psychology*]{}\ [*University of Leuven (K.U. Leuven)*]{}\ [*Tiensestraat 102 - box 3711, Leuven B-3000, Belgium*]{}\ [*E: [email protected]*]{}\ [*U: https://perswww.kuleuven.be/peter\_van\_der\_helm*]{} date:   title: | **Transparallel mind:\ Classical computing with quantum power** --- Abstract {#abstract .unnumbered} ======== Inspired by the extraordinary computing power promised by quantum computers, the quantum mind hypothesis postulated that quantum mechanical phenomena are the source of neuronal synchronization, which, in turn, might underlie consciousness. Here, I present an alternative inspired by a classical computing method with quantum power. This method relies on special distributed representations called hyperstrings. Hyperstrings are superpositions of up to an exponential number of strings, which – by a single-processor classical computer – can be evaluated in a transparallel fashion, that is, simultaneously as if only one string were concerned. Building on a neurally plausible model of human visual perceptual organization, in which hyperstrings are formal counterparts of transient neural assemblies, I postulate that synchronization in such assemblies is a manifestation of transparallel information processing. This accounts for the high combinatorial capacity and speed of human visual perceptual organization and strengthens ideas that self-organizing cognitive architecture bridges the gap between neurons and consciousness. Keywords {#keywords .unnumbered} ======== cognitive architecture; distributed representations; neuronal synchronization; quantum computing; transparallel computing by hyperstrings; visual perceptual organization. Highlights {#highlights .unnumbered} ========== - A neurally plausible alternative to the quantum mind hypothesis is presented. - This alternative is based on a classical computing method with quantum power. - This method relies on special distributed representations, called hyperstrings. - Hyperstrings allow many similar features to be processed as one feature. - Thereby, they enable a computational explanation of neuronal synchronization. Introduction ============ Mind usually is taken to refer to cognitive faculties, such as perception and memory, which enable consciousness. In this article at the intersection of cognitive science and artificial intelligence research, I do not discuss full-blown models of cognition as a whole, but I do aim to shed more light on the nature of cognitive processes. To this end, I review a powerful classical computing method – called transparallel processing by hyperstrings (van der Helm 2004) – which has been implemented in a minimal coding algorithm called PISA.[^1] The algorithm takes just symbol strings as input but my point is that transparallel processing might well be a form of cognitive processing (van der Helm 2012). This approach to neural computation has been developed in cognitive science – in research on human visual perceptual organization, in particular – and is communicated here to the artificial intelligence community. For both domains, the novel observations in this article are (a) that this classical computing method has the same computing power as that promised by quantum computers (Feynman 1982), and (b) that it provides a neurally plausible alternative to the quantum mind hypothesis (Penrose 1989). Next, to set the stage, five currently relevant ingredients are introduced briefly. Human visual perceptual organization ------------------------------------ Visual perceptual organization is the neuro-cognitive process that enables us to perceive scenes as structured wholes consisting of objects arranged in space (see Fig. 1). This process may seem to occur effortlessly and we may take it for granted in daily life, but by all accounts, it must be both complex and flexible. To give a gist (following Gray 1999): For a proximal stimulus, it usually singles out one hypothesis about the distal stimulus from among a myriad of hypotheses that also would fit the proximal stimulus (this is the inverse optics problem). To this end, multiple sets of features at multiple, sometimes overlapping, locations in a stimulus must be grouped in parallel. This implies that the process must cope simultaneously with a large number of possible combinations, which, in addition, seem to interact as if they are engaged in a stimulus-dependent competition between grouping criteria. This indicates that the combinatorial capacity of the perceptual organization process must be high, which, together with its high speed (it completes in the range of 100–300 ms), reveals its truly impressive nature. ![image](Fig01){width="11cm"} **Fig. 1** Visual perceptual organization. Both images at the top can be interpreted as 3D cubes and as 2D mosaics, but as indicated by “Yes” and “No”, humans preferably interpret the one at the left as a 3D cube and the one at the right as a 2D mosaic (after Hochberg and Brooks 1960) I think that this cognitive process must involve something additional to traditionally considered forms of processing (see also Townsend and Nozawa 1995). The basic form of processing thought to be performed by the neural network of the brain is parallel distributed processing (PDP), which typically involves interacting agents who simultaneously do different things. However, the brain also exhibits a more sophisticated processing mode, namely, neuronal synchronization, which involves interacting agents who simultaneously do the same thing – think of flash mobs or choirs going from cacophony to harmony. Neuronal synchronization ------------------------ Neuronal synchronization is the phenomenon that neurons, in transient assemblies, temporarily synchronize their firing activity. Such assemblies are thought to arise when neurons shift their allegiance to different groups by altering connection strengths, which may also imply a shift in the specificity and function of neurons (Edelman 1987; Gilbert 1992). Both theoretically and empirically, neuronal synchronization has been associated with various cognitive processes and 30–70 Hz gamma-band synchronization, in particular, has been associated with feature binding in visual perceptual organization (Eckhorn et al. 1988; Gray 1999; Gray and Singer 1989). Ideas about the meaning of gamma-band synchronization are, for instance, that it binds neurons which, together, represent one perceptual entity (Milner 1974; von der Malsburg 1981), or that it is a marker that an assembly has arrived at a steady state (Pollen 1999), or that its strength is an index of the salience of features (Finkel et al. 1998; Salinas and Sejnowski 2001), or that more strongly synchronized assemblies in a visual area in the brain are locked on more easily by higher areas (Fries 2005). These ideas sound plausible, that is, synchronization indeed might reflect a flexible and efficient mechanism subserving the representation of information, the regulation of the flow of information, and the storage and retrieval of information (Sejnowski and Paulsen 2006; Tallon-Baudry 2009). I do not challenge those ideas, but notice that they are about cognitive factors associated with synchronization rather than about the nature of the underlying cognitive processes. In other words, they merely express that synchronization is a manifestation of cognitive processing – just as the bubbles in boiling water are a manifestation of the boiling process (Bojak and Liley 2007; Shadlen and Movshon 1999). The question then still is, of course, of what form of cognitive processing it might be a manifestation. My stance in this article is that neuronal synchronization might well be a manifestation of transparallel information processing, which, as I explicate later on, means that many similar features are processed simultaneously as if only one feature were concerned. Apart from the above ideas about the meaning of synchronization, two lines of research into the physical mechanisms of synchronization are worth mentioning. First, research using methods from dynamic systems theory (DST) showed that the occurrence of synchronization and desynchronization in a network depends crucially on system parameters that regulate interactions between nodes (see, e.g., Buzsáki 2006; Buzsáki and Draguhn 2004; Campbell et al. 1999; van Leeuwen 2007; van Leeuwen et al. 1997). This DST research is relevant, because it investigates – at the level of neurons – how synchronized assemblies might go in and out of existence. Insight therein complements research, like that in this article, into what these assemblies do in terms of cognitive information processing. Second, the quantum mind hypothesis postulated that quantum mechanical phenomena, such as quantum entanglement and superposition, are the subneuron source of neuronal synchronization which, in turn, might underlie consciousness (Penrose 1989; Penrose and Hameroff 2011; see also Atmanspacher 2011). This hypothesis is controversial, mainly because quantum mechanical phenomena do not seem to last long enough to be useful for neuro-cognitive processing, let alone for consciousness (Chalmers 1995; Chalmers 1997; Searle 1997; Seife 2000; Stenger 1992; Tegmark 2000). Be that as it may, the quantum mind hypothesis had been inspired by quantum computing which is a currently relevant form of processing. Quantum computing ----------------- Quantum computing, an idea from physics (Deutsch and Jozsa 1992; Feynman 1982), is often said to be the holy grail of computing: It promises – rightly or wrongly – to be exponentially faster than classical computing. More specifically, the difference between classical computers and quantum computers is as follows. Classical computers, on the one hand, work with binary digits ([*bits*]{}) which each represent either a one or a zero, so that a classical computer with $N$ bits can be in only one of $2^N$ states at any one time. Quantum computers, on the other hand, work with quantum bits ([*qubits*]{}) which each can represent a one, a zero, or any quantum superposition of these two qubit states. A quantum computer with $N$ qubits can therefore be in an arbitrary superposition of $O(2^N)$ states simultaneously.[^2] A final read-out will give one of these states but, crucially, (a) the outcome of the read-out is affected directly by this superposition, which (b) effectively means that, until the read-out, all these states can be dealt with simultaneously as if only one state were concerned. The latter suggests an extraordinary computing power. For instance, many computing tasks (e.g., in string searching) require, for an input the size of $N$, an exhaustive search among $O(2^N)$ candidate outputs. A naive computing method, that is, one that processes each of the $O(2^N)$ options separately, may easily require more time than is available in this universe (van Rooij 2008), and compared with that, quantum computing promises an $O(2^N)$ reduction in the amount of work and time needed to complete a task. However, the idea of quantum computing also needs qualification. First, it is true that the quest for quantum computers progresses (e.g., by the finding of Majorana fermions which might serve as qubits; Mourik et al. 2012), but there still are obstacles, and thus far, no scalable quantum computer has been built. Second, it is true that quantum computing may speed up some computing tasks (see, e.g., Deutsch and Jozsa 1992; Grover 1996; Shor 1994), but the vast majority of computing tasks cannot benefit from it (Ozhigov 1999). This reflects the general tendency that more sophisticated methods have a more restricted application domain. Quantum computing, for instance, requires the applicability of unitary transformations to preserve the coherence of superposed states. Third, quantum computers are often claimed to be generally superior to classical computers, that is, faster than or at least as fast as classical computers for any computing task. However, there is no proof of that (Hagar 2011), and in this article, I in fact challenge this claim. To put this in a broader perspective, I next review generic forms of processing. Generic forms of processing --------------------------- In computing or otherwise, a traditional distinction is that between serial and parallel processing. Serial processing means that subtasks are performed one after the other by one processor, and parallel processing means that subtasks are performed simultaneously by different processors. In addition, however, one may define subserial processing, meaning that subtasks are performed one after the other by different processors (i.e., one processor handles one subtask, and another processor handles the next subtask). For instance, the whole process at a supermarket checkout is a form of multi-threading, but more specifically, (a) the cashiers work in parallel; (b) each cashier processes customer carts serially; and (c) the carts are presented subserially by customers. Compared to serial processing, subserial processing yields no reduction in the work and time needed to complete an entire task, while parallel processing yields reduction in time but not in work. Subserial processing as such is perhaps not that interesting, but taken together with serial and parallel processing – as in Fig. 2 – it calls for what I dubbed transparallel processing. Transparallel processing means that subtasks are performed simultaneously by one processor, that is, as if only one subtask were concerned. The next pencil selection metaphor may give a gist. To select the longest pencil from among a number of pencils, one or many persons could measure the lengths of the pencils in a (sub)serial or parallel fashion, after which the lengths can be compared to each other. However, a smarter – transparallel – way would be if one person gathers all pencils in one bundle upright on a table, so that the longest pencil can be selected in a glance.[^3] This example illustrates that, in contrast to (sub)serial and parallel processing, transparallel processing reduces the work – and thereby the time – needed to complete a task. ![image](Fig02){width="9cm"} [**Fig. 2**]{} Generic forms of processing, defined by the number of subtasks performed at a time (one or many) and the number of processors involved (one or many) In computing, transparallel processing may seem science-fiction and quantum computers indeed reflect a prospected hardware method to do transparallel computing. However, in Sect. 2, I review an existing software method to do transparallel computing on single-processor classical computers. This software method does not fit neatly in existing process taxonomies (Flynn 1972; Townsend and Nozawa 1995). For instance, it escapes the distinction between task parallelism and data parallelism. It actually relies on a special kind of distributed representations, which I next briefly introduce in general. Distributed representations --------------------------- In both classical and quantum computing, the term distributed processing is often taken to refer to a process that is divided over a number of processors. Also then, it refers more generally (i.e., independently of the number of processors involved) to a process that operates on a distributed representation of information. For instance, in the search for extraterrestrial intelligence (SETI), a central computer maintains a distributed representation of the sky: It divides the sky into pieces that are analyzed by different computers which report back to the central computer. Furthermore, PDP models in cognitive science often use the brain metaphor of activation spreading in a network of processors which operate in parallel to regulate interactions between pieces of information stored at different places in the network (e.g., Rumelhart and McClelland 1982). The usage of distributed representations in SETI is a form of data parallelism, which, just as task parallelism, reduces the time but not the work needed to complete an entire task. In PDP models, however, it often serves to achieve a reduction in work – and, thereby, also in time. Formally, a distributed representation is a data structure that can be visualized by a graph (Harary 1994), that is, by a set of interconnected nodes, in which pieces of information are represented by the nodes, or by the links, or by both. One may think of road maps, in which roads are represented by links between nodes that represent places, so that possibly overlapping sequences of successive links represent whole routes. Work-reducing distributed representations come in various flavors, but for $N$ nodes, they typically represent superpositions of $O(2^N)$ wholes by means of only $O(N^2)$ parts. To find a specific whole, a process then might confine itself to examining only the $O(N^2)$ parts. Well-known examples in computer science are the shortest path method (Dijkstra 1959) and methods using suffix trees (Gusfield 1997) and deterministic finite automatons (Hopcroft and Ullman 1979). In computer science, such classical computing methods are also called smart methods, because, compared to a naive method that processes each of those $O(2^N)$ wholes separately, they reduce an exponential $O(2^N)$ amount of work to a polynomial one – typically one between $O(N)$ and $O(N^2)$. Hence, their computing power is between that of naive methods and that of quantum computing, the latter reducing an exponential $O(2^N)$ amount of work to a constant $O(1)$ one. The main points in the remainder of this article now are, first, that hyperstrings are distributed representations that take classical computing to the level of quantum computing, and second, that they enable a neurally plausible alternative to the quantum mind hypothesis. Classical computing with quantum power ====================================== The idea that the brain represents information in a reduced, distributed, fashion has been around for a while (Hinton 1990). Fairly recently, ideas have emerged – in both cognitive science and computer science – that certain distributed representations allow for information processing that is mathematically analogous to quantum computing (Aerts et al. 2009), or might even allow for classical computing with quantum power (Rinkus 2012). Returning in these approaches is that good candidates for that are so-called holographic reduced representations (Plate 1991). To my knowledge, however, these approaches have not yet achieved actual classical computing with quantum power – which hyperstrings do achieve. Formally, hyperstrings differ from holographic reduced representations, but conceptually, they seem to have something in common: Hyperstrings are distributed representations of what I called holographic regularities (van der Helm 1988; van der Helm and Leeuwenberg 1991). Let me first briefly give some background thereof. Background ---------- The minimal coding problem, for which hyperstrings enable a solution, arose in the context of structural information theory (SIT), which is a general theory of human visual perceptual organization (Leeuwenberg and van der Helm 2013). In line with the Gestalt law of Prägnanz (Wertheimer 1912, 1923; Köhler 1920; Koffka 1935), it adopts the simplicity principle, which aims at economical mental representations: It holds that the simplest organization of a stimulus is the one most likely perceived by humans (Hochberg and McAlister 1953). To make quantifiable predictions, SIT developed a formal coding model for symbol strings, which, in the Appendix, is presented and illustrated in the form it has since about 1990. The minimal coding problem thus is the problem to compute guaranteed simplest codes of strings, that is, codes which – by exploiting regularities – specify strings by a minimum number of descriptive parameters. SIT, of course, does not assume that the human visual system converts visual stimuli into strings. Instead, it uses manually obtained strings to represent stimulus interpretations in the sense that such a string can be read as a series of instructions to reproduce a stimulus (much like a computer algorithm is a series of instructions to produce output). For instance, for a line pattern, a string may represent the sequence of angles and line segments in the contour. A stimulus can be represented by various strings, and the string with the simplest code is taken to reflect the prefered interpretation, with a hierarchical organization as described by that simplest code. In other words, SIT assumes that the processing principles, which its formal model applies to strings, reflect those which the human visual system applies to visual stimuli. The regularities considered in SIT’s formal coding model are repetition (juxtaposed repeats), symmetry (mirror symmetry and broken symmetry), and alternation (nonjuxtaposed repeats). These are mathematically unique in that they are the only regularities with a hierarchically transparent holographic nature (for details, see van der Helm and Leeuwenberg 1991, or its clearer version in van der Helm 2014). To give a gist, the string $ababbaba$ exhibits a global symmetry, which can be coded step by step, each step adding one identity relationship between substrings – say, from $aba\ S[(b)]\ aba$ to $ab\ S[(a)(b)]\ ba$, and so on until $S[(a)(b)(a)(b)]$. The fact that such a stepwise expansion preserves symmetry illustrates that symmetry is holographic. Furthermore, the argument $(a)(b)(a)(b)$ of the resulting symmetry code can be hierarchically recoded into $2*((a)(b))$, and the fact that this repetition corresponds unambiguously to the repetition $2*(ab)$ in the original string illustrates that symmetry is hierarchically transparent. The formal properties of holography and hierarchical transparency not only single out repetition, symmetry, and alternation, but are also perceptually adequate. They explain much of the human perception of single and combined regularities in visual stimuli, whether or not perturbed by noise (for details, see van der Helm 2014; van der Helm and Leeuwenberg 1996, 1999, 2004). For instance, they explain that mirror symmetry and Glass patterns are better detectable than repetition, and that the detectability of mirror symmetry and Glass patterns in the presence of noise follows a psychophysical law that improves on Weber’s law (van der Helm 2010). Currently relevant is their role in the minimal coding problem – next, this is discussed in more detail. Hyperstrings ------------ At first glance, the minimal coding problem seems to involve just the detection of regularities in a string, followed by the selection of a simplest code (see Fig. 3). However, these are actually the relatively easy parts of the problem – they can be solved by traditional computing methods (van der Helm 2004, 2012, 2014). Therefore, here, I focus not so much on the entire minimal coding problem, but rather on its hard part, namely, the problem that the argument of every detected symmetry or alternation has to be hierarchically recoded before a simplest code may be selected (repetition does not pose this problem). As spelled out in the Appendix, this implies that a string of length $N$ gives rise to a superexponential $O(2^{N \log N})$ number of possible codes. This has raised doubts about the tractability of minimal coding, and thereby, about the adequacy of the simplicity principle in perception (e.g., Hatfield and Epstein 1985). ![image](Fig03){width="14cm"} [**Fig. 3**]{} Minimal coding. Encoding the string $ababfabab$ involves an exhaustive search for subcodes capturing regularities in substrings (here, only a few are shown). These subcodes can be gathered in a directed acyclic graph, in which every edge represents a substring – with the complexity of the simplest subcode of the substring taken as its length, so that the shortest path through the graph represents the simplest code of the entire string Yet, the hard part of minimal coding can be solved, namely, by first gathering similar regularities in distributed representations (van der Helm 2004; van der Helm and Leeuwenberg 1991). For instance, the string [*ababfababbabafbaba*]{} exhibits, among others, the symmetry [*S\[(aba)(b)(f)(aba)(b)\]*]{}. Its argument [*(aba)(b)(f)(aba)(b)*]{} is represented in Fig. 4a by the path along vertices 1, 4, 5, 6, 9, and 10. In fact, Fig. 4a represents, in a distributed fashion, the arguments of all symmetries into which the string can be encoded. Because of the holographic nature of symmetry, such a distributed representation for a string of length $N$ can be constructed in $O(N^2)$ computing time, and represents $O(2^N)$ symmetry arguments. In Fig. 4b, the same has been done for the string [*ababfababbabafabab*]{}, but notice the difference: Though the input strings differ only slightly, the sets $\pi(1,5)$ and $\pi(6,10)$ of substrings represented by the subgraphs on vertices 1–5 and 6–10 are identical in Fig. 4a but disjoint in Fig. 4b. ![image](Fig04){width="16cm"} [**Fig. 4**]{} Distributed representations of similar regularities. (a) Graph representing the arguments of all symmetries into which the string [*ababfababbabafbaba*]{} can be encoded. (b) Graph representing the arguments of all symmetries into which the slightly different string [*ababfababbabafabab*]{} can be encoded. Notice that sets $\pi(1,5)$ and $\pi(6,10)$ of substrings represented by the subgraphs on vertices 1–5 and 6–10 are identical in (a) but disjoint in (b) The point now is that, if symmetry arguments (or, likewise, alternation arguments) are gathered this way, then the resulting distributed representations consist provably of one (as in Figs. 4a and 4b) or more independent hyperstrings (see van der Helm 2004, 2014, or the Appendix, for the formal definition of hyperstrings and for the proofs). As I clarify next, this implies that a single-processor classical computer can hierarchically recode up to an exponential number of symmetry or alternation arguments in a transparallel fashion, that is, simultaneously as if only one argument were concerned. ![image](Fig05){width="12cm"} [**Fig. 5**]{} A hyperstring. Every path from source (vertex 1) to sink (vertex 9) in the hyperstring at the top represents a normal string via its edge labels. The two hypersubstrings indicated by bold edges represent identical substring sets $\pi(1,4)$ and $\pi(5,8)$, both consisting of the substrings $abc$, $xc$, and $ay$. For the string $h_1 ... h_8$ at the bottom, with substrings defined as corresponding one-to-one to hypersubstrings, this implies that the substrings $h_1 h_2 h_3$ and $h_5 h_6 h_7$ are identical. This single identity relationship corresponds in one go to three identity relationships between substrings in the normal strings, namely, between the substrings $abc$ in string $abcfabcg$, between the substrings $xc$ in string $xcfxcg$, and between the substrings $ay$ in string $ayfayg$ As illustrated in Fig. 5, a hyperstring is an st-digraph (i.e., a directed acyclic graph with one source and one sink) with, crucially, one Hamiltonian path from source to sink (i.e., a path that visits every vertex only once). Every source-to-sink path in a hyperstring represents some normal string consisting of some number of elements, but crucially, substring sets represented by hypersubstrings are either identical or disjoint (as illustrated in Fig. 4). This implies that every identity relationship between substrings in one of the normal strings corresponds to an identity relationship between hypersubstrings, and that inversely, every identity relationship between hypersubstrings corresponds in one go to identity relationships between substrings in several of the normal strings. The two crucial properties above imply that a hyperstring can be searched for regularity as if it were a single normal string. For instance, the hyperstring in Fig. 5 can be treated as if it were a string $h_1 ... h_8$, with substrings that correspond one-to-one to hypersubstrings (see Fig. 5). The latter means that, in this case, the substrings $h_1h_2h_3$ and $h_5h_6h_7$ are identical, so that the string $h_1 ... h_8$ can be encoded into, for instance, the alternation $\langle(h_1h_2h_3)\rangle/\langle (h_4)(h_8)\rangle$. This alternation thus in fact captures the identity relationship between the substring sets $\pi(1,4)$ and $\pi(5,8)$, and thereby it captures in one go several identity relationships between substrings in different strings in the hyperstring. That is, it represents, in one go, alternations in three different strings, namely: > $\langle(abc)\rangle/\langle (f)(g)\rangle\quad$ = in the string $abcfabcg$\ > $\langle(xc)\rangle/\langle (f)(g)\rangle\quad$ in the string $xcfxcg$\ > $\langle(ay)\rangle/\langle (f)(g)\rangle\quad$ in the string $ayfayg$ ![image](Fig06){width="14cm"} [**Fig. 6**]{} Hyperstring encoding. The hyperstring represents the arguments of all symmetries into which the string [*ababfababbabafbaba*]{} can be encoded. The recursive regularity search yields subcodes capturing regularities in hypersubstrings (here, only a few are shown). Returning to hyperstrings of symmetry or alternation arguments, one could say that they represent up to an exponential number of hypotheses about an input string, which, as the foregoing illustrates, can be evaluated further simultaneously. That is, such a hyperstring can be encoded without having to distinguish explicitly between the arguments represented in it (see Fig. 6). At the front end of such an encoding, one, of course, has to establish the identity relationships between hypersubstrings, but because of the Hamiltonian path, this can be done in the same way as for a normal string. At the back end, one, of course, has to select eventually a simplest code, but by way of the shortest path method (Dijkstra 1959), also this can be done in the same way as for a normal string. Thus, in total, recursive hierarchical recoding yields a tree of hyperstrings, and during the buildup of this tree, parts of it can already be traced back to select simplest codes of (hyper)substrings – to select eventually a simplest code of the entire input string.[^4] Discussion ---------- Transparallel processing by hyperstrings has been implemented in the minimal coding algorithm PISA (see Footnote \[pisa\]). PISA, which runs on single-processor classical computers, relies fully on the mathematical proofs concerning hyperstrings (see the Appendix) and can therefore be said to provide an algorithmic proof that those mathematical proofs are correct. Hence, whereas quantum computers provide a still prospected hardware method to do transparallel computing, hyperstrings provide an already feasible software method to do transparallel computing on classical computers. To be clear, transparallel computing by hyperstrings differs from efficient classical simulations of quantum computing (Gottesman 1998). After all, it implies a truly exponential reduction from $O(2^N)$ subtasks to one subtask, that is, it implies that $O(2^N)$ symmetry or alternation arguments can be hierarchically recoded as if only one argument were concerned. This implies that it provides classical computers with the same extraordinary computing power as that promised by quantum computers. Thus, in fact, it can be said to reflect a novel form of quantum logic (cf. Dunn et al. 2013), which challenges the alleged general superiority of quantum computers over classical computers (see Hagar 2011). In general, the applicability of any computing method depends on the computing task at hand. This also holds for transparallel computing – be it by quantum computers or via hyperstrings by classical computers. Therefore, I do not dare to speculate on whether, for some tasks, hyperstrings might give super-quantum power to quantum computers. Be that as it may, for now, it is true that quantum computers may speed up some tasks but it is misleading to state in general terms that they will be exponentially faster than classical computers. As shown in this section, for at the least one computing task, hyperstrings provide classical computers with quantum power and it remains to be seen if quantum computers also can achieve this power for this task. As I next discuss more speculatively, this application also is relevant to the question – in cognitive neuroscience – of what the computational role of neuronal synchronization might be. Human cognitive architecture ============================ Cognitive architecture, or unified theory of cognition, is a concept from artificial intelligence research. It refers to a blueprint for a system that acts like an intelligent system – taking into account not only its resulting behavior but also physical or more abstract properties implemented in it (Anderson 1983; Newell 1990). Hence, it aims to cover not only competence (what is a system’s output?) but also performance (how does a system arrive at its output?), or in other words, it aims to unify representations and processes (Byrne 2012; Langley et al. 2009; Sun 2004; Thagard 2012). The minimal coding method in the previous section – developed in the context of a competence model of human visual perceptual organization – qualifies as cognitive architecture in the technical sense. In this section, I investigate if it also complies with ideas about neural processing in the visual hierarchy in the brain. From the available neuroscientific evidence, I gather the next picture of processing in the visual hierarchy. The visual hierarchy -------------------- The neural network in the visual hierarchy is organized with 10–14 distinguishable hierarchical levels (with multiple distinguishable areas within each level), contains many short-range and long-range connections (both within and between areas and levels), and can be said to perform a distributed hierarchical process (Felleman and van Essen 1991). This process comprises three neurally intertwined but functionally distinguishable subprocesses (Lamme and Roelfsema 2000; Lamme et al. 1998). As illustrated in the left-hand panel in Fig. 7, these subprocesses are responsible for (a) feedforward extraction of, or tuning to, features to which the visual system is sensitive, (b) horizontal binding of similar features, and (c) recurrent selection of different features. As illustrated in the right-hand panel in Fig. 7, these subprocesses together yield integrated percepts given by hierarchical organizations (i.e., organizations in terms of wholes and parts) of distal stimuli that fit proximal stimuli. Attentional processes then may scrutinize these organizations in a top-down fashion, that is, starting with global structures and, if required by task and allowed by time, descending to local features (Ahissar and Hochstein 2004; Collard and Povel 1982; Hochstein and Ahissar 2002; Wolfe 2007). ![image](Fig07){width="16cm"} [**Fig. 7**]{} Processing in the visual hierarchy in the brain. A stimulus-driven perception process, comprising three neurally intertwined subprocesses (left-hand panel), yields hierarchical stimulus organizations (right-hand panel). A task-driven attention process then may scrutinize these hierarchical organizations in a top-down fashion Hence, whereas perception logically processes parts before wholes, the top-down attentional scrutiny of hierarchical organizations implies that wholes are experienced before parts. Thus, this combined action of perception and attention explains the dominance of wholes over parts, as postulated in early twentieth century Gestalt psychology (Wertheimer 1912, 1923; Köhler 1920; Koffka 1935) and as confirmed later in a range of behavioral studies (for a review, see Wagemans et al. 2012). This dominance also has been specified further by notions such as global precedence (Navon 1977), configural superiority (Pomerantz et al. 1977), primacy of holistic properties (Kimchi 2003), and superstructure dominance (Leeuwenberg and van der Helm 1991, Leeuwenberg et al. 1994). Furthermore, notice that the combination of feedforward extraction and recurrent selection in perception is like a fountain under increasing water pressure: As the feedforward extraction progresses along ascending connections, each passed level in the visual hierarchy forms the starting point of integrative recurrent processing along descending connections (see also VanRullen and Thorpe 2002). This yields a gradual buildup from partial percepts at lower levels in the visual hierarchy to complete percepts near its top end. This gradual buildup takes time, so, it leaves room for attention to intrude, that is, to modulate things before a percept has completed (Lamme and Roelfsema 2000; Lamme et al. 1998). However, I think that, by then, the perceptual organization process already has done much of its integrative work (Gray 1999; Pylyshyn 1999), because, as I sustain next, its speed is high due to, in particular, transparallel feature processing. The transparallel mind hypothesis --------------------------------- The perceptual subprocess of feedforward extraction is reminiscent of the neuroscientific idea that, going up in the visual hierarchy, neural cells mediate detection of increasingly complex features (Hubel and Wiesel 1968). Furthermore, the subprocess of recurrent selection is reminiscent of the connectionist idea that a standard PDP process of activation spreading in the brain’s neural network yields percepts represented by stable patterns of activation (Churchland 1986). While I think that these ideas capture relevant aspects, I also think that they are not yet sufficient to account for the high combinatorial capacity and speed of the human perceptual organization process. I think that, to this end, the subprocess of horizontal binding is crucial. This subprocess may be relatively underexposed in neuroscience, but may well be the neuronal counterpart of the regularity extraction operations, which, in representational theories like SIT, are proposed to obtain structured mental representations. In this respect, notice that the minimal coding method in Sect. 2 relies on three intertwined subprocesses, which correspond to the three neurally intertwined subprocesses in the visual hierarchy (see Fig. 8). In the minimal coding method for strings, the formal counterpart of feedforward extraction is a search for identity relationships between substrings, by way of an $O(N^2)$ all-substrings identification method akin to using suffix trees (van der Helm 2014). Furthermore, it implements recurrent selection by way of the $O(N^3)$ all-pairs shortest path method (Cormen et al. 1994), which is a distributed processing method that simulates PDP (van der Helm 2004, 2012, 2014). Currently most relevant, it implements horizontal binding by gathering similar regularities in hyperstrings, which allows them to be recoded hierarchically in a transparallel fashion. ![image](Fig08){width="16cm"} **Fig. 8** (a) The three intertwined perceptual subprocesses in the visual hierarchy in the brain. (b) The three corresponding subprocesses implemented in the minimal coding method This correspondence between three neurally intertwined subprocesses in the visual hierarchy in brain and three algorithmically intertwined subprocesses in the minimal coding method is, to me, more than just a nice parallelism. Epistemologically, it substantiates that knowledge about neural mechanisms and knowledge about cognitive processes can be combined fruitfully to gain deeper insights into the workings of the brain (Marr 1982/2010). Furthermore, ontologically relevant, one of its resulting deeper insights is that transparallel processing might well be an actual form of processing in the brain. That is, horizontal binding of similar features is, in the visual hierarchy in the brain, thought to be mediated by transient neural assemblies, which signal their presence by synchronization of the neurons involved (Gilbert 1992). Synchronization is more than standard PDP and the hyperstring implementation of horizontal binding in the minimal coding method – which enables transparallel processing – therefore leads me to the following hypothesis about the meaning of neuronal synchronization. > [*The transparallel mind hypothesis*]{}\ > Neuronal synchronization is a manifestation of cognitive processing of similar features in a transparallel fashion. This hypothesis is, of course, speculative – but notice that it is based on (a) a perceptually adequate and neurally plausible model of the combined action of human perception and attention, and (b) a feasible form of classical computing with quantum power. Next, this hypothesis is discussed briefly in a broader, historical, context. Discussion ---------- In 1950, theoretical physicist Richard Feynman and cognitive psychologist Julian Hochberg met in the context of a colloquium they gave. Feynman would become famous for his work on quantum electrodynamics, and Hochberg was developing the idea that, among all possible organizations of a visual stimulus, the simplest one is most likely to be the one perceived by humans. They discussed parallels between their ideas and concluded that quantum-like cognitive processing might underlie such a simplicity principle in human visual perceptual organization (Hochberg 2012). This fits in with the long-standing idea that cognition is a dynamic self-organizing process (Attneave 1982; Kelso 1995; Koffka 1935; Lehar 2003). For instance, Hebb (1949) put forward the idea of phase sequence, that is, the idea that thinking is the sequential activation of sets of neural assemblies. Furthermore, Rosenblatt (1958) and Fukushima (1975) proposed small artifical networks – called perceptrons and cognitrons, respectively – as formal counterparts of cognitive processing units. More recently, the idea arose that actual cognitive processing units – or, as I call them, gnosons (i.e., fundamental particles of cognition) – are given by transient neural assemblies defined by synchronization of the neurons involved (Buzsáki 2006; Fingelkurts and Fingelkurts 2001, 2004; Finkel et al. 1998). In this article, I expanded on such thinking, by taking hyperstrings as formal counterparts of gnosons, and by proposing that neuronal synchronization is a manifestion of transparallel information processing. Because transparallel processing by hyperstrings provides classical computers with quantum power, it strengthens ideas that quantum-like cognitive processing does not have to rely on actual quantum mechanical phenomena at the subneuron level (de Barros and Suppes 2009; Suppes et al. 2012; Townsend and Nozawa 1995; Vassilieva et al. 2011). As I argued, it might rely on interactions at the level of neurons. It remains to be seen if the transparallel mind hypothesis is tenable too for synchronization outside the “visual” gamma band, but for one thing, it accounts for the high combinatorial capacity and speed of human visual perceptual organization. Conclusion ========== Complementary to DST research on how synchronized neural assemblies might go in and out of existence due to interactions between neurons, the transparallel mind hypothesis expresses the representational idea that neuronal synchronization mediates transparallel information processing. This idea was inspired by a classical computing method with quantum power, namely transparallel processing by hyperstrings, which allows up to an exponential number of similar features to be processed simultaneously as if only one feature were concerned. This neurocomputational account thus strengthens ideas that mind is mediated by transient neural assemblies constituting flexible cognitive architecture between the relatively rigid level of neurons and the still elusive level of consciousness. Acknowledgments {#acknowledgments .unnumbered} =============== I am grateful to Julian Hochberg and Jaap van den Herik for valuable comments on earlier drafts. 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The coding rules serve the extraction of the transparent holographic regularities repetition (or iteration I), symmetry (S), and alternation (A). They can be applied to any substring of an input string, and a code of the input string consists of a string of symbols and coded substrings, such that decoding the code returns the input string. Formally, SIT’s coding language and complexity metric are defined as follows. **Definition 1** A code $\overline{X}$ of a string $X$ is a string $t_1t_2...t_m$ of code terms $t_i$ such that $X = D(t_1)...D(t_m)$, where the decoding function $D : t \rightarrow D(t)$ takes one of the following forms: dd = = = = I-form: $n*(\overline{y})$ $\rightarrow\quad yyy...y$ ($n$ times $y$; $n \geq 2$)\ S-form: $S[\overline{(\overline{x_1})(\overline{x_2})...(\overline{x_n})},(\overline{p})]$ $\rightarrow\quad x_1x_2...x_n\ p\ x_n...x_2x_1$ ($n \geq 1$)\ A-form: $\langle(\overline{y})\rangle/\langle\overline{(\overline{x_1})(\overline{x_2})...(\overline{x_n})}\rangle$ $\rightarrow\quad yx_1\ yx_2\ ...\ yx_n$ ($n \geq 2$)\ A-form: $\langle\overline{(\overline{x_1})(\overline{x_2})...(\overline{x_n})}\rangle/\langle(\overline{y})\rangle$ $\rightarrow\quad x_1y\ x_2y\ ...\ x_ny$ ($n \geq 2$)\ Otherwise: $D(t) = t$ for strings $y$, $p$, and $x_i$ ($i = 1,2,...,n$). The code parts $(\overline{y})$, $(\overline{p})$, and $(\overline{x_i})$ are *chunks*. The chunk $(\overline{y})$ in an I-form or an A-form is a *repeat*, and the chunk $(\overline{p})$ in an S-form is a *pivot* which, as a limit case, may be empty. The chunk string $(\overline{x_1})(\overline{x_2})...(\overline{x_n})$ in an S-form is an *S-argument* consisting of *S-chunks* $(\overline{x_i})$, and in an A-form, it is an *A-argument* consisting of *A-chunks* $(\overline{x_i})$. **Definition 2** Let $\overline{X}$ be a code of string $X=s_1 s_2...s_N$. The [*complexity*]{} $I$ of $\overline{X}$ in structural information parameters (sip) is given by the sum of (a) the number of remaining symbols $s_i$ ($1 \leq i \leq N$) and (b) the number of chunks $(\overline{y})$ in which $y$ is neither a symbol nor an S-chunk. The last part of Definition 2 may seem somewhat ad hoc, but has a solid theoretical basis in terms of degrees of freedom in the hierarchical organization described by a code. Furthermore, Definition 1 implies that a string may be encodable into many different codes. For instance, a code may involve not only recursive encodings of strings inside chunks – that is, from $(y)$ into $(\overline{y})$ – but also hierarchically recursive encodings of S- or A-arguments $(\overline{x_1})(\overline{x_2})...(\overline{x_n})$ into $\overline{(\overline{x_1})(\overline{x_2})...(\overline{x_n})}$. The following sample of codes for one and the same string may give a gist of the abundance of coding possibilities: dd = = String: $X = abacdacdababacdacdab$ $I = 20$ sip\ Code 1: $\overline{X} = a\ b\ 2*(acd)\ S[(a)(b),(a)]\ 2*(cda)\ b$ $I = 14$ sip\ Code 2: $\overline{X} = \langle(aba)\rangle/\langle(cdacd)(bacdacdab)\rangle$ $I = 20$ sip\ Code 3: $\overline{X} = \langle(S[(a),(b)])\rangle/\langle(S[(cd),(a)])(S[(b)(a)(cd),(a)])\rangle$ $I = 15$ sip\ Code 4: $\overline{X} = S[(ab)(acd)(acd)(ab)]$ $I = 14$ sip\ Code 5: $\overline{X} = S[S[((ab))((acd))]]$ $I = 7$ sip\ Code 6: $\overline{X} = 2*(\langle(a)\rangle/\langle S[((b))((cd))]\rangle)$ $I = 8$ sip Code 1 is a code with six code terms, namely, one S-form, two I-forms, and three symbols. Code 2 is an A-form with chunks containing strings that may be encoded as given in Code 3. Code 4 is an S-form with an empty pivot and illustrates that, in general, S-forms describe broken symmetry; mirror symmetry then is the limit case in which every S-chunk contains only one symbol. Code 5 gives a hierarchical recoding of the S-argument in Code 4. Code 6 is an I-form in which the repeat has been encoded into an A-form with an A-argument that has been recoded hierarchically into an S-form. The computation of a simplest code for a string requires an exhaustive search for ISA-forms into which its substrings can be encoded, followed by the selection of a simplest code for the entire string. This also requires the hierarchical recoding of S- and A-arguments: A substring of length $k$ can be encoded into $O(2^k)$ S-forms and $O(k2^k)$ A-forms, and to pinpoint a simplest one, simplest codes of the arguments of these S- and A-forms have to be determined as well – and so on, with $O(\log N)$ recursion steps, because $k/2$ is the maximal length of the argument of an S- or A-form into which a substring of length $k$ can be encoded. Hence, recoding S- and A-arguments separately would require a superexponential $O(2^{N \log N})$ total amount of work. This combinatorial explosion can be nipped in the bud by gathering the arguments of S- and A-forms in distributed representations. The next definitions and proofs show that S-arguments and A-arguments group naturally into distributed representations consisting of one or more independent hyperstrings – which enable the hierarchical recoding of up to an exponential number of S- or A-arguments in a transparellel fashion, that is, simultaneously as if only one argument were concerned. Here, only A-forms $\langle(y)\rangle/\langle(x_1)(x_2)...(x_n)\rangle$ with repeat $y$ consisting of one element are considered, but Definition 4 and Theorem 1 below hold mutatis mutandis for other A-forms as well. [ ]{} Graph-theoretical definition of hyperstrings: [**Definition 3**]{} A *hyperstring* is a simple semi-Hamiltonian directed acyclic graph $(V,E)$ with a labeling of the edges in $E$ such that, for all vertices $i,j,p,q \in V$:\ either $\pi(i,j) = \pi(p,q)$ or $\pi(i,j) \cap \pi(p,q) = \emptyset$ \ where *substring set* $\pi(v_1,v_2)$ is the set of label strings represented by the paths between vertices $v_1$ and $v_2$; the subgraph on the vertices and edges in these paths is a *hypersubstring*. Definition of distributed representations called A-graphs, which represent all A-forms covering suffixes of strings, that is, all A-forms into which those suffixes can be encoded (see Fig. 9 for an example): [**Definition 4**]{} For a string $T = s_1s_2...s_N$, the *A-graph* $\mathcal{A}(T)$ is a simple directed acyclic graph $(V,E)$ with $V = \lbrace1,2,..,N+1\rbrace$ and, for all $1 \leq i < j \leq N$, edges $(i,j)$ and $(j,N+1)$ labeled with, respectively, the chunks $(s_i...s_{j-1})$ and $(s_j...s_N)$ if and only if $s_i = s_j$. Definition of diafixes, which are substrings centered around the midpoint of a string (this notion complements the known notions of prefixes and suffixes, and facilitates the explication of the subsequent definition of S-graphs): [**Definition 5**]{} A *diafix* of a string $T = s_1s_2...s_N$ is a substring $s_{i+1}...s_{N-i}$ ($0 \leq i < N/2$). Definition of distributed representations called S-graphs, which represent all S-forms covering diafixes of strings (see Fig. 10 for an example): [**Definition 6**]{} For a string $T = s_1s_2...s_N$, the *S-graph* $\mathcal{S}(T)$ is a simple directed acyclic graph $(V,E)$ with $V = \lbrace1,2,..,\lfloor N/2 \rfloor + 2\rbrace$ and, for all $1 \leq i < j < \lfloor N/2 \rfloor + 2$, edges $(i,j)$ and $(j,\lfloor N/2 \rfloor + 2)$ labeled with, respectively, the chunk $(s_i...s_{j-1})$ and the possibly empty chunk $(s_j...s_{N-j+1})$ if and only if $s_i...s_{j-1} = s_{N-j+2}...s_{N-i+1}$. ![image](Fig09){width="14cm"} **Fig. 9** The A-graph for string $T = akagakakag$, with three independent hyperstrings (connected only at vertex $11$) for the three sets of A-forms with repeats $a$, $k$, and $g$, respectively, which cover suffixes of $T$. An A-graph may contain so-called pseudo A-pair edges – like, here, edge $(10,11)$ – which do not correspond to actual repeat plus A-chunk pairs; they cannot end up in codes but are needed to maintain the integrity of hyperstrings during recoding ![image](Fig10){width="11cm"} **Fig. 10** The S-graph for string $T=\textit{ababfdedgpfdedgbaba}$, with two independent hyperstrings given by the solid edges, which represent S-chunks in S-forms covering diafixes of $T$. The dashed edges represent the pivots, which come into play after hyperstring recoding **Theorem 1** The A-graph $\mathcal{A}(T)$ for a string $T = s_1s_2...s_N$ consists of at most $N+1$ disconnected vertices and at most $\lfloor N/2 \rfloor$ independent subgraphs (i.e., subgraphs that share only the sink vertex $N+1$), each of which is a hyperstring.\ *Proof* (1) By Definition 4, vertex $i$ ($i \leq N$) in $\mathcal{A}(T)$ does not have incoming or outgoing edges if and only if $s_i$ is a unique element in $T$. Since $T$ contains at most $N$ unique elements, $\mathcal{A}(T)$ contains at most $N+1$ disconnected vertices, as required. \(2) Let $s_{i_1},s_{i_2},...,s_{i_n}$ ($i_p < i_{p+1}$) be a complete set of identical elements in $T$. Then, by Definition 4, the vertices $i_1,i_2,...,i_n$ in $\mathcal{A}(T)$ are connected with each other and with vertex $N+1$ but not with any other vertex. Hence, the subgraph on the vertices $i_1,i_2,...,i_n,N+1$ forms an independent subgraph. For every complete set of identical elements in $T$, $n$ may be as small as $2$, so that $\mathcal{A}(T)$ contains at most $\lfloor N/2 \rfloor$ independent subgraphs, as required. \(3) The independent subgraphs must be semi-Hamiltonian to be hyperstrings. Now, let $s_{i_1},s_{i_2},...,s_{i_n}$ ($i_p < i_{p+1}$) again be a complete set of identical elements in $T$. Then, by Definition 4, $\mathcal{A}(T)$ contains edges $(i_p,i_{p+1})$, $p = 1,2,...,n-1$, and it contains edge $(i_n,N+1)$. Together, these edges form a Hamiltonian path through the independent subgraph on the vertices $i_1,i_2,...,i_n,N+1$, as required. \(4) The only thing left to prove is that the substring sets are pairwise either identical or disjoint. Now, for $i < j$ and $k \geq 1$, let substring sets $\pi(i,i+k)$ and $\pi(j,j+k)$ in $\mathcal{A}(T)$ be not disjoint, that is, let them share at least one chunk string. Then, the substrings $s_{i}...s_{i+k-1}$ and $s_{j}...s_{j+k-1}$ of $T$ are necessarily identical and, also necessarily, $s_{i} = s_{i+k}$ and either $s_{j} = s_{j+k}$ or $j+k = N+1$. Hence, by Definition 4, these identical substrings of $T$ yield, in $\mathcal{A}(T)$, edges $(i,i+k)$ and $(j,j+k)$ labeled with the identical chunks $(s_{i}...s_{i+k-1})$ and $(s_{j}...s_{j+k-1})$, respectively. Furthermore, obviously, these identical substrings of $T$ can be chunked into exactly the same strings of two or more identically beginning chunks. By Definition 4, all these chunks are represented in $\mathcal{A}(T)$, so that each of these chunkings is represented not only by a path $(i,...,i+k)$ but also by a path $(j,...,j+k)$. This implies that the substring sets $\pi(i,i+k)$ and $\pi(j,j+k)$ are identical. The foregoing holds not only for the entire A-graph but, because of their independence, also for every independent subgraph. Hence, in sum, every independent subgraph is a hyperstring, as required. $\blacksquare$ **Lemma 1** Let the strings $c_1 = s_1s_2...s_k$ and $c_2 = s_1s_2...s_p$ ($k<p$) be such that $c_2$ can be written in the following two ways: $c_2 = c_1X$ with $X = s_{k+1}...s_p$\ $c_2 = Yc_1$ with $Y = s_1...s_{p-k}$ Then, $X=Y$ if $q=p/(p-k)$ is an integer; otherwise $Y=VW$ and $X=WV$, where $V = s_1...s_r$ and $W = s_{r+1}...s_{p-k}$, with $r= p- \lfloor q \rfloor (p-k)$.\ *Proof* (1) If $1<q<2$, then $c_2=c_1Wc_1$, so that $Y=c_1W$ and $X=Wc_1$. Then, too, $r=k$, so that $c_1=V$. Hence, $Y=VW$ and $X=WV$, as required. \(2) If $q=2$, then $c_2=c_1c_1$. Hence, $X=Y=c_1$, as required. \(3) If $q>2$, then the two copies of $c_1$ in $c_2$ overlap each other as follows: [lccccccccccc]{} c\_2 = c\_1 X = s\_1 & …& s\_[p-k]{} & s\_[p-k+1]{} & …& s\_k & s\_[k+1]{} & …& s\_p\ c\_2 = Y c\_1 = & Y & & s\_1 & …& s\_[2k-p]{} & s\_[2k-p+1]{} & …& s\_k Hence, $s_i=s_{p-k+i}$ for $i=1,2,...,k$. That is, $c_2$ is a prefix of an infinite repetition of $Y$. (3a) If $q$ is an integer, then $c_2$ is a $q$-fold repetition of $Y$, that is, $c_2=YY...Y$. This implies (because also $c_2=Yc_1$) that $c_1$ is a $(q-1)$-fold repetition of $Y$, so, $c_2$ can also be written as $c_2=c_1Y$. This implies $X=Y$, as required. (3b) If $q$ is not an integer, then $c_2$ is a $\lfloor q \rfloor$-fold repetition of $Y$ plus a residual prefix $V$ of $Y$, that is, $c_2=YY...YV$. Now, $Y=VW$, so that $c_2$ can also be written as $c_2=VWVW...VWV$. This implies (because also $c_2=Yc_1=VWc_1$) that $c_1=VW...VWV$, that is, $c_1$ is a $(\lfloor q \rfloor - 1)$-fold repetition of $Y = VW$ plus a residual part $V$. This, in turn, implies that $c_2$ can also be written as $c_2=c_1WV$, so that $X=WV$, as required. $\blacksquare$ **Lemma 2** Let $\mathcal{S}(T)$ be the S-graph for a string $T = s_1s_2...s_N$. Then:\ (1) If $\mathcal{S}(T)$ contains edges $(i,i+k)$ and $(i,i+p)$, with $k < p < \lfloor N/2 \rfloor + 2 - i$, then it also contains a path $(i+k,...,i+p)$.\ (2) If $\mathcal{S}(T)$ contains edges $(i-k,i)$ and $(i-p,i)$, with $k<p$ and $i < \lfloor N/2 \rfloor + 2$, then it also contains a path $(i-p,...,i-k)$.\ *Proof* (1) Edge $(i,i+k)$ represents S-chunk $(c_1) = (s_i...s_{i+k-1})$, and edge $(i,i+p)$ represents S-chunk $(c_2) = (s_i...s_{i+p-1})$. This implies that diafix $D = s_i...s_{N-i+1}$ of $T$ can be written in two ways: [ccccc]{} D & = & c\_2 & …& c\_2\ D & = & c\_1 & …& c\_1 This implies that $c_2$ (which is longer than $c_1$) can be written in two ways: $c_2 = c_1X$ with $X = s_{i+k}...s_{i+p-1}$\ $c_2 = Yc_1$ with $Y = s_i...s_{i+p-k-1}$ Hence, by Lemma 1, either $X = Y$ or $Y = VW$ and $X =WV$ for some $V$ and $W$. If $X = Y$, then $D = c_1Y...Yc_1$ so that, by Definition 6, $Y$ is an S-chunk represented by an edge that yields a path $(i+k,...,i+p)$ as required. If $Y = VW$ and $X =WV$, then $D=c_1WV...VWc_1$ so that, by Definition 6, $W$ and $V$ are S-chunks represented by subsequent edges that yield a path $(i+k,...,i+p)$ as required. \(2) This time, edge $(i-k,i)$ represents S-chunk $(c_1) = (s_{i-k}...s_{i-1})$, and edge $(i-p,i)$ represents S-chunk $(c_2) = (s_{i-p}...s_{i-1})$. This implies that diafix $D = s_{i-p}...s_{N-i+p+1}$ of $T$ can be written in two ways: [ccccc]{} D & = & c\_2 & …& c\_2\ D & = & Yc\_1 & …& c\_1X with $X$=$s_{i-p+k}...s_{i-1}$ and $Y$=$s_{i-p}...s_{i-k-1}$. Hence, as before, $c_2 = c_1X$ and $c_2 = Yc_1$, so that, by Lemma 1, either $X=Y$ or $Y = VW$ and $X =WV$ for some $V$ and $W$. This implies either $D=Yc_1...c_1Y$ or $D=VWc_1...c_1WV$. Hence, this time, Definition 6 implies that both cases yield a path $(i-p,...,i-k)$, as required. $\blacksquare$ **Lemma 3** In the S-graph $\mathcal{S}(T)$ for a string $T = s_1s_2...s_N$, the substring sets $\pi(v_1,v_2)$ ($1 \leq v_1 < v_2 < \lfloor N/2 \rfloor + 2$) are pairwise identical or disjoint.\ *Proof* Let, for $i < j$ and $k \geq 1$, substring sets $\pi(i,i+k)$ and $\pi(j,j+k)$ in $\mathcal{S}(T)$ be nondisjoint, that is, let them share at least one S-chunk string. Then, the substrings $s_{i}...s_{i+k-1}$ and $s_{j}...s_{j+k-1}$ in the left-hand half of $T$ are necessarily identical to each other. Furthermore, by Definition 6, the substring in each chunk of these S-chunk strings is identical to its symmetrically positioned counterpart in the right-hand half of $T$, so that also the substrings $s_{N-i-k+2}...s_{N-i+1}$ and $s_{N-j-k+2}...s_{N-j+1}$ in the right-hand half of $T$ are identical to each other. Hence, the diafixes $D_1=s_{i}...s_{N-i+1}$ and $D_2=s_{j}...s_{N-j+1}$ can be written as $D_1 = s_{i}...s_{i+k-1}\;\;p_1\;\;s_{N-i-k+2}...s_{N-i+1}$\ $D_2 = s_{i}...s_{i+k-1}\;\;p_2\;\;s_{N-i-k+2}...s_{N-i+1}$ with $p_1=s_{i+k}...s_{N-i-k+1}$ and $p_2=s_{j+k}...s_{N-j-k+1}$. Now, by means of any S-chunk string $C$ in $\pi(i,i+k)$, diafix $D_1$ can be encoded into the covering S-form $S[C,(p_1)]$. If pivot $(p_1)$ is replaced by $(p_2)$, then one gets the covering S-form $S[C,(p_2)]$ for diafix $D_2$. This implies that any S-chunk string in $\pi(i,i+k)$ is in $\pi(j,j+k)$, and vice versa. Hence, nondisjoint substring sets $\pi(i,i+k)$ and $\pi(j,j+k)$ are identical, as required. $\blacksquare$ **Theorem 2** The S-graph $\mathcal{S}(T)$ for a string $T = s_1s_2...s_N$ consists of at most $\lfloor N/2 \rfloor + 2$ disconnected vertices and at most $\lfloor N/4 \rfloor$ independent subgraphs that, without the sink vertex $\lfloor N/2 \rfloor + 2$ and its incoming pivot edges, form one disconnected hyperstring each.\ *Proof* From Definition 6, it is obvious that there may be disconnected vertices and that their number is at most $\lfloor N/2 \rfloor + 2$, so let us turn to the more interesting part. If $\mathcal{S}(T)$ contains one or more paths $(i,...,j)$ ($i<j<\lfloor N/2 \rfloor+2$) then, by Lemma 2, one of these paths visits every vertex $v$ with $i<v<j$ and $v$ connected to $i$ or $j$. This implies that, without the pivot edges and apart from disconnected vertices, $\mathcal{S}(T)$ consists of disconnected semi-Hamiltonian subgraphs. Obviously, the number of such subgraphs is at most $\lfloor N/4 \rfloor$, and if these subgraphs are expanded to include the pivot edges, they form one independent subgraph each. More important, by Lemma 3, these disconnected semi-Hamiltonian subgraphs form one hyperstring each, as required. $\blacksquare$ [^1]: \[pisa\] PISA is available at https://perswww.kuleuven.be/$\sim$u0084530/doc/pisa.html. Its worst-case computing time may be weakly-exponential (i.e., near-tractable), but in this article, the focus is on the special role of hyperstrings in it. The name PISA, by the way, was originally an acronym of Parameter load (i.e., a complexity metric), Iteration (i.e., repetition), Symmetry, and Alternation. [^2]: \[big-O\] Formally, for functions $f$ and $g$ defined on the positive integers, $f$ is $O(g)$ if a constant $C$ and a positive integer $n_0$ exist such that $f(n) \leq C*g(n)$ for all $n \geq n_0$. Informally, $f$ then is said to be in the order of magnitude of $g$. [^3]: \[spaghetti\] The pencil selection example is close to the spaghetti metaphor in sorting (Dewdney 1984) but serves here primarily to illustrate that, in some cases, items can be gathered in one bin that can be dealt with as if it comprised only one item (hyperstrings are such bins). [^4]: \[transient\] For a given input string, the tree of hyperstrings and its hyperstrings are built on the fly, that is, the hyperstrings are transient in that they bind similar features in the current input only. This contrasts with standard PDP modeling, which assumes that one fixed network suffices for many different inputs.
--- abstract: 'Since the 1960s, the question whether markets are efficient or not is controversially discussed. One reason for the difficulty to overcome the controversy is the lack of a universal, but also precise, quantitative definition of efficiency that is able to graduate between different states of efficiency. The main purpose of this article is to fill this gap by developing a measure for the efficiency of markets that fulfill all the stated requirements. It is shown that the new definition of efficiency, based on informational-entropy, is equivalent to the two most used definitions of efficiency from Fama and Jensen. The new measure therefore enables steps to settle the dispute over the state of efficiency in markets. Moreover, it is shown that inefficiency in a market can either arise from the possibility to use information to predict an event with higher than chance level, or can emerge from wrong pricing/ quotes that do not reflect the right probabilities of possible events. Finally, the calculation of efficiency is demonstrated on a simple game (of coin tossing), to show how one could exactly quantify the efficiency in any market-like system, if all probabilities are known.' author: - 'Roland Rothenstein[^1]$^{\ast}$ $\dag$ $\ddag$' bibliography: - 'finance\_2018.bib' title: 'Quantification of market efficiency based on informational-entropy' --- [*Keywords:* Market Efficiency, Information Theory, Econophysics, Information in Capital Markets, Market Prediction, Game Theory ]{} Introduction {#sec: Introduction} ============ Markets are an important part of our daily life. Markets (like supermarkets, the labor market or the stock market) influence nearly every aspect of human interactions. Therefore, their properties are subject to extensive studies. One of the most interesting claims with respect to markets (especially the stock market) is that markets are efficient. Thus, the so-called ’efficient market hypothesis’ (EMH) has become a very powerful and influencing concept. It is, for instance, the conceptional basis for the option pricing theory (see e.g. [@Black_1973]) and it supported the invention of exchange traded funds (see e.g. [@Gastineau_2002]). As a result, the Nobel committee acknowledged work that has been done on the efficient market hypothesis with the Nobel prize 2013 to Fama, Hansen and Shiller [@Fama_2013; @Hansen_2013; @Shiller_2013] ’for their empirical analysis of asset prices’.[^2] The surprising decision of the Nobel committee to admit the price to a supporter (Fama) and a skeptic (Shiller) of the EMH, however, shows that, despite of the work that had been done in this field since the 1960 [@Beechey_2000], the question whether markets are efficient or not is still unsolved. On the one hand, is the opinion that markets are overall efficient (see e.g. [@Fama_1991; @Malkiel_2003a; @Malkiel_2003b]), but, on the other hand, exists the view that efficiency is not a good description of market behavior (see e.g. [@Shleifer_1990; @Shiller_2003]). The main purpose of this paper is to overcome the discrepancies between the two positions on the efficiency of markets by developing a measure for the efficiency, as for instance Lo requested it [@Lo_1997]. Such a measure of efficiency leads to a new way of looking and analyzing markets. It changes the focus from the question “Are markets efficient?” to the question “How efficient are markets?” It enables the examination of sources of inefficiency and could lead to mechanisms to make markets more efficient. The reason why the efficiency of markets is still an unsolved problem lies in various difficulties connected to the problem. One difficulty in the discussion is that the thesis of efficiency as well as the thesis of inefficiency is supported by a wide range of different empirical results. The proponents of efficiency studied for instance fund manager performances, examined the random structure of price time series or performed event studies. One of the first was Alfred Cowels. He analyzed that fund manager performances indicate efficiency of markets [@Cowles_1933]. Which was further supported by studies of Jensen [@Jensen_1968],Treynor [@Treynor_1965] and Sharpe [@Sharpe_1966]. Fama (based on the fundamental work of Samuelson [@Samuelson_1965]) examined the random structure of price time series [@Fama_1965][^3] in context of efficiency of markets. Furthermore, event studies [@Fama_1969; @Ball_1968] supported the EMH empirically. But, the thesis of inefficiency of markets is also supported by empirical work. Starting with studies by Shiller [@Shiller_1981] comparing the volatility of markets to that of underlying information and by tests of the ability of markets to represent the information of future dividend structures (see also [@LeRoy_1981]), these studies showed evidence that markets are not overall efficient. Further studies showed the inefficiency of markets examining seasonal effects [@Ariel_1987], over-/under-reaction [@DeBondt_1985] or earning anomalies, which could be value-effects [@Fama_1992], size anomalies [@Banz_1981; @Chan_1991] or earning anomalies from technical trading [@Allen_1990]. This concentration on the verification or falsification of the EMH shows that the current discussion is mainly focused on the binarity of the thesis. Binarity of the thesis means that the EMH is bounded between the extremes of the statements that a market is efficient or inefficient. Even so most researchers know that neither is the case, it is still difficult to find an appropriate description for the state in between.[^4] In this paper it will be shown that it is possible to overcome this mostly binary discussion over the efficient market hypothesis by deriving a way to measure efficiency gradually. Doing so, it changes the focus of the description from a qualitative to a quantitative definition. The key to overcome the discrepancy in the discussion is to define a universal, mathematical precise definition. Another difficulty is based on the proposed definitions for efficiency. At the moment, there is no unique, general accepted definition for efficiency. Instead, different definitions were proposed in the past. The first article that defines efficiency is by Fama [@Fama_1965]. In his work, he summarizes different approaches studying efficiency and he refers to the true value of a stock to define efficiency (see section \[subsec: Informational-Efficiency\] for a more detailed discussion.) In the following, more empirical work on efficiency (as outlined in the previous paragraph) led to a more detailed view on efficiency and an alternative definition by Fama which focused more on information [@Fama_1970]. A few years later, Jensen [@Jensen_1978] emphasized a different aspect with his definition by introducing the impossibility of earning profits as relevant for efficiency (see section \[subsec: No-Profit-Efficiency\]). In 2004, Timmermann and Granger extended the definition with the aspect of informational costs, search technologies and forecasting models which brings the focus to more practical aspects in the efficiency discussion [@Timmermann_2004]. But this extension stays on the same conceptional basis as the definition of Jensen. Therefore, it is not further considered in this paper. Other definitions of efficiency where defined for special purposes, for example in the minority game [@Challet_1998], where efficiency is set equal to a minimum in volatility. Although, the definitions gave a good intuition about what is meant by efficiency, they were not able to combine mathematical precision with universality. Either the definition is universal, but lacking mathematical precision and manageability, like the first definition of efficiency by Fama (see section \[subsec: True-Value-Efficiency\]) that is mainly qualitative [@Fama_1965]. Therefore, the ability to use it on concrete cases is reduced. Or, the definition has a detailed mathematical description, but it lacks universality, like the Challet-Definition of efficiency on the minority game [@Challet_1998]. In this case efficiency can be shown only for special market assumptions. That leads to a situation in which it is ambiguous when a market is called efficient. This paper solves this problem by finding a universal, mathematical precise definition. Furthermore, I show with a theoretical analysis that inefficiency can be caused by two different mechanisms. The structure of the paper is as follows: In the next section, I describe the status quo of efficiency concepts. In section \[sec: Efficiency definition\], I introduce my new efficiency definition which covers the main ideas of the existing definitions (which will be shown in section \[sec: Relation to classical approaches\]). To derive the new measure, I consider developments on information theory of C.Shannon [@Shannon_1948] and C.L.Kelly [@Kelly_1956]. Shannon first introduced the concept of information entropy in the context of signal transfer. Kelly used the framework to a game theoretic environment to derive an optimal betting strategy for games. I use this framework to develop my idea of a measurement for efficiency of markets [@Rothenstein_2002]. In section \[sec: Sources of inefficiency\], I derive new insights regarding the sources of inefficiency enhancing the definition to pricing mechanisms. The paper concludes by using the new approach to a game of coin tossing, demonstrating how the new approach can work analyzing a real system. Different views on efficiency {#sec: Different views on efficiency} ============================= As described in the introduction, there is no unique, generally accepted definition for the efficiency of a market. Instead, existing definitions are not completely congruent but somehow overlapping and focus on different aspects of efficient markets. The different concepts of efficiency can be associated with different characteristics in the context of efficiency. Without being exhaustive I cluster the different aspects in three approaches. True-Value-Efficiency {#subsec: True-Value-Efficiency} --------------------- I call the first approach **true-value-efficiency**. It states that ’*actual prices at every point in time represent very good estimates of intrinsic values*’ [@Fama_1965]. It means the prices in a stock market reflect at every time the true values of the companies. Unfortunately, the term ’true value’ is not a well-defined term in context of efficient markets. One reason might be the model dependency of this approach, (as described in [@Shiller_1981] and [@LeRoy_1981]) which means that it always leaves the question open if reported inefficiencies are due to a not well defined model or due to real inefficiencies. ’*Thus, market efficiency perse is not testable.*’ [@Fama_1991] In fact, it is impossible to prove the true-value-efficiency of a system without superior knowledge of the underlying system. In the literature, the superior knowledge is incorporated through valuation models as Fama describes [@Fama_1991]. Therefore, efficiency needs a specification in its approach. Such a specification is often strongly connected to assumptions about stock market systems and the valuation of stocks. This means different models could lead to different results about the efficiency of the same market. Since a definition of efficiency should have its focus on a general approach, it is hardly appropriate to use such specific valuation approaches. To define a special model for each new system (for the calculation of a true value) would undermine the intention to define efficiency in a universal way. Therefore, true-value-efficiency is not further considered for a definition of efficiency. Informational-Efficiency {#subsec: Informational-Efficiency} ------------------------ Another characteristic of efficiency is described by Fama as ’*a market in which prices always fully reflect available information is called efficient*’ [@Fama_1970]. I refer to this as **informational-efficiency**. Informational-efficiency is one of the most referred aspects in the discussion about the efficient market hypothesis (see [@Malkiel_1992]). Informational-efficiency claims that any information that influences the price is already priced in the market, because otherwise traders could use this information to generate a profit.[^5] Different kinds of information, which are incorporated in the price, are thereby related to different kinds of efficiency. In his famous article from 1970 Fama states three kinds of efficiencies that should be distinguished in the discussion [@Fama_1970][^6]: 1. In a ’weak’ efficient market, prices only reflect all information given by past time series. 2. In a ’semi-strong’ efficient market, prices reflect all public available information. 3. In a ’strong’ efficient market, prices reflect all available information (e.g. also insider information). Throughout this paper, I will refer to strong efficiency (unless otherwise mentioned) to simplify the description of efficiency in the following. By using strong efficiency, a specification of underlying information is not necessary. Informational-efficiency is a less strong concept of efficiency in comparison to true-value-efficiency, because all true-value-efficient systems are also informational efficient but not vice versa. This can be shown with the following argument: If a true value exists, it clearly includes all information about an asset. So, if the system reflects the true value in the price, it also reflects all information about the asset in the price. Vice versa this is not the case, because it is not mandatory that the true value of an asset must be revealed by all information available.[^7] No-Profit-Efficiency {#subsec: No-Profit-Efficiency} -------------------- An alternative description to the informational-efficiency is given by Jensen: ’*A market is efficient with respect to information set $\Theta_t$ if it is impossible to make economic profits by trading on the basis of information set $\Theta_t$.*’ [@Jensen_1978] I will refer to this property as **no-profit-efficiency**.[^8] No-profit-efficiency means that no market participant is able to outperform the market in a systematic way [@Sharpe_1995]. The definition of no-profit-efficiency follows the same argument as the informational efficiency. Any profit opportunity is already taken before one can use it for a profit in the market.[^9] The no-profit-efficiency is, however, less strong than the informational efficiency, since all informational efficient markets are also no-profit-efficient but not vice versa. The argument for that is the following: In a market that contains all information, there is no external information on which a profit could be generated. Whereas, for instance, in a market generated by a complex dynamic, which is not foreseeable by traders, also no profit could be generated. But it does not inevitably reflect all information about the system. Efficiency definition {#sec: Efficiency definition} ===================== Based on the efficiency concepts described in the last section, it is now possible to introduce a unique and simple quantitative measure of efficiency, which I will call **informational efficiency definition**. Starting with a simplified version of the informational efficiency definition in this section,[^10] I show that the main properties of the EMH can be derived from these new definition. In section \[sec: Relation to classical approaches\], I show vice versa, that the definition of efficiency can also be derived from the main concepts described in section \[sec: Different views on efficiency\]. In section \[sec: Sources of inefficiency\], it is then possible to introduce an enhanced, more general definition. Let $X$ be a system with a given probability distribution $p(x)$ for an event $x$ in the system. I define the efficiency of the system $X$ in relation to an information $Y$ as follows: $$\label{eq:DefinitionEXY} \text{Eff}(X\vert Y)=\frac{H(X\vert Y)}{H(X)}$$ $H(X)$ is the informational entropy of the system $X$ given by[^11]: $$H(X)=-\sum_x p(x)\log_2 p(x)$$ $H(X\vert Y)$ is the conditional entropy of the system $X$ given the information $Y$. For a given information $Y=y$ it is: $$H(X\vert Y=y)=-\sum_y p(y)\bigg( \sum_x p(x\vert y)\log_2 p(x\vert y)\bigg)$$ $p(x\vert y)$ is the probability that a real event $x$ happens if the information of the event $y$ is given.[^12] $H(X)$ can be interpreted in different ways, which covers different aspects of a system: 1. It measures the amount of uncertainty in the system. 2. It is the mean informational amount (surprise value) contained in the system. Following Shannon [@Shannon_1948], this is the amount of information that can be transferred. 3. It is a measure of predictability of information content. For $H(X\vert Y)$ the following interpretations are common: 1. $H(X\vert Y)$ describes the uncertainty of the system $X$ given the information $Y$ is known. 2. The term $H(X\vert Y)$ can show how much predictability is in system $X$ when information $Y$ is known. Since all interpretations are mathematically identical I will follow all of them in the subsequent sections and refer to the aspect of the entropy that gives the best explanation in each situation. Starting with formula (\[eq:DefinitionEXY\]), I show that the main ideas from the qualitative definition of efficiency can be derived from the new efficiency definition: Imagine a system is fully efficient and therefore $\text{Eff}(X) = 1$. In this case $H(X\vert Y)=H(X)$. The predictability of the system hence is not influenced by an external information $Y$. The efficiency of the system to contain exclusive information and to remain unpredictable is maximal, which corresponds to common qualitative definitions of an efficient market. In case of $\text{Eff}(X)=0$ it follows that $H(X\vert Y)=0$. The efficiency of the system to contain exclusive information is, in this case, completely inefficient because all information of $X$ could be derived by knowing $Y$, which corresponds to common qualitative definitions of an inefficient market. Based on the definition, one can further see that any system with $0>\text{Eff}(X) > 1$, hence, $H(X)>H(X\vert Y)$ is not totally efficient. This is plausible since at least some information of the system $X$ can be predicted by using information $Y$. The grade of efficiency in such a case can be quantified between 0% and 100% which enables a graduation of efficiency of the system.[^13] As standard efficiency $\text{Eff}(X)$ of a system, I define the special case with the information set $Y^*$: $$\label{eq:DefinitionEX} \text{Eff} (X)=\text{Eff} (X\vert Y^*)=\frac{H(X\vert Y^*)}{H(X)}$$ Where $Y^*$ represents all available information about the system $X$. ’All’ information means the best possible information given to a certain moment in the given environment. This definition of $Y^*$ emphasizes a similarity to the definition of strong efficiency used by Fama (see section \[subsec: Informational-Efficiency\]). Relation to classical approaches {#sec: Relation to classical approaches} ================================ In this section, I show how the new definition can be derived from the two different concepts for EMH introduced in section \[sec: Different views on efficiency\], to demonstrate the universality of the informational efficiency definition. Informational-Efficiency {#subsec: Informational-Efficiency2} ------------------------ The main definition of informational efficiency, first given by Fama [@Fama_1970] is: - *Prices reflect all available information.* To be able to formalize the ideas behind the concept of efficiency to a new definition, I adjust some terms to a more appropriate context. My efficiency definition is not restricted to stock markets; the system to look at could be a system that generates prices (like markets), but it could also be a system that generates quotes (like the coin example in section \[sec: Example of tossing coin system\]) or it could be any other possible mechanism to make informational content explicit. To take that into account, I substitute the term “prices reflect” by the term “the system contains”. The term “containing” in the definition means, that information from outside the system cannot change the amount of information of the system $X$. This change in the formulation does not only maintain the essence of Fama’s definition of efficiency, but also fulfill the requirement for formalization. With this change, the informational efficiency definition becomes: - *The system contains all available information.* In the next step, I transfer this formulation into a formula. A system $X$ that contains information $Y$ can be described as a system where such an information does not change the informational content of the system given by $H(X)$. This can be easily expressed using the mutual information $M(X,Y)=H(X)-H(X\vert Y)$. Because the mutual information is the part of information one could deduce over the system $X$ by knowing information $Y$. Applying the mutual information to our formulation of efficiency it is described as a system where mutual information $$M(X,Y^*)=0$$ Where $Y^*$ is the same information used in formula (\[eq:DefinitionEX\]) in the last section. $M(X,Y^*)=0$ implies that $H(X|Y^*)=H(X)$. This means the knowledge of $Y^*$ does not change the amount of informational entropy of $H(X|Y^*)$ in comparison to $H(X)$. Thereby all available information $Y^*$ is contained in the system. To the contrary, a completely inefficient system is a system where all information of the system $X$ is revealed by knowing information $Y^*$. Such a system is not efficient because the system $H(X)$ contains information that could be completely derived from an information set outside the system. In this case, the conditional entropy $H(X|Y^*)=0$ and the mutual information $M(X,Y^* )=H(X)$. Based on these thoughts, $M(X,Y^* )$ can be used to derive a measure of efficiency of a system. To bound the values between zero and one the measure has to be scaled with $H(X)$ (the total information of this system). To get zero as figure for complete inefficiency and one as most efficient figure, the resulting term is subtracted from one. $$\text{Eff}(X\vert Y^*)=1-\frac{M(X,Y^* )}{H(X)} =1-\frac{H(X)-H(X\vert Y^* )}{H(X)} =\frac{H(X\vert Y^* )}{H(X)}$$ As a result, I get exactly the same definition as I have defined in section \[sec: Efficiency definition\]. So far, I only considered the strong efficiency from the definitions of Fama [@Fama_1970]. But semi-strong and weak efficiency can also be taken into account. The connection to semi-strong and weak efficiency can be made by altering the prior knowledge $Y$. In this case, efficiency is given by $\text{Eff}(X\vert Y_{semi} )$ and $\text{Eff}(X\vert Y_{weak} )$, respectively. $$\text{Eff} (X\vert Y_{weak} )=\frac{H(X\vert Y_{weak})}{H(X)}$$ $$\text{Eff} (X\vert Y_{semi} )=\frac{H(X\vert Y_{semi} )}{H(X)}$$ $$\text{Eff} (X\vert Y_{strong} )=\frac{H(X\vert Y_{strong})}{H(X)}=\frac{H(X\vert Y^*)}{H(X)}=\text{Eff}(X)$$ In contrast to the argument for strong efficiency (that the system contains the information already, as discussed above), the efficiency relative to a partial information set $Y$ can have an additional reason. A possible alternative explanation for a partial efficiency measure $\text{Eff}(X\vert Y)$ is that the information is simply not relevant for the information content of $X$. Taking $Y_{weak}$ as an example, it means that a market could either be 100% efficient because information about past time series is already contained, or it could be because past time series do not influence the system at all. In any case, $\text{Eff} (X\vert Y_{weak} )$ shows how efficient the system works in relation to the knowledge of $Y_{weak}$. In addition to the explanations in the last paragraph, one further difference of the new definition, in comparison to the definition given by Fama, is worth mentioning: It is the dependency from the informational basis. Within the new definition framework, it is for instance possible that a system is 100% efficient with respect to semi-strong information but only 60% efficient with respect to strong information. This can be interpreted as some sort of individual efficiency measure. If a participant perceives only a subset of information ($Y\subset Y^*$), it could be that the system is not 100% efficient in relation to $Y^*$, but on an individual level (in relation to $Y$) it seems 100% efficient to the participant. Of course, the other way around is not possible. A system that lacks efficiency on a subset of information could never be more efficient if one gets additional information about the system: $$\text{Eff} (X|Y)<1 \text{ with } Y\subset Y^*\Rightarrow \text{Eff}(X\vert Y^*)<1$$ No-Profit-Efficiency {#subsec: No-Profit-Efficiency2} -------------------- Another aspect of efficiency is the no-profit-property (see section \[subsec: No-Profit-Efficiency\]). To formalize this approach, I follow a work done by Kelly [@Kelly_1956], who defined an optimal winning strategy for a participant in a system (given a probabilistic outcome) without the risk of going bankrupt. Kelly showed, that in this case the maximal capital growth $$G_{max}=\lim_{n \to \infty} \log \frac{V_n}{V_0}$$ is given by the mutual information $$\label{eq:G_max} G_{max} (X|Y_i )=H(X)-H(X|Y_i)$$ This means, that based on the knowledge $Y_i$ of a participant $i$, the individual maximal capital growth is $G_{max} (X\vert Y_i )$. Following the efficiency definition that in an efficient market no profit is possible (see section \[subsec: No-Profit-Efficiency\]), the result is that $G_{max}$ should be zero in a completely efficient market. In a completely inefficient market the opposite is the case. This means $G_{max}$ should be maximal. Following equation (\[eq:G\_max\]) the maximal profit is equivalent to $H(X)$. To derive an efficiency measure between zero and one, I normalize $G_{max}$. To get zero as the result for an inefficient system and one as the result for the most efficient system I subtract the normalized $G_{max}$ from one. It follows: $$\text{Eff} (X\vert Y_i )=1-\frac{G_{max} (X\vert Y_i)}{H(X)} =1-\frac{H(X)-H(X\vert Y_i )}{H(X)} =\frac{H(X\vert Y_i)}{H(X)}$$ Again, I get the informational definition of efficiency. The difference between no-profit-efficiency and informational efficiency is only given by the informational set. The information set $Y^*$ in section \[subsec: Informational-Efficiency2\] describes all information about the system. Information $Y_i$ in this section refers only to the information of one participant and, therefore, the efficiency of the system with respect to only one participant. If however, the knowledge of one participant $i$ is $Y_i=Y^*$ (that means that one participant knows everything about the system), this participant could be seen as some ’all-knowing’ (ideal) participant. In this case, no-profit-efficiency and informational efficiency are equivalent. Therefore, the definition of efficiency could also be formulated as the ability of an ideal trader to gain profits in a market as is described in the paper of Rothenstein and Pawelzik [@Rothenstein_2005]. As argued in section \[subsec: No-Profit-Efficiency\], the no-profit-efficiency is less strong than the informational efficiency. The reason is, that usually no participant has access to all information and is therefore not able to make a profit, even if it is theoretically possible. Sources of inefficiency {#sec: Sources of inefficiency} ======================= In the previous section, I showed that a simplified version of the informational efficiency measure covers all aspects connected to classical definitions of efficiency. For further analysis, it is useful to enhance this definition to cover more complex and realistic situations. With the enhanced definition, it is possible to derive and discuss two main sources of inefficiency in markets. So far, I have only considered systems in which a given reward is in fair relation to the probabilities of the event. In such a case, the payed reward is $\frac{1}{p}$. This situation is a special case of a more general case where the reward is $\frac{1}{q}$ with $q$ as anticipated probability.[^14] The efficiency definition changes due to the maximal profit one can generate in such a system. As Kelly showed [@Kelly_1956], the maximal profit in this case is given by[^15] $$\label{eq: G_max} G_{max,q} (X\vert Y)=H(q)-H(X\vert Y)$$ with $$\label{eq: H_q} H(q)=-\sum_x p(x)\log_2 q(x)$$ where $p(x)$ is the probability of the event $x$ and $q(x)$ is the anticipated probability to determine the reward. $H(q)$ is the entropy of the system, including consideration for the probabilities of payoff payments. $H(q)$ represents the maximal profit that could be achieved if one has optimal information about such a system. From equation (\[eq: H\_q\]) one could see that $H(q)$ is minimal when the quotes are fair ($q(x)=p(x)$). It follows, that $G_{max,q} \ge G_{max} \forall q$ and $G_{max,q} = G_{max}$ if $q(x)=p(x)$. That means, a player with the optimal strategy can achieve more rewards in a system with unfair quotes than in one with fair quotes (see section \[subsec: Fair, unpredictable coin with unfair quotes\] and figure \[fig:H\_vs\_q\] for an example). Taken this into account, the simple efficiency definition: $$\text{Eff}(X\vert Y)=\frac{H(X\vert Y)}{H(X)}=1-\frac{G_{max}}{H(X)}$$ changes to the efficiency definition including unfair quotes: $$\text{Eff}_q (X\vert Y)=1-\frac{G_{max,q}}{H(q)} =1-\frac{H(q)-H(X\vert Y)}{H(q)} =\frac{H(X\vert Y)}{H(q)}$$ Just like in the definition with $p=q$, the efficiency definition is bounded between zero and one. But in the enhanced case, the denominator is H(q).[^16] With this extension of the definition, it is now possible to find sources of inefficiency. Therefore, I start with the enhanced definition of efficiency: $$\label{eq:DefinitionEqXY} \text{Eff}_q (X\vert Y)=\frac{H(X\vert Y)}{H(q)}$$ Equation (\[eq:DefinitionEqXY\]) shows that the efficiency is influenced by two different terms. The numerator, $H(X\vert Y)$, describes the possibility to predict the future of the system $X$ by knowing some information $Y$. $H(X\vert Y)$ is maximal (and therefore equal to $H(X)$) when there is no prediction-power in the information $Y$. In this case, the efficiency is maximal. But, if there is predictability it follows that $H(X\vert Y)<H(X)$ and the efficiency is reduced. The higher the predictability, the lower is $H(X\vert Y)$, and the lower is the efficiency. In the case of a perfect prediction, both $H(X\vert Y)$ and the efficiency would be zero. This represents inefficiency from predictability. The second term that influences the efficiency is the denominator $H(q)$. As mentioned above, $H(q)$ is minimal when the quotes are fair. For all other quotes, $q(x) \neq p(x)$ is $H(q)\ge H(X)$. Since $H(X\vert Y)\le H(X)$, it follows that $\text{Eff}= \frac{H(X\vert Y)}{H(q)} \le 1$. Therefore, for any case with $q(x)\neq p(x)$ such a system is necessarily inefficient. This is because in case of unfair quotes, the bookmaker is going to pay out more than he earns. The difference to inefficiency from predictability is that a participant can earn money (in such a system) without any knowledge about the realization of an event (see section \[subsec: Fair, unpredictable coin with unfair quotes\]). This represents inefficiency from wrong pricing. Of course, the second source of inefficiency could not in all cases be separated unanimously from the first, since the knowledge of the true distribution often also reveals knowledge of a single event. But, since the prices/quotes can change independently from the realization of one event, this source of inefficiency can be viewed as independent. Overall, the extension to unfair quotes shows that two nearly independent sources influence the efficiency of a system. One source of influence is on the level of real events and their predictability, the second is on the level of the price dynamics/quotes. Example of a tossing coin {#sec: Example of tossing coin system} ========================= In this section, I illustrate the functionality of the informational efficiency definition by using it to determine the efficiency of a game of coin tossing. The example has the advantage that all figures are computable and, therefore, the different consequences are visible. In such a case, it is possible to calculate exactly how efficient a system is.[^17] I show the calculation for different parametrizations: A fair coin with fair odds, a fair coin with unfair odds and an unfair coin with fair odds. Introduction of the game and the parameters {#subsec: Introduction of the system and the parameters} ------------------------------------------- The model works as follows: A player can bet an amount of one unit on an event $x$ (head ’h’ or tail ’t’ of a coin). Before the player bets, she gets information $y$ about the result. But, this information $y$ is correct only with a certain probability $p(x\vert y)$.[^18] If she bets correctly, she gets a reward $\alpha=1/q$, otherwise, if she bets incorrectly, she loses her stake. The following figures are needed to calculate the efficiency of this model: 1. Probabilities for head and tail: $$p(x='t')=1-p(x='h')$$ These figures describe the probabilities of a coin to show head or tail after tossing. 2. Probabilities for right knowledge of the result: $$p(y='h'\vert x='h')=1-p(y='t'\vert x='h')$$ $$p(y='t'\vert x='t')=1-p(y='h'\vert x='t')$$ with x representing the result of the event and y representing the information about the event. $p(y='h'\vert x='h')$ is the probability that a player gets the information (or has the belief) that the coin shows head if the result of the coin toss is head. $p(y='t'\vert x='t')$ is, accordingly, the same for the information (or belief) and probability for tails. Since there are no restrictions to the given information of head or tail, the conditional probabilities $p(y='t'\vert x='t')$ and $p(y='h'\vert x='h')$ are completely independent. However, to reduce the number of free parameters, I introduce the restriction to the conditional probability that the information of the player about the results is symmetric. $$p(y='h'\vert x='h')=p(y='t'\vert x='t')$$ This assumes that the player cannot predict the result of tail better than the result of head. 3. Probabilities to calculate the quotes: $$q(x='t')$$ is the expected probability for the event tail ’t’. From this probability, the quotes are derived via $\alpha(x='t')=\frac{1}{q(x='t')}$. As mentioned in section \[sec: Sources of inefficiency\], in case of no costs the expected probabilities have to sum up to one. Similar to the case of the real event it follows: $$q(x='t')=1-q(x='h')$$ For the quote follows: $$\alpha (x='h')=\frac{\alpha(x='t')}{\alpha(x='t')-1}$$ Given the above parameters, three different properties of this system can be distinguished: 1. Fairness of the result 2. Predictability 3. Fairness of the quote Each property has a direct relation to the probabilities and vice versa: 1. The coin is fair if $p(x='t')=p(x='h' )=0.5$. In all other cases the coin is rated as unfair. 2. The system will be unpredictable if $p(x\vert y)=p(x)$. In the case of a fair coin, this leads to $p(y)=0.5$. In all other cases, there is some sort of predictability in the system. 3. Quotes will be called fair if $q(x)=p(x)$ and therefore $\alpha=1/p(x)$. In case of a fair coin, it follows that $q(x)=0.5$ and therefore $\alpha=2$. Otherwise quotes are unfair. Fair, predictable coin with fair quotes {#subsec: Fair, predictable coin with fair quotes} --------------------------------------- First, I consider the case where the coin and the quotes are fair. In this case, the efficiency of the coin is determined only by its predictability $p(x|y)$. With the properties from section \[subsec: Introduction of the system and the parameters\], the efficiency of the coin can be calculated in dependence from $p(y\vert x)$. Starting with the definition of efficiency: $$\label{eq:DefinitionEXYdetail} \text{Eff}(X\vert Y)=\frac{H(X\vert Y)}{H(X)}=\frac{-\sum p(y)\left( \sum p(x\vert y) \log_2 p(x|y) \right)}{-\sum p(x) \log_2 p(x)}$$ It follows: $$\text{Eff}(X\vert Y)=(p(y\vert x)-1) \log_2 [1-p(y\vert x)] -p(y\vert x) \log_2 [p(y\vert x)]$$ In figure \[fig:Eff\_vs\_prob\], the curve of efficiency $E(X|Y)$ for a fair predictable coin with fair quotes is shown in dependence from the predictability $p(y|x)$. The figure shows that the higher the accuracy of the prediction is, the lower is the efficiency. The efficiency ranges from zero (for 100% and 0% accuracy of the prediction) to one (if a correct prediction is on chance level). It also shows that the efficiency is still around 50% even when the outcome can be predicted with nearly 90%. Furthermore, one can see that a difference of 10% from unpredictability (that means $p(x\vert y)=0.5\pm 0.05$) influences the efficiency by less than 3%. However, an inefficiency of 3% leads in case of coin tossing to 3% mean revenue per toss, without the risk of going bankrupt.[^19] Unfair, unpredictability coin with fair quotes {#subsec: Unfair, unpredictable coin with fair quotes} ---------------------------------------------- Second, I consider the case of an unfair, unpredictable coin where the quotes are fair in relation to the probabilities. If we use the respective relations from section \[subsec: Introduction of the system and the parameters\], in the efficiency definition shown in formula (\[eq:DefinitionEXYdetail\]) it follows that the system is always efficient $$\text{Eff} (X\vert Y)=1$$ Therefore, in case of unpredictability and fair quotes the efficiency is independent from the fairness of the coin. For instance, if 9 out of 10 times the outcome of the coin is head, but the reward is only 10/9 for this event, it is comprehensibly, that in this case no profit can be generated from the unfairness of the coin. If there is no further knowledge about the occurrence of this event (thus $p(x\vert y)=p(x)$), the efficiency is 100%. This result illustrates that the efficiency of the coin system does not depend on the fairness of the coin, but on the predictability and the quotes. Interesting in this case, is also the informational content $H(X)$ of the system, since it is not static as the efficiency is. $H(X)$ depends on the fairness of the coin in the following way: $$H (X)=(p(x)-1) \log_2[1-p(x)]-p(x) \log_2 p(x)$$ The curve for the dependence of $H(X)$ on the probabilities $p(x)$ is shown in figure \[fig:H\_vs\_prob\]. It is easy to understand that a coin that always shows tail $p(x='t' )=1$ has no informational content, and therefore $H(X)=0$. From this point the entropy rises to its maximum, the point where $p(x='t' )=0.5$. Fair, unpredictability coin with unfair quotes {#subsec: Fair, unpredictable coin with unfair quotes} ---------------------------------------------- Another interesting case is when the coin is fair and unpredictable but the quotes are not fair. In this case the efficiency (shown in figure \[fig:Eff\_vs\_q\]) is: $$E_q (X|Y)=-\frac{1}{0.5 \log_2 q(x)+0.5 \log_2[1-q(x)]}$$ And for the entropy (shown in figure \[fig:H\_vs\_q\]) follows $$\label{eq:H_q_unfair} H(q)=0.5 \log_2[1-q(x)]-0.5 \log_2 q(x)$$ In figure \[fig:Eff\_vs\_q\] it is shown that, although there is no predictability for a specific coin toss, the efficiency is not always one. As discussed in section \[sec: Sources of inefficiency\], the efficiency of the coin falls if the pricing is not correct even without any predictability of a single event. Figure \[fig:Eff\_vs\_q\] shows that the efficiency is around 60% even when the mispricing is about 90% in comparison to the quote for a fair coin, and that even for a mispricing of 99% the efficiency is around a relative high value of 30% efficiency. Discussion {#sec: Discussion} ========== In this paper, I introduced a definition of efficiency that enhances existing concepts in precision and universality. The new informational efficiency definition enables a substantial step in solving the puzzle, whether markets are efficient or not, by providing a measurement of how efficient markets are. I showed that the new definition based on informational entropy covers the essence of the definition of informational-efficiency (see section \[subsec: Informational-Efficiency2\]) as well as of no-profit-efficiency (see section \[subsec: No-Profit-Efficiency2\]). By expanding the definition to a wider range of applications (allowing an unfair pricing (see section \[sec: Sources of inefficiency\]), I showed that inefficiency can have two sources: predictability and unfair pricing. Finally, I demonstrated with a simple system of coin tossing, that, by using the informational efficiency definition, efficiency can be exactly quantified and I showed the efficiency for different cases of a coin. One may argue that the new definition only shifts the problem from the uncertainty of the market model to the uncertainty of the information. But this is not the case, as the results of the paper show. The new definition delivers far more advantages: First, the new definition is able to derive different grades of efficiency, which was not the case in the previously used definitions. The new definition further provided a new approach to actually measure efficiency quantitatively. With this approach, the question of whether markets are efficient or not can shift from a fundamental debate to a question of measurement, setting the concept of relative efficiency [@Lo_1997] in place. Therefore, my definition will help to overcome the binary focus in the discussion of efficiency and help to establish a more gradual view. Second, I show in section 5 that it is now possible to carry out theoretical analyses based on a clear quantitative definition deriving new insights, like the result that inefficiency can arise from two different sources. Third, it is now possible to exactly quantify the efficiency for a wide range of toy models.[^20] This introduction is only a first step solving the question of efficiency in markets and further steps have to follow. The time-dependency of the definition can be studied as well as the incorporation of some sort of price dynamic in $q(x)$. Also, I did not consider the case with costs included. But, all these extensions do not change the fundamental concept behind the new efficiency measure. This fundamental concept of a measure for efficiency shifts the focus of the EMH from a fundamental debate to a question of pure measurement. This transfers the efficient market hypothesis from the actual state of an economic theory into a quantitative science. Acknowledgment {#acknowledgment .unnumbered} ============== I would like to thank Franziska Becker, Tobias Basse, Rike Rothenstein and Wilfried Rickels, for their support in writing this article and Klaus Pawelzik and Mark Kirstein for helpful discussions. ![ The graph shows the efficiency of a fair coin in dependence from the conditional probability $p(y \vert x)$. It shows that the efficiency is one when the outcome is not predictable ($p(y \vert x)=0.5$). []{data-label="fig:Eff_vs_prob"}](Rplot04.eps){width="70.00000%"} \[ht!\] ![ The figure shows the entropy of the coin system in dependence from the probability of the event showing ’tail’. It shows that the entropy is maximal when the coin is fair and falls to zero if the probability of the coin to show ’tail’ is one. The graph for ’head’ is identical.[]{data-label="fig:H_vs_prob"}](Rplot03.eps "fig:"){width="70.00000%"} ![ This graph shows the efficiency of a fair coin that is unpredictable (but has unfair quotes) in dependence from the estimated probability $q(x)$ of an event $x$ determining the quote. It shows that the efficiency is one when the quote is based on the right probability of the event ($q(x)=p(x)=0.5$) and falls to zero if the probability $q$ goes to one and zero, respectively.[]{data-label="fig:Eff_vs_q"}](Rplot02.eps){width="70.00000%"} \[ht!\] ![ The graph shows the entropy of a fair coin when the quotes are variable. It shows the dependence from the estimated probability $q(x)$ of an event $x$ determining the quote. The entropy is rising if the quotes are rising. The minimum of the entropy is for a fair probability of the quote ($q(x)=p(x)=0.5$).[]{data-label="fig:H_vs_q"}](Rplot01.eps "fig:"){width="70.00000%"} [^1]: Email: [email protected] [^2]: All three laureates have explored different aspects of efficiency from theoretical ground work to analysis of information dynamics in markets [@Fama_1965; @Fama_1991; @Shiller_1981; @Shiller_2003; @Hansen_1980; @Hansen_1991]. [^3]: The random structure of price time series was also independently examined earlier by Mandelbrot [@Mandelbrot_1963a] [^4]: In 2004 Lo described an approach to overcome this problem, but he stayed qualitative in his description [@Lo_2004] [^5]: In later discussions this is weakened by the statement that only information up to a certain information level is incorporated [@Grossman_1980]. [^6]: In a later article, Fama connected these different kinds of efficiency to empirical studies [@Fama_1991]. He connected weak efficiency to the test of return predictability, semi-strong efficiency to studies about the influence of new information and strong efficiency to tests of private information. [^7]: Clearly, this argument depends on the interpretation of the term ’true value’. Since the term ’true value’ is open for interpretation (as discussed in section \[subsec: True-Value-Efficiency\]), the conclusion is not the only possible interpretation. [^8]: As indicated in the introduction, I interpret $\Theta_t$ in a broad view that includes search technologies and forecasting models as introduced by Timmermann [@Timmermann_2004] [^9]: Prices in a no-profit-efficient market have to be martingales [@Jensen_1978; @Samuelson_1965; @Mandelbrot_1966]. A martingale is given by the following formula: $$E(r)=E(r\vert \phi) \text{ with } r:\text{price, } \phi:\text{information}$$ This means that the actual price is the best predictor of the future price independent of information given. Since the concept of martingales is dedicated to time series and not to information in a system, I will not discuss this aspect in greater detail [^10]: In section \[sec: Sources of inefficiency\], I enhance the definition to a more general case. [^11]: Following the definition of Shannon [@Shannon_1948] [^12]: This probability can be calculated from the probability $p(y \vert x)$ through Bayes formula $$p(x \vert y) = \frac {p(y \vert x)} {p(y \vert x)+p(\bar{y} \vert x)}$$ where $p(y \vert x)$ is the probability to get the information y if the real event x happens and $p(\bar{y} \vert x)$ is the probability not getting the informations y if the real event x happens. [^13]: The graduation in efficiency should not be mixed up with the efficiency of a system based on different information sets as Fama introduced them (see section \[subsec: Informational-Efficiency\]). For the relation between these both see section \[subsec: Informational-Efficiency2\]. [^14]: In this paper, I only consider the case without costs in the system. This means, that $\sum q(x)=1$. [^15]: In the original paper, Kelly derives his formula not in dependence from the probability $q$ but from the quote $\alpha=\frac{1}{q}$ (fair quote without costs). In this case, the formula is: $$G_{max,\alpha} (X\vert Y)=H(\alpha)-H(X\vert Y)$$ where $$H(\alpha)=\sum_x p(x) \log_2 \alpha_x$$ with $\alpha$ as the reward payed if the participant predicts the event correct. [^16]: From equation (\[eq: G\_max\]) one can see that $G_{max}$ can be maximal $H(q)$ (in the case where $H(X|Y)=0$). Therefore, the maximal entropy H(q) is used as denominator in this definition, since $H(q)\ge H(X) \ge H(X \vert Y) \text{, this ensures that } 0 \ge \text{Eff}_q(X\vert Y) \le 1$. [^17]: The definition can be used for any information processing system where an informational entropy can be calculated (e.g. minority game [@Challet_1997] or the seesaw game [@Paetzelt_2013a]). [^18]: Alternatively, information could also be based on beliefs of the agent, which are correct with a certain probability $p(x\vert y)$. The mechanism of measuring efficiency is, therefore, not dependent on external information sources. [^19]: This follows from equation \[eq: G\_max\] with $H(X)=1$ [^20]: The range of toy models is only restricted by the possibility to determine all relevant probabilities (as I showed on a simple model in section 6).
--- abstract: | In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions. As an example we consider gamma Lévy process, $\alpha$-stable Lévy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of Lévy processes with randomly broken drift and on random intervals. MSC(2010): Primary 60G51; Secondary 60G70. author: - 'Zbigniew Michna[^1]' title: | **Ruin probabilities for two collaborating\ insurance companies** --- Introduction ============ In this paper we study the supremum distribution of a spectrally positive or negative Lévy process with a piecewise linear drift. We find exact formulas for the distribution of supremum which are expressed by one-dimensional densities of a given Lévy process. The results can be applied to find ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions (proportional reinsurance). Moreover the formulas can be used for a two-node tandem queue (see Lieshout and Mandjes [@li:ma:07]). Avram et al. [@av:pa:pi:08:a] investigates a spectrally positive Lévy process with a broken drift (reduction of the risk problem to one dimension) and they find the double Laplace transform of the infinite time survival probability. As an example they obtain an explicit analytical representation of the infinite time survival probability if the claims are exponentially distributed (the compound Poisson process with exponentially distributed claims). In Avram et al. [@av:pa:pi:08] the related problem is investigated if the accumulated claim amount is modeled by a Lévy process that admits negative exponential moments. They find exact formulas for ruin probabilities expressed by ordinary ruin probabilities when the accumulated claim amount process is spectrally negative or a compound Poisson process with exponential claims. Additionally they find asymptotic behavior of ruin probabilities under the Cramér assumption. In Foss et al. [@fo:ko:pa:ro:17] the same problem is investigated as in Avram et al. [@av:pa:pi:08] but the subexponential claims are admitted and an asymptotic behavior of ruin probabilities on finite and infinite time horizon is found. In the models analyzed in this contribution we assume that the accumulated claim amount process is a spectrally positive or a spectrally negative Lévy process with one-dimensional density functions. We find exact formulas for ruin probabilities expressed by one-dimensional densities of an underlying Lévy process. The main difference of our models and models of Avram et al. [@av:pa:pi:08] is that we admit heavy tailed claims and we provide explicit formulas of ruin probabilities both on finite and infinite time horizon unlike Avram et al. [@av:pa:pi:08:a] and Avram et al. [@av:pa:pi:08] where it is done only on the infinite time horizon. The layout of the rest of the article is the following. In this section we recall the formulas which will be used in the main results. The next section contains the main results that is the distribution of supremum of a Lévy process with a broken drift and examples. In Section \[sec3\] we outline how to apply the main results to ruin probabilities for two collaborating insurance companies. The last section deals with the identities for Laplace transform of the distribution for supremum of Lévy processes with a randomly broken drift and on random intervals. In Michna et al. [@mi:pa:pi:15] a joint distribution of the random variable $Y(T)$ and $\inf_{t< T}Y(t)$ was found where $Y$ is a spectrally negative Lévy process (we will consider real stochastic processes with time defined on the non-negative half real line). \[tacneginf\] If $Y$ is a spectrally negative Lévy process and the one-dimensional distributions of $Y$ are absolutely continuous then $${{\rm I\hspace{-0.8mm}P}}(\inf_{t< T} Y(t)< -u, \,Y(T)+u\in \td z)= \td z\int_0^T\frac{z}{T-s}\, p(z, T-s)\,p(-u,s)\td s,$$ where $T, u>0$, $z\geq 0$ and $p(x,s)$ is a density function of $Y(s)$ for $s>0$. We do not expose a linear drift of the process $Y$ but it can be incorporated in the process $Y$. If $X$ is a spectrally positive Lévy process then $X=-Y$ and we get the following corollary. \[jsp\] If $X$ is a spectrally positive Lévy process and the one-dimensional distributions of $X$ are absolutely continuous then $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} X(t)\leq u, \,X(T)\in \td z)=}\\ &&\left[f(z,T)-\int_0^T\frac{u-z}{T-s}\, f(z-u, T-s)\,f(u,s)\td s\right]\td z,\end{aligned}$$ where $T, u>0$, $z\in (-\infty, u]$ and $f(x,s)$ is a density function of $X(s)$ for $s>0$. Integrating the last formula with respect to $z$ we get the following theorem (see Michna et al. [@mi:pa:pi:15] and Michna [@mi:11]). \[mi\] If the one-dimensional distributions of $X$ are absolutely continuous then $$\label{mi1} {{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} X(t)>u) ={{\rm I\hspace{-0.8mm}P}}(X(T)>u)+\int_0^T\frac{{{\rm I\hspace{-0.8mm}E}}(X(T-s))^-}{T-s}\, f(u,s)\,{\rm d}s\,, $$ where $x^-=-\min\{x,0\}$ and $f(u,s)$ is a density function of $X(s)$ for $s>0$. The above formula extends the result of Takács [@ta:65] to Lévy processes with infinite variation. Let us now find the joint distribution of supremum and the value of the process for any spectrally negative Lévy process. It will easily follow from Corollary \[jsp\] and the duality lemma. \[refy\] If $Y$ is a spectrally negative Lévy process and the one-dimensional distributions of $Y$ are absolutely continuous then $${{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} Y(t)\leq u, \,Y(T)\in \td z)= \left[p(z,T)-u\int_0^T\frac{p(u, T-s)}{T-s}\,p(z-u,s)\td s\right]\td z\,,$$ where $T, u>0$, $z\in (-\infty, u]$ and $p(x,s)$ is a density function of $Y(s)$ for $s>0$. By the duality lemma (see e.g. Bertoin [@be:96]) we have that\ $X((T-t)-)-X(T)\stackrel{d}{=}Y(t)$ in the sense of finite dimensional distributions for $t\leq T$ ($X(t-)$ means the left-hand side limit at $t$). Thus we get $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} X(t)\leq u, \,X(T)\in \td z)=}\\ &&{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} X((T-t)-)\leq u, \,X(T)\in \td z)\\ &&={{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} (X((T-t)-)-X(T))\leq u-z, \,X(T)\in \td z)\\ &&={{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} Y(t)\leq u-z, \,-Y(T)\in \td z)\,.\end{aligned}$$ Substituting $u'=u-z$ and $z'=-z$ and using Corollary \[jsp\] we obtain the formula. [ $\Box$]{} Integrating the last formula with respect to $z$ we could get a similar result to eq. (\[mi1\]) for spectrally negative Lévy processes. However we obtain a simpler formula from Kendall’s identity (see Kendall [@ke:57]). The following theorem can be found in a more general form in Takács [@ta:65] (see also Michna [@mi:13] for the distribution of supremum for spectrally negative Lévy processes). \[supy\] If $Y$ is a spectrally negative Lévy process and the one-dimensional distributions of $Y$ are absolutely continuous then $${{\rm I\hspace{-0.8mm}P}}(\sup_{t< T} Y(t)>u)=u\int_0^T\frac{p(u,s)}{s}\,\td s\,,$$ where $p(u,s)$ is the density function of $Y(s)$. It follows directly from Kendall’s identity (see Kendall [@ke:57] or e.g. Sato [@sa:99] Th. 46.4). [ $\Box$]{} Main results and examples ========================= In this section we analyze the distribution of supremum for both $X(t)-c(t)$ and $Y(t)-c(t)$ where $X$ is a spectrally positive Lévy process and $Y$ is a spectrally negative Lévy process and $$\label{d} c(t)= \left\{\begin{array}{ll} c_1 t &\mbox{if }\, t\in[0, T]\\ c_2(t-T)+c_1 T&\mbox{if }\, t\in(T, \infty)\,, \end{array} \right.$$ where $c_1, c_2\geq 0$. Since we now expose the drift of the process we will assume that densities of $X(s)$ and $Y(s)$ are $f(x,s)$ and $p(x,s)$, respectively (unlike the previous section where a linear drift could be incorporated in the processes). \[main\] If $S>T$ ($S$ is finite or $S=\infty$) and $X(t)$ is absolutely continuous with density $f(x,t)$ then $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(X(t)-c(t))>u)=A+B\coloneqq}\\ &&{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(X(t)-c_1 t)>u)\\ &&+\,{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(X(t)-c_1 t)\leq u, \sup_{0<t<S-T}(X(t+T)-X(T)-c_2 t)>u-X(T)+c_1 T)\,,\end{aligned}$$ where $$A={{\rm I\hspace{-0.8mm}P}}(X(T)-c_1 T>u)+\int_0^T\frac{{{\rm I\hspace{-0.8mm}E}}(X(T-s)-c_1 (T-s))^-}{T-s}\, f(u+c_1 s,s)\,{\rm d}s$$ and $$\begin{aligned} B&=&\int_{0}^\infty {{\rm I\hspace{-0.8mm}P}}(\sup_{t< S-T}(X(t)-c_2 t)>z)f(-z+u+c_1 T,T)\td z\\ &&-\int_{0}^\infty z\,{{\rm I\hspace{-0.8mm}P}}(\sup_{t< S-T}(X(t)-c_2 t)>z)\td z\\ &&\,\,\,\,\,\,\cdot\int_{0}^T\frac{f(u+c_1 s,s)}{T-s}f(-z+c_1(T-s),T-s)\td s\,.\end{aligned}$$ The decomposition $A+B$ we get as follows $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(X(t)-c(t))>u)=A+B\coloneqq}\\ &&{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(X(t)-c_1 t)>u)\\ &&+\,{{\rm I\hspace{-0.8mm}P}}((\sup_{t< T}(X(t)-c_1 t)\leq u, \sup_{T<t<S}(X(t)-c_2(t-T)-c_1T)>u)\\ &&={{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(X(t)-c_1 t)>u)\\ &&+\,{{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(X(t)-c_1 t)\leq u, \sup_{0<t<S-T}(X(t+T)-X(T)-c_2 t)>u-X(T)+c_1 T)\,.\end{aligned}$$ The formula for $A$ we directly get from Theorem \[mi\]. Let $F(\td x, \td z)$ be the joint distribution of $(\sup_{t< T}(X(t)-c_1 t), X(T)-c_1 T)$. Then the formula for $B$ follows from the strong Markov property and Corollary \[jsp\] that is $$\begin{aligned} B&=&\int_0^u\int_{-\infty}^u{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S-T}(X(t)-c_2 t)>u-z)F(\td x, \td z)\\ &=&\int_{-\infty}^u{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S-T}(X(t)-c_2 t)>u-z)f(z+c_1 T, T)\td z\\ &&-\int_{-\infty}^u{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S-T}(X(t)-c_2 t)>u-z)\td z\\ &&\cdot\int_0^T\frac{u-z}{T-s}f(z-u+c_1 (T-s), T-s)f(u+c_1s, s)\td s\end{aligned}$$ and substituting $z'=u-z$ we obtain the final formula. [ $\Box$]{} Similarly we get a formula for spectrally negative Lévy processes. If $S>T$ ($S$ is finite or $S=\infty$) and $Y(t)$ is absolutely continuous with density $p(x,t)$ then ${{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(Y(t)-c(t))>u)=A+B$ where $$A={{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(Y(t)-c_1 t)>u)=u\int_0^T\frac{p(u+c_1s,s)}{s}\,\td s$$ and $$\begin{aligned} B&=&\int_{0}^\infty {{\rm I\hspace{-0.8mm}P}}(\sup_{t< S-T}(Y(t)-c_2 t)>z)p(-z+u+c_1 T,T)\td z\\ &&-u\,\int_{0}^\infty {{\rm I\hspace{-0.8mm}P}}(\sup_{t< S-T}(Y(t)-c_2 t)>z)\td z\\ &&\,\,\,\,\,\,\cdot\int_{0}^T\frac{p(-z+c_1s, s)}{T-s}\,p(u+c_1(T-s), T-s)\td s\,.\end{aligned}$$ Using Corollary \[refy\] and Th. \[supy\] we proceed the same way as in the proof of Th. \[main\]. [ $\Box$]{} The application of Th. \[main\] leads to the following example with Brownian motion (see Mandjes [@ma:04] and Lieshout and Mandjes [@li:ma:07] or Avram et al. [@av:pa:pi:08]). \[brexpl\] If $W$ is the standard Brownian motion then $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(W(t)-c(t))>u)=}\\ &&\Phi(-uT^{-1/2}-c_1\sqrt{T})+e^{-2c_1u}\Phi(-uT^{-1/2}+c_1\sqrt{T})\\ &&\,\,\,\,\,\,+e^{-2c_2(u+c_1T-c_2T)}\Phi(uT^{-1/2}+(c_1-2c_2)\sqrt{T})\\ &&\,\,\,\,\,\,-e^{2(c_2-c_1)u+2c_2^2T-2c_1c_2T}\Phi(-uT^{-1/2}+(c_1-2c_2)\sqrt{T})\,.\end{aligned}$$ Indeed using Theorem \[main\] and $${{\rm I\hspace{-0.8mm}P}}(\sup_{t< T}(W(t)-ct)>u)=\Phi(-uT^{-1/2}-c\sqrt{T})+e^{-2cu}\Phi(-uT^{-1/2}+c\sqrt{T})$$ and $${{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(W(t)-ct)>u)=e^{-2cu}$$ for $c\geq 0$ (see e.g. Dębicki and Mandjes [@de:ma:15]) we get $$\label{abbr} {{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(W(t)-c(t))>u)=A+B\,,$$ where $$\label{ab} A=A(c_1, T, u)\coloneqq\Phi(-uT^{-1/2}-c_1\sqrt{T})+e^{-2uc_1}\Phi(-uT^{-1/2}+c_1\sqrt{T})$$ and $$\begin{aligned} \lefteqn{B=}\\ &&e^{-2c_2(u+c_1T-c_2T)}\Phi(uT^{-1/2}+(c_1-2c_2)\sqrt{T})\\ &&\,\,\,\,\,-\,\frac{e^{-c_1u-c_1^2T/2}}{2\pi}\int_{0}^\infty ze^{(c_1-2c_2)z}\td z\int_0^T(T-s)^{-3/2}s^{-1/2}e^{-\frac{z^2}{2(T-s)}-\frac{u^2}{2s}}\td s\,.\end{aligned}$$ Let us take $c=c_1=c_2\geq 0$ in eq. (\[abbr\]). Then $A+B=e^{-2uc}$ and the second summand of $A$ and the first one of $B$ sum up to $e^{-2uc}$ thus we get $$\frac{e^{-cu-c^2T/2}}{2\pi}\int_{0}^\infty ze^{-cz}\td z\int_0^T(T-s)^{-3/2}s^{-1/2}e^{-\frac{z^2}{2(T-s)}-\frac{u^2}{2s}}\td s= \Phi(-uT^{-1/2}-c\sqrt{T})\,.$$ Thus using the last identity for $c=2c_2-c_1\geq 0$ we get the second therm of $B$. Similarly let us take $c=c_1$ and $c_2=0$ in eq. (\[abbr\]). Then $A+B=1$ and the first summand of $A$ and the first one of $B$ sum up to 1 thus we get $$\frac{e^{cu-c^2T/2}}{2\pi}\int_{0}^\infty ze^{cz}\td z\int_0^T(T-s)^{-3/2}s^{-1/2}e^{-\frac{z^2}{2(T-s)}-\frac{u^2}{2s}}\td s= \Phi(-uT^{-1/2}+c\sqrt{T})\,.$$ Thus using the last identity for $c=c_1-2c_2> 0$ we get the second therm of $B$. Let $0<T<S<\infty$ and $W$ be the standard Brownian motion then $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(W(t)-c(t))>u)=}\\ &&A(c_1, T, u) +\frac{1}{\sqrt{2\pi T}}\int_0^\infty A(c_2, S-T, z) e^{-\frac{(u+c_1T-z)^2}{2T}}\td z\\ &&-\frac{e^{-uc_1-\frac{c_1^2 T}{2}}}{2\pi}\int_0^\infty ze^{c_1 z} A(c_2, S-T, z)\td z \int_0^T s^{-1/2}(T-s)^{-3/2}\,e^{-\frac{z^2}{2(T-s)}-\frac{u^2}{2s}}\td s\,,\end{aligned}$$ where $A(c_1, T, u)$ is defined in eq. (\[ab\]). Let $X(t)$ be gamma Lévy process with the density $$f(x, t)=\frac{\delta^t}{\Gamma(t)}x^{t-1}e^{-\delta x}{{1\hspace{-1mm}{\rm I}}}_{\{x>0\}}$$ where $\delta>0$ and $c(t)$ be defined in eq. (\[d\]). Using Th. \[main\] we give the explicit formulas of ${{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(X(t)-c(t))>u)=A+B$ for both $T<S<\IF$ and $S=\IF$, respectively. For $T<S<\IF$, we have that $$\begin{aligned} A&=&\frac{\delta^T}{\Gamma(T)}\int_{u+c_1T}^\IF x^{T-1}e^{-\delta x}\td x\\ &&+\,\delta^T e^{-\delta u}\int_0^T\frac{(u+c_1s)^{s-1}e^{-c_1\delta s }}{\Gamma(s)\Gamma(T-s+1)}\td s\int_0^{c_1(T-s)}(c_1(T-s)-x)x^{T-s-1}e^{-\delta x} \td x\\ &\eqqcolon&A(c_1,T,u)\end{aligned}$$ and $$\begin{aligned} \lefteqn{B=}\\ &&\frac{\delta^S e^{-\delta (u+c_1T)}}{\Gamma(T)\Gamma(S-T)}\int_0^{u+c_1T} (u+c_1T-z)^{T-1}e^{\delta z}\td z\int_{z+c_2(S-T)}^\IF x^{S-T-1}e^{-\delta x}\td x\\ &&+\frac{\delta^S e^{-\delta (u+c_1T)}}{\Gamma(T)}\int_0^{u+c_1T} (u+c_1T-z)^{T-1}\td z\int_0^{S-T}\frac{(z+c_2s)^{s-1}e^{-c_2\delta s}}{\Gamma(s)\Gamma(S-T-s+1)}\td s\\ &&\quad \cdot \int_0^{c_2(S-T-s)}(c_2(S-T-s)-x)x^{S-T-s-1}e^{-\delta x}\td x\\ &&-\delta^T e^{-\delta (u+c_1T)}\int_0^{c_1T} z e^{\delta z}A(c_2,S-T,z)\td z\\ &&\cdot\int_0^{\frac{c_1T-z}{c_1}}\frac{(u+c_1s)^{s-1}(c_1(T-s)-z)^{T-s-1}}{\Gamma(s)\Gamma(T-s+1)}\td s\,.\end{aligned}$$ For $S=\IF$, we additionally assume that $c_2\delta >1$. In this case, since $X(t)$ has finite variation, in view of Th. 4 in Takács [@ta:65] we have $${{\rm I\hspace{-0.8mm}P}}(\sup_{t<\IF}(X(t)-c_2t)>z)=\frac{c_2\delta-1}{\delta}\,e^{-\delta z}\int_0^\IF\frac{\delta^s}{\Gamma(s)}(z+c_2 s)^{s-1}e^{-\delta c_2 s}\td s\,, \,z>0\,.$$ Let us notice that $A$ is the same as in the case $T<S<\IF$ and using the above expression we get $$\begin{aligned} \lefteqn{B=}\\ &&\frac{(c_2\delta-1)\delta^{T-1}e^{-\delta(u+c_1T)}}{\Gamma(T)}\int_0^{u+c_1T}(u+c_1T-z)^{T-1}\td z\\ &&\cdot\int_0^\IF\frac{\delta^s}{\Gamma(s)}(z+c_2s)^{s-1}e^{-\delta c_2 s}\td s\\ &&-\,(c_2\delta-1)\delta^{T-1} e^{-\delta (u+c_1T)}\int_0^{c_1T} z \td z\\ &&\cdot\int_0^{T-\frac{z}{c_1}}\frac{(u+c_1s)^{s-1}(c_1(T-s)-z)^{T-s-1}}{\Gamma(s)\Gamma(T-s+1)}\td s\int_0^\IF\frac{\delta^t}{\Gamma(t)}(z+c_2 t)^{t-1}e^{-\delta c_2 t}\td t.\end{aligned}$$ Let $Z(s)$ be an $\alpha$-stable Lévy process totally skewed to the right (that is with $\beta=1$ see e.g. Janicki and Weron [@ja:we:94] or Samorodnitsky and Taqqu [@sa:ta:94]) with $1<\alpha<2$ and expectation zero then its density function is the following $$f(x,s)=\frac{1}{\pi s^{1/\alpha}} \int_0^\infty e^{-t^\alpha}\cos\left(ts^{-1/\alpha}x-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t$$ (see e.g. Nolan [@no:97]). Then (see Michna et al. [@mi:pa:pi:15]) for $c>0$ $$\begin{aligned} \lefteqn{A(c,\infty,u)\coloneqq{{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(Z(t)-ct)>u)=}\\ &&\frac{c}{\pi} \int_0^\infty s^{-1/\alpha}\,\td s \int_0^\infty e^{-t^\alpha}\cos\left(ts^{-1/\alpha}(u+cs)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\end{aligned}$$ and for any $c$ and $T>0$ $$\begin{aligned} \lefteqn{A(c,T,u)\coloneqq {{\rm I\hspace{-0.8mm}P}}(\sup_{t<T}(Z(t)-ct)>u)=}\label{astable}\\ &&\frac{1}{\pi T^{1/\alpha}} \int_u^\infty\td x \int_0^\infty e^{-t^\alpha}\cos\left(tT^{-1/\alpha}(x+cT)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\nonumber\\ && +\frac{1}{\pi}\int_0^T \frac{{{\rm I\hspace{-0.8mm}E}}(Z(T-s)-c(T-s))^-}{(T-s)s^{1/\alpha}}\,\td s\nonumber\\ &&\,\,\,\,\,\,\,\,\cdot\int_0^\infty e^{-t^\alpha}\cos\left(ts^{-1/\alpha}(u+cs)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\nonumber\end{aligned}$$ where $${{\rm I\hspace{-0.8mm}E}}(Z(s)-cs)^-=\frac{-1}{\pi s^{1/\alpha}}\int_{-\infty}^0 x\, \td x\int_0^\infty e^{-t^\alpha}\cos\left(ts^{-1/\alpha}(x+cs)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\,.$$ Thus using Th. \[main\] for $S>T>0$ (allowing also $S=\infty$ and putting $\infty-T=\infty$) we get $${{\rm I\hspace{-0.8mm}P}}(\sup_{t<S}(Z(t)-c(t))>u)=A+B=A+B_1-B_2\,,$$ where $A=A(c_1, T, u)$ (see eq. (\[astable\])) and $$\begin{aligned} \lefteqn{B_1=}\\ &&\frac{1}{\pi T^{1/\alpha}}\int_0^\infty A(c_2,S-T,z)\,\td z\\ &&\cdot\int_0^\infty e^{-t^\alpha}\cos\left(tT^{-1/\alpha}(-z+u+c_1T)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\end{aligned}$$ and $$\begin{aligned} \lefteqn{B_2=}\\ &&\frac{1}{\pi^2}\int_0^\infty z\, A(c_2, S-T, z)\,\td z\int_0^T\frac{\td s}{(T-s)^{1/\alpha+1}s^{1/\alpha}}\\ &&\cdot\int_0^\infty e^{-t^\alpha}\cos\left(ts^{-1/\alpha}(u+c_1s)-t^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td t\\ && \cdot\int_0^\infty e^{-w^\alpha}\cos\left(w(T-s)^{-1/\alpha}(-z+c_1(T-s))-w^\alpha\tan{\frac{\pi\alpha}{2}}\right)\td w\,.\end{aligned}$$ Two collaborating insurance companies {#sec3} ===================================== Let us consider two insurance companies which split the amount they pay out of each claim in proportions $\delta_1> 0$ and $\delta_2> 0$ where $\delta_1+\delta_2=1$, and receive premiums at rates $p_1>0$ and $p_2>0$, respectively (see Avram et al. [@av:pa:pi:08]). Then the corresponding risk processes are $$R_i(t)=x_i+p_i t-\delta_i X(t)\,,$$ where $i=1,2$, $x_i>0$ and $X(t)$ is an accumulated claim amount up to time $t$. One can be interested in the instant when at least one insurance company is ruined $$\tau_{or}(x_1, x_2)=\inf\{t>0: R_1(t)<0\,\, \mbox{or}\,\, R_2(t)<0\}$$ and in the instant when both insurance companies are simultaneously ruined $$\tau_{sim}(x_1, x_2)=\inf\{t>0: R_1(t)<0\,\, \mbox{and}\,\, R_2(t)<0\}\,.$$ Let the ultimate ruin probabilities be $$\psi_{or}(x_1, x_2)={{\rm I\hspace{-0.8mm}P}}(\tau_{or}(x_1, x_2)<\infty)\,,\,\,\,\,\,\, \psi_{sim}(x_1, x_2)={{\rm I\hspace{-0.8mm}P}}(\tau_{sim}(x_1, x_2)<\infty)$$ and $$\psi_{1}(x_1)={{\rm I\hspace{-0.8mm}P}}(\tau_1(x_1)<\infty)\,,\,\,\,\,\,\, \psi_{2}(x_2)={{\rm I\hspace{-0.8mm}P}}(\tau_2(x_2)<\infty)\,,$$ where $\tau_i(x_i)=\inf\{t>0: R_i(t)<0\}$ for $i=1,2$. One can also be interested in the following ruin probability $$\psi_{and}(x_1, x_2)={{\rm I\hspace{-0.8mm}P}}(\tau_1(x_1)<\infty\,\,\mbox{and}\,\,\tau_2(x_2)<\infty)$$ and the following relation is easily to notice $$\psi_{and}(x_1, x_2)=\psi_1(x_1)+\psi_2(x_2)-\psi_{or}(x_1,x_2)\,.$$ Let us put $u_i=x_i/\delta_i$ and $c_i=p_i/\delta_i$ where $i=1,2$. Then we get $$\tau_{or}(x_1, x_2)=\inf\{t>0: X(t)>u_1+c_1 t\,\,\mbox{or}\,\,X(t)>u_2+c_2 t\}$$ and $$\tau_{sim}(x_1, x_2)=\inf\{t>0: X(t)>u_1+c_1 t\,\,\mbox{and}\,\,X(t)>u_2+c_2 t\}\,.$$ If the lines $y=u_1+c_1 t$ and $y=u_2+c_2 t$ do not cross each other in the first quadrant then the ruin probabilities $\psi_{or}(x_1, x_2)$ and $\psi_{sim}(x_1, x_2)$ reduce to ordinary ruin probabilities of a risk process with a linear drift. If they cross each other in the first quadrant and e.g. $u_1<u_2$ ($c_1>c_2$) then $$\label{orfor} \psi_{or}(x_1, x_2)={{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(X(t)-c(t))>u_1)\,,$$ where $c(t)$ is defined in eq. (\[d\]) with $T=(u_2-u_1)/(c_1-c_2)$ (we take $c(t)=\min(u_1+c_1 t, u_2+c_2 t)-u_1$). Similarly, if the lines have a common point in the first quadrant and e.g. $u_2<u_1$ ($c_2>c_1$) then $$\label{simfor} \psi_{sim}(x_1, x_2)={{\rm I\hspace{-0.8mm}P}}(\sup_{t<\infty}(X(t)-c(t))>u_1)\,,$$ where $c(t)$ is defined in eq. (\[d\]) with $T=(u_2-u_1)/(c_1-c_2)$ (we take $c(t)=\max(u_1+c_1 t, u_2+c_2 t)-u_1$). Let $X(t)$ be the standard Brownian motion. Then using eq. (\[orfor\]) and Example \[brexpl\] we get for $u_1<u_2$ and $c_1>c_2$ $$\begin{aligned} \lefteqn{\psi_{or}(x_1, x_2)=}\\ &&\Phi(a(-u_1,-c_1))+e^{-2c_1u_1}\Phi(a(-u_1, c_1))\\ &&\,\,\,+e^{-2c_2u_2}\Phi(a(u_1, c_1-2c_2))-e^{-2(c_1-2c_2)u_1-2c_2u_2}\Phi(a(-u_1, c_1-2c_2))\,,\end{aligned}$$ where $a(u,c)=uT^{-1/2}+c\sqrt{T}$, $T=(u_2-u_1)/(c_1-c_2)$, $u_i=x_i/\delta_i$ and $c_i=p_i/\delta_i$ for $i=1,2$. This formula recovers the result of Avram et al. [@av:pa:pi:08] Eq. (67). The same way we obtain the formula for $\psi_{sim}(x_1, x_2)$. In a similar way we can consider ruin probabilities on a finite time horizon. Randomly broken drift and random\ interval ================================= In the fluctuation theory there are many interesting identities for Lévy processes and an exponentially distributed time e.g. the distribution of supremum on an exponentially distributed time interval (see e.g. Bertoin [@be:96] Sec. VI. 2. and Sec. VII). Thus let us consider a spectrally positive Lévy process $X$ with a randomly broken drift that is let us assume that $T$ (see eq. (\[d\])) is an exponentially distributed random variable with mean $1/\lambda$ independent of the process $X$. Moreover let us investigate two cases $S=\infty$ (see Th. \[main\]) and $S-T=V$ is a positive random variable independent of the process $X$ and the random variable $T$. We put $$\varphi_i(\gamma)=\ln {{\rm I\hspace{-0.8mm}E}}\exp(-\gamma( X(1)-c_i)),\, i=1,2\,,$$ where $\gamma\geq 0$ and $\overleftarrow{\varphi}_i(\lambda), i=1,2$ is the inverse function of $\varphi_i$. \[mainlap\] If $X$ is a spectrally positive Lévy process and $T$ is an exponential random variable with mean $1/\lambda>0$ independent of $X$ then for any\ $\gamma>\overleftarrow{\varphi}_1(\lambda)$ $$\begin{aligned} \label{mainexp} \lefteqn{{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t< T+V} (X(t)-c(t))}=}\nonumber\\ &&{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_ {t< T} (X(t)-c_1t)}\\ &&+\,\frac{\gamma\lambda}{\varphi_1(\gamma)-\lambda}\left[\frac{1-{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{ t< V}(X(t)-c_2 t)}}{\gamma}-\frac{1-{{\rm I\hspace{-0.8mm}E}}e^{-\overleftarrow{\varphi}_1(\lambda)\sup_{t<V}(X(t)-c_2 t)}}{\overleftarrow{\varphi}_1(\lambda)}\right]\nonumber\end{aligned}$$ where $V$ is a positive random variable independent of $X$ and $T$. Observe that for $\gamma>0$ $${{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T+V} (X(t)-c(t))}=1-\gamma\int_{0}^\IF e^{-\gamma u}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T+V}X(t)-c(t)>u\right)}\td u$$ and $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T+V}(X(t)-c(t))>u\right)}}\\ &&={{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T}(X(t)-c_1 t)>u\right)}\\ &&+\,{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T}(X(t)-c_1 t)\leq u, \sup_{t<V}(X(t+T)-X(T)-c_2 t)>u-X(T)+c_1 T\right)}.\end{aligned}$$ Thus we have that $$\begin{aligned} \label{I123} \int_{0}^\IF e^{-\gamma u}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T+V}X(t)-c(t)>u\right)}\td u= I_1+I_2\,,\end{aligned}$$ where $$I_1\coloneqq\int_{0}^\IF e^{-\gamma u}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T}X(t)-c_1t>u\right)}\td u=\frac{1-{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T} (X(t)-c_1t)}}{\gamma}$$ and $$\begin{aligned} \lefteqn{I_2 \coloneqq\int_{0}^\IF e^{-\gamma u}}\\ &&\cdot{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<T}(X(t)-c_1 t)\leq u, \sup_{t<V}(X(t+T)-X(T)-c_2 t)>u-X(T)+c_1 T\right)}\td u\,.\end{aligned}$$ By the fact that $T$ is exponentially distributed and independent of $X$ and $V$ we have $$\begin{aligned} \lefteqn{I_2=}\\ &&\lambda\int_0^\IF e^{-\lambda s}\td s\,\int_{0}^\IF e^{-\gamma u}\\ &&\cdot{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<s}(X(t)-c_1 t)\leq u, \sup_{t<V}(X(t+s)-X(s)-c_2 t)>u-X(s)+c_1 s\right)}\td u\,.\end{aligned}$$ Moreover, by the independence of $X(t+s)-X(s)-c_2 t,\, t\geq 0$ and $X(s)-c_1s$ and the fact that $${{\rm I\hspace{-0.8mm}P}\left(\sup_{t<s}(X(t)-c_1 t)\leq u, u-X(s)+c_1 s\leq z\right)}=0\,, \quad z<0$$ we have $$\begin{aligned} I_2&=&\lambda\int_0^\IF e^{-\lambda s}\,\td s\int_{0}^\IF e^{-\gamma u}\,\td u\int_{0}^\IF {{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\\ &&\cdot{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<s}(X(t)-c_1 t)\leq u, u-X(s)+c_1 s\in \td z\right)}\\ &=&\lambda\int_{0}^\IF e^{-\gamma u}\,\td u\int_{0}^\IF e^{-\lambda s}\,\td s\int_0^\IF{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\\ &&\cdot {{\rm I\hspace{-0.8mm}P}\left(\inf_{t<s}(u-X(t)+c_1 t)>0, u-X(s)+c_1 s\in \td z\right)}\,.\end{aligned}$$ Due to Suprun [@su:76] (see also Bertoin [@be:97] Lemma 1) we have that $$\begin{aligned} &&\int_0^\IF e^{-\lambda s}{{\rm I\hspace{-0.8mm}P}\left(\inf_{t<s}(u-X(t)+c_1 t)>0, u-X(s)+c_1 s\in \td z\right)} \td s\\ &&\quad = \left[e^{-\overleftarrow{\varphi}_1(\lambda)z} W^{(\lambda)}(u)-{{1\hspace{-1mm}{\rm I}}}(u\geq z)W^{(\lambda)}(u-z)\right]\td z\,, \end{aligned}$$ where ${{1\hspace{-1mm}{\rm I}}}(\cdot)$ is the indicator function and $W^{(\lambda)}: [0,\IF)\rightarrow [0,\IF)$ is a continuous and increasing function such that $$\int_0^\IF e^{-\gamma x} \,W^{(\lambda)}(x)\td x=\frac{1}{\varphi_1(\gamma)-\lambda}\,, \quad \gamma>\overleftarrow{\varphi}_1(\lambda)\,.$$ Consequently, for $\gamma>\overleftarrow{\varphi}_1(\lambda)$ $$\begin{aligned} \lefteqn{I_2=}\\ &&\lambda\int_{0}^\IF\int_{0}^\IF e^{-\gamma u}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\\ &&\cdot\left[e^{-\overleftarrow{\varphi}_1(\lambda)z} W^{(\lambda)}(u)-\mathbb{I}(u\geq z)W^{(\lambda)}(u-z)\right]\td z\td u\\ &=&\lambda \int_{0}^\IF e^{-\overleftarrow{\varphi}_1(\lambda)z}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t< V}(X(t)-c_2 t)>z\right)}\td z\int_{0}^\IF e^{-\gamma u}W^{(\lambda)}(u)\td u\\ &&-\lambda \int_{0}^\IF {{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\td z\int_0^\IF \mathbb{I}(u\geq z)e^{-\gamma u}W^{(\lambda)}(u-z)\td u \\ &=& \frac{\lambda}{\varphi_1(\gamma)-\lambda}\left[\int_{0}^\IF e^{-\overleftarrow{\varphi}_1(\lambda)z}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\td z\right.\\ &&\left.- \int_{0}^\IF e^{-\gamma z}{{\rm I\hspace{-0.8mm}P}\left(\sup_{t<V}(X(t)-c_2 t)>z\right)}\td z\right]\\ &=& \frac{\lambda}{\varphi_1(\gamma)-\lambda}\left[\frac{1-{{\rm I\hspace{-0.8mm}E}}e^{-\overleftarrow{\varphi}_1(\lambda)\sup_{t<V}(X(t)-c_2 t)}}{\overleftarrow{\varphi}_1(\lambda)}-\frac{1-{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<V}(X(t)-c_2 t)}}{\gamma}\right].\end{aligned}$$ [ $\Box$]{} Under the assumption of Theorem \[mainlap\], if $V=\IF$, then $$\label{laplaceinf} {{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<\IF} (X(t)-c(t))} =\frac{\gamma\lambda \varphi_2'(0)[\varphi_2(\gamma)-\varphi_2(\overleftarrow{\varphi}_1(\lambda))]}{\varphi_2(\gamma)(\varphi_1(\gamma)-\lambda)\varphi_2(\overleftarrow{\varphi}_1(\lambda))}\,.$$ If $V$ is an exponentially distributed random variable with mean $1/\theta>0$ independent of $X$ and $T$ then $$\label{laplaceT} {{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T+V} (X(t)-c(t))} = \gamma \lambda\theta\,\frac{\frac{\overleftarrow{\varphi}_2(\theta)-\overleftarrow{\varphi}_1(\lambda)}{\theta-\varphi_2(\overleftarrow{\varphi}_1(\lambda))} -\frac{\overleftarrow{\varphi}_1(\lambda)[\overleftarrow{\varphi}_2(\theta)-\gamma]}{\gamma[\theta-\varphi_2(\gamma)]}}{\overleftarrow{\varphi}_1(\lambda)\overleftarrow{\varphi}_2(\theta)[\varphi_1(\gamma)-\lambda]}\,.$$ . It is well-known that $$\begin{aligned} \label{laplace1}{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T} (X(t)-c_1t)}=\frac{\lambda}{\lambda-\varphi_1(\gamma)}\left(1-\frac{\gamma}{\overleftarrow{\varphi}_1(\lambda)}\right)\,,\end{aligned}$$ where $\gamma\geq 0$ (see e.g. Bertoin [@be:96] eq. (3) p. 192 or Th. 4.1 in Dębicki and Mandjes [@de:ma:15]). Moreover, by Th. 3.2 in Dębicki and Mandjes [@de:ma:15] (or going with $\lambda$ to 0 in the previous identity), it follows that $${{\rm I\hspace{-0.8mm}E}}\exp\left(-\gamma\sup_{t<\IF}(X(t)-c_2 t)\right)=\frac{\gamma\varphi_2'(0)}{\varphi_2(\gamma)}\,.$$ Consequently, by (\[mainexp\]) for $\gamma>0$ $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<\IF} (X(t)-c(t))}=}\\ &&\frac{\lambda}{\lambda-\varphi_1(\gamma)}\left[1-\frac{\gamma}{\overleftarrow{\varphi}_1(\lambda)}\right] +\frac{\gamma\lambda}{\varphi_1(\gamma)-\lambda}\left[\frac{1-\frac{\gamma\varphi_2'(0)}{\varphi_2(\gamma)}} {\gamma}-\frac{1-\frac{\overleftarrow{\varphi}_1(\lambda)\varphi_2'(0)}{\varphi_2(\overleftarrow{\varphi}_1(\lambda))}}{\overleftarrow{\varphi}_1(\lambda)}\right]\\ &=&\frac{\gamma \lambda\varphi_2'(0)[\varphi_2(\gamma)-\varphi_2(\overleftarrow{\varphi}_1(\lambda))]}{\varphi_2(\gamma)(\varphi_1(\gamma)-\lambda)\varphi_2(\overleftarrow{\varphi}_1(\lambda))}\,.\end{aligned}$$ . Using (\[laplace1\]), for $\gamma\geq 0$ we have that $${{\rm I\hspace{-0.8mm}E}}\exp\left(-\gamma\sup_{t< V}(X(t)-c_2 t)\right)=\frac{\theta}{\theta-\varphi_2(\gamma)}\left(1-\frac{\gamma}{\overleftarrow{\varphi}_2(\theta)}\right).$$ Recalling (\[laplace1\]), for $\gamma>0$ it follows that $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T+V} X(t)-c(t)}=}\\ &&\frac{\lambda}{\lambda-\varphi_1(\gamma)}\left(1-\frac{\gamma}{\overleftarrow{\varphi}_1(\lambda)}\right)\\ &&+\frac{\gamma\lambda}{\varphi_1(\gamma)-\lambda}\left[\frac{1-\frac{\theta}{\theta-\varphi_2(\gamma)}\left[1-\frac{\gamma}{\overleftarrow{\varphi}_2(\theta)}\right]} {\gamma}-\frac{1-\frac{\theta}{\theta-\varphi_2(\overleftarrow{\varphi}_1(\lambda))}\left[1-\frac{\overleftarrow{\varphi}_1(\lambda)}{\overleftarrow{\varphi}_2(\theta)}\right]} {\overleftarrow{\varphi}_1(\lambda)}\right]\\ &=& \gamma \lambda\theta\,\frac{\frac{\overleftarrow{\varphi}_2(\theta)-\overleftarrow{\varphi}_1(\lambda)}{\theta-\varphi_2(\overleftarrow{\varphi}_1(\lambda))} -\frac{\overleftarrow{\varphi}_1(\lambda)[\overleftarrow{\varphi}_2(\theta)-\gamma]}{\gamma[\theta-\varphi_2(\gamma)]}}{\overleftarrow{\varphi}_1(\lambda)\overleftarrow{\varphi}_2(\theta)[\varphi_1(\gamma)-\lambda]}\,.\end{aligned}$$ [ $\Box$]{} Let $W$ be the standard Brownian motion. Then $$\varphi_1(\gamma)=\frac{1}{2}\gamma^2+c_1\gamma\,, \quad \varphi_2(\gamma)=\frac{1}{2}\gamma^2+c_2\gamma\,,$$ $$\overleftarrow{\varphi}_1(\lambda)=\sqrt{c_1^2+2\lambda}-c_1\,, \quad \overleftarrow{\varphi}_2(\lambda)=\sqrt{c_2^2+2\lambda}-c_2\,.$$ Consequently, for $\gamma>\sqrt{c_1^2+2\lambda}-c_1$ $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<\IF} (W(t)-c(t))}=}\\ && \frac{\gamma \lambda c_2 (\frac{1}{2}\gamma^2+c_2\gamma-c_1^2-\lambda -(c_2-c_1)\sqrt{c_1^2+2\lambda}+c_1c_2)}{(\frac{1}{2}\gamma^2+c_1\gamma-\lambda)(\frac{1}{2}\gamma^2+c_2\gamma)(c_1^2+\lambda +(c_2-c_1)\sqrt{c_1^2+2\lambda}-c_1c_2)}\\\end{aligned}$$ and $$\begin{aligned} \lefteqn{{{\rm I\hspace{-0.8mm}E}}e^{-\gamma\sup_{t<T+V} (W(t)-c(t))}=}\\ && \gamma\lambda\theta\, \frac{\frac{\sqrt{c_2^2+2\theta}-\sqrt{c_1^2+2\lambda}+c_1-c_2}{\theta-c_1^2-\lambda-(c_2-c_1)\sqrt{c_1^2+2\lambda}+c_1c_2}-\frac{(\sqrt{c_1^2+2\lambda}-c_1)(\sqrt{c_2^2+2\theta}-c_2-\gamma)}{\gamma(\theta-\frac{1}{2}\gamma^2-c_2\gamma)}}{(\sqrt{c_1^2+2\lambda}-c_1)(\sqrt{c_2^2+2\theta}-c_2)(\frac{1}{2}\gamma^2+c_1\gamma-\lambda)}\,.\end{aligned}$$ Acknowledgments {#acknowledgments .unnumbered} --------------- The author would like to express his sincere thanks to Professors Krzysztof Dębicki and Peng Liu for valuable comments and remarks and especially for pointing the problems with random time intervals. 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(1957) [*Some problems in the theory of dams*]{}. J. Royal Stat. Soc. Ser. B 19, pp. 207–212. Lieshout, P. and Mandjes, M. (2007) [*Tandem Brownian queues*]{}. Math. Meth. Oper. Res. 66, pp. 275–298. Mandjes, M. (2004) [*Packet models revisited: tandem and priority system*]{}. Queueing Systems 47, pp. 363–377. Michna, Z. (2011) [*Formula for the supremum distribution of a spectrally positive $\alpha$-stable Lévy process*]{}. Stat. Probabil. Lett. 81, pp. 231–235. Michna, Z. (2013) [*Explicit formula for the supremum distribution of a spectrally negative stable process*]{}. Electron. Commun. Probab. 18, pp. 1–6. Michna, Z., Palmowski, Z. and Pistorius, M. (2015) [*The distribution of the supremum for spectrally asymmetric Lévy processes*]{}. Electron. Commun. Probab. 20, pp. 1–10. Nolan, J.P. (1997) [*Numerical calculation of stable densities and distribution functions*]{}. Stochastic Models 13, pp. 759–774. Samorodnitsky, G. and Taqqu, M. 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--- abstract: 'In this paper, we consider the phase retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let $\{{\mathbf a}_j\}_{j=1}^m \subset {{\mathbb H}}^d$ and ${\mathbf b}=(b_1, \ldots, b_m)^\top\in{{\mathbb H}}^m$, where ${{\mathbb H}}={{\mathbb R}}$ or ${{\mathbb C}}$. We say $\{{\mathbf a}_j\}_{j=1}^m$ and ${\mathbf b}$ are affine phase retrievable for ${{\mathbb H}}^d$ if any ${{\mathbf x}}\in{{\mathbb H}}^d$ can be recovered from the magnitudes of the affine measurements $\{{\lvert{\langle {{{\mathbf a}}_j,{{\mathbf x}}} \rangle}+b_j\rvert},\, 1\leq j\leq m\}$. We develop general framework for affine phase retrieval and prove necessary and sufficient conditions for $\{{\mathbf a}_j\}_{j=1}^m$ and ${\mathbf b}$ to be affine phase retrievable. We establish results on minimal measurements and generic measurements for affine phase retrieval as well as on sparse affine phase retrieval. In particular, we also highlight some notable differences between affine phase retrieval and the standard phase retrieval in which one aims to recover a signal ${{\mathbf x}}$ from the magnitudes of its linear measurements. In standard phase retrieval, one can only recover ${{\mathbf x}}$ up to a unimodular constant, while affine phase retrieval removes this ambiguity. We prove that unlike standard phase retrieval, the affine phase retrievable measurements $\{{\mathbf a}_j\}_{j=1}^m$ and ${\mathbf b}$ do not form an open set in ${{\mathbb H}}^{m\times d}\times {{\mathbb H}}^m$. Also in the complex setting, the standard phase retrieval requires $4d-O(\log_2d)$ measurements, while the affine phase retrieval only needs $m=3d$ measurements.' address: - 'LSEC, Inst. Comp. Math., Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100091, China' - 'Department of Mathematics University of Central Florida Orlando, FL 32816, USA ' - 'Department of Mathematics, the Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong' - 'LSEC, Inst. Comp. Math., Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100091, China' author: - Bing Gao - Qiyu Sun - Yang Wang - Zhiqiang Xu title: Phase Retrieval From the Magnitudes of Affine Linear Measurements --- [^1] [^2] [^3] Introduction ============ Phase retrieval --------------- Phase retrieval is an active topic of research in recent years as it arises in many different areas of studies (see e.g. [@BCE06; @BoHa15; @CSV12; @CEHV13; @CCSW; @CCD; @FMW14; @HMW13] and the references therein). For a vector (signal) ${{\mathbf x}}\in {{\mathbb H}}^d$, where ${{\mathbb H}}={{\mathbb R}}$ or ${{\mathbb C}}$, the aim of phase retrieval is to recover ${{\mathbf x}}$ from ${\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}\rvert},\, j=1,\ldots,m$, where ${\mathbf a}_j\in {{\mathbb H}}^d$ and we usually refer to $\{{\mathbf a}_j\}_{j=1}^m$ as the [*measurement vectors*]{}. Since for any unimodular $c\in {{\mathbb H}}$, we have ${\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}\rvert}={\lvert{\langle {{\mathbf a}_j,c{{\mathbf x}}} \rangle}\rvert}$, the best outcome phase retrieval can achieve is to recover ${{\mathbf x}}$ up to a unimodular constant. We briefly overview some of the results in phase retrieval and introduce some notations. For the set of measurement vectors $\{{\mathbf a}_j\}_{j=1}^m$, we set ${\mathbf A}:=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb H}}^{m\times d}$ which we shall refer to as the [*measurement matrix*]{}. We shall in general identify the set of measurement vectors $\{{\mathbf a}_j\}_{j=1}^m$ with the corresponding measurement matrix ${\mathbf A}$, and often use the two terms interchangeably whenever there is no confusion. Define the map ${\mathbf M}_{{\mathbf A}}:{{\mathbb H}}^d\rightarrow {{\mathbb R}}^m_+$ by $${\mathbf M}_{{\mathbf A}}({{\mathbf x}})\,\,=\,\, ({\lvert{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle}\rvert},\ldots, {\lvert{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle}\rvert}).$$ We say ${\mathbf A}$ is [*phase retrievable* ]{} for ${{\mathbb H}}^d$ if ${\mathbf M}_{{\mathbf A}}({{\mathbf x}})={\mathbf M}_{{\mathbf A}}({{\mathbf y}})$ implies ${{\mathbf x}}\in \{c {{\mathbf y}}: c\in {{\mathbb H}}, {\lvertc\rvert}=1\}$. There have been extensive studies of phase retrieval from various different angles. For example many efficient algorithms to recover ${{\mathbf x}}$ from ${\mathbf M}_{\mathbf A}({{\mathbf x}})$ have been developed, see e.g. [@CSV12; @CESV12; @WF; @PN13] and their references. One of the fundamental problems on the theoretical side of phase retrieval is the following question: [*How many vectors in the measurement matrix ${\mathbf A}$ are needed so that ${\mathbf A}$ is phase retrievable?*]{} It is shown in [@BCE06] that for ${\mathbf A}$ to be phase retrievable for ${{\mathbb R}}^d$, it is necessary and sufficient that $m \geq 2d-1$. In the complex case ${{\mathbb H}}={{\mathbb C}}$, the same question becomes much more challenging, however. The minimality question remains open today. Balan, Casazza and Edidin [@BCE06] first show that ${\mathbf A}$ is phase retrievable if it contains $m\geq 4d-2$ generic vectors in ${{\mathbb C}}^d$. Bodmann and Hammen [@BoHa15] show that $m=4d-4$ measurement vectors are possible for phase retrieval through construction (see also Fickus, Mixon, Nelson and Wang [@FMW14]). Bandeira, Cahill, Mixon and Nelson [@BCMN] conjecture that (a) $m\geq 4d-4$ is necessary for ${\mathbf A}$ to be phase retrievable and, (b) ${\mathbf A}$ with $m\geq 4d-4$ generic measurement vectors is phase retrievable. Part (b) of the conjecture is proved by Conca, Edidin, Hering and Vinzant [@CEHV13]. They also confirm part (a) for the case where $d$ is in the form of $2^k+1, \, k\in {{\mathbb Z}}_+$. However, Vinzant in [@small] presents a phase retrievable ${\mathbf A}$ for ${{\mathbb C}}^4$ with $m=11=4d-5<4d-4$ measurement vectors, thus disproving the conjecture. The measurement vectors in the counterexample are obtained using Gröbner basis and algebraic computation. Phase retrieval from magnitudes of affine linear measurements ------------------------------------------------------------- Here we consider the affine phase retrieval problem, where instead of being given the magnitudes of linear measurements, we are given the magnitudes of affine linear measurements that include shifts. More precisely, instead of recovering ${{\mathbf x}}$ from $\{|{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}|\}_{j=1}^m$, we consider recovering ${{\mathbf x}}$ from the absolute values of the affine linear measurements $${\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert},\quad j=1,\ldots,m,$$ where ${\mathbf a}_j\in {{\mathbb H}}^d$, ${\mathbf b}=(b_1,\ldots,b_m)^\top\in {{\mathbb H}}^m$. Unlike in the classical phase retrieval, where ${{\mathbf x}}$ can only be recovered up to a unimodular constant, we will show that one can recover ${{\mathbf x}}$ [*exactly*]{} from $({\lvert{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle}+b_1\rvert},\ldots,{\lvert{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle}+b_m\rvert})$ if the vectors ${\mathbf a}_j$ and shifts $b_j$ are properly chosen. Let ${\mathbf A}= ({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb H}}^{m\times d}$ and ${\mathbf b}\in{{\mathbb H}}^m$. Define the map ${\mathbf M}_{{\mathbf A},{\mathbf b}}:{{\mathbb H}}^d\rightarrow {{\mathbb R}}^m_+$ by $$\label{eq:M1} {\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})=\left({\lvert{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle}+b_1\rvert},\ldots, {\lvert{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle}+b_m\rvert}\right).$$ We say the pair $({\mathbf A},{\mathbf b})$ (which can also be viewed as a matrix in ${{\mathbb H}}^{m \times (d+1)}$) is [*affine phase retrievable*]{} for ${{\mathbb H}}^d$, or simply [*phase retrievable*]{} whenever there is no confusion, if ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ is injective on ${{\mathbb H}}^d$. Note that sometimes it is more convenient to consider the map $$\label{eq:M2} {\mathbf M}^2_{{\bf A},{\bf b}}({{\mathbf x}}):=(|{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle} +b_1|^2, \ldots, |{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle} +b_m|^2).$$ Clearly $({\mathbf A},{\mathbf b})$ is affine phase retrievable if and only if ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}$ is injective on ${{\mathbb H}}^d$. The goal of this paper is to develop a framework of affine phase retrieval. There are several motivations for studying affine phase retrieval. It arises naturally in holography, see e.g. [@Lie03]. It could also arise in other phase retrieval applications, such as reconstruction of signals in a shift-invariant space from their phaseless samples [@CCSW], where some entries of ${{\mathbf x}}$ might be known in advance. In such scenarios, assume that the object signal is ${{\mathbf y}}\in {{\mathbb H}}^{d+k}$ and the first $k$ entries of ${{\mathbf y}}$ are known. We can write ${{\mathbf y}}=(y_1,\ldots,y_k, {{\mathbf x}})$, where $y_1,\ldots,y_k$ are known and ${{\mathbf x}}\in {{\mathbb H}}^d$. Suppose that ${\tilde {\mathbf a}}_j=(a_{j1},\ldots,a_{jk},{\mathbf a}_j)\in {{\mathbb H}}^{d+k},\, j=1,\ldots,m$ are the measurement vectors. Then $${\lvert{\langle {{\tilde {\mathbf a}}_j,{{\mathbf y}}} \rangle}\rvert}\,\,=\,\,{\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert},$$ where $b_j:=a_{j1}y_1+\cdots+a_{jk}y_k$. So if $(y_1,\ldots,y_k)$ is a nonzero vector, we can take advantage of knowing the first $k$ entries and reduce the standard phase retrieval in ${{\mathbb H}}^{d+k}$ to affine phase retrieval in ${{\mathbb H}}^d$. Our contribution ----------------- This paper considers affine phase retrieval for both real and complex signals. In Section 2, we consider the real case ${{\mathbb H}}={{\mathbb R}}$ and prove several necessary and sufficient conditions under which ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ is injective on ${{\mathbb R}}^d$. For an index set $T\subset \{1,\ldots,m\}$, we use ${\mathbf A}_T$ to denote the sub-matrix ${\mathbf A}_T:=({\mathbf a}_j: j\in T)^\top$ of ${\mathbf A}$. Let $ \#T $ denote the cardinality of $ T $, ${\rm span}({\mathbf A}_T)\subset {{\mathbb R}}^{\#T}$ denote the subspace spanned by the column vectors of ${\mathbf A}_T$. In particularly, we show that $({\mathbf A},{\mathbf b})$ is affine phase retrievable for ${{\mathbb R}}^d$ if and only if ${\rm span}\{{\mathbf a}_j:j\in S^c\}={{\mathbb R}}^d$ for any index set $S\subset \{1,\ldots,m\}$ satisfying ${\mathbf b}_S\in {\rm span}({\mathbf A}_S)$. Based on this result, we prove that the measurement vectors set ${\mathbf A}$ must have at least $m\geq 2d$ elements for $({\mathbf A},{\mathbf b})$ to be affine phase retrievable. Furthermore, we prove any generic ${\mathbf A}\in {{\mathbb R}}^{m\times d}$ and ${\mathbf b}\in{{\mathbb R}}^m$, where $m\geq 2d$ will be affine phase retrievable. The recovery of sparse signals from phaseless measurement also attracts much attention recently [@WaXu14; @GWaXu16]. In this section, we consider the real affine phase retrieval for sparse vectors. We turn to the complex case ${{\mathbb H}}={{\mathbb C}}$ in Section 3. First we establish equivalent necessary and sufficient conditions for $({\mathbf A},{\mathbf b})$ to be affine phase retrievable for ${{\mathbb C}}^d$. Using these conditions, we show that $({\mathbf A},{\mathbf b})\in {{\mathbb C}}^{m\times (d+1)}$ is [*not*]{} affine phase retrievable for ${{\mathbb C}}^d$ if $m<3d$. The result is sharp as we also construct an affine phase retrievable $({\mathbf A},{\mathbf b})$ for ${{\mathbb C}}^d$ with $m=3d$. This result shows that the nature of affine phase retrieval can be quite different from that of the standard phase retrieval in the complex setting, where it is known that $4d -O(\log_2d)$ measurements are needed for phase retrieval [@HMW13; @WaXu16]. Note that for $ j=1,\ldots,m$ we have $$\label{eq:shiftphase} {\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}\,\,=\,\,{\lvert{\langle {{\tilde {\mathbf a}}_j,{\tilde {{\mathbf x}}}} \rangle}\rvert}, ~~\mbox{where}~~ {\tilde {{\mathbf x}}}=\begin{pmatrix} {{\mathbf x}}\\ 1\end{pmatrix}, {\tilde {\mathbf a}}_j = \begin{pmatrix} {\mathbf a}_j \\ b_j\end{pmatrix}.$$ It shows that affine phase retrieval for ${{\mathbf x}}$ can be reduced to the classical phase retrieval for ${\tilde {{\mathbf x}}}\in {{\mathbb C}}^{d+1}$ from ${\lvert{\langle {{\tilde {\mathbf a}}_j,{\tilde {{\mathbf x}}}} \rangle}\rvert}, \,j=1,\ldots,m$. Because the last entry of $\tilde {{\mathbf x}}$ is $1$, it allows us to recover ${{\mathbf x}}$ without the unimodular constant ambiguity. Observe also from [@CEHV13] that $4(d+1)-4=4d$ generic measurements are enough to recover $\tilde {{\mathbf x}}$ up to a unimodular constant, and hence they are also enough to recover ${{\mathbf x}}$. In Section 3, we prove the stronger result that a generic $({\mathbf A},{\mathbf b})$ in ${{\mathbb C}}^{m\times (d+1)}$ with $m\geq 4d-1$ is affine phase retrievable. The classical phase retrieval has the property that the set of phase retrievable ${\mathbf A}\in {{\mathbb H}}^{m\times d}$ is an open set, and hence the phase retrievable property is stable under small perturbations [@RaduSt; @BaWaSt]. Surprisingly, viewing $({\mathbf A},{\mathbf b})$ as an element in ${{\mathbb H}}^{m\times (d+1)}$, we prove that the set of affine phase retrievable $({\mathbf A},{\mathbf b})$ is [*not*]{} an open set. As far as stability of affine phase retrieval is concerned, we prove several new results in Section 4. For the standard phase retrieval, one uses $\min_{{\lvert\alpha\rvert}=1}\|{{\mathbf x}}-\alpha {{\mathbf y}}\|$ to measure the distance between ${{\mathbf x}}$ and ${{\mathbf y}}$. The robustness of phase retrieval is established via the lower bound of the following bi-Lipschitz type inequalities for any phase retrievable ${\mathbf A}$, $$\label{eq:stability} c\min_{\alpha\in {{\mathbb C}}, {\lvert\alpha\rvert}=1}\|{{\mathbf x}}-\alpha {{\mathbf y}}\|\,\leq\, \|{\mathbf M}_{\mathbf A}({{\mathbf x}})-{\mathbf M}_{\mathbf A}({{\mathbf y}})\| \,\leq\, C\min_{\alpha\in {{\mathbb C}}, {\lvert\alpha\rvert}=1}\|{{\mathbf x}}-\alpha {{\mathbf y}}\|,$$ where $c, C>0$ depends only on ${\mathbf A}$ [@CCD]. More explicit estimate of the constant $c$ was given in[@BaWaSt]. For the affine phase retrieval, we use $\|{{\mathbf x}}-{{\mathbf y}}\|$ to measure the distance between ${{\mathbf x}}$ and ${{\mathbf y}}$ because it is possible to recover ${{\mathbf x}}$ exactly in the affine phase retrieval. For the affine phase retrieval, we show that both ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ and ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}$ are bi-Lipschitz continuous on any compact sets, but are not bi-Lipschitz on ${{\mathbb H}}^d$. Affine Phase Retrieval for Real Signals ======================================= We consider affine phase retrieval of real signals in this section. Several equivalent conditions for affine phase retrieval are established. We also study affine phase retrieval for sparse signals. In particular we answer the minimality question, namely what is the smallest number of measurements needed for affine phase retrievability for ${{\mathbb R}}^d$. Real affine phase retrieval --------------------------- Let $T\subset \{1,2, \ldots, m\}$. We first recall that for the measurement matrix ${\bf A}=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb R}}^{m\times d}$, we use ${\mathbf A}_T$ to denote the submatrix of ${\mathbf A}$ consisting only those rows indexed in $T$, i.e. ${\mathbf A}_T:=({\mathbf a}_j: j\in T)^\top$. Similarly we use ${\mathbf b}_T$ to denote the sub-vector of ${\mathbf b}$ consisting only entries indexed in $T$. For any matrix ${\bf B}$, we use ${\rm span} ({\bf B})$ to denote the subspace spanned by the [*columns*]{} of ${\bf B}$. Thus for any index subset $T$, the notation ${\rm span}({\bf A}_T)$ denotes the subspace of ${{\mathbb R}}^{\# T}$ spanned by the columns of ${\mathbf A}_T$. \[le:real\] Let ${\bf A}=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb R}}^{m\times d}$ and ${{\mathbf b}}=(b_1, \ldots, b_m)^\top\in {{\mathbb R}}^m$. Then the followings are equivalent: - $({\mathbf A},{\mathbf b})$ is affine phase retrievable for ${{\mathbb R}}^d$. - The map ${\mathbf M}^2_{{\bf A},{\bf b}}$ is injective on ${{\mathbb R}}^d$, where ${\mathbf M}^2_{{\bf A},{\bf b}}$ is defined in (\[eq:M2\]). - For any ${{\mathbf u}},{{\mathbf v}}\in{{\mathbb R}}^d$ and ${{\mathbf u}}\neq 0$, there exists a $k$ with $1\leq k \leq m$ such that $${\langle {{\mathbf a}_k,{{\mathbf u}}} \rangle}\bigl({\langle {{\mathbf a}_k,{{\mathbf v}}} \rangle} +b_k\bigr) \neq 0.$$ - For any $S\subset \{1,2, \ldots, m\}$, if ${\bf b}_S\in {\rm span}({\bf A}_S)$ then ${\rm span} ({\mathbf A}_{S^c}^\top)= {\rm span}\{{\mathbf a}_{j}:j\in S^c\}={{\mathbb R}}^d$. - The Jacobian $J({{\mathbf x}})$ of the map ${\mathbf M}^2_{{\bf A},{\bf b}}$ has rank $d$ for all ${{\mathbf x}}\in{{\mathbb R}}^d$. The equivalence of (A) and (B) have already been discussed earlier. We focus on the other conditions. \(A) $\Leftrightarrow$ (C).  Assume that ${\mathbf M}_{{\bf A},{\bf b}}({\bf x})={\mathbf M}_{{\bf A},{\bf b}}({\bf y})$ for some ${\bf x} \neq {\bf y}$ in ${{\mathbb R}}^d$. For any $j$, we have $$\label{eq:deng} {\lvert{\langle {{\mathbf a}_j,{\bf x}} \rangle}+b_j\rvert}^2-{\lvert{\langle {{\mathbf a}_j,{\bf y}} \rangle}+b_j\rvert}^2 ={\langle {{\mathbf a}_j, {\bf x}-{\bf y}} \rangle}({\langle {{\mathbf a}_j,{\bf x}+{\bf y}} \rangle}+2b_j).$$ Set $2{{\mathbf u}}= {{\mathbf x}}-{{\mathbf y}}$ and $2{{\mathbf v}}= {{\mathbf x}}+{{\mathbf y}}$. Then ${{\mathbf u}}\neq 0$ and for all $j$, $$\label{eq:innerp_zero} {\langle {{\mathbf a}_j,{{\mathbf u}}} \rangle}\bigl({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle} +b_j\bigr) = 0.$$ Conversely, assume that (\[eq:innerp\_zero\]) holds for all $j$. Let ${{\mathbf x}},{{\mathbf y}}\in{{\mathbb R}}^d$ be given by ${{\mathbf x}}-{{\mathbf y}}=2{{\mathbf u}}$ and ${{\mathbf x}}+{{\mathbf y}}=2{{\mathbf v}}$. Then ${{\mathbf x}}\neq {{\mathbf y}}$. However, we would have ${\mathbf M}^2_{{\bf A},{\bf b}}({\bf x})={\mathbf M}^2_{{\bf A},{\bf b}}({\bf y})$ and hence $({\mathbf A},{\mathbf b})$ cannot be affine phase retrievable. \(C) $\Leftrightarrow$ (D).  Assume that (C) holds. If for some $S\subset \{1,2, \ldots, m\}$ with ${\bf b}_S\in {\rm span}({\bf A}_S)$, we have ${\rm span}\{{\mathbf a}_{j}: ~j\in S^c\}\neq {{\mathbb R}}^d$, then we can find ${{\mathbf u}}\neq 0$ such that ${\langle {{\mathbf a}_j,{{\mathbf u}}} \rangle} = 0$ for all $j\in S^c$. Moreover, since ${\bf b}_S\in {\rm span}({\bf A}_S)$, we can find ${{\mathbf v}}\in{{\mathbb R}}^d$ such that $-b_j = {\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle} $ for all $j\in S$. Thus for all $1 \leq j \leq m$, we have $${\langle {{\mathbf a}_j,{{\mathbf u}}} \rangle}\bigl({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle} +b_j\bigr) = 0.$$ This is a contradiction. The converse clearly also holds. \(C) $\Leftrightarrow$ (E).   Note that the Jacobian $J({{\mathbf v}})$ of the map ${\mathbf M}^2_{{\bf A},{\bf b}}$ at the point ${{\mathbf v}}\in{{\mathbb R}}^d$ is precisely $$J({{\mathbf v}}) = \Bigl(({\langle {{\mathbf a}_1,{{\mathbf v}}} \rangle} +b_1){\mathbf a}_1, ({\langle {{\mathbf a}_2,{{\mathbf v}}} \rangle} +b_2){\mathbf a}_2, \ldots, ({\langle {{\mathbf a}_m,{{\mathbf v}}} \rangle} +b_m){\mathbf a}_m\Bigr),$$ i.e. the $j$-th column of $J({{\mathbf v}})$ is $({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle} +b_j){\mathbf a}_j$. Thus ${{\rm rank}}(J({{\mathbf v}}))\neq d$ if and only if there exists a nonzero ${{\mathbf u}}\in{{\mathbb R}}^d$ such that $${{\mathbf u}}^\top J({{\mathbf v}})=\Bigl({\langle {{\mathbf a}_1,{{\mathbf u}}} \rangle}\bigl({\langle {{\mathbf a}_1,{{\mathbf v}}} \rangle} +b_1\bigr), \ldots, {\langle {{\mathbf a}_m,{{\mathbf u}}} \rangle}\bigl({\langle {{\mathbf a}_m,{{\mathbf v}}} \rangle} +b_m\bigr)\Bigr)=0.$$ The equivalence of (C) and (E) now follows. As an application of Theorem \[le:real\], we show that the minimal number of affine measurements to recover all $d$-dimensional real signals is $2d$. \[realminimal.thm1\] Let ${\bf A}=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb R}}^{m\times d}$ and ${\bf b}\in {{\mathbb R}}^m$. If $m\leq 2d-1$, then $({\mathbf A},{\mathbf b})$ is not affine phase retrievable for ${{\mathbb R}}^d$. We divide the proof into two cases. [*Case 1*]{}: ${\rm rank}({\bf A})\leq d-1$. In this case, there exists a nonzero vector ${\bf u}\in {{\mathbb R}}^d$ such that $ \langle {\bf a}_j, {\bf u}\rangle =0, \ 1\le j\le m$. Thus for any ${\bf x}\in {{\mathbb R}}^d$, $${\lvert{\langle {{\mathbf a}_j,{\bf x}} \rangle}+b_j\rvert}^2={\lvert{\langle {{\mathbf a}_j,{\bf x}+{\bf u}} \rangle}+b_j\rvert}^2, \quad 1\le j\le m.$$ This means that ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ is not injective. [*Case 2*]{}: ${\rm rank}({\bf A})= d$. In this case, there exists an index set $S_0\subset \{1,\ldots,m\}$ with cardinality $d$ so that the square matrix ${\bf A}_{S_0}$ has full rank $d$, which implies $$\label{realminimal.thm1.pf.eq1} {\bf b}_{S_0}\in {\rm span}({\bf A}_{S_0}).$$ In other words, there exists ${{\mathbf v}}\in {{\mathbb R}}^d$ such that ${\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle}+b_j=0$ for all $j \in S_0$. Now since $m\le 2d-1$ and $\# S_0=d$, we have $\#S_0^c=m-d\leq d-1$. Hence there exists a nonzero ${{\mathbf u}}\in{{\mathbb R}}^d$ such that ${{\mathbf u}}\perp \{{\mathbf a}_j:~j\in S_0^c\}$. The non-injectivity follows immediately from Theorem \[le:real\] (C). We next consider generic measurements. There are various ways one can define the meaning of being generic. A rigorous definition involves the use of Zariski topology. In this paper, we adopt a simpler definition. An element ${{\mathbf u}}\in {{\mathbb H}}^N$ is generic, if ${{\mathbf u}}\in X$ for some dense open set $X$ in ${{\mathbb H}}^N$ such that $X^c$ is a null set. Sometimes in actual proofs, we obtain the stronger result where $X^c$ is a real algebraic variety. The following theorem on generic measurements also shows that the lower bound given in Theorem \[realminimal.thm1\] is optimal. \[realminimal.thm2\] Let $ m\geq 2d$. Then a generic $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ is affine phase retrievable. The theorem follows readily from Theorem \[le:real\] (D). Note that for a generic ${\mathbf A}\in {{\mathbb R}}^{m\times d}$, any $d$ rows are linearly independent, so that ${\rm span} ({\mathbf A}_{S^c}^\top) = {{\mathbb R}}^d$ as long as $\# S^c \geq d$. On the other hand, ${\rm span}({\bf A}_S)$ is a $d$ dimensional subspace in ${{\mathbb R}}^{\# S}$ and so ${\mathbf b}_S \not \in {\rm span}({\bf A}_S)$ if $\# S >d$. Thus if ${\bf b}_S\in {\rm span}({\bf A}_S)$, then $\# S \leq d$, which implies $\# S^c \geq d$. Consequently ${\rm span}\{{\bf a}_j:~j\in S^c\}={\rm span} ({\mathbf A}_{S^c}^\top)={{\mathbb R}}^d$. Hence $({\bf A}, {\bf b})$ is affine phase retrievable. The following theorem highlights a difference between the classical linear phase retrieval and the affine phase retrieval. \[th:nonopenReal\] Let $ m\geq 2d$. Then the set of affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ is not an open set. We only need to find an affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ such that for each $\epsilon>0$, there is a small perturbation $({\mathbf A}',{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ with $\|{\mathbf A}-{\mathbf A}'\|_F <\epsilon$ such that $({\mathbf A}',{\mathbf b})$ is not affine phase retrievable, where $\|\cdot\|_F$ denotes the $l^2$-norm (Frobenius norm). We first do so for $m=2d$. Set $${\mathbf A}=(I_d, I_d)^\top, {\hspace{2em}}{\mathbf b}=(b_{11}, \ldots, b_{d1}, b_{12}, \ldots, b_{d2})^\top.$$ Here we require that $b_{j1} \neq b_{j2}$ for all $j$ and specially suppose $ b_{12}=0 $. Then $({\mathbf A},{\mathbf b})$ is affine phase retrievable. To see this, assume that ${{\mathbf x}},{{\mathbf y}}\in{{\mathbb R}}^d$ such that ${\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})= {\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$. Then for each $j$, we must have $|x_j+b_{jk}|=|y_j+b_{jk}|$ for $k=1,2$. Since $b_{j1} \neq b_{j2}$, we must have $x_j=y_j$. Thus ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ is injective and hence $({\mathbf A},{\mathbf b})$ is phase retrievable. Now let $\delta>0$ be sufficiently small. We perturb ${\mathbf A}$ to $$\label{eq:perturbReal} {\mathbf A}' = \left(I_d+b_{11}\delta E_{21}, I_d\right)^\top,$$ where $E_{ij}$ denotes the matrix with the $(i,j)$-th entry being 1 and all other entries being 0. Now set ${{\mathbf x}}=(b_{11}, -1/\delta, 0,\ldots,0)^\top $ and ${{\mathbf y}}= (-b_{11}, -1/\delta, 0, \ldots, 0)^\top$. It is easy to see that $$|{\mathbf A}'{{\mathbf x}}+{\mathbf b}|=|{\mathbf A}'{{\mathbf y}}+{\mathbf b}|.$$ Hence $({\mathbf A}',{\mathbf b})$ is not affine phase retrievable. By taking $\delta$ sufficiently small, we will have $\|{\mathbf A}'-{\mathbf A}\|_F\leq \epsilon$. It follows that for $m=2d$, the set of affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ is not an open set. In general for $m> 2d$, we can simply take the above construction $({\mathbf A},{\mathbf b})\in {{\mathbb R}}^{2d\times (d+1)}$ and augment it to a matrix $(\tilde {\mathbf A}, \tilde{\mathbf b})\in {{\mathbb R}}^{m\times (d+1)}$ by appending $m-2d$ rows of zero vectors to form its last $m-2d$ rows. The $(\tilde{\mathbf A}, \tilde{\mathbf b})$ is clearly affine phase retrievable, and the same perturbation above applied to the first $2d$ rows of ${\mathbf A}$ now breaks the affine phase retrievability. Thus for any $m\geq 2d$, the set of affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ is not an open set. Real sparse affine phase retrieval ---------------------------------- Set $$\Sigma_s({{\mathbb H}}^d)\,\,:=\,\, \{{{\mathbf x}}\in {{\mathbb H}}^d: \|{{\mathbf x}}\|_0\leq s\}.$$ We say that $({\mathbf A},{\mathbf b})\in {{\mathbb H}}^{m\times (d+1)}$ is [*$s$-sparse affine phase retrievable for ${{\mathbb H}}^d$*]{} if ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ is injective on $\Sigma_s({{\mathbb H}}^d)$. In this subsection, we show that the minimal number of affine measurements to recover all $s$-sparse real signals is $2s+1$. -   Let $1 \leq s \leq d-1$ and $({\mathbf A},{\mathbf b})\in {{\mathbb R}}^{m\times (d+1)}$ be $s$-sparse affine phase retrievable for ${{\mathbb R}}^d$. Then $ m\geq 2s+1$. -   Let $m \geq 2s+1$ and $({\mathbf A},{\mathbf b})$ be a generic element in ${{\mathbb R}}^{m\times (d+1)}$. Then $({\bf A}, {\bf b})$ is $s$-sparse affine phase retrievable for ${{\mathbb R}}^d$. \(i) We first show that if $({\mathbf A},{\mathbf b})$ is $s$-sparse affine phase retrievable, then $m \geq 2s+1$. First we claim that the rank of ${\bf A}$ is at least $r=\min(d,2s)$. Indeed, suppose that the claim is false. Then there exists a nonzero vector ${\bf x}\in \Sigma_{r}({{\mathbb R}}^d)$, such that ${\bf A} {\bf x}={\bf 0}$. Write ${\bf x}={\bf u}-{\bf v}$ with ${\bf u},{\bf v}\in \Sigma_s({{\mathbb R}}^d)$. Then ${\bf u}\ne {\bf v}$ and ${\bf A}{\bf u}={\bf A}{\bf v}$. Hence for all $1 \leq j \leq m$, we have $$|\langle {\bf a}_j, {\bf u}\rangle +b_j|=|\langle {\bf a}_j, {\bf v}\rangle+b_j|,$$ which is a contradiction. Thus ${{\rm rank}}({\mathbf A}) \geq r=\min(d,2s)$. Assume that $ m \leq 2s$. We derive a contradiction. Since $s<d$, it follows that $r \geq s+1$. Thus there exists an index set $T\subset \{1, 2, \ldots, m\}$ with $\# T = s+1$, such that ${{\rm rank}}({\mathbf A}_{T})=s+1$. Without of loss of generality we may assume that $T=\{1, 2, \ldots, s+1\}$. Moreover, we may also without of loss of generality assume that the first $s+1$ columns of ${\mathbf A}_{T}$ are linearly independent. In other words, the $(s+1)\times (s+1)$ submatrix of ${\mathbf A}$ restricted to the first $s+1$ rows and columns is nonsingular. Call this matrix $B$. It follows that there exists a ${{\mathbf y}}\in{{\mathbb R}}^{s+1}$ such that $B{{\mathbf y}}= -{\mathbf b}_{T}$. Write ${{\mathbf y}}=(y_1, \ldots, y_{s+1})^\top$ and set $${{\mathbf v}}_0 = (y_1, \ldots, y_{s+1}, 0, \ldots, 0)^\top\in{{\mathbb R}}^{d}.$$ Then ${\mathbf A}_T{{\mathbf v}}_0 =-{\mathbf b}_T$. If $y_j=0$ for some $1 \leq j \leq s+1$, say $y_{s+1}=0$, we let ${{\mathbf u}}=(u_1, \ldots, u_s,0, \ldots, 0)^\top$. Since $\# T^c = m-(s+1) \leq s-1$, there exists such a ${{\mathbf u}}_0 \neq 0$ such that ${\langle {{\mathbf a}_j, {{\mathbf u}}_0} \rangle}=0$ for all $j\in T^c$. Now for ${{\mathbf x}}= {{\mathbf v}}_0+{{\mathbf u}}_0$ and ${{\mathbf y}}= {{\mathbf v}}_0-{{\mathbf u}}_0$, we have ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) = {\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$ and ${{\mathbf x}}\neq {{\mathbf y}}$. Furthermore, ${{\mathbf x}},{{\mathbf y}}\in\Sigma_s({{\mathbb R}}^d)$. This is a contradiction. Hence $y_j \neq 0$ for all $1 \leq j \leq s$. Now for any $1 \leq j_1<j_2 \leq s+1$ consider $$\label{eq:u-constrain} {{\mathbf u}}_{j_1,j_2} =( u_1, \ldots, u_{s+1}, 0, \ldots, 0)^\top\in{{\mathbb R}}^{d}, {\hspace{2em}}u_{j_1} = ty_{j_1},~u_{j_2} = -ty_{j_2}.$$ We view the other $u_j$’s and $t$ as unconstrained variables, so there are $s$ variables. Since $\# T^c = m-(s+1) \leq s-1$, it follows that there exists a $\tilde{{\mathbf u}}_{j_1,j_2} \neq 0$ satisfying (\[eq:u-constrain\]) such that ${\langle {{\mathbf a}_j, \tilde{{\mathbf u}}_{j_1,j_2}} \rangle}=0$ for all $j\in T^c$. If $t\neq 0$, then we may normalize $\tilde{{\mathbf u}}_{j_1,j_2}$ so that $t=1$. Set ${{\mathbf x}}= {{\mathbf v}}_0+\tilde{{\mathbf u}}_{j_1,j_2}$ and ${{\mathbf y}}= {{\mathbf v}}_0-\tilde{{\mathbf u}}_{j_1,j_2}$. It follows that ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) = {\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$ and $${{\rm supp}}({{\mathbf x}}) \subset \{1, 2,\ldots, s+1\}\setminus\{j_2\}, {\hspace{1em}}{{\rm supp}}({{\mathbf y}}) \subset \{1, 2,\ldots, s+1\}\setminus\{j_1\}.$$ This is a contradiction. To complete the proof of $m \geq 2s+1$, we finally need to consider the case that $t=0$ in $\tilde{{\mathbf u}}_{j_1,j_2} \neq 0$ for any two indices $1 \leq j_1<j_2 \leq s+1$. But if so, it implies that any $s-1$ columns among the first $s+1$ columns of ${\mathbf A}_{T^c}$ are linearly dependent. In particular, it means the $(m-s-1)\times s$ submatrix of ${\mathbf A}_{T^c}$ restricted to the first $s+1$ columns has rank at most $s-2$. Now because the $(s+1)\times (s+1)$ submatrix of ${\mathbf A}$ restricted to the first $s+1$ rows and columns is nonsingular, we may without loss of generality assume that $s\times s$ submatrix of ${\mathbf A}$ restricted to the first $s$ rows and columns is nonsingular, for otherwise we can make a simple permutation of the indices. The key now is to observe that $({\mathbf A},{\mathbf b})$ is not $s$-sparse affine phase retrievable, because $({\mathbf A},{\mathbf b})$ restricted to the first $s$ columns is not affine phase retrievable for ${{\mathbb R}}^s$. To see this, let ${\mathbf A}'$ be the submatrix of ${\mathbf A}$ consisting of only the first $s$ columns of ${\mathbf A}$. We show $({\mathbf A}',{\mathbf b})$ is not affine phase retrievable for ${{\mathbb R}}^s$. Note that for $S=\{1,2,\ldots,s\}$, we have ${\mathbf b}_S\in {\rm span}\,({\mathbf A}'_S)$ because by assumption ${\mathbf A}'_S$ is nonsingular. But we also know that the rows of ${\mathbf A}_{S^c}$ do not span ${{\mathbb R}}^s$ because it has ${{\rm rank}}({\mathbf A}_{S^c}) \leq s-1$. Hence $({\mathbf A}',{\mathbf b})$ is not affine phase retrievable by Theorem \[le:real\] (D). This completes the proof of $m \geq 2s+1$. \(ii)  Next we prove for $m \geq 2s+1$, a generic $({\mathbf A},{\mathbf b}) \in{{\mathbb R}}^{m \times (d+1)}$ is $s$-sparse affine phase retrievable. The set of all $({\mathbf A},{\mathbf b})\in {{\mathbb R}}^{m\times (d+1)}$ has real dimension $m(d+1)$. The goal is to show that the the set of $({\mathbf A},{\mathbf b})$ that are not $s$-sparse affine phase retrievable lies in a finite union of subsets of dimension strictly less than $m(d+1)$. Our result then follows. For any subset of indices $I,J\subset \{1,\ldots,m\}$ with $\#I ,\#J\leq s$, we say $({\mathbf A},{\mathbf b}) \in {{\mathbb R}}^{m\times (d+1)}$ is not $(I,J)$-sparse affine phase retrievable if there exist ${{\mathbf x}}\neq {{\mathbf y}}$ in ${{\mathbb R}}^d$ such that $$\label{eq:IJsparseReal} {\rm supp}({{\mathbf x}})\subset I, {\hspace{1em}}{\rm supp}({{\mathbf y}})\subset J, {\hspace{1em}}\mbox{and}{\hspace{1em}}{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) = {\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}}).$$ Let $ \mathcal{A}_{I,J} $ denote the set of all 4-tuples $({\mathbf A}, {\mathbf b}, {{\mathbf x}},{{\mathbf y}})$ satisfying (\[eq:IJsparseReal\]) and ${{\mathbf x}}\neq {{\mathbf y}}$. Then $$\mathcal{A}_{I,J} \subset {{\mathbb R}}^{m\times (d+1)}\times {{\mathbb R}}^{\# I}\times {{\mathbb R}}^{\# J}.$$ Then $ \mathcal{A}_{I,J} $ is a well-defined real quasi-projective variety ([@alge Page 18]). Write ${\mathbf A}=({\mathbf a}_1, {\mathbf a}_2, \ldots, {\mathbf a}_m)^\top$ and ${\mathbf b}=(b_1, \ldots, b_m)^\top$. Then by (\[eq:deng\]), ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) = {\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$ is equivalent to $$\label{eq:IJ-deng} {\langle {{\mathbf a}_j, {{\mathbf x}}-{{\mathbf y}}} \rangle}({\langle {{\mathbf a}_j,{\bf x}+{\bf y}} \rangle}+2b_j) = 0, {\hspace{2em}}j=1,2, \ldots, m.$$ Fix any $j$, the above equation holds if and only if $${\langle {{\mathbf a}_j, {{\mathbf x}}-{{\mathbf y}}} \rangle}=0 {\hspace{2em}}\mbox{or} {\hspace{2em}}{\langle {{\mathbf a}_j,{{\mathbf x}}+{{\mathbf y}}} \rangle}+2b_j = 0.$$ Thus for any ${{\mathbf x}}\neq {{\mathbf y}}$, the first condition requires ${\mathbf a}_j$ to lie on a hyperplane, which has co-dimension 1, while the second condition fixes $b_j$ to be $-{\langle {{\mathbf a}_j,{\bf x}+{\bf y}} \rangle}/2$. Overall, for any given ${{\mathbf x}}\neq {{\mathbf y}}$, these two conditions constraint the $j$-th row of $({\mathbf A},{\mathbf b})$ to lie on a real projective variety of codimension 1. We shall use $X_j({{\mathbf x}},{{\mathbf y}})$ to denote this variety. Now let $\pi_2: {\mathcal A}_{I,J} \longrightarrow {{\mathbb R}}^d \times {{\mathbb R}}^d$ be the projection $({\mathbf A},{\mathbf b},{{\mathbf x}},{{\mathbf y}}) \mapsto ({{\mathbf x}},{{\mathbf y}})$ onto the last two coordinates. Then for any ${{\mathbf x}}_0 \neq {{\mathbf y}}_0$ in ${{\mathbb R}}^d$, we have $$\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\} = X_1({{\mathbf x}}_0,{{\mathbf y}}_0) \times X_2({{\mathbf x}}_0,{{\mathbf y}}_0) \times \ldots \times X_m({{\mathbf x}}_0,{{\mathbf y}}_0)\times \{{{\mathbf x}}_0\}\times \{{{\mathbf y}}_0\}.$$ Hence the dimension of $\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\}$ is $$\dim\left(\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\} \right) = m(d+1) - m = md.$$ It follows that $\dim({\mathcal A}_{I,J}) \leq md +\# I+\# J \leq md +2s$. We now let $\pi_1: {\mathcal A}_{I,J} \longrightarrow {{\mathbb R}}^{m\times (d+1)}$ be the projection $({\mathbf A},{\mathbf b},{{\mathbf x}},{{\mathbf y}}) \mapsto ({\mathbf A},{\mathbf b})$. Since projections cannot increase the dimension of a variety, we know that $$\dim\left(\pi_1( {\mathcal A}_{I,J})\right) \leq md +2s = m(d+1) +2s-m<m(d+1).$$ However, $\pi_1( {\mathcal A}_{I,J} )$ contains precisely those $({\mathbf A},{\mathbf b})$ in ${{\mathbb R}}^{m\times (d+1)}$ that are not $(I,J)$-sparse affine phase retrievable. Thus a generic $({\mathbf A},{\mathbf b})\in{{\mathbb R}}^{m\times (d+1)}$ is $(I,J)$-sparse affine phase retrievable. Finally, there are only finitely many indices subsets $I,J$. Hence a generic $({\mathbf A},{\mathbf b})\in{{\mathbb R}}^{m\times (d+1)}$ ($m\geq 2s+1$) is $(I,J)$-sparse affine phase retrievable for any $I, J$ with $\# I, \# J \leq s$. The theorem is proved. Affine Phase Retrieval for Complex Signals ========================================== In this section, we consider affine phase retrieval for complex signals. Affine phase retrieval for complex signals, like in the case of the classical phase retrieval, poses additional challenges. Complex affine phase retrieval ------------------------------ We first establish the analogous of Theorem \[le:real\] for complex signals. \[th:PRScomplex\] Let ${\bf A}=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb C}}^{m\times d}$ and ${{\mathbf b}}=(b_1, \ldots, b_m)^\top\in {{\mathbb C}}^m$. Then the followings are equivalent: - $({\mathbf A},{\mathbf b})$ is affine phase retrievable for ${{\mathbb C}}^d$. - The map ${\mathbf M}^2_{{\bf A},{\bf b}}$ is injective on ${{\mathbb C}}^d$. - For any ${{\mathbf u}},{{\mathbf v}}\in{{\mathbb C}}^d$ and ${{\mathbf u}}\neq 0$, there exists a $1\leq k \leq m$ such that $$\Re\Bigl( {\langle {{{\mathbf u}}, {\mathbf a}_k} \rangle}\bigl({\langle {{\mathbf a}_k,{{\mathbf v}}} \rangle} +b_k\bigr)\Bigr) \neq 0.$$ - Viewing ${\mathbf M}^2_{{\bf A},{\bf b}}$ as a map ${{\mathbb R}}^{2d} \longrightarrow {{\mathbb R}}^m$, its (real) Jacobian $J({{\mathbf x}})$ has rank $2d$ for all ${{\mathbf x}}\in{{\mathbb R}}^{2d}$. The equivalence of (A) and (B) have already been discussed earlier. We focus on the other conditions. \(A) $\Leftrightarrow$ (C).  Assume that ${\mathbf M}^2_{{\bf A},{\bf b}}({\bf x})={\mathbf M}^2_{{\bf A},{\bf b}}({\bf y})$ for some ${\bf x} \neq {\bf y}$ in ${{\mathbb C}}^d$. Observe that for any $a,b\in{{\mathbb C}}$, we have $|a|^2-|b|^2 = \Re((\bar a-\bar b)(a+b))$. Thus for any $j$, we have $$\label{eq:diff} {\lvert{\langle {{\mathbf a}_j,{\bf x}} \rangle}+b_j\rvert}^2-{\lvert{\langle {{\mathbf a}_j,{\bf y}} \rangle}+b_j\rvert}^2 = \Re\Bigl( {\langle { {\bf x}-{\bf y}, {\mathbf a}_j} \rangle}({\langle {{\mathbf a}_j,{\bf x}+{\bf y}} \rangle}+2b_j)\Bigr).$$ Set $2{{\mathbf u}}= {{\mathbf x}}-{{\mathbf y}}$ and $2{{\mathbf v}}= {{\mathbf x}}+{{\mathbf y}}$. Then ${{\mathbf u}}\neq 0$ and for all $j$, $$\label{eq:C_innerp_zero} \Re\Bigl( {\langle { {{\mathbf u}}, {\mathbf a}_j} \rangle}({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle}+b_j)\Bigr)=0.$$ Conversely, assume that (\[eq:C\_innerp\_zero\]) holds for all $j$. Let ${{\mathbf x}},{{\mathbf y}}\in{{\mathbb C}}^d$ be given by ${{\mathbf x}}-{{\mathbf y}}=2{{\mathbf u}}$ and ${{\mathbf x}}+{{\mathbf y}}=2{{\mathbf v}}$. Then ${{\mathbf x}}\neq {{\mathbf y}}$. However, we would have ${\mathbf M}^2_{{\bf A},{\bf b}}({\bf x})={\mathbf M}^2_{{\bf A},{\bf b}}({\bf y})$ and hence $({\mathbf A},{\mathbf b})$ cannot be affine phase retrievable. \(C) $\Leftrightarrow$ (D).  The $k$-th entry of ${\mathbf M}^2_{{\bf A},{\bf b}}({\bf x})$ is $|{\langle {\mathbf a}_k, {{\mathbf x}}\rangle}+b_k|^2$. Since all variables here are complex, we shall separate them into the real and imaginary parts by adopting the notation ${{\mathbf x}}= {{\mathbf x}}_R+i{{\mathbf x}}_I$, ${\mathbf a}_k ={\mathbf a}_{k,R}+i{\mathbf a}_{k,I}$ and $b_k = b_{k,R}+ib_{k,I}$. The $k$-th entry of ${\mathbf M}^2_{{\bf A},{\bf b}}({\bf x})$ is now $$|{\langle {\mathbf a}_k, {{\mathbf x}}\rangle}+b_k|^2 = \left({\langle {{\mathbf a}_{k,R},{{\mathbf x}}_R} \rangle} + {\langle {{\mathbf a}_{k,I},{{\mathbf x}}_I} \rangle} + b_{k,R}\right)^2 + \left({\langle {{\mathbf a}_{k,R},{{\mathbf x}}_I} \rangle} - {\langle {{\mathbf a}_{k,I},{{\mathbf x}}_R} \rangle} - b_{k,I}\right)^2.$$ It follows that the (real) Jacobian of ${\mathbf M}^2_{{\bf A},{\bf b}}({{\mathbf x}}_R,{{\mathbf x}}_I)$ is $$J({{\mathbf x}}):=J({{\mathbf x}}_R,{{\mathbf x}}_I) = 2\begin{pmatrix} {\mathbf a}_{1,R}^\top\cdot \alpha_1({{\mathbf x}})-{\mathbf a}_{1,I}^\top\cdot \beta_1({{\mathbf x}}) &{\mathbf a}_{1,I}^\top\cdot \alpha_1({{\mathbf x}})+{\mathbf a}_{1,R}^\top\cdot \beta_1({{\mathbf x}})\\ {\mathbf a}_{2,R}^\top\cdot \alpha_2({{\mathbf x}})-{\mathbf a}_{2,I}^\top\cdot \beta_2({{\mathbf x}}) &{\mathbf a}_{2,I}^\top\cdot \alpha_2({{\mathbf x}})+{\mathbf a}_{2,R}^\top\cdot \beta_2({{\mathbf x}})\\ \vdots & \vdots \\ {\mathbf a}_{m,R}^\top\cdot \alpha_m({{\mathbf x}})-{\mathbf a}_{m,I}^\top\cdot \beta_m({{\mathbf x}}) & {\mathbf a}_{m,I}^\top\cdot \alpha_m({{\mathbf x}})+{\mathbf a}_{m,R}^\top\cdot \beta_m({{\mathbf x}}) \end{pmatrix},$$ where $\alpha_j({{\mathbf x}}):={\langle {{\mathbf a}_{j,R},{{\mathbf x}}_R} \rangle}+{\langle {{\mathbf a}_{j,I},{{\mathbf x}}_I} \rangle}+b_{j,R}$ and $\beta_j({{\mathbf x}}):={\langle {{\mathbf a}_{j,R},{{\mathbf x}}_I} \rangle}-{\langle {{\mathbf a}_{j,I},{{\mathbf x}}_R} \rangle}-b_{j,I}$ for all $0 \leq j\leq m$. Now assume that ${{\rm rank}}(J({{\mathbf x}}))$ is not $2d$ everywhere. Then there exist ${{\mathbf v}}={{\mathbf v}}_R+i{{\mathbf v}}_I$ and ${{\mathbf u}}={{\mathbf u}}_R+i{{\mathbf u}}_I \neq 0$, such that ${{\mathbf u}}$ as a vector in ${{\mathbb R}}^{2d}$ is in the null space of $J({{\mathbf v}})$, i.e., $$J({{\mathbf v}}) \begin{pmatrix} {{\mathbf u}}_R\\ {{\mathbf u}}_I\end{pmatrix} = 0.$$ It follows that for all $1 \leq k \leq m$, we have $$\label{null_u} C_k :={\langle {{\mathbf a}_{k,R},{{\mathbf u}}_R} \rangle} \alpha_k({{\mathbf v}})-{\langle {{\mathbf a}_{k,I},{{\mathbf u}}_R} \rangle} \beta_k({{\mathbf v}}) + {\langle {{\mathbf a}_{k,I},{{\mathbf u}}_I} \rangle} \alpha_k({{\mathbf v}})+{\langle {{\mathbf a}_{k,R},{{\mathbf u}}_I} \rangle} \beta_k ({{\mathbf v}}) =0.$$ But one can readily check that $C_k$ is precisely $$C_k=\Re\Bigl( {\langle { {{\mathbf u}}, {\mathbf a}_k} \rangle}({\langle {{\mathbf a}_k,{{\mathbf v}}} \rangle}+b_k)\Bigr).$$ Thus $({\mathbf A},{\mathbf b})$ cannot be affine phase retrievable by (C). The converse clearly also holds. Assume that (C) is false. Then there exists ${{\mathbf v}},{{\mathbf u}}\in{{\mathbb C}}^d$ and ${{\mathbf u}}\neq 0$ such that $$\Re\Bigl( {\langle { {{\mathbf u}}, {\mathbf a}_k} \rangle}({\langle {{\mathbf a}_k,{{\mathbf v}}} \rangle}+b_k)\Bigr) = 0$$ for all $1 \leq k \leq m$. It follows that (\[null\_u\]) holds for all $k$ and hence $$J({{\mathbf v}}) \begin{pmatrix} {{\mathbf u}}_R\\ {{\mathbf u}}_I\end{pmatrix} = 0.$$ Thus ${{\rm rank}}(J({{\mathbf v}})) < 2d$. Minimal measurement number -------------------------- We now show that the minimal number of measurements needed to be affine phase retrievable is $ 3d$. This is surprising compared to the classical affine phase retrieval, where the minimal number is $4d-O(\log_2d)$. \[le:3\] Let $z_1, z_2\in {{\mathbb C}}$. Suppose that $b_1,b_2,b_3\in {{\mathbb C}}$ are not collinear on the complex plane. Then $z_1=z_2$ if and only if ${\lvertz_1+b_j\rvert}={\lvertz_2+b_j\rvert},\, j=1,2,3$. We use $z_{j,R}$ and $z_{j,I}$ to denote the real and imaginary part of $z_j$, and similarly for $b_{j,R}$ and $b_{j,I}$. Assume the lemma is false, and that there exist $z_1, z_2$ with $z_1\neq z_2$ so that ${\lvertz_1+b_j\rvert}^2={\lvertz_2+b_j\rvert}^2,\, j=1,2,3$. Note that ${\lvertz_1+b_j\rvert}^2={\lvertz_2+b_j\rvert}^2$ implies that $$\label{eq:ling} (z_{2,R}-z_{1,R})\cdot b_{j,R}+(z_{2,I}-z_{1,I})\cdot b_{j,I}=\frac{|z_1|^2-|z_2|^2}{2}, \quad j=1,2,3.$$ The (\[eq:ling\]) together with $z_1\ne z_2$ implies that $b_1,b_2,b_3$ are collinear. This is a contradiction. \[th:m3d\] -  Suppose that $({\mathbf A},{\mathbf b})\in {{\mathbb C}}^{m\times (d+1)}$ is affine phase retrievable in ${{\mathbb C}}^d$. Then $m\geq 3d$. -  Let $B:=({\mathbf a}_1,\ldots,{\mathbf a}_d)\in {{\mathbb C}}^{d\times d}$ be nonsingular. Set ${\mathbf A}=(B,B,B)^\top\in {{\mathbb C}}^{3d\times d}$. Let $${\mathbf b}=(b_{11},\ldots,b_{d1},b_{12},\ldots,b_{d2},b_{13},\ldots,b_{d3})^\top\in {{\mathbb C}}^{3d}$$ such that $b_{j1},b_{j2},b_{j3}$ are not collinear in ${{\mathbb C}}$ for any $1\leq j\leq d$. Then $({\mathbf A},{\mathbf b})$ is affine phase retrievable in ${{\mathbb C}}^d$. \(i)   Write ${\mathbf A}=({\mathbf a}_1,\ldots,{\mathbf a}_m)^\top\in {{\mathbb C}}^{m\times d}$. Assume that $m <3d$. Clearly ${{\rm rank}}({\mathbf A})=d$, for otherwise we will have ${\mathbf A}{{\mathbf x}}=0$ for some ${{\mathbf x}}\neq 0$ and hence ${\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) = {\mathbf M}_{{\mathbf A},{\mathbf b}}(0)$. Hence there exists a $T\subset \{1, \ldots, m\}$ with $\# T=d$ such that ${{\rm rank}}(A_T)=d$, which means we can find ${{\mathbf v}}\in{{\mathbb C}}^d$ such that $${\langle {{\mathbf a}_k, {{\mathbf v}}} \rangle} + b_k = 0, {\hspace{2em}}k \in T.$$ Now, because $\# T^c =m-d <2d$, and the system of homogeneous linear equations for the variable ${{\mathbf u}}$ with ${{\mathbf v}}$ fixed, $$\Re\Bigl( {\langle { {{\mathbf u}}, {\mathbf a}_j} \rangle}({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle}+b_j)\Bigr)=0, {\hspace{2em}}j\in T^c$$ has $2d$ real variables ${{\mathbf u}}_R, {{\mathbf u}}_I$, it must have a nontrivial solution. The two vectors ${{\mathbf u}}\neq 0, {{\mathbf v}}$ combine to yield $$\Re\Bigl( {\langle { {{\mathbf u}}, {\mathbf a}_j} \rangle}({\langle {{\mathbf a}_j,{{\mathbf v}}} \rangle}+b_j)\Bigr)=0$$ for all $1 \leq j \leq m$. This contradicts with (C) in Theorem \[th:PRScomplex\]. (ii)  To prove $({\mathbf A},{\mathbf b})$ is affine phase retrievable, we prove that ${\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})={\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$ implies ${{\mathbf x}}={{\mathbf y}}$ in ${{\mathbb C}}^d$. The property ${\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})={\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})$ implies that $$\label{eq:bjdeng} {\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_{jk}\rvert}\,\,=\,\,{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_{jk}\rvert} , {\hspace{2em}}j=1,\ldots,d,{\hspace{1em}}k=1,2,3.$$ Thus by Lemma \[le:3\], for each fixed $j$ we have $${\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}={\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}.$$ This implies ${{\mathbf x}}={{\mathbf y}}$ since the matrix $B=({\mathbf a}_1,\ldots,{\mathbf a}_d)$ is nonsingular. It is well known that in the classical phase retrieval, the set of all phase retrievable ${\mathbf A}\in {{\mathbb C}}^{m\times d}$ is an open set in $ {{\mathbb C}}^{m\times d}$. But for affine phase retrieval, as with the real affine phase retrieval case, this property no longer holds. The following theorem shows that this property also doesn’t hold in the complex case when $m \geq 3d$. \[th:ins\] Let $m \geq 3d$. Then the set of affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m\times (d+1)}$ is not an open set in $ {{\mathbb C}}^{m\times (d+1)}$. In fact, there exists an affine phase retrievable $({\mathbf A},{\mathbf b})\in {{\mathbb C}}^{3d\times (d+1)}$, which satisfies the conditions in Theorem \[th:m3d\] (ii). Given any $\epsilon>0$, there exists $ ({\mathbf A}',{\mathbf b}) \in {{\mathbb C}}^{3d\times (d+1)} $ which does not have affine phase retrievable property such that $$\|{\mathbf A}'-{\mathbf A}\|_F\leq \epsilon,$$ where $\|\cdot \|_F$ denotes the Frobenius norm. Following the construction given in Theorem \[th:m3d\] (ii), we set ${\mathbf A}=(B,B,B)^\top$, where $B$ is nonsingular and $${\mathbf b}:=(\underbrace{{{\rm i}},\ldots,{{\rm i}}}_d,\underbrace{0,\ldots,0}_d,\underbrace{1,\ldots,1}_d)^\top\in {{\mathbb C}}^{3d}.$$ We will show that there exists an arbitrarily small perturbation ${\mathbf A}'$ such that $({\mathbf A}', {\mathbf b})$ is no longer affine phase retrievable. Making a simple linear transformation ${{\mathbf x}}= B^{-1}{{\mathbf y}}$, we see that all we need is to show that this property holds for ${\mathbf A}=(I_d, I_d, I_d)^\top$, where $I_d$ is the $d\times d$ identity matrix. Let $\delta>0$ be sufficiently small. We perturb ${\mathbf A}$ to $$\label{perturb} {\mathbf A}' = (I_d+{{\rm i}}\delta E_{21}, I_d, I_d)^\top,$$ where $E_{21}$ denotes the matrix with the $(2,1)$-th entry being 1 and all other entries being 0. Now set ${{\mathbf x}}=({{\rm i}},-1/\delta,0,\ldots,0)^\top$ and ${{\mathbf y}}= (-{{\rm i}}, -1/\delta, 0, \ldots, 0)^\top$. It is easy to see that $$|{\mathbf A}'{{\mathbf x}}+{\mathbf b}|=|{\mathbf A}'{{\mathbf y}}+{\mathbf b}|.$$ Thus $({\mathbf A}',{\mathbf b})$ is not affine phase retrievable. By taking $\delta$ sufficiently small we will have $\|{\mathbf A}'-{\mathbf A}\|_F\leq \epsilon$. In general for $m> 3d$, like the real case, we can simply take the above construction $({\mathbf A},{\mathbf b})\in {{\mathbb C}}^{3d\times (d+1)}$ and augment it to a matrix $(\tilde {\mathbf A}, \tilde{\mathbf b})\in {{\mathbb C}}^{m\times (d+1)}$ by appending $m-3d$ rows of zero vectors to form its last $m-3d$ rows. $(\tilde{\mathbf A}, \tilde{\mathbf b})$ is clearly affine phase retrievable, and the same perturbation above applied to the first $3d$ rows of $\tilde{\mathbf A}$ now breaks the affine phase retrievability. Thus for any $m\geq 3d$, the set of affine phase retrievable $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m\times (d+1)}$ is not an open set. We next consider complex affine phase retrieval for generic measurements. We have the following theorem: \[th:4d1\] Suppose that $m\geq 4d-1$. Then a generic $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m \times (d+1)} $ is affine phase retrievable in ${{\mathbb C}}^d$. Let $N=m+1$. Then $N \geq 4d = 4(d+1)-4$. Hence by [@CEHV13], there is an open dense set of full measure $X\subset {{\mathbb C}}^{N \times (d+1)} $, such that any ${\bf F}\in X $ is linear phase retrievable in the classical sense. Write ${\bf F}=({{\mathbf f}}_1, {{\mathbf f}}_2, \ldots, {{\mathbf f}}_N)^\top$, where each ${{\mathbf f}}_j \in {{\mathbb C}}^{d+1}$. For each ${{\mathbf g}}\in{{\mathbb C}}^{d+1}$, denote $X_{{{\mathbf g}}} :=\{{\bf F}=({{\mathbf f}}_1, {{\mathbf f}}_2, \ldots, {{\mathbf f}}_N)^\top \in X: ~{{\mathbf f}}_N = {{\mathbf g}}\}$. Then there exists a ${{\mathbf g}}_0\in{{\mathbb C}}^{d+1}$, such that the projection of $X_{{{\mathbf g}}_0}$ onto ${{\mathbb C}}^{(N-1)\times (d+1)}$ with the last row removed is a dense open set with full measure. Thus ${\bf F}=({{\mathbf f}}_1, \ldots, {{\mathbf f}}_{N-1},{{\mathbf g}}_0)^\top$ is phase retrievable in ${{\mathbb C}}^{d+1}$ in the classical sense for a generic $({{\mathbf f}}_1, \ldots, {{\mathbf f}}_{N-1})^\top \in {{\mathbb C}}^{(N-1)\times (d+1)}$. Now let $P_0\in{{\mathbb C}}^{(d+1)\times (d+1)}$ be nonsingular such that $P_0{{\mathbf g}}_0 = {\bf e}_{d+1}$. Then for any ${\bf F}\in X_{{{\mathbf g}}_0}$, we have $${\bf G}:={\bf F}P_0^\top =(P_0{{\mathbf f}}_1, \ldots, P_0{{\mathbf f}}_{N-1},{\bf e_{d+1}})^\top =: ({{\mathbf g}}_1, \ldots, {{\mathbf g}}_{N-1},{\bf e}_{d+1})^\top.$$ It is linear phase retrievable in the classical sense for generic ${{\mathbf g}}_1, \ldots, {{\mathbf g}}_{N-1}$. In particular, any vector ${{\mathbf y}}=(x_1, \ldots, x_d, 1)^\top$ can be recovered by ${\lvert{\bf G}{{\mathbf y}}\rvert}$, where ${\lvert\cdot\rvert}$ means the entry-wise absolute value. However, note that the last entry of ${{\mathbf y}}$ is 1, and the last row of ${\bf G}$ is ${\bf e}_{d+1}^\top$. So the measurement from last row provides no information. In other words, the above ${{\mathbf y}}$ can be recovered exactly from the measurements provided by the first $N-1 = m$ rows of ${\bf G}$. This means precisely that the first $N-1 = m$ rows of ${\bf G}$ are affine phase retrievable. Let $({\mathbf A}, {\mathbf b})$ denote the first $m$ rows of ${\bf G}$. It follows that $({\mathbf A},{\mathbf b})\in {{\mathbb C}}^{m \times (d+1)} $ is affine phase retrievable. Therefore a generic $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m \times (d+1)} $ ($ m\geq 4d-1$) is affine phase retrievable. Sparse complex affine phase retrieval ------------------------------------- We now focus on sparse affine phase retrieval by proving that generic $({\mathbf A},{\mathbf b})$ is $s$-sparse affine phase retrievable if $m\geq 4s+1$. \[th:4k1\] Let $m \geq 4s+1$. Then a generic $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m\times (d+1)}$ is $s$-sparse affine phase retrievable . The proof here is very similar to the proof in the real case. The set of all $({\mathbf A},{\mathbf b})$ has real dimension $\dim_{{\mathbb R}}({{\mathbb C}}^{m\times (d+1)})=2m(d+1)$. The goal is to show that the the set of $({\mathbf A},{\mathbf b})$’s that are not $s$-sparse affine phase retrievable lies in a finite union of subsets, each of which is a projection of real hypersurfaces of dimension strictly less than $2m(d+1)$. This would yield our result. For any subset of indices $I,J\subset \{1,\ldots,m\}$ with $\#I ,\#J\leq s$, we say $({\mathbf A},{\mathbf b}) \in {{\mathbb C}}^{m\times (d+1)}$ is not $(I,J)$-sparse affine phase retrievable if there exist ${{\mathbf x}}\neq {{\mathbf y}}$ in ${{\mathbb C}}^d$ such that $$\label{eq:IJsparse} {\rm supp}({{\mathbf x}})\subset I, {\hspace{1em}}{\rm supp}({{\mathbf y}})\subset J, {\hspace{1em}}\mbox{and}{\hspace{1em}}{\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}}) = {\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf y}}).$$ Let $ \mathcal{A}_{I,J} $ denote the set of all 4-tuples $({\mathbf A}, {\mathbf b}, {{\mathbf x}},{{\mathbf y}})$ satisfying (\[eq:IJsparse\]) and ${{\mathbf x}}\neq {{\mathbf y}}$. Then $$\mathcal{A}_{I,J} \subset {{\mathbb C}}^{m\times (d+1)}\times {{\mathbb C}}^{\# I}\times {{\mathbb C}}^{\# J},$$ where we view $({\mathbf A},{\mathbf b})$ as an element of $ {{\mathbb C}}^{m\times (d+1)}$. For our proof we shall identify ${{\mathbb C}}^{m\times (d+1)}\times {{\mathbb C}}^{\# I}\times {{\mathbb C}}^{\# J}$ with ${{\mathbb R}}^{m\times 2(d+1)}\times {{\mathbb R}}^{2\# I}\times {{\mathbb R}}^{2\# J}$. In this case $ \mathcal{A}_{I,J} $ is a well-defined real quasi-projective variety ([@alge Page 18]). Note that ${\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}}) = {\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf y}})$ yields $|{\langle {{\mathbf a}_j, {{\mathbf x}}} \rangle}+b_j|^2 = |{\langle {{\mathbf a}_j, {{\mathbf y}}} \rangle}+b_j|^2$ for all $1\leq j\leq m$, where ${\mathbf A}=({\mathbf a}_1, {\mathbf a}_2, \ldots, {\mathbf a}_m)^\top$ and ${\mathbf b}=(b_1, \ldots, b_m)^\top$. By (\[eq:diff\]), this is equivalent to $$\label{eq:IJ-diff} \Re\Bigl( {\langle { {\bf x}-{\bf y}, {\mathbf a}_j} \rangle}({\langle {{\mathbf a}_j,{\bf x}+{\bf y}} \rangle}+2b_j)\Bigr) = 0, {\hspace{2em}}j=1,2, \ldots, m.$$ Fix any $j$, the above equation holds if and only if - ${\langle { {\bf x}-{\bf y}, {\mathbf a}_j} \rangle}=0$; [or ]{} - ${\langle { {\bf x}-{\bf y}, {\mathbf a}_j} \rangle}\neq 0$ but (\[eq:IJ-diff\]) holds. Thus for any ${{\mathbf x}}\neq {{\mathbf y}}$, the first condition requires ${\mathbf a}_j$ to lie on a hyperplane, which has real co-dimension 2, while the second condition requires $b_j$ to be on a line in ${{\mathbb C}}$ (depending on ${{\mathbf x}},{{\mathbf y}},{\mathbf a}_j$). Overall, for any given ${{\mathbf x}}\neq {{\mathbf y}}$, these two conditions constraint the $j$-th row of $({\mathbf A},{\mathbf b})$ to lie on a real projective variety of codimension 1. We shall use $X_j({{\mathbf x}},{{\mathbf y}})$ to denote this variety. Now let $\pi_2: {\mathcal A}_{I,J} \longrightarrow {{\mathbb C}}^d \times {{\mathbb C}}^d$ be the projection $({\mathbf A},{\mathbf b},{{\mathbf x}},{{\mathbf y}}) \mapsto ({{\mathbf x}},{{\mathbf y}})$ onto the last two coordinates. Then for any ${{\mathbf x}}_0 \neq {{\mathbf y}}_0$ in ${{\mathbb C}}^d$, we have $$\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\} = X_1({{\mathbf x}}_0,{{\mathbf y}}_0) \times X_2({{\mathbf x}}_0,{{\mathbf y}}_0) \times \ldots \times X_m({{\mathbf x}}_0,{{\mathbf y}}_0)\times \{{{\mathbf x}}_0\}\times \{{{\mathbf y}}_0\}.$$ Hence the real dimension of $\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\}$ is $$\dim_{{\mathbb R}}\left(\pi_2^{-1}\{({{\mathbf x}}_0,{{\mathbf y}}_0)\} \right) = 2m(d+1) - m = 2md + m.$$ It follows that $\dim_{{\mathbb R}}({\mathcal A}_{I,J}) \leq 2md+m +2\# I+2\# J \leq 2md+m +4s$. We now let $\pi_1: {\mathcal A}_{I,J} \longrightarrow {{\mathbb C}}^{m\times (d+1)}$ be the projection $({\mathbf A},{\mathbf b},{{\mathbf x}},{{\mathbf y}}) \mapsto ({\mathbf A},{\mathbf b})$. Since projections cannot increase the dimension of a variety, we know that $$\dim_{{\mathbb R}}\left(\pi_1( {\mathcal A}_{I,J})\right) \leq 2md+m +4s = 2m(d+1) +4s-m<2m(d+1).$$ However, $\pi_1( {\mathcal A}_{I,J})$ contains precisely those $({\mathbf A},{\mathbf b})$ in ${{\mathbb C}}^{m\times (d+1)}$ that are not $(I,J)$-sparse affine phase retrievable. Thus a generic $({\mathbf A},{\mathbf b})\in{{\mathbb C}}^{m\times (d+1)}$ is $(I,J)$-sparse affine phase retrievable. Finally, there are only finitely many indices subsets $I,J$. Hence a generic $({\mathbf A},{\mathbf b})\in{{\mathbb C}}^{m\times (d+1)}$ ($m\geq 4s+1$) is $(I,J)$-sparse affine phase retrievable for any $I, J$ with $\# I, \# J \leq s$. The theorem is proved. Stability and Robustness of Affine Phase Retrieval ================================================== Stability and robustness are important properties for affine phase retrieval. For the standard phase retrieval, stability and robustness have been studied in several papers, see e.g. [@BaWaSt; @BCMN; @CCD; @GWaXu16]. In this section, we establish stability and robustness results for both maps ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ and ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}$. \[th:Stability\] Assume that $({\mathbf A},{\mathbf b}) \in {{\mathbb H}}^{m \times (d+1)}$ is affine phase retrievable. Assume that $\Omega\subset {{\mathbb H}}^d$ is a compact set. Then there exist positive constants $C_1, C_2, c_1, c_2$ depending on $({\mathbf A},{\mathbf b})$ and $\Omega$ such that for any ${{\mathbf x}}, {{\mathbf y}}\in \Omega $, we have $$\begin{aligned} \frac{c_1}{1+\|{{\mathbf x}}\|+\|{{\mathbf y}}\|} \,\|{{\mathbf x}}-{{\mathbf y}}\| &\leq \left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) - {\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\| \leq C_1 \|{{\mathbf x}}-{{\mathbf y}}\|,\label{eq:Lip1}\\ {c_2}\|{{\mathbf x}}-{{\mathbf y}}\| &\leq \left\|{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) - {\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\|\leq C_2({1+\|{{\mathbf x}}\|+\|{{\mathbf y}}\|}) \|{{\mathbf x}}-{{\mathbf y}}\|. \label{eq:Lip2}\end{aligned}$$ Write ${\mathbf A}=({\mathbf a}_1, \ldots, {\mathbf a}_m)^\top$ and ${\mathbf b}= (b_1, \ldots, b_m)^\top$. We first establish the inequality for the map ${\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}})$, where we recall $${\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}})=\left({\lvert{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle}+b_1\rvert}^2,\ldots, {\lvert{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle}+b_m\rvert}^2\right).$$ Denote the matrix $({\mathbf A}, {\mathbf b}) \in {{\mathbb H}}^{m \times (d+1)}$ by $({\mathbf A},{\mathbf b}) = (\tilde{\mathbf a}_1, \ldots, \tilde{\mathbf a}_m)^\top$, where ${\tilde {\mathbf a}}_j:=\left( \begin{array}{c} {\mathbf a}_j \\ b_j \end{array} \right), \,j=1,\ldots,m$. Similarly we augment ${{\mathbf x}}, {{\mathbf y}}\in{{\mathbb H}}^d$ into $\tilde{{\mathbf x}}, \tilde{{\mathbf y}}\in{{\mathbb H}}^{d+1}$ by appending 1 to the $(d+1)$-th entry. Now we have $$\begin{aligned} {\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}})&=({\lvert{\langle {{\mathbf a}_1,{{\mathbf x}}} \rangle}+b_1\rvert}^2,\ldots, {\lvert{\langle {{\mathbf a}_m,{{\mathbf x}}} \rangle}+b_m\rvert}^2)\\ &=\left({{\rm tr}}({\tilde{{\mathbf a}}_1\tilde{{\mathbf a}}_1^*\tilde {{\mathbf x}}\tilde {{\mathbf x}}^*}),\ldots, {{\rm tr}}({\tilde {{\mathbf a}}_m\tilde {{\mathbf a}}_m^* \tilde {{\mathbf x}}\tilde {{\mathbf x}}^*})\right) =:{\bf T} (\tilde {{\mathbf x}}\tilde {{\mathbf x}}^*),\end{aligned}$$ where ${\bf T}$ is a linear transformation from ${{\mathbb H}}^{(d+1)\times (d+1)}$ to ${{\mathbb R}}^m$. Let $X_\Omega =\{\tilde{{\mathbf x}}\tilde{{\mathbf x}}^*\in{{\mathbb H}}^{(d+1)\times (d+1)}: \ {{\mathbf x}}\in \Omega\}$, $$\Theta_\Omega=\{{\bf S} \in{{\mathbb H}}^{(d+1)\times (d+1)}: \ \|{\bf S}\|_F=1,\ t{\bf S}\in X_\Omega-X_\Omega\ {\rm for \ some} \ t>0\}$$ and $$\tilde \Theta_\Omega=\left\{ {\bf S}:= \left(\begin{array}{cc} {{\mathbf z}}{{\mathbf w}}^*+{{\mathbf w}}{{\mathbf z}}^* & {{\mathbf z}}\\ {{\mathbf z}}^* & 0\end{array}\right):\ {{\mathbf z}}\in {{\mathbb H}}^d, {{\mathbf w}}\in (\Omega+\Omega)/2\ {\rm and}\ \|{\bf S}\|_F=1 \right\},$$ where $\|\cdot \|_F$ denotes the $l^2$-norm (Frobenius norm) of a matrix. Then $$\label{eq110} \Theta_\Omega\subset \tilde \Theta_\Omega$$ because $${\bf S}=t^{-1} (\tilde {{\mathbf x}}\tilde {{\mathbf x}}^*-\tilde {{\mathbf y}}\tilde {{\mathbf y}}^*)= \left(\begin{array}{cc} {{\mathbf z}}{{\mathbf w}}^*+{{\mathbf w}}{{\mathbf z}}^* & {{\mathbf z}}\\ {{\mathbf z}}^* & 0\end{array}\right)\in \tilde \Theta_\Omega \ \ {\rm for \ all} \ {\bf S}\in \Theta_\Omega,$$ where the existence of $t>0, {{\mathbf x}}, {{\mathbf y}}\in \Omega$ in the first equality follows from the definition of $\Theta_\Omega$ and the second equality holds for ${{\mathbf z}}=({{\mathbf x}}-{{\mathbf y}})/t$ and ${{\mathbf w}}=({{\mathbf x}}+{{\mathbf y}})/2$. For any ${\bf S}= \left(\begin{array}{cc} {{\mathbf z}}{{\mathbf w}}^*+{{\mathbf w}}{{\mathbf z}}^* & {{\mathbf z}}\\ {{\mathbf z}}^* & 0\end{array}\right)\in \tilde\Theta_\Omega $, we have $$\label{eq111} {\bf T}({\bf S}) = {\bf T}\left(\begin{array}{cc} {{\mathbf x}}{{\mathbf x}}^*-{{\mathbf y}}{{\mathbf y}}^* & {{\mathbf x}}-{{\mathbf y}}\\ {{\mathbf x}}^*-{{\mathbf y}}^* & 0\end{array}\right) = {\bf T}( \widetilde {{{\mathbf x}}} \widetilde{{{\mathbf x}}}^*- \tilde {{{\mathbf y}}} \tilde{{{\mathbf y}}}^*) = {\bf M}_{{\bf A}, {\bf b}}^2({{\mathbf x}})-{\bf M}_{{\bf A},{\bf b}}^2({{\mathbf y}})\ne 0$$ by the affine phase retrievability of $({\bf A},{\bf b})$, where ${{\mathbf x}}= {{\mathbf w}}+{{\mathbf z}}/2$ and ${{\mathbf y}}={{\mathbf w}}-{{\mathbf z}}/2$. Clearly $\tilde \Theta_\Omega$ is a compact set. This together with [(\[eq111\])]{} implies that $$\label{eq1102} c_2:=\inf_{{\bf S}\in \tilde \Theta_\Omega}\|{\bf T} ({\bf S})\|>0.$$ Therefore $$\begin{aligned} \label{eq1103} & & \left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf y}})\right\| = \|{\bf T}({\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*)\|\nonumber\\ & \ge & \Big(\inf_{{\mathbf S}\in \Theta_\Omega} \|{\bf T}({\mathbf S})\|\Big) \|{\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*\|_F \ge c_2 \|{\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*\|_F, $$ where the last inequality holds by [(\[eq110\])]{}. Now for the unit vector ${\mathbf e}_{d+1}$, we have $$\|{\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*\|_F\ge \left\|({\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*){\mathbf e}_{d+1}\right\| =\|\tilde {{\mathbf x}}- \tilde{{\mathbf y}}\| = \|{{\mathbf x}}-{{\mathbf y}}\|.$$ This, together with [(\[eq1102\])]{} and [(\[eq1103\])]{}, establishes the lower bound in (\[eq:Lip2\]). Because ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}})$ is linear in $X={\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*$, we must also have $$\left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}^2({{\mathbf y}})\right\| \leq C_2' \|{\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*\|_F.$$ However using the standard estimate, we have $$\begin{aligned} \|{\tilde {{\mathbf x}}}{\tilde {{\mathbf x}}}^*-{\tilde {{\mathbf y}}}{\tilde {{\mathbf y}}}^*\|_F \leq \|\tilde{{\mathbf x}}\|\,\|\tilde{{\mathbf x}}-\tilde{{\mathbf y}}\| +\|\tilde{{\mathbf y}}\|\,\|\tilde{{\mathbf x}}-\tilde{{\mathbf y}}\| \leq 2(1+\|{{\mathbf x}}\| +\|{{\mathbf y}}\|)\,\|{{\mathbf x}}-{{\mathbf y}}\|.\end{aligned}$$ Here we have used the facts that $\|\tilde{{\mathbf x}}-\tilde{{\mathbf y}}\| =\|{{\mathbf x}}-{{\mathbf y}}\|$ and $\|\tilde{{\mathbf x}}\|\leq 1+\|{{\mathbf x}}\|$. Taking $C_2 = 2C_2'$ yields the upper bound in (\[eq:Lip2\]). We now prove the inequalities for ${\mathbf M}_{{\mathbf A},{\mathbf b}}$. The upper bound in (\[eq:Lip1\]) is straightforward. Note that $$\Bigl|{\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}-{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_j\rvert}\Bigr| \leq |{\langle {{\mathbf a}_j,{{\mathbf x}}-{{\mathbf y}}} \rangle}| \leq \|{\mathbf a}_j\|\,\|{{\mathbf x}}-{{\mathbf y}}\|.$$ It follows that $$\left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\| \leq \Bigl(\sum_{j=1}^m \|{\mathbf a}_j\|\Bigr)\,\|{{\mathbf x}}-{{\mathbf y}}\|.$$ The upper bound in (\[eq:Lip1\]) thus follows by letting $C_1 = \sum_{j=1}^m \|{\mathbf a}_j\|$. To prove the lower bound, we observe that $$\begin{aligned} \Bigl|{\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}^2-{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_j\rvert}^2\Bigr| & =\Bigl|{\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}-{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_j\rvert}\Bigr| ({\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}+{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_j\rvert}) \\ &\leq L(1+\|{{\mathbf x}}\| +\|{{\mathbf y}}\|) \Bigl|{\lvert{\langle {{\mathbf a}_j,{{\mathbf x}}} \rangle}+b_j\rvert}-{\lvert{\langle {{\mathbf a}_j,{{\mathbf y}}} \rangle}+b_j\rvert}\Bigr|,\end{aligned}$$ where $L>0$ is a constant depending only on $({\mathbf A},{\mathbf b})$. Hence $$\left\|{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\| \leq L(1+\|{{\mathbf x}}\| +\|{{\mathbf y}}\|) \left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\|.$$ It now follows from the lower bound $\|{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\| \geq c_2\|{{\mathbf x}}-{{\mathbf y}}\|$ and setting $c_2 = c_1/L$ that $$\left\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}}) - {\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\right\| \geq \frac{c_1}{1+\|{{\mathbf x}}\|+\|{{\mathbf y}}\|} \,\|{{\mathbf x}}-{{\mathbf y}}\|.$$ The theorem is proved. \[coro:bi-Lip\] Neither ${\mathbf M}_{{\mathbf A},{\mathbf b}}$ nor ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}$ is bi-Lipschitz on ${{\mathbb H}}^d$. The map ${\mathbf M}^2_{{\mathbf A},{\mathbf b}}({{\mathbf x}})$ is not bi-Lipschitz follows from the simple observation that it is quadratic in ${{\mathbf x}}$ (more precisely, in $\Re({{\mathbf x}})$ and $\Im({{\mathbf x}})$). No quadratic function can be bi-Lipschitz on the whole Euclidean space. To see ${\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})$ is not bi-Lipschitz, we fix a nonzero ${{\mathbf x}}_0\in {{\mathbb H}}^d$. Take ${{\mathbf x}}=r {{\mathbf x}}_0$ and ${{\mathbf y}}=-r {{\mathbf x}}_0$, where $r>0$. Note that $$\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\|=\Bigl(\sum_{j=1}^m ({\lvertr {\langle {{\mathbf a}_j,{{\mathbf x}}_0} \rangle}+b_j\rvert}-{\lvertr {\langle {{\mathbf a}_j,{{\mathbf x}}_0} \rangle}-b_j\rvert})^2\Bigr)^{1/2}$$ and $$\|{{\mathbf x}}-{{\mathbf y}}\|\,\,=\,\, 2 r\|{{\mathbf x}}_0\| .$$ Then $$\label{eq:rt} \frac{\|{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf x}})-{\mathbf M}_{{\mathbf A},{\mathbf b}}({{\mathbf y}})\|}{\|{{\mathbf x}}-{{\mathbf y}}\|}=\frac{1}{2\|{{\mathbf x}}_0\|} \Bigl(\sum_{j=1}^m ({\lvert {\langle {{\mathbf a}_j,{{\mathbf x}}_0} \rangle}+b_j/r\rvert}-{\lvert {\langle {{\mathbf a}_j,{{\mathbf x}}_0} \rangle}-b_j/r\rvert})^2\Bigr)^{1/2}.$$ A simple observation is that the right side of (\[eq:rt\]) tending to 0 as $r \rightarrow \infty$. 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--- abstract: 'We analyze the ideal gas like models of markets and review the different cases where a ‘savings’ factor changes the nature and shape of the distribution of wealth. These models can produce similar distribution of wealth as observed across varied economies. We present a more realistic model where the saving factor can vary over time (annealed savings) and yet produces Pareto distribution of wealth in certain cases. We discuss the relevance of such models in the context of wealth distribution, and address some recent issues in the context of these models.' author: - Arnab Chatterjee - 'Bikas K. Chakrabarti' title: 'Ideal-gas like market models with savings: quenched and annealed cases' --- Introduction ============ The study of wealth distribution [@cc:EWD05] in a society has remained an intriguing problem since Vilfredo Pareto who first observed [@cc:Pareto:1897] that the number of rich people with wealth $m$ decay following an inverse: $$P(m) \sim m^{-(1+\nu)}. \label{par}$$ $P(m)$ is number density of people possessing wealth $m$, and $\nu$ is known as the Pareto exponent. This exponent generally assumes a value between $1$ and $3$ in varied economies [@cc:realdatag; @cc:realdataln; @cc:Sinha:2006]. It is also known that for low and medium income, the number density $P(m)$ falls off much faster: exponentially [@cc:realdatag] or in a log-normal way [@cc:realdataln]. In recent years, easy availability of data has helped in the analysis of wealth or income distributions in various societies [@cc:EWD05]. It is now more or less established that the distribution has a power-law tail for the large (about 5% of the population) wealth/income [@cc:realdatag; @cc:realdataln; @cc:Sinha:2006], while the majority (around 95%) low income distribution fits well to Gibbs or log-normal form. There has been several attempts to model a simple economy with minimum trading ingredients, which involve a wealth exchange process [@cc:othermodels] that produce a distribution of wealth similar to that observed in the real market. We are particularly interested in microscopic models of markets where the (economic) trading activity is considered as a scattering process [@cc:marjit; @cc:Dragulescu:2000; @cc:Chakraborti:2000; @cc:Hayes:2002; @cc:Chatterjee:2004; @cc:Chatterjee:2003; @cc:Chakrabarti:2004; @cc:Slanina:2004] (see also Ref. [@cc:ESTP:KG] for a recent extensive review). We concentrate on models that incorporate ‘savings’ as an essential ingredient in a trading process, and reproduces the salient features seen across wealth distributions in varied economies (see Ref. [@cc:EWD:CC] for a review). Angle [@cc:Angle:1986] studied inequality processes which can be mapped to the savings wealth models is certain cases; see Ref. [@cc:Angle:2006] for a detailed review. These studies also show (and discussed briefly here) how the distribution of savings can be modified to reproduce the salient features of empirical distributions of wealth – namely the shape of the distribution for the low and middle wealth and the tunable Pareto exponent. In all these models [@cc:Chakraborti:2000; @cc:Hayes:2002; @cc:Chatterjee:2004; @cc:Chatterjee:2003; @cc:Chakrabarti:2004], ‘savings’ was introduced as an annealed parameter that remained invariant with time (or trading). Apart from a brief summary of the established results in such models, here we report some new results for annealed cases, where the saving factor can change with time, as one would expect in a real trading process. We report cases where the wealth distribution is still described by a Pareto law. We also forward some justification of the various assumptions in such models. Ideal-gas like models of trading ================================ We first consider an ideal-gas model of a closed economic system. Wealth is measured in terms of the amount of money possessed by an individual. Production is not allowed i.e, total money $M$ is fixed and also there is no migration of population i.e, total number of agents $N$ is fixed, and the only economic activity is confined to trading. Each agent $i$, individual or corporate, possess money $m_i(t)$ at time $t$. In any trading, a pair of agents $i$ and $j$ randomly exchange their money [@cc:marjit; @cc:Dragulescu:2000; @cc:Chakraborti:2000], such that their total money is (locally) conserved and none end up with negative money ($m_i(t) \ge 0$, i.e, debt not allowed): $$\label{consv} m_i(t) + m_j(t) = m_i(t+1) + m_j(t+1);$$ time ($t$) changes by one unit after each trading. The steady-state ($t \to \infty$) distribution of money is Gibbs one: $$\label{gibbs} P(m)=(1/T)\exp(-m/T);T=M/N.$$ No matter how uniform or justified the initial distribution is, the eventual steady state corresponds to Gibbs distribution where most of the people end up with very little money. This follows from the conservation of money and additivity of entropy: $$\label{prob} P(m_1)P(m_2)=P(m_1+m_2).$$ This steady state result is quite robust and realistic. Several variations of the trading, and of the ‘lattice’ (on which the agents can be put and each agent trade with its ‘lattice neighbors’ only) — compact, fractal or small-world like [@cc:EWD05], does not affect the distribution. Savings in Ideal-gas trading market: Quenched case ================================================== In any trading, savings come naturally [@cc:Samuelson:1980]. A saving factor $\lambda$ is therefore introduced in the same model [@cc:Chakraborti:2000] (Ref. [@cc:Dragulescu:2000] is the model without savings), where each trader at time $t$ saves a fraction $\lambda$ of its money $m_i(t)$ and trades randomly with the rest. In each of the following two cases, the savings fraction does not vary with time, and hence we call it ‘quenched’ in the terminology of statistical mechanics. Fixed or uniform savings ------------------------ For the case of ‘fixed’ savings, the money exchange rules are: $$\label{delm} m_{i}(t+1)=m_{i}(t)+\Delta m; \ m_{j}(t+1)=m_{j}(t)-\Delta m$$ where $$\label{eps} \Delta m=(1-\lambda )[\epsilon \{m_{i}(t)+m_{j}(t)\}-m_{i}(t)],$$ where $\epsilon$ is a random fraction, coming from the stochastic nature of the trading. $\lambda$ is a fraction ($0 \le \lambda < 1$) which we call the saving factor. The market (non-interacting at $\lambda =0$ and $1$) becomes ‘interacting’ for any non-vanishing $\lambda (<1)$: For fixed $\lambda$ (same for all agents), the steady state distribution $P_f(m)$ of money is sharply decaying on both sides with the most-probable money per agent shifting away from $m=0$ (for $\lambda =0$) to $M/N$ as $\lambda \to 1$. The self-organizing feature of this market, induced by sheer self-interest of saving by each agent without any global perspective, is very significant as the fraction of paupers decrease with saving fraction $\lambda$ and most people possess some fraction of the average money in the market (for $\lambda \to 1$, the socialists’ dream is achieved with just people’s self-interest of saving!). Although this fixed saving propensity does not give the Pareto-like power-law distribution yet, the Markovian nature of the scattering or trading processes (eqn. (\[prob\])) is lost and the system becomes co-operative. Indirectly through $\lambda$, the agents get to develop a correlation with (start interacting with) each other and the system co-operatively self-organizes [@cc:Bak:1997] towards a most-probable distribution. This model has been understood to a certain extent (see e.g, [@cc:Das:2003; @cc:Patriarca:2004; @cc:Repetowicz:2005]), and argued to resemble a gamma distribution [@cc:Patriarca:2004], and partly explained analytically. This model clearly finds its relevance in cases where the economy consists of traders with ‘waged’ income [@cc:Willis:2004]. Distributed savings ------------------- In a real society or economy, $\lambda$ is a very inhomogeneous parameter: the interest of saving varies from person to person. We move a step closer to the real situation where saving factor $\lambda$ is widely distributed within the population [@cc:Chatterjee:2004; @cc:Chatterjee:2003; @cc:Chakrabarti:2004]. The evolution of money in such a trading can be written as: $$\label{mi} m_i(t+1)=\lambda_i m_i(t) + \epsilon_{ij} \left[(1-\lambda_i)m_i(t) + (1-\lambda_j)m_j(t)\right],$$ $$\label{mj} m_j(t+1)=\lambda_j m_j(t) + (1-\epsilon_{ij}) \left[(1-\lambda_i)m_i(t) + (1-\lambda_j)m_j(t)\right]$$ One again follows the same rules as before, except that $$\label{lrand} \Delta m=\epsilon_{ij}(1-\lambda_{j})m_{j}(t)-(1-\lambda _{i})(1 - \epsilon_{ij})m_{i}(t)$$ here; $\lambda _{i}$ and $\lambda _{j}$ being the saving propensities of agents $i$ and $j$. The agents have fixed (over time) saving propensities, distributed independently, randomly and uniformly (white) within an interval $0$ to $1$ agent $i$ saves a random fraction $\lambda_i$ ($0 \le \lambda_i < 1$) and this $\lambda_i$ value is quenched for each agent ($\lambda_i$ are independent of trading or $t$). $P(m)$ is found to follow a strict power-law decay. This decay fits to Pareto law (\[par\]) with $\nu = 1.01 \pm 0.02$ for several decades. This power law is extremely robust: a distribution $$\label{lam0} \rho(\lambda) \sim |\lambda_0-\lambda|^\alpha, \ \ \lambda_0 \ne 1, \ \ 0 \le \lambda<1,$$ of quenched $\lambda$ values among the agents produce power law distributed $m$ with Pareto index $\nu=1$, irrespective of the value of $\alpha$. For negative $\alpha$ values, however, we get an initial (small $m$) Gibbs-like decay in $P(m)$. In case $\lambda_0 =1$, the Pareto exponent is modified to $\nu=1+\alpha$, which qualifies for the non-universal exponents in the same model [@cc:Chatterjee:2004; @cc:Mohanty:2006]. This model [@cc:Chatterjee:2004] has been thoroughly analyzed, and the analytical derivation of the Pareto exponent has been achieved in certain cases [@cc:Repetowicz:2005; @cc:Mohanty:2006; @cc:Chatterjee:2005]. The Pareto exponent has been derived to exactly $1$. In this model, agents with higher saving propensity tend to hold higher average wealth, which is justified by the fact that the saving propensity in the rich population is always high [@cc:Dynan:2004]. Savings in Ideal-gas trading market: Annealed case ================================================== ![ Distribution $P(m)$ of money $m$ in case of annealed savings $\lambda$ varying randomly in $[\mu,1)$. Here, $\zeta(\mu)$ has a uniform distribution. The distribution has a power law tail with Pareto index $\nu=1$. The simulation has been done for a system of $N=10^2$ agents, with $M/N=1$. $P(m)$ is the steady state distribution after $4 \times 10^4 \times N$ random exchanges, and averaged over an ensemble of $10^5$. []{data-label="fig:ann:1-lambda"}](ChakrabartiFig1.eps){width="8.5cm"} In a real trading process, the concept of ‘saving factor’ cannot be attributed to a quantity that is invariant with time. A saving factor always changes with time or trading. In earlier works, we reported the case of annealed savings, where the savings factor $\lambda_i$ changes with time in the interval $[0,1)$, but does not produce a power law in $P(m)$ [@cc:Chatterjee:2004]. We report below some special cases of annealed saving which produce a power law distribution of $P(m)$. If one allows the saving factor $\lambda_i$ to vary with time in $[0,1)$, the money distribution $P(m)$ does not produce a power law tail. Instead, we propose a slightly different model of an annealed saving case. We associate a parameter $\mu_i$ ($0 < \mu_i < 1$) with each agent $i$ such that the savings factor $\lambda_i$ randomly assumes a value in the interval $[\mu_i,1)$ at each time or trading. The trading rules are of course unaltered and governed by Eqns. (\[mi\]) and (\[mj\]). Now, considering a suitable distribution $\zeta(\mu)$ of $\mu$ over the agents, one can produce money distributions with power-law tail. The only condition that needs to hold is that $\zeta(\mu)$ should be non-vanishing as $\mu \to 1$. Figure \[fig:ann:1-lambda\] shows the case when $\zeta(\mu)=1$. Numerical simulations suggest that the behavior of the wealth distribution is similar to the quenched savings case. In other words, only if $\zeta(\mu) \propto |1-\mu|^\alpha$, it is reflected in the Pareto exponent as $\nu=1+\alpha$. Relevance of gas like models ============================ Al these gas-like models of trading markets are based on the assumption of (a) money conservation (globally in the market; as well as locally in any trading) and (b) stochasticity. These points have been criticized strongly (by economists) in the recent literature [@cc:Gallegati:2006]. In the following, we forward some of the arguments in favour of these assumptions (see also [@cc:ESOM:2006]). Money conservation ------------------ If we view the trading as scattering processes, one can see the equivalence. Of course, in any such ‘money-exchange’ trading process, one receives some profit or service from the other and this does not appear to be completely random, as assumed in the models. However, if we concentrate only on the ‘cash’ exchanged (even using Bank cards!), every trading is a money conserving one (like the elastic scattering process in physics!) It is also important to note that the frequency of money exchange in such models define a time scale in which the total money in the market does not change. In real economies, the total money changes much slowly, so that in the time scale of exchanges, it is quite reasonable to assume the total money to be conserved in these exchange models. This can also be justified by the fact that the average international foreign currency exchange rates change drastically (say, by more than 10%) very rarely; according to the Reserve Bank of India, the US Dollar remained at INR $45 \pm 3$ for the last eight years [@cc:Sarkar:2006]! The typical time scale of the exchanges considered here correspond to seconds or minutes and hence the constancy assumption cannot be a major problem. Stochasticity ------------- But, are these trading random? Surely not, when looked upon from individual’s point of view: When one maximizes his/her utility by money exchange for the $p$-th commodity, he/she may choose to go to the $q$-th agent and for the $r$-th commodity he/she will go to the $s$-th agent. But since $p \ne q \ne r \ne s$ in general, when viewed from a global level, these trading/scattering events will all look random (although for individuals these is a defined choice or utility maximization). Apart from the choice of the agents for any trade, the traded amount are considered to be random in such models. Some critics argue, this cannot be totally random as the amount is determined by the price of the commodity exchanged. Again, this can be defended very easily. If a little fluctuation over the ‘just price’ occurs in each trade due to the bargain capacities of the agents involved, one can easily demonstrate that after sufficient trading (time, depending on the amount of fluctuation in each trade), the distribution will be determined by the stochasticity, as in the cases of directed random walks or other biased stochastic models in physics. It may be noted in this context that in the stochastically formulated ideal gas models in physics (developed in late 1800/early 1900) one (physicists) already knew for more than a hundred years, that each of the constituent particle (molecule) follows a precise equation of motion, namely that due to Newton. The assumption of stochasticity here in such models, even though each agent might follow an utility maximizing strategy (like Newton’s equation of motion for molecules), is therefore, not very unusual in the context. Summary and conclusions ======================= We analyze the gas like models of markets. We review the different cases where a quenched ‘savings’ factor changes the nature and shape of the distribution of wealth. Suitable modification in the nature of the ‘savings’ distribution can simulate all observed wealth distributions. We give here some new numerical results for the annealed ‘savings’ case. We find that the more realistic model, where the saving factor randomly varies in time (annealed savings), still produce a Pareto distribution of wealth in certain cases. We also forward some arguments in favour of the assumptions made in such gas-like models. [99]{} *Econophysics of Wealth Distributions*, Eds. A. Chatterjee, S. Yarlagadda, B. K. Chakrabarti, Springer Verlag, Milan (2005) V. Pareto, *Cours d’economie Politique*, F. Rouge, Lausanne and Paris (1897) M. Levy, S. Solomon, Physica A **242** 90 (1997); A. A. Drăgulescu, V. M. Yakovenko, Physica A **299** 213 (2001); H. Aoyama, W. 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--- abstract: 'Ground state properties of the repulsive Hubbard model on a cubic lattice are investigated by means of the auxiliary-field quantum Monte Carlo method. We focus on low-density systems with varying on-site interaction $U/t$, as a model relevant to recent experiments on itinerant ferromagnetism in a dilute Fermi gas with contact interaction. Twist-average boundary conditions are used to eliminate open-shell effects and large lattice sizes are studied to reduce finite-size effects. The sign problem is controlled by a generalized constrained path approximation. We find no ferromagnetic phase transition in this model. The ground-state correlations are consistent with those of a paramagnetic Fermi liquid.' author: - 'Chia-Chen Chang${}^1$' - 'Shiwei Zhang${}^{1,2,3}$' - 'David M. Ceperley${}^4$' bibliography: - './ferro.bib' title: | Itinerant ferromagnetism in a Fermi gas with contact interaction:\ Magnetic properties in a dilute Hubbard model --- [*Introduction*]{} - The study of ferromagnetism has a long history in physics. At the microscopic level, the formation of ferromagnetic order is a consequence of strong interactions. Heisenberg first pointed out that an exchange interaction that lowers the energy of a pair of parallel spins would favor ferromagnetism. However a localized-spin mechanism cannot be fully responsible for ferromagnetism in transition metals, for instance iron and nickel, where electrons are extended. Both the interactions and the delocalized nature of electrons have to be taken into account at a more fundamental level. A generic description of itinerant ferromagnetism is given by the three-dimensional (3-D) Hubbard model [@Hubbard1963]: H=-t\_[ij]{}( c\_[i]{}\^c\_[j]{} + h.c. ) + U\_i n\_[i]{} n\_[i]{}. \[eq:Hubbard\] The operator $c_{i\sigma}^\dagger$ ($c_{i\sigma}$) creates (annihilates) an electron with spin $\sigma$ ($\sigma=\upa,\dna$ ), $i$ enumerates the sites in an $N=L^3$ lattice, and $\langle ij \rangle$ denotes a sum of nearest neighbor pairs. The parameter $t$ is the nearest-neighbor hopping amplitude, and $U>0$ is the on-site interaction strength. The total density is $n=(N_\uparrow+N_\downarrow)/N$. This simple Hamiltonian contains both the itinerant character and local repulsion. However, because neither of the two terms alone favors ferromagnetic ordering, the magnetic correlations in the Hubbard model is not obvious. The first evidence of ferromagnetism in the Hubbard model was discussed by Nagaoka [@Nagaoka1966] and by Thouless [@Thouless1965]. They showed that the ground state with a single hole in any finite bipartite lattice with $U\rightarrow \infty$ (and periodic boundary conditions) is fully polarized. Subsequent studies indicate that the stability of the state with more holes can depend on system size [@Takahashi1982; @*Doucot1989; @*Zhang1991; @*Puttika1992], and boundary conditions [@Riera1989; @*Barbieri1990]. The critical doping for the onset of ferromagnetism is still an open question [@Puttika1992; @Becca2001; @Carleo2010]. Away from infinite-$U$, the existence of ferromagnetism at non-zero density is less certain. Whether ferromagnetism is a generic property of the Hubbard model is still not answered. Rapid experimental progress in cold atoms has opened a new avenue for exploring the physics of itinerant ferromagnetism. In a recent experiment aimed to simulate the Stoner Hamiltonian (i.e. spin 1/2 fermions in continuous space interacting with a repulsive contact potential), a dilute gas of two hyperfine states of ${}^6$Li atoms are tuned to interact via large positive scattering lengths. Signatures of ferromagnetic instability [@Jo2009] have generated a lot of theoretical interest [@Zhai2009; @*Cui2010; @Pilati2010; @Chang2010; @Pekker2010]. The Hubbard Hamiltonian in Eq. (\[eq:Hubbard\]) gives a reasonable representation of the Stoner Hamiltonian on a lattice. As the density $n\rightarrow 0$ it seems clear that no ferromagnetism exists in the model in Eq. (\[eq:Hubbard\]) [@Tasaki1998], since the maximum scattering length is bounded by $\sim 1/3.173$ lattice spacing [@Castin2004; @Purwanto2005]. However, at low but not zero density, the magnetic properties and the phase diagram of the 3-D Hubbard model are not clear. We address this question here by very accurate many-body simulations with the constrained path Monte Carlo (CPMC) method. [*Method*]{} - The CPMC method [@Zhang1995; @Zhang1997; @Zhang2003; @Chang2008] projects the many-body ground state $\ket{\Psi_0}$ from a trial wave function $\ket{\Psi_T}$ by repeated application of an imaginary-time propagator $e^{-\Delta\tau H}$ ($\Delta\tau$ is the Trotter time step), provided that $\ket{\Psi_T}$ satisfies $\langle\Psi_0|\Psi_T\rangle\neq 0$. The propagator is decomposed into $e^{-\Delta\tau H}\approx e^{-\Delta\tau H_1/2}e^{-\Delta\tau H_2}e^{-\Delta\tau H_1/2} +\mathcal{O}(\Delta\tau^3)$, where $H_1$ and $H_2$ are one- and two-body parts of $H$ respectively. The two-body part $e^{-\Delta\tau H_2}$ is further decoupled into a sum over one-body projectors in Ising fields [@Hirsch1985]. This leads to a formally exact expression $ e^{-\Delta\tau H}=\sum_{\{\mathbf{x}\}}P(\{\mathbf{x}\})B(\{\mathbf{x}\})$, where $\{\mathbf{x}\}$ is a collection of $N$ Ising fields, $P(\{\mathbf{x}\})$ is their probability distribution, and $B(\{\mathbf{x}\})$ is a one-body projector. The multidimensional summation is carried out efficiently by importance-sampled random walks with non-orthogonal Slater determinants (SDs), where the one-body projectors $B(\{\mathbf{x}\})$ propagate one SD into another. The fermion sign problem is controlled approximately by the constrained path approximation.[@Zhang1995; @*Zhang1997] The many-body ground state is given by $|\Psi_0\rangle = \sum_\phi w(\phi) |\phi\rangle$, where $|\phi\rangle$ are SDs sampled by the QMC, and their probability distribution determines the weight factors $w(\phi)$. Because the Schrödinger equation is linear, $|\Psi_0\rangle$ is degenerate with $-|\Psi_0\rangle$. In a random walk, the SDs can move back and forth between the two sets of solutions. The appearance of the two sets with opposite signs in the Monte Carlo samples is the origin of the sign problem. To control the problem, the walker is required to satisfy the constraint $\langle\Psi_T|\phi\rangle >0$ in the course of the random walk. This is the only approximation in our method. More formal discussions of the theoretical basis of the generalized constrained path approximation and benchmarks can be found elsewhere [@Zhang1995; @*Zhang1997; @Zhang2003]. In the Hubbard model, the energy at $U=4t$ is typically within $< 0.5\%$ of the exact diagonalization result [@Chang2008]. Extensive benchmarks of this approach for molecules and solids are in Refs. [@AlSaidi2006; @*AlSaidi2007; @Purwanto2009]. The constrained path approximation is similar in spirit to the fixed-node approximation in the diffusion Monte Carlo (DMC) method [@Ceperley1980; @Foulkes2001], which has been used for all recent simulation work on the problem of itinerant ferromagnetism in the Stoner model [@Conduit2009a; @Pilati2010; @Chang2010]. In fixed-node DMC one uses a real-space trial function $\Psi_T(R)$ to determine the sign of the ground-state wave function. The random walks, which involve movements of electron coordinates $R$ (a $3\,(N_\uparrow+N_\downarrow)$-dimensional vector), are constrained to the region where $\Psi_T(R)>0$. Since, in CPMC the random walks take place in the space of SDs, where fermionic statistics are automatically maintained, the sign problem is reduced. As a result, the constrained path approximation is less sensitive to $|\Psi_T\rangle$ and typically has smaller systematic errors. In this work we apply twist-averaged boundary conditions (TABCs) [@Lin2001]. Under TABCs, the wave function gains a phase when electrons wind around the periodic boundary conditions: $ \Psi(\ldots,\mathbf{r}_j+\mathbf{L},\ldots) = e^{i\widehat{\mathbf{L}}\cdot{\bm\Theta}} \Psi(\ldots,\mathbf{r}_j,\ldots), $ where $\widehat{\mathbf{L}}$ is the unit vector along $\mathbf{L}$, and ${\bm\Theta}=(\theta_x,\theta_y,\theta_z)$ are random twists over which we average. A simple generalization of the CPMC method can be made to handle the overall phase that arises from TABC [@Zhang2003; @Chang2008]. As an additional benchmark for the present work, we studied several low-density $L=4$ systems in detail. For example, at $U=16t$ with $n=0.25$, the CPMC energy, averaged over $1000$ ${\bm\Theta}$-points, agrees to better than $0.2$% with exact diagonalization. ![(color online). Ground state energy per particle $e$ as a function of interaction strength $U/t$ at $n=0.25$ (left) and $n=0.0625$ (right). Symbols represent $e_{CPMC}$. Dashed (blue) line corresponds to the energy of a saturated ferromagnetic state ($e_{FM}$). $e_{MF}$ energy is represented by the thick solid (green) line. $e_P$ (perturbation theory [@Metzner1989]) is plotted by dot-dashed line. \[fig:ener-gr\] ](Fig1.eps) [*Energy*]{} - We first compare the ground-state energy of an unpolarized system ($N_\uparrow=N_\downarrow$) with that of a fully polarized state at the same total density. The results are summarized in Fig. \[fig:ener-gr\]. Because electrons of the same spin do not interact, the energy of the fully polarized state, $e_{FM}$, is purely kinetic and does not depend on $U$. In mean-field (MF) theory, the energy of a system with $n_\sigma=N_\sigma/N$ (with $n=n_\uparrow+n_\downarrow$) is $ e_{MF}(U,n) = [e_0(n_\uparrow) n_\uparrow+e_0(n_\downarrow) n_\downarrow + Un_\uparrow n_\downarrow]/n, $ where $e_0(n_\sigma)$ is the energy of the fully polarized system at density $n_\sigma$. At $n=0.25$, MF predicts a paramagnetic to ferromagnetic phase transition at $U = 13.9t$. This is to be compared to the corresponding transition point $k_F a\sim \pi/2$ in the continuum Stoner Hamiltonian, where $k_F = (3\pi^2 n)^{1/3}$ is the Fermi wave vector, and $a$ is the scattering length in continuum. When the system density is lowered to $n=0.0625$, the MF transition in the Hubbard model is at a larger interaction, $U = 29.3 t$. The equation of state has also been obtained from perturbation theory for an unpolarized system [@Metzner1989]: $ e_{P}(U,n) = e_{MF}(U,n) + e_c(U,n). $ The last term, $e_c(U,n)$, is the correlation energy estimated to $\mathcal{O}(U^2)$. The result is also included in Fig. \[fig:ener-gr\]. The CPMC result for the ground state energy $e_{CPMC}$ is obtained by averaging over twist-angles. The energies calculated from different lattice sizes are shown by different symbols in Fig. \[fig:ener-gr\]. It can be seen that our remaining finite-size errors are negligible on this scale. Free-electron trial wave functions are used for the constraint. In a few cases we have also checked with unrestricted Hartree-Fock trial wave functions, which gave statistically indistinguishable CPMC energies. The energies shown are for finite time steps, with $\Delta\tau$ satisfying $U\Delta\tau< 0.2$. The residual Trotter error is ${\cal{O}}(10^{-2})$, smaller than the symbol size. We see that MF theory, which gives a reasonable estimate of the energy at small $U$, quickly shows severe deviations as the interaction becomes stronger. The perturbation result, $e_{P}(U,n)$, gives an improved estimate of energy for small $U$, but deviates once the system enters the intermediate interaction regime $U \gtrsim 5t$. At the MF transition point, the CPMC energy is significantly lower than $e_{FM}$. Indeed the CPMC energy remains lower than $e_{FM}$ across the entire range of $U$ simulated. No indication of a ferromagnetic transition is seen. Individual components of the energy are shown in Fig. \[fig:ener-ek-ev\]. As $U$ increases, electrons in the unpolarized system occupy higher momentum states, outside the Fermi level, which increases the kinetic energy compared to the MF result. This enables the system to drastically decrease the interaction energy, by suppressing double occupancy. The net effect is that the total energy is greatly reduced and remains below $e_{MF}$ and $e_{FM}$. ![(color online). Kinetic (left panel) and interaction (right panel) energies as a function of interaction strength at density $n=0.0625$. Symbols are the CPMC data obtained on an $8^3$ lattice. Lines are defined in the same way as in Fig. \[fig:ener-gr\]. The inset on the right shows the double occupancy, normalized to $1$ at $U=0$. []{data-label="fig:ener-ek-ev"}](Fig2.eps) [*Correlation function*]{} - To probe the nature of the ground state, we examine the spin-dependent pair correlation function: g\_[’]{}()= \_[’]{} n\_[+’,]{}n\_[’,’]{} . \[eq:spin-corr\] The CPMC expectations are evaluated by the back-propagation technique [@Zhang1997; @Purwanto2004]. We average over different $\mathbf{r}$’s to obtain $g_{\sigma\sigma'}(r)$, with $r\equiv |\mathbf{r}|$ since the correlation function is primarily a function of distance in the paramagnetic or ferromagnetic phases. ![(color online). Left: Anti-parallel spin-spin pair correlation function of an unpolarized state at different interaction strengths. Right: Comparison of the parallel pair correlation functions of the fully polarized state (FM) and the unpolarized states at different interaction strengths. The inset shows $\Delta g_{\uparrow\uparrow}(r)=g_{\uparrow\uparrow}(r)-g^0_{\uparrow\uparrow}(r)$, where $g^0_{\uparrow\uparrow}(r)$ is the correlation function of the unpolarized non-interacting system. In both panels, the system is an $8^3$ lattice at density $n=0.0625$. []{data-label="fig:corr"}](Fig3.eps) The anti-parallel pair correlation $g_{\uparrow\downarrow}(r)$ is a constant in a non-interacting system or in the MF solution. In the presence of interaction, a correlation hole is created surrounding each electron. At $n=0.0625$, the size of the correlation hole is $r_{cor}\lesssim\sqrt{3}$. As $U$ is increased, the correlation hole becomes deeper, as illustrated in the left panel in Fig. \[fig:corr\]. Compared to $g_{\uparrow\downarrow}(r)$, the change in the parallel-spin pair correlation $g_{\uparrow\uparrow}(r)$ is less dramatic from the MF or non-interacting result. Strong interaction does appear to increase $g_{\uparrow\uparrow}(r)$ slightly at short distance. However, the correlation remains much less than that in the FM case. [*Momentum distribution*]{} - The creation of correlation hole is a result of minimizing the interaction energy. Electrons of opposite spins rearrange their relative positions to reduce the potential energy. The cost of the rearrangement is the kinetic energy increase, as discussed earlier. This can also be observed in Fig. \[fig:nk\] where the momentum distribution $n_\mathbf{k}$ is shown for different interaction strengths. We have plotted $n_\mathbf{k}$ as a function of the single-particle energy level $\varepsilon(\mathbf{k})=2\sum_{\alpha=x,y,z} [1-\cos (k_\alpha+\Theta_\alpha/L)]$, in units of the Fermi energy $\varepsilon_F$. Each curve contains the result of $n_\mathbf{k}$ from multiple $\Theta$-points. At $U=4t$, the distribution is very close to the non-interacting momentum distribution with only a few low lying excitations near the Fermi surface (FS). As $U$ is increased, more higher $\mathbf{k}$ states are populated outside the FS. In $n_\mathbf{k}$ a jump appears at $\varepsilon_F$ which can be read off directly in our finite size simulations. The jump indicates that the system is a normal Fermi liquid, with the value of the jump proportional to the renormalization factor $Z$. Its precise value can be determined with more extensive calculations and finite size scaling. [*Discussion*]{} - Although we have focused on the ground state of a homogeneous Fermi gas, it is not difficult to extend the results to the case with an external trap. For example, the kinetic energy results in Fig. \[fig:ener-ek-ev\] indicate that, with a trap, there would be a [*minimum*]{} in the curve of $e_K$ versus interaction strength, as observed in the experiment [@Jo2009] (see also discussion in Ref. [@Zhai2009]). Effects of confinement on the kinetic energy have been investigated in detail by CPMC for trapped Bose gases [@Purwanto2005]. The MF kinetic energy was shown to decrease monotonically because the gas expands in the trap as the scattering length $a$ is increased; on the other hand, correlation effects lead to an increase of $e_K$, similar to Fig. \[fig:ener-ek-ev\]. This competition results in a non-monotonic curve, with a [*minimum*]{} in the kinetic energy at a finite scattering length. ![The momentum distribution $n_\mathbf{k}$ for values of $U$ plotted as a function of the single particle energy. Two lattice sizes are shown, at the density $n=0.0625$. In each system, we average over 10 random twist angles. []{data-label="fig:nk"}](Fig4.eps) The Hubbard model, of interest in its own right, contains some of the same features (namely itinerant electrons and local interaction) as the continuum Stoner Hamiltonian. However, there are differences with respect to the experiment worth emphasizing. The experiment is in the continuum, using an attractive (negative) interaction with an effective positive scattering length for the excited state that describes the prepared state. (However, there are questions whether such an effective description is appropriate [@Pekker2010].) In our simulation, we use a discretized representation, with positive on-site interaction. As mentioned before, the lattice model leads to a scattering length bounded by roughly the lattice spacing. Using the above values for the maximum scattering length and $k_F$ in the unpolarized phase, we find that $k_F a < 1.03 n^{1/3}$. (For the transition to the ferromagnetic phase, $n \le 1$ since one cannot have two like-spin fermions on a single site. To focus on the dilute limit, we have done calculations for up to $n=0.5$.) Recently the problem of itinerant ferromagnetism in repulsive Fermi gases has been studied by several groups [@Conduit2009a; @Pilati2010; @Chang2010] using the DMC method with the fixed-node approximation. These calculations all found the existence of a ferromagnetic instability. The DMC calculations were all done in the continuum, while the present calculation is for the Hubbard (lattice) model. In the DMC simulations, the atomic interaction is modeled by a [*repulsive*]{} potential whose range is determined by the scattering length. We note that since the scattering length diverges near resonance, the range of potential (or the range of the node in the Jastrow when a negative interaction is used) can become very large. Note that the hard-sphere-like interaction is only between unlike spins. As the scattering length approaches the interparticle spacing, there is a strong tendency to separate into a spin-up and spin-down domains, to lower the interaction energy; i.e. it favors ferromagnetism. Of lesser importance, the DMC fixed-node errors in the calculated ground state energies bias the result in favor of ferromagnetism, since nodal surfaces for the ferromagnetic state are more accurate than the spin unpolarized state [@Zong2003]. Although the constrained path error from our calculations could also be biased, previous calculations indicate [@Zhang1997; @Zhang2003; @Purwanto2009] that the systematic error in CPMC is smaller than the fixed-node error from single determinant trial wave functions used in these calculations. [*Summary*]{} - We have examined the magnetic properties in the ground state of the dilute 3D Hubbard model, using the CPMC method and twist-averaged boundary conditions. Our simulation results indicate that there is no ferromagnetic instability in this model with strong on-site repulsions for densities up to $0.5$. The ground state appears to be a paramagnetic Fermi liquid. The total energy is effectively lowered by electron correlation which, while increasing the kinetic energy, can strongly suppress double occupancy to lower the interaction energy. In the presence of a trap, the kinetic energy can be decreased by the expansion of the gas due to repulsive interaction. A kinetic energy minimum, which was observed in the experiment, can be understood in terms of the competitions between these effects. We have also discussed the difference between our calculations and recent results from DMC simulations, as well as connections and differences with the Fermi gas itinerant ferromagnetism experiments. [*Acknowledgments*]{} - C.C. and S.Z. acknowledge support from ARO (56693-PH) and NSF (DMR-0535592) and D.M.C. from the OLE program. Computations were carried out at ORNL (Jaguar XT4) and William & Mary (CPD and SciClone clusters). We thank Jie Xu for help and T.L. Ho for useful discussions.
--- abstract: 'A quantum field theoretical approach, in which a quantum probe is used to investigate the properties generic non-flat FLRW space-times is discussed. The probe is identified with a conformally coupled massless scalar field defined on a space-time with horizon and the procedure to investigate the local properties is realized by the use of Unruh-DeWitt detector and by the evaluation of the regularized quantum fluctuations. In the case of de Sitter space, the coordinate independence of our results is checked, and the Gibbons-Hawking temperature is recovered. A possible generalization to the electromagnetic probe is also briefly indicated.' address: - | $^{(a)}$ Dipartimento di Fisica, Università di Trento,\ Via Sommarive 14, 38123 Povo, Italia - | $^{(b)}$ Centro INFN-TIFPA,\ Via Sommarive 14, 38123 Povo, Italia author: - 'Yevgeniya Rabochaya$\,^{(a)(b)}$[^1] and Sergio Zerbini$\,^{(a)(b)}$[^2]' title: 'Quantum detectors in generic non flat FLRW space-times' --- =5000 [*PACS numbers: 04.70.-s, 04.70.Dy*]{} Introduction ============ Hawking discovery of black hole radiation [@Haw] is considered one of the most important predictions of quantum field theory in curved space-time. The predicted effect is quite robust, see [@dewitt; @BD; @wald; @fulling; @igor; @U; @B]. Parikh and Wilczek [@PW](see also [@visser; @vanzo]) introduced a further approach, the so-called tunnelling method, for investigating Hawking radiation. Later, the Hamilton-Jacobi tunnelling method [@Angh; @tanu; @mann; @obr] was introduced. This method is covariant and can be generalized to the dynamical case [@bob07; @bob08; @sean09; @bob09; @bob10; @Vanzo12; @bin]. The aim of this paper is to continue the investigation of the local properties of a generic FLRW space-time by making use of suitable quantum probe, along the line presented in reference [@noi11]. We recall that FLRW space-times may be regarded spherically symmetric (dynamical) space-times, and a covariant formalism introduced by Hayward is at disposal [@sean09; @bob09; @kodama]. This approach will permit the introduction of Kodama vector and related observers. For the sake of completeness, first we review this general formalism. To begin with, let us recall that any spherically symmetric metric can locally be expressed in the form \[metric\] ds\^2 =\_[ij]{}(x\^i)dx\^idx\^j+ R\^2(x\^i) d\^2,i,j {0,1}, where the two-dimensional metric d\^2=\_[ij]{}(x\^i)dx\^idx\^j \[nm\] is referred to as the normal metric, $\{x^i\}$ are associated coordinates and $R(x^i)$ is the areal radius, considered as a scalar field in the two-dimensional normal space. Let us introduce the scalar quantity (x)=\^[ij]{}(x)\_i R(x)\_j R(x). \[sh\] The dynamical trapping horizon, if it exists, is located in correspondence of (x)\_H = 0, \[ho\] provided that $\partial_i\chi\vert_H \neq 0$. Hayward surface gravity associated with this dynamical horizon reads \_H=\_ R \_H. \[H\] In any spherical symmetric space-time there exists the Kodama vector field $\mathcal K$, defined by K\^i(x)=\^[ij]{}\_j R, K\^=0=K\^\[ko\] . Kodama observers are characterized by the condition $R=R_0$. Coming back to the Hamilton-Jacobi tunneling method, we recall the semi classical emission rate reads ||\^2 e\^[-2]{} . with $\Im$ standing for the imaginary part, the appearance of the imaginary part due to the presence of the horizon. The leading term in the WKB approximation of the tunnelling probability reads e\^[- ]{}, in which an energy $\omega$ of the particle, and the Hayward surface gravity evaluated at horizon appear. Here we recall the operational interpretation [@sean09]. We note that static observers in static BH space-times become in the dynamical case Kodama observers whose velocity v\^i\_K=, \_[ij]{}v\^i\_Kv\^j\_K=-1. Kodama observers are such that $R=R_0$, namely they have constant areal radius, and the energy measured by these Kodama observers at fixed $R_0$ is E=-v\^i\_K\_i I=-=. As a result, the tunneling rate can be rewritten as e\^[- E]{}e\^[-]{}, and the local quantity $T_0$ at radial radius $R_0$ is invariant, since it contains the invariant factor $\sqrt{\chi}$, and the Hayward surface gravity T\_0=, T\_H=. In the static case $\chi=g^{rr}=-g_{00}$ and recalling Tolman’s theorem: for a gravitational system at thermal equilibrium, $T\sqrt{-g_{00}}=\mbox{constant} $, it follows that $T_H=\frac{\kappa_H}{2\pi}$ is the intrinsic temperature of the BH, namely the Hawking temperature. As an important example that illustrates the gauge independence of the formalism is the de Sitter space-time. We shall consider three patches, or coordinate systems. The first is the static patch, namely ds\^2=-dt\_s\^2(1-H\_0\^2r\^2)++r\^2dS\^2. \[ds\] Here $R=r$, the horizon is located at $r_H=\frac{1}{H_0}$, and the surface gravity $\kappa_H=H_0$. The second patch is the one which describes a exponentially expanding FLRW flat space-time, ds\^2=-dt\^2+ e\^[2H\_0 t]{}(dr\^2+r\^2dS\^2 ). \[dsf\] Here $R=e^{H_0 t}r$, the dynamical horizon is $R_H=\frac{1}{H_0}$, there is no Killing vector, but the Hayward surface gravity is again $\kappa_H=H_0$. Finally, there exists the so called global dS patch, a non-flat FLRW space-time ds\^2=-dt\^2+ \^2( H\_0 t)(+r\^2dS\^2 ). \[dsc\] Here $R= \cosh( H_0 t) r$, and a straightforward calculation gives again $R_H=\frac{1}{H_0}$ and $\kappa_H=H_0$. In the dynamical case, but for slow changes in the geometry, the question is: could possibly the quantity $T_H=\frac{\kappa_H}{2\pi}$ be interpreted as a dynamical Hawking temperature? In our opinion a complete answer is still missing, see also [@bin] for a recent discussion. With regard to this issue, it should be important to have a quantum field theoretical confirmation of the tunnelling results (see, for example [@moretti]). As a first step toward this aim, we recall that in the flat FLRW case, a conformally coupled massless scalar has been used as a quantum probe in order to investigate such space-times (see [@obadia08; @noi11; @casadio] and references therein). The aim of this paper is to extend the investigation to the non flat FLRW space-times, since, at least in the case of De Sitter space-time, there exists the important example (\[dsc\]) of non flat FLRW space-time, namely the global de Sitter patch. The paper is organized as follows. In Section 2 we give a brief survey of a scalar quantization in a generic FLRW space-time. In Section 3, we present the main formula of Unruh-DeWitt response function detector. In Section 4, the computation of the renormalized quantum fluctuation is presented and the de Sitter case is discussed in detail. In Section 5, the conclusions are reported, and in the Appendix A, an elementary derivation of the Wightman function is given. Conformal quantum probe in FLRW space-times =========================================== In the following, we review the quantization of a conformally coupled massless scalar field in a generic FLRW space-time. First, it is convenient to introduce the conformal time $\eta$ by means $d\eta=\frac{d t}{a}$. Thus, we have $$ds^2=a^2(\eta)(-d\eta^2+d \Sigma_3^2)\,,\;, \label{cf}$$ where the metric of the spatial section may be written d\_3\^2= +r\^2dS\^2, with $k=0, \pm1$, and $h_0$ is a mass or inverse lenght scale, related to the Scalar Ricci curvature, which reads R=6( +k ), a’=. A useful and equivalent form for the spatial section is d\_3\^2= d\^2+ h\_k\^2()dS\^2, where $H_0 \xi= \sin^{-1} h_0 r$ and h()\_1=, h()\_0= , h()\_[-1]{}=. In the case of a free massless scalar field which is conformally coupled to gravity, the related Wightman function $W(x,x')=<\phi(x) \phi(x')>$ can be computed in an exact way. For the sake of completeness, we re-derive this well known result. The quantum field $\phi$ has the usual expansion (x)=\_f\_(x)a\_+f\^\*\_(x)a\^+\_ where the modes functions $f_{\alpha}(x)$ satisfy the conformally invariant equation ($\mathcal R$ being the Ricci curvature) $$\left(\Box -\frac{\mathcal R}{6}\right) f_{\alpha}(x)=0\;. \label{f}$$ Defining the vacuum by $a_{\alpha}|0>=0$, the Wightman function turns to be $$W(x,x')=\sum_{\alpha} f_{\alpha}(x)f^*_{\alpha}(x')\,, \label{w}$$ and it satisfies, with $x'$ fixed, $$\left(\Box -\frac{\mathcal R}{6}\right)W(x, x')=0\;. \label{f1}$$ An elementary derivation is presented in Appendix A. Here we give another derivation based on conformal invariance. First we recall that if we make a conformal transformation ds\^2=(x)\^2 ds\_0\^2, =\_0, the related Wightman $W(x.x')=<\phi(x)\phi(x')>$ transforms as W(x,x’)=W\_0(x.x’). \[wc\] We are interested in the non flat case. It is sufficient to consider only the $k=1$ case. The $k=-1$ may be obtained by the substitution $h_0 \rightarrow i h_0$. Recall that the metric on a non flat FLRW positive spatial curvature space-time is ds\^2=a\^2()( -d\^2+d\^2+\^2 h\_0dS\^2\_2 ). As a result, the above metric is conformally related to a static Einstein space $R \times S_3$, with metric ds\_E\^2= -d\^2+d\^2+\^2 h\_0dS\^2\_2 , On the other hand, it is well known that a static Einstein space-time is conformally related to Minkowski space-time, since ds\_E\^2=4\^2 (h\_0)4\^2 (h\_0)( -dt\^2+dr\^2+r\^2 dS\^2\_2 ) \[e\] with the Minkowski coordinates given by t r=(h\_0). Due to the homogeneity and isotropy of FLRW space-times, we may take $W(x,x')=W(x',x)=W(\eta-\eta', r-r')$, same radial separation. Thus, since the Minkowski Wightman function is known, making use of equations (\[wc\]) and (\[e\]), the Einstein space Wightman function turns out to be W\_E(x,x’)=. As a consequence, again making use of (\[wc\]), the Wightman function related to FLRW spherical spatial section is given by W(x,x’)=. Finally, the Wightman function related to FLRW hyperbolic spatial section can be obtained by the replacement $h_0\to ih_0$, and reads W(x,x’)=-. As a check, the Wightman function related to FLRW flat spatial section can be obtained taking the limit $h_0 \rightarrow 0$. Again for same radial separation W(x,x’)=. These results are in agreement with the ones obtained in Appendix A. The Unruh-DeWitt detector in non flat FLRW space-times ====================================================== The Unruh-DeWitt detector approach is a well known and used technique for exploring quantum field theoretical aspects in curved space-time. For a recent review see [@crispino]. Here, we review the basic formula following Ref. [@louko; @noi11]. The transition probability per unit proper time of the detector depends on the response function per unit proper time which, for radial trajectories, at finite time $\tau$ may be written as $\Delta \tau = \tau-\tau_0 >0$ (E,) =\_[0]{}\^ ds e\^[-i E s]{} W(, -s), where $\tau_0$ is the detector proper time at which we turn on the detector, and $E$ is the energy associated with the excited detector state (we are considering $E>0$). The flat case has been already considered in several places (see, [@noi11]), thus we shall consider the $k=1$ FLRW case, namely (E,) =\_[0]{}\^ where $\Delta \chi (s)=\chi(\tau)-\chi(\tau-s)$, and $\Delta \eta (s)=\eta(\tau)-\eta(\tau-s)$. The $i0$-prescription is necessary in order to deal with the second order pole at $s=0$. However, we will show in the next Section that the leading singularity in the coincidence limit, namely for small $s$, is W(, -s) -+ O(s\^0). As a result, one may try to avoid the awkward limit $\epsilon \rightarrow 0^+$ by omitting the $\epsilon$-terms but subtracting the leading pole at $s=0$ (see [@louko] for details), and introducing the quantity \^2(,s) a()a(-s)(h\_0((s))-h\_0 ((s))) , one can present the detector transition probability per unit time in the form (E,) =\_[0]{}\^ds ( E s) ( + ) + J\_(E) , \[L6\] where the ”tail” or finite time fluctuating term is given by J\_(E) := -\_\^ds. \[L00\] In the important stationary cases (examples are the static black hole, the FLRW de Sitter space), one has $\Sigma(\tau,s)^2=\Sigma^2(s)=\Sigma^2(-s)$, and Eq.  simply becomes (E,) =\_[-]{}\^ds e\^[-i E s]{}( +) + J\_=(E) +J\_(E) . \[Lr\] The first term is independent on $\tau$, and all the time dependence is contained only in the fluctuating tail. The de Sitter space in FLRW patches ----------------------------------- In order to check the coordinate independence (gauge-invariance) of our result, it is instructive to investigate the de Sitter space-time. With regard to the other two FLRW patches, they are the flat spatial section, physically relevant for inflation and the positive spatial curvature patch. The flat case has already been considered, see for example [@noi11]. For the spatial curved patch, we shall make use of the formula derived in the previous subsection. Recall that in a generic non- flat FLRW space-time, the Kodama observer is given by R(t)=R\_0= h\_0 , with constant $R_0$. For a radial trajectory, the proper time in the non-flat FLRW is d\^2=a\^2()(d\^2-d\^2). Thus, on the Kodama trajectory d \^2=dt\^2(1-). In the case of de Sitter, we put $h_0=H_0$ and the flat case is simple and one has an explicit expression for $\eta(\tau)$ [@noi11]. In the non-flat case, also for de Sitter, it is not easy to get an explicit expression of the conformal time as a function of $\tau$. For this reason we consider the $R_0=0$ case, comoving Kodama observer. Thus $d\tau=dt$. Furthermore, since $a(t)=\cosh H_0 t=\cosh H_0 \tau$, one has ()= e\^[H\_0 ]{}. We have to compute \[tetta\] \^2(,s)= (H\_0 ) H\_0 (-s)( ( H\_0 )-1 ) . \[i\] Making use of well known trigonometric identities, a direct calculation leads to following results H\_0 (, s)= -2 (), \^2 H\_0(,s)=, \^2(,s)= (H\_0 ) H\_0 (-s) \^2 (H\_0(,s)). As a consequence, the invariant distance (\[i\]) can be re-written in the form \[tetta1\] \^2(,s)=-\^2(). Since $\Sigma^2(\tau, s)= \Sigma^2(s)=\Sigma^2(-s)$, we may use (\[Lr\]) and obtain, for $E >0$ and making use of Residue Theorem (E)= , which shows that the Unruh-DeWitt detector in the non flat FLRW de Sitter space detects a quantum system in thermal equilibrium at a temperature $T_0=\frac{H_0}{2\pi}$, Gibbons-Hawking result is recovered [@GH]. This is an important check of the approach, since it shows the coordinate independence of the result for the important case of de Sitter space. The response function for unit proper time, in the stationary cases we have considered, gives information about the equilibrium temperature via the Planckian distribution. We also may argue as follow. In the stationary case, in the limit $\tau \rightarrow \infty$, one has (E)=\_[-]{}\^ds e\^[-i E s]{}( +)= . \[x\] Note also that in this case one has =e\^[-]{} ,(-E)-(E)=. Viceversa, if the above relations hold then $\dot{ F}(E)$ is the Plank distribution. Thus we may define the local equilibrium temperature by means of T\_0=. or T\_0= which shows in which sense a Unruh-DeWitt detector is a quantum thermometer. d-dimensional generalization ----------------------------- In the de Sitter case, it is possible to generalize the computation of the response function per unit time to the massive non conformally coupled d-dimensional scalar field [@proco]. In this case one may directly obtain \_[d,]{}(E)= |( ++ i ) ( -+i ) |\^2, \[dds\] where =, \[nu\] with $m^2$ mass of the scalar field and $\xi$ the coupling constant with the Ricci curvature, being the conformal coupling $\xi_c=\frac{d-2}{4(d-1)}$. As a consequence, one has \_[d,]{}(-E)= e\^ \_[d,]{}(E). Furthermore, the massless conformally coupled case in d dimension corresponds to $\nu=\frac{1}{2}$. In this case, for $d=4$ one recovers (E)= . Furthermore, making use of the identity (z)|\^2|(1-z)|\^2=, one also has for $d=3$ and $\nu=\frac{1}{2}$ \_[3,]{}(E)=(1+), in which the well known phenomenon of the inversion of the statistic for odd dimensional spaces is manifest. The other interesting case is the minimally coupled massless scalar field, for which $\nu=\frac{d-1}{2}$. In particular for $d=4$, one has \_[4,]{}(E)=(1+). In this physical important case, it mimics the graviton, the well known infrared problem associated with it appears in the bad behavior for small $E$. Quantum fluctuations ====================== Another proposal to detect local temperature associated with stationary space-time admitting an event horizon has been put forward by Buchholz and collaborators [@buchholz] (see also [@binosi; @Eme]). The idea may be substantiated by the following argument. Let us start with a free massless quantum scalar field $\phi(x)$ in thermal equilibrium at temperature $T$ in flat space-time. It is well known that finite temperature field theory effects of this kind may be investigated by considering the scalar field defined in the Euclidean manifold $S_1 \times R^3$, where one has introduced the imaginary time $\tau=-it$, compactified in the circle $S_1$, with period $\beta=\frac{1}{T}$ (see for example [@byt96]). Let us consider the local quantity $<\phi(x)^2>$, the quantum fluctuation. Formally this is a divergent quantity, since one is dealing with product of a valued operator distribution in the same point $x$, and regularization and renormalization are necessary. A simple and powerful way to deal with a regularized quantity is to make use of zeta-function regularization (see for example [@haw76; @eli94; @byt96], and references therein). Within zeta-function regularization, one has =(1|L\_)(x), \[M\] where $ \zeta(z|L_\beta)(x)$ is the analytic continuation of the local zeta-function associated with the operator $L_\beta$ L\_=-\^2\_-\^2, defined on $S_1 \times R^3$. The local zeta-function is defined with $\mbox{Re}\, z$ sufficiently large by means (z|L\_)(x)=\_0\^dt t\^[z-1]{}K\_t(x,x), \[M1\] where the heat-kernel on the diagonal is given by K\_t(x,x)=&lt;x|e\^[-tL\_]{}|x&gt; =\_n e\^[-n\^2]{} . In (\[M\]) the analytic continuation of the local $ \zeta(z|L_\beta)(x)$ appears and it is assumed that this analytical continuation is regular at $z=1$, which, as we shall see, is our case. If the analytic continuation has a simple pole in $z=1$, the prescription has to be modified (see [@iellici; @binosi]). A standard computation, which makes use of the Jacobi-Poisson formula leads to K\_t(x,x)=\_n e\^[-]{}. Let us plug this expression in (\[M1\]). The term $n=0$ leads to a formally divergent integral $\int_0^\infty dt (t^{z-3 })$, but this is zero in the sense of Gelfand analytic continuation, and it can be neglected. Thus, a direct computation gives the analytic continuation of the local zeta-function (z|L\_)(x)=()\^[z-2]{}\_R(4-2z), where $\zeta_R(z)$ is the Riemann zeta-function. It is easy to see that the analytic continuation of the local zeta-function is regular at $z=1$, and from (\[M\]), recalling that $\zeta_R(2)=\frac{\pi^2}{6}$, one has ==. \[M2\] Thus, the zeta-function renormalized vacuum expectation value of the observable $\phi^2$, the fluctuation, gives the temperature of the quantum field in thermal equilibrium, namely one is dealing with a quantum thermometer. Motivated by this argument, let us consider again a conformal coupled scalar field in a FLRW non flat space-time. We have seen that the off-diagonal Wigthman function is W(x,x’)=&lt;(x)(x’)&gt;=, where \^2(,-s)= a()a(-s)(-H\_0((s))+H\_0 ((s))) , with $a(\tau)$ being the conformal factor. In the limit $s \rightarrow 0$, formally one has =W(, ), but $W(\tau,\tau)$ is ill defined, and one has to regularize and then renormalize this object. In our case, we may make use of the simple point splitting regularization [@BD], namely we consider $W(\tau, \tau-s)$ and evaluate the limit $ s \rightarrow 0 $. To implement this limit procedure, one has to make use of several identities. For radial time-like separation, the starting point is a\^2()(\^2- \^2 )= 1. \[normaliz\] Taking first and second derivatives with respect the proper time, one has +a\^2( -)=0, and a\^2(-=-a\^2(\^2-\^2)-+3()\^2. Making use of these identities, a long but straightforward calculation leads to \^2=-s\^2(1+s\^2+O(s\^4) ). where B=H\^2+A\^2+2+(1-2\^2 ). In this expression $A^2$ is the square of the 4-acceleration along the time-like trajectory, given by A\^2=(+H( \^2-1) ). \[a\] In the above expression, the last term is the new one with respect the flat case. Thus, the point splitting gives W(,-s)=-++O(s\^2). With regard to the renormalization, we simply subtract the first divergent term for $s \rightarrow 0$. The physical meaning of this subtraction has been discussed in detail in reference [@noi11], and it amounts to subtract the contribution related to an inertial trajectory in Minkowski space-time. Thus, the renormalized quantum fluctuation reads \_R=(H\^2+A\^2+2+(1-2\^2 )). This result is the generalization of the one obtained in a flat FLRW space-time in [@noi11] and within Unruh-de Witt detector in [@obadia08]. As a first important check, let us consider again the de Sitter space-time in the global patch. In this stationary case, the fluctuation acts as a quantum thermometer. Recall that in this case, one has $h_0=H_0$, a(t)=H\_0 t ,H(t)=H\_0, = . As a consequence \_R=(H\_0\^2+A\^2). This is in agreement with the result obtained for the de Sitter space-time in reference [@thirr]. In fact, the acceleration can be computed, since, for a Kodama observer $R=R_0$, one has \^2-1=. Thus = H(t). Taking equation (\[a\]) into account, one gets A\^2=, which coincides with the Kodama de Sitter acceleration evaluated in the flat patch [@noi11]. Furthermore we also have \_R= . For a comoving Kodama observer $R_0=0$, and one gets again Gibbons-Hawking temperature associated with de Sitter space-time. Conclusion ========== In this paper, with the aim to better understand the temperature-versus-surface gravity paradigm, the asymptotic results obtained by tunnelling semi classical method have been tested with quantum field theory techniques like the Unruh-DeWitt detector and the evaluation of the quantum fluctuation associated with a massless conformally coupled scalar field. More precisely the results obtained in reference [@noi11] have been extended to a generic non spatial flat FLRW space-time. The de Sitter space-time in the global non flat FLRW patch has been used as important example, and the Gibbons-hawking temperature for the de Sitter space has been re-derived with our general fluctuation formula as well as the Unruh-de Witt detector technique. With regard to possible generalization, at least in the flat case, our approach may also be extended to the Maxwell field as soon as one makes use of the result obtained in reference [@re]. In fact there, quantizing the Maxwell field in a flat FLRW space-time and in the so called W gauge, a conformal lifting of the Lorenz gauge in Minkowski space-time, the Maxwell Wightman function has been obtained in the form W\_(x,x’)=-( + )W(x,x’), where $W(x,x')$ is the Whightman associated with massless conformally coupled scalar field in flat FLRW space-time. This strongly suggests that our conformally coupled scalar probe may mimic quite well the quantum Maxwell field. Of course, working with the Maxwell field, the gauge invariant has to be implemented and the relevant quantity is the Wightman function associated to (for example) the magnetic field. This is a very interesting issue with important cosmological applications, see for example the recent discussion appeared in [@campa; @du], and we hope to consider this issue in a future work. Acknowledgement {#acknowledgement .unnumbered} =============== We thank G. Cognola, M. Rinaldi, L. Vanzo for several discussions. Appendix A ========== In this Appendix an elementary derivation of Wightman function for a massless conformally coupled scalar field is presented. Due to the homogeneity and isotropy of FLRW space-times, we may take $W(x,x')=W(x',x)=W(\eta-\eta', r-r')$. It is convenient to introduce the auxiliary quantity Y(x,x’)=a()a(’)W(x,x’), and, from the equation of motion in FLRW space-time with the conformal time one has --kH\^2\_0Y+\_h Y=0, where $\Delta_h$ is the Laplace operator associated with the metric $d\Sigma_3^2$. We may take $x'=0$. Let us start with the flat case $k=0$. One has -++2 =0. The solution is Y= As a result, making use of the homogeneity and isotropy, and dealing with the distribution nature of $W$, $$W(x,x')= \frac{1}{4\pi^2 a(\eta)a(\eta')}\,\frac{1}{(r- r')^2-(\eta-\eta'-i\epsilon)^2 }\,. \label{wold}$$ Here we leave understood the limit as $\epsilon\to 0^{+}$. We may rewrite it in covariant form, according to Takagi [@T] and Schlicht [@S] We adapt Schlicht’s proposal to our conformally flat case, namely $$W(x,x')=\frac{1}{4\pi^2 a(\eta)a(\eta')}\,\frac{1}{[( x- x')-i\epsilon (\dot{x}+\dot{x}')]^2 }\,. \label{w3}$$ where an over dot stands for derivative with respect to proper time (see also [@noi11]). 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--- abstract: 'This report overviews a set of recent contributions in the field of path planning that were developed to enable the realization of the autonomous aerial robotic worker for inspection operations. The specific algorithmic contributions address several fundamental challenges of robotic inspection and exploration, and specifically those of optimal coverage planning given an a priori known model of the structure to be inspected, full coverage, optimized and fast inspection path planning, as well as efficient exploration of completely unknown environments and structures. All of the developed path planners support both holonomic and nonholonomic systems, and respect the on–board sensor model and constraints. An overview of the achieved results, followed by an integrating architecture in order to enable fully autonomous and highly–efficient infrastructure inspection in both known and unknown environments.' author: - 'Kostas Alexis[^1]' title: '**Realizing the Aerial Robotic Worker for Inspection Operations** ' --- INTRODUCTION {#sec:intro} ============ The vision of automated infrastructure monitoring and damage detection corresponds to the motivation of a body of work conducted by our research team and aims to develop the “Aerial Robotic Worker” (ARW) as a class of systems that –among others– are able to autonomously conduct structural inspection operations. Infrastructure is the foundation that connects our resources, energy flows, communities, and people, driving our economy and improving our quality of life. For our economies to be sustainable, a first class infrastructure system, in terms of quality, distribution, long–term operation, systematic inspection and maintenance is required. Within recent years, the scientific breakthroughs and technological developments in the area of civilian mobile robotics have progressively brought robotics closer to real–life challenging applications, and pioneering use cases have already shown a very promising potential. Indicative scenarios include those of monitoring of bridges, solar farms, power generation and distribution facilities, geothermal facilities, oil & gas industry infrastructure, water dams and more. As indicated in several reports such as the one provided by the American Society of Civil Engineers (ASCE) [@nsf_asce_card2013], our infrastructure is under the urgent need for a more systematic approach for the relevant inspection operations due to its degraded condition. Characteristically, the US dams are graded with $D$ by ASCE and among the $84000$ dams, $14000$ are characterized as high–hazard, while their average age is $52$ years. The $607380$ U.S. bridges are graded with $C+$, have an average age of $42$ years and $1$ out of $9$ of them are rated as structurally deficient. ![The concept of aerial robotic workers aims to realize a class of aerial robotic systems capable of autonomous inspection of infrastructure and other critical facilities. In the images depicted, instances from a large–scale simulation–based exploration scenario as well as photo and data from an experimental study on autonomous exploration are depicted. []{data-label="fig:intro"}](intro_arw_iros.eps){width="0.99\columnwidth"} But to automate the inspection process, an aerial robot should be something much more than a position–controlled camera in the sky. To develop the aerial robotic worker for inspection operations, significant challenges in perception and path planning have to be addressed. Within the framework of this work, we provide an overview of a set of algorithms that were recently proposed by our team and break new ground on how an aerial robot handles efficient exploration of unknown environments, volumetric mapping and full coverage, high fidelity structural 3D reconstruction and as a last step, possibly required contact–based inspection. The proposed algorithms specifically address the problems of a) optimal full coverage of a structure for which a known geometric model exists, b) fast, optimized (but suboptimal) full coverage inspection of structures with known geometric model, c) uniform coverage of such structures, d) autonomous exploration and inspection of completely unknown environments and the structures in them, and finally e) contact–based inspection of selected points of interest on the $3\textrm{D}$ structure. Each algorithm has its advantages and disadvantages, while an overall architecture is proposed such that their combination can lead to the fully autonomous execution of inspection tasks in either known or unknown environments with selective levels of computational cost and inspection precision. It is noted that, all the proposed methods have been experimentally verified. Figure \[fig:intro\] presents a subset of the relevant results. Within this work, we present highlights of the previously derived results but also new studies, including multi–robot exploration studies, and derive conclusions regarding the specific role of each algorithm and the considered future research directions. The report structure is as follows. In Section \[sec:sec2\] each of the exploration and inspection methods is overviewed. A discussion on the role of each method takes place in Section \[sec:sec4\], followed by the description of an architecture for their combination. Finally conclusions and remarks on future work takes place in Section \[sec:sec5\]. EXPLORATION & INSPECTION STRATEGIES {#sec:sec2} =================================== A set of algorithms addressing the problems of inspection and exploration were proposed and are summarized below. Optimal Inspection Planning --------------------------- In the literature, many contributions have been made towards addressing the challenges of coverage planning. Within the most recent contributions, those that employ a two–step optimization scheme proved to be more versatile with respect to the inspection scenario. In a first step, such algorithms compute the minimal set of viewpoints that cover the whole structure which corresponds to solving an Art Gallery Problem (AGP). As a second step, the shortest connecting tour over all these viewpoints has to be computed, which is the Traveling Salesman Problem (TSP). However, the general approach of breaking the problem of finding full coverage paths into that of finding a minimal set of viewpoints and only afterwards perform tour optimization does not –in general– guarantee inspection path optimality. Furthermore, in cases of nonholonomic vehicles and presence of obstacles, such methods can also lead to overall infreasible solutions. To overcome these limitations, a new algorithm called Rapidly–exploring Random Tree Of Trees (RRTOT) was proposed and employs sampling–based methods and a meta–tree structure consisting of multiple RRT$^{\star}$–trees to find admissible paths with decreasing cost. Using this approach, RRTOT does not suffer from the limitations of strategies that seperate the problem into that of finding the solution to the AGP and afterwards solving the derived TSP. Essentially, RRTOT relies on a sampling–based algorithm that generates trees that root from previously random sampled vertices of other trees. In that sense, this tree–of–trees structure (which is essentially a connected forest) has the capacity to arbitrarily vary coverage path topologies and from that ensures that the optimal solution can be found. In our relevant paper [@bircher_robotica], the principles of incremental solution derivation and asymptotic optimality are proven. The algorithm supports both holonomic as well as nonholonomic systems, and also further accounts for sensor model constraints. Figure \[fig:rrtot\_res\] presents one of the relevant experimental results for the case of a multirotor aerial robot. The recorded flight can be found at <https://youtu.be/e7ljyDM9h8o>. ![Indicative inspection path planning experimental result using the RRTOT method for the case of a hexacopter aerial robot considered to be flying holonomic paths. The path length is $9.1\textrm{m}$ and it was computed after several minutes of operation of the RRTOT algorithm. Video: `https://youtu.be/e7ljyDM9h8o` []{data-label="fig:rrtot_res"}](RRTOT_HOLONOMIC_vS.eps){width="0.99\columnwidth"} Efficient Structural Inspection Planning ---------------------------------------- Aiming towards *efficient* derivation of full coverage, admissible, optimized although not necessarily true optimal inspection path planning, our team further proposed the Structural Inspection Planner (SIP) [@SIP_AURO_2015; @BABOOMS_ICRA_15], an algorithm that retains a two–step optimization paradigm but contrary to trying to find a minimal set of viewpoints in the AGP, it rather tries to sample them such that the connecting path is short while ensuring coverage. This is driven by the idea that with a continuously sensing sensor, the number of viewpoints (and if this is minimal or not) is not necessarily important but mostly their configuration in space, which has to be such that short and full coverage paths are provided. To achieve its goal, SIP iterates between a step that samples a new set of viewpoint configurations and a second step within which it computes collision–free paths and performs tour optimization. One viewpoint is sampled for each subset of the structure (e.g. for each face of a mesh representation of the structure) and a convex optimization technique ensures the visibility of the associated subset of the overall manifold. As the algorithm iteratively executes these two steps, it manages to find improved solutions - a goal that is further assisted by a set of heuristics [@SIP_AURO_2015]. Both holonomic and nonholonomic vehicles are supported, the constraints of the sensor model are respected, while implementations for mesh–based and octomap–based representations of the structure are available. The overall implementation is open–sourced [@kOpt_CodeRelease] and available as a Robot Operating System (ROS) package. Figure \[fig:sip\_res\] presents an indicative inspection and $3\textrm{D}$ reconstruction result. The recorded flight video can be found at <https://youtu.be/5kI5ppGTcIQ>. ![Experimental study of the inspection of a subset of the ETH Polyterrasse truncated cones using SIP. The inspection path was computed based on a rough CAD model and the polyhedric obstacle was also included. The path cost is $167.3\textrm{s}$ for maximum forward velocity of $0.25\textrm{m/s}$ and maximum yaw rate equal to $0.5\textrm{rad/s}$. Video: `https://youtu.be/5kI5ppGTcIQ`[]{data-label="fig:sip_res"}](Bluebird_Polyterrasse_v2.eps){width="0.9\columnwidth"} Uniform Coverage Inspecetion Planning ------------------------------------- In real–life inspection operations, uniform coverage with equal focus on the details is one of the often desired properties. To aproach this problem, an algorithm that exploits the uniformity properties of Voronoi–based meshing techniques was proposed. The specific method of “uniform coverage $3\textrm{D}$ structural inspection path planning” (UC$3$D) iteratively loads lower–fidelity meshes of the structure to be inspected (by subsampling), computes a set of viewpoints with each one of them ensuring the inspection of one of the faces from a similar distance and perceiving angle and finds the optimal tour among them. Viewpoint derivation is achieved by randomly sampling within the subset of the configuration space that allows “uniform” inspection, while path feasibility is supported by verifying connectivity with the neighboring viewpoints subject to any nonholonomic constraints. As long as the algorithm cannot find an overall feasible solution at one of its iteration, the process is repeated until randomization leads to solution feasibility. Figure \[fig:uc3d\_res\] presents an indicative experimental result, while the recorded flight video is available at <https://youtu.be/Gg9qsF3y8IU>. ![Indicative uniform coverage inspection path planning experimental result using the UC$3$D method using a quadrotor aerial robot that relies on a monocular camera/RGB–D localization and mapping pipeline. The power transformed mesh has been subsampled to $134$ faces and $69$ vertices, while the camera mounting is considered to be with 15 degrees pitch down and the minimum inspection distance is set to $0.35\textrm{m}$. Video: `https://youtu.be/Gg9qsF3y8IU` []{data-label="fig:uc3d_res"}](uc3d_exp.eps){width="0.99\columnwidth"} Autonomous Exploration and Localizability ----------------------------------------- Autonomous exploration planning refers to the capacity of a robot to map a previously unknown environment. Early work includes [@connolly1985determination], where good “next–best–views” are determined in order to cover a given structure. Advanced versions were recently presented [@vasquez2014volumetric], while the method of frontiers–based planning corresponds to one of the most widely used exploration strategies. Within our work in [@NBVP_ICRA_16; @bircher2016receding], a receding horizon approach to the problem of Next–Best–Ciew Planning (NBVP) is proposed and experimentally verified. The views are sampled as nodes in a random tree, the edges of which directly give a path to follow such that the viewpoints are sequentially reached. At every step, a finite–depth tree of views is sampled but only the first step is executed by the robot, while the whole process is repeated at the next iteration. This receding horizon strategy improves and robustifies the exploratory behavior of the robot. Figure \[fig:nbvp\_res\] presents an indicative experimental result. It is noted that this planner is also open–sourced [@nbvpCodeRelease] and accompanied by an open dataset [@nbvp_DatasetRelease]. A relevant experiment is recorded and is available at <https://youtu.be/D6uVejyMea4>. ![Exploration experiment in a closed room. The colored voxels (color selected based on height) represent occupied parts of the occupancy map. The computed path is shown with black color, while the experimentally recorded path of the robot is shown with light blue. Video: `https://youtu.be/D6uVejyMea4` []{data-label="fig:nbvp_res"}](nbvp_leo_v2_vS.eps){width="0.99\columnwidth"} Furthermore, in our work in [@RHEM_ICRA_2017], the problems of autonomous exploration and robot localizability are addressed together. In particular, a localization uncertainty–aware Receding Horizon Exploration and Mapping (RHEM) planner is proposed. The RHEM planner relies on a two–step, receding horizon, belief space–based approach. At first, in an online computed random tree, the algorithm identifies the branch that optimizes the amount of new space expected to be explored. The first viewpoint configuration of this branch is selected, but the path towards it is decided through a second planning step. Within that, a new tree is sampled, admissible branches arriving at the reference viewpoint are found and the robot belief about its state and the tracked landmarks is propagated. As system state the concatenation of the robot states and tracked landmarks (visual features) is considered. Then, the branch that minimizes the localization uncertainty, as factorized using the D–optimality (D–opt) of the pose and landmarks covariance is selected. The corresponding path is conducted by the robot and the process is iteratively repeated. Figure \[fig:rhem\_steps\] illustrates the basic steps of this planner. Figure \[fig:arena\_experiment\] presents indicative experimental results and the video in <https://youtu.be/iveNtQyUut4> demonstrates the overall experiment. ![2D representation of the two–steps uncertainty–aware exploration and mapping planner. The first planning layer samples the path with the maximum exploration gain. The viewpoint configuration of the first vertex of this path becomes the reference to the second planning layer. Then this step, samples admissible paths that arrive to this configuration, performs belief propagation along the tree edges, and selects the one that provides minimum uncertainty over the robot pose and tracked landmarks. The video in `https://youtu.be/iveNtQyUut4` presents the overall experiment.[]{data-label="fig:rhem_steps"}](planning_layers_explanation.eps){width="0.99\columnwidth"} ![nstances of an exploration and mapping experiment in a closed room with a challenging geometry. The initial phase of the exploration is dominated by yawing motions. Especially when long paths are selected, the second planning layer identifies alternative paths that optimize the robot belief. Furthermore, as shown the probabilistic backend of octomap is maintained to allow the computation of the $\mathbf{ReobservationGain}$, while during belief propagation, visibility check for the tracked landmarks takes place. The result is a consistent $3\textrm{D}$ map despite the size and the challenges of the environment.[]{data-label="fig:arena_experiment"}](compiled_result_fig_v2.eps){width="0.99\columnwidth"} Contact–based Inspection ------------------------ Given that a structure is inspected and its $3\textrm{D}$ reconstruction has been succesfully derived, a next possible step within an infrastructure monitoring application may require contact–based inspection to conduct non–destructive testing for structural integrity aspects such as gas pipe wall thickness or measurements. These processes are conducted using sensors such as ultrasound probes which require physical contact with the structure. This fact, motivated the research efforts of our team and of the commmunity to address the problem of flight control during physical interaction. In response to this need, we developed a Hybrid Model Predictive Control (HMPC) approach that relies on a hybrid model of the aerial robot dynamics using essentially different dynamic modes during the free–flight and physical interaction phases of the operation [@DABS_ICRA_14; @AHS_ICRA_13]. The HMPC approach ensures stability during the mode switching, robust execution of physical interaction tasks and high–performance free–flight. Building on top of this capacity, a framework that allows the user to select a set of points to be inspected on the physical surface and then finding the optimal route among them was proposed [@ADBS_AURO_2015] and the overall method is called Contact–based Inspection Planning and Control (CIPC). This framework also allows to overcome an obstacle on the physical surface also by undocking from it and re–docking at the next point of interest. Figure \[fig:cip\_res\] presents an indicative result using a multicopter aerial robot, while the recorded flight video is available at <https://youtu.be/lDpHNEB66wE>. ![Contact–based inspection mission using the CIPC strategy. The robot has to visit the specified points of interest while avoiding any obstacles of the environment. As shown “obstacle” areas have been attached on the wall. The CIPC strategy successfully establishes contact and subsequently executes the optimized in–contact inspection path. Video: `https://youtu.be/lDpHNEB66wE` []{data-label="fig:cip_res"}](CIP_Env1_Data1d_vS.eps){width="0.80\columnwidth"} A UNIFYING ARCHITECTURE {#sec:sec4} ======================= A planning ensemble for inspection and exploration has been proposed and experimentally verified towards realizing the aerial robotic worker for inspection operations. Among the three algorithms for inspection, RRTOT is characterized by optimality but very expensive computations, UC$3$D focuses primarily on uniform coverage, while SIP provides a rather balanced solution characterized by optimized cost and limited computational cost and needs. All these three algorithms require that a geometrical model of the structure to be inspected is known a priori. On the contrary, NBVP and RHEM assume no prior knowledge of the environment and enable its autonomous exploration. While RRTOT, SIP and UC$3$D are global planners, NBVP and RHEM are local planning solutions that reactively compute the next–best–viewpoint of the robot given its online computed $3\textrm{D}$ reconstruction of the previously unknown environment. The RHEM planner goes further to identify the trajectory that visits the best exploration viewpoint while maintaining low localization uncertainty. Given the different role and features of these algorithms, they can correspond to a relatively complete, real–life, structural inspection solution through their combination. Figure \[fig:planning\_ensemble\] presents the proposed architecture for the utilization of the aerial robotic workers inspection planning ensemble. ![Proposed architecture for the combination of the different structural inspection and exploration algorithms of the proposed planning ensemble towards a complete solution that addresses the key end–user requirements regarding the operation in known or unknown environments, as well as the different operation requirements regarding the data to be used for the $3\textrm{D}$ reconstruction process. []{data-label="fig:planning_ensemble"}](ariw_current_architecture.eps){width="0.99\columnwidth"} OPEN SOURCE CONTRIBUTIONS {#sec:sec4} ========================= To accelerate the utilization of autonomous exploration and inspection technologies, support the community developments and overall lead to living contributions, a subset of these algorithms have been opens-sourced. This refers to the SIP planner [@kOpt_CodeRelease], NBVP [@nbvpCodeRelease] and RHEM [@rhemCodeRelease]. All repositories are also accompanied by experimental datasets. Further relevant data can be found in [@Oettershagen_FSR2015]. CONCLUSIONS & FUTURE WORK {#sec:sec5} ========================= A planning ensemble that enables the realization of the autonomous aerial robotic worker for inspection operations is presented. The set of algorithms contains solutions for the optimized inspection given a prior geometric model of the structure, as well as fully autonomous solutions that are futher localization belief uncertainty–aware. A summary of the functioning principle of each algorithm is presented, in combination with characteristic results and discussion on its main properties. Finally, an architecture of their combination is presented in order to solve real–life infrastructure inspection challenges for which a previous model might –or might not– be available. [10]{} \[1\][\#1]{} url@rmstyle \[2\][\#2]{} , “[ASCE 2013 Report Card on America’s Infrastructure]{}.” \[Online\]. Available: <http://www.infrastructurereportcard.org/> , “An incremental sampling-based approach to inspection planning: The rapidly-exploring random tree of trees,” 2015. , “,” **, pp. 1–25, 2015. , “Structural inspection path planning via iterative viewpoint resampling with application to aerial robotics,” in *IEEE International Conference on Robotics and Automation (ICRA)*, May 2015, pp. 6423–6430. \[Online\]. Available: <https://github.com/ethz-asl/StructuralInspectionPlanner> , “[Structural Inspection Planner Code Release]{}.” \[Online\]. Available: <https://github.com/ethz-asl/StructuralInspectionPlanner> C. Connolly *et al.*, “The determination of next best views,” in *Robotics and Automation. Proceedings. 1985 IEEE International Conference on*, vol. 2.1em plus 0.5em minus 0.4emIEEE, 1985, pp. 432–435. J. I. Vasquez-Gomez, L. E. Sucar, R. Murrieta-Cid, and E. Lopez-Damian, “Volumetric next best view planning for 3d object reconstruction with positioning error,” *Int J Adv Robot Syst*, vol. 11, p. 159, 2014. , “Receding horizon “next-best-view” planner for 3d exploration,” in *IEEE International Conference on Robotics and Automation (ICRA)*, May 2016. \[Online\]. Available: <https://github.com/ethz-asl/nbvplanner> A. Bircher, M. Kamel, K. Alexis, H. Oleynikova, and R. Siegwart, “Receding horizon path planning for 3d exploration and surface inspection,” *Autonomous Robots*, pp. 1–16, 2016. , “[Receding Horizon Next Best View Planner]{}.” \[Online\]. Available: <https://github.com/ethz-asl/nbvplanner> , “[Receding Horizon Next Best View Planner Dataset]{}.” \[Online\]. Available: <https://github.com/ethz-asl/nbvplanner/wiki/Example-Results> , “Uncertainty–aware receding horizon exploration and mapping using aerial robots,” in *IEEE International Conference on Robotics and Automation (ICRA)*, May 2017. \[Online\]. Available: <https://github.com/unr-arl/rhem_planner> G. Darivianakis, K. Alexis, M. Burri, and R. Siegwart, “Hybrid predictive control for aerial robotic physical interaction towards inspection operations,” in *Robotics and Automation (ICRA), 2014 IEEE International Conference on*, May 2014, pp. 53–58. K. Alexis, C. Huerzeler, and R. Siegwart, “Hybrid modeling and control of a coaxial unmanned rotorcraft interacting with its environment through contact,” in *2013 International Conference on Robotics and Automation (ICRA)*, Karlsruhe, Germany, 2013, pp. 5397–5404. K. Alexis, G. Darivianakis, M. Burri, and R. Siegwart, “,” **, pp. 1–25, 2015. \[Online\]. Available: <http://dx.doi.org/10.1007/s10514-015-9485-5> , “[Uncertainty–aware Receding Horizon Exploration and Mapping Planner]{}.” \[Online\]. Available: <https://github.com/unr-arl/rhem_planner> , “Long-endurance sensing and mapping using a hand-launchable solar-powered uav,” June 2015. [^1]: The author is with the Autonomous Robots Lab, University of Nevada, Reno, 1664 N. Virginia Street, Reno, NV 89557, USA [[email protected]]{}
--- abstract: 'We propose several novel physical phenomena based on nano-scale helical wires. Applying a static electric field transverse to the helical wire induces a metal to insulator transition, with the band gap determined by the applied voltage. Similar idea can be applied to “geometrically" constructing one-dimensional systems with arbitrary external potential. With a quadrupolar electrode configuration, the electric field could rotate in the transverse plane, leading to a quantized dc charge current proportional to the frequency of the rotation. Such a device could be used as a new standard for the high precession measurement of the electric current. The inverse effect implies that passing an electric current through a helical wire in the presence of a transverse static electric field can lead to a mechanical rotation of the helix. This effect can be used to construct nano-scale electro-mechanical motors. Finally, our methodology also enables new ways of controlling and measuring the electronic properties of helical biological molecules such as the DNA.' author: - 'Xiao-Liang Qi$^{1,2}$ and Shou-Cheng Zhang$^{1}$' bibliography: - 'chargepumping.bib' title: 'Field-induced Gap and Quantized Charge Pumping in Nano-helix' --- Helical nanostructures occur ubiquitously in self-assembled form in both inorganic materials[@zhang2002; @mcilroy2001; @zhang2003; @yang2004] and in the biological world[@xu2004; @endres2006]. In this paper, we propose several novel physical phenomena based on nano-scale helix wires. Firstly, when a uniform electric field is applied perpendicular to the helical direction, the electrons moving in the nanowire experience a periodical potential due to the potential energy difference. Consequently, such a uniform electric field can induce a gap in the electron energy spectrum, which drives the nanowire from a metallic state to an insulating state if the electron density is commensurate. The general principle behind such a simple phenomenon is that a uniform electric field can generate a nonuniform potential acting on a quantum wire if the quantum wire itself has a curved shape. More generally, one can obtain a quasi one-dimensional system in an arbitrary electric potential by applying an uniform electric field to a quantum wire with proper shape. Recent advances in nano-technology enables this design principle. A more interesting phenomenon occurs when the applied electric field is slowly rotated. When the system has a commensurate filling and stays in the insulator state, a slow enough rotation of the electric field satisfies the adiabatic condition and the system will stay in the instantaneous ground state. During each period $T$ of the electric field rotation, integer number of charge will be pumped through the nanohelix, thus generating a quantized charge current. In such a way, the nanohelix in a rotating electric field provides a new realization of the quantum charge pumping effect proposed by Thouless[@thouless1983]. The principle behind this charge pumping effect is exactly the quantum analog of the celebrated Archimedean screw invented more than two millenniums ago. By making use of such an effect, one can design a quantized and controllable current source. The reverse effect can also be studied, leading to the possibility of a quantum nano-motor driven by electric current. ![(a) Schematic picture of a nanohelix. (b) The definition of length scales $d,R,P,L$ shown for one pitch period of a helical wire. \[schematic\]](fig1.eps){width="3in"} ![(a) Definition of coordinates and direction of transverse electric field. (b) Lowest 4 subbands under transverse field $eER=0.02\hbar^2/2ma^2$ and $L=30a$, with $a$ being the lattice constant. The dashed line shows the band structure in the reduced zone scheme when $E=0$.[]{data-label="chiralwire"}](fig2.eps){width="3in"} To begin with, we consider a helical wire (as shown in Fig. \[schematic\]) where the diameter of the wire is $d$, the radius of the helix is $R$, the helical angle is $\alpha$, the pitch length of the helix is $P=2\pi R\tan\alpha$ and the net length of one period is $L=2\pi R/\cos\alpha$. The carrier density of the wire is denoted by $n$. In the present work, we will focus on the case when the helical wire is sufficiently thin so that the electron motion can be considered as one dimensional, and then discuss later in detail the conditions imposed on the one-dimensionality. In the continuum limit, the one-dimensional Hamiltonian of a single electron in the nano-helix is simply written as $$\begin{aligned} H_0=-\frac{\hbar^2}{2m}\partial_\rho^2\label{H0}\end{aligned}$$with $m$ the effective mass and $\rho$ the length coordinate of the helix. When a transverse electric field is applied, a potential energy term is induced in the Hamiltonian. To write it down explicitly, one can define the cylindrical coordinate system $(r,\theta,z)$ as shown in Fig. \[chiralwire\] (a), with the $z$ axis defined as the axis of the helix. The coordinate of a point with length coordinate $\rho$ is $$\begin{aligned} r=R,\text{ }\theta=\frac{2\pi \rho}{L}({\rm mod}2\pi),\text{ }z=\rho\sin\alpha.\end{aligned}$$ [*—The Quantum Helical Transistor (QHT).*]{} When a transverse uniform electric field ${\bf E}=E\left(\cos\phi\hat{\bf x}+\sin\phi\hat{\bf y}\right)$ is applied, as shown in Fig. \[chiralwire\] (a), the potential energy is given by $$\begin{aligned} H_{\rm el}=e{\bf E\cdot r}(\rho)=eER\cos\left(\frac{2\pi \rho}L-\phi\right).\label{Hamiltonian}\end{aligned}$$ Thus the combined single electron Hamiltonian takes the form of $H=H_0+H_{\rm el}$. In this case, the continuous quadratic energy spectrum will split into energy bands, with first Brillouin zone $k\in[-\pi/L,\pi/L)$. The typical band structure is shown in Fig. \[chiralwire\] (b). The gap between the $n$th and $(n+1)$th bands can be calculated by perturbation theory as $E_{g}(n)\propto eER\left(eER/E_0\right)^{n-1}$ in the limit $eER\ll E_0\equiv \hbar^2\pi^2/2mL^2$. For concreteness, we shall focus on the first gap $n=1$, since it is the largest, corresponding to $E_g\simeq eER$. The transverse electric field can be generated by applying a voltage $V_g=V_{g1}-V_{g2}$ on the gate electrodes as shown in Fig. \[switch\] (a). Assuming that the helical wire just fits into the gate electrodes, the resulting electric field is $E=V_g/2R$, and therefore the band gap is simply given by $E_g\simeq eV_g/2$, which is independent of the radius $R$ of the helix. On the other hand, the average potential $V_a=(V_{g1}+V_{g2})/2$ relative to the source-drain potential $(V_s+V_d)/2$ can be used to tune the chemical potential of the wire and thus the electron density $n$. When the chemical potential lies in the first gap, the system is an insulator and the corresponding one-dimensional filling fraction is $n_{\rm 1d}=2/L$, that is, two electrons per helical period. The factor of two arises from the spin degeneracy. Since the system with such a filling is gapless and conducting before applying electric field, the transverse electric field leads to a metal-insulator transition in the nano-helix, and thus defines a new type of nano-scale transistor switch, the status of which is “on" when the electric field is turned off, and “off" when electric field is turned on, as illustrated in Fig. \[switch\] (a). Such a switch can work under a source-drain voltage $V_{sd}<V_g/2$, so that the chemical potential of both leads lie inside the gap. ![(a) Illustration of the on and off state of the nano-helix switch. The gate voltage is given by $V_g=V_{g1}-V_{g2}$ and source-drain voltage is given by $V_{sd}=V_s-V_d$. (b) DC conductance of the nano-helix connected with two metallic leads. The blue and red line stands for the case with $eER=0$ and $eER=0.2t$, respectively, where the error bar stands for the impurity effect with the impurity potential $W=0.05t$. The dashed line is the conductance without impurity under the same electric field $eER=0.2t$. The arrow marks the first gap induced by electric field. All the calculations are done under the temperature $T=0.01t$ for the helix system with $N=200,~L=20,~\Gamma/2\pi=0.1t$. The inset shows the temperature dependence of conductance with the electric field $eER=0$ (blue line) or $eER=0.2t$ (red line).[]{data-label="switch"}](fig3.eps){width="2.8in"} To make our discussions here and below more concretive, we can regularize the Hamiltonian (\[H0\]) to a tight-binding model: $$\begin{aligned} H_{\rm TB}&=&-t\sum_{i=1}^{N-1}\left(c_i^\dagger c_{i+1}+h.c.\right)\label{Htb}\\ & &+eER\sum_{i=1}^N\cos\left(\frac{2\pi a}{L}i-\phi\right)c_i^\dagger c_i+\sum_{i=1}^N\epsilon_ic_i^\dagger c_i\nonumber\end{aligned}$$ in which $a$ is the lattice constant and $N$ is the total number of sites. The last term stands for quenched disorder, with the random potential $\left\langle{\left\langle{\epsilon_i\epsilon_j}\right\rangle}\right\rangle=\delta_{ij}W^2$. To study the transport properties in such a mesoscopic system, we need to include the effect of leads, which in this one-dimensional model can be described by the terms below: $$\begin{aligned} H_{\rm Lead}&=&\frac{V}{\sqrt{\Omega}}\sum_{\bf k}\left(c_{1}^\dagger a_{L{\bf k}}+c_{N}^\dagger a_{R{\bf k}}+h.c.\right)\nonumber\\ & &+\sum_{{\bf k}}\sum_{\alpha=L,R}\left(\epsilon({\bf k})-\mu_\alpha\right)a_{\alpha {\bf k}}^\dagger a_{\alpha {\bf k}}\label{Hlead}\end{aligned}$$ with $a_{L{\bf k}}$ and $a_{R{\bf k}}$ stands for the annihilation operators of electron in the left and right leads, respectively. Then the conductance can be calculated by[@landauer1957; @meir1992] $$\begin{aligned} G(E)&=&\frac{2e^2}{h}\int dE\left(-\frac{\partial f(E)}{\partial E}\right)\left|{t_{1,-1}(E)}\right|^2\\ t_{\alpha\beta}(E)&=&\Gamma\left\langle i_\beta\right|G^r(E)\left| i_\alpha\right\rangle\end{aligned}$$ in which $\Gamma=2\pi V^2n_F$ with $n_F$ the density of state on the fermi surface of each lead, and a factor $2$ from spin degeneracy has been included. For later convenience, the scattering amplitude $t_{\alpha\beta}$ is defined, in which $\left|i_1\right\rangle=\left|1\right\rangle,~\left|i_{-1}\right\rangle=\left|N\right\rangle$ are the local Weinner states on the left and right end site of the nanowire, respectively. $G^r(E)$ is the retarded Green function of the nanohelix, $G^r(E)=\left[E+i\delta-H_{\rm TB}-\Sigma\right]^{-1}$. Under wide band approximation, the self-energy is $\Sigma=-\frac i2\Gamma\left(\left|1\right\rangle\left\langle 1\right|+\left|N\right\rangle\left\langle N\right|\right)$. The typical behavior of conductance is shown in Fig. \[switch\] (b), in which the conductance with and without external electric field is compared. The metal-insulator transition induced by the electric field can be seen explicitly from the temperature dependence of the conductance, as shown in the inset of fig. \[switch\] (b). Another important information from this calculation is that weak impurity $W\ll eER$ can further widen the insulating region induced by electric field, since the electric-field induced subband is much narrower than the original energy band in the helix, and thus much easier to be localized by disorder. As shown in fig. \[switch\] (b), under the same disorder strength, the first subband of the system with electric field is fully localized, while the one without electric field remains metallic. However, a strong disorder $W\gtrsim eER$ can dominate the effect of electric field and thus kill this metal-insulator transition. Another important issue in this system is the elecron-electron interaction. According to the Luttinger liquid theory, repulsive interaction will make periodical potential more relevant and thus further stabilize this switch effect[@kane1992]. In summary, the terms that may harm this effect are attractive interaction and strong impurity. Although we won’t involve more quantitative discussion in the present paper, a lower-limit estimate to the stability of the present effect can be given as $W\ll E_g$, $V\ll E_g$, with $W, V$ the characteristic energy scale of impurity random potential and phonon-induced attractive interaction, respectively. Under such a condition, the electric field-induced potential scattering dominant the interaction and impurity effect and thus the switching effect (and also the charge pumping and motor effect shown below) remains robust. The device concepts discussed so far depend only on the periodically curved nature of the helical wire, and does not depend on the net helicity of the wire. Therefore, these concepts can be equally well implemented by patterning a quasi-1D wire in a periodically curved form, e.g. a sine wave form, on a plane, and by applying a transverse voltage. The helical wire perhaps has the advantage of being self-assembled and can be more easily realized in the nano-scale. In principle, the same idea can be generalized to design an arbitrary potential in a quasi-one-dimensional system. Consider a planar quantum wire with the shape of function $y(x)$ in an uniform electric field ${\bf E}=E{\bf \hat{y}}$, then the effective one-dimensional potential $V(r)$ is determined by $$\begin{aligned} \frac{Ey'(x)}{\sqrt{1+y'(x)^2}}=-\frac{dV}{dr}\end{aligned}$$ in which $r$ is the arc length of the wire. In this way, one can obtain a quasi-one-dimensional system in any potential $V(r)$ by choosing a proper shape $y(x)$, as shown in Fig. \[geodesign\]. Such a “geometrical design” of one-dimensional systems takes the advantage of tunable strength and shape of potential energy, and thus can help to produce artificial one-dimensional materials with highly controllable electronic and optical properties. In particular, our device can realize a light-emitting-diode (LED) with tunable bandgap and color, controlled purely by the external gate voltage. ![Schematic illustration of the geometrical design of one-dimensional potentials. (a) A straight quantum wire in a sine wave potential (blue line) is equivalent to (b) a periodically curved quantum wire in a uniform electric field ${\bf E}$. (c) More generally, a straight quantum wire in an arbitrary potential $V(r)$ (blue line) is equivalent to (d) a curved quantum wire in a uniform electric field ${\bf E}$. []{data-label="geodesign"}](fig4.eps){width="2.5in"} [*–The Quantum Helical Pump (QHP).*]{} We now consider an adiabatical rotation of the transverse electric field, when a more interesting effect emerges in the nano-helix system. Experimentally, the rotation of electric field can be realized by a set of quadrupolar electrodes, as shown in Fig. \[Gpump\] (a). A rotating electric field with angular frequency $\omega$ is described by a time-dependent $\phi(t)=\omega t$ in the Hamiltonian (\[Hamiltonian\]). In the adiabatical limit $\hbar\omega\ll E_g$, the gapped system with commensurate filling $n_{\rm 1d}=2/L$ will stay in the time-dependent ground state. Similar to what Thouless[@thouless1983] proposed by using a sliding linear periodic potential, such an adiabatical translation of periodical potential on a gapped electron system can in general lead to a quantized charge pumping current $$\begin{aligned} J=2Ne\frac{\omega}{2\pi}, N\in\mathbb{Z},\label{current}\end{aligned}$$ in the zero temperature limit, which means $N$ electrons per spin component are pumped through the wire system during each period $T=2\pi/\omega$. Intuitively, such a quantized charge pumping can be easily understood as a quantum version of Archimedean screw. Due to the electric force, the electron density in the lower subband is larger on the side nearer to positive electrode, and a charge-density-wave (CDW) is induced by the transverse electric field. Consequently, the high-density region will follow the rotation of electric field and thus the coordinate of each electron shifts by one pitch distance during one period of electric field. More quantitatively, the current $J_{\rm pump}$ induced by a time-dependent electric field can be calculated in the tight-binding model (\[Htb\]) and (\[Hlead\]) in a similarly way as the DC conductance:[@brouwer1998; @aharony2002] $$\begin{aligned} G_{\rm pump}&\equiv&\frac{J_{\rm pump}}{e\omega}\nonumber\\ &=&2\int \frac{dE}{2\pi}\int_0^{2\pi}\frac{d\phi}{2\pi}\left(-\frac{\partial f(E)}{\partial E}\right)\nonumber\\ & &\cdot\sum_{\alpha,\beta=\pm 1}\beta{\rm Im}\left[t_{\alpha\beta}^*(E,\phi)\frac{\partial t_{\alpha\beta}(E,\phi)}{\partial \phi}\right]\label{Gpump}\end{aligned}$$ A typical result of this calculation is shown in Fig. \[Gpump\] (b). As expected by topological protection, random disorder can only induce fluctuation of $G$ for gapless system, and leaves the quantized plateaus unchanged. Actually, under zero temperature such a quantized adiabatical charge pumping is robust under any deformation of the Hamiltonian, as long as the subband gap $E_g$ is not closed. In particular, even if the two AC voltages applied to the quadrupolar electrodes are not perfectly harmonic but with some deformations or noises, as long as the electric field vector ${\bf E}(t)$ still encircles the $(0,0)$ point once each period, the quantization of pumping conductance (in the zero temperature limit) remains robust [*without any correction*]{}. In the same way it will remain robust when the nano-helix has a different shape as shown in Fig.\[schematic\] (a) but with the same helical topology. The pumping conductance $G_{\rm pump}$ at finite temperature is simply a convolution of the zero temperature result $G_{\rm pump}(T=0)$ with the thermo factor $-\partial f(E)/\partial E$. Consequently, $G_{\rm pump}$ will deviate from the quantized value. However, for a quantized plateu $G_{\rm pump}(T=0)=2N/2\pi$ with width $E_g$, the deviation $\delta G=G_{\rm pump}(T)-G_{\rm pump}(T=0)$ at the middle-point of the plateu can be estimated by $\delta G/G_{\rm pump}(T=0)\simeq -\frac{2}{e^{\beta E_g/2}+1}$, which is exponentially small when $k_BT\ll E_g$. Compared with the earlier experiments to realize Thouless’s charge pumping effect, like those involving surface accoustic wave[@shilton1996; @aharony2002] or deformation potential on a quantum dot[@switkes1999], the present realization has the advantage of “coding" the topological information directly into the geometrical structure of the self-assembled nano-helix, whose long periodic structure makes the effect more intrinsic and robust. Our device could have higher precision of the current quantization and potentially lead to a new standard of current definition.[@niu1990] ![(a) Illustration of the quantized charge pumping effect, with four electrodes causing a rotating electric field. (b) Pumping conductance $G_{\rm pump}=J_{\rm pump}/e\omega$ under zero temperature(red solid line) and finite temperature $T=0.01t$ (blue dashed line). The error bar shows the fluctuation induced by the disorder strength $W=0.1t$. The parameters of the tight-binding system are taken as $N=100,~L=20,~eER=0.2t,\Gamma/2\pi=0.1t$.[]{data-label="Gpump"}](fig5.eps){width="2.5in"} [*—The Quantum Helical Motor (QHM)*]{} As a direct inverse effect of the topological charge pumping, a nano-helix in a transverse electric field can work as a quantum motor, where a longitudinal current can lead to an uniform mechanical rotation with the frequency, as shown in Fig. \[motor\] $$\begin{aligned} \omega=\frac{\pi J}{Ne},N\in \mathbb{Z}.\label{frequency}\end{aligned}$$ This is a direct quantum analog of a propeller or a windmill. In order to realize this effect, both ends of the helical wire should be attached to some kind of molecular swivel, similar to those described in Ref.[@bryant2003], which enables the uni-axial rotation of the helix. It is also possible to drive an AC current $J(t)$ through a helical wire with fixed ends, which will cause an AC oscillation of the helix. However, the AC effect is not as robust as DC effect, since the relation between AC oscillation and AC current is generally not protected by topological reason. The relation (\[frequency\]) is generally true under any friction or other perturbations, as long as $\hbar\omega\ll E_g$ and $k_BT\ll E_g$ so that the adiabatical evolution condition is satisfied. When there are more frictions, it will be harder to inject a current, but the relation between frequency and current remains valid. In the extreme case, if the nano-helix is fixed, then $\omega=0$ and at the same time $J=0$, which recovers the switch effect. Suppose there is a frictional torque $\mathcal{T}=-\eta \omega$ acting on the helix, then the energy cost per unit time is $P=-\mathcal{T}\omega=\eta\omega^2$. Consequently, one needs a finite voltage $V$ to drive a constant current in this helix. The voltage is determined by the energy equilibrium condition $P=\eta\omega^2=VJ$, which implies that the power of the voltage cancels the friction energy cost. Thus we get the relation $$\begin{aligned} \eta\left(\frac{\pi J}{Ne}\right)^2=VJ\Rightarrow R=\frac{V}{J}=\frac{\pi^2\eta}{N^2e^2},\label{resistance}\end{aligned}$$ which relates friction to a resistivity. As has been discussed in switch effect, the source-drain voltage $V$ must satisfy $V<E_g/e$ so as to keep the adiabatical evolution. Consequently, for a given friction $\eta$, the rotating frequency of such a nano-motor is restricted by $\omega<\frac{\pi}{Ne}\frac{V_{\rm max}}R=\frac{E_gN}{\pi\eta}$ and also by the adiabatical condition $\hbar\omega\ll E_g$. ![Illustration of the quantum helical nano-motor. []{data-label="motor"}](motor.eps){width="1.5in"} [*—More discussions on experimental realizations*]{} After proposing these three effects, we now analyze the detailed experimental conditions for their realizations: 1. [The system is quasi one-dimensional, which requires $E_g\ll E_\perp$, $E_F\ll E_\perp$, with $E_\perp$ the transverse excitation gap and $E_F=k_F^2/2m$ the fermi energy. ]{} 2. [The electron (or hole) filling is commensurate $n_{\rm 1d}=2N/L, N\in \mathbb{N}$. ]{} 3. [The adiabatical approximation is applicable, which requires (i) temperature $k_BT\ll E_g$; (ii) impurity and attractive interaction energy scale $W,U\ll E_g$; (iii) rotation frequency of electric field or nano-helix $\omega\ll E_g/\hbar$. ]{} 4. [The total length of the nanohelix $L_{\rm tot}\gg\xi= \frac{\hbar v_F}{E_g}$, so as to prevent the direct tunneling between the two ends and protect the topological transport. ]{} To satisfy the requirements above, an ideal nano-helix for our purpose should have thin diameter $d$, large helix radius $R$, long length $L_{\rm tot}$ and also be very clean. Experimentally, two most possible candidates for this effect are helical nano-wires made from ZnO, SiC, CB, etc.[@zhang2002; @mcilroy2001; @zhang2003; @yang2004] and chiral biological molecules such as RNA, DNA and some proteins. To make the discussion more concrete, here we give an estimate of the present effects in the deformation-free ZnO nano-helix realized in Ref.[@yang2004]. The size of that nano-helix is reported as $d\simeq 12{\rm nm}$, $\alpha\simeq 40^\circ$, $R=15{\rm nm}$, $L=6R/\cos\alpha\simeq 123{\rm nm}$ (the estimate of $L$ is a little different from the previous one since the intercept of ZnO helix here is hexagonal rather than round.) If we approximate the electron effective mass by the bulk ZnO value $m\simeq 0.24m_e$[@karpina2004], then the transverse excitation gap can be estimated as $E_\perp\sim \frac{h^2}{2m d^2}\simeq 44{\rm meV}$. The filling corresponding to first gap is $n_{\rm 1d}=2/L\sim 1.6\times 10^5/{\rm cm}$, which in 3-d unit gives $n_{\rm 3d}=2/L\pi(d/2)^2\simeq 1.4\times 10^{17}/{\rm cm}^3$. The corresponding $E_F=\hbar^2 k_F^2/2m\simeq 0.1{\rm meV}$. Thus the condition $E_F\ll E_\perp$ is always satisfied, and condition $E_g\ll E_\perp\simeq 44{\rm meV}$ requires the electric field $E\ll3\times 10^6{\rm V/m}$ or gate voltage $V_g\ll 88{\rm mV}$. The $\xi$ in condition (4) is $\xi=\hbar^2 k_F/mE_g\simeq 1.6{\rm nm}$, thus the condition (4) is always satisfied. If we take an electric field $E=3.3\times 10^5{\rm V/m}$ or $E_g=eER=5{\rm meV}$, then the conditions 3 requires i) $T\ll 60K$; ii) in condition 3, $W,U\ll 5{\rm meV}$; iii) $\omega\ll 7.6\times 10^{12}{\rm Hz}$. In summary, such an effect should be observable in a wide temperature range for the ZnO nanowire in Ref.[@yang2004], if it is clean enough and the doping can be controlled well. (To avoid impurity effect, the filling should be controlled by gate voltage rather than chemical doping. ) Compared to the inorganic nano-helixes, chiral biological molecules such as RNA, DNA or protein may have the advantage of better one-dimensionality, which implies a larger transverse gap $E_\perp$, since they can be much thinner than the nano-wires. For the effects proposed here to be observed, one needs to find molecules which are semi-conducting and have a good one-dimensional energy band. In a recent review article[@endres2006], transport properties of various DNA molecules are summarized, some of which show semiconducting behavior. If for some molecules $E_\perp\gtrsim 1{\rm eV}$, then it’s possible to observe the effect at room temperature with voltage $V_g\gtrsim 200{\rm meV}\gg 2k_BT\simeq 60{\rm meV}$. In summary, in this paper we proposed three related effects in quantum helical systems under a transverse electric field. Under a slowly rotating electric field, a nanohelix with commensurate filling works as a quantum Archimedean screw. The experimental conditions to realize such effects are shown to be feasible for present experimental techniques. Since helical structures occur naturally in the biological world, the principles discussed here also provides new methods to control and detect biological molecules. [**Acknowledgement.**]{} The authors wish to thank B. A. Bernevig, D. Cox, Y. Cui, S. Doniach, C. Huang, T. Hughes, S. Kivelson, C.-X. Liu, P. Wong and B.-H. Yan for useful discussions. This work is supported by the NSF under the grant No. DMR-0342832 and the US Department of Energy, Office of Basic Energy Sciences under contract No. DE-AC03-76SF00515.
--- abstract: 'Stellar surface processes represent a fundamental limit to the detection of extrasolar planets with the currently most heavily-used techniques. As such, considerable effort has gone into trying to mitigate the impact of these processes on planet detection, with most studies focusing on magnetic spots. Meanwhile, high-precision photometric planet surveys like CoRoT and [*Kepler*]{} have unveiled a wide variety of stellar variability at previously inaccessible levels. We demonstrate that these newly revealed variations are not solely magnetically driven but also trace surface convection through light curve “flicker.” We show that “flicker” not only yields a simple measurement of surface gravity with a precision of $\sim$0.1 dex, but it may also improve our knowledge of planet properties, enhance radial velocity planet detection and discovery, and provide new insights into stellar evolution.' author: - 'Fabienne A. Bastien$^{1,}$$^{2}$' title: 'Convection in Cool Stars, as Seen Through [*Kepler*]{}’s Eyes' --- Introduction ============ Most planets are observed only indirectly, through their influence on their host star. The planet properties we infer therefore strongly depend on how well we know those of the stars. Our ability to determine the surface gravity ([$\log g$]{}) of field stars, however, is notoriously limited: broadband photometry, while efficient, yields errors of $\sim$0.5 dex; spectroscopy suffers from well known degeneracies between [$\log g$]{}, [$T_{\rm eff}$]{}and metallicity [@torres10] while having [$\log g$]{} errors of 0.1–0.2 dex [@ghezzi10]; and asteroseismology, the gold standard for stellar parameter estimation with [$\log g$]{} errors of $\sim$0.01 dex [@chaplin11; @chaplin14], is time and resource intensive and, particularly for dwarfs, is limited to the brightest stars. Meanwhile, high precision photometric surveys like CoRoT and [*Kepler*]{} have surveyed over $\sim$200 000 Sun-like stars in their hunt for exoplanets, revealing stellar variations that have previously only been robustly observed in the Sun and a handful of bright Sun-like stars — and also variations that were previously unknown but, as we show, encode a simple measure of stellar [$\log g$]{}. In what follows, I describe our analysis of the newly unveiled high frequency photometric variations, which we term “flicker” (or [$F_8$]{}) and which enable us to measure [$\log g$]{} with an accuracy of $\sim$0.1 dex. I summarize our work thus far in using [$F_8$]{} to study granulation in Sun-like stars, to examine the impact of granulation in radial velocity planet detection, and to improve size estimates of transiting exoplanets. Photometric “Flicker:” a Tracer of Granulation and a Simple Measure of Stellar Surface Gravity ============================================================================================== Using light curves from NASA’s [*Kepler*]{} mission, we discovered that stellar [$\log g$]{} reveals itself through [$F_8$]{}– a measure of photometric variations on timescales of $<$ 8hr — and may hence be used to measure [$\log g$]{} with errors of $\sim$0.1 dex, even for stars too faint for asteroseismology [@bastien13 Fig. \[fig:bastien13\]]. The measurement of [$\log g$]{} from [$F_8$]{} only requires the discovery light curves, and this measurement not only yields a result with an accuracy that rivals spectroscopy, it also does so very quickly and efficiently, requiring only a simple routine that can be executed by anyone in just a few seconds per star. ![\[fig:bastien13\] Stellar surface gravity manifests in a simple measure of brightness variations. Asteroseismically determined [$\log g$]{} shows a tight correlation with [$F_8$]{}. Color represents the amplitude of the stars’ brightness variations; outliers tend to have large brightness variations. Excluding these outliers, a cubic-polynomial fit through the Kepler stars and through the Sun (large star symbol) shows a median absolute deviation of 0.06 dex and a r.m.s. deviation of 0.10 dex. To simulate how the solar [$\log g$]{} would appear in data we use to measure [$\log g$]{} for other stars, we divide the solar data into 90-d “quarters”. Our [$F_8$]{}–[$\log g$]{} relation measured over multiple quarters then yields a median solar [$\log g$]{} of 4.442 with a median absolute deviation of 0.005 dex and a r.m.s. error of 0.009 dex (the true solar [$\log g$]{} is 4.438). From @bastien13.](bastien13_fig2.eps) In @bastien13, we ascribed [$F_8$]{} to granulation power, which is known to depend on the stellar [$\log g$]{} [@kjeldsen11; @mathur11]. Recent independent simulations and asteroseismic studies have examined the expected photometric manifestations of granulation [@samadi13a; @samadi13b; @mathur11], nominally through the Fourier spectrum from which it can be difficult to extract the granulation signal. We used the simulations to predict the granulation-driven [$F_8$]{}, and we find excellent agreement with our observed [$F_8$]{}, demonstrating that the [$F_8$]{} is indeed granulation-driven [@cranmer14]. We also determined an empirical correction to the granulation models, particularly for F stars which have the shallowest convective outer layers. Indeed, our results suggest that these models must include the effects of the magnetic suppression of convection in F stars in order to reproduce the observations. This work can ultimately help to develop our technique of “granulation asteroseismology,” enabling the precise determination of a larger number of stellar, and hence planetary, parameters. Stellar “Flicker” Suggests Larger Radii for Bright [*Kepler*]{} Planet Host Stars ================================================================================= The speed and efficiency with which one can determine accurate [$\log g$]{} solely with the discovery light curves translates directly into a rapid assessment of the distribution of bulk planet properties — in particular, with greater accuracy and fewer telescopic and computational resources than similar studies [@batalha13; @burke14] that of necessity relied on broadband photometric measurements to determine stellar properties. We therefore applied our [$F_8$]{} technique to a few hundred bright ([*Kepler*]{} magnitudes between 8 and 13) planet candidates in the [*Kepler*]{} field, and we find that these stars are significantly more evolved than previous studies suggest [@bastien14b]. As a result, the planet radii are 20–30% larger than previously estimated. In addition, we find that the high proportion of subgiants we derive (48%) is consistent with predictions from galactic models of the underlying stellar population (45%), whereas previous analyses heavily bias stellar parameters towards the main sequence and hence yield a low subgiant fraction (27%; Figs. \[fig:bastien14b\_1\],\[fig:bastien14b\_2\]). ![\[fig:bastien14b\_1\] Distributions of [$\log g$]{} for the TRILEGAL simulated sample (black) and KOI host stars with [$F_8$]{}-based [$\log g$]{} (red) and broadband photometry/spectroscopy-based [$\log g$]{} (“NEA”; cyan curve). We limit the [$T_{\rm eff}$]{} range here to 4700–6500 K, for which the [*Kepler*]{} targets should be representative of the field. Vertical lines indicate the range of [$\log g$]{} corresponding to subgiants. We find that [$F_8$]{} reproduces the expected underlying distribution, and, in particular, recovers the expected population of subgiants, while the NEA parameters are preferentially pushed towards the main sequence. Adapted from @bastien14b.](bastien14b_fig4.eps) ![\[fig:bastien14b\_2\] H-R diagram of KOI host stars with [$\log g$]{} derived from [$F_8$]{} (middle) and broadband photometry/spectroscopy (bottom), and as predicted by a TRILEGAL [@girardi05] simulation (top). Colored curves represent the theoretical evolutionary tracks (masses labeled in [M$_\odot$]{}). Vertical lines demarcate the range of stellar [$T_{\rm eff}$]{}considered in this study. The horizontal lines demarcate the range of [$\log g$]{} for subgiants (3.5 $<$ [$\log g$]{} $<$ 4.1). A representative error bar on [$\log g$]{} for each stellar sample is in the upper right of each panel. We find that the [$F_8$]{}-based [$\log g$]{} distribution more closely matches expectation than previous [$\log g$]{} measurements, particularly in the subgiant domain, perhaps because [$F_8$]{}involves no main-sequence prior on the [$F_8$]{}-based [$\log g$]{} values. From @bastien14b.](bastien14b_fig3.ps) We expand upon this work by tailoring our initial [$F_8$]{} relation to be more directly useful to the exoplanet community by deriving a relationship between [$F_8$]{} and stellar density [@kipping14]. This relation, which can yield the stellar density with an uncertainty of $\sim$30%, can help to constrain exoplanet eccentricities and enable the application of techniques like astrodensity profiling to hundreds of exoplanet host stars in the [*Kepler*]{} field alone. RV Jitter in Magnetically Inactive Stars is Linked to High Frequency “Flicker” in Light Curves ============================================================================================== RV planet detection, particularly of small planets, requires precise Doppler measurements, and only a few instruments are able to achieve the precision needed to observe them. Key to the success of RV planet campaigns is the avoidance of “RV loud” stars — those likely to exhibit large levels of RV jitter that can impede and sometimes even mimic planetary signals [@queloz01]. Most RV surveys therefore focus their attention on magnetically quiet stars, as magnetic spots tend to drive the largest amount of RV jitter. Nonetheless, magnetically inactive stars can exhibit unexpectedly high levels of RV jitter [@wright05; @galland05], and even low jitter levels can impede the detection of small planets. The drivers of RV jitter in inactive stars remain elusive [@dumusque11a; @dumusque11b; @boisse12], continuing to plague RV planet detection and, in the case of F dwarfs, resulting in the outright avoidance of whole groups of notoriously RV noisy stars, even in transit surveys with large ground-based follow-up efforts like [*Kepler*]{} [@brown11]. ![\[fig:bastien14a\] Comparison between RV jitter (RV RMS) and [$F_8$]{}-based [$\log g$]{}: RV jitter shows a strong anti-correlation with [$F_8$]{}-based [$\log g$]{}, with a statistical confidence of 97% derived from a survival analysis. A similar trend was found by @wright05. [$F_8$]{} measures granulation power [@bastien13], indicating that the RV jitter of magnetically inactive stars is driven by convective motions on the stellar surface whose strength increases as stars evolve. Adapted from @bastien14a.](bastien14a_fig9.ps) Given the breadth of stellar photometric behavior newly revealed by ultra-high precision light curves, and the new insights that they are giving into stellar surface processes, we compared different ways of characterizing this photometric behavior with RV jitter for all stars with both ultra-high precision light curves and high precision, long term RV monitoring [@bastien14a]. These stars have very low photometric amplitudes (less than 3 ppt), a previously unexplored regime of both photometric variability and RV jitter. We find that the RV jitter of these stars, ranging from 3 m s$^{-1}$ to 135 m s$^{-1}$, manifests in the light curve Fourier spectrum, which we then use to develop an empirical predictor of RV jitter. We also find that spot models grossly under-predict the observed jitter by factors of 2–1000. Finally, we demonstrate that [$F_8$]{} itself is a remarkably clean predictor of RV jitter in magnetically quiet stars (Fig. \[fig:bastien14a\]), suggesting that the observed jitter is driven by convective motions on the stellar surface and is strongly tied to [$\log g$]{}. Summary ======= We find that surface convection in cool stars manifests as the high frequency “flicker” observed in high precision, long time-baseline light curves, such as those from [*Kepler*]{}. We show that it yields a simple measure of stellar surface gravity and density, and we use it to place empirical constraints on granulation models. We use it to perform an ensemble analysis of exoplanet host stars, finding that the exoplanet radii are larger than previous studies suggested. Finally, we find that it is a clean predictor of RV jitter in magnetically inactive stars and can hence be used to identify promising targets for RV follow-up campaigns and RV planet searches. More generally, we show that stellar variability — traditionally considered a major noise source and nuisance, particularly in exoplanet detection — can be used to enhance both exoplanet science and our understanding of stellar evolution. 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--- author: - 'Celine Degrande,' - 'Fabio Maltoni,' - 'Ken Mimasu,' - 'Eleni Vryonidou,' - Cen Zhang bibliography: - 'refs.bib' title: 'Single-top associated production with a $Z$ or $H$ boson at the LHC: the SMEFT interpretation' --- Introduction ============ The study of the top-quark, gauge and Higgs boson interactions is one of the main goals of the exploration of the TeV scale at colliders. The golden era of precision physics at the LHC started after the discovery of the Higgs boson in Run I and a coordinated theoretical and experimental effort is ongoing to detect deviations and/or constrain new physics with sensitivities that go up to the multi-TeV scales. A powerful and general framework to analyse and parametrise deviations from the Standard Model (SM) predictions is the one of SM Effective Field Theory (SMEFT) [@Weinberg:1978kz; @Buchmuller:1985jz; @Leung:1984ni], where the SM is augmented by a set of higher-dimension operators $$\mathcal{L}_\mathrm{SMEFT}=\mathcal{L}_\mathrm{SM}+ \sum_i\frac{C_{i}}{\Lambda^2}{\ensuremath{\mathcal{O}}}_{i}+\mathcal{O}(\Lambda^{-4}). \label{eq:smeft}$$ Within the SMEFT, predictions can be systematically improved by computing higher-order corrections. Significant progress in this direction has been achieved in both the top-quark [@Zhang:2013xya; @Zhang:2014rja; @Degrande:2014tta; @Franzosi:2015osa; @Zhang:2016omx; @Bylund:2016phk; @Maltoni:2016yxb; @Rontsch:2014cca; @Rontsch:2015una] and Higgs sectors [@Hartmann:2015oia; @Ghezzi:2015vva; @Hartmann:2015aia; @Gauld:2015lmb; @Mimasu:2015nqa; @Degrande:2016dqg]. Among the least known interactions between the heaviest particles of the standard model are the neutral gauge and Higgs top-quark interactions. These interactions can be probed directly for the first time at the LHC through the associated production of a Higgs, $Z$ or $\gamma$ with a top-quark pair. In this case the leading production mechanisms are through QCD interactions (at order $\alpha_S^2$ at the Born level) and both theoretical studies and experimental ones exist that establish the present and future sensitivities  [@Degrande:2012gr; @Rontsch:2014cca; @Rontsch:2015una; @Bylund:2016phk; @Schulze:2016qas; @Maltoni:2016yxb] to new couplings as parametrised in the SMEFT. An intrinsic limitation of this strategy is the fact that a plethora of operators enter these processes some of which are of QCD nature or involve four fermions. Therefore they need to be constrained very well (through, for example, $t \bar t$ production) before being able to access the electroweak ones. A promising alternative, discussed in this work, is to consider the corresponding set of associated production processes of neutral heavy bosons with a single top. At the LHC top quarks can be produced singly via electroweak interactions, the leading process being $t$-channel production ($tj$), $qb \to q't$, which features a total single top and anti-top rate which is about $220$ pb at $\sqrt{S}=13$ TeV, [*i.e.*]{}, one fourth of strong $t \bar t$ production. The cross section probes a limited set of top-quark electroweak couplings, [*i.e.*]{}, at leading order, two four-quark interactions and three operators which induce a modification of the top electroweak couplings. Considering also the top decay one can additionally probe top-quark four-fermion operators involving leptons. Requiring a $Z$ or a $H$ boson in association with single-top significantly extends the sensitivity of $tj$, opening up the rather unique possibility of accessing top-Higgs, top-gauge, triple gauge, gauge-Higgs interactions in the same final state.[^1] The fact that these processes can play an important role in the search for new neutral top-quark interactions has been already noted at the theory level [@Maltoni:2001hu; @Biswas:2012bd; @Farina:2012xp; @Demartin:2015uha] (even though not yet analysed in the context of the SMEFT) and motivated experimental activities, such as the measurements of the associated production of a $Z$ with a single top quark by ATLAS [@Aaboud:2017ylb] and CMS [@CMS-PAS-TOP-16-020; @Sirunyan:2017nbr] at 13 TeV, as well as the searches for $tHj$ production, which are also underway [@CMS-PAS-HIG-16-019; @CMS-PAS-HIG-17-005]. In addition, asking for just one top-quark (or anti-top-quark) in the final state implies no QCD interactions at the leading order (LO) and therefore makes this class of processes ‘purely’ electroweak with two important consequences. First, SM QCD corrections are typically small and under control. Second, dim-6 modifications of QCD interactions enter only at NLO with a weak sensitivity that does not spoil that of the EW couplings. ![\[fig:eftdiagram\]Schematic representation of the interplay between operators and processes, focussing on single-top production and associated channels. Six (five at LO and one at NLO in QCD) operators enter single-top production ($tj$, blue square), and are therefore also present in $Z$ boson ($tZj$, red square) and in Higgs ($tHj$, purple square) associated production. Operators exist that contribute to either $tZj$ or $tHj$ and also to both processes without contributing to $tj$. The operators entering in diboson ($VV$) production are a subset (green square) of those contributing to $tZj$, while some of the operators contributing to Higgs associated production ($VH$) and Vector Boson Fusion (VBF, orange dashed square) are shared between $tHj$ and $tZj$.](Figures/eftdiagram){width="80.00000%"} In this work, we consider the $t$-channel $tZj$ and $tHj$ production at the LHC, providing predictions at NLO accuracy in QCD in the general framework of the SMEFT, including all relevant operators up to dimension six. This is the first time NLO in QCD corrections are calculated for processes that involves all possible types of dim-6 operators, [*i.e.*]{} bosonic, two-fermion and four-fermion ones in a fully automatic way. We perform a complete study of the sensitivity to new interactions of these processes, highlighting the interplay and complementarity among $tj$, $tZj$ and $tHj$ in simultaneously constraining top-quark, triple gauge, and gauge-Higgs interactions in the current and future runs at the LHC (see Fig. \[fig:eftdiagram\]). We first study the energy dependence of relevant $2\to2$ sub-amplitudes to identify the set of operators that may induce deviations in each process and characterise the expected energy growth in each case. We then compute the complete dependence of the inclusive rates on these operators at NLO in QCD, including estimates of the scale uncertainty due to the running of the Wilson coefficients where applicable. Our approach is based on the [MadGraph5\_aMC@NLO]{} ([MG5\_aMC]{}) framework [@Alwall:2014hca], and is part of the ongoing efforts of automating NLO SMEFT simulations for colliders [@Zhang:2016snc]. Using these results, we perform sensitivity studies of current and future inclusive measurements of the two processes, contrasting them with existing limits on the operators of interest. Finally, we present differential distributions for a number of selected benchmark values of the Wilson coefficients inspired by current limits, highlighting the possibility of large deviations in the high energy regime of both processes. This paper is organised as follows. In Section \[section:Operators\] we establish the notation and the conventions, we identify the set of operators entering $tj$, $tZj$ and $tHj$ and we establish which ones can lead to an energy growth. In Section \[section:constraints\] a summary of the current constraints available on the Wilson coefficients of the corresponding operators is given. In Section \[sec:setup\] results for total cross sections as well as distributions are presented, operator by operator and the prospects of using $tZj$ and $tHj$ to constrain new interactions are discussed. The last section presents our conclusions and the outlook. Top-quark, electroweak and Higgs operators in the SMEFT {#section:Operators} ======================================================= The processes that we are studying lie at the heart of the electroweak symmetry breaking sector of the SM. They involve combinations of interactions between the Higgs boson and the particles to which it is most strongly coupled: the top quark and the EW gauge bosons. The measurement of these processes is therefore a crucial test of the nature of EW symmetry breaking in the SM and any observed deviations could reveal hints about the physics that lies beyond. We adopt the SMEFT framework to parametrise the deviations of the interactions in question from SM expectations. Dim-6 operators suppressed by a scale, $\Lambda$, are added to the SM Lagrangian as in Eq. (\[eq:smeft\]). Specifically, we employ the Warsaw basis [@Grzadkowski:2010es] dim-6 operators relevant for the $tHj$ and $tZj$ processes. To this end, it is convenient to work in the limit of Minimal Flavour Violation (MFV) [@DAmbrosio:2002vsn], in which it is assumed that the only sources of departure from the global $U(3)^5$ flavour symmetry of the SM arise from the Yukawa couplings. By assuming a diagonal CKM matrix and only keeping operators with coefficients proportional to the third generation Yukawas, we retain all operators in the top-quark sector, as well as all the light-fermion operators that are flavour-universal [@AguilarSaavedra:2018nen]. Assuming in addition $CP$ conservation, we are left with a well-defined set of operators that can directly contribute to the processes, summarised in Table \[tab:operators\] where all Yukawa and gauge coupling factors are assumed to be absorbed in the operator coefficients. We adopt the following definitions and conventions: $$\begin{aligned} {\ensuremath{\varphi}}^\dag {\overleftrightarrow D}_\mu {\ensuremath{\varphi}}&={\ensuremath{\varphi}}^\dag D^\mu{\ensuremath{\varphi}}-(D_\mu{\ensuremath{\varphi}})^\dag{\ensuremath{\varphi}}\nonumber\\ {\ensuremath{\varphi}}^\dag \tau_{{\scriptscriptstyle}K} {\overleftrightarrow D}^\mu {\ensuremath{\varphi}}&= {\ensuremath{\varphi}}^\dag \tau_{{\scriptscriptstyle}K}D^\mu{\ensuremath{\varphi}}-(D^\mu{\ensuremath{\varphi}})^\dag \tau_{{\scriptscriptstyle}K}{\ensuremath{\varphi}}\nonumber \\ W^{{\scriptscriptstyle}K}_{\mu\nu} &= \partial_\mu W^{{\scriptscriptstyle}K}_\nu - \partial_\nu W^{{\scriptscriptstyle}K}_\mu + g \epsilon_{{\scriptscriptstyle}IJ}{}^{{\scriptscriptstyle}K} \ W^{{\scriptscriptstyle}I}_\mu W^{{\scriptscriptstyle}J}_\nu \nonumber\\ B_{\mu\nu} &= \partial_\mu B_\nu - \partial_\nu B_\mu \\ D_\rho W^{{\scriptscriptstyle}K}_{\mu\nu} &= \partial_\rho W^{{\scriptscriptstyle}K}_{\mu\nu} + g \epsilon_{{\scriptscriptstyle}IJ}^{{\scriptscriptstyle}K} W^{{\scriptscriptstyle}I}_\rho W_{\mu\nu}^{{\scriptscriptstyle}J}\nonumber \\ D_\mu{\ensuremath{\varphi}}=& (\partial_\mu - i g \frac{\tau_{{\scriptscriptstyle}K}}{2} W_\mu^{{\scriptscriptstyle}K} - i\frac12 g^\prime B_\mu){\ensuremath{\varphi}},\nonumber\end{aligned}$$ where $\tau_I$ are the Pauli matrices. We compute predictions for on-shell top quark, Higgs and $Z$ bosons, ignoring operators that could mediate the same decayed final state through a contact interaction such as the $\bar{t}t\bar{\ell}\ell$ four-fermion operators. This contribution is expected to be suppressed, as the experimental analyses typically apply a cut on the invariant mass of the lepton pair around the $Z$ mass. It can nevertheless be straightforwardly included as was done in [@Durieux:2014xla]. Two four-quark operators that also mediate single-top production do affect these processes. These operators $ {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}Qq}}^{{\scriptscriptstyle}(3,1)}$ and $ {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}Qq}}^{{\scriptscriptstyle}(3,8)}$ listed in Table \[tab:operators\] contribute at $1/\Lambda^2$ and $1/\Lambda^4$ respectively, the latter not interfering with the SM processes at LO due to colour. While the measurement of the single-top process already constrains these operators, the higher kinematic thresholds of the associated production may enhance the dependence on the Wilson coefficients. In addition, the following operators contribute indirectly, by affecting the muon decay and consequently the relation between the Fermi constant and the Higgs vacuum-expectation-value: $$\begin{aligned} {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}ll}}^{{\scriptscriptstyle}(3)}& = (\bar{l}_i\,\gamma_\mu\tau_{{\scriptscriptstyle}I}\,l_i) (\bar{l}_j\,\gamma^\mu\tau^{{\scriptscriptstyle}I}\,l_j),\\ {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}l}}^{{\scriptscriptstyle}(3)}& = i\big({\ensuremath{\varphi}}^\dagger\lra{D}_\mu\,\tau_{{\scriptscriptstyle}I}{\ensuremath{\varphi}}\big) \big(\bar{l}_{i}\,\gamma^\mu\,\tau^{{\scriptscriptstyle}I}l_{i}\big) + \text{h.c.}\,. \end{aligned}$$ Some of these operators are constrained by Electroweak Precision Observables EWPO [@Han:2004az]. These include the two previous operators and those involving light-fermion fields, [*i.e.*]{}, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}q}}^{(1)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}q}}^{{\scriptscriptstyle}(3)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}u}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}d}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}l}}^{{\scriptscriptstyle}(3)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}l}}^{{\scriptscriptstyle}(1)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}e}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}ll}}^{{\scriptscriptstyle}(3)}$, where &[[$\mathcal{O}$]{}\_[[$\varphi$]{}d]{}]{}=i([$\varphi$]{}\^\_[$\varphi$]{}) (|[d]{}\_i\^d\_i) +\ &[[$\mathcal{O}$]{}\_[[$\varphi$]{}l]{}]{}\^[(1)]{} =i([$\varphi$]{}\^\_[$\varphi$]{}) (|[l]{}\_i\^l\_i) +\ &[[$\mathcal{O}$]{}\_[[$\varphi$]{}e]{}]{}=i([$\varphi$]{}\^\_[$\varphi$]{}) (|[e]{}\_i\^e\_i) + , as well as the operators that are often identified with the S and T parameters $$\begin{aligned} {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}WB}} &= ({\ensuremath{\varphi}}^\dagger \tau_{{\scriptscriptstyle}I}{\ensuremath{\varphi}})\,B^{\mu\nu}W_{\mu\nu}^{{\scriptscriptstyle}I}\,\text{ and }\\ {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}D}} &= ({\ensuremath{\varphi}}^\dagger D^\mu{\ensuremath{\varphi}})^\dagger({\ensuremath{\varphi}}^\dagger D_\mu{\ensuremath{\varphi}}).\end{aligned}$$ It is well-known that among these 10 basis operators, only 8 degrees of freedom are tightly constrained [@Falkowski:2014tna], leaving two flat directions that are constrained only by diboson production processes. This effect has been discussed in the literature [@Grojean:2006nn; @Alonso:2013hga; @Brivio:2017bnu]. These two directions correspond exactly to the two basis-operators in the HISZ parametrisation [@Hagiwara:1993ck]: $$\begin{aligned} {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}HW}}&=&(D^\mu{\ensuremath{\varphi}})^\dagger\tau_{{\scriptscriptstyle}I}(D^{\nu}{\ensuremath{\varphi}})W^{{\scriptscriptstyle}I}_{\mu\nu}\\ {{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}HB}}&=&(D^\mu{\ensuremath{\varphi}})^\dagger(D^{\nu}{\ensuremath{\varphi}})B_{\mu\nu}.\end{aligned}$$ Apart from modifying the Higgs couplings, the coefficients of these two operators are often used to parametrise the triple-gauge-boson (TGC) couplings, together with the coefficient of ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}W}}$ [@Degrande:2012wf]. They can be determined by di-boson and tri-boson production processes. Since one interesting application of this work is to determine the sensitivity of the $tZj$ and $tHj$ processes to TGC couplings relative to the di-boson processes, we include these two additional operators to cover all possible Lorentz structures in TGC modifications from dim-6 SMEFT. With this choice we can safely exclude the 10 Warsaw basis operators that enter the EWPO measurements. We also neglect the operator $(\varphi^\dagger\varphi)\square(\varphi^\dagger\varphi)$, which universally shifts all Higgs couplings. This operator does not lead to any different energy-dependent behaviour, and is likely to be better constrained by other Higgs measurements. We briefly mention here that the complete RG structure of the SMEFT has been given in [@Jenkins:2013zja; @Jenkins:2013wua; @Alonso:2013hga]. In this work we will consider the QCD induced running of the Wilson coefficients, which is relevant for our calculation, [*i.e.*]{} $\mathcal{O}(\alpha_s)$ terms with our normalisation. The only operators from our set that run under QCD are $({{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}},{{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}},{{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}})$. The mixing matrix has a diagonal form: $$\frac{dC_i(\mu)}{d\log\mu}=\frac{\alpha_s}{\pi}\gamma_{ij}C_j(\mu),\quad \gamma= \left( \begin{array}{ccc} -2 & 0 & 0 \\ 0 & 2/3 & 0 \\ 0 & 0 & 2/3 \end{array} \right)\,. \label{eq:rg}$$ The chromomagnetic operator, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tG}}$, also mixes into the weak dipole operators at NLO in QCD and therefore contributes to our two processes at one-loop. While this is an interesting effect, we do not expect to obtain significant additional information from $tZj$ or $tHj$ given the current constraints from top measurements and the fact that it enters at higher order in $\alpha_{{\scriptscriptstyle}S}$. We nevertheless compute its contribution to our processes for completeness. In summary, the operators to be considered in this work are: \[oplist1\] & [[$\mathcal{O}$]{}\_[[$\varphi$]{}W]{}]{},[[$\mathcal{O}$]{}\_[W]{}]{},[[$\mathcal{O}$]{}\_[HW]{}]{},[[$\mathcal{O}$]{}\_[HB]{}]{},\ \[oplist2\] & [[$\mathcal{O}$]{}\_[[$\varphi$]{}Q]{}]{}\^[(3)]{},[[$\mathcal{O}$]{}\_[[$\varphi$]{}Q]{}]{}\^[(1)]{},[[$\mathcal{O}$]{}\_[[$\varphi$]{}t]{}]{},[[$\mathcal{O}$]{}\_[tW]{}]{},[[$\mathcal{O}$]{}\_[tB]{}]{},[[$\mathcal{O}$]{}\_[tG]{}]{},[[$\mathcal{O}$]{}\_[tb]{}]{},[[$\mathcal{O}$]{}\_[t[$\varphi$]{}]{}]{},\ \[oplist3\] & [[$\mathcal{O}$]{}\_[Qq]{}]{}\^[(3,1)]{},[[$\mathcal{O}$]{}\_[Qq]{}]{}\^[(3,8)]{}. Figure \[fig:eftdiagram\] gives a visual representation of how different operators contribute to the set of processes $tj$, $tZj$ and $tHj$, and also $VV$ and $VH$,VBF production. As mentioned already, an interesting feature of $tZj$ and $tHj$ is that they are affected by the same operators that enter $ttZ$ and $ttH$, respectively, yet they are entangled in a non-trivial way. The connection of different sectors by these two processes is required by the nature of SMEFT [@Corbett:2013pja; @Falkowski:2015jaa; @Maltoni:2016yxb] and makes these processes a unique testing ground for operators at the heart of the EW symmetry breaking sector. Figure \[fig:feyndiags\] shows a selection of representative Feynman diagrams for the $tHj$ process in which the SMEFT modifications can enter. ![\[fig:feyndiags\] Representative LO Feynman diagrams for $tHj$ production in the SMEFT. The operator insertions (black dots) correspond to operators involving either electroweak gauge boson or third generation fermions interactions. These can modify existing SM interactions such as the top Yukawa or Higgs-$W$-$W$ interaction, induce new Lorentz structures, [*e.g.*]{}, with the weak dipole operators or mediate new contact interactions between fermion currents and two EW bosons. Equivalent diagrams for the $tZj$ process can be obtained by replacing the Higgs with a $Z$ boson and keeping in mind that the $Z$ boson can also couple to the light-quark line. ](Figures/thj_Wtb "fig:"){width="20.00000%"} ![\[fig:feyndiags\] Representative LO Feynman diagrams for $tHj$ production in the SMEFT. The operator insertions (black dots) correspond to operators involving either electroweak gauge boson or third generation fermions interactions. These can modify existing SM interactions such as the top Yukawa or Higgs-$W$-$W$ interaction, induce new Lorentz structures, [*e.g.*]{}, with the weak dipole operators or mediate new contact interactions between fermion currents and two EW bosons. Equivalent diagrams for the $tZj$ process can be obtained by replacing the Higgs with a $Z$ boson and keeping in mind that the $Z$ boson can also couple to the light-quark line. ](Figures/thj_HWW "fig:"){width="20.00000%"} ![\[fig:feyndiags\] Representative LO Feynman diagrams for $tHj$ production in the SMEFT. The operator insertions (black dots) correspond to operators involving either electroweak gauge boson or third generation fermions interactions. These can modify existing SM interactions such as the top Yukawa or Higgs-$W$-$W$ interaction, induce new Lorentz structures, [*e.g.*]{}, with the weak dipole operators or mediate new contact interactions between fermion currents and two EW bosons. Equivalent diagrams for the $tZj$ process can be obtained by replacing the Higgs with a $Z$ boson and keeping in mind that the $Z$ boson can also couple to the light-quark line. ](Figures/thj_yuk "fig:"){width="20.00000%"} ![\[fig:feyndiags\] Representative LO Feynman diagrams for $tHj$ production in the SMEFT. The operator insertions (black dots) correspond to operators involving either electroweak gauge boson or third generation fermions interactions. These can modify existing SM interactions such as the top Yukawa or Higgs-$W$-$W$ interaction, induce new Lorentz structures, [*e.g.*]{}, with the weak dipole operators or mediate new contact interactions between fermion currents and two EW bosons. Equivalent diagrams for the $tZj$ process can be obtained by replacing the Higgs with a $Z$ boson and keeping in mind that the $Z$ boson can also couple to the light-quark line. ](Figures/thj_cont "fig:"){width="20.00000%"} Energy growth and sub-amplitudes \[sec:energybehaviour\] -------------------------------------------------------- One of the characteristic ways in which anomalous interactions between SM particles manifest themselves is through the energy growth of the scattering amplitudes. An enhancement can arise through two basic mechanisms. The first is due to vertices involving higher dimension Lorentz structures, [*i.e.*]{}, with additional derivatives or four-fermion interactions. The second, more subtle, can come from deformations induced by operators that do not feature new Lorentz structures, yet spoil delicate unitarity cancellations that might take place in the SM amplitudes. In general, higher dimensional operators involving Higgs fields can contribute to either of these effects, given that insertions of the Higgs vacuum-expectation-value can lower the effective dimension of a higher -dimesion operator down to dim-4. A concrete example of this phenomenon can be found in $tHj$, where both the diagram featuring the top-quark Yukawa coupling and the one with the $W$-Higgs interaction (see the second and third diagrams of Fig. \[fig:feyndiags\]), grow linearly with energy, and yet this unitarity-violating dependence exactly cancels in the SM [@Bordes:1992jy; @Maltoni:2001hu; @Farina:2012xp]. The rate of this process is therefore sensitive to the deviations in the Higgs couplings to the top quark and $W$-boson. This can be understood by factorising the process into the emission of an on-shell $W$-boson from the initial light quark, weighted by an appropriate distribution function, times the $b\,W\to t\,h$ sub-amplitude. The sub-amplitudes of the two diagrams in question, involving a longitudinally polarised $W$, both display an unacceptable energy growth which cancels in the SM limit. Similarly for the $tZj$ process, the $b\,W\to t\,Z$ sub-amplitude for longitudinally polarised gauge bosons can suffer from such behaviour away from the SM limit. We note here that whilst $tHj$ essentially always proceeds through the $b\,W\to t\,h$ sub-amplitude, the $b\,W\to t\,Z$ sub-amplitude is not the only one contributing to $tZj$ as the $Z$ can be emitted from light quark lines. In the framework of SMEFT, the high energy behaviour of the $2\to2$ sub-amplitudes as a function of the Wilson coefficients for $b\,W\to t\,h$ and $b\,W\to t\,Z$ for the operators in Eqs. (\[oplist1\]-\[oplist3\]) are shown in Tables \[tab:bwth\] and \[tab:bwtz\] respectively. One can see that the energy growth due to higher dimensional operators can arise from sources other than additional derivatives, [*i.e*]{}, from the top Yukawa and Higgs-fermion current operators. The former operator only modifies the SM Higgs-top coupling and its energy dependence is a manifestation of the previously discussed unitarity violating behaviour. The latter operators both modify the SM gauge boson coupling to fermions and induce a $f\bar{f}VH$ contact term as in the last diagram of Fig. \[fig:feyndiags\]. A more complete study of the full set of top & EW $2\to2$ subamplitudes in the SMEFT and associated LHC processes is on going and will appear in future work. In the meantime we keep these tables for reference and to help put into context the energy dependence of the results of our predictions. Overall, the possible energy enhancements in these channels suggest that, although $tZj$ and particularly $tHj$ are rare processes in the SM, such behaviour might nevertheless lead to interesting constraints on the operators studied, especially at differential level. Constraints on dim-6 operators {#section:constraints} ============================== In order to examine the sensitivity of our processes to SMEFT operators we first consider the current limits on the dim-6 operators of interest. We briefly summarise the current constraints in Table \[tab:constraints\]. Firstly, all top-quark operators can be constrained using collider measurements. For example, the TopFitter collaboration has performed a global fit (excluding ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$) at LO using both the Tevatron and the LHC data [@Buckley:2015lku]. Individual limits are given for each operator, by setting other operator coefficients to zero. Marginalised constraints are provided for ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}\varphi Q}}^{{\scriptscriptstyle}(3)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$, and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tG}}$, while the remaining operator constraints are too weak due to large uncertainties in $pp\to t\bar tZ$ and $pp\to t\bar t\gamma$ measurements. One can see that ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tG}}$ is already significantly better constrained than its weak counterparts. In addition, the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}\varphi tb}}$ operator gives rise to right handed $Wtb$ coupling, which is constrained at tree-level by top decay measurements and indirectly at loop-level by $B$ meson decay and $h\to b\bar b$ [@Alioli:2017ces]. The electroweak and top-quark Yukawa operators ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}W}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}\varphi W}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t\varphi}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}HW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}HB}}$ are constrained by a combined fit including Higgs data and TGC measurements at both LEP and LHC, presented in Ref. [@Butter:2016cvz]. For the Yukawa operator $\mathcal{O}_{t\varphi}$, we follow the approach in Ref. [@Maltoni:2016yxb], and update the analysis with the recent $t\bar{t}H$ measurements at 13 TeV in Refs. [@CMS:2017lgc; @CMS:2017vru; @Aaboud:2017jvq; @Aaboud:2017rss], obtaining a confidence interval of ${c_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}\subset[-6.5,1.3]$. Note that we do not use the $gg\to H$ process. Even though this process could impose strong constraints on the coefficient of $\mathcal{O}_{t\varphi}$, the effect is loop-induced, and so we consider it as an indirect constraint. The constraints on the color singlet and octet four-fermion operators are obtained from single-top and $t\bar{t}$ measurements [@Zhang:2017mls] respectively. Although the color octet operator interferes with the SM $q\bar{q}\to t\bar{t}$ amplitude, the sensitivity of the process to this operator is diluted by the dominantly $gg$-induced SM component. Even though this operator does not interfere with the SM single-top amplitude, the sensitivity from the pure EFT squared contribution is still significantly better than that of $t\bar{t}$. Combining a set of LHC measurements of single top (and anti-top) production [@Chatrchyan:2012ep; @Khachatryan:2014iya; @Aad:2014fwa; @Aad:2015upn; @Khachatryan:2016ewo; @Aaboud:2016ymp; @CMS:2016ayb; @Aaboud:2017pdi], we obtain a significant improvement on the confidence interval, ${c_{{\scriptscriptstyle}Qq}}^{{\scriptscriptstyle}(3,8)}\subset[-1.40,1.20]$. The cross-section dependence is obtained from our model implementation at NLO in QCD. Finally, the precision electroweak measurements provide indirect limits on top-quark operators at the one-loop level. Electroweak operators to which they mix under RG running are required to be included in a global fit, but constraints on top-quark operators can be obtained by marginalising over these operators [@Zhang:2012cd]. Calculation setup and numerical results {#sec:setup} ======================================= Our computation is performed within the [MG5\_aMC]{} framework [@Alwall:2014hca] with all the elements entering the NLO computations available automatically starting from the SMEFT Lagrangian [@Alloul:2013bka; @deAquino:2011ub; @Degrande:2011ua; @Degrande:2014vpa; @Hirschi:2011pa; @Frederix:2009yq]. In addition to the SM-like scale and PDF uncertainties, we also compute the uncertainties due to missing higher orders in the $\alpha_s$ expansion of the EFT operators, following the procedure described in [@Maltoni:2016yxb]. Therein, a second renormalisation scale, $\mu_{EFT}$, is introduced such that the EFT renormalisation scale can be varied independently from the QCD one. The cross section can be parametrised as: =\_[SM]{}+\_iC\_i\_i +\_[ij]{} C\_iC\_j\_[ij]{}. \[eq:xsecpara\] We provide results for $\sigma_i$ and $\sigma_{ij}$ for the LHC at 13 TeV in the 5-flavour scheme. Results are obtained with NNPDF3.0 LO/NLO PDFs [@Ball:2014uwa], for LO and NLO results respectively; input parameters are &m\_t=172.5 , m\_H=125 , m\_Z=91.1876 ,\ &\_[EW]{}\^[-1]{}=127.9, G\_F=1.1663710\^[-5]{} \^[-2]{}. \[eq:input\] Central scales for $\mu_R,\mu_F,\mu_{EFT}$ are chosen as $(m_t+m_H)/4$ for the $tHj$ process following the discussion in [@Demartin:2015uha], and correspondingly $(m_t+m_Z)/4$ for the $tZj$. Three types of uncertainties are computed. The first is the standard scale uncertainty, obtained by independently setting $\mu_R$ and $\mu_F$ to $\mu/2$, $\mu$ and $2\mu$, where $\mu$ is the central scale, obtaining nine $(\mu_R,\mu_F)$ combinations. The second uncertainty comes from the NNPDF3.0 sets. The third one is the EFT scale uncertainty, representing the missing higher-order corrections to the operators, obtained by varying $\mu_{EFT}$, taking into account the effect of running of the Wilson coefficients from the central scale up to this new scale. This uncertainty is obtained for contributions involving the $({{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}},{{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}},{{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}})$ operators which are the ones that run under QCD as discussed in Section \[section:Operators\]. Inclusive results \[sec:inclusive\] ----------------------------------- Results for the $tHj$ and $tZj$ cross section from individual operators are shown in Tables \[tHj:sigma\] and \[tZj:sigma\] along with the corresponding uncertainties and $K$-factors. We note here that our results refer to the sum of the top and anti-top contributions. Central values for the cross-terms between the different operators are reported in Tables \[tHj:sigma\_crs\] and  \[tZj:sigma\_crs\]. Several observations are in order. First we notice that the $K$-factors vary a lot between operator contributions. As we work in the 5-flavour scheme, the $b$-quark is massless, and therefore ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$ does not interfere with the SM or any of the other operators. We also find that typically the relative EFT contributions to $tHj$ are larger than for $tZj$, as the Higgs always couples to the top or the gauge bosons, whilst the $Z$ can be also be emitted from the light quark lines thus being unaffected by modifications of the top-$Z$ and triple gauge boson interactions. For $tZj$ some interferences between operators are suppressed and our results can suffer from rather large statistical errors as these contributions are extracted from Monte Carlo runs which involve all relevant SM, $\mathcal{O}\left(1/\Lambda^2 \right)$ and $\mathcal{O}\left(1/\Lambda^4 \right)$ terms arising from a given combination of couplings. In general, we see that the NLO corrections reduce the theory uncertainties and that the EFT scale uncertainty is typically subdominant. One striking case stands out in which the scale uncertainty for the inclusive interference contribution from ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ to $tHj$ grows significantly. This can be understood by looking at the differential level and noticing that there is a very strong cancellation over the phase space such that the contribution to the total rate coming from the interference almost cancels. Figure \[fig:top\_pt\_sqint\] shows the top $p_T$ distributions of the interference and squared contributions at LO and NLO. Clearly, the cancellation is even more exact at NLO and leads to large scale uncertainties in the inclusive result and the unusual $K$-factor of 0.2. A partial cancellation effect is also present for the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ interference contribution at LO, which is reduced at NLO, leading to the correspondingly large $K$-factor. This is best seen from the top-Higgs invariant mass distribution also shown in Figure \[fig:top\_pt\_sqint\]. ![\[fig:top\_pt\_sqint\] Differential cross-section contributions to $tHj$ from ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ and similarly for the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ contribution to $tZj$, all for values of 1 TeV$^{-2}$ of the corresponding Wilson coefficient. Hatched and solid bars represent the LO and NLO predictions respectively. The subplots show the relative theory uncertainty from scale variation and PDFs of each contribution. ](Figures/top_pt_sqint_atphi.pdf "fig:"){width="0.45\linewidth"} ![\[fig:top\_pt\_sqint\] Differential cross-section contributions to $tHj$ from ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ and similarly for the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ contribution to $tZj$, all for values of 1 TeV$^{-2}$ of the corresponding Wilson coefficient. Hatched and solid bars represent the LO and NLO predictions respectively. The subplots show the relative theory uncertainty from scale variation and PDFs of each contribution. ](Figures/mth_sqint_a3phidQL.pdf "fig:"){width="0.45\linewidth"} ![\[fig:top\_pt\_sqint\] Differential cross-section contributions to $tHj$ from ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ and similarly for the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ contribution to $tZj$, all for values of 1 TeV$^{-2}$ of the corresponding Wilson coefficient. Hatched and solid bars represent the LO and NLO predictions respectively. The subplots show the relative theory uncertainty from scale variation and PDFs of each contribution. ](Figures/mtz_sqint_tzj_atW.pdf "fig:"){width="0.45\linewidth"} As for $tZj$, we observe qualitatively similar results moving from LO to NLO. In some cases, for numerically very small contributions coming from interference terms between operators, the theory uncertainties are inflated due to lack of MC stats. The main unexpected result is the $K$-factor of 5 for the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ interference contribution. The top-$Z$ invariant mass distribution in Figure \[fig:top\_pt\_sqint\], does indicate a cancellation over the full phase space which disappears at NLO. This is in part due to cancellations in the interference contributions to $tZj$ and $\bar{t}Zj$, which are summed over in our results. Considering the existing limits on the Wilson coefficients summarised in Table \[tab:constraints\] in combination with the information in Tables \[tHj:sigma\] and \[tZj:sigma\] suggests that there is still much room for observable deviations in both processes and therefore that they may be used to further constrain the SMEFT parameter space. For example, saturating the current limits on the weak dipole operators, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$, leads to 20% deviations in the inclusive $tZj$ cross section at NLO while for $tHj$, the corresponding effects of ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and the top-Yukawa operator, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$, are around 300%. Deviations to $tZj$ are generally possible within current limits at the level of up to 20% while, for $tHj$, order one effects can additionally be accommodated for the right handed charged current operator. Given the weak limits on the operators in question the large cross-section contributions are dominated by the EFT-squared term. It is instructive to put these calculations into context by comparing to the $t$-channel single top production process, which is a common sub-process of both processes studied in this work. Table \[tab:sensitivity\] compares the interferences and squared contributions at NLO, relative to the SM, of the operators common to the $tHj$, $tZj$ and $t$-channel single top processes. We observe the expected enhancement of the relative contribution of the four fermion operators with respect to single top due to the higher kinematic thresholds involved. This is confirmed by adding a minimum $p_T^{t}$ such that the cross sections of $tj$ ($tZj$) becomes comparable to that of $tZj$ ($tHj$), which shows that $tj$ is likely to provide tighter constraints for these operators once the high $p_T^{t}$ regime is measured at 13 TeV. The behaviour of the sensitivity between the inclusive and high energy regions of each operator in $tZj$ is in line with the expectations from the $2\to2$ sub-amplitudes shown in Table \[tab:bwtz\]. One can confirm that the left handed quark current operator ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ is significantly enhanced both at interference and squared level by the $p_T^t$ cut. Interestingly, this is in contrast to the case of single top production or top decay, where this operator only shifts the SM $Wtb$ vertex, not leading to any energy growth. The $tZj$ process provides a new source of energy dependence and therefore sensitivity to this operator that, as we will show in Section \[sec:sensitivity\], may lead to potentially improved constraints in the near future. In the case of the weak dipole and RHCC operators, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$, the leading energy growth is confirmed to arise from the squared contribution. For ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$, the high energy behaviour is enhanced with respect to single top. As discussed in Section \[sec:energybehaviour\], the interferences of the configurations that have energy growth from ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$, are counterbalanced by an inverse dependence in the corresponding SM amplitudes, leading to the expected result that these pieces do not grow with energy. Differential distributions \[sec:differential\] ----------------------------------------------- Given the promising effects observed in the inclusive cross-section predictions as well as Table \[tab:sensitivity\], one expects even more striking deviations at differential level. This allows us to further investigate the energy dependence of the contributions from the various operators, comparing this to the expectations from the $2\to2$ helicity sub-amplitude calculations summarised in Tables \[tab:bwth\] and \[tab:bwtz\]. In order to showcase this, we present differential results in top $p_T$ and top-Higgs/$Z$ invariant mass for a number of benchmark scenarios, switching on one operator at a time to a value roughly saturating the tree-level, individual limits presented in Table \[tab:constraints\]. Individual limits are chosen for a fair representation since we are only switching on one operator at a time while indirect, loop-level limits are not taken into account since we are quantifying direct effects from SMEFT operators to these LHC processes. A selection of distributions are shown in Figures \[fig:thj\_distributions\] and \[fig:tzj\_distributions\]. The already large effects at inclusive level are amplified in the tails of the $p_T$ distributions, with significant energy growth present in all distributions shown. The $tHj$ deviations reach factors of many in the tails, while for $tZj$, the 20% inclusive effects become a factor of a few in the high energy bins. There is therefore a complementarity between the two processes since, although the largest effects are present in $tHj$, the process is comparatively rare and may not be probed differentially at the LHC, at least until the late high-luminosity phase. $tZj$, however has a ten times larger cross section and could therefore gather enough statistics for differential measurements and an enhanced sensitivity to the operators in question. ![\[fig:thj\_distributions\] Differential distributions of the top $p_T$ and top-Higgs system invariant mass for the $tHj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{top_pt_thj_atW+2.5}.pdf "fig:"){width="0.45\linewidth"} ![\[fig:thj\_distributions\] Differential distributions of the top $p_T$ and top-Higgs system invariant mass for the $tHj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{mth_atW+2.5}.pdf "fig:"){width="0.45\linewidth"}\ ![\[fig:thj\_distributions\] Differential distributions of the top $p_T$ and top-Higgs system invariant mass for the $tHj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{top_pt_thj_atphi-6.5}.pdf "fig:"){width="0.45\linewidth"} ![\[fig:thj\_distributions\] Differential distributions of the top $p_T$ and top-Higgs system invariant mass for the $tHj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{mth_atphi-6.5}.pdf "fig:"){width="0.45\linewidth"} ![\[fig:tzj\_distributions\] Differential distributions of the top $p_T$ and top-$Z$ system invariant mass for the $tZj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{top_pt_tzj_atB-7}.pdf "fig:"){width="0.45\linewidth"} ![\[fig:tzj\_distributions\] Differential distributions of the top $p_T$ and top-$Z$ system invariant mass for the $tZj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{mtz_atB-7}.pdf "fig:"){width="0.45\linewidth"}\ ![\[fig:tzj\_distributions\] Differential distributions of the top $p_T$ and top-$Z$ system invariant mass for the $tZj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{top_pt_tzj_aphitb+5}.pdf "fig:"){width="0.45\linewidth"} ![\[fig:tzj\_distributions\] Differential distributions of the top $p_T$ and top-$Z$ system invariant mass for the $tZj$ process for given values of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$ operator coefficients roughly saturating current individual, direct limits. The lower insets show the scale and PDF uncertainty bands, the ratio over the SM prediction and finally the corresponding $K$-factor. ](Figures/{mtz_aphitb+5}.pdf "fig:"){width="0.45\linewidth"} Current and future sensitivity {#sec:sensitivity} ------------------------------ The two most recent measurements of the $tZj$ process [@Aaboud:2017ylb; @Sirunyan:2017nbr] allow for a first sensitivity assessment of this process to the EFT coefficients of interest at the inclusive level. The experiments perform fits to the signal strength, $\mu$, with respect to the SM expectation in this channel to extract the measured cross section. In order to eliminate some dependence on the overall normalisation and reduce scale uncertainties, we construct confidence intervals on the Wilson coefficients by performing a $\Delta\chi^2$ fit to the signal strength directly rather than the measured cross section. The ratio of the $tZj$ cross section over the SM one as a function of the Wilson coefficients is taken from the results of Table \[tZj:sigma\] and compared to the observed values of $\mu=0.75\pm0.27$ and $1.31\pm0.47$ reported by CMS and ATLAS respectively, where the uncertainty is taken to be the sum in quadrature of the statistical and systematic components. Both measurements are made searching for the electron and muon decay modes of the $Z$-boson on-shell, [*i.e.*]{}, including a cut on the dilepton invariant mass. We therefore take into account the modification of these branching fractions in the presence of the ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(1)}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ operators. Note that this procedure is rather simplistic and uncertain given the complexity of the $tZj$ measurement the LHC. Firstly, due to the relatively small rates and large potential background contributions, multivariate analysis methods are employed to improve the signal to background ratio. The efficiency and acceptance factors that are used in the extrapolation to the full phase space apply strictly to the SM kinematics and may be different in general for the EFT. One is only truly sensitive to enhancements of the cross section in the observed fiducial region after selection requirements. Furthermore, the signal yields are fitted using templates for the multivariate classifier output, which may also differ between the SM and EFT. Finally, many of the backgrounds considered in this analysis would also be affected non-negligibly by the presence of the same operators. The dominant di-boson background, for example would be modified by ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}W}}$ while several others, such as $t\bar{t}V$, $t\bar{t}H$ and $tWZ$ would get affected by a combination of top and EW operators. Our confidence intervals are obtained neglecting all of these effects and should therefore be viewed as approximate sensitivity estimates. Figure \[fig:tzj\_sensitivity\] (a) reports the obtained confidence intervals compared to the existing individual limits from Table \[tab:constraints\]. In most cases, the current inclusive measurement does not probe the operators beyond existing limits. The single exception is in the case of the weak dipole operator, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$. The enhanced relative squared dependence on this operator leads to a slightly improved sensitivity over the individual limit obtained from a combination of LHC Run 1 single-top and $W$ helicity fraction measurements. The differential results of Section \[sec:differential\] indicate that more information may be provided by a future measurement of this process, particularly at high $p_T$. In order to test this, we consider a hypothetical future measurement of $tZj$ in the high energy region, in which the top transverse momentum is required to be above 250 GeV. In the SM, the predicted cross section at NLO in this phase space region is 69 fb, roughly a factor 10 smaller compared to the inclusive prediction. Remaining agnostic about the nature of a future analysis, we assume that such a cross section should be attainable with the same precision as the current measurement with about 10 times more data. This suggests that one could expect this level of sensitivity in the early stages of the high-luminosity LHC run. Our projected sensitivities, shown in Figure \[fig:tzj\_sensitivity\] (c), are obtained assuming the SM prediction, $\mu=1$, observed by both experiments and taking the same uncertainties as for the inclusive measurement. As expected, we see significant improvements, particularly for ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tW}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}tB}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}tb}}$, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}Qq}}^{{\scriptscriptstyle}(3,1)}$ and ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}Qq}}^{{\scriptscriptstyle}(3,8)}$, that may reach beyond the current limits summarised in Table \[tab:constraints\]. Considering the high energy growth of the sub amplitudes of Table \[tab:bwtz\], one can see that the large relative gains in sensitivity all occur for operators with the strongest energy growths while for operators without many enhanced helicity configurations such as ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}t}}$ or ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(1)}$ do not benefit at all. We note in particular the improvement on the limit on ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ due to the unitarity violating behaviour of the amplitude at high-energy, a feature not present in single top production where ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$ uniformly rescales the cross section, as discussed in Section \[sec:energybehaviour\]. Although the four-fermion operators can be constrained significantly better than from Run 1 single top, we expect that forthcoming Run 2 single-top measurements will constrain such operators better. As of today, the $tHj$ process has yet to be measured in isolation at the LHC. However, several searches have been performed in which this process is a part of the signal selection [@CMS-PAS-HIG-16-019; @CMS:2017uzk]. The former sets an upper limit of 113 times the SM prediction on the combination of $tHj$ and $tHW$ processes with 2.3 fb$^{-1}$ of integrated luminosity while the latter additionally includes the $t\bar{t}H$ process and obtains a combined signal strength for the SM hypothesis of $\mu=1.8\pm0.67$ with 35.9 fb$^{-1}$. Since the former analysis lacks sensitivity due to the small dataset used, we use the second measurement to estimate current sensitivity to the $tHj$ process, accepting a large amount of pollution from $t\bar{t}H$. In this case we assume that only the $tHj$ process is modified apart from the contribution to $t\bar{t}H$ from the top Yukawa operator, ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}t{\ensuremath{\varphi}}}}$, obtained from [@Maltoni:2016yxb]. This operator affects the dominant, QCD-induced component of $t\bar{t}H$, while the other operators that we consider would only contribute to the EW component, which in the SM is more than two orders of magnitude below the QCD one. Similarly, the $tHW$ process is about five times smaller than $tHj$ in the SM. Furthermore, since the measurement targets the $\tau\tau$, $WW$ and $ZZ$ decay modes of the Higgs, we also take into account the effect of the modified branching ratios due to ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}W}}$ at LO. The sensitivity estimates from this measurement are shown in Figure \[fig:tzj\_sensitivity\] (b), and suggest that a significant improvement is needed to obtain relevant constraints on the operators of interest. Phenomenological studies on future $tHj$ prospects in the SM have been performed for the high-luminosity LHC run [@Biswas:2012bd; @Chang:2014rfa], concluding that it may be possible to access this mode with the full design integrated luminosity of 3 ab$^{-1}$. For our purposes, we consider the possibly optimistic scenario in which the process is measured with the same sensitivity as the current $tZj$ measurement, just to highlight the gain that would occur in this hypothetical case. Figure \[fig:tzj\_sensitivity\] (d) clearly shows a marked improvement. In the case of the dipole and RHCC, the potential sensitivity goes beyond that of the high-$p_T$ $tZj$, while for the four-fermion operators, the benefit of looking at the kinematic tails of $tZj$ outweighs the strong dependence of the inclusive $tHj$ cross section. Overall, the interesting individual sensitivity prospects concerning the operators included in our study mainly cover the weak dipole, RHCC and single-top four-fermion operators, with the sensitivity to most of the current-current, triple gauge and gauge-Higgs operators remaining below the existing limits from other measurements of less rare and already established processes such as single-top, diboson and Higgs production/decay. The main exception to this is with ${{\ensuremath{\mathcal{O}}}_{{\scriptscriptstyle}{\ensuremath{\varphi}}Q}}^{{\scriptscriptstyle}(3)}$, for which a new, interfering, energy growth arises and will lead to significant improvement on current sensitivities through high energy $tZj$ measurements. Nevertheless, when performing a global analysis and marginalising over the various operators, these processes may well provide some additional constraining power also in these directions towards the latter stages of the LHC lifetime. \ Conclusions =========== Electroweak production of a single top quark in association with a $Z$ or Higgs boson provides a natural opportunity to constrain possible deviations of the neutral couplings of the top quark with respect to the SM predictions. The motivations and interest for this class of processes are multifold. First, being mediated only by electroweak interactions at LO, they can be predicted accurately in perturbative QCD, already at NLO accuracy, and they are not affected by possible deviations in the QCD interactions (at LO). Second, these processes feature an enhanced sensitivity, appearing as a non-trivial energy dependence, also for operators that, per se, do not necessarily lead to interactions that grow with energy. This is due to the spoiling of delicate gauge cancellations that take place in the SM, when anomalous interactions are present. Last, but not least, these processes are of phenomenological interest, as they are already being studied at the LHC. In this work we have considered for the first time $tHj$ and $tZj$ in the context of the standard model effective field theory, in the presence of all the relevant dim-6 operators. We have included NLO QCD corrections and studied the relevant theoretical uncertainties on our predictions. As expected, while not very large in general, QCD corrections typically reduce the theoretical uncertainties and can lead to non-flat $K$-factors for differential observables. Using the measurements of the signal strengths of these processes at the LHC we have performed a first sensitivity study allowing one non-zero operator coefficient at a time. This study can be therefore considered the first necessary step before performing a global fit. Whilst at the moment the constraints from $tZj$ measurements cannot compete with the already existing limits on the operators of interest, there is enough evidence that complementary constraints could be obtained within the projected experimental accuracies. Given the promising signs found already at the inclusive level, we have examined the impact of the dim-6 operators on differential observables such as the top-quark transverse momentum and the invariant mass of the top-quark-$H/Z$ system. We have found that the effects on the total cross section are typically amplified at the tails of distributions leading to allowed deviations from the SM predictions of a factor of a few. We have argued that this behaviour is directly related to the energy behaviour of the relevant sub-amplitudes $b\,W\to t\,h$ and $b\,W\to t\,Z$ involved in $tHj$ and $tZj$, respectively, which we have also reported in detail. New sources of energy growth not present in, [*e.g.*]{}, single top production are identified and exploited in our sensitivity studies. Our findings support extracting useful constraints from inclusive and/or differential measurements of the $tHj$ and $tZj$ processes, which are expected at the high-luminosity LHC. For example, given the current constraints on the weak dipole and right handed charged current operators, very large deviations can be still expected in both $tZj$ and $tHj$. In addition, the information that could be extracted on the Yukawa operator could also become competitive with enough integrated luminosity. Whilst not discussed in this work, we have also verified that $t \gamma j$ displays similar sensitivities as $tZj$ to the same class of dim-6 operators. A dedicated study of this process with the goal of motivating a measurement at the LHC, which to our knowledge is not being pursued yet, is ongoing. In summary, we have proposed to use measurements of $tZj$ and $tHj$ at the LHC to constrain the least known operators in the SMEFT, [*i.e.*]{}, those involving top-quark, gauge and Higgs interactions. We have computed $tHj$ and $tZj$ cross sections in the SMEFT at NLO in QCD, achieving for the first time such an accuracy for processes where the three types of operators, namely, purely gauge, two-fermion and four-fermion operators, can contribute. This work proves that it is now possible to obtain NLO accurate predictions automatically for any dim-6 operator and process involving top-quarks, weak bosons and Higgs final states and therefore paves the way to performing global SMEFT fits at the LHC. We would like to thank Liam Moore and Ambresh Shivaji for discussions. FM has received fundings from the European Union’s Horizon 2020 research and innovation programme as part of the Marie Skłodowska-Curie Innovative Training Network MCnetITN3 (grant agreement no. 722104) and by the F.R.S.-FNRS under the ‘Excellence of Science‘ EOS be.h project n. 30820817. EV and KM are supported by a Marie Skłodowska-Curie Individual Fellowship of the European Commission’s Horizon 2020 Programme under contract numbers 704187 and 707983, respectively. CZ is supported by IHEP under Contract No. Y7515540U1. Computational resources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI). [^1]: We have explicitly verified that $t\gamma j$ production displays, in fact, similar sensitivities to new neutral gauge and top-quark interactions as $tZj$ and that the corresponding predictions at the LHC can be automatically obtained at NLO in QCD in our framework. As no dedicated experimental analysis of this process is available yet, we defer a detailed study to the future.
--- abstract: 'We give a systematic treatment of a spin $1/2$ particle in a combined electromagnetic field and a weak gravitational field that is produced by a slowly moving matter source. This paper continues previous work on a spin zero particle, but it is largely self-contained and may serve as an introduction to spinors in a Riemann space. The analysis is based on the Dirac equation expressed in generally covariant form and coupled minimally to the electromagnetic field. The restriction to a slowly moving matter source, such as the earth, allows us to describe the gravitational field by a gravitoelectric (Newtonian) potential and a gravitomagnetic (frame-dragging) vector potential, the existence of which has recently been experimentally verified. Our main interest is the coupling of the orbital and spin angular momenta of the particle to the gravitomagnetic field. Specifically we calculate the gravitational gyromagnetic ratio as $\text{ g}_\text{ g}=1$ ; this is to be compared with the electromagnetic gyromagnetic ratio of $\text{g}_e=2$ for a Dirac electron.' author: - 'Ronald J. Adler' - Pisin Chen - Elisa Varani title: Gravitomagnetism and spinor quantum mechanics --- Introduction ============= Classical systems in external gravitational fields have been studied for centuries, and recently the existence of the gravitomagnetic (or frame-dragging) field caused by the earth’s rotation has been observed by the Gravity Probe B (GPB) satellite [@Adler2006; @WILL1993; @Ohanian1976; @Everitt2011]. GPB verified the prediction of general relativity for the gravitomagnetic precession of a gyroscope in earth orbit ($42$ mas/yr) to better than $20\%$ [@Will43]. Previously, observations of the LAGEOS satellites also indicated the existence of the gravitomagnetic interaction via its effect on the satellite orbits [@Will43; @Ciufolini1997]. Analysis of the LAGEOS data involves modeling classical effects to very high accuracy in order to extract the gravitomagnetic effect, and the accuracy of the results has been questioned by some authors [@Iorio2004]. Analysis of the GPB data also requires highly accurate modeling of classical effects [@Will43]. While gravitomagnetic effects are generally quite small in the solar system it is widely believed that they may play a large role in jets from active galactic nuclei, so their experimental verification is of more than theoretical interest [@Throne2009]. At the other end of the interest spectrum extensive theoretical work has been done on quantum fields in classical background spaces, the most well known being related to Hawking radiation from black holes [@Hawking1970; @Birell1982; @Adler2001; @Adler2006]. However it is important to keep in mind that Hawking radiation has never been observed. Interesting experimental work has also been done on quantum systems in the earth’s gravitational field, such as neutrons interacting with the earthÕs Newtonian field and atom interferometer experiments aimed at accurately testing the equivalence principle and other subtle general relativistic effects [@Nesvizhevdky2002; @Dimopoulos2008; @Weinberg1972]. There has been some discussion of attempts to see gravitomagnetic effects with these devices but such experiments would be quite difficult due to the small size of the effects and the similarity to classical effects of rotation; this is to be expected since gravitomagnetism manifests itself in a way that is quite similar to rotation, hence the appellation “frame dragging.” Laboratory detection of gravitomagnetic effects on a quantum system would clearly be of fundamental interest. In this work we give a systematic treatment of a spin $1/2$ particle in a combined electromagnetic field and weak gravitational field; this continues the work of reference [@Adler2010]. We describe the particle with the generally covariant Dirac equation in a Riemann space, minimally coupled to the electromagnetic field in the standard gauge invariant way [@Bjorken1964; @Lawrie1990]. The weak gravitational field is naturally treated according to linearized general relativity theory, and we also assume a slowly moving matter source, such as the earth [@Misner1970; @Adler1999; @Adler2000]. Within this approximation the gravitational field is described by a gravitoelectric (or Newtonian) potential and a gravitomagnetic (or frame-dragging) vector potential, and the field equations are quite analogous to those of classical electromagnetism. We thus refer to it as the gravitoelectromagnetic (GEM) approximation. Our special emphasis throughout this paper is on the gravitomagnetic interaction. The paper is organized as follows. After brief review comments on the GEM approximation (section 2) and the Dirac equation in flat space (section 3) we give a detailed discussion of generally covariant spinor theory and the Dirac equation, using the standard approach based on tetrads (sections 4 and 5). We then obtain the limit of the Dirac Lagrangian and the Dirac equation for a weak gravitational field and discuss its interpretation in terms of an energy-momentum tensor (section 6). Our discussion of generally covariant spinors and the generally covariant Dirac equation is largely self-contained, and may serve as an introduction to the subject for uninitiated readers. In section 6 we also observe that the non-geometric or “flat space gravity” approach of Feynman, Weinberg and others does not appear to be completely equivalent to linearized general relativity theory in its coupling to spin [@Feynman1965]. We have not found this discussed in the literature. Using the weak gravitational field results we then obtain the non-relativistic limit of the theory (section 7). We do this by integrating the interaction Lagrangian to obtain the interaction energy of the spinor particle with the electromagnetic and the GEM fields, and from that obtain the non-relativistic interaction energies. This allows us to read off, in a simple and intuitive way, the interaction terms that one could use in a non-relativistic Hamiltonian treatment. In particular we obtain (section 8) the usual anomalous g-factor of the electron $\text{g}_e=2$ and the analogous result for the gravitomagnetic g-factor of a spinor, which is $\text{ g}_\text{ g}=1$. Section 8 also contains brief comments on the numerical value of some interesting and conceivably observable quantities such as the precession of a spinning particle in the earth’s gravitomagnetic field and its relation to the precession of a macroscopic gyroscope; such precession appears to be universal for bodies with angular momentum. The phase shift in an atom interferometer is also mentioned as an experiment that could, in principle, show the existence of the gravitomagnetic field. Lastly it is worth noting what we do [*not*]{} do in this paper. We study the effect of the gravitational field on a quantum mechanical spinor but not the effect of the spinor on the gravitational field; thus the work does not relate to quantum gravity or quantum spacetime [@Oriti2009]. Similarly we do not consider torsion, in which the affine connections have an anti-symmetric part and are not equal to the Christoffel symbols. Torsion does not prove necessary in our discussion, but some authors believe it is necessary in describing the effects of spin on gravity [@Trautman2010]. The gravitoelectromagnetic (GEM) approximation ============================================== In previous work we discussed linearized general relativity theory for slowly moving matter sources like the earth [@Adler2010; @Adler1999; @Adler2000]. Here we summarize the results very briefly. The metric may be written as the Lorentz metric plus a small perturbation, $$\begin{aligned} \text{g}_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.\end{aligned}$$ We use coordinate freedom to impose the Lorentz gauge condition $$\begin{aligned} (h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h)^{|\nu}=0,\end{aligned}$$ where the single slash denotes an ordinary derivative. Then the field equations of general relativity tell us that the metric perturbation may be written as $$\begin{aligned} h_{\mu\nu}= \left( \begin{array}{cccc} 2\phi& h^1 & h^2 & h^3 \\ h^1 & 2\phi & 0 & 0 \\ h^2 & 0 & 2\phi & 0 \\ h^3 & 0 & 0 & 2\phi \end{array} \right),\text{\space\space} h_{00}=2\phi,\text{\space\space} h_{0k}=h^k,\end{aligned}$$ where $\phi$ is the Newtonian or gravitoelectric potential and $h^k \leftrightarrow\vec{h}$ is the gravitomagnetic potential. For slowly moving sources the field equations and the Lorentz condition become$$\begin{aligned} \nabla^2\phi=4\pi G\rho,\text{\space\space}\nabla^2h^j=-16\pi G \rho v^j,\text{\space\space}4\dot{\phi}-\nabla\cdot\vec{h}=0,\text{\space\space}\dot{\vec{h}}=0,\end{aligned}$$ where $\rho$ is the source mass-energy density and $v^j$ is its velocity. The physical fields, which exert forces on particles, are the gravitoelectric (or Newtonian) field and the gravitomagnetic (or frame-dragging) field, which are defined by $$\begin{aligned} \vec{\text{g}}=\nabla\phi,\text{\space\space}\vec{\Omega}=\nabla \times\vec{h}.\end{aligned}$$ We call this equation system the gravitoelectromagnetic or GEM limit because of its close similarity to classical electromagnetism. Flat space Dirac equation and the non-relativistic limit ======================================================== In this section we discuss the Dirac equation in the flat space of special relativity and recast it into a Schroedinger equation form (SEF), which provides one convenient way to obtain the non-relativistic limit [@Bjorken1964]. The SEF is exact and involves only the upper two components of the spinor wave function - the relevant components for positive energy solutions in the non-relativistic limit. One reason for doing this is to serve as a basis of comparison for the alternative method we will use in section 6 when we discuss gravitational interactions. Throughout this section $\gamma^\mu$ denotes the flat space Dirac matrices [@Peskin1995; @Bjorken1964]. The Dirac Lagrangian and the Euler-Lagrange equations that follow from it are $$\begin{aligned} &L=a\bar{\psi}(i\gamma^\mu\vec{\partial_\mu}-m)\psi+b\bar{\psi}(-i\gamma^\mu\overleftarrow{\partial}_\mu-m)\psi-eA_\mu\bar{\psi}\gamma^\mu\psi,\\ &(i\gamma^\mu\partial_\mu-m)\psi=eA_\mu\gamma^\mu\psi,\text{\space\space}\bar{\psi}(-i\gamma^\mu\overleftarrow{\partial}_\mu m)=eA_\mu\bar{\psi}\gamma^\mu.\end{aligned}$$ The spinor and its adjoint are considered independent in obtaining (3.1b). The constants $a$ and $b$ are arbitrary, so long as $a+b\neq0$ . The $\gamma^\mu$ obey the flat space Dirac algebra, $$\begin{aligned} \{\gamma^\mu,\gamma^\alpha\}=2\eta^{\mu\nu}I.\end{aligned}$$ The adjoint spinor is assumed to be related to the spinor by a linear metric relation, $\bar{\psi}=\psi^\dagger M$ where M is to be determined; consistency of the equations (3.1b) is then assured if M obeys $$\begin{aligned} M^{-1}\hat{\gamma}^{\mu^\dagger}M=\hat{\gamma}^\mu,\text{\space\space}M^{-1}=M=\hat{\gamma}^0,\text{\space\space}\bar{\psi}=\psi^\dagger\gamma^0.\end{aligned}$$ Eq. (3.2) is easy to verify for the choice of gamma matrices given below in (3.4). The Hamiltonian form of the Dirac equation will be useful for studying interaction energies in this section. It is gotten by multiplying (3.1) by $\gamma^0$ to obtain $$\begin{aligned} i\partial_t\psi=\beta m\psi+V+\vec{\alpha}\cdot\vec{\Pi}\psi,\text{\space\space}\beta\equiv\gamma^0,\text{\space\space}\alpha\equiv\gamma^0\gamma^k,\text{\space\space}\vec{p}\equiv-i\nabla.\end{aligned}$$ Pauli’s choice of gamma matrices is natural for our later discussion of the non-relativistic limit, $$\begin{aligned} \beta=\gamma^0=\left( \begin{array}{cc} I & 0 \\ 0 & I \end{array} \right), \text{\space\space}\gamma^i=\left( \begin{array}{cc} 0 & \sigma^i \\ -\sigma^i & 0 \end{array} \right),\text{\space\space}\vec{\alpha}\equiv\left( \begin{array}{cc} 0 & \sigma \\ \sigma & 0 \end{array} \right).\end{aligned}$$ Next we break the 4-component wave function $\psi$ into two 2-component Pauli spinor wave functions and also factor out the time dependence due to the rest mass by substituting $$\begin{aligned} \psi=e^{-imt}\left( \begin{array}{c} \Psi \\ \varphi \end{array} \right),\end{aligned}$$ which leads to the coupled equations, $$\begin{aligned} i\partial_t\Psi=V\Psi+(\vec{\sigma}\cdot\vec{\Pi})\varphi,\text{\space\space}i\partial_t\phi+2m\varphi-V\varphi=(\vec{\sigma}\cdot\vec{Pi})\Psi.\end{aligned}$$ We are interested in $\Psi$ so we solve for $\varphi$ , and obtain symbolically, $$\begin{aligned} i\partial_t\Psi=V\Psi+(\vec{\sigma}\cdot\vec{\Pi})(2m-V+i\partial_t)^{-1}(\vec{\sigma}\cdot\vec{\Pi})\Psi,\\ \varphi=(2m-V+i\partial_t)^{-1}(\vec{\sigma}\cdot\vec{\Pi})\Psi.\end{aligned}$$ The inverse operator $(2m-V+i\partial_t)^{-1}$ may be defined by its expansion in the time derivative, as discussed in Appendix A. Note that Eq. (3.7a) is an exact equation for $\Psi$, although it is of infinite order in the time derivative. For the special case of a free particle the operator factors on the right side of (3.7a) commute and it becomes simply $$\begin{aligned} i\partial_t\psi=(i\partial_t+2m)^{-1}\vec{p}^2\psi.\end{aligned}$$ However the operators on the right side of (3.7a) will not in general commute unless the field $A_{\mu}$ is constant. In a low velocity system the time variations of $\Psi$ and $V$ are associated with non-relativistic energies, which are much less than the rest energy $m$, so we may approximate (3.7a) by $$\begin{aligned} i\partial_t\Psi=V\psi+\frac{(\vec{\sigma}\cdot\vec{\Pi})^2}{2m}\Psi.\end{aligned}$$ This is the Schroedinger equation for spin $1/2$ particles, often called the Pauli equation. The Pauli equation shows clearly how the spin and orbital angular momentum interact with the magnetic field. Pauli spin matrix algebra leads to an illuminating form for (3.9): to lowest order in ${e}$, $$\begin{aligned} i\partial_t\Psi&=V\Psi+\frac{\vec{\Pi}^2}{2m}\Psi-\frac{e\vec{B}\cdot\sigma}{2m}\Psi\notag\\ &=V\Psi+\frac{\vec{p}^2}{2m}\Psi-\frac{e\vec{A}\cdot\vec{p}}{m}\Psi-\frac{e\vec{B}\cdot\sigma}{2m}\Psi,\end{aligned}$$ where we have used the Lorentz gauge in which $\nabla\cdot\vec{A}=-\dot{A}^0$ and assumed the Coulombic $A^0$ has negligible time dependence. The gyromagnetic ratio or g-factor of a particle or system is defined in terms of its magnetic moment $\vec{\mu}$ and angular momentum $\vec{J}$ by $\vec{\mu}=\text{g}_e(e/2m)\vec{J}$ ; thus, from (3.10), the fact that the energy is $-\vec{\mu}\cdot\vec{B}$, and the electron spin of $\vec{S}=\sigma/2$ it is evident that the electron g-factor is g$_e=2$. The relative coupling of the spin and orbital magnetic moments is made most clear if we consider a magnetic field that is approximately constant over the size of the system, in which case we can choose $\vec{A}=(\vec{B}\times\vec{r})/2$ and find from (3.10) $$\begin{aligned} i\partial_t\Psi&=V\Psi+\frac{\vec{p}^2}{2m}\Psi-\frac{e\vec{B}}{2m}(2\vec{S}+\vec{L})\Psi,\notag\\ \vec{S}&=\sigma/2,\text{\space\space}\vec{L}=\vec{r}\times\vec{p}.\end{aligned}$$ That is g$_e=2$ for the electron spin and g$_e=1$ for the orbital angular momentum. Equation (3.7a) may be expanded to higher order to study such things as hyperfine structure and relativistic corrections in the hydrogen atom spectrum [@Shankar1994]. That is $$\begin{aligned} i\partial_t\Psi=V\Psi+\frac{(\vec{\sigma}\cdot\vec{\Pi})^2}{2m}\Psi-\frac{(\vec{\sigma}\cdot\vec{\Pi})(i\partial_t-V)(\vec{\sigma}\cdot\vec{\Pi})}{4m^2}\Psi,\end{aligned}$$ However an important problem and caveat is that the wave function $\Psi$ in (3.12) is only the upper half of the Dirac wave function, so the quantity that must be normalized is $|\Psi|^2+|\varphi|^2$ rather than $|\psi_s|^2$ for a Schroedinger or Pauli wave function $\psi_s$ . Thus to insure Hermiticity and conserve probability one must renormalize the wave function as discussed in detail in ref. [@Shankar1994]. It is for this reason that we will adopt an alternative and conceptually simpler approach to the non-relativistic limit in section 7. Generally covariant spinor theory ================================= The gravitational interaction of a spinor may be obtained most easily by making the Dirac Lagrangian (3.1a) and Dirac equation (3.1b) generally covariant. To do this we adopt the standard approach of using a tetrad of basis vectors in order to relate the generally covariant theory to the special relativistic theory in Lorentz coordinates [@Weinberg1972; @Lawrie1990]. This is a most natural, almost inevitable, approach since Dirac spinors transform by the lowest dimensional representation $S$ of the Lorentz group; that is $\psi'=S\psi$ . Two properties of the Dirac Lagrangian and Dirac equation must be modified to obtain a generally covariant theory: the Dirac algebra in (3.2) must be made covariant and the derivative of the spinor in (3.1) must be made into a covariant derivative. We will discuss both in detail. The Dirac algebra (3.1c) is easily made covariant by replacing the Lorentz metric $\eta^{\mu\nu}$ by the Riemannian metric g$_{\mu\nu}$ , $$\begin{aligned} \{\gamma^\mu,\gamma^{\alpha}\}=2\text{g}^{\mu\nu}I.\end{aligned}$$ A set of $\gamma^\mu$ matrices that satisfy (4.1) is easily constructed by using a set of constant $\hat{\gamma}^b$ that satisfies the special relativistic relation (3.2) and a tetrad field $e^\mu_b$ normalized by the usual tetrad relations $$\begin{aligned} e^\mu_b e^\nu_\text{a}\text{g}_{\mu\nu}=\eta_{\text{a}b},\text{\space\space}\text{g}^{\alpha\beta}=e^{\alpha}_ce^{\beta}_d\eta^{cd}.\end{aligned}$$ Here the Greek indices label components of the tetrad vectors and Latin indices label the vectors. In terms of a convenient set of constant Dirac matrices $\hat{\gamma}^b$, such as those in (3.4), we define the $\gamma^\mu$ by $$\begin{aligned} \gamma^\mu=e^\mu_b\hat{\gamma}^b.\end{aligned}$$ It then follows from (3.1c) and (4.2) that the $\gamma^\mu$ satisfy $$\begin{aligned} \{\gamma^\mu,\gamma^\nu\}=e^\mu_be^\nu_\text{a}\{\hat{\gamma}^b,\hat{\gamma}^\text{a}\}=e^\mu_be^\nu_\text{a}2\eta^{\text{a}b}I=2\text{g}^{\mu\nu}I.\end{aligned}$$ The covariant derivative of a spinor is defined so as to transform as a vector under general coordinate transformations and as a spinor under Lorentz transformation of the tetrad basis. As with the covariant derivative of a vector we define a rule for transplanting a spinor from $x$ to a nearby point $x+dx$ , $$\begin{aligned} \psi^*(x+dx)=\psi(x)-\Gamma_\mu\psi(x)dx^\mu.\end{aligned}$$ The matrices $\Gamma_a$ are variously called spin connections, affine spin connections, or Fock-Ivanenko coefficients. The covariant derivative is then defined in terms of the difference between the value of the spinor and the value it would have if transplanted to the nearby point. That is $$\begin{aligned} \psi(x)_{||\nu}dx^\nu&=[\psi(x)+\psi(x)_{|\nu}dx^\nu]-[\psi(x)-\Gamma_\nu(x)\psi(x)dx^\nu]\notag\\ &=[\psi(x)_{|\nu}+\Gamma_\nu(x)\psi(x)]dx^\nu,\notag\\ \psi_{||\nu}&=\psi_{|\nu}+\Gamma_\nu\psi=(\partial_\nu+\Gamma_\nu)\psi\equiv D_\nu\psi. \end{aligned}$$ Here the double slash denotes a covariant derivative. Since the spinor covariant derivative must transform as a vector under coordinate transformations and as a spinor under Lorentz transformations of the tetrad basis, we have $$\begin{aligned} \psi'_{||\mu}=\frac{\partial x^\nu}{\partial x'^\mu}S\psi_{||\nu}, \end{aligned}$$ It follows from (4.6) and (4.7) that the spin connections must transform according to $$\begin{aligned} \Gamma'_\nu=\frac{\partial x^\nu}{\partial x'^\mu}[S\Gamma_\nu S^{-1}-S_{|\nu}S^{-1}].\end{aligned}$$ The transformation (4.8) is formally similar to that of the affine connections used for vector covariant derivatives. The covariant derivative of an adjoint spinor follows easily from that of a spinor in (4.6); we ask that the inner product $\bar{\psi}\chi$ of a spinor $\chi$ and an adjoint spinor $\bar{\psi}$ be a scalar and thus have a covariant derivative $(\bar{\psi}\chi)_{||\mu}$ equal to the ordinary derivative $(\bar{\psi}\chi)_{|\mu}$, and we also ask that the product rule hold for both the ordinary and the covariant derivatives. The result is $$\begin{aligned} \bar{\psi}_{||\mu}=\bar{\psi}_{|\mu}-\bar{\psi}\Gamma_\mu.\end{aligned}$$ The same idea leads to the covariant derivative of a gamma matrix, with only a bit more algebra; that is we ask that the expression $(\bar{\psi}\gamma^\mu\chi)_{||\alpha}$ be a second rank tensor and that it obey the product rule of differentiation, and find from (4.6) and (4.9) $$\begin{aligned} \gamma^\mu_{||\omega}=\gamma^\mu_{|\omega}+\left\{ \begin{array}{c} \mu\\ \omega\sigma \end{array} \right\}+[\Gamma_\omega,\gamma^\mu].\end{aligned}$$ This expression plays an important role in obtaining the spin connections in the next section. Covariant Dirac Lagrangian and Dirac equation ============================================== In this section we give a covariant Lagrangian and obtain the covariant Dirac equation. In the process we get a relation between the spinor and its adjoint (i.e. a spin metric) and evaluate the spin connections. The choice of a covariant Dirac Lagrangian $L$, and its associated Lagrangian density $\mathcal{L}$, is rather obvious from the flat space Lagrangian in (3.1), $$\begin{aligned} L=a\bar{\psi}(i\gamma^\mu\psi_{||\mu}-m\psi)+b(-i\bar{\psi}_{||\mu}\gamma^\mu-\bar{\psi}m)\psi,\text{\space\space}\mathcal{L}=\sqrt{g}L.\end{aligned}$$ Coupling to the electromagnetic field will be included later. The $\gamma^\mu$ denotes the [*covariant*]{} Dirac matrices (4.3) throughout this section. The Dirac equations for the spinor and the adjoint spinor follow directly as the Euler-Lagrange equations of the Lagrangian density $\mathcal{L}$ with $\psi$ and $\bar{\psi}$ treated as independent variables, $$\begin{aligned} (a+b)(i\gamma^\mu\psi_{||\mu}-m\psi)+ib\gamma^\mu_{\text{\space}||\mu}\psi=0\\ (a+b)(\bar{\psi}_{||\mu}i\gamma^\mu+m\psi)+ia\bar{\psi}\gamma^\mu_{\text{\space}||\mu}=0.\end{aligned}$$ For simplicity we assume that the spin connections, unspecified up to this point, may be chosen so that the divergence of $\gamma^\alpha$ that appears in (5.2) vanishes, $\gamma^\mu_{\text{\space}||\mu}=0$. The covariant Dirac equation is then the obvious generalization of the flat space equations (3.1). The spin connections will be obtained below. Also for simplicity and symmetry we choose henceforth $a=b=1/2$; this will prove convenient later. Next, as in flat space in section 3, we ask that there be a relation between the adjoint and the spinor, $\bar{\psi}=\psi^\dagger M$, such that the two equations (5.2) are consistent. Manipulating (5.2a) we get for the adjoint, $$\begin{aligned} -i\bar{\psi}_{|\mu}\tilde{\gamma}^\mu-i\bar{\psi}M^{-1}_{\text{\space}|\mu}M\tilde{\gamma}^\mu-i\bar{\psi}\tilde{\Gamma}_\mu\tilde{\gamma}^\mu-\bar{\psi}m=0,\notag\\ \tilde{\gamma}^\mu\equiv M^{-1}\gamma^{\mu^\dagger}M,\text{\space\space}\tilde{\Gamma}_\mu\equiv M^{-1}\Gamma_\mu^{\text{\space}\dagger}M.\end{aligned}$$ We then compare (5.3) with (5.2b), written as $$\begin{aligned} -i\bar{\psi}_{|\mu}\gamma^\mu+i\bar{\psi}\Gamma_\mu\gamma^\mu-\bar{\psi}m=0,\end{aligned}$$ and see that $M$ must satisfy the following two equations $$\begin{aligned} \gamma^\mu=\tilde{\gamma}^\mu=M^{-1}\gamma^{\mu^\dagger}M,\\ -\Gamma_\mu=\tilde{\Gamma}_\mu=M^{-1}\Gamma_\mu^{\text{\space\space}\dagger}M+M^{-1}_{\text{\space\space\space}|\mu}M.\end{aligned}$$ Eq. (5.5a) may be written in terms of flat space $\hat{\gamma}^b$ as $$\begin{aligned} e^\mu_b\hat{\gamma}^b=e^\mu_bM^{-1}\hat{\gamma}^{b^\dagger}M.\end{aligned}$$ Thus it is obvious that we should ask $\hat{\gamma}^b=M^{-1}\hat{\gamma}^{b^\dagger}M$, which is the same as in the case of flat space and special relativity (3.2), so we may choose $M^{-1}=M=\hat{\gamma}^0$. Then the derivative of $M$ is zero, and it is easy to verify that the choice $M^{-1}=M=\hat{\gamma}^0$ also satisfies (5.5b). Our remaining task is to obtain specific spin connections $\Gamma_\alpha$. To do this we make the natural demand that $\gamma^\mu$ have a null covariant derivative, so from (4.10) $$\begin{aligned} \gamma^\mu_{||\alpha}=\gamma^\mu_{|\alpha}+\left\{ \begin{array}{c} \mu\\ \alpha\beta \end{array} \right\}\gamma^\beta+[\Gamma_\alpha,\gamma^\mu]=0.\end{aligned}$$ This guarantees that the divergence vanishes, $\gamma^\mu_{\text{\space}||\mu}=0$, as we have already mentioned. However it is a stronger demand analogous to the standard demand in general relativity that the metric have a null covariant derivative, which forces the affine connections to be the Christoffel symbols. Note also that $\Gamma_\alpha$ is obviously arbitrary up to a multiple of the identity, which we will suppress henceforth. To solve (5.7) we express $\gamma^\mu$ in terms of flat space gammas $\hat{\gamma}^b$ as in (4.3) and rewrite (5.7) as $$\begin{aligned} e^\mu_{b||\alpha}\hat{\gamma}^b+[\Gamma_\alpha,\hat{\gamma}^b]e^\mu_b=0.\end{aligned}$$ Multiplying this by the inverse tetrad matrix we get $$\begin{aligned} [\Gamma_\alpha,\hat{\gamma}^c]=-e^c_\mu e^\mu_{b||\alpha}\hat{\gamma}^b.\end{aligned}$$ We next note the well-known commutation relation on the sigma matrices, which are defined as $\hat{\sigma}^{\text{a}b}\equiv(i/2)[\hat{\gamma}^\text{a},\hat{\gamma}^b]$, $$\begin{aligned} [\hat{\sigma}^{\text{a}b},\hat{\gamma}^c]=2i(\hat{\gamma}^\text{a}\eta^{bc}-\hat{\gamma}^b\eta^{\text{a}c}),\end{aligned}$$ From (5.10) it is evident that we should seek a solution that is proportional to $\hat{\sigma}^{\text{a}b}$ times a product of the tetrad and its derivatives. It is easy to verify that the specific choice $$\begin{aligned} \Gamma_\alpha=\frac{i}{4}e_{b\mu}e^\mu_{\text{a}||\alpha}\hat{\sigma}^{\text{a}b},\end{aligned}$$ satisfies (5.9) and thus serves as the spin connection. We thus have obtained a generally covariant theory in which the Lagrangian, the Dirac equations, the relation of the spinor to its adjoint, and the spin connections are generally covariant and consistent. Finally we include coupling to the electromagnetic field in the usual minimal coupling way, that is by substituting $iD_\mu \rightarrow iD_\mu-eA_\mu$; this gives the complete covariant Lagrangian $$\begin{aligned} L=\frac{1}{2}\bar{\psi}(i\gamma^\mu\psi_{||\mu}-m\psi)+\frac{1}{2}(-i\bar{\psi}_{||\mu}\gamma^\mu-\bar{\psi}m)\psi-eA_\mu\bar{\psi}\gamma^\mu\psi.\end{aligned}$$ We will study the weak gravitational field limit of this in the next section. Linearized theory for weak gravity ================================== In this section we use the results of section 5 for covariant spinor theory to work out the weak field linearized theory. This is done by setting up an appropriate tetrad and using it to expand the Lagrangian (5.12) to lowest order in the metric perturbation. The result is that there are three interaction terms in the Lagrangian, one associated with the spin coefficients and the second with the alteration in the $\gamma^\mu$ caused by gravity. Remarkably the first vanishes in the linearized theory, while the second corresponds to an interaction via the energy momentum tensor, as intuition should suggest. The third term is a cross term between the weak gravity and electromagnetic fields. In a space with a nearly Lorentz metric (2.1) it is natural to choose a tetrad that lies nearly along the coordinate axes, $$\begin{aligned} e^\mu_a=\delta^\mu_a+w^\mu_a,\text{\space\space}e^b_\nu=\delta^b_\nu-w^b_\nu,\end{aligned}$$ where $w^\mu_a$ is a small quantity to be determined. From the fundamental tetrad relation (4.2) it follows that we should choose a symmetric $w_{\mu\nu}=-(1/2)h_{\mu\nu}$ and thus have a tetrad and $\gamma^\mu$ matrices given by $$\begin{aligned} e^\mu_a&=\delta^\mu_a-(1/2)h^\mu_a,\notag\\ \gamma^\mu&=[\delta^\mu_a-(1/2)h^\mu_a]\hat{\gamma}^a=\hat{\gamma}^\mu-(1/2)h^\mu_a\hat{\gamma}^a.\end{aligned}$$ Since Greek tensor indices and Latin tetrad indices are intimately mixed in the linearized theory we will not distinguish between them in this section. To evaluate the spin connections (5.11) with the tetrad (6.2) we need the Christoffel symbols and the covariant derivative of the tetrad to first order in $h_{\mu\nu}$, $$\begin{aligned} \left\{ \begin{array}{c} \nu\\ \mu\omega \end{array} \right\}=(1/2)(h_{\omega\text{\space}|\mu}^{\text{\space}\nu}+h_{\mu\text{\space}|\omega}^{\text{\space}\nu}-h_{\mu\omega}^{\text{\space\space\space}|\nu}),\notag\\ e^\nu_{\text{a}||\mu}=(1/2)(h_{\mu\text{\space}|\text{a}}^{\text{\space}\nu}-h_{\mu\text{a}}^{\text{\space\space\space}|\nu}).\end{aligned}$$ From (5.11), (6.2) and (6.3) we obtain the spin connections, $$\begin{aligned} \Gamma_\mu=\frac{i}{4}e_{b\nu}e^\nu_{\text{a}||\mu}\hat{\sigma}^{\text{a}b}\cong\frac{i}{4}h_{\mu b|\text{a}}\hat{\sigma}^{\text{a}b}.\end{aligned}$$ Thus the Dirac Lagrangian (5.12) becomes, $$\begin{aligned} L=\frac{1}{2}\bar{\psi}(i\gamma^\mu\psi_{|\mu}-m\psi)+\frac{1}{2}(-i\bar{\psi}_{|\mu}\gamma^\mu-\bar{\psi}m)\psi-eA_\mu\bar{\psi}\gamma^\mu\psi\notag\\ +\frac{i}{2}\bar{\psi}\{\hat{\gamma},\Gamma_\mu\}\psi-\frac{i}{4}h^\mu_{\text{\space}\alpha}[\bar{\psi}\hat{\gamma}^\alpha\psi_{|\mu}-\bar{\psi}_{|\mu}\hat{\gamma}^\alpha\psi]+\frac{1}{2}h^\mu_{\text{\space}\alpha}A_\mu\bar{\psi}\hat{\gamma}^\alpha\psi,\end{aligned}$$ with $\Gamma_\mu$ given in (6.4). The first line is the Dirac Lagrangian in flat space (3.1a), and the other three terms are gravitational interactions that we now address. The first interaction term in the second line of (6.5), due to the spin connections, contains the anti-commutator $\{\hat{\gamma}^\mu,\Gamma_\mu\}$. With the use of the symmetry of $h_{\mu\nu}$, the Dirac algebra (3.1c), and the operator identity $[AB,C]=A\{B,C\}-\{A,C\}B$ it is straightforward to verify the following two expressions, $$\begin{aligned} h_{\mu b|a}\hat{\gamma}^\mu\hat{\sigma}^{ab}=i(h^a_{\text{\space}b|a}-h_{|b})\hat{\gamma}^b,\notag\\ h_{\mu b|a}\hat{\sigma}^{ab}\hat{\gamma}^\mu=i(h_{|b}-h^a_{\text{\space} b|a})\hat{\gamma}^b,\end{aligned}$$ and thereby see that $$\begin{aligned} \{\hat{\gamma}^\mu,\Gamma_\mu\}=\frac{i}{4}h_{\mu b|a}\{\hat{\gamma}^\mu,\hat{\sigma}^{ab}\}=0.\end{aligned}$$ Thus the interaction term containing the spin connections in (6.5) vanishes, which is a remarkable simplification. It should be stressed that this is only true to first order, and the spin connections will generally be of interest in the full theory. There remains in the Lagrangian (6.5) only interactions due to the modification of the $\hat{\gamma}^\mu$ by gravity in (6.2); $L$ may now be written as $$\begin{aligned} L=\frac{1}{2}\bar{\psi}(i\hat{\gamma}\psi_{|\mu}-m\psi)+\frac{1}{2}(-i\bar{\psi}_{|\mu}\hat{\gamma}-\bar{\psi}m)\psi-eA_\mu\bar{\psi}\hat{\gamma}^\mu\psi\notag\\ -\frac{1}{2}h^\mu_{\text{\space}\alpha}[\frac{1}{2}\bar{\psi}\hat{\gamma}^\alpha(i\psi_{|\mu}-eA_\mu\psi)-\frac{1}{2}(i\bar{\psi}_{|\mu}+eA_\mu\bar{\psi})\hat{\gamma}^\alpha\psi]\end{aligned}$$ The quantity in brackets in (6.8) is the appropriately symmetrized energy-momentum tensor $T^a_{\text{\space}\mu}$ for the Dirac field interacting with the electromagnetic field; that is, the gravitational interaction Lagrangian may be expressed as $$\begin{aligned} L_{IG}=-\frac{1}{2}h^\mu_{\text{\space}\alpha}[\frac{1}{2}\bar{\psi}\hat{\gamma}^\alpha(i\psi_{|\mu}-eA_\mu\psi)-\frac{1}{2}(i\bar{\psi}_{|\mu}+eA_\mu\bar{\psi})\hat{\gamma}^\alpha\psi]\notag\\ =-\frac{1}{2}h_{\mu\alpha}T^{\mu\alpha}.\end{aligned}$$ The energy momentum tensor is discussed further in Appendix B. The interaction (6.9) consists of the inner product of the field $h_{\mu\nu}$ with the conserved energy-momentum tensor $T^{\mu\nu}$; this coupling is in close analogy with the electromagnetic coupling between the field $A_\mu$ and the conserved current $j^\mu=e\bar{\psi}\gamma^\mu\psi$ in (6.8). Feynman has emphasized this analogy and developed a complete “flat space” gravitational theory, with gravity treated as an “ordinary” two index (spin 2) field and formulated by analogy with electromagnetism, at least to lowest order [@Feynman1965]. The geometric interpretation of gravity is thereby suppressed or ignored. Weinberg has similarly stressed that the geometric interpretation of gravity is not essential [@Weinberg1972; @Feynman1965]. Schwinger also has used a similar and probably equivalent non-geometric methodology called source theory to obtain the standard results of general relativity theory, including the precession of a gyroscope due to the gravitomagnetic field [@Schwinger1976]. However there is a problem with relating the geometric and non-geometric viewpoints, in that the Euler-Lagrange field equations are based on the Lagrangian $\it{density}$ $\mathcal{L}\sqrt{\text{g}}L\cong(1+h/2)L$ and not the Lagrangian $L$, so there is an additional interaction term $(h/2)L$ in the geometric theory that is not present in the non-geometric theory; the equivalence of the Feynman approach to the linearized geometric approach is thus spoiled whenever the additional term does not vanish. The difference between the Dirac equation per our geometric development and that which one would obtain from the non-geometric approach is easy to see. The Dirac equation that follows from (6.8) with $\mathcal{L}=\sqrt{\text{g}}L\cong(1+h/2)L$ is $$\begin{aligned} \gamma^\mu(i\psi_{|\mu}&-eA_\mu)-m\psi\notag\\ &=\frac{1}{2}h_{\mu\nu}\hat{\gamma}^\mu(i\psi^{|\nu}-eA^\nu\psi)+\frac{1}{4}(h^\mu_{\text{\space}\nu|\mu}-h_{|\nu})i\hat{\gamma}^\nu\psi.\end{aligned}$$ The last term on the right containing $h_{|\nu}$ would not be present in the non-geometric approach. This will be discussed further in section 7. In summary of this section, the Lagrangian (6.8) contains the interaction of the Dirac field with the electromagnetic field to all orders and the interaction with the gravitational field only to lowest order; (6.10) is the corresponding Dirac equation. We will discuss the interaction energies further in the following section in which we consider the non-relativistic or low velocity limit of the theory. Non-relativistic limit ====================== We wish to use the results of the previous sections to obtain a non-relativistic limit of the theory and calculate in a simple way some interesting properties of a spin $1/2$ particle such as the electromagnetic g-factor and its gravitomagnetic analogue. The most familiar approach to this problem is to work with the upper two components of the Dirac wave function as we did in section 3, and take the non-relativistic limit [@Bjorken1964; @Shankar1994]. However the alternative approach we use in this section is conceptually simpler and avoids the problems of renormalization and Hermiticity that occur in the approach of sec. 3. The basic idea is to integrate the interaction Lagrangian over 3-space to get the interaction energy, then put the energy expression with Dirac 4-spinor wave functions, into a form using Pauli 2-spinor wave functions, all in the low velocity limit. In this section we will always work in nearly flat space with Lorentz coordinates; the Dirac $\gamma^\mu$ will be those of flat space and no hat will be used. Moreover for simplicity we will work in the Lorentz gauge for both the electromagnetic and GEM fields, and take both the Coulomb potential $A^0$ and the Newtonian potential $\phi$ to have negligible time dependence; that is $\dot{A}^0=-\nabla\cdot\vec{A}=0$ and $4\dot{\phi}=\nabla\cdot\vec{h}=0$. This is quite appropriate, for example, for electromagnetic interactions in atoms and GEM interactions on the earth. To illustrate the method we first consider only the electromagnetic interaction in flat space; the results will be the same as those in section 3, in particular $\text{g}_e=2$. The interaction Lagrangian and the interaction energy are, from (6.8), $$\begin{aligned} L_{IEM}=-eA_\mu(\bar{\psi}\gamma^\mu\psi)=-A_\mu j^\mu,\\ \Delta E_{EM}=-\int L_{IEM}d^3x.\end{aligned}$$ For the Dirac $\psi$ we use a convenient device, an expansion in terms of free positive energy Dirac wave functions on the mass shell. That is $$\begin{aligned} \psi=\sum_{s=1,2}\limits\int\frac{d^3p}{(2\pi)^3}f(p,s)[e^{ip_\alpha}x^\alpha u(p,s)],\notag\\ E^2=(p^0)^2=p^2+m^2.\end{aligned}$$ The positive energy wave functions do not form a complete set, but the approximation (7.2) should be quite good for distances much larger than the Compton wavelength, $\hbar/m$; (7.2) is our fundamental assumption. A key idea in the calculation is to express the Dirac 4-spinor $u(p,s)$ in terms of a Pauli 2-spinor $\chi_s$ [@Adler1966], $$\begin{aligned} e^{-ip_\alpha x^\alpha}u(p,s)=e^{-ip_\alpha x^\alpha}\sqrt{\frac{E+M}{2m}}\left( \begin{array}{c} I\\ \frac{\vec\sigma\cdot\vec{p}}{E+M} \end{array} \right)\chi_s.\end{aligned}$$ Correspondingly we express the non-relativistic Pauli wave function as $$\begin{aligned} \Psi=\sum_{s=1,2}\limits\int\frac{d^3p}{(2\pi)^3}f(p,s)e^{ip_\alpha x^\alpha}\chi_s.\end{aligned}$$ In terms of the above expressions (7.2) and (7.3) the interaction energy (7.1b) is $$\begin{aligned} \Delta E_{EM}=\sum_{s,s'=1,2}\limits&\int\frac{d^3p}{(2\pi)^3}\frac{d^3p'}{(2\pi)^3}f^*(p',s')f(p,s)\notag\\ &[e\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}\bar{u}(p',s')\gamma^\mu u(p,s)A_\mu].\end{aligned}$$ The bracket in (7.5) corresponds to scattering of a free Dirac spinor by an external field, which is equivalent to scattering by an infinitely heavy source particle. It contains all the information about the spin interaction and corresponds to the diagram in fig. 7.1: the particle leaves the wave function blob with 3-momentum $\vec{p}$, scatters from the external field via the QED vertex amplitude into momentum $\vec{p'}$, and then reenters the wave function blob. The electron remains on the mass shell, corresponding to zero energy transfer, which is consistent with a non-relativistic wave function. We denote the 4-momentum transfer by $q_\mu=p'_\mu-p_\mu$, with $q_0=0$ . The magnitude of the allowed 3-momentum transfer $\vec{q}$ is limited by the width of the function $f(p,s)$ in momentum space. ![The electron in the wave function scatters from the field and back into the wave function. ](1) It is now straightforward to calculate the bracket in (7.5). We split it into 2 parts, $\mu=0$ for the electric interaction and $\mu=j$ for the magnetic interaction. For the electric part we have $$\begin{aligned} e\int d^3xA_0 &e^{iq_\alpha x^\alpha}\bar{u}(p',s')\gamma^0u(p,s)\notag\\ =e\int d^3x &e^{iq_\alpha x^\alpha}A_0\left(\frac{E+m}{2m}\right)\chi^\dagger_{s'}\left[I,\frac{\vec\sigma\cdot \vec p'}{E+M}\right]\left[ \begin{array}{c} I\\\frac{\vec\sigma\cdot \vec p}{E+m} \end{array} \right]\chi_s\notag\\ =e\int d^3 x &e^{iq_\alpha x^\alpha}\notag\\ &A_0\chi^\dagger_{s'}\left[\frac{E}{m}+\frac{\vec{q}\cdot\vec{p}}{2m(E+m)}+\frac{i\vec{q}\times\vec{p}\cdot\vec\sigma}{2m(E+m)}\right]\chi_s.\end{aligned}$$ The first term in the bracket in (7.6) is the obvious charge coupling to the Coulomb field. The second and third terms may be simplified. First, because there is no energy transferred $\vec{p}^2=\vec{p'}^2$, from which it follows that $\vec{p}\cdot\vec{q}=-\vec{q}^2/2$. Secondly the vector $\vec{q}$ multiplying the exponential may be replaced by $i\nabla$ operating on the exponential, after which integration by parts allows us to replace it with $-i\nabla$ operating on the function $A_0$; that is we may replace $\vec{q}A_0\rightarrow-i\nabla A_0$. Thus the second term vanishes since $\nabla^2A_0=0$ in a charge free region for the Lorentz gauge. What remains is, to order $1/m^2$, $$\begin{aligned} e&\int d^3 x A_0 e^{iq_\alpha x^\alpha}\bar{u}(p',s')\gamma^0 u(p,s)\notag\\ &=\int d^3 e^{iq_\alpha x^\alpha}\left[e(\chi^\dagger_{s'}\chi_s)+\frac{e}{4m^2}\nabla\phi_c\times\vec{p}\cdot(\chi^\dagger_{s'}\vec\sigma\chi_s)\right].\end{aligned}$$ The second term in (7.7) is clearly a nonlocal fine structure correction, which we mentioned in sec. 4 and which will not concern us further in the present work [@Shankar1994]. The $\mu=j$ magnetic part of the interaction (7.5) is handled in exactly the same way as the electric part. We have $$\begin{aligned} e\int d^3x A_j e^{iq_\alpha x^\alpha}&\bar{u}(p',s')\gamma^ju(p,s)\notag\\ =e\int d^3xe^{iq_\alpha x^\alpha}&A_j\left(\frac{E+m}{2m}\right)\chi^\dagger_{s'}\notag\\ &\left[I,\frac{\vec\sigma\cdot \vec p'}{E+m}\right]\left[ \begin{array}{cc} 0 &\sigma^j \\ \sigma^j &0 \end{array} \right]\left[ \begin{array}{c} I\\ \frac{\vec\sigma\cdot \vec p}{E+m} \end{array} \right]\chi_s\notag\\ =\int d^3xe^{iq_\alpha x^\alpha}&A_j\left(\frac{e}{2m}\right)\chi^\dagger_{s'}[\sigma^j\vec\sigma\cdot \vec p+\vec\sigma\cdot \vec p'\sigma^j]\chi_s\notag\\ =-\int d^3 xe^{i\vec q_\alpha x^\alpha}&\left(\frac{e}{2m}\right)\chi^\dagger_{s'}[2\vec{p}\cdot\vec{A}+\vec{q}\cdot\vec{A}+i\vec q\times\vec{A}\cdot\vec\sigma]\chi_s.\end{aligned}$$ We then replace $\vec{q}\rightarrow-i\nabla$ as discussed above and see that the second term in the bracket vanishes in a gauge with $\nabla\cdot\vec{A}=0$, and we are left with $$\begin{aligned} &e\int d^3 x A_j e^{iq_\alpha x^\alpha}\bar{u}(p',s')\gamma^j u(p,s)\notag\\ &=-\int d^3x e^{iq_\alpha x^\alpha}\left(\frac{e}{2m}\right)\chi^\dagger_{s'}[2\vec{p}\cdot\vec{A}+\nabla\times\vec{A}\cdot\vec\sigma]\chi_s\notag\\ &=-\int d^3xe^{iq_\alpha x^\alpha}\left[\frac{e}{m}\vec{p}\cdot\vec{A}(\chi^\dagger_{s'}\chi_s)+\frac{e}{2m}\vec{B}\cdot(\chi^\dagger_{s'}\vec\sigma\chi_s)\right].\end{aligned}$$ Finally we combine (7.7) and (7.9) and substitute into (7.5) to obtain, to order $1/m$, $$\begin{aligned} \Delta E_{EM}=&\sum_{s,s'=1,2}\limits\int\frac{d^3p}{(2\pi)^3}\frac{d^3p'}{(2\pi)^3}f^*(p',s')f(p,s)\notag\\ &\int d^3xe^{-i(\vec{p'}-\vec{p})\cdot\vec{x}}\chi^\dagger_{s'}[eA_0-\frac{e}{m}\vec{p}\cdot\vec{A}\frac{e}{2m}\vec{B}\cdot\vec\sigma]\chi_s\notag\\ =&\int d^3 x\Psi^\dagger[eA_0-\frac{e}{m}\vec{p}\cdot\vec{A}-\frac{e}{2m}\vec{B}\cdot\vec\sigma]\Psi.\end{aligned}$$ This is the same result that we discussed in section 3, so we have thus verified that our present approach reproduces the usual result for the electron g factor, g$_e=2$. We now work out the non-relativistic limit of the gravitational interaction in (6.8), following the same procedure as for the electromagnetic interaction; we will not include the product of the electromagnetic and gravitational fields, that is the cross term in (6.8). The algebra is a bit lengthier but equally straightforward. As with the Lagrangian and energy for the electromagnetic case in (7.1) we have for the gravitational case $$\begin{aligned} L_{IG}=-\frac{1}{2}h_{\mu\nu}T^{\mu\nu},\text{\space\space}\Delta E_G =-\int L_{IG}d^3x,\end{aligned}$$ where $T^{\mu\nu}$ is given in (6.9). It is convenient to write $T^{\mu\nu}$ in close analogy with the electromagnetic current, as $$\begin{aligned} T^{\mu\alpha}=\bar{\psi}\hat{\gamma}^\alpha(\frac{1}{2}i\overleftrightarrow{\partial}^\mu)\psi.\end{aligned}$$ Note the relation between the electromagnetic and the gravitational interactions, $$\begin{aligned} A_\mu\leftrightarrow h_{\mu\nu}/2,\text{\space\space}\gamma^\mu\leftrightarrow\gamma^\mu(\frac{i}{2}\overleftrightarrow{\partial}^\nu).\end{aligned}$$ Then $\Delta E_G$ is, in analogy with (7.5), $$\begin{aligned} \Delta E_G&=\sum_{s,s'=1,2}\limits\int\frac{d^3p}{(2\pi)^3}\frac{d^3p'}{(2\pi)^3}f^*(p',s')f(p,s)\notag\\ &[\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}\bar{u}(p',s')\gamma^\mu(p^\nu+q^\nu/2)u(p,s)(h_{\mu\nu}/2)].\end{aligned}$$ As with the electromagnetism calculation we split the gravitational interaction into two parts, the gravitoelectric for $h_{00}=h_{ii}=2\phi$ and the gravitomagnetic for $h_{0j}=h_{j0}=h^j$. The gravitoelectric part of the bracket in (7.14) involves the same spin products as encountered with the electromagnetic calculation in (7.7) and (7.9), and after some algebra we obtain, to order $1/m^2$, $$\begin{aligned} [\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}&\bar{u}(p',s')\{\gamma^0E+(p^j+\frac{q^j}{2})\gamma^j\}u(p,s)\phi]\notag\\ =\int d^3 x e^{i(p'_\alpha-p_\alpha)x^\alpha}&\chi^\dagger_{s'}[\left(\frac{E^2}{m}\phi+\frac{E}{4m^2}\nabla\phi\times \vec p\cdot\vec\sigma\right)\notag\\ &+\left(\frac{\vec{p}^2}{m}\phi+\frac{1}{2m}\nabla\phi\times \vec p\cdot\vec\sigma\right)]\chi_s\notag\\ =\int d^3xe^{i(p'_\alpha-p_\alpha)x^\alpha}&m\chi^\dagger_{s'}[(1+\frac{2\vec{p}^2}{m^2})\phi+\frac{3}{4m^2}\nabla\phi\times \vec p\cdot\vec\sigma]\chi_s.\end{aligned}$$ A word is in order about the physical interpretation of the gravitoelectric result (7.15). The term $m\phi$ is of course the expected Newtonian energy; the factor $(1+2\vec{p}^2/m^2)$ occurs also in the analysis of a spin zero system in ref. [@Adler2010], and is approximately the Lorentz transformation factor between the potential in the lab frame and the moving frame of the particle; thus $(1+2\vec{p}^2/m^2)\phi$ is the Newtonian potential seen by the moving particle. The last term in the bracket has the same form and is the gravitational analog of the fine structure term in the electromagnetic energy (7.7), except of course for the different coefficient. We will not be concerned further with the higher order terms in (7.15) and will henceforth keep only the lowest order term $\phi$ in the bracket. We turn finally to the gravitomagnetic part of the interaction (7.14), which is our main interest in this work. The gravitomagnetic part of the bracket, proportional to $h^j$, is $$\begin{aligned} [\int d^3xe^{i(p'_\alpha-p_\alpha)x^\alpha}\bar{u}(p',s')\{\gamma^0(p^j+q^j/2)&+E\gamma^j\}\notag\\ u(p,s)&(h^j/2)]\notag\\ =[\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}u^\dagger(p',s')\{(p^j+q^j/2)(&h^j/2)\notag\\ +E(h^j/2)\alpha^j&\}u(p,s)].\end{aligned}$$ Note that the term $\vec{q}\cdot\vec{h}$ will vanish by gauge choice, just as the $\vec{q}\cdot\vec{A}$ term vanished for the electromagnetic case. Then, using the same manipulations as previously on the spin products we reduce this to $$\begin{aligned} [\int d^3xe^{i(p'_\alpha-p_\alpha)x^\alpha}\bar{u}(p',s')\{\gamma^0(p^j+q^j/2)&+E\gamma^j\}\notag\\ u(p,s)&(h^j/2)]\notag\\ =[\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}\chi^\dagger_{s'}\{\vec{p}\cdot\vec{h}+\frac{1}{4}\nabla\times&\vec{h}\cdot\vec\sigma\}\chi], \end{aligned}$$ where we have neglected terms of higher order, that is $1/m^2$. Finally we combine (7.15) and (7.17) to obtain the total energy $$\begin{aligned} \Delta E_G&=\sum_{s,s'=1,2}\limits\int\frac{d^3p}{(2\pi)^3}\frac{d^3p'}{(2\pi)^3}f^*(p',s')f(p,s)\notag\\ &[\int d^3x e^{i(p'_\alpha-p_\alpha)x^\alpha}\chi^\dagger_{s'}(m\phi+\vec{p}\cdot\vec{h}+\frac{1}{4}\nabla\times\vec{h}\cdot\vec\sigma)\chi]\notag\\ &=\int d^3x \Psi^\dagger(m\phi+\vec{p}\cdot\vec{h}+\frac{1}{4}\vec{\Omega}\cdot\vec\sigma)\Psi.\end{aligned}$$ (Recall that the gravitomagnetic field is $\vec{\Omega}=\nabla\times\vec{h}$.) This is the main result of this section and is consistent with the result of ref. [@Adler2010] for a scalar particle. Finally we note that since we have expanded the wave function in terms of a free Dirac particle on the mass shell (7.2) the free Dirac Lagrangian is zero and the extra geometric interaction term $(h/2)L$ discussed in section 6 vanishes. Gravitomagnetic physical effects ================================ The result (7.18) is to be compared with the analogous electromagnetic result (7.10). We see, of course, that the Newtonian potential is the analog of the Coulomb potential $eA^0$ and the gravitomagnetic potential is the analog of the vector potential according to $$\begin{aligned} eA^0\leftrightarrow\phi,\text{\space\space}(-e/m)\vec{A}\leftrightarrow\vec{h}.\end{aligned}$$ We also see that the coupling of the spin to the gravitomagnetic field $\vec{\Omega}$ is only half the analogous electromagnetic coupling. To make this most obvious we consider a gravitomagnetic field $\vec\Omega$ that is approximately constant over the system so that we may choose $\vec{h}=(\vec{\Omega}\times \vec r)/2$. Then $$\begin{aligned} \Delta E_G&=\int d^3x\Psi^\dagger(m\phi+\frac{1}{2}\vec{\Omega}\times\vec{r}\cdot\vec{p}+\frac{1}{4}\vec{\Omega}\cdot\vec\sigma)\Psi\notag\\ &=\int d^3x\Psi^\dagger(m\phi+\frac{1}{2}\vec{\Omega}\cdot\vec{r}\times\vec{p}+\frac{1}{2}\vec{\Omega}\cdot\frac{\vec\sigma}{2})\Psi\notag\\ &=\int d^3x\Psi^\dagger[m\phi+\frac{1}{2}\vec{\Omega}\cdot(\vec L+\vec S)]\Psi\end{aligned}$$ Both orbital and spin angular momenta couple in the same way to the gravitomagnetic field, so there is no anomalous g factor for gravitomagnetism; that is g$_\text{g}=1$ for both orbital and spin angular momenta. From the above correspondence it is clear that since a magnetic moment due to orbital angular momentum, $(e/2m)\vec{L}$, precesses at the Larmor frequency $(eB/2m)$ in a magnetic field $B$, the gravitomagnetic moment due to both orbital and spin angular momenta will precess in a gravitomagnetic field $\Omega$ with frequency $\Omega$, but in the opposite direction. Thus quantum precession should be the same as that observed in the classical gyroscope systems of the GPB satellite experiment [@Everitt2011]. It thus seems very likely that the precession rate is universal for any angular momentum system, whether the angular momentum is classical or quantum mechanical, orbital or spin. For the surface of the earth the magnitude of the gravitomagnetic field is quite small, as estimated in ref. [@Adler2010] The field and the associated quantum energy are of order $$\begin{aligned} \Omega\approx10^{-13}rad/s,\text{\space\space}E_\Omega=\hbar\Omega\approx10^{-28}eV.\end{aligned}$$ Experimental detection of such small quantum gravitomagnetic effects in an earth-based lab would obviously be difficult. Such an experiment might be performed with an atomic interferometer. The atomic beam could be split into two components with angular momenta differing by $\Delta L=\hbar$. Then, according to (8.2) the two components would have energies differing by about $\Delta E\approx\Omega\Delta L\approx\Omega\hbar$ and thus suffer phase shifts differing by about $\Delta\varphi\approx\Delta Et/\hbar\approx\Omega t$, where $t$ is the time of flight. For a typical $t=1s$ this implies a phase shift of order $10^{-13}rad$, which is orders of magnitude less than presently detectable [@Adler2011]. In addition to the small size of gravitomagnetic effects one might see in the laboratory there is a serious further inherent difficulty in almost any such experiment; a rotation of the apparatus would in general have similar effects and swamp the gravitomagnetic effects, so such rotations would have to be controlled and compensated to very high accuracy as mentioned in the introduction and in ref. [@Adler2010]. The results of the GPB experiment and the theoretical results of this paper and ref.[@Adler2010] are probably most important in establishing the validity and consistency of general relativity and the gravitomagnetic effects that it implies. Such gavitomagnetic effects are quite small in earth-based labs and satellite systems, as is clear from (8.3), but may play a large role in astrophysical phenomena such as the jets observed in active galactic nuclei, for which the gravitomagnetic fields are much stronger [@Throne2009]. Summary and conclusions ======================= We have developed the theory of a spin $1/2$ Dirac particle in a Riemann space and its weak field limit in considerable detail. In the low velocity limit for the particle the energies due to the Newtonian or gravitoelectric field and the Òframe-draggingÓ or gravitomagnetic field take simple and intuitive forms. Detection of the small gravitomagnetic effects in earth-based or satellite experiments is quite difficult, but such effects are expected to be large and of great interest in astrophysical systems such as jets from active galactic nuclei and black holes.\ The inverse differential operator ================================= We briefly study the type of differential operator that appears in (3.7) by solving the differential equation $$\begin{aligned} Af+\partial f=(A+\partial)f=F,\text{\space\space}f=f(x),\text{\space\space}F=F(x),\end{aligned}$$ where $F(x)$ is a given function that may be expanded as a power series in the region of interest and $A$ is a constant. The solution of the homogeneous equation is $$\begin{aligned} f_h=Ce^{-Ax}\text{\space\space}(C=\text{arbitrary constant}).\end{aligned}$$ The general solution of (A.1) is $f_h$ plus any particular solution $f_p$; for the particular solution we solve (A.1) symbolically as, $$\begin{aligned} f_p=(A+\partial)^{-1}F=\frac{1}{A}\left(1-\frac{\partial F}{A}+\frac{\partial^2F}{A^2}...\right).\end{aligned}$$ Operating on (A.3) with $(A+\partial)$ obviously gives $F$. To further justify the above formal operations we may solve (A.1) in a different way. An integrating factor is easily seen to be $e^{Ax}$, so $$\begin{aligned} \partial(e^{Ax}f)=e^{Ax}(A+\partial)f=e^{Ax}F.\end{aligned}$$ Integration then gives the general solution $$\begin{aligned} f=e^{-Ax}\int^x\limits e^{-Ax'}F(x')dx'+Ce^{-Ax}.\end{aligned}$$ Since (A.1) is linear and $F$ is assumed to be expandable in a power series we need only consider powers, $F=x^n$. Then we easily evaluate (A.5) using integration by parts, to obtain $$\begin{aligned} f=\frac{1}{A}\left(\frac{x^n}{A}-\frac{nx^{n-1}}{A^2}+\frac{n(n-1)x^{n-2}}{A^3}...+1\right)+Ce^{-Ax}.\end{aligned}$$ This agrees with the power series for $f_p$ given in (A.3). Energy momentum tensor for the Dirac field ========================================== We wish to obtain the energy momentum tensor for a Dirac field in flat space, which occurs in (6.8) and (6.9)[@Bjorken1964]. We begin with the Lagrangian (3.1) for the free Dirac field and work out the canonical energy momentum tensor according to the Noether theorem; it is, up to a constant multiplier $C$, $$\begin{aligned} T^\mu_{\text{\space}\nu}=C[\frac{\partial L}{\partial\psi_{|\mu}}\psi_{|\nu}+\frac{\partial L}{\partial\bar{\psi}_{|\mu}}\bar{\psi}_{|\nu}-\delta^\mu_\nu L]\notag\\ =C[a\bar{\psi}i\gamma^\mu\psi_{|\nu}-b\bar{\psi}_{|\nu}i\gamma^\mu\psi].\end{aligned}$$ where we have omitted the term proportional to $L$ since it is zero for a solution of the free Dirac equation. Using the fact that the Dirac and the Klein-Gordon equations are obeyed by $\psi$ we calculate the two divergences of this tensor to be $$\begin{aligned} T^{\mu\nu}_{\space\space\space}|{\mu}=0,\text{\space\space}T^{\mu\nu}_{\space\space\space}|{\nu}=C(b-a)[m^2(\bar{\psi}i\gamma^\mu\psi)-(\bar{\psi}^{|\nu}i\gamma^\mu\psi_{|\nu})]\end{aligned}$$ If we choose $b=a$, as in the text, both divergences are zero and the tensor has symmetry in $\psi$ and $\bar{\psi}$. Moreover we may then consistently symmetrize $T^{\mu\nu}$ and have $$\begin{aligned} T^{\mu\nu}=\frac{1}{4}[\bar{\psi}i\gamma^\mu\psi^{|\nu}-\bar{\psi}^{|\nu}i\gamma^\mu\psi+\bar{\psi}i\gamma^\nu\psi^{|\mu}-\bar{\psi}^{|\mu}i\gamma^\nu\psi]\end{aligned}$$ This has now been normalized so that in the low velocity limit $$\begin{aligned} T^{00}\approx m\bar{\psi}\psi\end{aligned}$$ Finally, to include the electromagnetic field we use the minimal substitution recipe $i\partial_\mu\rightarrow i\partial_\mu-eA_\mu$ to get $$\begin{aligned} T^{\mu\nu}&=\frac{1}{4}[\bar{\psi}i\gamma^\mu\psi^{|\nu}-\bar{\psi}^{|\nu}i\gamma^\mu\psi+\bar{\psi}i\gamma^\nu\psi^{|\mu}-\bar{\psi}^{|\mu}i\gamma^\nu\psi]\notag\\ &-\frac{1}{2}[e\bar{\psi}A^\nu\gamma^\mu\psi+e\bar{\psi}A^\mu\gamma^\nu\psi]\end{aligned}$$ To verify the result (B.5) we may calculate the divergence of $T^{\mu\nu}$ to find, after some algebra, that it gives the correct Lorentz force, $$\begin{aligned} T^{\mu\nu}_{\text{\space\space\space}|\mu}=-j_\alpha F^{\mu\alpha}=-(\bar{\psi}\gamma_\alpha\psi)F^{\mu\alpha}\end{aligned}$$ In the interaction Lagrangian (6.8) the energy momentum tensor is contracted with the symmetric $h_{\mu\nu}$ so the symmetrization in (B.5) is not relevant. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by NASA grant 8-39225 to Gravity Probe B and by NSC of Taiwan under Project No. NSC 97-2112-M-002-026-MY3. Pisin Chen thanks Taiwan’s National Center for Theoretical Sciences for their support. Thanks go to Robert Wagoner, Francis Everitt, and Alex Silbergleit and other members of the Gravity Probe B theory group for useful discussions, and to Mark Kasevich of the Stanford physics department for interesting comments on atomic beam interferometry and equivalence principle experiments. Kung-Yi Su provided valuable help with the manuscript. Elisa Varani thanks Cavallo Pacific for encouragement and support. [99]{} R. J. Adler, “Gravity,” chapter 2 of [*The New Physics for the Twenty-first Century*]{}, edited by Gordon Fraser, (Cambridge University Press, Cambridge UK, 2006). C. 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--- abstract: 'We discuss the time-series behavior of 8 extragalactic 3FGL sources away from the Galactic plane (i.e., $\mid b\mid \geq 10^{\circ}$) whose uncertainty ellipse contains a single X-ray and one radio source. The analysis was done using the standard Fermi *ScienceTools*, package of version v10r0p5. The results show that sources in the study sample display a slight indication of flux variability in $\gamma$-ray on monthly timescale. Furthermore, based on the object location on the variability index versus spectral index diagram, the positions of 4 objects in the sample were found to fall in the region of the already known BL Lac positions.' --- Introduction {#intro} ============ The study of variability is particularly important in $\gamma$-ray astronomy primarily due to different advantages such as assisting in the identification of the correct radio/optical/X-ray source within the $\gamma$-ray position box, with the observations at other wavelengths ([@De2015 De Cicco et al. 2015]; [@ferrara2015 Ferrara et al. 2015]). For unidentified sources, variability characteristics can also support the recognition of the correct source class ([@Nolan2003 Nolan et al. 2003]). Fortunately, the Large Area Telescope (LAT) aboard the *Fermi Gamma-ray Space Telescope* has revolutionised the field of $\gamma$-ray astronomy by detecting a wealth of new $\gamma$-ray sources and allowing the study of previously known sources with unprecedented details ([@Zechlin2015 Zechlin & Horns. 2015]). Previous studies show that most of the sources detected by the *Fermi*-LAT are blazars ([@Ackermann2015 Ackermann et al. 2015]). The 3FGL ([@Acero Acero et al. 2015]) and the 4FGL ([@Collabo2019 The Fermi-LAT collaboration 2019]) catalogs reported a significantly large fraction of sources compared to the previous ones. However, the majority of 3FGL and 4FGL sources remain unassociated with low-energy counterparts, hence understanding their nature is an open question in high-energy astrophysics. In addition, it seems plausible that most of the unassociated high-latitude $\gamma$-ray sources are expected to be faint AGN, which may include blazar sub-class ([@2012Mirabal Mirabal et al. 2012]; [@Massaro Massaro et al. 2012]; [@Ackermann Ackermann et al. 2012]). These unidentified $\gamma$-ray sources represent a discovery area for the new source classes or new members of existing source classes which may include different types of AGN. For instance, previous studies show a combined effort to isolate potential blazar candidates among this large population (e.g. [@Massaro Massaro et al. 2012]; [@Zechlin2015 Zechlin & Horns 2015]; [@2017Paiano Paiano et al. 2017]). Some studies used the analysis of the multiwavelength Spectral Energy Distribution (SED) through detecting a double peaked spectrum. This indicated that the radiation among the selected sample originates mainly from synchrotron and the inverse-Compton emission in the so-called synchrotron-Compton blazars ([@Mbarubucyeye Mbarubucyeye J.D., Krauß F., and Nkundabakura P. 2019 in prep...]), though the SED alone is not enough to fully characterise the blazar nature based on their broad band properties. Since blazars display intrinsic variability and more significantly in the $\gamma$-ray energy band ([@Ulrich Ulrich et al. 1997]), it is needed to use this property to characterise individual synchrotron-Compton blazar candidates that may be present in the $\gamma$-ray unidentified and unassociated population. In this paper, we discuss the time-series behavior of 8 extragalactic 3FGL sources away from the Galactic plane (i.e., $\mid b\mid \geq 10^{\circ}$) which were carefully selected among the Unidentified *Fermi*-LAT sources with the purpose to detect any sign of variability which can be linked to the blazar nature of these sources.\ Sample selection {#sect.2} ================ The following selection criteria were used to obtain a study sample: i. Being unidentified sources at high Galactic latitudes, $\mid b\mid \geq 10^{\circ}$, ii. Being unidentified sources which have a single X-ray and one radio source in its uncertainty region, iii. Being unidentified sources that were reported in the 4FGL catalog. Applying all cuts to the population of unidentified sources listed in 3FGL, a sample of 8 unidentified sources thought to be potential blazar candidates was isolated. Data analysis {#sect.4} ============= The astrophysical data analysis of LAT begins with a list of counts detected. This list results from processing made by the LAT instrument team, which reconstructs events for the signals from different parts of LAT. Two principal types of analysis were applied in this study, they were performed in a systematic way such that the results from the first analysis became the input of the next one. The types of analysis performed are: - Global analysis which was performed using *Fermi* *ScienceTools* v10r0p5. This provided the fluxes and spectral parameters of all objects in our study sample. The photon counts within a region of interest of 25 degree radius were taken into account. We selected events within the energy range 100 MeV–300 GeV, a maximum zenith angle of 90 degrees and event type 3. - Time-series analysis (light-curve analysis & variability analysis). This provided the $\gamma$-ray light-curves for the period of 9 years and the variability indices of target sources, together with the significance of the observed variability in light-curves. The observed variability was obtained using the following equation as in [@Nolan2003 Nolan et al. 2003]: $$\label{eq:0} TS_{var}=2\sum_{i} \frac{\Delta F_i^2}{\Delta F_i^2 +f^2 F_{const}^2} \ln\left(\frac{ \mathcal{L}_{i}(F_{i})}{\mathcal{L}_{i}(F_{const})}\right),$$ where $f = 0.02$, i.e., a $2\% $, is a systematic correction factor, $F_{i}$ and $\Delta F_i$ are the flux and error in flux in the $i^{th}$ bin, respectively. $\mathcal{L}_{i}(F_{const})$ is the value of the likelihood in the $i^{th}$ bin under the null hypothesis where the source flux is constant across the full period and $F_{const}$ is the constant flux for this hypothesis. $\mathcal{L}_{i}(F_{i})$ is the value of the likelihood in the $i^{th}$ bin under the alternate hypothesis where the flux in the $i^{th}$ bin is optimised. Results and discussion {#sect.5} ====================== Average results --------------- Before performing the light-curve analysis, a fit of the entire 108 months LAT data using a power-law model was performed through binned likelihood analysis for each target source. This provided the sample average results fluxes and spectral properties in a period of 9 years. Sources in the sample were found to be faint $\gamma$-ray emitters with $\gamma$-ray spectral index, $\Gamma \sim 2$ (see Table  \[Global\]), which is consistent with the previous studies (e.g., [@Ackermann2015 Ackermann et al. (2015)]). ---- ---------------- -------------------------- ------------------------------- ---------- -- -- -- No 3FGL Name $\centering{F_{\gamma}}$ $\centering{\Gamma_{\gamma}}$ $\sigma$ (1) (2) (3) (4) 1 J0049.0+4224 0.93 $\pm$ 0.43 1.81 $\pm$ 0.14 7.02 2 J1119.8$-$2647 2.99 $\pm$ 0.81 1.94 $\pm$ 0.11 10.03 3 J1132.0$-$4736 3.72 $\pm$ 1.21 2.00 $\pm$ 0.09 10.11 4 J1220.0$-$2502 7.62 $\pm$ 1.26 2.16 $\pm$ 0.20 7.46 5 J1220.1$-$3715 3.44 $\pm$ 0.81 1.96 $\pm$ 0.09 10.24 6 J1619.1+7538 0.85 $\pm$ 0.23 1.78 $\pm$ 0.10 10.01 7 J1923.2$-$7452 8.61 $\pm$ 0.68 2.04 $\pm$ 0.10 14.05 8 J2015.3$-$1431 4.63 $\pm$ 1.52 2.23 $\pm$ 0.19 5.03 ---- ---------------- -------------------------- ------------------------------- ---------- -- -- -- : Gamma-Ray average fluxes and spectral characteristics of the sample sources. []{data-label="Global"} \ Note: Column 1, 2, 3 and 4 show the source 3FGL name, the average $\gamma$-ray flux for 108 months LAT data in scale of $10^{-9} \, \rm{ph} \, \rm{cm^{-2}} \, \rm{s^{-1}}$, the $\gamma$-ray spectral index corresponding to column (2), the significance in sigma units corresponding to column (2), respectively. Monthly $\gamma$-ray light-curves --------------------------------- To determine the trends of flux change and variability of sources for a period of 9 years, light-curve analysis was performed. This was done through extracting the monthly fluxes along this period, and plotting light-curves.\ Generally, we found that sources in our sample do not show significant signal in their light-curves, which is an indication that they are relatively faint in $\gamma$-ray. This was also suggested by [@Acero Acero et al. (2015)]. Therefore, the light-curves indicate that signals are not significantly detectable in many monthly bins (represented as an upper limit). This implies that their fluxes are close to zero, hence summing them over the full period (9 years) tends to lower the source average flux as shown in Figure \[lch1\]. The sample light-curves display behaviours commonly shown by blazars such as: non periodic flux change characterised by undefined and no specific trends, associated with unpredictable and sudden flux rise seen across the whole period of 9 years (see Figure \[lch1\]). However, the large error bars on the data points does not allow to firmly establish such sharp flux rises. ![The 100 MeV–300 GeV monthly light-curves for 3FGL J1132.0$-$4736. The horizontal solid line along with two dashed lines present the 9-years average flux and its 1$\sigma$ error range derived in the global analysis, respectively. The blue points represent flux with its $1\sigma$ error bar, while the downward arrows together with black points represent the 95% upper limits.[]{data-label="lch1"}](./Figures/lc_4736){width="9.2cm" height="6cm"} Variability indices {#vari} ------------------- The light-curves presented in this study show many upper limits that correspond to the time when the signal in monthly bins was not significant enough to characterise a source. It is also clear that the error bars corresponding to the significant flux points are relatively large. Therefore, we used the ‘*variability index*’ defined in Equation  \[eq:0\] to quantify the observed variability in light-curves, in which the information of the upper limits is properly considered. To compare the already known classification in 3FGL with our sources that are lacking classifications, variability-spectral index diagram for all 3FGL sources including the study sample was plotted. [@Ackermann2015 Ackermann et al. (2015)] observed that blazars are located in different zones on the variability-spectral index diagram, according to their subtypes (FSRQs and BL Lacs), though there is also a large recovery zone (see Figure \[TSvarE\]). ![Variability index ($TS_{var}$) versus power law spectral index (PL index) diagram of all 3FGL extragalactic sources. Red points: Flat Spectrum Radio Quasars, blue crosses: BL Lacs, and sources in the study sample. Circles in different colours indicate data of sources in the selected sample, while solid squares in same colours present results of the same sources obtained in this study. The black horizontal dotted line indicates the 3FGL variability index threshold (72.44), while the green horizontal dashed line shows the 9-years variability index threshold (143.94) obtained in this study. The black vertical dashed line shows the spectral index (1.931) below which the region is populated by BL Lacs.[]{data-label="TSvarE"}](./Figures/pyfits_TSvar_vs_Index_for_Kamanzi.eps){width="0.6\linewidth"} The significance of the observed variability from sample light-curves was estimated by using a $\chi^2$ distribution. This provided the variability index ($TS_{var}$) threshold at which we assigned the source a 99% probability of being variable (on a timescale of $\gtrsim 1 \, \rm{month}$). For 9-years data, we found that variability is considered significant with 99% confidence level if the variability index is greater than 143.94. However, sources in our study sample have $TS_{var}$ values much lower than the threshold (i.e., $TS_{var} < 143.94$). This implies that we can not conclude at 99% confidence level that our target sources are variable due to lack of statistics. The variability significance of all sources in the study sample was found to be in the range of 0.5% to 12%. The variability significance of 3FGL J2015.3-1431 was found to be the highest compared to other sources in the study sample. Conclusions {#sec.6} =========== Although probing the $\gamma$-ray variability of blazar candidate sources is of definite interest in the study of AGN properties towards a better classification of the sources, definite classification is expected to be properly achieved by multiwavelength studies of their spectral energy distribution together with their optical spectra. Indeed, variability is well understood when it is studied across the electromagnetic spectrum (Radio, Optical/UV and X-rays), and on different timescales. This contributes to checking the variability correlation in different energy bands and testing whether variability exists for all timescales. Therefore, future studies are expected to consider multi-waveband variability and on different timescales. The Variability can be applied to estimate physical parameters of AGN such as the size of the emitting region, timescale of variability, magnetic field in the jets, mass of the central engine (blackhole), etc. However, the estimation of all these parameters requires primarily to know the object’s redshift, which can be obtained through spectroscopic studies. Therefore, future studies through the analysis of optical spectra of sources listed in our studied sample should be considered. Such observations from ground-based optical telescopes (such as a 10-meter class telescope) would be the ideal program to determine the nature of blazar candidates. Acknowledgements {#sec.7} ================ We acknowledge the useful contribution of Richard J.G. Britto, University of the Free State - South Africa. Financial support from the Swedish International Development Cooperation Agency (SIDA) through the International Science Programme (ISP) is also gratefully acknowledged. 2015, *ApJS*, 218, 23 2012 *ApJ*, 753, 83 2015, *ApJ*, 810, 14 1995, *A&A*, 574, A112 2015, *in AmericanAstronomical Society Meeting Abstracts \# 225. p.* 336.02 2019, in prep..., *MNRAS* 2012 *ApJ*, 752, 61 2012 *MNRAS*, 424, L64 2003, *ApJ*, 597,615 2017 *ApJ*, 851, 135 2019, *arXiv e-prints, arXiv:* 1902.10045 2015, *J. Cosmology Astropart. Phys.*, 2, E01 1995, *PASP*, 107, 803
--- abstract: 'We present a novel analytical framework for the evaluation of important second order statistical parameters, as the level crossing rate (LCR) and the average fade duration (AFD) of the amplify-and-forward multihop Rayleigh fading channel. More specifically, motivated by the fact that this channel is a cascaded one, which can be modelled as the product of $N$ fading amplitudes, we derive novel analytical expressions for the average LCR and AFD of the product of $N$ Rayleigh fading envelopes, or of the recently so-called $N*$Rayleigh channel. Furthermore, we derive simple and efficient closed-form approximations to the aforementioned parameters, using the multivariate Laplace approximation theorem. It is shown that our general results reduce to the specific dual-hop case, previously published. Numerical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed approximations.' author: - - - title: | Level Crossing Rate and Average Fade Duration\ of the Multihop Rayleigh Fading Channel --- [^1] Introduction ============ communications, a viable option for providing broader and more efficient coverage, can be categorized as either non-regenerative (amplify-and-forward, AF) or regenerative decode-and-forward, DF) depending on the relay functionality [@1]-[@Patel]. In DF systems, each relay decodes its received signal and then re-transmits this decoded version. In AF systems, the relays just amplify and re-transmit their received signal. Furthermore, a system with AF relays can use channel state information (CSI)-assisted relays [@1] or fixed-gain relays [@2] (also known as blind or semi-blind relays [@9]). A (CSI)-assisted relay uses instantaneous CSI of the channel between the transmitting terminal and the receiving relay to adjust its gain, whereas a fixed-gain relay just amplifies its received signal by a fixed gain [@2][@9]. Systems with fixed-gain relays perform close to systems with (CSI)-assisted relays [@2], while their easy deployment and low complexity make them attractive from a practical point of view. Several works in the open literature have provided performance analysis of AF or DF systems in terms of bit error rate (BER) and outage probability under different assumptions of the amplifier gain [@1]-[@Patel]. Among them, only two works dealt with the dynamic, time-varying nature of the underlying fading channel, [@Yang], [@Patel], despite the fact that it is necessary for the system’s design or rigorous testing. In [@Yang], the level crossing rate (LCR) and the average fade duration (AFD) of multihop DF communication systems over generalized fading channels was studied, both for noise-limited and interference-limited systems, while Patel et. al in [@Patel] provide useful exact analytical expressions for the AF channel’s temporal statistical parameters such as the auto-correlation and the LCR. However, the approach presented in [@Patel] is limited only to the dual-hop fixed-gain AF Rayleigh fading channel. In this paper, we study the second order statistics of the fixed-gain AF multihop Rayleigh fading channel. More specifically, motivated by the fact that this channel is a cascaded one, which can be modeled as the product of $N$ fading amplitudes, we derive a novel analytical framework for the evaluation of the average LCR and the AFD of the product of $N$ Rayleigh fading envelopes. Furthermore, we derive simple and efficient closed-form approximations using the multivariate Laplace approximation theorem \[16, Chapter IX.5\], [@18]. These important theoretical results are then applied to investigate the second order statistics of the multihop Rayleigh fading channel. Numerical and computer simulation examples verify the accuracy of presented mathematical analysis and show the tightness of the proposed approximations. Level Crossing Rate and Average Fade Duration of The Product of $N$ Rayleigh Envelopes ======================================================================================= Let $\{{X_i(t)}\}_{i=1}^N$ be $N$ independent and not necessarily identically distributed (i.n.i.d.) Rayleigh random processes, each distributed according to [@11]-[@12], $$\label{1} f_{X_i}(x)=\frac{2x}{\Omega_i} \exp \left( -\frac{x^2}{\Omega_i}\right) , \qquad x \geq 0,$$ in an arbitrary moment $t$, where $\Omega_i=E\{X_i^2(t)\}$ is the mean power of the $i$-th random process ($1\leq i \leq N$). If $\{{X_i(t)}\}_{i=1}^N$ represent received signal envelopes in an isotropic scattering radio channel exposed to the Doppler Effect, they must be considered as time-correlated random processes with some resulting Doppler spectrum. This Doppler spectrum differs depending on whether fixed-to-mobile channel [@11]-[@12] or mobile-to-mobile channel [@13]-[@14] appears in the wireless communications system. In both cases, it was found that time derivative of $i$-th envelope is independent from the envelope itself, and follows the Gaussian PDF [@11]-[@14] $$\label{2} f_{\dot X_i}(\dot x )=\frac{1}{\sqrt{2\pi}\sigma_{\dot X_i }} \exp\Big(-\frac{\dot x^2 }{2\sigma_{\dot X_i }^2}\Big) ,$$ with variance calculated as $$\label{3} \sigma_{\dot X_i }^2=\pi^2\Omega_i f_i^2 \,.$$ If envelope $X_i$ is formed on a fixed-to-mobile channel, then $f_i=f_{mi}$ where $f_{mi}$ is the maximum Doppler frequency shift induced by the motion of the mobile station [@11]-[@12]. If envelope $X_i$ is formed on a mobile-to-mobile channel, then $$\label{3a} f_i=\sqrt{f_{mi}^{'2}+f_{mi}^{''2}} \,. \vspace{-1.0mm}$$ where $f_{mi}^{'}$ and $f_{mi}^{''}$ are the maximum Doppler frequency shifts induced by the motion of both mobile stations (i.e., the transmitting and the receiving stations, respectively) [@14]. It is important to underline that the maximum Doppler frequency in a fixed-to-mobile channel is $f_{d \max}=f_{mi}$, whereas the maximum Doppler frequency in a mobile-to-mobile channel is $f_{d \max}=f_{mi}^{'}+f_{mi}^{''}$. The above results are essential in deriving the second-order statistical parameters of individual envelopes, as the LCR and the AFD [@11], [@12], [@14]. Below, we derive exact and approximate solutions for both of the above parameters for product of $N$ Rayleigh envelopes, $$\label{4} Y(t)=\prod_{i=1}^N X_i(t) \,. \vspace{-1.0mm}$$ We denote $Y(t)$ as $N*$Rayleigh random process or, at any given moment $t$, $N*$Rayleigh random variable, following the definition given in [@15]. For some specified value $\{X_i\}_{i=1}^N = \{x_i\}_{i=1}^N$, the product $Y$ is fixed to the specific value $y=\prod_{i=1}^N x_i$. The LCR of $Y$ at threshold $y$ is defined as the rate at which the random process crosses level $y$ in the negative direction [@11]. To extract LCR, we need to determine the joint probability density function (PDF) between $Y$ and $\dot Y$, $f_{Y\dot Y}(y,\dot y)$, and to apply the Rice’s formula \[11, Eq. (2.106)\], $$\label{6} N_Y(y)=\int_0^\infty \dot y f_{Y\dot Y} (y,\dot y) d\dot y \,. \vspace{-0.0mm}$$ Our method does not require explicit determination of $f_{Y\dot Y}(y,\dot y)$ in order to determine analytically the LCR of the $N*$Rayleigh random process, as presented below. First, we need to find the time derivative of (\[4\]), which is $$\label{7} \dot Y=Y\sum_{i=1}^N\frac{\dot X_i}{X_i} \,. \vspace{-0.0mm}$$ Conditioning on the first $N-1$ envelopes $\{X_i\}_{i=1}^{N-1} = \{x_i\}_{i=1}^{N-1}$, we have the conditional joint PDF $Y$ and $\dot Y$ written as $f_{Y\dot Y|X_1\cdot\cdot\cdot X_{N-1}}(y,\dot y|x_1,...,x_{N-1})$. This conditional joint PDF can be averaged with respect to the joint PDF of the $N-1$ envelopes $\{X_i\}_{i=1}^{N-1}$ to produce the required joint PDF, $$\begin{aligned} \label{8} f_{Y\dot Y}(y,\dot y) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\, \nonumber \\ = \int_{x_1=0}^\infty \cdots\int_{x_{N-1}=0}^\infty f_{Y\dot Y | X_1\cdots X_{N-1}}(y,\dot y | x_1,..., x_{N-1}) \nonumber \\ \times \, f_{X_1}(x_1)\cdots f_{X_{N-1}}( x_{N-1})dx_1\cdots dx_{N-1} \vspace{-0.0mm}\end{aligned}$$ where to derive (\[8\]) the mutual independence of the $N-1$ envelopes is used. The conditional joint PDF $f_{Y\dot Y|X_1\cdot\cdot\cdot X_{N-1}}(y,\dot y|x_1,...,$ $x_{N-1})$ can be further simplified by setting $Y = y$ and using the total probability theorem, $$\begin{aligned} \label{9} f_{Y\dot Y | X_1\cdots X_{N-1}}(y,\dot y | x_1,..., x_{N-1}) \qquad \qquad \qquad \qquad \nonumber \\ =f_{\dot Y | Y X_1\cdots X_{N-1}}(\dot y |y, x_1,..., x_{N-1}) \qquad \qquad \quad \nonumber \\ \times \, f_{Y| X_1\cdots X_{N-1}}(y | x_1,..., x_{N-1}) \,\,,\end{aligned}$$ where each of the two multipliers in (\[9\]) can be determined from the above defined individual PDFs and their parameters. Based on (\[7\]), the conditional PDF $f_{\dot Y|Y X_1\cdot\cdot\cdot X_{N-1}}(\dot y|y,x_1,$ $...,x_{N-1})$ can be easily established to follow the Gaussian PDF with zero mean and variance $$\begin{aligned} \label{10} \sigma_{\dot Y | Y X_1\cdots X_{N-1}}^2=\left( y^2\sum_{i=1}^{N-1}\frac{\sigma_{\dot X_i}^2}{x_i^2}+ \sigma_{\dot X_N}^2\prod_{i=1}^{N-1}x_i^2\right) \qquad \quad \nonumber \\ =\sigma_{\dot X_N}^2\left[ 1+y^2\left(\prod_{i=1}^{N-1} \frac{1}{x_i^2}\right) \sum_{i=1}^{N-1}\frac{\sigma_{\dot X_i}^2}{\sigma_{\dot X_N}^2} \frac{1}{x_i^2}\right]\prod_{i=1}^{N-1}x_i^2 \,.\end{aligned}$$ The conditional PDF of $Y$ given $\{X_i\}_{i=1}^{N-1} = \{x_i\}_{i=1}^{N-1}$ that appears in (\[9\]) is easily determined in terms of the PDF of the remaining $N$-th envelope, $$\begin{aligned} \label{11} f_{Y|X_1\cdots X_{N-1}}(y|x_1,..., x_{N-1}) \qquad \qquad \qquad \qquad \qquad \nonumber \\ =f_{X_N}\left(y \prod_{i=1}^{N-1} \frac{1}{x_i}\right) \prod_{i=1}^{N-1}\frac{1}{x_i} \qquad\end{aligned}$$ Introducing (\[9\]) and (\[11\]) into (\[8\]), then (\[8\]) into (\[6\]) and changing the orders of the integration, we obtain $$\begin{aligned} \label{12} N_Y(y) = \int_{x_1=0}^\infty \cdots\int_{x_{N-1}=0}^\infty \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\ \left(\int_{\dot y=0}^\infty \dot y f_{\dot Y | Y X_1\cdots X_{N-1}}(\dot y |y, x_1,..., x_{N-1})d\dot y \right) \prod_{i=1}^{N-1}\frac{1}{x_i} \qquad \nonumber\\ f_{X_N}\left (y\prod_{i=1}^{N-1}\frac{1}{x_i}\right ) f_{X_1}(x_1)\cdots f_{X_{N-1}}(x_{N-1})dx_1\cdots dx_{N-1}\end{aligned}$$ The bracketed integral in (\[12\]) is found using (\[10\]) as $$\begin{aligned} \label{13} \int_{0}^\infty \dot y f_{\dot Y | Y X_1\cdots X_{N-1}}(\dot y |y, x_1, \cdots , x_{N-1})d\dot y =\frac{\sigma_{\dot Y | Y X_1\cdots X_{N-1}}} {\sqrt{2\pi}}\end{aligned}$$ By substituting (\[1\]) and (\[13\]) into (\[12\]), we obtain the exact formula for the LCR as $$\begin{aligned} \label{14} N_Y(y)= \frac{\sigma_{\dot X_N}}{\sqrt{2\pi}}\frac{2^Ny}{\Phi} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\ \times \int_{x_1=0}^{\infty} \cdots\int_{x_{N-1}=0}^\infty\left[ 1+y^2\left(\prod_{i=1}^{N-1} \frac{1}{x_i^2}\right)\sum_{i=1}^{N-1}\frac{\sigma_{\dot X_i}^2}{\sigma_{\dot X_N}^2} \frac{1}{x_i^2}\right]^{1/2} \nonumber \\ \times\exp\left[-\left(\frac{y^2}{\Omega_N}\prod_{i=1}^{N-1} \frac{1}{x_i^2}+\sum_{i=1}^{N-1}\frac{x_i^2}{\Omega_i}\right)\right]dx_1\cdots dx_{N-1}, \qquad\end{aligned}$$ where $$\label{15} \Phi =\prod_{k=1}^{N}\Omega_k$$ In principle, (\[14\]) together with (\[15\]) provide an exact analytical expression for the LCR of the product of the product of $N$ Rayleigh envelopes (i.e., $N*$Rayleigh random process [@15]). However, (\[14\]) becomes computationally attractive only for small values of $N$, where it is possible to apply a numerical computation method (as Gaussian-Hermite quadrature). Note that, (\[14\]) is transformed into a single integral when $N=2$, which, after introducing (\[3\]) for $i = 1, 2$ and changing integration variable $x$ with new variable $t$ according $x=y/t$, reduces to the known result \[9, Eq. (17)\]. The AFD of $Y$ at threshold $y$ is defined as the average time that the $N*$Rayleigh random process remains below level $y$ after crossing that level in the downward direction, $$\label{16} T_Y(y)=\frac{F_Y(y)}{N_Y(y)} ,$$ where $F_Y(\cdot)$ denotes the cumulative distribution function (CDF) of $Y$. Fortunately, $F_Y(\cdot)$ was derived recently in closed-form \[14, Eq. (7)\], as $$\label{17} F_Y(y)=G_{1,N+1}^{N,1} \left[\frac{y^2}{\Phi}\Bigg | \begin{array}{cc} \qquad 1 \\ \underbrace{1,1,\cdots ,1}_N, 0 \end{array} \right],$$ where $G[\cdot]$ is the Meijer’s $G$-function \[15, Eq. (9.301)\]. An Approximate Solution for the LCR ----------------------------------- Next, we present a tight closed-form approximation of (\[14\]) using the multivariate Laplace approximation theorem \[16, Chapter IX.5\], [@18] for the Laplace-type integral $$\label{18} J(\lambda)=\int_{\textbf{x} \in D} u(\textbf{x}) \exp(-\lambda h(\textbf{x}))d\textbf{x} ,$$ where $u$ and $h$ are real-valued multivariate functions of $\mathbf{x}=[x_1, \cdots , x_{N-1}]$, $\lambda$ is a real parameter and $D$ is unbounded domain in the multidimensional space $R ^{N-1}$. A comparison of (\[14\]) and (\[18\]) yields $$\label{19} u(\textbf{x})=\left[1+y^2\left(\prod_{i=1}^{N-1}\frac{1}{x_i^2}\right)\sum_{i=1}^{N-1}\frac{\sigma_{\dot X_i}^2}{\sigma_{\dot X_N }^2}\frac{1}{x_i^2}\right]^{1/2} ,$$ $$\label{20} h(\textbf{x})=\frac{y^2}{\Omega_N}\prod_{i=1}^{N-1}\frac{1}{x_i^2}+\sum_{i=1}^{N-1}\frac{x_i^2}{\Omega_i},$$ and $\lambda=1$. Note, that in the case of (\[14\]), all the applicability conditions of the theorem are fulfilled. Namely, within the domain of interest $D$, the function $h(\mathbf{x})$ has a single interior critical point $\tilde \textbf{x}=[\tilde x_1,\cdots ,\tilde x_{N-1}]$, where $$\label{21} \tilde{x_i}=y^{1/N}\frac{\Omega_i^{1/2}}{\Phi^{1/(2N)}} ,\qquad 1\leq i\leq N-1 ,$$ which is obtained from solving the set of equations $\partial h /\partial x_i =0$, where $1\leq i\leq N-1$. The Hessian $(N-1)\times (N-1)$ square matrix $\mathbf A$, defined by \[15, Eq. (14.314) \], is written as $$\label{22} \mathbf{A}=\left[ \begin{array}{cccc} 8/\Omega_1 & 4/\sqrt{\Omega_1\Omega_2} & \cdots & 4/\sqrt{\Omega_1\Omega_{N-1}}\\ 4/\sqrt{\Omega_2\Omega_1} & 8/\Omega_2 & \cdots & 4/\sqrt{\Omega_2\Omega_{N-1}}\\ . & . & \cdots & .\\ 4/\sqrt{\Omega_{N-1}\Omega_1} & 4/\sqrt{\Omega_{L-1}\Omega_2} & \cdots & 8/\Omega_{N-1} \end{array} \right ]$$ By using induction, it is easy to determine that the $N-1$ eigenvalues of $\mathbf A$ are calculated as $\mu_i=4/\Omega_i$ for $1\leq i \leq N-2$, and $\mu_{N-1}=4N/\Omega_{N-1}$. Thus, all eigenvalues of $\mathbf A$ are positive, which, by definition, means that the matrix $\mathbf A$ is positive definite. By means of the second derivative test, since the Hessian matrix $\mathbf A$ is positive definite at point $\tilde \mathbf x$, $h(\mathbf x)$ attains a local minimum at this point (which in this case is the absolute minimum in the entire domain $D$). At this interior critical point $\tilde \mathbf x$, $$\label{23} u(\mathbf{\tilde{x}})=\left (1+\sum_{i=1}^{N-1}\frac{\sigma_{\dot X_i}^2}{\sigma_{\dot X_N}^2}\frac{\Omega_N}{\Omega_i}\right)^{1/2}= \left ( 1+\sum_{i=1}^{N-1}\frac{f_i^2}{f_N^2}\right)^{1/2} ,$$ $$\label{24} h(\mathbf{\tilde{x}})=N\left (\frac{y^2}{\Phi}\right)^{1/N} \, ,$$ where (\[23\]) is obtained using (\[3\]). Now, it is possible to approximate (\[18\]) for large $\lambda$ as $$\begin{aligned} \label{25} J(\lambda)\approx \left( \frac{2\pi}{\lambda}\right)^{(N-1)/2}\left[ \frac{1}{\det (\mathbf{A})}\left(1+\sum_{i=1}^{N-1}\frac{f_i^2}{f_N^2}\right )\right ]^{1/2} \nonumber \\ \times \exp\left (-\lambda N\frac{y^{2/N}}{\Phi^{1/N}}\right) .\end{aligned}$$ It is well-know that the determinant of the square matrix is equal to the product of its eigenvalues. Hence, $\mathbf A$ can be written as $$\label{26} \det(\mathbf A)=\frac{N2^{2(N-1)}}{\prod_{k=1}^{N-1}\Omega_k} =\frac{\Omega_N N2^{2(N-1)}}{\Phi} .$$ Although approximation (\[25\]) is proven for large $\lambda$ [@17]-[@18], it is often applied when $\lambda$ is small and is observed to be very accurate as well. Similarly to [@19], we apply the theorem for $\lambda = 1$. Therefore, the approximate closed-form solution for the LCR of $N*$Rayleigh random process $Y$ at threshold $y$ is $$\begin{aligned} \label{27} N_Y(y) \approx \frac{\sigma_{\dot X_{N}}}{\sqrt{2\pi}}\frac{2^Ny}{\Phi}J(1) = \frac{2y (2\pi)^{N/2-1} \sigma_{\dot X_N}}{\Omega_N^{1/2} \Phi^{1/2}} \qquad \qquad \qquad \nonumber \\ \times \left[\frac{1}{N} \left( 1+\sum_{i=1}^{N-1}\frac{f_i^2}{f_N^2}\right)\right]^{1/2} \exp\left(-N\frac{y^{2/N}}{\Phi^{1/N}}\right) \qquad \nonumber\\ %= f_N\left[\frac{1}{N}\left ( %1+\sum_{i=1}^{N}\frac{f_i^2}{f_N^2}\right)\right]^{1/2}\frac{(2\pi)^{N/2}y}{\Phi^{1/2}} %\exp\left(-N\frac{y^{2/N}}{\Phi^{1/N}}\right)\end{aligned}$$ $$\label{27} = \left(\frac{1}{N}\sum_{i=1}^N f_i^2\right)^{1/2}\frac{(2\pi)^{N/2}y}{\Phi^{1/2}} \exp\left(-N\frac{y^{2/N}}{\Phi^{1/N}}\right) \,. \quad$$ The numerical results presented in Section IV validate the high accuracy of the Laplace approximation applied for our particular case. Combining (\[17\]) and (\[27\]) into (\[16\]), the AFD of the $N*$Rayleigh random process $Y$ at threshold $y$ is approximated as $$\begin{aligned} \label{27a} T_Y(y)\approx\left(\frac{1}{N}\sum_{i=1}^{N}f_i^2\right)^{-1/2}\frac{\Phi^{1/2}}{(2\pi)^{N/2}}\frac{1}{y} \qquad \qquad \qquad \qquad \nonumber \\ \times G_{1,N+1}^{N,1} \left[\frac{y^2}{\Phi}\Bigg | \begin{array}{cc} & \qquad 1 \\ & \underbrace{1,1,\cdots ,1}_N, 0 \end{array} \right]\exp\left(N\frac{y^{2/N}}{\Phi^{1/N}}\right).\end{aligned}$$ Second Order Statiscs of Multihop Transmission ============================================== Next, we apply the important theoretical result of the previous Section to analyze the second order statistics of the multihop relay fading channel. System Model ------------ We now consider a multihop wireless communications system, operating over i.n.i.d flat fading channels. Source station $S$ communicates with destination station $D$ through $N-1$ relays $T_1$, $T_2$,..., $T_{N-1}$, which act as intermediate stations from one hop to the next. These intermediate stations are employed with non-regenerative relays with fixed gain $G_i$ given by $$\label{30} G_i^2=\frac{1}{C_iW_{0,i}}$$ with $G_0=1$ and $C_0=1$ for the source $S$. In (\[30\]), $W_{0,i}$ is the variance of the Additive White Gaussian Noise (AWGN) at the output of the $i$-th relay, and $C_i$ is a constant for the fixed gain $G_i$. Assume that terminal $S$ is transmitting a signal $s(t)$ with an average power normalized to unity. Then, the received signal at the first relay, $T_1$, at moment $t$, can be written as $$\label{31} r_1(t)=\alpha_1(t)s(t)+w_1(t) \,,$$ where $\alpha_1(t)$ is the fading amplitude between $S$ and $T_1$, and $w_1(t)$ is the AWGN at the input of $T_1$ with variance $W_{0,1}$. The signal $r_1$ is then multiplied by the gain $G_1$ of the relay $T_1$ and re-transmitted to relay $T_2$. Generally, the received signal at the $k$-th relay $T_k$ ($k=1, 2,..., N-1$) is given by $$\label{33} r_k(t)=G_{k-1}\alpha_k(t)r_{k-1}(t)+w_k(t) \,,$$ resulting in a total fading amplitude at the destination node $D$, given by $$\label{34} \alpha(t)=\prod_{i=1}^N\alpha_i(t)G_{i-1} \,.$$ LCR and AFD of Multihop Transmissions ------------------------------------- If the fading amplitude received at node $T_i$, $\alpha_i(t)$, is a time-correlated (due to mobility of $T_{i-1}$ and/or $T_i$) Rayleigh random process, distributed according to (\[1\]) with mean power $\hat \Omega_i=E\{\alpha_i^2(t)\}$, then the $i$-th element of the product in (\[34\]), $X_i(t)=\alpha_i(t)G_{i-1}$, is again a time-correlated Rayleigh random process, distributed according to (\[1\]) with mean power $\Omega_i=\hat \Omega_i\, G_{i-1}^2$. Comparing (\[4\]) and (\[34\]), we realize that the total fading amplitude at the destination station $D$ (i.e., the received desired signal without the AWGN) is described as the $N*$Rayleigh random process $Y(t)=\alpha(t)$, whose average LCR and AFD are determined in the previous Section. If all stations are assumed mobile with maximum Doppler frequency shifts $f_{mS}, f_{mD}$, $f_{mi} (1\leq i\leq N-1)$ for the source $S$, destination $D$ and relays, respectively, then for the $i$-th hop $f_i^2=f_{m(i-1)}^2+f_{mi}^2$ with $f_{m0}=f_{mS}$ and $f_{mN}=f_{mD}$, and $$\label{35} \sum_{i=1}^N f_i^2=f_{mS}^2+2\sum_{i=1}^{N-1} f_{mi}^2+f_{mD}^2 \,.$$ Combining (\[27\]) and (\[35\]), we obtain approximate solution for the average LCR of the total fading amplitude $\alpha$ at the output of a multihop non-regenerative relay transmission system, $$\begin{aligned} \label{36} N_{\alpha}(\alpha)\approx \left[\frac{1}{N}\left(f_{mS}^2+2\sum_{i=1}^{N-1} f_{mi}^2+f_{mD}^2\right)\right]^{1/2} \qquad \quad \nonumber \\ \times \frac{(2\pi)^{N/2}\alpha}{\Phi^{1/2}} \exp\left(-N\frac{\alpha^{2/N}}{\Phi^{1/N}}\right) \,,\end{aligned}$$ where $\Phi$ is given by (\[15\]). We see that (\[36\]) approximates the average LCR of the total fading amplitude for arbitrary mean power of the fading amplitudes $\hat \Omega_i$, arbitrary relay gains $G_i$ and arbitrary maximal Doppler shifts $f_{mi}$. Note that, for $N=2$, (\[36\]) is an efficient closed-form alternative to the corresponding one \[9, Eq. (17)\] for the dual-hop case, which is shown in next section to be highly accurate. Numerical Results and Discussion ================================ In this section, we provide some illustrative examples for the average LCR and AFD of the fading gain process of the received desired signal at the destination of the multihop non-regenerative relay transmission system model from Section III. The numeric examples obtained from the derived approximate solutions are validated by extensive Monte-Carlo simulations. We considered a multihop system consisted of a source terminal $S$, 4 relays, and a destination terminal $D$. The fixed-gain relays are assumed semi-blind with gains in Rayleigh fading channel calculated as \[2, Eq. (15)\] and \[6, Eq. (19)\] $$\label{38a} G_{i,sb}^2=\frac{1}{\hat \Omega_i}\exp\left(\frac{1}{\bar\gamma_i}\right)\Gamma\left(0,\frac{1}{\bar\gamma_i}\right),$$ where $\bar \gamma_i=\hat \Omega_i/W_{0,j}$ is the mean SNR on the i-th hop, and $\Gamma(\cdot,\cdot)$ is the incomplete Gamma function. Relay gain calculated according to (\[38a\]) assures mean power consumption equal to that of a CSI-assisted relay, whose gain inverts the fading effect of the previous hop while limiting the output power at moments with deep fading. Depending on the stations’ mobility, we used two different 2D isotropic scattering models for the Rayleigh radio channel on each hop of the multihop transmission system. For the fixed-to-mobile channel (hop), we used the classic Jakes channel model [@11]-[@12]. For the mobile-to-mobile channel (hop), we used the Akki and Habber’s channel model [@13]-[@14]. The Monte-Carlo simulations of the latter were realized by using the sum-of-sinusoids method proposed in [@20]-[@21]. More precisely, all mobile stations are assumed to induce same maximal Doppler shifts $f_m$, while the destination $D$ is fixed. For all hops, $\hat \Omega_i=\hat \Omega$ and $W_{0,i}=W_0$. Thus, $\bar\gamma_{i}=\bar\gamma$, $G_{i,sb}=G_{sb}$, and the mean of Rayleigh random process $X_i(t)=\alpha_i(t)G_{i-1,sb}$ is calculated as $$\label{38b} \Omega_i = \exp\left(\frac{1}{\bar\gamma}\right)\Gamma\left(0,\frac{1}{\bar\gamma}\right)= \Omega \,\,, 2\leq i \leq N$$ whereas $\Omega_1=\hat \Omega$ is selected independently from the AWGN, since $G_0=1$. In this case, $$\label{38c} \Phi=\hat\Omega\,\exp\left(\frac{N-1}{\bar\gamma}\right)\left[\Gamma\left(0,\frac{1}{\bar\gamma}\right)\right]^{N-1}$$ Note that, when introducing above scenario into (\[36\]), $\alpha$ and $\hat \Omega$ appear together as $\alpha/\sqrt{\hat \Omega}$. ![Average LCR, $\hat \gamma_i=\hat \gamma = 5$ dB[]{data-label="fig_1"}](Fig1){width="3.2in"} ![AFD, $\hat \gamma_i=\hat \gamma = 5$ dB[]{data-label="fig_2"}](Fig2){width="3.2in"} Figs. 1-4 depict the received signal’s normalized LCR ($N_{\alpha}/f_m$) or normalized AFD ($T_{\alpha}f_m$) versus the normalized threshold ($\alpha/\sqrt{\hat \Omega}$) at 3 different stations along the multihop transmission system: at relay $T_2$ (curve denoted by $N = 2$), at relay $T_3$ (curve denoted by $N = 3$) and at the destination $D$ (curve denoted by $N = 5$). All comparative curves show an excellent match between the approximate solution and the Monte-Carlo simulations. [1]{} J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative Diversity in Wireless Networks: Efficient Protocols and Outage Behavior," *IEEE Trans. Inform. Theory*, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. M. O. Hasna, and M. S. Alouini, “A performance study of dual-hop transmissions with fixed gain relays," *IEEE Trans. Wireless Commun.*, vo. 3, no. 6, pp. 1963-1968, Nov. 2004. N. C. Beaulieu, and J. Hu, “A closed-form expression for the outage probability of decode-and-forward relaying in dissimilar Rayleigh fading channels," *IEEE Commun. Lett.*, vol. 10, no. 12, pp. 813-815, Dec. 2006. I. -H. Lee, and D. Kim, “Symbol error probabilities for general cooperative links," *IEEE Trans. Wireless Commun.*, vol. 4, no. 3, pp. 1264-1273, May 2005. M. O. Hasna, and M. S. Alouini, “Outage probability of multi-hop transmission over Nakagami fading channels," *IEEE Commun. Lett.*, vol. 7, no. 5, pp. 216-218, May 2003. G. K. Karagiannidis, “Performance bounds of multihop wireless communications with blind relays over generalized fading channels," *IEEE Trans. Wireless Commun.*, vol. 5, no. 3, pp. 498-503, March 2006. G. K. Karagiannidis, T. Tsiftsis, and R. K. Malik, “Bounds for multihop relayed communications in Nakagami-m fading," *IEEE Trans. Commun.*, vol. 54, no. 1, Jan. 2006. L. Yang, M. O. Hasna, and M.-S. Alouini, “Average Outage Duration of Multihop Communication Systems With Regenerative Relays," *IEEE Trans. Wireless Commun.*, vol. 4, no. 4, pp. 1366-1371, July 2005 C.S. Patel, G.L. Stuber and T.G. Pratt, “Statistical Properties of Amplify and Forward Relay Fading Channels," *IEEE Trans. Veh. Tech.*, vo. 55, no. 1, Jan. 2006 W. C. Jakes, *Microwave Mobile Communications*, Piscataway, NJ: IEEE Press, 1994. G. L. Stuber, *Principles of Mobile Communications*, Boston: Kluwer Academic Publishers, 1996. A. S. Akki and F. Haber, “A Statistical Model for Mobile-To-Mobile Land Communication Channel," *IEEE Trans. Veh. Technol.*, vol. VT-35, no. 1, pp. 2-7, Feb. 1986. A. Akki, “Statistic Properties of Mobile-to-Mobile Land Communication Channels," *IEEE Trans. Veh. Tech.*, vol. 43, no. 4, pp. 826-831, Nov. 1994 G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N\*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels," *IEEE Trans. Commun.*, vol. 55, no. 8, pp. Aug. 2007 I. S. Gradshteyn and I.M. Ryzhik, *Table of Integrals, Series, and Products*, 6th ed. New York: Academic, 2000. R. Wong, *Asymptotic Approximations of Integrals*, SIAM: Society for Industrial and Applied Mathematics, New edition, 2001. L. C. Hsu, “A Theorem on the Asymptotic Behavior of a Multiple Integral," *Duke Mathematical Journal*, 1948, pp. 623-632. R. Butler and A. T. A. Wood, “Laplace Approximations for Hypergeometric Functions of Matrix Argument," *The Annals of Statistics*, vol. 30, pp. 1155-1177, 2001. C.S. Patel, G.L. Stüber, and T.G. Pratt, “Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels," *IEEE Trans. Commun.*, vol. 53, no. 11, pp. 1876-1884, Nov. 2005 A.G. Zajic and G.L. Stuber, “A New Simulation Model for Mobile-to-Mobile Rayleigh Fading Channels," *Proc. IEEE WNCN 2006* ![Average LCR, $\hat \gamma_i=\hat \gamma = 20$ dB[]{data-label="fig_3"}](Fig3){width="3.2in"} ![AFD, $\hat \gamma_i=\hat \gamma = 20$ dB[]{data-label="fig_4"}](Fig4){width="3.2in"} [^1]: Accepted at IEEE ICC 2008
--- title: '$SU(n)$ and $U(n)$ Representations of Three-Manifolds with Boundary ' --- Sylvain E. Cappell  and  Edward Y. Miller
--- abstract: 'The out-of-plane intercalate phonons of superconducting YbC$_6$ have been measured with inelastic x-ray scattering. Model fits to this data, and previously measured out-of-plane intercalate phonons in graphite intercalation compounds (GICs), reveal surprising trends with the superconducting transition temperature. These trends suggest that superconducting GICs should be viewed as electron-doped graphite.' author: - 'M. H. Upton' - 'T. R. Forrest' - 'A. C. Walters' - 'C. A. Howard' - 'M. Ellerby' - 'A. H. Said' - 'D. F. McMorrow' bibliography: - 'ArXivYbC6.bib' title: 'Phonons and superconductivity in YbC$_6$ and related compounds' --- Interest in superconductivity in graphite intercalation compounds (GICs) has been reignited by the discovery of relatively high transition temperatures in YbC$_6$ (6.5 K) and CaC$_6$ (11.4 K) [@Weller; @Emery]. Because neither graphite nor the intercalate exhibit a high transition temperature independently the superconductivity must result from the combination of the graphene and intercalate sheets. The major question in GIC physics is the nature and strength of the graphene-intercalate interactions, which, despite a great deal of theoretical and experimental work, are still not clearly understood [@Weller; @Emery; @Csanyi; @MarkDean; @Valla; @Pan; @Mazin1; @Mazin; @Walters]. There are now two competing views of the phenomenon. Superconductivity may arise mainly from the interaction of the electronic states and in-plane phonons of the electron-doped graphene. Alternately, it may be a result of coupling between interlayer electronic bands and both in-plane intercalate (I$_\textrm{xy}$) and out-of-plane carbon (C$_\textrm{z}$) phonons. The question addressed in this paper is whether the dominant contribution to superconductivity in GICs is phonon coupling with the electron-doped graphite band structure or the intercalate band structure. Angle resolved photoemission (ARPES) experiments suggest that GICs are best viewed as electron-doped graphite, with the intercalate relegated to the role of electron donor. Electron doping raises the Fermi level above the Dirac point and thus changes the Fermi surface. The measured electron-phonon coupling to the in-plane carbon C$_\textrm{xy}$ phonons is sufficient to explain the superconductivity in these compounds [@Valla; @Pan] although this interpretation of the data is not universally accepted [@Calandra_epc; @Park]. Furthermore, Raman measurements show greater electron-phonon interaction with the C$_\textrm{xy}$ phonons than predicted by theory [@MarkDean]. The electron doping of the carbon atoms as a function of T$_\textrm{c}$ has been measured and a largely monotonic dependence found [@Pan]. The alternate view is that the intercalate is more directly involved and that interlayer electrons couple to both I$_\textrm{xy}$ and C$_\textrm{z}$ phonons and that this coupling is responsible for superconductivity [@Mauri]. This model has a number of variations because the origin of the interlayer electronic band is controversial. The interlayer electrons may come from either the graphite bandstructure [@Csanyi], the intercalate bandstructure [@Mauri] or both [@Mazin1]. There is strong experimental evidence that the intercalate atoms must be important to superconductivity in GICs. In CaC$_6$, the isotope effect for Ca is measured to be $\alpha(\textrm{Ca})=-(d\log T_c/d\log M_{\textrm{Ca}})=0.40$, where $M_{\textrm{Ca}}$ is the mass of the calcium atom [@HinksAPS]. This value is even greater than the substantial calculated value of $\alpha(\textrm{Ca})=0.24$ [@Calandra_epc]. A similarly high value was reported for $\alpha(\textrm{Yb})$ [@HinksAPS]. Furthermore, DFT calculations suggest coupling to the intercalate Fermi surface is responsible for superconductivity in some GICs [@Mauri]. YbC$_6$ is a critical material for understanding GIC superconductivity since it is one of only a few GICs superconduct between 11.4 K and 1 K. Previous papers have often relied on comparing CaC$_6$ ($\textrm{T}_\textrm{c}=11.4$ K) to GICs which superconduct below 1 K or do not exhibit any superconductivity. Therefore, understanding superconductivity in YbC$_6$ may provide a missing link in understanding the role of the intercalate in GICs superconductivity. This paper presents the dispersion of out-of-plane intercalate (I$_\textrm{z}$) phonons in YbC$_6$. The intercalate [I$_\textrm{z}$ ]{}phonons are a suggestive probe of the intercalate-graphene interaction because they reflect the forces between the intercalate and graphene planes. Additionally, soft phonons in YbC$_6$ are interesting in their own right because calculations predict a considerable density of Yb states at the Fermi level, suggesting the possibility of superconductivity resulting from electron-phonon coupling between soft Yb phonons and electrons [@Mazin_YbC6]. Model fits to the measured [I$_\textrm{z}$ ]{}YbC$_6$ phonons may be compared to new fits of a number of previously measured GICs: CaC$_6$ [@Upton], KC$_8$, CsC$_8$ and RbC$_8$ [@Magerl]. Comparing the fitted results from the YbC$_6$ to previously measured compounds shows a surprising trend correlated with the superconducting transition temperature. These trends suggest that the superconducting transition temperature is a function of charge transfer from the intercalate to the graphene sheets. Samples were synthesized using the vapor transport method from natural Madagascan graphite flake, as described in Weller et al. [@Weller]. The sample dimensions were 3 x 3 x 0.7 mm with a post-intercalation *c*-axis mosaic of $5^\circ$. The space group of YbC$_6$ is P$6_3$/*mmc*. The structure is single graphene layers separated by ordered Yb layers, with a=4.32 $\textrm{\AA}$ and c=9.14 $\textrm{\AA}$ [@ElMarkini]. The graphene layers are not staggered, that is, all the carbon atoms in successive layers are superimposed and their stacking is AA. Successive Yb layers, however, are staggered and have a stacking $\alpha \beta$. Thus, the entire crystal has a periodicity of A$\alpha$A$\beta$. A picture of the structure is shown in Fig. \[fig:ResultsUC\]. \[fig:Pics\] After synthesis, the samples were mounted in a beryllium dome in an argon atmosphere to prevent exposure to oxygen and water, which degrade the samples. The diffraction pattern of the sample was checked before, after and at several times during the experiment to ensure that their quality did not diminish. The sample was composed of two regions: a region of YbC$_6$ and an unintercalated graphite region. The phonons were measured in the region with the largest YbC$_6$ signal which also contained a small amount of unintercalated graphite. It was possible to exclude graphite signal from this analysis by examining the periodicity of the measured phonons and by comparing the results to the well known graphite phonon dispersion [@Mohr]. Inelastic x-ray scattering (IXS) experiments were carried out at the high-energy resolution IXS spectrometer (HERIX) at sector 30 at the Advanced Photon Source of Argonne National Laboratory. The incident synchrotron beam is monochromatized to 1 meV and focused by a Kirkpatrik-Baez mirror to a spot size at the sample of $35 \times 20 \mu\textrm{m}$. To collect energy loss spectra the incident energy is scanned while the measured energy of the scattered radiation was held constant at 23.724 keV. Data were collected using Si(12 12 12) analyzer reflections. The instrument had an overall energy resolution of $\sim1.5$ meV. Nine analyzer crystals and nine independent detectors allowed data collection at nine momentum transfers simultaneously. The instrumental momentum resolution was $0.13 \textrm{\AA}^{-1}$. The spectra were normalized by a beam intensity monitor immediately before the sample. A series of spectra were measured at room temperature in reflection geometry, one typical spectrum is shown in Fig. \[fig:ResultsDisp\]. The data is shown as black open circles. A number of spectral features are noticeable. The large peak at zero energy loss is the elastic line; the three smaller peaks at non-zero energy loss are phonons. The data is fitted on a log scale to reduce the influence of the elastic line relative to the phonons. A sample fit is shown as a blue line. The elastic line is fitted as a pseudo-Voigt while each phonon peak is fitted as a Lorentzian. Corresponding stokes and anti-stokes phonons are constrained to have identical energy losses, widths and the theoretical intensity ratio. The intensity, width and peak position are fit. The dispersion of YbC$_6$ phonons is shown in Figs. \[fig:SpringFIT\] and \[fig:ResultsFIT\]. The data are shown as open symbols in a reduced zone scheme with spectra from different Brillouin zones folded back into the first Brillouin zone. Three bands are measured, two acoustic and one optical. The spread in measured phonon energies comes from fitting error, inexact sample alignment and integration over the experimental momentum resolution. By comparison with the calculations of Calandra and Mauri [@Calandra_epc] the lower acoustic band, shown as open squares, is attributed to in-plane vibration while the upper acoustic and optical branches, shown as open circles, are attributed to out-of-plane vibration. It is surprising that the lower acoustic band is observed because the direction of momentum transfer is perpendicular to the predicted direction of vibration and therefore IXS selection rules predict that the phonon would not be observed in this geometry. Despite this apparent prohibition, however, the lower acoustic mode has been observed in CaC$_6$ [@Upton]. An explanation for this phenomena has been advanced by d’Astuto et al. [@Matteo]. d’Astuto et al. have suggested that the lower [I$_\textrm{z}$ ]{}band does not reach zero at $\Gamma$ in CaC$_6$, but instead interacts with a lower energy mode and has a finite energy at $\Gamma$ [@Matteo]. Although our data lacks the resolution necessary to observe or eliminate this suggestion, we believe the results of the our model fits are valid. First, when the optical mode are fit alone, without the acoustic modes, the trends discussed in this paper remain. Second, the suggestion by d’Astuto, while intriguing, is clearly a higher order effect and involves the details of the interactions in CaC$_6$, not the more general (but less detailed picture) presented in this paper. In YbC$_6$, no phonon peak widths greater than the instrumental resolution were measured. Phonon peak broadening beyond the instrumental resolution might have been indicative of electron-phonon coupling, presumably with the interlayer electrons. The absence of measured broadening does not eliminate the possibility of electron-phonon coupling. Measurements of the phonon dispersion were repeated at 10 K, just above the superconducting transition temperature. The low temperature dispersion was identical to the dispersion measured at room temperature. Furthermore, no change in phonon peak width were observed. Had either effect been observed it would have been suggestive of electron-phonon coupling. \ \[fig:Results\] In an attempt to understand the physics of GIC interplanar interactions a number of different models were applied to YbC$_6$ and to previously measured GIC dispersions, as in [@Zabel_long]. Three models are discussed here: a simple spring mode, a bond charge model and a shell model. The best fit comes from the bond charge model and its implications are discussed. These models are all one-dimensional and the graphene and intercalate planes are treated as mass densities. The mass densities are calculated from the atomic masses of C and Yb and knowledge of the structure of the compounds. In all of these models, the in-plane structure of the GICs is ignored. While in-plane phonons are, potentially, very important to superconductivity in GICs [@Valla; @Calandra_epc], they are ancillary to a model designed to illuminate the interactions between the graphene and intercalate planes. A simple spring models with nearest neighbor and second nearest neighbor springs did not fit the data well. In this model, neighboring graphene and intercalate planes are connected by springs and each graphene (Yb) plane is connected to the two nearest graphene (Yb) planes. In total, three spring constants are fit. A cartoon of the second nearest neighbor spring model is shown in Fig. \[fig:Springmodel\] and the fitting results are shown as solid lines in Fig. \[fig:SpringFIT\]. In particular the model did not fit the phonon gap at L, the edge of the Brillouin zone, and did not reproduce the acoustic mode dispersion. A fit to the YbC$_6$ data with the bond charge model is shown as black solid lines in Fig. \[fig:ResultsFIT\]. A cartoon of this model is shown in Fig. \[fig:Resultsmodel\]. The bond charge model assumes some electrons (the bond charge) bond to both the graphene and intercalate planes. The bond charge is partially localized away from either plane and both the graphene and intercalate planes are coupled to both planes by separate springs. The mass of the bond charge is assumed to be zero (adiabatic approximation). The fitting parameters are the two spring constants - one between the graphene and bond charge, called K$_\textrm{e-c}$, and one between the intercalate and bond charge, K$_\textrm{e-i}$. Notice that with just two parameters the model is in excellent agreement with the data. The bond charge model was previously used to fit GIC phonons [@Zabel_long]. Finally, while the bond charge is shown as physically dividing the space between the graphene and intercalate planes in Fig. \[fig:Resultsmodel\] the model does not require this exact separation but only requires that the net force from the bond charge attraction be perpendicular to the plane. It is possible to identify the bond charge with the nearly free electron band, proposed by Mazin and others [@Mazin1; @Boeri; @Csanyi; @Mazin_YbC6; @Mauri], however, other possible interpretations exist. In a simple shell model [@Bruesch], a cartoon of which is shown in Fig. \[fig:Shellmodel\], a spherical electronic shell isotropically couples to its rigid ion-core by a spring constant, the shell is coupled to the neighboring electronic shell by an additional spring constant. The model has three fitting parameters: the spring constant between the intercalate ion core and its electronic shell, K$_\textrm{i}$; the spring constant between the carbon ion core and its electronic shell, K$_\textrm{c}$; and the spring constant between the shells of the intercalate and carbon, K$_\textrm{s-s}$. The fit generated by this model is indistinguishable from the fit generated by the bond charge model. However, because it has three parameters, the quality of shell model fit is lower than quality of the bond charge model fit, a two parameter model. In general models which allowed electron motion separate from nuclear motion were more successful than models which treated the atoms as points without intermediaries. This provides supporting, though not conclusive, evidence for the existence of the interlayer band. The bond charge model is able to fit the measured dispersions of many GICs in addition to YbC$_6$, as seen in Fig. \[fig:fit\] [@Upton; @Walters; @Zabel_long; @Magerl; @Zabel], in agreement with the results of Zabel [@Zabel_long]. The fitted spring constants for a series of GICs is shown in Fig. \[fig:fit\]. The fitted spring constant between the graphene and electron shell, K$_\textrm{e-c}$, is shown in Fig. \[fig:fitEC\]. The overall magnitude of the attraction between the graphene and the bond charge does not change much, but there is a noticeable drop in the superconducting transition temperature value as the spring constant increases. The fitted spring constant between the intercalate layer and bond charge, K$_\textrm{e-i}$, shown in Fig. \[fig:fitEI\], has a wide range of values. There is over a factor of four difference between the highest and lowest spring constants. Higher superconducting transition temperatures are associated with a high intercalate-bond charge spring constant (K$_\textrm{e-i}$). Both trends in the superconducting transition temperature as a function of spring constant are consistent with the idea of superconductivity coming from electronic doping of the graphene layers by the intercalate. In the case of greater transfer of electrons from the intercalate to the graphene there will be a stronger attraction (higher spring constant) between the intercalate and the bond charge. Similarly, if more electrons are transferred to graphene, the attractive interaction between the graphene and bond charge will be reduced. Previous ARPES measurements also suggested that GICs are best viewed as electron-doped graphene [@Valla; @Pan]. We note, however, that IXS measurements reflect the bulk of the material, while ARPES measurements are sensitive only to the surface. Therefore, the present measurements provide critical support for this understanding of the system. The present measurements can be readily compared to previous Raman measurements of Dean et al., which show an interesting, and related, trend linking the out-of-plane carbon phonons and the superconducting transition temperature [@MarkDean]. The zone-center out-of-plane carbon phonons are measured by Raman to be softer in GICs with a high superconducting transition temperature. These results have been interpreted as showing a correlation between electron-doping of the graphene sheets by the intercalate and the superconducting transition temperature. In conclusion, we have measured the out-of-plane intercalate phonons of YbC$_6$ and fitted them with the bond charge model. We have applied the model to a series of compounds and, in agreement with previous results [@Zabel_long], found that the bond charge model fits the \[00L\] intercalate phonons of many first stage GICs very well. The model fits are consistent with the understanding of superconductivity in GICs arising from phonon coupling to the electron-doped graphene Fermi surface rather than exclusively to the intercalate Fermi surface. The model can not, however, rule out a contribution to the interlayer band from intercalate atoms. Additionally, no phonon lifetimes shorter than instrumental resolution were measured in YbC$_6$ at 300 K and no change in the material was observed upon cooling to 10 K. Although this does not eliminate the possibility of coupling between electrons and intercalate phonons, it does not suggest it either. Use of the Advanced Photon Source was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The construction of HERIX was partially supported by the NSF under Grant No. DMR-0115852. Work in London was supported by the EPSRC and a Wolfson Royal Society Award. We thank Diego Casa for comments on the manuscript.
--- abstract: 'Standard semi-classical models of decoherence do not take explicit account of the classical information required to specify the system - environment boundary. I show that this information can be represented as a finite set of reference eigenvalues that must be encoded by any observer, including any apparatus, able to distinguish the system from its environment. When the information required for system identification is accounted for in this way, decoherence can be described as a sequence of entanglement swaps between reference and pointer components of the system and their respective environments. Doing so removes the need for the a priori assumptions of ontic boundaries required by semi-classical models.' author: - | [Chris Fields]{}\ \ [23 Rue des Lavandières, 11160 Caunes Minervois, France]{}\ [[email protected]]{}\ [ORCID: 0000-0002-4812-0744]{} title: '**Decoherence as a sequence of entanglement swaps**' ---  \ **Keywords:** Decomposition; Einselection; Measurement; Predictability sieve; Reference observable\ \ **PACS:** 03.65.Yz; 03.65.Ta; 03.65.Ud\ Introduction ============ While decoherence theory is of enormous practical importance for estimating the effective lifetimes of manipulable quantum states, standard models of decoherence and methods for calculating decoherence times remain semi-classical (for reviews, see [@zurek:98; @zurek:03; @schloss:07]). The physical mechanism of decoherence likewise remains controversial, with the environment functioning as an information sink in some formulations [@zeh:73; @joos-zeh:85; @tegmark:00; @tegmark:12] and as an information channel in others [@zurek:04; @zurek:05; @zurek:06; @zurek:09]. Hence while the timecourse of decoherence can be observed experimentally [@brune:96; @myatt:00; @brune:08], the underlying information dynamics are not yet fully characterized. Here I show that when the observations and hence information transfer required to identify the system undergoing decoherence are taken into account, decoherence can be represented as a sequence of entanglement swaps between distinct “reference” and “pointer” components of the system and their respective environments. The exponential decay of phase coherence for pointer observables that is derived as an approximation using semi-classical methods and observed experimentally emerges naturally in this representation. Consistent with Landauer’s principle [@landauer:61; @landauer:99], an “observation” in this setting is an exchange of energy for classical information in the form of observational outcomes, where information becomes effectively classical when it is recorded on some physical medium in a way that is thermodynamically irreversible over times much longer than the time required for the observation. An “observer” in this setting is thus any system that records observational outcomes as effectively classical states. As Krechmer [@krechmer:18] has emphasized, standard treatments of measurement within quantum theory, including all standard formulations of decoherence, ignore the operations required for apparatus calibration and hence ignore the calibration-relativity of observational outcomes. The calibration process is treated here as a change in system (typically apparatus) “settings” that enables specific calibrating pointer-state outcomes to be obtained; see [@fields:18] for discussion of this approach. Reference observables and system identification =============================================== In standard models of decoherence, the decompositional boundary between the system of interest $S$ and the decohering environment $E$, and hence the interaction $H_{SE}$ and the self-Hamiltonians $H_S$ and $H_E$ are considered to be given a priori [@zurek:98; @zurek:03; @schloss:07]. The assumption of an $S - E$ boundary is, effectively, an assumption of classical information sufficient to specify this boundary, and hence to specify the Hilbert space $\mathcal{H}_{S}$ of $S$. Any complete description of information flow in decoherence must account for this system-identifying information. In practice, systems of interest are only “given” by observation [@fields:18]. Reading the “pointer state” of an apparatus, for example, requires first identifying the apparatus by observing degrees of freedom other than the pointer degrees of freedom: the size, shape, layout of knobs and dials, and location of an apparatus in the laboratory are commonplace examples (Fig. 1). These identifying or *reference* degrees of freedom [@fields:18] must have observational outcome values, e.g. the particular shape, size, and layout of an apparatus, that are invariant over sufficiently long times to allow re-identification of the system of interest across multiple cycles of observations. The observational outcome values of these degrees of freedom must also remain invariant during “preparation” operations, including calibration. Degrees of freedom that are sufficiently stable to serve as reference degrees of freedom are generally shared by many systems other than the particular system of interest $S$; all macroscopic apparatuses, for example, have fixed sizes, shapes, and layouts. To identify and, later, re-identify $S$, an observer must search for something having the specific, invariant values of each of these reference degrees of freedom that identify $S$: the specific shape, size, layout, etc. to be $S$ and not something else. Finding something that satisfies all of the observational criteria to be $S$ typically requires interacting with many things besides $S$; one must, for example, typically look at many things in a cluttered laboratory to find the particular apparatus one is interested in. ![Identifying a system of interest requires measuring a set $\{ M^{(R)}_i \}$ of reference observables with time-invariant (or only slowly-varying) outcome values $\{ x^{(R)}_i \}$, such as the particular size, shape, and layout of the system, that are distinct from the set $\{ M^{(P)}_j \}$ of pointer observables with outcomes $\{ x^{(P)}_j \}$ that indicate the system’s time-varying state. Note that the $\{ x^{(P)}_j \}$ include the values of any manipulable settings that “prepare” the system to be in some particular subset of pointer states. Adapted from [@fields:12] Figs. 2 and 3. \[fig1\]](swaps-fig1-rev.eps){width="5.0in"} To make this process of searching, by observation, for the system of interest $S$ precise, consider an observer $O$ equipped with a finite collection $\{ M^{(R)}_i \}$ of Hermitian operators that act on collections of degrees of freedom within the “world” $W$ comprising everything other than $O$. Assume these operators act on reference degrees of freedom as described above, i.e. on degrees of freedom that are invariant or very slowly varying over time, and can hence be considered *reference observables*. The $\{ M^{(R)}_i \}$ can, without loss of generality, be considered to pose yes/no questions and hence to have binary outcome values $\{ x^{(R)}_i \}$ in some fixed, specified basis; the observable $M^{(R)}_j$, for example, may correspond to the question “is the color of what I am looking at black” while $M^{(R)}_k$ may correspond to “does it have linear dimensions $37 \times 25 \times 2.5$ cm?” Clearly such operators can act on many different systems within $W$. The system of interest $S$ is identified whenever all of a specified set $\{ x^{(R)}_i \}$ of reference outcome values are simultaneously obtained. Note that in this picture, systems that yield the same set of reference outcome values when probed with the $\{ M^{(R)}_i \}$ are indistinguishable. This inevitable [@moore:56] ambiguity of system identification by observers restricted to finite observational resources is decreased as the set $\{ M^{(R)}_i, x^{(R)}_i \}$ of reference observables and specified outcome values becomes large; this decrease in ambiguity is paid for by an increase in the thermodynamic resources required for system identification [@fields:18]. The reference outcome values employed to identify $S$ must, obviously, remain invariant not only during the time that $S$ is being observed, but long enough to allow re-identification of $S$ across multiple cycles of re-preparation and re-observation. They cannot, therefore, contribute to the characterization of the time-varying state of $S$ or specify any apparatus “settings” or other “preparation” procedures, including calibration, that alter the state but not the identity of $S$. In order to detect the time-varying state of $S$, including settings or other preparation outcomes, $O$ must also be equipped with a second finite set $\{ M^{(P)}_j \}$ of *pointer observables* with binary outcome values $\{ x^{(P)}_j (t) \}$ that specify the time-varying pointer state $|P \rangle$ of $S$, including any adjustable settings or preparation outcomes. These $\{ M^{(P)}_j \}$ must be assumed to act on whatever system is identified by the $\{ M^{(R)}_i \}$. In practice, the number of pointer observables is typically much smaller than the number of reference observables; the number of pointers, readouts, and settings on a typical apparatus, for example, is much smaller than the number of observable properties that would need to be specified to allow a novice observer to find that apparatus in the laboratory. Let $\{ M_k \} = \{ M^{(R)}_i \} \cup \{ M^{(P)}_j \}$ and assume that $O$ interacts with $W$ only by deploying the $M_k$ one at a time in some order. Assuming a total of $N$ observables, we can then represent the $O - W$ interaction as: $$\label{HOW} H_{OW} (t) = \sum_{k=1}^N \alpha_k (t) M_k$$ subject to the constraints that at all $t$, $$\sum_{k=1}^N \alpha_k (t) = 1$$ and assuming a constant time interval $\Delta t$ to make any single observation, $$\sum_{k=1}^N \int_t^{t + \Delta t} dt ~\alpha_k (t) M_k = c^{(O)} k_B T \Delta t \label{diss}$$ where $k_B$ is Boltzmann’s constant, $T$ is temperature, and $c^{(O)} \geq$ ln2 is a measure of $O$’s thermodynamic efficiency [@fields:18]. The function $\alpha_k (t)$ is naturally interpreted as the probability of deploying the measurement operator $M_k$ at $t$. The sequence of outcomes obtained depends on the $\alpha_k (t)$; however, the incremental heat dissipation of the measurements does not. The system of interest $S$ is, in this formulation, defined entirely implicitly: $S$ is whatever is identified by the $M^{(R)}_i$ returning the specified outcomes $x^{(R)}_i$ and its pointer state $|P \rangle$ is given by the outcomes $x^{(P)}_j$ returned by the $M^{(P)}_j$. Note that there is no choice of measurement basis in this formulation. The environment $E$ comprises, in this case, all degrees of freedom of $W$ except those of $S$. With these definitions of $S$ and $E$, embodies the assumption of an unobserved environment that justifies tracing over the state of $E$ in the environment as information sink formulation of decoherence [@zeh:73; @joos-zeh:85; @tegmark:00; @tegmark:12]; the interpretation of in the alternative environment as witness formulation is considered in §5 below. If the $\{ M^{(R)}_i \}$ are to return invariant outcome values $\{ x^{(R)}_i \}$ over multiple rounds of measurement as required for system identification, they must satisfy Zurek’s “predictability sieve” requirement. With $H_{OW}$ given by , this requirement can be written (cf. [@zurek:03] Eq. 4.41), for each operator $M^{(R)}_i$: $$[H_W + H_{OW}, M^{(R)}_i] = 0 \label{commute1}$$ where $H_W$ is the self-Hamiltonian of $W$. It is satisfied provided: $$[M^{(R)}_i, M^{(R)}_j] = 0 \quad \mathrm{and} \quad [M^{(R)}_i, M^{(P)}_j] = 0 \label{commute2}$$ for all $i, j$. Nothing, however, requires the pointer measurements $M^{(P)}_j$ to all mutually commute, as they typically do not in practice. Making the information dynamics of system identification explicit in this way moves the physics of decoherence from the $S - E$ boundary, where it is placed in semi-classical models [@zurek:98; @zurek:03; @schloss:07], to the $O - W$ boundary; the interaction $H_{SE}$ here remains unspecified, while $H_{OW}$ is given by . Decoherence is, therefore, observer-relative by definition in this representation. As shown in the next section, it is the process of observation itself, i.e. the interaction between $O$ and $W$, not an observer-independent interaction between $S$ and a semi-classical environment, that removes coherence from $S$ in this setting. Coarse-grained measurements as entanglement swaps ================================================= To make the information flow given by more explicit, suppose the observer $O$ has the structure shown in Fig. 2. Each of the operators $M_k$ is implemented by a qubit $q_k$, a classical, time-indexed outcome memory $x_k (t)$, and a single-bit recoding device that travels to the $k^{th}$ position with probability $\alpha_k$. The rest of the observer, not represented explicitly in Fig. 2, handles energy acquisition to drive the recording process and the dissipation of waste heat. No assumptions are made about the observer’s ability to *read* the memories it records; we can assume, however, that they have some effect on its behavior. ![Model observer $O$ comprising a qubit register $[q_k]$, a classical memory register $[x_k(t)]$, and a single-bit memory recording device.\[fig2\]](swaps-fig2.eps){width="5.0in"} Assume now that the system $S$ can be represented as $S = R \oplus P$, where $R$ and $P$ are the reference and pointer components, respectively, of $S$ and separability (indicated by $\oplus$) is guaranteed by . Consider both $R$ and $P$ to be implemented by qubit arrays $[r_i]$ and $[p_j]$ respectively, with numbers of qubits of at least the cardinality of $\{ M^{(R)}_i \}$ and $\{ M^{(P)}_j \}$ respectively. Define the “environments” of $R$ and $P$, respectively, by $E_R \otimes R = E_P \otimes P = W$, where $W$ as before is everything but $O$. Note that these definitions are consistent with $W$ being non-separable. With these definitions, the environment $E_P$ of $P$ contains $R$, as standardly assumed when the environment of the pointer state of an apparatus, for example, is assumed to include the rest of the apparatus [@tegmark:00; @tegmark:12]. With these assumptions, a reference-state measurement operator $M^{(R)}_i$ monogamously entangles the observer qubit $q_i$ with the reference qubit $r_i$ and a pointer-state measurement operator $M^{(P)}_j$ monogamously entangles the observer qubit $q_j$ with the pointer qubit $p_j$. Both operations also entail recording of the selected classical bit; completion of this recording step is the completion of the measurement operation, after which the next operation can take place. Recording requires a time increment $\Delta t$ and dissipates $c^{(O)} k_B T$ into either $E_R$ (for reference measurements) or $E_P$ (for pointer measurements) as specified by . Provided the recorded memory bits remain unread by $O$ or any third party, the order of measurements remains unknown, the description of the measurement process remains purely quantum mechanical, and no decoherence takes place [@zurek:18]. This situation corresponds to placing the “von Neumann cut” somewhere outside the joint $O - S$ system and its immediately-surrounding environment, e.g. to $O$ being an isolated, unexamined apparatus into which $S$ has been embedded. Either $O$ or some third party reading the recorded memory bits reveals the order of recording, and allows each measurement operation to be classified as either an operation on $R$ or an operation on $P$. A human observer or an apparatus “reporting” an observational outcome requires such a memory-read operation. This classification of observations by target subsystem (i.e. $R$ or $P$) effectively coarse-grains the entanglement process from the qubit scale to the multi-qubit subsystem scale. Equivalently, third-party observations of the measurement process that distinguish operations on $R$ from operations on $P$ are coarse-grainings from the qubit scale to the subsystem scale. Coarse-grained operations on $R$ imply monogamous $O - R$ entanglement, while coarse-grained operations on $P$ imply monogamous $O - P$ entanglement. Alternating between system-identifying reference measurements and state-determining pointer measurements is, therefore, executing an entanglement swap between $R$ and $P$. To see this in a simplified setting, consider $O$ to comprise just one “observer” qubit with state $| o \rangle$, and similarly consider $R$, $P$, $E_R$, and $E_P$ to comprise single qubits with states $| r \rangle$, $| p \rangle$, $| e_R \rangle$, and $| e_P \rangle$, respectively. Measuring $R$, then $P$, then $R$ again corresponds, given the $R - P$ separability required by , to the swap: $$| o \otimes r \rangle \oplus | e_R \rangle \longrightarrow | o \otimes p \rangle \oplus | e_P \rangle \longrightarrow | o \otimes r \rangle \oplus | e_R \rangle \label{swap}$$ where again $\oplus$ indicates separability. This swap corresponds to an alternation of decompositional boundaries, in which $P$ and $R$ are alternately entangled with the shared “external” environment component $E$ comprising those degrees of freedom shared by $E_P$ and $E_R$. Exponential decay of phase coherence ==================================== The presence of phase coherence in a system can be measured by violations of the Leggett-Garg inequality; mapping each binary outcome from $\{0, 1 \}$ to $\{ -1, 1 \}$, the Leggett-Garg inequality can be written $C_{21} + C_{32} - C_{31} \leq 1$ for $C_{ij}$ the classical correlation of measurements $i$ and $j$, and the indices referring to consecutive measurements at $t_1, t_2$ and $t_3$ [@emary:14]. Sequential measurements violating the Leggett-Garg inequality are indicative of an (at least approximately) pure state of an (at least approximately) isolated system; sequential measurements satisfying the Leggett-Garg inequality are, on the other hand, indicative of mixed states and thus sampling from a classical ensemble of identically-prepared but otherwise independent and hence separable systems. Acting with the $M^{(R)}_i$ to identify $S$ can be regarded as “preparing” $S$ by fixing the values of its non-pointer degrees of freedom. Similarly, any finite set $\{ x^{(R)}_i \}$ of time-invariant reference outcome values is consistent with a classical sampling process that selects a different element of an ensemble of distinct and independent but identically-prepared systems, each characterized by the constant $\{ x^{(R)}_i \}$ [@moore:56]. Evidence that a single system is being acted upon at multiple times with both the reference operators $M^{(R)}_i$ and the pointer operators $M^{(P)}_j$ is only obtained if the pointer outcomes $x^{(P)}_j$ obtained exhibit Leggett-Garg inequality violations [@fields:18]. Suppose now an ensemble of systems $< S_i > = < R_i \oplus P_i>$ that are indistinguishable by the reference operators $ M^{(R)}_i$, and for each of which a single, fixed pointer operator $M^{(P)}$ acts on a single qubit of each $P_k$ to yield an outcome $x^{(P)}$ and project a state $|p_k \rangle$. Suppose further that the $M^{(R)}_i$ are all executed once, in a fixed sequence, followed by $M^{(P)}$ in each cycle of measurements, with each operation requiring a time interval $\Delta t$ as before. The standard “picture” of measurement in which some particular system $S = S_k$ is given a priori, and hence system identification by the $M^{(R)}_i$ is unnecessary, corresponds in this case to a sequence of executions of $M^{(P)}$: $$M^{(P)} (t) \dots M^{(P)} (t + \Delta t) \dots M^{(P)} (t + 2 \Delta t) \dots M^{(P)} (t + 3 \Delta t) \dots$$ Here the projection postulate implies that each action of $M^{(P)}$ re-prepares the state $| p_k \rangle$ of the single pointer qubit of the given $S_k$; hence if $\Delta t$ is small compared to the timescale of interactions between $P_k$ and its environment $E_{P_k}$, all executions of $M^{(P)}$ return the same outcome value $x^{(P)}$. This is the familiar quantum Zeno effect [@misra:77]. Note that it assumes that repeated interactions with the observer, i.e. executions of $M^{(P)}$, maintain phase coherence and so do not decohere $P_k$. This corresponds to the swaps to $| r \rangle$ being removed from , leaving the fixed state $| o \otimes p_k \rangle \oplus | e_{P_k} \rangle$. If, however, $n - 1$ reference operators $M^{(R)}_i$ are executed between each execution of $M^{(P)}$, the delay between executions of $M^{(P)}$ is increased: $$M^{(P)} (t) \dots M^{(P)} (t + n \Delta t) \dots M^{(P)} (t + 2n \Delta t) \dots M^{(P)} (t + 3n \Delta t) \dots$$ This increased delay between pointer measurements allows more time for $P_k - E_{P_k}$ interactions. Coupling a degree of freedom $\xi$ of $E_{P_k}$ to $P_k$ is, effectively, exchanging the pointer component $P_k$ for the pointer component $P_k \otimes \xi$. Provided the $M^{(R)}_i$ do not (detectably) act on $\xi$, the commutativity conditions remain satisfied and this new pointer component $P_k \otimes \xi$ remains the pointer component of some element of the ensemble $< S_i >$ of systems that are indistinguishable by the $M^{(R)}_i$. If the probability of such a coupling to a degree of freedom of $E_{P_k}$ is $P_{int} > 0$ in any interval $\Delta t$ during which a reference operator is executed, the probability that the same pointer component $P_k$ and hence the same single element $S_k$ of $< S_i >$ is being probed on every cycle of measurement is, after $m$ cycles: $$Prob (\mathrm{pure}) = (1 - P_{int})^{m(n - 1)} \longrightarrow 0 ~\mathrm{for} ~m >> 1 ~\mathrm{or} ~n >> 1 \label{prob}$$ In semi-classical models of decoherence as resulting from otherwise-unobserved scattering of environmental particles, this exchange of $P_k$ for $P_k \otimes \xi$ is implemented by scattering events, with $n - 1$ corresponding to the particle number density in the scattering constant and $mn \Delta t$ to the total elapsed time [@schloss:07]. The practical implication of can be seen by considering Schrödinger’s cat. If $n = 1$, there are no swaps to $R$ and the cat state $(|\mathrm{dead} \rangle + |\mathrm{alive} \rangle) / \sqrt{2}$ can be maintained indefinitely by the quantum Zeno effect. In this case, however, no reference degrees of freedom are ever measured, so nothing identifies $(|\mathrm{dead} \rangle + |\mathrm{alive} \rangle) / \sqrt{2}$ as the state of a *cat*. Introducing reference operators $M^{(R)}_i$ to identify the cat introduces swaps to $R$ and the probability of operating repeatedly on a pure state of a single element of the ensemble $< S_i >$ of cat-like entities indistinguishable by the $M^{(R)}_i$ exponentially decreases as specified by , i.e. the cat state decoheres. These reference operators can act on whatever is singled out to function as a reference component $R$; identifying the exterior of the box and measuring heat radiated from it equally decoheres $(|\mathrm{dead} \rangle + |\mathrm{alive} \rangle) / \sqrt{2}$ [@nielsen:00]. What tell us, therefore, is that state purity can be maintained only if system identification is abandoned or at least minimized (i.e. $n$ is kept very small). The states of strongly-identified systems - e.g. macroscopic objects with well-defined sizes, shapes, and positions for which $n >> 1$ - decohere almost immediately. Einselection and the $S - E$ boundary ===================================== The complete set of observational outcomes $\{ x_k \} = \{ x^{(R)}_i \} \cup \{ x^{(P)}_j \}$ is also the set of eigenvalues of the interaction $H_{OW}$ between the observer and the observed world defined by . These eigenvalues encode, by assumption, all of the information that $O$ can obtain about $W$. This set can clearly be made arbitrarily large by including reference observables and sets of specified outcome values that identify and pointer observables that characterize many different systems; such an increase in observational capability must, as noted earlier, be paid for by an increase in thermodynamic resources. In the “environment as witness” formulation of decoherence, the external environment $E$ serves as an information channel from $S$ to $O$ [@zurek:04; @zurek:05]; this formulation is extended to multiple, non-interacting observers in quantum Darwinism [@zurek:06; @zurek:09]. The information transmitted by $E$ in these formulations is an encoding of the eigenvalues of the interaction $H_{SE}$, an interaction defined at the $S - E$ boundary. As noted earlier, this boundary and hence $H_{SE}$ are taken as given a priori. Decoherence is implemented at this boundary by the predictability sieve requirement , a process termed “environment-induced superselection” or einselection [@zurek:98]. If $E$ is treated as an observer and hence identified with $O$ and $S$ is identified with $W$, einselection is decoherence as described by , , and . The invariant eigenvalues $x^{(R)}_i$ in this case specify the $S - E$ boundary to $E$ and the pointer eigenvalues $x^{(P)}_j$ are the information einselected on that boundary. This remains true when an external observer is considered and both $S$ and $E$ become components of $W$ as above (Fig. 3). ![(a) If the external environment $E$ is treated as an observer, einselection is decoherence at the $S - E$ boundary. (b) Interposing $E$ as a channel between $O$ and $S$ changes neither the definition of $H_{OW}$ nor the specification of the $S - E$ boundary by the reference eigenvalues $x^{(R)}_i$. \[fig3\]](swaps-fig3.eps){width="6.0in"} The present, fully observation-based formulation of decoherence thus clarifies two points of ambiguity present in traditional, environment as information sink formulations but made more obvious in the environment as witness formulation. First, it shows that the classical information specifying the location of the $S - E$ boundary is encoded by $O$, however $O$ is defined. This information is encoded, specifically, by the invariant $S$-identifying eigenvalues $x^{(R)}_i$. This answers Zurek’s question of “how one can define systems given an overall Hilbert space ‘of everything’ and the total Hamiltonian” ([@zurek:98] p. 1794) without the need to postulate objectively-bounded, ontic systems in contravention of the associativity of Hilbert-space decomposition [@zanardi:01; @zanardi:04; @dugic:06; @dugic:08]. Second, it shows that the encoding redundancy required by quantum Darwinism is implemented by redundant encoding of the $x^{(R)}_i$ by multiple observers. There is no need to assume an objective, ontic redundancy of encoding by the environment [@fields:10]. Conclusion ========== Standard semi-classical models of decoherence do not take explicit account of the classical information required to specify the system - environment boundary. It is shown here that when this is done, decoherence can be described as a sequence of entanglement swaps between reference and pointer components of the system and their respective environments. The classical information specifying the boundary is encoded by the set $\{ x^{(R)}_i \}$ of reference eigenvalues that must be encoded in memory by any observer, including any apparatus, able to distinguish the system of interest $S$ from the external environment $E$. This formulation of decoherence renders both the $S - E$ boundary and the decoherence process itself observer-relative. It thus represents the physics of decoherence in a way that is broadly consistent with the relational view of quantum theory introduced by Rovelli [@rovelli:96]. In this picture, $S - E$ interactions that induce decoherence, e.g. scattering by environmental particles in semi-classical models, are equivalent to observer-relative exchanges between observationally-indistinguishable systems [@fields:12b]. “Systems” in this picture are, therefore, not ontic but rather FAPP [@bell:90]. 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--- abstract: 'In this Letter, we investigated theoretically the Mott-insulating phase of a deficient spinel chalcogenide GaV$_4$S$_8$, which is known to form a tetrahedral V$_4$S$_4$ cluster unit that results in molecular orbitals (MOs) with a narrow bandwidth in the noninteracting limit. We used a cluster extension of charge self-consistent embedded dynamical mean-field theory to study the impact of strong intra-cluster correlations on the spectral properties as well as the structural degrees of freedom of the system. We found that the strong tetrahedral clustering renders the atomic Mott picture ineffective, and that the resulting MO picture is essential to describe the Mott phase. It was also found that, while the spectral properties can be qualitatively described by the truncation of the Hilbert space down to the lowest-energy MO, a proper description of the structural degrees of freedom requires the inclusion of multi-MO correlations that span a larger energy window. Specifically, we found that the lowest-energy MO description overemphasizes the clustering tendency, while the inclusion of the Hund’s coupling between the lower- and higher-energy MOs corrects this tendency, bringing the theoretically predicted crystal structure into good agreement with the experiment.' author: - 'Heung-Sik Kim' - Kristjan Haule - David Vanderbilt bibliography: - 'GVS\_CDMFT.bib' title: ' Molecular Mott state in the deficient spinel GaV$_4$S$_8$ ' --- \#1 Intermetallic covalency in transition-metal chalcogenides or oxides often leads to the formation of density waves or transition-metal clustering [@Whangbo1992]. While this typically results in a reduction of Fermi surface, more dramatic changes may happen in correlated systems such as VO$_2$ [@Morin1959; @Qazilbash2007; @Biermann2004; @Brito2016] or 1$T$-phases of TaS$_2$ and NbSe$_2$ [@Wilson1975; @DISALVO1977; @Fazekas1979; @Nakata2016; @Calandra2018]. Another interesting class of materials is ternary deficient spinel chalcogenides $AM_4X_8$ ($A$ = Al, Ga, Ge; $M$ = Ti, V, Nb, Mo, Ta; $X$ = S, Se), where the four $M$ sites form a tetrahedral cluster and drive the system to be Mott insulating [@YAICH19849; @Pocha2000; @Pocha2005; @Johrendt1999; @Helen2006CM; @CHUDO2006; @Vaju2008; @Dorolti2010]. Among this family, GaV$_4$S$_8$ has been actively studied recently because of the existence of a rhombohedral polar ([*i.e.*]{}, with nonzero bulk electric polarization) phase with significant magnetoelectric coupling and the formation of a skyrmion crystal below $T_C$ = 13 K [@Kezsmarki2015; @Ruffe1500916; @Widmann2017]. However, the nature of the Mott-insulating phase, which remains robust even in the room-temperature paramagnetic cubic phase, has remained elusive; a simple electron counting per V site ($d^{1.75}$/V) suggests a metallic phase, while the actual system is insulating. Based on this observation and the strong V$_4$ clustering, this system has been suggested to be a Mott insulator with the V$_4$ molecular orbitals (MO) comprising the correlated subspace. It seems likely that, as in the example of VO$_2$, the electron-lattice coupling in GaV$_4$S$_8$ can be modified by electron correlations in a non-trivial manner, which may affect the nature of the low-temperature multiferroic phase [@Kezsmarki2015; @Ruffe1500916; @Widmann2017]. However, the difficulty of treating multisite correlated clusters has hindered a proper [*ab-initio*]{} theoretical treatment of this MO-based Mott phase. Dynamical mean-field theory (DMFT) has become a standard tool for tackling such correlated materials in an [*ab-initio*]{} manner [@DMFTreview1; @DMFTreview2; @DMFTreview3]. The cluster extension of the conventional single-site DMFT [@DMFTreview2] can be used to systematically increase the range of spatial correlations, extending the notion of locality from an atomic site to a cluster. The real-space version of cluster DMFT, *i.e.*, cellular DMFT (CDMFT) [@KotliarCDMFT2001], is particularly suitable for systems with strong clustering, as it allows one to compute all the intra-cluster self-energies from a corresponding quantum cluster-impurity model. However, the exponentially growing computational cost as a function of system size becomes an issue at this point; the number of cubic $t_{\rm 2g}$-orbitals in the V$_4$ cluster is 12, while state-of-the-art impurity solvers such as continuous-time Monte Carlo [@HauleCTQMC; @PatrickCTQMC; @CTQMCreview] cannot treat more than about 10 correlated orbitals. Because of this difficulty, a proper [*ab-initio*]{} study of the Mott phase of GaV$_4$S$_8$, fully incorporating lattice and charge degrees of freedom, has not yet appeared. Hence, in this Letter, we have studied the Mott phase of GaV$_4$S$_8$ in the high-temperature cubic (non-polar) phase above $T$ = 45 K, specifically focusing on the occurrence of the Mott phase via the MO formation and its impact on the structural degrees of freedom. We employed fully charge self-consistent CDMFT applied to the tetrahedral cluster of four V sites, starting from the simplest model containing only the lowest-energy MO ($T^2$ in Fig. \[fig:dft\]) and progressively enlarging the correlated Hilbert space to include the majority of $t_{\rm 2g}$ states in the $V_4$ cluster ($T^2+E+T^1_a$). The effects of such an extended model on the predictions of spectroscopic and structural properties were studied. Our CDMFT prediction was compared to most standard as well as advanced density functional theory (DFT) exchange-correlation functionals, including SCAN meta-GGA [@SCAN] and HSE hybrid functionals [@HSE06; @HSE06-2]. While these all fail to predict an insulating phase, our cluster calculation opens a gap very naturally, thus demonstrating that the MO picture is essential for describing the Mott phase. Surprisingly, the V$_4$S$_4$ clustering is strongly affected by the strength of the Hund’s coupling at the V sites. The CDMFT approach applied to this compound yields qualitatively different results compared to those obtained from DFT or DFT+$U$ [@Sieberer2007], demonstrating the power of DMFT in tackling correlated systems with multisite clusters. ![ (a) Crystal structure of the deficient spinel GaV$_4$S$_8$ in the cubic phase, in comparison with a fictitious perfect spinel Ga$_2$V$_4$S$_8$ illustrated in (b). Note the inter-cluster V-V bonds depicted in red dashed lines in (a), and white Ga sites in (b) which are absent in deficient spinel structure (a). (c) Splitting of 12 atomic $t_{\rm 2g}$ orbitals at 4 V sites in the V$_4$S$_4$ cluster into the molecular-orbital (MO) states. Seven electrons in the (V$_4$)$^{13+}$ cluster occupy the singlet $A^1$, doublet $E$, and triplet $T^2$ states, as shown the diagram. (d) MO-projected fat-band representation and density of state (PDOS) plots of GaV$_4$S$_8$ from the DFT results (without $U$). []{data-label="fig:dft"}](./Fig1.pdf){width="45.00000%"} [*Computational tools.*]{} To incorporate the electronic and structural degrees of freedom on an equal footing, we employed a state-of-the-art DFT+embedded DMFT code [@Dmft; @eDMFT] which allows relaxation of internal atomic coordinates. In CDMFT the experimental lattice parameter reported in Ref.  was employed, and optimizations of internal atomic coordinates were done using DMFT forces [@Force2016]. The hybridization-expansion continuous-time quantum Monte Carlo method [@HauleCTQMC; @PatrickCTQMC] was employed as the impurity solver. The atomic on-site Coulomb interactions were unitarily transformed and projected onto the MO basis, where the impurity hybridization function has a more appropriate form for the impurity solver. Details of this transformation and its implementation in the DFT+embedded DMFT code are discussed in the Supplementary Information (SI). The Vienna [*ab-initio*]{} Simulation Package ([vasp]{}) [@VASP1; @VASP2] was used for independent structural optimizations at the DFT level. ![ (a) A plot of the single-site DMFT spectral function with atomic V $t_{\rm 2g}$ states chosen as the correlated subspace ($U_d$ = 6 eV, $J_{\rm H}$ = 0.8 eV, T = 232K), showing a robust metallic character. (b) CDMFT spectral function and PDOS with the MO-$T^2$ states as the correlated subspace ($U_d$ = 6 eV, T = 232K). The red hue in the spectral function plot depicts the character of the MO-$T^2$ states. []{data-label="fig:MOT1"}](./Fig2.pdf){width="47.50000%"} [*Crystal structure and MO formation.*]{} Fig. \[fig:dft\](a) shows the crystal structure of cubic GaV$_4$S$_8$. Compared to the fictitious non-deficient spinel Ga$_2$V$_4$S$_8$ shown in Fig. \[fig:dft\](b), half of the Ga sites (white Ga$_2$ sites in the figure) are missing in GaV$_4$S$_8$, which breaks the inversion symmetry (space group $F\bar{4}3m$) and allows the clustering of V and half of S (S$_1$ sites in the figure). This gives rise to MOs formed out of the 12 atomic $t_{\rm 2g}$ orbitals in the V$_4$ cluster, as depicted in Fig. \[fig:dft\](c), where the 12 orbitals are split into 5 irreducible representations of the cubic $T_d$ point group, specifically $A^1 \oplus E \oplus T^{2} \oplus 2T^{1}$ (two $2T^{1}$ denoted as $T^{1}_{a,b}$ in the diagram). Note that the charge configuration is (V$_4$)$^{13+}$, so there are 7 electrons left in the cluster, fully occupying the singlet $A^1$ and doublet $E$ and filling one electron in the $T^2$ triplet, as shown in Fig. \[fig:dft\](c). The result of a DFT calculation (without including $U$) is shown in Fig. \[fig:dft\](d), showing MO-projected fat bands and partial density of states (PDOS) where blue, green, and red colors depict the MO-$A^1$, $E$, and $T^{2}$ orbital characters respectively. The MOs can be seen to be well separated in energy because of the strong clustering, implying that the MO orbitals can be a reasonable basis set for the following CDMFT calculations. ![image](./Fig3.pdf){width="98.00000%"} [*Single-site vs. CDMFT.*]{} Fig. \[fig:MOT1\] shows the comparison between the results from the conventional single-site DMFT and the simplest $T^2$-CDMFT calculations ($T$ = 232 K) [^1]. In the latter scheme, one treats the partially-filled $T^2$ triplet MO as the correlated subspace. Note that choosing the $T^2$ only as the correlated subspace is the simplest cluster-type approximation, but it already yields a completely different result compared to the single-site DMFT. Fig. \[fig:MOT1\](a) shows the $k$-dependent spectral function from the single-site DMFT calculation, employing the atomic V $t_{\rm 2g}$-orbitals as the correlated subspace with an on-site Coulomb repulsion of $U$ = 6 eV, appropriate for the V $t_{\rm 2g}$ set of quasi-atomic orbitals. A metallic band structure is clearly visible around the Fermi level, similar to the DFT result (Fig. \[fig:dft\](d)), due to the strong hybridization between the intra-cluster V sites and the mixed valence occupancy ($d^{1.75}$ per V). Increasing the $U$ value within the single-site DMFT did not induce a qualitative change. While the single-site DMFT cannot open the Mott gap for any physical value of $U$, the CDMFT yields a qualitatively correct result even when applied to the simplest $T^2$-triplet MO as shown in Fig. \[fig:MOT1\](b). Therein the splitting of the $T^2$ states into the lower and upper Hubbard bands can be seen, depicted in red hue in the spectral function plot (and the red curve in the PDOS), which leads to the opening of a charge gap. Note that since the $T^2$ triplet is 1/6-filled, it is not possible to obtain an insulating phase in the band picture without breaking both the cubic and time-reversal symmetries [@Sieberer2007], while in the Mott phase both symmetries can be kept. Hence we conclude that the cluster-MO description is indeed crucial in describing the Mott physics of GaV$_4$S$_8$, at least in its cubic and paramagnetic phase. Note that a similar result was previously reported on GaTa$_4$Se$_8$ by employing maximally-localized Wannier functions for the $T^2$ triplet and solving the Hubbard model via DMFT [@Camjayi2014]. However, as we will show below, this approach overestimates the tendency toward V$_4$ clustering since it ignores the important effect of the Hund’s coupling between the $T^2$ and other MOs on the structural degrees of freedom. [*$T^2 \oplus E$ subspace and Hund’s coupling.*]{} Despite the appearance of the Mott phase within the simplest $T^2$-CDMFT calculation, this is a crude approximation because other MO states are separated from the $T^2$ manifold by less than a fraction of an eV, and the Coulomb repulsion as well as the Hund’s coupling are larger or comparable to this separation. Therefore it is important to check what is the effect of including the next set of orbitals into the correlated space. Recently it was shown that the Hund’s coupling can have a very strong effect on the strength of correlations by promoting the local high-spin state and consequently allowing spins to decouple from the orbitals, thus allowing strong orbital differentiation [@Haule_njp; @Yin-nm11; @powerlaws; @Hundreview]. Such physics is completely absent in the $T^2$ model, as we assumed that the $E$ MOs are completely filled and inert, leaving a single electron in the $T^2$ MO set. We next treat the combination of $T^2 \oplus E$ MOs as our correlated subset. Fig. \[fig:MOT2\](a-c) shows the orbital-projected spectral functions from calculations with $J_{\rm H}$ = 0.5, 1.0, and 1.5 eV, respectively ($T$ = 232 K, $U$ = 8 eV). The red and green colors represent the $T^2$ and $E$ characters respectively. The signature of a low-to-high spin crossover, from the $S$ = 1/2 to 5/2 configuration, can be noticed in the plots where the fully occupied $E$ doublet (at $J_{\rm H}$ = 0.5 eV) begins to lose spectral weight as $J_{\rm H}$ is enhanced. Tracking the Monte Carlo probabilities for the $S_z$ = 1/2 and 5/2 states, plotted in Fig. \[fig:MOT2\](d), shows the same tendency that the $S_z$ = 1/2 probability decreases and collapses almost to zero around $J_{\rm H}$ $\sim$ 1 eV. Note that we report $S_z$ values rather than $S$ values, because of our choice of an Ising-type approximation of the Coulomb interaction in the CDMFT impurity solver [^2]. For $J_{\rm H} \gtrsim$ 1 eV, it can be seen that the $E$ doublet becomes half-filled (see Fig. \[fig:MOT2\](c) and (e)), showing that the crossover to the high-spin state is almost complete. Note that even a moderate $J_{\rm H} \lesssim$ 1 eV, appropriate for 3$d$ transition-metal compounds [@Vaugier2012], induces substantial mixing between the low-spin and high-spin states. Therefore one may suspect a potential role of the Hund’s coupling physics in the high-temperature cubic phase of GaV$_4$S$_8$. Unexpectedly, it turns out that the Hund’s coupling significantly weakens the degree of the V$_4$S$_4$ clustering, in contrast with the Coulomb repulsion $U$ which enhances the clustering, as shown in the following. ![ (a) Definitions of the intra- and inter-cluster V-V bond lengths $d^{\rm V}_{\rm int}$ and $d^{\rm V}_{\rm ic}$ respectively. (b) Schematic representations of the nonmagnetic (NM), low-spin (L-FM, $S$ = 1/2), and high-spin ferromagnetic (H-FM, $S$ = 5/2 or 7/2) configurations, where the dots and arrows depict nonmagnetic and magnetic electrons respectively. (c) $d^{\rm V}_{\rm int} / d^{\rm V}_{\rm ic}$ from DFT results with different choices of exchange-correlation potentials: LDA [@LDA], PBE [@PBE], PBEsol [@PBEsol], SCAN meta-GGA functional [@SCAN], DFT+$U$ [@Dudarev], and HSE06 hybrid functional [@HSE06; @HSE06-2]. In the DFT+$U$ results, the L-FM and H-FM configurations are obtained by employing $U_{\rm eff}$ = 2 and 4 eV in the simplified rotationally-invariant DFT+$U$ scheme [@Dudarev]. Horizontal gray dashed and black dotted lines show the values of $d^{\rm V}_{\rm int} / d^{\rm V}_{\rm ic}$ from experimental structures measured at $T$ = 295 and 20 K respectively [@Pocha2000]. []{data-label="fig:dftrx"}](./Fig4.pdf){width="48.00000%"} [*V$_4$S$_4$ clustering from DFT.*]{} A parameter quantifying the size of the V$_4$S$_4$ clustering is the ratio between the nearest-neighbor V-V distances, $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$, where $d^{\rm V}_{\rm int}$ and $d^{\rm V}_{\rm ic}$ denote the inter- and intra-cluster V-V distances respectively as shown in Fig. \[fig:dftrx\](a). $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ is unity in the ideal spinel structure, while in GaV$_4$S$_8$ the value was reported to be 1.35 at $T$ = 295 K and 1.37 at 20 K respectively (see the horizontal dashed/dotted lines in Fig. \[fig:dftrx\](c)) [^3]. Fig. \[fig:dftrx\](c) shows the ratios obtained from DFT calculations with different choices of exchange-correlation functionals [@LDA; @PBE; @PBEsol; @SCAN; @Dudarev; @HSE06; @HSE06-2], which have been reported to yield different values of lattice parameters. Three distinct magnetic configurations were considered: a nonmagnetic configuration (NM), a low-spin ferromagnetic configuration (L-FM) with $S$ = 1/2, and high-spin ferromagnetic configurations (H-FM) with $S$ = 5/2 or 7/2. These are schematically illustrated in Fig. \[fig:dftrx\](b). Note that because the $V_4$ cluster is believed to host a cluster spin moment, FM configurations were considered in our DFT calculations as appropriate for systems with local moments. Remarkably, the values of $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ shown in Fig. \[fig:dftrx\](c) are almost identical, at about 1.4, for all the results on the NM or L-FM configurations, despite different optimized lattice parameters (except HSE, see below). Thus, the degree of clustering is consistently overestimated compared to experimental values. On the other hand, the H-FM solutions with the DFT+$U$ or HSE06 hybrid functionals severely underestimate the clustering, as shown in Fig. \[fig:dftrx\](c). We notice that in H-FM solutions the lowest occupied MO bonding states ($E$, $A^1$) have been emptied at the expense of occupying higher nonbonding- or antibonding-like states. Therefore it is natural that H-FM solutions show a reduced tendency to clustering. Hence it appears that the small but significant discrepancy between the theoretical (in NM or L-FM) and experimental $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ values results from the small admixture of the high-spin configurations to the dominant low-spin configuration in the electronic states of GaV$_4$S$_8$, which cannot be captured in the framework of conventional DFT. Note that even though the HSE06 results with NM or L-FM configurations seem to reproduce reasonable $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ values, those states are much higher in energy by 1.5 eV/f.u. compared to the $S$ = 7/2 H-FM phase. Also, all of the DFT results (NM, L-FM, and H-FM) fail to reproduce the insulating phase, signifying the failure of the DFT methods in this system. ![ $d^{\rm V}_{\rm int} / d^{\rm V}_{\rm ic}$ from DFT results as a function of $J_{\rm H}$. Note that MO-$T^2$ and MO-$\{T^2 \oplus T^1_a\}$ configurations are not affected by $J_{\rm H}$ because of the single occupancy, and that the MO-$\{T^2 \oplus E \oplus T^1_a\}$ reaches the experimental $d^{\rm V}_{\rm int} / d^{\rm V}_{\rm ic}$ near $J_{\rm H}$ = 0.5 eV. []{data-label="fig:dV"}](./Fig5.pdf){width="48.00000%"} [*V$_4$S$_4$ clustering from CDMFT.*]{} Figure \[fig:dV\] shows the evolution of the $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ values from the DMFT results. As explained above, within the single-site DMFT the correlations appear to be weak, so that the predicted structure is very close to the DFT prediction. As the intra-cluster correlations are considered via the $T^2$ MO, the local Hubbard $U$ enhances the clustering tendency, which is clear from the predicted values at $J_{\rm H}$ = 0. It can be seen that the clustering tendency is substantially overemphasized when the $T^2\oplus E$ are considered as correlated, due to the bonding nature of the $E$ MO. When the antibonding $T^1_a$ MO is also included, the degree of clustering reverts back to similar value as for the $T^2$-only calculation. Still, the value of $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ is larger than the DFT-optimized one at $J_{\rm H}$ = 0, showing the role of $U$ in enhancing the clustering. Once the Hund’s coupling is turned on, the degree of clustering is quickly reduced (except for the $T^2$-only case where there is only one electron) as shown in Fig. \[fig:dV\]. We then obtain the experimental $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ values around $J_{\rm H}$ = 0.5 eV, which is a reasonable value for our model, in which $e_{\rm g}$ states (as well as $A^1$ and $T^1_b$) are screening the interaction. This observation is consistent with the spectroscopic tendency mentioned above, where $J_{\rm H}$ promotes the high-spin state so that spin moments can be more localized on each V site. We thus find, quite surprisingly, that in cases with strong clustering the Coulomb $U$ and Hund’s $J_{\rm H}$ can play opposite roles: the former promotes non-local correlations and formation of the bonding molecular orbital state, while the latter promotes local atom-centered high-spin states. This Janus-faced effect of $U$ and $J_{\rm H}$ is a central result of this study. Note also that the reduction of $d^{\rm V}_{\rm int}$/$d^{\rm V}_{\rm ic}$ is significant already at $J_{\rm H}$ = 0.5 eV, where the mixture of the high-spin configurations is quite small as shown in Fig. \[fig:MOT2\](d). This implies an unusual strong coupling between the electronic configuration and the V$_4$ clustering, which may be exploited to tune the spin configuration by employing optical pumping techniques as done in VO$_2$ [@Zheng2011ARMR]. [*Discussion and Summary.*]{} Our results demonstrate the promise of the MO-CDMFT approach employed in this work. With a careful choice of the MO correlated subspace, this approach can tackle systems with large-sized clusters that are not amenable to solution using conventional cluster DMFT approaches, also yielding much improved results compared to conventional DFT or single-site DMFT. Hence, with proper caution, it should be capable of treating other systems in which large clusters appear, such as $1T$-TaS$_2$. Overall, in this work we have clarified the significance of electron correlations in describing the MO Mott physics and structural properties of GaV$_4$S$_8$, especially the Janus-faced role of $U$ and $J_{\rm H}$ in its crystal structure. It should be emphasized that this is the first [*ab-initio*]{} study on the Mott phase of this compound, which can be extended to investigate the low-temperature ferroelectric and multiferroic phases [@Kezsmarki2015; @Ruffe1500916; @Widmann2017] and possible unconventional electron-lattice couplings therein. [*Acknowledgments*]{}: This work was supported by NSF DMREF DMR-1629059. Supplementary Information {#supplementary-information .unnumbered} ========================= Density functional theory calculations -------------------------------------- For unit cell optimizations (cell volume and shape) and relaxations of initial internal coordinates, the Vienna [*ab-initio*]{} Simulation Package ([vasp]{}), which employs the projector-augmented wave (PAW) basis set [@VASP1; @VASP2], was used for density functional theory (DFT) calculations in this work. 330 eV of plane-wave energy cutoff (PREC=high) and 15$\times$15$\times$15 $\Gamma$-centered $k$-grid sampling were employed. For the treatment of electron correlations within DFT, several exchange-correlation functional were employed, including Ceperley-Alder (CA) parametrization of local density approximation [@LDA], Perdew-Burke-Ernzerhof generalized gradient approximation (PBE) [@PBE] and its revision for crystalline solids (PBEsol) [@PBEsol], SCAN meta-GGA functional [@SCAN], DFT+$U$ [@Dudarev] on top of LDA, PBE, and PBEsol, and HSE06 hybrid functional [@HSE06; @HSE06-2]. $10^{-4}$ eV/Å of force criterion was employed for structural optimizations. Cluster dynamical mean-field theory calculations ------------------------------------------------ A fully charge-self-consistent dynamical mean-field method[@Dmft], implemented in DFT + Embedded DMFT (eDMFT) Functional code (<http://hauleweb.rutgers.edu/tutorials/>) which is combined with [wien2k]{} code[@wien2k], is employed for computations of electronic properties and optimizations of internal coordinates[@Force2016]. At the DFT level the Perdew-Wang (PW) local density approximation is employed, which was argued to yield the best agreement of lattice properties when combined with DMFT[@Haule2015FE]. 15$\times$15$\times$15 $\Gamma$-centered $k$-grid was used to sample the first Brillouin zone with $RK_{\rm max}$ = 7.0. A force criterion of 10$^{-4}$ Ry/Bohr was adopted for optimizations of internal coordinates. The cubic lattice parameter was fixed to be the experimental value reported in Ref. . A continuous-time quantum Monte Carlo method in the hybridization-expansion limit (CT-HYB) was used to solve the auxiliary quantum impurity problem[@HauleQMC]. For the CT-HYB calculations, up to $3 \times 10^{10}$ Monte Carlo steps were employed for each Monte Carlo run. In most runs temperature was set to be 232K, but in calculations with 8 molecular orbitals (MOs) ($T^2 \oplus E \oplus T^1_a$ in Fig. 1 in the main text) as the correlated subspace it was increased up to 1160K because of the increased computational cost. -10 to +10 eV of hybridization window (with respect to the Fermi level) was chosen, and the on-site Coulomb interaction parameters $U$ and $J_{\rm H}$ for V $t_{\rm 2g}$ orbitals were varied within the range of 6 $\sim$ 8 eV and 0 $\sim$ 1.5 eV, respectively. A simplified Ising-type (density-density terms only) Coulomb interaction was employed in this work, and it was tested that the use of full Coulomb interaction yields only quantitative difference in results with MO-$T^2$ and $T^2\oplus E$ (not tested for MO-$T^2\oplus E \oplus T^1_a$ case due to the high cost). A nominal double counting scheme was used, with the MO occupations for double counting corrections for for the V$_4$ cluster were chosen to be 1 or 5, depending on the choice of correlated subspace; 1 for MO-$T^2$ and $T^2\oplus T^1_a$, and 5 for other cases with including $E$ in the correlated subspace. In the CT-HYB calculations of the $T^2 \oplus E \oplus T^1_a$ MO subspace, MO multiplet states with the occupancy $n\leq 7$ were kept (26,333 states out of $4^8$ = 65,536 states in the 8 orbital Fock space) to reduce the computational cost, where the average impurity occupancy was $\sim$ 5. It was checked that the sum of probabilities for $n\geq 8$ configurations are less than 1 percent. The high-frequency tail of the Green’s function was calculated via the Hubbard-I approximation. Projecting the on-site Coulomb interactions onto the MO subspace ---------------------------------------------------------------- Note that the $U$ and $J_{\rm H}$ are parameters defined for the atomic orbitals, which should be unitary transformed and projected onto the MOs for the impurity solver. More generally, the Coulomb repulsion matrix elements $U_{m_1, m_2, m'_1, m'_2}$ at an atomic site have the form, $$\begin{aligned} U_{m_1, m_2, m'_1, m'_2} &= \sum_{m,k} \frac{4}{2\pi+1} \langle Y_{lm_1} \vert Y_{km} \vert Y_{lm'_1} \rangle \langle Y_{lm_2} \vert Y^*_{km} \vert Y_{lm'_2} \rangle F^k,\end{aligned}$$ where $F^k$ are nonzero only for $k$ = 0, 2, 4 for $d$-orbitals ($l$ = 2) and $\langle Y_{lm_1} \vert Y_{km} \vert Y_{lm'_1} \rangle$ are Clebsch-Gordan coefficients. We introduce the MO states $$\begin{aligned} \vert D_\alpha \rangle &= \sum_{im} (Q^\dag)^{im}_\alpha \vert Y^i_{lm} \rangle,\end{aligned}$$ where $Q$ is the unitary transform between the MO and the atomic orbitals, and $\alpha$ and $i=1,\cdots,4$ are the MO orbital and atomic site indices respectively. Then the Coulomb repulsion matrix elements for the MO states $U_{\alpha_1, \alpha_2, \alpha'_1, \alpha'_2}$ can be written as $$\begin{aligned} U_{\alpha_1, \alpha_2, \alpha'_1, \alpha'_2} &= \sum_{i,m,k} \frac{4}{2\pi+1} \langle D_{\alpha_1} \vert Y^i_{km} \vert D_{\alpha'_1} \rangle \langle D_{\alpha_2} \vert Y^{i*}_{km} \vert D_{\alpha'_2} \rangle F^k \\ &\sim (QQQ^\dag Q^\dag)^{i\{m\}}_{\{\alpha\}} U^i_{\{m\}}. \end{aligned}$$ Note that the inter-site Coulomb interactions were ignored here, which can be considered insignificant in 3$d$ transition metal compounds. Below we show explicitly how the on-site Coulomb interactions projected onto the $T^2$ triplet subspace should look like. As shown in Fig. 1 in the main text, electronic structure near the Fermi level (\[-1eV, 1eV\] window with respect to the Fermi level) is dominated by the atomic $t_{\rm 2g}$ orbitals of V due to the distorted but prevalent cubic VS$_6$ octahedral environment. Therefore choosing 12 $t_{\rm 2g}$ orbitals as our main interest is a reasonable choice. For simplicity we chose the Kanamori form of the Coulomb interaction, which is written in a normal-ordered form as follows; $$\begin{aligned} \hat{H}_K = -\sum_i \Big[ (U-2J) & \sum_{mm'} \hat{d}^\dag_{im\uparrow} \hat{d}^\dag_{im'\downarrow} \hat{d}_{im\uparrow} \hat{d}_{im'\downarrow} \nonumber \\ + 2J & \sum_{m} \hat{d}^\dag_{im\uparrow} \hat{d}^\dag_{im\downarrow} \hat{d}_{im\uparrow} \hat{d}_{im\downarrow} \nonumber \\ + \frac{U-3J}{2} & \sum_{m \neq m',\sigma} \hat{d}^\dag_{im\sigma} \hat{d}^\dag_{im'\sigma} \hat{d}_{im\sigma} \hat{d}_{im'\sigma} \nonumber \\ -J & \sum_{m \neq m'} \hat{d}^\dag_{im\uparrow} \hat{d}^\dag_{im'\downarrow} \hat{d}_{im\downarrow} \hat{d}_{im'\uparrow} \nonumber \\ -J & \sum_{m \neq m'} \hat{d}^\dag_{im\uparrow} \hat{d}^\dag_{im\downarrow} \hat{d}_{im'\downarrow} \hat{d}_{im'\uparrow} \Big].\end{aligned}$$ Here $i$, $\sigma$, and $m$, $m'$ are site, spin, and orbital indices for Cartesian $t_{\rm 2g}$ orbitals ($d_{xz,yz,xy}$) respectively. Now we introduce the MO creation/annihilation operators; $$\begin{aligned} \hat{d}_{im\sigma} &= \sum_{\alpha} Q^\alpha_{im} \hat{D}_{\alpha\sigma} \\ \hat{d}^\dag_{im\sigma} &= \sum_{\alpha} (Q^\dag)^{im}_\alpha \hat{D}^\dag_{\alpha\sigma}\end{aligned}$$ where $\alpha$ runs over the 12 molecular orbitals and we are ignoring spin-orbit coupling (SOC) at this stage. $Q^\alpha_{im}$ is the 12$\times$12 transformation matrix from the atomic $t_{\rm 2g}$ to the MO spaces. In terms of [*global*]{} coordinates (using the same cartesian coordinates for all $V$ sites) it is tabulated in Table \[tab:Q\]. Note that in actual calculations, since the four V sites are equivalent to each other up to a symmetry operation, $Q$ should be unitarily transformed to a local coordinate system at each V site. ----------------- ----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- Irreps No. $d_{xy}$ $d_{yz}$ $d_{xz}$ $d_{xy}$ $d_{yz}$ $d_{xz}$ $d_{xy}$ $d_{yz}$ $d_{xz}$ $d_{xy}$ $d_{yz}$ $d_{xz}$ $A$ 1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 \[7pt\] $E$ 1 +1 +$w^1$ +$w^2$ -1 +$w^1$ -$w^2$ +1 -$w^1$ -$w^2$ -1 -$w^1$ +$w^2$ 2 +1 +$w^2$ +$w^1$ -1 +$w^2$ -$w^1$ +1 -$w^2$ -$w^1$ -1 -$w^2$ +$w^1$ \[7pt\] $T^2$ 1 +1 0 0 +1 0 0 +1 0 0 +1 0 0 2 0 +1 0 0 +1 0 0 +1 0 0 +1 0 3 0 0 +1 0 0 +1 0 0 +1 0 0 +1 \[7pt\] $T^1_a$ 1 0 +1 -1 0 -1 -1 0 -1 +1 0 +1 +1 2 +1 0 -1 -1 0 +1 -1 0 -1 +1 0 +1 3 +1 -1 0 +1 +1 0 -1 -1 0 -1 +1 0 \[7pt\] $T^1_b$ 1 0 +1 +1 0 -1 +1 0 -1 -1 0 +1 -1 2 +1 0 +1 -1 0 -1 -1 0 +1 +1 0 -1 3 +1 +1 0 +1 -1 0 -1 +1 0 -1 -1 0 ----------------- ----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- Plugging them into $\hat{H}_{K}$ yields, $$\begin{aligned} \hat{H}_K = -\sum_{\alpha\beta\gamma\delta} \Big[ (U-2J)& \sum_{i} \left\{ \sum_{mm'} (Q^\dag)^{im}_\alpha (Q^\dag)^{im'}_\beta Q^\gamma_{im} Q^\delta_{im'} \right\} \hat{D}^\dag_{\alpha\uparrow} \hat{D}^\dag_{\beta\downarrow} \hat{D}_{\gamma\uparrow} \hat{D}_{\delta\downarrow} \nonumber \\ +2J & \sum_{i} \left\{ \sum_{m} (Q^\dag)^{im}_\alpha (Q^\dag)^{im}_\beta Q^\gamma_{im} Q^\delta_{im} \right\} \hat{D}^\dag_{\alpha\uparrow} \hat{D}^\dag_{\beta\downarrow} \hat{D}_{\gamma\uparrow} \hat{D}_{\delta\downarrow} \nonumber \\ +\frac{U-3J}{2} & \sum_{i} \left\{ \sum_{m\neq m'} (Q^\dag)^{im}_\alpha (Q^\dag)^{im'}_\beta Q^\gamma_{im} Q^\delta_{im'} \right\} \sum_\sigma \hat{D}^\dag_{\alpha\sigma} \hat{D}^\dag_{\beta\sigma} \hat{D}_{\gamma\sigma} \hat{D}_{\delta\sigma} \nonumber \\ -J & \sum_{i} \left\{ \sum_{m\neq m'} (Q^\dag)^{im}_\alpha (Q^\dag)^{im'}_\beta Q^\gamma_{im} Q^\delta_{im'} \right\} \hat{D}^\dag_{\alpha\uparrow} \hat{D}^\dag_{\beta\downarrow} \hat{D}_{\gamma\downarrow} \hat{D}_{\delta\uparrow} \nonumber \\ -J & \sum_{i} \left\{ \sum_{m\neq m'} (Q^\dag)^{im}_\alpha (Q^\dag)^{im}_\beta Q^\gamma_{im'} Q^\delta_{im'} \right\} \hat{D}^\dag_{\alpha\uparrow} \hat{D}^\dag_{\beta\downarrow} \hat{D}_{\gamma\downarrow} \hat{D}_{\delta\uparrow} \Big].\end{aligned}$$ In the above expression, product of $Q$s can be rewritten as $$\begin{aligned} \left( Q^\dag \otimes Q^\dag \right)^{imm'}_{\alpha\beta} &\equiv (Q^\dag)^{im}_\alpha (Q^\dag)^{im'}_\beta \\ \left( Q \otimes Q \right)_{imm'}^{\gamma\delta} &\equiv Q_{im}^\gamma Q_{im'}^\delta,\end{aligned}$$ and, since we are considering [*local*]{} Coulomb interactions, we are taking direct products of $i$-subsections ($i$=1,$\cdots$,4) of $Q$ and $Q^\dag$ matrices, so that $Q \otimes Q$ (and $Q^\dag \otimes Q^\dag$) has dimension of 9$\times$144 for each $i$ when we are considering the full 12-dimensional molecular orbital space.\ Since we don’t include SOC and the transformation matrices does not have spin indices, all $(Q^\dag \otimes Q^\dag) \cdot (Q \otimes Q)$ terms are free of spin components and can be classified into four different kinds; i) $\sum_{mm'} \left( Q^\dag \otimes Q^\dag \right)^{imm'}_{\alpha\beta} \left( Q \otimes Q \right)_{imm'}^{\gamma\delta}$, ii) $\sum_{m} \left( Q^\dag \otimes Q^\dag \right)^{imm}_{\alpha\beta} \left( Q \otimes Q \right)_{imm}^{\gamma\delta}$, iii) $\sum_{m \neq m'} \left( Q^\dag \otimes Q^\dag \right)^{imm'}_{\alpha\beta} \left( Q \otimes Q \right)_{imm'}^{\gamma\delta}$, and iv) $\sum_{m \neq m'} \left( Q^\dag \otimes Q^\dag \right)^{imm}_{\alpha\beta} \left( Q \otimes Q \right)_{im'm'}^{\gamma\delta}$. Here case iii) is just the subtraction of ii) from i).\ Computation of the transformation matrix is straightforward, but now all different molecular orbitals can mix even in a simple density-density interaction form (the first three terms in $\hat{H}_K$). However, things become much simpler in the most basic case of considering only the $T^2$ irrep as the correlated subspace. In that case, all $Q_i$ (and $Q^{\dag,i}$) become 3$\times$3 identity matrix (with normalization factor 1/2), so that all $Q \otimes Q$ and $Q^\dag \otimes Q^\dag$ become 9$\times$9 identity matrix with a prefactor 1/4, so that $$\begin{aligned} {\rm i)} & \sum_{imm'} \left( Q^\dag \otimes Q^\dag \right)^{imm'}_{\alpha\beta} \left( Q \otimes Q \right)_{imm'}^{\gamma\delta} \rightarrow \frac{1}{4} \delta_{\alpha\gamma} \delta_{\beta\delta}, \\ {\rm ii)} & \sum_{im} \left( Q^\dag \otimes Q^\dag \right)^{imm}_{\alpha\beta} \left( Q \otimes Q \right)_{imm}^{\gamma\delta} \rightarrow \frac{1}{4} \delta_{\alpha\gamma} \delta_{\beta\delta} \delta_{\alpha\beta}, \\ {\rm iii)} & \sum_{i,m \neq m'} \left( Q^\dag \otimes Q^\dag \right)^{imm'}_{\alpha\beta} \left( Q \otimes Q \right)_{imm'}^{\gamma\delta} \rightarrow \frac{1}{4} \delta_{\alpha\gamma} \delta_{\beta\delta} (1-\delta_{\alpha\beta}), \\ {\rm iv)} & \sum_{m \neq m'} \left( Q^\dag \otimes Q^\dag \right)^{imm}_{\alpha\beta} \left( Q \otimes Q \right)_{im'm'}^{\gamma\delta} \rightarrow \frac{1}{4} \delta_{\alpha\beta} \delta_{\gamma\delta} (1-\delta_{\alpha\gamma}).\end{aligned}$$ Hence $\hat{H}_{K}$, projected onto the MO-$T^2$ subspace, becomes $$\begin{aligned} \hat{H}^{\rm MO}_K = -\frac{1}{4}\Big[ (U-2J) & \sum_{mm'} \hat{D}^\dag_{m\uparrow} \hat{D}^\dag_{m'\downarrow} \hat{D}_{m\uparrow} \hat{D}_{m'\downarrow} \nonumber \\ + 2J & \sum_{m} \hat{D}^\dag_{m\uparrow} \hat{D}^\dag_{m\downarrow} \hat{D}_{m\uparrow} \hat{D}_{m\downarrow} \nonumber \\ + \frac{U-3J}{2} & \sum_{m \neq m',\sigma} \hat{D}^\dag_{m\sigma} \hat{D}^\dag_{m'\sigma} \hat{D}_{m\sigma} \hat{D}_{m'\sigma} \nonumber \\ -J & \sum_{m \neq m'} \hat{D}^\dag_{m\uparrow} \hat{D}^\dag_{m'\downarrow} \hat{D}_{m\downarrow} \hat{D}_{m'\uparrow} \nonumber \\ -J & \sum_{m \neq m'} \hat{D}^\dag_{m\uparrow} \hat{D}^\dag_{m\downarrow} \hat{D}_{m'\downarrow} \hat{D}_{m'\uparrow} \Big]. \label{eq:HK}\end{aligned}$$ Note that $\hat{H}^{\rm MO}_K$ has the exactly same form with the atomic $\hat{H}_K$, except the prefactor 1/4 because of the equidistribution of the MO-$T^2$ wavefunctions all over the four V sites. On-site and inter-site self-energies ------------------------------------ In this section the role of the Hund’s coupling is discussed in terms of the real space representation of the self-energy. Here we focus on the $T^2 \oplus E$ subspaces and their self-energies. Similar analysis can be done with other MO subspaces, however, for the purpose of discussing the role of $J_{\rm H}$ it seems that $T^2 \oplus E$ should suffice. In our calculations the cluster self-energies are diagonalized within the MO representation. When back-transformed into the atomic orbital basis representation, on-site (local) and inter-site (non-local) self-energies within the V$_4$ tetrahedron can be obtained. In the simplest case with the correlated MO-$T^2$ triplet only, the form of the self-energy in the atomic representation becomes simple; Namely, in the four-site real-space representation (four sites $\otimes$ atomic $t_{\rm 2g}$), all the on-site and inter-site blocks are enforced to be identical due to the choice of the $T^2$ correlated orbitals when the cubic and time-reversal symmetries are present, so that $$\begin{aligned} \boldsymbol{\Sigma}\left[ T^2 \right] (\omega) \equiv \frac{1}{4} \hat{\Sigma}^{T^2} (\omega) \left( \begin{array}{cccc} 1 & 1 &1 &1 \\ 1 & 1 &1 &1 \\ 1 & 1 &1 &1 \\ 1 & 1 &1 &1 \end{array} \right), \label{eq:self1}\end{aligned}$$ where each $3 \times 3$ block $\hat{\Sigma}^{T^2} = \Sigma^{T^2} \times \hat{I}_{3\times 3}$ in the atomic $t_{\rm 2g}$ space ($d_{xy}$, $d_{yz}$, and $d_{xz}$), $\hat{I}_{3\times 3}$ is an identity matrix of dimension 3, and the frequency $\omega$ can be either real or imaginary. Note that $\Sigma^{T^2}$ is the diagonal self-energy in the $T^2$-MO representation, and that the prefactor $\frac{1}{4}$ in Eq. (\[eq:self1\]) is the one appearing in Eq. (\[eq:HK\]). Here we choose the same global coordinate in defining the $t_{\rm 2g}$ orbitals at all sites, and proper coordinate transforms should be applied to each block when represented in local coordinates ($\hat{\Sigma}^{T^2}_{ij} \rightarrow (Q^\dag)_{gi} \hat{\Sigma}^{T^2}_{ij} Q_{jg}$, where the transformation $Q_{ig}$ is made from the global to the site-$i$ local coordinates). Fig. \[figS:T2self\] plots the real and imaginary parts of ${\Sigma}^{T^2}$ in the real frequency space, showing a pole in the imaginary part inside the Mott gap. ![ Real (blue) and imaginary (red) part of ${\Sigma}^{T^2}(\omega)$ after the analytic continuation from the imaginary frequency space. []{data-label="figS:T2self"}](./FigS1.png){width="45.00000%"} From this real-space representation of the self-energy, the implication of choosing only the $T^2$ MO as correlated orbitals becomes clearer; i) it introduces the inter-site self-energy in addition to the on-site counterpart, and ii) it prevents the correlations from becoming more local by enforcing the on-site and inter-site self-energies to be identical. The latter, especially, can be a serious issue when the size of the correlations that favor the formation of the local moments, [*e.g.*]{} the Hund’s coupling, becomes comparable to that of inter-site hopping. Next, the form of self-energy in the $T^2\oplus E$ is as presented below: $$\begin{aligned} \boldsymbol{\Sigma}\left[ T^2 \oplus E \right] (\omega) &= \boldsymbol{\Sigma}\left[ T^2 \right] (\omega) + \boldsymbol{\Sigma}\left[ E \right] (\omega),\end{aligned}$$ where the $T^2$-part of the self-energy is shown in Eq. (\[eq:self1\]). $\boldsymbol{\Sigma}\left[ E \right] (\omega)$ is as follows; $$\begin{aligned} \boldsymbol{\Sigma}\left[ E \right] (\omega) &\equiv \left( \begin{array}{cccc} \hat{\Sigma}_{11} & \hat{\Sigma}_{12} &\hat{\Sigma}_{13} &\hat{\Sigma}_{14} \\ \hat{\Sigma}^T_{12} & \hat{\Sigma}_{22} &\hat{\Sigma}_{23} &\hat{\Sigma}_{24} \\ \hat{\Sigma}^T_{13} & \hat{\Sigma}^T_{23} &\hat{\Sigma}_{33} &\hat{\Sigma}_{34} \\ \hat{\Sigma}^T_{14} & \hat{\Sigma}^T_{24} &\hat{\Sigma}^T_{34} &\hat{\Sigma}_{44} \end{array} \right). \label{eq:self2}\end{aligned}$$ Here the on-site parts $\hat{\Sigma}_{ii}$ are $$\begin{aligned} \hat{\Sigma}_{ii} &\equiv \Sigma^E (\omega) \left( \frac{1}{6} \hat{I}_{3\times 3} + \frac{1}{12} \hat{\Delta}_{ii} \right),\end{aligned}$$ where $\Sigma^E (\omega)$ is the self-energy for the $E$ doublet in the MO representation, and $\hat{\Delta}_{ii}$ determines the direction of the “[*trigonal crystal fields*]{}" to $t_{\rm 2g}$ orbitals at each V site, exerted by $\frac{1}{12}\Sigma^E (\omega) \hat{\Delta}_{ii}$. Namely, if the VS$_6$ octahedron surrounding site 1 is trigonally distorted along the cubic \[111\] direction with respect to the global Cartesian coordinate ([*i.e.*]{}, if the site 1 and the center of the V$_4$ cluster are on the same \[111\] line), then $$\begin{aligned} \hat{\Delta}_{11} &= \left( \begin{array}{ccc} 0 & -1 & -1 \\ -1 & 0 & -1 \\ -1 & -1 & 0 \end{array} \right). \end{aligned}$$ Other $\hat{\Delta}_{ii}$, for a coordinate choice, should be as follows, $$\begin{aligned} \hat{\Delta}_{22} &= \left( \begin{array}{ccc} 0 & +1 & -1 \\ +1 & 0 & +1 \\ -1 & +1 & 0 \end{array} \right), \hat{\Delta}_{33} = \left( \begin{array}{ccc} 0 & +1 & +1 \\ +1 & 0 & -1 \\ +1 & -1 & 0 \end{array} \right), \hat{\Delta}_{44} = \left( \begin{array}{ccc} 0 & -1 & +1 \\ -1 & 0 & +1 \\ +1 & +1 & 0 \end{array} \right).\end{aligned}$$ Note that this is the coordinate choice that was adopted in this work. ![image](./FigS2.pdf){width="100.00000%"} The inter-site component $\hat{\Sigma}_{ij}$ has a similar form; $\hat{\Sigma}_{ij} \equiv \frac{1}{12} \Sigma^E (\omega) \hat{O}_{ij} $, where $$\begin{aligned} \hat{O}_{12} &= \left( \begin{array}{ccc} -2 & -1 & +1 \\ +1 & +2 & +1 \\ +1 & -1 & -2 \end{array} \right), \hat{O}_{13} = \left( \begin{array}{ccc} +2 & +1 & +1 \\ -1 & -2 & +1 \\ -1 & +1 & -2 \end{array} \right), \hat{O}_{23} = \left( \begin{array}{ccc} -2 & +1 & -1 \\ +1 & -2 & -1 \\ +1 & +1 & +2 \end{array} \right), \nonumber \\ \label{eq:offdiag} \hat{O}_{23} &= \left( \begin{array}{ccc} -2 & -1 & -1 \\ -1 & -2 & +1 \\ +1 & -1 & +2 \end{array} \right), \hat{O}_{24} = \left( \begin{array}{ccc} +2 & -1 & +1 \\ +1 & -2 & -1 \\ -1 & -1 & -2 \end{array} \right), \hat{O}_{34} = \left( \begin{array}{ccc} -2 & +1 & -1 \\ -1 & +2 & +1 \\ -1 & -1 & -2 \end{array} \right).\end{aligned}$$ Combining (\[eq:self1\]-\[eq:offdiag\]), the site-orbital resolved self-energies in the $T^2 \oplus E$ case is as follows. i) [*On-site (diagonal blocks), between same orbitals*]{}: $\left [ \frac{1}{4}\Sigma^{T^2} (\omega) + \frac{1}{6} \Sigma^E (\omega) \right ] \hat{I}_{3 \times 3}$, ii) [*On-site (diagonal blocks), between different orbitals*]{}: $\frac{1}{12} \Sigma^E (\omega) \hat{\Delta}_{ii}$, iii) [*Inter-site ($i \neq j$ blocks)*]{}: $\frac{1}{4}\Sigma^{T^2} (\omega) \hat{I}_{3 \times 3} + \frac{1}{12} \Sigma^E(\omega) \hat{O}_{ij}$. Here, we note in passing that $\frac{1}{12} \Sigma^E$ is small compared to other terms when $J_{\rm H}$ is not large ($<$ 1 eV), so that terms i) and iii) are dominant contributions, and that the balance between the terms i) and iii) determines whether it is locally (on-site) or non-locally (inter-site) correlated. Plugging (\[eq:offdiag\]) into the case iii) above yields an explicit expression of the $ij$-block of $\boldsymbol{\Sigma} \left[ T^2 \oplus E \right]$. For example, the block between the site 1 and 2 is as follows, $$\begin{aligned} &\boldsymbol{\Sigma} \left[ T^2 \oplus E \right]_{12} = \nonumber \\ &\left( \begin{array}{ccc} \frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6} \Sigma^E & -\frac{1}{12} \Sigma^E & +\frac{1}{12} \Sigma^E \\ +\frac{1}{12} \Sigma^E & \frac{1}{4}\Sigma^{T^2}{\color{blue}\boldsymbol{+}}\frac{1}{6} \Sigma^E & +\frac{1}{12} \Sigma^E \\ +\frac{1}{12} \Sigma^E & -\frac{1}{12} \Sigma^E & \frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6} \Sigma^E \end{array} \right), \end{aligned}$$ where the plus and minus signs in the diagonal components are colored in blue and red to emphasize terms where $\Sigma^{T^2}$ and $\Sigma^E$ are adding up and cancelling out, respectively. Among the three diagonal components, the central term ($\frac{1}{4}\Sigma^{T^2}{\color{blue}\boldsymbol{+}}\frac{1}{6}\Sigma^{E}$) is between the $d_{yz}$ orbitals at V site 1 and 2, which are forming a strong $\sigma$-type direct overlap, while the other two $\frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6} \Sigma^E$ are contributing to the $\delta$-like weak overlap between the $d_{xy,xz}$ orbitals. Interestingly, the inclusion of $\Sigma^E$ (and $J_{\rm H}$) affects the inter-site self-energies in an opposite way depending on the orbitals; while the imaginary part of $\frac{1}{4}\Sigma^{T^2}{\color{blue}\boldsymbol{+}}\frac{1}{6}\Sigma^{E}$ is enhanced by the nonzero $\Sigma^{E}$ (because causal self-energies should always have negative imaginary parts), it is canceled out in $\frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6}\Sigma^{E}$. This implies that the presence of $\Sigma^{E}$ selectively enhances the singlet moment formation within the stronger $\sigma$-bonding, while reducing inter-site correlations in other bondings. In addition, depending on the sign of the real parts of $\Sigma^{T^2}$ and $\Sigma^{E}$, one can either enhance or suppress the real part of the self-energy. Fig. \[figS:T2Eself\] show the evolution of ${\Sigma}^{T^2,E}(\omega)$ as a function of the Hund’s coupling $J_{\rm H}$. Note that the relative signs of the real part of ${\Sigma}^{T^2,E}(\omega)$ tend to be opposite when $J_{\rm H}$ is small, but increasing $J_{\rm H}$ drives them to be the same. Just after the crossover to the high-spin state happens ($J_{\rm H}$ = 1.1 eV), both the ${\rm Re} {\Sigma}^{T^2,E}(\omega)$ show very similar behavior. This is because of the development of the pole in ${\Sigma}^{E}$, signaling the formation of the $E$ local moments, as shown in the lower panels of Fig. \[figS:T2Eself\]. As the system goes into the high-spin configuration, both the ${\rm Im} {\Sigma}^{T^2,E}$ should similarly show a well-defined pole, then the shapes of ${\rm Re} {\Sigma}^{T^2,E} (\omega)$ should become similar to each other because of the Kramers-Kronig relation. Hence $\frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6} \Sigma^E$ within $\boldsymbol{\Sigma} \left[ T^2 \oplus E \right]_{ij}$ tends to cancel better as $J_{\rm H}$ becomes larger. Since the diagonal parts of the inter-site self energies are most dominant contributions, and we have two $\frac{1}{4}\Sigma^{T^2}{\color{red}\boldsymbol{-}}\frac{1}{6} \Sigma^E$ terms compared to just one $\frac{1}{4}\Sigma^{T^2}{\color{blue}\boldsymbol{+}}\frac{1}{6} \Sigma^E$, the overall self-energy correction to the inter-site hopping terms becomes weaker as the Hund’s coupling becomes enhanced. This is consistent with the observation in the main text that increasing $J_{\rm H}$ suppresses the degree of V$_4$ clustering, and that while $U$ enhanced the inter-site correlation via $\Sigma^{T^2}$, $J_{\rm H}$ reduces it by introducing $\Sigma^E$ that cancels $\Sigma^{T^2}$ out. [^1]: See SI for the details of the single-site DMFT. [^2]: This approximation leads to some mixing between half-integer spin states, but is not expected to change qualitative aspects of the results [^3]: Note that at $T$ = 20 K, the compound has a rhombohedral distortion. The value 1.37 is obtained by averaging $d^{\rm V}_{\rm int}$ and $d^{\rm V}_{\rm ic}$ separately and taking the ratio between them.
--- author: - 'Vincent Vennin,' - Kazuya Koyama - and David Wands bibliography: - 'curvevid.bib' date: today title: | Inflation with an extra light scalar field\ after Planck --- Introduction {#sec:intro} ============ The recent high-quality measurements [@Adam:2015rua; @Ade:2015lrj; @Ade:2015ava] of the Cosmic Microwave Background (CMB) anisotropies have shed new light on the physical processes that took place in the very early Universe. These data clearly support inflation [@Starobinsky:1980te; @Sato:1980yn; @Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Linde:1983gd] as the leading paradigm for explaining this primordial epoch. At present, the full set of observations can be accounted for in a minimal setup, where inflation is driven by a single scalar inflaton field $\phi$ with canonical kinetic term, minimally coupled to gravity, and evolving in a flat potential $V(\phi)$ in the slow-roll regime. Because inflation proceeds at very high energy where particle physics remains evasive, a variety of such potentials have been proposed in the literature so far. They have been recently mapped in where $\sim 80$ potentials are identified and analyzed under the slow-roll framework. The Bayesian evidence of the corresponding $\sim 200$ inflationary models was then derived in , which was used to identify the best single-field scenarios, mostly of the “Plateau” type. From a theoretical point of view however, as already said, inflation takes place in a regime that is far beyond the reach of accelerators. The physical details of how the inflaton is connected with the standard model of particle physics and its extensions are still unclear. In particular, most physical setups that have been proposed to embed inflation contain extra scalar fields that can play a role either during inflation or afterwards. This is notably the case in string theory models where extra light scalar degrees of freedom are usually considered [@Turok:1987pg; @Damour:1995pd; @Kachru:2003sx; @Krause:2007jk; @Baumann:2014nda]. A natural question [@Langlois:2008mn; @Clesse:2008pf; @Sugiyama:2011jt; @Peterson:2011yt; @Biagetti:2012xy; @Battefeld:2012qx; @Levasseur:2013tja; @Burgess:2013sla; @Turzynski:2014tza; @Price:2014xpa] is therefore whether single-field model predictions are robust under the introduction of these additional fields, and whether these fields change the potentials for which the data show the strongest preference. In this paper, we address this issue using Bayesian inference techniques. We present the results of a systematic analysis of single-field slow-roll models of inflation when an extra light (relative to Hubble scale) scalar field $\sigma$ is introduced and plays a role both during inflation and afterwards. In the limit where this added field $\sigma$ is entirely responsible for the observed primordial curvature perturbations, the class of models this describes is essentially the curvaton scenarios [@Linde:1996gt; @Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct; @Bartolo:2002vf]. Here however, we address the generic setup where both $\phi$ and $\sigma$ can a priori contribute to curvature perturbations [@Dimopoulos:2003az; @Langlois:2004nn; @Lazarides:2004we; @Moroi:2005np].[^1] In particular, while we require that $\phi$ becomes massive at the end of inflation, we do not make any assumption as to the ordering of the three events: $\sigma$ becomes massive, $\phi$ decays and $\sigma$ decays. Nor do we restrict the epochs during which $\sigma$ can dominate the energy content of the Universe. This leaves us with 10 possible cases (including situations where $\sigma$ drives a secondary phase of inflation [@Langlois:2004nn; @Moroi:2005kz; @Ichikawa:2008iq; @Dimopoulos:2011gb]). These ten “reheating scenarios” are listed and detailed in but, for convenience, they are sketched in the of appendix \[sec:ReheatingCases\]. The usual curvaton scenario corresponds to case number 8 but one can see that a much wider class of models is covered by the present analysis. An important aspect of this work is also that reheating kinematic effects are consistently taken into account. In practice, this means that the number of elapsed between the Hubble exit time of the CMB pivot scale and the end of inflation is not a free parameter but is given by an explicit function of the inflaton potential parameters, the mass of the extra scalar field, its *vev* at the end of inflation and the decay rates of both fields. As a consequence, there is no free reheating parameter. This is particularly important for curvaton-like scenarios, since in these cases the same parameters determine the statistical properties of perturbations and the kinematics of reheating. It is therefore crucial to properly account for the interplay between these two physical effects and the suppression of degeneracies it yields. When the inflaton has a quadratic potential, extra light scalar fields have recently been studied in and it has been shown [@Enqvist:2013paa] that the fit of quartic chaotic inflation can be significantly improved in the curvaton limit. In , a Bayesian analysis was carried out for the quadratic inflaton + curvaton models assuming instantaneous reheating (corresponding to our reheating case number 8), and these models were found not to be disfavoured with respect to standard quadratic inflation. In this work however, we build a pipeline that incorporates all $\sim 200$ single-field models mapped in , in all $10$ reheating cases. In practice, this means that the Bayesian evidence and complexity of $\sim 2000$ scenarios are addressed, which corresponds to an important step forward in the current state of the art of early Universe Bayesian analysis. This paper is organised as follows. In section \[sec:method\], we explain the method that we have employed. We introduce the physical systems under consideration and briefly recall how their predictions were calculated in . The theory of Bayesian inference is summarised and it is explained how the numerical pipeline [@aspic] was extended to implement scenarios with extra light scalar fields. We then present our results in section \[sec:results\]. Reheating cases are analysed one by one for a few prototypical inflaton potentials in section \[sec:PrototypicalPotentials\]. Bayesian complexity is introduced as a measure of the number of unconstrained parameters in section \[sec:complexity\], so that the effective number of added parameters compared to single-field setups is quantified. In section \[sec:priorweight\], a procedure of averaging over reheating scenarios is presented, which allows us to derive the Bayesian evidence of categories of models and to discuss the observational status of inflation with an extra light scalar field with respect to standard single-field inflation. Dependence on the prior chosen for the *vev* of the extra light field at the end of inflation is also studied in section \[sec:sigmaendprior\]. Finally, in section \[sec:conclusion\], we summarise our results and present a few concluding remarks. Method {#sec:method} ====== The method employed in this paper combines the analytical work of with the numerical tools developed in . In this section, we describe its main aspects. Inflation with an extra light scalar field ------------------------------------------ In the present work, we investigate the situation where an extra light scalar field is present during inflation and (p)reheating. The potentials under scrutiny are of the form $$\begin{aligned} V(\phi)+\frac{m_\sigma^2}{2}\sigma^2,\end{aligned}$$ where $\sigma$ is taken to be lighter than $\phi$ at the end of inflation. Both fields are assumed to be slowly rolling during inflation, and eventually decay into radiation fluids with decay rates respectively denoted $\Gamma_\phi$ and $\Gamma_\sigma$, during reheating. The parameters describing the inflationary and reheating sectors of the theory are therefore given by $$\begin{aligned} \theta_{\mathrm{inf}+\mathrm{reh}}=\lbrace \theta_V, m_\sigma, \sigma_\uend,\Gamma_\phi,\Gamma_\sigma\rbrace\, , \label{eq:params}\end{aligned}$$ where $\lbrace \theta_V \rbrace$ are the parameters appearing in the inflaton potential $V(\phi)$, $\sigma_\uend$ is the *vev* of $\sigma$ evaluated at the end of inflation, and $m_\sigma$, $\Gamma_\phi$ and $\Gamma_\sigma$ have been defined before. It is important to stress that, as mentioned in the introduction, reheating kinematics is entirely fixed by these parameters, so that the number of between Hubble exit of the CMB pivot scale and the end of inflation, $\Delta N_*$, only depends on the parameters listed in . In , it is explained how one can make use of the $\delta N$ formalism to relate observables of such models to variations in the energy densities of both fields at the decay time of the last field. This allows us to calculate all relevant physical quantities by only keeping track of the background energy densities. Analytical expressions have been derived for all $10$ reheating cases, that have been implemented in the publicly available library [@aspic]. For a given inflaton potential, and from the input parameters of , this code returns the value of the three first slow-roll parameters (or equivalently, of the scalar spectral index $\nS$ and its running, and of the tensor-to-scalar ratio $r$) and of the local-type non-Gaussianity parameter $f_{\mathrm{NL}}$. Bayesian Approach to Model Comparison {#sec:bayes} ------------------------------------- The next step is to compare the performances of the different inflationary scenarios under consideration. One way to carry out this program is to make use of the Bayesian approach to model comparison [@Cox:1946; @Jeffreys:1961; @deFinetti:1974; @Box:1992; @Bernardo:1994; @Jaynes:2003; @Berger:2003; @Trotta:2005ar; @Trotta:2008qt]. Bayesian inference uses Bayes theorem to express the posterior probabilities of a set of alternative models $\mathcal{M}_i$ given some data set $\mathcal{D}$. It reads $$p\left(\mathcal{M}_i\vert\mathcal{D}\right) =\frac{\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i \right) \pi\left(\mathcal{M}_i\right)}{p\left(\mathcal{D}\right)}\, .$$ Here, $\pi\left(\mathcal{M}_i\right)$ represents the prior belief in the model $\mathcal{M}_i$, $p\left(\mathcal{D}\right) =\sum_{i}\mathcal{E}(\mathcal{D}\vert\mathcal{M}_i)\pi(\mathcal{M}_i)$ is a normalisation constant and $\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i \right)$ is the Bayesian evidence of $\mathcal{M}_i$, defined by $$\label{eq:evidence:def} \mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i \right) = \int\dd\theta_{ij}\mathcal{L} \left(\mathcal{D}\vert\theta_{ij},\mathcal{M}_i\right) \pi\left(\theta_{ij}\vert \mathcal{M}_i\right)\, ,$$ where $\theta_{ij}$ are the $N$ parameters defining the model $\mathcal{M}_i$ and $\pi\left(\theta_{ij}\vert \mathcal{M}_i\right)$ is their prior distribution. The quantity $\mathcal{L}\left(\mathcal{D}\vert\theta_{ij},\mathcal{M}_i\right)$, called likelihood function, represents the probability of observing the data $\mathcal{D}$ assuming the model $\mathcal{M}_i$ is true and $\theta_{ij}$ are the actual values of its parameters. Assuming model $\mathcal{M}_i$, the posterior probability of its parameter $\theta_{ij}$ is then expressed as $$\begin{aligned} \label{eq:posterior:def} p\left(\theta_{ij}\vert\mathcal{D},\mathcal{M}_i\right)=\frac{\mathcal{L} \left(\mathcal{D}\vert\theta_{ij},\mathcal{M}_i\right) \pi\left(\theta_{ij}\vert \mathcal{M}_i\right)}{\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i \right) } \, .\end{aligned}$$ The posterior odds between two models $\mathcal{M}_i$ and $\mathcal{M}_j$ are given by $$\frac{p\left(\mathcal{M}_i\vert\mathcal{D}\right)} {p\left(\mathcal{M}_j\vert\mathcal{D}\right)} =\frac{\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i\right)} {\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_j\right)} \frac{\pi\left(\mathcal{M}_i\right)}{\pi\left(\mathcal{M}_j\right)}\equiv B_{ij }\frac{\pi\left(\mathcal{M}_i\right)}{\pi\left(\mathcal{M}_j\right)}\, ,$$ where we have defined the Bayes factor $B_{ij}$ by $B_{ij}=\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_i\right) /\mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_j\right)$. Under the principle of indifference, one can assume non-committal model priors, $\pi(\mathcal{M}_i)=\pi\left(\mathcal{M}_j\right)$, in which case the Bayes factor becomes identical to the posterior odds (see however section \[sec:priorweight\] where another approach is used). With this assumption, a Bayes factor larger (smaller) than one means a preference for the model $\mathcal{M}_i$ over the model $\mathcal{M}_j$ (a preference for $\mathcal{M}_j$ over $\mathcal{M}_i$). In practice, the “Jeffreys’ scale” gives an empirical prescription for translating the values of the Bayes factor into strengths of belief. When $\ln(B_{ij})>5$, $\mathcal{M}_j$ is said to be “strongly disfavoured” with respect to $\mathcal{M}_i$, “moderately disfavoured” if $2.5<\ln(B_{ij})<5$, “weakly disfavoured” if $1<\ln(B_{ij})<2.5$, and the situation is said to be “inconclusive” if $\vert \ln(B_{ij})\vert<1$. Bayesian analysis allows us to identify the models that achieve the best compromise between quality of the fit and simplicity. In other words, more complicated descriptions are preferred only if they provide an improvement in the fit that can compensate for the larger number of parameters. In the rest of this section, we illustrate how this idea works in practice on a simple example [@Vennin:2015eaa]. ![Sketch of the likelihood for the toy model $\mathcal{M}_1 $ discussed in section \[sec:bayes\] (pale blue surface). The solid blue line corresponds to the likelihood of model $\mathcal{M}_2$, which is a sub-model of $\mathcal{M}_1 $ with $\theta_2=0$.[]{data-label="fig:likelihood"}](figs/likelihood){width="6cm"} Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two competing models aiming at explaining some data $\mathcal{D}$. Two parameters $\theta_1$ and $\theta_2$ describe the first model $\mathcal{M}_1$, and we assume that the likelihood function is a Gaussian centred at $(\bar{\theta}_1,\bar{\theta}_2)$ with standard deviations $\sigma_1$ and $\sigma_2$, $$\label{eq:likelihood:M1} \mathcal{L}\left(\mathcal{D}\vert\theta_{1},\theta_2,\mathcal{M}_1\right)= \mathcal{L}_1^\mathrm{max} e^{-\frac{\left(\theta_1-\bar{\theta}_1\right)^2}{2\sigma_1^2} -\frac{\left(\theta_2-\bar{\theta}_2\right)^2}{2\sigma_2^2}} .$$ This likelihood is represented in . We assume that the prior distribution on $\theta_1$ and $\theta_2$ is also a Gaussian, with standard deviations $\Sigma_1$ and $\Sigma_2$, and that the likelihood is much more peaked than the prior, that is to say $\Sigma_1\gg \sigma_1$ and $\Sigma_2\gg \sigma_2$. In this limit, gives rise to a simple expression for the evidence of model $\mathcal{M}_1$, namely $$\label{eq:evid:M1} \mathcal{E}\left(\mathcal{D}\vert\mathcal{M}_1 \right) = \frac{\sigma_1\sigma_2}{\Sigma_1\Sigma_2} \mathcal{L}_1^\mathrm{max}\, .$$ One can readily see that the higher the best fit, $\mathcal{L}_1^\mathrm{max}$, the better the Bayesian evidence, which is of course expected. On the other hand, the ratio $\sigma_1\sigma_2/(\Sigma_1\Sigma_2)$ stands for the volume reduction in parameter space induced by the data $\mathcal{D}$ and, therefore, quantifies how much the parameters $\theta_1$ and $\theta_2$ must be fine tuned around the preferred values $\bar{\theta}_1$ and $\bar{\theta}_2$ to account for the data. From , it is thus clear that the larger this fine tuning, the worse the Bayesian evidence, the final result being a trade-off between both effects. Now, let us imagine that the second parameter $\theta_2$ is associated with some extra, non-minimal, feature (such as, say, isocurvature perturbations, non-Gaussianities, oscillations in the power spectrum, *etc*). We want to determine whether $\theta_2$ (and, hence, the associated feature) is, at a statistically significant level, required by the data. To this end, we introduce the model $\mathcal{M}_2$ that is a sub-model of $\mathcal{M}_1$ where we choose $\theta_2=0$. This new model $\mathcal{M}_2$ has a single parameter $\theta_1$. By definition, its prior distribution is Gaussian with standard deviation $\Sigma_1$ and its likelihood function is given by $$\begin{aligned} \mathcal{L}\left(\mathcal{D}\vert\theta_{1},\mathcal{M}_2\right)&=& \mathcal{L}\left(\mathcal{D}\vert\theta_{1},0,\mathcal{M}_1\right) =\mathcal{L}_1^\mathrm{max}e^{-\frac{\bar{\theta}_2^2}{2\sigma_2^2}} e^{-\frac{\left(\theta_1-\bar{\theta}_1\right)^2}{2\sigma_1^2}} \equiv \mathcal{L}_2^\mathrm{max} e^{-\frac{\left(\theta_1-\bar{\theta}_1\right)^2}{2\sigma_1^2}}\, ,\end{aligned}$$ where we have defined the maximum likelihood for model $\mathcal{M}_2$ by $\mathcal{L}_2^\mathrm{max}=\mathcal{L}_1^\mathrm{max} \exp[-\bar{\theta}_2^2/(2\sigma_2^2)]$. This likelihood is displayed as the solid blue line in , and simply corresponds to the intersection of the full likelihood (\[eq:likelihood:M1\]) with the plane $\theta_2=0$. In the same limit $\Sigma_1\gg \sigma_1$ as before, the evidence for the model $\mathcal{M}_2$ is given by an expression similar to , namely $\mathcal{E}(\mathcal{D}\vert\mathcal{M}_2) = \sigma_1 \mathcal{L}_2^\mathrm{max}/\Sigma_1$. The Bayes factor between models $\mathcal{M}_1$ and $\mathcal{M}_2$ therefore reads $$B_{12}=\frac{\mathcal{L}_1^\mathrm{max}}{\mathcal{L}_2^\mathrm{max}} \frac{\sigma_2}{\Sigma_2} = e^{\frac{\bar{\theta}_2^2}{2\sigma_2^2}}\frac{\sigma_2}{\Sigma_2}\,.$$ The first term, $\mathcal{L}_1^\mathrm{max}/\mathcal{L}_2^\mathrm{max}$ represents the change in the best fit due to the fact that we have added new parameters. Obviously, this ratio is always larger than one since adding more degrees of freedom to describe the data can only improve the quality of the fit. On the other hand, the second term $\sigma_2/\Sigma_2$ represents the amount of fine tuning required for this new parameter $\theta_2$ and is smaller than one. As a consequence, if the improvement of the fit quality is not large enough to beat fine tuning, one concludes that the parameter $\theta_2$ is not required by the data. In the opposite case, one concludes that there is a statistically significant indication that $\theta_2\neq 0$. Finally, let us notice that if the data is completely insensitive to $\theta_2$, $\mathcal{L}_1^\mathrm{max}=\mathcal{L}_2^\mathrm{max}$ and $\sigma_2=\Sigma_2$, the two models $\mathcal{M}_1$ and $\mathcal{M}_2$ have the same Bayesian evidence. Bayesian evidence is therefore insensitive to unconstrained parameters (in such a case, $\mathcal{M}_1$ and $\mathcal{M}_2$ can be differentiated using Bayesian complexity, see section \[sec:complexity\]). Fast Bayesian Evidence Computation ---------------------------------- The computation of Bayesian evidence is a numerically expensive task, since it requires one to evaluate the multi-dimensional integral of . A typical analysis based on the Planck likelihood coupled with an exact inflationary code to integrate the perturbations typically requires more than 3 CPU years of computing time on standard modern processors. Given the large number of models over which we want to carry out the Bayesian programme, this means that these conventional methods cannot be employed in a reasonable amount of time. This is why we resort to two important simplifications. First, perturbations are calculated making use of the slow-roll formalism during inflation, and the $\delta N$ formalism (for curvature perturbations) afterwards. These two approaches provide analytical expressions for the power spectra of scalar curvature and tensor fluctuations, and for the local non-Gaussianity level. They exempt us from using a numerical integrator of the mode equations. Second, we use the “effective likelihood via slow-roll reparametrisation” approach proposed by Christophe Ringeval in . The method relies on the determination of an effective likelihood for inflation, which is a function of the primordial amplitude of the scalar perturbations complemented with the necessary number of slow-roll parameters to reach the desired accuracy. The effective likelihood is obtained by marginalisation over the standard cosmological parameters, viewed as “nuisance” from the early Universe point of view. Machine-learning algorithms are then used to reproduce the multidimensional shape of the likelihood, and Bayesian inference is carried out with the nested sampling algorithm . The high accuracy of the method, which increases by orders of magnitude the speed of performing Bayesian inference and parameter estimation in an inflationary context, has been confirmed in . We use this procedure [@Ringeval:PC] with the Planck 2015 $TT$ data combined with high-$\ell$ $C_\ell^{TE}+C_\ell^{EE}$ likelihood and low-$\ell$ temperature plus polarization likelihood (PlanckTT,TE,EE+lowTEB in the notations of , see Fig. 1 there), together with the BICEP2-Keck/Planck likelihood described in . Prior Choices {#sec:priors} ------------- The priors encode physical information one has a priori on the values of the parameters (\[eq:params\]) that describe the models. For the parameters of the potential $\lbrace \theta_V \rbrace$, we use the same priors as the ones proposed in , which are based on the physical, model-building related, considerations of . Because the extra field $\sigma$ is supposed to be still light at the end of inflation, its mass $m_\sigma$ must be smaller than the Hubble scale at the end of inflation, $H_\uend$. The same condition applies to the two decay rates, $\Gamma_\phi,\ \Gamma_\sigma<H_\uend$, since both fields decay after inflation. On the other hand, we want the Universe to have fully reheated before Big Bang Nucleosynthesis (BBN), which means that the two decay rates are also bounded from below by $H_{\mathrm{BBN}}\simeq (10\MeV)^2/\Mp$. The same lower bound applies to $m_\sigma$ since, assuming perturbative decay, $m_\sigma>\Gamma_\sigma$. Between these two values, the order of magnitude of $m_\sigma$ and of the two decay rates is unknown, which is why a logarithmically flat prior (or “Jeffreys prior”) is chosen: $$\begin{aligned} \ln H_{\mathrm{BBN}} < \ln \Gamma_\phi,\,\ln\Gamma_\sigma,\,\ln m_\sigma < \ln H_\uend\, , \label{eq:prior:massscales}\end{aligned}$$ where, for a given reheating case, the extra-ordering conditions given in are further imposed. Two kinds of priors are then considered for $\sigma_\uend$. A first approach corresponds to stating that the order of magnitude of $\sigma_\uend$ is unknown, and that a logarithmically flat prior on $\sigma_\uend$ should be employed $$\begin{aligned} \label{eq:sigmaend:LogPrior} \ln\sigma_\uend^\umin < \ln\sigma_\uend < \ln \sigma_\uend^\umax \, .\end{aligned}$$ Here, $\sigma_\uend^\umin$ and $\sigma_\uend^\umax$ are the boundary values given for each reheating case in . A second approach relies on the equilibrium distribution[^2] of long wavelength modes of a light spectator field in de Sitter [@Starobinsky:1986fx; @Enqvist:2012] $$\begin{aligned} P\left(\sigma_\uend\right) \propto \exp\left(-\frac{4\pi^2 m_\sigma^2\sigma_\uend^2}{3H_\uend^4}\right)\, . \label{eq:sigmaend:GaussianPrior}\end{aligned}$$ This distribution is often referred to as the “Gaussian prior” for $\sigma_\uend$. However, one should note that the Hubble scale at the end of inflation, $H_\uend$, that appears in , is in fact a function of $\sigma_\uend$. Indeed, it depends on the mass scale of the inflaton potential, which is fixed to reproduce a given value of the curvature primordial power spectrum amplitude $P_*$. Since $\sigma$ contributes to the total amount of scalar perturbations, $P_*$ explicitly depends on $\sigma_\uend$, and so does $H_\uend$. This is why, in , one should write $H_\uend(\sigma_\uend)$ and the corresponding distribution is not, strictly speaking, a Gaussian. Its physical interpretation should therefore be handled carefully. In this work, we thus take a more pragmatic approach and interpret only as the idea that physically acceptable values of the parameters should be such that $m_\sigma\sigma_\uend\sim H_\uend^2$, in agreement with what one would expect for a light scalar field in de Sitter. In practice, we implement this requirement by simply rejecting realisations for which the argument of the exponential function in is smaller than $1/10$ or larger than $10$ (we have checked that when changing these arbitrary values to, say, $1/100$ and $100$, very similar results are obtained). Results {#sec:results} ======= In this section we cast our results in a series of a few tables. Let us first explain which quantities are displayed. In the following tables, the first column is an acronym for the name of the inflationary scenario under consideration that follows the same conventions as in . The index appearing after “MC” refers to the reheating case number, while the second part of the acronym stands for the name of the inflaton potential, following . For example, $\mathrm{MC}_3\mathrm{LFI}_2$ corresponds to the case where the inflaton potential is of the large field, quadratic type, and where the reheating scenario is of the third kind (see ). In the second column is given the logarithm of the Bayesian evidence, defined in section \[sec:bayes\], normalised to the Bayesian evidence of (the single-field version of) Higgs Inflation (the Starobinsky model). Normalisation choice is anyway arbitrary and what only makes physical sense is ratios of Bayesian evidence, but we chose Higgs Inflation to be the reference model in order to match the convention of and to make comparison with this work easier. The third and fourth columns respectively stand for the number of input parameters and the number of unconstrained parameters, that will be defined and commented on in section \[sec:complexity\]. Finally, the fifth and last column gives the maximal value of the likelihood (“best fit”), still normalised to the Bayesian evidence of Higgs Inflation. This quantity is irrelevant from a purely Bayesian perspective (it would only need to be considered in a frequentist analysis), but we display it for indicative purpose and as it allows one to check consistency with the results presented in where exploration in parameter space in performed. For comparison, in all tables displayed below, the first line corresponds to the single-field version of the models under consideration, in the case where the mean equation of state parameter during reheating vanishes, $\bar{w}_\ureh=0$, and the energy density at the end of reheating, $\rho_\ureh$, has the same logarithmically flat prior as in , $\ln\rho_{{}_\mathrm{BBN}}<\ln\rho_\ureh<\ln\rho_\uend$. The reason is that, in this work, the inflaton is assumed to oscillate around a quadratic minimum of its potential after inflation ends (in which case its energy density redshits as matter), and we want the limit where $\sigma_\uend\rightarrow 0$ to match the single-field version of the model (even if subtleties regarding this limit are to be noted, see section \[sec:PrototypicalPotentials\]). Finally, let us mention that apart from section \[sec:sigmaendprior\], the results given here are obtained from the logarithmically flat prior (\[eq:sigmaend:LogPrior\]) on $\sigma_\uend$. Prototypical Inflaton Potentials {#sec:PrototypicalPotentials} -------------------------------- As already mentioned, the inclusion of an extra light scalar field in the pipeline, for all ten reheating scenarios, gives rise to $\sim 2000$ models of inflation for which the Bayesian programme can be carried out. In this paper, for conciseness, we choose not to display all corresponding Bayesian evidence and select four prototypical inflaton potentials that will allow us to discuss the main generic trends that we have more generically observed. These four examples are also discussed in great detail in , where, in all ten reheating cases, plots in the $(\nS,r)$, $(\nS,f_{_\mathrm{NL}})$ and $(f_{_\mathrm{NL}},r)$ planes are provided. Together with the table 1 of this same reference where the main properties of these figures are summarised, they constitute useful prerequisites to properly interpret the following results. - Large-field inflation (LFI) is a typical example of a “large-field” model. Its potential is given by $$\begin{aligned} \label{eq:lfi:pot} V\left(\phi\right) = M^4\left(\frac{\phi}{\Mp}\right)^p\, .\end{aligned}$$ Here, $p$ is the free parameter of the potential. In this section, we present the results obtained for $p=2$ (i.e. for a quadratic potential) but the results obtained for $p=2/3$, $p=1$, $p=3$ and $p=4$, as well as for marginalising over $p\in [0.2,5]$, are given in appendix \[sec:lfi:otherp\] (and in appendix \[sec:Gaussian\] for a Gaussian prior on $\sigma_\uend$). Large-field models are well-known for yielding a value for $r$ that is too large in their single-field versions. Since the introduction of light scalar fields typically reduces the predicted value of $r$, at least in reheating cases 4, 5, 7 and 8, we expect the Bayesian evidence of these models to be modified. - Higgs inflation (HI, the Starobinsky model) is a typical example of a “Plateau model” for which the single-field version of the model already provides a very good fit to the data (its Bayesian evidence can therefore only decrease). Its potential is given by $$\begin{aligned} V\left(\phi\right) = M^4 \left[1-\exp\left(-\sqrt{\frac{2}{3}}\frac{\phi}{\Mp}\right)\right]^2\, .\end{aligned}$$ - Natural Inflation (NI) has a potential given by $$\begin{aligned} V\left(\phi\right) = M^4\left[1+\cos\left(\frac{\phi}{f}\right)\right]\, .\end{aligned}$$ When $f$ is not super-Planckian, it is a typical example that yields a value for $\nS$ that is too small in the single-field version of the model. The introduction of a light scalar field tends to drive $\nS$ towards $1$ so that the predictions of the models intersect the region that is preferred by the data, but only for fine-tuned parameters. How the Bayesian evidence will change is therefore difficult to predict. As in , a logarithmically flat prior is chosen on $f$, $0<\log(f/\Mp)<2.5$. - Power-Law Inflation (PLI) is a typical potential yielding a too large value of $\nS$, or a too large value of $r$. Since extra light scalar fields tend to decrease $r$ but to increase $\nS$, it is also difficult to predict how the Bayesian evidence will evolve. Its potential is given by $$\begin{aligned} V\left(\phi\right) = M^4\exp\left(-\alpha\frac{\phi}{\Mp}\right)\, .\end{aligned}$$ Since this potential has the specific feature to be conformally invariant, its predictions do not depend on the number of $\Delta N_*$ elapsed between Hubble exit and the end of inflation, hence do not depend on the reheating kinematics. This is another reason why this example is interesting, since it isolates the effects of $\sigma$ coming from its contribution to the total amount of scalar perturbations, and is not sensitive to the role it plays in the reheating dynamics. As in , a logarithmically flat prior is chosen on $\alpha$, $-4<\log\alpha<-1$. The results for Higgs Inflation and Natural Inflation are presented in table \[table:HINI\] and the results for Large-Field (quadratic) Inflation and Power-Law Inflation are given in table \[table:LFI2PLI\]. A first remark concerns the single-field limit. The first reheating scenario (see ) corresponds to values of the parameters such that the extra light field never dominates the energy budget of the Universe, and decays before the inflaton field does. In this case, it does not contribute to curvature perturbations neither does it play a role in the reheating dynamics. As a consequence, the first reheating scenario can be viewed as the single-field limit of the models under consideration, and in , it is shown that the same predictions as for the single-field models are indeed obtained in this case. However, in tables \[table:HINI\] and \[table:LFI2PLI\], one can see that the Bayesian evidence for the first reheating scenarios (“$\mathrm{MC}_1$”) and the single-field models are not exactly the same. The reason is that reheating is effectively implemented with different priors. In the single-field models, as explained in the beginning of section \[sec:results\], the total energy density at the end of reheating is drawn according to a logarithmically flat prior $\ln \rho_{{}_\mathrm{BBN}}<\ln \rho_\ureh< \ln \rho_\uend$. In the first reheating scenario on the other hand, $\rho_\ureh\simeq 3\Mp^2\Gamma_\phi^2$ which is drawn according to the same distribution, with the difference that the condition $\Gamma_\phi < \Gamma_\sigma$ is further implemented. This means that, when marginalised over all other parameters, the effective prior distribution for $\Gamma_\phi$ alone is biased towards smaller values (for which more values of $\Gamma_\sigma$ are allowed). In the first reheating scenario, $\bar{w}_\ureh\simeq 0$, which means that [@Martin:2010kz; @Easther:2011yq] $\Delta N_* \propto \ln \Gamma_\phi$ shows preference for smaller values too. As a consequence, steeper portions of the inflaton potential are preferentially sampled in the first reheating scenario, hence slightly lower Bayesian evidence are obtained (an important exception being Power-Law Inflation which is, as explained above, insensitive to reheating kinematic effects). This again shows the importance of properly accounting for reheating kinematic effects. The second reheating scenario (“$\mathrm{MC}_2$”) is similar in the sense that the extra light field decays before the inflaton does, and is subdominant when this happens. In , it is shown that almost the same predictions as for the single-field models are obtained in this case too. However, reheating dynamics is more complicated and contrary to case $1$ where reheating is only made of a matter phase, case $2$ contains a matter phase, a second inflation phase, an other matter phase, a radiation phase and then a matter phase again. In practice, it turns out that the values of $\Delta N_*$ it yields are closer to the single-field ones, which is why the effect described here is smaller in these cases. For the other reheating scenarios, predictions can be very different from the single-field ones, and we now review our four prototypical potentials one by one. For Higgs Inflation (table \[table:HINI\], left panel), since the single-field version of the model already provides one of the best possible fits to the data, as expected, including an extra light scalar field leads to decreasing the Bayesian evidence in all reheating scenarios. However, according to the Jeffrey scale, this change is “inconclusive” in most cases (and only “moderately disfavoured” in reheating scenarios $3$ and $10$). This means that, at the Bayesian level, Plateau models such as Higgs Inflation are robust under the introduction of extra light degrees of freedom. In reheating scenarios $4$, $5$, $7$ and $8$, $\nS$ varies between the single-field prediction and $1$, which explains why the evidence decreases. The main differences between these cases is that in scenarios $4$ and $7$, large non-Gaussianities can also be produced while in scenarios $5$ and $8$, non-Gaussianities always remain within the observational bounds. For this reason, it may seem counter-intuitive that the evidence of cases $5$ and $8$ is slightly lower than the one of cases $4$ and $7$. This means that non-Gaussianities play a negligible role in constraining these models, and that most parameters that are excluded because of too large non-Gaussianities are already excluded because of too large values of $\nS$. Non-Gaussianities observational constrains are therefore still too weak to really constrain these models. Finally, reheating scenarios $3$, $6$, $9$ and $10$ contain a second phase of inflation which means that the scales observed in the CMB exit the Hubble radius closer to the end of inflation where the potential is steeper (hence smaller values of $\nS$ and larger values for $r$), which is why these models have lower evidence, the most disfavoured scenario being case $10$. For Natural Inflation (table \[table:HINI\], right panel), let us again note that no radical change is observed when introducing a light scalar field. The single-field version of the model produces a too small value for $\nS$ which is why it is “moderately disfavoured” compared to, say, Higgs Inflation. In reheating scenarios $4$, $5$, $7$ and $8$, $\nS$ varies continuously between the single-field prediction and $1$, crossing the “sweet spot” of the data. However, this only yields a small increase of the Bayesian evidence of these four cases compared to reheating scenario $\mathrm{MC}_1$, confirming that, as already mentioned, parameters achieving the right scalar tilt are fine-tuned. In fact, only cases $5$ and $8$ are preferred to the single-field version of the model (although at an “inconclusive” level). Indeed, only in these cases, non-Gaussianities remain within observational constrains. This means that contrary to Higgs Inflation, current observational constraints on non-Gaussianities are already relevant for these models. Finally, cases $3$, $6$, $9$ and $10$ produce values of $\nS$ that are even smaller than the single-field predictions, and are therefore disfavoured. The worst scenario is again $10$, but it is only weakly disfavoured with respect to the single-field version of the model. For large-field models such as Large-Field Inflation, depending on the power index $p$ of the potential (\[eq:lfi:pot\]), larger modifications in the Bayesian evidence can be observed. The values for a quadratic potential are displayed in the left panel of table \[table:LFI2PLI\] while other values of $p$ are dealt with in appendix \[sec:lfi:otherp\]. The single-field versions of the large-field models suffer from a value for $r$ which is too large compared with the current observational constrains. In scenarios $4$, $5$, $7$ and $8$, the corresponding value decreases. This is why, in cases $5$ and $8$ where no large non-Gaussianities are produced, one obtains larger Bayesian evidence than in the single-field case. For a quadratic potential, the difference in only “inconclusive” (with respect to single-field) or “weak” (with respect to $\mathrm{MC}_1$) according to the Jeffreys scale, and the best scenario ($\mclfiFIVETWO$) still remains “moderately disfavoured” with respect to the best single-field models such as Higgs inflation. For a quartic potential however (see appendix \[sec:lfi:otherp\]), the effect is much larger as cases $5$ and $8$ are “strongly favoured” with respect to their single-field counterpart. The best scenario, $\mclfiFIVEFOUR$, is even in the “inconclusive” zone of Higgs inflation. From a Bayesian perspective, this scenario therefore belongs to the best models of inflation. On the contrary, because large non-Gaussianities are produced in cases $4$ and $7$, the decrease in $r$ is not enough to “rescue” these models that even have slightly worse Bayesian evidence than their single-field versions. This means that, as in Natural Inflation but contrary to Higgs Inflation, non-Gaussianities already are a constraining observable for these models. In cases $3$, $6$, $9$ and $10$, smaller values of $r$ are obtained when $p>2$, but $\nS$ reaches unacceptably small values in these regimes. This is why these scenarios are disfavoured, even compared to their single-field counterpart. Finally, Power-Law Inflation is displayed in the right panel of table \[table:LFI2PLI\]. Being conformally invariant, its predictions do not depend on $\Delta N_*$, and there is no reheating kinematic effects in these models. This is why, as already noticed, the reheating scenarios $\mathrm{MC}_1$ and $\mathrm{MC}_2$ have exactly the same Bayesian evidence as the single-field version of the model. In practice, the Bayesian evidence of all reheating scenarios are extremely low and close to the numerical limit of our code, so it is not easy to well resolve the differences between them. However, one can see that all scenarios are within the same “inconclusive” zone. This means that they are all equally strongly disfavoured. This also reinforces the statement that, for the models considered in the present paper in particular and for inflation in general, reheating kinematic effects play an important role, to which the data are now sensitive. Complexity and Number of Unconstrained Parameters {#sec:complexity} ------------------------------------------------- As shown in section \[sec:bayes\], Bayesian evidence is not sensitive to unconstrained parameters. Concretely, this means that if one adds a new parameter to a model, if this parameter does not change any of the model predictions, the Bayesian evidence remains the same. In this case, the model with less parameters may be considered more “simple”. This is why the idea of “complexity” naturally arises in Bayesian analysis. At first sight, the models considered here are more “complex” than pure single-field scenarios as they contain more parameters. However, the data may be more efficient in constraining these parameters and the relevant question is rather how many unconstrained parameters these models have. A well-suited measure of the number of free parameters that the data can actually constrain in a model is provided by the relative entropy between the prior and posterior distributions [@Spiegelhalter] (the Kullback-Leibler divergence). In , it is shown that such an effective number of parameters, called Bayesian complexity $\mathcal{C}$, can be written as $$\begin{aligned} \mathcal{C}_i = -2 \left\langle \ln\mathcal{L}\left(\theta_{ij}\right) \right\rangle_p + 2 \ln \mathcal{L}\left(\theta_{ij}^\mathrm{ML}\right)\, ,\end{aligned}$$ where $\langle\cdot \rangle_p$ denotes averaging over the posterior $p\left(\theta_{ij}\vert\mathcal{D},\mathcal{M}_i\right)$ and $\theta_{ij}^\mathrm{ML}$ is the parameters values where the likelihood is maximal. Bayesian complexity therefore assesses the constraining power of the data with respect to the measure provided by the prior. ![Toy model described in section \[sec:complexity\]. In the left panel, the flat prior $\Pi$ with standard deviation $\Sigma$, and the Gaussian likelihood $\mathcal{L}$ that peaks at $\theta^\mathrm{ML}$ and has standard deviation $\sigma$, are displayed. In the right panel, the Bayesian complexity $\mathcal{C}$ of the model is given in terms of $\Sigma/\sigma$.[]{data-label="fig:complexity:toymodel"}](figs/ComplexityToyModel_likelihood "fig:"){width="6cm"} ![Toy model described in section \[sec:complexity\]. In the left panel, the flat prior $\Pi$ with standard deviation $\Sigma$, and the Gaussian likelihood $\mathcal{L}$ that peaks at $\theta^\mathrm{ML}$ and has standard deviation $\sigma$, are displayed. In the right panel, the Bayesian complexity $\mathcal{C}$ of the model is given in terms of $\Sigma/\sigma$.[]{data-label="fig:complexity:toymodel"}](figs/ComplexityToyModel_complexity "fig:"){width="6cm"} To see how this works in practice, let us consider the toy example depicted in the left panel of . This model has a single parameter $\theta$ with a flat prior between $-\sqrt{2}\Sigma$ and $\sqrt{2}\Sigma$ ($\Sigma$ is the prior standard deviation). Let us assume the likelihood to be Gaussian, centred over $\theta^\mathrm{ML}$ and with standard deviation $\sigma$. For simplicity, in the expressions given below, let us assume that $\theta^\mathrm{ML}=0$ so that the prior and likelihood are centred over the same value. Making use of , the Bayesian evidence of this model is given by $$\begin{aligned} \label{eq:evidence:toymodel:complexity} \mathcal{E}=\frac{\sqrt{\pi}}{2}\mathcal{L}_\mathrm{max} \frac{\sigma}{\Sigma}\erf\left(\frac{\Sigma}{\sigma}\right)\, ,\end{aligned}$$ and one recovers the features commented on in section \[sec:bayes\], namely the fact that the Bayesian evidence increases with $\mathcal{L}_\mathrm{max}$ but decreases with $\Sigma/\sigma$. Using , the posterior distribution of the model can then be calculated, and one obtains for the Bayesian complexity $$\begin{aligned} \mathcal{C}= 1-\frac{2}{\sqrt{\pi}}\frac{\Sigma}{\sigma}\frac{\ee^{-\Sigma^2/\sigma^2}}{\erf\left(\Sigma/\sigma\right)}\, .\end{aligned}$$ One can see that, contrary to the Bayesian evidence in , the complexity only depends on the ratio $\Sigma/\sigma$ and not on $\mathcal{L}_\mathrm{max}$. It is displayed in the right panel of . When $\Sigma\gg \sigma$, $\theta$ is well measured and accordingly, the complexity (i.e. the number of constrained parameters) goes to one. On the contrary, when $\Sigma\ll\sigma$, the data does not constrain the parameter $\theta$ and the complexity vanishes. From this example, it is clear that Bayesian complexity allows us to quantify the number of unconstrained parameters $$\begin{aligned} \UnconstrainedParams = N - \mathcal{C}\, . \label{eq:unconParam}\end{aligned}$$ The models considered in the present work have the same number of inflaton potential parameters as their single-field counterpart. As already mentioned, see , their predictions also depend on $\lbrace m_\sigma,\sigma_\uend,\Gamma_\phi,\Gamma_\sigma\rbrace$, while their single-field analogues only rely on $\rho_\mathrm{reh}$ to describe the reheating sector. This is why $3$ more parameters are involved, which can be checked in the third column of the tables displayed above. However, the number of unconstrained parameters differs in general by a different amount when comparing single-field models to cases where an extra light field is present. In practice, one can check that this amount is always larger than $0$ (except for Power-Law potentials, but see below), hence there are more unconstrained parameters, but also always smaller than $3$, hence more parameters are constrained. For $\mclfiFIVETWO$ for example, more parameters are constrained with respect to the single-field version of the model. This means that the data is more efficient in constraining parameters when an extra light scalar field is added. This should be related to the fact that, as already stressed, the added parameters not only contribute to the reheating kinematic description (which, at the effective level, boils down to one single parameter), but also to the statistical properties of the perturbations themselves to which the added field contributes. The interplay between these two effects lead to a suppression of degeneracies that Bayesian complexity is therefore quantifying. Finally, let us notice that for Power-Law Inflation, as well as for some reheating scenarios of Natural Inflation, negative numbers of unconstrained parameters can be obtained. This is because, when the best-fit likelihood of a model is very poor, its Bayesian complexity can be arbitrarily large, hence $\UnconstrainedParams$ in becomes negative. In some sense, it means that the model is so strongly disfavoured that the data calls for more parameters than it can offer. Averaging over Reheating Scenarios {#sec:priorweight} ---------------------------------- So far, Bayesian evidence of inflationary models where an extra light scalar field is present have been given for each reheating scenario individually. However, in order to determine whether predictions of purely single-field models of inflation are robust under the introduction of light scalar fields, and whether the inflaton potentials for which the data show the strongest preference change once these extra fields are accounted for, one may want to derive “consolidated” Bayesian evidence associated to the inflaton potential only, regardless of the reheating scenario. This can be done by averaging over reheating scenarios in the following manner. For the purpose of illustration, let us consider two toy models $\mathcal{M}_1$ and $\mathcal{M}_2$, that both depend on the same parameter $\theta$. In model $\mathcal{M}_1$, $\theta$ is assumed to lie within the range $[a,b]$ with a flat prior distribution, while in model $\mathcal{M}_2$, $\theta$ lies within the range $[b,c]$ with a flat prior distribution too. The model $\mathcal{M}_{1+2}$ is defined to be the “union” of $\mathcal{M}_1$ and $\mathcal{M}_2$, where $\theta$ lies in $[a,c]$ with a flat prior distribution, so that $\mathcal{M}_1$ and $\mathcal{M}_2$ are simply sub-models of $\mathcal{M}_{1+2}$. From the definition (\[eq:evidence:def\]), one readily obtains $$\begin{aligned} \mathcal{E}\left(\mathcal{D}\vert \mathcal{M}_{1+2}\right) = \frac{b-a}{c-a}\mathcal{E}\left(\mathcal{D}\vert \mathcal{M}_{1}\right) + \frac{c-b}{c-a}\mathcal{E}\left(\mathcal{D}\vert \mathcal{M}_{2}\right)\, .\end{aligned}$$ In other words, the Bayesian evidence of model $\mathcal{M}_{1+2}$ is obtained by averaging the evidence of models $\mathcal{M}_1$ and $\mathcal{M}_2$, weighted by the relative fraction of the prior volume of $\mathcal{M}_{1+2}$ that falls into their respective domains. Somehow, these prior volume fractions can be viewed as priors for the sub-models $\mathcal{M}_1$ and $\mathcal{M}_2$ themselves. This is the strategy we adopt here. Notice that it requires the prior spaces associated with the different reheating scenarios to be disjoint and one can check that, from , it is indeed the case. In practice, starting from the global priors (\[eq:prior:massscales\]), and using a fiducial, constant likelihood in our Bayesian inference code, we compute the fraction of attempts that fall into each of the ten reheating scenarios. The corresponding ten relative weights are given in the second columns of tables \[table:HINI:Average\] and \[table:LFI2PLI:Average\] for the four potentials considered in this section (the analysis for other monomial large-field potentials is presented in appendix \[sec:largefield:average\]). One can see that these weights are roughly independent of the inflaton potential, small differences being due to hard prior conditions (notably the requirement that the scalar power spectrum can properly be normalised) that slightly depend on the inflaton potential. When a logarithmically flat prior is used for $\sigma_\uend$, the reheating scenarios that are mostly populated, at the prior level, are $1$, $4$ and $7$. Let us recall that, up to different reheating parameters priors, case $1$ corresponds to the single-field limit of the models under consideration. Cases $4$ and $7$, on the other hand, are usually associated with larger $\nS$, smaller $r$ and larger $f_{{}_\mathrm{NL}}$. These are the scenarios that mostly contribute to the averaged Bayesian evidence. In the third column of tables \[table:HINI:Average\] and \[table:LFI2PLI:Average\] is given the Bayesian evidence corresponding to situations where $\Gamma_\phi<\Gamma_\sigma<m_\sigma$ (cases $1$, $2$ and $3$), $\Gamma_\sigma<\Gamma_\phi<m_\sigma$ (cases $4$, $5$ and $6$) and $\Gamma_\sigma<m_\sigma<\Gamma_\phi$ (cases $7$, $8$, $9$ and $10$). The last column of these tables contain the global, consolidated Bayesian evidence of the models under consideration. For Higgs inflation, introducing an extra light scalar field decreases the logarithm of the Bayesian evidence of the model by less than $\sim 0.5$, at a level that is “inconclusive” according to the Jeffreys scale. This is why we conclude that the best single-field models, of the Plateau type, are generally robust under the introduction of such light scalar degrees of freedom. For models that predict a value for $\nS$ that is too low such as Natural Inflation, one can see that scenarios with an extra light scalar field remain, on average, in the “inconclusive” zone of their single-field counterpart and are even very slightly disfavoured. The same holds for large-field models predicting a value for $r$ which is too large such as when the potential is quadratic. When the potential is quartic however, models containing an extra light scalar field are moderately favoured compared to their single-field version, and become only moderately disfavoured compared to the best single-field models such as Higgs Inflation. For Power-Law inflation, the situation is rather unchanged, and all versions of the model remain strongly disfavoured. Prior on ${\bm \sigma_{\mathbf{end}}}$ {#sec:sigmaendprior} -------------------------------------- The results of the Bayesian analysis when the Gaussian prior (\[eq:sigmaend:GaussianPrior\]) is used for $\sigma_\uend$ are given in appendix \[sec:Gaussian\]. The averaged Bayesian evidence is also displayed in tables \[table:HINI:AverageGaussian\] and \[table:LFI2PLI:AverageGaussian\], and in appendix \[sec:largefield:average:Gaussian\] for other large-field models. At the prior level, one can see that the most populated scenarios are $1$, $2$, $3$ and $6$. As already mentioned, scenarios $1$ and $2$ are close to the single-field limit of the models under consideration. Cases $3$ and $6$, on the other hand, correspond to situations where a second phase of inflation takes place. It is therefore interesting to notice that scenarios corresponding to the original “curvaton” setup [@Linde:1996gt; @Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct] (scenarios 5 and 8) only represent a few percents of the prior space and may not seem very natural, if a Gaussian prior is adopted for $\sigma_\uend$ (see however the caveats about such a prior choice mentioned in section \[sec:priors\]). This also emphasises the importance of not restricting the present work to pure curvaton scenarios but investigating all possibilities. Since situations with extra phases of inflation probe steeper parts of the inflaton potential, they are always disfavoured. This is why we find that the averaged Bayesian evidence of all potentials considered in this section is decreased when an extra light scalar field is present, and that the corresponding models are “weakly disfavoured” compared to their single-field counterpart. Conclusion {#sec:conclusion} ========== Let us now summarise our main results. In this paper, we have used Bayesian inference techniques to investigate situations where an extra light scalar field is present during inflation and reheating. Combining the analytical work of with the numerical tools developed in , we have designed a numerical pipeline where $\sim 200$ inflaton setups $\times\, 10$ reheating scenarios $= 2000$ models are implemented. For simplicity, we have presented the results obtained for a few prototypical inflaton potentials only, but they are representative of the generic trends that can be more generally observed. We have found that plateau models, that already provide a good fit to the data in their single-field version, are very robust under the introduction of a light scalar field, which only decreases the Bayesian evidence at an inconclusive level. For hilltop potentials, single-field scenarios usually lead to values of $\nS$ that are too low when the width of the hill is not super-Planckian. When a light scalar field is added, the right value of $\nS$ can be obtained, but this happens for very fine-tuned values of the extra field parameters and/or when large non-Gaussianities are produced. As a consequence, the Bayesian status of these models is not improved. The same holds for most large-field models, that give rise to values of $r$ that are too large in their single-field version. The only exception is quartic potentials, for which we have found that reheating scenarios $5$ and $8$ are “strongly favoured” with respect to their single-field counterpart, and that scenario $5$ even lies in the inconclusive zone of (the single-field version of) Higgs Inflation (the Starobinsky model). To summarise, we have thus showed that the best models after Planck are of two kinds: plateau potentials, regardless of whether an extra field is added or not, and quartic large-field inflation with an extra light scalar field, in reheating scenarios $5$ and $8$. The role non-Gaussianities play in constraining these models [@Valiviita:2006mz; @Huang:2008ze; @Enqvist:2008gk; @Nakayama:2009ce; @Byrnes:2010xd; @Enomoto:2012uy; @Fonseca:2012cj; @Hardwick:2015tma] was also discussed. We have seen that current bounds on non-Gaussianities are already sufficient to constrain hilltop and large-field potentials, but not plateau potentials. In the future, an important question is therefore whether the most efficient way to constrain these models is to improve measurements on $\nS$ and $r$ or measurements on non-Gaussianities. We plan to address this issue in a separate paper. Different priors have also been compared for the *vev* of the added field at the end of inflation. When it is set to its expected value from quantum dispersion effects, we have found that the most natural scenarios are the ones containing extra phases of inflation, that are generally disfavoured. This also means that the results of our analysis are sensitive to the distribution chosen for the *vev* of the light field, and that given a model, such a distribution can therefore be constrained. This may be relevant to the question [@Starobinsky:1986fx; @Enqvist:2012; @Burgess:2015ajz] whether observations can give access to scales beyond the observational horizon. Finally, we have used Bayesian complexity to quantify the number of parameters that are left unconstrained by the present analysis. We have found that, while the models studied here have more unconstrained parameters than their single-field versions, they also allow us to constrain more parameters. This is due to the fact that the added parameters not only contribute to the kinematic description of reheating but also to the statistical properties of the perturbations themselves, to which the added field contributes. The interplay between these two effects lead to a suppression of degeneracies that is responsible for having more constrained parameters. This also means that non trivial constraints on the reheating temperatures can be obtained in these models, which we plan to investigate in a future publication. Acknowledgments {#acknowledgments .unnumbered} =============== It is a pleasure to thank Christophe Ringeval for sharing his effective likelihood code with us and Robert Hardwick for careful reading of this manuscript. This work was realised using the ICG Sciama HPC cluster and we thank Gary Burton for constant help. This work is supported by STFC grants ST/K00090X/1 and ST/L005573/1. Reheating Cases {#sec:ReheatingCases} =============== ![image](./figs/10cases){width="99.00000%"} Bayesian Results for Other Large-Field Models {#sec:lfi:otherp} ============================================= In the following tables, we display the results of the Bayesian analysis when the inflaton potential is given by with $p=2/3$, $p=1$, $p=2$, $p=3$ and $p=4$, as well as for marginalising over $p\in [0.2,5]$. [2]{} $\lfi$ & $\Elfi$ & $\NPlfi$ & $\NUPlfi$ & $\BElfi$\ $\mclfiONE$ & $\EmclfiONE$ & $\NPmclfiONE$ & $\NUPmclfiONE$ & $\BEmclfiONE$\ $\mclfiTWO$ & $\EmclfiTWO$ & $\NPmclfiTWO$ & $\NUPmclfiTWO$ & $\BEmclfiTWO$\ $\mclfiTHREE$ & $\EmclfiTHREE$ & $\NPmclfiTHREE$ & $\NUPmclfiTHREE$ & $\BEmclfiTHREE$\ $\mclfiFOUR$ & $\EmclfiFOUR$ & $\NPmclfiFOUR$ & $\NUPmclfiFOUR$ & $\BEmclfiFOUR$\ $\mclfiFIVE$ & $\EmclfiFIVE$ & $\NPmclfiFIVE$ & $\NUPmclfiFIVE$ & $\BEmclfiFIVE$\ $\mclfiSIX$ & $\EmclfiSIX$ & $\NPmclfiSIX$ & $\NUPmclfiSIX$ & $\BEmclfiSIX$\ $\mclfiSEVEN$ & $\EmclfiSEVEN$ & $\NPmclfiSEVEN$ & $\NUPmclfiSEVEN$ & $\BEmclfiSEVEN$\ $\mclfiEIGHT$ & $\EmclfiEIGHT$ & $\NPmclfiEIGHT$ & $\NUPmclfiEIGHT$ & $\BEmclfiEIGHT$\ $\mclfiNINE$ & $\EmclfiNINE$ & $\NPmclfiNINE$ & $\NUPmclfiNINE$ & $\BEmclfiNINE$\ $\mclfiONEZERO$ & $\EmclfiONEZERO$ & $\NPmclfiONEZERO$ & $\NUPmclfiONEZERO$ & $\BEmclfiONEZERO$\ $\lfiTWOTHREE$ & $\ElfiTWOTHREE$ & $\NPlfiTWOTHREE$ & $\NUPlfiTWOTHREE$ & $\BElfiTWOTHREE$\ $\mclfiONETWOTHREE$ & $\EmclfiONETWOTHREE$ & $\NPmclfiONETWOTHREE$ & $\NUPmclfiONETWOTHREE$ & $\BEmclfiONETWOTHREE$\ $\mclfiTWOTWOTHREE$ & $\EmclfiTWOTWOTHREE$ & $\NPmclfiTWOTWOTHREE$ & $\NUPmclfiTWOTWOTHREE$ & $\BEmclfiTWOTWOTHREE$\ $\mclfiTHREETWOTHREE$ & $\EmclfiTHREETWOTHREE$ & $\NPmclfiTHREETWOTHREE$ & $\NUPmclfiTHREETWOTHREE$ & $\BEmclfiTHREETWOTHREE$\ $\mclfiFOURTWOTHREE$ & $\EmclfiFOURTWOTHREE$ & $\NPmclfiFOURTWOTHREE$ & $\NUPmclfiFOURTWOTHREE$ & $\BEmclfiFOURTWOTHREE$\ $\mclfiFIVETWOTHREE$ & $\EmclfiFIVETWOTHREE$ & $\NPmclfiFIVETWOTHREE$ & $\NUPmclfiFIVETWOTHREE$ & $\BEmclfiFIVETWOTHREE$\ $\mclfiSIXTWOTHREE$ & $\EmclfiSIXTWOTHREE$ & $\NPmclfiSIXTWOTHREE$ & $\NUPmclfiSIXTWOTHREE$ & $\BEmclfiSIXTWOTHREE$\ $\mclfiSEVENTWOTHREE$ & $\EmclfiSEVENTWOTHREE$ & $\NPmclfiSEVENTWOTHREE$ & $\NUPmclfiSEVENTWOTHREE$ & $\BEmclfiSEVENTWOTHREE$\ $\mclfiEIGHTTWOTHREE$ & $\EmclfiEIGHTTWOTHREE$ & $\NPmclfiEIGHTTWOTHREE$ & $\NUPmclfiEIGHTTWOTHREE$ & $\BEmclfiEIGHTTWOTHREE$\ $\mclfiNINETWOTHREE$ & $\EmclfiNINETWOTHREE$ & $\NPmclfiNINETWOTHREE$ & $\NUPmclfiNINETWOTHREE$ & $\BEmclfiNINETWOTHREE$\ $\mclfiONEZEROTWOTHREE$ & $\EmclfiONEZEROTWOTHREE$ & $\NPmclfiONEZEROTWOTHREE$ & $\NUPmclfiONEZEROTWOTHREE$ & $\BEmclfiONEZEROTWOTHREE$\ [2]{} $\lfiONE$ & $\ElfiONE$ & $\NPlfiONE$ & $\NUPlfiONE$ & $\BElfiONE$\ $\mclfiONEONE$ & $\EmclfiONEONE$ & $\NPmclfiONEONE$ & $\NUPmclfiONEONE$ & $\BEmclfiONEONE$\ $\mclfiTWOONE$ & $\EmclfiTWOONE$ & $\NPmclfiTWOONE$ & $\NUPmclfiTWOONE$ & $\BEmclfiTWOONE$\ $\mclfiTHREEONE$ & $\EmclfiTHREEONE$ & $\NPmclfiTHREEONE$ & $\NUPmclfiTHREEONE$ & $\BEmclfiTHREEONE$\ $\mclfiFOURONE$ & $\EmclfiFOURONE$ & $\NPmclfiFOURONE$ & $\NUPmclfiFOURONE$ & $\BEmclfiFOURONE$\ $\mclfiFIVEONE$ & $\EmclfiFIVEONE$ & $\NPmclfiFIVEONE$ & $\NUPmclfiFIVEONE$ & $\BEmclfiFIVEONE$\ $\mclfiSIXONE$ & $\EmclfiSIXONE$ & $\NPmclfiSIXONE$ & $\NUPmclfiSIXONE$ & $\BEmclfiSIXONE$\ $\mclfiSEVENONE$ & $\EmclfiSEVENONE$ & $\NPmclfiSEVENONE$ & $\NUPmclfiSEVENONE$ & $\BEmclfiSEVENONE$\ $\mclfiEIGHTONE$ & $\EmclfiEIGHTONE$ & $\NPmclfiEIGHTONE$ & $\NUPmclfiEIGHTONE$ & $\BEmclfiEIGHTONE$\ $\mclfiNINEONE$ & $\EmclfiNINEONE$ & $\NPmclfiNINEONE$ & $\NUPmclfiNINEONE$ & $\BEmclfiNINEONE$\ $\mclfiONEZEROONE$ & $\EmclfiONEZEROONE$ & $\NPmclfiONEZEROONE$ & $\NUPmclfiONEZEROONE$ & $\BEmclfiONEZEROONE$\ $\lfiTWO$ & $\ElfiTWO$ & $\NPlfiTWO$ & $\NUPlfiTWO$ & $\BElfiTWO$\ $\mclfiONETWO$ & $\EmclfiONETWO$ & $\NPmclfiONETWO$ & $\NUPmclfiONETWO$ & $\BEmclfiONETWO$\ $\mclfiTWOTWO$ & $\EmclfiTWOTWO$ & $\NPmclfiTWOTWO$ & $\NUPmclfiTWOTWO$ & $\BEmclfiTWOTWO$\ $\mclfiTHREETWO$ & $\EmclfiTHREETWO$ & $\NPmclfiTHREETWO$ & $\NUPmclfiTHREETWO$ & $\BEmclfiTHREETWO$\ $\mclfiFOURTWO$ & $\EmclfiFOURTWO$ & $\NPmclfiFOURTWO$ & $\NUPmclfiFOURTWO$ & $\BEmclfiFOURTWO$\ $\mclfiFIVETWO$ & $\EmclfiFIVETWO$ & $\NPmclfiFIVETWO$ & $\NUPmclfiFIVETWO$ & $\BEmclfiFIVETWO$\ $\mclfiSIXTWO$ & $\EmclfiSIXTWO$ & $\NPmclfiSIXTWO$ & $\NUPmclfiSIXTWO$ & $\BEmclfiSIXTWO$\ $\mclfiSEVENTWO$ & $\EmclfiSEVENTWO$ & $\NPmclfiSEVENTWO$ & $\NUPmclfiSEVENTWO$ & $\BEmclfiSEVENTWO$\ $\mclfiEIGHTTWO$ & $\EmclfiEIGHTTWO$ & $\NPmclfiEIGHTTWO$ & $\NUPmclfiEIGHTTWO$ & $\BEmclfiEIGHTTWO$\ $\mclfiNINETWO$ & $\EmclfiNINETWO$ & $\NPmclfiNINETWO$ & $\NUPmclfiNINETWO$ & $\BEmclfiNINETWO$\ $\mclfiONEZEROTWO$ & $\EmclfiONEZEROTWO$ & $\NPmclfiONEZEROTWO$ & $\NUPmclfiONEZEROTWO$ & $\BEmclfiONEZEROTWO$\ [2]{} $\lfiTHREE$ & $\ElfiTHREE$ & $\NPlfiTHREE$ & $\NUPlfiTHREE$ & $\BElfiTHREE$\ $\mclfiONETHREE$ & $\EmclfiONETHREE$ & $\NPmclfiONETHREE$ & $\NUPmclfiONETHREE$ & $\BEmclfiONETHREE$\ $\mclfiTWOTHREE$ & $\EmclfiTWOTHREE$ & $\NPmclfiTWOTHREE$ & $\NUPmclfiTWOTHREE$ & $\BEmclfiTWOTHREE$\ $\mclfiTHREETHREE$ & $\EmclfiTHREETHREE$ & $\NPmclfiTHREETHREE$ & $\NUPmclfiTHREETHREE$ & $\BEmclfiTHREETHREE$\ $\mclfiFOURTHREE$ & $\EmclfiFOURTHREE$ & $\NPmclfiFOURTHREE$ & $\NUPmclfiFOURTHREE$ & $\BEmclfiFOURTHREE$\ $\mclfiFIVETHREE$ & $\EmclfiFIVETHREE$ & $\NPmclfiFIVETHREE$ & $\NUPmclfiFIVETHREE$ & $\BEmclfiFIVETHREE$\ $\mclfiSIXTHREE$ & $\EmclfiSIXTHREE$ & $\NPmclfiSIXTHREE$ & $\NUPmclfiSIXTHREE$ & $\BEmclfiSIXTHREE$\ $\mclfiSEVENTHREE$ & $\EmclfiSEVENTHREE$ & $\NPmclfiSEVENTHREE$ & $\NUPmclfiSEVENTHREE$ & $\BEmclfiSEVENTHREE$\ $\mclfiEIGHTTHREE$ & $\EmclfiEIGHTTHREE$ & $\NPmclfiEIGHTTHREE$ & $\NUPmclfiEIGHTTHREE$ & $\BEmclfiEIGHTTHREE$\ $\mclfiNINETHREE$ & $\EmclfiNINETHREE$ & $\NPmclfiNINETHREE$ & $\NUPmclfiNINETHREE$ & $\BEmclfiNINETHREE$\ $\mclfiONEZEROTHREE$ & $\EmclfiONEZEROTHREE$ & $\NPmclfiONEZEROTHREE$ & $\NUPmclfiONEZEROTHREE$ & $\BEmclfiONEZEROTHREE$\ $\lfiFOUR$ & $\ElfiFOUR$ & $\NPlfiFOUR$ & $\NUPlfiFOUR$ & $\BElfiFOUR$\ $\mclfiONEFOUR$ & $\EmclfiONEFOUR$ & $\NPmclfiONEFOUR$ & $\NUPmclfiONEFOUR$ & $\BEmclfiONEFOUR$\ $\mclfiTWOFOUR$ & $\EmclfiTWOFOUR$ & $\NPmclfiTWOFOUR$ & $\NUPmclfiTWOFOUR$ & $\BEmclfiTWOFOUR$\ $\mclfiTHREEFOUR$ & $\EmclfiTHREEFOUR$ & $\NPmclfiTHREEFOUR$ & $\NUPmclfiTHREEFOUR$ & $\BEmclfiTHREEFOUR$\ $\mclfiFOURFOUR$ & $\EmclfiFOURFOUR$ & $\NPmclfiFOURFOUR$ & $\NUPmclfiFOURFOUR$ & $\BEmclfiFOURFOUR$\ $\mclfiFIVEFOUR$ & $\EmclfiFIVEFOUR$ & $\NPmclfiFIVEFOUR$ & $\NUPmclfiFIVEFOUR$ & $\BEmclfiFIVEFOUR$\ $\mclfiSIXFOUR$ & $\EmclfiSIXFOUR$ & $\NPmclfiSIXFOUR$ & $\NUPmclfiSIXFOUR$ & $\BEmclfiSIXFOUR$\ $\mclfiSEVENFOUR$ & $\EmclfiSEVENFOUR$ & $\NPmclfiSEVENFOUR$ & $\NUPmclfiSEVENFOUR$ & $\BEmclfiSEVENFOUR$\ $\mclfiEIGHTFOUR$ & $\EmclfiEIGHTFOUR$ & $\NPmclfiEIGHTFOUR$ & $\NUPmclfiEIGHTFOUR$ & $\BEmclfiEIGHTFOUR$\ $\mclfiNINEFOUR$ & $\EmclfiNINEFOUR$ & $\NPmclfiNINEFOUR$ & $\NUPmclfiNINEFOUR$ & $\BEmclfiNINEFOUR$\ $\mclfiONEZEROFOUR$ & $\EmclfiONEZEROFOUR$ & $\NPmclfiONEZEROFOUR$ & $\NUPmclfiONEZEROFOUR$ & $\BEmclfiONEZEROFOUR$\ Averaging over Reheating Scenarios for Other Large-Field Models {#sec:largefield:average} =============================================================== Bayesian Results with a Gaussian Prior on $\sigma_\uend$ {#sec:Gaussian} ======================================================== In this appendix, we give the Bayesian evidence and number of unconstrained parameters in the case where a “Gaussian” (in the sense defined in section \[sec:priors\]) prior for $\sigma_\uend$ is chosen. The value of the maximum likelihood $\mathcal{L}_\umax$ is not reported since it is independent of the prior, and is therefore the same as the one given in the tables of section \[sec:results\] and appendix \[sec:lfi:otherp\]. [2]{} $\hi$ & $\EhiGaussian$ & $\NPhi$ & $\NUPhiGaussian$\ $\mchiONE$ & $\EmchiONEGaussian$ & $\NPmchiONE$ & $\NUPmchiONEGaussian$\ $\mchiTWO$ & $\EmchiTWOGaussian$ & $\NPmchiTWO$ & $\NUPmchiTWOGaussian$\ $\mchiTHREE$ & $\EmchiTHREEGaussian$ & $\NPmchiTHREE$ & $\NUPmchiTHREEGaussian$\ $\mchiFOUR$ & $\EmchiFOURGaussian$ & $\NPmchiFOUR$ & $\NUPmchiFOURGaussian$\ $\mchiFIVE$ & $\EmchiFIVEGaussian$ & $\NPmchiFIVE$ & $\NUPmchiFIVEGaussian$\ $\mchiSIX$ & $\EmchiSIXGaussian$ & $\NPmchiSIX$ & $\NUPmchiSIXGaussian$\ $\mchiSEVEN$ & $\EmchiSEVENGaussian$ & $\NPmchiSEVEN$ & $\NUPmchiSEVENGaussian$\ $\mchiEIGHT$ & $\EmchiEIGHTGaussian$ & $\NPmchiEIGHT$ & $\NUPmchiEIGHTGaussian$\ $\mchiNINE$ & $\EmchiNINEGaussian$ & $\NPmchiNINE$ & $\NUPmchiNINEGaussian$\ $\mchiONEZERO$ & $\EmchiONEZEROGaussian$ & $\NPmchiONEZERO$ & $\NUPmchiONEZEROGaussian$\ $\nati$ & $\EniGaussian$ & $\NPni$ & $\NUPniGaussian$\ $\mcniONE$ & $\EmcniONEGaussian$ & $\NPmcniONE$ & $\NUPmcniONEGaussian$\ $\mcniTWO$ & $\EmcniTWOGaussian$ & $\NPmcniTWO$ & $\NUPmcniTWOGaussian$\ $\mcniTHREE$ & $\EmcniTHREEGaussian$ & $\NPmcniTHREE$ & $\NUPmcniTHREEGaussian$\ $\mcniFOUR$ & $\EmcniFOURGaussian$ & $\NPmcniFOUR$ & $\NUPmcniFOURGaussian$\ $\mcniFIVE$ & $\EmcniFIVEGaussian$ & $\NPmcniFIVE$ & $\NUPmcniFIVEGaussian$\ $\mcniSIX$ & $\EmcniSIXGaussian$ & $\NPmcniSIX$ & $\NUPmcniSIXGaussian$\ $\mcniSEVEN$ & $\EmcniSEVENGaussian$ & $\NPmcniSEVEN$ & $\NUPmcniSEVENGaussian$\ $\mcniEIGHT$ & $\EmcniEIGHTGaussian$ & $\NPmcniEIGHT$ & $\NUPmcniEIGHTGaussian$\ $\mcniNINE$ & $\EmcniNINEGaussian$ & $\NPmcniNINE$ & $\NUPmcniNINEGaussian$\ $\mcniONEZERO$ & $\EmcniONEZEROGaussian$ & $\NPmcniONEZERO$ & $\NUPmcniONEZEROGaussian$\ [2]{} $\lfiTWO$ & $\ElfiTWOGaussian$ & $\NPlfiTWO$ & $\NUPlfiTWOGaussian$\ $\mclfiONETWO$ & $\EmclfiONETWOGaussian$ & $\NPmclfiONETWO$ & $\NUPmclfiONETWOGaussian$\ $\mclfiTWOTWO$ & $\EmclfiTWOTWOGaussian$ & $\NPmclfiTWOTWO$ & $\NUPmclfiTWOTWOGaussian$\ $\mclfiTHREETWO$ & $\EmclfiTHREETWOGaussian$ & $\NPmclfiTHREETWO$ & $\NUPmclfiTHREETWOGaussian$\ $\mclfiFOURTWO$ & $\EmclfiFOURTWOGaussian$ & $\NPmclfiFOURTWO$ & $\NUPmclfiFOURTWOGaussian$\ $\mclfiFIVETWO$ & $\EmclfiFIVETWOGaussian$ & $\NPmclfiFIVETWO$ & $\NUPmclfiFIVETWOGaussian$\ $\mclfiSIXTWO$ & $\EmclfiSIXTWOGaussian$ & $\NPmclfiSIXTWO$ & $\NUPmclfiSIXTWOGaussian$\ $\mclfiSEVENTWO$ & $\EmclfiSEVENTWOGaussian$ & $\NPmclfiSEVENTWO$ & $\NUPmclfiSEVENTWOGaussian$\ $\mclfiEIGHTTWO$ & $\EmclfiEIGHTTWOGaussian$ & $\NPmclfiEIGHTTWO$ & $\NUPmclfiEIGHTTWOGaussian$\ $\mclfiNINETWO$ & $\EmclfiNINETWOGaussian$ & $\NPmclfiNINETWO$ & $\NUPmclfiNINETWOGaussian$\ $\mclfiONEZEROTWO$ & $\EmclfiONEZEROTWOGaussian$ & $\NPmclfiONEZEROTWO$ & $\NUPmclfiONEZEROTWOGaussian$\ $\pli$ & $\EpliGaussian$ & $\NPpli$ & $\NUPpliGaussian$\ $\mcpliONE$ & $\EmcpliONEGaussian$ & $\NPmcpliONE$ & $\NUPmcpliONEGaussian$\ $\mcpliTWO$ & $\EmcpliTWOGaussian$ & $\NPmcpliTWO$ & $\NUPmcpliTWOGaussian$\ $\mcpliTHREE$ & $\EmcpliTHREEGaussian$ & $\NPmcpliTHREE$ & $\NUPmcpliTHREEGaussian$\ $\mcpliFOUR$ & $\EmcpliFOURGaussian$ & $\NPmcpliFOUR$ & $\NUPmcpliFOURGaussian$\ $\mcpliFIVE$ & $\EmcpliFIVEGaussian$ & $\NPmcpliFIVE$ & $\NUPmcpliFIVEGaussian$\ $\mcpliSIX$ & $\EmcpliSIXGaussian$ & $\NPmcpliSIX$ & $\NUPmcpliSIXGaussian$\ $\mcpliSEVEN$ & $\EmcpliSEVENGaussian$ & $\NPmcpliSEVEN$ & $\NUPmcpliSEVENGaussian$\ $\mcpliEIGHT$ & $\EmcpliEIGHTGaussian$ & $\NPmcpliEIGHT$ & $\NUPmcpliEIGHTGaussian$\ $\mcpliNINE$ & $\EmcpliNINEGaussian$ & $\NPmcpliNINE$ & $\NUPmcpliNINEGaussian$\ $\mcpliONEZERO$ & $\EmcpliONEZEROGaussian$ & $\NPmcpliONEZERO$ & $\NUPmcpliONEZEROGaussian$\ [2]{} $\lfi$ & $\ElfiGaussian$ & $\NPlfi$ & $\NUPlfiGaussian$\ $\mclfiONE$ & $\EmclfiONEGaussian$ & $\NPmclfiONE$ & $\NUPmclfiONEGaussian$\ $\mclfiTWO$ & $\EmclfiTWOGaussian$ & $\NPmclfiTWO$ & $\NUPmclfiTWOGaussian$\ $\mclfiTHREE$ & $\EmclfiTHREEGaussian$ & $\NPmclfiTHREE$ & $\NUPmclfiTHREEGaussian$\ $\mclfiFOUR$ & $\EmclfiFOURGaussian$ & $\NPmclfiFOUR$ & $\NUPmclfiFOURGaussian$\ $\mclfiFIVE$ & $\EmclfiFIVEGaussian$ & $\NPmclfiFIVE$ & $\NUPmclfiFIVEGaussian$\ $\mclfiSIX$ & $\EmclfiSIXGaussian$ & $\NPmclfiSIX$ & $\NUPmclfiSIXGaussian$\ $\mclfiSEVEN$ & $\EmclfiSEVENGaussian$ & $\NPmclfiSEVEN$ & $\NUPmclfiSEVENGaussian$\ $\mclfiEIGHT$ & $\EmclfiEIGHTGaussian$ & $\NPmclfiEIGHT$ & $\NUPmclfiEIGHTGaussian$\ $\mclfiNINE$ & $\EmclfiNINEGaussian$ & $\NPmclfiNINE$ & $\NUPmclfiNINEGaussian$\ $\mclfiONEZERO$ & $\EmclfiONEZEROGaussian$ & $\NPmclfiONEZERO$ & $\NUPmclfiONEZEROGaussian$\ $\lfiTWOTHREE$ & $\ElfiTWOTHREEGaussian$ & $\NPlfiTWOTHREE$ & $\NUPlfiTWOTHREEGaussian$\ $\mclfiONETWOTHREE$ & $\EmclfiONETWOTHREEGaussian$ & $\NPmclfiONETWOTHREE$ & $\NUPmclfiONETWOTHREEGaussian$\ $\mclfiTWOTWOTHREE$ & $\EmclfiTWOTWOTHREEGaussian$ & $\NPmclfiTWOTWOTHREE$ & $\NUPmclfiTWOTWOTHREEGaussian$\ $\mclfiTHREETWOTHREE$ & $\EmclfiTHREETWOTHREEGaussian$ & $\NPmclfiTHREETWOTHREE$ & $\NUPmclfiTHREETWOTHREEGaussian$\ $\mclfiFOURTWOTHREE$ & $\EmclfiFOURTWOTHREEGaussian$ & $\NPmclfiFOURTWOTHREE$ & $\NUPmclfiFOURTWOTHREEGaussian$\ $\mclfiFIVETWOTHREE$ & $\EmclfiFIVETWOTHREEGaussian$ & $\NPmclfiFIVETWOTHREE$ & $\NUPmclfiFIVETWOTHREEGaussian$\ $\mclfiSIXTWOTHREE$ & $\EmclfiSIXTWOTHREEGaussian$ & $\NPmclfiSIXTWOTHREE$ & $\NUPmclfiSIXTWOTHREEGaussian$\ $\mclfiSEVENTWOTHREE$ & $\EmclfiSEVENTWOTHREEGaussian$ & $\NPmclfiSEVENTWOTHREE$ & $\NUPmclfiSEVENTWOTHREEGaussian$\ $\mclfiEIGHTTWOTHREE$ & $\EmclfiEIGHTTWOTHREEGaussian$ & $\NPmclfiEIGHTTWOTHREE$ & $\NUPmclfiEIGHTTWOTHREEGaussian$\ $\mclfiNINETWOTHREE$ & $\EmclfiNINETWOTHREEGaussian$ & $\NPmclfiNINETWOTHREE$ & $\NUPmclfiNINETWOTHREEGaussian$\ $\mclfiONEZEROTWOTHREE$ & $\EmclfiONEZEROTWOTHREEGaussian$ & $\NPmclfiONEZEROTWOTHREE$ & $\NUPmclfiONEZEROTWOTHREEGaussian$\ [2]{} $\lfiONE$ & $\ElfiONEGaussian$ & $\NPlfiONE$ & $\NUPlfiONEGaussian$\ $\mclfiONEONE$ & $\EmclfiONEONEGaussian$ & $\NPmclfiONEONE$ & $\NUPmclfiONEONEGaussian$\ $\mclfiTWOONE$ & $\EmclfiTWOONEGaussian$ & $\NPmclfiTWOONE$ & $\NUPmclfiTWOONEGaussian$\ $\mclfiTHREEONE$ & $\EmclfiTHREEONEGaussian$ & $\NPmclfiTHREEONE$ & $\NUPmclfiTHREEONEGaussian$\ $\mclfiFOURONE$ & $\EmclfiFOURONEGaussian$ & $\NPmclfiFOURONE$ & $\NUPmclfiFOURONEGaussian$\ $\mclfiFIVEONE$ & $\EmclfiFIVEONEGaussian$ & $\NPmclfiFIVEONE$ & $\NUPmclfiFIVEONEGaussian$\ $\mclfiSIXONE$ & $\EmclfiSIXONEGaussian$ & $\NPmclfiSIXONE$ & $\NUPmclfiSIXONEGaussian$\ $\mclfiSEVENONE$ & $\EmclfiSEVENONEGaussian$ & $\NPmclfiSEVENONE$ & $\NUPmclfiSEVENONEGaussian$\ $\mclfiEIGHTONE$ & $\EmclfiEIGHTONEGaussian$ & $\NPmclfiEIGHTONE$ & $\NUPmclfiEIGHTONEGaussian$\ $\mclfiNINEONE$ & $\EmclfiNINEONEGaussian$ & $\NPmclfiNINEONE$ & $\NUPmclfiNINEONEGaussian$\ $\mclfiONEZEROONE$ & $\EmclfiONEZEROONEGaussian$ & $\NPmclfiONEZEROONE$ & $\NUPmclfiONEZEROONEGaussian$\ $\lfiTHREE$ & $\ElfiTHREEGaussian$ & $\NPlfiTHREE$ & $\NUPlfiTHREEGaussian$\ $\mclfiONETHREE$ & $\EmclfiONETHREEGaussian$ & $\NPmclfiONETHREE$ & $\NUPmclfiONETHREEGaussian$\ $\mclfiTWOTHREE$ & $\EmclfiTWOTHREEGaussian$ & $\NPmclfiTWOTHREE$ & $\NUPmclfiTWOTHREEGaussian$\ $\mclfiTHREETHREE$ & $\EmclfiTHREETHREEGaussian$ & $\NPmclfiTHREETHREE$ & $\NUPmclfiTHREETHREEGaussian$\ $\mclfiFOURTHREE$ & $\EmclfiFOURTHREEGaussian$ & $\NPmclfiFOURTHREE$ & $\NUPmclfiFOURTHREEGaussian$\ $\mclfiFIVETHREE$ & $\EmclfiFIVETHREEGaussian$ & $\NPmclfiFIVETHREE$ & $\NUPmclfiFIVETHREEGaussian$\ $\mclfiSIXTHREE$ & $\EmclfiSIXTHREEGaussian$ & $\NPmclfiSIXTHREE$ & $\NUPmclfiSIXTHREEGaussian$\ $\mclfiSEVENTHREE$ & $\EmclfiSEVENTHREEGaussian$ & $\NPmclfiSEVENTHREE$ & $\NUPmclfiSEVENTHREEGaussian$\ $\mclfiEIGHTTHREE$ & $\EmclfiEIGHTTHREEGaussian$ & $\NPmclfiEIGHTTHREE$ & $\NUPmclfiEIGHTTHREEGaussian$\ $\mclfiNINETHREE$ & $\EmclfiNINETHREEGaussian$ & $\NPmclfiNINETHREE$ & $\NUPmclfiNINETHREEGaussian$\ $\mclfiONEZEROTHREE$ & $\EmclfiONEZEROTHREEGaussian$ & $\NPmclfiONEZEROTHREE$ & $\NUPmclfiONEZEROTHREEGaussian$\ [2]{} $\lfiFOUR$ & $\ElfiFOURGaussian$ & $\NPlfiFOUR$ & $\NUPlfiFOURGaussian$\ $\mclfiONEFOUR$ & $\EmclfiONEFOURGaussian$ & $\NPmclfiONEFOUR$ & $\NUPmclfiONEFOURGaussian$\ $\mclfiTWOFOUR$ & $\EmclfiTWOFOURGaussian$ & $\NPmclfiTWOFOUR$ & $\NUPmclfiTWOFOURGaussian$\ $\mclfiTHREEFOUR$ & $\EmclfiTHREEFOURGaussian$ & $\NPmclfiTHREEFOUR$ & $\NUPmclfiTHREEFOURGaussian$\ $\mclfiFOURFOUR$ & $\EmclfiFOURFOURGaussian$ & $\NPmclfiFOURFOUR$ & $\NUPmclfiFOURFOURGaussian$\ $\mclfiFIVEFOUR$ & $\EmclfiFIVEFOURGaussian$ & $\NPmclfiFIVEFOUR$ & $\NUPmclfiFIVEFOURGaussian$\ $\mclfiSIXFOUR$ & $\EmclfiSIXFOURGaussian$ & $\NPmclfiSIXFOUR$ & $\NUPmclfiSIXFOURGaussian$\ $\mclfiSEVENFOUR$ & $\EmclfiSEVENFOURGaussian$ & $\NPmclfiSEVENFOUR$ & $\NUPmclfiSEVENFOURGaussian$\ $\mclfiEIGHTFOUR$ & $\EmclfiEIGHTFOURGaussian$ & $\NPmclfiEIGHTFOUR$ & $\NUPmclfiEIGHTFOURGaussian$\ $\mclfiNINEFOUR$ & $\EmclfiNINEFOURGaussian$ & $\NPmclfiNINEFOUR$ & $\NUPmclfiNINEFOURGaussian$\ $\mclfiONEZEROFOUR$ & $\EmclfiONEZEROFOURGaussian$ & $\NPmclfiONEZEROFOUR$ & $\NUPmclfiONEZEROFOURGaussian$\ Averaging over Reheating Scenarios for Other Large-Field Models with a Gaussian Prior on $\sigma_\uend$ {#sec:largefield:average:Gaussian} ======================================================================================================= [^1]: In this work, we assume that all particles are in full thermal equilibrium after $\phi$ and $\sigma$ decay; thus there are no residual isocurvature modes [@Lyth:2002my; @Weinberg:2004kf]. Any isocurvature modes surviving after reheating would provide very strong additional constraints, but are dependent on the specific process of decay and thermalisation [@Langlois:2004nn; @Lemoine:2006sc; @Langlois:2008vk; @Lemoine:2008qj; @Smith:2015bln] which we do not consider here. [^2]: In , it is shown that the timescale of equilibration depends on $m_\sigma$ in practice, but can be surprisingly large (even more than thousands of $e$-folds). This implies that the initial conditions for spectator fields are not automatically erased during inflation, and that time variation of $H$ in (even slow-roll) inflation could play a role as well. Therefore, distributions different from may also be relevant and this is another reason why, in this work, only the rough condition $m_\sigma\sigma_\uend\sim H_\uend^2$ is imposed.
--- abstract: 'While the radio detection of cosmic rays has advanced to a standard method in astroparticle physics, the radio detection of neutrinos is just about to start its full bloom. The successes of pilot-arrays have to be accompanied by the development of modern and flexible software tools to ensure rapid progress in reconstruction algorithms and data processing. We present NuRadioReco as such a modern Python-based data analysis tool. It includes a suitable data-structure, a database-implementation of a time-dependent detector, modern browser-based data visualization tools, and fully separated analysis modules. We describe the framework and examples, as well as new reconstruction algorithms to obtain the full three-dimensional electric field from distributed antennas which is needed for high-precision energy reconstruction of particle showers.' author: - Christian Glaser - Anna Nelles - Ilse Plaisier - Christoph Welling - 'Steven W. Barwick' - 'Daniel García-Fernández' - Geoffrey Gaswint - Robert Lahmann - Christopher Persichilli bibliography: - 'BIB.bib' date: 'Received: date / Accepted: date' title: 'NuRadioReco: A reconstruction framework for radio neutrino detectors' --- Introduction ============ In this article, we present a novel modular framework for the detector simulation and data reconstruction of radio detectors for neutrinos and cosmic rays along with the corresponding algorithms. For neutrino detection, the radio technique allows to significantly extend the energy range of current experiments of a few times [@IceCubePRL; @IceCubePRL_E], which is required to reach the next major milestone in astroparticle physics: the discovery of cosmogenic neutrinos [@1966PhRvL..16..748G; @1966JETPL...4...78Z; @1969PhLB...28..423B]. In high-energy cosmic-ray physics, the radio technique has already been established as a competitive detection method during recent years [@reviewHuege; @reviewSchroder]. In particular its excellent sensitivity to the cosmic-ray mass [@LofarNature] and energy [@AERAPRD; @AERAPRL] make this technique very attractive. While rivalling in accuracy it is less sensitive to atmospheric conditions than optical methods and has a duty cycle of close to 100% [@LOFARrefractiveindex; @Gottowik_2018]. Many aspects of data processing, detector simulation and reconstruction are similar between radio detectors for cosmic rays and neutrinos. Radio neutrino detectors such as ARIANNA are even cosmic-ray detectors themselves [@ARIANNACRs]. Hence, many analysis methods and strategies from the cosmic-ray community can be transferred to neutrino detectors allowing to benefit from the maturity of radio cosmic-ray observations. Consequently, it was the obvious choice to develop a framework suitable for both neutrinos and cosmic rays. This framework builds on extensive experience with both Monte-Carlo studies and data analysis of cosmic-ray as well as neutrino detectors [@LofarNature; @AERAPRD; @AERAPRL; @ARIANNACRs; @GlaserErad2016; @LOFAREnergy]. It is also based on many years of experience with the software needs of large radio cosmic-ray experiments, in particular the experience with existing software projects such as Offline [@OfflineSoftware; @RadioOffline], the reconstruction framework of the Pierre Auger Observatory and its radio extension AERA, the LOFAR cosmic-ray software , the Physics eXtension Library (PXL) for high-energy physics and the web analysis framework VISPA [@VISPA]. NuRadioReco combines their strengths while addressing shortcomings of the existing projects for radio detection. NuRadioReco was developed in the context of the ARIANNA [@ARIANNA2015], a pilot-array for the detection of high-energy neutrinos with energies above . It consists of an array of autonomous stations located close to the surface on the Antarctic ice sheet. Each station has multiple spatially separated antennas with different orientations to reconstruct the incoming signal direction and polarization. Radio signals are produced via the Askaryan effect [@Askaryan] from particle cascades generated in the ice by interactions of these neutrinos. The Antarctic ice is transparent to MHz–GHz radio signals which allows for a cost-effective instrumentation of large volumes [@barwick_besson_gorham_saltzberg_2005]. Therefore, the radio technique is the method of choice with two fully-operational pilot detectors [@ARA; @ARIANNA2015] and larger experimental efforts planned for the near future [@RNO; @COSPAR]. Established cosmic-ray detectors such as AERA [@AERAPRD], LOFAR [@LOFAR] and Tunka-Rex [@TunkaRex] also consist of many autonomous detector stations. Here, each station has just one dual-polarized antenna and a coincident measurement of multiple stations is required for data analysis, while neutrino detectors typically only have radio signal data from one station, however with multiple antennas. The necessary flexibility to account for either is part of NuRadioReco. This article serves two purposes: First, to document NuRadioReco and second, to describe the algorithms required to reconstruct data from radio neutrino detectors, in particular the algorithms to recover a radio signal from multiple spatially displaced antennas with different polarization responses, a problem not yet addressed in literature. An accurate reconstruction of the electric field is the foundation for a high-precision measurement of the energy contained in a neutrino or cosmic ray induced particle shower using its radio emission. NuRadioReco is written in Python, open-source and publicly available on github [@NuRadioReco]. The design goals are to be easy-to-use with a user friendly interface and a maximum amount of flexibility. It follows a modular design with a strict differentiation between event data, detector description and processing modules depicted in Fig. \[fig:structure\]. The breakdown of the data processing into independent steps (the processing modules) fosters collaborative development, enforces a clear structure and allows for an easy modification of a processing pipeline. Each module is independent of each other, as modules only interact with the event data, and thus can be exchanged easily. Through the consistent use of numpy [@numpy] for all operations on arrays, the code is sufficiently fast while offering all advantages of Python: Easy software installation as no compilation is required which is often cumbersome on different systems and platforms. The flexible Python steering allows arbitrary loops around modules, complex if/else branches, stop criteria and the possibility to call a module several times with different arguments. At the same time, more complex calculations can be included in optimized compiled languages such as C++, as long as Python wrappers are provided. ![image](figures/Three_Components.pdf){width="80.00000%"} In the following paragraphs, we briefly discuss the main advantages of NuRadioReco. The details are given in the individual sections of this paper. We also describe the properties of all default modules and provide an end-to-end example of a signal simulation, full detector simulation, and data reconstruction. The paper concludes by describing two new algorithms used in signal processing for single-station detectors. #### I/O In NuRadioReco, the default input and output file format is the same. The i/o modules allow to save the current state of the event data to disk after each processing step, and to read it back in. Hence, a data processing pipeline can be split up into consecutive steps. It is also straight forward to implement modules to read instrument generated data into the event structure. #### Time-dependent detector description In NuRadioReco we use an SQL database designed to store a time-dependent detector description. While SQL is the method of choice to store the description of a large experiment, SQL has its limitations in usability and for queries from parallel processing on clusters. Therefore, we implemented a database export into a human readable JSON text file. This also allows for a simple setup of new detectors for simulation studies. #### Data visualization - event browser Data is visualized using state-of-the art web technologies. The GUI is platform independent as the only requirement is a web browser. This design also allows for a remote deployment such that data can be inspected over the internet. This is particularly useful for outreach activities and easy collaborative sharing of data and results. #### Default system of units Keeping track of units is a must for physics analyses. NuRadioReco employs the same concept as [@OfflineSoftware]: Every time a variable is defined, it is multiplied by its unit, and every time a variable is plotted or printed out, it is divided by the unit of choice, such as [python]{} from NuRadioReco.utilities import units time = 132. \* units.ms \# define 132 milli seconds d = 5. \* units.mm \# define 5 mm v = d/time \# calculate speed print(“the speed is [:.2f]{} km/h”.format(v/units.km\* units.hour)) \# the speed is 0.14 km/h In this way, all internal computations can be done without the need for the user to worry about the correct units. We have chosen to not import an existing unit system (such as astropy or pypi), as we need access to units from all modules, including those not written in Python. #### Link to simulations Simulations of the radio emission following a neutrino interaction are currently performed with limited flexibility in detector design and using simple signal parameterizations only (e. g. [@ShelfMC; @ARASim]). In parallel to NuRadioReco, NuRadioMC is being developed as community-driven simulation code that addresses the short-comings of previous codes [@NuRadioMC]. As it shares certain characteristics of NuRadioReco, such as the data-format, there will be a seamless integration of signal simulation, detector simulation and reconstruction for future experiments. Data structure ============== All measured, simulated and reconstructed quantities are saved in a hierarchical class structure that also supports the simple storage of analysis quantities and is designed having the (for radio experiments) natural representation in time- and frequency-domain in mind. Event structure --------------- The class structure fits the requirements of multi- and single-station detectors and both cosmic-ray and neutrino reconstruction. Askaryan neutrino detectors, such as ARIANNA [@ARIANNA2015] and ARA [@ARA], consist of independent detector stations, i.e., the design foresees that measurement, identification and reconstruction of a neutrino properties are done using data from a single station. In contrast, typical cosmic ray detectors, such as AERA [@AERAPRD] or LOFAR [@LOFAR], consist of many stations that collectively measure the cosmic-ray signal. Although multi-station coincidences are not typical for Askaryan neutrino detectors, very high energy events or ‘double-bang’ tau events might be observed in multiple stations. Hence, the data structure is flexible enough to accommodate both cases. However, the focus of this paper lies on single station events. The treatment of radio signals from air showers (cosmic rays) and in-ice showers (neutrinos) is slightly different. The air-shower signal is measured at large geometrical distances to the shower maximum and thus extends over a large area. It can safely be assumed that the signal does not change over the small lateral extent of a compact station of a few meters[^1]. Hence, all antennas of one station observe the same signal. This is not the case for the Askaryan signal of in-ice showers. Here, the showers are observed at closer distances in an inhomogeneous medium. Thus, the signal can be significantly different, especially for antennas displaced in depth, because of its strong dependence on the viewing angle of the Askaryan signal. Furthermore, each antenna may detect two pulses from the same in-ice shower from different directions and propagation paths through the ice [@KelleyARENA2018]. This is because the upper ice layer is a non-uniform medium where the signal trajectory is bent, leading to two distinct solutions, either a direct path, a refracted path or path where the signal gets reflected off the ice-air interface at the surface. Also more exotic emission models might lead to even more pulses per antenna with different incoming directions. Consequently, there is the need to store an arbitrary number of signals that arrive at the same channel at different times and from different directions. All these requirements can be mapped into the event structure depicted in Fig. \[fig:eventstructure\], also showing the definition of hierarchical levels. First, the *event* level that includes all simulated, measured and reconstructed data of all stations that have detected a signal. Second, the *station* level that includes all antennas of a single radio detector station, and third, the *channel* level, one for each antenna, storing the measured signal. All simulated quantities are stored in the *SimStation* class. Furthermore, we differentiate between voltage traces $V_i(t)$, i.e., the signal as a function of time measured in an antenna $i$, and electric-field traces $\vec{E}(t)$ that refer to the three-dimensional electromagnetic pulse before being measured by the antenna. Electric fields are stored in a dedicated *electric field* class. Apart from storing the time series, it also stores the incoming signal direction and the information for which channel(s) it is valid. This allows to cover the cosmic-ray case, i.e., a single electric field is valid for all channels, as well as the neutrino case where it might be necessary to store an electric field for every channel separately. For simulated neutrino events, we can have multiple electric fields per antenna from different signal paths that are treated by adding a second electric field for the same channel. ![image](figures/event_structure.pdf){width="65.00000%"} Parameter storage ----------------- The data structure offers a flexible mechanism to save parameters on event, station and channel level, e.g., the reconstructed air-shower direction on station level and the signal-to-noise ratio on channel level. All parameters are defined in an enumerated type *enum* and can be accessed via a generic *setter* and *getter* function. We also allow to save both an uncertainty for each parameter and the covariances between any pair of parameters. To add a new parameter, the parameter simply needs to be added to the *enum* table. Then, this parameter can be accessed from each module and is automatically included in the input/output file. This is an advantage in time-efficiency compared to the standard way of adding a new member variable for each parameter, because in the latter case additional *getter* and *setter* methods need to be implemented and the variable needs to be manually included into the i/o data stream. Explicitly defining all parameters in an *enum* ensures that all parameters are well defined and that users know which parameters exist. Therefore, we chose not to use a Python dictionary to store parameters but implemented a dictionary-like usage: [python]{} from NuRadioReco.framework.parameters import stationParameters as stnp from NuRadioReco.utilities import units \# set parameters via generic setter function station.set\_parameter(stnp.nu\_energy, 7e8 \* units.GeV) \# or via dictionary like interface station\[stnp.nu\_energy\] = 7e8 \* units.GeV \# set uncertainty of neutrino energy station.set\_parameter\_error(stnp.nu\_energy, 1e6 \* units.GeV) \# access of parameters nu\_energy = station.get\_parameter(stnp.nu\_energy) \# or nu\_energy = station\[stnp.nu\_energy\] Time and frequency domain ------------------------- The voltage and electric-field traces can be represented in the time or frequency domain. The two representations can be used interchangeably as depending on the processing step one representation may be more convenient to work with. For example, a bandpass filter is implemented easiest in the frequency domain, whereas a pulse finding algorithm is naturally implemented in the time domain. Therefore, the event structure offers the functions [python]{} \# access trace in time domain time\_trace = channel.get\_trace() times = channel.get\_times() \# or access trace in frequency domain frequency\_spectrum = channel.get\_frequency\_spectrum() frequencies = channel.get\_frequencies() to transparently obtain the time or frequency domain representation depending on the needs of a processing module. Internally, it is kept track of which representation was last modified and a Fourier transform is performed if necessary. Similarly, we provide functions to define a new trace either in time or frequency domain. Technically this functionality is implemented once in a generic *base trace* class from which the *channel* and *electric field* classes inherit. This approach avoids typical errors using Fourier transforms and their normalization. We chose to normalize the transforms as such that Parseval’s theorem is observed and the physical quantity of signal power is conserved. Input/Output ============ NuRadioReco provides several input modules for different sources (CoREAS simulations [@Coreas], ARIANNA raw data format, etc.) but also has its own file format, ending by default in *\*.nur*. Philosophy of *.nur* files -------------------------- The main advantage of NuRadioReco’s own *.nur* file format is that it was designed to save/read the current state of the event data to/from disk after every modular processing step. Therefore a reconstruction can be split up into multiple steps without complications. For example, computationally expensive low level processing only needs to be done once and secondary reconstruction can be started from the pre-processed files without having to save data in an intermediate file format. A practical application for the ARIANNA experiment is the following. Most triggers are caused by thermal noise fluctuation with typical rates of a few events per minute per detector station. The rate of cosmic rays is only one or two per day. Hence, in a first processing step, cosmic-ray candidate events are identified and only those events are saved to disk. This largely reduces the data volume to an easily manageable file size and serves as the starting point of high-level analyses. Technical implementation ------------------------ The NuRadioReco data format is implemented through a serialization and deserialization function in each event data class, a concept adapted from PXL [@VISPA]. In other words, the data structure knows how to (de)serialize itself. The (de)serialization is performed recursively per event, i.e., calling the serialization function of the event class will call the serialization function of all stations that are part of this event, which will call the serialization function of its channels and so on. Another advantage is that new properties can be added to the event data structure without the need to also modify the i/o-modules. A new property only needs to be added to the (de)serialization function. This implementation also allows for backward compatible additions to the file format. If a certain property is not present in an older file version, it can be initialized with an appropriate default value during the deserialization. Internally, it is made use of the *pickle* module to create a binary representation of most data members, but the data file itself is a custom binary format and the use of *pickle* could be replaced in the future. The data format is compatible across different computing systems and Python versions. The only requirement to read the data is a Python installation and the NuRadioReco modules but no installation is required. We note that we also considered other file format options, e.g., to build our data format on top of HDF5 but didn’t find it suitable for our case (see e.g. the discussion in [@HDF5discussion] and [@HDF5discussion2]). In addition to the full storage of the event structure, the high-level parameters on station level are saved in an additional event header. This enables quick parsing of data files, and access and plotting of high-level quantities. During the initialization of the i/o class, the headers of all events are parsed and the high-level parameters are stored in numpy arrays. This allows for a quick inspection and plotting of analysis results. With just a couple of lines of code one can plot the maximum pulse amplitude as a function of time, or a histogram of the reconstructed signal directions as: [python]{} import NuRadioReco.modules.io.NuRadioRecoio as NuRadioRecoio from NuRadioReco.utilities import units from NuRadioReco.framework.parameters import stationParameters as stnp import matplotlib.pyplot as plt nurio = NuRadioRecoio.NuRadioRecoio(“my\_file.nur”) header = nurio.get\_header() station\_id = 51 station\_header = header\[station\_id\] \# get numpy arrays of reconstructed direction zeniths\_rec = station\_header\[stnp.zenith\] azimuths\_rec = station\_header\[stnp.azimuth\] \# plot zenith vs azimuth in degrees plt.plot(zeniths\_rec/units.deg, azimuths\_rec/units.deg) plt.show() Another feature of the NuRadioReco i/o class is that the amount of data written to disk can be controlled on an event-by-event basis. The vast majority of the disc space is typically occupied by signal data as voltage or electric field trace. NuRadioReco offers three output modes: - ‘full’ (default): the full event content is written to disk - ‘mini’: only electric-field traces are written to disc, but no channel traces - ‘micro’: no traces are written to disc that are specified in the *eventWriter*’s run method, e,g., . Another feature is that output files can automatically be split up into several files by specifying a maximum file size. Correspondingly, the *eventReader* has the functionality to read in a list of files transparent to the user. Time dependent detector description {#sec:detector} =================================== Any larger experimental effort requires a complete detector description that provides all information relevant for data analysis in a machine readable form. This includes the position and orientation of each antenna of each station, the details of the analog signal chain of each channel such as cable lengths, amplifier responses and ADC (analog-to-digital converter) details, and so on. Furthermore, a detector layout might change over time, detector stations might be reconfigured or certain components might be replaced. Hence, the detector description needs to be time-dependent. The requirement is that the user can request the exact configuration of a station at any time. Database structure ------------------ The method of choice is to store the detector description in a database. We use MySQL and present the database structure in Fig. \[fig:mysql\]. We have followed standard database design rules, in particular that no information is ever duplicated. Similar to the event data structure, the database has a hierarchical table structure. Different tables are related to each other by their unique ids. For example, to add a channel to a station, a new row needs to be inserted into the *channels* table with the unique id of the respective station. Each *channel* entry contains a reference to an antenna, cable, amplifier (amp) and ADC, which need to be defined in the corresponding tables. ![image](mysql_structure.png){width="90.00000%"} This design has a number of practical advantages: Each channel can have the same reference amplifier without re-specifying what *reference amplifier* means. If the reference measurement changes, it needs to be changed only at one place. If a more detailed description is demanded for analysis, individual measurements of the amplifier response can be added to the *amps* table and referenced from the respective channels. The time dependent nature is implemented at two places. The *stations* and the *channels* table contain a commission and decommission time. A typical use case is that one or multiple channels of a station are reconfigured with a new antenna. To put this into the database, first the new antenna properties need to be added (as a new row) into the *antennas* table. Then, the decommission time of the current channel is set to the time of the hardware change and a new channel is inserted into the *channels* table with the same properties as the previous channel but with the antenna id pointing to the new antenna and with the proper commission and decommission time. An alternative way to implement the time dependence would have been to give each *antennas*, *cables*, *amps* and *adcs* entry a time dependence, and to remove the time dependence of the channel. And instead of referencing the channel components from the *channels* entry, the *antennas*, *cables*, *amps* and *adcs* entries would reference back to the channel, similar to the *stations* - *channels* relationship. Such a structure would have the advantage that, for the above mentioned antenna replacement, only the antenna table needed to be altered. However, it comes with the large disadvantage that no default detector components can be specified. Suppose most ADCs are so similar that we can use the same ADC reference description for most channels. In the latter structure, a reference ADC entry needs to be added for every channel which is a huge duplication of information. Consequently, we have chosen the former design. The tables can be combined via ’JOIN’ statements. The following code, for example, retrieves the position of the antenna of channel 2 of station 10 on November 5th 2018 at noon: [mysql]{} SELECT position\_x, position\_y, position\_z FROM stations AS st JOIN channels AS ch USING(station\_uid) JOIN antennas USING(antenna\_uid) WHERE CAST(’2018-10-05 12:00’ AS DATETIME) between ch.commission\_time and ch.decommission\_time AND CAST(’2018-10-05 12:00’ AS DATETIME) between st.commission\_time and st.decommission\_time AND st.station\_id = 10 AND ch.channel\_id = 2; User friendly implementation ---------------------------- Although a central database is the method of choice to keep track of the time-dependent detector description, it comes with several disadvantages for the user: - A (internet) connection to the MySQL server is required when running the software. - Queries to a remote MySQL server are relatively slow. - In MySQL the number of simultaneous connections to database is limited, which precludes parallel processing on computing cluster. - It is difficult to make local changes to the detector description for testing purposes. Therefore, we have implemented additional options. Either the database is buffered at the beginning of the processing, i.e., we connect only once to the database, or the database is exported into a simple JSON text file The total amount of data required for the detector description is small, e.g., the complete detector description of the current ARIANNA detector is less than . Hence, we can buffer the complete database and store all information in memory. Internally, we use the *TinyDB* Python package [@tinydb] to buffer the database. *TinyDB* provides a convenient interface to the data and supports ’WHERE’ statements to access the information of a specific station and channel at a specific time. However, *TinyDB* does not support relationships between tables which we need to properly setup the detector description. Hence, we linearize the database and combine all channel related tables into one single table to store it in *TinyDB*, which results in a duplication of data. At this point though, this is not a cause for concern as the master is always the MySQL database. *TinyDB* also allows us to save the database in a simple JSON text file. This is the method of choice for most users and the default in NuRadioReco. A simple detector description is shipped as part of the software so that everything works out-of-the box. In this way, all the advantages of a MySQL database of storing a complex detector description are combined with the user-friendly usage of human readable text files. When running on a large cluster, the number of MySQL connections typically limits the number of parallel compute nodes. Through the usage of JSON files, this can be avoided by copying the relevant JSON files to individual nodes. Due to the limited size, i/o will also not be an issue. We note that queries through *TinyDB* are relatively slow even if all data is present in memory. Therefore, we have added an additional layer of buffering such that database access is a negligible fraction of the total processing time. Usage for simulation studies ---------------------------- NuRadioReco is not only used for data reconstruction of existing experiments but also used to simulate future experiments. As setting up a MySQL database has to be considered too much overhead for most simulation studies, the JSON text file representation offers a convenient method to quickly define arbitrary station configurations. A few examples of detector descriptions are included in NuRadioReco [@NuRadioReco] that can be adopted to the users needs. It should be noted that it is sufficient to specify only the relevant fields, e.g., if a simulation study does not simulate the ADC digitization, the ADC related tables can be left empty or can be removed completely from the JSON file. Custom detector descriptions are specified during the initialization of the detector: [python]{} import NuRadioReco.detector.detector as detector det = detector.Detector( json\_filename=“/path/to/my\_detector.json”) Handling of antenna sensitivities --------------------------------- NuRadioReco provides a convenient interface to antenna models and provides a library of commonly used antennas for neutrino and cosmic-ray detection. Most antenna models available in NuRadioReco have been simulated with WIPL-D [@Kolundzija2011], but also antenna simulations using XFDTD [@XFDTD] or NEC-2 [@Nec2] are available. This is handled technically by pre-processing the raw antenna simulation output to the same data structure, stored in a pickle file. This also significantly reduces the file size from raw simulation output that can easily exceed for a fine sampling in frequency and incoming signal direction. The antenna response files are provided on a central server and are downloaded automatically when needed. The conversion scripts for WIPL-D and XFDTD are provided. The antenna response is quantified as the vector effective length which is a complex quantity that depends on frequency and incoming signal direction and can be thought of as proportionality constant between the incident electric field and the voltage output of the antenna (cf. Eq.  for more details). However, this quantity is typically not a direct output of antenna simulation software, with the output differing from software to software. In \[sec:Antenna\_effective\] we detail how the output of the different simulations are converted into the relevant vector effective length. NuRadioReco’s antenna model class then provides a user-friendly interface. The requested antenna model is buffered in memory, it is interpolated to the required frequencies and angles, and all coordinate rotations are handled internally to match the orientation of the antenna in the detector description. Coordinate system ----------------- We differentiate between the relative coordinates of the components of a detector station and its absolute position. The positions of the components (e.g. the antennas) are expressed in a local Cartesian coordinate system with the coordinate origin in the horizontal center of the station and with $z = 0$ at the ice surface. The positive x-axis is oriented into the Easting direction. For stations at the South Pole we use a special coordinate system that moves with the ice (the ice drifts by approximately each year) such that the station coordinates remain constant with time with respect to each other. For other locations we use the UTM coordinate system. Both absolute coordinate system are local Cartesian projections onto a 2D surface. Hence, standard euclidean geometry can be used to calculate e.g. distances between stations. Given the typical distance between stations of , the Earth curvature can be neglected. However, it is foreseeable to introduce more a more refined treatment of coordinates in the future. Data analysis and processing modules ==================================== In this section, we describe the setup of the data analysis modules and briefly discuss the standard processing steps to extract the physics properties from radio data. This is illustrated by a full example of reading in a cosmic ray simulation, a detector simulation and the reconstruction. New techniques for data analysis beyond what is currently used in radio detection of cosmic rays and neutrinos are discussed in Sec. \[sec:Reco\_algo\]. Format of data analysis modules ------------------------------- A detector simulation and event reconstruction using NuRadioReco is split into several modules that are executed in sequence, with each module fulfilling one specific purpose. In principle, modules can be arranged in any order, including loop or if/else branching, though some may require a certain module to be executed beforehand; for example, using a module to apply a filter to an electric-field trace is only sensible if another module has reconstructed the field. \[sec:example\_rec\] ![image](figures/example.pdf){width="80.00000%"} Each module consists of four components: - A constructor to create the module instance. This is called before the reconstruction loop over all events. - A *begin* function to set parameters that remain constant for each event, such as a list of input files for the event reader module. - A *run* function that is executed for each event and in which the module fulfills the task it was designed for. - An *end* function that may be executed after the last event was processed. Full simulation and reconstruction cycle ---------------------------------------- For hands-on understanding of NuRadioReco, this section describes a full reconstruction cycle for a signal from an air shower simulated with CoREAS. A schematic overview of all the modules used is shown in Fig. \[fig:example\_schema\]. The full Python code, as well as a more detailed description are available at [@NuRadioReco_example]. ### Event generation The event generation starts by reading the electric-field traces that were simulated using CoREAS [@Coreas]. All events are selected that have signal in the frequency band of 100-500 MHz[^2]. In order to improve the timing accuracy, the simulated electric-field is up-sampled before the voltage traces per channel are calculated. This up-sampling is also needed to shift the signals in time according to the geometric time delays to the antennas with high precision. The ARIANNA hardware response and noise with an RMS of is added to fully simulate data-like events. An example of the simulated electric-field traces, the calculated voltage trace for a channel, and the voltage trace with added noise are shown in Fig. \[fig:example\_traces\]. ![image](figures/example_traces.pdf){width="80.00000%"} ### Event selection For the event selection, first a trigger is simulated, i.e., if a certain event would be recorded by the detector. Only the events marked as triggered are selected. In order for an event to trigger the amplitude must exceed in at least two of the channels in this simple example. The voltage traces are filtered again in the frequency range of 80-500 MHz to reduce contamination by noise outside of the pass-band. ### Event reconstruction The direction of the incoming cosmic ray is fitted using the direction-fitting modules. This reconstructed arrival direction is used to convert the voltage traces per channel to the fitted reconstructed electric-field. The electric-field reconstruction is done using the *forward folding technique*, which uses an analytic description of the electric-field pulse and is discussed in detail in Sec. \[sec:Reco\_algo\]. The simulated voltage traces and the reconstructed electric-field traces are down-sampled to the original detector bandwidth before storing to disk in order to reduce the file size. Overview of relevant default modules ------------------------------------ A number of modules are considered default in NuRadioReco and many are used in the example. They are meant as illustration in the same way as starting point for more complex analyses. #### Event reading/writing The two modules *eventReader* and *eventWriter* allow to read and write from/into the NuRadioReco file format. #### Reading CoREAS files CoREAS is the state-of-the art simulation code for radio emission from air showers [@Coreas] and NuRadioReco provides a direct interface to the HDF5 output of CoREAS. Two different modules are provided serving different user requirements. The *readCoREASStation* module creates a new event with a single station for each simulated observer in the CoREAS file, which is suitable for a detector like ARIANNA where air showers are measured independently by each station. The other module called *readCoREAS* is optimized to read in ’star pattern’ simulations for single station detectors such as ARIANNA. The spatial distribution of the radio emission can be sampled efficiently in a special coordinate system in the shower plane where one axis is oriented perpendicular to the air-shower axis and the geomagnetic field (see e.g. [@LOFARLDF; @Alvarez-Muniz:2014wna; @GlaserErad2016; @Glaser_2019] for more details). Hence, instead of running a new time-consuming CoREAS simulation for different locations of the air shower on the ground, called *shower core* in the following, we can just use one star pattern simulation and pick the closest station. This allows to reuse the same CoREAS simulation many times. The *readCoREAS* module will generate a definable number of core positions randomly distributed on the ground. The module then determines the closest simulated station (measured in the shower plane) and creates and returns a corresponding event object. In this way, a realistic distribution of cosmic-ray events is obtained. #### Re-sampling Recorded traces are usually up-sampled at the beginning of the reconstruction process in order to improve accuracy and down-sampled after the analysis to reduce file size. This is done by using the modules *channelResampler* to re-sample voltage traces and *electricFieldResampler* to re-sample electric-field traces. #### Band-pass filtering There are two modules available to filter signals: The *channelBandPassFilter* for voltage traces and the *electricFieldBandPassFilter* for electric field traces. Both support several different filter types, like a simple rectangular filter that cuts off all frequencies outside of its pass-band or a Butterworth filter modelling a hardware filter. Both can be applied to any frequency band. #### Converting electric fields to voltages In order to perform simulation studies, it is necessary to calculate the waveform that an electric field produces in each channel, which is done with the *efieldToVoltageConverter*. Since the radio pulse from a particle shower in the ice may reach the antenna via several different ways, each channel may have multiple electric fields associated with it, each resulting from a different ray path. Therefore, the first step is to calculate the minimal trace length necessary to store all radio pulses and create an empty voltage trace of that length, plus some padding before and after the pulse. Subsequently, the electric field is convolved with the antenna response retrieved from the detector description. The result is then added to the voltage trace, whereby the different trace start times of each electric field trace and the channel’s cable delay are taken into account. If the signal stems from an air shower where only one electric field per station is simulated as the signal does not change over such small spacial extents, the differences in signal travel times between channels are also calculated and corrected for. #### Adjusting trace lengths If so desired, the lengths of the channel traces can be adjusted using the *channelLengthAdjuster*. If the trace is longer than needed, it is cut to size after the pulse position is determined to ensure it is not accidentally removed. If the trace is too short, it is appended by zeros. #### Accounting for amplifiers, filters and cable effects Characteristics from the time-dependent detector description (see Sec. \[sec:detector\]), are included in different processing modules that convert data to/from ideal voltage traces from/to instrument data. NuRadioReco contains both simplified models, such as ideal filters, and true implementations of the complex behavior of amplifiers and cables. In all these modules, gain and phase-delay is applied to the data. Measurements or simulations of components are interpolated such that they match the sampling rate of the data at that point in the processing step. Also, via the units utilities, the use of different units at different steps of the processing chain is automatically accounted for.. #### Noise generator For a realistic simulation of signals recorded by the antennas it is essential to add noise. This can be done in the module *channelGenericNoiseAdder*. In the module, simple white noise with a normal amplitude distribution for a frequency band specified by the user is calculated in the frequency domain. The user also specifies the required RMS voltage of the noise in the time domain. In most cases, it will be desirable to remove the zero-frequency component, in which case the RMS of the voltage values will be identical to their standard deviation. When calculating the corresponding amplitudes of each frequency bin, it is taken into account that only the specified frequency band contributes to the signal power. To obtain the desired noise distribution, the phase of each frequency bin is drawn from a uniform random distribution in the range $[0,2\pi)$. The amplitude can be chosen either to be perfectly flat over the specified frequency range or to follow a Rayleigh distribution for each frequency bin. In the time domain, the former yields the specified RMS voltage *exactly*, while in most cases providing a reasonable approximation of the noise background. This mode is hence ideally suited for developing and debugging new code. The Rayleigh distribution is expected for the absolute value of complex numbers, where the real and imaginary components are uncorrelated and each follow a normal distribution with equal variance and zero mean. This provides for a more realistic noise model with small statistical deviations from the specified RMS voltage in the time domain. #### Trigger simulations The last step in generating real sensitivities is the simulation of a trigger. Per default, several options are included in NuRadioReco. A simplified threshold trigger, an ARIANNA-style dual-threshold trigger, as well as an ARA-style tunnel-diode trigger. The framework can also account for a trigger from the phasing of several antennas, and is capable of calculating coincidence requirements across multiple channels. #### Template correlation In order to distinguish desired signals from background noise, recorded voltage traces can be correlated with one or more neutrino or cosmic-ray waveform templates using the *channelTemplateCorrelation* module. The templates are voltage traces generated from simulated radio pulses, which have to be calculated beforehand. The *channelTemplateCorrelation* module re-samples the template to match the sampling rate of the recorded signal and calculates the correlation $$\rho (V_{sig}, V_{tmp}, \Delta n) = \frac{\sum_i (V_{sig})_i\cdot (V_{tmp})_{i-\Delta n}}{\sqrt{\sum_i(V_{sig})_i^2}\cdot\sqrt{\sum_i(V_{tmp})_i^2}}$$ where $V_{sig}$ and $V_{tmp}$ are the voltage traces of the recorded signal and the template, respectively, which are shifted by $\Delta n$ samples relative to each other. The denominator normalizes the expression to $-1 \leq \rho \leq 1$. The $\Delta n$ yielding the highest correlation is found and both the correlation and the time offset corresponding to $\Delta n$ are saved in the parameter storage. If the channel was compared to multiple templates, the average value over all templates is also calculated, as well as the maximal correlation. #### Directional reconstruction Both modules providing a reconstruction of the signal direction, the *correlationDirectionFitter* and the *templateDirectionFitter* use the same principle: Assuming a plane-wave, a signal coming from the direction $\vec{e}_r(\theta,\phi)$ will arrive at an antenna positioned at $\vec{x}$ at the time $$t_{exp} = \vec{e}_r(\theta,\phi) \cdot \vec{x} \cdot \frac{n}{c} + t_0 \, ,$$ where $c$ is the speed of light in vacuum and $n$ is the index of refraction of the medium surrounding the antennas. The *templateDirectionFitter* uses the relative time shift $t_{corr}$ for which the *channelTemplateCorrelation* module found the best correlation to a template and minimizes the $\chi^2$-function $$\chi^2 =\sum_i \frac{((t_{exp, i}(\vec{e}_{signal}) - \langle t_{exp}\rangle) - (t_{corr,i} - \langle t_{corr}\rangle))^2}{\sigma_t^2}$$ where $<t_{exp}>$ and $<t_{corr}>$ are the averages of $t_{exp}$ and $t_{corr}$, respectively. The *correlationDirectionFitter* takes two pairs of channels that measure the same polarization and correlates them with each other. It then finds the direction for which correcting for the time difference between channels results in the best correlation between channel pairs. The advantage of this method is that it is independent of the description of the antenna response as only the time differences of parallel channels are considered where the antenna response is the same and thus cancels out as systematic uncertainty. This method is tailor-made for an ARIANNA-like detector with parallel channels. A more general direction-fitting routine can easily be adapted from the existing modules. Similarly, modules for spherical or hyperbolic arrival times are not in the default repository. #### Converting voltages to electric fields After having identified signals and having obtained their arrival direction, a typical task is to reconstruct the electric field from the measured data, essentially inverting the *efieldToVoltageConverter*. In the presence of noise, this is however not a straight-forward inversion and we present two novel algorithms to obtain this from spatially distributed antennas in Sec. \[sec:Reco\_algo\]. Data visualization ================== ![image](figures/EventBrowserScreenshot.png){width="90.00000%"} NuRadioReco uses state-of-the-art web technologies for data visualization. Using web technologies comes with a number of advantages: - The GUI is platform independent and the only requirement is a web browser. Hence, computers and laptops with all operating systems can be used as well as tablets or smartphones. - Data can easily be visualized and made available on the internet, opening up new possibilities for outreach activities and collaborative sharing. - HTML templates and CSS spreadsheets provide an efficient way to design responsive user interfaces. - The layout and behavior of the user interface can easily be extended by external libraries, such as Bootstrap [@bootstrap]. The NuRadioReco EventBrowser is based on the Dash package [@dash]. While Dash itself is written in Python, it creates an HTML/JavaScript template that is rendered by the web browser and can be extended using custom JavaScript or CSS add-ons. Thanks to this framework, the EventBrowser is responsive and customizable (see Fig. \[fig:eventbrowser\]). For example, any graph can be zoomed into or out of. Additional quantities, such as overview quantities as function of time can be visualized and a mouse click on a point representing a specific event will immediately show the details of the event in the EventBrowser. The EventBrowser can be used locally by starting a webserver via [bash]{} python NuRadioReco/eventbrowser/index.py /path/files with the last argument passed to the python command specifying the location of the data files files to be viewed. The EventBrowser can then be accessed by opening any web browser and going to the address provided in the terminal output, which is http://127.0.0.1:8080/ by default. The EventBrowser lists all .nur files in the specified location in a drop down menu, from which they can be selected for viewing. New reconstruction algorithms for the electric field {#sec:Reco_algo} ==================================================== In this section we present two novel reconstruction algorithms for the electric field, developed for the station layout of radio neutrino detectors. The incident electric field is a central quantity as many other properties such as the signal polarization, the energy fluence and the frequency spectrum are directly calculated from it, which are then used to determine the neutrino or cosmic-ray properties. For example, to reconstruct the arrival direction of a neutrino, one naturally needs the signal arrival direction, but also the frequency slope and polarization to determine on which part of the Cherenkov cone the signal was detected. The polarization breaks the degeneracy around the shower axis and the frequency slope determines the angle to the shower axis. Without information about polarization and frequency slope, the signal could have been detected anywhere on the Cherenkov cone leaving a large uncertainty on the neutrino arrival direction. Also for the energy reconstruction of neutrinos, knowing on which part of the Cherenkov cone the signal was detected is crucial as the Askaryan signal amplitude depends on it. For cosmic-rays, analyses recovering the full electric-field provide the most precise reconstruction of the energy to-date. Hence, an accurate electric-field reconstruction is a crucial parameter for the overall event reconstruction. Here, we will concentrate on the cosmic-ray case. There is no standard-approach to neutrino reconstruction yet and discussing such strategies in detail would go beyond the scope of this paper. However, one can easily envision how the concept of the forward folding (Sec. \[sec:forwardfolding\]) can be applied to neutrino pulses. Cosmic-ray detectors typically are built with dual-polarized antennas, i.e., the radio signal is measured at the same point in space and time in two orthogonal polarizations which allows for a straight-forward reconstruction of the three-dimensional electric field [@RadioOffline; @AERAPolarization; @LofarPolarization2014]. In contrast, radio neutrino detector stations typically consist of multiple spatially separated antennas of different orientations to maximize the effective volume for neutrino detection and to minimize the antenna costs. Cosmic-ray reconstruction in neutrino detectors therefore required the development of a new method to reconstruct the incident three-dimensional electric-field pulse. Standard reconstruction {#sec:standard_reconstruction} ----------------------- For antenna separations within a detector station of less than (such as the dimensions of a current ARIANNA station) it can safely be assumed that all antennas observe the same pulse generated by an air shower. Using the reconstructed signal arrival direction $(\varphi_0,\vartheta_0)$, we correct for the time delays and combine the measurements of all antennas into an over-determined system of equations that is solved for the electric field using a chi-square minimization per frequency bin. Mathematically, this is expressed in the frequency domain as $$\begin{pmatrix} \mathcal{V}_1(f) \\ \mathcal{V}_2(f) \\ ...\\ \mathcal{V}_n(f)\end{pmatrix} = \begin{pmatrix} \mathcal{H}_1^\theta (f)& \mathcal{H}_1^\phi (f)\\ \mathcal{H}_2^\theta (f) & \mathcal{H}_2^\phi (f)\\ ... \\ \mathcal{H}_n^\theta (f)& \mathcal{H}_n^\phi (f)\end{pmatrix} \begin{pmatrix} \mathcal{E}^\theta(f) \\ \mathcal{E}^\phi(f)\end{pmatrix} \, , \label{eq:H_full}$$ where $\mathcal{V}_i$ is the Fourier transform of the measured voltage trace of antenna $i$, $\mathcal{H}_i^{\theta, \phi}$ represents the response of antenna $i$ to the $\phi$ and $\theta$ polarization of the electric field $\mathcal{E}^{\theta, \phi}$ from the direction $(\varphi_0,\vartheta_0)$. This system of equations is then solved for $\mathcal{E}^{\theta, \phi}$. Due to the typical noise contribution on measured waveforms, there is no perfect solution. Hence, we determine the electric field values $\mathcal{E}^{\theta}(f_i)$, $\mathcal{E}^{\phi}(f_i)$ for each frequency bin $f_i$ that minimize the sum of the squared differences of $\mathcal{V}_i(f_i)$. We refer to this technique as the *standard technique* as it is an extension of the method used in dedicated cosmic-ray detectors from two orthogonal channels to many. In comparison to the former, it has the advantages that the signal-to-noise ratio is reduced by adding more antennas to the reconstruction, and, more importantly, it allows for more flexible station designs. For example, instead of building a complicated 3D antenna that measures all three electric-field polarizations at the same point in space, one could simply place a dedicated vertical antenna at a few meters distance from a dual-polarized antenna and combine the signals in software rather than hardware. The standard method has proven to work reliably as long as at least two orthogonal antennas have a good signal-to-noise ratio in all frequency bins. However, there are two shortcomings. The general assumption of this deconvolution method is that all measured voltages originate only from the incident electric field, while in reality it is a sum of the electric-field signal and recorded noise. In a scenario, where an electric-field pulse has no high-frequency content and the upper half of the bandwidth is therefore dominated by noise, this reconstruction method produces incorrect results at high frequencies. A second limitation of this method occurs if the signal is only measured in two of three orthogonal polarization components, which is the case for most radio cosmic-ray detectors that only measure the two horizontal (east-west and north-south) components. Although this is in principle sufficient information to determine the full electric field, using the signal arrival direction, the algorithm leads to incorrect results for horizontal air showers [@GlaserPhD2017]: Here, the $\vec{e}_\theta$ component of the electric field has a strong vertical and only a small horizontal component. If the antennas are only sensitive to the small horizontal component, the signal is often below the noise level. In the reconstruction it is assumed that the measured noise level is identical to the horizontal component of a much larger $\vec{e}_\theta$ component, leading to a vast overestimation of this polarization component. Thus, current analyses only use the horizontal components of the reconstructed electric field [@AERAHorizontal2018], which does not allow for a proper determination of the polarization or the total signal strength. Forward folding technique {#sec:forwardfolding} ------------------------- ![image](figures/forwad_folding_example.png){width="100.00000%"} We developed the *forward folding technique* to reconstruct the electric field from multiple channel measurements and address the shortcomings of the *standard method*. It improves the reconstruction for small signal-to-noise ratios, can be used for horizontal showers and prevents spurious results for cases in which the bandwidth of the signal is smaller than the detector bandwidth. Early versions of this technique were already presented in [@GlaserPhD2017; @GlaserARENACR2018]. Instead of recovering numerically the incident electric field, frequency bin by frequency bin, we fit an analytic model of the electric-field pulse directly to the measured voltages in the time domain. Here, we concentrate on the application of this technique for cosmic-ray signals. An extension to neutrino Askaryan pulses should be straight forward and will be subject to forthcoming studies. In a typical experimental bandwidth, a cosmic-ray radio pulse can be described sufficiently well with just four parameters in the frequency domain: the signal amplitude of both polarization components $A_{\theta, \phi}$, the frequency slope $m_f$ and a phase offset $\Delta$[^3] (see Eq. ). It should be noted that this assumes that the pulse is fully linearly polarized. Should one want to study the small component of circular polarization [@2016PhRvD..94j3010S], an adaptation is needed. For additional discussion refer to [@GlaserPhD2017]. We forward fold the such parameterized pulse with the antenna responses of the different channels by multiplying $\mathcal{E}^{\theta, \phi}(A_{\theta, \phi}, m_f, \Delta)$ with the antenna response according to Eq. . The resulting voltage traces are compared to the measurement. The optimal parameters of the electric-field pulse $(A_{\theta, \phi}, m_f, \Delta)$ are determined in a chi-square minimization in the time-domain of all channels simultaneously. By applying the antenna response in forward direction to a noiseless waveform, the effects of noise are minimized and an artificial overestimation of the signal is reduced. The fundamental principle of comparing prediction to measurement in the instrumental voltages and not in the physical quantity, the electric field, has already been used in the LOFAR analysis [@LOFARPRD]. Their approach of using many dedicated CoREAS simulations per event, is however significantly more computationally expensive and only used to reconstruct high quality detections. One example of an electric-field reconstruction presented in Fig. \[fig:deconvolution\] illustrates the two main advantages of the forward folding technique: A small amplitude in one of the polarization components is correctly identified (the $\theta$- component in this example), and high frequency components, where the signal amplitude is smaller and the antenna has a reduced sensitivity, are not overestimated. The recovered electric field is less biased by noise. Implementation details ---------------------- The electric field pulse of a cosmic ray is described in the frequency domain as $$\begin{pmatrix} \mathcal{E}_\theta \\ \mathcal{E}_\phi \end{pmatrix} = \begin{pmatrix} A_\theta \\ A_\phi \end{pmatrix} 10^{f \cdot m_f} \, \exp(\Delta\, j) \, \label{eq:pulse}$$ where $f$ is the frequency and $j$ stands for the imaginary unit. The four parameters that describe the electric-field pulse are: the amplitudes of the $\theta$ and $\phi$ components $A_{\theta, \phi}$, the frequency slope $m_f$ and the phase offset $\Delta$. We do not consider a (linear) phase slope because it corresponds to a shift of the pulse position in the time domain and we assume that time differences have been removed by correcting for the reconstructed arrival direction. We found that the following incremental fitting procedure leads to stable results: First only the frequency slope $m_f$ is determined. We use the sum of the maximum cross correlation of all participating channels as objective function. $$\sum_i -\max\left(\rho\left[V_i, \mathrm{IFFT}\left(\begin{pmatrix} \mathcal{H}^\theta_i & \mathcal{H}^\phi_i \end{pmatrix} \begin{pmatrix} \mathcal{E}_\theta \\ \mathcal{E}_\phi \end{pmatrix}\right)\right]\right) \, ,$$ where $V_i$ is the measured voltage trace of channel $i$, $\mathcal{H}^{\theta, \phi}_i$ is the antenna vector effective length, $\mathrm{IFFT}$ represents an inverse fast Fourier transform and $\rho$ is the normalized Pearson correlation defined as $$\rho(x, y)_k = \frac{\sum_n x_{n+k} y_n}{\sqrt{\sum_k x_k^2} \sqrt{\sum_k y_k^2}}$$ for each time bin $k$. This objective function conveniently removes the dependence on the amplitude and determines the time shift of the analytic pulse with respect to the measured voltages which is given by the value $k$ that maximizes $\rho(x, y)$. In this first optimization step, we set $\Delta = 0$, $A_\phi = 1$ and $A_\theta$ = 0, because cosmic-ray signals are mostly polarized in $\vec{e}_\phi$-direction due to the orientation of the geomagnetic field in polar regions. In the next iteration, the amplitude is determined by minimizing the following objective function: $$\chi^2 = \sum_i \left(\sum_k \frac{\left|V_{i,k} - \mathrm{IFFT}\left(\begin{pmatrix} \mathcal{H}^\theta_i & \mathcal{H}^\phi_i \end{pmatrix} \begin{pmatrix} \mathcal{E}_\theta \\ \mathcal{E}_\phi\end{pmatrix} \right)_k\right|}{V_\mathrm{RMS}}\right)^2 \, , \label{eq:obj2}$$ where the index $i$ runs over the channels, $k$ runs over the time bins of each channel and $V_\mathrm{RMS}$ is the RMS of the measured voltage traces. In a first step, only $A_\phi$ is determined, $A_\theta$ and $\Delta$ are set to zero, and $m_f$ is fixed to the previous fit result. In the second step, both $A_\phi$ and $A_\theta$ are optimized simultaneously. In the final step, the amplitudes and the slope parameter are optimized simultaneously using the objective function of Eq.  but using the Hilbert envelope of the voltage traces instead of the voltage traces directly. ![image](figures/polarization_reconstruction.png){width="80.00000%"} Performance of electric field reconstruction -------------------------------------------- ![image](figures/energy_fluence_hist.png){width="70.00000%"} ![image](figures/slopes_hist.png){width="70.00000%"} We evaluate the performance of the electric-field reconstruction using the standard and the forward folding technique in a Monte Carlo study with CoREAS air shower pulses. We calculate the polarization, the energy fluence and the frequency slope from the reconstructed electric field and compare it to the Monte Carlo truth. In case of the forward folding technique, the polarization is given by $\arctan(A_\theta/A_\phi)$, defining it as the angle between $\vec{e}_\phi$ and the electric field vector. The frequency slope is given by the parameter $m_f$ (see Eq. \[eq:pulse\]). We perform the Monte Carlo study for a detector layout consisting of four upward facing LPDA antennas where one pair is oriented along the North-South direction and the other pair is oriented along the East-West direction. All antennas are placed at a distance of to the center of the station. This corresponds to the station layout of a dedicated cosmic-ray station of the ARIANNA detector. We note that the method works with any kind of antenna but we focus on LPDAs here because the cosmic-ray signal is mostly horizontally polarized and vertically aligned dipoles typically do not add signal. We present the resolution for the reconstructed polarization for a set of 100 simulated air showers in Fig. \[fig:polarization\_reconstruction\]. The data set is generated to resemble a realistic cosmic-ray distribution. We also use the module *readCoREAS*, which randomly picks shower core positions. For each simulated shower 150 randomly chosen shower core positions are used. The energy distribution for the events follows a power-law with spectral index –2 and the arrival directions are isotropically distributed. We apply a full detector simulation and event reconstruction as detailed in the example of Sec. \[sec:example\_rec\], including the simulation of noise with an RMS amplitude of . We consider only events that exceed a trigger threshold of at least in two of the channels. An additional cut is made on the signal-to-noise ratio[^4] for all channels, which needs to be higher than 4. This cut is needed because the current arrival direction reconstruction fails when there is no detectable signal in one of the two channel-pairs. It selects 423 pulses out of 603 pulses obtained by the event selection. The event set selected by this cut still contains 95% of the events for which the directions are considered to be well reconstructed, meaning a direction reconstruction within 5 degrees of the simulated direction. For the electric-field reconstruction, a frequency range of 80 - 500 MHz is used. The obtained resolution of the polarization, energy fluence and frequency slope are presented in Figs. \[fig:polarization\_reconstruction\] and \[fig:frequency\_slope\_hist\]. For the polarization reconstruction using the standard method, we obtain a median resolution of $(6.05^{+5.28}_{-6.02})^{\circ}$. For the forward folding method, we obtain a median of $(0.93^{ +0.84}_{-3.66})^{\circ}$. The uncertainties represent the 68% quantiles. The better performance for the forward folding method is especially visible for the low signal-to-noise ratio events, which are indicated by the orange histogram in the figure (SNR &lt;10). Overall, the method indeed addresses the shortcomings observed earlier. In the reconstruction of the energy fluence of the radio signal, standard and forward folding methods perform equally well, as shown in Fig. \[fig:frequency\_slope\_hist\], which shows the relative accuracy of the methods. With the standard methods, we obtain a median relative uncertainty of $0.03^{+0.10}_{-0.04 }$ and with the forward folding a median of $0.04^{+0.07}_{-0.04}$. The reason for a similar performance is that the average noise contribution is subtracted from the energy fluence that is calculated as integrated quantity around the signal pulse, making the energy fluence a robust estimator in the presence of noise [@AERAPRD]. The quality of the reconstruction of the frequency slope parameter $m_f$ is shown in Fig. \[fig:frequency\_slope\_hist\]. For the simulated electric field and the one reconstructed using the standard method, $m_f$ was determined by a linear fit to $\log_{10}(|\vec{E}(f)|)$, while for the forward folding method the parameter was a result of the fitting process. The forward folding method shows a solid performance, resulting in a relative uncertainty with a median of $0.04^{+0.13}_{-0.26}$, while the standard method performs much worse. Especially for signals with a low SNR, $m_f$ is overestimated. As described in Sec. \[sec:standard\_reconstruction\], noise can cause the electric field strength at higher frequencies to be overestimated, leading to a value of $m_f>0$, while the actual $m_f$ is almost always negative in this frequency range. For this reason, the resulting distribution has a median of $0.63^{+0.54}_{-0.50}$, where the uncertainties represent the 68% quantiles. To summarize, in all three quantities tested, polarization, energy fluence, and frequency slope, the forward folding performs on-par or better than the standard method. Especially for reconstructions sensitive to the polarization and the frequency slope, we recommending using the forward folding method to recover the electric field and to overcome biases and reduce the contribution of noise. Conclusions =========== We have presented a new Python-based reconstruction framework for particle radio detectors. The framework has been designed to analyze data from current radio neutrino detectors, such as ARIANNA, and to prepare the reconstruction for a planned large radio array. Due to the design-goal of high flexibility it is also usable for cosmic-ray radio detectors. The framework provides both a native data structure and a time-dependent detector description, which is designed to account for large and complex detectors. Data visualization relies on web-based tools, allowing for the easy separation of a server-based data analysis and remote inspection. In its current version, the framework provides the algorithms for all steps necessary to reconstruct the full electric field for incoming cosmic-ray signals and basic event identification for neutrino detectors. A well-documented example is provided in the code. Due to the strict modularity it is straight-forward to design additional modules to complete the reconstruction for neutrinos. Also, previously unpublished reconstruction algorithms are made available to the community in the code and have been described in this article. As this framework builds on experience gained with all currently used software for the radio detection of cosmic rays and neutrinos, we were able to anticipate a number of complexities and avoid them in software design. We expect the library of standard algorithms available in NuRadioReco to grow in time along with their development in the neutrino community. Acknowledgements ================ We would like to thank the members of the InIceMC working group, consisting of members from the ARA and ARIANNA collaborations, for helpful discussions regarding simulations of the radio signal of neutrinos that helped shape our reconstruction framework. In particular, we would like to thank Simon Archambault and Keiichi Mase from Chiba University for discussions concerning the handling of antenna responses in different simulation tools. We acknowledge funding from the German research foundation (DFG) under grants GL 914/1-1 and NE 2031/2-1, and the U.S. National Science Foundation-Physics Division (grant NSF-1607719). Extracting the realized effective length from antenna simulations {#sec:Antenna_effective} ================================================================= For completeness we provide the equations used to calculate antenna quanitities for the detector simulation. This hopefully allows the reader to compare different antenna simulations programs such as WIPL-D and XFDTD. The vector effective length --------------------------- The vector effective length $\vec{\mathcal{H}}$ relates the incident electric field $\vec{\mathcal{E}}$ to the open circuit voltage at the antenna terminals $\mathcal{V}_\mathrm{OC}$ (in the Fourier domain) as $$\mathcal{V}_\mathrm{OC} = \vec{\mathcal{H}}\cdot \vec{\mathcal{E}} = (\mathcal{H}_\theta, \mathcal{H}_\phi) \cdot (\mathcal{E}_\theta, \mathcal{E}_\phi)^T$$ where $\vec{\mathcal{H}}$ and $\vec{\mathcal{E}}$ are vectors in spherical coordinates, having $\mathcal{H}_r = \mathcal{E}_r = 0$, thus only having signal in the $\theta$ and $\phi$ polarization. In a measurement setup the antenna will be read out at a load impedance. For a simple measurement setup we get $$\mathcal{V}_L = \frac{Z_L}{Z_A + Z_L} \mathcal{V}_\mathrm{OC} \,,$$ where $Z_L$ is the load impedance which is typically and $Z_L$ is the antenna impedance. The realized VEL is then given by $$\vec{\mathcal{H}}_{rl} = \frac{Z_L}{Z_A + Z_L} \vec{\mathcal{H}} \, . \label{eq:Hr2}$$ and we can relate the incident electric field to the measured voltage by $$\mathcal{V}_\mathrm{L} = \vec{\mathcal{H}}_{rl}\, \vec{\mathcal{E}} \, .$$ Simulation of the effective height ---------------------------------- Most antenna simulation software computes the far field electric field generated by an antenna, i.e., they simulate an emitting antenna and due to reciprocity the receiving antenna case can also be calculated from such a simulation. The vector electric field $\vec{\mathcal{E}}(\omega)$ emitted by an antenna is related to the vector effective length (VEL) as (see [@AntennaPaper] Eq. 5.1 or [@Kravchenko2007] Eq. 6) $$\vec{\mathcal{E}}(\omega) = -i Z_0 \frac{1}{2 \lambda R} \mathcal{I}_0 \vec{\mathcal{H}} \exp(-i \omega R/c) \, . \label{eq:1}$$ where $Z_0$ is the free space impedance, $\lambda$ is the wavelength, $R$ is the distance to the antenna and $\mathcal{I}_0$ is the current at the feedpoint of the antenna. This equation can be simplified by normalizing the electric field to a unit distance at , which removes the distance dependence from the equation. The normalized electric field (also called complex voltage) is given by $$\vec{\mathcal{E}}'(\omega)=\vec{\mathcal{E}}(\omega) \, R \, \exp(i \omega R/c) \, . \label{eq:I}$$ Then, solving for the effective length gives: $$\vec{\mathcal{H}} = \frac{2 \lambda \mathcal{I}_0}{-i Z_0} \vec{\mathcal{E}}'(\omega) \, . \label{eq:H3}$$ In the case of WIPL-D, not a current $\mathcal{I}_0$ is simulated at the antenna feedpoint but a perfect voltage generator of $V_{OC}$ = 1 Volt. Then, $$\mathcal{I}_0 = V_{OC} / Z_A$$ and Eq. (\[eq:H3\]) becomes $$\vec{\mathcal{H}} = \frac{2 \lambda Z(\omega)}{-i Z_0 V_{OC}} \vec{\mathcal{E}}'(\omega) \, . \label{eq:H}$$ Then, using Eq. (\[eq:Hr2\]) and exploiting the identity $Z_A = \frac{1 + S11}{1-S11} Z_L$ we find the following simplified formula for the realized effective length $$\vec{\mathcal{H}_{rl}} = \frac{\lambda (1 + S11) Z_L}{-i Z_0 V_{OC}} \vec{\mathcal{E}}'(\omega) \, . \label{eq:Hr}$$ Relation between realized vector effective length and realized gain ------------------------------------------------------------------- Gain and realized gain are related via (from Eq. 6.8 of [@IEEE_ht]) $$G_{rl}(\omega) = G(\omega)\left[1-|S11(\omega)|^2\right]\, . \label{eq:GGr}$$ The vector effective length (not $\vec{\mathcal{H}_{rl}}$ but $\vec{\mathcal{H}}$) is related to the gain via (from Eq. (A.8) from [@AntennaPaper]) $$|\vec{\mathcal{H}}|^2 = \frac{c_0^2}{f^2\, n} \frac{\Re(Z_A)}{\pi Z_0} G(\omega) \, . \label{eq:HG}$$ Finally, the realized vector effective length is related to the realized gain via $$|\vec{\mathcal{H}}_{rl}|^2 = \frac{c_0^2}{f^2\, 4 \, n} \frac{\SI{50}{\Omega}}{\pi Z_0} G_{rl}(\omega)\, . \label{eq:HrGr}$$ Impact of embedding an antenna in a medium ------------------------------------------ We characterize a medium (e.g. ice) by its relative permittivity ($\epsilon_r$), its relative permeability ($\mu_r$) and its conductivity ($\sigma$). In the WIPL-D simulation we set the conductivity to $\sigma$ = and assume that $\mu_r = 1$. Then, $$\epsilon_r = n^2 \, ,$$ where $n$ is the index of refraction. Also the wave impedance changes accordingly $$Z = \sqrt{\frac{\mu}{\epsilon}} = \sqrt{\frac{\mu_0 \mu_r}{\epsilon_0 \epsilon_r}} \stackrel{\mu_r = 1}{=} Z_0 \sqrt{\frac{1}{\epsilon_r}} = \frac{Z_0}{n} \, \label{eq:Z}$$ where $Z_0$ is the free space impedance. Thus, the formula to calculate the vector effective height from the simulation output (Eq. \[eq:H\]) does not change: The wavelength changes with $n$ by $\lambda = \lambda_0 / n$ but at the same time Eq. \[eq:Z\] adds another factor of $n$ that cancels the first. In Fig. \[fig:bicone\_media\], the effective height of a bicone antenna in different media is presented. The complete curve shifts to lower frequencies according to $f = f_{air} / n$ whereas the magnitude of the effective height remains essentially the same. This is in accordance with our intuition that the antenna should behave essentially the same and only the resonance frequency should change according to the change in wavelength. ![The (open circuit) vector effective length of a WIPL-D simulation of the ’birdcage’ bicone antenna [@ARA] in different media (air and ice with different refractive indices (n)). []{data-label="fig:bicone_media"}](figures/bicone_media.png){width="47.00000%"} Useful identities ----------------- The S11 parameter (measured in a system) is related to the antenna impedance by $$Z = \frac{1 + S11}{1-S11} \SI{50}{\Omega} \iff S11 = \frac{Z - \SI{50}{\Omega}}{Z + \SI{50}{\Omega}} \label{eq:S11Z}$$ The voltage standing wave ratio (VSWR) is related to S11 by $$\text{VSWR} = \frac{1 + |S11|}{1-|S11|}$$ [^1]: This is technically only true at low frequencies, but an elaborate discussion of the emission at high frequencies would go beyond the scope of this paper. [^2]: CoREAS simulations show a mixture of numerical noise and incoherent signal at high frequencies that can mimic signal in the time-domain [@LOFARLDF]. Events with coherent signal in the relevant band are selected. [^3]: As all signals in the time domain are real-valued, only the positive frequencies are considered. The amplitudes of the negative frequencies are just the complex conjugates of the amplitudes of the positive frequencies. Hence, the phase offset is $-\Delta$ for negative frequencies. [^4]: SNR is defined as half of the peak-to-peak amplitude divided by the noise RMS
[**An evidence of mass dependent differential kinetic freeze-out scenario observed in Pb-Pb collisions at 2.76 TeV**]{} 1.0cm Hai-Ling Lao$^{a}$, Hua-Rong Wei$^{a}$, Fu-Hu Liu$^{a,}$[[^1]]{}, and Roy A. Lacey$^{b,}$[[^2]]{} *$^a$Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China* $^b$Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA 1.0cm [**Abstract:**]{} Transverse momentum spectra of different particles produced in mid-rapidity interval in lead-lead (Pb-Pb) collisions with different centrality intervals, measured by the ALICE Collaboration at center-of-mass energy per nucleon pair $\sqrt{s_{NN}}=2.76$ TeV, are conformably and approximately described by the Tsallis distribution. The dependences of parameters (effective temperature, entropy index, and normalization factor) on event centrality and particle rest mass are obtained. The source temperature at the kinetic freeze-out is obtained to be the intercept in the linear relation between effective temperature and particle rest mass, while the particle (transverse) flow velocity in the source rest frame is extracted to be the slope in the linear relation between mean (transverse) momentum and mean moving mass. It is shown that the source temperature increases with increase of particle rest mass, which exhibits an evidence of mass dependent differential kinetic freeze-out scenario or multiple kinetic freeze-out scenario.\ [**Keywords:**]{} Source temperature, kinetic freeze-out scenario, mass dependent differential kinetic freeze-out scenario\ [**PACS:**]{} 12.38.Mh, 25.75.Dw, 24.10.Pa 1.0cm Introduction ============ High energy nucleus-nucleus (heavy ion) collisions at the large hadron collider (LHC) \[1–5\] have been providing another excellent environment and condition of high temperature and density, where the new state of matter, namely the quark-gluon plasma (QGP) \[6–8\], is expected to form and to live for a longer lifetime than that at the relativistic heavy ion collider (RHIC) \[9\]. Although the RHIC is scheduled to run at lower energies which are around the critical energy of phase transition from hadronic matter to QGP, the LHC is expected to run at higher energies. Presently, the LHC has provided three different types of collisions: proton-proton ($pp$), proton-lead ($p$-Pb), and lead-lead (Pb-Pb) collisions at different collision energies. The former two are not expected to form the QGP due to small system, though the deconfinement of quarks and gluons may appear. The latter one is expected to form the QGP due to large system and high energy. It is believed that the QGP is formed in Pb-Pb collisions at the LHC and in nucleus-nucleus collisions at lower energy till dozens of GeV at the RHIC \[10, 11\]. If mesons are produced in the participant region where violent collision had happened and the QGP is formed, nuclear fragments such as helium or heavier nuclei are expected to emit in spectator region where non-violent evaporation and fragmentation had happened \[12–14\]. The ALICE Collaboration \[15–18\] measured together positive pions $\pi^+$, positive kaons $K^+$, protons $p$, deuterons $d$, and one of helium isotopes $^3$He in Pb-Pb collisions with different centrality intervals at the LHC. It gives us a chance to describe uniformly different particles. In particular, we are interested in the uniform description of transverse momentum spectra of $\pi^+$, $K^+$, $p$, $d$, and $^3$He, so that we can extract the kinetic freeze-out (KFO) temperature of interacting system (i.e. source temperature at KFO). From source temperature at KFO, we can draw a KFO scenario. There are three different KFO scenarios discussed in literature \[3, 19–22\]. The single KFO scenario \[19\] uses one set of parameters for both the spectra of strange and non-strange particles. The double KFO scenario \[3, 20\] uses a set of parameters for the spectra of strange particles, and another set of parameters for the spectra of non-strange particles. The multi-KFO scenario \[21, 22\] uses different sets of parameters for different particles with different masses. Naturally, the mass dependent differential KFO scenario \[22\] belongs to the multi-KFO scenario. It is an open question which KFO scenario describes correctly. We are interested in the study of KFO scenario in the present work. As can be seen from the following sections, our analysis provides an evidence of mass dependent differential KFO scenario. To extract source temperature at the KFO, we have to describe transverse momentum spectra. More than ten functions are used in the descriptions of transverse momentum spectra. In the present work, we select the Tsallis distribution \[23–25\] that covers the sum of two or three standard distributions \[26, 27\] and describes temperature fluctuations among different local equilibrium states. Based on the descriptions of the experimental data of the ALICE Collaboration \[15, 16\] on Pb-Pb collisions at center-of-mass energy per nucleon pair $\sqrt{s_{NN}}=2.76$ TeV, the source temperature at the KFO is obtained to be the intercept in the linear relation between effective temperature and particle rest mass, while the particle (transverse) flow velocity in the source rest frame is extracted to be the slope in the linear relation between mean (transverse) momentum and mean moving mass. If we use other functions, the method is in fact the same. Because of no difference between positive and negative spectra being reported \[16\], we are just fitting the available positive data in the analysis. The structure of the present work is as followings. The model and method are shortly described in section 2. Results and discussion are given in section 3. In section 4, we summarize our main observations and conclusions.\ The model and method ==================== We discuss the collision process in the framework of the multisource thermal model \[28–30\]. According to the model, many emission sources are formed in high energy nucleus-nucleus collisions. We can choose different distributions to describe the emission sources and particle spectra. These distributions include, but are not limited to, the Tsallis distribution \[23–25\], the standard (Boltzmann, Fermi-Dirac, and Bose-Einstein) distributions \[26\], the Tsallis + standard distributions \[31–36\], the Erlang distribution \[28\], and so forth. The Tsallis distribution can be described by two or three standard distributions. The Tsallis + standard distributions can be described by two or three Tsallis distributions \[27\]. It is needless to choose the standard distributions due to multiple sources (temperatures). It is also needless to choose the Tsallis + standard distributions due to not too many sources (temperatures). A middle way is to choose the Tsallis distribution which describes the temperature fluctuation in a few sources to give an average value. These sources with different excitation degrees can be naturally described by the standard distributions with different effective temperatures, which result from the multisource thermal model \[28–30\]. The Tsallis distribution has more than one function forms \[23–25, 31–38\]. In the rest frame of a considered source, we choose a simplified form of the joint probability density function of transverse momentum ($p_T$) and rapidity ($y$), $$f(p_T,y) \propto \frac{d^2N}{dydp_T}=\frac{gV}{(2\pi)^2} p_T \sqrt{p^2_T+m^2_0}\cosh y \bigg[ 1+\frac{q-1}{T} \Big( \sqrt{p^2_T+m^2_0}\cosh y-\mu \Big) \bigg]^{-q/(q-1)},$$ where $N$ is the particle number, $g$ is the degeneracy factor, $V$ is the volume of emission sources, $T$ is the temperature which describes averagely a few sources (local equilibrium states), $q$ is the entropy index which describes the degree of non-equilibrium among different states, $\mu$ is the chemical potential which is related to $\sqrt{s_{NN}}$ \[39\] and can be regarded as 0 at the LHC, $m_0$ is the rest mass of the considered particle. Generally, the 4-parameter ($T$, $q$, $\mu$, and $V$) form of Eq. (1) is capable of reproducing the particle spectra, where $T$, $q$, and $\mu$ are fitted independently for the considered particle species, and $V$ is related to other parameters. Eq. (1) results in the transverse momentum probability density function which is an alternative representation of the Tsallis distribution as follows $$f_{p_T}(p_T)=\frac{1}{N} \frac{dN}{dp_T} = \int^{y_{\max}}_{y_{\min}} f(p_T,y) dy,$$ where $y_{\max}$ and $y_{\min}$ denote the maximum and minimum rapidities, respectively. Similarly, Eq. (1) results in the rapidity probability density function in the source rest frame as follows $$f_y(y)=\frac{1}{N} \frac{dN}{dy} = \int^{p_{T\max}}_0 f(p_T,y) dp_T,$$ where $p_{T\max}$ denotes the maximum transverse momentum. Under the assumption of isotropic emission in the source rest frame, we have the polar angle probability density function to be $$f_{\theta}(\theta)=\frac{1}{2} \sin \theta.$$ Let $r_1$ and $r_2$ denote the random numbers distributed uniformly in \[0,1\] respectively. We can use the Monte Carlo method to obtain a series of $p_T$ which satisfies $$\int_0^{p_T} f_{p_T}(p_T) dp_T <r_1< \int_0^{p_T+dp_T} f_{p_T}(p_T) dp_T.$$ The Monte Carlo method results in $$\theta=2\arcsin \sqrt{r_2}$$ due to Eq. (4). Thus, we can obtain a series of values of momentum and energy due to the momentum $p=p_T/\sin \theta$ and the energy $E=\sqrt{p^2+m_0^2}$. The energy $E$ is in fact equal to the moving mass $m$ in the natural unit system. Then, we have the mean moving mass $\overline{m}$ to be the mean energy $\overline{E}$.\ Results and discussion ====================== -1.0cm ![image](fig1.eps){width="12.0cm"} .0cm Figure 1. Transverse momentum spectra of $\pi^+$, $K^+$, $p$, $d$, and $^3$He produced in mid-rapidity interval ($|y|<0.5$) in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. The symbols represent the experimental data of the ALICE Collaboration \[16\] in centrality interval 0–20%, which are scaled by different amounts marked in the panel. The curves are our results fitted by using the Tsallis distribution based on Eq. (1). In the fitting, the method of least squares is used to obtain the values of related parameters. Figure 1 presents the transverse momentum spectra, $(1/N_{EV}) d^2N/(2\pi p_T dy dp_T)$, of $\pi^+$, $K^+$, $p$, $d$, and $^3$He produced in mid-rapidity interval ($|y|<0.5$) in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV, where $N_{EV}$ denotes the number of events. The symbols represent the experimental data of the ALICE Collaboration \[16\] in centrality interval 0–20%, which are scaled by different amounts marked in the panel. The curves are our results fitted by using the Tsallis distribution based on Eq. (1) at mid-rapidity ($y=0$). In the fitting, the method of least squares is used. The values of related parameters $T$, $q$, and $N_0$ are listed in Table 1 with values of $\chi^2$ per degree of freedom ($\chi^2$/dof), where $$N_0=\frac{gV}{(2\pi)^3} \int_{0}^{p_{T\max}} \int_{y_{\min}}^{y_{\max}} \sqrt{p^2_T+m^2_0}\cosh y \bigg[ 1+\frac{q-1}{T} \Big( \sqrt{p^2_T+m^2_0}\cosh y-\mu \Big) \bigg]^{-q/(q-1)} dy dp_T$$ is the normalization factor which is used to compare the normalized curve with experimental data. One can see that the Tsallis distribution describes conformably and approximately $\pi^+$, $K^+$, $p$, $d$, and $^3$He spectra. The effective temperature increases with increase of particle rest mass.\ \ [ ]{} 1.0cm -1.0cm ![image](fig2.eps){width="12.0cm"} .0cm Figure 2. The same as for Figure 1, but showing the results of $d$ in Pb-Pb collisions with different centrality intervals and in $pp$ collisions. -1.0cm ![image](fig3.eps){width="12.0cm"} .0cm Figure 3. The same as for Figure 1, but showing the results of $^3$He in Pb-Pb collisions with two centrality intervals. Figure 2 is similar to Figure 1, but it shows the results for $d$ in different centrality intervals, which are scaled by different amounts marked in the panels. At the same time, the result in $pp$ collisions at $\sqrt{s}=7$ TeV is presented for comparison, where $\sqrt{s}$ is a simplified form of $\sqrt{s_{NN}}$ for $pp$ collisions. Figure 3 is similar to Figure 1, but it shows the results for $^3$He in centrality intervals 0–20% and 20–80%, where the data set for 0–20% is the same as that for $^3$He in Figure 1 \[16\]. The related parameter values are listed in Table 1 with values of $\chi^2$/dof. One can see that the Tsallis distribution describes approximately the experimental data of $d$ produced in Pb-Pb collisions with different centrality intervals at the LHC. The effective temperature extracted from $d$ spectra decreases with decrease of centrality (or increase of centrality percentage). To study the change trends of parameters with centrality interval ($C$) of event and rest mass of particle, Figure 4 gives the dependences of (a) $T$ on $C$ for $d$ in events with different centrality intervals and (b) $T$ on $m_0$ for particles in events with centrality interval 0–20%, where only the result for $d$ in Figure 4(a) is available due to the experimental result \[16\]. The symbols represent the parameter values extracted from Figures 1 and 2 and listed in Table 1, and the curves are our results fitted by the method of least squares. The curve in Figure 4(a) is described by $$T=-(0.000059\pm0.000005)C^2-(0.0012\pm0.0002)C+(0.668\pm0.005)$$ with $\chi^2$/dof=0.927, where $T$ is in the units of GeV. The solid, dotted, and dashed curves in Figure 4(b) are linear fittings for i) $\pi^+$, $K^+$, and $p$; ii) $\pi^+$, $K^+$, $p$, and $d$; and iii) $\pi^+$, $K^+$, $p$, $d$, and $^3$He, which are described by $$T=(0.091\pm0.009)+(0.345\pm0.014)m_0,$$ $$T=(0.115\pm0.017)+(0.291\pm0.016)m_0,$$ and $$T=(0.148\pm0.032)+(0.241\pm0.020)m_0,$$ with $\chi^2$/dof=0.429, 2.091, and 5.869, respectively, where $m_0$ is in the units of GeV/$c^2$. The intercept in Eq. (9) is regarded as the KFO temperatures \[40–43\] of emission source, which is 0.091 GeV corresponding to massless particles, when the source produces $\pi^+$, $K^+$, and $p$. Including $d$ causes a large intercept (0.115 GeV) in Eq. (10), while including $d$ and $^3$He causes a larger intercept (0.148 GeV) in Eq. (11). Although the errors in intercepts are large, these results render that the KFO temperature increases with increase of particle rest mass. This is an evidence of mass dependent differential KFO scenario or multiple KFO scenario \[21, 22\]. The blast-wave model \[44\] gives the KFO temperature extracted from $d$ spectra being 0.077–0.124 GeV and from $^3$He spectra being 0.101 GeV \[16\] which are comparable with the present work. In particular, the blast-wave model gives the KFO temperature in central collisions being less than that in peripheral collisions \[16\], which is inconsistent with Figure 4(a) which shows an opposite result on correlation between effective temperature and centrality. Although the result of blast-wave model can be explained as that the interacting system in central collisions undergoes a longer kinetic evolution which results in a lower KFO temperature comparing with peripheral collisions, the present result can be explained as that the interacting system in central collisions stays in a higher excitation state comparing with peripheral collisions. On the other hand, we have used an alternative method to extract indirectly the KFO temperature based on the linear relation between effective temperature and rest mass \[40–43\]. The evidence coming from similar analyses in RHIC and LHC experiments \[45, 46\], where the fit parameters have been studied also against centrality, even down to data of $d$-nucleus or $pp$ collisions, confirms that the KFO temperature in central collisions is less than that in peripheral collisions, which is inconsistent with the present work. For central and peripheral collisions, the relative size of KFO temperature obtained in the present work is similar to those of the chemical freeze-out temperature and effective temperature. We would like to point out that the present work is qualitatively consistent with ref. \[19\], where the Tsallis + blast-wave model is used at RHIC energy. Although the interpretation of the Tsallis distribution is still controversial, at least in the field of concern here, it could be interesting to learn the behavior of non-additive entropy. In Figure 5, the dependences of (a) $q$ on $C$ for $d$ in events with different centrality intervals and (b) $q$ on $m_0$ for different particles in events with centrality interval 0–20% are given, where only the result for $d$ in Figure 5(a) is available due to the experimental result \[16\]. The symbols represent the parameter values extracted from Figures 1 and 2 and listed in Table 1. For $d$ in events with different centrality intervals, the values of $q$ are foregone to be consistent with one \[Figure 5(a)\]. For the events with centrality interval 0–20%, $\pi^+$ corresponds to a larger $q$ than others \[Figure 5(b)\]. This renders that the production of pions is more polygenetic than others. Because of the most values of $q$ being small, the interacting system stays approximately in an equilibrium state. In fact, the ranges of most $p_T$ spectra considered in Pb-Pb collisions in the present work are narrow, which result mainly from the soft process which is a single source and can be described by the standard distribution. If we study wide $p_T$ spectra, both the soft and hard processes have to be considered. We need two or three standard distributions, the standard distribution + a power law, or the Tsallis distribution with large $q$ to describe the wide spectra. The situation for $pp$ collisions is similar to Pb-Pb collisions. The advantage of Tsallis distribution will appear in description of the wide spectra. For the narrow spectra, both the standard distribution and the Tsallis distribution with small $q$ are satisfied. However, we use the Tsallis distribution due to its potential application in wide $p_T$ spectra. In addition, the STAR experiment already tried a Tsallis-like study, publishing also a Tsallis + blast-wave model-based interpretation of their data \[9\]. The ALICE data, on the other hand, have been compared to a blast-wave + thermal-based fit \[16\]. In both cases the agreement is very good. These facts render that the Tsallis distribution has a wide application in high energy physics. In Figure 6, the dependences of (a) $N_0$ on $C$ for $d$ in events with different centrality intervals and (b) $N_0$ on $m_0$ for different particles in events with centrality interval 0–20% are given, where only the result for $d$ in Figure 6(a) is available due to the experimental result \[16\]. The symbols represent the parameter values extracted from Figures 1 and 2 and listed in Table 1. The curves are our results fitted by the method of least squares, which are described by $$N_0=(0.0166\pm0.0004)\exp[-(0.019\pm0.001)C]-(0.0033\pm0.0002)$$ and $$N_0=(580.207\pm62.425)\exp[-(5.912\pm0.118)m_0]$$ with $\chi^2$/dof=0.572 and 1.712 respectively. It is shown that $N_0$ decreases with decrease of centrality. The larger the particle rest mass is, the lower the production probability is. Although $N_0$ is only a normalization factor and the data are not cross-section, they are proportional to the volumes of sources producing different particles. Therefore, studying $N_0$ dependence is significative. -1.0cm ![image](fig4.eps){width="15.0cm"} .0cm Figure 4. Dependences of (a) $T$ on $C$ for $d$ in events with different centrality intervals and (b) $T$ on $m_0$ for particles in events with centrality interval 0–20%. The symbols represent the parameter values listed in Table 1. The curves and lines are our results fitted by the method of least squares. -1.0cm ![image](fig5.eps){width="15.0cm"} .0cm Figure 5. Dependences of (a) $q$ on $C$ for $d$ in events with different centrality intervals and (b) $q$ on $m_0$ for particles in events with centrality interval 0–20%. The symbols represent the parameter values listed in Table 1. -1.0cm ![image](fig6.eps){width="15.0cm"} .0cm Figure 6. Dependences of (a) $N_0$ on $C$ for $d$ in events with different centrality intervals and (b) $N_0$ on $m_0$ for particles in events with centrality interval 0–20%. The symbols represent the parameter values listed in Table 1. The curve and line are our results fitted by the method of least squares. -1.0cm ![image](fig7.eps){width="15.0cm"} .0cm Figure 7. Dependences of (a) $\langle p_T \rangle$ on $\overline{m}$ and (b) $\langle p \rangle$ on $\overline{m}$ for particles in events with centrality interval 0–20%. The symbols represent the values of $\langle p_T \rangle$ and $\langle p \rangle$ at different $\overline{m}$, which are calculated by using the Monte Carlo method in the source rest frame. The lines are our results fitted by the method of least squares. To extract the transverse flow velocity, we display the dependence of mean transverse momentum ($\langle p_T \rangle$) on mean moving mass ($\overline{m}$) in Figure 7(a). The symbols represent the values of $\langle p_T \rangle$ and $\overline{m}$ for different particles calculated by using the Monte Carlo method in the source rest frame. The solid, dotted, and dashed curves in Figure 7(a) are linear fittings for i) $\pi^+$, $K^+$, and $p$; ii) $\pi^+$, $K^+$, $p$, and $d$; and iii) $\pi^+$, $K^+$, $p$, $d$, and $^3$He, which are described by $$\langle p_T \rangle=(0.108\pm0.006)+(0.502\pm0.004)\overline{m},$$ $$\langle p_T \rangle=(0.124\pm0.008)+(0.490\pm0.003)\overline{m},$$ and $$\langle p_T \rangle=(0.162\pm0.029)+(0.468\pm0.009)\overline{m},$$ with $\chi^2$/dof=0.009, 0.025, and 0.155, respectively, where $\langle p_T \rangle$ and $\overline{m}$ are in the units of GeV/$c$ and GeV/$c^2$ respectively. From the consideration of dimension, the slopes in Eqs. (14)–(16) are regarded as the (average) transverse flow velocity, which is close to $0.5c$. Including $d$ or $d$ and $^3$He, one can see a small decrease in transverse flow velocity. The blast-wave model \[44\] gives the transverse flow velocity for $d$ is 0.38–0.63$c$ and for $^3$He is 0.56–0.57$c$ \[16\] which is comparable with the present work. Figure 7(b) is the same as for Figure 7(a), but showing the dependence of mean momentum ($\langle p \rangle$) on $\overline{m}$. The values of $\langle p \rangle$ are calculated by using the Monte Carlo method in the source rest frame, too. The solid, dotted, and dashed curves in Figure 7(b) are linear fittings for i) $\pi^+$, $K^+$, and $p$; ii) $\pi^+$, $K^+$, $p$, and $d$; and iii) $\pi^+$, $K^+$, $p$, $d$, and $^3$He, which are described by $$\langle p \rangle=(0.170\pm0.010)+(0.786\pm0.006)\overline{m},$$ $$\langle p \rangle=(0.195\pm0.013)+(0.768\pm0.005)\overline{m},$$ and $$\langle p \rangle=(0.254\pm0.046)+(0.733\pm0.015)\overline{m},$$ with $\chi^2$/dof=0.009, 0.024, and 0.156, respectively, where $\langle p \rangle$ is in the units of GeV/$c$. From the consideration of dimension, the slopes in Eqs. (17)–(19) are regarded as the (average) flow velocity, which is close to $(\pi/2)0.5c$ which confirms our very recent work \[47\]. Including $d$ or $d$ and $^3$He, one can see a small decrease in flow velocity. In the above discussions, although KFO temperatures and (transverse) flow velocities are only extracted by us by indirect methods and we have not obtained the straightforward KFO, the present work provides anyhow alternative methods to describe $p_T$ spectra and to extract indirectly KFO temperature and (transverse) flow velocity. In fact, different functions result in different KFO temperatures. To extract the absolute temperature at KFO, the standard distribution is the best choice. However, we have to use a multi-component standard distribution. Because the mean (transverse) momentum and mean moving mass are independent of models. The (transverse) flow velocity extracted from the slope in the linear relation between mean (transverse) momentum and mean moving mass should be independent of models, too. In our very recent work \[47\], the linear relations between $T$ and $m_0$, $T$ and $\overline{m}$, $\langle p_T \rangle$ and $m_0$, $\langle p_T \rangle$ and $\overline{m}$, $\langle p \rangle$ and $m_0$, as well as $\langle p \rangle$ and $\overline{m}$ are studied. It is shown that the intercept in the linear relation between $T$ and $m_0$ can be regarded as the KFO temperature, the slope in the linear relation between $\langle p_T \rangle$ and $\overline{m}$ can be regarded as the transverse flow velocity, and the slope in the linear relation between $\langle p \rangle$ and $\overline{m}$ can be regarded as the flow velocity. In the present work, we use the same treatment to obtain the KFO temperature, transverse flow velocity, and flow velocity in the source rest frame. The present work shows that light particles correspond to low KFO temperature, which reflects that light particles freeze later than heavy particles. At the same time, due to small mass, light particles have larger (transverse) flow velocity than heavy particles.\ Conclusions =========== We summarize here our main observations and conclusions. a\) The transverse momentum distributions of $\pi^+$, $K^+$, $p$, $d$, and $^3$He produced in Pb-Pb collisions at 2.76 TeV with different centrality intervals are conformably analyzed by using the Tsallis distribution. The results calculated by us can fit approximately the experimental data of the ALICE Collaboration. The values of parameters such as the effective temperature, entropy index, and normalization factor are obtained. Small sizes of entropy index show that the interacting system considered in the present work stays approximately in an equilibrium state. The effective temperature extracted from transverse momentum spectra increases with increase of particle rest mass, and decreases with decrease of centrality. b\) We have used an alternative method to extract the kinetic freeze-out temperature of the interacting system based on the linear relation between the effective temperature and particle rest mass. The kinetic freeze-out temperature is regarded as the intercept in the linear relation by us, which shows the same or similar tendency as those of effective temperature and chemical freeze-out temperature in the case of studying their dependences on centrality. The values in central collisions are larger than those in peripheral collisions, which renders that the interacting system in central collisions stays in a higher excitation state comparing with peripheral collisions at kinetic (or chemical) freeze-out. c\) The kinetic freeze-out temperature extracted by us in 0–20% Pb-Pb collisions at 2.76 TeV for including $\pi^+$, $K^+$, and $p$ is 0.091 GeV. Including $d$ causes the kinetic freeze-out temperature increasing to 0.115 GeV, while including $d$ and $^3$He causes the kinetic freeze-out temperature increasing to 0.148 GeV. The particle mass effects of kinetic freeze-out temperature for $d$ and $^3$He are obvious. We think that we have observed an evidence of mass dependent differential kinetic freeze-out scenario or multiple kinetic freeze-out scenario. d\) We have also used an alternative method to extract the transverse flow velocity and flow velocity of the produced particles in the source rest frame based on the slopes in the linear relation between the mean transverse momentum and mean moving mass, as well as the mean momentum and mean moving mass. The particle mass effects of (transverse) flow velocity for $d$ and $^3$He are not obvious, though light particles have larger (transverse) flow velocity. 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--- author: - | Keiji Matsumoto\ National Institute of Informatics\ 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan\ [e-mail: [email protected]]{}\ Susumu Osawa\ Faculty of Mathematics, Kyushu University\ 744 Motooka, Nishi-ku, Fukuoka-city, Fukuoka 819-0385, Japan\ [e-mail: [email protected]]{} title: '[**CONSIDERATIONS IN THE TIME-ENERGY UNCERTAINTY RELATION FROM THE VIEWPOINT OF HYPOTHESIS TESTING**]{}' --- INTRODUCTION ============ The purpose of this study is to investigate time-energy uncertainty relation from the viewpoint of hypothesis testing. There are various derivations of time-energy uncertainty relation, and interpretation of $\Delta t$ is also various. The most acceptable derivation is that the relation is derived from the condition that the state of a system can hardly be distinguished from the initial state. For example, it is derived in the explanation of the sudden approximation in Messiah $\cite{Messiah}$. The outline is as follows. We suppose the Hamiltonian to change-over in a continuous way from a certain initial time $t_0$ to a certain final time $t_1$. We put $$\Delta t = t_1-t_0$$ and denote by $H(t)$ the value taken by the Hamiltonian at time $t$. Let $|0 \rangle $ denote the state vector of the system at time $t_0$ , $Q_0$ the projector onto the space of the vectors orthogonal to $|0\rangle$, and $U(t_1, t_0)$ the time evolution operator from $t_0$ to $t_1$. Supposing $|0\rangle$ to be of norm 1, we have $$Q_0 = 1- |0\rangle \langle 0 |.$$ The sudden approximation consists in writing $$U(t_1,t_0)|0\rangle \approx |0\rangle .$$ Messiah regarded a probability $w$ as that of finding the system in a state other than the initial state and interpreted it to be a measure of the error involved in this approximation: $$w= \langle 0 | U^ {\dag}(t_1, t_0) Q_0 U(t_1, t_0)|0 \rangle \label{eq:w}.$$ One obtains the expansion of $w$ in powers of $\Delta t$ by the perturbation method. Put $$\overline {H} = \frac{1}{ \Delta t} \int_{t_0}^{t_1} H(t) dt.$$ We then have $$w= \frac {\Delta t^2}{\hbar^2}\langle 0 |\overline{H}Q_0 \overline{H} |0 \rangle + O(T^3).$$ And since $$\langle 0 |\overline{H}Q_0 \overline{H} |0 \rangle =\langle 0|\overline{H}^2|0 \rangle - \langle 0|\overline{H}|0 \rangle ^2=(\Delta \overline{H})^2$$ where $\Delta \overline{H}$ is the root mean squre deviation of the ovservable $\overline{H}$ in the state $|0\rangle$, one has $$w= \frac{\Delta t^2 (\Delta \overline{H})^2 }{\hbar^2}+ O(T^3).$$ Thus the condition for the validity of the sudden approximation, $w\ll1$, requires that $$\Delta t \ll \frac{\hbar}{\Delta \overline{H}}$$ We can point out some questions about the derivation of the relation. Messiah remarked that $w$ is “the probability of finding the system in a state other than the initial state” and the condition that the state of a system can hardly be distinguished from the initial state is $w \ll 1$. The first question is that the physical meaning of “finding the system in a state other than the initial state” is so ambiguous that the above condition cannot have a firm basis. We can find the state of the system only through measurements. Therefore, the degree of discernibility between the two states is dependent on the way of detection of the system. The second question is that the detection scheme is not shown in Messiah’s discussion and the indicator of discernibility is not shown from this point of view. In this study, we investigate these questions from the viewpoint of hypothesis testing. TIME-ENERGY UNCERTAINTY RELATION FROM THE VIEWPOINT OF HYPOTHESIS TESTING ========================================================================= Appropriate Indicator of Discernibility --------------------------------------- We investigate pure state in the following discussion. The scheme of detection of the system should be constructed from a viewpoint of measurement and the decision rule of measurement outcomes. Here, we propose an appropriate indicator of discernibility by constructing the best detection scheme. Put $n$ copies of state $\rho_t$, where $t$ is a time parameter. Consider the following hypothesis teting problem about a parameter $t$. $$\begin{aligned} \label{quantum test} & & H_0: \quad \rho_ t=\rho_{t_0}\quad \quad \mbox{( null hypothesis)} \\ & & H_1: \quad \rho_t=\rho_{t_1} \quad \quad\mbox{ (alternative hypothesis)} \end{aligned}$$ From hypothesis testing theory, the power of this test could represent discernibility between the states. Therefore, we define an indicator of discernibility between $\rho_{t_0}$ and $ \rho_{t_1}$ as a maximum power of test. Then let us construct the test that maximizes the power of test $\gamma$. Since the probability distribution of measured value is determined by parameter $t$ and measurement $M$, two steps is needed to maximize $\gamma$ in the test. The first step is to select the most powerful test based on Neyman-Pearson’s theorem subject to a fixed measurement. The second step is to select measurement in order to maximize $\gamma$ of the most powerful test dependent on measurement. These processes are called optimization of the test. The selected test and measurement by optimization are called optimum test and optimum measurement respectively. Thus, the indicator of discernibility is the power of the optimum test. Asymptotic Behavior of The maximum Power of Test ------------------------------------------------ Let us consider the power of test and the optimum measurement when $\Delta t=t_1-t_0$ is very small and $n$ is very large. To begin with, consider the first step. From stein’s lemma ( see Appendix), the maximum power of test subject to a fixed measurement $M$ is $$\gamma_M \approx 1- \exp[-nD(p_{t_0} \| p_{t_1})], \label{gamma of kullback}$$ where $D(p_{t_0} \| p_{t_1})$ is Kullback divergence defined by $(\ref{kullback})$ in appendix, $p_{t_0}$ and $p_{t_1}$ probability distriibution of measured value at time $t_0$ and $t_1$. Because of $(\ref{gamma of kullback})$ and $(\ref{kullback and fisher})$, the power of test is written as $$\gamma_M \approx 1-\exp[-\frac{n}{2}J_M(t_0)(\Delta t)^2]+o((\Delta t)^2)\qquad(\Delta t \ll 1), \label{kensyuturyoku}$$ where $J_M(t_0)$ is classical Fisher information for the classical model $p(x|t_0)= {\rm Tr}\rho_{t_0}M(x)$ with a measurement $M$ defined as follows: $$J_M(t_0) \stackrel{\rm def}{=} \lim_{t \rightarrow t_0} \Sigma_x \frac{\dot p(x|t)^2}{ p(x|t)}.$$ Then consider the second step. We select the measurement which maximize $ \gamma_M$. Because of $(\ref{kensyuturyoku})$, the optimum measurement maximizes classical Fisher information $J_M(t_0)$. From the relation between classical and quantum Fisher information $(\ref{quantum and classical fisher})$ in appendix, the optimum measurement $M_{opt}$ is one which satisfies $$J_{M_{opt}}(t_0)= J^s(t_0),$$ where $J^s(t_0)$ is quantum Fisher information defined as follows: $$J^s(t_0) \stackrel{\rm def}{=} 4{\rm Tr}\rho_{t_0}(\frac{d \rho_{t_0}}{d t})^2.$$ According to the pure state quantum estimation theory [@Fujiwara], we have $$J^s(t_0)= \frac{4}{\hbar^2}\Delta H^2.$$ Thus we have $$J_{M_{opt}}(t_0) =\frac{4}{\hbar^2}\Delta H^2.\label{JM}$$ From $(\ref{kensyuturyoku})$ and $(\ref{JM})$, the power of the optimum test is $$\gamma_{max}=1-\exp(-\frac{2n}{\hbar^2}\Delta t^2\Delta H^2)+o(\Delta t^2)\qquad(\Delta t \ll 1).$$ If $\frac{2n}{\hbar^2}\Delta t^2\Delta H^2\ll1$ holds, $$\gamma_{max} \approx \frac{2n}{\hbar^2}\Delta t^2\Delta H^2.$$ Now we can show the condition that $\rho_{t_1}$ can hardly be distingished from $\rho_{t_0}$ using $n$ data when $\Delta t \ll1$ and $n \gg 1$ are satisfied. As it means $\gamma_{max} \ll 1$, we have $$1-\exp(-\frac{2n}{\hbar^2}\Delta t^2\Delta H^2)+o(\Delta t^2) \ll 1 \qquad(\Delta t \ll 1),$$ or $$\frac{2n \Delta t^2\Delta H^2 }{\hbar^2}\ll 1.$$ The Optimum Measurement ----------------------- Denoting by $\Pi$ the measurement which is made up of operators $Q_0$ and $1-Q_0$, we can easily prove that $\Pi$ is one of the optimum measurements as follows. By fixing a state $\rho_t$ and a measurement $\Pi$, measured value follows the probability function $ p_i(t) \; (i=1,2)$: $$\begin{aligned} p_1(t)&=&{\rm Tr}[\rho_t(1-Q_0)],\\ p_2(t)&=&{\rm Tr} [\rho_tQ_0]\\ &=&1-p_1(t).\end{aligned}$$ Therefore, classical Fisher information is $$J_\Pi(t_0)= \lim_{t \to t_0} [\frac{\dot{p_1}(t)^2}{p_1(t)}+\frac{\dot{p_2}(t)^2}{p_2(t)}].$$ This limit is intermediate form, but $p_1(t)$ is easily expanded as follows: $$\begin{aligned} p_1(t)&=&1+\dot{p_1}(t_0)(t-t_0)+\frac{1}{2}\ddot{p_1}(t_0)(t-t_0)^2+ \cdots\\ &=&1-\frac{1}{\hbar^2}[\langle 0|H^2|0\rangle-(\langle 0|H|0\rangle)^2](t-t_0)^2+\cdots.\end{aligned}$$ Hence, $$\begin{aligned} J_{\Pi}(t_0)&=& \lim_{t\to t_0} \frac{(\dot{-p_1}(t))^2}{1-p_1(t)}\nonumber \\ &=&-2\ddot{p_1}(t_0) \nonumber \\ &=&\frac{4}{\hbar^2}\Delta H^2. \label{JP}\end{aligned}$$ From $(\ref{JM})$ and $(\ref{JP})$, $\Pi$ is one of the optimum measurements. A probability $w$ is that of a measured value of this measurement which supports $H_1$. **CONCLUSION AND DISCUSSION** ============================= A maximum power of test in the hypothesis testing $H_0: \; \rho_ t=\rho_{t_0}\quad H_1: \; \rho_t=\rho_{t_1}$ can be regarded as an indicator of discernibility between the states. The condition that $\rho_{t_1}$ can hardly be distinguished from $\rho_{t_0}$ using $n$ data is $$1-\exp(-\frac{2n}{\hbar^2}\Delta t^2\Delta H^2)+o(\Delta t^2)\ll 1 \qquad(\Delta t \ll 1),$$ or if $ \frac{2n}{\hbar^2}\Delta t^2\Delta H^2\ll1$, $$\frac{2n \Delta t^2\Delta H^2 }{\hbar^2}\ll 1.$$ This condition represensts time-energy uncertainty relation from the viewpoint of hypothesis testing. Measurement $\Pi$ made up of opetators $Q_0$ and $1-Q_0$ is one of the optimum measurements. A probability $w$ is that of a measured value of this measurement which supports $H_1$. It is remarcable that the previous study has suggested the optimum measurement that maximizes the power of test. Here we give a brief summary of the conventional hypothesis teting theory and related fields. Suppose that random variables $X_i \;(i=1 \cdots, n)$ obey the probability distribution $p(x|\theta)$ with a given parameter $\theta \in\Theta\subset {\rm {\bf R}}$ . Simple hypothesis testing about parameter $\theta$ is as follows: $$\begin{aligned} & & H_0: \quad \theta=\theta_0\quad \quad \mbox{( null hypothesis)} \\ & & H_1: \quad \theta=\theta_1 \quad \quad\mbox{ (alternative hypothesis)} \end{aligned}$$ We consider nonrandomized test based on $n$ data. Random variables $X_1,X_2, \cdots ,X_n$ are independent and obey identical probability distribution $p(x|\theta)$. $(X_1,X_2, \cdots , X_n)$ is denoted by $X$. A hypothesis testing rule is a partition of the measurement space into two disjoint sets $U_0$ and $U_1=U_0^c$ . If observation value $x$ is an element of $U_0$, we decide that $H_0$ is true; if $x$ is an element of $U_1$, we decide $H_1$ is true. Accepting hypothesis $H_1$ when $H_0$ actually is true is called a type I error, and the probability of this event is denoted by $\alpha$. Accepting hypothesis $H_0$ when $H_1$ actually is true is called a type II error, and the probability of this event is denoted by $\beta$. The problem is to specify $(U_0, U_1)$ so that $\alpha$ and $\beta$ are as small as possible. This is not yet a well-defined problem because $\alpha$ generally can be made smaller by reducing $U_1$, although $\beta$ thereby increases. The Neyman-Pearson point of view assumes that a maximum value of $\alpha$ given by $\alpha^*$ is specified and $(U_0, U_1)$ must be determined so as to minimize $\beta$ subject to the constraint that $\alpha$ is not larger than $\alpha^*$. We call $\gamma=1-\beta $ power of test, and the test with the maximum power of test subject to the above constraint is called the most powerful test. A method for finding the optimum decision regions is given by the following theorem. Denote joint density function of random variables $X=(X_1, X_2, \cdots , X_n)$ by $$p_n(x|\theta)= \Pi_{i=1}^n p(x_i|\theta), \quad x=(x_1,x_2,\cdots x_n),$$ and put $$\Lambda_n \equiv \frac{ p_n(x|\theta_1)}{ p_n(x|\theta_0)}.$$ When a constant $k$ is set so that $$\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \phi^*(x)p_n(x)dx = \alpha^*$$ holds, the regions of the most powerful test are determined as $$\begin{aligned} & & U_0= \{ x: \quad \Lambda_n \le k \} \\ & & U_1 =\{ x: \quad \Lambda_n > k \},\end{aligned}$$ where $\phi^*(x)$ is the function which is defined as $$\phi^*(x) = \left\{ \begin{array}{@{\,}ll} 1 & (\Lambda >k)\\ 0 & (\Lambda \le k). \end{array} \right.$$ The asymptotic behavior can be described in the following lemma. Let $\alpha^* \in (0,1)$be given. Suppose that observaton consists of n independent measurements. Let $\beta^*$ be the smallest probability of type II error over all decision rules such that the probability of type I does not exceed $\alpha^*$. Then all $\alpha^* \in (0,1)$, $$\lim _{n \to \infty }(\beta_n^*)^{\frac{1}{n}}= \exp[-D(p_{\theta_0}\|p_{\theta_1})].$$ Here, $ D(p \| q)$ is called Kullback divergence and defined as $$D(p \| q) \stackrel{\rm def}{=} E_p[ \log \frac{q}{p}], \label{kullback}$$ where $p$ and $q$ are probability distributions and $E_p$ means expectation by $p$. On the other hand, the following relation between Fisher information in classical information theory ( we call it classical Fisher information) and Kullback divergence holds$(\cite{Nagaoka})$ $$D(p_{\theta+\Delta \theta}\| p_\theta)=\frac{1}{2}J(\theta)(\Delta \theta)^2+o((\Delta \theta)^2), \label{kullback and fisher}$$ where $J(\theta)$ is classical Fisher information for the classical model $p_{\theta}$. Generally, the maximum value of classical Fisher information of a given state $\rho_{\theta}$ equals quantum Fisher information [@Nagaoka87]: $$J^s(\theta)=\max_M J_M(\theta), \label{quantum and classical fisher}$$ where $J^s(\theta)$ is quantum Fisher information and $J_M(\theta)$ is classical Fisher information for the classical model $p(x|\theta)= {\rm Tr}[\rho_{\theta}M(x)]$ with a measurement $M$. [99]{} Fujiwara, A. and H. Nagaoka, “Quantum Fisher metric and estimation for pure state models,” Phys. lett, 201A, 119-124(1995). Messiah, A., “MECANIQUE QUANTIQUE,” Dunod, Paris 1959 Nagaoka, H., “ On Fisher information of quantum statistical models,” SITA’87, 241-246,(1987). Nagaoka, H., “ On the relation between Kullback divergence and Fisher information - from calssical systems to quantum systems - ,”(1991).
--- abstract: 'We introduce Quantum Graph Neural Networks (<span style="font-variant:small-caps;">qgnn</span>), a new class of quantum neural network ansatze which are tailored to represent quantum processes which have a graph structure, and are particularly suitable to be executed on distributed quantum systems over a quantum network. Along with this general class of ansatze, we introduce further specialized architectures, namely, Quantum Graph Recurrent Neural Networks (<span style="font-variant:small-caps;">qgrnn</span>) and Quantum Graph Convolutional Neural Networks (<span style="font-variant:small-caps;">qgcnn</span>). We provide four example applications of <span style="font-variant:small-caps;">qgnn</span>s: learning Hamiltonian dynamics of quantum systems, learning how to create multipartite entanglement in a quantum network, unsupervised learning for spectral clustering, and supervised learning for graph isomorphism classification.' author: - | Guillaume Verdon\ X, The Moonshot Factory\ Mountain View, CA\ `[email protected]`\ Trevor McCourt\ Google Research\ Venice, CA\ `[email protected]`\ Enxhell Luzhnica, Vikash Singh,\ **Stefan Leichenauer, Jack Hidary**\ X, The Moonshot Factory\ Mountain View, CA\ `{enxhell,singvikash,`\ `sleichenauer,hidary}@x.team`\ bibliography: - 'references.bib' title: Quantum Graph Neural Networks --- Introduction ============ Variational Quantum Algorithms are a promising class of algorithms that are rapidly emerging as a central subfield of Quantum Computing [@mcclean2016theory; @farhi2014quantum; @farhi2018classification]. Similar to parameterized transformations encountered in deep learning, these parameterized quantum circuits are often referred to as Quantum Neural Networks (QNNs). Recently, it was shown that QNNs that have no prior on their structure suffer from a quantum version of the no-free lunch theorem [@mcclean2018barren] and are exponentially difficult to train via gradient descent. Thus, there is a need for better QNN ansatze. One popular class of QNNs has been Trotter-based ansatze [@farhi2014quantum; @hadfield2019quantum]. The optimization of these ansatze has been extensively studied in recent works, and efficient optimization methods have been found [@verdon2019learning; @li2019quantum]. On the classical side, graph-based neural networks leveraging data geometry have seen some recent successes in deep learning, finding applications in biophysics and chemistry [@kearnes2016molecular]. Inspired from this success, we propose a new class of Quantum Neural Network ansatz which allows for both quantum inference and classical probabilistic inference for data with a graph-geometric structure. In the sections below, we introduce the general framework of the <span style="font-variant:small-caps;">qgnn</span> ansatz as well as several more specialized variants and showcase four potential applications via numerical implementation. Background ========== Classical Graph Neural Networks ------------------------------- Graph Neural Networks (<span style="font-variant:small-caps;">gnn</span>s) date back to [@sperduti1997supervised] who applied neural networks to acyclic graphs. [@gori2005new] and [@scarselli2008graph] developed methods that learned node representations by propagating the information of neighbouring nodes. Recently, <span style="font-variant:small-caps;">gnn</span>s have seen great breakthroughs by adapting the convolution operator from <span style="font-variant:small-caps;">cnn</span>s to graphs [@bruna2013spectral; @henaff2015deep; @defferrard2016convolutional; @kipf2016semi; @niepert2016learning; @hamilton2017inductive; @monti2017geometric]. Many of these methods can be expressed under the message-passing framework [@gilmer2017neural]. Let graph $\gG=(\mA,\mX)$ where $\mA \in \sR^{n \times n}$ is the adjacency matrix, and $\mX \in \sR^{n \times d}$ is the node feature matrix where each node has $d$ features. $$\mH^{(k)} = P(\mA, \mH^{(k-1)}, \mW^{(k)})$$ where $\mH^{(k)} \in \sR^{n \times d}$ are the node representations computed at layer $k$, $P$ is the message propagation function and is dependent on the adjacency matrix, the previous node encodings and some learnable parameters $\mW^{(k)}$. The initial embedding, $\mH^{(0)}$, is naturally $\mX$. One popular implementation of this framework is the <span style="font-variant:small-caps;">gcn</span> [@kipf2016semi] which implements it as follows: $$\mH^{(k)} = P(\mA,\mH^{(k-1)}, \mW^{(k)}) = \operatorname{ReLU}(\Tilde{\mD}^{-\frac{1}{2}}\Tilde{\mA}\Tilde{\mD}^{-\frac{1}{2}}\mH^{(k-1)}\mW^{(k-1)})$$ where $\Tilde{\mA} = \mA + \mI$ is the adjacency matrix with inserted self-loops and $\Tilde{\mD} = \sum_{j} \Tilde{\mA}_{ij}$ is the renormalization factor (degree matrix). Networked Quantum Systems ------------------------- Consider a graph $\mathcal{G} = \{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}$ is the set of vertices (or nodes) and $\mathcal{E}$ the set of edges. We can assign a quantum subsystem with Hilbert space $\mathcal{H}_v$ to each vertex in the graph, forming a global Hilbert space $\mathcal{H}_\mathcal{V} \equiv \bigotimes_{v\in\mathcal{V}} \mathcal{H}_v$. Each of the vertex subsystems could be one or several qubits, a qudit, a qumode [@weedbrook2012gaussian], or even an entire quantum computer. [One may also define a Hilbert space for each edge and form $\mathcal{H}_\mathcal{E} \equiv \bigotimes_{e\in\mathcal{E}} \mathcal{H}_e$. The total Hilbert space for the graph would then be $\mathcal{H}_\mathcal{E}\otimes \mathcal{H}_\mathcal{V}$. For the sake of simplicity and feasibility of numerical implementation, we consider this to be beyond the scope of the present work, so for us the total Hilbert space consists only of $\mathcal{H}_\mathcal{V}$.]{} The edges of the graph dictate the communication between the vertex subspaces: couplings between degrees of freedom on two different vertices are allowed if there is an edge connecting them. This setup is called a quantum network [@kimble2008quantum; @qian2019heisenberg] with topology given by the graph $\mathcal{G}$. Quantum Graph Neural Networks ============================= General Quantum Graph Neural Network Ansatz ------------------------------------------- The most general Quantum Graph Neural Network ansatz is a parameterized quantum circuit on a network which consists of a sequence of $Q$ different Hamiltonian evolutions, with the whole sequence repeated $P$ times: $$\label{eq:QGNN} \hat{U}_{\textsc{qgnn}}(\bm{\eta}, \bm{\theta}) = \prod_{p=1}^P \left[ \prod_{q=1}^Q e^{-i\eta_{pq}\hat{H}_q(\bm{\theta})}\right],$$ where the product is time-ordered [@poulin2011quantum], the $\bm{\eta}$ and $\bm{\theta}$ are variational (trainable) parameters, and the Hamiltonians $\hat{H}_q(\bm{\theta})$ can generally be any parameterized Hamiltonians whose topology of interactions is that of the problem graph: $$\label{eq:Ham} \hat{H}_q(\bm{\theta}) \equiv \sum_{\{j,k\}\in \mathcal{E}} \sum_{r\in\mathcal{I}_{jk}} W_{qrjk} \hat{O}^{(qr)}_{j}\otimes \hat{P}^{(qr)}_{k}+ \sum_{v\in \mathcal{V}} \sum_{r\in\mathcal{J}_{v}} B_{qrv} \hat{R}^{(qv)}_{j}.$$ Here the $W_{qrjk}$ and $B_{qrv}$ are real-valued coefficients which can generally be independent trainable parameters, forming a collection $\bm{\theta} \equiv \cup_{q,j,k,r}\{W_{qrjk}\}\cup_{q,v,r}\{B_{qrjk}\}$. The operators $\hat{R}^{(qv)}_{j}, \hat{O}^{(qr)}_{j}, \hat{P}^{(qr)}_{j}$ are Hermitian operators which act on the Hilbert space of the $j^\mathrm{th}$ node of the graph. The sets $\mathcal{I}_{jk}$ and $\mathcal{J}_{v}$ are index sets for the terms corresponding to the edges and nodes, respectively. To make compilation easier, we enforce that the terms of a given Hamiltonian $\hat{H}_q$ commute with one another, but different $\hat{H}_q$’s need not commute. In order to make the ansatz more amenable to training and avoid the barren plateaus (quantum parametric circuit no free lunch) problem [@mcclean2018barren], we need to add some constraints and specificity. To that end, we now propose more specialized architectures where parameters are tied spatially (convolutional) or tied over the sequential iterations of the exponential mapping (recurrent). Quantum Graph Recurrent Neural Networks (<span style="font-variant:small-caps;">qgrnn</span>) --------------------------------------------------------------------------------------------- We define quantum graph recurrent neural networks as ansatze of the form of where the temporal parameters are tied between iterations, $\eta_{pq} \mapsto \eta_q$. In other words, we have tied the parameters between iterations of the outer sequence index (over $p=1,\ldots, P$). This is akin to classical recurrent neural networks where parameters are shared over sequential applications of the recurrent neural network map. As $\eta_q$ acts as a time parameter for Hamiltonian evolution under $\hat{H}_q$, we can view the <span style="font-variant:small-caps;">qgrnn</span> ansatz as a Trotter-based [@lloyd1996universal; @poulin2011quantum] quantum simulation of an evolution $e^{-i\Delta \hat{H}_{\mathrm{eff}}}$ under the Hamiltionian $\hat{H}_{\mathrm{eff}} = \Delta^{-1}\sum_q \eta_q\hat{H}_q$ for a time step of size $\Delta =\lVert\bm{\eta}\rVert_1 = \sum_q |\eta_q|$. This ansatz is thus specialized to learn effective quantum Hamiltonian dynamics for systems living on a graph. In Section \[sec:dynamics\] we demonstrate this by learning the effective real-time dynamics of an Ising model on a graph using a <span style="font-variant:small-caps;">qgrnn</span> ansatz. Quantum Graph Convolutional Neural Networks (<span style="font-variant:small-caps;">qgcnn</span>) ------------------------------------------------------------------------------------------------- Classical Graph Convolutional neural networks rely on a key feature: that of permutation invariance. In other words, the ansatz should be invariant under permutation of the nodes. This is analogous to translational invariance for ordinary convolutional transformations. In our case, permutation invariance manifests itself as a constraint on the Hamiltonian, which now should be devoid of *local* trainable parameters, and should only have global trainable parameters. The $\bm{\theta}$ parameters thus become tied over indices of the graph: $W_{qrjk} \mapsto W_{qr}$ and $B_{qrv}\mapsto B_{qr}$. A broad class of graph convolutional neural networks we will focus on is the set of so-called Quantum Alternating Operator Ansatze [@hadfield2019quantum], the generalized form of the Quantum Approximate Optimization Algorithm ansatz [@farhi2014quantum]. Quantum Spectral Graph Convolutional Neural Networks (<span style="font-variant:small-caps;">qsgcnn</span>) {#sec:qsgnn} ----------------------------------------------------------------------------------------------------------- We can take inspiration from the continuous-variable quantum approximate optimization ansatz introduced in [@verdon2019quantum] to create a variant of the <span style="font-variant:small-caps;">qgcnn</span>: the Quantum Spectral Graph Convolutional Neural Network (<span style="font-variant:small-caps;">qsgcnn</span>). We show here how it recovers the mapping of Laplacian-based graph convolutional networks [@kipf2016semi] in the Heisenberg picture, consisting of alternating layers of message passing, node update, and nonlinearities. Consider an ansatz of the form from with four different Hamiltonians ($Q=4$) for a given graph. First, for a weighted graph $\mathcal{G}$ with edge weights $\Lambda_{jk}$, we define the *coupling Hamiltonian* as $$\hat{H}_C \equiv \tfrac{1}{2}\textstyle\sum_{\{j,k\}\in \mathcal{E}} \Lambda_{jk}(\hat{x}_j - \hat{x}_k)^2.$$ The $\Lambda_{jk}$ here are the *weights* of the graph $\mathcal{G}$, and are *not* trainable parameters. The operators denoted here by $\hat{x}_j$ are quantum continuous-variable position operators, which can be implemented via continuous-variable (analog) quantum computers [@weedbrook2012gaussian] or emulated using multiple qubits on digital quantum computers [@somma2015quantum; @verdon2018universal]. After evolving by $\hat{H}_C$, which we consider to be the message passing step, one applies an exponential of the *kinetic* Hamiltonian, $ \hat{H}_K \equiv \tfrac{1}{2}\sum_{j\in \mathcal{V}} \hat{p}_j^2 $. Here $\hat{p}_j$ denotes the continuous-variable momentum (Fourier conjugate) of the position, obeying the canonical commutation relation $[\hat{x}_j,\hat{p}_j]=i\delta_{jk}$. We consider this step as a node update step. In the Heisenberg picture, the evolution generated by these two steps maps the position operators of each node according to $$e^{-i\gamma \hat{H}_K }e^{-i\alpha \hat{H}_C }:\hat{x}_j\mapsto \hat{x}_j +\gamma\hat{p}_j -\alpha\gamma \textstyle\sum_{k\in \mathcal{V}} L_{jk}\hat{x}_k,$$ where $$L_{jk} =\textstyle \delta_{jk}\left(\sum_{v\in \mathcal{V}} \Lambda_{jv}\right) -\Lambda_{jk}$$ is the *Graph Laplacian* matrix for the weighted graph $\mathcal{G}$. We can recognize this step as analogous to classical spectral-based graph convolutions. One difference to note here is that *momentum* is free to accumulate between layers. Next, we must add some non-linearity in order to give the ansatz more capacity.[^1] The next evolution is thus generated by an *anharmonic* Hamiltonian $\hat{H}_A = \sum_{j\in \mathcal{V}} f(\hat{x}_j),$ where $f$ is a nonlinear function of degree greater than 2, e.g., a quartic potential of the form $f(\hat{x}_j) =((\hat{x}_j-\mu)^2 - \omega^2)^2$ for some $\mu, \omega$ hyperparameters. Finally, we apply another evolution according to the kinetic Hamiltonian. These last two steps yield an update $$e^{-i\beta \hat{H}_K }e^{-i\delta \hat{H}_A }:\hat{x}_j\mapsto \hat{x}_j +\beta\hat{p}_j -\delta\beta f'(\hat{x}_j),$$ which acts as a nonlinear mapping. By repeating the four evolution steps described above in a sequence of $P$ layers, i.e., $$\hat{U}_{\textsc{qsgcnn}}(\bm{\alpha},\bm{\beta},\bm{\gamma}, \bm{\delta}) = \prod_{j=1}^P e^{-i\beta_j \hat{H}_K }e^{-i\delta_j \hat{H}_A }e^{-i\gamma_j \hat{H}_K }e^{-i\alpha_j \hat{H}_C }$$with variational parameters $\bm{\theta} = \{\bm{\alpha},\bm{\beta},\bm{\gamma}, \bm{\delta}\}$, we then recover a quantum-coherent analogue of the node update prescription of  [@kipf2016semi] in the original graph convolutional networks paper.[^2] Applications & Experiments ========================== Learning Quantum Hamiltonian Dynamics with Quantum Graph Recurrent Neural Networks {#sec:dynamics} ---------------------------------------------------------------------------------- ![Left: Batch average infidelity with respect to ground truth state sampled at 15 randomly chosen times of quantum Hamiltonian evolution. We see the initial guess has a densely connected topology and the <span style="font-variant:small-caps;">qgrnn</span> learns the ring structure of the true Hamiltonian. Right: Ising Hamiltonian parameters (weights & biases) on a color scale.[]{data-label="fig:qgrnn"}](qrnn_graph_only.pdf "fig:"){width="55.00000%"}![Left: Batch average infidelity with respect to ground truth state sampled at 15 randomly chosen times of quantum Hamiltonian evolution. We see the initial guess has a densely connected topology and the <span style="font-variant:small-caps;">qgrnn</span> learns the ring structure of the true Hamiltonian. Right: Ising Hamiltonian parameters (weights & biases) on a color scale.[]{data-label="fig:qgrnn"}](qrnn_params.pdf "fig:"){width="25.00000%"} Learning the dynamics of a closed quantum system is a task of interest for many applications [@wiebe2014hamiltonian], including device characterization and validation. In this example, we demonstrate that a Quantum Graph Recurrent Neural Network can learn effective dynamics of an Ising spin system when given access to the output of quantum dynamics at various times. Our target is an Ising Hamiltonian with transverse field on a particular graph, $$\textstyle\hat{H}_{\text{target}} = \sum_{\{j,k\}\in \mathcal{E}} J_{jk}\hat{Z}_j\hat{Z}_k + \sum_{v\in\mathcal{V}}Q_v\hat{Z}_v + \sum_{v\in\mathcal{V}} \hat{X}_j.$$ We are given copies of a fixed low-energy state $\ket{\psi_0}$ as well as copies of the state $\ket{\psi_T} \equiv \hat{U}(T)\ket{\psi_0} = e^{-i T \hat{H}_{\text{target}}}$ for some known but randomly chosen times $T\in[0,T_{\text{max}}]$. Our goal is to learn the target Hamiltonian parameters $\{J_{jk}, Q_v\}_{j,k,v\in\mathcal{V}}$ by comparing the state $\ket{\psi_T}$ with the state obtained by evolving $\ket{\psi_0}$ according to the <span style="font-variant:small-caps;">qgrnn</span> ansatz for a number of iterations $P \approx T/\Delta$ (where $\Delta$ is a hyperparameter determining the Trotter step size). We achieve this by training the parameters via Adam [@kingma2014adam] gradient descent on the average infidelity $\mathcal{L}(\bm{\theta}) = 1-\tfrac{1}{B}\sum_{j=1}^B |\braket{\psi_{T_j}|U^j_{\textsc{qgrnn}}( \Delta,\bm{\theta})\ket{\psi_0}}|^2$ averaged over batch sizes of 15 different times $T$. Gradients were estimated via finite difference differentiation with step size $\epsilon =10^{-4}$. The fidelities (quantum state overlap) between the output of our ansatz and the time-evolved data state were estimated via the quantum swap test [@cincio2018learning]. The ansatz uses a Trotterization of a random densely-connected Ising Hamiltonian with transverse field as its initial guess, and successfully learns the Hamiltonian parameters within a high degree of accuracy as shown in Figure 1a. Quantum Graph Convolutional Neural Networks for Quantum Sensor Networks ----------------------------------------------------------------------- ![Left: Stabilizer Hamiltonian expectation and fidelity over training iterations. A picture of the quantum network topology is inset. Right: Quantum phase kickback test on the learned GHZ state. We observe a $7$x boost in Rabi oscillation frequency for a 7-node network, thus demonstrating we have reached the Heisenberg limit of sensitivity for the quantum sensor network.[]{data-label="fig:ghz"}](GHZ_loss.pdf "fig:"){width="40.00000%"}![Left: Stabilizer Hamiltonian expectation and fidelity over training iterations. A picture of the quantum network topology is inset. Right: Quantum phase kickback test on the learned GHZ state. We observe a $7$x boost in Rabi oscillation frequency for a 7-node network, thus demonstrating we have reached the Heisenberg limit of sensitivity for the quantum sensor network.[]{data-label="fig:ghz"}](GHZ_phase.pdf "fig:"){width="40.00000%"} Quantum Sensor Networks are a promising area of application for the technologies of Quantum Sensing and Quantum Networking/Communication [@kimble2008quantum; @qian2019heisenberg]. A common task considered where a quantum advantage can be demonstrated is the estimation of a parameter hidden in weak qubit phase rotation signals, such as those encountered when artificial atoms interact with a constant electric field of small amplitude [@qian2019heisenberg]. A well-known method to achieve this advantange is via the use of a quantum state exhibiting multipartite entanglement of the Greenberger–Horne–Zeilinger kind, also known as a GHZ state [@greenberger1989going]. Here we demonstrate that, without global knowledge of the quantum network structure, a <span style="font-variant:small-caps;">qgcnn</span> ansatz can learn to prepare a GHZ state. We use a <span style="font-variant:small-caps;">qgcnn</span> ansatz with $\hat{H}_1 = \sum_{\{j,k\}\in\mathcal{E}} \hat{Z}_j\hat{Z}_k$ and $\hat{H}_2= \sum_{j\in\mathcal{V}} \hat{X}_j$. The loss function is the negative expectation of the sum of stabilizer group generators which stabilize the GHZ state [@toth2005entanglement], i.e., $$\mathcal{L}(\bm {\eta}) = -\braket{\textstyle \bigotimes_{j=0}^n\hat{X} +\sum_{j=1}^{n-1}\hat{Z}_j\hat{Z}_{j +1}}_{\bm{\eta}}$$ for a network of $n$ qubits. Results are presented in Figure 1b. Note that the advantage of using a <span style="font-variant:small-caps;">qgnn</span> ansatz on the network is that the number of quantum communication rounds is simply proportional to $P$, and that the local dynamics of each node are independent of the global network structure. In order to further validate that we have obtained an accurate GHZ state on the network after training, we perform the quantum phase kickback test on the network’s prepared approximate GHZ state [@wei2019verifying].[^3] We observe the desired frequency boost effect for our trained network preparing an approximate GHZ state at test time, as displayed in Figure \[fig:ghz\]. Unsupervised Graph Clustering with Quantum Graph Convolutional Networks ----------------------------------------------------------------------- As a third set of applications, we consider applying the <span style="font-variant:small-caps;">qsgcnn</span> from Section \[sec:qsgnn\] to the task of spectral clustering [@ng2002spectral]. Spectral clustering involves finding low-frequency eigenvalues of the graph Laplacian and clustering the node values in order to identify graph clusters. In Figure \[fig:spectral\] we present the results for a <span style="font-variant:small-caps;">qsgcnn</span> for varying multi-qubit precision for the representation of the continuous values, where the loss function that was minimized was the expected value of the anharmonic potential $\mathcal{L}(\bm{\eta}) = \braket{\hat{H}_C + \hat{H}_A}_{\bm{\eta}}$. Of particular interest to near-term quantum computing with low numbers if qubits is the single-qubit precision case, where we modify the <span style="font-variant:small-caps;">qsgcnn</span> construction as $\hat{p}^2_j\mapsto \hat{X}_j$, $\hat{H}_A\mapsto I $ and $\hat{x}_j\mapsto \ket{1}\!\bra{1}_j$ which transforms the coupling Hamiltonian as $$\label{eq:1qb_cost} \hat{H}_C \mapsto \tfrac{1}{2}\textstyle\sum_{\{j,k\}\in \mathcal{E}} \Lambda_{jk}(\ket{1}\!\bra{1}_j - \ket{1}\!\bra{1}_k)^2 = \sum_{jk} L_{jk} \ket{1}\!\bra{1}_j\otimes \ket{1}\!\bra{1}_k,$$ where $\ket{1}\!\bra{1}_k = (\hat{I}-\hat{Z}_k)/2$. We see that using a low-qubit precision yields sensible results, thus implying that spectral clustering could be a promising new application for near-term quantum devices. ![<span style="font-variant:small-caps;">qsgcnn</span> spectral clustering results for 5-qubit precision (top) with quartic double-well potential and 1-qubit precision (bottom) for different graphs. Weight values are represented as opacity of edges, output sampled node values as grayscale. Lower precision allows for more nodes in the simulation of the quantum neural network. The graphs displayed are the most probable (populated) configurations, and to their right is the output probability distribution over potential energies. We see lower energies are most probable and that these configurations have node values clustered.[]{data-label="fig:spectral"}](CV_graph_clusters.pdf "fig:"){width="0.63\linewidth"}![<span style="font-variant:small-caps;">qsgcnn</span> spectral clustering results for 5-qubit precision (top) with quartic double-well potential and 1-qubit precision (bottom) for different graphs. Weight values are represented as opacity of edges, output sampled node values as grayscale. Lower precision allows for more nodes in the simulation of the quantum neural network. The graphs displayed are the most probable (populated) configurations, and to their right is the output probability distribution over potential energies. We see lower energies are most probable and that these configurations have node values clustered.[]{data-label="fig:spectral"}](CV_energy_spectral.pdf "fig:"){width="0.35\linewidth"} ![<span style="font-variant:small-caps;">qsgcnn</span> spectral clustering results for 5-qubit precision (top) with quartic double-well potential and 1-qubit precision (bottom) for different graphs. Weight values are represented as opacity of edges, output sampled node values as grayscale. Lower precision allows for more nodes in the simulation of the quantum neural network. The graphs displayed are the most probable (populated) configurations, and to their right is the output probability distribution over potential energies. We see lower energies are most probable and that these configurations have node values clustered.[]{data-label="fig:spectral"}](qb_graph_clusters.pdf "fig:"){width="0.62\linewidth"}![<span style="font-variant:small-caps;">qsgcnn</span> spectral clustering results for 5-qubit precision (top) with quartic double-well potential and 1-qubit precision (bottom) for different graphs. Weight values are represented as opacity of edges, output sampled node values as grayscale. Lower precision allows for more nodes in the simulation of the quantum neural network. The graphs displayed are the most probable (populated) configurations, and to their right is the output probability distribution over potential energies. We see lower energies are most probable and that these configurations have node values clustered.[]{data-label="fig:spectral"}](qb_energy_spectral.pdf "fig:"){width="0.35\linewidth"} Graph Isomorphism Classification via Quantum Graph Convolutional Networks ------------------------------------------------------------------------- Recently, a benchmark of the representation power of classical graph neural networks has been proposed [@xu2018powerful] where one uses classical <span style="font-variant:small-caps;">gcn</span>s to identify whether two graphs are isomorphic. In this spirit, using the <span style="font-variant:small-caps;">qsgcnn</span> ansatz from the previous subsection, we benchmarked the performance of this Quantum Graph Convolutional Network for identifying isomorphic graphs. We used the single-qubit precision encoding in order to order to simulate the execution of the quantum algorithms on larger graphs. Our approach was the following, given two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$, one applies the single-qubit precision <span style="font-variant:small-caps;">qsgcnn</span> ansatz $\prod_{j=1}^P e^{i\eta_j \hat{H}_K}e^{i\gamma_j \hat{H}_C}$ with $\hat{H}_K= \sum_{j\in\mathcal{V}} \hat{X}_j$ and $\hat{H}_C$ from in parallel according to each graph’s structure. One then samples eigenvalues of the coupling Hamiltonian $\hat{H}_C$ on both graphs via standard basis measurement of the qubits and computation of the eigenvalue at each sample of the wavefunction. One then obtains a set of samples of “energies” of this Hamiltonian. By comparing the energetic measurement statistics output by the <span style="font-variant:small-caps;">qsgcnn</span> ansatz applied with identical parameters $\bm{\theta} = \{\bm{\eta},\bm{\gamma}\}$ for two different graphs, one can then infer whether the graphs are isomorphic. We used the Kolmogorov-Smirnoff test [@lilliefors1967kolmogorov] on the distribution of energies sampled at the output of the <span style="font-variant:small-caps;">qsgcnn</span> to determine whether two given graphs were isomorphic. In order to determine the binary classification label deterministically, we considered all KS statistic values above $0.4$ to indicate that the graphs were non-isomorphic. For training and testing purposes, we set the loss function to be $\mathcal{L}(y, \operatorname{KS}) = (1-y)(1-\operatorname{KS}) + y \operatorname{KS}$, where $y=1$ if graphs are isomorphic, and $y=0$ otherwise. For the dataset, graphs were sampled uniformly at random from the Erdos-Renyi distribution $G(n,p)$ with $p=0.5$ at fixed $n$. In all of our experiments, we had 100 pairs of graphs for training, 50 for validation, and 50 for testing, always balanced between isomorphic and non-isomorphic pairs. Moreover, we only considered graphs that were connected. The networks were trained via a Nelder-Mead optimization algorithm. Presented in Figure \[fig:iso\_loss\] is the training and testing losses for various graph sizes and numbers of energetic samples. In Tables 1 and 2, we present the graph isomorphism classification accuracy for the training and testing sets using the trained <span style="font-variant:small-caps;">qgcnn</span> with the previously described thresholded KS statistic as the label. We see we get highly accurate performance even at low sample sizes. This seems to imply that the <span style="font-variant:small-caps;">qgcnn</span> is fully capable of identifying graph isomorphism, as desired for graph convolutional network benchmarks. ![Graph isomorphism loss curves for training and validation for various numbers of samples. Left is for 6 node graphs and right is for 15 node graphs. The loss is based on the Kolmogorov-Smirnov statistic comparing the sampled distribution of energies of the <span style="font-variant:small-caps;">qgcnn</span> output on two graphs. []{data-label="fig:iso_loss"}](training_info_6nodes_losses.pdf "fig:"){width="50.00000%"}![Graph isomorphism loss curves for training and validation for various numbers of samples. Left is for 6 node graphs and right is for 15 node graphs. The loss is based on the Kolmogorov-Smirnov statistic comparing the sampled distribution of energies of the <span style="font-variant:small-caps;">qgcnn</span> output on two graphs. []{data-label="fig:iso_loss"}](training_info_15nodes_losses.pdf "fig:"){width="50.00000%"} \[sample-table\] Conclusion & Outlook ==================== Results featured in this paper should be viewed as a promising set of first explorations of the potential applications of <span style="font-variant:small-caps;">qgnn</span>s. Through our numerical experiments, we have shown the use of these <span style="font-variant:small-caps;">qgnn</span> ansatze in the context of quantum dynamics learning, quantum sensor network optimization, unsupervised graph clustering, and supervised graph isomorphism classification. Given that there is a vast set of literature on the use of Graph Neural Networks and their variants to quantum chemistry, future works should explore hybrid methods where one can learn a graph-based hidden quantum representation (via a <span style="font-variant:small-caps;">qgnn</span>) of a quantum chemical process. As the true underlying process is quantum in nature and has a natural molecular graph geometry, the <span style="font-variant:small-caps;">qgnn</span> could serve as a more accurate model for the hidden processes which lead to perceived emergent chemical properties. We seek to explore this in future work. Other future work could include generalizing the <span style="font-variant:small-caps;">qgnn</span> to include quantum degrees of freedom on the edges, include quantum-optimization-based training of the graph parameters via quantum phase backpropagation [@verdon2018universal], and extending the <span style="font-variant:small-caps;">qsgcnn</span> to multiple features per node. ### Acknowledgments {#acknowledgments .unnumbered} Numerics in this paper were executed using a custom interface between Google’s Cirq [@cirq] and TensorFlow [@abadi2016tensorflow]. The authors would like to thank Edward Farhi, Jae Yoo, and Li Li for useful discussions. GV, EL, and VS would like to thank the team at X for the hospitality and support during their respective Quantum@X and AI@X residencies where this work was completed. X, formerly known as Google\[x\], is part of the Alphabet family of companies, which includes Google, Verily, Waymo, and others (www.x.company). GV acknowledges funding from NSERC. [^1]: From a quantum complexity standpoint, adding a nonlinear operation (generated by a potential of degree superior to quadratic) creates states that are *non-Gaussian* and hence are non efficiently simulable on classical computers [@bartlett2002efficient], in general composing layers of Gaussian and non-Gaussian quantum transformations yields quantum computationally universal ansatz [@lloyd1999quantum]. [^2]: For further physical intuition about the behaviour of this ansatz, note that the sum of the coupling and kinetic Hamiltonians $\hat{H}_K+\hat{H}_C$ is equivalent to the Hamiltonian of a network of quantum harmonic oscillators coupled according to the graph weights and network topology. By adding a quartic $\hat{H}_A$, we are thus emulating parameterized dynamics on a harmonically coupled network of anharmonic oscillators. [^3]: For this test, one applies a phase rotation $\bigotimes_{j\in\mathcal{V}} e^{-i\varphi \hat{Z}_j}$ on all the qubits in paralel, then one applies a sequence of CNOTs (quantum adder gates) to concentrate the phase shifts onto a single collector node, $m\in\mathcal{V}$. Given that one had a GHZ state initially, one should then observe a phase shift $e^{-in\varphi \hat{Z}_m}$ where $n = |\mathcal{V}|$. This boost in frequency of oscillation of the signal is what gives quantum multipartite entanglement its power to increase sensitivity to signals to super-classical levels [@degen2017quantum].
--- abstract: 'We prove equiconsistency results concerning gaps between a singular strong limit cardinal $\kappa$ of cofinality $\aleph_0$ and its power under assumptions that $2^\kap =\kap^{+\del +1}$ for $\del <\kap$ and some weak form of the Singular Cardinal Hypothesis below $\kap$. Together with previous results this basically completes the study of consistency strength of various gaps between such $\kap$ and its power under GCH type assumptions below.' author: - | Moti Gitik\ School of Mathematical Sciences\ Tel Aviv University\ Tel Aviv 69978, Israel\ gitik@@ math.tau.ac.il title: On Gaps under GCH Type Assumptions --- Introduction ============ Our first result deals with cardinal gaps. We continue \[Git-Mit\] and show the following: [**Theorem 1.**]{} Together with \[Git-Mag\] and \[Git1\] this provides the equiconsistency result for cardinal gaps of uncountable cofinality. Surprisingly the proof uses very little of the indiscernibles theory for extenders developed in \[Git-Mit\]. Instead, basic results of the Shelah pcf-theory play the crucial role. Building on the analysis of indiscernibles for uncountable cofinality of \[Git-Mit\] and pcf-theory we show the following: [**Theorem 2.**]{} Using this result, we extend Theorem 1 to ordinal gaps: [**Theorem 3.**]{} If the pcf structure between $\kap$ and $2^\kap$ is not “wild" (thus, for example, if there is no measurable of the core model between $\kap$ and $2^\kap$), then the result holds also for $\del=\aleph_1$. These theorems and related results are proved in Section 1 of the paper. Actually more general results (1.20, 1.21) are proved for ordinal gaps but the formulations require technical notions “Kinds" and “Kinds$^*$" and we will not reproduce them here. In Section 2 we sketch some complimentary forcing constructions based on \[Git1\]. Thus we are able to deal with cardinal gaps of cofinality $\aleph_0$ and show the following which together with Theorem 1 provides the equiconsistency for the cases of cofinality $\aleph_0$. [**Theorem 4.**]{} The Rado-Milner paradox is used to show the following: [**Theorem 5.**]{} A more general result (2.6) of the same flavor is obtained for ordinal gaps. In the last section, we summarize the situation and discuss related open questions and some further directions. A knowledge of the basic $pcf$-theory results is needed for Section 1. We refer to the Burke-Magidor \[Bur-Mag\] survey paper or to Shelah’s book \[Sh-g\] on these matters. Results on ordinal gaps and the strength of “$|pcf a|>|a|$" require in addition familiarity with basics of indiscernible structure for extenders. See Gitik-Mitchell \[Git-Mit\] on this subject. Results of Sections 2 are built on short extender based Prikry forcings, mainly those of \[Git1\]. [**Acknowledgement.**]{}We are grateful to Saharon Shelah for many helpful conversations and for explanations that he gave on the pcf-theory. On the Strength of Gaps ======================= Let $SSH^\del_{<\kap}$ ($SSH^{\le\del}_{<\kap}$) denote the Shelah Strong Hypothesis below $\kap$ for cofinality $\del$ $(\le\del)$ which means that for every singular cardinal $\tau <\kap$ of cofinality $\del(\le\del)$ $\ pp(\tau)=\tau^+$. We assume that there is no inner model with a strong cardinal. First we will prove the following: [**Theorem 1.1.**]{} [**Remark 1.2.**]{}(1) in either case we have in the core model a cardinal $\alp$ carrying an extender of the length $\alp^{+\del +1}$. \(2) By \[Git-Mag\] or \[Git1\] it is possible to force, using (i) or (ii), the situation assumed in the theorem. So this provides equiconsistency result. [**Proof.**]{}If $\del$ is a regular cardinal then let $A$ be the set of cardinals $\kap^{+\tau +1}$ so that $\tau <\del$ and either $o(\alp)<\kap^{+\tau}$ for every $\alp <\kap^{+\tau}$ or else $\kap^{+\tau}$ is above every measurable of the core model smaller than $\kap^{+\del}$. The set $A$ is unbounded in $\kap^{+\del}$ since there is no overlapping extenders in the core model. If $cf\del <\del$ then we fix $\l\del_i\mid i<cf\del\r$ an increasing sequence of regular cardinals with limit $\del$. For every $i<cf\del$ define $A_i$ to be the set of cardinals $\kap^{+\tau +1}$ so that $\tau <\del_i$ and either $o(\alp)<\kap^{+\tau}$ for every $\alp <\kap^{+\tau}$ or else $\kap^{+\tau}$ is above every measurable of the core model smaller than $\kap^{+\del_i}$. Again, each of $A_i$’s will be unbounded in $\kap^{+\del_i}$ since there is no overlapping extenders in the core model. The following fact was proved in \[Git-Mit, 3.24\]: [**Claim 1.3.**]{}If $B\subseteq A$ in case $cf\del =\del$ or $B\subseteq A_i$ for some $i<cf\del$, in case $cf\del <\del$ then $|B|<\inf B$ implies $\max(pcf(B))=(\sup B)^+$. Now for every $\kap^{+\alp +1}\in A$ or $\kap^{+\alp +1}\in\bigcup_{i<cf\del}A_i$ (if $cf\del <\del$) we pick a set $\{c^\alp_n\mid n<\ome\}$ of regular cardinals below $\kap$ so that $\kap^{+\alp+1}\in pcf\{c^\alp_n \mid n<\ome\}$. Set $$a=\{c^\alp_n\mid n<\ome\ ,\quad\kap^{+\alp +1}\in A \quad\hbox{if}\quad cf\del =\del\quad\hbox{or}\quad \kap^{+\alp+1}\in\bigcup_{i <cf\del}A_i\quad \hbox{otherwise}\}\ .$$ Removing its bounded part, if necessary, we can assume that $\min a>|a|^+$. [**Claim 1.4.**]{}For every $b\subseteq a\mid A\cap pcf (b)|\le |b|$ or $|A_i\cap pcf (b)|\le |b|$, for every $i<cf\del$, if $cf\del <\del$. [**Proof.**]{}It follows from Shelah’s Localization Theorem \[Sh-g\] and Claim 1.3.$\square$ In particular, $|a|=\del$. Let $b_{\kap^+}[a]$ be the pcf-generator corresponding to $\kap^+$. Consider $a^*=a\bks b_{\kap^+}[a]$. For every $\alp >0$, if $\kap^{+\alp +1}\in A$ or $\bigcup_{i<cf\del}A_i$ then $\kap^{+\alp +1}\in pcf(a^*)$. Hence, $|(pcf a^*)\cap A|=\del$ or $|pcf(a^*)\cap A_i\mid =\del_i$ for each $i<cf\del$ and by Claim 1.4, then $|a^*|=\del$. [**Claim 1.5.**]{}Let $\l\tau_n\mid n<\ome\r$ be an increasing unbounded in $\kap$ sequence of limit points of $a^*$ of cofinality $cf\del$. Then for every ultrafilter $D$ on $\ome$ including all cofinite sets$cf\Big(\prodl_{n<\ome}\tau^+_n/D\Big)>\kap^+$. [**Proof.**]{}For every $n<\ome$, $\tau_n$ is a singular cardinal of cofinality $cf\del$. So, by the assumption $pp(\tau_n)=\tau^+_n$. Then $\tau^+_n=cf (\prod t/E)$, for every unbounded in $\tau_n$ set of regular cardinals with $|t|<\tau_n$ and an ultrafilter $E$ on it including all cobounded subset of $t$. In particular, $\tau^+_n\in pcf (a^*\cap\tau_n)$ since $\tau_n$ is a limit point of $a^*$. So $\{\tau^+_n\mid n<\ome\}\subseteq pcf a^*$. By \[Sh-g\], then $pcf \{\tau^+_n\mid n<\ome\}\subseteq pcf (pcf a^*)=pcf a^*$. But by the choice of $a^*$, $\kap^+\notin pcf a^*$. Hence for every ultrafilter $D$ on $\ome$, cf $\Big(\prodl_{n<\ome}\tau^+_n/D\Big)\not= \kap^+$. $\square$ Now, $|a^*|=\del,\cup a^*=\kap,\ cf\del >\aleph_0$ and $cf\kap =\aleph_0$. Hence there is an increasing unbounded in $\kap$ sequence $\l\tau_n\mid n <\ome\r$ of limit points of $a^*$ so that for every $n>0$ $|a^*\cap (\tau_{n-1},\tau_n)|=\del$ and $|(a^*\cap\tau_n)\bks \bet|=\del$ for every $\bet <\tau_n$. By Claim 1.5, $\l\tau^+_n\mid n<\ome\r$ are limits of indiscernibles. We refer to \[Git-Mit\] for basic facts on this matter used here. There is a principal indiscernible $\rho_n\le\tau^+_n$ for all but finitely many $n$’s. By the Mitchell Weak Covering Lemma, $\tau^+_n$ in the sense of the core model is the real $\tau^+_n$, since $\tau_n$ is singular. This implies that $\rho_n\le\tau_n$, since a principal indiscernible cannot be successor cardinal of the core model. Also, $\rho_n$ cannot be $\tau_n$, since again $\tau^+_n$ computed in the core model correctly and so there is no indiscernibles between measurable now $\tau_n$ and its successor $\tau^+_n$. Hence $\rho_n<\tau_n$. By the choice of $\tau_n$, the interval $(\rho_n,\tau_n)$ contains at least $\del$ regular cardinals. So $\rho_n$ is a principal indiscernible of extender including at least $\del +1$ regular cardinals which either seats over $\kap$ or below $\kap$. This implies that either $o(\kap)\ge\kap^{+\del +1}+1$ or $\{\alp <\kap\mid o(\alp)\ge\alp^{+\del +1}+1\}$ is unbounded in $\kap$.$\square$ Using the same ideas, let us show the following somewhat more technical result: [**Theorem 1.6.**]{} [**Remark 1.7.**]{}The theorem implies results of the following type proved in \[Git-Mit\]: if $2^\kap =\kap^{+m}$ $(2<m<\ome)$ and GCH below $\kap$, then $o(\kap)\ge\kap^{+m}+1$, provided that for some $k<\ome$ the set of $\nu<\kap$ such that $o(\nu)>\nu^{+k}$ is bounded in $\kap$. [**Proof.**]{}Suppose otherwise. Collapsing if necessary $2^\kap$ to $\kap^{++}$, we can assume that $2^\kap =\kap^{++}$. Let $\l\rho_{n,i}\mid n <\ome$, $i<\ome_1\r$ and $\l\del_{n,i}\mid n<\ome, i<\ome_1\r$ witness the failure of the theorem. We can assume that for every $n<\ome$ and $i<j<\ome_1$ $$\rho_{n,i}\le\del_{n,i}<\rho_{n,j}\le \del_{nj}\ .$$ Let $a=\l\del_{n,i}\mid n<\ome, i<\ome_1\r$. Consider $a^*=a\bks b_{\kap^+}[a]$. Then for every $i<\ome_1$ the set $c_i=a^*\cap\{\del_{n,i}\mid n<\ome\}$ is infinite, since $cf\Big(\prodl_{n<\ome}\del_{n,i}\Big/D_i\Big)=\kap^{++}$ for some $D_i$. The following is obvious. [**Claim 1.8.**]{}There is an infinite set $d\subseteq\ome$ such that for every $n\in d$ there are uncountably many $i$’s with $\del_{n,i}\in c_i$. For every $n\in d$ let $$\tau_n=\sup\{\del_{n,i}\mid\del_{n,i}\in C_i\}\ .$$ Then each such $\tau_n$ is a singular cardinal of uncountable cofinality. Also, $\tau^+_n\in pcf a^*$ for every $n\in d$, since $pp\tau_n=\tau^+_n$. But then $pcf\{\tau^+_n\mid n\in d\}\subseteq pcf a^*$. Hence $\kap^+\not\in pcf \{\tau^+_n\mid n\in d\}$. Now, this implies as in the proof of 1.1 that $\tau^+_n$’s are indiscernibles and there are principal indiscernibles for $\tau^+_n$’s below $\tau_n$. Here this is impossible since then there should be overlapping extenders. Contradiction.$\square$ We will use 1.6 further in order to deal with ordinal gaps. As above, we show the following assuming that there is no inner model with a strong cardinal. [**Proposition 1.9.**]{} [**Proof.**]{}Here we apply the analysis of indiscernibles of \[Git-Mit\] for uncountable cofinality. Let $\l\nu_\bet\mid\bet\le\tet\r$ be the increasing enumeration of the closure of $\l\tau_\alp\mid\alp <\tet\r$. Let $A\subseteq\tet$ be the set of indexes of all principal indiscernibles for $\nu_\tet$ among $\nu_\bet$’s $(\bet <\tet)$. Then $A$ is a closed subset of $\tet$. Now split into two cases. [**Case 1.**]{}$A$ is bounded in $\tet$. Let $\bet^*=\sup A$. We have a club $C\subseteq\tet$ so that for every $\alp\in C$, $\bet\in (\bet^*,\alp)$ if $\nu_\bet$ is a principal indiscernible, then it is a principal indiscernible for an ordinal below $\nu_\alp$. Now let $\alp$ be a limit point of $C$ of uncountable cofinality. Then by results of \[Git-Mit\], $pp\nu_\alp=\nu^+_\alp$ and moreover $tcf\Big(\prodl_{\bet <\alp}\nu_\bet\Big/J^{bd}_{\nu_\alp} \Big)=\nu^+_\alp$. So we are done. [**Case 2.**]{}$A$ is bounded in $\tet$. Let $\tilA$ be the set of limit points of $A$. For every $\alp\in\tilA$ we consider $\nu_{\alp +1}$. Let $\nu^*_{\alp +1}$ be the principal indiscernible of $\nu_{\alp +1}$. Then $\nu_\alp\le\nu^*_{\alp +1}\le\nu_{\alp +1}$. The following is the main case: [**Subcase 2.1.**]{}For every $\alp$ in an unbounded set $S\subseteq\tet$, $\nu^*_{\alp +1}$ is a principal indiscernible for $\nu_{\tet}$ and $\nu_{\alp +1}$ is an indiscernible belonging to some $\onu_{\alp +1}$ over $\nu_{\tet}$ of cofinality $\ge\nu_{\ome_1}$ in the core model. We consider the set $B=\{\onu_{\alp +1}|\alp\in S\}$. If $|B|<\tet$, then we can shrink $S$ to set $S'$ of the same cardinality such that for every $\bet, \alp\in S'$ $\onu_{\alp +1}=\onu_{\bet +1}$. Now projecting down to limit points of $S'$ of uncountable cofinality we will obtain (\*) of the conclusion of the theorem. So, suppose now that $|B|=\tet$. W.l. of g., we can assume that $\alp <\bet$ implies $\onu_{\alp +1}<\onu_{\bet +1}$. Now, by \[Git-Mit\], $B$ (or at least its initial segments) is contained in the length of an extender over $\nu_\tet$ in the core model. There is no overlapping extenders, hence $$tcf\Big(\prodl_{\alp\in S}\onu_{\alp +1}\Big/J^{bd}_\tet \Big)=\Big(\sup (\{\onu_{\alp+1}\mid\alp\in S\})\Big)^+$$ where the successor is in sense of the core model or the universe which is the same by the Mitchell Weak Covering Lemma. Also, for every $\alp$ which is a limit point of $S$ of uncountable cofinality $$tcf\Big(\prodl_{\bet\in S\cap_\alp}\onu_{\bet +1}\Big/ J^{bd}_{S\cap\alp}\Big)=\Big(\sup \{\onu_{\bet +1}\mid\bet\in S\cap\alp\}\Big)^+\ .$$ Projecting down we obtain (\*). [**Subcase 2.2.**]{}Starting with some $\alp^* <\tet$ each $\nu^*_{\alp +1}$ is not a principal indiscernible for $\nu_\tet$ or it is but $\nu_{\alp +1}$ corresponds over $\nu_\tet$ to some $\onu_{\alp +1}$ which has cofinality $<\nu_\tet$ in the core model. Suppose for simplicity that $\alp^*=0$. If $\nu^*_{\alp +1}$ is not a principal indiscernible for $\nu_{\tet}$, then we can use functions of the core model to transfer the structure of indiscernibles over $\nu^*_{\alp +1}$ to the interval $[\nu_\alp$, length of the extender used over $\nu_\alp]$. This will replace $\nu_{\alp +1}$ be a member of the interval. So let us concentrate on the situation when $\nu^*_{\alp +1}$ is a principal indiscernible for $\nu_\tet$ but $\onu_{\alp +1}$ has cofinality $\le\nu_\tet$ $(\alp <\tet)$. Let us argue that this situation is impossible. Thus we have increasing sequences $\langle\alp_i\mid i\le\tet\rangle$, $\langle \rho_i\mid i <\tet\rangle$ and $\langle\rho'_i\mid i<\tet\rangle$ such that for every $i<\tet$ $\rho_i$ is between $\nu_{\alp_i}$ and the length of the extender used over $\nu_{\alp_i}$, $cf\rho_i\ge\nu_{\alp_i}$ in the core model, $\rho'_i$ is the image of $\rho_i$ over $\nu_{\alp_i+1}$ and $cf\rho'_i<\nu_{\alp_i+1}$ in the core model. Then $cf\rho'_i<\nu_{\alp_i}$ again in the core model since $\rho'_i$ is the image of $\rho_i$ in the ultrapower and $\nu_{\alp_i+1}$ the image of $\nu_{\alp_i}$ which is the critical point of the embedding. Fix for every $i<\tet$ a sequence $c_i$ unbounded in $\rho'_i$, in the core model and of cardinality $cf\rho'_i$ there. Take a precovering set including $\{c_i\mid i<\tet\}$. By \[Git-Mit\], assignment functions can change for this new precovering set only on a bounded subset of $\nu_{\alp_i}$’s. Pick $i<\tet$ such that $\nu_{\alp_i}$ is above supremum of this set. Again, consider the ultrapower used to move from $\nu_{\alp_i}$ to $\nu_{\alp_i+1}$. Now we have $c_i$ in this ultrapower and its cardinality is $<\nu_{\alp_i}$. Let $j:M\to M'$ be the embedding. $c_i\in M'$ and $M'$ is an ultrapower by extender. Hence for some $\tau$ and $f$ $c_i=j(f)(\tau)$. Let $U_\tau=\{X\subseteq\nu_{\alp_i}\mid\tau\in j(X)\}$ and $\tilj:M\to\tilM$ be the corresponding ultrapower. Denote $\tilj(\nu_{\alp_i})$ by $\tilnu_{a_i +1}$, $\tilj(\rho_i)=\tilc_i$ and $\tilj(f)([id])=\tilc_i$. Let $\tilc_i=\langle j(f_\xi)([id])\mid \xi <\xi^*=cf\rho'_i=cf\tilrho_i \rangle$ be increasing enumeration (everything in the core model). Then for most $\bet$’s $(\MOD U_\tau)$ $f(\bet)=\langle f_\xi(\bet)\mid \xi <\xi^*\rangle$ will be a sequence in $M$ cofinal in $\rho_i$ of order type $\xi$. Which contradicts the assumption that $cf\rho_i\ge\nu_i$.$\square$ Let us use 1.9 in order to deduce the following: [**Theorem 1.10.**]{} [**Remark.**]{}If $a$ is an interval then $|pcf a|=|a|$ by \[Git-Mit, 3.24\]. [**Proof.**]{}Suppose that for some $a$ as in the statement of the theorem $|pcf a|>|a|+\aleph_1$. Let $\tet =|a|^++\aleph_2$. Then $|pcf a|\ge \tet$. Pick an increasing sequence $\langle\tau_\alp\mid\alp <\tet\rangle$ inside $pcf (a)$. By 1.9 we can find an unbounded subset $S$ of $\tet$ satisfying the conclusion (\*) of 1.9. Let $D$ be an ultrafilter on $\tet$ including all cobounded subsets of $S$. Let $\tau = cf(\prodl_{\alp <\tet}\tau_\alp/D)$. Then, clearly, $\tau\ge(\bigcup_{\alp <\tet}\tau_\alp)^+$. By the Localization Theorem \[Sh-g\], then there is $a_0\subseteq\{ \tau_\alp\mid\alp\in S\}$, $|a_0|\le |a|$ with $\tau\in pcf a_0$. Consider $S\bks\sup a_0$. $S\bks\sup a_0\in D$ since $a_0$ is bounded in $S$. Hence $cf\Big(\prodl_{\alp\in S\bks\sup a_0}\tau_\alp\Big/D \Big)=\tau$. Again by the Localization Theorem, there is $a_1\subseteq S\bks\sup a_0$, $|a_1|\le |a|$ and $\tau\in pcf a_1$. Continue by induction and define a sequence $\langle a_\alp \mid \alp <\ome_1\rangle$ such that for every $\alp <\ome_1$ the following holds: - $a_\alp\subseteq S$ - $|a_\alp|\le |a|$ - $\tau\in pcf a_\alp$ - $\min a_\alp >\sup a_\bet$ for every $\bet<\alp$. Let $\del =\bigcup_{\alp <\ome_1}\sup a_\alp$. Then $\del$ is a limit of points of $S$ and $cf \del =\aleph_1$. Hence (\*) of 1.9 applies. Thus $tcf\Big(\prodl_{\alp\in\del\cap S}\tau_\alp / J^{bd}_{\del\cap S}\Big)$ exists is below $\tau_{\del +1}$ and is equal to $tcf\Big(\prodl_{\alp\in\del\cap S}\tau_\alp/F\Big)$ for every ultrafilter $F$ on $\del\cap S$ including all cobounded subsets of $\del\cap S$. Denote $tcf\Big(\prodl_{\alp\in\del\cap S} \tau_\alp/J^{bd}_{\del\cap S}\Big)$ by $\mu$. Let $c=pcf (a)$ and $\langle b_\xi[c]\mid\xi\in pcf (a)=c\r$ be a generating sequence. Clearly both $\mu$ and $\tau$ are in $c$ and $\mu <\tau$. Consider $b=b_\tau [c]\bks b_\mu [c]$. For every $\alp <\ome_1$, $b\cap a_\alp\not=\emptyset$, since $\tau\in pcf (a_\alp)$. Hence, $b\cap\del\cap S$ is unbounded in $\del$ (by (d) of the choice of $a_\alp$’s). Let $F$ be an ultrafilter on $\del\cap S$ including $b\cap\del\cap S$ And all cobounded subsets of $\del \cap S$. Then $tcf \Big(\prodl_{\alp\in\del\cap S}\tau_\alp/ F\Big) =\mu$ but this means that $\mu\in pcf b$, which is impossible by the choice of $b$, see for example \[Bur-Mag, 1.2\].$\square$ The proof of 1.10 easily gives a result related to the strength of the negation of the Shelah Weak Hypothesis (SWH). (SWH says that for every cardinal $\lam$ the number of singular cardinals $\kap <\lam$ with $pp\kap\ge\lam$ is at most countable). [**Theorem 1.10.1.**]{} Now we continue the task started in 1.1. and deal with ordinal gaps. Let us start with technical definitions. [**Definition 1.11.**]{}Let $$\text{Kinds}=\Big\{\del^{\ell_0}_0\cdot\del^{\ell_1}_1 \cdots\del^{\ell_{k-1}}_{k-1}\Big| k <\ome, 1\le\ell_0\nek \ell_{k-1} <\ome, \del_0 >\del_1 >\cdots\del_{k-1}\quad\text{are cardinals}$$ of uncountable cofinality $\Big\}\cup\{ 0\}$, where the operations used are the ordinals operations. [**Remark 1.12.**]{}The only kinds around $\ome_1$ are $\ome_1$ itself, $\ome^2_1\nek\ome^n_1 \cdots (n <\ome)$. But already with $\ome_2$ we can generate in addition to $\ome_2,\ome^2_2\nek \ome^n_2\cdots (n<\ome)$ also $\ome_2\cdot\ome^5_1$, $\ome^{19}_2\cdot\ome^3_1$ etc. Note that between $\ome^\ome_1$ and $\ome_2$ there are no new kinds. Using the Rado-Milner paradox we will show in the next section that the consistency strength of the length the gap does not change in such an interval. [**Definition 1.13.**]{}Let $\gam$ be an ordinal - $\gam$ is of kind $0$ if $\gam$ is a limit ordinal. - $\gam$ is of kind $\del_0$ for a cardinal $\del_0\in {\rm Kinds}$ if $\gam$ is a limit of an increasing sequence of length $\del_0$. In particular, if $\del_0$ is regular this means that $cf\gam =\del_0$. - $\gam$ is of kind $\del_0^{\ell_0}\cdot \del_1^{\ell_1}\cdots \del^{\ell_{k-1}}_{k-1}\in {\rm Kinds}$, with $\ell_0>1$, if $\gam$ is a limit of an increasing sequence of $\del_{k-1}$ ordinals of kind $\del^{\ell_0}_0\cdot\del^{\ell_1}_1\cdots \del^{\ell_{k-1}-1}_{k-1}$. [**Lemma 1.14.**]{} [**Remark 1.15.**]{} - The lemma provides a bit more information then will be needed for deducing the strength of $2^\kap =\kap^{+\xi +1}$. - The condition (2) is not very restrictive since we are interested in small $(<\kap)$ gaps between $\kap$ and its power. [**Proof.**]{}We prove the statement by induction on $\xi$. Fix $\alp <\del^+$. Let $\xi =\del^{\ell_0}_0\cdots\del^{\ell_{k-1}}_{k-1}$, where $\del_0=\del$. Set for each $\sig <\del_{k-1}$ $$\kap(\sig)=\kap^{+\alp +\del_0^{\ell_0}\cdots \del^{\ell_{n-2}}\cdot\del^{\ell_{n-1}-1}_{k-1}\cdot\sig +\del_0^{\ell_0}\cdots\del^{\ell_{k-2}}_{k-2}\cdot \del^{\ell_{k-1}-1}_{k-1}+1}$$ if $(k>1)$ or $(k=1$ and $\ell_0 >1$) and $$\kap (\sig)=\kap^{+\alp +\sig +1}$$ if $k=1$ and $\ell_0=1$, i.e. $\xi=\del$. For every $\sig <\del_{k-1}$, if $\xi\not=\del$ then by induction $\kap(\sig)\in pcf(\{\tau^{+\nu +1}_{n,i}|\nu$ is an ordinal of kind $\del_0^{\ell_0}\cdots \del_{k-2}^{\ell_{k-2}}\cdot\del_{k-1}^{\ell_{k-1}-1}, i<i(n), n<\ome\})$. Let $E$ be the set consisting of all regular cardinals of blocks $B_{n,i} (n<\ome, i<i(n))$ together with all regular cardinals between $\kap$ and $\min\Big(\kap^{+\del^+} ,2^\kap)$. Set $E^*=pcf E$. Then $\kap >|pcf E^*|$, since $\kap$ is strong limit. We can assume also that $\min E^*>|pcf E^*|$. By \[Sh-g\], then $pcf E^*=E^*$ and there is a set $\langle b_\chi [E^*]\mid\chi\in E^*\rangle$ of $pcf E^*$ generators which is smooth and closed, i.e. $\tau\in b_\chi [E^*]$ implies $b_\tau [E^*]\subseteq b_\chi [E^*]$ and $pcf (b_\chi [E^*])=b_\chi [E^*]$. The assumption (2) of the lemma implies that for every unbounded in $\kap^{+\alp+\xi}$ set $B$ consisting of regular cardinals above $\kap$ and below $\kap^{+\alp +\xi}$ $\max pcf(B)=\kap^{+\alp +\xi +1}$. In particular $\max pcf \{\kap (\sig)|\sig <\del_{k-1}\})=\kap^{+\alp +\xi +1}$. Denote $\kap^{+\alp +\xi +1}$ by $\mu$. Let $$A^*=b_\mu [E^*]\cap \{\kap (\sig)\mid\sig <\del_{k-1}\}\ .$$ Then, $|A^*|=\del_{k-1}$ and for every $\lam\in A^*$ $b_\lam [E^*]\subseteq b_\mu[E^*]$. For every $\lam\in A^*$, fix a sequence $\langle \rho^\lam_n\mid n<\ome\rangle\in\prodl_{n<\ome}\kap^+_{n+1}$ inside $b_\lam[E^*]$ such that - $\rho^\lam_n\in B_{n,i}$ for some $i<i(n)$ and, if $\xi\not=\del$ then also - $\rho^\lam_n$ is of kind $\del_0^{\ell_0}\cdots\del_{k-2}^{\ell_{k-2}}\cdot \del_{k-1}^{\ell_{k-1}-1}$. It is possible to find $\rho^\lam_n$’s of the right kind using the inductive assumption, as was observed above. [**Claim 1.16.**]{}There are infinitely many $n<\ome$ such that $$|\{\rho^\lam_n\mid\lam\in a^*\}|=\del_{k-1}$$ [**Proof.**]{}Otherwise by removing finitely many $n$’s or boundedly many $\rho^\lam_n$’s we can assume that for every $n$ $|\{\rho^\lam_n\mid\lam\in A^*\}|<\del_{k-1}$. But $cf\del_{k-1} >\aleph_0$. Hence, the total number of $\rho^\lam_n$’s is less than $\del_{k-1}$. Now, $pcf\{\rho^\lam_n\mid n<\ome\ ,\ \lam\in A^*\}\supseteq A^*$. So, $|A^*\cap pcf\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}|\ge |A^*|=\del_{k-1}$. By (2) of the statement of the lemma this situation is impossible. $\square$ of the claim. Suppose for simplicity that each $n<\ome$ satisfies the conclusion of the claim. If not then we just can remove all the “bad" $n$’s. This will effect less than $\del_{k-1}$ of $\rho$’s which in turn effects less than $\del_{k-1}$ of $\lam$’s. Let us call a cardinal $\tau$ reasonable, if for some $n<\ome$ $\tau$ is a limit of $\del_{k-1}$-sequence of elements of $\{\rho^\lam_n\mid \lam\in A^*\}$. Clearly, a reasonable $\tau$ is of kind $\del_0^{\ell_0}\cdots \del_{k-1}^{\ell_{k-1}}$, since $\rho^\lam_n$’s are of kind $\del_0^{\ell_0}\cdot\del_1^{\ell_1}\cdots \del_{k-2}^{\ell_{k-2}}\cdot\del_{k-1}^{\ell_{k-1}-1}$. The successor of such $\tau$ is in $pcf\{\rho^\lam_n \mid\lam\in A^*\}$ since $cf\tau=cf\del_{k-1}$ and we assumed $SSH^{cf\del_{k-1}}_{<\kap}$, i.e. $pp\tau =\tau^+$. Also $pp\tau=\tau^+$ implies that the set $\{\rho^\lam_n\mid\lam\in A^*\}\bks b_{\tau^+}[E^*]$ is bounded in $\tau$. [**Claim 1.17.**]{}$pcf\{\tau^+\mid\tau$ is reasonable$\}\subseteq b_\mu[E^*]$. [**Proof.**]{}$\{\rho^\lam_n\mid n<\ome\}\subseteq b_\lam [E^*]$ for every $\lam\in A^*$. Also, $b_\lam [E^*]\subseteq b_\mu [E^*]$. By the above, for every reasonable $\tau$, $\tau^+\in pcf\{\rho^\lam_n\mid\lam\in A^*\}$ for some $n<\ome$. But $pcf(b_\mu[E^*])=b_\mu [E^*]$ and $pcf\{\rho^\lam_n| n<\ome,\lam\in A^*\}\subseteq pcf (b_\mu [E^*])$ since the pcf generators are closed and $\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}\subseteq b_\mu[E^*]$. So, $\{\tau^+\mid\tau\ \hbox{is reasonable}\}\subseteq b_\mu [E^*]$ and again using closedness of $b_\mu[E^*]$, we obtain the desired conclusion. $\square$ of the claim. [**Claim 1.18.**]{}For every $\mu'\in pcf\{\tau^+\mid\tau\ \hbox{is reasonable}\}$, $b_{\mu'}[E^*]\subseteq b_\mu[E^*]$. [**Proof.**]{}By the smoothness of the generators $b_{\mu'}[E^*]\subseteq b_\mu[E]$ for every $\mu'\in pcf\{\tau^+\mid\tau\ \hbox{is reasonable}\}$. $\square$ of the claim. In order to conclude the proof we shall argue that there should be $\mu'\in pcf\{\tau^+\mid\tau$ is reasonable$\}$ such that $\mu\in b_{\mu'}[E^*]$. This will imply $b_\mu [E^*]=b_{\mu'}[E^*]$ and hence $\mu=\mu'$. Let us start with the following: [**Claim 1.19.**]{} $|\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}\bks\bigcup \{b_{\tau^+}[E^*]|\tau\ \hbox{is reasonable}\}|<\del_{k-1}$. [**Proof.**]{}Suppose otherwise. Let $S=\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}\bks \bigcup\{b_{\tau^+}[E^*]|\tau\ \hbox{is reasonable}\}$ and $|S|=\del_{k-1}$. Then for some $n<\ome$ also $\{\rho^\lam_n\mid\rho^\lam_n\in S\}$ has cardinality $\del_{k-1}$, since $cf\del_{k-1} >\aleph_0$. Fix such an $n$ and denote $\{\rho^\lam_n\mid\rho^\lam_n\in S\}$ by $S_n$. But now there is a reasonable $\tau$ which is a limit of elements of $S_n$. $pp\tau =\tau^+$ implies that the set $\{\rho^\lam_n\mid\lam\in A^*\}\bks b_{\tau^+}[E^*]$ is bounded in $\tau$. In particular, $S_n\cap b_{\tau^+}[E^*]$ is unbounded. Contradiction, since $S_n\subseteq S$ which is disjoint to every $b_{\tau^+}[E^*]$ with $\tau$ reasonable. $\square$ of the claim. Now, removing if necessary less than $\del$ elements, we can assume that $\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}$ is contained in $\cup\{b_{\tau^+} [E^*]\mid\tau\ \hbox{is reasonable}\}$. Recall that this can effect only less than $\del$ of $\lam$’s in $A^*$ which has no influence on $\mu$. Let $b=pcf\{\tau^+\mid\tau\ \hbox{is reasonable}\}$. Then $pcf b=b$ and $b\subseteq E^*$. By \[Sh-g\], there are $\mu_1\nek\mu_\ell\in pcf b=b$ such that $b\subseteq b_{\mu_1}[E^*]\cup\cdots\cup b_{\mu_\ell} [E^*]$. Using the smoothness of generators, we obtain that for every reasonable $\tau$ there is $k$, $1\le k\le\ell$ such that $b_{\tau^+}[E^*]\subseteq b_{\mu_k}[E^*]$. Now, $\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}\subseteq\cup \{b_{\tau^+}[E^*]\mid\tau\ \hbox{is reasonable}\}$. Hence, $\{\rho^\lam_n\mid n<\ome,\lam\in A^*\}\subseteq\bigcup^\ell_{k=1}b_{\mu_k}[E^*]$. For every $\lam\in A^*$ fix an ultrafilter $D_\lam$ on $\ome$ including all cofinite sets so that $tcf\Big(\prodl_{n<\ome}\rho^\lam_n\Big/D_\lam \Big)=\lam$. Let $\lam\in A^*$. There are $x_\lam\in D_\lam$ and $k(\lam)$, $1\le k(\lam)\le\ell$ such that for every $n\in x_\lam$ $\rho^\lam_n\in b_{\mu_{k(\lam)}}[E^*]$. Then $\lam\in pcf\Big(b_{\mu_{k(\lam)}}[E^*]\Big)= b_{\mu_{k(\lam)}}[E^*]$. Finally, we find $A^{**}\subseteq A^*$ of cardinality $\del_{k-1}$ (or just unbounded in $\mu$) and $k^*$, $1\le k^*\le\ell$ such that for every $\lam\in A^{**}$ $k(\lam)=k^*$. Then $A^{**}\subseteq b_{\mu_{k^*}}[E^*]$. But, recall that $\mu =\max pcf (B)$ for every unbounded subset $B$ of $A^*$. In particular, $\mu =\max pcf (A^{**})$. Hence, $\mu\in pcf A^{**}\subseteq pcf\Big(b_{\mu_{k^*}}[E^*] \Big)=b_{\mu_k^*}[E^*]$. $\square$ Lemma 1.14 implies the following: [**Theorem 1.20.**]{} [**Proof.**]{}By 1.14, for infinitely many $n$’s for some $i_k <i(n)$ the length of the block $B_{n,i*_n}$ will be at least $\tau^{+\xi +1}_{n,i_n}$, since it should contain some $\tau^{+\nu +1}_{n,i_n}$ for an ordinal $\nu$ of kind $\xi$. Clearly, $\nu\ge\xi$ since $\xi$ is the least ordinal of kind $\xi$. $\square$ We like now outline a way to remove (2) of 1.20 by cost of restricting possible $\xi$’s. First change Definitions 1.11 and 1.13. Thus in 1.11 we replace uncountable by “above $\aleph_1$". Denote by Kinds$^*$ the resulting class. Then define kind$^*$ of ordinal as in 1.13 replacing Kinds by Kinds$^*$. [**Theorem 1.21.**]{} The theorem, as in the case of 1.20, will follow from the following: [**Lemma 1.22.**]{} Let us first deal with a special case – $\xi$ is a cardinal. We split it into two cases: (a) $\xi$ is regular and (b) $\xi$ is singular. The result will be stronger than those of 1.22. [**Lemma 1.23.**]{} [**Proof.**]{}Let $\mu =\kap^{+\alp +\del +1}$. We choose $E^*$ and $\langle b_\chi[E^*]\mid\chi\in E^*\rangle$ as in the proof of 1.14. Measurables of a core model between $\kap$ and $2^\kap$ are allowed here. So in contrast to 1.14 we cannot claim anymore for every unbounded $B\subseteq [\kap,\kap^{+\alp +\del})$ consisting of regulars $\max pcf (B)=\kap^{+\alp +\del +1}$. Hence the choice of $A^*$ (the crucial for the proof set in 1.14) will be more careful. Set $A$ to be the set of cardinals $\kap^{+\alp +\tau +1}\in [\kap^{+\alp +1}, \kap^{+\alp+\del})$ such that either $o(\bet)< \kap^{+\alp +\tau}$ for every $\bet <\kap^{+\alp +\tau}$ or else $\kap^{+\alp +\tau}$ is above every measurable of the core model smaller than $\kap^{+\alp +\del}$. Clearly, $|A|=\del$, since there is no overlapping extenders and as in 1.1 $|(pcf b)\cap A|\le |b|$ for every set of regular cardinals $b\subseteq\kap$, $|b|\le\del$. By 1.3, $\max pcf (B)= \kap^{+\alp +\del +1}$ for every unbounded $B\subseteq A$. This implies that $A\bks b_\mu [E^*]$ is bounded in $\kap^{+\alp +\del +1}$. Define $A^*=A\cap b_\mu [E^*]$. The rest of the proof completely repeats 1.14. $\square$ [**Lemma 1.24.**]{} [**Proof.**]{}Let $\langle\del_i\mid i<cf\del\rangle$ be an increasing continuous sequence of limit cardinals unbounded in $\del$. Consider the set $$B=\{\kap^{+\alp +\del_i+\nu}\mid i<cf\del\ ,\ i\quad\hbox{limit and}\quad \nu <\del_i\} .$$ Since $cf\del >\aleph_0$, the analysis of indiscernibles of \[Git-Mit, Sec. 3.4\] can be applied to show that $\{cf (\prod B/D)|D$ is an ultrafilter over $B$ extending the filter of cobounded subsets of $B\}\subseteq\{\kap^{+\alp +\nu +1}\mid\del \le\nu\le\del +\del \}$. We cannot just stick to $\kap^{+\alp +\del +1}$ alone since we like to have $\del$ cardinals below $\kap^{+\alp +\del}$. But once measurable above $\kap$ allowed, it is possible that $\max pcf(\{\kap^{+\alp +\rho +1}\mid\rho <\del \})>\kap^{+\alp +\del +1}$. Still by \[Sh-g\], for a club $C\subseteq cf\del\quad tcf (\prodl_{\nu\in C}\kap^{+\alp +\nu +1}/$ cobounded $\upharpoonright C)=\kap^{+\alp+\del +1}$. Unfortunately, this provided only $cf\del$ many cardinals $\kap^{+\alp +\nu +1}$ and not $\del$-many. Define a filter $D$ over $B$: $X\in D$ iff $\{i <cf\del|i$ is limit and $\{ j<i|\{\nu <\del^+_j|\kap^{\alp +\del_i+\xi +1}\in X\}$ is cobounded in $\del^+_j\}$ is cobounded in $i\}$ contains a club. Let $D^*$ be an ultrafilter extending $D$. Set $\mu =cf (\prod B/D^*)$. By the choice of $D$, for every $C\subseteq B$ of cardinality less than $\del$ $B\bks C\in D$. So, $\mu\in [\kap^{+\alp +\del +1}, \kap^{+\alp +\del +\del +1}]$. Define $E^*$ as before. Set $A^*=B\cap b_\mu[E^*]$. [**Claim 1.25.**]{}If $A^*\in D^*$. [**Proof.**]{}Otherwise the compliment of $A^*$ is in $D^*$. Let $A'=B\bks b_\mu[E^*]$. Clearly, $D^*\cap J_{<\mu}[E^*] =\emptyset$. By \[Bur-Mag, 1.2\], then there is $S\in D^*$ $S\in J_{<\mu^*}[E^*]\bks J_{<\mu}[E^*]$. But $b_\mu [E^*]$ generates $J_{<\mu}[E^*]$ over $J_{<\mu}[E^*]$. So, $S\subseteq b_\mu[E^*] \cup c$ for some $c\in J_{<\mu}[E^*]$. Hence, $S\cap b_\mu [E^*]\in D^*$. But $A'\in D^*$ and $A'\cap B\cap (S\cap b_\mu [E^*])=\emptyset$. Contradiction. $\square$ of the claim. Now we continue as in the proof of 1.14. In order to eliminate possible effects of less than $\del$ cardinals, we use 1.10. At the final stage of the proof a set $A^{**}$ was defined. Here we pick it to be in $D^*$. This insures that $\mu\in pcf A^{**}$ and we are done. $\square$ Now we turn to the proof of 1.22. [**Proof.**]{}As in 1.14, we prove the statement by induction on $\xi$. Fix $\alp <\del^+$. Let $\xi =\del_0^{\ell_0}\cdots \del_{k-1}^{\ell_{k-1}}$. The case $k=1$ and $\ell_0=1$ (i.e. $\xi =\del$) was proved in 1.23, 1.24. So assume that $k>1$ or $(k=1$ and $\ell_0 >1)$. For each $\sig <\del_{k-1}$ let $$\kap (\sig)\in pcf (\{\tau^{+\nu +1}_{n,i}\mid i<i(n), n<\ome\ ,\quad\text{and}\quad\nu\quad\text{is an ordinal of kind}^*\quad \del_0^{\ell_0}\cdots \del_{k-2}^{\ell_{k-2}}\cdot\del_{k-1}^{\ell_{k-1}-1}\}) \cap$$ $$[\kap^{+\alp+\xi^-\cdot\sig +\xi^-+1},\kap^{+\alp +\xi^-\cdot\sig+ \xi^-+ \xi^-+1}]\ ,\quad {\rm where}$$ $$\xi^-= \begin{cases} \del_0^{\ell_0}\cdots\del_{k-2}^{\ell_{k-2}}\cdot \del_{k-1}^{\ell_{k-1}-1}\ ,if&\xi =\del_0^{\ell_0} \cdots\del_{k-1}^{\ell_{k-1}}\quad\text{and}\quad (k>1\quad {\rm or}\quad (k=1\quad {\rm and}\quad \ell_0>1)\\ 0,&if\quad k=1\quad\text{and}\quad\ell_0=1 \end{cases}$$ In the last case the inductive assumption insures the existence of such $\kap (\sig)$. Define $E^*$ and $\langle b_\chi[E^*]|\chi\in E^*\rangle$ as in the proof of 1.14. We do not know now if for every unbounded in $\kap^{+\alp +\xi}$ set $B\subseteq [\kap,\kap^{+\alp +\xi})$ consisting of regular cardinals $\max pcf (B)=\kap^{+\alp +\xi +1}$. We may consider the set $\{\kap^{+\alp +\xi^-\cdot\nu +1}\mid\nu <\del_{k-1}\}$. If for club many $\nu$’s $\kap^{+\alp +\xi^-\cdot\nu +1}$ is not a principle indiscernible then by \[Git-Mit\] $cf (\prod B/{\rm bounded})=\kap^{+\alp +\xi +1}$ for any unbounded subset $B$ of $\kap^{+\alp +\xi}$ consisting of regular cardinals. Note that $cf\del_{k-1}>\aleph_0$ is crucial here. In this case we define $A^*=\{\kap(\sig)\mid\sig <\del_{k-1}\}\cap b_{\kap^{+\alp +\xi +1}}[E^*]$ and proceed as in the proof of 1.14. The only difference will be the use of 1.10 to eliminate a possible influence of $<\del_{k-1}$ cardinals. Here the assumption $\del_{k-1}>\aleph_1$ comes into play. In the general case it is possible to have $\{\kap (\sig )\mid \sig <\del_{k-1}\}\cap b_{k^{+\alp+\xi+1}}[E^*]$ empty. But once for a club of $\nu$’s below $\del_{k-1}$ $\kap^{\alp +\xi^-\cdot\nu +1}$’s are principal indiscernibles, by \[Git-Mit\] we can deduce that $$pcf(\{\kap (\sig)\mid\sig <\del_{k-1}\})\bks \kap^{+\alp+\xi}\subseteq$$ $$[\kap^{+\alp+\xi +1},\kap^{+\alp +\xi +\xi^-+\xi^-+1}]\subseteq [\kap^{+\alp +\xi +1},\kap^{+\alp +\xi +\xi +1}]\ .$$ Let $D$ be an ultrafilter on the set $\{\kap (\sig)\mid\sig <\del_{k+1}\}$ containing all cobounded subsets. Set $$\mu =cf (\prod \{\kap (\sig)\mid \sig <\del_{k-1}\}/D)\ .$$ Define $A^*=b_\mu [E^*]\cap \{\kap (\sig)\mid \sig <\del_{k-1}\}$. By Claim 1.25, then $A^*\in D$. From now we continue as in 1.14 only using 1.10 in a fashion explained above and at the final stage picking $A^{**}$ inside $D$. $\square$ [**Remark 1.26.**]{}The use of Kinds$^*$ and not of Kinds in 1.21 (or actually in 1.22) is due only to our inability to extend 1.10 in order to include the case of a countable set. Still in view of 1.1 and also 1.23, 1.24, the first unclear case will not be $\ome_1$ but rather $\ome_1+\ome_1$. Some Related Forcing Constructions ================================== In this section we like to show that (1) it is impossible to remove SSH assumptions from Theorem 1.6; (2) the conclusion of Theorem 1.11 is optimal, namely, starting with $\kap =\bigcup_{n<\ome}\kap_n$, $\kap_0 <\kap_1<\cdots <\kap_n <\cdots$ and $o(\kap_n) =\kap^{+\del^n+1}_n +1$ we can construct a model satisfying $2^\kap \ge\kap^{+\alp}$ for every $\alp <\del^+$, where $\del$ as in 1.9 is a cardinal of uncountable cofinality; (3) the forcing construction for $\del$’s of cofinality $\aleph_0$ will be given. All these results based on forcing of \[Git1\] and we sketch them modulo this forcing. [**Theorem 2.1**]{} [**Proof.**]{}Without loss of generality we can assume that $\del$ is a regular cardinal. We pick an increasing sequence $\langle \kap_n\mid n<\ome\rangle$ converging to $\kap$ so that for every $n<\ome$ $o(\kap_n)=\kap^{+n+2}_n+\del +1$. Fix at each $n$ a coherent sequence of extenders $\langle E^n_i\mid i\le\del\rangle$ with $E^n_i$ of the length $\kap^{+n+2}_n$. We like to use the forcing of \[Git1, Sec. 2\] with the extenders sequence $\langle E^n_\del \mid n<\ome\rangle$ to blow power of $\kap$ to $\kap^{++}$ together with extender based Magidor forcing changing cofinality of the principal indiscernible of $E^n_\del$ to $\del$ (for every $n<\ome)$ simultaneously blowing its power to the double plus. We refer to M. Segal \[Seg\] or C. Merimovich \[Mer\] for generalizations of the Magidor forcing to the extender based Magidor forcing. The definitions of both of these forcing notions are rather lengthy and we would not reproduce them here. Instead let us emphasize what happens with indiscernibles and why (iii) of the conclusion of the theorem will hold. Fix $n<\ome$. A basic condition of \[Git1, Sec. 2\] is of the form $\langle a_n,A_n,f_n\rangle$, where $a_n$ is an order preserving function from $\kap^{++}$ to $\kap_n^{+n+2}$ of cardinality $<\kap_n$, $A_n$ is a set of measure one for the maximal measure of $rng a_n$ which is in turn a measure of the extender $E^n_\del$ over $\kap_n$. The function of $f_n$ is an element of the Cohen forcing over a $\kap^+$. Each $\alp\in dom a_n$ is intended to correspond to indiscernible which would be introduced by the measure $a_n(\alp)$ of $E^n_\del$. In present situation we force over the principal indiscernible $\del_n$, i.e. one corresponding to the normal measure of $E^n_\del$. The extender based Magidor forcing changes its cofinality to $\del$ and adds for every $\gam$ $\rho_n\le\gam\le\rho_n^{+n+2}$ a sequence $t_{n\gam}$ of order type $\del$ cofinal in $\rho_n$. Actually, $t_{n,\rho_n^{+n+2}}(i)= \rho_{n,i}^{+n+2}(i<\del)$, where $\langle\rho_{ni}|i<\del \rangle$ is the sequence $t_{n\rho_n}$. Now, if $\gam <\rho_n^{+n+2}$ is produced by $a_n(\alp)$, then we connect $\alp$ with the sequence $t_{n\gam}$ in addition to its connection with $\gam$. Using standard arguments about Prikry type forcing notions, it is not hard to see that $cf\Big(\prodl_{n<\ome}\rho^{+n+2}_{n,i},{\rm finite}\Big)=\kap^{++}$ for every $i<\del$ as witnessed by $t_{n\gam}(i)'s$.$\square$ [**Remark 2.2**]{}Under the assumptions of the theorem, one can obtain $2^\kap\ge\kap^{+\alp}$ for any countable $\alp$. But we do not know whether it is possible to reach uncountable gaps. See also the discussion in the final section. [**Theorem 2.3**]{} [**Remark 2.4**]{} By the results of the previous section, this is optimal if $\alp\in [\bigcup_{n<\ome}\del^n,\del^+)$, at least if one forces over the core model. [**Proof.**]{}Fix an increasing sequence $\kap_0<\kap_1<\cdots <\kap_n<\cdots$ converging to $\kap$ so that each $\kap_n$ carries an extender $E_n$ of the length $\kap_n^{+\del^n}$. W.l. of $g$. $\alp\ge \bigcup_{n<\ome}\del^n$. We use the Rado-Milner Paradox (see K. Kunen \[Kun, Ch. 1, ex. 20\]) and find $X_n\subset\alp (n\in\ome)$ such that $\alp =\bigcup_{n<\ome} X_n$ and $otp (X_n)\le\del^n$. W.l. of $g$. we can assume that each $X_n$ is closed and $X_n\subseteq X_{n+1}(n<\ome)$. Now the forcing similar to those of \[Git1, 5.1\] will be applied. Assign cardinals below $\kap$ to the cardinals $\{\kap^{+\bet +1}\mid 1\le\bet\le\alp\}$ as follows: at level $n$ elements of the set $\{\kap^{+\bet +1}\mid\bet +1\in X_n\}$ will correspond to elements of the set $\{\kap^{+n+\gam+1}_n\mid \gam <\del^n\}$. The next definition repeats 5.2 of \[Git1\] with obvious changes taking in account the present assignment. [**Definition 2.5**]{}The forcing noting $\cal{P}(\alp)$ consists of all sequences $\l\l A^{0\nu},A^{1\nu},F^\nu\r\mid\nu\le\alp\rangle$ so that 1. $\l\l\langle A^{0\nu},A^{1\nu}\rangle\mid\nu\le \alp\rangle$ is as in 4.14 of \[Git1\]. 2. for every $\nu\le\alp$ $F^\nu$ consists of $p=\langle p_n\mid n<\ome\rangle$ and for every $n\ge\ell (p)$, $p_n=\langle a_n,A_n,f_n\rangle$ as in 4.14 of \[Git1\] with the following changes related only to $a_n$; <!-- --> 1. $a_n(\kap^{+\nu})= \kap_n^{+\varphi_n(\nu)}$ where $\varphi_n$ is some fixed in advance order preserving function from successor ordinals in $X_n$ to successor ordinals of $[n+2,\del^n)$. 2. only of cardinalities $\kap^{+\nu}$ for $\nu\in X_n\cap$ Successors can appear in dom $a_n$. The rest of the argument repeats those of \[Git1\]. The following is a more general result that deals with all kinds (i.e. elements of Kinds) of ordinals and not only with $\del^n$’s. [**Theorem 2.6.**]{} Again, this is optimal by results of the previous section, if $$\alp\in [\bigcup_{n<\ome}\del_0^{\ell_0}\cdots \del_{k-1}^{\ell_{k-1}}\cdot\del^n\ ,\ \del_0^{\ell_0} \cdots \del_{k-1}^{\ell_{k-1}}\cdot\del^+)$$ at least if one forces over the core model in case $\del =\aleph_1$. The construction is parallel to those of 2.3, only we use the following version of Rado-Milner Paradox: For every $\alp\in [\bigcup_{n<\ome}\del_0^{\ell_0}\cdots \del_{k-1}^{\ell_{k-1}}\cdot\del^n$, $\del_0^{\ell_0}\cdots \del_{k-1}^{\ell_{k-1}}\cdot\del^+)$ there are $X_n\subseteq \alp (n<\ome)$ such that $\alp =\bigcup_{n<\ome}X_n$ and $otp (X_n)\le\del_0^{\ell_0}\cdots\del_{k-1}^{\ell_{k-1}} \cdot\del^n$. $\square$ Under the same lines we can deal with gaps of size of a cardinal of countable cofinality below $\kap$. Thus the following result which together with the results of the previous section provides the equiconsistency holds: [**Theorem 2.7**]{} The proof is similar to those of 2.3. Only notice that we can present $\alp$ as an increasing union of sets $X_n(n<\ome)$ with $|X_n|<\del$ since $\alp <\del^+$, $cf\del =\ome$ and there is a function from $\del$ onto $\alp$. Concluding Remarks and Open Questions ===================================== Let us first summarize in the table below the situation under $SSH_{<\kap}$ (i.e. for every singular $\mu <\kap$ $pp\mu=\mu^+$) assuming that $2^\kap\ge\kap^{+\del}$ for some $\del$, where $\kap$ as usual here in a strong limit cardinal of cofinality $\aleph_0$. For $\del =\aleph^\ell_1$, for $2\le\ell <\ome$, in the cases dealing with ordinals in ${\rm Kinds}\bks{\rm Kinds}^*$ we assume in addition that there is no measurable of the core model between $\kap$ and $\kap^{+\del}$. -------------------------------- ------------------------ ---------------------------------------------------------- ---------------------------------------------------------- $\delta = 2$ $2 < \delta < \aleph_0$ $cf|\delta| =\aleph_0$ $\delta$ is a cardinal $o(\kappa) \ge \kappa^{+\delta + 1} + 1$ $\kappa > \delta \ge \aleph_0$ $cf |\delta | or > \aleph_0$ $\{\alpha < \kappa | o(\alpha) \ge \alpha^{+\delta + 1} + 1\}$ is unbounded in $\kappa$ $\delta = |\delta|^\ell$, $o(\kappa) \ge \kappa^{+|\delta|^\ell + 1} + 1$ for some or $1 < \ell < \omega$ $\{\alpha < \kappa | o(\alpha) \ge^{+|\delta|^\ell + 1} + 1\}$ is unbounded in $\kappa$ $\delta\ge\bigcup\limits_{\ell < \omega} $\forall \ell < \omega |\delta|^\ell$ \{\alpha < \kappa|o(\alpha)\ge\alpha^{+|\delta|^\ell}\}$ [is unbounded in $\kappa$]{} $\delta_0^{\ell_0}\cdots\delta_{k-1}^{\ell_{k-1}}\cdot $\forall \delta^\omega_k{\le}\del{<}\del_0^{\ell_0}\cdots n{<}\ome\{\alp{<}\kappa\mid o(\alp){\ge} \delta_{k-1}^{\ell_{k-1}}\cdot\delta^+_k$ \alp^{+\delta_0^{\ell_0}\cdots\delta_{k-1}^{\ell_{k-1}} \cdot\delta^n_k}\}$ for some $\delta_0^{\ell_0}\cdots is unbounded in $\kappa$ \delta_{k-1}^{\ell_{k-1}}\cdot\del_k\in$ Kinds $\delta_0^{\ell_0}\cdots\delta_k^{\ell_k}\le\del $o(\kap)\ge\kap^{+\del_0^{\ell_0}\cdots <\del_0^{\ell_0}\cdots\del_k^{\ell_k} \del_k^{\ell_k}+1}+1$ \cdot\ome_1$ for some $\del_0^{\ell_0}\cdots\del_k^{\ell_k}\in$ Kinds or $\{\alp <\kap\mid o(\alp)\ge\alp^{+\del_0^{\ell_0}\cdots\del_k^{\ell_k}+1} +1\}$ is unbounded in $\kap$ $\del\ge\kap$ -------------------------------- ------------------------ ---------------------------------------------------------- ---------------------------------------------------------- The proofs are spread through the papers \[Git1,2,3,4,5\], \[Git-Mag\], \[Git-Mit\] and the present paper. The forcing constructions in these papers give GCH below $\kap$. Let us finish with some open problems. [**Question 1.**]{}Let $a$ be a countable set of regular cardinals. Does “$|pcf a|>|a|=\aleph_0$" imply an inner model with a strong cardinal? In view of 1.10, it is natural to understand the situation for countable $a$. Recall that the consistency of “$|pcfa|>|a|$" is unknown and it is a major question of the cardinal arithmetic. The next question is more technical. [**Question 2.**]{}Can the assumption that there are no measurables in the core model between $\kap$ and $2^\kap$ be removed in 1.11? It looks like this limitation is due only to the weakness of the proof. But probably there is a connection with “$|pcf a|>|a|"$. The simplest unclear case is $2^\kap\ge\kap^{+\ome^2_1}$. The situation without $SSH_{<\kap}$ is unclear. In view of 2.1 probably weaker assumptions then those used in the case of $SSH_{<\kap}$ may work. A simplest question in this direction is as follows. [**Question 3.**]{}Is “$\{\alp\mid o(\alp)\ge\alp^{+n}\}$ unbounded in $\kap$ for each $n<\ome$" sufficient for “$\kap$ strong limit, $cf\kap =\aleph_0$ and $2^\kap\ge\kap^{+\ome_1}$"? If the answer is affirmative, then the construction will require a new forcing with short extenders, which will be interesting by itself. We then conjecture that the same assumption will work for arbitrary gap as well. For uncountable cofinalities (i.e. $cf\kap >\aleph_0$), as far as we are concerned with consistency strength, the only unknown case is the case of cofinality $\aleph_1$. We restate a question of \[Git-Mit\]: [**Question 4.**]{}What is the exact strength of “$\kap$ is a strong limit, $cf\kap =\aleph_1$ and $2^\kap\ge\lam$ for a regular $\lam>\kap^+$? It is known that the strength lies between $o(\kap)=\lam$ and $o(\kap)=\lam +\ome_1$, see \[Git-Mit\]. [2]{} M. Burke and M. Magidor, Shelah’s pcf theory and its applications, APAL 50(1990), 207-254. M. Gitik, Blowing power of a singular cardinal-wider gaps, submitted to APAL. M. Gitik, Wide gaps with short extenders, math. LO/9906185. M. Gitik, The negation of SCH from $o(\kap) =\kap^{++}$, APAL 43(3) (1989), 209-234. M. Gitik, The strength of the failure of SCH, APAL 51(3) (1991), 215-240. M Gitik, There is no bound for the power of the first fixed point. M. Gitik and M. Magidor, The singular cardinals problem revisited in: H. Judah, W. Just and H. Woodin eds., Set Theory of the Continuum (Springer, Berlin, 1992) 243-279. M. Gitik and W. Mitchell, Indiscernible sequences for extenders and the singular cardinal hypothesis, APAL 82 (1996) 273-316. K. Kunen, Set Theory: An introduction to independence proofs, North-Holland Publ. Co., 1983. W. Mitchell, The core model for sequences of measures, Math. Proc. Cambridge Philos, Soc. 95(1984), 41-58. M. Segal, Master’s thesis, The Hebrew University, 1993. C. Merimovich, Extender Based Radin Forcing, submitted to Trans. AMS. S. Shelah, Cardinal arithmetic, Oxford Logic Guides 29 (1994). S. Shelah, The Singular Cardinal Problem. Independence Results, in: A. Mathias ed., vol. 87, London Math. Soc. Lecture Note Series, Surveys in Set Theory, 116-133.
--- bibliography: - 'GPbiblio4.bib' --- [**** ]{}\ Luca Ambrogioni^1^, Marcel A. J. van Gerven^1^, Eric Maris^1^\ **[1]{} Radboud University, Donders Institute for Brain, Cognition and Behaviour, Nijmegen, The Netherlands\ [email protected]** Abstract {#abstract .unnumbered} ======== Neural signals are characterized by rich temporal and spatiotemporal dynamics that reflect the organization of cortical networks. Theoretical research has shown how neural networks can operate at different dynamic ranges that correspond to specific types of information processing. Here we present a data analysis framework that uses a linearized model of these dynamic states in order to decompose the measured neural signal into a series of components that capture both rhythmic and non-rhythmic neural activity. The method is based on stochastic differential equations and Gaussian process regression. Through computer simulations and analysis of magnetoencephalographic data, we demonstrate the efficacy of the method in identifying meaningful modulations of oscillatory signals corrupted by structured temporal and spatiotemporal noise. These results suggest that the method is particularly suitable for the analysis and interpretation of complex temporal and spatiotemporal neural signals. Introduction {#introduction .unnumbered} ============ Human neocortex has an impressively complex organization. Cortical electrical activity is determined by dynamic properties of neurons that are wired together in large cortical networks. These neuronal networks generate measurable time series with characteristic spatial and temporal structure. In spite of the staggering complexity of cortical networks, electrophysiological measurements can often be properly described in terms of a few relatively simple dynamic components. By dynamic components we mean signals that exhibit characteristic properties such as rhythmicity, time scale and peak frequency. For example, neural oscillations at different frequencies are extremely prominent in electroencephalographic (EEG) and magnetoencephalographic (MEG) measurements and have been related to a wide range of cognitive and behavioral states [@cheyne2013meg; @roux2014working; @basar2013review]. Neural oscillations have been the subject of theoretical and experimental research as they are seen as a way to connect the dynamic properties of the cortex to human cognition [@tallon1999oscillatory; @engel2010beta; @kirschfeld2005alpha; @klimesch2007eeg; @jensen2010shaping]. Importantly, an oscillatory process can be described using simple mathematical models in the form of linearized differential equations [@bressloff2011spatiotemporal]. In this paper, we introduce a framework to integrate prior knowledge of neural signals (both rhythmic and broadband) into an analysis framework based on Gaussian process (GP) regression [@rasmussen2006gaussian]. The aim is to decompose the measured time series into a set of dynamic components, each defined by a linear stochastic differential equation (SDE). These SDEs determine a prior probability distribution through their associated GP covariance functions. The covariance function specifies the prior correlation structure of the dynamic components, i.e. the correlations between the components’ activity at different time points. Using this prior, a mathematical model of the signal dynamics is incorporated into a Bayesian data analysis procedure. The resulting decomposition method is able to separate linearly mixed dynamic components from a noise-corrupted measured time series. This is conceptually different from blind decomposition methods such as ICA and PCA [@comon1994independent; @tipping1999probabilistic] that necessarily rely on the statistical relations between sensors and are not informed by a prior model of the underlying signals. In particular, since each component extracted using the GP decomposition is obtained from an explicit model of the underlying process, these components are easily interpretable and can be naturally compared across different participants and experimental conditions. The GP decomposition can be applied to spatiotemporal brain data by imposing a spatial smoothness constraint at the level of the cortical surface. We will show that the resulting spatiotemporal decomposition is related to well-known source reconstruction methods [@hamalainen1994interpreting; @pascual2002functional; @tarantola2005inverse; @petrov2012harmony] and allows to localize the dynamic components across the cortex. The connections between EEG/MEG source reconstruction and GP regression have recently been shown by Solin et al. [@solin2016gprec]. Our approach complements and extends their work by introducing an explicit additive model of the underlying neural dynamics. Through computer simulations and analysis of empirical data, we show that the GP decomposition method allows to quantify subtle modulations of the dynamic components, such as oscillatory amplitude modulations, and does so more reliably than conventional methods. We also demonstrate that the output of the method is highly interpretable and can be effectively used for uncovering reliable spatiotemporal phenomena in the neural data. Therefore, when applied to the data of a cognitive experiment, this approach may give rise to new insights into how cognitive states arise from neural dynamics. Results {#results .unnumbered} ======= In the following, we will show how to construct a probabilistic model of the neural dynamics that captures the main dynamical features of the electrophysiological signals. The temporal dynamics of the neural sources are modeled using linear SDEs, and these in turn determine a series of GP prior distributions. These priors will be used to decompose the signal into several dynamic components with a characteristic temporal correlation structure. Building from the temporal model, we introduce a spatiotemporal decomposition method that can localize the dynamic components on the cortical surface. Decomposing a signal using temporal covariance functions {#decomposing-a-signal-using-temporal-covariance-functions .unnumbered} -------------------------------------------------------- #### Modeling neural activity with stochastic differential equations {#modeling-neural-activity-with-stochastic-differential-equations .unnumbered} We start our exposition by considering a single sensor that measures the signal produced by the synchronized subthreshold dynamics of some homogeneous neuronal population. Neural activity is defined for all possible time points. However, it is only observed through discretely-sampled and noise-corrupted measurements $y_t$. We assume the observation noise $\xi(t)$ to be Gaussian but not necessarily white. The effect of the discrete and noise-corrupted sampling is exemplified in Fig. \[fig1\]A, which shows a simulation of a continuous-time process sampled at regular intervals and corrupted by white noise. Modeling the neural signal as a continuous (rather than a discrete) time series has the advantage of being invariant under changes of sampling frequency and can also accommodate non-equidistant samples. Our prior of the temporal dynamics of the neural activity is specified using linear SDEs. For example, we model the neural oscillatory process $\varphi(t)$ using the following equation: $$\frac{d^{2}}{dt^{2}}\varphi(t)+b\frac{d}{dt}\varphi(t)=-\omega_{0}^{2}\varphi(t)+w(t)\,.\label{eq:oscillator SDE}$$ This differential equation describes a damped harmonic oscillator, which responds to input by increasing its oscillatory amplitude. The parameter $b$ regulates the exponential decay of these input-driven excitations. The frequency w of these excitations is equal to $\sqrt{\omega_{0}^{2}-\frac{1}{2}b^{2}}$. Clearly, this frequency is only defined for $\omega_{0}^{2}>\frac{1}{2}b^{2}$. For larger values of $b$, the system ceases to exhibit oscillatory responses and is said to be overdamped. These dynamical states are referred to as an oscillator in case $\omega_{0}^{2}>\frac{1}{2}b^{2}$ and an integrator in case $\omega_{0}^{2}<\frac{1}{2}b^{2}$ [@izhikevich2007dynamical]. We assume the process to be driven by a random input $w(t)$ (also denoted as *perturbation*). This random function models the combined effect of the synaptic inputs to the neuronal population that generates the signal. Fig. \[fig1\]B shows the expected value (black) and a series of samples (coloured) of the process, starting from an excited state $(\varphi(0)=0.4)$ and decaying back to its stationary dynamics. Note that the expected value converges to zero whereas the individual samples do not; this is due to the continued effect of the random input. Also note that the samples gradually become phase inconsistent, with the decay of phase consistency being determined by the damping parameter $b$. Thus, the damping parameter also determines the decay of the temporal correlations. In general, we model the measured time series as a mixture of four processes, which we will now describe. Of these four, one reflects rhythmic brain activity (i.e., an oscillation), two reflect non-rhythmic brain activity, and one accounts for the residuals: - *Damped harmonic oscillator.* Oscillations are a feature of many electrophysiological recordings [@gray1989stimulus; @lubenov2009hippocampal], and they are thought to be generated by synchronized oscillatory dynamics of the membrane potentials of large populations of pyramidal neurons [@silva1991intrinsic]. We model the neural oscillatory process as a stochastic damped harmonic oscillator as defined in Eq. (\[eq:oscillator SDE\]) with damping coefficient $b < \sqrt{2\omega_{0}^{2}}$. This linear differential equation can be obtained by linearizing a model of the neuronal membrane potential that is characterized by sub-threshold oscillations [@izhikevich2007dynamical]. - *Second order integrator.* We model the smooth non-oscillatory component of the measured time series using an equation of the same form as Eq. (\[eq:oscillator SDE\]) but in the overdamped state. We will denote this dynamic component as $\chi(t)$. In the overdamped regime, the equation has smooth, non-rhythmic solutions (see Fig. \[fig1\]C). Equations like these emerge by linearizing neuronal models around a non-oscillatory fixed point [@izhikevich2007dynamical]. - *First order integrator.* Most neurophysiological signals have a significant amount of energy in very low frequencies. We model this part of the signal with a simple first order SDE of which the covariance function decays exponentially. This process captures some of the qualitative features of the measured time series, such as roughness and non-rhythmicity. The model is determined by the following first order SDE: $$\frac{d}{dt}\psi(t) = -c\psi(t)+w(t)\,.\label{eq:first order SDE, results}$$ The positive number $c$ determines the exponential relaxation of the process, i.e. how fast its mean decays to zero after a perturbation. For a compact neuron this is a good model of the sub-threshold membrane potential under random synaptic inputs [@dayan2001theoretical]. See Fig. \[fig1\]D for some samples of this process. - *Residuals.* Finally, we account for the residuals $\xi(t)$ of our model using a process with temporal covariance that decays as $e^{-\frac{t^{2}}{2\delta^{2}}}$, where $\delta$ is a small time constant. This noise is characterized by short-lived temporal autocorrelations (see Fig. \[fig1\]E). As $\delta$ tends to zero, the process tends to Gaussian white noise. The temporal covariance of this component was not derived from a stochastic differential equation. #### From stochastic differential equations to Gaussian processes regression {#from-stochastic-differential-equations-to-gaussian-processes-regression .unnumbered} In our dynamical model, the random input is Gaussian and the dynamics are linear. The linearity implies that the value of the process at any time point is a linear combination of the random input at the past time points. As a consequence, because every linear combination of a set of Gaussian random variables is still Gaussian, the solutions of the SDEs are Gaussian. The Gaussian Process (GP) distribution is the generalization of a multivariate Gaussian for infinitely many degrees of freedom, where the covariance function of the former is analogous to the covariance matrix of the latter. As a zero-mean multivariate Gaussian distribution is fully specified by a covariance matrix, a zero-mean GP $\alpha (t)$ can be completely determined by its covariance function: $$k_{\alpha}(t,t')=\textnormal{cov}(\alpha(t),\alpha(t')) \label{eq:covariance function, results}$$ which captures the temporal correlation structure of the stochastic process $\alpha (t)$. In our case, the covariance function of the dynamical component $\varphi(t)$, $\chi(t)$ and $\psi(t)$ can be obtained analytically from Eq. (\[eq:oscillator SDE\]) and (\[eq:first order SDE, results\]). This allows to derive a GP distribution for each linear SDE. Moreover, a sum of independent GPs is again a GP, but now with a covariance function that is the sum of the covariance functions of each of its components. This decomposition of the covariance function is exemplified in Fig. \[fig1\]F, which shows the decomposition of the covariance function of a complex signal into several component-specific covariance functions, together with examples of the corresponding dynamic component time series. For visual clarity, the second order integrator component has been omitted from this figure. With these GPs as prior distributions, we can use Bayes’ theorem for estimating the time course of the dynamic components from the measured time series $y$. In particular, we assume that $y$ is generated by the sum of all dynamic components and corrupted by Gaussian noise $\xi(t)$. The aim is to individually estimate the posterior marginal expectations of $\varphi(t)$, $\chi(t)$ and $\psi(t)$. These marginal expectations are estimates of a dynamic component time course obtained by filtering out from $y$ all the contributions of the other components plus the noise. Since both the prior distributions and the observation model are Gaussian, the posterior distribution is itself Gaussian and its marginal expectations can be computed exactly (see Eq. (\[eq:additive covariance, methods\]) in Materials and Methods). ![**Stochastic processes and covariance functions.**](GPpaper_Figure1.png) A\) Example of a continuous-time oscillatory process (blue line) sampled at discrete equally-spaced time points though noise corrupted measurements (red dots). B–E) Samples (colored) and expected values (black) of the stochastic processes. The processes are a damped harmonic oscillator, second order integrator, first order integrator and residuals respectively. The samples start from an excited state and decay back to their respective stationary distribution. F) Illustration of the decomposition of a complex signal’s covariance function into simpler additive components. This corresponds to an additive decomposition of the measured time series. The second order integrator process has been excluded from this panel for visualization purposes. \[fig1\] Spatiotemporal GP decomposition {#spatiotemporal-gp-decomposition .unnumbered} ------------------------------- So far, we have shown how SDE modeling of dynamic components can be used for analyzing a neural time series through GP regression. Here, we complement this temporal model by introducing a spatial correlation structure. In this way, we define a full spatiotemporal model. We define the total additive spatiotemporal neural signal as follows: $$\rho(\boldsymbol{x},t) = \varphi(\boldsymbol{x},t) + \chi(\boldsymbol{x},t) + \psi(\boldsymbol{x},t)\,,$$ where $\boldsymbol{x}$ denotes a cortical location. Strictly speaking, $\rho(\boldsymbol{x},t)$ should be a vector field because the neural electrical activity at each cortical point is modeled as an equivalent current dipole. However, for simplicity, we present the methods for the case in which the dipole orientation is fixed and $\rho(\boldsymbol{x},t)$ can be considered as a scalar field. All formulas for the vector-valued case are given in Appendix IV. #### Modeling spatial correlations {#modeling-spatial-correlations .unnumbered} Correlations between different cortical locations can be modeled using a spatial covariance function $s(\boldsymbol{x},\boldsymbol{x}')$. Since the localization of an electric or magnetic source from a sensor array is in general an ill-posed problem, the specification of a prior covariance function is required in order to obtain a unique solution [@tarantola2005inverse]. We do not model the spatial correlation structure directly using spatial SDEs. Instead, we impose a certain degree of spatial smoothness, and this is motivated by the fact that fine details of the neural activity cannot be reliably estimated from the MEG or EEG measurements. This procedure has been shown to reduce the localization error and attenuate some of the typical artifacts of source reconstruction [@petrov2012harmony; @pascual2002functional]. Modeling the spatial correlations between measurements of neural activity requires a proper definition of distance between cortical locations. The conventional Euclidean distance is likely to be inappropriate because cortical gyri can be nearby according to the Euclidean distance in three-dimensional space, but far apart in terms of the intrinsic cortical geometry that is determined by the synaptic connectivity between grey matter areas. Surface reconstruction algorithms such as Freesurfer [@fischl2012freesurfer] allow to map each of the cortical hemispheres onto a sphere in a way that preserves this intrinsic cortical geometry. Building this spherical representation, we can make use of the so-called spherical harmonics. These are basis functions that generalize sines and cosines on the surface of a sphere and are naturally ordered according to their spatial frequency. Using the spherical harmonics we define a spatial covariance function $s(\boldsymbol{x},\boldsymbol{x'})$ between cortical locations, and choose a particular covariance function by discounting high spatial-frequency harmonics. This operation smooths out the fast-varying neural activity and thereby induces spatial correlations. This can be interpreted as a low-pass spatial filter on the cortical surface. The amount of spatial smoothing is regulated by a smoothing parameter $\upsilon$ and a regularization parameter $\lambda$, where the former controls the prior spatial correlations and the latter the relative contribution of the prior and the observed spatial correlation (see Eqs. (\[eq:spherical filter, methods\]) and (\[eq:posterior mean spatiotemporal, methods\]) in the Materials and Methods). #### Decomposing spatiotemporal signals using separable covariance functions {#decomposing-spatiotemporal-signals-using-separable-covariance-functions .unnumbered} We combine the spatial and temporal model by making a separability assumption, namely we assume that the covariance between $\rho(\boldsymbol{x},t)$ and $\rho(\boldsymbol{x}',t')$ is given by the product $k_\rho(t,t') s(\boldsymbol{x},\boldsymbol{x}')$. Using this spatiotemporal GP prior we compute the marginal expectations of the spatiotemporal dynamic components (see Eq. (\[eq:posterior mean spatiotemporal, methods\]) in Materials and Methods). We refer to this approach as spatiotemporal GP decomposition (SGPD). Estimating the model parameters {#estimating-the-model-parameters .unnumbered} ------------------------------- The covariance functions of the dynamic components have parameters that can be directly estimated from the data. Instead of using a full hierarchical model, we estimate the parameters by fitting the total additive covariance function of the model to the empirical auto-covariance matrix of the measured time series using a least-squares approach. This procedure allows to infer the parameters of the prior directly from the data, thereby tuning the dynamical model on the specific features of each participant/experimental condition. Specifically, the parameters of the prior are estimated from the data of all trials, and these parameters in turn determine the GP prior distribution that is used for the analysis of the trial-specific data. The details of the cost function are described in the Materials and Methods section. Because this optimization problem is not convex, it can have several local minima. For that reason, we used a gradient-free simulated annealing procedure [@kirkpatrick1984optimization] to find a good approximate solution to the global optimization problem. Analyzing oscillatory amplitude using GP decomposition, a simulation study {#analyzing-oscillatory-amplitude-using-gp-decomposition-a-simulation-study .unnumbered} --------------------------------------------------------------------------- #### Spectral analysis with temporal GP decomposition {#spectral-analysis-with-temporal-gp-decomposition .unnumbered} We now investigate the performance of the temporal GP separation of dynamic components for the purpose of evaluating modulations of oscillatory amplitude. Such amplitude modulations have been related to many cognitive processes. For example, in tasks that require attentional orienting to some part of the visual field, alpha oscillations are suppressed over the corresponding brain regions [@foxe2011role; @kelly2006increases]. Because the spectral content of electrophysiological measurements is almost always broadband, when there is an interest in oscillations, it makes sense to isolate these oscillations from the rest of the measured time series. The resulting procedure involves a separation of the oscillatory components of interest from the interfering non-rhythmic components. In the GP decomposition framework, this separation can be achieved by modeling both the oscillatory component $\varphi(t)$ and the interfering processes. We use the symbol $\boldsymbol{m}_{\varphi|y}$ for the marginal expectation of the process $\varphi(t)$ at the sample points. The average amplitude can be obtained from $\boldsymbol{m}_{\varphi|y}$ by calculating its root mean square deviation: $$A=\sqrt{\frac{1}{N} \text{\ensuremath{\sum}}_{j}\left(\left[m_{\varphi|y}\right]_{j}-\bar{{m}}\right)^{2}}\label{eq:mean squarred deviation} $$ with $\bar{{m}} = \frac{1}{N} \sum_{i}\left[\boldsymbol{m}_{\varphi|y}\right]_{i}$. Here, we compared the sensitivity of the GP method with DPSS multitaper spectral estimation [@percival1993spectral], a widely used non-parametric technique. In the simulation study, the methods had to estimate a simulated experimental modulation of the amplitude of a 10 Hz oscillatory process. For each of two conditions, we generated oscillatory time series from a non-Gaussian oscillatory process. The choice for a non-Gaussian process was motivated by our objective not to bias our evaluation in favor of the GP method. The oscillatory time series was then corrupted by a first order integrator and residuals. The simulation design involved 16 levels that covered an amplitude modulation range from 15% to 60% in equidistant steps. For each level, per experimental condition, we generated 150,000 trials of 2 s. The effect size was defined as follows: $$f=\frac{\langle A_{1}\rangle - \langle A_{2} \rangle}{\text{var}(A)}\,, \label{eq:effect size, results}$$ where $\langle A_{j}\rangle$ is the mean oscillatory amplitude in the j-th experimental condition and $\mathrm{var}(A)$ is its variance. Mean and variance were calculated over the trials. The GP method does not have free parameters, since the parameters of the covariance functions are estimated from the data. In contrast, the spectral smoothing of a multitaper analysis is determined by the number of tapers, which is a parameter that can be chosen freely. We selected the number of tapers that maximizes the effect size in order not to bias the evaluation in favor of the GP method. In addition, we reported the effect sizes for the multitaper analysis with a fixed smoothing of 0.6 Hz. Fig. \[fig2\]A shows the effect sizes for the GP and the multitaper method as a function of the true between-condition amplitude difference. The Gaussian process consistently outperforms the non-parametric method. Fig. \[fig2\]B shows the ratio between the GP and the optimal multitaper effect size as a function of the true amplitude difference. Here we can see that the superior performance is more pronounced when the amplitude difference is smaller, corresponding to a situation with a lower signal-to-noise ratio. #### SGPD improves accuracy and sharpness of source reconstruction {#sgpd-improves-accuracy-and-sharpness-of-source-reconstruction .unnumbered} We now investigate how SGPD compares to existing methods with respect to the spatial localization of an oscillatory amplitude modulation in the presence of noise sources with both spatial and temporal structure. We compare our method to the Harmony source reconstruction technique [@petrov2012harmony], which has been shown to outperform several commonly used linear source reconstruction methods. For this, we set up a simulation study in which the performance was evaluated by the extent to which a spatially focal amplitude modulation could be detected. We modeled the brain activity as generated by three cortical patches, each with a constant spatial profile and a time course generated in the same way as in the single sensor simulation. The patches had a radius of approximately one centimeter and were localized in the right temporal, right occipital, and left parietal cortex (Fig. \[fig3\]A). All three patches exhibited oscillatory activity, but the one in the right temporal lobe had an amplitude that was modulated by the simulated conditions. The source activity was projected to the sensors by a forward model that was obtained using a realistic head model [@nolte2003magnetic]. The sensor level activity for the first trial is shown in Fig. \[fig3\]B. The regularization parameter l of both Harmony and SGPD were identified using leave-one-out cross validation [@stone1974cross], while the smoothing parameter $\upsilon$ was set by hand and had the same value of 3 in both models. The spectral smoothing of the DPSS multitaper spectral estimation was set to 0.6 Hz. The value was chosen because, on average, this gave the highest effect size of the amplitude modulation. We assessed the quality of the reconstructed effects using two indices, one for accuracy and one for sharpness. The accuracy index is obtained by dividing the estimated effect in the center of the amplitude-modulated patch (more specifically, the sum over the points in a sphere with 1 cm radius) by the maximum of the estimated effects in the centers of the other two patches (again, by summing over the points in a sphere). The accuracy index will be high if it localizes the effect in the right patch but not in the interfering ones. The sharpness index evaluates how much the effect maps are focused around the center of the effect. It is computed by dividing the summed estimated effect in the center of the amplitude-modulated patch by the summed estimated effect outside that region. Figs. \[fig3\]C&D show the results of the simulation. Each disc in the scatter plot represents the outcome of SGPD and Harmony for a single simulation. The median accuracy and sharpness were respectively 33% and 28% higher for SGPD as compared to the Harmony approach. ![**Results of the single sensor simulation.**](GPpaper_Figure2.png) A\) Effect size of temporal GP and DPSS multitaper spectral analysis as function of mean percentage amplitude difference between simulated conditions. The parameters of the temporal GP decomposition (blue line) were estimated from the raw simulated time series. The spectral smoothing of the multitaper method (green line) was chosen for each to maximize the effect size. The red line is the effect size for a multitaper method with constant spectral smoothing of 0.6 Hz. B) Effect size ratio between temporal GP and (optimized) multitaper method as function of the mean amplitude difference between conditions. \[fig2\] ![**Results of the source level simulation.**](GPpaper_Figure3.png) A\) Spatial maps of the simulated brain sources. The left map shows the spatial extent of the amplitude-modulated source while the two right maps show the interfering sources. The dipole orientation was set to be orthogonal to the mesh surface. B) Visualization of sensor activity as a mixing of the three sources. The dots represent MEG sensors. The color of the dots show the sign (red for positive and blue for negative) together with the magnitudes. The time series was taken from an occipital sensor. C) Scatter plot of the accuracy of SGPD and Harmony. The index was computed by dividing the total reconstructed effect within the amplitude-modulated cortical patch by the sum of total effects in the non-modulated patches. D) Scatter plot of the sharpness of SGPD and Harmony. The sharpness index was obtained by dividing the total reconstructed effect within the amplitude-modulated cortical patch by the total effect elsewhere. For the purpose of visualization, in both scatterplots, we excluded some outliers ($> 5 \times \text{median}$). These outliers arise when the denominator of one of the indices becomes too small. The outliers have been removed from the figure but they were involved in the calculation of the medians for the two methods. \[fig3\] Gaussian process analysis of example MEG data {#gaussian-process-analysis-of-example-meg-data .unnumbered} ---------------------------------------------- We tested the temporal GP decomposition on an example MEG dataset that was collected from 14 participants that performed a somatosensory attention experiment [@van2012beyond]. We will use this dataset for different purposes, and start by using it for evaluating the performance of our parameter estimation algorithm. Fig. \[fig4\] shows the empirical auto-covariance functions and the least squares fit for two participants. To make them comparable, we normalized these auto-covariance functions by dividing them by their variance. The fitted auto-covariance functions capture most features of the observed auto-covariance functions. The comparison shows some individual differences: First, Participant 1 has a higher amplitude alpha signal relative to the other dynamic components, but the correlation peaks are separated only by about three cycles. Second, the auto-covariance of Participant 2 is dominated more by a signal component with a high temporal correlation for nearby points, and the rhythmic alpha component decays much more slowly. The latter is a signature of a longer phase preservation. We quantified the goodness-of-fit as the normalized total absolute deviation from the model: $$g=\frac{\sum_{i,j}|c_{ij}-k(t_{i},t_{j})|}{\sum_{i,j}|c_{ij}|}\,,\label{eq:goodness of fitt, results}$$ where $c_{ij}$ is the empirical auto-covariance between $y_{t_{i}}$ and $y_{t_{j}}$, and $k(t_{i},t_{j})$ is the auto-covariance predicted by our dynamical model. We evaluated the goodness-of-fit by computing this deviation measure for each participant. The median goodness-of-fit was 0.06, meaning that the median deviation from the empirical auto-covariance was only 6% of the sum of its absolute values. The goodness-of-fit for the two example participants one and two in Fig. \[fig4\] are 0.04 and 0.02, respectively. Next, we inspect the reconstructed spatiotemporal dynamic components obtained from the resting state MEG signal of Participant 1 (with auto-covariance as shown in Fig. \[fig4\]A), as obtained by SGPD. Fig. \[fig5\]A shows an example of time courses of the dynamic components for an arbitrarily chosen cortical vertex situated in the right parietal cortex. The first order integrator time series (upper-left panel) tends to be slow-varying but also exhibits some fast transitions. The second order integrator (lower-left panel) is equally slow but smoother. In this participant, the alpha oscillations, as captured by the damped harmonic oscillator, are quite irregular (upper-right panel), and this is in agreement with its covariance function (see Fig. \[fig4\]A). Finally, the residuals (lower-right panel) are very irregular, as is expected from the signals short-lived temporal correlations. Fig. \[fig5\]B shows an example of the spatiotemporal evolution of alpha oscillations for a period of 32 milliseconds in a resting-state MEG signal. For the purpose of visualization, we only show the value of the dipole along an arbitrary axis. The pattern in the left hemisphere has a wavefront that propagates through the parietal cortex. Conversely, the alpha signal in the right hemisphere is more stationary. ![**Estimation of the model covariance functions.**](GPpaper_Figure4.png) Parametric fit of the MEG auto-covariance functions of Participant 1 and Participant 2. The red lines refer to the estimated parametric model and the blue lines reflect the empirical auto-covariance of the measured time series. A single auto-covariance was obtained from the multi-sensor data by performing a principal component analysis and averaging the empirical auto-covariance of the first 50 components, weighted by their variance. The parameters of the model were estimated using a least-squares simulated annealing optimization method. The graphs have been scaled between 0 and 1 by dividing them by the maximum of the individual empirical auto-covariance. \[fig4\] ![**Estimated dynamic components.**](GPpaper_Figure5.png) Reconstructed source-level neural activity of Participant 1. A) Reconstructed time series of the four dynamic processes localized in a right parietal cortical vertex. B) Reconstructed spatiotemporal dynamics of alpha oscillations along the $x$ axis. This choice of axis is arbitrary and has been chosen solely for visualization purposes. The source-reconstructed activity has been normalized by dividing it by the maximum of the absolute of the spatiotemporal signal. \[fig5\] Attention-induced spatiotemporal dynamics of oscillatory amplitude {#attention-induced-spatiotemporal-dynamics-of-oscillatory-amplitude .unnumbered} ------------------------------------------------------------------ Next, we applied the SGPD source reconstruction method to the example MEG data that were collected in a cued tactile detection experiment. Identifying the neurophysiological mechanisms underlying attentional orienting is an active area of investigation in cognitive neuroscience [@foxe2011role; @jensen2010shaping; @van2011orienting; @van2012beyond]. Such mechanisms could involve neural activity of which the spatial distribution varies over time (i.e., neural activity with dynamic spatial patterns), and GP source reconstruction turns out to be highly suited for identifying such activity, as we will demonstrate now. In the cued tactile detection experiment an auditory stimulus (high or low pitch pure tone) cued the location (left or right hand) of a near-threshold tactile stimulus in one-third of the trials. This cue was presented 1.5 s before the target. The remaining two-thirds of the trials were uncued. In the following, we compare the pre-target interval between the cued and the uncued conditions in terms of how the alpha amplitude modulation develops over time. In the analysis, we made use of the fact that the experiment involved two recording sessions, separated by a break. We explored the data of the first session in search for some pattern, and then used the data of the second session to statistically test for the presence of this pattern. Thus, the spatiotemporal details of the null hypothesis of this statistical test were determined by the data of the first session, and we used the data of the second session to test it. Figure \[fig6\]A shows the group-averaged alpha amplitude modulation as a function of time. An amplitude suppression for the cued relative to the uncued condition originates bilaterally in the parietal cortex and gradually progresses caudal to rostral until it reaches the sensorimotor cortices. The time axes are expressed in terms of the distance to the target. Similar patterns can be seen in individual participants (see Fig. \[fig6\]B&C for representative participants 1 and 2). Participant 1 has a suppressive profile that is almost indistinguishable from the group average. On the other hand, participant 2 shows an early enhancement of sensorimotor alpha power accompanied by a parietal suppression, and the latter then propagates forward until it reaches the sensorimotor areas. Thus, in the grand average and in most of the participants, there is a clear caudal-to-rostral progression in the attention-induced alpha amplitude suppression. We characterized this progression by constructing cortical maps of the linear dependence (slope) between latency and amplitude modulation. The group average of the slope maps for the first session is shown in Fig. \[fig6\]D. This figure shows that the posterior part of the brain has positive slopes, reflecting the fact that the effect tended to become less negative over time. Conversely, the sensorimotor regions have positive slopes, reflecting the fact that the effect tended to become more negative over time. To evaluate the reliability of this pattern, we build on the reasoning that, if this pattern in the slope map is due to chance, then it must be uncorrelated with the slope map for the second session. To evaluate this, for every participant, we calculated the dot product between the normalized slope maps for the two sessions and tested whether the average dot product was different from zero. The one-sample t-test showed that the effect was significantly different from zero ($p<0.05$), supporting the claim that the caudal-to-rostral progression in the attention-induced alpha amplitude suppression is genuine. Thus, we have shown that, during the attentional preparation following the cue, the alpha modulation progresses from the parietal to the sensorimotor cortex. ![**Caudal-to-rostral progression of alpha amplitude attentional modulation.**](GPpaper_Figure6.png) A\) Group average of alpha amplitude attentional modulation as function of time. B,C) Alpha amplitude attentional modulation for participants 1 and 2, respectively. D) Spatial map obtained by computing the slope of the average alpha difference between cued and non-cued conditions as a function of time for each cortical vertex. \[fig6\] Discussion {#discussion .unnumbered} ========== In this paper, we introduced a new signal decomposition technique that incorporates explicit dynamical models of neural activity. We showed how dynamical models can be constructed and integrated into a Bayesian statistical analysis framework based on GP regression. The resulting statistical model can be used for decomposing the measured time series into a set of temporal or spatiotemporal dynamic components that reflect different aspects of the neural signal. We validated our method using simulations and real MEG data. The simulations demonstrate that the use of the dynamical signal and noise model improves on the current state-of-the-art (non-parametric spectral estimation) with respect to the identification of amplitude modulations between experimental conditions. Different from non-parametric spectral estimation, our method first separates the oscillatory signal from the interfering components on the basis of the autocorrelation structures of both, and only in the second step calculates the amplitude of the signal of interest. A spatiotemporal version of the decomposition method was obtained by decomposing the neural processes in spherical harmonics. Our simulations show that, in the presence of spatially and temporally correlated noise, spatiotemporal GP decomposition localizes amplitude modulations more accurately than a related method that does not make use of the temporal decomposition of the signal of interest [@petrov2012harmony]. Lastly, using the spatiotemporal decomposition on real MEG data from a somatosensory detection task, we demonstrated its usefulness by identifying an intriguing anterior-to-posterior propagation in the attention-induced suppression of oscillatory alpha power. Generality, limitations, and robustness {#generality-limitations-and-robustness .unnumbered} --------------------------------------- Although we used a specific set of SDEs, the method is fully general in that it can be applied to any linearized model of neuronal activity. Therefore, it establishes a valuable connection between data analysis and theoretical modeling of neural phenomena. For example, neural masses models and neural field equations (see, e.g. [@deco2008dynamic]) can be linearized around their fixed points and the resulting SDEs form the basis for a GP analysis that extracts the theoretically defined components. Furthermore, the GP decomposition method could be used as an analytically solvable starting point for the statistical analysis of non-linear and non-Gaussian phenomena through methods such as perturbative expansion, where the initial linear Gaussian model is corrected by non-linear terms that come from the Taylor expansion of the non-linear couplings between the neural activity at different spatiotemporal points [@dickman2003path]. The method’s limitations pertain to the model’s assumptions of linearity and its specific spatial correlation structure. Specifically, the linear SDEs cannot account for the complex non-linear effects that are found in both experimental [@freyer2009bistability; @osorio2010epileptic] and modeling work [@golos2015multistability; @ghosh2008noise; @breakspear2006unifying]. In addition, the assumed homegeneous spatial correlation structure solely depends on the distance between cortical locations and therefore does not account for the rich connectivity structure of the brain [@greicius2009resting; @stam2004functional; @sporns2005human; @van2011rich]. Nevertheless, the method has some robustness against the violations of the underlying assumptions. This robustness follows from the fact that the model specifies the prior distribution but does not constrain the marginal expectations to have a specific parametric form. The temporal prior affects the estimation of a dynamic component to a degree that depends on the ratio between its variance and the cumulative variance of all other components. Specifically, the smaller the prior variance of a component relative to the combined variance of all the others, the more the pattern in the prior covariance matrix will affect the posterior. Since we estimate all these prior variances directly from the measured time series, our method is able to reconstruct complex non-linear effects in components that have a relatively high SNR while it tends to ”linearize” components with low SNR. As a consequence, the more pronounced the non-linear effects in the observed signal, the more these will be reflected in the posterior, gradually dominating the linear structure imposed by the prior. Importantly, because our temporal prior is based on a larger data set, it will be adequate, on average, over all epochs while still allowing strong components in individual epochs to dominate the results. The situation is similar but not identical for our spatial prior. Contrary to our temporal prior, this spatial prior is not derived from an empirically fitted dynamical model but on the basis of our prior belief that source configurations with high spatial frequencies are unlikely to be reliably estimated from MEG measurements. Since the problem of reconstructing source activity from MEG measurements is generally ill-posed, the choice of the spatial prior will bias the inference even for very high SNR. Nevertheless, it has been shown that the discounting of high spatial frequencies leads to reduced localization error and more interpretable results [@petrov2012harmony]. Connections with other methods {#connections-with-other-methods .unnumbered} ------------------------------ The ideas behind the GP decomposition derive from a series of recent developments in machine learning, connecting GP regression to stochastic dynamics [@sarkka2012infinite; @solin2013infinite]. The approach is closely connected with many methods in several areas of statistical data-analysis. We will now review some of these links, focusing on methods that are commonly used in neuroscience. #### Spectral analysis {#spectral-analysis .unnumbered} Throughout the paper, we showed that the GP decomposition can be profitably used to estimate amplitude modulations in an oscillatory signal, which is an important application of spectral analysis. There are two classes of spectral analysis methods: parametric and non-parametric [@percival1993spectral]. Non-parametric methods mostly rely on the discrete Fourier transform applied to a tapered signal, as for example in DPSS multitaper spectral estimation [@percival1993spectral; @bronez1992performance]. These methods are non-parametric because they do not explicitly model the process that generates the signal. Parametric methods [*do*]{} depend on an explicit model, and typically this is an autoregressive (AR) model [@muthuswamy1998spectral; @pace1998spatiotemporal]. AR models are closely related to GPs as they are typically formulated as discrete-time Gaussian processes driven by stochastic [*difference* ]{} equations. In this sense, the GP prior distributions used in this paper are continuous-time versions of an AR process. However, the usual AR approach to spectral estimation is different from our approach. AR models are usually parametrized in terms of a series of matrices whose entries (the so-called model coefficients) describe the statistical dependencies within and between the channels of a multivariate signal. The coefficients that describe the statistical dependencies within a univariate signal are related to the inverse of the impulse response function in our approach (see Materials and Methods for a description of the impulse response function). Spectral analysis based on AR models has the disadvantage that a very large number of parameters may have to be estimated. Specifically, a spatiotemporal AR model for multi-sensor EEG/MEG data can easily have hundreds of model coefficients that must be estimated from the measured time series. This great flexibility in the analytic form of the AR model is required as the spectrum is directly obtained from the model coefficients. Compared to spatiotemporal AR modeling, the GP decomposition model is much more constrained by the underlying theory, having an explicit additive structure with few parameters for each dynamic component. The rigidity of the model is compensated by the fact that the oscillatory amplitude is not obtained from the fitted model covariance function. Instead, it is computed from the marginal expectation of the oscillatory component, which is obtained by applying Bayes’ rule. Therefore, while the prior has a parametric form, the posterior mean obtained from the GP decomposition method is actually non-parametric. #### Signal decomposition {#signal-decomposition .unnumbered} In the neuroscience literature, the most widely used signal separation techniques are the blind source separation methods known as principal component analysis (PCA) and independent component analysis (ICA), together with their extensions [@comon1994independent; @tipping1999probabilistic; @zou2006sparse; @scholkopf1998nonlinear; @kolda2009tensor; @van2012phase; @van2015uncovering]. These methods rely on the statistical properties of multi-sensor data (maximum variance for PCA and statistical independence for ICA) and produce components whose associated signals are linear combinations of the sensor-level signals. Specifically, these statistical properties pertain to the resulting component-level signals. Importantly, whereas GP decomposition depends on a specific model of the neural signal, neither PCA nor ICA makes use of prior knowledge of the component-level signals. Also, both PCA and ICA require multi-sensor data, whereas GP decomposition can be applied to a single time series. It is important to note that GP decomposition is not a tool for separating statistically independent or uncorrelated components. Instead, its goal is to decompose the measured signal into several processes characterized by different autocorrelation structures. Hence, the method does not discriminate between two independent processes generated by two sources with the same dynamics, such as a frontal and an occipital alpha oscillation. Therefore, the GP decomposition is complementary to blind source separation. In fact, the latter can be used to extract interesting temporal and spatiotemporal patterns from the dynamic components obtained from GP decomposition. #### Source reconstruction {#source-reconstruction .unnumbered} A general framework for GP source analysis has recently been introduced [@solin2016gprec]. In this work, the authors show that several well-known source reconstruction methods are special cases of GP regression with appropriate covariance functions. In particular, the spatial filter of techniques such as minimum norm estimation [@hamalainen1994interpreting] and exact Loreta [@pascual2002functional] are obtained as a discretization of a spatial GP analysis with an appropriate spatial covariance function. The authors also introduced a general framework for GP spatiotemporal analysis using separable covariance functions designed to localize averaged neural activity (e.g. evoked fields). This GP spatiotemporal source reconstruction is formally similar to several other spatiotemporal source reconstruction methods [@darvas2001spatio; @friston2008multiple; @trujillo2008bayesian; @dannhauer2013spatio]. Our approach improves on these works by using informed temporal covariance functions that explicitly model the temporal dynamics of the ongoing neural signal. The additive structure of the temporal covariance function allows to individually source localize signal components with specific dynamic properties. In particular, the spatial configuration of these components are analyzed in the spherical harmonics domain, as this greatly reduces the dimensionality of the source space. As shown in the Materials and Methods section, the resulting spatial filter is closely related to the Harmony source reconstruction method [@petrov2012harmony]. Benefits of GP Decomposition for Cognitive Neuroscience {#benefits-of-gp-decomposition-for-cognitive-neuroscience .unnumbered} ------------------------------------------------------- In two simulation studies, we showed that the GP decomposition improves the quantification and source localization of oscillatory amplitude modulations. These improvements are particularly noticeable in the presence of the kind of temporally and spatially structured noise that is ubiquitous in neural measurements. Additionally, the method is particularly suited for data-driven exploration of complex spatiotemporal data as it decomposes the signal into a series of more interpretable dynamic components. As a demostration, we used the SGDP to investigate the modulation of alpha oscillations associated with attentional preparation to a tactile stimulus. Several previous works demonstrated that alpha amplitude is reduced prior to a predicted stimulus [@van2012beyond; @van2011orienting; @foxe2011role]. These amplitude modulations have been associated to modality specific preparatory regulations of the sensory cortices [@van2008prestimulus; @klimesch2007eeg; @van2011orienting; @volberg2009eeg; @rohenkohl2011alpha]. While the attentional role of alpha oscillations in the primary sensory cortices is well established, it is still unclear how this generalizes to supramodal areas. Although the parietal cortex is known to play a role in the top-down control of attention [@culham2001neuroimaging; @corbetta2002control], parietal alpha oscillations have typically been considered as closely related to the visual system [@foxe2011role]. The involvement of the parietal cortex in the somatosensory detection task went unnoticed in the first analysis of the data that have been reanalyzed in the present paper [@van2012beyond]. In our new analysis, we used the SGDP to more effectively explore the data, looking for interesting spatiotemporal effects. This led to the identification of a suppression of alpha amplitude that originates from the parietal cortex and then propagates to the somatosensory regions. This effect turned out to be statistically robust when tested in a second independent dataset that was collected in the same experiment. The results suggest a hierarchical organization of the reconfiguration of alpha amplitude following an attentional cue. In particular, the initial reduction of parietal alpha amplitude could reflect the activation of a supramodal attentional network that paves the way for later sensorimotor-specific cortical reconfiguration. While we mainly restricted our attention to the analysis of alpha oscillations, we believe that the GP decomposition can be useful for the study of other neural oscillations as well as non-rhythmic components. Several experimental tasks are related to effects in multiple dynamic components. For example, perception of naturalistic videos induces modulations in several frequency bands [@betti2013natural]. Studying the interplay between these differential modulations requires an appropriate decomposition of the measured signals that can be effectively performed using GP decomposition. Conclusions {#conclusions .unnumbered} ------------ Our dynamic decomposition method starts from a precise mathematical model of the dynamics of the neural fields. The formalism of GP regression allows translation of linear stochastic dynamics into a well-defined Bayesian prior distribution. In this way, the method establishes a connection between mathematical modeling and data analysis of neural phenomena. On the one hand, the experimentalist and the data-analyst can benefit from the method as it allows to isolate the dynamic components of interest from the interfering noise. These components are interpretable and visualizable, and their study can lead to the identification of new temporal and spatiotemporal neural phenomena that are relevant for human cognition. On the other hand, the theorist can use this formalism for obtaining a probabilistic formulation of dynamical models, thereby relating them to the experimental data. Materials and Methods {#materials-and-methods .unnumbered} ===================== In this section we will explain the mathematical underpinnings of the GP decomposition. Following the lines of the Results section, the exposition begins from the connection between SDEs and Gaussian processes and continues with the exposition of the temporal and spatiotemporal GP decomposition. In order to improve the readability and to not overshadow the main ideas, we left some technical derivations to the appendices. From SDEs to GPs {#from-sdes-to-gps .unnumbered} ---------------- At the core of our method is the connection between Gaussian processes and SDEs. This connection leads to the definition of the covariance functions of the dynamic components that will be used for determining the prior of the GP regression. In the Results section, we introduced the SDE (Eq. (\[eq:oscillator SDE\])) $$\frac{d^{2}}{dt^{2}}\varphi(t)+b\frac{d}{dt}\varphi(t)=-\omega_{0}^{2}\varphi(t)+w(t)$$ to model an oscillatory signal. In fact, this SDE can be interpreted as a damped harmonic oscillator when $b<\sqrt{2\omega_0^2}$. As initial conditions, we set $\varphi(-\infty) = \frac{d\varphi}{dt}(-\infty) = 0$. This choice implies that the (deterministic) effects of the initial conditions are negligible. Given these initial conditions, the solution of Eq. (\[eq:oscillator SDE\]) is fully specified by the random input $w(t)$ that follows a temporally uncorrelated normal distribution. Since the equation is linear, the solution, given a particular instantiation of $w(t)$, can be obtained by convolving $w(t)$ with the impulse response function of the SDE (see Appendix I for more details): $$\varphi(t) = \int_{-\infty}^{\infty} G_\varphi(t - s) w(s)ds.\label{eq:green function, methods}$$ Intuitively, the impulse response function $G_\varphi(t)$ determines the response of the system to a localized unit-amplitude input. Consequently, Eq. (\[eq:green function, methods\]) states that the process $\varphi(t)$ is generated by the infinite superposition of responses to $w(t)$ at every time point. This proves that the resulting stochastic process $\varphi(t)$ is Gaussian, since it is a linear mixture of Gaussian random variables. The impulse response function of Eq. (\[eq:oscillator SDE\]) is $$G_\varphi (t)= \vartheta(t) e^{-b/2 t} \sin \omega t, \label{eq:Green oscillator, methods}$$ where $\vartheta(t)$ is a function equal to zero for $t<0$ and 1 otherwise. This function assures that the response cannot precede the input impulse. From this formula, we see that the system responds to an impulse by oscillating at frequency $\omega = \sqrt{\omega_0^2 - 1/4b^2}$ and with an amplitude that decays exponentially with time scale $b/2$. The covariance function of the process $\varphi(t)$ can be determined from its impulse response function using Eq. (\[eq:deriv. prior cov ||, methods\]) (see Appendix I) and is given by $$k_\varphi(t_i,t_j) = k_\varphi(\tau) = \frac{\sigma_\varphi^2} {2b} e^{-b/2 |\tau|} \left( \cos \omega \tau + \frac {b} {\omega} \sin \omega |\tau| \right). \label{eq:Cov oscillator, methods}$$ where $\tau$ denotes the time difference $t_i - t_j$. In the case of the second order integrator, the parameter $\omega_0$ is smaller than $b/2$ and the system is overdamped. In this case, the response to an impulse is not oscillatory, the response initially rises and then decays to zero with time scale $b/2$. This behavior is determined by the impulse response function $$G_\chi (t)= \vartheta(t) e^{-b/2 t} \sinh z t \label{eq:Green integrator, methods}$$ in which $z$ is equal to $\sqrt{1/4 b^2-\omega_0^2}$. The covariance function is given by $$k_\chi(\tau) = \frac{\sigma_\chi^2} {2b} e^{-b/2 |\tau|} \left( \cosh z \tau + \frac {b} {z} \sinh z |\tau| \right). \label{eq:Cov integrator, methods}$$ Finally, the first order integrator (Eq. (\[eq:first order SDE, results\])) $$\ \frac{d}{dt}\psi(t)-c\psi(t)+w(t)$$ has a discontinuous impulse response function that decays exponentially: $$G_{\psi} (t)= \vartheta(t) e^{-c t}\,. \label{eq:Green first order, methods}$$ The discontinuity of the impulse response at $t = 0$ implies that the process is not differentiable as it reacts very abruptly to the external input. The covariance function of this process is given by: $$k_{\psi}(\tau) = \frac{\sigma_{\psi}^2} {2c} e^{-c |\tau|}\,. \label{eq:Cov first order, methods}$$ #### Covariance function for the residuals The stochastic differential equations are meant to capture the most important (linear) qualitative features of the neural signal. Nevertheless, the real underlying neural dynamics are much more complex than can be captured by any simple model. Empirically, we found that the residuals of our model have short-lived temporal correlations. We decided to account for these correlations by introducing a residuals process $\xi(t)$ with covariance function $$k_{\xi}(\tau) = \sigma_{\xi}^2 e^{- \frac {\tau^2} {2 \delta^2}} \label{eq:Cov res, methods}$$ in which the small time constant $\delta$ is the signal’s characteristic time scale and $\sigma_{\xi}$ is its standard deviation. This covariance function is commonly called the squared exponential and is one of the most used in the machine learning literature [@rasmussen2006gaussian]. As $k_{\xi}(\tau)$ decays to zero much faster than our SDE-derived covariance functions for $\tau$ tending to $\infty$, this covariance function is appropriate for modeling short-lived temporal correlations. Analysing neural signals using Gaussian process regression {#analysing-neural-signals-using-gaussian-process-regression .unnumbered} ----------------------------------------------------------- In this section, we show how to estimate the value of a dynamic component such as $\varphi (t)$ in the set of sample points $t_1,…,t_N$ using GP regression. To this end, it is convenient to collect all the components other than $\varphi (t)$ in a total residuals process $\zeta (t) = \chi (t) + \psi (t) + \xi (t)$. In fact, in this context, they jointly have the role of interfering noise. The vector of data points $\boldsymbol{y}$ is assumed to be a sum of the signal of interest and the noise: $$y_j = \varphi (t_j) + \zeta (t_j)\,. \label{eq:Bayesian model likelihood I, methods}$$ In order to estimate the values of $\varphi (t)$ using Bayes’ theorem we need to specify a prior distribution over the space of continuous-time signals. In the previous sections, we saw how to construct such probability distributions from linear SDEs. In particular, we found that those distributions were GPs with covariance functions that can be analytically obtained from the impulse response function of the SDEs. These prior distributions can be summarized in the following way: $$\begin{aligned} \varphi (t) &\sim GP(0,k_\varphi (t_1, t_2) ) \\ \zeta (t) &\sim GP(0,k_{\zeta} (t_1, t_2) ) \nonumber \label{eq:Bayesian model prior I, methods}\end{aligned}$$ where the symbol $\sim$ indicates that the random variable on the left-hand side follows the distribution on the right-hand side and $GP(\mu(t),k(t_1,t_2))$ denotes a GP with mean function $\mu(t)$ and covariance function $k(t_1,t_2)$. Note that, in this functional notation, expressions such as $\mu(t)$ and $k(t_1,t_2)$ denote whole functions rather than just the values of these functions at specific time points. We will now derive the marginal expectation of $\varphi(t)$ under the posterior distribution. Since we are interested in the values of $\varphi(t)$ at sample points $t_1,\ldots,t_N$, it is convenient to introduce the vector $\boldsymbol{\varphi}$ defined by the entries $\varphi_j = \varphi(t_j )$. Any marginal distribution of a GP for a finite set of sample points is a multivariate Gaussian whose covariance matrix is obtained by evaluating the covariance function at every pair of time points: $$[K_\varphi]_{ij} = k_\varphi(t_i, t_j). \label{eq:Cov matrix, methods}$$ Using Bayes’ theorem and integrating out the total residual $\zeta(t)$, we can now write the marginal posterior of $\boldsymbol{\varphi}$ as $$p(\boldsymbol{\varphi} \mid \boldsymbol{y} ) \propto \int p(\boldsymbol{y}\mid \boldsymbol{\varphi}, \boldsymbol{\zeta}) p(\boldsymbol{\zeta}) d\boldsymbol{\zeta}\ p(\boldsymbol{\varphi}) = N(\boldsymbol{y} \mid \boldsymbol{\varphi},K_{\zeta} )N(\boldsymbol{\varphi} \mid 0,K_\varphi) \label{eq:Bayes theorem, methods}$$ in which $K_{\zeta}$ is the temporal covariance matrix of $\zeta (t)$. As a product of two Gaussian densities, the posterior density is a Gaussian distribution itself. The parameters of the posterior can be found by writing the prior and the likelihood in canonical form. From this form, it is easy to show that the posterior marginal expectation is given by the vector $\boldsymbol{m}_{\varphi|y}$ (see [@rasmussen2006gaussian] for more details about this derivation): $$\boldsymbol{m}_{\varphi|y} = K_{\varphi} (K_{\varphi} + K_{\zeta})^{-1} \boldsymbol{y}. \label{eq:temporal posterior mean, methods}$$ Furthermore, if we assume that $\chi (t)$, $\psi (t)$ and $\xi (t)$ are independent, the noise covariance matrix reduces to $$K_{\zeta} = K_{\chi} + K_{\psi} + K_{\xi}. \label{eq:additive covariance, methods}$$ GP analysis of spatiotemporal signals {#gp-analysis-of-spatiotemporal-signals .unnumbered} ------------------------------------- In the following, we show how to generalize GP decomposition to the spatiotemporal setting. This requires the construction of a source model and the definition of an appropriate prior covariance between cortical locations. In fact, the problem of localizing brain activity from MEG or EEG sensors becomes solvable once we introduce prior spatial correlations by defining a spatial covariance $s(\boldsymbol{x}_i,\boldsymbol{x}_j )$ between every pair of cortical locations $\boldsymbol{x}_i$ and $\boldsymbol{x}_j$. In this paper, we construct $s(\boldsymbol{x}_i,\boldsymbol{x}_j)$ by discounting high spatial frequencies in the spherical harmonics domain, thereby limiting our reconstruction to spatial scales that can be reliably estimated from the sensor measurements. However, prior to the definition of the covariance function, we need to specify a model of the geometry of the head and the brain cortex. #### The source model {#the-source-model .unnumbered} In order to define a source model, we construct a triangular mesh of the cortex from a structural MRI scan using Freesurfer [@fischl2012freesurfer]. The cortical boundary is morphed into a spherical hull in a way that maximally preserves the intrinsic geometry of the cortex. This allows to parameterize the surface $C$ using the spherical coordinates $\alpha$ and $\theta$, respectively azimuth and elevation. For notational simplicity, we collect the spherical coordinates into the coordinate pair $\boldsymbol{x}=(\alpha,\theta)$ that refers to a spatial location in the cortex. Furthermore, we denote the finite set of $M$ points in the mesh as $\mathcal{X} = \{\boldsymbol{x}_1,\ldots,\boldsymbol{x}_M\}$. We define our source model as a vector field of current dipoles on the cortical surface. We first consider GP source reconstruction of the total neural activity $\vec{\rho}(\boldsymbol{x}, t)$, without differentiating between spatiotemporal dynamic components such as $\vec{\varphi}(\boldsymbol{x}, t)$,$\vec{\chi}(\boldsymbol{x}, t)$ and $\vec{\psi}(\boldsymbol{x}, t)$. The vector field $\vec{\rho}(\boldsymbol{x}, t)$ is characterized by the three Cartesian coordinates $\rho_1 (\boldsymbol{x},t)$, $\rho_2 (\boldsymbol{x},t)$, and $\rho_3 (\boldsymbol{x},t)$. In all the analyses contained in this paper, we estimate the full vector field. However, since we do not assume any prior correlations between the dipole coordinates, in the following we will simplify the notation by describing the source decomposition method for a dipole field $\rho (\boldsymbol{x},t) \vec{v}(\boldsymbol{x})$, where the unit-length vector field $\vec{v}(\boldsymbol{x})$ of dipole orientations is assumed to be known. Appendix IV explains how to adapt all the formulas to the vector-valued case using matrices with a block diagonal form. #### Spatial Gaussian processes source reconstruction in the spherical harmonics domain {#spatial-gaussian-processes-source-reconstruction-in-the-spherical-harmonics-domain .unnumbered} The linearity of the electromagnetic field allows to model the spatiotemporal data matrix $Y$ as the result of a linear operator acting on the neural activity $\rho (\boldsymbol{x},t)$ [@nolte2003magnetic]: $$Y_{ij} = \int_C \mathcal{L}_i (\boldsymbol{x}) \rho (\boldsymbol{x}, t_j) d \boldsymbol{x}~, \label{eq:leadifield, methods}$$ in which the component $\mathcal{L}_i (\boldsymbol{x})$ describes the effect of a source located at $\boldsymbol{x}$ on the $i$-th sensor. Note that $\mathcal{L}_i (\boldsymbol{x})$ implicitly depends on the orientation $\vec{v}(\boldsymbol{x})$ since different dipole orientations generate different sensor measurements. We refer to $\mathcal{L}_i (\boldsymbol{x})$ as the forward model relative to the $i$-th sensor, note that this is a function of the spatial location on the cortical surface. In this section, we ignore the prior temporal correlations induced by the temporal covariance functions, i.e. we implicitly assume a prior for $\rho(\boldsymbol{x}, t)$ that is temporally white. In a GP regression setting, the spatial smoothing can be implemented by using a spatially homogeneous covariance function, i.e. a covariance function that only depends on the cortical distance between the sources. To define this covariance function, we make use of the so-called spherical Fourier transform. Whereas the ordinary Fourier transform decomposes signals into sinusoidal waves, the spherical Fourier transform decomposes spatial configurations defined over a sphere into the spherical harmonics $\mathcal{H}_l^m(\boldsymbol{x})$. These basis functions are characterized by a spatial frequency number $l$ and a ”spatial phase” number $m$. Fig. \[fig7\]A shows the spherical harmonics corresponding to the first three spatial frequencies morphed on the cortical surface. For notational convenience, we assign an arbitrary linear indexing to each $(l,m)$ couple that henceforth will be denoted as $(l_k,m_k)$. It is convenient to represent the neural activity $\rho(\boldsymbol{x},t)$ in the spherical harmonics domain. Specifically, we will use the symbol $\tilde{\rho} (l_k,m_k; t)$ to denote the Fourier coefficient of the spherical harmonic indexed by $(l_k,m_k)$ (see Eq. (\[eq:direct Fourier, methods\]) in Appendix II). We assume that the spherical Fourier coefficients $\tilde{\rho} (l_k,m_k; t)$ are independent Gaussian random variables. Under this assumption, we just need to define the prior variance of the coefficients $\tilde{\rho} (l_k,m_k; t)$. Since we aim to reduce the effect of noise with high spatial frequencies, we define these prior variances using a frequency damping function $f(l_k)$ that monotonically decreases as a function of the spatial frequency number $l_k$. This effectively discounts high spatial frequencies and therefore can be seen as a spherical low-pass filter. The variance of the spherical Fourier coefficients is given by the following variance function $$\tilde{s} (l_k,m_k; t) = f(l_k), \label{eq:fourier variance, methods}$$ where, as damping function, we use a spherical version of the truncated Butterworth low-pass filter: $$f(l_k) = \begin{cases} {\Big( 1 + (\frac{l_k}{\upsilon})^{2k} \Big) }^{-1/2} & \text{for} \ l_k \leq L\\ 0 & \text{for} \ l_k > L \end{cases} \label{eq:spherical filter, methods}$$ with smoothing parameter $\upsilon$, order $k$, and cut-off frequency $L$. This filter has been shown to have good properties in the spatial domain [@devaraju2012performance]. Note that, under the covariance function defined by Eq. (\[eq:fourier variance, methods\]) and (\[eq:spherical filter, methods\]), the spherical Fourier coefficients with frequency number larger than $L$ have zero variance and are therefore irrelevant. Although the analysis is carried out in the spherical harmonics domain, it is informative to be able to visualize the covariance function in the spatial domain. By applying the inverse spherical Fourier transform, the function $s(\boldsymbol{x}_i,\boldsymbol{x}_j)$ can be explicitly obtained as follows: $$s(\boldsymbol{x}_i,\boldsymbol{x}_j) = \sum_{l,m} \mathcal{H}_l^m(\boldsymbol{x}_i) \mathcal{H}_l^m(\boldsymbol{x}_j) f(l) \,. \label{eq:spatial covariance, methods}$$ Figs. \[fig7\]B and \[fig7\]C show the correlations induced by our spatial covariance function. In order to formulate the spatial GP regression in the spherical harmonics domain, we rewrite the integral in Eq. (\[eq:leadifield, methods\]) using the inverse spherical Fourier transform (see Eq. (\[eq:inverse Fourier, methods\]) in Appendix II) and interchanging the order of summation and integration: $$Y_{ij} = \int_C \mathcal{L}_i (\boldsymbol{x}) \Bigg({\sum_{k} \tilde{\rho} (l_k,m_k; t_j) \mathcal{H}_{l_k}^{m_k}(\boldsymbol{x})}\Bigg) d\boldsymbol{x} = \sum_{k} \tilde{\mathcal{L}}_i (l_k,m_k)\tilde{\rho} (l_k,m_k; t_j) , \label{eq:fourier leadifield, methods}$$ where $$\tilde{\mathcal{L}}_i (l_k,m_k) = \int_C \mathcal{L}_i (\boldsymbol{x}) \mathcal{H}_{l_k}^{m_k}(\boldsymbol{x}) d\boldsymbol{x}$$ is the spherical Fourier transform of $\mathcal{L}_i (\boldsymbol{x})$. Therefore, the spherical Fourier transform converts the forward model (which is a function of the cortical location) from the spatial to the spherical harmonics domain. We can simplify Eq. (\[eq:fourier leadifield, methods\]) by organizing the spherical Fourier coefficients $\tilde{\rho} (l_k,m_k; t_j)$ in the matrix $\tilde{R}$, whose element $\tilde{R}_{kj}$ is $\tilde{\rho} (l_k,m_k; t_j)$. Analogously, the spherical Fourier transform of the forward model can be arranged in a matrix $\Lambda$ with elements $\Lambda_{ik} = \tilde{\mathcal{L}}_i (l_k,m_k)$. Using this notation, we can write the observation model for the spatiotemporal data matrix $Y$ in a compact way: $$Y = \Lambda \tilde{R} + \boldsymbol{\xi}~, \label{eq:spatial observation model, methods}$$ where $\boldsymbol{\xi}$ are Gaussian residuals with spatial covariance matrix $\Sigma$. We can now combine this observation model with the spherical harmonics domain spatial GP prior, as determined by the variance function given by Eq. (\[eq:fourier variance, methods\]), and from this we obtain the posterior of the neural activity $\tilde{R}$ given the measured signal $Y$. Because the spatial process is Gaussian, the prior distribution of the spherical Fourier coefficients is normal and, because we assumed that the spherical Fourier coefficients are independent, their covariance matrix $D$ is diagonal with entries specified by the variance function $D_{kk} = f(l_k)$ (see Eq. (\[eq:fourier variance, methods\])). Alltogether, the prior and the observation model specify a Gaussian linear regression. The posterior expectation of the regression coeffcients $\tilde{R}$ can be shown to be [@tarantola2005inverse]: $$M_{\tilde{R}|Y} = D \Lambda^{T} ( \Lambda D \Lambda^{T} + \Sigma)^{-1} Y. \label{eq:spatial fourier posterior mean, methods}$$ In this formula, $\Lambda D \Lambda^{T}$ is the sensor level covariance matrix induced by the spatially smooth brain activity and $\Sigma$ is the residual covariance matrix of the sensors. This expression can be recast in terms of the original cortical locations $\mathcal{X}$ using the inverse spherical Fourier transform (Eq. (\[eq:inverse Fourier, methods\])). In matrix form, this can be written as $$M_{R|Y} = H M_{\tilde{\rho}|Y}, \label{eq:spatial posterior mean, methods}$$ where the matrix $H$ is obtained by evaluating the spherical harmonics at the discrete spatial grid-points $\mathcal{X}$: $$H_{lk} = \mathcal{H}_{l_k}^{m_k}(\boldsymbol{x}_l)\,. \label{eq:harmonic matrix, methods}$$ This formula gives the Harmony source reconstruction solution as presented in [@petrov2012harmony]. We can reformulate this expression by introducing the Harmony spatial filter $$P = H D \Lambda^{T} ( \Lambda D \Lambda^{T} + \Sigma)^{-1}. \label{eq:spatial filter, methods}$$ Using this matrix, the posterior expectation of the neural activity at the cortical locations $\mathcal{X}$ can be written as follows: $$M_{R|Y} = P Y. \label{eq:spatial posterior mean II, methods}$$ #### Spatiotemporal GP decomposition {#spatiotemporal-gp-decomposition-1 .unnumbered} The temporal and spatial GP regression can be combined by assigning a temporal covariance function to each spherical Fourier coefficient. In other words, we model the time series of each coefficient as an independent temporal Gaussian process. These processes have the same prior temporal correlation structure as specified in our additive temporal model. However, as in the spatial model, their prior variance is discounted as a function of the spatial frequency $l_k$. Using functional notation, this can be written as follows: $$\tilde{\rho} (l,m;t) \sim GP(0,f(l) k_{\rho} (t_1, t_2)). \label{eq:spatiotemporal process, methods}$$ Considering the prior distributions of the processes $\tilde{\rho} (l,m;t)$ at the sample points, the matrix-valued random variable $\tilde{R}$, when vectorized, follows a multivariate Gaussian distribution with covariance matrix $K_{\rho} \otimes D$, where $\otimes$ denotes the Kronecker product (see Appendix III). This Kronecker product form follows from the fact that the covariance function of $\tilde{\rho} (l,m;t)$ is the product of a spatial and a temporal part. Multivariate Gaussian distributions with this Kronecker structure can be more compactly reformulated as a matrix normal distribution (see [@dawid1981some]): $$\tilde{R} \sim MN(0, D, K_{\rho})\,, \label{eq:matrix normal prior, methods}$$ where the matrix parameters $D$ and $K_{\rho}$ determine the covariance structure across, respectively, the spherical harmonics and time. We define a spatiotemporal observation model in which the residuals have a spatiotemporal covariance structure of the form $K_{\xi} \otimes (\Lambda D \Lambda^T)$. This implies that the spatial covariance matrix of the residuals (previously denoted as $\Sigma$) has the form $\Lambda D \Lambda^T$. Thus, it is assumed that the residuals have the same spatial covariance as the brain activity of interest (see Eq. (\[eq:spatial fourier posterior mean, methods\])) but a different temporal covariance. Hence, $\xi(\boldsymbol{x},t)$ should be interpreted as brain noise [@mivsic2010brain]. This assumption greatly simplifies the derivation of the posterior distribution. Under this observation model, the probability distribution of the spatiotemporal data matrix can be written as follows: $$Y \sim MN(\Lambda \tilde{R}, \Lambda D \Lambda^T, K_{\xi})~. \label{eq:matrix normal likelihood, methods}$$ The posterior expectation for this model can be obtained using the properties of Kronecker product matrices. This derivation is slightly technical and is reported in Appendix III. In this derivation, to enhance numerical stability, we introduce a Tikhonov regularization parameter $\lambda$. This allows us to deal with the fact that the matrix $\Lambda D \Lambda^{T}$ (which must be inverted), is usually close to singular for an MEG or EEG forward model. The resulting posterior expectation is the following: $$M_{\tilde{R}|Y} = D \Lambda^T ( \Lambda D \Lambda^{T} + \lambda I)^{-1} Y (K_{\rho} + K_\xi)^{-1} K_{\rho} \,. \label{eq:posterior mean spatiotemporal, methods}$$ Besides regularizing the matrix inversion, the $\lambda I$ term contributes to filtering out the spatially non-structured observation noise. This is consistent with the fact that the regularization matrix replaces the noise spatial covariance matrix in Eq. (\[eq:spatial fourier posterior mean, methods\]) and, being diagonal, corresponds to spatially white noise. In the spatial domain, Eq. (\[eq:posterior mean spatiotemporal, methods\]) becomes: $$M_{R|Y} = P Y (K_{\rho} + K_\xi)^{-1} K_{\rho}\,. \label{eq:posterior mean spatiotemporal II, methods}$$ Therefore, the spatiotemporal expectation is obtained by applying the Harmony spatial filter (with $\Sigma = \lambda I$) to the expectation of the temporal model given by Eq. (\[eq:temporal posterior mean, methods\]). We can now apply this to the situation in which we want to estimate some component of interest, such as $\varphi(\boldsymbol{x},t)$, in the presence of other components $\zeta(\boldsymbol{x},t) = \chi(\boldsymbol{x},t) + \psi(\boldsymbol{x},t) + \xi(\boldsymbol{x},t)$. In analogy with Eq. (\[eq:temporal posterior mean, methods\]), the marginal expectation of the spatiotemporal component $\varphi(\boldsymbol{x},t)$ is given by $$M_{\Phi|Y} = P Y (K_{\varphi} + K_{\zeta})^{-1} K_{\varphi}\,, \label{eq:posterior mean spatiotemporal III, methods}$$ where $K_{\zeta}$ is the temporal covariance matrix of $\zeta(\boldsymbol{x},t) = \chi(\boldsymbol{x},t) + \psi(\boldsymbol{x},t) + \xi(\boldsymbol{x},t)$. This formula allows to individually reconstruct the dynamic components. ![**Spherical harmonics and covariance functions.**](GPpaper_Figure7.png) Visualization of the spherical harmonics morphed onto the cortex and the resulting spatial correlation structure. A) Example of spherical harmonics on the brain cortex for frequency numbers from 0 to 2. For each frequency number $l$ there are $2 l + 1$ harmonics with ”phase” number $m$ ranging from $-l$ to $l$. As clear from the picture, the spatial frequency increases as a function of the frequency number. In all our analyses we truncated the harmonic expansion after the 11th frequency number. B,C) Prior correlation structure induced by Eq. (\[eq:spatial covariance, methods\]). Panel B shows the prior correlations on the cortical surface from a cortical point identified by a red dot. Panel C shows the same function on the spherical hull. The spatial correlations are determined by the frequency discount function $f(l)$; here we used the same smoothing parameters as all analyses in the paper: $k =2$ and $\upsilon = 3$. \[fig7\] Estimating the model parameters {#estimating-the-model-parameters-1 .unnumbered} ------------------------------- We estimate the parameters of the covariance functions from all the data of each participant using an empirical Bayes method. This produces a prior distribution that is both informed by the participant-specific signal dynamics and flexible enough to account for the variability across different epochs. Specifically, given $K$ trials, the parameters are estimated from the empirical autocovariance matrix $S$ of the total measured time series: $$S = \sum_{k=1}^K Y_{k} Y_{k}^T \label{eq:empiric covariance, methods}$$ where $Y_{k}$ denotes the demeaned (mean-subtracted) spatiotemporal data matrix of an experimental trial $k$. For notational convenience, we organize all the parameters of the model covariance function in the vector $\boldsymbol{\vartheta}$. Furthermore, we make the dependence on the parameters explicit by denoting the total covariance function of the total additive model as $$k_{\rho}(t,t'; \boldsymbol{\vartheta}) = k_{\varphi}(t,t'; \boldsymbol{\vartheta}) + k_{\chi}(t,t'; \boldsymbol{\vartheta}) + k_{\psi}(t,t'; \boldsymbol{\vartheta}) + k_{\xi}(t,t'; \boldsymbol{\vartheta}). \label{eq:total covariance, methods}$$ As the objective function to be minimized, we use the sum of the squared deviations of the measured time series’ auto-covariance from the covariance function of our model: $$C(\boldsymbol{\vartheta}) = \sum_{i,j} \bigg(S_{ij} - k_{\rho}(t_i, t_j ; \boldsymbol{\vartheta})\bigg)^2 \label{eq:cost function, methods}$$ This objective function is, in general, multimodal and requires the use of a robust optimization technique. Gradient-based methods can be unstable since they can easily lead to sub-optimal local-minima. For that reason we used a gradient-free simulated annealing strategy. The details of the simulated annealing algorithm are described in [@kirkpatrick1984optimization]. As proposal distribution we used $$p(\vartheta^{(k+1)}_j) = t(\vartheta^{(k+1)}_j|\vartheta^{(k)}_j,\gamma_j,1)\,, \label{eq:proposal distribution, methods}$$ where $t(x|a,b,c)$ denotes a univariate Student’s t-distribution over $x$ with mean $a$, scale $b$ and $c$ degrees of freedom. We chose this distribution because the samples can span several order of magnitudes, thereby allowing both a quick convergence to the low cost region and an effective fine tuning at the final stages. We used the following annealing schedule: $$T(n + 1) = 0.8 \cdot T(n)\,, \label{eq:temperature schedule, methods}$$ where $T(0)$ was initialized at $10$ and the algorithm stopped when the temperature was smaller than $10^{-8}$. We estimated all the temporal parameters of the model. Specifically, the estimated parameters were the following: (a) the alpha frequency $\omega = \sqrt{\omega_0^2 - 1/4b^2}$, phase decay $\beta_{\varphi} = 1/2b_{\varphi}$, and amplitude $\mathcal{A}_{\varphi} = \sigma_{\varphi} / \sqrt{2b_{\varphi}}$, (b) the second order integrator parameters $z$, $\beta_{\chi} = 1/2b_{\chi}$, and its amplitude $\mathcal{A}_{\chi} = \sigma_{\chi} / \sqrt{2 b_{\chi}}$, (c) the first order integrator decay constant $c$ and its amplitude $\mathcal{A}_{\psi} = \sigma_{\psi} / \sqrt{2 b_{\psi}}$, and (d) the residual’s time scale $\delta$, and standard deviation $\sigma_{\xi}$. The parameters were initialized at plausible values (e.g. 10 Hz for the oscillator frequency) and were constrained to stay within realistic intervals ( 6–15 Hz for alpha frequency, positive for $\beta_{\varphi}$, $\beta_{\chi}$, $c$, $\delta$ and all the amplitudes). Details of the simulation studies {#details-of-the-simulation-studies .unnumbered} --------------------------------- #### Single sensor simulation {#single-sensor-simulation .unnumbered} The simulation was composed of two ”experimental” conditions that differed only with respect to the mean of the oscillatory amplitude. The simulation design involved 16 levels, with amplitude differences ranging from 15% to 60%. For each level, we generated 150,000 trials per experimental condition, which gave us very reliable estimates of the effect size. The trials were 2 s long. In order to not give an unfair advantage to our method (which is based on Gaussian processes), the trial time series were generated as a non-Gaussian random process according to the following formula: $$y(t)= \sqrt{a^2(t)+1} \cos(\omega(t)t+ \gamma)+\xi(t)+\psi(t) \,. \label{eq:syntethic signal I, methods}$$ The random initial phase $\gamma$ in this formula was drawn from a uniform distribution, and the functions $a(t)$ and $\omega(t)$ are Gaussian processes with a squared exponential covariance function (see Eq. (\[eq:Cov res, methods\])). The mean of the angular frequency $\omega(t)$ was equal to $2\pi \cdot 10$ (the typical frequency of alpha oscillations) for both experimental conditions. The noise processes $\xi(t)$ and $\psi(t)$ were generated by, respectively, a first order integrator and a residual Gaussian process (see Eq. (\[eq:Cov res, methods\]) and (\[eq:Cov first order, methods\])). We used the temporal GP decomposition to extract the oscillatory component from the simulated time series. The effect sizes were quantified as the between-condition differences between the trial-averaged amplitudes divided by the across-trials standard deviation of the amplitudes. We compared the sensitivity of the GP decomposition with the non-parametric spectral estimation using DPSS multitaper spectral analysis as described in [@percival1993spectral]. For every trial, the mean oscillatory amplitude was obtained by averaging over the amplitude estimates for the orthogonal tapers. In this method, the number of tapers is a free parameter that determines the degree of spectral smoothing. For each cell of the simulation design, we chose the number of tapers that maximizes the effect size. This selection procedure is biased in favor of the multitaper method since it tends to overfit the data and therefore produces larger effect sizes. #### Source level simulation {#source-level-simulation .unnumbered} A template cortical surface mesh was created using Freesurfer [@fischl2012freesurfer], down-sampled using the MNE toolbox [@gramfort2014mne], and aligned to a template MEG sensor configuration. We ran 500 trials, each involving two conditions that differed only with respect to the oscillatory amplitude of one cortical location. Sources were generated at three locations in the brain: one in the right parieto-temporal, one in the right occipital and one in the left parietal cortex. For each trial and condition, we generated three time series with the same temporal structure as those generated in the single sensor simulation study. The three time series were localized in cortical mesh with a spatial profile that is proportional to a Fisher-von Mises distribution. These spatial profiles can model a localized patch of activity. The dipole orientation was set to be orthogonal to the mesh surface. While all patches of activity contained the oscillatory component, only one patch involved an amplitude modulation between the two experimental conditions, and this was set at 20%. The activity was projected to the MEG sensors using a forward model obtained from a realistic head model [@nolte2003magnetic]. The effect was computed for each cortical vertex as the difference in average oscillatory amplitude between the two conditions. The oscillatory signal was first reconstructed at each cortical vertex using the spatiotemporal GP decomposition. Next, as in the simulation study for the single sensor, the GP estimate of average oscillatory amplitude was obtained as the standard deviation of the estimate of the oscillatory component. We compared the spatiotemporal GP decomposition with the Harmony source reconstruction of the estimated cross-spectral density matrix. Using the DPSS multitaper spectral analysis, we first estimated the sensor-level cross-spectral density matrix $F$. Next, we projected this matrix to the source level by sandwiching it between the Harmony spatial filters (see Eq. (\[eq:spatial posterior mean, methods\])): $ F_H = P F P^T $. The source level amplitude is obtained by taking the square root of the diagonal elements of $F_H$. The spectral smoothing was kept fixed at 0.6 Hz since we found this value to be optimal given the simulation parameters. Details of the application to an MEG study on anticipatory spatial attention {#details-of-the-application-to-an-meg-study-on-anticipatory-spatial-attention .unnumbered} ---------------------------------------------------------------------------- #### Participants and data collection {#participants-and-data-collection .unnumbered} We tested the spatiotemporal GP source reconstruction method on a cued tactile detection experiment in which the magneto-encephalogram (MEG) was recorded \[26\]. The study was conducted in accordance with the Declaration of Helsinki and approved by the local ethics committee (CMO Regio Arnhem-Nijmegen). Informed written consent was obtained from all participants. Fourteen healthy participants (5 male; 22–49 yr) participated in the study. The MEG system (CTF MEG; MISL, Coquitlam, British Columbia, Canada) had 273 axial gradiometers and was located in a magnetically shielded room. The head position was determined by localization coils fixed to anatomic landmarks (nasion and ears). The data were low-pass filtered (300-Hz cutoff), digitized at 1,200 Hz and stored for offline analysis. #### Experimental design {#experimental-design .unnumbered} The experiment was a tactile detection task in which the location and timing of the targets were either cued or not. A short auditory stimulus (50 ms, white noise) was presented together with an electrotactile stimulus (0.5-ms electric pulse close to threshold intensity) in half of the trials. In the other half the auditory stimulus was presented alone. Participants were asked to indicate if a tactile stimulus was presented. In one-third of the trials, an auditory cue (150 ms, pure tone) informed the participants about the timing and the location at which the tactile stimulus might occur. In particular, the target auditory signal was always presented 1.5 s after the cue. Two independent sessions were collected for each participant. More details can be found in [@van2012beyond]. #### MEG preprocessing {#meg-preprocessing .unnumbered} Third-order synthetic gradients were used to attenuate the environmental noise [@vrba2001signal]. In addition, extra-cerebral physiological sources such as heartbeat and eye movements were detected using independent component analysis [@comon1994independent] and regressed out from the signal prior to the spatiotemporal GP decomposition. #### Details of the GP spatiotemporal data analysis {#details-of-the-gp-spatiotemporal-data-analysis .unnumbered} We started the GP analysis by learning the parameters of the additive dynamical model for each individual participant using the simulated annealing method. To reduce the contribution of low-amplitude noise, we estimated this matrix from the first 50 principal components of the total empirical temporal cross-covariance matrix averaged over all channels. A template cortical surface mesh was created using Freesurfer [@fischl2012freesurfer], downsampled using the MNE toolbox [@gramfort2014mne], and aligned to the MEG sensors using the measured head position. The Tikhonov regularization parameter $\lambda$ was identified for each participant using leave-one-out cross-validation [@stone1974cross]. The spatial smoothing parameters $k$ and $\upsilon$ were set to, respectively, 2 and 3. The spatiotemporal GP decomposition was applied to 1.8 s long segments, starting ten milliseconds before the presentation of the cue and ending ten milliseconds after the target stimulus. The alpha amplitude envelope $A(t,x)$ was obtained for all cortical vertices and dipole directions by performing a Hilbert transform on the estimated alpha signal and taking the absolute value of the resulting analytic signal \[61\]. For each cortical location, the total amplitude was obtained by summing the amplitude envelopes for the three independent dipole directions $\varphi_1 (\boldsymbol{x},t)$, $\varphi_2 (\boldsymbol{x},t)$, and $\varphi_3 (\boldsymbol{x},t)$. The individual topographic maps of the attention-induced alpha amplitude suppression were obtained by computing the mean amplitude difference between cued and non-cued trials, separately for each vertex and time point. These individual maps were then averaged across participants, again for each vertex and time point. #### Statistical analysis {#statistical-analysis .unnumbered} For each cortical vertex, the dynamic effect was quantified as the rate of change of the attention-induced alpha amplitude suppression as a function of elapsed time from cue onset. Specifically, we used linear regression to estimate the slope of the relation between attention-induced alpha amplitude suppression and time. We did this separately for every vertex. The cortical maps of regression coefficients were constructed from the first experimental session of every participant and then averaged across participants. This map was subsequently used as data-driven hypothesis which was tested using the data from the second session. As a test statistic, we used the dot product between the individual regression coefficients maps, computed from the second sessions, and the group-level map. Under the null hypothesis that the group-level map is not systematic (i.e., is driven by noise only), the expected value of this test statistic is zero. Therefore we tested this null hypothesis using a one-sample t-test. Appendix I: Covariance functions defined by linear SDEs {#sec: appendix I .unnumbered} ======================================================= Consider a general linear SDE of the form $$\sum_{k}^K c_{k} \frac {d^k\alpha(t)} {dt^k} = w(t)\label{eq:general SDE, methods}$$ where the coefficients $c_{k}$ are chosen in a way to have stable solutions. An important tool for analyzing a linear differential equation is the impulse response function $G(t)$. This function is defined as the response of the system to a unit-amplitude impulse $\delta(t)$: $$\sum_{k}^K c_{k} \frac {d^k G(t)} {dt^k} = \delta(t)\label{eq:green function definition, methods}$$ Using the impulse response function, a solution of the linear SDE driven by an arbitrary random input $w(t)$ can be written as follows: $$\alpha(t) = \int_{-\infty}^{\infty} G (t- s) w(s)ds.\label{eq:green function, appendix}$$ This means that the stochastic process $\alpha(t)$ is an infinite linear superposition of responses to the random uncorrelated input $w(s)$. Using Eq. (\[eq:green function, appendix\]) we can derive the mean and covariance function of $\alpha(t)$. The mean function is defined as $$m_{\alpha}(t)= \langle \alpha (t) \rangle,\label{eq:prior expectation,appendix}$$ where the triangular brackets $\langle \cdot \rangle$ denote the expectation with respect of the distribution of the random input $w(s)$. Using (\[eq:green function, appendix\]) in (\[eq:prior expectation,appendix\]), we obtain: $$m_{\alpha}(t) = \int_{-\infty}^{\infty} G (t - s) \langle w(s) \rangle ds =0. \label{eq:deriv. prior mean, appendix}$$ Here, we used the fact that the order of expectation and integration can be interchanged and that the expectation of the white noise process is equal to zero. Analogously, we can obtain the covariance function as follows: $$k_{\alpha} (t,t' )= \langle \alpha(t)\alpha(t') \rangle = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} G (t- s) G (t'- s') \langle w(s)w(s' \rangle ds ds'. \label{eq:deriv. prior cov |, appendix}$$ Since $w(s)$ is white, its covariance $\langle w(s)w(s') \rangle$ is given by the delta function $\sigma_\alpha^2 \delta(s-s')$, where $\sigma_\alpha^2$ is the variance of the random input. The integral over $s'$ can be solved by using the translation property of the delta function: $$\int_{-\infty}^{\infty} \delta(s - s') G (t'- s') ds' = G (t' - s). \label{eq:delta function |, appendix}$$ Using this formula and introducing the new integration variable $s^*$ equal to $t' -s$, the covariance function becomes $$k_\alpha(t,t') = \sigma_\alpha^2 \int_{-\infty}^{\infty} G (t - t' + s^*) G (s^*) ds^* . \label{eq:deriv. prior cov ||, methods}$$ Since the covariance function depends on $t$ and $t'$ only though their difference $\tau = t - t'$, we denote it as $k_\varrho (\tau)$. Appendix II: Spherical harmonics and spherical Fourier transform {#sec: appendix IIs .unnumbered} ================================================================= Spherical harmonics are the generalization of sine and cosine on the surface of a sphere. They are parametrized by the integers $l$ and $m$, of which $l$ is a positive integer and $m \in \{-l,\ldots,l\}$. These two parameters determine, respectively, the angular frequency and the spatial orientation. Spherical harmonics are defined by the following formula: $$\mathcal{H}_l^m(\boldsymbol{x})= \mathcal{H}_l^m(\alpha, \theta) = \sqrt{\frac{(2l+1)(l - |m|)!}{4 \pi (l + |m|)!}} P_l^{|m|} \cos \alpha \begin{cases} 1, & \text{for} \ m = 0\\ \sqrt{2} \cos m \theta & \text{for} \ m > 0 \\ \sqrt{2} \sin |m| \theta & \text{for} \ m < 0 \end{cases} \, , \label{eq:Spherical harmonics, methods}$$ where $P_l^{|m|}$ is a Legendre polynomial [@groemer1996geometric]. Spherical harmonics form a set of orthonormal basis functions and, consequently, we can use them to define a spherical Fourier analysis [@mohlenkamp1999fast]. Specifically, the spatiotemporal process $\alpha (x,t)$ can be expressed as a linear combination of spherical harmonics $$\alpha (x,t) = \sum_{l,m} \tilde{\alpha} (l,m;t) \mathcal{H}_l^m(\boldsymbol{x}), \label{eq:inverse Fourier, methods}$$ where $\tilde{\alpha} (l,m;t)$ is the $l,m$-th spherical Fourier coefficient as a function of time, defined as $$\tilde{\alpha} (l,m; t) = \int_C \alpha (l,m;t) \mathcal{H}_l^m(\boldsymbol{x})d\boldsymbol{x}. \label{eq:direct Fourier, methods}$$ Eqs. (\[eq:inverse Fourier, methods\]) and (\[eq:direct Fourier, methods\]) are the equivalent of respectively inverse and direct Fourier transform for functions defined on the surface of a sphere. Appendix III: Properties of the Kronecker product and GP regression with separable covariance matrices {#sec: appendix III .unnumbered} ====================================================================================================== In order to derive the posterior expectations of the spatiotemporal GP regression, it is useful to introduce some of the properties of the Kronecker product between matrices. The Kronecker product between two $N \times N$ matrices is defined by the block form: $$A \otimes B = \begin{bmatrix} a_{11} B & \cdots & a_{1N} B \\ \vdots & \ddots & \vdots \\ a_{N1} B & \cdots & a_{NN} B \end{bmatrix} \, . \label{eq:Kronecker I, app I}$$ The following formula relates the regular matrix product with the Kronecker product: $$(A \otimes B) (C \otimes D) = (AC) \otimes (BD) \label{eq:Kronecker II, app I}$$ The inverse and transpose of a Kronecker product are respectively $$(A \otimes B)^{-1} = A^{-1} \otimes B^{-1} \, . \label{eq:Kronecker III, app I}$$ and $$(A \otimes B)^{T} = A^{T} \otimes B^{T} \label{eq:Kronecker IV, app I}\,.$$ The following formula relates the Kronecker product to the vectorization of a matrix: $$(A \otimes B) \text{vec} (C) = \text{vec} (B^T C A). \label{eq:Kronecker V, app I}$$ Using these formulas, we can now derive the posterior expectation (\[eq:posterior mean spatiotemporal, methods\]) of the spatiotemporal GP regression. Combining the spatiotemporal prior (\[eq:matrix normal prior, methods\]) and the observation model (\[eq:matrix normal likelihood, methods\]) using Bayes’ theorem, we obtain the posterior $$p(\text{vec} (\tilde{R}) | \text{vec} (Y)) \propto N( \text{vec} (Y)| (\Lambda \otimes I) \text{vec} (\tilde{R}), \Sigma \otimes K_\xi) N(\text{vec}(\tilde{R})|0, D \otimes K_{\rho} ) \label{eq:Derivation I, app III}$$ This is the product of two multivariate Gaussian densities and it is therefore a multivariate Gaussian itself. Its expectation is given by $$\text{vec} (M_{\tilde{\rho}|y}) = (K_{\rho} \otimes D) (I \otimes \Lambda)^T \Bigg( (I \otimes \Lambda) (K_{\rho} \otimes D) (I \otimes \Lambda)^T + (K_\xi \otimes \Sigma) \Bigg)^{-1} \text{vec} (Y). \label{eq:Derivation II, app III}$$ Using (\[eq:Kronecker II, app I\]) and (\[eq:Kronecker IV, app I\]), the expression simplifies to: $$\text{vec} (M_{\tilde{\rho}|y}) = \big( K_{\rho} \otimes (D \Lambda^T) \big) \Bigg( K_{\rho} \otimes (\Lambda D \Lambda^T) + (K_\xi \otimes \Sigma) \Bigg)^{-1} \text{vec} (Y). \label{eq:Derivation III, app III}$$ This formula involves the inversion of a matrix that is the sum of two Kronecker product components. Inverting this matrix would be computationally impractical. We simplify the problem by imposing $\Sigma = \Lambda D \Lambda^T$. In this case, Eq. (\[eq:Kronecker III, app I\]) allows to invert the spatial and temporal covariance matrices separately: $$\text{vec} (M_{\tilde{\rho}|y}) = \big( K_{\rho} \otimes (D \Lambda^T) \big) \Bigg( \big( K_{\rho} + K_\xi \big)^{-1} \otimes \big(\Lambda D \Lambda^T \big)^{-1} \Bigg) \text{vec} (Y). \label{eq:Derivation IV, app III}$$ In most realistic cases, the MEG observation model $\Lambda$ will not be full rank, therefore we introduced a Tikhonov regularization parameter $\lambda$. $$\big( \Lambda D \Lambda^T \big)^{-1} ~ \rightarrow ~ \big( \Lambda D \Lambda^T + \lambda I\big)^{-1} \label{regularization, app III}$$ Using Eq. (\[eq:Kronecker V, app I\]), we finally arrive at Eq. (\[eq:posterior mean spatiotemporal, methods\]): $$M_{\tilde{\rho}|Y} = D \Lambda^T ( \Lambda D \Lambda^{T} + \lambda I)^{-1} Y (K_{\rho} + K_\xi)^{-1} K_{\rho}\,.$$ Appendix IV: Modeling vector-valued sources using block matrices {#sec: appendix IV .unnumbered} ================================================================ The source reconstruction formulae (\[eq:spatial fourier posterior mean, methods\]) and (\[eq:spatial posterior mean, methods\]) are expressed for fixed dipole directions $\vec{v}(\boldsymbol{x})$. The solution for the general case, in which the dipole direction is estimated from the data, is obtained by introducing an independent set of spherical harmonics for each of the orthogonal spatial directions $\vec{v}_1$, $\vec{v}_2$, and $\vec{v}_3$. In this Appendix, we refer to the (spherical harmonics domain) forward model matrix relative to the $k$-th direction as $\Lambda_k$. Using this notation, we can define the total forward model matrix with the following block form: $$\Lambda_{tot} = \begin{bmatrix} \Lambda_1 \\ \Lambda_2 \\ \Lambda_3 \end{bmatrix} \label{eq: total forwardf model} .$$ Using an analogous notation, the total spherical harmonics covariance matrix can be written in the following block diagonal form: $$D_{tot} = \begin{bmatrix} D_1&&0&&0 \\ 0&&D_2&&0 \\ 0&&0&&D_3 \end{bmatrix} \label{eq: total harmonic covariance function} .$$ Hence, the general source reconstruction formula in the spherical harmonics domain is obtained from Eq. (\[eq:spatial fourier posterior mean, methods\]) by replacing $D$ and $\Lambda$ with $D_{tot}$ and $\Lambda_{tot}$ respectively. This solution can be mapped back to the spatial domain using the total spherical harmonics matrix $$H_{tot} = \begin{bmatrix} H&&0&&0 \\ 0&&H&&0 \\ 0&&0&&H \end{bmatrix} \, , \label{eq: total spherical harmonics matrix}$$ where $H$ is defined as in Eq. (\[eq:harmonic matrix, methods\]).
--- abstract: 'We studied the spatial distributions of multiple stellar populations (MPs) in a sample of 20 globular clusters (GCs) spanning a broad range of dynamical ages. The differences between first-population (FP) and second-population (SP) stars were measured by means of the parameter $A^+$, defined as the area enclosed between their cumulative radial distributions. We provide the first purely observational evidence of the dynamical path followed by MPs from initial conditions toward a complete FP-SP spatial mixing. Less dynamically evolved clusters have SP stars more centrally concentrated than FPs, while in more dynamically evolved systems the spatial differences between FP and SP stars decrease and eventually disappear. By means of an appropriate comparison with a set of numerical simulations, we show that these observational results are consistent with the evolutionary sequence expected by the long-term dynamical evolution of clusters forming with an initially more centrally concentrated SP sub-system. This result is further supported by the evidence of a trend between $A^+$ and the stage of GC dynamical evolution inferred by the ratio between the present-day and the initial mass of the cluster.' author: - Emanuele Dalessandro - 'M. Cadelano' - 'E. Vesperini' - 'S. Martocchia' - 'F. R. Ferraro' - 'B. Lanzoni' - 'N. Bastian' - 'J. Hong' - 'N. Sanna' title: 'A family picture: tracing the dynamical path of the structural properties of multiple populations in globular clusters' --- Introduction {#sec:intro} ============ The presence of sub-populations differing in terms of their light-element abundances (e.g. He, C, N, O, Na, Mg, Al) while having the same iron (and iron-peak) content (hereafter multiple stellar populations - MPs) is a key general property of globular clusters (GCs; see @bastian18 for a recent review). In fact, MPs are observed in nearly all old ($t>2$ Gyr) and relatively massive systems ($M>10^4 M_{\odot}$), both in the Milky Way and in external galaxies (e.g., @mucciarelli08 [@larsen14; @dalessandro16]). MPs are characterized by specific light-element chemical abundance patterns like C-N, Na-O, Mg-Al anti-correlations. Stars sharing the same chemical abundances as the surrounding field stars (Na-poor/O-rich, CN-weak) are commonly classified as first-population (FP), while Na-rich/O-poor, CN-strong stars are referred to as second-population (SP). Light-element chemical abundance variations can have an impact on both the stellar structure and atmosphere thus producing a variety of features (such as broadening or splitting of different evolutionary sequences) in color-magnitude diagrams (CMDs) when appropriate optical and near-UV bands are used [@sbordone11; @piotto15; @milone17]. It has been shown that the fraction of SP stars and the amplitude of the light-element anti-correlations depends on the present-day cluster mass (e.g. @carretta10 [@schiavon13; @milone17]), with relatively small systems ($M<10^5 M_{\odot}$) typically having a fraction of $\sim 40\%-50\%$ of SP stars, which then increases to $\sim 90\%$ for the most massive ones. Light-element inhomogeneities appear to decrease also as a function of cluster age, becoming undetectable for cluster younger than $\sim 2$ Gyr (@martocchia18a), although the exact role of age is currently not clear yet. MPs are believed to form during the very early epochs of GC formation and evolution ($\sim 10-100$ Myr, but see @martocchia18b for recent observational constraints on this aspect). A number of scenarios have been proposed over the years to explain their formation, however their origin is still strongly debated [@decressin07; @dercole08; @bastian13; @denissenkov14; @gieles18; @calura19]. The kinematical and structural properties of MPs can provide key insights into the early epochs of GC evolution and formation. In fact, one of the predictions of MP formation models (see e.g. @dercole08) is that SP stars form a centrally segregated stellar sub-system possibly characterized by a more rapid internal rotation [@bekki11] than the more spatially extended FP system. Although the original structural and kinematical differences between FP and SP stars are gradually erased during GC long-term dynamical evolution (see e.g. @vesperini13 [@henault15; @miholics15; @tiongco19]), some clusters are expected to still retain some memory of these initial differences in their present-day properties. Indeed, sparse and inhomogeneous observations show that MPs are characterized by quite remarkable differences in their relative structural parameters/radial distributions [@lardo11; @dalessandro16; @massari16; @simioni16], different degrees of orbital anisotropy [@richer13; @bellini15], different rotation amplitudes [@cordero17] and significantly different binary fractions [@lucatello15; @dalessandro18a]. However, so far the lack of a homogeneous and self-consistent study of MP kinematical and structural properties for a statistically representative sample of clusters has hampered our ability to build an observational picture to test and constrain models for the formation and evolutionary history of GCs. In this Letter we use the $A^+$ parameter (originally introduced for blue straggler star studies; @alessandrini16 [@lanzoni16]) to quantify the differences in the radial distributions of FP and SP stars for a large sample of GCs in different stages of their dynamical evolution measured here by the ratio $N_h = t/t_{rh}$ between the cluster age $t$ and its current half-mass relaxation times ($t_{rh}$). A comparison of our results with those of numerical simulations following the dynamical evolution and spatial mixing of MPs allows us to draw, for the first time, an observational picture of the evolutionary path of FP and SP structural properties. Sample definition and population selection ========================================== For the present analysis we mainly used the publicly available photometric catalogs of Galactic GCs presented in @nardiello18 [see also @piotto15] and observed through proposals GO-13297, GO-12605 and GO-12311 (PI: Piotto) with the HST WFC3/UVIS camera in the F275W, F336W and F438W bands and with the HST ACS/WFC under proposal GO-10775 (PI: Sarajedini) in the F606W and F814W filters. We limited our analysis only to systems for which the available HST catalogs cover at least 2 cluster half-light radii ($r_h$) allowing us to probe a region large enough to capture possible differences between the SP and FP spatial distributions. With the adopted selection we are able to include in our sample 15 GCs, most of which have $N_h>7-8$. To further extend our analysis and include clusters with smaller values of $N_h$, which is essential for the goals of our study, we complemented our data-set with the wide-field photometric catalog (that includes U, B, V and I bands) published by @stetson19 for the low-mass cluster NGC288, the Stromgren photometry of NGC5272 (M3) presented by @massari16 and the combined HST and ground-based wide-field catalog of NGC6362 published in @dalessandro14. Finally, we included also two extra-galactic systems, namely NGC121 in the Small Magellanic Cloud and NGC1978 in the Large Magellanic Cloud. The HST photometry of these two clusters was presented in @dalessandro16 and @martocchia18a respectively. It is important to stress that to make the MP separation and selection as straightforward/clear as possible, only clusters with intermediate-high metallicity[^1], low reddening, relatively low field contamination and with a well populated red giant branch were added to the initial list of 15 GCs. With such a combination our sample counts 20 GCs covering (see Table 1) a wide range in metallicity ($-0.4<[Fe/H]<-2$) and present-day mass ($3.6\times 10^4 M_{\odot}<M<1.4\times10^6 M_{\odot}$), which are well representative of the population of Galactic and Magellanic Cloud GCs, with the exception of the lower mass systems in the Clouds. More importantly to the present analysis, the sample covers the full range of dynamical stages derived for Galactic and Magellanic Cloud clusters ($1<N_h<80$). For the clusters for which we used the photometric catalogs published by @nardiello18, MPs were selected along the red giant branch (RGB) in the ($\Delta_{F275W,F814W}, \Delta_{F275W,F336W,F438W}$) diagram, the so called “chromosome map”, following the same approach used by @milone17 and schematically shown in Figure 1 (panels a). Briefly, we verticalized the distribution of RGB stars in the ($m_{F814W}, C_{F275W,F336W,F438W}$) (where $C_{F275W,F336W,F438W}=(m_{F275W}-m_{F336W}) - (m_{F336W}m_{F438W})$) and ($m_{F814W}, m_{F275W}-m_{F814W}$) diagrams with respect to two fiducial lines at the blue and red edges of the RGB in both CMDs (Figure 1 panels a1 and a2). The combination of the two verticalized distributions ($\Delta_{F275W,F814W}$ and $\Delta_{F275W,F336W,F438W}$) gives the “chromosome map" (Figure 1 panel a3). Only stars with a membership probability $>75\%$ and with quality flags $>0.9$ in all bands were used (see @nardiello18 for details). For NGC121, NGC6362, M3 and NGC1978 we adopted the same sub-population selections described in @dalessandro14 [@dalessandro16; @massari16; @martocchia18a] respectively. For the case of NGC288, we used a two-step approach. For stars at a cluster-centric distance $R<100\arcsec$ we used the HST catalog published by @nardiello18 and the selection criteria described before. For the external region we first matched the ground-based catalog with Gaia DR2 data. Cluster bona-fide stars were selected based on their Gaia proper motions. We assumed ($\mu_{\alpha}=4.24, \mu_{\delta}=5.65)$ mas/yr as cluster mean motion [@helmi18] and we selected stars at distance $d<1.5$ mas yr$^{-1}$ in the vector-point diagram. RGB likely cluster members were verticalized in the ($U, (U-B)-(B-I)$) CMD with respect to a fiducial line on the blue edge of the RGB (Figure 1 panel b1; see also @monelli13). The resulting distribution is clearly bimodal (panel b2). Stars redder/bluer than $\Delta_{(U-B)-(B-I)}=-0.55$ were selected as FP/SP stars. It is important to note that, while in general, the adoption of different filter combinations for FP and SP classifications can introduce some bias, this is not the case for the specific targets in our sample for which both ground-based photometry and the HST “chromosome-map” are available, namely NGC288, NGC6362 and M3. In fact, we have verified, by using the stars in common between the available HST and wide-field catalogs, that there is a nice match between the two sub-population selections thus ensuring homogeneity of the different samples. Radial distribution of multiple populations and empirical derivation of the parameter $A^+$ =========================================================================================== We derived the cumulative radial distributions of the selected sub-populations by using the cluster centers reported in @ferraro12 and @lanzoni16 and references therein for the clusters in common, and those listed in @goldsbury10 for the other Galactic GCs. For NGC121 and NGC1978 we used the centers derived by @dalessandro16 and @martocchia18a respectively. In order to obtain a homogeneous measure of the differences between the SP and FP spatial distributions we have used the $A^+$ parameter introduced by @alessandrini16 and @lanzoni16 in the context of the study the spatial segregation of blue straggler stars. In our study $A^+$ is calculated as the area enclosed between the cumulative radial distributions of FP and SP stars, $\phi_{FP}(R)$ and $\phi_{SP}(R)$, respectively: $$A^+(R)=\int_{R_{min}}^R (\phi_{FP}(R^{'})-\phi_{SP}(R^{'}))dR^{'}$$ where R is the distance from the cluster center. With such a definition, a more centrally concentrated SP yields negative values of $A^+$. By construction $A^+$ depends on the considered cluster-centric distance and therefore a meaningful cluster-to-cluster comparison requires that the parameter is measured over equivalent radial portions in every system. As shown in numerical studies (see e.g. @vesperini13), spatial mixing is achieved first in a cluster’s inner regions and later in the cluster’s outskirts. Therefore capturing a complete dynamical picture of the mixing process in a given cluster would require a wide radial coverage possibly extending to the cluster’s outermost regions, which retain memory of the initial spatial differences for a longer time. With this in mind, we decided to measure $A^+$ within 2 $r_h$ from the cluster center ($A^+_{2}$). This limit represents a compromise between radial coverage and cluster sample size. We adopted the values of $r_h$ reported by (@harris96 - 2010 version) for all the Galactic clusters, while we used @glatt11 for NGC121. For NGC1978 we derived $r_h=31.5\arcsec$ by fitting its number count density profile (derived by using the HST catalog) with a single-mass @king66 model. Uncertainties on the derived values of $A^+$ have been obtained by applying a jackknife bootstrapping technique [@lupton93]. The results are reported in Table 1. Results ======= The MP radial distributions in the targeted clusters appear to be quite different from one case to the other. However, in general we can identify two main behaviors: in about half of the sample, SP stars are more centrally concentrated than FPs, in the other clusters there is no significant difference between the FP and SP distributions. As a result, the derived values of $A^+_{2}$ cover a quite large range, from a minimum of $\sim-0.107\pm 0.006$ for NGC6715 (M54) to $\sim0.080\pm0.016$ for NGC6717 (Table 1). The cumulative radial distributions for three systems with different behaviors are shown in Figure 2 as an example. For every cluster we determined $N_h$ by adopting the ages derived by @dotter10 for Galactic GCs and by @martocchia18a and @glatt11 for NGC1978 and NGC121 respectively, while the values of $t_{rh}$ are taken from @harris96 and @glatt11 for NGC121. For NGC1978 we derived Log$(t_{rh})=9.02$ (where $t_{rh}$ has been calculated as in @harris96). Figure 3 shows the distribution of $A^+_2$ as a function of $N_h$. The $A^+_2$ parameter increases almost linearly up to $N_h \sim 10$, reaching values close to 0 where FP and SP stars are (almost) fully radially mixed, then it shows an almost constant distribution for older dynamical ages up to $N_h\sim80$. The general trend shown in Figure 3 suggests that SP stars are significantly more concentrated than FPs in systems with $N_h< 8-10$, while MP radial distributions do not show significant differences for clusters in more advanced stages of their dynamical evolution (with $N_h>10$). The only two exceptions are NGC 6093 (M80) and NGC 6717, which are the systems in the sample characterized by most positive values of $A^+$. The MP radial distribution of M80 has been analyzed in detail and extensively discussed in @dalessandro18b. To illustrate the expected evolution of $A^+_2$ as a function of $N_h$, in Figure 3 (bottom panel) we show the time evolution of $A^+_2$ obtained from $N$-body simulations following the long-term dynamical evolution of two MP clusters in which the SP is initially 5 and 10 times more centrally concentrated than the FP one. The simulations start with 50000 stars equally split between FP and SP and follow a cluster internal evolution and mass loss due to the combined effects of two-body relaxation and tidal truncation. The simulations have been presented in @vesperini18 and @dalessandro18b and we refer to those papers for further details. Here we use these simulations to explore the role of internal two-body relaxation and the interaction of the external tidal field of the host galaxy in the evolution of $A^+_2$ as a function of $N_h$. We point out that the simulations presented here are still idealized and not meant to model any specific cluster in detail, but they serve to illustrate the general evolutionary trend expected for $A^+_2$ as the SP and the FP mix. Detailed models aimed at reproducing the properties of specific clusters would require more realistic simulations. Since the $N$-body models start with a more centrally concentrated SP radial distribution, the simulations have initially negative values of $A^+_2$. As the FP and SP stellar sub-systems evolve (i.e. $N_h$ increases) the two populations gradually mix and, as a consequence, $A^+_2$ increases evolving toward zero, which represents the value corresponding to a fully radially mixed configuration. Although the simulations are still simplified, they follow the general $A^+_2$ trends. This suggests that the different shapes of MP radial distributions and the trend found in this study are the result of the effects of the long-term dynamical evolution in clusters formed with an initially more centrally concentrated SP stellar sub-system. It is important to note that in this comparison FP and SP are assumed to have the same He abundance or only small mean variations ($\Delta Y<0.01-0.02$). Indeed, this is observed to be the case in the vast majority of GCs (see for example @dalessandro13) with only a few exceptions in our sample, such as NGC2808 [@piotto07], M80 [@dalessandro18b], NGC7078 (M15) and M54 [@milone18]. In Figure 4 we show the dependence of $A^+_2$ on the ratio between the present-day and the initial cluster mass ($M_{\rm PD}/M_{\rm ini}$), as estimated by @baumgardt19. Although it is important to emphasize that much caution should be used in taking $M_{\rm PD}/M_{\rm ini}$ ratios at face value because of the underlying strong assumptions made to derive them, and the possible missing contribution of related effects[^2], they nevertheless provide a measure of the evolutionary stage of a cluster and its degree of mass loss due to two-body relaxation and the interaction with the Galactic potential. Our data show a significant correlation (Spearman’s rank correlation coefficient $r\sim-3.7$) between $A^+_2$ and $M_{\rm PD}/M_{\rm ini}$: clusters with small value of $M_{\rm PD}/M_{\rm ini}$ (i.e. systems that lost a larger fraction of their original mass) tend to have their MPs spatially mixed. Interestingly, such a behavior is also reproduced (at least qualitatively) by our $N$-body models, thus demonstrating that the fraction of mass lost is a key ingredient of the MP spatial mixing process (see the discussion on this issue in @vesperini13 [@henault15; @miholics15]). Not surprisingly (because of the known dependence with the dynamical parameters used before) we find that $A^+_2$ nicely anti-correlates with the present-day mass ($M_{PD}$ from @baumgardt18). Conclusions =========== The variations of the MP radial distributions as a function of the evolutionary stage in the clusters’ dynamical evolution shown in this Letter provides [ *the first observational evidence of the dynamical path followed by MPs from their initial conditions toward a complete spatial mixing.*]{} Our study has revealed a clear trend of the difference between the SP and FP spatial radial distributions ($A^+_2$) and globular cluster dynamical evolution, as constrained by both the ratio of a cluster’s age to its half-mass relaxation timescale and the ratio of a cluster’s present-day to its initial mass. This is the first time that observational constraints on the evolutionary path of the MP structural differences are set and put in the framework of star cluster dynamical evolution. Although additional work is needed to constrain in detail the initial physical properties of MPs both observationally and in the context of different theoretical formation models, our results provide a global view of the evolution of the MP structural properties. They lend support to an interpretation of the different degrees of spatial mixing observed in various clusters in terms of dynamical evolution of systems in which the SP formed more centrally concentrated than the FP. At the same time, the empirical evolutionary sequence found in our analysis also provides a key constraint for models exploring the long-term dynamics of MPs, which is an important aspect of the study of MP clusters. The result presented here has important implications also for the interpretation of other kinematical features observed in MPs, such as their rotation patterns and anisotropy profiles, and therefore is key to shed light on the physical initial conditions that brought to the formation of MPs. An extension of the present analysis, mainly including a larger sample of less dynamically evolved clusters, is needed to further confirm and sharpen the picture emerging from our study. In addition, a systematic combination of structural and kinematic information of MPs is an essential step to properly interpreting observational data, as well as testing the key elements of theoretical scenarios of cluster formation and evolution. [ccccccccc]{} NGC121 & -0.047 & 0.001 & 10.021 & 9.53 & 3.42 & 27.0 & 61.9 & -1.28\ NGC288 & -0.045 & 0.002 & 10.097 & 9.32 & 1.16 & 133.8 & 12.0 & -1.32\ NGC362 & -0.040 & 0.001 & 10.061 & 8.93 & 3.45 & 49.2 & 9.4 & -1.26\ NGC1261 & 0.023 & 0.001 & 10.061 & 9.12 & 1.67 & 40.8 & 18.1 & -1.27\ NGC1851 & -0.032 & 0.001 & 10.079 & 8.82 & 3.02 & 30.6 & 16.6 & -1.18\ NGC1978 & -0.081 & 0.003 & 9.301 & 9.02 & 2.00 & 31.1 & 49.6 & -0.35\ NGC2808 & -0.029 & 0.001 & 10.079 & 9.15 & 7.42 & 49.0 & 11.1 & -1.14\ NGC5272 & -0.059 & 0.001 & 10.097 & 9.79 & 3.94 & 186.0 & 12.0 & -1.5\ NGC5286 & -0.013 & 0.001 & 10.114 & 9.11 & 4.01 & 43.8 & 8.9 & -1.69\ NGC6093 & 0.056 & 0.001 & 10.130 & 8.80 & 2.49 & 36.6 & 3.8 & -1.75\ NGC6101 & -0.003 & 0.001 & 10.114 & 9.22 & 1.27 & 63.0 & 11.2 & -1.98\ NGC6362 & -0.010 & 0.002 & 10.097 & 9.20 & 1.47 & 123.0 & 5.1 & -0.99\ NGC6584 & 0.033 & 0.002 & 10.088 & 9.02 & 0.91 & 43.8 & 7.0 & -1.50\ NGC6624 & 0.016 & 0.002 & 10.114 & 8.71 & 0.73 & 49.2 & 1.2 & -0.44\ NGC6637 & -0.028 & 0.001 & 10.097 & 8.82 & 2.45 & 50.4 & 1.7 & -0.64\ NGC6652 & 0.029 & 0.003 & 10.122 & 8.39 & 0.57 & 28.8 & 2.7 & -0.81\ NGC6681 & -0.031 & 0.003 & 10.114 & 8.65 & 1.13 & 42.6 & 2.2 & -1.62\ NGC6715 & -0.107 & 0.001 & 10.079 & 9.93 & 14.1 & 49.2 & 18.9 & -1.49\ NGC6717 & 0.080 & 0.004 & 10.114 & 8.22 & 0.36 & 45.0 & 2.4 & -1.26\ NGC6934 & 0.000 & 0.002 & 10.079 & 9.04 & 1.17 & 41.4 & 12.8 & -1.47\ The authors thank the anonymous referee for the careful reading of the paper and the useful comments that improved the presentation of this work E.D. acknowledges support from The Leverhulme Trust Visiting Professorship Programme VP2- 2017-030. E.D. warmly thank Holger Baumgardt for providing the $M_{PD}/M_{ini}$ values. E.D. also thank Francesco Calura and Michele Bellazzini for useful discussions. The research is funded by the project Light-on-Dark granted by MIUR through PRIN2017-000000 contract (PI: Ferraro). N.B. gratefully acknowledges financial support from the Royal Society (University Research Fellowship) and the European Research Council (ERC-CoG-646928, Multi-Pop). 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Thus, photometric broadenings or splittings of the evolutionary sequences in the CMD are harder to detect in metal-poor systems. [^2]: Examples of the missing contribution are early time-variation of the external potential or other mechanisms related to a cluster’s response to early evolutionary processes (e.g. gas expulsion, mass loss due to stellar evolution, interactions with giant molecular clouds)
--- abstract: 'The valence band of a variety of few-layer, two-dimensional materials consists of a ring of states in the Brillouin zone. The energy-momentum relation has the form of a ‘Mexican hat’ or a Rashba dispersion. The two-dimensional density of states is singular at or near the band edge, and the band-edge density of modes turns on nearly abruptly as a step function. The large band-edge density of modes enhances the Seebeck coefficient, the power factor, and the thermoelectric figure of merit ZT. Electronic and thermoelectric properties are determined from ab initio calculations for few-layer III-VI materials GaS, GaSe, InS, InSe, for Bi$_{2}$Se$_{3}$, for monolayer Bi, and for bilayer graphene as a function of vertical field. The effect of interlayer coupling on these properties in few-layer III-VI materials and Bi$_{2}$Se$_{3}$ is described. Analytical models provide insight into the layer dependent trends that are relatively consistent for all of these few-layer materials. Vertically biased bilayer graphene could serve as an experimental test-bed for measuring these effects.' author: - Darshana Wickramaratne - Ferdows Zahid - 'Roger K. Lake' title: 'Mexican Hat and Rashba Bands in Few-Layer van der Waals Materials' --- Introduction ============ The electronic bandstructure of many two-dimensional (2D), van der Waals (vdW) materials qualitatively changes as the thickness is reduced down to a few monolayers. One well known example is the indirect to direct gap transition that occurs at monolayer thicknesses of the Mo and W transition metal dichalcogenides (TMDCs)[@MoS2_Mak_Heinz]. Another qualitative change that occurs in a number of 2D materials is the inversion of the parabolic dispersion at a band extremum into a ‘Mexican hat’ dispersion.[@Fermi_ring_Neto_PRB07; @zolyomi_GaX; @zolyomi_InX] Mexican hat dispersions are also referred to as a Lifshiftz transition [@graphene_trilayer_Macdonald; @zolyomi_GaX; @Falko_BLG_Lifshitz_PRL14], an electronic topological transition [@blanter_ETT_theory] or a camel-back dispersion [@PbTe_camelback_thermo; @Te_camelback_thermo]. In a Mexican hat dispersion, the Fermi surface near the band-edge is approximately a ring in $k$-space, and the radius of the ring can be large, on the order of half of the Brillouin zone. The large degeneracy coincides with a singularity in the two-dimensional (2D) density of states close to the band edge. A similar feature occurs in monolayer Bi due to the Rashba splitting of the valence band. This also results in a valence band edge that is a ring in $k$-space although the diameter of the ring is generally smaller than that of the Mexican hat dispersion. Mexican hat dispersions are relatively common in few-layer two-dimensional materials. Ab-initio studies have found Mexican hat dispersions in the valence band of many few-layer III-VI materials such as GaSe, GaS, InSe, InS [@zolyomi_GaX; @zolyomi_InX; @GaS_photodetector_AnPingHu; @Hennig_GroupIII_ChemMat; @SGLouie_GaSe_arxiv; @WYao_GaS_GaSe_arxiv]. Experimental studies have demonstrated synthesis of monolayers and or few layers of GaS, GaSe and InSe thin films.[@GaSe_Ajayan_NL13; @Ajayan_InSe; @GaS_photodetector_AnPingHu; @Xiao_GaSe_nanosheets; @Dravid_GaS_GaSe_AdvMat; @GaSe_Geohagen_SciRep; @GaSe_Geohagen_ACSNano; @CZhou_GaS_ACSNano]. Monolayers of Bi$_2$Te$_3$ [@Zahid_Lake], and Bi$_{2}$Se$_{3}$ [@Udo_Bi2Se3] also exhibit a Mexican hat dispersion in the valence band. The conduction and valence bands of bilayer graphene distort into approximate Mexican hat dispersions, with considerable anisotropy, when a a vertical field is applied across AB-stacked bilayer graphene. [@MacDonald_bi_gap_PRB07; @Fermi_ring_Neto_PRB07; @Falko_BLG_Lifshitz_PRL14] The valence band of monolayer Bi(111) has a Rashba dispersion. [@Bi_ARPES_Rashba_PRL11] The large density of states of the Mexican hat dispersion can lead to instabilities near the Fermi level, and two different ab initio studies have recently predicted Fermi-level controlled magnetism in monolayer GaSe and GaS [@SGLouie_GaSe_arxiv; @WYao_GaS_GaSe_arxiv]. The singularity in the density of states and the large number of conducting modes at the band edge can enhance the Seebeck coefficient, power factor, and the thermoelectric figure of merit ZT. [@Mahan:1996; @heremans_PbTe_thermo; @heremans_distortionDOS_Thermo] Prior studies have achieved this enhancement in the density of states by using nanowires [@Dressel:1D:PRB:1993; @Dresselhaus_New_Directions07], introducing resonant doping levels [@heremans_PbTe_thermo; @heremans_distortionDOS_Thermo], high band degeneracy [@Snyder_degenerate_thermo; @Kanatzidis_New_Old_AChemie09], and using the Kondo resonance associated with the presence of localized $d$ and $f$ orbitals [@Kondo_thermo_Cava; @Steglich_FeSb2_thermo; @Gang_Chen_Kondo_NW11]. The large increase in ZT predicted for monolayer Bi$_2$Te$_3$ resulted from the formation of a Mexican hat bandstructure and its large band-edge degeneracy [@Zahid_Lake; @Lundstrom_Jesse_Bi2Te3]. This work theoretically investigates the electronic and thermoelectric properties of a variety of van der Waals materials that exhibit a Mexican hat dispersion or Rashba dispersion. The Mexican hat and Rashba dispersions are first analyzed using an analytical model. Then, density functional theory is used to calculate the electronic and thermoelectric properties of bulk and one to four monolayers of GaX, InX (X = Se, S), Bi$_{2}$Se$_{3}$, monolayer Bi(111), and bilayer graphene as a function of vertical electric field. Figure \[fig:structure\] illustrates the investigated structures that have either a Mexican hat or Rashba dispersion. ![(Color online) Atomic structures of van-der Waals materials with a Mexican hat or Rashba dispersion: (a) Bilayer III-VI material. The $\beta$ phase stacking geometry is shown at right. (b)Bi$_{2}$Se$_{3}$, (c) Bilayer Graphene and (d) Bi(111) monolayer []{data-label="fig:structure"}](Mats_All.eps){width="5in"} The analytical model combined with the numerically calculated orbital compositions of the conduction and valence bands explain the layer dependent trends that are relatively consistent for all of the few-layer materials. While numerical values are provided for various thermoelectric metrics, the emphasis is on the layer-dependent trends and the analysis of how the bandstructure affects both the electronic and thermoelectric properties. The metrics are provided in such a way that new estimates can be readily obtained given new values for the electrical or thermal conductivity. Models and Methods ================== Landauer Thermoelectric Parameters ---------------------------------- In the linear response regime, the electronic and thermoelectric parameters are calculated within a Landauer formalism. The basic equations have been described previously [@Lundstrom_Jesse_Bi2Te3; @Klimeck_DOM_thermoelectric_JCE; @Darshana_MX2_Thermo], and we list them below for convenience. The equations for the electronic conductivity ($\sigma$), the electronic thermal conductivity ($\kappa_{e}$), and the Seebeck coefficient (S) are $$\begin{aligned} \sigma &= (2q^{2}/h)I_{0}\quad (\mathrm{\Omega^{-1} m^{2-D}}), \label{eq:sigma}\\ \kappa_{e} &= (2Tk_{B}^{2}/h)(I_{2} - I_{1}^{2}/I_{0}) \quad (\mathrm{W m^{2-D} K^{-1}}), \label{eq:Ke}\\ S &= -(k_{B}/q)\frac{I_{1}}{I_{0}}\quad (\mathrm{V/K}), \label{eq:S}\\ \mathrm{with} \nonumber \\ I_{j} &= L \int_{-\infty}^{\infty} \left(\frac{E-E_{F}}{k_{B}T}\right)^{j} \bar{T}(E) \left(-\frac{\partial f}{\partial E}\right)dE \label{eq:Ij}\end{aligned}$$ where $L$ is the device length, $D$ is the dimensionality (1, 2, or 3), $q$ is the magnitude of the electron charge, $h$ is Planck’s constant, $k_B$ is Boltzmann’s constant, and $f$ is the Fermi-Dirac factor. The transmission function $\bar{T}$ is $$\bar{T}(E) = T(E)M(E) \label{eq:TE}$$ where M(E) as the density of modes. In the diffusive limit, $$T(E)=\lambda(E)/L , \label{eq:TE_diff}$$ where $\lambda(E)$ is the electron mean free path. The power factor ($PF$) and the thermoelectric figure of merit ($ZT$) are given by $PF = S^2 \sigma$ and $$ZT = S^2 \sigma T / (\kappa_{l} + \kappa_{e}) \label{eq:ZT}$$ where $\kappa_l$ is the lattice thermal conductivity. Analytical Models {#sec:analytical} ----------------- The single-spin density of modes for transport in the $x$ direction is [@Datta_book05; @Lundstrom:TE:TB:JAP:2010] $$M(E)= \frac{2\pi}{L^D} \sum_{{{\bf k}}} \delta(E - \epsilon({{\bf k}})) \frac{\partial {\epsilon}}{\partial k_x} \label{eq.ME}$$ where $D$ is the dimensionality, $E$ is the energy, and ${\epsilon}({{\bf k}})$ is the band dispersion. The sum is over all values of ${{\bf k}}$ such that $\frac{\partial {\epsilon}}{\partial k_x} > 0$, i.e. all momenta with positive velocities. The dimensions are $1/L^{D-1}$, so that in 2D, $M(E)$ gives the number of modes per unit width at energy $E$. If the dispersion is only a function of the magnitude of ${{\bf k}}$, then Eq. (\[eq.ME\]) reduces to $$M(E)= \frac{N_D}{\left(2\pi\right)^{D-1}} \sum_{b} k_b^{D-1}(E) \label{eq.MEk}$$ where $N_D = \pi$ for $D=3$, $N_D = 2$ for $D=2$, and $N_D = 1$ for $D=1$. $k_b$ is the magnitude of ${{\bf k}}$ such that $E = {\epsilon}(k_b)$, and the sum is over all bands and all values of $k_b$ within a band. When a band-edge is a ring in $k$-space with radius $k_0$, the single-spin 2D density of modes at the band edge is $$M(E_{\rm edge}) = N \frac{k_0}{\pi} , \label{eq:M_edge}$$ where $N$ is either 1 or 2 depending on the type of dispersion, Rashba or Mexican hat. Thus, the 2D density of modes at the band edge depends only on the radius of of the $k$-space ring. For a two dimensional parabolic dispersion, $E = \frac{\hbar^2k^2}{2m^*}$, the radius is 0, and Eq. (\[eq.MEk\]) gives a the single-spin density of modes of [@Lundstrom:TE:analytic:JAP:2009] $$M_{\rm par}(E)=\frac{\sqrt{2m^*E}}{\pi\hbar}. \label{eq:DOM_2D}$$ In real III-VI materials, there is anisotropy in the Fermi surfaces, and a 6th order, angular dependent polynomial expression is provided by Zólyomi et al. that captures the low-energy anisotropy [@zolyomi_GaX; @zolyomi_InX]. To obtain physical insight with closed form expressions, we consider a 4th order analytical form for an isotropic Mexican hat dispersion $$\epsilon(k) = \epsilon_{0} - \frac{\hbar^{2} k^2}{2m^*} + \frac{1}{4\epsilon_{0}} \left( \frac{\hbar^{2} k^2}{2m^*} \right)^2 \label{eq:MexHat_Ek}$$ where ${\epsilon}_0$ is the height of the hat at $k=0$, and $m^*$ is the magnitude of the effective mass at $k=0$. A similar quartic form was previously used to analyze the effect of electron-electron interactions in biased bilayer graphene [@Fermi_ring_Neto_PRB07]. The function is plotted in Figure \[fig:MexicanHat\](a). ![(Color online) (a) Comparison of a Mexican hat dispersion (red) and a Rashba dispersion (green). The band edges are rings in $k$-space with radius $k_0$ illustrated for the Mexican hat band by the orange dotted circle. The height of the Mexican hat band at $k=0$ is $\epsilon_{0} = 0.111$ eV. The Rashba parameter is 1.0 eV Å, and the effective mass for both dispersions is the bare electron mass $m_0$. (b) Density of modes of the Mexican hat dispersion (red) versus parabolic band (blue). The parabolic dispersion also has an effective mass of 1.0. (c) Room temperature Seebeck coefficients (solid lines) and carrier concentrations (broken lines) of the Mexican hat band (red) and the parabolic band (blue) as a function of Fermi level position, E$_{F}$. (d) Room temperature ballistic power factor of the Mexican hat band (red) and the parabolic band (blue) calculated from Eqs. (\[eq:sigma\]), (\[eq:S\]), and (\[eq:Ij\]) with $T(E) = 1$. []{data-label="fig:MexicanHat"}](Analytical_Ek_MHat_Parabolic.eps){width="5in"} The band edge occurs at ${\epsilon}= 0$, and, in $k$-space, in two dimensions (2D), it forms a ring in the $k_x-k_y$ plane with a radius of $$k_0^{\rm MH} = \frac{2 \: \sqrt{m^* {\epsilon}_0}}{\hbar} . \label{eq:k0_MHat}$$ For the two-dimensional Mexican hat dispersion of Eq. (\[eq:MexHat\_Ek\]), the single-spin density of modes is $$M_{\rm MH}(E)= \left\{ \begin{array}{ll} \frac{k_0^{\rm MH}}{\pi} \left( \sqrt{1+\sqrt\frac{E}{\epsilon_{0}}} + \sqrt{1-\sqrt\frac{E}{\epsilon_{0}}} \; \right) \;\;\;\;\; & \left( 0 \leq E \leq {\epsilon}_0 \right) \\ \frac{k_0^{\rm MH}}{\pi} \left( \sqrt{1+\sqrt\frac{E}{\epsilon_{0}}} \; \right) & \left( {\epsilon}_0 \leq E \right) . \end{array} \right. \label{eq.ME_hat}$$ Figure \[fig:MexicanHat\](b) shows the density of mode distributions plotted from Eqs. (\[eq:DOM\_2D\]) and (\[eq.ME\_hat\]). At the band edge ($E=0$), the single-spin density of modes of the Mexican hat dispersion is finite, $$M_{\rm MH}(E=0) = \frac{2 k_0^{\rm MH}}{\pi} . \label{eq.ME_bandedge}$$ The Mexican hat density of modes decreases by a factor of $\sqrt{2}$ as the energy increases from 0 to ${\epsilon}_0$, and then it slowly increases. The step-function turn-on of the density of modes is associated with a singularity in the density of states. The single-spin density of states resulting from the Mexican hat dispersion is $$D_{\rm MH}(E)= \left\{ \begin{array}{ll} \frac{m^*}{\pi \hbar^2} \sqrt{ \frac{{\epsilon}_0}{E} } \;\;\;\; & (0 \leq E \leq {\epsilon}_0)\\ \frac{m^*}{2 \pi \hbar^2 } \sqrt{ \frac{{\epsilon}_0}{E} } & \left( {\epsilon}_0 < E \right) \; . \end{array} \right. \label{eq.DE_hat}$$ Rashba splitting of the spins also results in a valence band edge that is a ring in $k$-space. The Bychkov-Rashba model with linear and quadratic terms in $k$ gives an analytical expression for a Rashba-split dispersion, [@Bychkov_Rashba_JPhysC] $${\epsilon}({{\bf k}}) = {\epsilon}_0 + \frac{h^{2}k^{2}}{2m^*} \pm \alpha_{R} k \label{eq:MexHat_Rashba}$$ where the Rashba parameter, $\alpha_{R}$, is the strength of the Rashba splitting. In Eq. (\[eq:MexHat\_Rashba\]), the bands are shifted up by ${\epsilon}_0 = \frac{\alpha_R^2 m^*}{2 \hbar^2}$ so that the band edge occurs at ${\epsilon}= 0$. The radius of the band edge in $k$-space is $$k_0^{\rm R} = \frac{m^* \alpha_R}{\hbar^2} = \frac{\sqrt{2 m^* {\epsilon}_0}}{\hbar}. \label{eq:k0_Rashba}$$ The energy dispersion of the split bands is illustrated in Figure \[fig:MexicanHat\](a). The density of modes, [*including both spins*]{}, resulting from the dispersion of Eq. (\[eq:MexHat\_Rashba\]) is $$M_{\rm R}^{\rm 2 \: spins}(E) = \left\{ \begin{array}{ll} \frac{2 k_0^{\rm R}}{\pi} & \;\;\;\;\; \left( 0 \leq E \leq {\epsilon}_0 \right)\\ \frac{2 k_0^{\rm R}}{\pi} \sqrt{ \frac{E}{{\epsilon}_0} } & \;\;\;\;\; \left( {\epsilon}_0 \leq E \right) \end{array} \right. \label{eq:MexHat_DOM}$$ For $0 \leq E \leq {\epsilon}_0$, the density of modes is a step function and the height is determined by $\alpha_R$ and the effective mass. Values for $\alpha_{R}$ vary from 0.07 eVÅ in InGaAs/InAlAs quantum wells to 0.5 eVÅ in the Bi(111) monolayer.[@BiTeI_Rashba_NatMat11] The density of states including both spins is $$D_{\rm R}(E)= \left\{ \begin{array}{ll} \frac{m^*}{\pi \hbar^2} \sqrt{ \frac{{\epsilon}_0}{E} } \;\;\;\; & (0 \leq E \leq {\epsilon}_0)\\ \frac{m^*}{\pi \hbar^2 } & \left( {\epsilon}_0 < E \right) \end{array} \right. \label{eq.DE_R}$$ In general, we find that the diameter of the Rashba $k$-space rings are less than the diameter of the Mexican hat $k$-space rings, so that the enhacements to the thermoelectric parameters are less from Rashba-split bands than from the inverted Mexican hat bands. In the real bandstructures considered in the Sec. \[sec:num\_results\], there is anisotropy to the $k$-space Fermi surfaces. The band extrema at K and M have different energies. For the III-VIs, Bi$_2$Se$_3$, and monolayer Bi, this energy difference is less than $k_BT$ at room temperature. In the III-VIs, the maximum energy difference between the valence band extrema at K and M is 6.6 meV in InS. In Bi$_2$Se$_3$, it is 19.2 meV, and in monolayer Bi, it is 0.5 meV. The largest anisotropy occurs in bilayer graphene under bias. At the maximum electric field considered of 0.5 V/Å, the energy difference of the extrema in the conduction band is 112 meV, and the energy difference of the extrema in the valence band is 69 meV. The anisotropy experimentally manifests itself in the quantum Hall plateaus.[@Falko_BLG_Lifshitz_PRL14] Anisotropy results in a finite slope to the turn-on of the density of modes and a shift of the singularity in the density of states away from the band edge. The energy of the singularity in the density of states lies between the two extrema [@zolyomi_GaX; @zolyomi_InX]. Figure \[fig:MexicanHat\](c) compares the Seebeck coefficients and the electron densities calculated from the Mexican hat dispersion shown in Fig. \[fig:MexicanHat\](a) and a parabolic dispersion. The quantities are plotted versus Fermi energy with the conduction band edge at $E=0$. The bare electron mass is used for both dispersions, $m^* = m_0$, and, for the Mexican hat, $\epsilon_{0} = 0.111$ eV which is the largest value for $\epsilon_{0}$ obtained from our ab-initio simulations of the III-VI compounds. The temperature is $T = 300$ K. The Seebeck coefficients are calculated from Eqs. (\[eq:S\]), (\[eq:Ij\]), and (\[eq:TE\]) with $T(E) = 1$. The electron densities are calculated from the density of state functions given by two times Eq. (\[eq.DE\_hat\]) for the Mexican hat dispersion and by $m^*/\pi \hbar^2$ for the parabolic dispersion. Over the range of Fermi energies shown, the electron density of the Mexican hat dispersion is approximately 6 times larger than that of the parabolic dispersion. To gain further insight, consider the itegrals of the density of states for low energies near the band edges, $n = \int_0^E dE' D(E')$. For the parabolic dispersion, $n_P = \frac{m^*}{\pi \hbar^2}E$, and for the Mexican hat dispersion, $n_{MH} = \frac{4m^*}{\pi \hbar^2} \sqrt{{\epsilon}_0 E}$. The ratio is $n_{MH} / n_P = 4 \sqrt{{\epsilon}_0/E}$. One factor of 2 results from the two branches of the Mexican hat dispersion at low energies ($E<{\epsilon}_0$) and a second factor of 2 results from integrating $1/\sqrt{E}$. In this case, ${\epsilon}_0 = 0.111$ eV, so that at $E=0.05$ eV, $\sqrt{{\epsilon}_0/E}$ gives a factor of 1.5 resulting in a total factor of 6 in the ratio $n_{MH} / n_P$ which is consistent with the numerical calculation at finite temperature shown in Fig. \[fig:MexicanHat\](c). There are two important points to take away from this plot. At the same electron density, the Fermi level of the Mexican hat dispersion is much lower than that of the parabolic dispersion. At the same electron density, the Seebeck coefficient of the Mexican dispersion is much larger than the Seebeck coefficient of the parabolic dispersion. Figure \[fig:MexicanHat\](d) compares the ballistic power factors calculated from the Mexican hat dispersion shown in Fig. \[fig:MexicanHat\](a) and the parabolic dispersion, again with $m^* = m_0$ for both dispersions. The temperature is $T = 300$ K. The ballistic power factor is calculated from Eqs. (\[eq:sigma\]), (\[eq:S\]), (\[eq:Ij\]), and (\[eq:TE\]) with $T(E) = 1$. Eqs. (\[eq:DOM\_2D\]) and (\[eq.ME\_hat\]) for the density of modes are used in Eq. (\[eq:TE\]). The peak power factor of the Mexican hat dispersion occurs when $E_F = -30.0$ meV, i.e. 30 meV below the conduction band edge. This is identical to the analytical result obtained by approximating the density of modes as an ideal step function. The peak power factor of the parabolic dispersion occurs when $E_F = -7.5$ meV. At the peak power factors, the value of $I_{1}$ of the Mexican hat dispersion is 3.5 times larger than $I_1$ of the parabolic dispersion, and $I_0$ of the Mexican hat dispersion is 3.2 times larger than $I_0$ of the parabolic dispersion. The reason for the larger increase in $I_1$ compared to $I_0$ is that, at the maximum power factor, the Fermi level of the Mexican hat dispersion is further below the band edge. Thus, the factor $(E - E_F)$ in the integrand of $I_1$ increases, and the average energy current referenced to the Fermi energy given by $I_1$ increases more than the average particle current given by $I_0$. Since the ratio $I_1 / I_0$ gives the Seebeck coefficient, this translates into an increase of the Seebeck coefficient at the peak power factor. At the peak power factors, the Seebeck coefficient of the Mexican hat dispersion is enhanced by 10$\%$ compared to the parabolic dispersion. We consistently observe a larger increase in $I_1$ compared to that of $I_0$ at the peak power factor when comparing monolayer structures with Mexican hat dispersions to bulk structures with parabolic dispersions. For the III-VI materials, at their peak power factors, GaSe shows a maximum increase of the Seebeck coefficient of 1.4 between a monolayer with a Mexican hat dispersion and bulk with a parabolic dispersion. The power factor is proportional to $I_1^2/I_0 \propto S I_1$. Since the increase in $S$ at the peak power factor lies between 1 and 1.4, the large increase in the maximum power factor results from the large increase in $I_1$. Since the increase in $I_1$ is within a factor of 1 to 1.4 times the increase in $I_0$, one can also view the increase in the power factor as resulting from an increase in $I_0$ which is simply the particle current or conductivity. The increase of both of these quantities, $I_1$ or $I_0$, results from the increase in the density of modes near the band edge available to carry the current. Over the range of integration of several $k_BT$ of the band edge, the density of modes of the Mexican hat dispersion is significantly larger than the density of modes of the parabolic dispersion as shown in Fig. \[fig:MexicanHat\](c). From the Landauer-B[ü]{}ttiker perspective of Eq. (\[eq:TE\]), the increased conductivity results from the increased number of modes. From a more traditional perspective, the increased conductivity results from an increased density of states resulting in an increased charge density $n$. At their peak power factors, the charge density of the Mexican hat dispersion is $5.05 \times 10^{12}$ cm$^{-2}$, and the charge density of the parabolic dispersion is $1.57 \times 10^{12}$ cm$^{-2}$. The charge density of the Mexican hat dispersion is 3.2 times larger than the charge density of the parabolic dispersion even though the Fermi level for the Mexican hat dispersion is 22.5 meV less than the Fermi level of the parabolic dispersion. Since the peak power factor always occurs when $E_F$ is below the band edge, the charge density resulting from the Mexican hat dispersion will always be significantly larger than that of the parabolic dispersion. This, in general, will result in a higher conductivity. When the height of the Mexican hat ${\epsilon}_0$ is reduced by a factor of 4 ($k_0$ is reduced by a factor of 2), the peak power factor [*decreases*]{} by a factor of 2.5, the Fermi level at the peak power factor [*increases*]{} from -30 meV to -20.1 meV, and the corresponding electron density [*decreases*]{} by a factor 2.3. When $\epsilon_{0}$ is varied with respect to the thermal energy at 300K using the following values, 5k$_{B}$T, 2k$_{B}$T, k$_{B}$T and 0.5k$_{B}$T the ratios of the Mexican hat power factors with respect to the parabolic band power factors are 3.9, 2.2, 1.5 and 1.1, respectively. The above analytical discussion illustrates the basic concepts and trends, and it motivates the following numerical investigation of various van der Waals materials exhibiting either Mexican hat or Rashba dispersions. Computational Methods {#sec:comp_methods} ===================== Ab-initio calculations of the bulk and few-layer structures (one to four layers) of GaS, GaSe, InS, InSe, Bi$_{2}$Se$_{3}$, Bi(111) surface, and bilayer graphene are carried out using density functional theory (DFT) with a projector augmented wave method [@PAW] and the Perdew-Burke-Ernzerhof (PBE) type generalized gradient approximation [@perdew:1996:PBE; @ernzerhof:1999:PBE_test] as implemented in the Vienna ab-initio Simulation Package (VASP). [@VASP1; @VASP2] The vdW interactions in GaS, GaSe, InS, InSe and Bi$_{2}$Se$_{3}$ are accounted for using a semi-empirical correction to the Kohn-Sham energies when optimizing the bulk structures of each material.[@Grimme_DFT_D2] For the GaX, InX (X = S,Se), Bi(111) monolayer, and Bi$_{2}$Se$_{3}$ structures, a Monkhorst-Pack scheme is used for the integration of the Brillouin zone with a k-mesh of 12 x 12 x 6 for the bulk structures and 12 x 12 x 1 for the thin-films. The energy cutoff of the plane wave basis is 300 eV. The electronic bandstructure calculations include spin-orbit coupling (SOC) for the GaX, InX, Bi(111) and Bi$_{2}$Se$_{3}$ compounds. To verify the results of the PBE band structure calculations of the GaX and InX compounds, the electronic structures of one to four monolayers of GaS and InSe are calculated using the Heyd-Scuseria-Ernzerhof (HSE) functional.[@HSE_VASP] The HSE calculations incorporate 25$\%$ short-range Hartree-Fock exchange. The screening parameter $\mu$ is set to 0.2 Å$^{-1}$. For the calculations on bilayer graphene, a 32 $\times$ 32 $\times$ 1 k-point grid is used for the integration over the Brillouin zone. The energy cutoff of the plane wave basis is 400 eV. 15Å of vacuum spacing was added to the slab geometries of all few-layer structures. The ab-initio calculations of the electronic structure are used as input into a Landauer formalism for calculating the thermoelectric parameters. The two quantities requred are the density of states and the density of modes. The density of states is directly provided by VASP. The density of modes calculations are performed by integrating over the first Brillouin zone using a converged k-point grid, $51 \times 51 \times 10$ k-points for the bulk structures and $51 \times 51 \times 1$ k-points for the III-VI, Bi$_{2}$Se$_{3}$ and Bi(111) thin film structures. A $101 \times 101 \times 1$ grid of k-points is required for the density of mode calculations on bilayer graphene. The details of the formalism are provided in several prior studies.[@Lundstrom_Jesse_Bi2Te3; @Klimeck_DOM_thermoelectric_JCE; @Darshana_MX2_Thermo] The temperature dependent carrier concentrations for each material and thickness are calculated from the density-of-states obtained from the ab-initio simulations. To obtain a converged density-of-states a minimum k-point grid of 72$\times$72$\times$36 (72$\times$72$\times$1) is required for the bulk (monolayer and few-layer) III-VI and Bi$_{2}$Se$_{3}$ structures. For the density-of-states calculations on bilayer graphene and monolayer Bi(111) a 36$\times$36$\times$1 grid of k-points is used. The calculation of the conductivity, the power factor, and $ZT$ requires values for the electron and hole mean free paths and the lattice thermal conductivity. Electron and hole scattering are included using a constant mean free path, $\lambda_{0}$ determined by fitting to experimental data. For GaS, GaSe, InS and InSe, $\lambda_{0}$ = 25 nm gives the best agreement with experimental data. [@GaSe_sigma_PSSb; @GaS_sigma_JAP; @InSe_sigma_TSF; @InS_sigma_JPhysChem] The room temperature bulk n-type electrical conductivity of GaS, GaSe, InS and InSe at room temperature was reported to be 0.5 $\Omega^{-1}$m$^{-1}$, 0.4 $\Omega^{-1}$m$^{-1}$, 0.052 $\Omega^{-1}$m$^{-1}$ and 0.066 $\Omega^{-1}$m$^{-1}$ respectively at a carrier concentration of 10$^{16}$ cm$^{-3}$. Using $\lambda_{0}$ = 25 nm for bulk GaS, GaSe and InSe we obtain an electrical conductivity of 0.58 $\Omega^{-1}$m$^{-1}$, 0.42 $\Omega^{-1}$m$^{-1}$, 0.058 $\Omega^{-1}$m$^{-1}$ and 0.071 $\Omega^{-1}$m$^{-1}$, respectively at the same carrier concentration. For the Bi(111) monolayer surface, the relaxation time for scattering in bulk Bi is reported to be 0.148 ps at 300K.[@Bi_1L_thermo_JPC] Using the group velocity of the conduction and valence bands ($\sim 6.7 \times 10^{4}$ m/sec for electrons and holes) from our ab-initio simulations, an electron and hole mean free path of 10 nm is used to determine the thermoelectric parameters of the Bi(111) monolayer. Prior theoretical studies of scattering in thin films of Bi$_{2}$Se$_{3}$ ranging from 2 QLs to 4 QLs give a scattering time on the order of 10 fs.[@GYin_TI_DRC; @GYin_TI_APL14; @Udo_Bi2Se3] Using a scattering time of $\tau$ = 10 fs and electron and hole group velocities from the ab-initio simulations of 3 $\times$10$^{5}$ m/s and 2.4$\times$10$^{5}$ m/s, respectively, electron and hole mean free paths of $\lambda_{e}$=3 nm and $\lambda_{p}$=2.4 nm are used to extract the thermoelectric parameters for bulk and thin film Bi$_{2}$Se$_{3}$. For bilayer graphene, $\lambda_{0}$ = 88 nm gives the best agreement with experimental data on conductivity at room temperature.[@bilayergraphene_thermopower_PRL11] Values for the lattice thermal conductivity are also taken from available experimental data. The thermal conductivity in defect-free thin films is limited by boundary scattering and can be up to an order of magnitude lower than the bulk thermal conductivity.[@Lundstrom_Si_thermal] As the thickness of the film increases, $\kappa_{l}$ approaches the Umklapp limited thermal conductivity of the bulk structure. Hence, the values of $\kappa_{l}$ obtained from experimental studies of bulk materials for this study are an upper bound approximation of $\kappa_{l}$ in the thin film structures. The experimental value of 10 Wm$^{-1}$K$^{-1}$ reported for the in-plane lattice thermal conductivity $\kappa_{l}$ of bulk GaS at room temperature is used for the gallium chalcogenides.[@GaS_kappa_exp] The experimental, bulk, in-plane, lattice thermal conductivities of 7.1 Wm$^{-1}$K$^{-1}$ and 12.0 Wm$^{-1}$K$^{-1}$ measured at room temperature are used for InS and InSe, respectively. [@Spitzer_kappaL_JPhysChem] For monolayer Bi(111), the calculated $\kappa_{l}$ from molecular dynamics [@Bi_1L_thermo_JPC] at 300K is 3.9 Wm$^{-1}$K$^{-1}$. For Bi$_{2}$Se$_{3}$, the measured bulk $\kappa_{l}$ value at 300K is 2 Wm$^{-1}$K$^{-1}$.[@goldsmid2009thermo; @Bi2Se3_Cava_PRB] A value of 2000 Wm$^{-1}$K$^{-1}$ is used for the room temperature in-plane lattice thermal conductivity of bilayer graphene. This is consistent with a number of experimental measurements and theoretical predictions on the lattice thermal conductivity of bilayer graphene. [@balandin2011thermal; @KWKim_graphene_kappa] When evaluating $ZT$ in Eq. (\[eq:ZT\]) for the 2D, thin film structures, the bulk lattice thermal conductivity is multiplied by the film thickness. When tabulating values of the electrical conductivity and the power factor of the 2D films, the calculated conductivity from Eq. (\[eq:sigma\]) is divided by the film thickness. Much of the experimental data from which the values for $\lambda_0$ and $\kappa_l$ have been determined are from bulk studies, and clearly these values might change as the materials are thinned down to a few monolayers. However, there are presently no experimental values available for few-layer III-VI and Bi$_2$Se$_3$ materials. Our primary objective is to obtain a qualitative understanding of the effect of the bandstructure in these materials on their thermoelectric properties. To do so, we use the above values for $\lambda_0$ and $\kappa_l$ to calculate $ZT$ for each material as a function of thickness. We tabulate these values and provide the corresponding values for the electron or hole density, Seebeck coefficient, and conductivity at maximum $ZT$. It is clear from Eqs. (\[eq:S\]) and (\[eq:Ij\]) that the Seebeck coefficient is relatively insensitive to the value of the mean free path. Therefore, when more accurate values for the conductivity or $\kappa_l$ become available, new values for $ZT$ can be estimated from Eq. (\[eq:ZT\]) using the given Seebeck coefficient and replacing the electrical and/or thermal conductivity. Numerical Results {#sec:num_results} ================= III-VI Compounds GaX and InX (X = S, Se) {#sec:GaX_InX} ---------------------------------------- The lattice parameters of the optimized bulk GaX and InX compounds are summarized in Table \[tab:mat\_params\]. For the GaX and InX compounds the lattice parameters and bulk bandgaps obtained are consistent with prior experimental [@Kuhn_GaS_experiment; @Kuhn_GaSe_experiment] and theoretical studies [@zolyomi_GaX; @zolyomi_InX; @GaSe_GaS_1L_PCCP] of the bulk crystal structure and electronic band structures. ---------------------------- ------------ ------------ --------- --------------- ------------------- ------------------- ---------------- ------------- -------------------- $a_{0}$(Å) $c_{0}$(Å) $d$ (Å) $d_{vdW}$ (Å) $a_{0}^{expt}$(Å) $c_{0}^{expt}$(Å) $d^{expt}$ (Å) $E_{g}$(eV) $E_{g}^{expt}$(eV) \[0.5ex\] GaS 3.630 15.701 4.666 3.184 3.587 15.492 4.599 1.667 - \[0.5ex\] GaSe 3.755 15.898 4.870 3.079 3.752 15.950 4.941 0.870 2.20 \[0.5ex\] InS 3.818 15.942 5.193 2.780 … … … 0.946 - \[0.5ex\] InSe 4.028 16.907 5.412 3.040 4.000 16.640 5.557 0.48 1.20 \[0.5ex\] Bi$_{2}$Se$_{3}$ 4.140 28.732 7.412 3.320 4.143 28.636 … 0.296 0.300 \[0.5ex\] BLG 2.459 - 3.349 3.349 2.460 - 3.400 - - \[0.5ex\] Bi(111) 4.34 - 3.049 - 4.54 - - 0.584 - \[0.5ex\] ---------------------------- ------------ ------------ --------- --------------- ------------------- ------------------- ---------------- ------------- -------------------- In this study, the default stacking is the $\beta$ phase illustrated in Fig. \[fig:MexicanHat\]a. The $\beta$ phase is isostructural to the AA’ stacking order in the 2H polytypes of the molybdenum and tungsten dichalcogenides.[@Franchini_stacking_PRB14] The bandgap of the one to four monolayer structures is indirect for GaS, GaSe, InS and InSe. Figure \[fig:Ek\_GaS\] illustrates the PBE band structure for one-layer (1L) through four-layers (4L), eight-layer (8L) and bulk GaS. ![(Color online) PBE SOC band structure of GaS: (a) 1L, (b) 2L, (c) 3L and (d) 4L, (e) 8L and (f) bulk GaS. []{data-label="fig:Ek_GaS"}](Ek_GaS2_all.eps){width="5in"} The PBE SOC band gaps and energy transitions for each of the III-VI materials and film thicknesses are are listed in Table \[tab:GaX\_InX\_Egap\]. For GaS, the HSE SOC values are also listed. The effective masses extracted from the PBE SOC electronic bandstructure are listed in Table \[tab:eff\_mass\]. [c | c | c c c c]{} Structure & Transition & GaS & GaSe & InS & InSe\ 1L & E$_{v}$ to $\Gamma_{c}$ & 2.563 (3.707)& **2.145** & **2.104** & **1.618**\ \[0.5ex\] & E$_{v}$ to $K_{c}$ & 2.769 (3.502) & 2.598 & 2.684 & 2.551\ \[0.5ex\] &E$_{v}$ to $M_{c}$ & **2.549 (3.422)** & 2.283 & 2.520 & 2.246\ \[0.5ex\] 2L &E$_{v}$ to $\Gamma_{c}$ & **2.369 (3.156)** & **1.894** & **1.888** & **1.332**\ \[0.5ex\] &E$_{v}$ to $K_{c}$ & 2.606 (3.454) & 2.389 & 2.567 & 2.340\ \[0.5ex\] &E$_{v}$ to $M_{c}$ & 2.389 (3.406) & 2.065 & 2.353 & 2.025\ \[0.5ex\] 3L &E$_{v}$ to $\Gamma_{c}$ & **2.288 (3.089)** & **1.782** & **1.789** & **1.152**\ \[0.5ex\] &E$_{v}$ to $K_{c}$ & 2.543 (3.408) & 2.302 & 2.496 & 2.201\ \[0.5ex\] &E$_{v}$ to $M_{c}$ & 2.321 (3.352) & 1.967 & 2.273 & 1.867\ \[0.5ex\] 4L &E$_{v}$ to $\Gamma_{c}$ & **2.228 (3.011)** & **1.689** & **1.749** & **1.086**\ \[0.5ex\] &E$_{v}$ to $K_{c}$ & 2.496 (3.392) & 2.224 & 2.471 & 2.085\ \[0.5ex\] &E$_{v}$ to $M_{c}$ & 2.267 (3.321) & 1.879 &2.242 & 1.785\ \[0.5ex\] Bulk &$\Gamma_{v}$ to $\Gamma_{c}$ & 1.691 (2.705) & **0.869** & **0.949** & **0.399**\ \[0.5ex\] &$\Gamma_{v}$ to $K_{c}$ & 1.983 (2.582) & 1.435 & 1.734 & 1.584\ \[0.5ex\] &$\Gamma_{v}$ to $M_{c}$ & **1.667** (**2.391**) & 0.964 & 1.400 & 1.120\ \[0.5ex\] [c | p[1.5cm]{} p[1.5cm]{} p[1.5cm]{} p[1.5cm]{}| p[2.7cm]{} p[1.5cm]{} p[1.5cm]{} p[1.5cm]{}]{} Structure & GaS & GaSe & InS & InSe & GaS & GaSe & InS & InSe\ & &\ 1L & 0.409 & 0.544 & 0.602 & 0.912 & 0.067 (0.698) & 0.053 & 0.080 & 0.060\ \[0.5ex\] 2L & 0.600 & 0.906 & 0.930 & 1.874 & 0.065 (0.699) & 0.051 & 0.075 & 0.055\ \[0.5ex\] 3L & 0.746 & 1.439 & 1.329 & 6.260 & 0.064 (0.711) & 0.050 & 0.074 & 0.053\ \[0.5ex\] 4L & 0.926 & 2.857 & 1.550 & 3.611 & 0.064 (0.716) & 0.049 & 0.073 & 0.055\ \[0.5ex\] The conduction bands of GaSe, InS, and InSe are at $\Gamma$ for all layer thicknesses, from monolayer to bulk. The conduction band of monolayer GaS is at M. This result is consistent with that of Zólyomi et al.[@zolyomi_GaX]. However, for all thicknesses greater than a monolayer, the conduction band of GaS is at $\Gamma$. Results from the PBE functional give GaS conduction valley separations between M and $\Gamma$ that are on the order of $k_BT$ at room temperature, and this leads to qualitatively incorrect results in the calculation of the electronic and thermoelectric parameters. For the three other III-VI compounds, the minimum PBE-SOC spacing between the conduction $\Gamma$ and M valleys is 138 meV in monolayer GaSe. For InS and InSe, the minimum conduction $\Gamma$-M valley separations also occur for a monolayer, and they are 416 eV and 628 eV, respectively. For monolayer GaS, the HSE-SOC conduction M valley lies 80 meV below the K valley and 285 meV below the $\Gamma$ valley. At two to four layer thicknesses, the order is reversed, the conduction band edge is at $\Gamma$, and the energy differences between the valleys increase. For the electronic and thermoelectric properties, only energies within a few $k_BT$ of the band edges are important. Therefore, the density of modes of n-type GaS is calculated from the HSE-SOC bandstructure. For p-type GaS and all other materials, the densities of modes are calculated from the PBE-SOC bandstructure. The orbital composition of the monolayer GaS conduction $\Gamma$ valley contains 63% Ga $s$ orbitals and 21% S $p_z$ orbitals. The orbital compositions of the other III-VI conduction $\Gamma$ valleys are similar. As the film thickness increases from a monolayer to a bilayer, the conduction $\Gamma$ valleys in each layer couple and split by 203 meV as shown in Fig. \[fig:Ek\_GaS\]b. Thus, as the film thickness increases, the number of low-energy $\Gamma$ states near the conduction band-edge remains the same, or, saying it another way, the number of low-energy $\Gamma$ states per unit thickness decreases by a factor of two as the the number of layers increases from a monolayer to a bilayer. This affects the electronic and thermoelectric properties. The Mexican hat feature of the valence band is present in all of the 1L - 4L GaX and InX structures, and it is most pronounced for the monolayer structure shown in Fig. \[fig:Ek\_GaS\]a. For monolayer GaS, the highest valence band at $\Gamma$ is composed of 79% sulfur $p_z$ orbitals ($p_z^S$). The lower 4 valence bands at $\Gamma$ are composed entirely of sulfur $p_x$ and $p_y$ orbitals ($p_{xy}^S$). When multiple layers are brought together, the $p_z^S$ valence band at $\Gamma$ strongly couples and splits with a splitting of 307 meV in the bilayer. For the 8-layer structure in Fig. \[fig:Ek\_GaS\]e, the manifold of 8 $p_z^S$ bands touches the manifold of $p_{xy}^S$ bands, and the bandstructure is bulklike with discrete $k_z$ momenta. In the bulk shown in Fig. \[fig:Ek\_GaS\]f, the discrete energies become a continuous dispersion from $\Gamma$ to $A$. At 8 layer thickness, the large splitting of the $p_z^S$ valence band removes the Mexican hat feature, and the valence band edge is parabolic as in the bulk. The nature and orbital composition of the bands of the 4 III-VI compounds are qualitatively the same. [l | c | c]{} Material & ${\epsilon}_0$ (meV) & $k_0$ (nm$^{-1}$)\ (Theory/Stacking Order) & 1L, 2L, 3L, 4L & 1L, 2L, 3L, 4L\ GaS & 111.2, 59.6, 43.8, 33.0 & 3.68, 2.73, 2.52, 2.32\ GaS (no-SOC) & 108.3, 60.9, 45.1, 34.1 & 3.16, 2.63, 2.32, 2.12\ GaS (HSE) & 97.9, 50.3, 40.9, 31.6 & 2.81, 2.39, 2.08, 1.75\ GaS (AA) & 111.2, 71.5, 57.1, 47.4 & 3.68, 2.93, 2.73, 2.49\ GaSe & 58.7, 29.3, 18.1, 10.3 & 2.64, 2.34, 1.66, 1.56\ GaSe ($\epsilon$) & 58.7, 41.2, 23.7 , 5.1 & 2.64, 1.76, 1.17 , 1.01\ InS & 100.6, 44.7, 25.8, 20.4 & 4.03, 3.07, 2.69, 2.39\ InSe & 34.9, 11.9, 5.1, 6.1 & 2.55, 1.73, 1.27, 1.36\ InSe (HSE) & 38.2, 15.2, 8.6, 9.2 & 2.72, 2.20, 1.97, 2.04\ Bi$_2$Se$_3$ & 314.7, 62.3, 12.4, 10.4 & 3.86, 1.23, 1.05, 0.88\ Bi$_2$Se$_3$ (no-SOC) & 350.5, 74.6, 22.8, 20.1 & 4.19, 1.47, 1.07, 1.02\ In the few-layer structures, the Mexican hat feature of the valence band can be characterized by the height, $\epsilon_{0}$, at $\Gamma$ and the radius of the band-edge ring, $k_0$, as illustrated in Figure \[fig:MexicanHat\](b). The actual ring has a small anisotropy that has been previously characterized and discussed in detail [@zolyomi_GaX; @zolyomi_InX; @SGLouie_GaSe_arxiv]. For all four III-VI compounds of monolayer and few-layer thicknesses, the valence band maxima (VBM) of the inverted Mexican hat lies along $\Gamma -K$, and it is slightly higher in energy compared to the band extremum along $\Gamma - M$. In monolayer GaS, the valence band maxima along $\Gamma - K$ is 4.7 meV above the band extremum along $\Gamma - M$. In GaS, as the film thickness increases from one layer to four layers the energy difference between the two extrema decreases from 4.7 meV to 0.41 meV. The maximum energy difference of 6.6 meV between the band extrema of the Mexican hat occurs in a monolayer of InS. In all four III-VI compounds the energy difference between the band extrema is maximum for the monolayer structure and decreases below 0.5 meV in all of the materials for the four-layer structure. The tabulated values of $k_0$ in Table \[tab:e0\_k0\_III\_VIs\] give the distance from $\Gamma$ to the VBM in the $\Gamma - K$ direction. Results calculated from PBE and HSE functionals are given, and results with and without spin-orbit coupling are listed. The effects of AA’ versus AA stacking order of GaS and AA’ versus $\epsilon$ stacking order of GaSe [@GaSe_topological_Udo; @Zheng_GaSe_topological_JCP] are also compared. Table \[tab:e0\_k0\_III\_VIs\] shows that the valence band Mexican hat feature is robust. It is little affected by the choice of functional, the omission or inclusion of spin-orbit coupling, or the stacking order. A recent study of GaSe at the G$_0$W$_0$ level found that the Mexican hat feature is also robust against many-electron self-energy effects.[@SGLouie_GaSe_arxiv] For all materials, the values of $\epsilon_{0}$ and $k_0$ are largest for monolayers and decrease as the film thicknesses increase. This suggests that the height of the step function density of modes will also be maximum for the monolayer structures. ![(Color online) Distribution of valence band modes per unit width versus energy for (a) GaS, (b) GaSe, (c) InS and (d) InSe for 1L (blue), 2L (red), 3L (green) and 4L (purple) structures. The midgap energy is set to E=0. []{data-label="fig:GaX_DOM"}](GaX_InX_DOM_all_new.eps){width="5in"} Figure \[fig:GaX\_DOM\] illustrates the valence band density of modes for 1L, 2L, 3L and 4L GaS, GaSe, InS and InSe. The valence band density of modes is a step function for the few-layer structures, and the height of the step function at the valence band edge is reasonably approximated by Eq. (\[eq.ME\_bandedge\]). The height of the numerically calculated density of modes step function for monolayer GaS, GaSe, InS and InSe is 4.8 nm$^{-1}$, 5.2 nm$^{-1}$, 5.1 nm$^{-1}$ and 3.4 nm$^{-1}$ respectively. Using the values for $k_0$ and Eq. (\[eq.ME\_bandedge\]) and accounting for spin degeneracy, the height of the step function for monolayer GaS, GaSe, InS and InSe is 4.1 nm$^{-1}$, 3.4 nm$^{-1}$, 5.1 nm$^{-1}$ and 3.2 nm$^{-1}$. The height of the numerically calculated density of modes in GaS decreases by $\sim30\%$ when the film thickness increases from one to four monolayers, and the value of $k_0$ decreases by $\sim38\%$. The height of the step function using Eq. (\[eq.ME\_bandedge\]) and $k_0$ is either underestimated or equivalent to the numerical density of modes. For all four materials GaS, GaSe, InS and InSe, decreasing the film thickness increases $k_0$ and the height of the step-function of the band-edge density of modes. A larger band-edge density of modes gives a larger power factor and ZT compared to that of the bulk. ![(Color online) Seebeck coefficient, power factor and thermoelectric figure-of-merit, ZT, of p-type (solid line) and n-type (broken line) 1L (blue), 2L (red), 3L (green), 4L (purple) and bulk (black) (a)-(c) GaS, (d)-(f) GaSe, (g)-(i) InS and (j)-(l) InSe at room temperature.[]{data-label="fig:III_VI_thermo"}](III_VI_thermo_all.eps){width="6.5in"} The p-type Seebeck coefficients, the p-type and n-type power factors, and the thermoelectric figures-of-merit (ZT) as functions of carrier concentration at room temperature for GaS, GaSe, InS and InSe are shown in Figure \[fig:III\_VI\_thermo\]. The thermoelectric parameters at $T=300$ K of bulk and one to four monolayers of GaS, GaSe, InS and InSe are summarized in Tables \[tab:GaS\_thermo\] - \[tab:InSe\_thermo\]. For each material the peak p-type ZT occurs at a monolayer thickness. The largest room temperature p-type ZT occurs in monolayer InS. At room temperature, the peak p-type (n-type) ZT values in 1L, 2L, 3L and 4L GaS occur when the Fermi level is 42 meV, 38 meV, 34 meV and 30 meV (22 meV, 17 meV, 11 meV, and 7 meV) above (below) the valence (conduction) band edge, and the Fermi level positions in GaSe, InS and InSe change in qualitatively the same way. The p-type hole concentrations of monolayer GaS, GaSe, InS and InSe at the peak ZT are enhanced by factors of 9.7, 10.8, 7.2 and 5.5 compared to those of their respective bulk structures. At the peak p-type room-temperature ZT, the Seebeck coefficients of monolayer GaS, GaSe, InS and InSe are enhanced by factors of 1.3, 1.4, 1.3, and 1.3, respectively, compared to their bulk values. However, the monolayer and bulk peak ZT values occur at carrier concentrations that differ by an order of magnitude. At a fixed carrier concentration, the monolayer Seebeck coefficients are approximately 3.1 times larger than the bulk Seebeck coefficients. The p-type power factor (PF) at the peak ZT for 1L GaS is enhanced by a factor of 17 compared to that of bulk GaS. The p-type ZT values of monolayer GaS, GaSe, InS and InSe are enhanced by factors of 14.3, 16.9, 8.7 and 7.7, respectively, compared to their bulk values. At the peak p-type ZT, the contribution of $\kappa_{e}$ to $\kappa_{tot}$ is minimum for the bulk structure and maximum for the monolayer structure. The contributions of $\kappa_{e}$ to $\kappa_{tot}$ in bulk and monolayer GaS are 5% and 24%, respectively. The increasing contribution of $\kappa_{e}$ to $\kappa_{tot}$ with decreasing film thickness reduces the enhancement of ZT relative to that of the power factor. [l c c c c | c c c c]{} Thickness & & & & & & & &\ & & & & & & & &\ 1L & 3.19 & 251.6 & 1.41 & 2.01 & 1.02 & 237.0 & .348 & .431\ \[0.5ex\] 2L & 1.51 & 222.9 & .776 & 1.02 & .621 & 219.6 & .229 & .218\ \[0.5ex\] 3L & 1.13 & 213.2 & .530 & .630 & .595 & 200.9 & .206 & .147\ \[0.5ex\] 4L & .922 & 211.2 & .390 & .421 & .545 & 191.9 & .195 & .111\ \[0.5ex\] Bulk & .330 & 187.6 & .149 & .140 & .374 & 210.8 & .116 & .095\ \[0.5ex\] The increases in the Seebeck coefficients, the charge densities, and the electrical conductivities with decreases in the film thicknesses follow the increases in the magnitudes of $I_{0}$ and $I_{1}$ as discussed at the end of Sec. \[sec:analytical\]. For bulk p-type GaS, the values of $I_0$ ($I_1$) at peak ZT are 0.94 (1.85), and for monolayer GaS, they are 8.87 (23.4). They increase by factors of 9.4 (12.6) as the film thickness decreases from bulk to monolayer. In 4L GaS, the values of $I_0$ ($I_1$) are 2.45 (5.38), and they increase by factors of 3.6 (5.4) as the thickness is reduced from 4L to 1L. For all four of the III-VI compounds, the increases in $I_{1}$ are larger than the increases in $I_0$ as the film thicknesses decrease. As described in Sec. \[sec:analytical\], these increases are driven by the transformation of the dispersion from parabolic to Mexican hat with an increasing radius of the band edge $k$-space ring as the thickness is reduced from bulk to monolayer. [l c c c c | c c c c]{} Thickness & & & & & & & &\ & & & & & & & &\ 1L & 5.81 & 256.1 & 1.28 & 1.86 & 2.71 & 202.9 & .310 & .321\ \[0.5ex\] 2L & 2.70 & 225.3 & .711 & .870 & 1.20 & 201.4 & .152 & .162\ \[0.5ex\] 3L & 2.09 & 221.2 & .450 & .561 & .79 & 194.0 & .103 & .110\ \[0.5ex\] 4L & 1.49 & 210.2 & .352 & .391 & .69 & 186.4 & .085 & .082\ \[0.5ex\] Bulk & .541 & 180.9 & .121 & .112 & .29 & 127.9 & .033 & .132\ \[0.5ex\] While the focus of the paper is on the effect of the Mexican hat dispersion that forms in the valence band of these materials, the n-type thermoelectric figure of merit also increases as the film thickness is reduced to a few layers, and it is also maximum at monolayer thickness. The room temperature, monolayer, n-type thermoelectric figures of merit of GaS, GaSe, InS and InSe are enhanced by factors of 4.5, 2.4, 3.8 and 5.3, respectively, compared to the those of the respective bulk structures. The largest n-type ZT occurs in monolayer GaS. In a GaS monolayer, the 3-fold degenerate M valleys form the conduction band edge. This large valley degeneracy gives GaS the largest n-type ZT among the 4 III-VI compounds. As the GaS film thickness increases from a monolayer to a bilayer, the conduction band edge moves to the non-degenerate $\Gamma$ valley so that the number of low-energy states near the conduction band edge decreases. With an added third and fourth layer, the M valleys move higher, and the $\Gamma$ valley continues to split so that the number of low-energy conduction states does not increase with film thickness. Thus, for a Fermi energy fixed slightly below the band edge, the electron density and the conductivity decrease as the number of layers increase as shown in Tables \[tab:GaS\_thermo\] - \[tab:InSe\_thermo\]. As a result, the maximum n-type ZT for each material occurs at a single monolayer and decreases with each additional layer. [l c c c c | c c c c]{} Thickness & & & & & & & &\ & & & & & & & &\ 1L & 9.30 & 244.2 & 1.26 & 2.43 & 3.75 & 210.8 & .210 & .350\ \[0.5ex\] 2L & 4.20 & 228.7 & .610 & 1.12& 1.63 & 200.0 & .113 & .181\ \[0.5ex\] 3L & 2.32 & 229.5 & .361 & .701 & 1.25 & 196.9 & .078 & .120\ \[0.5ex\] 4L & 1.91 & 222.0 & .292 & .532 & 1.02 & 198.1 & .059 & .094\ \[0.5ex\] Bulk & 1.30 & 195.1 & .180 & .280 & 1.21 & 179.8 & .070 & .092\ \[0.5ex\] [l c c c c | c c c c]{} Thickness & & & & & & & &\ & & & & & & & &\ 1L & 9.71 & 229.8 & .981 & 1.08 & 2.34 & 200.5 & .192 & .180\ \[0.5ex\] 2L & 4.04 & 219.8 & .430 & .471 & 1.22 & 194.7 & .111 & .090\ \[0.5ex\] 3L & 4.18 & 204.2 & .471 & .292 & .781 & 189.1 & .067 & .059\ \[0.5ex\] 4L & 2.45 & 201.0 & .261 & .252 & .610 & 186.8 & .053 & .045\ \[0.5ex\] Bulk & 1.75 & 179.1 & .181 & .142 & .652 & 160.9 & .054 & .034\ \[0.5ex\] Bi$_{2}$Se$_{3}$ {#sec:Bi2Se3} ---------------- Bi$_{2}$Se$_{3}$ is an iso-structural compound of the well known thermoelectric, Bi$_{2}$Te$_{3}$. Both materials have been intensely studied recently because they are also topological insulators.[@Bi2Te3_bulkARPES_Science09; @GYin_TI_JAP13; @Hasan:BiSe:Nature:2008] Bulk Bi$_{2}$Se$_{3}$ has been studied less for its thermoelectric properties due to its slightly higher thermal conductivity compared to Bi$_{2}$Te$_{3}$. The bulk thermal conductivity of Bi$_{2}$Se$_{3}$ is 2 W-(mK)$^{-1}$ compared to a bulk thermal conductivity of 1.5 W-(mK)$^{-1}$ reported for Bi$_{2}$Te$_{3}$. [@goldsmid2009thermo; @goyal_Balandin] However, the thermoelectric performance of bulk Bi$_{2}$Te$_{3}$ is limited to a narrow temperature window around room temperature because of its small bulk band gap of approximately 160 meV.[@Bi2Te3_bulkARPES_Science09] The band gap of single quintuple layer (QL) Bi$_{2}$Te$_{3}$ was previously calculated to be 190 meV.[@Zahid_Lake] In contrast, the bulk bandgap of Bi$_{2}$Se$_{3}$ is $\sim$300 meV [@Bi2Se3_bulkgap_Cava_PRL] which allows it to be utilized at higher temperatures. ![ (Color online) Ab-initio band structure including spin-orbit interaction of Bi$_{2}$Se$_{3}$: (a) 1 QL, (b) 2 QL, (c) 3 QL and (d) 4 QL. []{data-label="fig:Ek_Bi2Se3"}](Ek_Bi2Se3_all.eps){width="5in"} The optimized lattice parameters for bulk Bi$_{2}$Se$_{3}$ are listed in Table \[tab:mat\_params\]. The optimized bulk crystal structure and bulk band gap is consistent with prior experimental and theoretical studies of bulk Bi$_{2}$Se$_{3}$. [@nakajima_Bi2Se3_structure_expt; @Udo_Bi2Se3] Using the optimized lattice parameters of the bulk structure, the electronic structures of one to four quintuple layers of Bi$_{2}$Se$_{3}$ are calculated with the inclusion of spin-orbit coupling. The electronic structures of 1 to 4 QLs of Bi$_{2}$Se$_{3}$ are shown in Figure \[fig:Ek\_Bi2Se3\]. The band gaps for one to four quintuple layers of Bi$_{2}$Se$_{3}$ are 510 meV, 388 meV, 323 meV and 274 meV for the 1QL, 2QL, 3QL and 4QL films, respectively. The effective masses of the conduction and valence band at $\Gamma$ for 1QL to 4QL of Bi$_{2}$Se$_{3}$ are listed in Table \[tab:bi2se3\_mass\]. For each of the thin film structures, the conduction bands are parabolic and located at $\Gamma$. The conduction band at $\Gamma$ of the 1QL structure is composed of 13$\%$ Se $s$, 24$\%$ Se $p_{xy}$, 16$\%$ Bi $p_{xy}$, and 39$\%$ Bi $p_{z}$. The orbital composition of the $\Gamma$ valley remains qualitatively the same as the film thickness increases to 4QL. The orbital composition of the bulk conduction band is 79$\%$ Se $p_{z}$ and 16$\%$ Bi $s$. As the film thickness increases above 1QL, the conduction band at $\Gamma$ splits, as illustrated in Figs. \[fig:Ek\_Bi2Se3\](b)-(d). In the 2QL, 3QL and 4QL structures the conduction band splitting varies between 53.9 meV and 88.2 meV. As with the III-VIs, the number of low-energy conduction band states per unit thickness decreases with increasing thickness. The valence bands have slightly anistropic Mexican hat dispersions. The values of $\epsilon_{0}$ and $k_0$ used to characterize the Mexican hat for the 1QL to 4QL structures of Bi$_{2}$Se$_{3}$ are listed in Table \[tab:e0\_k0\_III\_VIs\]. The radius k$_{0}$ is the distance from $\Gamma_{v}$ to the band extremum along $\Gamma_{v} - M_{v}$, which is the valence band maxima for the 1QL to 4QL structures. The energy difference between the valence band maxima and the band extremum along $\Gamma_{v} - K_{v}$ decreases from 19.2 meV to 0.56 meV as the film thickness increases from 1QL to 4QL. The Mexican hat dispersion in 1QL of Bi$_{2}$Se$_{3}$ is better described as a double brimmed hat consisting of two concentric rings in $k$-space characterized by four points of extrema that are nearly degenerate. The band extremum along $\Gamma_{v} - M_{v}$ adjacent to the valence band maxima, is 36 meV below the valence band maxima. Along $\Gamma_{v} - K_{v}$ the energy difference between the two band extrema is 4.2 meV. At $\Gamma_{v}$, the orbital composition of the valence band for 1QL of Bi$_{2}$Se$_{3}$ is 63$\%$ p$_{z}$ orbitals of Se, 11$\%$ p$_{xy}$ orbitals of Se and 18$\%$ s orbitals of Bi, and the orbital composition remains qualitatively the same as the film thickness increases to 4QL. As the thickness increases above a monolayer, the energy splitting of the valence bands from each layer is large with respect to room temperature $k_BT$ and more complex than the splitting seen in the III-VIs. At a bilayer, the highest valence band loses most of the outer $k$-space ring, the radius $k_0$ decreases by a factor 3.1 and the height (${\epsilon}_0$) of the hat decreases by a factor of 5.1. This decrease translates into a decrease in the initial step height of the density of modes shown in Figure \[fig:Bi2Se3\_thermo\](a). The second highest valence band retains most of the shape of the original monolayer valence band, but it is now too far from the valence band edge to contribute to the low-energy electronic or thermoelectric properties. Thus, Bi$_2$Se$_3$ follows the same trends as seen in Bi$_2$Te$_3$; the large enhancement in the thermoelectric properties resulting from bandstructure are only significant for a monolayer [@Lundstrom_Jesse_Bi2Te3]. [c | c | c ]{} Structure & $\Gamma_{v}$ (m$_{0}$) & $\Gamma_{c}$ (m$_{0}$)\ 1L & 0.128 & 0.132\ \[0.5ex\] 2L & 0.436 & 0.115\ \[0.5ex\] 3L & 1.435 & 0.176\ \[0.5ex\] 4L & 1.853 & 0.126\ \[0.5ex\] ![(Color online) (a) Distribution of modes per unit width versus energy for Bi$_{2}$Se$_{3}$. The midgap energy is set to E=0. Thermoelectric properties of p-type (solid line) and n-type (broken line) Bi$_{2}$Se$_{3}$: (b) Seebeck coefficient, (c) power factor and (d) thermoelectric figure-of-merit, ZT, at room temperature for 1L (blue), 2L (red), 3L (green), 4L (purple) and bulk (black) []{data-label="fig:Bi2Se3_thermo"}](Bi2Se3_DOM_thermo.eps){width="5in"} [l c c c c | c c c c]{} Thickness & & & & & & & &\ & & & & & & & &\ 1L & 7.66 & 279.3 & .371 & 2.86 & 4.63 & 210.1 & .067 & .411\ \[0.5ex\] 2L & 4.65 & 251.3 & .282 & 1.17 & 3.38 & 208.2 & .049 & .271\ \[0.5ex\] 3L & 2.77 & 259.4 & .172 & 1.12 & 2.96 & 198.3 & .043 & .232\ \[0.5ex\] 4L & 2.58 & 237.8 & .161 & .942 & 2.56 & 185.8 & .037 & .190\ \[0.5ex\] Bulk & 1.95 & 210.7 & .095 & .521 & 1.23 & 191.9 & .020 & .123\ \[0.5ex\] The p-type and n-type Seebeck coefficient, electrical conductivity, power factor and the thermoelectric figure-of-merit (ZT) as a function of carrier concentration at room temperature for Bi$_{2}$Se$_{3}$ are illustrated in Figure \[fig:Bi2Se3\_thermo\]. The thermoelectric parameters at $T=300$ K of bulk and one to four quintuple layers for Bi$_{2}$Se$_{3}$ are summarized in Table \[tab:Bi2Se3\_thermo\]. The p-type ZT for the single quintuple layer is enhanced by a factor of 5.5 compared to that of the bulk film. At the peak ZT, the hole concentration is 4 times larger than that of the bulk, and the position of the Fermi energy with respect to the valence band edge ($E_F - E_V$) is 45 meV higher than that of the bulk. The bulk and monolayer magnitudes of $I_0$ ($I_1$) are 0.88 (2.14) and 3.45 (11.2), respectively, giving increases of 3.9 (5.2) as the thickness is reduced from bulk to monolayer. As the film thickness is reduced from 4 QL to 1 QL, the magnitudes of $I_{0}$ and $I_{1}$ at the peak ZT increase by factors of 2.4 and 2.8, respectively. The peak room temperature n-type ZT also occurs for 1QL of Bi$_{2}$Se$_{3}$. In one to four quintuple layers of Bi$_{2}$Se$_{3}$, two degenerate bands at $\Gamma$ contribute to the conduction band density of modes. The higher $\Gamma$ valleys contribute little to the conductivity as the film thickness increases. The Fermi levels at the peak n-type, room-temperature ZT rise from 34 meV to 12 meV below the conduction band edge as the film thickness increases from 1 QL to 4 QL while the electron density decreases by a factor of 1.8. This results in a maximum n-type ZT for the 1QL structure. A recent study on the thickness dependence of the thermoelectric properties of ultra-thin Bi$_{2}$Se$_{3}$ obtained a p-type ZT value of 0.27 and a p-type peak power factor of 0.432 mWm$^{-1}$K$^{-2}$ for the 1QL film. [@Udo_Bi2Se3] The differences in the power factor and the ZT are due to the different approximations made in the relaxation time (2.7 fs) and lattice thermal conductivity (0.49 W/mK) used in this study. Using the parameters of Ref.\[\] in our density of modes calculation of 1QL of Bi$_{2}$Se$_{3}$ gives a peak p-type ZT of 0.58 and peak p-type power factor of 0.302 mWm$^{-1}$K$^{-2}$. We also compare the thermoelectric properties of single quintuple layer Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$. In both materials, the valence band of the single quintuple film is strongly deformed into a Mexican hat. The radius $k_{0}$ for 1QL of Bi$_{2}$Se$_{3}$ is a factor of $\sim$2 higher than $k_{0}$ for 1QL Bi$_{2}$Te$_{3}$. The peak p-type ZT of 7.15 calculated for Bi$_{2}$Te$_{3}$ [@Zahid_Lake] is a factor of 2.5 higher than the peak p-type ZT of 2.86 obtained for a single quintuple layer of Bi$_{2}$Se$_{3}$. This difference in the thermoelectric figure of merit can be attributed to the different approximations in the hole mean free path chosen for Bi$_{2}$Se$_{3}$ ($\lambda_{p}$=2.4 nm) and Bi$_{2}$Te$_{3}$ ($\lambda_{p}$=8 nm) [@Zahid_Lake] and the higher lattice thermal conductivity of Bi$_{2}$Se$_{3}$ ($\kappa_{l}$=2 W/mK) compared to Bi$_{2}$Te$_{3}$ ($\kappa_{l}$=1.5 W/mK). Bilayer Graphene {#sec:BiGraphene} ---------------- AB stacked bilayer graphene (BLG) is a gapless semiconductor with parabolic conduction and valence bands that are located at the $K$ ($K'$) symmetry points. Prior experimental [@Wang_exp_bi_gap_Nat09; @Falko_BLG_Lifshitz_PRL14] and theoretical [@MacDonald_bi_gap_PRB07] studies demonstrated the formation of a bandgap in BLG with the application of a vertical electric field. The vertical electric field also deforms the conduction and valence band edges at $K$ into a Mexican-hat dispersion [@Fermi_ring_Neto_PRB07; @Falko_BLG_Lifshitz_PRL14]. Using ab-initio calculations we compute the band structure of bilayer graphene subject to vertical electric fields ranging from 0.05 V/Å to 0.5 V/Å. The lattice parameters for the bilayer graphene structure used in our simulation are given in Table \[tab:mat\_params\]. The ab-initio calculated band gaps are in good agreement with prior calculations. [@MacDonald_bi_gap_PRB07; @McCann_Eg_bilayer_PRB06] The bandgap increases from 144.4 meV to 277.3 meV as the applied field increases from 0.05 V/Å to 0.5 V/Å. For each applied field ranging from from 0.05 V/Å to 0.5 V/Å both the valence band and the conduction band edges lie along the path $\Gamma - K$, and the radius $k_{0}$ is the distance from $K$ to the band edge along $\Gamma - K$. The magnitude of $k_{0}$ increases linearly with the electric field as shown in Figure \[fig:Bilayergraphene\_Ek\](a). The dispersions of the valence band and the conduction band quantitatively differ, and $k_0$ of the valence band is up to 20$\%$ higher than $k_0$ of the conduction band. The anisotropy of the conduction and valence Mexican hat dispersions increase with increasing vertical field. The extremum point along $K - M$ of the valence (conduction) band Mexican hat dispersion is lower (higher) in energy compared to the band extremum along $\Gamma - K$. As the field increases from 0.05 V/Å to 0.5 V/Å the energy difference between the two extrema points increases from 5.2 meV to 69.4 meV in the valence band and 7.7 meV to 112.3 meV in the conduction band. This anisotropy in the Mexican hat of the valence and conduction band leads to a finite slope in the density of modes illustrated in Figure \[fig:Bilayergraphene\_Ek\](b). As the applied field is increased from 0.05 V/Å to 0.5 V/Å the height of the density of modes step function in the valence and conduction band increases by a factor of 5.7. Figure \[fig:Bilayergraphene\_Ek\](b) illustrates the density of modes distribution for the conduction and valence band states for the lowest field applied (0.05 V/Å) and the highest field applied (0.5 V/Å). The p-type thermoelectric parameters of bilayer graphene subject to vertical electric fields ranging from 0.05 V/Å to 0.5 V/Å are summarized in Table \[tab:BiGraphene\_thermo\]. The p-type and n-type thermoelectric parameters are similar. Figure \[fig:Bilayergraphene\_Ek\](d) compares the calculated ZT versus Fermi level for bilayer graphene at applied electric fields of 0.05 V/Å to 0.5 V/Å. For an applied electric field of 0.5 V/Å the p-type and n-type ZT is enhanced by a factor of 6 and 4 in bilayer graphene compared to the ZT of bilayer graphene with no applied electric field. ![(Color online) (a) Evolution of the radius of the Mexican hat, k$_{0}$ in bilayer graphene as a function of an applied vertical electric field. (b) Density of modes per unit width for two different vertical fields of 0.05 V/Å (blue) and 0.5 V/Å (red). (c) Seebeck coefficients (solid lines) and carrier concentrations (broken lines) for two different vertical fields. (d) ZT of bilayer graphene as a function of the Fermi level for two different vertical fields. []{data-label="fig:Bilayergraphene_Ek"}](BiGraphene_k0_DOM_Seebeck_ZT.eps){width="5in"} [l c c c c]{} Field & p & $S_{p}$ & $\sigma_{p}$ & ZT$_{p}$\ (V/Å) & ($\times 10^{12}$ cm$^{-2}$) & $(\mu VK^{-1})$ & ($\times 10^{7}\Omega m)^{-1}$) &\ 0.0 & .12 & 138.4 & .83 & .0230\ \[0.5ex\] 0.05 & .11 & 154.9 & .77 & .0270\ \[0.5ex\] 0.1 & .16 & 192.1 & 1.1 & .0281\ \[0.5ex\] 0.2 & .19 & 190.7 & 1.3 & .0693\ \[0.5ex\] 0.3 & .21 & 179.8 & 1.4 & .0651\ \[0.5ex\] 0.4 & .27 & 196.4 & 1.8 & .1001\ \[0.5ex\] 0.5 & .31 & 188.0 & 2.1 & .1401\ \[0.5ex\] Bi Monolayer {#sec:Bi} ------------ The large spin-orbit interaction in bismuth leads to a Rasha-split dispersion of the valence band in a single monolayer of bismuth. The lattice parameters for the Bi(111) monolayer used for the SOC ab-initio calculations are summarized in Table \[tab:mat\_params\]. The bandgap of the bismuth monolayer is 503 meV with the conduction band at $\Gamma_{c}$. The inclusion of spin-orbit interaction splits the two degenerate bands at $\Gamma_{v}$ by 79 meV and deforms the valence band maxima into a Rashba split band. The calculated band structure of the Bi(111) monolayer is shown in Figure \[fig:Bi\_Ek\_thermo\](a,b). ![(Color online) Electronic structure and thermoelectric properties of Bi(111) monolayer. (a) Valence band, (b) Conduction band of Bi(111) monolayer with spin-orbit interaction. (c) Density of modes with SOC interactions included, (c) Thermoelectric figure of merit, ZT, at room temperature.[]{data-label="fig:Bi_Ek_thermo"}](Bi_Ek_DOM_ZT.eps){width="5in"} The Rashba parameter for the bismuth monolayer is extracted from the ab-initio calculated band structure. The curvature of the valence band maxima of the Rashba band gives an effective mass of $m^* = 0.1351$. The vertical splitting of the bands at small $k$ gives an $\alpha_{R} = 2.142$ eVÅ. Prior experimental and theoretical studies on the strength of the Rashba interaction in Bi(111) surfaces demonstrate $\alpha_R$ values ranging from 0.55 eVÅ$^{-1}$ to 3.05 eVÅ$^{-1}$ . [@BiTeI_Rashba_NatMat11] A slight asymmetry in the Rashba-split dispersion leads to the valence band maxima lying along $\Gamma_{v} - M_{v}$. The band extremum along $\Gamma_{v} - K_{v}$ is 0.5 meV below the valence band maxima. The radius of the valence band-edge $k_0$, which is the distance from $\Gamma_{v}$ to the band extremum along $\Gamma_{v} - M_{v}$ is 1.40 nm$^{-1}$ similar to 4L InSe. The valence band-edge density of modes shown in Fig. \[fig:Bi\_Ek\_thermo\](c) is a step function with a peak height of 0.96 nm$^{-1}$. Figure \[fig:Bi\_Ek\_thermo\](d) shows the resulting thermoelectric figure of merit ZT as a function of Fermi level position at room temperature. The thermoelectric parameters at $T=300$ K are summarized in Table \[tab:Bi\_thermo\]. [ &gt;p[1.1cm]{} p[0.9cm]{} p[1.1cm]{} p[0.9cm]{} |&gt; p[1.1cm]{} p[0.9cm]{} p[1.1cm]{} p[0.9cm]{}]{} & & & & & & &\ & & & & & & &\ .61 & 239.7 & .39 & 1.38 & .35 & 234.1 & .19 & .61\ \[0.5ex\] Using mean free paths of $\lambda_{e}$=50nm for electrons and $\lambda_{p}$=20nm for holes, our peak ZT values are consistent with a prior report on the thermoelectric properties of monolayer Bi.[@Bi_1L_thermo_JPC]. The peak p-type (n-type) ZT and Seebeck values of 2.3 (1.9) and 786 $\mu$V/K (-710 $\mu$V/K) are consistent with reported values of 2.4 (2.1) and 800 $\mu$V/K (-780 $\mu$V/K) in Ref.\[\]. Summary and Conclusions ======================= Monolayer and few-layer structures of III-VI materials (GaS, GaSe, InS, InSe), Bi$_{2}$Se$_{3}$, monolayer Bi, and biased bilayer graphene all have a valence band that forms a ring in $k$-space. For monolayer Bi, the ring results from Rashba splitting of the spins. All of the other few-layer materials have valence bands in the shape of a ‘Mexican hat.’ For both cases, a band-edge that forms a ring in $k$-space is highly degenerate. It coincides with a singularity in the density of states and a near step-function turn-on of the density of modes at the band edge. The height of the step function is approximately proportional to the radius of the $k$-space ring. The Mexican hat dispersion in the valence band of the III-VI materials exists for few-layer geometries, and it is most prominent for monolayers, which have the largest radius $k_0$ and the largest height ${\epsilon}_0$. The existence of the Mexican hat dispersions and their qualitative features do not depend on the choice of functional, stacking, or the inclusion or omission of spin-orbit coupling, and recent calculations by others show that they are also unaffected by many-electron self-energy effects.[@SGLouie_GaSe_arxiv] At a thickness of 8 layers, all of the III-VI valence band dispersions are parabolic. The Mexican hat dispersion in the valence band of monolayer Bi$_{2}$Se$_{3}$ is qualitatively different from those in the monolayer III-VIs. It can be better described as a double-brimmed hat characterized by 4 points of extrema that lie within $\sim k_BT$ of each other at room temperature. Futhermore, when two layers are brought together to form a bilayer, the energy splitting of the two valence bands in each layer causes the highest band to lose most of its outer ring causing a large decrease in the density of modes and reduction in the thermoelectric properties. These trends also apply to Bi$_{2}$Te$_{3}$. [@Lundstrom_Jesse_Bi2Te3] The valence band of monolayer Bi also forms a $k$-space ring that results from Rashba splitting of the bands. The diameter of the ring is relatively small compared to those of monolayer Mexican hat dispersions. However, the ring is the most isotropic of all of the monolayer materials considered, and it gives a very sharp step function to the valence band density of modes. As the radius of the $k$-space ring increases, the Fermi level at the maximum power factor or ZT moves higher into the bandgap away from the valence band edge. Nevertheless, the hole concentration increases. The average energy carried by a hole with respect to the Fermi energy increases. As a result, the Seebeck coefficient increases. The dispersion with the largest radius coincides with the maximum power factor provided that the mean free paths are not too different. For the materials and parameters considered here, the dispersion with the largest radius also results in the largest ZT at room temperature. Bilayer graphene may serve as a test-bed to measure these effects, since a cross-plane electric field linearly increases the diameter of the Mexican hat ring, and the features of the Mexican hat in bilayer graphene have recently been experimentally observed.[@Falko_BLG_Lifshitz_PRL14] With the exception of monolayer GaS, the conduction bands of few-layer n-type III-VI and Bi$_2$Se$_3$ compounds are at $\Gamma$ with a significant $p_z$ orbital component. In bilayers and multilayers, these bands couple and split pushing the added bands to higher energy above the thermal transport window. Thus, the number of low-energy states per layer is maximum for a monolayer. In monolayer GaS, the conduction band is at M with 3-fold valley degeneracy. At thicknesses greater than a monolayer, the GaS conduction band is at $\Gamma$, the valley degeneracy is one, and the same splitting of the bands occurs as described above. Thus, the number of low-energy states per layer is also maximum for a monolayer GaS. This results in maximum values for the n-type Seebeck coefficients, power factors, and ZTs at monolayer thicknesses for all of these materials. This work is supported in part by the National Science Foundation (NSF) Grant Nos. 1124733 and the Semiconductor Research Corporation (SRC) Nanoelectronic Research Initiative as a part of the Nanoelectronics for 2020 and Beyond (NEB-2020) program, FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575. [80]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.105.136805>. , , , , ****, (). , , , ****, (). , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.89.205416>. , , , , ****, (). , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevLett.113.116602>. , , , , ****, (). , , , , ****, (). , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , (). , , , , , , (). , , , , , , , , , ****, (). , , , , , , , , , , , (). , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , **** (). , , , , , , , , , , , (). , , , , ****, (). , ****, (pages ) (), <http://link.aip.org/link/?APL/97/212102/1>. , , , ****, (). , , , , ****, (). , , , , ****, (). , ** (, , ), vol. , pp. . , , , , , , , , ****, (). , , , ****, (). , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.47.16631>. , , , , , , , , , ****, (), ISSN , <http://dx.doi.org/10.1002/adma.200600527>. , , , , , , ****, (). , , , ****, (). , , , , , ****, (), <http://scitation.aip.org/content/aip/journal/aplmater/1/6/10.1063/1.4833055>. , , , , , ****, (), <http://stacks.iop.org/1882-0786/2/i=9/a=091102>. , , , , , ****, (). , ****, (pages ) (), <http://link.aip.org/link/?APL/102/093103/1>. , , , , , ****, (), ISSN , <http://dx.doi.org/10.1007/s10825-011-0379-2>. , , , ****, (). , ** (, , ). , , , , , ****, (pages ) (), <http://link.aip.org/link/?JAP/107/023707/1>. , , , ****, (pages ) (), <http://link.aip.org/link/?JAP/105/034506/1>. , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (), ISSN , <http://dx.doi.org/10.1002/jcc.20495>. , , , ****, (), <http://scitation.aip.org/content/aip/journal/jcp/118/18/10.1063/1.1564060>. , , , ****, (), ISSN , <http://dx.doi.org/10.1002/pssb.19660170269>. , , , , ****, (). , ****, (), ISSN , <http://www.sciencedirect.com/science/article/pii/004060908390202X>. , ****, (). , , , , , , , , (). , , , in ** (, ), pp. . , , , , ****, (), <http://scitation.aip.org/content/aip/journal/apl/105/3/10.1063/1.4891574>. , , , , , , , , ****, (). , , , ****, (). , , , ****, (). , ****, (). , **, vol. (, ). , , , , , , , , , ****, (), <http://link.aps.org/doi/10.1103/PhysRevB.79.195208>. , ****, (). , , , , ****, (). , , , ****, (). , , , ****, (). , , , , , ****, (). , , , , ****, (). , ****, (). , , , , , ****, (). , , , ****, (). , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (), , <http://www.sciencemag.org/content/325/5937/178.abstract>. , , , ****, (). , , , , , , , ****, (), <http://www.nature.com/nature/journal/v452/n7190/suppinfo/nature06843_S1.html>. , , , ****, (pages ) (), <http://link.aip.org/link/?APL/97/133117/1>. , , , , ****, (). , , , , , , , , , ****, (). , ****, (), [dx.doi.org/10.1103/PhysRevB.74.161403](dx.doi.org/10.1103/PhysRevB.74.161403).
--- abstract: 'We show that information about scattering data of a quantum field theory can be obtained from studying the system at finite density and low temperatures. In particular we consider models formulated on the lattice which can be exactly dualized to theories of conserved charge fluxes on lattice links. Apart from eliminating the complex action problem at nonzero chemical potential $\mu$, these dualizations allow for a particle world line interpretation of the dual fluxes from which one can extract data about the 2-particle wave function. As an example we perform dual Monte Carlo simulations of the 2-dimensional O(3) model at nonzero $\mu$ and finite volume, whose non-perturbative spectrum consists of a massive triplet of particles. At nonzero $\mu$ particles are induced in the system, which at sufficiently low temperature give rise to sectors of fixed particle number. We show that the scattering phase shifts can be obtained either from the critical chemical potential values separating the sectors or directly from the wave function in the 2-particle sector. We find that both methods give excellent agreement with the exact result. We discuss the applicability and generality of the new approaches.' author: - Falk Bruckmann - Christof Gattringer - Thomas Kloiber - Tin Sulejmanpasic bibliography: - 'bibliography.bib' title: | Grand canonical ensemble, multi-particle wave functions, scattering data,\ and lattice field theories --- In many quantum field theories the low energy excitations are very different from the field content of the elementary Hamiltonian or Lagrangian. Indeed excitations, rather than fundamental degrees of freedom, are responsible for many important phenomena, ranging from superconductivity in condensed matter to confinement and dynamical mass generation in Quantum Chromodynamics (QCD). These excitations are often referred to as particles (or quasi-particles in condensed matter literature) with particular masses, charges and interactions, although their internal structure may be rather complicated. For a satisfactory theoretical description non-perturbative methods are needed, often complemented with numerical simulations. In the approach discussed here we consider lattice field theories with a chemical potential $\mu$. In recent years there was quite some progress with finding so-called dual representations (see [@Chandrasekharan:2008gp; @Gattringer:2014nxa] for reviews), which provide an exact mapping to new (dual) degrees of freedom, which are fluxes for matter and surfaces for gauge fields. Initially dualization was motivated to overcome the complex action problem (at nonzero $\mu$ the action $S$ is complex and the Boltzmann factor $e^{-S}$ cannot be used as a probability in a Monte Carlo simulation). Here we show that very precise information about scattering phases and interactions can be obtained based on two simple observations: 1. At finite volume and zero temperature the one, two, three, etc. charge sectors are separated by finite energy steps, and hence these charges will appear in the system at certain chemical potential thresholds $\mu_1,\mu_2,\mu_3,...$ the difference of which contains information about the interaction energy of particles carrying charges. 2. The dual variables allow a particle world line interpretation from which the multi-particle ground state wave function can be obtained. The latter in principle contains all the information about the interactions. It is these two observations which are the cornerstone of the two methods we develop in this work: the [*charge condensation method*]{} and the [*dual wave function method*]{}, respectively. We develop the ideas using the 2-dimensional O(3) model, and discuss their generality in the end. In addition to being an important condensed matter system, the O(3) model is a widely used toy theory for QCD because of common features such as asymptotic freedom, a dynamically generated mass gap, and topological excitations. Moreover, there exist exact results [@Zamolodchikov:1977nu] for the scattering phase shifts, i.e., the physical observables which we aim at in our new approach. Suitable dual representations at nonzero $\mu$ are known [@Bruckmann:2015sua; @Bruckmann:2014sla] and basics of the finite density behavior were discussed for the related case of the O(2) model in [@Banerjee:2010kc]. The Lagrangian of a quantum field theory typically shows some continuous symmetries resulting in corresponding conserved Noether charges. These symmetries and charges play a two-fold role in our approach: When transforming the theory to the dual representation the symmetries give rise to constraints of the dual variables making flux conservation explicit. The resulting “world lines of flux” transport the corresponding charge in space and time, and the particle number turns into a topological invariant: the winding number of the corresponding flux around the compact Euclidean time. The second important aspect of the symmetries and the corresponding charges is that a chemical potential can be coupled to the associated charge, which then can be used to populate the corresponding charge sector. In the dual representation the chemical potentials introduce an additional weight for the winding number of the corresponding flux lines. At sufficiently low temperature $T$ and finite volume one can control the particle number with the chemical potential and systematically probe the system in the different charge sectors, which are separated by the aforementioned critical values $\mu_i, i = 1,2, 3 ...$ of the chemical potential. The critical values $\mu_i$ are clearly marked by steps when plotting the expectation value of the charge $Q$ at low $T$ as a function of $\mu$ (see [@Banerjee:2010kc] and Fig. \[fig:donau\] below). In the presence of a mass gap $m$, the (‘Silver blaze’) range from $\mu = 0$ to $\mu = \mu_1$ corresponds to the charge-0 sector, no particle is present, and for sufficiently large spatial volume one has $\mu_1 = m$. The interval $\mu\in(\mu_1,\mu_2)$ delimits the charge-1 sector, where $\mu_2$ corresponds to the energy of the lowest charge-2 state, i.e., containing two particle masses plus their interaction energy [^1]. At finite volume, it is exactly this 2-particle energy that can be related to scattering data using the Lüscher formula [@Luscher:1986pf; @Luscher:1990ux; @Luscher:1990ck], and analyzing the 2-particle condensation threshold $\mu_2$ as a function of the volume constitutes our first approach to extract scattering data from nonzero density and finite volume. We refer to this approach as the [*“charge condensation method”*]{}. Our second approach – which we refer to as the [*“dual wave function method”*]{} – is based on a direct analysis of the dual fluxes in the 2-particle sector, i.e., the interval $\mu\in(\mu_2,\mu_3)$, where $\mu_3$ marks the onset of the charge-3 sector. In this interval we analyze the two winding flux loops that characterize the charge-2 sector and determine the distribution of their spatial distance which we relate to the 2-particle wave function, from which we again compute scattering data. Both methods are presented in detail for the 2-D O(3) model. We begin with discussing the dual representation as derived in [@Bruckmann:2015sua] where the details of the dualization and also the conventional form of the model are presented. We consider the O(3) model with a single chemical potential coupled to one of the conserved charges. The conventional degrees of freedom are O(3) rotors and the chemical potential is coupled to the 3-component of the corresponding angular momentum. In the dual form the partition sum $Z$ is exactly rewritten into a sum over configurations $\{m,l,k\}$ of 3 sets of dual variables $m_{x,\nu} \in \mathds{Z}$, $l_{x,\nu} , k_{x,\nu} \in \mathds{N}$ on the links $(x,\nu)$ of a $N_t \times N_s$ lattice (periodic boundary conditions, lattice constant $a$): $$\begin{aligned} \label{dualz} Z(\mu) & = & \sum_{\{m,l,k\!\}} B_J[m,l,k] \; \; e^{-a\mu\sum_x \! m_{x,0}} \\ && \qquad \times \prod_x \, \delta\Big(\sum_\nu[m_{x,\nu}-m_{x-\hat{\nu},\nu}]\Big). \nonumber\end{aligned}$$ Each configuration of the variables $m_{x,\nu}, l_{x,\nu}, k_{x,\nu}$ comes with a real non-negative weight $B_J[m,l,k]$ which depends on the coupling $J$. $B_J[m,l,k]$ is given explicitly in [@Bruckmann:2015sua], but irrelevant for the discussion here. A second weight factor $e^{-a\mu\sum_x \! m_{x,0}}$ originates from the chemical potential $\mu$ which couples to the temporal component of the current $m_{x,\nu}$ (in our notation $\nu = 0,1$ and $\nu = 0$ denotes Euclidean time). Obviously all weights are real and positive such that the complex action problem is solved in the dual formulation for arbitrary $\mu$. The dual variables $m_{x,\nu}$, which correspond to one of the O(2) subgroups of O(3), obey constraints at each site $x$. The product of Kronecker deltas $\delta(..)$ enforces $\nabla \vec{m}_x \equiv \sum_\nu[m_{x,\nu}-m_{x-\hat{\nu},\nu}] = 0 \; \forall x$, which is a discrete version of a vanishing divergence condition. Thus the flux $m_{x,\nu}$ is conserved and the admissible configurations of $m_\nu$-flux are closed loops. Since the $m_\nu$-flux must form loops, the term in (\[dualz\]) that multiplies $\mu$ can be written as $a\sum_x \! m_{x,0} = a N_t \, w[m_\nu]$, where $w[m_\nu]$ is the total winding number of $m_\nu$-flux around the compact time direction with extent $a N_t \equiv 1/T$ (we use natural units with $k_B = \hbar = c = 1$). Thus we identify the winding number $w[m_\nu]$ as the particle number in the dual formulation. ![The charge $Q$ as a function of $\mu$ (in units of the mass $m$). We compare different temperatures $T$ at fixed spatial extent $Lm = 4.4$ ($N_t = 100, 200, 400, 1000$ at $N_s = 20$ using $J = 1.3$).[]{data-label="fig:donau"}](o3_2dim_dual_j1.3_ns20_partnr_vs_mu_){width="\linewidth"} ![image](o3_2dim_dual_j1.3_nt1000_chipartnr_vs_mu_){width="0.48\linewidth"} ![image](o3_2dim_dual_j1.3_mucrit_vs_ns_){width="0.48\linewidth"} The constrained $m_\nu$-fluxes can be updated with a generalization of the worm algorithm [@Prokofev:2001zz], while for the other fluxes local Monte Carlo updates are sufficient. Most of the Monte Carlo results presented in this letter were computed at fixed coupling $J = 1.3$ (with some scaling checks performed also at $J = 1.4$ and $J = 1.5$, i.e., closer to the continuum). The temperature $T = 1/aN_t$ was varied by changing the temporal extent $N_t$ of the lattice, the spatial size $L = a N_s$ by changing $N_s$. Dimensionful quantities are expressed in units of the mass $m$ of the lowest excitation, which was determined from propagators in the conventional representation, and at, e.g., $J = 1.3$ is $am = 0.222$. To substantiate the above discussion of sectors with fixed $Q$, we show in Fig. \[fig:donau\] our $J = 1.3$ results for $Q = \langle w[m_\nu] \rangle$ as function of $\mu$ for several low temperatures, at fixed $mL = 4.4$. Decreasing the temperature we indeed find the expected formation of plateaus in Fig. \[fig:donau\], which correspond to the sectors of fixed charge, and we can read off the critical values $\mu_i$. In a practical calculation one actually determines the $\mu_i$ from the peaks of the corresponding particle number susceptibility $\chi = \langle (w[m_\nu] - Q)^2 \rangle/L$. These susceptibilities are shown as a function of $\mu$ in the left panel of Fig. \[fig:inn\_mur\], and we compare results for different spatial extents $L$ in units of $m$. The susceptibilities show pronounced peaks which we can use to determine the values $\mu_i$. We remark that the position $\mu_1$ of the first peak is independent of $L$, while the second peak $\mu_2$ shifts to smaller values when increasing $L$ (see the discussion below). We now make quantitative the above arguments connecting the critical chemical potentials $\mu_1$ and $\mu_2$ with the mass $m$ of the lightest particle and the energy of the 2-particle states. We write the grand-canonical partition sum $Z(\mu)$ with a grand potential $\Omega(\mu)$ in the form ($\hat{H}$ and $\hat{Q}$ denote Hamiltonian and charge operator) $$Z(\mu) \; =\; \text{tr }e^{-(\hat{H}-\mu\, \hat{Q})/T} \; \equiv \; e^{-\Omega(\mu)/T} \, .$$ The small-$T$ limit is governed by the minimal exponents, i.e., in each sector with charge $Q$ the corresponding minimal energy $E_{\text{min}}^{Q}$ dominates and the grand potential in the different sectors is $$\Omega(\mu)\stackrel{T\to 0}{\longrightarrow} \left\{ \begin{array}{ll} E_{\text{min}}^{(Q=0)} = 0 & \; \mbox{for} \; \mu\in[0,\mu_1) \; ,\\ E_{\text{min}}^{(Q=1)}-\mu = m-\mu & \; \mbox{for} \; \mu\in(\mu_1,\mu_2) \; , \\ E_{\text{min}}^{(Q=2)}-2\mu = W-2\mu & \; \mbox{for} \; \mu\in(\mu_2,\mu_3) \; , \\ \ldots \; \; ,& \end{array} \right. \label{omega}$$ where we introduced the minimal 2-particle energy $W$. Note that in the charge-1 sector we assumed $\mu_1 = m$, i.e., we neglected possible finite volume corrections (see below for a cross check). The 2-particle energy $W$ can be calculated from $\mu_1$ and $\mu_2$: The first two transitions between neighboring sectors occur when $0=m-\mu_1$ and $m-\mu_2=W-2\mu_2$, respectively, and thus we find $$W \; = \; m+\mu_2 \; = \; \mu_1+\mu_2 \; . \label{eq:arber}$$ So far we have not discussed the role of nonzero spatial momenta of the states, which for our 2-dimensional model are given by $2\pi n/L = 2\pi n /a N_s, \, n = 0,1 \, ...\, N_s\!-\!1$. We stress again that at very low $T$ only states with vanishing total momentum contribute to the partition sum. Thus in the charge-1 sector we have one particle at rest, in the charge-2 sector two particles with opposite spatial momentum et cetera. We can combine this with the discreteness of the momenta to understand the $L$-dependence of the $\mu_i$, which is documented in the right panel of Fig. \[fig:inn\_mur\], where we plot the values of $\mu_1$ and $\mu_2$ as a function of $L$. For $\mu_1$ we expect a dependence on $L$ only when $L$ becomes smaller than the Compton wave length of the lightest excitation [^2] and self interactions around periodic space alter the mass. This is indeed what we observe in the right panel of Fig. \[fig:inn\_mur\]. For $\mu_2$ a-nonvanishing dependence on $L$ is expected throughout, since the box size $L$ controls the allowed relative momentum. Also such a non-trivial dependence is obvious from Fig. \[fig:inn\_mur\]. For a short ranged potential one can write the two particle energy $W$ as twice the energy of an asymptotically free particle, $$W \; = \; 2\sqrt{m^2+k^2} \; , \label{eq:rachel}$$ where $k$ denotes the relative momentum which is shifted from the values $2\pi n/L$ of the free case. In a finite volume, only certain quantized values of $k$ can account for the scattering phase shift $\delta(k)$ of the interaction and the periodicity, as expressed by the Lüscher formula [@Luscher:1986pf] $$e^{2 i\delta(k)} \; = \; e^{-ikL} \; . \label{eq:lusen}$$ Varying $L$ allows one to scan a whole range of momenta $k$. The temperature must be low enough for pronounced plateaus to form (see Fig. \[fig:donau\]), which gives rise to the following two conditions: $T\ll m$ and $T/m\ll 1/(Lm)^2$. Our numerical results for this extraction are given in Fig. \[fig:isar\_elbe\]. In the top panel we show our data for $k$ as a function of $L$ and compare to the exact result [@Zamolodchikov:1977nu] for “isospin 2”, which is the relevant case for our choice of the chemical potential which excites the 3-component of the O(3) angular momentum. We also include results from the numerical spectroscopy calculation in the 2-particle channel [@Luscher:1990ck] and find excellent agreement of our results with the analytical and numerical reference data. In the bottom panel we give the results for the phase shift $\delta(k)$ as a function of $k$, and again find excellent agreement with the reference data. ![Results from the [*charge condensation method*]{} (blue symbols, labelled as CC) and the [*dual wave function method*]{} (red, DWF) using different $J$ and $N_s$. We compare our results to the analytic solution [@Zamolodchikov:1977nu] (full curves) and the analysis based on 2-particle spectroscopy [@Luscher:1990ck] (blue, LW). In the top panel we show the momenta $k$ as a function of $L$ and in the bottom panel the phase shift $\delta(k)$ versus $k$ ($k$ and $L$ in units of $m$).[]{data-label="fig:isar_elbe"}](o3_2dim_dual_nt1000_km_vs_lm_ "fig:"){width="\linewidth"}\ ![Results from the [*charge condensation method*]{} (blue symbols, labelled as CC) and the [*dual wave function method*]{} (red, DWF) using different $J$ and $N_s$. We compare our results to the analytic solution [@Zamolodchikov:1977nu] (full curves) and the analysis based on 2-particle spectroscopy [@Luscher:1990ck] (blue, LW). In the top panel we show the momenta $k$ as a function of $L$ and in the bottom panel the phase shift $\delta(k)$ versus $k$ ($k$ and $L$ in units of $m$).[]{data-label="fig:isar_elbe"}](o3_2dim_dual_nt1000_phaseshift2_vs_km_ "fig:"){width="\linewidth"} Our second approach for the determination of scattering data, the [*“dual wave function method”*]{}, is based on a direct analysis of the flux variables $m_{x,\nu}$ which carry the charge: In the charge-2 sector we identify the two flux lines which wind around the compact time and interpret their temporal flux segments $m_{x=(x_0,x_1),\nu=0}=1$ as the spatial position $x_1$ of the charge at time $x_0$. Thus for a given time $x_0$ we obtain two positions $x_1^{(1)}$, $x_1^{(2)}$ and identify $\Delta x = | (x_1^{(1)}-x_1^{(2)}) |$ as the distance of the two charges at that time. Sampling over time and many configurations we obtain the probability distribution of the distance $\Delta x$, and the square root of this distribution can be identified as the relative wave function $\psi(\Delta x)$ of the two charges. In Fig. \[fig:regen\] we show $\psi(\Delta x)$ for different spatial extents $L$. ![The relative wave function $\psi(\Delta x)$ as determined from analyzing the flux lines in the charge-2 sector. We show results for $J = 1.3$ at $T/m = 0.0045$ $(N_t = 1000)$ and different spatial extents, $Lm=4.4, 6.6, 8.8, 11, 13.2$ ($N_s=20,30,40,50,60$). The wave functions are fit according to Eq. (\[eq:dachstuhl\]) and in the legend we give the corresponding fit results for $k$.[]{data-label="fig:regen"}](o3_2dim_dual_j1.3_nt1000_2particlewavefunction_vs_x_fit_){width="\linewidth"} Except for very small $\Delta x$ the wave functions are very well described by shifted cosines, $$\psi(\Delta x) \; \propto \; \cos(k(\Delta x-L/2)) \; . \label{eq:dachstuhl}$$ This finding confirms the applicability of Eq. (\[eq:rachel\]): Outside the interaction range the wave functions for the relative motion of the two particles are standing waves with momenta $k$ which are related to the scattering phase shifts via (\[eq:lusen\]). Thus we can fit the data for $\psi(\Delta x)$ with the cosines (\[eq:dachstuhl\]) and obtain the momenta $k$ from that fit. The fit results are also shown in Fig. \[fig:regen\] and in the legend we give the corresponding momenta $k$. From these one again obtains the phase shift $\delta(k)$ via (\[eq:lusen\]). The results from the [*dual wave function method*]{} are included in Fig. \[fig:isar\_elbe\] and agree very well with the [*charge condensation method*]{} and the reference data. Let us summarize the two approaches for obtaining scattering data presented here and comment on their generality: Both methods are based on simulations at nonzero chemical potential $\mu$ at very low temperatures. The [*charge condensation method*]{} relates the critical chemical potentials $\mu_1$ and $\mu_2$ to the 2-particle energy, which in turn can be related to the scattering phase shift via the Lüscher formula. Technically this method assumes that nonzero chemical potential simulations are feasible, which for example is the case for the isospin potential in QCD. The basic concept of the [*charge condensation method*]{} is generalizable also to higher dimensions, in particular the relation of the condensation thresholds $\mu_1$, $\mu_2$ to the 2-particle energy. In higher dimensions a full partial wave analysis would be necessary for complete information about scattering, but at least the scattering length can be extracted from the 2-particle energy [@Luscher:1986pf; @Luscher:1990ux]. For the second approach, the [*dual wave function method*]{}, the chemical potential is adjusted such that the system is in the charge-2 sector. This method is rather general in arbitrary dimensions. The relative 2-particle wave function can be determined from the fluxes in the charge-2 sector and in principle this gives access to the complete scattering information. We remark that both methods can be straightforwardly generalized to three and more particles. A second remark concerns the possibility to couple different chemical potentials to different conserved charges. Using such a setting one can also populate 2-particle sectors with two different particle species and study their scattering properties. While the [*charge condensation method*]{} is independent of the representation of the system, we emphasize that the [*dual wave function method*]{} requires a suitable dualization and makes explicit use of the wordline interpretation. The latter method opens the possibility to study particles and antiparticles, i.e., charges of opposite sign. This exploratory paper is a first step towards a full understanding of the relation between low temperature multi-particle sectors and scattering data. We believe that this is an interesting connection to explore, and understanding the physics involved clearly goes beyond the development of a new method for determining scattering data in lattice simulations. FB is supported by the DFG (BR 2872/6-1) and TK by the Austrian Science Fund, FWF, DK [*Hadrons in Vacuum, Nuclei, and Stars*]{} (FWF DK W1203-N16). Furthermore this work is partly supported by the Austrian Science Fund FWF Grant I 1452-N27 and by DFG TR55, [*“Hadron Properties from Lattice QCD”*]{}. We thank G. Bali, J. Bloch, G. Endrődi, C.B. Lang, J. Myers, A. Schäfer and T. Schäfer for discussions. [^1]: The interpretation of the second critical $\mu$ differs if the system contains particles of different charge, as is the case in e.g. QCD with gauge group $G_2$ [@Wellegehausen:2013cya]. [^2]: The QCD-like theories at nonzero $\mu$ with $L\Lambda_{\text{QCD}}\ll 1$ studied in [@Hands:2010zp; @Hands:2010vw] are in that regime (and perturbative).
--- abstract: | We present a measurement of the time-dependent -violating asymmetries in  decays based on 124 million $\Y4S\to\BB$ decays collected with the  detector at the PEP-II asymmetric-energy $B$ Factory at the Stanford Linear Accelerator Center. In a sample containing $105\pm 14$ signal decays, we measure $\skstargamma = 0.25 \pm 0.63 \pm 0.14$ and $\ckstargamma = -0.57 \pm 0.32 \pm 0.09$, where the first error is statistical and the second systematic. author: - 'B. Aubert' - 'R. Barate' - 'D. Boutigny' - 'F. Couderc' - 'J.-M. Gaillard' - 'A. Hicheur' - 'Y. Karyotakis' - 'J. P. Lees' - 'V. Tisserand' - 'A. Zghiche' - 'A. Palano' - 'A. Pompili' - 'J. C. Chen' - 'N. D. Qi' - 'G. Rong' - 'P. Wang' - 'Y. S. Zhu' - 'G. Eigen' - 'I. Ofte' - 'B. Stugu' - 'G. S. Abrams' - 'A. W. Borgland' - 'A. B. Breon' - 'D. N. Brown' - 'J. Button-Shafer' - 'R. N. Cahn' - 'E. Charles' - 'C. T. Day' - 'M. S. Gill' - 'A. V. Gritsan' - 'Y. Groysman' - 'R. G. Jacobsen' - 'R. W. Kadel' - 'J. Kadyk' - 'L. T. Kerth' - 'Yu. G. Kolomensky' - 'G. Kukartsev' - 'G. Lynch' - 'L. M. Mir' - 'P. J. Oddone' - 'T. J. Orimoto' - 'M. Pripstein' - 'N. A. Roe' - 'M. T. Ronan' - 'V. G. Shelkov' - 'W. A. Wenzel' - 'M. Barrett' - 'K. E. Ford' - 'T. J. Harrison' - 'A. J. Hart' - 'C. M. Hawkes' - 'S. E. Morgan' - 'A. T. Watson' - 'M. Fritsch' - 'K. Goetzen' - 'T. Held' - 'H. Koch' - 'B. Lewandowski' - 'M. Pelizaeus' - 'M. Steinke' - 'J. T. Boyd' - 'N. Chevalier' - 'W. N. Cottingham' - 'M. P. Kelly' - 'T. E. Latham' - 'F. F. Wilson' - 'T. Cuhadar-Donszelmann' - 'C. Hearty' - 'N. S. Knecht' - 'T. S. Mattison' - 'J. A. McKenna' - 'D. Thiessen' - 'A. Khan' - 'P. Kyberd' - 'L. Teodorescu' - 'A. E. Blinov' - 'V. E. Blinov' - 'V. P. Druzhinin' - 'V. B. Golubev' - 'V. N. Ivanchenko' - 'E. A. Kravchenko' - 'A. P. Onuchin' - 'S. I. Serednyakov' - 'Yu. I. Skovpen' - 'E. P. Solodov' - 'A. N. Yushkov' - 'D. Best' - 'M. Bruinsma' - 'M. Chao' - 'I. Eschrich' - 'D. Kirkby' - 'A. J. Lankford' - 'M. Mandelkern' - 'R. K. Mommsen' - 'W. Roethel' - 'D. P. Stoker' - 'C. Buchanan' - 'B. L. Hartfiel' - 'S. D. Foulkes' - 'J. W. Gary' - 'B. C. Shen' - 'K. Wang' - 'D. del Re' - 'H. K. Hadavand' - 'E. J. Hill' - 'D. B. MacFarlane' - 'H. P. Paar' - 'Sh. Rahatlou' - 'V. Sharma' - 'J. W. Berryhill' - 'C. Campagnari' - 'B. Dahmes' - 'S. L. Levy' - 'O. Long' - 'A. Lu' - 'M. A. Mazur' - 'J. D. Richman' - 'W. Verkerke' - 'T. W. Beck' - 'A. M. Eisner' - 'C. A. Heusch' - 'W. S. Lockman' - 'G. Nesom' - 'T. Schalk' - 'R. E. Schmitz' - 'B. A. Schumm' - 'A. Seiden' - 'P. Spradlin' - 'D. C. Williams' - 'M. G. Wilson' - 'J. Albert' - 'E. Chen' - 'G. P. Dubois-Felsmann' - 'A. Dvoretskii' - 'D. G. Hitlin' - 'I. Narsky' - 'T. Piatenko' - 'F. C. Porter' - 'A. Ryd' - 'A. Samuel' - 'S. Yang' - 'S. Jayatilleke' - 'G. Mancinelli' - 'B. T. Meadows' - 'M. D. Sokoloff' - 'T. Abe' - 'F. Blanc' - 'P. Bloom' - 'S. Chen' - 'W. T. Ford' - 'U. Nauenberg' - 'A. Olivas' - 'P. Rankin' - 'J. G. Smith' - 'J. Zhang' - 'L. Zhang' - 'A. Chen' - 'J. L. Harton' - 'A. Soffer' - 'W. H. Toki' - 'R. J. Wilson' - 'Q. L. Zeng' - 'D. Altenburg' - 'T. Brandt' - 'J. Brose' - 'M. Dickopp' - 'E. Feltresi' - 'A. Hauke' - 'H. M. Lacker' - 'R. Müller-Pfefferkorn' - 'R. Nogowski' - 'S. Otto' - 'A. Petzold' - 'J. Schubert' - 'K. R. Schubert' - 'R. Schwierz' - 'B. Spaan' - 'J. E. Sundermann' - 'D. Bernard' - 'G. R. Bonneaud' - 'F. Brochard' - 'P. Grenier' - 'S. Schrenk' - 'Ch. Thiebaux' - 'G. Vasileiadis' - 'M. Verderi' - 'D. J. Bard' - 'P. J. Clark' - 'D. Lavin' - 'F. Muheim' - 'S. Playfer' - 'Y. Xie' - 'M. Andreotti' - 'V. Azzolini' - 'D. Bettoni' - 'C. Bozzi' - 'R. Calabrese' - 'G. Cibinetto' - 'E. Luppi' - 'M. Negrini' - 'L. Piemontese' - 'A. Sarti' - 'E. Treadwell' - 'R. Baldini-Ferroli' - 'A. Calcaterra' - 'R. de Sangro' - 'G. Finocchiaro' - 'P. Patteri' - 'M. Piccolo' - 'A. Zallo' - 'A. Buzzo' - 'R. Capra' - 'R. Contri' - 'G. Crosetti' - 'M. Lo Vetere' - 'M. Macri' - 'M. R. Monge' - 'S. Passaggio' - 'C. Patrignani' - 'E. Robutti' - 'A. Santroni' - 'S. Tosi' - 'S. Bailey' - 'G. Brandenburg' - 'M. Morii' - 'E. Won' - 'R. S. Dubitzky' - 'U. Langenegger' - 'W. Bhimji' - 'D. A. Bowerman' - 'P. D. Dauncey' - 'U. Egede' - 'J. R. Gaillard' - 'G. W. Morton' - 'J. A. Nash' - 'M. B. Nikolich' - 'G. P. Taylor' - 'M. J. Charles' - 'G. J. Grenier' - 'U. Mallik' - 'J. Cochran' - 'H. B. Crawley' - 'J. Lamsa' - 'W. T. Meyer' - 'S. Prell' - 'E. I. Rosenberg' - 'J. Yi' - 'M. Davier' - 'G. Grosdidier' - 'A. Höcker' - 'S. Laplace' - 'F. Le Diberder' - 'V. Lepeltier' - 'A. M. Lutz' - 'T. C. Petersen' - 'S. Plaszczynski' - 'M. H. Schune' - 'L. Tantot' - 'G. Wormser' - 'C. H. Cheng' - 'D. J. Lange' - 'M. C. Simani' - 'D. M. Wright' - 'A. J. Bevan' - 'C. A. Chavez' - 'J. P. Coleman' - 'I. J. Forster' - 'J. R. Fry' - 'E. Gabathuler' - 'R. Gamet' - 'R. J. Parry' - 'D. J. Payne' - 'R. J. Sloane' - 'C. Touramanis' - 'J. J. Back' - 'C. M. Cormack' - 'P. F. Harrison' - 'F. Di Lodovico' - 'G. B. Mohanty' - 'C. L. Brown' - 'G. Cowan' - 'R. L. Flack' - 'H. U. Flaecher' - 'M. G. Green' - 'P. S. Jackson' - 'T. R. McMahon' - 'S. Ricciardi' - 'F. Salvatore' - 'M. A. Winter' - 'D. Brown' - 'C. L. Davis' - 'J. Allison' - 'N. R. Barlow' - 'R. J. Barlow' - 'M. C. Hodgkinson' - 'G. D. Lafferty' - 'A. J. Lyon' - 'J. C. Williams' - 'A. Farbin' - 'W. D. Hulsbergen' - 'A. Jawahery' - 'D. Kovalskyi' - 'C. K. Lae' - 'V. Lillard' - 'D. A. Roberts' - 'G. Blaylock' - 'C. Dallapiccola' - 'K. T. Flood' - 'S. S. Hertzbach' - 'R. Kofler' - 'V. B. Koptchev' - 'T. B. Moore' - 'S. Saremi' - 'H. Staengle' - 'S. Willocq' - 'R. Cowan' - 'G. Sciolla' - 'F. Taylor' - 'R. K. Yamamoto' - 'D. J. J. Mangeol' - 'P. M. Patel' - 'S. H. Robertson' - 'A. Lazzaro' - 'F. Palombo' - 'J. M. Bauer' - 'L. Cremaldi' - 'V. Eschenburg' - 'R. Godang' - 'R. Kroeger' - 'J. Reidy' - 'D. A. Sanders' - 'D. J. Summers' - 'H. W. Zhao' - 'S. Brunet' - 'D. Côté' - 'P. Taras' - 'H. Nicholson' - 'F. Fabozzi' - 'C. Gatto' - 'L. Lista' - 'D. Monorchio' - 'P. Paolucci' - 'D. Piccolo' - 'C. Sciacca' - 'M. Baak' - 'H. Bulten' - 'G. Raven' - 'H. L. Snoek' - 'L. Wilden' - 'C. P. Jessop' - 'J. M. LoSecco' - 'T. A. Gabriel' - 'T. Allmendinger' - 'B. Brau' - 'K. K. Gan' - 'K. Honscheid' - 'D. Hufnagel' - 'H. Kagan' - 'R. Kass' - 'T. Pulliam' - 'A. M. Rahimi' - 'R. Ter-Antonyan' - 'Q. K. Wong' - 'J. Brau' - 'R. Frey' - 'O. Igonkina' - 'C. T. Potter' - 'N. B. Sinev' - 'D. Strom' - 'E. Torrence' - 'F. Colecchia' - 'A. Dorigo' - 'F. Galeazzi' - 'M. Margoni' - 'M. Morandin' - 'M. Posocco' - 'M. Rotondo' - 'F. Simonetto' - 'R. Stroili' - 'G. Tiozzo' - 'C. Voci' - 'M. Benayoun' - 'H. Briand' - 'J. Chauveau' - 'P. David' - 'Ch. de la Vaissière' - 'L. Del Buono' - 'O. Hamon' - 'M. J. J. John' - 'Ph. Leruste' - 'J. Malcles' - 'J. Ocariz' - 'M. Pivk' - 'L. Roos' - 'S. T’Jampens' - 'G. Therin' - 'P. F. Manfredi' - 'V. Re' - 'P. K. Behera' - 'L. Gladney' - 'Q. H. Guo' - 'J. Panetta' - 'F. Anulli' - 'M. Biasini' - 'I. M. Peruzzi' - 'M. Pioppi' - 'C. Angelini' - 'G. Batignani' - 'S. Bettarini' - 'M. Bondioli' - 'F. Bucci' - 'G. Calderini' - 'M. Carpinelli' - 'F. Forti' - 'M. A. Giorgi' - 'A. Lusiani' - 'G. Marchiori' - 'F. Martinez-Vidal' - 'M. Morganti' - 'N. Neri' - 'E. Paoloni' - 'M. Rama' - 'G. Rizzo' - 'F. Sandrelli' - 'J. Walsh' - 'M. Haire' - 'D. Judd' - 'K. Paick' - 'D. E. Wagoner' - 'N. Danielson' - 'P. Elmer' - 'Y. P. Lau' - 'C. Lu' - 'V. Miftakov' - 'J. Olsen' - 'A. J. S. Smith' - 'A. V. Telnov' - 'F. Bellini' - 'G. Cavoto' - 'R. Faccini' - 'F. Ferrarotto' - 'F. Ferroni' - 'M. Gaspero' - 'L. Li Gioi' - 'M. A. Mazzoni' - 'S. Morganti' - 'M. Pierini' - 'G. Piredda' - 'F. Safai Tehrani' - 'C. Voena' - 'S. Christ' - 'G. Wagner' - 'R. Waldi' - 'T. Adye' - 'N. De Groot' - 'B. Franek' - 'N. I. Geddes' - 'G. P. Gopal' - 'E. O. Olaiya' - 'R. Aleksan' - 'S. Emery' - 'A. Gaidot' - 'S. F. Ganzhur' - 'P.-F. Giraud' - 'G. Hamel de Monchenault' - 'W. Kozanecki' - 'M. Langer' - 'M. Legendre' - 'G. W. London' - 'B. Mayer' - 'G. Schott' - 'G. Vasseur' - 'Ch. Yèche' - 'M. Zito' - 'M. V. Purohit' - 'A. W. Weidemann' - 'J. R. Wilson' - 'F. X. Yumiceva' - 'D. Aston' - 'R. Bartoldus' - 'N. Berger' - 'A. M. Boyarski' - 'O. L. Buchmueller' - 'R. Claus' - 'M. R. Convery' - 'M. Cristinziani' - 'G. De Nardo' - 'D. Dong' - 'J. Dorfan' - 'D. Dujmic' - 'W. Dunwoodie' - 'E. E. Elsen' - 'S. Fan' - 'R. C. Field' - 'T. Glanzman' - 'S. J. Gowdy' - 'T. Hadig' - 'V. Halyo' - 'C. Hast' - 'T. Hryn’ova' - 'W. R. Innes' - 'M. H. Kelsey' - 'P. Kim' - 'M. L. Kocian' - 'D. W. G. S. Leith' - 'J. Libby' - 'S. Luitz' - 'V. Luth' - 'H. L. Lynch' - 'H. Marsiske' - 'R. Messner' - 'D. R. Muller' - 'C. P. O’Grady' - 'V. E. Ozcan' - 'A. Perazzo' - 'M. Perl' - 'S. Petrak' - 'B. N. Ratcliff' - 'A. Roodman' - 'A. A. Salnikov' - 'R. H. Schindler' - 'J. Schwiening' - 'G. Simi' - 'A. Snyder' - 'A. Soha' - 'J. Stelzer' - 'D. Su' - 'M. K. Sullivan' - 'J. Va’vra' - 'S. R. Wagner' - 'M. Weaver' - 'A. J. R. Weinstein' - 'W. J. Wisniewski' - 'M. Wittgen' - 'D. H. Wright' - 'A. K. Yarritu' - 'C. C. Young' - 'P. R. Burchat' - 'A. J. Edwards' - 'T. I. Meyer' - 'B. A. Petersen' - 'C. Roat' - 'S. Ahmed' - 'M. S. Alam' - 'J. A. Ernst' - 'M. A. Saeed' - 'M. Saleem' - 'F. R. Wappler' - 'W. Bugg' - 'M. Krishnamurthy' - 'S. M. Spanier' - 'R. Eckmann' - 'H. Kim' - 'J. L. Ritchie' - 'A. Satpathy' - 'R. F. Schwitters' - 'J. M. Izen' - 'I. Kitayama' - 'X. C. Lou' - 'S. Ye' - 'F. Bianchi' - 'M. Bona' - 'F. Gallo' - 'D. Gamba' - 'C. Borean' - 'L. Bosisio' - 'C. Cartaro' - 'F. Cossutti' - 'G. Della Ricca' - 'S. Dittongo' - 'S. Grancagnolo' - 'L. Lanceri' - 'P. Poropat' - 'L. Vitale' - 'G. Vuagnin' - 'R. S. Panvini' - 'Sw. Banerjee' - 'C. M. Brown' - 'D. Fortin' - 'P. D. Jackson' - 'R. Kowalewski' - 'J. M. Roney' - 'R. J. Sobie' - 'H. R. Band' - 'S. Dasu' - 'M. Datta' - 'A. M. Eichenbaum' - 'M. Graham' - 'J. J. Hollar' - 'J. R. Johnson' - 'P. E. Kutter' - 'H. Li' - 'R. Liu' - 'A. Mihalyi' - 'A. K. Mohapatra' - 'Y. Pan' - 'R. Prepost' - 'A. E. Rubin' - 'S. J. Sekula' - 'P. Tan' - 'J. H. von Wimmersperg-Toeller' - 'J. Wu' - 'S. L. Wu' - 'Z. Yu' - 'M. G. Greene' - 'H. Neal' title: ' [ **Measurement of Time-dependent -violating Asymmetries in  Decays** ]{} ' --- [^1] The recent data[@Sin2betaObs] from the $B$ factory experiments have provided strong evidence that the quark mixing mechanism in the Standard Model (SM), encapsulated in the Cabibbo-Kobayashi-Maskawa (CKM) matrix[@CKM], is the dominant source of  violation in the quark sector. Nonetheless, decays which originate from radiative loop processes, such as $b\to s \gamma$, may exhibit significant deviations from the SM due to new physics contributions. In this letter we report the first measurement of time-dependent -violating (CPV) asymmetries in a $b\to s \gamma$ process through the exclusive decay $B^0\to K^{*0}\gamma$, where $\Kstar\to\KS\piz$[@ref:cc]. D. Atwood, M. Gronau and A. Soni were the first to point out that such a measurement probes the polarization of the photon [@soni], which is dominantly left-handed (right-handed) for $b\to s\gamma$ ($\bar b\to \bar s\gamma$) in the SM, but is mixed in various new physics scenarios. The exclusive decays $B^0\to (\KS\piz)\gamma_R$ and $\bar{B}^0\to (\KS\piz)\gamma_L$ are orthogonal transitions and are the dominant decays in the SM. Therefore the CPV asymmetry due to interference between decays with or without mixing is expected to be very small, $\approx 2 (m_s/m_b) \sin 2\beta$ ($\beta\equiv\arg (-V_{cd}V^*_{cb}/V_{td}V^*_{tb})$). Any significant deviation would indicate phenomena beyond the SM. The  decays have been previously explored by the CLEO[@CLEOBRPRL], [@BabarBRPRL], and Belle collaborations [@BelleBR], who reported measurements of branching fractions and the direct  and isospin asymmetries. The measurements reported in this letter are based on 124 million $\Y4S\to\BB$ decays collected in 1999-2003 at the PEP-II $e^+e^-$ collider at the Stanford Linear Accelerator Center with the detector, which is fully described in Ref. [@ref:babar]. For the extraction of the time dependence of  decays, we adopt an analysis approach that closely follows our recently published measurement of CPV asymmetries in the decay $\Bz\to\KS\piz$ [@BaBarKsPi0]. There we established a technique of vertex reconstruction for $B$ decay modes to final states containing a $\KS\ra\pip\pim$ decay and other neutral particles, but no primary charged particles at the $B$ decay vertex. We search for  decays in hadronic events, which are selected based on charged particle multiplicity and event topology. We reconstruct $\KS\to\pip\pim$ candidates from pairs of oppositely charged tracks, detected in the silicon vertex detector (SVT) and/or the central drift chamber (DCH). We require that these tracks originate from a vertex which is more than $0.3$ cm from the primary vertex and that the resulting candidates have a $\pip\pim$ invariant mass between $487$ and $508$[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}. We form $\piz\to\gamma\gamma$ candidates from pairs of photon candidates in ’s electromagnetic calorimeter (EMC) which are not associated with any charged tracks, carry a minimum energy of 30[$\mathrm{\,Me\kern -0.1em V}$]{}, and possess the expected lateral shower shape. We require that the $\gamma\gamma$ combination has an energy greater than $200$[$\mathrm{\,Me\kern -0.1em V}$]{}and an invariant mass between $115$ and $155$[${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}. We reconstruct candidate $\Kstar\to \KS\piz$ decays from $\KS\piz$ combinations with invariant mass in the range $0.8 <M(\KS\piz)<1.0$[${\mathrm{\,Ge\kern -0.1em V\!/}c^2}$]{}. For photons originating from the $B$ decay, we select clusters in the EMC which are isolated by 25 cm from all other energy deposits and are inconsistent with $\piz\to \gamma\gamma$ or $\eta\to \gamma\gamma$ decays. We identify  decays in $\Kstar\gamma$ combinations using two nearly independent kinematic variables: the energy-substituted mass $\mes=\sqrt{(s/2+{\bf p}_i\cdot{\bf p}_B)^2/E_i^2-p^2_B}$ and the energy difference $\DeltaE=E^*_B-\sqrt{s}/2$. Here $(E_i,{\bf p}_i)$ and $(E_B,{\bf p}_B)$ are the four-vectors of the initial $e^+e^-$ system and the $B$ candidate, respectively, $\sqrt{s}$ is the center-of-mass energy, and the asterisk denotes th center-of-mass (CMS) frame. For signal decays, the  distribution peaks near the $B$ mass with a resolution of $\approx3.5{\ensuremath{{\mathrm{\,Me\kern -0.1em V\!/}c^2}}\xspace}$ and  peaks near 0[$\mathrm{\,Me\kern -0.1em V}$]{}with a resolution of $\approx50$[$\mathrm{\,Me\kern -0.1em V}$]{}. Both  and  exhibit a low-side tail from energy leakage in the EMC. For the study of CPV asymmetries, we consider candidates within $5.2<\mes<5.3{\ensuremath{{\mathrm{\,Ge\kern -0.1em V\!/}c^2}}\xspace}$ and $|\DeltaE|<300{\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$, which includes the signal as well as a large “sideband” region for background estimation. When more than one candidate is found in an event, we select the combination with the $\piz$ mass closest to the nominal $\piz$ value, and if ambiguity persists we select the combination with the $\KS$ mass closest to the nominal  value. The sample of candidate events selected by the above requirements contains significant background contributions from continuum $e^+e^- \to q\bar{q}$ $(q=\{u,d,s,c\})$, as well as random combinations from generic  decays. We suppress both of these backgrounds by taking advantage of the expected angular distribution of the decay products of these processes. Angular momentum conservation restricts the  meson in the  decay to transversely polarized states, which leads to an angular distribution of $\sin^2\theta_H$ for the decay products, where $\theta_H$ is the angle between the  and the $B$ meson directions in the  rest frame. Monte Carlo studies show that the background candidates peak near $\cos\theta_H =-1$. We require $\cos\theta_H>-0.6$, resulting in rejection of $68\%$ of $\BB$ and $48\%$ of continuum background candidates, while retaining $91\%$ of the signal. We exploit topological variables to further suppress the continuum backgrounds, which in the CMS frame tend to retain the jet-like features of the $q\bar{q}$ fragmentation process, as opposed to spherical  decays. In the CMS system we calculate the angle $\theta^*_S$ between the sphericity axis of the $B$ candidate and that of the remaining particles in the rest of the event (ROE). While $|\cos\theta^*_S|$ is highly peaked near 1 for continuum background, it is nearly uniformly distributed for $B\bar{B}$ events. We require $|\cos\theta^*_S|<0.9$, eliminating $58\%$ of the continuum events. We also employ an event-shape Fisher discriminant in the maximum-likelihood fit (described below) from which we extract the CPV measurements. This variable is defined as ${\cal F}=0.53 - 0.60 L_0 + 1.27 L_2$, where $L_j\equiv\sum_{i \in {\rm ROE}} |{\bf p}^*_i| |\cos \theta^*_i|^j$, $\bf p^*_i$ is the momentum of particle $i$ in the CMS system and $\theta^*_i$ is the angle between ${\bf p}^*_i$ and the sphericity axis of the $B$ candidate. ![Distribution of (a) $\mes$ and (b) $M_{\Kstar}$ for events enhanced in signal decays. The dashed and solid curves represent the background and signal-plus-background contributions, respectively, as obtained from the maximum likelihood fit to the full data sample. The selection technique is described in the text.[]{data-label="fig:prplots"}](mes-projplot.eps "fig:"){width="0.49\linewidth"} ![Distribution of (a) $\mes$ and (b) $M_{\Kstar}$ for events enhanced in signal decays. The dashed and solid curves represent the background and signal-plus-background contributions, respectively, as obtained from the maximum likelihood fit to the full data sample. The selection technique is described in the text.[]{data-label="fig:prplots"}](mass-projplot.eps "fig:"){width="0.49\linewidth"} The above selections yield 1916 $\TheDecay$ candidates. We extract our measurements from this sample using an unbinned maximum-likelihood fit to kinematic (, , and  mass), event shape (), flavor tag, and time structure variables (described below). As input to the fit, we parameterize the probability distribution functions (PDF) describing the observables of signal and $\BB$ background events using either more copious fully-reconstructed $B$ decays in data or simulated samples. For the continuum background, we select the functional form of the PDFs describing each fit variable in data using the sideband regions of the other observables where the $q\bar{q}$ background dominates. We include these regions in the fitted sample and simultaneously extract the parameters of the background PDFs along with the CPV measurements. We fit $105\pm14$ signal and $19\pm15$ other $B$ decays in the selected sample. This signal yield is consistent with expectations from the previous measurements of the branching fractions[@CLEOBRPRL; @BabarBRPRL; @BelleBR]. Figure \[fig:prplots\] displays the and $M_{\Kstar}$ distributions for signal-enhanced sub-samples of these events, selected using the PDFs employed in the fit (see below). For each  candidate, we examine the remaining tracks and neutral particles in the event to determine if the other $B$ in the event, , decayed as a  or a  (flavor tag). Time-dependent CPV asymmetries are determined by reconstructing the distribution of the proper decay time difference, $\deltat\equiv t_{\CP}-t_{\rm tag}$. At the $\Upsilon(4S)$ resonance, the distribution of  follows $$\begin{aligned} P^{\Bz}_{\Bzb}(\deltat)= &&\frac{e^{-|\deltat|/\tau}}{4\tau}[1\pm \label{eqn:td}\\ &&(S_f \sin{(\deltat \deltamd)}-C_f\cos{(\deltat \deltamd)})], \nonumber\end{aligned}$$ where the upper (lower) sign corresponds to  decaying as (), $\tau$ is the  lifetime,  is the mixing frequency, and $S_f$ and $C_f$ are the magnitude of the mixing-induced and direct CPV asymmetries, respectively. As stated above, in the SM we expect $\skstargamma\approx 2 (m_s/m_b) \sin 2\beta \approx 0.05$. We expect $\ckstargamma=-A_{\Kstar\gamma}$, the direct asymmetry measured in the self-tagging and more copious $\Bztokstargamma (\Kstar \ra \Kp\pim)$ decay. We use a neural network to determine the flavor, $T$, of the $B_{\rm tag}$ meson from kinematic and particle identification information[@ref:Sin2betaPRD]. Each event is assigned to one of five mutually exclusive tagging categories, designed to combine flavor tags with similar performance and $\deltat$ resolution. We parameterize the performance of this algorithm in a data sample ($B_{\rm flav}$) of fully reconstructed $\Bz\to D^{(*)-} \pip/\rho^+/a_1^+$ decays. The average effective tagging efficiency obtained from this sample is $Q=\Sigma_c\epsilon^c_S(1-2w^c)^2=0.288\pm 0.005$, where $\epsilon^c_S$ and $w^c$ are the efficiency and mistag probabilities, respectively, for events tagged in category $c$. In each tagging category, we extract the fraction of events ($\epsilon^c_{q\bar{q}}$) and the asymmetry in the rate of $\Bz$ and $\Bzb$ tags in the continuum background events in the fit to the data. We compute the proper time difference from the known boost of the system and the measured $\deltaz={\ensuremath{z_{\CP}}}-{\ensuremath{z_\mathrm{tag}}}$, the difference between the reconstructed decay vertex positions of the and candidate along the boost direction ($z$). A description of the inclusive reconstruction of the vertex using tracks in ROE is given in [@ref:Sin2betaPRD]. Replicating the vertexing technique developed for  decays[@BaBarKsPi0], we determine the decay point [$z_{\CP}$]{} for  candidates from the intersection of the trajectory with the interaction region. This is accomplished by constraining the vertex to the interaction point (IP) in the plane transverse to the beam, which is determined in each run from the spatial distribution of vertices from two-track events. We combine the uncertainty in the IP position, which follows from the size of the interaction region (about horizontal and vertical), with the RMS of the transverse flight length distribution (about ) to assign an uncertainty to the IP constraint. Simulation studies indicate that  decays exhibit properties which are characteristic of the IP vertexing technique, namely that the per-event estimate of the error on $\deltat$, $\sigma_{\deltat}$, reflects the expected dependence of the [$z_{\CP}$]{} resolution on the $\KS$ flight direction and the number of SVT layers traversed by its decay daughters. Though the fit extracts  from all flavor tagged signal decays, we only allow $68\%$ of these events contribute to the measurement of . This subset consists of candidates which are composed of  decays with at least one hit in the SVT on both tracks and pass the quality requirements of and . For $66\%$ of this subset, both tracks have hits in the inner three SVT layers, which results in a mean  resolution that is comparable to decays with the vertex directly reconstructed from charged particles originating at the $B$ decay point [@ref:Sin2betaPRD]. In the remainder of the subset, the resolution is nearly two times worse. We obtain the PDF for the time-dependence of signal decays from the convolution of Eq. \[eqn:td\] with a resolution function ${\cal R}(\delta t \equiv \deltat -\deltat_{\rm true},\sigma_{\deltat})$. The resolution function is parameterized as the sum of a ‘core’ and a ‘tail’ Gaussian function, each with a width and mean proportional to the reconstructed $\sigma_{\deltat}$, and a third Gaussian centered at zero with a fixed width of  [@ref:Sin2betaPRD]. Using simulated data, we have verified that the parameters of ${\cal R}(\delta t, \sigma_{\deltat})$ for  decays and the $\BB$ backgrounds are similar to those obtained from the $B_{\rm flav}$ sample, even though the distributions of $\sigma_{\deltat}$ differ considerably. Therefore, we extract these parameters from a fit to the $B_{\rm flav}$ sample. We find that the distribution of continuum background candidates is well described by a delta function convoluted with a resolution function with the same functional form as used for signal events. We determine the parameters of the background function in the fit to the  dataset. To extract the CPV asymmetries we maximize the logarithm of the likelihood function [$$\begin{aligned} {\cal L}(\sf,\cf,N_h,f_h,\epsilon_{q\bar{q}}^c, \vec{\alpha})= & \frac{e^{-(N_S+N_{\BB}+N_{q\bar{q}})}}{(N_S+N_{\BB}+N_{q\bar{q}})\,!} \times \mbox{}\nonumber\\ & \prod_{i \in \mathrm{w/\,} \deltat} [ N_S f_S \epsilon^{c}_S{\cal P}_S(\vec{x}_i,\vec{y}_i;\sf,\cf) + \mbox{} \nonumber\\ & N_{\BB} f_{\BB} \epsilon^{c}_{\BB}{\cal P}_{\BB}(\vec{x}_i,\vec{y}_i) + \mbox{}\nonumber\\ & N_{q\bar{q}} f_{q\bar{q}} \epsilon^{c}_{q\bar{q}} {\cal P}_{q\bar{q}}(\vec{x}_i,\vec{y}_i;\vec{\alpha}) ] \times \mbox{} \nonumber\\ & \prod_{i \in \mathrm{w/o\,} \deltat} [ N_S (1-f_S) \epsilon^{c}_S {\cal P}'_S(\vec{y}_i;\cf) + \mbox{}\nonumber\\ & N_{\BB} (1-f_{\BB}) \epsilon^{c}_{\BB} {\cal P}'_{\BB}(\vec{y}_i) + \mbox{}\nonumber\\ & N_{q\bar{q}} (1-f_{q\bar{q}}) \epsilon_{q\bar{q}}^{c} {\cal P}'_{q\bar{q}}(\vec{y}_i;\vec{\alpha}) ], \mbox{} \nonumber \end{aligned}$$]{} where the second (third) factor on the right-hand side is the contribution from events with (without) $\deltat$ information. The vectors $\vec{x}_i$ and $\vec{y}_i$ represent the time-structure and remaining observables, respectively, for event $i$. The PDFs ${\cal P}_h(\vec{x}_i,\vec{y}_i)=P_h(\mes_i)P_h(\DeltaE_i)P_h({\cal F}_i)P_h(M_{\Kstar,i}) P_h^{c_i}(\deltat_i|\sigma_{\deltat,i},T_i)$ and ${\cal P'}_h(\vec{y}_i)=P_h(\mes_i)P_h(\DeltaE_i)P_h({\cal F}_i)P_h(M_{\Kstar,i}) P_h^{c_i}(T_i)$ are the products of the PDFs described above for hypothesis $h$ of signal ($S$), $\BB$ background ($\BB$), and continuum background ($q\bar{q}$). Along with the CPV asymmetries  and , the fit extracts the yields $N_S$, $N_{\BB}$, and $N_{q\bar{q}}$, the fractions of events with $\deltat$ information $f_S$ and $f_{q\bar{q}}$, and the parameters $\vec{\alpha}$ which describe the background PDFs. We determine $\epsilon^c_B$ and $f_{\BB}$ in simulated generic  decays. The fit to the data sample yields and , where the uncertainties are statistical and systematic, respectively. The fit reports a correlation of $1\%$ between these parameters. The result for $\ckstargamma$ is consistent with a fit that does not employ $\deltat$ information. Since the present measurements of $A_{\Kstar\gamma}$[@BabarBRPRL; @BelleBR] are consistent with zero, we also fit the data sample with $\ckstargamma$ fixed to zero and obtain $\skstargamma=0.25\pm 0.65\pm 0.14$. The event selection criteria employed to isolate signal-enhanced samples displayed in Figure \[fig:prplots\] are based on a cut on the likelihood ratio $R={\cal P}_S/({\cal P}_{S}+{\cal P}_{BB}+{\cal P}_{q\bar{q}})$ calculated without the displayed observable. The dashed and solid curves indicate background and signal-plus-background contributions, respectively, as obtained from the fit, but corrected for the selection efficiency of $R$. Figure \[fig:dtplot\] shows distributions of $\deltat$ for $\Bz$- and $\Bzb$-tagged events, and the asymmetry ${\cal A}_{\Kstar\gamma}(\deltat) = \left[N_{\Bz} - N_{\Bzb}\right]/\left[N_{\Bz} + N_{\Bzb}\right]$ as a function of $\deltat$, also for a signal-enhanced sample. ![ Distributions of $\deltat$ for events enhanced in signal decays with $B_{\rm tag}$ tagged as (a) $\Bz$ or (b) $\Bzb$, and (c) the resulting asymmetry ${\cal A}_{\Kstar\gamma}(\deltat)$. The dashed and solid curves represent the fitted background and signal-plus-background contributions, respectively, as obtained from the maximum likelihood fit. The raw asymmetry projection corresponds to approximately $38$ signal and $19$ background events.[]{data-label="fig:dtplot"}](dtasymmetryproj.eps){width="0.9\linewidth"} We consider several sources of systematic uncertainties related to the level and possible asymmetry of the background contribution from generic $B\bar {B}$ decays. We estimate the impact of potential biases in the determination of the  background rate to lead to a systematic uncertainty of $0.04$ ($0.05$) on (). We estimate an uncertainty of $0.12$ ($0.03$) due to potential CPV asymmetries in the $\BB$ backgrounds and $0.02$ ($0.06$) due to possible asymmetries in the rate of versus tags in continuum backgrounds. We quantify possible systematic effects due to the vertexing method in the same manner as Ref.[@BaBarKsPi0], estimating systematic uncertainties of $0.04$ ($0.02$) due to the choice of resolution function, $0.04$ ($<0.01$) due to the vertexing technique, and $0.03$ ($0.01$) due to possible misalignments of the SVT. Finally, we include a systematic uncertainty of $0.02$ ($0.02$) due to tagging asymmetries in the signal and $0.02$ ($0.02$) due to imperfect knowledge of the PDFs used in the fit. In summary, we have performed a measurement of the time-dependent CPV asymmetry  and the direct- violating asymmetry  from  decays. Our measurement is consistent with the SM expectation of very small CPV asymmetries. We are grateful for the excellent luminosity and machine conditions provided by our 2 colleagues, and for the substantial dedicated effort from the computing organizations that support . The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), and PPARC (United Kingdom). Individuals have received support from CONACyT (Mexico), A. P. Sloan Foundation, Research Corporation, and Alexander von Humboldt Foundation. [99]{} B. Aubert [*et al.*]{} \[ Collaboration\], [[Phys. Rev. Lett.]{} [****87****]{}]{}, 091801 (2001). K. Abe [*et al.*]{} \[BELLE Collaboration\], [[Phys. Rev. Lett.]{} [**87**]{}]{}, 091802 (2001). N. Cabibbo, [[Phys. Rev. Lett.]{} [****10****]{}]{}, 531 (1963); M. Kobayashi, and T. Maskawa, [[Prog. Th. Phys. [****49****]{}]{}]{}, 652 (1973). Unless explicitly stated, charge conjugate decay modes are assumed throughout this paper. D. Atwood, M. Gronau and A. Soni, Phys. Rev. Lett.  [**79**]{}, 185 (1997) \[arXiv:hep-ph/9704272\]. T.E. Coan , \[CLEO Collaboration\], , 5283 (2000). B. Aubert , \[ Collaboration\], Phys. Rev. Lett. 88, 101805 (2002) M. Nakao , \[Belle Collaboration\], arXiv:hep-ex/0402042. B. Aubert [*et al.*]{}, \[ Collaboration\], [[Nucl. Instr. and Methods]{} A [**479**]{}]{}, 1 (2002). B. Aubert [*et al.*]{}, \[ Collaboration\], arXiv:hep-ex/0403001. Submitted to Phys. Rev. Lett. B. Aubert [*et al.*]{}, \[ Collaboration\] , [[Phys. Rev.]{} D [****66****]{}]{}, 032003 (2002). K. Hagiwara [*et al.*]{}, \[Particle Data Group\], [[Phys. Rev.]{} D [**66**]{}]{}, 010001 (2002). [^1]: Deceased
--- abstract: 'All stars in the [*Kepler*]{} field brighter than 12.5 magnitude have been classified according to variability type. A catalogue of $\delta$ Scuti and $\gamma$ Doradus stars is presented. The problem of low frequencies in $\delta$ Sct stars, which occurs in over 98percent of these stars, is discussed. Gaia DR2 parallaxes were used to obtain precise luminosities, enabling the instability strips of the two classes of variable to be precisely defined. Surprisingly, it turns out that the instability region of the $\gamma$ Dor stars is entirely within the $\delta$ Sct instability strip. Thus $\gamma$Dor stars should not be considered a separate class of variable. The observed red and blue edges of the instability strip do not agree with recent model calculations. Stellar pulsation occurs in less than half of the stars in the instability region and arguments are presented to show that this cannot be explained by assuming pulsation at a level too low to be detected. Precise Gaia DR2 luminosities of high-amplitude $\delta$ Sct stars (HADS) show that most of these are normal $\delta$ Sct stars and not transition objects. It is argued that current ideas on A star envelopes need to be revised.' author: - | L. A. Balona\ South African Astronomical Observatory, P.O. Box 9, Observatory, Cape 2735, South Africa bibliography: - 'pulaf.bib' date: 'Accepted .... Received ...' title: 'Gaia luminosities of pulsating A-F stars in the Kepler field' --- \[firstpage\] stars: oscillations - stars: variables: $\delta$ Scuti - parallaxes Introduction ============ The $\delta$ Sct stars are A and early F dwarfs and giants with multiple frequencies in the range 5–50d$^{-1}$ while $\gamma$ Dor stars are F dwarfs and giants pulsating in multiple frequencies in the range 0.3– 3d$^{-1}$. The two types of variable have been considered as two separate classes of pulsating star driven by different mechanisms: the opacity-driven $\kappa$ mechanism in $\delta$ Sct stars and the convective blocking mechanism in $\gamma$ Dor stars as described by @Guzik2000. Simultaneous low-frequency $\gamma$ Dor and high-frequency $\delta$ Sct pulsations in the same star were first discovered by @Handler2002b in the A9/F0V star HD209295. Until then none of the hundreds of known $\delta$ Sct stars were found to contain low frequencies. The discovery was at variance with the predictions of models using the $\kappa$ mechanism in which frequencies below 5d$^{-1}$ are stable. Before the advent of the [*CoRoT*]{} and [*Kepler*]{} missions, only six $\delta$ Sct/$\gamma$ Dor hybrids had been discovered. They all lie roughly at the high-temperature end of the known $\gamma$ Dor instability strip, but extending beyond the strip to higher temperatures. With the first release of the [*Kepler*]{} data, it became clear that the hybrids were not rare at all [@Grigahcene2010]. From only 50 days of data, at least one-quarter of $\delta$ Sct stars were found to be hybrids. In this paper it is found that significant low-frequency peaks are present in at least 98percent of stars. Hybrid behaviour is the norm and occurs even among the hottest stars, as shown by @Balona2015d. The reason why so few hybrids were discovered from the ground must be partly attributed to the fact that ground-based photometry is greatly affected by variations in atmospheric extinction and daily data gaps, masking the low amplitudes of the long-period pulsations. The few hybrids discovered from the ground appear to be located in the region of instability where largest amplitudes tend to occur [@Balona2014a]. Recent pulsation models using time-dependent perturbation theory show that there is a complex interplay of driving and damping processes which cannot be reduced to just the $\kappa$ or convective blocking mechanisms. @Xiong2015 and @Xiong2016 show that factors such as turbulent dissipation, turbulent diffusion and anisotropy of turbulent convection need to be considered. Furthermore, in a region with a radiative flux gradient, the flux will itself be modulated by the oscillations. This is called “radiative modulation excitation” (RME) by @Xiong1998b. These additional factors co-exist and appear to explain the low frequencies seen in these stars. As @Xiong2016 point out, from this point of view both $\delta$ Sct and $\gamma$ Dor stars may be regarded as a single class of pulsating variable. High-precision [*Kepler*]{} photometry has enabled a large number of pulsating variables to be detected in the [*Kepler*]{} field. In this paper a list of $\delta$ Sct and $\gamma$ Dor stars, complete to [*Kepler*]{} magnitude $K_p = 12.5$mag but including fainter stars, is presented. Using photometric or spectroscopic estimates of effective temperature and luminosities determined from the parallaxes in the second data release of Gaia (Gaia DR2; @Gaia2018), these stars are precisely located in the Hertzsprung-Russell (H-R) diagram. Comparison between the observed and calculated red and blue edges are made. In addition to the problem of low frequencies in $\delta$ Sct stars, it is shown that less than half the stars in the instability strip pulsate. It is argued that this cannot be a result of pulsations at a level too low to be detected. Finally, it is shown that most of the high-amplitude $\delta$ Sct stars (HADS) are normal $\delta$ Sct stars and not objects intermediate between $\delta$ Sct and Cepheid variables. The problem of the low frequencies ================================== Models using the $\kappa$ mechanism show pulsational instability in main sequence and giant stars of intermediate mass only for frequencies higher than about 5d$^{-1}$. Lower frequencies are all stable. It may be possible to account for low frequencies as a result of rotational splitting. One can introduce rotational splitting in a simple way by using the know distribution of equatorial velocities among A/F dwarfs and giants using frequencies obtained from non-rotating models. The frequency distributions obtained in this way can be compared to observations, but they do not agree with the observed distributions [@Balona2015d]. It seems that low frequencies cannot be explained as a result of rotation. It is also possible that inertial modes, in particular r modes [@Papaloizou1978], might account for the low frequencies. These modes consist of predominantly toroidal motions which do not cause compression or expansion and hence no light variations. However, in a rotating star, the toroidal motion couples with spheroidal motion caused by the Coriolis force, leading to temperature perturbations and hence light variations. These modes have been recently proposed as an explanation of the broad hump that appears just below the rotation frequency in many A stars and also period spacings in some $\gamma$ Dor stars [@Saio2018a]. If r modes are responsible for the low frequencies in $\delta$ Sct stars, then all rotating stars within the instability strip should show such frequencies. This is not the case; in fact the majority of stars in the $\delta$ Sct instability strip do not seem to pulsate at all. For this reason, inertial modes can be ruled out as the cause of the low frequencies in $\delta$ Sct stars. @Balona2015d examined the possibility that the opacities in the outer layers of A stars may be underestimated. Artificially increasing the opacities by a factor of two does lead to instability of some low-degree modes at low frequencies, but also decreases the frequency range of $\delta$ Sct pulsations to some extent. An increase in opacities by such a large factor is unlikely and at present cannot be regarded as a possible solution to this problem. A fundamental obstacle to our understanding of stellar pulsations is that we lack a suitable theory of convection. The treatment of convection in pulsating stars has progressed quite considerably since the description of the $\gamma$ Dor pulsation mechanism in terms of “convective blocking” [@Guzik2000]. Convective blocking uses the simplest description of convection and does not take into account the interaction between pulsation and convection. Such “frozen-in” convection precludes the possibility of predicting the red edge of the $\delta$ Sct and $\gamma$ Dor instability strips. More recent treatments of pulsation use time-dependent perturbation theory [@Dupret2005; @Dupret2005b]. This allows the interaction between pulsation and other processes, such as turbulent pressure and turbulent kinetic energy dissipation, to be included. According to the time-dependent convection model of @Houdek2000, the damping of pulsations at the red edge of the $\delta$ Sct instability strip appears to be mostly due to fluctuations of the turbulent pressure which oscillates out of phase with the density fluctuations. However, in the models of @Xiong1989 and @Dupret2005, turbulent pressure driving and turbulent kinetic energy dissipation damping cancel near the red edge and stability is determined by the perturbations of the convective heat flux. All models are able to predict the red edge, but further research is necessary to identify the correct processes. A more detailed discussion can be found in [@Houdek2015]. Existing theories of convection rely on unknown parameters to characterize the effects of turbulent pressure and turbulent kinetic energy dissipation. The parameters are adjusted to obtain best agreement with observations. For example, three parameters are introduced by @Xiong2015. In order to fix these parameters the observed red and blue edges of the $\delta$ Sct and $\gamma$ Dor instability strips need to be determined. At present these are poorly know due to the large error in the luminosities. This problem can now be solved using distances derived from Gaia DR2 parallaxes [@Gaia2016]. The data ======== The [*Kepler*]{} observations consist of almost continuous photometry of many thousands of stars over a four-year period. The vast majority of stars were observed in long-cadence (LC) mode with exposure times of about 30min. [*Kepler*]{} light curves are available as uncorrected simple aperture photometry (SAP) and with pre-search data conditioning (PDC) in which instrumental effects are removed [@Stumpe2012; @Smith2012]. Most stars in the [*Kepler*]{} field have been observed by multicolour photometry, from which effective temperatures, surface gravities, metal abundances and stellar radii can be estimated. These stellar parameters are listed in the [*Kepler Input Catalogue*]{} (KIC, @Brown2011a). Subsequently, @Pinsonneault2012a and @Huber2014 revised these parameters for stars with $T_{\rm eff} < 6500$K. For hotter stars, the KIC effective temperatures were compared with those determined from high-dispersion spectroscopy by @Balona2015d. It was found that the KIC temperatures are very well correlated with the spectroscopic temperatures, but 144K cooler. Adding 144K to the KIC temperatures reproduces the spectroscopic temperatures with a standard deviation of about 250K. This can be taken as a realistic estimate of the true standard deviation since the KIC and spectroscopic effective temperatures are independently determined. In this paper the values of $T_{\rm eff}$ given by @Huber2014 are used for stars with $T_{\rm eff} < 6500$K. For hotter stars, the KIC effective temperatures, increased by 144K, are used. To determine the luminosities of [*Kepler*]{} $\delta$ Sct and $\gamma$ Dor stars requires knowledge of the apparent magnitude, interstellar extinction, bolometric correction and the parallax. A table of the bolometric correction, BC, in the Sloan photometric system as a function of $T_{\rm eff}$ and $\log g$ is presented in @Castelli2003. For this purpose, the small corrections described by [@Pinsonneault2012a] are applied to the [*Kepler*]{} $griz$ magnitudes to bring them into agreement with the Sloan system. Correction for interstellar extinction was applied to the $r$ magnitude using $r_0 = r - 0.874A_V$ [@Pinsonneault2012a]. The value of $A_V$ listed in the KIC is from a simple reddening model which depends only on galactic latitude and distance. A three-dimensional reddening map with a radius of 1200pc around the Sun and within 600pc of the galactic midplane has been calculated by @Gontcharov2017. This is likely to produce more accurate values of $A_V$ and is used in this paper. For more distant stars, the simple reddening model is used but adjusted so that it agrees with the 3D map at 1200pc. A comparison shows that the KIC values of $A_V$ are typically 0.017 mag higher than those given by the 3D map. From the Gaia DR2 parallax, $\pi$, the absolute magnitude is calculated using $r_{\rm abs} = r_0 + 5(\log_{10}\pi + 1)$. The absolute bolometric magnitude is then given by $M_{\rm bol} = r_{\rm abs} + {\rm BC}_r - M_{\rm bol\odot}$ with the solar absolute bolometric magnitude $M_{\rm bol\odot} = 4.74$. Finally, the luminosity relative to the Sun is found using $\log L/L_\odot = -0.4M_{\rm bol}$. Classification and light curves =============================== Stars in the [*Kepler*]{} field were observed almost continuously for 17 quarters covering a period of just over 4 years. Light curves and periodograms of all short-cadence observations (4827 stars) were visually examined. All long-cadence data brighter than magnitude 12.5, but including many more stars fainter than this limit (20784 stars in total) were visually examined as well. Detection of $\delta$ Sct stars is relatively easy since the periodograms show peaks at high frequencies. The $\beta$ Cep variables and some types of compact stars also show high frequencies, but these can be distinguished from $\delta$ Sct using the KIC effective temperatures and surface gravities. Among the 20784 stars examined, 1740 $\delta$ Sct stars were discovered. The distinction between $\delta$ Sct and $\gamma$ Dor stars was based purely on the absence of peaks with significant amplitudes having frequencies in excess of around 5d$^{-1}$. These stars have multiple peaks below this frequency. It is sometimes difficult to distinguish between $\gamma$ Dor and rotating variables which are the very common. Classification as a $\gamma$ Dor star was made only if the frequencies were too widely spread to be due to differential rotation. It should be noted that surface differential rotation reaches a maximum in the F stars [@Balona2016b] where most $\gamma$ Dor stars are to be found. It is possible that at least some frequency peaks in $\gamma$ Dor stars may be due to rotation. $\gamma$ Dor stars are distinguished from the slowly pulsating B (SPB) stars and some compact objects using the KIC effective temperatures and surface gravities. Among the 20784 stars examined, 820 $\gamma$ Dor stars were discovered. During the course of examination of the periodograms, instances were noted of the presence of low-frequency peaks in $\delta$ Sct stars, excluding peaks which might be attributed to binarity or rotation. The number of stars without significant low frequencies was found to be very low, probably less than 2percent of the $\delta$ Sct stars. Thus nearly all $\delta$ Sct stars are hybrids. Even a cursory examination of the [*Kepler*]{} light curves reveals a set of stars with beating and highly asymmetric minima and maxima. Maximum light amplitudes far exceeded those of minimum light. In many cases sudden high-amplitude excursions can easily be mistaken for flares. The morphology of these light curves is striking and quite unlike any other type of variable. These $\gamma$ Dor stars were first described by @Balona2011f who named them the ASYM (asymmetric) type of $\gamma$ Dor variable. ![Top panels: [*Kepler*]{} light curve of KIC8180361 showing asymmetrical light curve with occasional large excursions, one of which is depicted in the middle panel. The periodogram is shown in the bottom panel.[]{data-label="gdora"}](gdora.ps) Fig,\[gdora\] shows an example of the light curve and periodogram of the ASYM type. It should be noted that the [*Kepler*]{} PDC light curve flags most of the flare-like excursions as bad points. The light curve shown in the figure was reconstructed from the raw data using the good points of the PDC data to determine the necessary corrections. One of the questions that need to be asked is whether the large excursions arise as a result of beating of sinusoidal components. This can be answered by clipping the light curve to eliminate the large excursions. If the sudden high maxima are a simple result of beating of pure sinusoids, clipping these maxima should not introduce new frequencies. The periodogram of the clipped data shows significantly fewer low-amplitude peaks, indicating that these additional frequency components are required and that the excursions are simply due to a highly non-linear physical process. The light curves of most $\gamma$ Dor stars show characteristic beating, but with symmetric minima and maxima (the SYM type). The beating may be traced to two or more dominant closely-spaced frequencies in the periodograms. Another type of $\gamma$ Dor star shows no obvious beating in the light curve and an even frequency spread of peaks in the periodogram with comparable amplitudes (the MULT type). Because the distinction between the three groups may be important, each star was classified as either GDORA,GDORS or GDORM corresponding to the ASYM, SYM and MULT types. Among the 820 $\gamma$ Dor stars there are 137 GDORA and 447 GDORS stars. There are 16 stars which are GDORS for some of the time and GDORA at other times. There are 215 GDORM stars. A few stars which are difficult to classify into the three groups are labeled simply as GDOR. --------- ------- -------- ------- ----- ------ 1571152 DSCT 9.268 7192 149 1.88 2568519 GDORS 11.258 6299 182 0.18 2572386 DSCT 13.278 7345 275 2856756 DSCT 10.250 10477 365 2.39 2975832 DSCT 12.610 6824 249 --------- ------- -------- ------- ----- ------ : An extract of the catalogue of $\delta$ Sct and $\gamma$ Dor stars in the [*Kepler*]{} field for which Gaia DR2 luminosities could not be obtained. The [*Kepler*]{} magnitude, $K_p$, the effective temperature, $T_{\rm eff}$, and its error and the luminosity, $\log L/L_\odot$, determined from $T_{\rm eff}$ and the KIC radii are given. The error in $\log(L/L_\odot)$ is about 0.4 dex.[]{data-label="table2"} A catalogue of the $\delta$ Sct and $\gamma$ Dor stars is available in electronic form. An extract from the catalogue is shown in Table\[table1\]. In this table the effective temperatures for $T_{\rm eff} < 6500$K are from @Huber2014. For hotter stars, 144K has been added to the KIC effective temperature as discussed above. The interstellar absorption, $A_V$, is obtained from the 3D map of @Gontcharov2017. If the KIC values of $A_V$ are used, the higher absorption leads to a slight increase in $\log L/L_\odot$ of only 0.006 dex. There are 1680 $\delta$ Sct stars and 796 $\gamma$ Dor stars with luminosities estimated from Gaia DR2 parallaxes. For some stars Gaia DR2 parallaxes do not exist, no effective temperature is available or the interstellar extinction cannot be estimated. These 94 stars are listed separately (see Table\[table2\] for an extract). The mean difference between the photometrically estimated luminosities, $\log(L/L_\odot)_P$, and the luminosities from Gaia DR2, $\log(L/L_\odot)_G$, is $<\log(L/L_\odot)_P -\log(L/L_\odot)_G> = -0.22 \pm 0.01$. The photometrically estimated luminosities have a standard error of 0.39 dex. When deriving the location of the instability strips, the surface gravity, $\log g$, as a function of $T_{\rm eff}$ is sometimes used instead instead of $\log L/L_\odot$ as a function of $T_{\rm eff}$ (eg. @Uytterhoeven2011). This is simply because $\log g$ can be directly obtained from the observations. However, for comparison with theoretical models, $\log L/L_\odot$ is to be preferred. In this paper, $\log L/L_\odot$ is the natural choice because it can be obtained directly from the parallax as described above. Moreover, the Gaia DR2 parallaxes result in luminosities with very high accuracy, so that the instability strip can be determined far more precisely than the use of $\log g$. Many stars are binaries. If a star is a binary with components of equal luminosity, the luminosity calculated from the parallax will be twice as large as a single star of the same luminosity. Thus $\log L/L_\odot$ will be too high by about 0.3 dex. We do not know which stars are binaries in the [*Kepler*]{} field and it is possible that the estimated $\log L/L_\odot$ may be too large for some stars. If the components have different temperatures, this will also affect the $T_{\rm eff}$ of the combined stars. Unfortunately without detailed spectroscopic observations of each star it is impossible to correct for these effects. The $\delta$ Sct stars ====================== In Fig.\[dsct\] the $\delta$ Sct stars are shown in the H-R diagram together with the zero-age main sequence from models with solar abundances ($Z = 0.017$) and helium abundance $Y = 0.26$ by @Bertelli2008. The dashed polygon is a visual estimate of the location of the majority of the $\delta$ Sct stars and includes 94percent of these stars. Most of the stars are within the temperature range $6580 < T_{\rm eff} < 9460$K. The typical standard deviation is about 260K in $T_{\rm eff}$ (0.015 in $\log T_{\rm eff}$) and about 0.052 in $\log(L/L_\odot)$. These errors are shown by the cross in Fig.\[dsct\]. The main uncertainty in the luminosities is the effect of interstellar light absorption. The extinction values used here are interpolated from the table by @Gontcharov2017. Reduced extinction will lead to smaller luminosities. Also shown in the figure are the red and blue edges from @Xiong2016. The effective temperatures of the red and blue edges are clearly too cool. ![[*Kepler*]{} $\delta$ Sct stars in the H-R diagram (dots) using Gaia DR2 parallaxes and @Gontcharov2017 values of $A_V$. The solid lines are the zero-age main sequence (solar abundance from @Bertelli2008) and the nonradial red and blue edges from @Xiong2016. The dashed polygon defines the region which includes the majority of $\delta$ Sct stars. The cross on the bottom left shows the 1-$\sigma$ error bars.[]{data-label="dsct"}](dsct.ps) ![Examples of stars which show both solar-like oscillations (indicated by the arrow) and $\delta$ Sct high frequencies. These are composite stars containing a cool giant and a $\delta$ Sct star.[]{data-label="dsol"}](dsol.ps) There are a number of outliers on both the hot and cool sides of the instability region which need further study. It is possible that their effective temperatures are in error, but a study by @Balona2016c suggests that there is evidence for a class of variables with multiple high frequencies characteristic of $\beta$ Cep and $\delta$ Sct stars which lie between the red edge of the $\beta$ Cep and and the blue edge of the $\delta$ Sct instability regions. These have been called Maia variables. The few outliers below the ZAMS may be evolved objects, though the KIC surface gravities seem to be normal. @Qian2018 observed a group of 131 cool multiperiodic variable stars that are much cooler than the red edge of the $\delta$ Sct instability strip. Many of these are in the [*Kepler*]{} field. Inspection of their periodograms show the typical Gaussian amplitude envelope characteristic of solar-like oscillations, so it is possible that @Qian2018 have mis-classified solar-like pulsations in red giants as $\delta$ Sct stars. There is no indication of a cool population of $\delta$ Sct stars among the [*Kepler*]{} data examined. There are, however, composite objects consisting of a cool giant and a normal $\delta$ Sct star. Fig.\[dsol\] shows periodograms of three of these stars where the Gaussian-like amplitude envelope characteristic of solar-like oscillations in a cool giant and the pulsations in a $\delta$ Sct star are clearly visible. There are other cool stars that can be classified as $\delta$ Sct variables. An example is KIC4142768 where the effective temperature from several sources (including the KIC) is only about 5400K. The star is a heartbeat variable [@Balona2018b] and the LAMOST spectrum is A9V with no sign of a cool giant. In this case the explanation may be additional reddening caused by gas and dust associated with the binary. ![Example of periodograms of four $\delta$ Sct stars which have practically the same stellar parameters. The top panel shows their location in the H-R diagram, all within the single filled circle. The ZAMS and the $\delta$ Sct instability region are shown.[]{data-label="simlc"}](simlc.ps) ![The location of the $\gamma$ Dor stars in the H-R diagram (dots) using Gaia DR2 parallaxes and @Gontcharov2017 values of $A_V$. The outer dashed polygon defines the instability region of $\delta$ Sct stars. The smaller polygon defines the approximate location of $\gamma$ Dor stars. The solid lines show the zero-age main sequence and the red and blue edges of the instability strip calculated by @Xiong2016.[]{data-label="gdor"}](gdor.ps) ------- ------- ------- ------- 3.818 0.770 3.818 1.600 3.818 1.800 3.890 1.350 3.901 2.190 3.890 0.867 3.976 1.870 3.845 0.651 3.976 1.250 3.818 0.770 3.845 0.651 3.818 1.400 3.818 0.770 ------- ------- ------- ------- : Coordinates of the vertices of the $\delta$ Sct and $\gamma$ Dor instability regions as shown in Figs.\[dsct\] and \[gdor\].[]{data-label="boxes"} Fig.\[simlc\] is an illustration of the disparity in frequencies and amplitudes among $\delta$ Sct stars with practically the same stellar parameters. All four stars have $\log T_{\rm eff} = 7630$K and $\log L/L_\odot = 1.012$ within 20K and 0.002 dex respectively. The disparity in the general appearance in frequency peaks is remarkable. It is possible that rotation may be an important factor or that some of the four stars may be composite which will affect the derived luminosity. Also, the observational error of around 250K in $T_{\rm eff}$ could modify the expected pulsation frequencies somewhat. Nevertheless, the general impression obtained from visual inspection is that the periodogram of each star is unique. The $\gamma$ Dor stars ====================== In Fig.\[gdor\] the $\gamma$ Dor stars are shown in the H-R diagram together with the zero-age main sequence from models with solar abundance [@Bertelli2008]. The figure shows the instability polygon of the $\delta$ Sct stars as reference. The smaller nested polygon contains 89percent of the $\gamma$ Dor stars. The coordinates of the vertices in the polygonal regions of the $\delta$ Sct and $\gamma$ Dor stars are listed in Table\[boxes\]. As mentioned above, a star was classified as a $\gamma$ Dor variable only if all the peaks in the periodogram are below 5d$^{-1}$ (with some leeway if there are a few peaks of low amplitude above this frequency). In addition, a very important qualification is added: the frequencies must exclude rotational modulation. If, for example, a peak and its harmonic is seen, then it is a rotational variable and not a $\gamma$ Dor. It is this criterion more than anything else which is responsible for refining the $\gamma$ Dor instability strip. Of course, the luminosities derived from Gaia DR2 also assist in this refinement. The location of $\delta$ Sct and $\gamma$ Dor stars using KIC radii and effective temperatures to determine the luminosities is shown in Fig.2 of @Balona2014a. These can be compared with Fig.\[dsct\] and \[gdor\] in this paper. It is interesting that the $\gamma$ Dor instability region lies completely within the $\delta$ Sct instability region. In the $\gamma$ Dor box there are 711 $\gamma$ Dor stars, but there are also 815 $\delta$ Sct stars (and 994 other objects) in the same box. It seems that the $\gamma$ Dor variables are simply a subset of the $\delta$ Sct stars and not an independent class, as suggested by @Xiong2016. There is no difference in the locations of the GDORA, GDORS and GDORM subtypes within the $\gamma$ Dor box. The red and blue edges of the $\gamma$ Dor instability strip calculated by @Xiong2016 are shown in the figure, but do not agree with observations. There are quite a number of hot outliers which have been studied by @Balona2016c. The spectra confirm that many of these stars are indeed hot $\gamma$ Dor stars. Whether or not these deserve a separate classification or whether these stars, like the $\gamma$ Dor stars, may just be due to unusual mode selection processes remains to be seen. @Mowlavi2013 found a large population of new variable stars between the red edge of the SPB stars and the blue edge of the $\delta$ Sct stars, a region in the H-R diagram where no pulsation is predicted to occur based on standard stellar models. Their periods range from 0.1–0.7d, with amplitudes between 1 and 4mmag. It is possible that these could be identified with the hot $\gamma$ Dor stars in the [*Kepler*]{} field. The three coolest $\gamma$ Dor stars (KIC4840401, 8264287, and 12218727) are not solar-like variables. No known variable class in this temperature range resembles the $\gamma$ Dor class. They could be composite objects, but merit further study. The stars lying below the ZAMS also merit further study. They may perhaps be evolved compact objects. Fraction of pulsating stars =========================== ![Top panel: the fraction of $\delta$ Sct stars in the instability box relative to all stars in the box as a function of effective temperature. Bottom panel: the fraction of $\gamma$ Dor stars relative to all stars in the $\gamma$ Dor box as a function of effective temperature.[]{data-label="dsctdis"}](dsctdis.ps) In order to determine whether or not all stars in the $\delta$ Sct instability region pulsate, it is necessary to count the number of pulsating and non-pulsating stars within the same region. To assure completeness, the sample is limited to stars with $K_p < 12.5$mag because all stars in the [*Kepler*]{} field down to this brightness level have been classified according to variability class. It is found that 2881 stars for which luminosities can be determined from Gaia DR2 lie within the $\delta$ Sct instability box (excluding known evolved objects). Of these, 874 are $\delta$ Sct stars and 281 are $\gamma$ Dor stars. The remainder are non-pulsating as far as can be ascertained. Most are rotational variables or eclipsing systems. There are 1759 stars within the $\gamma$ Dor instability region, of which 260 are $\gamma$ Dor stars and 401 are $\delta$ Sct stars. The number of $\delta$ Sct stars relative to the total number of stars within the $\delta$ Sct instability box varies as a function of effective temperature (Fig.\[dsctdis\], top panel). The largest fraction of $\delta$ Sct stars occurs around $T_{\rm eff} \approx 7600$K. The bottom panel of the same figure shows the relative number of $\gamma$ Dor stars in the $\gamma$ Dor instability box as a function of temperature. ![Top panel: the distribution of the maximum amplitude (ppm) in [*Kepler*]{} $\delta$ Sct stars with $K_p < 12.5$mag in the instability box. Bottom panel: As above, but including all stars with $K_p < 12.5$mag in the $\delta$ Sct instability box on the assumption that these “constant” stars actually pulsate at below the detection limit. The peak at $A_{\rm max} < 10$ppm reaches 1781 stars.[]{data-label="ampdist"}](ampdist.ps) These results strongly indicate that both pulsating and non-pulsating stars co-exist within the instability strip. This poses a problem because it is difficult to understand why some stars with closely similar parameters, and with presumably very similar driving and damping regions, should pulsate while others do not pulsate. It is possible to argue that the non-pulsating stars do pulsate, but at a level below the detection limit, but statistics of the amplitude distribution argue against this [@Balona2011g]. In the top panel of Fig\[ampdist\], the distribution of maximum amplitude for $\delta$ Sct stars with $K_p < 12.5$mag is shown. As might be expected, the number of stars increases as the amplitude decreases. If we assume that the non-pulsating stars are actually pulsating below the detection level, they should also be included in this distribution. They must then be added to the number in the bin with the lowest amplitude covering amplitudes between 0 and 10ppm. If that is done, there is a discontinuous jump in the amplitude distribution, as seen in the bottom panel of Fig.\[ampdist\], which does not appear to be physical. The only way of avoiding this strange behaviour in the amplitude distribution is to assume that the non-pulsating stars belong to a different population and should not be included in the amplitude distribution of the pulsating stars. In other words, the simplest explanation is that the non-pulsating stars are not pulsating below the detection limit and do not pulsate at all. It can be concluded that the $\delta$ Sct instability strip is not pure. As already mentioned, this poses a serious problem. We seem to have an incomplete understanding of the outer layers of A stars. The conclusion here differs from that of @Murphy2015. From a study of only 54 stars, they found that all stars within the $\delta$ Sct instability strip pulsate. @Guzik2014 found that most stars pulsate, but a few constant stars remain. It could be argued that most of the constant stars are outside the instability strip due to errors in the effective temperature. The typical error in $T_{\rm eff}$ is about 200–300 K, while the width of the instability strip is about 3000K. To move a star in the middle of the instability strip to the edges of the strip requires that$T_{\rm eff}$ be in error by about 5 standard deviations (a probability less than $10^{-5}$). Of course, the probability will be higher if the star is closer to the edge of the instability strip, but it means that the probability that all 1781 non-pulsating stars are outside the instability strip is the product of the individual probabilities which is essentially zero. Furthermore, one has to assume that (for some unknown reason) the values of $T_{\rm eff}$ for $\delta$ Sct stars are much more accurate, otherwise many of these $\delta$ Sct stars would also be moved out of the instability region. For these reasons it is a certainty that non-pulsating stars exist in the instability region unless the discontinuity in the amplitude distribution can be understood in some other way. High-amplitude $\delta$ Sct stars ================================= The high-amplitude $\delta$ Sct stars (HADS) are a well-known group characterized by high photometric amplitude (generally higher than 0.3mag) and fairly simple frequency spectra, but with many combination frequencies. None of the stars in the [*Kepler*]{} field attain such a large amplitude, but several are known in the general field. They have been assumed to be transition objects between Cepheids and $\delta$ Sct stars - in fact they were originally called “dwarf Cepheids”. It is interesting to locate the field HADS in the H-R diagram to determine their evolutionary status. Using the catalogue of @Rodriguez2000, all stars with amplitudes exceeding 0.3 mag were selected. Gaia DR2 parallaxes were obtained for those stars which have effective temperatures from Apsis-Priam [@Bailer-Jones2013]. These effective temperatures are available in the Gaia DR2 catalogue. The interstellar absorption was estimated using the 3D map of @Gontcharov2017. Results are shown in Table\[tabhads\]. Some of the HADS appear to be evolved stars belonging to Population II on the basis of their high proper motions and low metallicities. These are called SX Phe variables. ![Location of the HADS in the H-R diagram. The filled circles are HADS of the $\delta$ Sct type and the open circles are SX Phe stars. The $\delta$ Sct box, the ZAMS and evolutionary tracks (masses labeled) from @Bertelli2008 are shown.[]{data-label="hads"}](hads.ps) Fig.\[hads\] shows the HADS in the H-R diagram. They appear to lie in the middle of the instability strip except for V0567 Oph which has a very low effective temperature. The SX Phe group also lie well within the instability box. If HADS were transition objects between $\delta$ Sct and Cepheids, one would expect all of them to have high luminosities, intermediate between the two groups of variables. However, most HADS appear to be normal $\delta$ Sct stars. ![Contours showing maximum pulsation amplitudes of $\delta$ Sct stars in the H-R diagram with polygonal region of instability. The contours show amplitudes from 500 to 7000 ppm.[]{data-label="amp"}](amp.ps) Fig.\[amp\] shows how the typical maximum amplitude in $\delta$ Sct stars varies within the instability strip. Largest amplitudes tend to occur around $T_{\rm eff} \approx 7000$K among the more luminous cool $\delta$ Sct stars. The mean effective temperature of the HADS is $7200 \pm 200$ which suggests that the high amplitudes in HADS are in line with what might be expected in normal $\delta$ Sct stars. The reason why the amplitude is so much larger in HADS compared to normal $\delta$ Sct stars can only be answered by nonlinear, nonradial pulsation models which do not yet exist. Conclusions =========== Using the full four-year light curves and periodograms of stars in the [*Kepler*]{} field, all stars with [*Kepler*]{} magnitude $K_p < 12.5$mag, as well as many more fainter stars, were classified according to variability type. A catalog of $\delta$ Sct and $\gamma$ Dor stars with Gaia DR2 parallaxes is presented. From these data, luminosities, $\log L/L_\odot$, with a standard deviation of 0.04 dex. By contrast, previous luminosity estimates based on multicolour photometry have typical errors of 0.4 dex. The most surprising result is that the $\gamma$ Dor variables do not occupy a separate instability strip, but lie entirely within the $\delta$ Sct instability region. There are, in fact, more $\delta$ Sct stars inside the $\gamma$ Dor instability region than $\gamma$ Dor stars. No two classes of pulsating star are known to share the same instability region. This must be the case if the driving and damping mechanisms differ, as they do in the conventional explanation for the two classes: the opacity $\kappa$ mechanism for $\delta$ Sct stars and the convective blocking mechanism for $\gamma$ Dor stars. It seems that the frequencies in $\gamma$ Dor stars may be just an effect of mode selection rather than reflecting different driving and damping mechanisms. The presence of a large variety of light curves among the $\gamma$ Dor stars, classified here as GDORA, GDORS and GDORM is probably an indication of the sensitivity of mode selection to the conditions in the outer layers of the star. The GDORA type, for example, shows extreme non-linear effects. Why such nonlinearity exists only in a subset of these stars is not known. This effect is not seen in $\delta$ Sct stars, but it might be masked by the presence of other frequencies of higher amplitudes. The suggestion by @Xiong2016 that $\gamma$ Dor stars should not be seen as a separate class has great merit. Nevertheless, the distinction is still a useful one for classification purposes. Our understanding of stellar pulsation has evolved quite considerably over the last few decades. It is now recognized that multiple driving and damping mechanisms occur in the same star. The most important recent works in this respect are that of @Xiong2015 and @Xiong2016 where it is demonstrated that the interplay of different processes can account for the instability at low frequencies seen in practically all $\delta$ Sct stars. In fact, inspection of the [*Kepler*]{} light curves shows that low frequencies are present in at least 98percent of $\delta$ Sct stars. The work of @Xiong2016 offers a very attractive explanation for the low frequencies in $\delta$ Sct stars. Unfortunately, the predicted red and blue edges from @Xiong2016 do not agree with the limits of the instability regions determined in this paper. It is also not clear whether an explanation for the co-existence of $\gamma$ Dor, $\delta$ Sct and non-pulsating stars can be found by tuning available free parameters in the theory. Nevertheless, this work shows great promise for a better understanding of these stars. Another well-known group among the $\delta$ Sct stars are the HADS. In this paper it is shown that most HADS are normal $\delta$ Sct stars. The data analyzed in this paper indicates that the majority of stars within the $\delta$ Sct instability region do not pulsate. If it is assumed that these stars actually pulsate below the detectable level, then they should be included in the calculation of the distribution of maximum amplitude. In that case, the very large number of apparently non-pulsating stars introduces a nonphysical discontinuity in the distribution of maximum amplitudes. This suggests that these stars do not pulsate. This introduces yet another unresolved issue because a reason needs to be found for the high damping of pulsations in the majority of stars in the instability region. It appears that one or more unknown damping and driving processes are operating in the outer layers, rendering the pulsation amplitudes and mode selection very sensitive to conditions in these layers. These problems are perhaps not too surprising in view of the fact that starspots are present in most A stars [@Balona2017a]. None of the current models of A stars provide a possible explanation for the presence of starspots. The problems discussed here add to the need for a revision in the current view of A star atmospheres. Acknowledgments {#acknowledgments .unnumbered} =============== LAB wishes to thank the National Research Foundation of South Africa for financial support. This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. \[lastpage\]
--- abstract: 'Visual perception entails solving a wide set of tasks, e.g., object detection, depth estimation, etc. The predictions made for multiple tasks from the same image are not independent, and therefore, are expected to be ‘consistent’. We propose a broadly applicable and fully computational method for augmenting learning with **Cross-Task Consistency**.[^1] The proposed formulation is based on **inference-path invariance** over a graph of arbitrary tasks. We observe that learning with cross-task consistency leads to more accurate predictions and better generalization to out-of-distribution inputs. This framework also leads to an informative unsupervised quantity, called **Consistency Energy**, based on measuring the intrinsic consistency of the system. Consistency Energy correlates well with the supervised error ($r{=}0.67$), thus it can be employed as an unsupervised confidence metric as well as for detection of out-of-distribution inputs ($\text{ROC-AUC=}0.95$). The evaluations are performed on multiple datasets, including Taskonomy, Replica, CocoDoom, and ApolloScape, and they benchmark cross-task consistency versus various baselines including conventional multi-task learning, cycle consistency, and analytical consistency.' author: - | Amir R. Zamir$^{\dagger*}$ Alexander Sax$^{\ddagger*}$ Teresa Yeo$^{\dagger}$ Oğuzhan Kar$^{\dagger}$ Nikhil Cheerla$^{\S}$\ Rohan Suri$^{\S}$ Zhangjie Cao$^{\S}$ Jitendra Malik$^{\ddagger}$ Leonidas Guibas$^{\S}$\ $^\dagger$ Swiss Federal Institute of Technology (EPFL) $^\S$ Stanford University $^\ddagger$ UC Berkeley\ bibliography: - 'egbib.bib' title: 'Robust Learning Through Cross-Task Consistency' --- \[sec:intro\] [**What**]{} is consistency: suppose an object detector detects a ball in a particular region of an image, while a depth estimator returns a flat surface for the same region. This presents an issue – at least one of them has to be wrong, because they are *inconsistent*. More concretely, the first prediction domain (objects) and the second prediction domain (depth) are not independent and consequently enforce some constraints on each other, often referred to as *consistency constraints*. [**Why**]{} is it important to incorporate consistency in learning: first, desired learning tasks are usually predictions of different aspects of one underlying reality (the scene that underlies an image). Hence inconsistency among predictions implies contradiction and is inherently undesirable. Second, consistency constraints are informative and can be used to better fit the data or lower the sample complexity. Also, they may reduce the tendency of neural networks to learn “surface statistics” (superficial cues) [@jo2017measuring], by enforcing constraints rooted in different physical or geometric rules. This is empirically supported by the improved generalization of models when trained with consistency constraints (Sec. \[sec:results\]). [**How**]{} can we design a learning system that makes consistent predictions: this paper proposes a method which, given an arbitrary dictionary of tasks, augments the learning objective with explicit constraints for cross-task consistency. The constraints are learned from data rather than apriori given relationships.[^2] This makes the method applicable to any pairs of tasks as long as they are not statistically independent; *even if their analytical relationship is unknown, hard to program, or non-differentiable*. The primary concept behind the method is ‘inference-path invariance’. That is, the result of inferring an domain from an domain should be the same, regardless of the domains mediating the inference (e.g., $\shortrightarrow$ and $\shortrightarrow$$\shortrightarrow$ and $\shortrightarrow$$\shortrightarrow$ are expected to yield the same normals result). When inference paths with the same endpoints, but different intermediate domains, yield similar results, this implies the intermediate domain predictions did not conflict as far as the output was concerned. We apply this concept over paths in a graph of tasks, where the nodes and edges are prediction domains and neural network mappings between them, respectively (Fig. \[fig:graph\](d)). Satisfying this invariance constraint over *all* paths in the graph ensures the predictions for all domains are in global cross-task agreement.[^3] To make the associated large optimization job manageable, we reduce the problem to a ‘separable’ one, devise a tractable training schedule, and use a ‘perceptual loss’ based formulation. The last enables mitigating residual errors in networks and potential ill-posed/one-to-many mappings between domains; this is crucial as one may not be able to always infer one domain from another with certainty (Sec. \[sec:method\]). [Interactive visualizations](https://consistency.epfl.ch/visuals), [trained models](https://consistency.epfl.ch/#models), [code](https://consistency.epfl.ch/#models), and a [live demo](https://consistency.epfl.ch/demo) are available at <http://consistency.epfl.ch/>. The concept of consistency and methods for enforcing it are related to various topics, including structured prediction, graphical models [@Koller:2009:PGM:1795555], functional maps [@Ovsjanikov:2012:FMF:2185520.2185526], and certain topics in vector calculus and differential topology [@guillemin1974differential]. We review the most relevant ones in context of computer vision. **Utilizing consistency:** Various consistency constraints have been commonly found beneficial across different fields, e.g., in language as ‘back-translation’ [@BrislinCrossCultural; @Artetxe18Multilingual; @Lample19crosslingual; @backtranslation2018] or in vision over the temporal domain [@WangCycleConsistency19; @DwibediTimeConsistency19], 3D geometry [@godard2017unsupervised; @Geonet18; @garg2016unsupervised; @hickson2019floors; @ZhouKAHE16; @zhang2017physically; @Huang:2013:CSM:2600289.2600314; @yin2019enforcing; @zou2018df; @Zhang18PathInvariant; @kusupati2019normal; @cosmo2017consistent], and in recognition and (conditional/unconditional) image translation [@hertzmann2001image; @mirza2014conditional; @pix2pix; @cycleGan17; @HoffmancyCADA17; @choi2018stargan]. In computer vision, consistency has been extensively utilized in the cycle form and often between two or few domains [@cycleGan17; @HoffmancyCADA17]. In contrast, we consider consistency in the more general form of arbitrary paths with varied-lengths over a large task set, rather than the special cases of short cyclic paths. Also, the proposed approach needs *no prior explicit knowledge* about task relationships [@Geonet18; @kusupati2019normal; @yin2019enforcing; @zou2018df]. **Multi-task learning:** In the most conventional form, multi-task learning predicts multiple output domains out of a shared encoder/representation for an input. It has been speculated that the predictions of a multi-task network may be automatically cross-task consistent as the representation from which the predictions are made are shared. This has been observed to not be necessarily true in several works [@Kokkinos16; @Zhang17MTL; @xu2018pad; @standley2019], as consistency is not directly enforced during training. We also make the same observation (see visuals [here](http://consistency.epfl.ch/visuals/)) and quantify it (see Fig. \[fig:energy\_over\_time\]), which signifies the need for explicit augmentation of consistency in learning. **Transfer learning** predicts the output of a target task given another task’s solution as a source. The predictions made using transfer learning are sometimes assumed to be cross-task consistent, which is often found to not be the case [@taskonomy18; @sharif2014cnn], as transfer learning does not have a specific mechanism to impose consistency by default. Unlike basic multi-task learning and transfer learning, the proposed method includes explicit mechanisms for learning with general data-driven consistency constraints. **Uncertainty metrics:** Among the existing approaches to measuring prediction uncertainty, the proposed Consistency Energy (Sec. \[sec:energy\_method\]) is most related to Ensemble Averaging [@lakshminarayanan2017simple], with the key difference that the estimations in our ensemble are from *different cues/paths*, rather than retraining/reevaluating the same network with different random initializations or parameters. Using multiple cues is expected to make the ensemble more effective at capturing uncertainty. \[sec:method\] We define the problem as follows: suppose ${{\scriptstyle \mathcal{X}}}$ denotes the query domain (e.g., RGB images) and ${{\scriptstyle \mathcal{Y}}}{=}\{{{\scriptstyle \mathcal{Y}}}_1,$...$,{{\scriptstyle \mathcal{Y}}}_n\}$ is the set of $n$ desired prediction domains (e.g., normals, depth, objects, etc). An individual datapoint from domains $({{\scriptstyle \mathcal{X}}},{{\scriptstyle \mathcal{Y}}}_1,$...$,{{\scriptstyle \mathcal{Y}}}_n)$ is denoted by $(x,y_1,$...$,y_n)$. The goal is to learn functions that map the query domain onto the prediction domains, i.e. $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{=} \{f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_j} | {{\scriptstyle \mathcal{Y}}}_j {\in} {{\scriptstyle \mathcal{Y}}}\}$ where $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_j}(x)$ outputs $y_j$ given $x$. We also define $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}{=} \{f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_j} | {{\scriptstyle \mathcal{Y}}}_i, {{\scriptstyle \mathcal{Y}}}_j {\in} {{\scriptstyle \mathcal{Y}}}, i{\neq}j \}$, which is the set of ‘cross-task’ functions that map the prediction domains onto each other; we use them in the consistency constraints. For now assume $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}$ is given apriori and frozen; in Sec. \[sec:full\_graph\] we discuss all functions $f$s are neural networks in this paper, and we learn $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}$ just like $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}$. Triangle: The Elementary Consistency Unit {#sec:triangle} ----------------------------------------- The typical supervised way of training the neural networks in $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}$, e.g., $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)$, is to find parameters of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ that minimize a loss with the general form $|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) \text{-} {y}_1|$ using a distance function as $|.|$, e.g., $\ell_1$ norm. This standard *independent* learning of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}$s satisfies various desirable properties, including cross-task consistency, if given infinite amount of data, but not under the practical finite data regime. This is qualitatively illustrated in Fig. \[fig:qualitative\_consistency\] (upper). Thus we introduce additional constraints to guide the training toward cross-task consistency. We define the loss for predicting domain ${{\scriptstyle \mathcal{Y}}}_1$ from ${{\scriptstyle \mathcal{X}}}$ *while enforcing consistency with domain ${{\scriptstyle \mathcal{Y}}}_2$* as a directed triangle depicted in Fig. \[fig:graph\](b): $$\hspace{-3pt} \mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}} {\raisebox{-0.15\totalheight}{$\triangleq$}}|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-2pt}(x) \text{-} {y}_1| \text{+} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-2pt}(x) \text{-} f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}\hspace{-2pt}(x)| \text{+} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}\hspace{-2pt}(x) \text{-} {y}_2|. \label{eq:triangle}$$ The first and last terms are the standard *direct* losses for training $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ and $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$. The middle term is the *consistency term* which enforces that predicting ${{\scriptstyle \mathcal{Y}}}_2$ out of the predicted ${{\scriptstyle \mathcal{Y}}}_1$ yields the same result as directly predicting ${{\scriptstyle \mathcal{Y}}}_2$ out of ${{\scriptstyle \mathcal{X}}}$ (done via the given cross-task function $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$).[^4] Thus learning to predict ${{\scriptstyle \mathcal{Y}}}_1$ and ${{\scriptstyle \mathcal{Y}}}_2$ are not independent anymore. The triangle loss \[eq:triangle\] is the smallest unit of enforcing cross-task consistency. Below we make two improving modifications on it via function ‘separability’ and ‘perceptual losses’. ### Separability of Optimization Parameters The loss $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}}$ involves *simultaneous* training of two networks $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ and $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$, thus it is resource demanding. We show $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}}$ can be reduced to a ‘separable’ function [@stewart2012essential] resulting in two terms that can be optimized independently. From the triangle inequality we can derive: $$|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-1pt}(x) \text{-} f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}\hspace{-1pt}(x)|{\leq}|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-1pt}(x) \text{-} {y}_2| \text{+} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}\hspace{-1pt}(x) \text{-} {y}_2|, \label{eq:separableineq}$$ which after substitution in Eq. \[eq:triangle\] yields: $$\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}} {\leq} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) \text{-} {y}_1| \text{+} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) \text{-} {y}_2| \text{+} 2|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}(x) \text{-} {y}_2|. \label{eq:separableineqtriangle}$$ The upper bound for $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}}$ in inequality \[eq:separableineqtriangle\] can be optimized in lieu of $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{triangle}}}$ itself, as they both have the same minimizer.[^5] The terms of this bound include either $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ or $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$, but not both, hence we now have a loss separable into functions of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ or $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$, and they can be optimized independently. The part pertinent to the network $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ is: $$\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}} {\raisebox{-0.15\totalheight}{$\triangleq$}}|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - {y}_1| + |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - {y}_2|, \label{eq:pathloss}$$ named *separate*, as we reduced the closed triangle objective ${ \mathop{\vphantom{\triangle}}_{{{\scriptstyle \mathcal{X}}}}\hspace{-0.17em}{\stackrel{{{\scriptstyle \mathcal{Y}}}_1}{\triangle}}_{{{\scriptstyle \mathcal{Y}}}_2}}$in Eq. \[eq:triangle\] to two separate path objectives [${{{\scriptstyle \mathcal{X}}}}{\shortrightarrow}{{{\scriptstyle \mathcal{Y}}}_1}{\shortrightarrow}{{{\scriptstyle \mathcal{Y}}}_2}$]{} and [${{{\scriptstyle \mathcal{X}}}}{\shortrightarrow}{{{\scriptstyle \mathcal{Y}}}_2}$]{}. The first term of Eq. \[eq:pathloss\] enforces the general correctness of predicting ${{\scriptstyle \mathcal{Y}}}_1$, and the second term enforces its consistency with ${{\scriptstyle \mathcal{Y}}}_2$ domain. ### Reconfiguration into a “Perceptual Loss” {#sec:perceploss} \[sec:sep\_paired\] Training $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ using the loss $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}}$ requires a training dataset with multi domain annotations for one input: $(x,{y_1},{y_2})$. It also relies on availability of a *perfect* function $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$ for mapping ${{\scriptstyle \mathcal{Y}}}_1$ onto ${{\scriptstyle \mathcal{Y}}}_2$; i.e. it demands ${y_2}{=}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y_1})$. We show how these two requirements can be reduced. Again, from triangle inequality we can derive: $$\begin{gathered} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - {y}_2| {\leq} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1)| + \\ |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1) - {y}_2|, \end{gathered}$$ which after substitution in Eq. \[eq:pathloss\] yields: $$\begin{gathered} \mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}} {\leq} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - {y}_1| + |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1)| + \\ |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1) - {y}_2|. \label{eq:percepinequality} \end{gathered}$$ Similar to the discussion for inequality \[eq:separableineqtriangle\], the upper bound in inequality \[eq:percepinequality\] can be optimized in lieu of $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}}$ as both have the same minimizer.[^6] As the last term is a constant w.r.t. $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$, the final loss for training $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ subject to consistency with domain ${{\scriptstyle \mathcal{Y}}}_2$ is: $$\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}} {\raisebox{-0.15\totalheight}{$\triangleq$}}|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - {y}_1| + |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) - f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1)|. \label{eq:percep_loss}$$ The loss $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ no longer includes ${y_2}$, hence it admits pair training data $(x,{y_1})$ rather than triplet $(x,{y_1},{y_2})$.[^7] Comparing $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ and $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}}$ shows the modification boiled down to replacing ${y_2}$ with $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({{y_1}})$. This makes intuitive sense too, as ${y_2}$ is the match of ${y_1}$ in the ${{\scriptstyle \mathcal{Y}}}_2$ domain. **Ill-posed tasks and imperfect networks:** If $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$ is a *noisy* estimator, then $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y_1}){=}{y_2{+}{noise}}$ rather than $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y_1}){=}{y_2}$. Using a noisy $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$ in $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{separate}}}$ corrupts the training of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ since the second loss term does not reach 0 if $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)$ correctly outputs ${y_1}$. That is in contrast to $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ where both terms have the same global minimum and are always 0 if $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)$ outputs ${y_1}$ – even when $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y_1}){=}{y_2{+}{noise}}$. Thus $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ enables a robust training of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)$ w.r.t. imperfections in $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$. This is crucial since neural networks are almost never perfect estimators, e.g., due to lacking an optimal training process for them or potential ill-posedness of the task $y_1{\shortto}y_2$. Further discussion and experiments are available in [supplementary material](http://consistency.epfl.ch/supplementary_material). **Perceptual Loss:** The process that led to Eq. \[eq:percep\_loss\] can be generally seen as using the loss $|g{\circ}f(x){-}g(y)|$ instead of $|f(x){-}y|$. The latter compares $f(x)$ and $y$ in their explicit space, while the former compares them via the lens of function $g$. This is often referred to as “perceptual loss” in super-resolution and style transfer literature [@DBLP:journals/corr/JohnsonAL16]–where two images are compared in the *representation space* of a network pretrained on ImageNet, rather than in *pixel space*. Similarly, the consistency constraint between the domains ${{\scriptstyle \mathcal{Y}}}_1$ and ${{\scriptstyle \mathcal{Y}}}_2$ in Eq. \[eq:percep\_loss\] (second term) can be viewed as judging the prediction $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)$ against ${y_1}$ via the lens of the network $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$; here $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$ is a “perceptual loss” for training $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$. However, unlike the ImageNet-based perceptual loss [@DBLP:journals/corr/JohnsonAL16], this function has the specific and interpretable job of enforcing consistency with another task. We also use multiple $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}$s simultaneously which enforces consistency of predicting ${{\scriptstyle \mathcal{Y}}}_1$ against multiple other domains (Sections \[sec:multitriangle\] and \[sec:full\_graph\]). Consistency of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ with ‘Multiple’ Domains {#sec:multitriangle} ---------------------------------------------------------------------------------------------------------------------------------------------------------- The derived $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ loss augments learning of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ with a consistency constraint against *one* domain ${{\scriptstyle \mathcal{Y}}}_2$. Straightforward extension of the same derivation to enforcing consistency of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ against *multiple* other domains (i.e. when $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ is part of multiple simultaneous triangles) yields: $$\hspace{-5.5pt}\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{{\hspace{-1pt}\scriptscriptstyle {Y}}}}}^{\textit{\tiny{perceptual}}} {\raisebox{-0.15\totalheight}{$\triangleq$}}|Y|{{{\hspace{-1pt}\scriptstyle {\times}\hspace{-1pt}}}}|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-2.5pt}(x) \text{-} {y}_1|\text{+}\hspace{-5pt}\sum_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i \in Y}\hspace{-3pt} |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}\hspace{-1.5pt}{\circ}\hspace{-1.5pt}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}\hspace{-2.5pt}(x) \text{-} f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}\hspace{-2pt}({y}_1)|, \hspace{-2.5pt}\label{eq:setloss}$$ where $Y$ is the *set* of domains with which $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ must be consistent, and $|Y|$ is the cardinality of $Y$. Notice that $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}^{\textit{\tiny{perceptual}}}$ is a special case of $\mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{{\hspace{-1pt}\scriptscriptstyle {Y}}}}}^{\textit{\tiny{perceptual}}}$ where $Y{=}\{{{{\scriptstyle \mathcal{Y}}}_2}\}$. Fig. \[fig:all\_losses\] summarizes the derivation of losses for $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$. Fig. \[fig:qualitative\_method\] shows qualitative results of learning $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ with and without cross-task consistency for a sample query. Beyond Triangles: Globally Consistent Graphs {#sec:full_graph} -------------------------------------------- The discussion so far provided the loss for the cross-task consistent training of *one* function $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ using elementary *triangle* based units. We also assumed the functions $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}$ were given apriori. The more general multi-task setup is: given a *large set of domains*, we are interested in learning functions that map the domains onto each other in a *globally cross-task consistent* manner. This objective can be formulated over a graph $\mathcal{G} {=} (\mathcal{D},\mathcal{F})$ with nodes representing all of the domains $\mathcal{D} {=} ({{\scriptstyle \mathcal{X}}}\cup {{\scriptstyle \mathcal{Y}}}$) and edges being neural networks between them $\mathcal{F} {=} (\mathcal{F}_{{\scriptstyle \mathcal{X}}}\cup \mathcal{F}_{{\scriptstyle \mathcal{Y}}})$; see Fig.\[fig:graph\](c). **Extension to Arbitrary Paths**: The transition from three domains to a large graph $\mathcal{G}$ enables forming more general consistency constraints using *arbitrary-paths*. That is, two *paths* with same endpoint should yield the same results – an example is shown in Fig.\[fig:graph\](d). The triangle constraint in Fig.\[fig:graph\](b,c) is a special case of the more general constraint in Fig.\[fig:graph\](d), if paths with lengths 1 and 2 are picked for the green and blue paths. Extending the derivations done for a triangle in Sec. \[sec:triangle\] to paths yields: $$\begin{gathered} \mathcal{L}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2...{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}^{\textit{\tiny{perceptual}}} = |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) {-} {y}_1| + \\ |f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_{k-1}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}{\circ}...{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x) {-} f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_{k-1}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}{\circ}...{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1)|, \label{eq:longpathloss} \end{gathered}$$ which is the loss for training $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$ using the arbitrary consistency path ${{\scriptstyle \mathcal{X}}}{\shortrightarrow}{{\scriptstyle \mathcal{Y}}}_1{\shortrightarrow}{{\scriptstyle \mathcal{Y}}}_2...{\shortrightarrow}{{\scriptstyle \mathcal{Y}}}_k$ with length $k$ (full derivation provided in [supplementary material](http://consistency.epfl.ch/supplementary_material)). Notice that Eq. \[eq:percep\_loss\] is a special case of Eq. \[eq:longpathloss\] if $k{=}2$. Equation \[eq:longpathloss\] is particularly useful for incomplete graphs; if the function ${{\scriptstyle \mathcal{Y}}}_1{\shortrightarrow}{{\scriptstyle \mathcal{Y}}}_k$ is missing, consistency between domains ${{\scriptstyle \mathcal{Y}}}_1$ and ${{\scriptstyle \mathcal{Y}}}_k$ can still be enforced via transitivity through other domains using Eq. \[eq:longpathloss\]. Also, extending Eq. \[eq:longpathloss\] to *multiple simultaneous paths* (as in Eq. \[eq:setloss\]) by summing the path constraints is straightforward. **Global Consistency Objective:** We define reaching *global* cross-task consistency for graph $\mathcal{G}$ as satisfying the consistency constraint for *all* feasible paths in $\mathcal{G}$. We can write the global consistency objective for $\mathcal{G}$ as $ \mathcal{L}_{\mathcal{G}} = \sum_{p \in \mathcal{P}} \mathcal{L}_{p}^{\textit{\tiny{perceptual}}}$, where $p$ represents a path and $\mathcal{P}$ is the set of all feasible paths in $\mathcal{G}$. Optimizing the objective $\mathcal{L}_{\mathcal{G}}$ directly is intractable as it would require simultaneous training of all networks in $\mathcal{F}$ with a massive number of consistency paths[^8]. In Alg.\[algorithm\] we devise a straightforward training schedule for an approximate optimization of $\mathcal{L}_{\mathcal{G}}$. This problem is similar to inference in graphical models, where one is interested in marginal distribution of unobserved nodes given some observed nodes by passing “messages” between them through the graph until convergence. As exact inference is usually intractable for unconstrained graphs, often an approximate message passing algorithm with various heuristics is used. Train each $f {\in} \mathcal{F}$ independently.\ \[algo:initline\] Instead of optimizing all terms in $\mathcal{L}_{\mathcal{G}}$, Alg.\[algorithm\] selects one network $f_{ij}{\in}\mathcal{F}$ to be trained, selects consistency path(s) $p{\in}\mathcal{P}$ for it, and trains $f_{ij}$ with $p$ for a fixed number of steps using loss \[eq:longpathloss\] (or its multi path version if multiple paths selected). This is repeated until all networks in $\mathcal{F}$ satisfy a convergence criterion. \[fig:qualitative\_accuracy\] A number of choices for the selection criterion in *SelectNetwork* and *SelectPath* is possible, including round-robin and random selection. While we did not observe a significant difference in the final results, we achieved the best results using *maximal violation* criterion: at each step select the network and path with the largest loss[^9]. Also, Alg.\[algorithm\] starts from shorter paths and progressively opens up to longer ones (up to length $L$) only after shorter paths have converged. This is based on the observation that the benefit of short and long paths in terms of enforcing cross-task consistency overlap, while shorter paths are computationally cheaper^\[seesupmat\]^. For the same reason, all of the networks are initialized by training using the standard direct loss (Op.\[algo:initline\] in Alg.\[algorithm\]) before progressively adding consistency terms. Finally, Alg.\[algorithm\] does not distinguish between $\mathcal{F}_x$ and $\mathcal{F}_x$ and can be used to train them all in the same pool. This means the selected path $p$ may include networks not fully converged yet. This is not an issue in practice, because, first, all networks are pre-trained with their direct loss (Op.\[algo:initline\] in Alg.\[algorithm\]) thus they are not wildly far from their convergence point. Second, the perceptual loss formulation makes training $f_{ij}$ robust to imperfections in functions in $p$ (Sec. \[sec:perceploss\]). However, as practical applications primarily care about $\mathcal{F}_x$, rather than $\mathcal{F}_y$, one can first train $\mathcal{F}_y$ to convergence using Alg.\[algorithm\], then start the training of $\mathcal{F}_x$ with well trained and converged networks $\mathcal{F}_y$. We do the latter in our experiments.[^10] Please see [supplementary material](http://consistency.epfl.ch/supplementary_material) for how to normalize and balance the direct and consistency loss terms, as they belong to different domains with distinct numerical properties. \[sec:energy\_method\] We quantify the amount of cross-task consistency in the system using an energy-based quantity [@lecun2006tutorial] called *Consistency Energy*. For a single query $x$ and domain ${{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k$, the consistency energy is defined to be the standardized average of pairwise inconsistencies: $$\text{Energy}_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_{k}}\hspace{-2pt}{(x)} ~{\raisebox{-0.15\totalheight}{$\triangleq$}}~ \hspace{-0.5pt} \raisebox{-1pt}{$ \frac{ {1} }{ {|{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}|{{-}1}} } $} \hspace{-6pt} \raisebox{0pt}{$\displaystyle \sum_{\substack{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i \in {{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}, i\neq k}} $} \hspace{-9pt} \raisebox{-1pt}{$ { \frac{ {|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}(x){-}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}(x)| {-} \mu_{i}} }{ {\sigma_{i}} } } $}, \label{eq:energy}$$ where $\mu_{i}$ and $\sigma_{i}$ are the average and standard deviation of $|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}{\circ}f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_i}(x){-} f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_k}(x)|$ over the dataset. Eq. \[eq:energy\] can be computed per-pixel or per-image by average over its pixels. Intuitively, the energy can be thought of as the amount of variance in predictions in the lower row of Fig. \[fig:qualitative\_consistency\] – the higher the variance, the higher the inconsistency, and the higher the energy. The consistency energy is an intrinsic quantity of the system and needs no ground truth or supervision. In Sec. \[sec:energy\_results\], we show this quantity turns out to be quite informative as it can indicate the reliability of predictions (useful as a confidence/uncertainty metric) or a shift in the input domain (useful for domain adaptation). This is based on the fact that if the query is from the same data distribution as the training and is unchallenging, all inference paths of a system trained with consistency path constraints work well and yield similar results (as they were trained to); whereas under a distribution shift or for a challenging query, different paths break in different ways resulting in dissimilar predictions, and therefore, creating a higher variance. In other words, usually *correct* predictions are *consistent* while *mistakes* are *inconsistent*. (Plots \[fig:correlation\], \[fig:id\_ood\_energy\_dist\], \[fig:error\_energy\_blur\].) \[fig:normalqualitative\_accuracy\] \[sec:results\] The evaluations are organized to demonstrate the proposed approach yields predictions that are **I.** more *consistent* (Sec.\[sec:results\_consistency\]), **II.** more *accurate* (Sec.\[sec:results\_accuracy\]), and **III.** more *generalizable* to out-of-training-distribution data (Sec.\[sec:results\_generalization\]). We also **IV.** quantitatively analyze the *Consistency Energy* and report its utilities (Sec.\[sec:energy\_results\]). **Datasets:** We used the following datasets in the evaluations: **Architecture & Training Details:** We used a UNet [@RonnebergerFB15] backbone architecture. We benchmarked alternatives, e.g., ResNet [@resnet], and found UNets to yield superior pixel-wise predictions. All networks in $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}$ and $\mathcal{F}_{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}$ have a similar architecture. The networks have 6 down and 6 up sampling blocks and were trained using AMSGrad [@reddi2019convergence] and Group Norm [@wu2018group] with learning rate $3{\times}10^{-5}$, weight decay $2{\times}10^{-6}$, and batch size 32. Input and output images were linearly scaled to the range $[0,1]$ and resized down to $256\times256$. We used $\ell_1$ as the norm in all losses and set the max path length $L\text{=}3$. We experimented with different loss normalization methods and achieved the best results when the loss terms are weighted negative proportional to their respective gradient magnitude (details in [supplementary material](http://consistency.epfl.ch/supplementary_material)). \[table\_joint\] **Baselines:** The main baseline categories are described below. To prevent confounding factors, our method and all baselines were implemented using the *same UNet network* when feasible and were *re-trained on Taskonomy dataset*. Consistency of Predictions {#sec:results_consistency} -------------------------- Fig. \[fig:energy\_over\_time\] (blue) shows the amount of inconsistency in test set predictions (Consistency Energy) successfully decreases over the course of training. The convergence point of the network trained with consistency constraints is well below baseline independent learning (orange) and multi-task learning (green)–which shows consistency among predictions *does not naturally emerge* in either case without explicit constraining. Plots of individual loss terms similarly show minimizing the direct term does not lead to automatic minimization of consistency terms (provided in [supplementary](http://consistency.epfl.ch/supplementary_material)). Accuracy of Predictions {#sec:results_accuracy} ----------------------- Figures \[fig:qualitative\_accuracy\] and  \[fig:normalqualitative\_accuracy\] compare the prediction results of networks trained with cross-task consistency against the baselines in different domains. The improvements are considerable particularly around the difficult *fine-grained details*. Quantitative evaluations are provided in Tab. \[table\_joint\] for Replica dataset and Taskonomy datasets on depth, normal, reshading, and pixel-wise semantic prediction tasks. Learning with consistency led to large improvements in most of the setups. As most of the pixels in an image belong to easy to predict regions governed by the room layout (e.g., ceiling, walls), the standard pixel-wise error metrics (e.g., $\ell_1$) are dominated by them and consequently insensitive to fine-grained changes. Thus, besides standard *Direct* metrics, we report *Perceptual* error metric (e.g., *normal$\shortrightarrow$curvature*) that evaluate the same prediction, but with a non-uniform attention to pixel properties.[^11] Each perceptual error provides a different angle, and the optimal results would have a low error for *all* metrics. The corresponding *Standard Error* for the reported numbers are provided in [supplementary material](http://consistency.epfl.ch/supplementary_material), which show the trends are statistically significant. Tab. \[table\_joint\] also includes evaluation of the networks when trained with little data (0.25% subset of Taskonomy dataset), which shows the consistency constraints are useful under low-data regime as well. We adopted normals as the canonical task for more extensive evaluations, due to its practical value and abundance of baselines. The conclusions remained the same regardless. **Using Consistency with Unsupervised Tasks**: Unsupervised tasks can provide consistency constraints, too. Examples of such tasks are 2D Edges and 2D Keypoints (SURF[@bay2006surf]), which are included in our dictionary. Such tasks have fixed operators that can be applied on any image to produce their respective domains without any additional supervision. Interestingly, we found enforcing consistency with these domains is still useful for gaining better results (see [supplementary material](http://consistency.epfl.ch/supplementary_material) for the experiment). The ability to utilize unsupervised tasks further extends the applicability of our method to single/few task datasets. Utilities of [[Consistency Energy]{}]{} {#sec:energy_results} --------------------------------------- Below we quantitatively analyze the Consistency Energy. The energy is shown (per-pixel) for sample queries in Fig. \[fig:qualitative\_accuracy\]. **Consistency Energy as a Confidence Metric (Energy vs Error):** Plot \[fig:correlation\] shows the energy of predictions has a strong positive correlation with the error computed using ground truth (Pearson corr. 0.67). This suggests the energy can be adopted for confidence quantification and handling uncertainty. This experiment was done on Taskonomy test set thus images had no domain shift from the training data. ![image](figures/baseline_vs_percep_mse.pdf){width=".56\columnwidth"} \[fig:ood\_quantative\] -------------- ----------- --------------- ---------- ------------- ---------- Novel Domain \# images Consistency Baseline Consistency Baseline 128 17.4 (+14.7%) 20.4 16 22.3 (+8.6%) 24.4 128 18.5 (+19.2%) 22.9 16 27.1 (+24.5%) 35.9 ApolloScape 8 40.5 (+11.9%) 46.0 -------------- ----------- --------------- ---------- ------------- ---------- \[table:ood\_quantative\] **Consistency Energy as a Domain Shift Detector:** Plot \[fig:id\_ood\_energy\_dist\] shows the energy distribution of in-distribution (Taskonomy) and out-of-distribution datasets (ApolloScape, CocoDoom). Out-of-distribution datapoints have notably higher energy values, which suggests that energy can be used to detect anomalous samples or domain shifts. Using the per-image energy value to detect out-of-distribution images achieved $\text{ROC-AUC}\text{=}0.95$; the out-of-distribution detection method OC-NN [@chalapathy2018anomaly] scored $0.51$. Plot \[fig:error\_energy\_blur\] shows the same concept as \[fig:id\_ood\_energy\_dist\] (energy vs domain shift), but when the shift away from the training data is smooth. The shift was done by applying a progressively stronger Gaussian blur with kernel size 6 on Taskonomy test images. The plot also shows the error computed using ground truth which has a pattern similar to the energy. We find the reported utilities noteworthy as handling uncertainty, domains shifts, and measuring prediction confidence in neutral networks are open topics of research [@ovadia2019trust; @guo2017calibration] with critical values in, e.g., active learning [@sener2017active], real-world decision making [@Kochenderfer:2015:DMU:2815660], and robotics [@proctor2017tolerances]. ![](figures/ood_qual_updated2.jpeg){width="1\columnwidth"} \[fig:ood\_qualitative\] Generalization & Adaptation to New Domains {#sec:results_generalization} ------------------------------------------ To study: **I.** how well the networks generalize to new domains without any adaptation and quantify their resilience, and **II.** how efficiently they can adapt to a new domain given a few training examples by fine-tuning, we test the networks trained on Taskonomy dataset on various new domains. The experiment were conducted on smooth (blurring [@jo2017measuring]) and discrete (Doom [@researchdoom], ApolloScape [@appoloscape]) shifts. For (**II**), we use a small number (16-128) of images from the new domain to fine-tune the networks with and without consistency constraints. The original training data (Taskonomy) is retained during fine-tuning so prevent the networks from forgetting the original domain [@learningWithoutForgetting]. Models trained with consistency constraints generally show more robustness against domain shifts (see Fig. \[fig:ood\_quantative\] and pre-adaptation numbers in Table \[table:ood\_quantative\]) and a better adaptation with little data (see post-adaptation numbers in Table \[table:ood\_quantative\] and Fig. \[fig:ood\_qualitative\]). The challenging external queries shown in Figures \[fig:qualitative\_accuracy\]&\[fig:normalqualitative\_accuracy\]&\[fig:motiv\] similarly denote a good generalization. **Supplementary Material:** We defer additional discussions and experiments, particularly analyzing different aspects of the optimization, stability analysis of the experimental trends, and proving qualitative results at scale to the [supplementary material](http://consistency.epfl.ch/supplementary_material) and the [project page](http://consistency.epfl.ch/). We presented a general and data-driven framework for augmenting standard supervised learning with cross-task consistency. The evaluations showed learning with cross-task consistency fits the data better yielding more accurate predictions and leads to models with improved generalization. The Consistency Energy was found to be an informative intrinsic quantity with utilities toward confidence estimation and domain shift detection. Below we briefly discuss some of the limitations and assumptions: **Path Ensembles**: We used the various inference paths only as a way of enforcing consistency. Aggregation of *multiple* (comparably weak) inference paths into a *single strong* estimator (e.g., in a manner similar to boosting) is a promising direction that this paper did not address. Performing the aggregation in a probabilistic manner seems viable, as we found the errors of different paths are sufficiently uncorrelated, suggesting possibility of assembling a strong estimator. **Unlabeled/Unpaired Data**: The current framework requires paired training data. Extending the concept to unlabeled/unpaired data, e.g., as in [@cycleGan17], appears feasible and remains open for future work. **Categorical/Low-Dimensional Tasks**: We primarily experimented with pixel-wise tasks. Classification tasks, and generally tasks with low-dimensional outputs, will be interesting to experiment with, especially given the more severely ill-posed cross-task relationships they induce. **Optimization Limits**: The improvements gained by incorporating consistency are bounded by the success of available optimization techniques, as addition of consistency constrains at times makes the optimization job harder. Also, implementing cross-task functions as neural networks makes them subject to certain **output artifacts** similar to those seen in image synthesis with neural networks. **Adversarial Robustness**: Lastly, if learning with cross-task consistency indeed reduces the tendency of neural networks to learn surface statistics [@jo2017measuring] (Sec. \[sec:intro\]), studying its implications in defence against adversarial attacks will be worthwhile. **Energy Analyses**: We performed post-hoc analyses on the Consistency Energy. More concrete understanding of the properties of the energy and potentially using it actively for network modification, e.g, in unsupervised domain adaptation, requires further focused studies. [^1]: Abbreviated [**X-TC**]{}, standing for **Cross**-**T**ask **C**onsistency.\ \*Equal. [^2]: For instance, it is not necessary to encode that surface normals are the 3D derivative of depth or occlusion edges are discontinuities in depth. [^3]: inference-path invariance was inspired by **Conservative Vector Fields** in vector calculus and physics that are (at a high level) fields in which integration along *different paths yield the same results, as long as their endpoints are the same* [@guillemin1974differential]. Many key concepts in physics are ‘conservative’, e.g., gravitational force: the work done against gravity when moving between two points is independent of the path taken. [^4]: Operator ${\circ}$ denotes function composition: $g{\circ}h(x){{\raisebox{-0.15\totalheight}{$\triangleq$}}}g(h(x))$. [^5]: Both sides of inequality \[eq:separableineqtriangle\] are ${\geq}0$ and $\text{=}0$ for the minimizer $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x)\text{=} {y}_1$ & $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}(x)\text{=} {y}_2$. [^6]: Both sides of inequality \[eq:percepinequality\] are ${\geq}0$ and ${=}0$ for the minimizer $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}(x){=} {y}_1$. The term $|f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}({y}_1) - {y}_2|$ is a constant and ${\sim}0$, as it is exactly the training objective of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$. The non-zero residual should be ignored and assumed 0 as the non-zero part is irrelevant to $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{X}\hspace{.5pt}}}{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1}$, but imperfections of $f_{{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_1{{\hspace{-1pt}\scriptscriptstyle \mathcal{Y}}}_2}$. [^7]: Generally for $n$ domains, this formulation allows using datasets of *pairs* among $n$ domains, rather than one *$n$-tuple* multi annotated dataset. [^8]: For example, a complete $\mathcal{G}$ with $n$ nodes includes $n(n-1)$ networks and $\sum_{k=2}^{L}{\binom{n}{k+1}}(k+1)!$ feasible paths, with path length capped at $L$. [^9]: \[seesupmat\]See [supplementary material](http://consistency.epfl.ch/supplementary_material) for an experimental comparison. [^10]: A further cheaper alternative is applying cross-task consistent learning only on $\mathcal{F}_x$ and training $\mathcal{F}_y$ using standard independent training. This is significantly cheaper and more convenient, but still improves $\mathcal{F}_x$ notably. [^11]: For example, evaluation of normals via the *normal$\shortrightarrow$curvature* metric is akin to paying more attention to where normals change, hence reducing the domination of flat regions, such as walls, in the numbers.
--- abstract: | Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed $v>1$. How large must $v$ be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve $\mbox{FF}_v$ that develops when the fighter keeps building, at speed $v$, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function $(e^{w Z} - s \, Z)^{-1}$, where $w$ and $s$ are real functions of $v$. For $v>v_c=2.6144 \ldots$ all zeroes are complex conjugate pairs. If $\phi$ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs $\Theta( 1/\phi)$ rounds before the fire is contained. As $v$ decreases towards $v_c$ these two zeroes merge into a real one, so that argument $\phi$ goes to 0. Thus, curve $\mbox{FF}_v$ does not contain the fire if the fighter moves at speed $v=v_c$. (That speed $v>v_c$ is sufficient for containing the fire has been proposed before by Bressan et al. [@bbfj-bsfcp-08], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that any curve that visits the four coordinate half-axes in cyclic order, and in inreasing distances from the origin, needs speed $v>1.618\ldots$, the golden ratio, in order to contain the fire. [**Keywords:**]{} Motion Planning, Dynamic Environments, Spiralling strategies, Lower and upper bounds author: - Rolf Klein$^1$ - Elmar Langetepe$^1$ - Christos Levcopoulos$^2$ title: 'On a Fire Fighter’s Problem ' --- Introduction ============ Fighting wildfires and epidemics has become a serious issue in the last decades. Professional fire fighters need models and simulation tools on which strategic decisions can be based; for example see [@faoun-ihffp]. Thus, a good understanding of the theoretical foundations seems necessary. In theoretical computer science, substantial work has been done on the fire fighting problem in graphs; see, e.g., the survey article [@fm-fpsr-09]. Here, initially one vertex is on fire. Then an immobile firefighter can be placed at one of the other vertices. Next, the fire spreads to each adjacent vertex that is not defended by a fighter, and so on. The game continues until the fire cannot spread anymore. The objective, to save a maximum number of vertices from the fire, is NP-hard to achieve, even for trees of degree three; see [@fkmr-fpgmd-07]. Optimal strategies are known for special graphs, i.e., for grid graphs [@wm-fcg-02]. The problem can also be interpreted as an intruder search game. The total extension of the fire in the graph represents the current possible location of the intruder. Some algorithms and lower bounds have been given for the problem of finding an intruder in special graphs; see[@bffs-cima-02; @bggk-hmlnc-09; @bkns-eosdi-07]. A more geometric setting has been studied in [@kll-aagfb-14]. Suppose that inside a simple polygon $P$ a candidate set of pairwise disjoint diagonal barriers has been defined. If a fire starts at some point inside $P$ one wants to build a subset of these barriers in order to save a maximum area from the fire. But each point on a barrier must be built before the fire arrives there. This problem is a special case of a hybrid scheduling and coverage problem, for which an 11.65 approximation algorithm exists. Bressan et al. [@b-dicff-07; @b-dbpmf-12; @bbfj-bsfcp-08; @bdl-eosfc-09; @bw-msbhp-09; @bw-efnaf-10; @bw-gocdb-12; @bw-osibp-12] introduced a purely geometric model where a fire spreads in the plane and one or more fire fighters are tasked to block it by building barriers. They cover a wide range of possible scenarios, including fires whose shapes can change under the influence of wind, and provide a wealth of results, among them upper and lower bounds for the speed the fire fighter(s) need, and existence theorems for solutions that contain the fire and minimize the area burned. A very interesting case has been studied in Bressan et al. [@bbfj-bsfcp-08]. A circular fire centered at the origin spreads at unit speed, and a single fire fighter can build a barrier at speed $v$. Their construction is based on the following observation. Let $P$ be a point on the logarithmic spiral $S^\alpha =(\varphi, e^{\varphi \cot\alpha})$ of excentricity $\alpha$, and let $Q$ denote the next point on $S^\alpha$ touched by a tangent at $P$; see Figure \[logspiral-fig\], (i). ![(i) A logarithmic spiral $S_\alpha$ of excentricity $\alpha$; angle $\varphi$ ranges from $-\infty$ to $\infty$. (ii) Curve $\mbox{FF}_v$ results when the fighter moves at speed $v$ along the fire’s expanding boundary.[]{data-label="logspiral-fig"}](Fig1Part1.png "fig:")![(i) A logarithmic spiral $S_\alpha$ of excentricity $\alpha$; angle $\varphi$ ranges from $-\infty$ to $\infty$. (ii) Curve $\mbox{FF}_v$ results when the fighter moves at speed $v$ along the fire’s expanding boundary.[]{data-label="logspiral-fig"}](Fig1Part2.png "fig:") Then the spiral’s length to $Q$ is at most $2.6144\ldots$ times the sum of its length to $P$ plus the length $|PQ|$ of the tangent, for all values of $\alpha$. In other words, if the fighter builds such a barrier at some speed $v>v_c:=2.6144\ldots$ she will always reach $Q$ before the fire does, which crawls around the spiral’s outside to point $P$ and then runs straight to $Q$. In [@bbfj-bsfcp-08] the fighter uses this leeway to build a sequence of logarithmic spiral segments of increasing excentricities $\alpha_i$ that stay away from the fire, plus one final line segment that closes this barrier curve onto itself. Logarithmic spiral movements have also been used for shortest paths amidst growing circles [@vo-psspa-06; @mnrj-oacsp-07] and in the context of search games [@l-ooss-10]. In this paper we study the rather natural approach where the fire fighter keeps building a barrier right along the boundary of the expanding fire, at constant speed $v$. Let $\mbox{FF}_v$ denote the resulting barrier curve. At each point $Q$ both fighter and fire arrive simultaneously, by definition. As we shall see below, tangents form a constant angle $\alpha = \cos^{-1}(1/v)$ with the curve; see Figure \[logspiral-fig\], (ii). While the fighter keeps building $\mbox{FF}_v$, the fire is coming after her along the outside of the barrier, as shown in Figure \[fire-fig\]. Intuitively, the fighter can only win this race, and contain the fire, if the last coil of the barrier hits the previous coil. ![The race between the fire and the fighter. When the fighter arrives at point $p_2$, having constructed a barrier from $p_0$ to $p_2$, the fire has expanded along the outer side of the barrier up to point $q$. []{data-label="fire-fig"}](FireRace4.png) This is equivalent to saying that the length, $F$, of the tangent from the fighter’s current position $Q$ back to point $P$ becomes zero. First, we deduce two structural properties of barrier curve $\mbox{FF}_v$. They lead to a recursive system of linear differential equations that allow us to describe the values of $F$ in the $i$-th round. It turns out that we need only check the signs of the values $F_i$ at the end of round $i$, in order to see if the fighter is successful. Therefore, we look at the generating function $F(Z) = \sum_{i=0}^\infty {F_i Z^i}$ and obtain, from the recursions, the equation $$\begin{aligned} \frac{F(Z)}{F_0} \ = \ \frac{ e^{v Z} \ - \ r \, Z}{ e^{w Z} \ - \ s \, Z} \label{equa}\end{aligned}$$ where $v, r, w, s$ are real functions of speed $v$. Singularities can only arise from zeroes of the denominator that do not cancel out with the numerator. It turns out that for $v > v_c = 2.6144\ldots$ only conjugate pairs of complex zeroes of the denominator exist [@f-ansdd-95]. A theorem of Pringsheim’s directly implies that not all coefficients $F_i$ of power series $F(Z)$ can be positive, showing that barrier curve $\mbox{FF}_v$ does close on itself at some time. To find out after how many rounds this happens we look at the conjugate pair of zeroes of smallest modulus and let $\phi_v$ denote their (positive) argument. Residue analysis shows that $\Theta(1/\phi_v)$ rounds are necessary before the fire is contained. As speed $v$ decreases towards $v_c$, the two conjugate zeroes merge into a real zero. Therefore, argument $\phi_v$ tends to 0, proving that the fire fighter cannot succeed at speed $v=v_c$. In addition to the results on curve $\mbox{FF}_v$, we obtain the following lower bound. Let us call a curve “spiralling” if it visits the four coordinate half-axes in cyclic order, and at increasing distances from the origin. (Note that curve $\mbox{FF}_v$ is spiralling even though the fighter’s distance to the origin may be decreasing: the barrier’s intersection points with any ray from $0$ are of increasing order since the curve does not self-intersect.) We prove that a fire fighter who follows such a spiralling curve can only be successful if her speed exceeds $\frac{1+\sqrt{5}}{2} \, \approx \, 1.618$, the golden ratio. Acknowledgement =============== A preliminary version of part of this paper has appeared at SoCG’15 [@kll-ffp-15]. We thank all anonymous referees for their valuable suggestions and, in particular, for pointing out to us the work by Bressan et al. [@b-dicff-07; @b-dbpmf-12; @bbfj-bsfcp-08; @bdl-eosfc-09; @bw-msbhp-09; @bw-efnaf-10; @bw-gocdb-12; @bw-osibp-12]. The barrier curve $\mbox{FF}_v$ {#FF-sec} =============================== The first rounds ---------------- Let $p$ be a point on the barrier curve’s first round, as depicted in Figure \[Spirals-fig\]. If $\alpha$ denotes the angle between the fighter’s velocity vector at $p$ and the ray from 0 through $p$, the fighter moves at speed $v \cos\alpha$ away from 0. This implies $v \cos\alpha =1$, because the fire expands at unit speed and the fighter stays on its frontier, by definition. Since the fighter is operating at constant speed $v$, angle $\alpha$ is constant, and given by $\alpha = \cos^{-1}(1/v)$. Consequently, the first part of the barrier curve, between points $p_0$ and $p_1$ shown in Figure \[Spirals-fig\], (i), is part of a logarithmic spiral of excentricity $\alpha$ centered at 0. ![The barrier curve starts with two parts of logarithmic spirals of excentricity $\alpha$, centered at 0 and $p_0$, respectively.[]{data-label="Spirals-fig"}](Part1Fig3.png "fig:")![The barrier curve starts with two parts of logarithmic spirals of excentricity $\alpha$, centered at 0 and $p_0$, respectively.[]{data-label="Spirals-fig"}](Part2Fig3.png "fig:") In polar coordinates, this segment can be desribed by $(\varphi, A\cdot e^{\varphi\cot\alpha})$, where $\varphi\in[0,2\pi]$, and $A$ denotes the distance from the origin to $p_0$, i.e., the fire’s intitial radius. In general, the curve length of a logarithmic spiral of excentricity $\alpha$ between two points at distance $d_1<d_2$ to its center is known to be $\frac{1}{\cos\alpha}\left(d_2-d_1\right)$. Thus, we have for the length $l_1$ of the barrier curve from $p_0$ to $p_1$ the equation $$\begin{aligned} l_1 \, = \, \frac{A}{\cos\alpha} \cdot (e^{2\pi \cot\alpha} - 1). \label{l1-eq}\end{aligned}$$ From point $p_1$ on, the geodesic shortest path, along which the fire spreads from 0 to the fighter’s current position, $p$, is no longer straight. It starts with segment $0 p_0$, followed by segment $p_0 p$, until, for $p=p_2$, segment $p_0p$ becomes tangent to the barrier curve at $p_0$; see Figure \[Spirals-fig\], (ii). By the same arguments as above, between $p_1$ and $p_2$ barrier curve $\mbox{FF}_v$ is also part of a logarithmic spiral of excentricity $\alpha$, but now centered at $p_0$. This spiral segment starts at $p_1$ at distance $A'=A(e^{2\pi \cot\alpha}-1)$ from its center $p_0$. Since $p_2$ and $p_1$ form an angle $\alpha$ at $p_0$, the distance from $p_2$ to $p_0$ equals $A' e^{\alpha \cot\alpha}$. Thus, the curve length from $p_1$ to $p_2$ is given by $l_2'=\frac{A'}{\cos\alpha} (e^{\alpha \cot\alpha}-1)= \frac{A}{\cos\alpha} (e^{2\pi \cot\alpha} -1) (e^{\alpha \cot\alpha}-1)$. Consequently, the overall curve length $l_2$ from $p_0$ to $p_2$ equals $$\begin{aligned} l_2 \, = \, l_1 + l'_2 \, = \, \frac{A}{\cos\alpha} (e^{2\pi \cot\alpha} -1) e^{\alpha \cot\alpha}. \label{l2-eq}\end{aligned}$$ ![From point $p_2$ on the barrier curve results from wrapping around the barrier already constructed. The last segment, *free string* $F$, of the shortest path from the fire source to the current barrier point $p$ shrinks, by wrapping, and simultaneously grows by $\cos\alpha$. The fighter will be successful if, and only if, $F$ ever shrinks to zero.[]{data-label="String-fig"}](Part3Fig4.png) From point $p_2$ on, the geodesic shortest path from 0 to the fighter’s current position, $p$, starts wrapping around the existing spiral part of the curve, beginning at $p_0$; see Figure \[String-fig\]. The last segment of this path is tangent to the previous round of the curve. As mentioned in the Introduction, we shall endeavor to determine its length, $F$, because the fire will be contained if and only if $F$ ever attains the value 0. One could think of this tangent as a string (named the [*free string*]{}) at whose endpoint, $p$, a pencil is attached that draws the barrier curve. But unlike an involute, here the string is not normal to the outer layer. Rather, its extension beyond $p$ forms an angle $\alpha$ with the barrier’s tangent at $p$. This causes the string to grow in length by $\cos\alpha$ for each unit drawn. At the same time, the inner part of the string gets wrapped around the previous coil of the barrier. It is this interplay between growing and wrapping we need to analyze. One can show that after $p_2$ no segment of positive length of $\mbox{FF}_v$ is part of a logarithmic spiral. Structural properties {#struct-subsec} --------------------- In this subsection we assume that the fighter has built quite a few rounds of the barrier curve without yet containing the fire. That the first two rounds of the curve involve two different spiral segments, around $0$ and around $p_0$, influences the subsequent layers. The structure of the curve can be described as follows. Let $l_1$ and $l_2$ denote the curve lengths from $p_0$ to $p_1$ and $p_2$, respectively, as in Equations \[l1-eq\] and \[l2-eq\]. For $l \in [0,l_1]$ let $F_0(l)$ denote the segment connecting $0$ to the point of curve length $l$; see the sketch given in Figure \[linkage-fig\]. ![Repeatedly constructing backwards tangents may end in 0 or in $p_0$. This way, two types of linkages are defined.[]{data-label="linkage-fig"}](linkage.pdf) At the endpoint of $F_0(l)$ we construct the tangent and extend it until it hits the next layer of the curve, creating a segment $F_1(l)$, and so on. This construction gives rise to a “linkage” connecting adjacent layers of the curve. Each edge of the linkage is turned counterclockwise by $\alpha$ with respect to its predecessor. The outermost edge of a linkage is the free string mentioned above. As parameter $l$ increases from $0$ to $l_1$, edge $F_0(l)$, and the whole linkage, rotate counterclockwise. While $F_0(0)$ equals the line segment from the origin to $p_0$, edge $F_0(l_1)$ equals segment $0p_1$. Analogously, let $l' \in [l_1, l_2]$, and let $\phi_0(l')$ denote the segment from $p_0$ to the point at curve length $l'$ from $p_0$. This segment can be extended into a linkage in the same way. We observe that $$\begin{aligned} F_{j+1}(l_1) &=& \phi_{j+1}(l_1) \label{l1} \\ F_{j+1}(0) &= &\phi_{j}(l_2) \label{l2} \end{aligned}$$ hold (but initially, we have $F_0(l) = A + \cos(\alpha) \, l$ and $\phi_0(l') = \cos(\alpha) \, l'$, so that $F_0(l_1)\not= \phi_0 ( l_1)$). Clearly, each point on the curve can be reached by a unique linkage, as tangents can be constructed backwards. We refer to the two types of linkages by $F$-type and $\phi$-type. As Figure \[linkage-fig\] illustrates, points of the same linkage type form alternating intervals along the barrier curve. If $p$’s linkage is of $F$-type then $p$ is uniquely determined by the index $j \geq 0$ and parameter $l \in [0,l_1] $ such that $p$ is the outer endpoint of edge $F_j(l)$. Now we will derive two structural properties of $F$-linkages on which our analysis will be based; analogous facts hold for $\phi$-linkages, too. To this end, let $L_j(l)$ denote the length of the barrier curve from $p_0$ to the outer endpoint of edge $F_j(l)$, and let $F_j(l)$ also denote the length of edge $F_j(l)$. \[grow-lem\] We have $L_{j-1}(l) + F_j(l) = \cos\alpha \, L_j(l)$. Both, fire and fire fighter, reach the endpoint of $F_j(l)$ at the same time. The fire has travelled a geodesic distance of $L_{j-1}(l) + F_j(l)$ at unit speed, the fighter a distance of $L_j(l)$ at speed $1/\cos\alpha$. The second property is related to the wrapping of the free string. Intuitively, it says that if we turn an $F$-linkage, the speed of each edge’s endpoint is proportional to its length. \[links-lem\] As functions in $l$, $L_j$ and $F_j$ satisfy the following equation. $$\frac{L'_{j-1}(l)}{L'_{j}(l)} \ = \ \frac{F_{j-1}(l)}{F_{j}(l)}.$$ We will derive Lemma \[links-lem\] from two general facts on smooth curves stated in Lemma \[wrap-lem\] and Lemma \[turn-lem\]. \[wrap-lem\] Suppose a string of length $F$ is tangent to a point $t$ on some smooth curve $C$. Now the end of the string moves a distance of $\epsilon$ in the direction of $\alpha$, as shown in Figure \[wrapApp-fig\]. Then for the curve length ${C^{t_\epsilon}_t}$ between $t$ and the new tangent point, $t_\epsilon$, we have $$\lim_{\epsilon \to 0} \ \frac{{C^{t_\epsilon}_t}}{\epsilon} \ = \ \frac{\sin\alpha \ r}{F}$$ where $r$ denotes the radius of the osculating circle at $t$. This fact is quite intuitive. The more perpendicular the motion of the string’s endpoint, and the larger the radius of curvature, the more of the string gets wrapped. But if the string is very long, the effect of the motion decreases. ![A string wrapping around a curve.[]{data-label="wrapApp-fig"}](wrap2.pdf) The center of the osculating circle at $t$ is known to be the limit of the intersections of the normals of all points near $t$ with the normal at $t$. Lemma \[turn-lem\] shows what happens if, instead of the normals, we consider the lines turned by the angle $\pi/2-\alpha$. \[turn-lem\] Let $t$ be a point on a smooth curve $C$, whose osculating circle at $t$ is of radius $r$. Consider the lines $L_{s}$ resulting from turning the normal at points $s$ by an angle of $\pi/2-\alpha$. Then their limit intersection point with $L_{t}$ has distance $\sin\alpha \, r$ to $t$. A simple example is shown in Figure \[wrap-fig\] for the case where curve $C$ itself is a circle. Now we can prove Lemma \[links-lem\]. ![Intersection of turned normals.[]{data-label="wrap-fig"}](wrap4.pdf) By Lemma \[wrap-lem\], applied to the innermost point $t$ of edge $F_j(L_j)$, we have $$\frac{L'_{j-1}(L_j)}{L'_{j}(L_j)} \, = \, L'_{j-1}(L_j) \, = \, \frac{\sin\alpha \ r}{F_j(L_j)}.$$ Lemma \[turn-lem\] implies that $\sin\alpha \ r$ equals the distance between $t$ and the limit intersection point of the normals turned by $\pi/2 - \alpha$ near $t$. But for the barrier curve $\mbox{FF}_v$, these turned normals are the tangents to the previous coil, so that $\sin\alpha \ r = F_{j-1}(L_j)$ holds. As we substitute variable $L_j$ with $L_j(l)$, the derivatives of the inner functions cancel out and we obtain Lemma \[links-lem\]. Using the notations in Figure \[wrapApp-fig\], the following hold. From $r \, \sin(\phi/2) = s = a \, \cos(\phi/2)$ we obtain $a = r \, \tan(\phi/2)$. For short, let $c:=C^{t_\epsilon}_t.$ By l’Hospital’s rule, $$\frac{c}{2a} \ = \ \frac{r \, \phi}{2r \, \tan(\phi/2)} \ \approx \ \cos^2(\phi/2) \ \to \ 1$$ as $\epsilon$, hence $\phi$, go to $0$. Thus, $2a$ is a good approximation of $c={C^{t_\epsilon}_t}$. By the law of sines, $$\frac{\epsilon \, \sin(\alpha)}{\sin(\phi)} \ = \ \frac{F_\epsilon + a}{\sin(\pi/2)},$$ hence $$\frac{\sin(\phi)}{\epsilon} \ = \ \frac{\sin(\alpha)}{F_\epsilon + a} \ \to \ \frac{\sin(\alpha)}{F}$$ This implies $\sin(\phi/2)/\epsilon \ \to \ \sin(\alpha)/(2F)$, and we conclude $$\begin{aligned} \frac{C^{t_\epsilon}_t}{\epsilon} \ &=& \ \frac{c}{2a} \, \frac{2a}{\epsilon} \ \approx \ \frac{2r \, \tan(\phi/2)}{\epsilon} = \frac{2r \, \sin(\phi/2)}{\epsilon \, \cos(\phi/2)} \ \to \ \frac{r \sin(\alpha)}{F}.\end{aligned}$$  Let us assume that $C$ is locally parameterized by $Y=f(X)$ and that $t=(x_0,f(x_0))$. Then the tangent in $t$ is $$Y \ = \ f'(x_0) X \ - \ f'(x_0) x_0 \ + \ f(x_0),$$ and line $L_t$, the tangent turned counterclockwise by $\alpha$, is given by $$Y \ = \ \tan(\arctan(f'(x_0)) + \alpha) X \ - \ \tan(\arctan(f'(x_0)) + \alpha) x_0 \ + \ f(x_0).$$ Now let $(v,w)$ denote the point of intersection of $L_t$ and $L_s$, where $s=(x_0+\epsilon, f(x_0 + \epsilon))$. Equating the two line equations we obtain $$\big( h(x_0 +\epsilon) - h(x_0) \big) \, v \ = \ g(x_0 +\epsilon) - g(x_0) + f(x_0) - f(x_0+\epsilon)$$ where $$h(x) \, := \, \tan(\arctan(f'(x)) + \alpha) \ \mbox{ and } \ g(x) \, := \, h(x) x$$ After dividing by $\epsilon$ and taking limits, we have $$h'(x_0) \, v_0 = g'(x_0) - f'(x_0) = h'(x_0) \, x_0 + h(x_0) - f'(x_0),$$ which results in $$\begin{aligned} v_0 \ &=& \ x_0 \, + \, \frac{h(x_0)-f'(x_0)}{h'(x_0)} \\ w_0 \ &=& \ h(x_0) \, v_0 -g(x_0) + f(x_0) \\ &=& f(x_0) \, + \, \frac{h^2(x_0) - h(x_0) f'(x_0)}{h'(x_0)}.\end{aligned}$$ In other words, $$\begin{aligned} (v_0,w_0) - (x_0, f(x_0)) \ &=& \ \frac{h(x_0) - f'(x_0)}{h'(x_0)} \, (1, h(x_0)) \\ |(v_0,w_0) - (x_0, f(x_0))| \ &=& \ | \frac{h(x_0) - f'(x_0)}{h'(x_0)} | \, \sqrt{1+h^2(x_0)}\end{aligned}$$ Using the addition formula for $\tan$, $$h(x) \ = \ \tan(\arctan(f'(x)) + \alpha) \ = \ \frac{f'(x) + \tan(\alpha)}{1- f'(x) \tan(\alpha)},$$ we obtain $$h(x_0) - f'(x_0) \ = \ \frac{1 + (f'(x_0))^2 + \tan(\alpha)}{1- f'(x_0) \tan(\alpha)}.$$ and $$1+h^2(x_0) \ = \ \frac{( 1+(f'(x_0))^2) \, (1+\tan^2(\alpha))}{(1- f'(x_0) \tan(\alpha))^2}.$$ Moreover, $$h'(x_0) \ = \ \frac{f''(x)\, (1+\tan^2(\alpha)) }{(1- f'(x_0) \tan(\alpha))^2}.$$ Putting expressions together we obtain $$|(v_0,w_0) - (x_0, f(x_0))| \ = \ |\frac{\big( 1+(f'(x_0))^2 \big)^{3/2}}{f''(x_0)}| \ \frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}.$$ The first term is known to be the radius of the osculating circle, $r$, and the second equals $\sin(\alpha)$. Recursive differential equations {#recdgl-sec} ================================ In this section we turn the structural properties observed in Subsection \[struct-subsec\] into differential equations. By multiplication, Lemma \[links-lem\] generalizes to non-consecutive edges. Thus, $$\begin{aligned} \frac{F_j(l)}{F_0(l)} \ = \ \frac{L'_j(l)}{l'} \ = \ L'_j(l) \label{quots}\end{aligned}$$ holds. On the other hand, taking the derivative of the formula in Lemma \[grow-lem\] leads to $$\begin{aligned} F'_j(l) \ + \ L'_{j-1}(l) \ = \ \cos\alpha \, L'_j(l) \label{deriv}.\end{aligned}$$ We substitute in \[deriv\] both $L'_j(l)$ and $L'_{j-1}(l)$ by the expressions we get from \[quots\] and obtain a linear differential equation for $F_j(l)$, $$\begin{aligned} F'_j(l) \ - \ \frac{\cos(\alpha)}{F_0(l)} \, F_j(l) \ = \ - \, \frac{F_{j-1}(l)}{F_0(l)}\,. \end{aligned}$$ The solution of $y'(x) + f(x)y(x) = g(x)$ is $$y(x)=\exp(-a(x)) \left(\int{g(t)\exp(a(t))} \, \mathrm{d}t + \kappa \right),$$ where $a = \int f$ and $\kappa$ denotes a constant that can be chosen arbitrarily. In our case, $$a(l) \, = \, \int -\frac{ \cos(\alpha) }{A + \cos(\alpha) \, l} \, = \, - \ln(F_0(l))$$ because of $F_0(l) = A + \cos(\alpha) \, l$, and we obtain $$\begin{aligned} F_j(l) \ = \ F_0(l) \Big( \kappa_j \ - \ \int{ \frac{ F_{j-1}(t) } {F^2_0(t)} \, \mathrm{d}t } \Big). \label{Fdgl}\end{aligned}$$ Next, we consider a linkage of $\phi$-type, for parameter $l \in [l_1, l_2]$, and obtain analogously $$\begin{aligned} \phi_j(l) \ = \ \phi_0(l) \Big( \lambda_j \ - \ \int{ \frac{ \phi_{j-1}(t) } {\phi^2_0(t)} \, \mathrm{d}t } \Big). \label{phidgl}\end{aligned}$$ Now we determine the constants $\kappa_j, \lambda_j$ such that the solutions \[Fdgl\] and \[phidgl\] describe a contiguous curve. To this end, we must satisfy conditions \[l1\] and \[l2\]. We define $\kappa_0 := 1$ and $$\kappa_{j+1} := \frac{\phi_j(l_2)}{F_0(0)} \ + \ \int{ \frac{F_j(t)}{F^2_0(t)} \mathrm{d}t }|_{l=0}$$ so that \[Fdgl\] becomes $$\begin{aligned} F_{j+1}(l) \ = \ F_0(l) \, \Big( \frac{\phi_j(l_2)}{F_0(0)} \, - \, \int_{0}^l { \frac{ F_{j}(t) }{F^2_0(t)} } \, \mathrm{d}t \Big), \label{FRec}\end{aligned}$$ which, for $l=0$, yields $F_{j+1}(0) = \phi_j(l_2)$ (satisfying condition \[l2\]). Similarly, we set $\lambda_0 :=1$ and $$\lambda_{j+1} := \frac{F_{j+1}(l_1)}{\phi_0(l_1)} \ + \ \int{ \frac{\phi_j(t)}{\phi^2_0(t)} \mathrm{d}t }|_{l=l_1}$$ so that \[phidgl\] becomes $$\begin{aligned} \phi_{j+1}(l) \ = \ \phi_0(l) \, \Big( \frac{F_{j+1}(l_1)}{\phi_0(l_1)} \, - \, \int_{l_1}^l { \frac{ \phi_{j}(t) }{\phi^2_0(t)} } \, \mathrm{d}t \Big), \label{phiRec}\end{aligned}$$ and for $l=l_1$ we get $F_{j+1}(l_1) = \phi_{j+1}(l_1)$ (satisfying condition \[l1\]). For simplicity, let us write $$\begin{aligned} G_j(l) \, := \, \frac{F_j(l)}{F_0(l)} \ \mbox{ and } \ \chi_j(l) \, := \, \frac{\phi_j(l)}{\phi_0(l)}, \label{subst1}\end{aligned}$$ which leads to $$\begin{aligned} G_{j+1}(l) \ &=& \ \frac{\phi_0(l_2)}{F_0(0)} \, \chi_j(l_2) \ - \ \int_{0}^l { \frac{ G_{j}(t) }{F_0(t)} } \, \mathrm{d}t \label{G1} \\ \chi_{j+1}(l) \ &=& \ \frac{F_0(l_1)}{\phi_0(l_1)} \, G_{j+1}(l_1) \ - \ \int_{l_1}^l { \frac{ \chi_{j}(t) }{\phi_0(t)} } \, \mathrm{d}t \label{chi1}. \end{aligned}$$ The integrals in \[G1\] and \[chi1\] are increasing in $l$ provided that $G_j(t)>0$ and $\chi_j(t)>0$ hold for all $t$. This leads to a useful observation. In order to find out if the fire fighter is successful, (that is, if there exists an index $j$ such that $F_{j}(l) = 0$ holds for some $l \in [0,l_1]$, or $\phi_j(l)=0$ for some $l \in [l_1,l_2]$), we need to check only the values $F_j(l_1)$ at the end of each round. \[suff1-lem\] The curve encloses the fire if and only if there exists an index $j$ such that $F_{j}(l_1) \leq 0$ holds. Clearly, $G_j$ and $F_j$ have identical signs, as well as $\chi_j$ and $\phi_j$ do. Suppose that $G_j >0$ and $G_{j+1}(l) =0$, for some $j$ and some $l \in [0,l_1]$. By \[G1\], function $G_{j+1}$ is decreasing, therefore $G_{j+1}(l_1) \leq 0$. Now assume that $G_i >0$ holds for all $i$, and that we have $\chi_{j-1} >0$ and $\chi_{j}(l) =0$ for some $j$ and some $l \in [l_1,l_2]$. By \[chi1\] this implies $\chi_{j}(l_2) \leq 0$, and from \[G1\] we conclude $G_{j+1} \leq 0$, in particular $G_{j+1}(l_1) \leq 0$. Next, we make the integrals in \[G1\] and \[chi1\] disappear by iterated substitution, and replace variable $l$ with the concrete values of $l_1$ (resp. $l_2$) given in \[l1-eq\] and \[l2-eq\]. In equation \[G1\] iterated substitution yields $$\begin{aligned} G_{j+1}(l) \ = \ \frac{\phi_0(l_2)}{F_0(0)} \, \sum_{\nu = 0}^j (-1)^\nu \, I_\nu(l) \, \chi_{j-\nu}(l_2) \, + \, (-1)^{j+1} I_{j+1}(l) \label{Gl}\end{aligned}$$ where $$I_n(x_n) \ = \ \int_{0}^{x_n} \frac{1}{F_0(x_{n-1})} \int_0^{x_{n-1}} \frac{1}{F_0(x_{n-2})} \ldots \int_0^{x_1} \frac{1}{F_0(x_0)} \, \mathrm{d}x_0 \ldots \, \mathrm{d}x_{n-1}.$$ By induction on $n$ we derive $$I_n(x_n) \ = \ \frac{1}{n!} \, \frac{1}{\cos^n\alpha} \, \big(\ln (\frac{A+\cos(\alpha) x_n}{A})\big)^n$$ since $F_0(x)=A+\cos(\alpha) \, x$. By definition of $l_1$, we have $\ln (\frac{A+\cos(\alpha) l_1}{A}) = 2 \pi \cot\alpha$, so that setting $l=l_1$ in formula \[Gl\] leads to $$\begin{aligned} G_{j}(l_1) \ = \ \frac{\phi_0(l_2)}{F_0(0)} \, \sum_{\nu = 0}^j{ \frac{(-1)^\nu}{\nu!} \, \big( \frac{2 \pi}{\sin\alpha} \big)^\nu \, \chi_{j-1-\nu}(l_2)}\end{aligned}$$ where, for convenience, $\chi_{-1}(l_2):=\frac{F_0(0)}{\phi_0(l_2)}$. We observe that this formula is also true for $j=0$. Multiplying both sides by $F_0(l_1)$, and re-substituting \[subst1\], results in $$\begin{aligned} F_{j}(l_1) \ = \ \frac{F_0(l_1)}{F_0(0)} \, \sum_{\nu = 0}^j{ \frac{(-1)^\nu}{\nu!} \, \big( \frac{2 \pi}{\sin\alpha} \big)^\nu \, \phi_{j-1-\nu}(l_2)} \label{finF}\end{aligned}$$ where $\phi_{-1}(l_2)=F_0(0)$. In a similar way we solve the recursion in \[chi1\], using $$\int_{0}^{x_n} \frac{1}{\phi_0(x_{n-1})} \int_0^{x_{n-1}} \ldots \int_0^{x_1} \frac{1}{\phi_0(x_0)} \, \mathrm{d}x_0 \ldots \, \mathrm{d}x_{n-1} \ = \ \frac{1}{n!} \, \frac{1}{\cos^n\alpha} \, \big(\ln (\frac{ x_n}{l_1})\big)^n$$ and $\ln(\frac{l_2}{l_1}) = \alpha \cot\alpha$. One obtains, after substituting $l=l_2$, $$\begin{aligned} \phi_{j}(l_2) \ = \ \frac{\phi_0(l_2)}{\phi_0(l_1)} \, \sum_{\nu = 0}^j{ \frac{(-1)^\nu}{\nu!} \, \big( \frac{\alpha}{\sin\alpha} \big)^\nu \, \hat{F}_{j-\nu}(l_1)} \label{finphi}\end{aligned}$$ where $\hat{F}_{0}(l_1):=\phi_0(l_1)$ and $\hat{F}_{i+1}(l_1):=F_{i+1}(l_1)$. Generating functions and singularities ====================================== The cross-wise recursions \[finF\] and \[finphi\] are convolutions. In order to solve them for the numbers $F_j(l_1)$ we are interested in, we define the generating functions $$F(Z) := \sum_{j=0}^\infty F_j \, Z^j\ \mbox{ and } \ \phi(Z) := \sum_{j=0}^\infty \phi_j \, Z^j$$ where $F_j:=F_j(l_1)$ and $\phi_j:= \phi_{j}(l_2)$, for short. From \[finF\] we obtain $$\begin{aligned} F(Z) \ = \ \frac{F_0}{F_0(0)} \, e^{-\frac{2 \pi}{\sin\alpha} Z} \, \big( Z \, \phi(Z) + F_0(0) \big), \label{Fsum1}\end{aligned}$$ and from \[finphi\], $$\begin{aligned} \phi(Z) \ = \ \frac{\phi_0}{\phi_0(l_1)} \, e^{-\frac{\alpha}{\sin\alpha} Z} \, \big( Z \, F(Z) - F_0 + \phi_0(l_1) \big) \label{phisum1}.\end{aligned}$$ Both equalities can be verified by plugging in expansions of the exponential functions, using $e^W=\sum_{j=0}^\infty {\frac{W^j}{j!}}$, computing the products, and comparing coefficients. Now we substitute \[phisum1\] into \[Fsum1\], solve for $F(Z)$, divide both sides by $F_0$ and expand by $e^{\frac{2 \pi + \alpha}{\sin\alpha}}$ to obtain $$\begin{aligned} \frac{F(Z)}{F_0} \ = \ \frac{ e^{v Z} \ - \ r \, Z}{ e^{w Z} \ - \ s \, Z}, \label{efunc}\end{aligned}$$ where $v, r, w, s$ are the following functions of $\alpha = \cos^{-1}(1/v)$: $$\begin{aligned} v \ &=& \ \frac{\alpha}{\sin\alpha} \ \ \ \ \, \mbox{ and } \ r \ = \ e^{\alpha \cot\alpha}\nonumber \\ w \ &=& \ \frac{2 \pi +\alpha}{\sin\alpha} \ \ \mbox{ and } \ s \ = \ e^{(2 \pi + \alpha) \cot\alpha}. \label{sw}\end{aligned}$$ Singularities of $F(Z)$ can arise only from zeroes of the denominator, $$\begin{aligned} \label{denominator} e^{w Z} - s Z. \end{aligned}$$ As the fighter’s speed $v$ increases from 1 to $\infty$, angle $\alpha=\cos^{-1}(1/v)$ of the fighter’s velocity vector increases from 0 to $\pi/2$, causing $s/w$ to decrease from $\infty$ to 0. Precisely at $v=v_c=2.6144\ldots$ does $s/w=e$ hold; then $1/w$ is a real root of \[denominator\], as direct computation shows. For $v>v_c$ we have $s/w < e$. The following lemma can be inferred from Falbo [@f-ansdd-95]. A complete proof is given in Subsection \[Falbo-subsec\] below. \[FalboApp-lem\] For $v>v_c$ function $F(Z)$ has infinitely many discrete conjugate complex poles of order 1. The pair $z_0, \overline{z_0}$ nearest to the origin have absolut values $< 0.31$, all other poles have moduli $\geq 1$. As $v$ decreases to $v_c$, poles $z_0, \overline{z_0}$ converge towards a real pole $1/w \approx 0.124$. Now we directly obtain the following. \[qual-theo\] At speed $v>v_c=2.6144\ldots$ curve $\mbox{FF}_v$ contains the fire. From Lemma \[FalboApp-lem\] we know that $F(Z)$ has a radius of convergence $R$ in $(0,0.31)$. If the fighter were unsuccessful then all coefficients $F_j$ of $F(Z)$ would be positive. By a theorem of Pringsheim’s (see, e.g., [@fs-ac-09] p. 240), this would imply that $R$ is a singularity of $F(Z)$. But there are only complex singularities, due to Lemma \[FalboApp-lem\]. Proof of Lemma \[FalboApp-lem\] {#Falbo-subsec} ------------------------------- The equation $e^{wZ} - sZ = 0$ has received some attention in the field of delay differential equations, see, e.g., Falbo [@f-ansdd-95]. With the following claim our main interest will be in case (i) and its transition into case (ii). [**Claim 1**]{} \[zero-lem\] $\mbox{}$\ (i) If $\frac{s}{w} < e$ then the equation $e^{wZ} - sZ = 0$ has an infinite number of non-real, discrete conjugate pairs of complex roots.\ (ii) As $\frac{s}{w}$ increases to $e$, the pair of complex roots $z_0$ and $\overline{z_0}$ of minimum imaginary part converge to the real zero $x_0=1/w$.\ (iii) For $\frac{s}{w} > e$, the real zero $x_0$ splits into two different real zeros. Let $z=a+i \, b$ be a complex zero of $e^{wZ} - sZ \, = \, 0$, for real parameters $w,s \not= 0$, that is, $$\begin{aligned} e^{wa} \, \big( \cos(wb) + i\, \sin(wb) \big) \, = \, sa+i \, sb. \label{zero}\end{aligned}$$ If the imaginary part $b$ of $z$ equals zero then $e^{wa}=sa$, hence $$\frac{e^{wa}}{wa} \, = \, \frac{s}{w}.$$ This implies $\frac{s}{w} \geq e$; see Figure \[exdx-fig\]. ![The real function $e^X/X$ takes on only values $\geq e$, and those $>e$ exactly twice.[]{data-label="exdx-fig"}](exdx.pdf) Now suppose that $b\not=0$ holds. Then \[zero\] implies $$\begin{aligned} e^{wa} \, \cos(wb) \, &=& \, sa \\ e^{wa} \, \sin(wb) \, &=& \, sb ,\end{aligned}$$ hence $\cot(wb) = \frac{a}{b}$ and $$\begin{aligned} e^{wb \, \cot(wb)} \, \sin(wb) \, = \, \frac{s}{w} \, wb. \label{imfunc} \end{aligned}$$ The graph of the real function $h(X):=e^{X \cot X} \sin X$ intersects the line $q \, X$ in an infinite number of discrete points; see Figure \[ke-fig\]. ![As slope $q$ of line $qX$ grows to $e$, its intersections with the center part of $e^{X \cot X} \sin X$ disappear.[]{data-label="ke-fig"}](ke.pdf) Each intersection point $p$ with abscissa $x$ corresponds to a zero $\cot(x)\frac{x}{w}+\frac{x}{w} \, i$ of $e^{wZ} - sZ = 0$, of absolute value $\frac{1}{\sin^2 x} \, \frac{x^2}{w^2}$. Function $h(X)$ has poles at the integer multiples of $\pi$. As shown in Figure \[poles-fig\], the first intersection point to the right of $0$ has abscissa $x_0<\pi$, the following ones, $x_k>2k\pi$. As slope $q$ of the line $qX$ increases beyond $e$, its two innnermost intersections $\not=0$ with the graph of $h(X)$ disappear. Thus, the imaginary parts of $z_0, \overline{z_0}$ become zero, causing a “double” real zero at $1/w$. As $q$ grows beyond $e$, this zero splits into two simple zeroes $a/w$ and $a'/w$; compare Figure \[exdx-fig\]. For later use we note the following. While slope $q$ is less than $e$ we can write the zero $z_0$ of positive imaginary part associated with $p_0$ as $$\begin{aligned} z_0 \, = \, a + b \, i \, = \, \rho \, (\cos(\phi) + \sin(\phi)) \, i. \label{z0}\end{aligned}$$ This representation yields $\cot(\phi) = a/b$. Since we have also derived $\cot(wb) = a/b$ it follows that angle $\phi$ of $z_0$ and $wb$ are congruent modulo $\pi$. Since $wb=x_0 < \pi$ and $\phi < \pi$ because of $b>0$ we conclude that $$\begin{aligned} \phi \, = \, wb \, = \, w \rho \sin(\phi) \label{phival}\end{aligned}$$ is the smallest positive solution of $e^{X \cot X} \sin X = qX$. ![The first intersection points of $h(X):=e^{X \cot X} \sin X$ with line $qX$ to the right of $0$. Here, $q=s/w$ is a function of angle $\alpha$. The numbers $\mbox{abs}i$ denote the absolute values of the zeroes of $e^{wZ} - sZ = 0$ that correspond to the intersection points $p_i$, for $i=0,\ldots,3$. The value of $\mbox{abs}0$ is decreasing towards $0.30563$, as $\alpha$ tends to $\pi/2$. We have $\mbox{abs}1=1$ because of the zero in Lemma \[common-lem\] (i). All other zeroes have absolute values substantially larger than 1.[]{data-label="poles-fig"}](poles.pdf) First we show that only poles can arise from these zeroes. [**Claim 2**]{} \[pole-lem\] $\mbox{}$\ Each zero $u=a+b \, i$ (except $1/w$ in case (ii) of Claim 1) is a pole of order one of the function $$d(Z)\, :=\, \frac{1}{e^{w Z} - s \, Z}$$ with residue $\mu=((wu-1)s)^{-1}$. We have $$\frac{Z-u}{e^{wZ}-sZ} \, = \, \frac{Z-u}{e^{wZ}-e^{wu}+su-sZ} \, = \, \frac{1}{\frac{e^{wZ}-e^{wu}}{Z-u}-s}.$$ As $Z$ tends to $u$, the differential quotient in the denominator tends to the finite number $(e^{wZ})'(u) = we^{wu} = wsu$. Hence, $u$ is a pole of order 1, and $d(Z)$ has local expansion $$d(Z) \ = \ \frac{\mu}{Z-u} \, + \, \sum_{i=0}^\infty w_i \, (Z-u)^i.$$ One can show that, in case (ii), zero $1/w$ gives rise to a pole of order two of $d(Z)$. Thus, function $d(Z)$, and therefore $$\begin{aligned} f(Z) \ := \ \frac{ e^{v Z} \ - \ r \, Z}{ e^{w Z} \ - \ s \, Z} \label{cfunc}\end{aligned}$$ are meromorphic. Next, we consider which of the poles of $d(Z)$ cancel out in the numerator of $f(Z)$. From now on, the parameters $v,r,w,s$ are no longer considered independent but functions of angle $\alpha$. [**Claim 3**]{} \[common-lem\] $\mbox{}$\ Numerator and denominator of function $f(Z)$ in \[cfunc\] have the following zeroes in common:\ (i) $\cos(\alpha) + \sin(\alpha) \, i$\ (ii) $\cos(\alpha) + (q+1)\sin(\alpha) \, i$ for each integer $q$ satisfying $\alpha=\frac{2p}{q} \pi$, for some integer $p$.\ Let $z=a+b \,i$ be a common zero of $e^{vZ}-rZ$ and $e^{wZ}-sZ$. As in the proof of Claim 1 we have $$\begin{aligned} e^{wb \, \cot(wb)} \, \sin(wb) \, &=& \, sb \\ e^{wb \, \cot(wb)} \, \cos(wb) \, &=& \, sa \label{s}\end{aligned}$$ and, analogously, $$\begin{aligned} e^{vb \, \cot(vb)} \, \sin(vb) \, &=& \, rb \\ e^{vb \, \cot(vb)} \, \cos(vb) \, &=& \, ra. \label{r}\end{aligned}$$ This implies $\cot(wb)=a/b=\cot(vb)$, hence $wb=vb+k\pi$ for some integer $k$. Because $s$ and $r$ are positive for all $\alpha$, the expressions in \[s\] and \[r\] must have the same sign, and we conclude that $k=2h$ is even. This implies $\sin(wb)=\sin(vb)$ and from $$\frac{2\pi}{\sin\alpha} b = (w-v)b =2h\pi$$ follows $b=h\sin\alpha$. Moreover, we have $$e^{\frac{2\pi}{\sin\alpha} a} = e^{(w-v)a} = \frac{e^{wa}\sin(wb)}{e^{va}\sin(vb)} = \frac{sb}{rb} =\frac{s}{r} = e^{2\pi \cot\alpha},$$ and we obtain $a=\cos\alpha$. This yields $$r \cos(vb) = e^{v \cos\alpha}\cos(vb) = e^{va}\cos(vb) = ra = r\cos\alpha,$$ hence $vb=\alpha + 2p\pi$ for some integer $p$, and from $h\alpha = hv\sin\alpha=vb$ follows $$\alpha \, = \, \frac{2p}{h-1} \pi.$$ From now on we consider only case (i) of Claim 1. Since $\frac{s}{w}$ is a strictly decreasing function in $\alpha$, we have $$\begin{aligned} \frac{e^{(2 \pi + \alpha) \cot\alpha}}{\frac{2 \pi + \alpha}{\sin\alpha}} \ = \ \frac{s}{w} \ &<& \ e \label{alfac} \\ \Longleftrightarrow \ \alpha \, &>& \, \alpha_c \, := \, 1.17830\ldots.\end{aligned}$$ The critical angle $\alpha_c$ corresponds to a speed $v_c = 1/\cos(\alpha_c) = 2.61440\ldots$. For $\alpha \in (\alpha_c, \pi/2)$ we can summarize our findings as follows. [**Claim 4**]{} \[summa-lem\] $\mbox{}$\ For $\alpha \in (\alpha_c, \pi/2)$, function $f(Z)$ in \[cfunc\] has only non-real, first-order poles as singularities. A conjugate pair $z_0, \overline{z_0}$ is situated at distance $<0.31$ from the origin. All other poles are of absolute value $>1$. For $\alpha \rightarrow \alpha_c$ both $z_0,\overline{z_0}$ converge to the real pole $(1/w,0)$ where $1/w \approx 0.12383.$ By Claim 2, and by Claim 1 (i), function $f(Z)$ has only poles of order one for singularities, none of which are real. Zero $\cos\alpha+ \sin\alpha \, i$ of the denominator is of absolute value 1, but it is not a pole of $f(Z)$, by Claim 3 (i). All other zeroes of the denominator canceling out must be of absolute value $>1$, by Claim 3 (ii). Hence, $z_0$ and $\overline{z_0}$ are in fact poles of $f(Z)$. The bounds on the absolute values can be obtained by numerical evaluation; see Figure \[poles-fig\]. This concludes the proof of Lemma \[FalboApp-lem\]. Residue analysis ================ From now on let $v>v_c$. In order to find out how many rounds barrier curve $\mbox{FF}_v$ runs before it closes down on itself we employ a technique of Flajolet’s; see [@fs-ac-09] p. 258 ff. Instead of $F(Z)=F_0\, f(Z)$ we consider the function $$\begin{aligned} g(Z) \ &:=& \ \frac{1}{Z^{j+1}} \, \frac{F(Z)}{F_0} \ = \ \frac{1}{Z^{j+1}} \, \frac{ e^{v Z} \ - \ r \, Z}{ e^{w Z} \ - \ s \, Z} \\ \ &=& \ \frac{\frac{F_0}{F_0}}{Z^{j+1}} \, + \, \frac{\frac{F_1}{F_0}}{Z^{j}} \, + \, \frac{\frac{F_2}{F_0}}{Z^{j-1}} + \, \ldots \, \frac{{\color{blue} \frac{F_j}{F_0}}}{Z} \ + \ \sum_{i=0}^\infty \frac{F_{j+i+1}}{F_0} \, Z^i \label{zeropole}\end{aligned}$$ By Lemma \[FalboApp-lem\], function $g(Z)$ has complex poles at $z_0$ and $\overline{z_0}$, and a real pole at the origin, and these are the only singularities inside the circle $\Gamma$ of radius $\gamma:=0.9$; see Figure \[poles3-fig\]. Let $\mu$ and $\overline{\mu}$ denote the residues of the poles at $z_0$ and $\overline{z_0}$. By \[zeropole\], the pole at 0 has residue $F_j/F_0$, the coefficient we are interested in. ![The poles of minimum modulus of $F(Z)$.[]{data-label="poles3-fig"}](poles3.pdf) By Cauchy’s Residue Theorem, $$\begin{aligned} \frac{F_j}{F_0} \, &=& \, - (\mu + \overline{\mu}) \ + \ \frac{1}{2 \pi \, i} \, \int_\Gamma g(u) \, \mathrm{d}u . \label{CRT}\end{aligned}$$ The integral can be upper bounded by $$\begin{aligned} {\left\lvert\frac{1}{2 \pi \, i} \, \int_\Gamma\frac{f(z)}{z^{j+1}} \, \mathrm{d}z\right\rvert} \ \leq \ \frac{1}{2 \pi} \, D \, \int_\Gamma\frac{1}{{\left\lvertz^{j+1}\right\rvert}} \, \mathrm{d}z \ &=& \ \frac{1}{2 \pi} \, D \, \Big(\frac{1}{\gamma}\Big)^{j+1} \, 2 \pi \, \gamma \\ &=& \ D \, \gamma^{-j} \label{int}\end{aligned}$$ because all $z$ on $\Gamma$ have absolute value $\gamma=0.9$. Here, $D$ denotes the maximum value function ${\left\lvertf(z)\right\rvert}$ attains on the compact set $\Gamma \times [\alpha_c,\pi/2]$. Now let $$\begin{aligned} z_0 \, = \, \rho (\cos\phi + \sin\phi \, i) \, = \, a + b \, i \label{coord}\end{aligned}$$ be the pole different from zero whose imaginary part, $b$, is positive. Let us recall from \[phival\] in the proof of Claim 1 that $\phi$, the angle of $z_0$, is the smallest positive real number solving $e^{x \cot(x)} \sin(x) = \frac{s}{w} x$, with $s, w$ as defined in \[sw\]. We also have shown that $$\begin{aligned} \phi \, = \, w b \, = \, w \rho \, sin\phi \label{philab}\end{aligned}$$ holds. Furthermore, $\rho={\left\lvertz_0\right\rvert}$. Let $x_0=(1/w,0)$ denote the real pole to which $z_0$ and $\overline{z_0}$ converge as $\alpha$ decreases to $\alpha_c$; compare Claim 4. We obtain the following identities. $$\begin{aligned} \ w {\left\lvertz_0-x_0\right\rvert} \ &=& \ \sqrt{w^2 \rho^2 \, - \, 2w \rho \cos(\phi) +1} \\ &=& \ \sqrt{\frac{\phi^2}{\sin^2\phi} -2\phi \frac{\cos\phi}{\sin\phi} +1} \\ &=& \ \sqrt{(wa-1)^2+w^2b^2} \label{wurz}\end{aligned}$$ The sum of residues can be written as follows. \[ressum-lem\] We have $$\begin{aligned} - (\mu + \overline{\mu}) \ = \ \Big(\ \frac{e^{va}}{\rho^2}\, \cos((j+1)\phi - vb) \, &-& \, \frac{e^{va}w}{\rho}\, \label{major} \cos((j+2)\phi -vb) \\ - \ \big( \ \frac{r}{\rho} \, \cos(j \phi) \, &-& \ rw \, \cos((j+1)\phi) \ \big) \ \Big) \label{minor}\\ \ \cdot \ \frac{2}{s\big( (wa-1)^2+w^2b^2 \big)} \, &\cdot& \, \frac{1}{\rho^{j-1}} \label{power}\end{aligned}$$ Using $z_0=a + b \, i$ we obtain $$\begin{aligned} \mu + \overline{\mu} \ &=& \ \frac{e^{v z_0} - r z_0}{s (w z_0 - 1) \, z_0^{j+1}} \ + \ \frac{e^{v \overline{z_0}} - r \overline{z_0}}{s (w \overline{z_0} - 1) \, \overline{z_0}^{j+1}} \\ &=& \ \frac{1}{s} \ \frac{w e^{v z_0} \, \overline{z_0}^{j+2} - e^{v z_0} \, \overline{z_0}^{j+1} - r w \big( a^2 + b^2 \big) \, \overline{z_0}^{j+1} + r \big( a^2 + b^2 \big) \, \overline{z_0}^{j} }{\Big((w a -1)^2 + w^2 b^2 \Big) \, \big( a^2 + b^2 \big)^{j+1}} \\ &+& \ \frac{1}{s} \ \frac{w e^{v \overline{z_0}} \, {z_0}^{j+2} - e^{v \overline{z_0}} \, {z_0}^{j+1} - r w \big( a^2 + b^2 \big) \, {z_0}^{j+1} + r \big( a^2 + b^2 \big) \, {z_0}^{j} }{\Big((w a -1)^2 + w^2 b^2 \Big) \, \big( a^2 + b^2 \big)^{j+1}}.\end{aligned}$$ With $e^{v z_0}=e^{va} \, (\cos(v b) + \sin(v b) \, i)$ and $z_0^j = \rho^j (\cos(j \phi) + \sin(j \phi) \, i)$ one gets $$\begin{aligned} \mbox{Re}\big(e^{v {z_0}} \, \overline{z_0}^{j+2} \big) \ &=& \ \mbox{Re}\big(e^{va} (\cos(vb)+\sin(vb) \, i) \, \rho^{j+2}(\cos((j+2)\phi ) - \sin((j+2)\phi) \, i \big) \\ &=& \ e^{va} \rho^{j+2} \big( \cos(vb) \cos((j+2)\phi) + \sin(vb) \sin((j+2)\phi) \big) \\ &=& \ e^{va} \rho^{j+2} \cos(vb - (j+2)\phi),\end{aligned}$$ and substituting $a^2+b^2=\rho^2$ and $z + \overline{z} = 2 \mbox{Re}(z)$ shows that $\mu + \overline{\mu}$ equals $$\begin{aligned} \frac{2}{s} \ \frac{w e^{va} \rho^{j+2}\cos(vb-(j+2)\phi) - e^{va}\rho^{j+1}\cos(vb-(j+1)\phi)-rw\rho^{j+3}\cos((j+1)\phi) + r\rho^{j+2}\cos(j\phi)}{\big((w a -1)^2 + w^2 b^2 \big) \, \rho^{2{j+2}}}.\end{aligned}$$ The sign of $- (\mu + \overline{\mu})$ in Lemma \[ressum-lem\] is determined by the four cosine terms. If we substitute $j$ with a real “time” variable $t$, we can consider them as sine waves of the same frequency, $\phi$, but different amplitudes and phases. A finite sum of such waves is again a sine wave of frequency $\phi$, so that $$\begin{aligned} \Big(\ \frac{e^{va}}{\rho^2}\, \cos((t+1)\phi - vb) \, &-& \, \frac{e^{va}w}{\rho}\, \cos((t+2)\phi -vb) \label{strongterm} \\ - \ \big( \ \frac{r}{\rho} \, \cos(t \phi) \, &-& \ rw \, \cos((t+1)\phi) \ \big) \ \Big) \label{weakterm}\\ &=& \ L \, \sin(t\phi + p) \label{ampphase}\end{aligned}$$ holds, with some amplitude $L$ and some phase $p$. \[sine-lem\] We have $$\begin{aligned} L \ \ge \ L_0 \, := \, \sqrt{w^2 \rho^2 \, - \, 2w \rho \cos(\phi) +1} \, \Big( \frac{e^{va}}{\rho^2} \, - \, \frac{r}{\rho} \Big). \label{L0}\end{aligned}$$ In general, one has $$a_1 \, \sin(t \phi + p_1) \ + \ a_2 \, \sin(t \phi + p_2) \ = \ \sqrt{a^2_1 + a^2_2 + 2 a_1 a_2 \cos(p_1-p_2)} \, \sin(t \phi +p),$$ where phase $p$ depends on $a_1,a_2,p_1,p_2$. These formulae for the sum of two waves of identical frequency can be found in textbooks or, for example, in *Bronstein et al., Taschenbuch der Mathematik, 1993*. This yields $$\begin{aligned} \frac{e^{va}}{\rho^2}\, \cos((t &+& 1)\phi - vb) \, - \, \frac{e^{va}w}{\rho}\, \cos((t+2)\phi -vb) \\ &=& \, \frac{e^{va}}{\rho^2}\, \sin(t\phi +\phi- vb +\frac{\pi}{2}) \, + \, \frac{e^{va}w}{\rho}\, \sin(t\phi + 2\phi-vb + \frac{3\pi}{2}) \\ &=& \, \sqrt{\frac{e^{2va}}{\rho^4} + \frac{e^{2va}w^2}{\rho^2} + 2 \frac{e^{2va}w}{\rho^3} \cos(-\phi-\pi) } \, \sin(t\phi + p) \\ &=& \, \frac{e^{va}}{\rho^2} \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1} \, \sin(t\phi + p). \label{strong}\end{aligned}$$ Similarly, $$\begin{aligned} - \ \big( \, \frac{r}{\rho} \, \cos(t \phi) \, &-& \ rw \, \cos((t+1)\phi) \, \big) \\ &=& \, \frac{r}{\rho} \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1} \, \sin(t\phi + q). \label{weak}\end{aligned}$$ Thus, the sum of these two sine waves has amplitude $$\begin{aligned} \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1} \, \sqrt{\frac{e^{2va}}{\rho^4} + \frac{r^2}{\rho^2} + 2 \frac{e^{va}}{\rho^2} \frac{r}{\rho} \, \cos(p - q) } \label{sumamp} \\ \geq \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1} \, \Big( \frac{e^{va}}{\rho^2} \, - \, \frac{r}{\rho} \Big).\end{aligned}$$ Now we can prove a quantitative version of Theorem \[qual-theo\] showing that at speed $v>v_c$ the fire fighter is successfull after at most $O(1/\phi)$ many rounds, where $\phi$ denotes the complex argument of the smallest zero of $e^{wZ}-sZ$. \[quant-theo\] Let $\alpha > \alpha_c$. Then there is an index $j \in O(\frac{1}{\phi}) $ such that $F_j < 0$ holds. The function $h(t) := L \, \sin(t\phi + p)$ of \[ampphase\] attains its minima $-L$ at arguments $t^*$ where $t^* \phi + p \, \equiv \, \frac{3 \pi}{2} \mod 2\pi$. For an integer $j$ at most $1/2$ away from $t^*$ we have $$h(j) \, \leq \, h(t^*+\frac{1}{2}) \, = \, L \, \sin(\frac{3\pi+\phi}{2}) \, = \, - \, L \, \cos(\frac{\phi}{2});$$ these terms are negative because of $\phi < 2.09< \pi$. This implies $$\begin{aligned} \frac{F_j}{F_0} \ < \ - \ \frac{2}{s} \, \Big( \frac{e^{va}}{{\left\lvertz_0\right\rvert}} \, - \, r \Big) \, \frac{1}{w} \, \frac{1}{{\left\lvertz_0-x_0\right\rvert}} \, \cos(\frac{\phi}{2}) \, {\left\lvertz_0\right\rvert}^{-j} \ + \ D \, \cdot \, 0.9^{-j}, \label{upbound}\end{aligned}$$ summarizing \[CRT\], \[major\] to \[power\], \[ampphase\], Lemma \[sine-lem\], and \[int\]. We observe that such integers $j$ occur (at least) once in every period of length $2\pi/\phi$ of function $h(t)$. Since ${\left\lvertz_0\right\rvert} < 0.31 < 0.9$, the powers ${\left\lvertz_0\right\rvert}^{-j}$ grow in $j$ much faster than $0.9^{-j}$ does. All coefficients in \[upbound\] are positive, and lower bounded by independent constants on $[\alpha_c, \pi/2]$. Indeed, we have $\frac{2}{s} \geq 0.091$ with a minimum at $\alpha_c$, $\frac{e^{va}}{{\left\lvertz_0\right\rvert}} - r \geq 1.581$ with its minimum at $\alpha=\pi/2$, and $w {\left\lvertz_0-x_0\right\rvert} \leq 0.33$ with a maximum at $\alpha=\pi/2$. Hence, after a constant number of periods the value of \[upbound\] becomes negative. This completes the proof of Theorem \[quant-theo\]. Numerical inspection shows that a suitable integer $j$ can already be found in the first period of function $h$, so that $j \leq \frac{2\pi}{\phi} +1$. Now we let speed $v$ decrease to the critical value $v_c$, and prove that the first index $j$, for which $F_j$ becomes negative, grows with $\pi/\phi$ to infinity. To this end we prove that the sine wave in \[ampphase\] starts, at zero, near the beginning of its positive half-cycle, so that it takes half a period before negative values can occur. The graph of $\sin(t \phi +p)$ is shifted, along the $t$-axis, by $p/\phi$ to the left, as compared to the graph of $\sin(t)$. As $\alpha$ tends to $\alpha_c$, frequency $\phi$ goes to zero, and so does phase $p$. But, surprisingly, their ratio rapidly converges to a small constant. \[shift-lem\] We have $$\begin{aligned} \sigma \ := \ \lim_{\alpha \rightarrow \alpha_c} \ \frac{p}{\phi} \ = \ \Big(\frac{r}{e^{\frac{v}{w}} w -r} +1\Big) \, \big(1-\frac{v}{w}\big) \ + \ \frac{1}{3} \approx \ 1.351. \end{aligned}$$ Figure \[wave-fig\] shows $-(\mu + \overline{\mu}) \rho^{j-1}$ as a function of time parameter $j=t$; see  \[major\] to \[power\]. ![The shift to the left is almost constant, as $\alpha$ tends to $\alpha_c$ and period $2 \pi/\phi$ goes to infinity.[]{data-label="wave-fig"}](wave.pdf) Crucial in the proof is the following geometric fact. \[tria-lem\] Consider the triangle shown in Figure \[fulltria-fig\], which has a base of length 1, a base angle of $\phi$, and height $\phi$. As $\phi$ goes to zero, the ratio $\frac{\tau}{\phi}$ tends to $1/3$, and $\gamma$ converges to $\pi/2$. ![Ratio $\tau / \phi$ tends to $1/3$, as $\phi$ goes to zero.[]{data-label="fulltria-fig"}](fulltria.pdf) From $C \cos\tau = \phi$ and $C \sin\tau = 1-\phi \cot\phi$ we obtain $$\tan\tau \, = \, \frac{\sin\phi - \phi \cos\phi}{\phi \sin\phi},$$ and because $\tau$ must go to zero as $\phi$ does, we have $$\frac{\tau}{\phi} \, = \, \cos\tau \, \frac{\tau}{\sin\tau} \, \frac{\tan\tau}{\phi} \, \sim \, \frac{\sin\phi - \phi \cos\phi}{\phi^2 \sin\phi}.$$ A twofold application of l’Hospital’s rule shows that the last term has the same limit as $$\frac{\sin\phi}{2 \sin\phi + \phi \cos\phi} \, \sim \, \frac{\cos\phi}{3 \cos\phi - \phi \sin\phi},$$ which converges to $1/3$. Moreover, we have $$\sin\gamma \, = \, \sin(\pi/2 - \phi) \, = \, \cos(\phi)$$ which tends to 1, so that $\gamma$ converges to $\pi/2$. Now we give the proof of Lemma \[shift-lem\]. As in the proof of Lemma \[sine-lem\] one generally has $$a_1 \, \sin(t \phi + p_1) \ + \ a_2 \, \sin(t \phi + p_2) \ = \ a_3 \, \sin(t \phi +p_3),$$ where the new amplitude, $a_3$, is given by $$a_3 \ = \ \sqrt{a^2_1 + a^2_2 + 2 a_1 a_2 \cos(p_1-p_2)},$$ and the new phase, $p_3$, fulfills $$p_3 \ = \ \arcsin(\frac{a_2 \, \sin(p_2-p_1)}{a_3}) \ + \ p_1.$$ First, we are applying this formula to \[strongterm\], $$\begin{aligned} \frac{e^{va}}{\rho^2}\, \cos((t+1)\phi - vb) \, &-& \, \frac{e^{va}w}{\rho}\, \cos((t+2)\phi -vb) \\ = \ \frac{e^{va}}{\rho^2}\, \sin(t\phi + \phi - vb + \pi/2) \, &+& \, \frac{e^{va}w}{\rho}\, \sin(t\phi + 2\phi -vb + 3\pi/2) \end{aligned}$$ and obtain, as in \[strong\], $$a_{\ref{strongterm}} \ = \ \frac{e^{va}}{\rho^2} \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1},$$ and, for the new phase, $$\begin{aligned} p_{\ref{strongterm}} \ &=& \ \arcsin(-\frac{w \rho \sin\phi}{\sqrt{w^2 \rho^2 - 2w \rho \cos\phi +1}}) \ + \ \phi - vb + \pi/2 \\ &=& \ \arcsin(-\frac{\phi}{\sqrt{\frac{\phi^2}{\sin^2\phi} -2\phi \frac{\cos\phi}{\sin\phi} +1}}) \ + \ \phi - vb + \pi/2 \\ &=& \ \arcsin( - \cos\tau) \ + \ \phi - vb + \pi/2 \\ &=& \ \arcsin(\sin(-\pi/2 + \tau)) \ + \ \phi - vb + \pi/2 \\ &=& \ \tau \, + \, \phi - vb\end{aligned}$$ using \[philab\] and the triangle in Figure \[fulltria-fig\]. We conclude that the value of $\arcsin$ goes to $-\pi/2$, so that $p_{\ref{strongterm}}$ converges to zero. For the resulting shift we obtain $$\begin{aligned} \frac{p_{\ref{strongterm}}}{\phi} \ &=& \ \frac{\tau}{\phi} \ + \ 1 \ - \frac{v}{w} \\ &\rightarrow& \ \frac{1}{3} \, + \, 1 - \frac{v}{w}\end{aligned}$$ by Lemma \[tria-lem\]. Next, we consider \[weakterm\], $$\begin{aligned} \ rw \, \cos((t+1)\phi) \, &-& \, \frac{r}{\rho} \, \cos(t \phi) \\ &=& \ rw \, \sin(t \phi +\phi + \pi/2) \, + \, \frac{r}{\rho} \, \sin(t \phi + 3\pi/2)\end{aligned}$$ As in \[weak\], $$a_{\ref{weakterm}} \ = \, \frac{r}{\rho} \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1},$$ and for the phase, $$\begin{aligned} p_{\ref{weakterm}} \ &=& \ \arcsin\Big(\frac{\sin\phi}{\sqrt{w^2 \rho^2 - 2w \rho \cos\phi +1}}\Big) \ + \ \phi + \pi/2 \\ &=& \ \pi/2 \, + \, \gamma \, + \, \tau \, + \, \phi,\end{aligned}$$ observing that the argument of $\arcsin$ equals $$\frac{\sin\phi}{C} \, = \, \frac{\sin(\gamma+\tau)}{1},$$ applying the law of sines to the triangle shown in Figure \[fulltria-fig\]. We see that $p_{\ref{weakterm}}$ goes to $\pi$; in this case, the shift does not converge. Now we consider the sum of \[strongterm\] and \[weakterm\]. We know from \[sumamp\] the final amplitude, $$\begin{aligned} a \ = \ \sqrt{w^2 \rho^2 \, - \, 2w \rho \, \cos(\phi) +1} \, \sqrt{\frac{e^{2va}}{\rho^4} + \frac{r^2}{\rho^2} + 2 \frac{e^{va}}{\rho^2} \frac{r}{\rho} \, \cos( p_{\ref{weakterm}} - p_{\ref{strongterm}} ) },\end{aligned}$$ and obtain for the phase $$\begin{aligned} p \ = \ \arcsin\Biggl( \frac{r}{\rho} \, \frac{\sin(p_{\ref{weakterm}} - p_{\ref{strongterm}} )}{\sqrt{\frac{e^{2va}}{\rho^4} + \frac{r^2}{\rho^2} + 2 \frac{e^{va}}{\rho^2} \frac{r}{\rho} \, \cos( p_{\ref{weakterm}} - p_{\ref{strongterm}}) }} \Biggr) \ + \ p_{\ref{strongterm}}.\end{aligned}$$ For short, let $\overline{p}$ denote the $\arcsin$ term, and let $R$ be the square root in the denominator. Since $p_{\ref{weakterm}} - p_{\ref{strongterm}}$ tends to $\pi$ we conclude that $\overline{p}$ goes to zero. Thus, we obtain $$\begin{aligned} \frac{p}{\phi} \ &=& \ \frac{\overline{p}}{\sin \overline{p}} \, \frac{\sin \overline{p}}{\phi} \, + \, \frac{p_{\ref{strongterm}}}{\phi} \\ \ &=& \ \frac{\overline{p}}{\sin \overline{p}} \, \frac{r}{\rho} \, \frac{1}{R} \, \frac{\sin(p_{\ref{weakterm}} - p_{\ref{strongterm}} )}{\pi - (p_{\ref{weakterm}} - p_{\ref{strongterm}} )} \ \cdot \ \frac{\pi - (p_{\ref{weakterm}} - p_{\ref{strongterm}} )}{\phi} \ + \ \frac{p_{\ref{strongterm}} }{\phi} \\ &\sim& \ \frac{r}{\rho} \, \frac{1}{R} \ \cdot \ \frac{\pi/2 - \gamma - vb}{\phi} \ + \ \frac{p_{\ref{strongterm}} }{\phi} \\ &=& \ \frac{r}{\rho} \, \frac{1}{R} \ \cdot \ \frac{\phi - vb}{\phi} \ + \ \frac{p_{\ref{strongterm}} }{\phi} \\ &\rightarrow& \ \frac{rw}{e^{v/w}w^2 - rw}\, \Big(1-\frac{v}{w}\Big) \ + \ 1/3 + 1 - \frac{v}{w}.\end{aligned}$$ This concludes the proof of Lemma \[shift-lem\]. Now let $j$ be an integer satisfying $$j \, \leq \, \frac{\pi}{\phi} \, - \, 2 \, \sigma.$$ Lemma \[shift-lem\] implies that the sign of $\sin(j \phi +p)$ in \[ampphase\] is positive, and, even more, that $$\begin{aligned} \sin(j \phi +p) \, > \, \sin(p) > \sin(1.35 \, \phi)\end{aligned}$$ holds. For such integers $j$ we obtain, similarly to \[upbound\], $$\begin{aligned} \frac{F_j}{F_0} \ > \ \frac{2}{s} \, \Big( \frac{e^{va}}{{\left\lvertz_0\right\rvert}} \, - \, r \Big) \, \frac{1}{w} \, \frac{1}{{\left\lvertz_0-x_0\right\rvert}} \, \sin(1.35 \, \phi) \, {\left\lvertz_0\right\rvert}^{-j} \ - \ 0.1 \, \cdot \, 0.9^{-j} \label{lobound}\end{aligned}$$ Here we have used the following estimate for $D$ in \[int\]. \[D-lem\] With the radius of $\Gamma$ equal to $\gamma=0.9$, we have ${\left\lvertf(z)\right\rvert} \leq 0.1$ for all $z$ on $\Gamma$, if $\alpha$ is close enough to $\alpha_c$. Let $z=\gamma \, e^{i \psi}$ be a parameterization of circle $\Gamma$ for $\psi \in [0 \ldots 2\pi]$. By multiplication with complex conjugates, $$\begin{aligned} {\left\lvertf(z)\right\rvert}_{z \in \Gamma} \ &=& \ {\left\lvert\frac{ e^{v \gamma e^{i \psi}} \ - \ r \, \gamma e^{i \psi}}{ e^{w \gamma e^{i \psi}} \ - \ s \, \gamma e^{i \psi}} \right\rvert} \\ &=& \sqrt{\frac{ e^{2v \gamma \cos\psi} - 2 e^{v \gamma \cos\psi} \, r \gamma \, \cos(v \gamma \sin\psi - \psi) + r^2 \gamma^2} {e^{2w \gamma \cos\psi} - 2 e^{w \gamma \cos\psi} \, s \gamma \, \cos(w \gamma \sin\psi - \psi) + s^2 \gamma^2} }.\end{aligned}$$ The maximum is attained at $\psi=\pi$, and it grows monotonically from $0.09\ldots$ for $\alpha=\alpha_c$ to $1.269\ldots$ for $\alpha = \pi/2$. Now we can state the lower bound. \[lowbo-theo\] As angle $\alpha$ decreases to the critical value $\alpha_c$, the number $j$ of rounds necessary to contain the fire is at least $j > \frac{\pi}{\phi} - 2.71$. This lower bound grows to infinity. By the preceeding discussion, estimate \[lobound\] holds for each $j$ that stays below this bound. As $\phi$ tends to 0 we get $$\begin{aligned} \frac{1}{w} \, \frac{1}{{\left\lvertz_0-x_0\right\rvert}} \, \sin(1.35 \phi) \ &=& \ \frac{\sin(1.35 \phi)}{\sqrt{w^2 \rho^2 -2w \rho \cos(\phi) +1}} \\ &=& \ \frac{\sin(1.35 \phi)}{\sqrt{\frac{\phi^2}{\sin^2\phi} -2\phi \frac{\cos\phi}{\sin\phi} +1}} \\ &\sim& \ 1.35 \, \frac{\sin\phi}{\sqrt{\frac{\phi^2}{\sin\phi^2} -2\phi \frac{\cos\phi}{\sin\phi} +1}} \\ &=& \ 1.35 \, \sin(\gamma + \tau) \\ &\sim& \, 1.35.\end{aligned}$$ Here, the first equality follows from \[wurz\] and the second, from \[philab\]. Then we have applied l’Hospital’s rule, and the next line follows from Lemma \[tria-lem\]. Indeed, the square root is equal to $C$ in Figure \[fulltria-fig\], and we can apply the law of sines together with the fact that $\gamma$ goes to $\pi/2$, and $\tau$ to zero. Substituting in \[lobound\] the other limit values (non of which is critical) we find $$\begin{aligned} \frac{F_j}{F_0} \ &>& \ 0.091 \,\cdot \, 7.82 \, \cdot \, 1.35 \, \cdot \, {\left\lvertz_0\right\rvert}^{-j} \ - \ 0.1 \, \cdot \, 0.9^{-j} \\ &\geq& \ 0.96 \, \cdot \, 0.1239^{-j} \ - \ 0.1 \, \cdot \, 0.9^{-j} \\ &>& 0.\end{aligned}$$ Here 7.82 is the limit of $\frac{e^{va}}{{\left\lvertz_0\right\rvert}} -r$ as $\alpha$ tends to $\alpha_c$. This completes the proof of Theorem \[lowbo-theo\]. Figure \[j-fig\] shows how many rounds the fighter needs to contain the fire, depending on her speed $v$. ![The approximate number of rounds, $j$, barrier curve $\mbox{FF}_v$ needs before closing on itself.[]{data-label="j-fig"}](JversusV.pdf) Lower bound {#lobowrap-sec} =========== In this section a barrier curve $S$ is called [*spiralling*]{} if it starts on the boundary of a fire of radius $A$, and visits the four coordinate half-axes in counterclockwise order and at increasing distances from the origin. We are proving the following. \[lowbound-theo\] In order to contain a fire by a spiralling barrier, the fighter needs speed $$v \ > \ \frac{1+\sqrt{5}}{2} \, \approx \, 1.618,$$ the golden ratio. Now let $S$ be a spiralling curve, and assume that the fighter proceeds at maximum speed $v \leq (1 +\sqrt 5)/2$. Let $p_0, p_1, p_2, \ldots$ denote the points on the coordinate axes visited, in this order, by $S$. The following lemma shows that $S$ cannot succeed because there is still fire burning outside the barrier on the axis previously visited. ![Proof of Lemma \[inv-lem\].[]{data-label="lowbo-fig"}](lowbo.pdf) \[inv-lem\] Let $A$ be the initial fire radius. When $S$ visits point $p_{i+1}$, the interval $[p_i,p_i+{\tt sign}(p_i)A]$ on the axis visited before is on fire. The proof is by induction on $i$. Suppose barrier $S$ is of length $x$ between $p_0$ and $p_1$, as shown in Figure \[lowbo-fig\] (i). While this part is under construction, the fire advances $x/v$ along the positive $X$-axis, so that $A+x/v \leq p_1 \leq x$ must hold, or $$\frac{x}{v} \, \geq \, \frac{1}{v-1} A \, > \, A;$$ the last inequality follows from $v<2$. Thus, the fire has enough time to move a distance of $A$ from $p_0$ downwards along the negative $Y$-axis. Now let us assume that the fighter builds a barrier of length $y$ between $p_i$ and $p_{i+1}$, as shown in Figure \[lowbo-fig\] (ii). By induction, the interval of length $A$ below $p_{i-1}$ is on fire. Also, when the fighter moves on from $p_i$, there must be a burning interval of length at least $A+x/v$ on the positive $Y$-axis which is not bounded by a barrier from above. This is clear if $p_{i+1}$ is the first point visited on the positive $Y$-axis, and it follows by induction, otherwise. Thus, we must have $A+x/v +y/v \leq p_{i+1} \leq y$, hence $$\frac{y}{v} \, \geq \, \frac{1}{v-1} A \, + \, \frac{1}{v(v-1)} x \, > \, A+x.$$ The rightmost inequality follows since $v$ is supposed to be smaller than the golden ratio, which satisfies $X^2 -X -1 =0$; hence, $v^2-v <1$. This shows that the fire has time to crawl along the barrier from $p_{i-1}$ to $p_i$, and a distance $A$ to the right, as the fighter moves to $p_{i+1}$, completing the proof of Lemma \[inv-lem\] and of Theorem \[lowbound-theo\]. [50]{} L. Barri[è]{}re, P. Flocchini, P. Fraigniaud and N. Santoro. Capture of an intruder by mobile agents. 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--- author: - 'G. Mellema' - 'J. D. Kurk' - 'H. J. A. Röttgering' bibliography: - 'special.bib' date: 'Received 31 July 2002 / Accepted 6 September 2002' title: Evolution of clouds in radio galaxy cocoons --- Introduction ============ Regions of luminous optical line and continuum emission near high redshift radio galaxies ($z > 0.6$) are often found to be extended along the direction of the radio axis [@Chambers87; @McCarthy87]. One obvious explanation for these alignments, is that star formation takes place in regions where the shock bounding the radio jet, has passed. Recent observations seem to support this idea. Deep spectra of the radio galaxy 4C41.17 at $z = 3.8$, show that the bright, spatially extended rest-frame UV continuum emission is unpolarized and contains P Cygni-like absorption features, indicating the presence of a large population of young, hot stars [@Deyetal97]. @bicknelletal00 argue that this can best be understood if the shock associated with the radio jet has triggered star formation within the emission line clouds. A nearby example where stars might be formed under the influence of a radio source is the case of Cen A. Here, young stars are found near filaments of ionized gas in a radio lobe [@mouldetal2000]. @Rees89 and @Begelmanetal89 analytically explored the evolution of intergalactic medium (IGM) clouds, overtaken by shocks from the cocoon of a radio jet. They argue that these clouds would be compressed and then gravitationally contract to form stars. However, @Icke99 claimed that the destructive aspects of the interaction between the expanding cocoon and the clouds would dominate the evolution of the clouds. In his scenario the clouds evaporate and their material mixes into the jet cocoon. Given the complexity of the interaction between the clouds and the jet cocoon, numerical studies are a good tool to investigate this problem. Although the ‘shock-cloud interaction’ problem was studied numerically before, none of these studies addresses the effects of radiative cooling, important for intergalactic clouds. Here we present new results of a numerical hydrodynamic study of the shock-cloud interaction problem, including the effects of radiative cooling. In Sect. 2 we describe the general problem of shock-cloud interaction and the application to IGM clouds. Section 3 deals with the numerical method, and Sect. 4 contains the results, which we further discuss in the fifth section. We sum up the conclusions in Sect. 6. Shock-Cloud Interactions ======================== Many numerical studies of single shock-cloud interactions have been carried out, [@Woodward76] being one of the first. Various others followed, of which we will only mention two more recent studies: [@Kleinetal94], who provided a thorough analysis of the problem, and [@Poludnenkoetal01], who studied the case of a shock running over a system of clouds; see these two papers for an overview of the literature. It is notable that in nearly all numerical studies to date, radiative cooling was either neglected or had little effect. For work considering the large scale effects of the passage of radio jets, see @Steffenetal97 and @Reynoldsetal01. The evolution of a single, non-cooling cloud, which is run over by a strong shock wave, consists of three phases. Initially, the shock runs over the cloud. The time scale for this is the shock passing time, $t_{\rm sp} = 2R_{\rm cl}/v_{\rm shock}$, where $R_{\rm cl}$ is the cloud radius, and $v_{\rm shock}$ the velocity of the passing shock. The second phase is the compression phase, in which the cloud finds itself inside the high pressure cocoon. It is now underpressured compared to its environment, and shock waves start to travel into the cloud from all sides. This phase lasts for a time $t_{\rm cc}=R_{\rm cl}/v_{\rm s,cl}$, the cloud crushing time, where $v_{\rm s,cl}$ is the velocity of the shock travelling into the cloud. For a strong shock this velocity is of order $v_{\rm s,cl}=v_{\rm shock}/\sqrt{\chi}$, in which $\chi$ is the ratio of the $n_{\rm cl}$ to $n_{\rm env}$, the densities of the cloud and the environment, respectively; see @Kleinetal94 for a better estimate. The third phase starts when the shocks travelling into the cloud, meet and interact. This produces a rarefaction wave travelling through the shocked cloud material. The cloud, which was compressed by the shock waves, now starts expanding again, and soon afterwards is destroyed and mixes in with the surrounding flow. This typically happens in a few cloud crushing times. Cloud properties ---------------- Following @Rees89, @Begelmanetal89, and @McCarthy93, we assume the undisturbed clouds to be the cooler and denser phase of an ionized two-phase IGM, of which the low density phase has a temperature of $T_{\rm ig}=10^7$ K and a density of $n_{\rm ig}=10^{-2}$ cm$^{-3}$. Assuming pressure equilibrium between the two phases, a cloud temperature of $10^4$ K gives a density of $n_{\rm cl}=10$ cm$^{-3}$. We choose an initial radius of $3\times 10^{20}$ cm ($\sim 100$ pc) and hence the cloud mass is $9.5\times 10^5$ $M_\odot$. Following the analysis of Cygnus A by @Begelmanetal89, we take the Mach number of the shock bounding the jet cocoon to be 10, yielding $v_{\rm shock}$= 3500 km s$^{-1}$ (0.01$c$). With these parameters we obtain $t_{\rm sp}=5\times 10^4$ years, $t_{\rm cc}=8\times 10^5$ years, and $v_{\rm cl,s}=120$ km s$^{-1}$. The cooling time can be estimated from $t_{\rm cool}=Cv_{\rm s,cl}^3/\rho_{\rm cl}$ [see e.g. @Kahn76], where $C$ is a constant depending on the cooling processes, with a value of $6.0\times 10^{-35}$ g cm$^{-6}$ s$^4$ for a gas in collisional ionization equilibrium at solar abundances. With the numbers above one finds $t_{\rm cool}=2\times 10^2$ years. This is the shortest time scale thus far, showing that cooling will dominate the evolution of the shocked cloud. It is instructive to derive a condition for which cooling will dominate. Using the expressions for $t_{\rm cc}$ and $t_{\rm cool}$, we find that the condition $t_{\rm cc}>10t_{\rm cool}$ can be rewritten as $$\begin{aligned} M_{\rm cl} &>& 10^{-9} M_\odot \times\nonumber\\ && \left({v_{\rm shock}\over 10^3 {\rm km~s}^{-1}}\right)^{12} \left({\chi \over 10^3}\right)^{-8} \left({n_{\rm e}\over 10^{-2} {\rm cm}^{-3}}\right)^{-2}\,.\end{aligned}$$ This shows that cooling dominates for a large range of values for $M_{\rm cl}$, $v_{\rm shock}$, $\chi$, and $n_{\rm e}$. For our values of $M_{\rm cl}$ and $n_{\rm e}$, the shock velocity needs to be above 17000 km s$^{-1}$, or the density ratio $\chi$ below 15, for cooling [*not*]{} to dominate the evolution. Numerical method ================ The calculations were performed with a two-dimensional hydrodynamics code based on the Roe solver method, an approximate Riemann solver [@Roe81; @EulMel]. Second order accuracy was achieved with the [*superbee*]{} flux limiter, which was made less steep by lowering the coefficients from 2.0 to 1.2; taking 1.0 would correspond to using the [*minmod*]{} flux limiter, see Sect. 20.2 in @laney. Better two-dimensional behaviour was implemented by using the transverse waves method as described by @LeVeque. In order to include the effects of cooling, we used a cooling curve [@DalgarnoMcCray], which gives the cooling as function of temperature, for a low density plasma in collisional ionization equilibrium. This is a reasonable approximation of the real cooling processes of astrophysical gases. The radiative terms were implemented using operator splitting, where the appropriate radiative losses and gains were added as a separate source term every time step. The heating rate is proportional to the density, and was set so that for the initial conditions, heating and cooling in the cloud are balanced. In order to deal with short cooling times, we subdivided the time steps into smaller fractions of the order of the cooling time when applying the cooling. We imposed a minimum temperature of 10 K. This approximately corresponds to the cosmic microwave background temperature at the redshifts we are considering. We did not follow the ionization state of the gas, but assumed the material to always be in collisional ionization equilibrium. The geometry of the grid was either cylindrical $(R,z)$, assuming cylindrical symmetry, or cartesian $(x,y)$, assuming slab symmetry. The use of two different coordinate systems helps in understanding the true three-dimensional nature of the flow. Cylindrical coordinates are the proper choice as long as the flow pattern retains its large scale character, [i.e.]{}during the initial phase of the interaction. However, when the cloud starts to fragment, off-axis pieces are represented by ring-shaped structures. Furthermore, there is a strictly imposed symmetry axis at the centre of the cloud. In cartesian coordinates the initial conditions do not describe a spheroid, but rather a cylinder. On the other hand, the fragmentation is more properly followed, and no symmetry axis is imposed. We ran two simulations: in run A the shock wave interacted with a spherical cloud (with the parameters from Sect. 2.1) on cylindrical coordinates, and in run B with an elliptical cloud (with a semi-major axis of 100 pc, axis ratio 1.5, the major axis at an angle of $45^\circ$ with respect to the incoming shock, and all other properties the same as in run A) on cartesian coordinates. Using an elliptical cloud, rather than a spherical cloud, further reduces the symmetry. For both runs the cell sizes were $0.486\times 0.486$ pc, using $800\times 1600$ (A) and $1600\times 1600$ (B) computational cells. Results of the simulations ========================== Figure \[logdensA\] shows the logarithm of the density for run A at times $0.79\times 10^6$ and $1.1\times 10^6$ years. Figure \[logdensB\] shows the same for run B[^1]. The cocoon shock wave came from the left, and passed the entire cloud at $t=5\times 10^4$ years. At $t=0.79\times 10^6$ years the shock waves travelling into the cloud have just merged (compare with the estimate for $t_{\rm cc}$ in Sect. 2.1). In the non-cooling case this is followed by a re-expansion of the shocked cloud (due to the extra heating generated in the merging of the shocks), but here the excess energy is radiated away, and the merging of the front- and back-side shocks leads to the formation of a dense, cool, elongated, but fragmented structure (‘sheet’) perpendicular to the flow direction in run A, and more parallel to the major axis orientation in run B. In both cases there is a concentration near the centre of the former cloud. The two righthand boxes of Figs. \[logdensA\] and \[logdensB\] show how this sheet fragments further. In run A, the imposed symmetries lead to an elongated concentration of material on the axis, which we measured to contain approximately 10% of the original cloud mass. The rest of the cloud material is compressed into dense structures, spread out over a volume which is 30% of the original cloud size (part of the outer contour of the original cloud boundary is indicated in Fig. \[logdensA\]). In run B the cloud develops into an ensemble of dense small fragments, filling an area of approximately the same diameter as the original cloud. Without imposing symmetry, the largest and densest fragment is found near the centre of the ensemble. The integrated density of this largest fragment was measured to be some 30% of the integrated density of the original cloud. In both runs, at the end of the simulation, less than a percent of the original cloud material has been mixed into the cocoon, showing that the evaporation process is slow, as is expected for high density contrasts. The ensemble does spatially disperse since the velocities of the fragments range from 90 to 500 km s$^{-1}$, the leftmost fragments having the lowest velocities. Discussion ========== The simulations presented here show a completely new behaviour compared to the scenarios presented in @Rees89 (compression), or @Icke99 (disruption). In our simulations, instead of being simply compressed or disrupted, the cloud breaks up into many small dense fragments, spread out over a certain volume, and which evaporate only slowly. This has not been seen in numerical simulations before. It is completely due to the introduction of cooling, which prevents the ‘third phase’ or re-expansion (see Sect. 2). We expect that full three-dimensional simulations will show a result lying somewhat in between what we found in runs A and B. It would definitely enhance rather than suppress the fragmentation, since there is one extra degree of freedom available for instabilities . There are a number of processes which could work against the cooling, and hence slow down the compression. These are for example heating by the UV and X-ray photons from the AGN and the presence of a magnetic field in the clouds. Simulations of magnetized flows in three dimensions, as reported by @gregorietal, show that if the magnetic field is strong enough, it will actually enhance the fragmentation of the cloud, and presumably aid evaporation rather than compression. However, these simulations did not include the effects of cooling, so it is difficult to compare their results to ours. Note that whenever the cooling time is substantially shorter than the cloud crushing time, we expect an evolution similar to the one above. Equation 1 shows that this holds for a wide range of cloud parameters. For example, the [*interstellar*]{} clouds from [@Poludnenkoetal01] should strongly cool, be compressed, and develop into a long-lived mass loading flow, something which these authors failed to achieve in their non-cooling simulations, where the clouds are destroyed within a few $t_{\rm cc}$. The further evolution of our fragments will be dominated by two processes: gravitational collapse, and further acceleration and erosion by the passing flow. All fragments found in our simulations will collapse under their own gravity, which makes them smaller, and even harder to disrupt and/or accelerate. As pointed out in Sect. 4, nearly all of the original cloud material ends up in these dense fragments, and would be available for star formation. This implies that the estimate for the induced star formation rate from @Begelmanetal89, is still valid. For a cloud filling factor (by volume) of $10^{-3}$, and a relativistic jet, they find an induced star formation rate of $\sim 100$ M$_\odot$ yr$^{-1}$, in rough agreement with the observations. Conclusions =========== We have for the first time simulated the cooling dominated evolution of an intergalactic cloud which is overrun by the cocoon of a passing radio jet. Previous analytical studies conjectured that the cloud would either be compressed, or be completely destroyed and evaporate into the cocoon. We instead find a new picture. Radiative cooling is so rapid, that nearly all of the cloud mass is compressed into many small and dense fragments with a long hydrodynamical survival time. These fragments are likely to collapse and form stars, in line with the scenario of jet induced star formation. This type of fragmentation is expected whenever the cooling time is much shorter than the cloud crushing time. Evaluating this condition, shows this to be case for a wide range of parameters, stretching from intergalactic to interstellar conditions, see Eq. 1. The collapse-and-fragment sequence we find, may well be the way to create long lived mass loading flows inside post-shock regions [@HartquistDyson]. These simulations are only a first step, and definitely more work is needed. In future papers we plan to explore the effects three-dimensionality and self-gravity have on the fragmentation process. This work was sponsored by the National Computing Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for Scientific Research (NWO). The research of GM has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. [^1]: Movies of the entire density evolution of the two runs are available with the electronic version of this letter.
--- abstract: 'In this study, we apply the monitoring master equation describing decoherence of internal states to an optically active molecule prepared in a coherent superposition of non-degenerate internal states in interaction with thermal photons at low temperatures. We use vibrational Raman scattering theory up to the first chiral-sensitive contribution, i.e., the mixed electric-magnetic interaction, to obtain scattering amplitudes in terms of molecular polarizability tensors. The resulting density matrix is used to obtain elastic decoherence rates.' author: - Farhad Taher Ghahramani - Afshin Shafiee title: Emergence of Molecular Chirality by Vibrational Raman Scattering --- *Introduction.* Chirality is a fundamental concept in molecular physics and chemistry. Chiral molecules are stable, but not found in symmetric stationary states. In 1927, Hund explained stability of chiral states (and hence instability of their superposition) by a double-well potential model [@Hun]. In Hund’s model, chiral states are assumed to be localized in two minima of the potential. The superposition of chiral states is realized by tunneling between these two minima. However, Hund’s approach seems unsatisfactory for some stable chiral molecules [@Jan]. The problem can be addressed by introducing parity-violating terms in the molecular Hamiltonian [@Rei; @Heg; @Let; @Har; @Qua; @Wes] or non-linear terms due to the interaction with the environment, known as Decoherence program [@Joo; @Sim; @Pfe; @Fai; @Sil; @Has; @Par; @Ber; @Hst1; @Hst2; @Hst3; @Tro] (for a rather complete treatment see [@sch]). The former, despite its small effect, can stabilize chiral states, if it would be larger than the inversion frequency, which is the case for many biologically stable chiral molecules, but the latter has received many attentions. According to the decoherence theory, properties relating to molecular structure like chirality emerge after the interaction of the molecule with the environment [@Sim; @Hst1; @Hst2; @Hst3; @Joo]. A molecule is generally described by translational and internal states, and environment is often modeled as a background gas or thermal photons. The theory of collisional decoherence for a particle with internal states is an extension of the positional decoherence of a particle without internal states [@Gal; @Dio; @Dod; @Hor1; @Hor2; @Hor3; @Hor4; @Vac1; @Hor5; @Vac2]. Hornberger derived a master equation describing internal quantum dynamics of an immobile system [@Hor3] in the so-called monitoring approach (hereafter monitoring master equation), and Vacchini considered decoherence of translational and internal states of a system interacting with an inert gas [@Vac1]. Trost and Hornberger applied the monitoring master equation to decoherence of chiral states of optically active molecules affected by a background gas [@Tro]. Their basic idea is that an initially chiral molecule is blocked in that state through repeated scattering by a host gas. Here, by using monitoring approach of collisional decoherence, we study the chiral stabilization of optically active molecules with internal states by thermal photons. The intermolecular effects are assumed to be negligible, which is the case for a dilute chiral media. Our discussion is limited to low temperatures at which two first states of contortional vibration (responsible for transforming between chiral configurations) are available. This is valid in most cases of interest. The initial state of the molecule is expressed by a coherent superposition of contortional states. Unlike Trost and Hornberger approach [@Tro], we assume that the initial superposition does not necessarily correspond to any chiral configuration. Then, we show that chirality of the molecule emerges due to the interaction with the beam of photons. After a brief introduction to monitoring approach, we derive the monitoring master equation for an immobile two-state system (hereafter implicit master equation). The differential cross-sections appeared in the implicit master equation can be related to the vibrational Raman cross-sections. The theory of vibrational Raman scattering of optically active molecules was first presented by Atkins and Barron, based on polarizability tensors, as an extension of Kramers-Heisenberg formula [@Atk]. We calculate the vibrational Raman cross-sections up to the discriminatory mixed electric-magnetic interaction. The resulting master equation (hereafter explicit master equation) is used to obtain elastic scattering rates. *Monitoring Master Equation.* Let us first explain the most widely used form of incorporating the environment i.e., the weak coupling approach. Long before and long after the collision, particles are well-separated and then evolution of the whole system is governed by the Hamiltonian $\hat {H}_{\mbox{\tiny$\circ$}}=\hat {H}^S_{\mbox{\tiny$\circ$}}+\hat {H}^E_{\mbox{\tiny$\circ$}}$ where $\hat {H}^S_{\mbox{\tiny${\mbox{\tiny$\circ$}}$}}$ and $\hat {H}^E_{\mbox{\tiny$\circ$}}$ are Hamiltonians of the system and the environment, respectively. Then, the total state at time $t$ after scattering is obtained by $$\label{Eq:1} |\psi(t)\rangle=\hat {U}|\psi_{sca}\rangle =\hat {U}\hat {S}|\psi_{inc}\rangle$$ where $\hat {U}=\exp(-{\frac{\imath \hat {H}_{\mbox{\tiny$\circ$}} t}{\hbar}})$ and $\hat{S}$ is the scattering operator. The $S$ matrix is characterized by the interaction Hamiltonian. In the weak coupling approach, interaction is weak, so that a perturbative treatment of the interaction is permissible. The monitoring approach, on the other hand, describes the environmental coupling non-perturbatively by picturing the environment as monitoring the system continuously, i.e., by sending probe particles which scatter off the system at random times. The temporal change of the system is obtained by multiplying the rate of collisions to the state transformation due to a single scattering. In this approach, the time evolution of the density matrix of the system is characterized by [@Hor3] $$\begin{aligned} \label{Eq:2} \partial_{t}\rho^{S} &=\frac{1}{\imath\hbar}[\hat {H}^S_{\mbox{\tiny$\circ$}},\rho^S]+Tr_{E}\Big(\hat {T}\hat\Gamma^\frac{1}{2}\rho^S\otimes\rho^E\hat\Gamma^\frac{1}{2}\hat {T}^{\dagger}\Big )\nonumber \\ & \quad +\frac{\imath}{2}Tr_{E}\Big[\hat\Gamma^\frac{1}{2}Re(\hat {T})\hat\Gamma^\frac{1}{2},\rho^S\otimes\rho^E\Big] \nonumber \\ & \quad -\frac{1}{2}Tr_{E}\Big \{\hat\Gamma^\frac{1}{2}\hat {T}^{\dagger}\hat {T}\hat\Gamma^\frac{1}{2},\rho^S\otimes\rho^E\Big \}\end{aligned}$$ where $[\:]$ and $\{\}$ stand for commutation and anti-commutation relations, respectively. The operator $\hat T$ is the nontrivial part of the two-particle $\hat S$ operator, $\hat {S}=\hat {I}+\imath \hat {T}$ describing the effect of a single collision between environmental particle and system. The operator $\hat\Gamma$ specifies the rate of collisions. In the next section, we apply this master equation to the scattering of a beam of photons from chiral molecules. *Implicit Master Equation.* A chiral molecule transforms between two chiral configurations by a long-amplitude vibration known as contortional vibration. To characterize this vibration, we employ a two-dimensional approach, which is valid for most molecular systems at low temperatures [@Leg]. In this approach, chiral molecule is effectively described by a symmetric double-well potential with two minima. If we denote the small-amplitude vibration in each well by ${\omega}_{\mbox{\tiny$\circ$}}$, and the potential height by ${V}_{\mbox{\tiny$\circ$}}$, in the limit ${{V}_{\mbox{\tiny$\circ$}}}\gg\hbar{\omega}_{\mbox{\tiny$\circ$}}\gg k_{\mbox{\tiny$B$}}T$ (where $T$ is the temperature of the bath and $k_{\mbox{\tiny$B$}}$ is the Boltzmann constant), the first two states of the contortional vibration energy are available. The monitoring master equation is considerably made simple under this assumption. Here, we assume that the interaction of the chiral molecule with the beam of photons does not lead to any recoil of momentum of the molecule, only internal states are changed. This would be the case for massive molecules, in which translational degrees of freedom are fully decohered and therefore dynamics of them can be neglected. Then, the initial state of the molecule $\rho^{M}_{inc}$ for the relevant dynamics can be written as a superposition of first two states of contortional vibration $$\label{Eq:3} \rho^{M}_{inc}=\sum^{2}_{\nu,\nu'=1}c_{\nu}c^{\ast}_{\nu'}|\nu \rangle\langle \nu'|$$ where $|\nu \rangle$ is the energy eigenstate, or a “channel” in the language of standard scattering theory. Note that chiral states are the maximal superposition of two corresponding channels. The diagonal representation of the density matrix of the incident photons can be expressed as $$\label{Eq:4} \rho^{P}_{inc}=\frac {(2\pi\hbar)^{3}}{V}\int d\textbf{k} \mu(\textbf{k})|\eta (\textbf{k},n) \rangle\langle \eta (\textbf{k},n)|$$ where $|\eta (\textbf{k},n) \rangle$ denotes the eigenstate of $\eta$ photons in the mode of momentum $\textbf{k}$ (normalized over box volume $V$) and circular polarization $n$. The momentum state of the incoming photons can be written as a phase space integration over projectors onto minimum uncertainty Gaussian wavepackets. Assuming black-body radiation, the momentum probability distribution of photons in unit volume could be expressed as $$\label{Eq:5} \mu(\textbf{k})d\textbf{k}=\frac{1}{4\pi^{3}\hbar^{3}N}\left(\frac{k^{2}}{e^{ck/k_{\mbox{\tiny$B$}}T}-1}\right)dkd\hat n$$ with $N$ as the number of photons, $c$ as the speed of light and $d\hat n$ as a solid angle differential in momentum space. In the channel basis, time evolution of the reduced density matrix of the molecule is obtained as $$\label{Eq:6} \partial_{t}\rho^{M}=\sum_{\nu'',\nu'''}\partial_{t}\rho^{M}_{\nu''\nu'''}|\nu''\rangle\langle\nu'''|$$ with matrix elements $$\begin{aligned} \label{Eq:7} \partial_{t}\rho^{M}_{\nu''\nu'''} &=\Lambda_{\nu''\nu'''}\rho^M_{\nu'',\nu'''}+\sum_{\nu,\nu'}\rho^M_{\nu,\nu'}M^{\nu\nu'}_{\nu''\nu'''} \nonumber \\ &-\frac {1}{2}\bigg(\sum_{\nu}\rho^M_{\nu,\nu'''}\sum_{\nu^{(4)}}M^{\nu\nu''}_{\nu^{(4)}\nu^{(4)}}\nonumber \\ &+\sum_{\nu'}\rho^M_{\nu'',\nu'}\sum_{\nu^{(4)}}M^{\nu'''\nu'}_{\nu^{(4)}\nu^{(4)}}\bigg)\end{aligned}$$ where $$\label{Eq:8} \Lambda_{\nu''\nu'''}=\frac {E_{\nu''}+\varepsilon_{\nu''}-(E_{\nu'''}+\varepsilon_{\nu'''})}{\imath\hbar}$$ and $\varepsilon_{\nu''}$ is the energy shift of the molecule from energy $E_{\nu''}$ to $E_{\nu''}+\varepsilon_{\nu''}$. The rate coefficients are defined as $$\begin{aligned} \label{Eq:9} M^{\nu\nu'}_{\nu''\nu'''}&=\int d\textbf{k}'\langle\eta(\textbf{k}',n')|\langle\nu''|\hat {T}\hat\Gamma^\frac{1}{2}|\nu\rangle\nonumber \\ & \qquad \qquad \qquad \rho^P_{inc}\langle\nu'|\hat\Gamma^\frac{1}{2}\hat {T}^{\dagger}|\nu'''\rangle|\eta(\textbf{k}',n')\rangle\end{aligned}$$ The rate operator $\hat\Gamma$ is given by $$\label{Eq:10} \hat\Gamma=\sum_{\nu}|\nu\rangle\langle\nu|\otimes n_{\mbox{\tiny$P$}}c\sigma(k,\nu)$$ where $n_{\mbox{\tiny$P$}}$ is the number density of photons, and $\sigma(k,\nu)$ is the total scattering cross section. The elements of $T$-matrix are conveniently defined in terms of the multi-channel scattering amplitude $f$ as $$\begin{aligned} \label{Eq:11} \hat {T}^{kk',nn'}_{\nu\nu'}=\frac{\imath c}{2\pi\hbar}\frac{f_{\nu\nu'}(\textbf{k},n;\textbf{k}',n')}{k} \delta(E_{\nu,k}-E_{\nu',k'})\end{aligned}$$ where $E_{\nu,k}=ck+E_{\nu}$. At first sight, this leads to an ill-defined expression in terms of a squared delta function. However, conservation of the probability current implies a simple rule to deal with the squared matrix element  [@Hor3; @Hor4]. So, we have $$\begin{aligned} \label{Eq:12} &\frac {(2\pi\hbar)^{3}}{V}\hat {T}^{kk',nn'}_{\nu\nu''}\hat {T}^{\dagger kk',nn'}_{\nu'\nu'''}\rightarrow c\chi^{\nu\nu'}_{\nu''\nu'''}\nonumber \\ & \frac{f_{\nu''\nu}(\textbf{k},n;\textbf{k}',n')f^{\ast}_{\nu'''\nu'}(\textbf{k},n;\textbf{k}',n')}{k^{2}\sqrt {\sigma(k,\nu)\sigma(k,\nu')}}\delta(E_{\nu'',k'}-E_{\nu,k})\end{aligned}$$ with $$\label{Eq:13} \chi^{\nu\nu'}_{\nu''\nu'''} = \left\{ \begin{array}{rl} 1 & \text{if } E_{\nu''}-E_{\nu}=E_{\nu'''}-E_{\nu'} \\ 0 & \text{otherwise} \end{array} \right.$$ Then, one obtains the rate coefficients as $$\begin{aligned} \label{Eq:14} M^{\nu\nu'}_{\nu''\nu'''}&=n_{P}c^{2}\chi^{\nu\nu'}_{\nu''\nu'''}\int d\textbf{k}\mu({\textbf{k}})d\textbf{k}'\nonumber \\ & \qquad\frac{f_{\nu''\nu}(\textbf{k},n;\textbf{k}',n')f^{\ast}_{\nu'''\nu'}(\textbf{k},n;\textbf{k}',n')}{k^{2}}\nonumber \\ &\qquad \delta(E_{\nu'',k'}-E_{\nu,k})\end{aligned}$$ Inserting rate coefficients into the density matrix of Eq. (\[Eq:6\]), after some mathematics we obtain $$\begin{aligned} \label{Eq:15} \partial_{t}\rho^{M} &=\frac{n_{P}c^{2}}{2\pi^{3}\hbar^{3}}\int dkd\hat{n}dk'd\hat{n}'\frac{k'^{2}}{e^{ck/k_{\mbox{\tiny$B$}}T}-1}\nonumber \\ & \qquad \sum_{\nu\neq\nu'}\Big[(\rho^M_{\nu'\nu'}-\rho^M_{\nu\nu})|f_{\nu\nu'}|^{2}|\nu\rangle\langle\nu|\nonumber \\ & \qquad \quad -\rho^M_{\nu\nu'}\Big(|f_{\nu\nu}|^2+|f_{\nu\nu'}|^{2}\Big)|\nu\rangle\langle\nu'|\Big]\end{aligned}$$ where $d\textbf{k}'=k'^{2}dk'd\hat {n}'$, and the right side is multiplied by $N$, the number of independent scattering events. Here, photon dependence of scattering amplitudes and corresponding energy conservations are implied for brevity. In the case of elastic scattering, coherences are found to decay exponentially $$\label{Eq:16} \partial_{t}|\rho_{\nu\nu'}|=-\gamma^{ela}_{\nu\nu'}|\rho_{\nu\nu'}|$$ The corresponding scattering rates are determined by the difference of scattering amplitudes $$\label{Eq:17} \gamma^{ela}_{\nu\nu'}=\frac{n_{P}c^{2}}{4\pi^{3}\hbar^{3}}\int dkd\hat{n}dk'd\hat{n}'\frac{k'^{2}}{e^{\frac{ck}{k_{\mbox{\tiny$B$}}T}}-1}|f_{\nu\nu}-f_{\nu'\nu'}|^{2}$$ In the next section, we calculate the corresponding scattering amplitudes. *Scattering amplitudes.* The squared modulus of each scattering amplitude can be related to the corresponding Raman differential scattering cross-section as $$\label{Eq:18} |f_{\nu\nu'}(\textbf{k},n;\textbf{k}',n')|^2=4\pi^2\Big(\frac{k^2}{k'^2}\Big)\Big(\frac{d\sigma_{\nu\nu'}}{dn'}\Big)_{R}$$ Since molecule transforms between two chiral configurations by a vibration, the scattering amplitudes correspond to the vibrational Raman scattering, in which the interaction between chiral molecule and photon changes the vibrational state of the molecule (the electronic state of the molecule being unchanged) corresponding to the change of momentum and polarization of the photon. The corresponding contribution of the scattering amplitude is of the second order with two types of intermediate states, where there is absorption of one photon with momentum $\textbf{k}$ and circular polarization $n$, and emission of one photon with momentum $\textbf{k}'$ and polarization $n'$. Then, initial and final states can be written as $|\nu;\eta(\textbf{k},n)\rangle$ and $|\nu';(\eta -1)(\textbf{k},n),1(\textbf{k}',n')\rangle$. There are two types of scattering amplitudes, one-channel amplitudes ($f_{\nu\nu}$) and two-channel amplitudes ($f_{\nu\nu'}$). It is convenient to develop one-channel amplitudes of Rayleigh scattering first and then convert them to two-channel amplitudes of Raman scattering. In Rayleigh scattering, final state of the molecule is the same as initial state. The matrix element corresponding to the second order is obtained by $$\label{Eq:19} R_{\nu\nu}=\sum_{I}\frac{\langle f|I\rangle\langle I|\hat H_{int}|I\rangle\langle I|i\rangle}{E_{\nu}-E_{I}}$$ where $|i\rangle$ and $|f\rangle$ are initial and final states, summation is over all possible intermediate states $I$, and $\hat H_{int}$ is the molecule-photon interaction Hamiltonian. The leading contribution of the scattering amplitude is purely electric in essence, occurring via electric dipole coupling, by which one cannot recognize optical activity. For chiral molecules, however, it is necessary to include the magnetic dipole coupling, leading to a relatively small chiral-sensitive mixed electric-magnetic contribution in the interaction Hamiltonian. The absolute value of the matrix element corresponding to the interaction of electric- ($\mu$) and magnetic-dipole moment ($m$) of the molecule with the corresponding fields of the light is obtained as [@Cra] $$\begin{aligned} \label{Eq:20} \big|R_{\nu\nu}\big|&=\Big(\frac{\hbar k}{2\varepsilon_{\mbox{\tiny$\circ$}}V}\Big)\eta^{\frac{1}{2}}\big|c {\hat n}'^{\ast}_{i}\hat n_{j}\alpha^{\nu\nu}_{ij}(k)\nonumber \\ & \qquad +{\hat n}'^{\ast}_{i}{(\hat k\times \hat n)}_{j}\beta^{\nu\nu}_{ij}(k)\mp \imath {\hat n}'^{\ast}_{i}\hat n_{j} {\beta}^{\nu\nu\ast}_{ji}(k)\big|\end{aligned}$$ where unit vectors $\hat n$ and $\hat n'$ are incident and scattered polarization vectors and $\hat k$ is direction of momentum of the incident photon. Here, upper and lower minus/plus signs refer to left- and right-circular polarizations of the incident photon. Frequency-dependent electric polarizability and mixed electric-magnetic polarizability are defined as $$\begin{aligned} \label{Eq:21} \alpha^{\nu\nu}_{ij}(k)=\sum_{r}\Big (\frac{\mu^{r\nu}_{i}\mu^{r\nu}_{j}}{E_{r\nu}-\hbar ck}+\frac{\mu^{\nu r}_{j}\mu^{\nu r}_{i}}{E_{r\nu}+\hbar ck}\Big)\nonumber \\ \beta^{\nu\nu}_{ij}(k)=\sum_{r}\Big (\frac{\mu^{r\nu}_{i}m^{r\nu}_{j}}{E_{r\nu}-\hbar ck}+\frac{m^{\nu r}_{j}\mu^{\nu r}_{i}}{E_{r\nu}+\hbar ck}\Big)\end{aligned}$$ where $\mu^{r\nu}=\langle r|\mu|\nu\rangle$ and $m^{r\nu}=\langle r|m|\nu\rangle$, and $E_{r\nu}$ stands for the energy difference. Unlike the electric polarizability tensor $\boldsymbol{\alpha}$ (and its magnetic analogue), mixed electric-magnetic polarizability tensor $\boldsymbol{\beta}$ is parity-variant. Therefore, it can discriminate two chiral configurations. The transition rates can be obtained by Fermi rule as $$\label{Eq:22} \Gamma_{\nu\nu}=\frac{2\pi}{\hbar}\rho|R_{\nu\nu}|^2$$ where $\rho$ is the density of final states $$\label{Eq:23} \rho=\frac{Vk'^{2}dn'}{(2\pi)^{3}\hbar c}$$ with $dn'$ as a solid angle around $k'$. Since in fluids, molecules are randomly oriented, transition rate is obtained by taking a rotational average [@Cra] $$\label{Eq:24} \langle\Gamma_{\nu\nu}\rangle=\frac{4\pi\rho c\eta}{\hbar}\Big(\frac{\hbar k}{2\varepsilon_{\mbox{\tiny$\circ$}}V}\Big )^{2}A_{\nu\nu}$$ with $$\begin{aligned} \label{Eq:25} A_{\nu\nu} &=\mp Re\Big [\frac{\imath}{2}\Big(\delta_{jl}-\hat k_{j}\hat k_{l}\mp\imath\varepsilon_{jlm}\hat k_{m}\Big)\nonumber \\ &\qquad \qquad \qquad {\hat n}'^{\ast}_{i}{\hat n}'_{k}\Big(\langle\alpha^{\nu\nu}_{ij}\beta^{\ast\nu\nu}_{kl}\rangle+\langle\alpha^{\nu\nu}_{ij}\beta^{\nu\nu}_{lk}\rangle\Big)\Big]\end{aligned}$$ where brackets denote the rotational average. Here, squared terms $\alpha^2$ and $\beta^2$ are vanished for random orientations, and elements of polarization of the incident photon was simplified as $$\label{Eq:26} \hat n_{i}\hat n^{\ast}_{j}=\frac{1}{2}\Big(\delta_{ij}-\hat k_{i}\hat k_{j}\mp\imath\varepsilon_{ijl}\hat k_{l}\Big)$$ The rotational averaging of the fourth-rank tensor $\langle\alpha_{ij}\beta_{kl}\rangle$ is calculated by [@Cra] $$\label{Eq:27} \langle\alpha_{ij}\beta_{i'j'}\rangle=I^{(4)}\alpha_{\mu\mu'}\beta_{\lambda\lambda'}$$ with $$\label{Eq:28} I^{(4)}=\frac{1}{30} \begin{bmatrix} \delta_{ij}\delta_{i'j'}\\ \delta_{ii'}\delta_{jj'}\\ \delta_{ij'}\delta_{i'j} \end{bmatrix} ^{T} \begin{bmatrix} 4 & -1 & -1 \\ -1 & 4 & -1 \\ -1 & -1 & 4 \end{bmatrix} \begin{bmatrix} \delta_{\mu\mu'}\delta_{\lambda\lambda'}\\ \delta_{\mu\lambda}\delta_{\mu'\lambda'}\\ \delta_{\mu\lambda'}\delta_{\mu'\lambda} \end{bmatrix}$$ where $T$ means transpose, and Latin and Greek indices refer to space-fixed and molecular-fixed frames, respectively. Summation over repeated tensor suffices is implied. For the non-degenerate molecular states, polarizabilities $\boldsymbol\alpha$ and $\boldsymbol\beta$ can be chosen to be real and imaginary, respectively. So, inserting Eq. (\[Eq:28\]) into Eq. (\[Eq:27\]), and then into Eq. (\[Eq:25\]) one gets $$\begin{aligned} \label{Eq:29} A_{\nu\nu} &=\pm\frac{1}{30}\Big[\big(|\hat n'.\hat k|^{2}\pm5|\hat k.\hat k'|-7\big)\alpha^{\nu\nu}_{\lambda\mu}\beta^{\nu\nu}_{\lambda\mu}+\nonumber \\ &\qquad \qquad \big(3|\hat n'.\hat k|^{2}\mp5|\hat k.\hat k'|+1\big)\alpha^{\nu\nu}_{\mu\mu}\beta^{\nu\nu}_{\lambda\lambda}\Big]\end{aligned}$$ The differential cross-section is obtained by dividing the transition rate to the incident flux of the photons $\eta c/V$ $$\label{Eq:30} \Big(\frac{d\sigma_{\nu\nu}}{dn'}\Big)_{R}=\frac{k^{2}k'^{2}}{8\pi^{2}\varepsilon_{\mbox{\tiny$\circ$}}^{2}c}A_{\nu\nu}$$ To obtain two-channel cross-sections, we extend the results to the case of Raman scattering. As in conventional Raman experiments, we assume that frequency of the incident photon is not near-resonance, i.e. $|E_{r\nu}-\hbar ck|\gg0$. Then, corresponding polarization tensors for the case of Raman scattering after factoring out rotational transitions is obtained by $$\label{Eq:31} \alpha^{\nu\nu'}_{ij}=\langle \nu|\alpha^{\nu\nu}_{ij}|\nu'\rangle, \qquad \beta^{\nu\nu'}_{ij}=\langle \nu|\beta^{\nu\nu}_{ij}|\nu'\rangle$$ where $\alpha^{\nu\nu}_{ij}$ and $\beta^{\nu\nu}_{ij}$ are the usual Rayleigh polarizability tensors which depends on the normal coordinates of nuclei for the relevant vibration. Then, Raman scattering cross-section for optically active molecules is obtained by substituting usual Rayleigh tensors with corresponding Raman tensors. *The Explicit Master Equation.* Inserting the differential cross-sections in Eq. (\[Eq:30\]) into Eq. (\[Eq:18\]) and then into Eq. (\[Eq:15\]), dynamics of density matrix of the molecule is obtained as $$\begin{aligned} \label{Eq:32} \partial_{t}\rho^{M} &=\frac{n_{P}c}{4\pi^{3}\hbar^{3}\varepsilon_{\mbox{\tiny$\circ$}}^{2}}\int dkd\hat{n}dk'd\hat{n}'\frac{k^{4}}{e^{ck/k_{\mbox{\tiny$B$}}T}-1}\nonumber \\ &\quad \sum_{\nu\neq\nu'}\Big[(\rho^M_{\nu'\nu'}-\rho^M_{\nu\nu})A_{\nu\nu'}|\nu\rangle\langle\nu|\nonumber \\ & \qquad -\rho^M_{\nu\nu'}\big(A_{\nu\nu}+A_{\nu\nu'}\big)|\nu\rangle\langle\nu'|\Big]\end{aligned}$$ Here, the fourth power dependence on $k$ (Rayleigh’s law) is appeared. The polarization of the scattered photon can be written as the linear superposition of linear polarizations $\hat n'=1/\sqrt{2}\big(\hat n'^{\parallel}\pm\imath\hat n'^{\perp}\big)$ where $\hat n'^{\parallel}$ and $\hat n'^{\perp}$ are linearly polarized basis vectors. Then, if $\theta$ is the angle between $\hat k$ and $\hat k'$, we have $$\begin{aligned} \label{Eq:33} A_{\nu\nu} &=\pm\frac{1}{30}\Big[\big(\frac{1}{\sqrt{2}}\sin^{2}\theta\pm5\cos \theta-7\big)\alpha^{\nu\nu}_{\lambda\mu}\beta^{\nu\nu}_{\lambda\mu}\nonumber \\ &\qquad \qquad +\big(\frac{3}{\sqrt{2}}\sin^{2}\theta\mp5\cos\theta+1\big)\alpha^{\nu\nu}_{\mu\mu}\beta^{\nu\nu}_{\lambda\lambda}\Big]\end{aligned}$$ The squared of the amplitudes are isotropic, depending only on the magnitude of $\textbf{k}$ and the scattering angle $\theta$. Then, we can carry out the angular integrations by $$\label{Eq:34} \int d\hat {n}d\hat {n}'\rightarrow 8\pi^{2}\int d(\cos \theta)$$ and the momentum integral can be computed using the definition of the Riemann $\zeta$-function for integer $n$ $$\label{Eq:35} \zeta (n)=\frac{1}{(n-1)!}\int^{\infty}_{0}dx\frac{x^{n-1}}{e^{x}-1}$$ So, we finally obtain $$\begin{aligned} \label{Eq:36} \partial_{t}\rho^{M}&=\frac{8n_{p}k^{5}_{\mbox{\tiny$B$}}T^{5}}{5\pi\hbar^{3}c^{4}\varepsilon_{\mbox{\tiny$\circ$}}^{2}} \sum_{\nu\neq\nu'}\Big[(\rho^M_{\nu'\nu'}-\rho^M_{\nu\nu})B_{\nu\nu'}|\nu\rangle\langle\nu|\nonumber \\ & \qquad \qquad \quad -\rho^M_{\nu\nu'}\big(B_{\nu\nu}+B_{\nu\nu'}\big)|\nu\rangle\langle\nu'|\Big]\end{aligned}$$ with $$\label{Eq:37} B_{\nu\nu}=\mp\big[\frac{38}{3\sqrt{2}}\alpha^{\nu\nu}_{\lambda\mu}\beta^{\nu\nu}_{\lambda\mu}-\frac{6}{\sqrt{2}}\alpha^{\nu\nu}_{\mu\mu}\beta^{\nu\nu}_{\lambda\lambda}\big]$$ Inserting corresponding amplitudes into Eq. (\[Eq:17\]), and assuming no phase difference between $f_{\nu\nu}$ and $f_{\nu'\nu'}$, the elastic decoherence rates are obtained as $$\label{Eq:38} \gamma^{ela}_{\nu\nu'}=\frac{4n_{p}k^{5}_{\mbox{\tiny$B$}}T^{5}}{5\pi\hbar^{3}c^{4}\varepsilon_{\mbox{\tiny$\circ$}}^{2}} \Big|B^{\frac{1}{2}}_{\nu\nu}-B^{\frac{1}{2}}_{\nu'\nu'}\Big|^{2}$$ In order to estimate an order of magnitude for the decoherence rate, one should calculate electric polarizability $\alpha$ and electric-magnetic polarizability $\beta$ tensors for the two-state contortional vibration mode in Rayleigh scattering from a chiral media. The quantum mechanical calculations of Rayleigh optical activity can be employed for our purpose. Results quoted in the literature are usually expressed in terms of mean $(\alpha\beta)^{\nu\nu}$ and anisotropic $(\gamma^{2})^{\nu\nu}$ invariant observables [@Bar] $$\begin{aligned} \label{Eq:39} (\alpha\beta)^{\nu\nu}&=\frac{1}{9}\alpha^{\nu\nu}_{\lambda\lambda}\beta^{\nu\nu}_{\mu\mu} \nonumber \\ (\gamma^{2})^{\nu\nu}&=\frac{1}{2}\big(3\alpha^{\nu\nu}_{\lambda\mu}\beta^{\nu\nu}_{\lambda\mu}-\alpha^{\nu\nu}_{\lambda\lambda}\beta^{\nu\nu}_{\mu\mu}\big)\end{aligned}$$ The calculations show that the mean invariant $(\alpha\beta)^{\nu\nu}$ is usually 1-3 orders of magnitude smaller than the anisotropic invariant $\gamma^{2}$ [@Zub], and polarizability of molecules at their vibrational excited states goes smoothly to larger values [@Mar]. Then, polarization-dependent term in decoherence rate would be at the order of $(\gamma^{2})^{00}$. The order of magnitude of a typical $(\gamma^{2})^{00}/c$ ($c$ is the speed of light) is about $10^{-83}C^{2}V^{-2}m^{4}$ [@Zub]. So, after making explicit the temperature dependence of number density of photons, the order of magnitude of decoherence rate could estimated as $10^{-95}(T/K)^{8}s^{-1}$. This shows clearly that the environmental photons cannot cause any suppression of interference between ground and excised states. At low temperatures, the maximal superpositions of first two relevant molecular states are chiral states. Then, according to einselection rule, decoherence of molecular states is equivalent to the stabilization of chiral states. *Conclusion.* Chemists and some physicists are using chirality in the classical sense, i.e., by presupposing a molecule to be in one particular chiral state. However, according to decoherence theory, classical properties like chirality emerge out as a consequence of the interaction with the environment. Based on this approach, we have explored the collisional decoherence of a chiral molecule prepared in a coherent superposition of non-degenerate internal states in interaction with thermal photons. The temperature is assumed low, so that first two states of the relevant vibration of the molecule would be available. The reduced density matrix of the molecule are obtained in Eq. (\[Eq:15\]) as an extension of monitoring master equation. The appeared differential scattering amplitudes are calculated using vibrational Raman scattering theory to obtain the final master equation in Eq. (\[Eq:36\]). The corresponding elastic decoherence rates are calculated in Eq. (\[Eq:38\]). According to its estimated value, one can claim that the chirality of a molecule is an emergent property resulted due to the interaction of the molecule with a beam of photons. *Acknowledgement.* We acknowledge the financial support of Iranian National Science Foundation (INSF) for this work. 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--- bibliography: - 'misc.bib' - 'hqet.bib' - 'nrqcd-qed.bib' - 'hbchpt.bib' - 'pipi.bib' --- [Matching in Nonrelativistic Effective Quantum Field Theories]{} Inauguraldissertation\ der Philosophisch-naturwissenschaftlichen Fakultät\ der Universität Bern vorgelegt von [Alexander Gall]{} von Laupen Leiter der Arbeit: ------------------------------------------------------------------------ Introduction ============ \[chap:introduction\] Relativistic quantum field theories (RQFTs) describe the interaction of particles at energies accessible in today’s experiments. In most cases, exact solutions are not known and one has to resort to perturbation theory or lattice calculations. The former is only valid at energies where the interaction is small (in the asymptotic region). In an infrared free theory like QED, there can be an additional problem when the energies and momenta of the process under consideration become small. Consider, for example, the differential cross section of $e^+e^-$ scattering in the center of mass system to leading order. Expanding in the relative velocity $v$ one finds that it diverges like $1/v^4$. The reason for this nonsensical result is the fact that perturbation theory breaks down at a scale of the order of $m_e\alpha^2$ – one would have to sum infinitely many graphs that all give contributions of the same order of magnitude. Non-perturbative effects of this kind are notoriously difficult to handle in a RQFT. In this particular process, the particles can form a bound state which shows up as an isolated pole of the fermionic four-point function in the center of mass momentum, which cannot be seen to any finite order in ordinary perturbation theory. The tool to study this object is the so-called homogeneous Bethe-Salpeter equation. It is a fourth-order integro-differential equation for the “wave function”, which is essentially the residue of the pole. This is a rather complicated object and no methods are known to solve this difficult mathematical problem exactly. All approaches to solve the Bethe-Salpeter equation perturbatively take advantage of the fact that the scale $m_e\alpha^2$ is much smaller than $m_e$, suggesting that a non-relativistic approximation is a good starting point. It turns out that this procedure suffers from numerous technical problems and despite the long history of the topic, there is to date no truly systematic perturbation theory available. Caswell and Lepage [@caswell-lepage] pointed out that the traditional approach is not well adapted to the non-relativistic nature of the problem. After all, simple quantum mechanics gives the energy levels of positronium quite accurately. They recognized that the source of all the problems is the existence of a hierarchy of physical scales: the electron mass $m_e$, the typical bound state momentum $m_e\alpha$ and the bound state energy $m_e\alpha^2$. In the relativistic treatment, all of these scales are present in the integral kernel of the Bethe-Salpeter equation and it is very difficult to expand it systematically. They suggest that one should first construct an effective theory, in which the physics that takes place at scales of the order of $m_e$ or higher are represented by local interactions of the fields, which are suppressed by powers of $1/m_e$. The coefficients of these terms are determined by comparing scattering amplitudes with those of the full theory at energies where bound states can be neglected. With the information about high energies encoded in the effective couplings, one can then perform bound state calculations with the effective theory. The point is that the remaining physical scales are much smaller than $m_e$. As a consequence, no additional heavy particles (of mass $m_e$) can be created[^1] and the theory is confined to a subspace of the Fock space in which their number is conserved. This is precisely the setting one has in quantum mechanics: As long as we don’t try to resolve processes taking place at a scale of the order of the Compton wavelength $1/m_e$, the description in terms of a wave function that obeys a Schrödinger equation is perfectly adequate. The concept of a non-relativistic quantum field theory (NRQFT) outlined above is the bridge between field theory and quantum mechanics. It is equivalent to the full theory below the heavy scale but takes advantage of the non-relativistic character of some degrees of freedom by incorporating relativistic effects in a systematic expansion in inverse powers of some heavy scale $M$. The interaction of heavy particles is described by a Schrödinger equation whose Hamilton operator is obtained from the Lagrangian of the NRQFT. What seemed so hard to do in the RQFT, namely the summation of the non-perturbative part of the theory, simply amounts to solving a lowest order approximation of this Schrödinger equation exactly. One can then use standard methods of quantum mechanics to perform a systematic perturbation theory from there. The formalism has been applied to various processes with considerable success (the following references are only a selection and by no means complete). NRQCD, the low-energy version of QCD, was used to study bound states of heavy quarks by Bodwin, Braaten and Lepage [@lepage-bodwin-braaten]. Muonium and Positronium hyperfine splitting was already considered in [@caswell-lepage] and later extended to higher order corrections [@labelle-thesis; @labelle-lepage; @labelle-zebarjad-burgess; @hoang-labelle-zebarjad; @kinoshita-nio-1; @kinoshita-nio-2]. Another system where a NRQFT approach can be useful is the bound state formed by $\pi^+$ and $\pi^-$. Because the binding energy is of the order of keV, it probes the $\pi\pi$ interaction practically at threshold. The decay width of this atom is related to the $\pi\pi$ scattering lengths and will be measured soon in the DIRAC experiment at CERN [@DIRAC], providing a high precision test of low energy QCD. The leading term of the lifetime was given by Uretsky and Palfrey [@uretsky-palfrey]. Recently, corrections have been calculated using different techniques to solve the Bethe-Salpeter equation in the relativistic framework [@jallouli-sazdjian; @rusetsky1; @rusetsky2; @rusetsky3; @rusetsky4]. Another approach, based on non-relativistic potential models, was pursued by the authors of refs. [@gashi; @badertscher-1; @badertscher-2]. First attempts using a NRQFT approach have been published [@labelle-buckley; @kong-ravndall] but need further clarification. Let us also mention that there is a different branch of NRQFTs, where there is only one heavy particle involved. In this case, the scales $m_e\alpha$, $m_e\alpha^2$ are absent and with them the non-perturbative effects. The heavy particle can be considered to be static and power counting becomes very simple. This version of a NRQFT is used for the description of mesons containing one heavy quark under the name of heavy quark effective theory (HQET) and also for the pion-nucleon system where it is called heavy baryon chiral perturbation theory (HBCHPT). See refs. [@hqet-review; @hbchpt-review] for reviews on these subjects. A crucial step in the construction of a NRQFT is the matching with the fundamental theory, where the coupling constants are adjusted such that the scattering amplitudes agree to some order in inverse powers of the heavy scale. In order to do this, one needs to renormalize both theories, i.e. introduce a regularization scheme that allows to absorb the divergences of Green’s functions into the coupling constants in a systematic way. Also, one has to express the physical mass of the heavy particle in terms of the parameters of the theory and determine the effective normalization of the field (the “wave function renormalization”) due to self energy effects. This is certainly no problem in the RQFT, which is expressed in a Lorentz covariant form. The NRQFT is not covariant and it is not a priori clear how these tasks should be performed there. The fact that it took some time for people to realize that in some versions of HQET and HBCHPT the fields were incorrectly normalized even at tree level [@balk:1993; @hbchpt-wave-function-ecker] shows that this question is not as innocent as it may seem. Unfortunately, the discussions are often obscured by the formalism of the particular model under consideration. However, as in a RQFT, the procedure of mass and wave function renormalization is independent of a particular model and can be treated once and for all in the language of the one-particle irreducible two-point function. To the best of the author’s knowledge, such a discussion is not available in the literature. The present work tries to fill this gap by studying how this mechanism works in the case of a heavy scalar field. We only consider Yukawa-type couplings to other scalar fields to avoid complications due to gauge symmetry and spin. This work is organized as follows. In chapter \[chap:matching\] we show how amplitudes and Green’s functions of a generic Lagrangian with one heavy scalar field can be matched with the corresponding effective theory. In chapter \[chap:toy-model\], we consider a toy-model and explicitly construct two non-local effective Lagrangians that are equivalent to the full theory in the pure particle- and anti-particle sectors, verifying the general statements made in chapter \[chap:matching\]. Finally, the $1/M$ expansion of tree-level Green’s functions and Amplitudes in the full theory is discussed in chapter \[chap:M-expansion\] and it is shown how they can be reproduced by the effective theory order by order in powers of the inverse heavy scale. Matching in the Particle Sector =============================== \[chap:matching\] Transition Amplitudes {#sec:transition-amplitudes} --------------------- To have a specific example and to keep things simple at the same time, we consider a theory of the form $$\begin{aligned} \label{eq:L-gener} {\mathcal L}&= {\mathcal L}^0+ \bar{\mathcal L}^0 + {\mathcal L}^{\text{int}} \notag \\ {\mathcal L}^0 &= \partial_\mu{H}^*\partial^\mu{H}-M^2{H}^*{H}.\end{aligned}$$ Here, $\bar{\mathcal L}^0$ contains the kinetic part of all the fields that interact with ${H}$ through the interaction Lagrangian ${\mathcal L}^{\text{int}}$. We assume that the masses of these fields are all much smaller than $M$, i.e. ${H}$ is the only heavy degree of freedom. As such, they appear unaltered in the effective theory that describes physics at a scale much smaller than $M$. Therefore, we first concentrate on processes among heavy particles alone. The free Lagrangian of the heavy field has a $U(1)$ symmetry and the particles carry a charge which is conserved in all processes if ${\mathcal L}^{\text{int}}$ respects this symmetry. We shall refer to the two types of field quanta as particle- and anti-particle. They enter the free Lagrangian symmetrically and can only be distinguished by the interaction with an external field. Scattering processes which are related by crossing are described by the same invariant amplitude. ### Relativistic Theory {#sec:full-theory} The fundamental objects we have to study are the connected Green’s functions $$\label{eq:G-conn} G^{(2n)}(x,y) = {\langle0|T\hat{H}(x)\hat{H}^\dag(y)|0\rangle}_c.$$ Here, $x,y$ are vectors $(x_1,\dots,x_n)$, $(y_1,\dots,y_n)$ and we use the notation $\hat f(x)\equiv f(x_1)\dots f(x_n)$. Further notation is given in appendix \[app:notation\]. To each external momentum corresponds a two-point function $G^{(2)}$ and we define the truncated function $G_{tr}^{(2n)}$ by $$\label{eq:def-truncation} G^{(2n)}(p,q) = \hat G^{(2)}(p)\hat G^{(2)}(q)G_{tr}^{(2n)}(p,q).$$ Each of the factors $G^{(2)}(p_i)$ has a pole when the momentum is on the mass shell $p_i^2={M_{\text{p}}}^2$, where ${M_{\text{p}}}$ is the physical mass of the particle. The scattering amplitude, involving $2n$ heavy particles in this case, is related to the residue of the multiple pole when all momenta are put on their mass shells. The precise relation is given by the LSZ formalism summarized in appendix \[app:reduction\] for the case at hand. Applied to the process with $n$ heavy particles in the initial and final states $$\begin{gathered} {\langlep_1,\dots,p_n;\text{out}|q_1,\dots,q_n;\text{in}\rangle} = {\langlep_1,\dots,p_n;\text{in}|q_1,\dots,q_n;\text{in}\rangle} \\ + i(2\pi)^4\delta^4\left(P-Q\right) T_{n\rightarrow n},\end{gathered}$$ where $P=\sum_{i=1}^n p_i$ and $Q=\sum_{i=1}^n q_i$, we find $$T_{n\rightarrow n} = \frac{1}{i}Z_{H}^n \left.G_{tr}^{(2n)}(p,q)\right|_{\text{on-shell}}.$$ “On-shell” means $p_i^0={\omega_{\text{p}}}(\mathbf p_i) = \sqrt{{M_{\text{p}}}^2+\mathbf p_i^2}, q_i^0={\omega_{\text{p}}}(\mathbf q_i)$ and $Z_{H}$ is the residue of the two-point function $G^{(2)}$. Note that, due to the manifest covariance of the theory, this quantity transforms as a scalar under the Lorentz group. In such a process, heavy anti-particles are only involved as virtual states. Therefore, it should be possible to remove them as an explicit degree of freedom and incorporate them into the interaction. ### Separating Particles and Anti-Particles {#sec:free-fields} The first step towards this goal is to separate particles and anti-particles in the free field. Consider the equation of motion $$\label{eq:KG-eq} (\Box + M^2){H}= 0$$ obtained from ${\mathcal L}^0$. The most general solution is a superposition of plane waves $${H}(x) = \int \frac{d^3p}{(2\pi)^32\omega(\mathbf p)}\left(a(p)e^{-ipx}+b^*(p)e^{ipx}\right).$$ To separate the positive and negative frequency contributions, we define the differential operators (see also appendix \[app:KG\]) $$\begin{aligned} D_\pm &= \pm i\partial_t - \sqrt{M^2-\Delta} \\ d &= (2\sqrt{M^2-\Delta})^{-\frac{1}{2}}\end{aligned}$$ and set $$\label{eq:def-hpm} {H}_\pm = -D_\mp d{H}.$$ With this choice we have $$\begin{aligned} {{H}_+}(x) &= \int\frac{d^3p}{(2\pi)^3\sqrt{2\omega(\mathbf p)}} a(p) e^{-ipx} \\ {{H}_-}(x) &= \int\frac{d^3p}{(2\pi)^3\sqrt{2\omega(\mathbf p)}} b^*(p) e^{ipx}\end{aligned}$$ and $${H}= d({{H}_+}+ {{H}_-}).$$ The operator $d$ is not really necessary for this decomposition and was only introduced for later convenience. The fields ${H}_\pm$ satisfy the equations $$D_\pm{H}_\pm(x) = 0,$$ which are the Euler-Lagrange equations of the Lagrangians $${\mathcal L}_\pm^0 = {H}_\pm^*D_\pm{H}_\pm.$$ After canonical quantization, the operators ${{H}_+}^\dag$ and ${{H}_-}$ create a particle and an anti-particle state, respectively (see appendix \[app:canonical-quantisation\]). ### Effective Theory in the Particle Sector {#sec:eff-particle-theory} A theory for the particle sector should therefore be of the form $$\label{eq:Lp-gener} {{\mathcal L}_+}= {{\mathcal L}_+}^0 + \bar{\mathcal L}^0 + {{\mathcal L}_+}^{\text{int}}.$$ In appendix \[app:reduction\] it is shown that the Fock space of the free heavy particles, in which the incoming and outgoing particles live, is the same as in the relativistic theory. This is obviously a necessary condition for the existence of an interpolating field that should reproduce transition amplitudes of a relativistic theory. The interaction Lagrangian is a local function of the fields and their derivatives and can be written as $${{\mathcal L}_+}^{\text{int}} = \sum_{\nu=1}^\infty \frac{1}{M^\nu}{{\mathcal L}_+}^\nu,$$ where ${{\mathcal L}_+}^\nu$ contains $\nu$ space or time derivatives. This means that we deal here with an effective field theory in which $M$ is considered to be a hard scale. It can only describe processes in which all relevant scales are much smaller than that. In practice, one always truncates the Lagrangian at some power in $1/M$ but for the sake of the following arguments, let us assume that we have summed up the contributions to all orders and postpone the discussion of this issue. $U(1)$ symmetry of the Lagrangian insures that the heavy field only occurs in the combination ${{H}_+}^\dag{{H}_+}$, which means that the number of heavy particles is conserved at each vertex (${{H}_+}$ destroys an incoming particle and ${{H}_+}^\dag$ creates an outgoing one) and therefore for any process (this is simply the consequence of charge conservation when there is only one type of charge). The theory is thus naturally confined to a subspace of the Fock space in which the number of heavy particles is fixed. We can start by writing down the most general interaction Lagrangian which respects the symmetries of ${\mathcal L}$. However, we can immediately see that Lorentz symmetry is already violated by ${{\mathcal L}_+}^0$. The question is then, how much of this symmetry we have to incorporate into ${{\mathcal L}_+}$ to be able to calculate a transition amplitude with the correct transformation properties under the Lorentz group. Let us formulate a pragmatic approach to the problem. Due to the lack of knowledge of the transformation properties of ${{H}_+}$ under the Lorentz group[^2], we only require rotational invariance of the Lagrangian. We can then calculate the connected Green’s functions $$G_+^{(2n)}(x,y) = {\langle0|T\hat{{H}_+}(x)\hat{{H}_+}^\dag(y)|0\rangle}_c.$$ Next, we can try to derive a reduction formula for this theory, relating transition amplitudes to poles of these Green’s functions. As shown in appendix \[app:reduction\], this involves one non-trivial assumption about the structure of the two-point function, namely that it permits the definition of a physical mass ${M_{\text{p}}}$ so that $$G_+^{(2)}(p) = \frac{1}{i}\frac{Z_+(\mathbf p^2)}{{\omega_{\text{p}}}(\mathbf p)-p^0-i\epsilon} + \dots$$ This implies that not all of the coupling constants of the original Lagrangian are independent. No additional assumptions are needed to define the object $$T_{n\rightarrow n}^+ = \frac{1}{i}\prod_{i=1}^nZ_+(\mathbf p_i^2)^\frac{1}{2} Z_+(\mathbf q_i^2)^\frac{1}{2} \left.G_{+,tr}^{(2n)}(p,q)\right|_{\text{on-shell}},$$ where the truncated function is defined by $$G_+^{(2n)}(p,q) = \hat G_+^{(2)}(p)\hat G_+^{(2)}(q)G_{tr}^{(2n)}(p,q).$$ $T_{n\rightarrow n}^+$ does not yet transform as a scalar under the Lorentz group as it should if it is supposed to reproduce $T_{n\rightarrow n}$. ### Matching {#sec:amplitude-matching} Symmetry only fixes each term in the Lagrangian up to a factor. These low energy constants (LEC) are at our disposal and can be chosen in such a way that all scattering amplitudes considered above are identical. This procedure is called matching. Before we formulate it, we should say a word about the normalization of one-particle states, because the amplitudes clearly depend on them. Although arbitrary, there is still a most natural choice of normalization (see also appendix \[app:canonical-quantisation\]). In the full theory, we chose it to be Lorentz invariant $${\langlep|p'\rangle} = {\langle\bar p|\bar p'\rangle} = 2{\omega_{\text{p}}}(\mathbf p) (2\pi)^3 \delta^3(\mathbf p-\mathbf p'),$$ where as for ${{\mathcal L}_+}$ we chose $${\langlep|p'\rangle} =(2\pi)^3 \delta^3(\mathbf p-\mathbf p').$$ Therefore, before we try to match amplitudes, we must make up for this difference in normalization by replacing, say, the states used in the effective theory by $${|p\rangle} \rightarrow \sqrt{2\omega_{\text{p}}(\mathbf p)}{|p\rangle}.$$ The matching condition then reads $$\label{eq:T-matching} T_{n\rightarrow n} = \prod_{i=1}^n\sqrt{2{\omega_{\text{p}}}(\mathbf p_i)} \sqrt{2{\omega_{\text{p}}}(\mathbf q_i)} T_{n\rightarrow n}^+,$$ which automatically restores Lorentz symmetry for the transition amplitudes. In the last section we have seen that the effective theory is actually an expansion in inverse powers of the heavy scale $M$. The matching can only make sense if the relativistic amplitude possesses such an expansion in the region of phase space we are interested in. Green’s Functions {#sec:greens-functions} ----------------- The matching of scattering amplitudes involves only Green’s functions evaluated on the mass shell of all particles involved. They are, however, also interesting in the unphysical region because they reflect general properties of quantum field theories like unitarity in their non-trivial analytic structure. It is interesting to see how the Green’s functions of the fundamental theory compare to the ones of the effective theory. Being unphysical quantities, off-shell Green’s functions have no unique definition. Redefinitions of the fields that do not change the classical field theory give, in general, different off-shell results while describing the same physics. Suppose we have chosen a particular off-shell extrapolation in the fundamental theory. Naively, one may be tempted to identify the truncated functions $G_{tr}^{(2n)}$ with $G_{+,tr}^{(2n)}$, i.e. consider the latter to be the $1/M$ expansion of the former. One would then expect that they differ only by a polynomial in the momenta which can be absorbed by a proper choice of coupling constants in the effective theory. However, this is not true, as we will show now. Two remarks about the following statements are in order. First, we suppose that renormalization was performed in both theories and that everything is finite and well defined. Second, as mentioned before, the effective theory is an expansion in $1/M$. Therefore, the matching is actually performed order by order in $1/M$ and we assume that the relativistic Green’s functions can be expanded in this way. Let us start with eq. . It can be written in terms of the truncated Green’s functions as $$\begin{gathered} \label{eq:T-matching-2} Z_{H}^n \left.G_{tr}^{(2n)}(p,q)\right|_{\text{on-shell}}= \\ \prod_{i=1}^n (Z_+(\mathbf p_i^2){\omega_{\text{p}}}(\mathbf p_i))^\frac{1}{2} (Z_+(\mathbf q_i^2){\omega_{\text{p}}}(\mathbf p_i))^\frac{1}{2} \left.G_{+,tr}^{(2n)}(p,q)\right|_{\text{on-shell}}.\end{gathered}$$ Without knowing the relationship between the residues $Z_{H}$ and $Z_+$, we cannot express, say, $G_{tr}^{(2n)}$ in terms of quantities that can be calculated with ${{\mathcal L}_+}$ alone. In appendix \[app:2pf\] it is shown how such a relation emerges from the matching of the two-point functions. The statement is that when the irreducible parts $\Sigma$, $\Sigma_+$ defined by $$\begin{aligned} G^{(2)}(p) &= \frac{1}{i}\frac{1}{M^2-p^2 + i\Sigma(p^2) -i\epsilon} \\ G_+^{(2)}(p) &= \frac{1}{i}\frac{1}{\omega(\mathbf p)-p^0 + i\Sigma_+(p^0,\mathbf p^2) - i\epsilon},\end{aligned}$$ are matched according to $$\Sigma_+(p^0,\mathbf p^2) = \frac{\Sigma(p^2)}{2\omega(\mathbf p)+ \frac{i\Sigma(p^2)}{\omega(\mathbf p)+p^0}},$$ the physical masses defined by $$\begin{aligned} {M_{\text{p}}}&= M^2 + i\Sigma({M_{\text{p}}}^2) \\ {\omega_{\text{p}}}(\mathbf p) &= \sqrt{{M_{\text{p}}}^2+\mathbf p^2} = \omega(\mathbf p) + i\Sigma_+({\omega_{\text{p}}}(\mathbf p), \mathbf p^2)\end{aligned}$$ are identical and the residues of $$\begin{aligned} G^{(2)}(p) &= \frac{1}{i}\frac{Z_{H}}{{M_{\text{p}}}^2-p^2-i\epsilon} + \text{regular},p^2\rightarrow{M_{\text{p}}}^2 \\ G_+^{(2)}(p) &= \frac{1}{i}\frac{Z_+(\mathbf p^2)} {{\omega_{\text{p}}}(\mathbf p)-p^0-i\epsilon} + \text{regular},p^0\rightarrow{\omega_{\text{p}}}(\mathbf p)\end{aligned}$$ are related by $$Z_+(\mathbf p^2) = \frac{(\omega(\mathbf p) + {\omega_{\text{p}}}(\mathbf p))^2}{4{\omega_{\text{p}}}(\mathbf p)\omega(\mathbf p)} Z_{H}.$$ If we plug this into eq. , we find $$\label{eq:T-matching-3} \left. G^{(2n)}_{tr}(p,q)\right|_{\text{on-shell}} = \prod_{i=1}^n \frac{\omega(\mathbf p_i)+{\omega_{\text{p}}}(\mathbf p_i)}{\sqrt{2\omega(\mathbf p_i)}} \frac{\omega(\mathbf q_i)+{\omega_{\text{p}}}(\mathbf q_i)}{\sqrt{2\omega(\mathbf q_i)}} \left.G_{+,tr}^{(2n)}(p,q)\right|_{\text{on-shell}}$$ and all quantities on the r.h.s. can be calculated with the Lagrangian ${{\mathcal L}_+}$. Let us extend this relation to off-shell Green’s functions. For this purpose we define a new truncation procedure $$\label{eq:def-trunc-bar} G_+^{(2n)}(p,q) = \hat \mathcal G_+(p) \hat \mathcal G_+(q) \bar G_{+,tr}^{(2n)}(p,q)$$ with $$\label{eq:eff-ext-line} \mathcal G_+(p) \doteq \frac{G_+^{(2)}(p)}{\sqrt{2\omega}} \left(1-\frac{i\Sigma_+(p^0,\mathbf p^2)}{\omega(\mathbf p) + p^0}\right)$$ and impose the off-shell matching condition $$\label{eq:gf-matching} G_{tr}^{(2n)}(p,q) = \bar G_{+,tr}^{(2n)}(p,q),$$ which indeed reduces to eq  on the mass shell. The functions $G_{+,tr}^{(2n)}$ and $\bar G_{+,tr}^{(2n)}$ differ essentially by the self-energy $\Sigma_+$, which is a non-trivial function of momentum. This is the reason why a matching between the “naturally” truncated functions $G_{tr}^{(2n)}$ and $G_{+,tr}^{(2n)}$ is impossible - they differ by more than just a polynomial. This point will be illustrated in section \[sec:p-a-sector\] in a simple toy-model. Construction of the Effective Lagrangian for a simple Model {#chap:toy-model} =========================================================== The Model --------- The model we are considering is given by $$\begin{aligned} \label{eq:L-toy-model} \bar{\mathcal L}^0 &\equiv {\mathcal L}_{l}^0 = \frac{1}{2}\partial_\mu {l}\partial^\mu {l}-\frac{m^2}{2}{l}^2 \notag \\ {\mathcal L}^{\text{int}} &= e{H}^*{H}{l}\end{aligned}$$ in the notation of section \[sec:transition-amplitudes\]. To stay in the scope of that section we chose $m\ll M$ and refer to $l$ as the light field. It will always keep its relativistic form. In the following, we will explicitly construct an effective theory of the form given in eq.  that can be proven to reproduce the scattering amplitudes in the sector where there is a fixed number of heavy particles and an arbitrary number of light particles. Interaction with an External Field {#sec:external-field} ---------------------------------- In this section, the light field ${l}$ is a given function of space and time and we consider the Lagrangian $$\label{eq:L-h-l-ext} {\mathcal L}^{ext} = {\mathcal L}^0 + {\mathcal L}^{\text{int}}+ j^*{H}+ {H}^*j.$$ The equation of motion $$\label{eq:eom-h-l-ext} D_e {H}\doteq (D_M-e{l}){H}= j,$$ where $D_M = \Box+M^2$, has the formal solution $$\label{eq:sol-h-l-ext} {H}= D_e^{-1}j.$$ $D_e^{-1}$ is the complete two-point function of this theory and can be expressed in terms of the free propagator $D_M^{-1}$ defined in appendix \[app:KG\] as $$\label{eq:inv-De-as-inv-D} D_e^{-1} = D_M^{-1}\frac{1}{1-e{l}D_M^{-1}}.$$ We define the truncated two-point function $T$ by $$\label{eq:def-T} D_e^{-1} = D_M^{-1} + D_M^{-1}TD_M^{-1}.$$ In perturbation theory, it is simply a string of free propagators with insertions of the external field $$\label{eq:T-perturbative} T = e{l}+ e^2 {l}D_M^{-1}{l}+O(e^3).$$ All information about a particle moving in the external field is contained in this operator. The possible physical processes are the scattering of a particle or an anti-particle (including the formation of bound states if the external field allows them), pair-annihilation and pair-creation. We are about to construct two independent non-local theories that can reproduce the scattering processes for particles and anti-particles separately. To this end, we define the fields ${H}_\pm$ as in eq.  and introduce the vectors $$\begin{aligned} \vec{H}&= \begin{pmatrix} {{H}_+}\\ {{H}_-}\end{pmatrix} & \vec j &= \begin{pmatrix} j \\ j \end{pmatrix}\end{aligned}$$ and the operator $$\begin{aligned} D &= \begin{pmatrix} A & eB \\ eB & C \end{pmatrix} \notag \\ A &= D_+ + eB \notag \\ C &= D_- + eB \notag \\ B &= d{l}d.\end{aligned}$$ It is easy to check that $\vec{H}$ obeys $$\label{eq:eom-H-l-ext} D \vec{H}= -d\vec j.$$ Writing $D^{-1}$ as $$\label{eq:def-inf-DFW} D^{-1} = \begin{pmatrix} G_1 & G_2 \\ G_3 & G_4 \end{pmatrix}$$ and using the fact that $j$ is arbitrary, we find the operator identity $$\label{eq:De-decomposition} D_e^{-1} = -\sum_{n=1}^4 d G_n d.$$ To explore the significance of this, we investigate the structure of the $G_n$. They can be expressed in terms of the $A,B,C$ defined above by solving the equation $D D^{-1} = \mathbf 1$. Their structure in terms of the Green’s functions $D_\pm^{-1}$ allows for a definition of truncated objects $T_{\pm\pm}$ just like in (\[eq:def-T\]) $$\begin{aligned} \label{eq:def-G1} G_1 &= (A-e^2BC^{-1}B)^{-1} \doteq D_+^{-1} - D_+^{-1}T_{++}D_+^{-1} \\ G_2 &= -eA^{-1}BG_4 \doteq -D_+^{-1}T_{+-}D_-^{-1} \\ G_3 &= -eC^{-1}BG_1 \doteq -D_-^{-1}T_{-+}D_+^{-1} \\ G_4 &= (C-e^2BA^{-1}B)^{-1} \doteq D_-^{-1} - D_-^{-1}T_{--}D_-^{-1}.\end{aligned}$$ It is straight forward to show that the $T_{\pm\pm}$ are all essentially equal to $T$ (see appendix \[app:T\]). More precisely, we find that $$\label{eq:Tpm-T-equivalence} T_{++}=T_{+-}=T_{-+}=T_{--}=dTd$$ holds to all orders in perturbation theory. We have therefore found a decomposition of the r.h.s. of eq.  in which each of the four pieces contains the complete truncated function $T$. We define a non-local Lagrangian for each of the fields ${H}_\pm$ by $$\begin{aligned} \label{eq:L-hp-ext} \mathcal L_\pm^{ext} &=& {H}_\pm^*\mathcal D_\pm{H}_\pm \nonumber \\ \mathcal D_+ &=& A -e^2BC^{-1}B \nonumber \\ \mathcal D_- &=& C -e^2BA^{-1}B.\end{aligned}$$ The associated two-point functions $$\begin{aligned} {\langle0|T{{H}_+}(x){{H}_+}^\dag(y)|0\rangle} &= iG_1(x,y) \\ {\langle0|T{{H}_-}(x){{H}_-}^\dag(y)|0\rangle} &= iG_4(x,y)\end{aligned}$$ contain all the information about the interaction of one particle and one anti-particle with the external field, respectively. Note that pair creation or annihilation processes are not included: the fields ${H}_\pm$ do not talk to each other. Let us illustrate the connection between the original Lagrangian and these two effective Lagrangians for the case of scattering in a static field ${l}={l}(\mathbf x)$. In the notation for in- and out states introduced in appendix \[app:reduction\], the transition amplitudes $T_\pm$ for particle- and anti-particle scattering are defined by $$\begin{aligned} {\langlep;\text{out}|q;\text{in}\rangle} &= {\langlep;\text{in}|q;\text{in}\rangle}+i2\pi\delta(p^0-q^0) T_+(p,q) \\ {\langle\bar p;\text{out}|\bar q;\text{in}\rangle} &= {\langle\bar p;\text{in}|\bar q;\text{in}\rangle}+i2\pi\delta(p^0-q^0) T_-(p,q).\end{aligned}$$ Fourier transformation is defined as in appendix \[app:notation\] with the difference that only the energy is conserved $$2\pi\delta(p^0-q^0)T(p,q) = \int d^4x d^4y e^{i(px-qy)}T(x,y).$$ The physical momenta of incoming particles (anti-particles) and outgoing particles (anti-particles) are given by $q(-q)$ and $p(-p)$, respectively. Applying the reduction formula of appendix \[app:reduction\], we find in the full theory $$T_\pm(p,q) = \left. T(\pm p,\pm q)\right|_{p^0= q^0=\omega(\mathbf p)},$$ whereas the effective theories give $$T_\pm(p,q) = \left. \frac{1}{\sqrt{2\omega(\mathbf p)}} \frac{1}{\sqrt{2\omega(\mathbf q)}} T(\pm p,\pm q)\right|_{p^0= q^0=\omega(\mathbf p)}.$$ The additional kinematical factors $1/\sqrt{2\omega}$ are due to the different normalizations of free one-particle states. We have thus verified that the Lagrangians ${\mathcal L}_\pm^{ext}$ produce scattering amplitudes that automatically satisfy the matching condition stated in eq. . Non-local Lagrangians in the Particle and Anti-Particle Sector {#sec:p-a-sector} -------------------------------------------------------------- We return to the original Lagrangian defined in eq. , where ${l}$ represents a dynamical degree of freedom. The results of the last section can be used to construct two non-local Lagrangians that are equivalent to the original theory in the pure particle- and anti-particle sectors of the heavy field including any number of light fields. Ultimately, these Lagrangians will be brought to a local form by expanding in $1/M$. It is the expanded version that is a true [*effective*]{} theory in the sense that it reproduces the fundamental theory only at low energies. The non-local version still contains the complete information about truncated Green’s functions as we are about to show now. ### Green’s Functions {#sec:p-q-greens-functions} We consider the generating functional $Z$ of all Green’s functions and perform the integration over the heavy field. In appendix \[app:det\] it is shown that it can be written in the form $$\begin{aligned} \label{eq:Z-integrated} Z[j,j^*,J] &= \frac{1}{\mathcal Z} \int [d{l}](\det D_+^{-1}\mathcal D_+)^{-1}e^{i\int \mathcal L_{l}^0 + j^*D_e^{-1}j+J{l}} \\ \mathcal Z &= \int [d{l}](\det D_+^{-1}\mathcal D_+)^{-1}e^{i\int \mathcal L_{l}^0},\end{aligned}$$ with $D_e$ and $\mathcal D_+$ given in and , respectively. The determinants are evaluated in $D\neq 4$ dimensions where they are finite to all orders in perturbation theory, i.e. we deal here with a regularized but not renormalized theory. The statements derived in this section are a priori only valid within this framework. In appendix \[app:1-loop-renormalization\], we determine the counter terms necessary to render all Green’s functions finite in $D=4$ to one loop (i.e. $O(e^2)$). By working only to this order in perturbation theory, the results of this section can be proven to hold also in $D=4$. Now consider the theory defined by $$\label{eq:L-hp} \mathcal L_+ = {H}_+^*\mathcal D_+{H}_+ + \mathcal L_{l}^0.$$ Its generating functional after integration over ${H}_+$ is $$\label{eq:Z-hp-integrated} Z_+[j,j^*,J] = \frac{1}{\mathcal Z} \int [d{l}] (\det D_+^{-1}\mathcal D_+)^{-1} e^{i\int \mathcal L_{l}^0 - j^*\mathcal D_+^{-1}j + J{l}}.$$ This is simply $Z$ with $D_e^{-1}$ replaced by $-\mathcal D_+^{-1}$. In the last section we have found that they can be written as $$\begin{aligned} \label{eq:p-a-T-def} D_e^{-1} &= D_M^{-1}\left( 1 + T D_M^{-1}\right) \\ \label{eq:p-a-D-eff} \mathcal D_+^{-1} &= D_+^{-1}\left(1 - dTd D_+^{-1}\right).\end{aligned}$$ The first equation is the definition of $T$ which is to be considered as a functional of ${l}$ within the path integrals above. Let us first consider the $n$-point functions (the tilde distinguishes them from the connected functions defined below) $$\begin{aligned} \label{eq:orig-n-point-f} \tilde G^{(a,b)}(x,y,z) &= \langle 0|T\,\hat{H}(x)\hat{H}^+(y)\hat{l}(z)|0\rangle \nonumber \\ &= \left. \frac{1}{i^n}\frac{\delta^n Z}{\widehat{\delta j}^*(x) \widehat{\delta j}(y)\widehat{\delta J}(z)}\right|_{j=j^*=J=0},\end{aligned}$$ where $(a,b)$ is a pair of integers with $2a+b=n$ and $x,y,z$ are vectors $(x_1,\dots,x_a)$, $(y_1,\dots,y_a)$, $(z_1,\dots,z_b)$. We recall that we use the shorthand notation for the product of fields and the definition of the Fourier transform as given in appendix \[app:notation\]. The functions $\tilde G_+^{(a,b)}$ of the effective theory are defined through $Z_+$ in an analogous manner. The derivatives with respect to the sources $j$,$j^*$ bring down factors of $D_e^{-1}$ and $\mathcal D_+^{-1}$ in $Z$ and $Z_+$, respectively. It is clear that the free parts $D_M^{-1}$ and $D_+^{-1}$ of eqns.  and  only contribute to disconnected Green’s functions (except for the two-point functions, see below) and we ignore them for the moment. Denoting a permutation $P$ of the coordinates $y_i$ by $$P(y_1,\dots,y_a) = (y_{P_1},\dots,y_{P_a}),$$ the remaining contributions to $\tilde G^{(a,b)}$ and $\tilde G_+^{(a,b)}$ can then be written as the sum over all permutations of the term $$\frac{1}{i^a}\frac{1}{\mathcal Z} \int [d{l}](\det D_+^{-1}\mathcal D_+)^{-1}\prod_{i=1}^a f(x_i,y_{P_i})\prod_{j=1}^b {l}(z_j)e^{i\int {\mathcal L}_{l}^0}.$$ For $Z$, the function $f$ is given by $$f(u,v) = \int d^Ds d^Dt \Delta_M(u-s)T(s,t)\Delta_M(t-v)$$ and for $Z_+$ by $$f(u,v) = -\int d^Ds d^Dt \Delta_+(u-s)d_s T(s,t) d_t \Delta_+(t-v).$$ The point is that ${l}$ only occurs in $T$, which is the same in both expressions. The free propagators, which form the endpoints of external legs corresponding to heavy particles, and the differential operators $d$ can be taken out of the remaining path integral. Since we have already discarded some disconnected pieces, it is useful to consider only connected Green’s functions denoted by $G^{(a,b)}$ and $G_+^{(a,b)}$, generated by the functionals $iW$ and $iW_+$ defined by $$\begin{aligned} e^{iW[j,j^*,J]} &\doteq Z[j,j^*,J] \\ e^{iW_+[j,j^*,J]} &\doteq Z_+[j,j^*,J]\end{aligned}$$ in analogy with eq. . What we have found above is that these functions differ only by the outermost parts of their external heavy lines. More precisely, if we write ($u$,$v$,$w$ are vectors like $x$,$y$,$z$ and $\Delta_m$ is the propagator of the light field obtained from $\Delta_M$ by replacing $M$ by $m$) $$\begin{aligned} G^{(a,b)}(x,y,z) =&\; \frac{1}{i^{n}}\int d^Du d^Dv d^D w \prod_{i=1}^a\prod_{j=1}^b \Delta_M(x_i-u_i) \\ & S(u,v,w)\Delta_M(v_i-y_i)\Delta_m(z_j-w_j) \\ G_+^{(a,b)}(x,y,z) =&\; \frac{(-1)^{2a}}{i^{n}}\int d^Du d^Dv d^D w \prod_{i=1}^a\prod_{j=1}^b \Delta_+(x_i-u_i) \\ & d_{u_i}S_+(u,v,w)d_{v_i}\Delta_+(v_i-y_i)\Delta_m(z_j-w_j),\end{aligned}$$ for $2a+b>2$ we have $S=S_+$ to any order in perturbation theory. In particular, $G_+^{(a,b)}$ has the full loop structure of $G^{(a,b)}$. Let us consider the two-point functions of the heavy fields in detail. In momentum space we find $$\begin{aligned} \label{eq:G2-orig} G^{(1,0)}(p) &= \frac{1}{i}\Delta_M(p)\left( 1 + S(p)\frac{1}{i}\Delta_M(p) \right) \\ \label{eq:G2-eff} G_+^{(1,0)}(p) &= i\Delta_+(p)\left( 1 + \frac{S(p)}{2\omega(\mathbf p)} i\Delta_+(p) \right)\end{aligned}$$ with $S(p)$ being the Fourier transform of $$S(x-y) = \frac{i}{\mathcal Z}\int [dl](\det D_+^{-1}\mathcal D_+)^{-1} T(x,y) e^{i\int {\mathcal L}_{l}^0}.$$ The interesting thing about this is that the irreducible two-point functions $\Sigma$, $\Sigma_+$ defined by $$\begin{aligned} G^{(1,0)}(p) &= \frac{1}{i}\frac{1}{M^2-p^2+i\Sigma(p^2)-i\epsilon} \\ G_+^{(1,0)}(p) &= \frac{1}{i}\frac{1}{\omega(\mathbf p)-p^0 + i\Sigma_+(p^0,\mathbf p) - i\epsilon}\end{aligned}$$ automatically obey the equation $$\Sigma_+(p^0,\mathbf p^2) = \frac{\Sigma_(p^2)}{2\omega(\mathbf p) + \frac{i\Sigma(p^2)}{\omega(\mathbf p)+p^0}}$$ that was [*imposed*]{} as a matching condition in the general discussion of the of two-point functions of a relativistic theory and a non-relativistic effective theory in appendix \[app:2pf\]. Based on this matching, we have discussed in section \[sec:greens-functions\] how off-shell truncated Green’s functions can be matched. The statements made there are true in this model and we conclude that if we truncate external lines through the function $$\label{eq:p-a-new-G-def} \mathcal G_+(p) \doteq \frac{G_+^{(1,0)}(p)}{\sqrt{2\omega}} \left(1-\frac{i\Sigma_+(p^0,\mathbf p^2)}{\omega(\mathbf p) + p^0}\right)$$ according to $$\label{eq:p-a-truncation-rule} G_+^{(a,b)}(p,q,k) = \hat \mathcal G_+(p) \hat \mathcal G_+(q) \hat G_+^{(0,2)}(k) \bar G_{+,tr}^{(a,b)}(p,q,k),$$ the equation $$\label{eq:p-a-trunc-relation} G_{tr}^{(a,b)}(p,q,k) = \bar G_{+,tr}^{(a,b)}(p,q,k)$$ is true to all orders in perturbation theory. Furthermore, the residues $Z_{H}$ and $Z_+$ of $G^{(1,0)}$ and $G_+^{(1,0)}$ are related by $$\label{eq:p-a-residues} Z_+(\mathbf p^2) = \frac{(\omega(\mathbf p) + {\omega_{\text{p}}}(\mathbf p))^2}{4{\omega_{\text{p}}}(\mathbf p)\omega(\mathbf p)} Z_{H}.$$ ### Amplitudes As a consequence of eqns.  and , the on-shell relation $$\begin{gathered} \label{eq:on-shell-equiv-particle} Z_{H}^a Z_{l}^{\frac{b}{2}} \left. G_{tr}^{(a,b)}(p,q,k)\right|_{\text{on-shell}} = \\ \prod_{i=1}^{a}\left(Z_+(\mathbf p_i^2) 2{\omega_{\text{p}}}(\mathbf p_i)\right)^{\frac{1}{2}} \left(Z_+(\mathbf q_i^2) 2{\omega_{\text{p}}}(\mathbf q_i)\right)^{\frac{1}{2}} Z_{l}^{\frac{b}{2}} \left. G_{+,tr}^{(a,b)} (p,q,k)\right|_{\text{on-shell}},\end{gathered}$$ where $p_i^0=\omega_{\text{p}}(\mathbf p_i), q_i^0=\omega_{\text{p}}(\mathbf q_i)$ and $k_i^0=\sqrt{m_{\text{p}}^2+\mathbf k^2}$ is also true. According to the LSZ formalism, the l.h.s. is related to the amplitude of the process where $a$ heavy particles scatter into $a$ heavy and $b$ light particles[^3] $$\begin{gathered} {\langlep_1,\dots,p_a,k_1,\dots,k_b;\text{out}|q_1,\dots,q_a;\text{in}\rangle} = \\ {\langlep_1,\dots,p_a,k_1,\dots,k_b;\text{in}|q_1,\dots,q_a;\text{in}\rangle}\\ + i(2\pi)^4\delta^4\left(P+K-Q\right) T_{a\rightarrow a+b},\end{gathered}$$ where $P=\sum_{i=1}^a p_i$ etc. ,through $$\label{eq:Taab-rel} T_{a\rightarrow a+b} = \frac{1}{i}Z_{H}^a Z_{l}^{\frac{b}{2}} \left. G_{tr}^{(a,b)}(p,q,k)\right|_{\text{on-shell}}.$$ The same amplitude in the effective theory is given by $$\label{eq:Taab-eff} T_{a\rightarrow a+b}^+ = \frac{1}{i}\prod_{i=1}^aZ_+(\mathbf p_i)^\frac{1}{2} Z_+(\mathbf q_i)^\frac{1}{2} Z_{l}^{\frac{b}{2}} \left. G_{+,tr}^{(a,b)}(p,q,k)\right|_{\text{on-shell}}$$ and eq. \[eq:on-shell-equiv-particle\] is simply the statement that $$T_{a\rightarrow a+b} = \prod_{i=1}^a\sqrt{2{\omega_{\text{p}}}(\mathbf p_i)} \sqrt{2{\omega_{\text{p}}}(\mathbf q_i)} T_{a\rightarrow a+b}^+,$$ which is nothing but the matching condition stated in section \[sec:amplitude-matching\]. We can repeat this procedure with the Lagrangian $$\label{eq:L-hm} {{\mathcal L}_-}= {{H}_-}^*\mathcal D_-{{H}_-}+ {\mathcal L}_{l}^0,$$ describing the anti-particle sector of the theory. In the relativistic theory, the amplitude for the process where all particles are replaced by anti-particles is obtained by a simple change of sign of the momenta $p$ and $q$ as a consequence of crossing symmetry. In the effective theory, however, the crossed process is described by its own amplitude $G_{-,tr}^{(a,b)}$ and we get (on-shell has the same meaning as above) $$\begin{gathered} \label{eq:on-shell-equiv-anti-particle} Z_{H}^a Z_{l}^{\frac{b}{2}} \left. G_{tr}^{(a,b)}(-p,-q,k)\right|_{\text{on-shell}} = \\ \prod_{i=1}^{a}\left(Z_-(\mathbf p_i^2) 2{\omega_{\text{p}}}(\mathbf p_i)\right)^{\frac{1}{2}} \left(Z_-(\mathbf q_i^2) 2{\omega_{\text{p}}}(\mathbf q_i)\right)^{\frac{1}{2}} Z_{l}^{\frac{b}{2}} \left. G_{-,tr}^{(a,b)} (-p,-q,k)\right|_{\text{on-shell}}.\end{gathered}$$ The connection with the amplitudes $T_{\bar a\rightarrow\bar a+b}$,$T_{\bar a\rightarrow\bar a+b}^+$ of the scattering of $a$ anti-particles into $a$ anti-particles and $b$ light particles is analogous to eqns.  and  and we arrive at the same conclusions as above. We have demonstrated in this section that the non-local Lagrangians ${\mathcal L}_\pm$ defined in eqns.  and  generate scattering amplitudes in the pure particle- and anti-particle sector (including any number of light particles) that are related to the corresponding quantities in the full theory by the matching condition described in section \[sec:amplitude-matching\]. Strictly speaking, the expressions given in eqns.  and  are valid to all orders in perturbation theory only in the presence of a regulator that renders all loops finite. However, the non-local theory is related so closely to the original one that it is evident that once the full theory is renormalized to some order in $e$, these expressions are valid up to the same order, because the very same counter terms render both theories finite at the same time (see appendix \[app:1-loop-renormalization\] for the explicit renormalization to one loop). ### Comment on the Structure of Green’s Functions The seemingly complicated relation   between the Green’s functions of the relativistic and the effective theory is in fact quite simple. Let us illustrate this with the 3-point functions $G^{(1,1)}$ and $G_+^{(1,1)}$ to $O(e^3)$. The former can be depicted as the sum of the graphs[^4] of figure \[fig:G3-rel\]. ![The graphs contributing to the 3-point function $G^{(1,1)}$ to $O(e^3)$. The solid and dashed lines represent propagators $\Delta_M$ and $\Delta_m$, respectively.[]{data-label="fig:G3-rel"}](figure1.ps){width="\textwidth"} The corresponding function $G_+^{(1,1)}$ can be obtained from these graphs by the following simple rules. - Replace all internal propagators $\Delta_M(p)$ by the sum $$\Delta_M(p) = -\frac{1}{2\omega(\mathbf p)}\left( \Delta_+(p) + \Delta_-(p) \right).$$ - Replace all external heavy propagators by particle propagators according to $$\Delta_M(p) \rightarrow \frac{1}{\sqrt{2\omega(\mathbf p)}} \Delta_+(p).$$ The resulting graphs are shown in figure \[fig:G3-eff\]. It is convenient to display the decomposition of $\Delta_M$ only for the lines that connect 1-particle irreducible subgraphs. ![The graphs contributing to the 3-point function $G_+^{(1,1)}$ to $O(e^3)$. The solid and dashed lines represent propagators $\Delta_M$ and $\Delta_m$, the double line particle propagators $1/(2\omega)\Delta_+$ and the thick solid line anti-particle propagators $1/(2\omega)\Delta_-$. External heavy lines are multiplied with an additional factor of $\sqrt{2\omega}$ so that they effectively correspond to $1/\sqrt{2\omega}\Delta_+$.[]{data-label="fig:G3-eff"}](figure2.ps){width="\textwidth"} The meaning of the truncation rule in eq.  becomes now apparent. The function $G_{tr}^{(1,1)}$ is given by the sum of graphs a) and e) of figure \[fig:G3-rel\] with external lines removed. The “naturally” truncated function $G_{+,tr}^{(1,1)}$, however, is the sum of graphs a), e), f) and g) with external factors of $\Delta_+$ and $\Delta_m$ removed. The point is that some parts that belong to insertions on the heavy external lines in the relativistic theory are now considered to belong to the irreducible vertex function because the anti-particle propagator $\Delta_-$ is considered to be irreducible. The modified truncation rule, involving $\mathcal G$ defined in eq. , on the other hand gives the truncated function $\bar G_{+,tr}^{(1,1)}$ which only contains graphs a) and e). We have thus verified explicitly the equation $$G_{tr}^{(1,1)} = \bar G_{+,tr}^{(1,1)}$$ to $O(e^3)$. $1/M$ Expansion =============== \[chap:M-expansion\] The Lagrangians constructed in the preceding chapter are non-local, i.e. they depend on the entire configuration space. The explicit expression for $\mathcal L_+$ is $$\begin{gathered} \label{eq:L-hp-non-local} \mathcal L_+(x) = \int d^4y{{H}_+}^*(x)\left(\delta^4(x-y)(D_{+,y}-eB(y))- \right. \\ \left. e^2B(x)C^{-1}(x,y)B(y){{H}_+}(y)\right).\end{gathered}$$ We showed that this theory contains the same truncated Green’s functions as the original local field theory. The whole purpose of the construction of $\mathcal L_\pm$ is to pave the way for the expansion of these Green’s functions in the region where all energies and momenta are small compared to the mass $M$. This expansion turns the non-local Lagrangians into local ones, which should be able to reproduce the expansion of relativistic Green’s functions. In this chapter, we first look at a few simple processes in the relativistic theory and discuss their $1/M$ expansion at tree level. Then we perform the expansion in the non-local Lagrangian and discuss how perturbation theory works. Finally, we check the method in the case of the scattering of a heavy and a light particle at tree level. Expansion of Relativistic Amplitudes at Tree Level {#sec:on-shell-expansion} -------------------------------------------------- We consider the truncated Green’s functions $G_{tr}^{(2,0)}$ and $G_{tr}^{(1,2)}$ on the mass shell, i.e. the heavy momenta obey $p^2={M_{\text{p}}}^2$ and the light momenta $k^2=m_{\text{p}}^2$. ### Heavy-Heavy Scattering The function $G_{tr}^{(2,0)}(p_1,p_2,q_1,q_2)$ involves only heavy external particles. With the convention for the Fourier transform of Green’s functions given in appendix \[app:notation\], $q_1$,$q_2$ are the physical momenta of incoming particles and $p_1$,$p_2$ those of outgoing ones. Therefore, $q_1+q_2$ is the total energy in the CMS of particle-particle scattering. We define the Mandelstam variables $$\begin{aligned} s &= (q_1 + q_2)^2 \notag \\ t &= (q_1 - p_1)^2 \notag \\ u &= (q_1 - p_2)^2\end{aligned}$$ related by $$s+t+u = 4{M_{\text{p}}}^2.$$ The invariant amplitude $$\label{eq:heavy-heavy-amplitude} A(s,t,u) = \frac{1}{i} Z_{H}^2 \left. G_{tr}^{(2,0)}(p_1,p_2,q_1,q_2)\right|_{\text{on-shell}}$$ describes several physical processes in different regions of momentum space (cf. figure \[fig:heavy-heavy-scattering\]). We define the amplitudes belonging to the various channels by $$\begin{aligned} \label{eq:heavy-heavy-channels} A_s(s,t,u) &= \left.A(s,t,u)\right|_{q_1^0,q_2^0,p_1^0,p_2^0>0} \notag \\ A_t(t,s,u) &= \left.A(s,t,u)\right|_{q_1^0,p_2^0>0;q_2^0,p_1^0<0} \notag \\ A_u(u,t,s) &= \left.A(s,t,u)\right|_{q_1^0,p_1^0>0;q_2^0,p_2^0<0},\end{aligned}$$ writing the energy in the CMS and the momentum transfer as the first and second arguments, respectively. In the $s$-channel, $A(s,t,u)$ describes particle-particle scattering and in the $t$- and $u$-channels particle-anti-particle scattering. ![Physical processes associated with the amplitude $A(s,t,u)$ defined in eq. . In the $s$-channel it describes the scattering of two heavy particles and in the $t$- and $u$-channels the scattering of a particle and an anti-particle. The lines are labeled by the physical momenta in the respective channels.[]{data-label="fig:heavy-heavy-scattering"}](figure3.ps){width="\textwidth"} The presence of identical particles is reflected in the crossing symmetry $$A(s,t,u) = A(s,u,t).$$ In perturbation theory we write $$A(s,t,u) = e^2 A^{(2)}(s,t,u) + O(e^4)$$ and find that the lowest order is given by the two tree-level Feynman diagrams shown in figure \[fig:heavy-heavy-tree\] $$\label{eq:A2} A^{(2)}(s,t,u) = \frac{1}{m^2-t} + \frac{1}{m^2-u}.$$ ![The graphs that contribute to $A(s,t,u)$ at tree-level.[]{data-label="fig:heavy-heavy-tree"}](figure4.ps) Let us expand this quantity for the case when all three-momenta as well as $m$ are much smaller than the heavy scale $M$. This expansion has to be performed separately in each channel and we start with the $s$-channel. It is convenient to work in the CMS, where $q_1=(\sqrt{s}/2,\mathbf q)$, $q_2=(\sqrt{s}/2,-\mathbf q)$, $p_1=(\sqrt{s}/2,\mathbf p)$ and $p_2=(\sqrt{s}/2,-\mathbf p)$. The CM energy $s$ is of the order of $4M^2$ and thus represents a hard scale, where as the momentum transfer $t$ and $u = 4M^2 - s-t$ are soft $$\begin{aligned} t &= -(\mathbf q - \mathbf p)^2 \\ u &= -(\mathbf q + \mathbf p)^2\end{aligned}$$ Therefore, both denominators in eq.  are small and $$A_s^{(2)}(s,t,u) = \frac{1}{m^2+(\mathbf q - \mathbf p)^2} + \frac{1}{m^2+(\mathbf q + \mathbf p)^2}.$$ In the $t$-channel, $t$ is the hard CM energy. In the CMS, where $q_1=(\sqrt{t}/2,\mathbf q)$, $p_1=(-\sqrt{t}/2,\mathbf q)$, $q_2=(-\sqrt{t}/2,\mathbf p)$ and $p_2=(\sqrt{t}/2,\mathbf p)$, we have $$t = 4(M^2 + \mathbf q^2).$$ The momentum transfer $s$ and $$u = -(\mathbf q - \mathbf p)^2$$ are still soft. In this channel, the first graph of figure \[fig:heavy-heavy-tree\] represents an annihilation process, where the particle and anti-particle convert into a light particle which is then considerably off its mass shell, followed by pair production. The leading term of the expanded propagator is of $O(1/M^2)$ and indicates that this process looks essentially point-like in configuration space on a scale much larger than $1/M$. The second term involves only the exchange of soft momenta and has a leading piece that is not suppressed by powers of $1/M$ $$\label{eq:At2} A_t^{(2)}(t,s,u) = \frac{1}{m^2+(\mathbf q - \mathbf p)^2} - \frac{1}{4M^2}\left(1-\frac{4\mathbf q^2-m^2}{4M^2} + O(\frac{1}{M^4})\right).$$ ### Heavy-Light Scattering Let us chose the momentum assignment in the Fourier transform of $G^{(1,2)}$ as follows $$\begin{gathered} (2\pi)^4\delta^4(p+k_2-q-k_1)G^{(1,2)}(p,q,k_1,k_2) = \\ \int d^4xd^4yd^4z_1d^4z_2 e^{ipx - iqy + ik_2z_2 - ik_1z_1}G^{(1,2)}(x,y,z_1,z_2).\end{gathered}$$ With this choice, $q$,$k_1$ are the physical momenta of incoming particles and $p$,$k_2$ those of outgoing ones. Therefore, $q+k_1$ is the total energy in the CMS of the process where a light particle scatters off a heavy one and we chose the Mandelstam variables $$\begin{aligned} s &= (q + k_1)^2 \notag \\ t &= (q - p)^2 \notag \\ u &= (q - k_2)^2\end{aligned}$$ with $$s+t+u = 2({M_{\text{p}}}^2 + m_{\text{p}}^2).$$ The different processes represented by $$\label{eq:heavy-light-amplitude} B(s,t,u) = \frac{1}{i} Z_{H}Z_{l}\left. G_{tr}^{(1,2)}(p,q,k_1,k_2)\right|_{\text{on-shell}}$$ ![Physical processes associated with the amplitude $B(s,t,u)$ defined in eq. . Solid and dashed lines represent heavy and light particles, respectively. In the $s$- and $u$- channels it describes scattering and in the $t$-channel pair-annihilation. The lines are labeled by the physical momenta in the respective channels.[]{data-label="fig:heavy-light-scattering"}](figure5.ps){width="\textwidth"} are shown in figure \[fig:heavy-light-scattering\] and the amplitudes in the different channels are defined in analogy with eq.  $$\begin{aligned} \label{eq:heavy-light-channels} B_s(s,t,u) &= \left.B(s,t,u)\right|_{q^0,p^0,k_1^0,k_2^0>0} \notag \\ B_t(t,s,u) &= \left.B(s,t,u)\right|_{q^0,k_2^0>0;p^0,k_1^0<0} \notag \\ B_u(u,t,s) &= \left.B(s,t,u)\right|_{q^0,p^0>0;k_1^0,k_2^0<0}.\end{aligned}$$ We refer to the $s$- and $u$-channels as Compton scattering and the $t$-channel as pair-annihilation. Because of the crossing symmetry $$B(s,t,u) = B(u,t,s),$$ we can again restrict the analysis to the $s$- and $t$-channels. Let us set $$B(s,t,u) = e^2 B^{(2)}(s,t,u) + O(e^4)$$ where $B^{(2)}$ is given by the Feynman diagrams displayed in figure \[fig:heavy-light-tree\] $$B^{(2)}(s,t,u) = \frac{1}{M^2-s} + \frac{1}{M^2-u}.$$ ![The graphs that contribute to $B(s,t,u)$ at tree-level.[]{data-label="fig:heavy-light-tree"}](figure6.ps) In contrast to the processes considered above, this amplitude explicitly depends on the heavy scale through the propagator $$\Delta_M(p) = \frac{1}{M^2-p^2}.$$ The construction of the non-local Lagrangians ${\mathcal L}_\pm$ relied essentially on the decomposition $$\label{eq:prop-decomposition} \Delta_M(p) = \frac{1}{2\omega(\mathbf p)}\left( \frac{1}{\omega(\mathbf p)-p^0} + \frac{1}{\omega(\mathbf p)+p^0} \right).$$ of this function, representing the propagation of a particle and an anti-particle separately. The important point is that when $p^0$ is in the vicinity of $+\omega(\mathbf p)$, the first term dominates where as the second one can be expanded in powers of $1/M$ and vice versa if $p^0$ is in the vicinity of $-\omega(\mathbf p)$. In configuration space, the first graph of figure \[fig:heavy-light-tree\] may be depicted as the sum of the two graphs in figure \[fig:heavy-light-z-graph\]. ![Decomposition of the first graph of figure \[fig:heavy-light-tree\] according to eq.  in configuration space (an integration over the internal points $u$,$v$ is implied).[]{data-label="fig:heavy-light-z-graph"}](figure7.ps) In these diagrams, the internal propagators correspond to factors $d^2\Delta_-(v-u)$ and $d^2\Delta_+(u-v)$, respectively (cf. appendix \[app:KG\] for the definition of these objects). Due to its shape, the first graph is called a “Z” graph in the language of old-fashioned (non-covariant) perturbation theory. In the $s$-channel, the incident light particle pushes the incident heavy particle only slightly off the mass shell, so that the internal anti-particle propagator in the $Z$ graph is far away from its pole at $p^0=-\omega(\mathbf p)$ and is suppressed relative to the other graph. The $Z$ graph looks like an effective local four-particle interaction $$\includegraphics{figure8.ps}$$ Let us work in the rest frame of the incoming heavy particle where $q=(M,0)$, $k_1=(\Omega(\mathbf k_1),\mathbf k_1)$, $p=(\omega(\mathbf p),\mathbf p)$, $k_2=(\Omega(\mathbf k_2),\mathbf k_2)$ and $\Omega(\mathbf k)=\sqrt{m^2+\mathbf k^2}$. The contribution to the amplitude $B_s^{(2)}(s,t,u)$ of the Z graph is $$\begin{gathered} \frac{1}{2\omega(\mathbf k_1)} \frac{1}{\omega(\mathbf k_1) + \Omega(\mathbf k_1) + M} = \frac{1}{4M^2}\left( 1 - \frac{\Omega(\mathbf k_1)}{2M} + O(\frac{1}{M^2}) \right). \end{gathered}$$ The other part of the diagram gives the leading contribution $$\begin{gathered} \frac{1}{2\omega(\mathbf k_1)} \frac{1}{\omega(\mathbf k_1) - \Omega(\mathbf k_1) - M} = \\ \frac{-1}{2M\Omega(\mathbf k_1)}\left( 1 + \frac{\mathbf k_1^2}{2M\Omega(\mathbf k_1)} + O(\frac{1}{M^2})\right)\end{gathered}$$ and $$\begin{aligned} \label{eq:Bs2} B_s^{(2)}(s,t,u) =&\; \frac{-1}{2M\Omega(\mathbf k_1)}\left( 1 + \frac{\mathbf k_1^2}{2M\Omega(\mathbf k_1)}\right) + \frac{1}{4M^2}\left( 1 - \frac{\Omega(\mathbf k_1)}{2M}\right) \notag \\ &\; + (\mathbf k_1 \rightarrow -\mathbf k_2,\Omega(\mathbf k_1) \rightarrow -\Omega(\mathbf k_2) ) + O(\frac{1}{M^2}).\end{aligned}$$ In the $t$-channel, things are different again. Let us chose the CMS and go to the threshold, where $q=(M,0)$, $p=(-M,0)$, $k_1=(-M,\mathbf k)$ and $k_2=(M,\mathbf k)$ (remember that the physical momenta are $q$, $-p$, $-k_1$ and $k_2$). The invariants have the values $s=u=m^2-M^2$ and $$\label{eq:annihilation-expansion} B_t(u,t,s) = \frac{2}{2M^2-m^2} = \frac{1}{M^2}\left(1+\frac{m^2}{2M^2} + O(\frac{1}{M^4}) \right).$$ This means that pair-annihilation has no soft component: the entire process looks local on a scale much larger than $1/M$. To summarize, we may group all processes we have just discussed into three categories. If the initial and final states contain exclusively either heavy particles or anti-particles, we call it a [*soft*]{} process (charge conservation implies that the number of particles is conserved). If the initial and final states contain both types of particles but their number is separately conserved, we call it a [*semi-hard*]{} process. Finally, if the numbers of particles and anti-particles are not conserved separately, we call it a [*hard*]{} process. The number of light particles is not important for this classification. - Soft processes. This category comprises the $s$-channel of the amplitude $A$ and the $s$- and $u$-channels of the amplitude $B$ (i.e. particle-particle and Compton scattering). They have in common that at each vertex of the tree-level diagrams, only energies and momenta that are much smaller than $M$ are transferred. This means that all virtual light particles are not far from the mass shell, mediating the interaction over distances that are not small compared to $1/M$, and all virtual heavy particles are in the vicinity of the particle mass shell, i.e. the energy component of its momentum is close to $\omega$. Therefore, only the anti-particle components of these propagators represent a local interaction. As a consequence, no more than two heavy lines are attached to a local effective vertex. - Semi-hard processes. The $t$- and $u$ channels, describing particle-anti-particle scattering, of the amplitude $A$ are the only members of this category. In the annihilation channel (the first graph of figure \[fig:heavy-heavy-tree\] in the $t$-channel and the second graph in the $u$-channel) a heavy particle annihilates with a heavy anti-particle, emitting a virtual light particle that is well off its mass shell and travels only a distance of the order of $1/M$, giving rise to local interactions with more than two heavy particles involved. The other contribution to the process is soft in the sense described above. - Hard processes. The pair-annihilation (the $t$-channel of the amplitude $B$) is completely local because the virtual particles are always far away from the mass shell. This can be traced to the fact that at least one of the emerging light particles must be hard: even at threshold, the energy released by the annihilating particles is of the order of $M$. Let us discuss the hard processes in more detail. In the terminology just established, the process where two heavy particles annihilate into, say, 100 light particles is still considered to be hard. One may think that this is not adequate, because each of the light particles can be very soft. However, there are still some regions of phase space where a sizeable fraction of the energy is distributed among a few of them, which are then hard. Thus, the expansion of internal heavy lines depends on the configuration of the final states and it seems that there is no expansion that is valid everywhere in phase space. One may say that some pieces of the amplitude require one to treat both, particle and anti-particle as heavy degrees of freedom. The point is that neither ${{\mathcal L}_+}$ nor ${{\mathcal L}_-}$ are valid in this region. Looking at equation , one might be tempted to simply add a local interaction of the type ${{H}_+}{{H}_-}^*{l}^2$ (and its hermitian conjugate), since the entire process is local. Such a term contributes also to the two-point function ${\langle0|T{{H}_+}(x){{H}_+}^\dag(y)|0\rangle}$ at $O(e^4)$. Now, this Green’s function is already correctly described by ${{\mathcal L}_+}$ alone, as we have seen in section \[sec:p-a-sector\], and there arises the problem of double counting: by adding the mentioned local term, we must change the coefficients of ${{\mathcal L}_+}$ already fixed by a matching in the particle sector. It is a priori not clear if this procedure can be implemented systematically. In addition, unitarity tells us that the tree-level amplitudes of figure \[fig:heavy-light-tree\] in the $t$ channel are related to the imaginary part of the diagram $$\includegraphics{figure9.ps}$$ in the particle-anti-particle channel. Being a semi-hard process, we expect that the box is represented as a string of local four-particle (two particle and two anti-particle) interactions. This is again in conflict with a term of the form ${{H}_+}{{H}_-}^*{l}^2$, because two of these vertices essentially generate the box itself. These are the reasons why annihilation processes are usually excluded from the effective Lagrangian. Attempts have been made to include them in order to describe positronium decay [@labelle-thesis] or heavy quarkonium decay [@hqqbaret]. Clearly, this subject deserves further investigation. Off-Shell Expansion {#sec:off-shell-expansion} ------------------- In section \[sec:p-a-sector\] we have seen that we can reproduce the truncated off-shell Green’s functions of the relativistic theory if we use a special truncation prescription in the effective theory, which amounts to multiply with an additional factor $\sqrt{2\omega}$ for every external heavy line at tree level. The local effective theory is expected to produce a $1/M$ expansion of Green’s functions. Let us therefore extend the expansion discussed in the last section to off-shell momenta, i.e. we go back to the functions $G_{tr}^{(2,0)}$ and $G_{tr}^{(1,2)}$, treating the energy components of the momenta as independent variables. ### Heavy-Heavy scattering We keep the notation with Mandelstam variables but discard the on-shell conditions. Strictly speaking, we cannot talk about different channels any more because we are outside of the physical region. However, to stay in the scope of a $1/M$ expansion, we cannot move too far away from the mass shell so that the notion of channels still has some meaning. In the $s$-channel, for example, we restrict the energies of the particles to be much smaller than $M$ in the sense that $|q_i^0-M|$, $|p_1^0-M| \ll M$. It is convenient to introduce new variables ($i=1,2$) $$\begin{aligned} E_{q_i} &\doteq q_i^0-M \\ E_{p_i} &\doteq p_i^0-M. \end{aligned}$$ The Green’s function depends on several small dimensionless quantities $E_{q_i}/M$, $|\mathbf q_i|/M$, …and we must decide what their relative magnitude is. At the moment, we do not have any preference and simply consider all of them to be of equal magnitude, which is the same as counting powers of $1/M$ as before. More about this issue will be said below. In this framework, the function $$\begin{gathered} \frac{1}{i} G_{tr}^{(2,0)}(p_1,p_2,q_1,q_2) = \frac{1}{m^2 - (E_{q_1}-E_{p_1})^2 + (\mathbf q_1 - \mathbf p_1)^2} \\ + \frac{1}{m^2 - (E_{q_1}-E_{p_2})^2 + (\mathbf q_1 - \mathbf p_2)^2}\end{gathered}$$ cannot be expanded at all. In the $t$-channel, $|p_1^0-M|$ and $|q_2^0-M|$ are of the order of $M$. The good variables are in this case $$\begin{aligned} \bar E_{p_1} &\doteq p_1^0+M \\ \bar E_{q_2} &\doteq q_2^0+M\end{aligned}$$ in the sense that $|\bar E_{p_1}|$, $|\bar E_{q_2}|\ll M$. We find $$\begin{gathered} \label{eq:h-h-t-off-shell} \frac{1}{i}G_{tr}^{(2,0)}(p_1,p_2,q_1,q_2) = \frac{1}{m^2 - (E_{q_1}-E_{p_2})^2 + (\mathbf q_1 - \mathbf p_2)^2} \\ - \frac{1}{4M^2}\left(1 - \frac{E_{q_1}-\bar E_{p_1}}{M} + \frac{3(E_{q_1}-\bar E_{p_1})^2 + (\mathbf q_1 - \mathbf p_1)^2 + m^2}{4M^2} + O(\frac{1}{M^3}) \right).\end{gathered}$$ ### Heavy-Light scattering Using the same energy variables as before and considering the energy components of the light momenta to be of the same order, we find in the $s$-channel $$\begin{gathered} \label{eq:Gtr12-HM-expanded} \frac{1}{i} G_{tr}^{(1,2)}(p,q,k_1,k_2) = \frac{-1}{2M(E_q+k_1^0)}\left( 1 + \frac{(\mathbf q + \mathbf k_1)^2}{2M(E_q+k_1^0)} + \right. \\ \left. + \frac{(\mathbf q + \mathbf k_1)^4}{4M^2(E_q+k_1^0)^2} - \frac{(\mathbf q + \mathbf k_1)^2}{4M^2} O(\frac{1}{M^3}) \right) \\ + \frac{1}{4M^2}\left(1-\frac{E_q+k_1^0}{2M} + O(\frac{1}{M^2}) \right) + (k_1\rightarrow - k_2).\end{gathered}$$ Effective Local Lagrangians for Soft Processes {#sec:M-expansion-of-L} ---------------------------------------------- The non-local theories constructed in section \[sec:p-a-sector\] are naturally restricted to soft processes in the particle and anti-particle sectors and we have proven that they reproduce the relativistic theory exactly at tree level. The effective local Lagrangians are obtained by expanding the non-local pieces, which are the anti-particle propagator $\Delta_-$ and the particle propagator $\Delta_+$ for ${{\mathcal L}_+}$ and ${{\mathcal L}_-}$, respectively. Let us first concentrate on ${{\mathcal L}_+}$. We find $$\begin{aligned} \Delta_-(x) =\; & -\frac{1}{2M}\left(1-\frac{i\partial_t-M}{2M} + \frac{\Delta}{4M^2} \right. \notag \\ & \left. + \frac{(i\partial_t-M)^2}{4M^2} + O(\frac{1}{M^3}) \right) \delta^4(x)\end{aligned}$$ and, expanding the operator $d=(2\sqrt{M^2-\Delta})^{-1/2}$ as well, we can write the Lagrangian in the form $${{\mathcal L}_+}= {{H}_+}^*D_+{{H}_+}+ \sum_{n=1}^\infty \frac{1}{(2M)^n}{{\mathcal L}_+}^{(n)},$$ where $$\begin{aligned} {{\mathcal L}_+}^{(1)} =&\; e{{H}_+}^*{l}{{H}_+}\\ {{\mathcal L}_+}^{(2)} =&\; 0 \\ {{\mathcal L}_+}^{(3)} =&\; {{H}_+}^*\left\{ e({l}\Delta + \Delta{l}) + e^2{l}^2 \right\}{{H}_+}\\ {{\mathcal L}_+}^{(4)} =&\; -e^2{{H}_+}^*{l}(i\partial_t-M){l}{{H}_+}\\ {{\mathcal L}_+}^{(5)} =&\; {{H}_+}^*\left\{ e\left(\Delta{l}\Delta + \frac{5}{2}{l}\Delta^2 + \frac{5}{2}\Delta^2{l}\right) \right. \notag \\ & \left. + e^2\left({l}^2\Delta + \Delta{l}^2 + 3{l}\Delta{l}+ {l}[i\partial_t - M]^2{l}\right)\right. \notag \\ & \biggr. + e^3{l}^3 \biggr\}{{H}_+}\end{aligned}$$ and the differential operators act on everything on their right. In the anti-particle sector, the Lagrangian is of the same form and the ${{\mathcal L}_-}^{(n)}$ are obtained from the ${{\mathcal L}_+}^{(n)}$ by replacing ${{H}_+}$ by ${{H}_-}$ and $i\partial_t-M$ by $-i\partial_t-M$. ### Including Semi-Hard Processes The semi-hard processes contain virtual pair-annihilation and creation processes, represented by local effective interactions of several heavy particles. It is clear that a candidate for the effective theory that should include these reactions must contain both types of heavy particles. Consider the Lagrangian $${\mathcal L}= {{\mathcal L}_+}+ {{\mathcal L}_-}.$$ It clearly contains the pure particle- and anti-particle sectors as well as the soft part of the semi-hard particle-anti-particle processes but not the hard part of the latter. To include those, we must supplement the Lagrangian with contact interactions between particles and anti-particles of the form $${\mathcal L}_c = \sum_{n=1}^\infty e^{2n} {\mathcal L}_c^{(n)},$$ where ${\mathcal L}_c^{(n)}$ contains $n$ factors of the fields ${{H}_+}$,${{H}_+}^*$,${{H}_-}$ and ${{H}_-}^*$. Each of these terms is itself an expansion in $1/M$ $${\mathcal L}_c^{(n)} = \sum_{m=0}^\infty \frac{1}{(2M)^{4n+m}} {\mathcal L}_c^{(n,m)}.$$ The first two terms of ${\mathcal L}_c^{(1)}$ can be read off from the second term of eq.  $$\begin{aligned} {\mathcal L}_c^{(1,0)}(x) =&\; {{H}_+}^*(x){{H}_+}(x){{H}_-}^*(x){{H}_-}(x) \\ {\mathcal L}_c^{(1,1)}(x) =&\; -2{{H}_+}^*(x)\left( [(i\partial_{x^0}-M){{H}_+}(x)]{{H}_-}^*(x) \right. \notag \\ & \left. + {{H}_+}(x)[(i\partial_{x^0}-M){{H}_-}^*(x)]\right){{H}_-}(x).\end{aligned}$$ Power Counting Schemes {#sec:power-counting} ---------------------- In the relativistic theory, there is only one expansion parameter: the coupling constant $e$. The effective theory contains many more small parameters, namely the energies and momenta of the process of interest, which are considered to be small compared to $M$. In such a multiple expansion, the question of ordering arises, i.e. what is the relative magnitude of the expansion parameters, which determines what terms in the expansion should be grouped together. We refer to a particular ordering as a [*power counting scheme*]{}. In the following we discuss the two schemes which are of practical importance. Because the effective theory should reproduce quantities of the fundamental theory, the primary expansion parameter is the coupling $e$. If we go to the mass shell, the energies of the particles are expressed in terms of their momenta and the number of independent expansion parameters is reduced. In section \[sec:on-shell-expansion\], we have expanded some on-shell amplitudes to a fixed order in $e$ and some power of $1/M$, i.e. we have collected terms with the same powers of $e$ and $1/M$. Formally, we may introduce a small number $v$ as a bookkeeping device and assign powers of it to the expansion parameters after making them dimensionless by dividing with appropriate powers of $M$. To the momentum $\mathbf p$ of a heavy or a light particle we assign $$\frac{|\mathbf p|}{M} = O(v).$$ The energy $\Omega(\mathbf k) = \sqrt{m^2 + \mathbf k^2}$ of a light particle is counted as $$\frac{\Omega(\mathbf k)}{M} = O(v).$$ This implies that, formally, $m/M$ is considered to be of the same order as $|\mathbf k|/M$. As a consequence of these assignments, $|\mathbf p|/\Omega(\mathbf k)$ is of order one. In this language, we would say, for example, that the amplitude $B_s^{(2)}$ in eq.  is correct up to terms of $O(v^2)$. In the off-shell expansion performed in section \[sec:off-shell-expansion\], we have simply counted powers of $1/M$. This is equivalent to setting $$\begin{aligned} \frac{|E|}{M} &= O(v) & \frac{|k^0|}{M} &= O(v),\end{aligned}$$ where $E$ is the energy component of the four vector of a heavy particle with the mass $M$ subtracted and $k^0$ the energy of a light particle. Clearly, the assignment of $E/M$ is not compatible with the one of $|\mathbf p|/M$ if we go on-shell, because $$E = p^0-M = \frac{\mathbf p^2}{2M} + O(M v^4),$$ i.e. $E$ becomes a quantity of $O(v^2)$. However, no harm is done, because we formally consider $E$ to be larger than it actually is on-shell. This can be seen, for example, in the amplitude $A_t^{(2)}$. The $1/M$ suppressed contribution to the on-shell function is given by[^5] (eq. ) $$-\frac{1}{4M^2}\left(1-\frac{4\mathbf q^2-m^2}{4M^2} + O(v^4)\right),$$ where as the off shell expansion yields[^6] (eq. ) $$\begin{gathered} -\frac{1}{4M^2}\left(1 - \frac{E_{q_1}-\bar E_{p_1}}{M} + \frac{3(E_{q_1}-\bar E_{p_1})^2 + (\mathbf q_1 - \mathbf p_1)^2 + m^2}{4M^2} + O(v^3) \right).\end{gathered}$$ On-shell we have $E_{q_1}=\frac{\mathbf q_1^2}{2M}+O(Mv^4)$ and $\bar E_{p_1} = - \frac{\mathbf p_1^2}{2M} + O(Mv^4)$ and, going to the CMS, both expressions agree to $O(v^2)$. Of course, we could just as well have performed the off-shell expansion by setting $$\frac{|E|}{M} = O(v^2).$$ In this case, we get $$-\frac{1}{4M^2}\left(1 - \frac{4M(E_{q_1}-\bar E_{p_1}) - (\mathbf q_1 - \mathbf p_1)^2 - m^2}{4M^2} + O(v^4) \right),$$ which also agrees with the previous expressions on-shell and to $O(v^2)$. From this discussion, we can learn two things - The expansion of on-shell amplitudes is naturally associated with an expansion in $1/M$. - There is no natural choice for the expansion of off-shell amplitudes (or Green’s functions). Counting $E/M$ the same as $|\mathbf p|/M$ conserves the strict $1/M$ expansion but the orders get mixed if we go on-shell (the terms $E/M$ will contribute to all higher orders). If we count $E/M$ as $|\mathbf p|^2/M^2$, we do not expand simply in $1/M$ but the energies are considered to be of the order they actually are on-shell. Also in this case does a term $E/M$ contribute to all higher orders if we go on-shell. Different counting schemes are possible but not of practical importance. To conclude, we define two power counting schemes ($p$ and $k$ denote the four-momenta of a heavy and a light particle, respectively): 1. Heavy-Meson (HM) scheme. Its defining feature is that the three-momentum and the energy variable (with the heavy mass subtracted) are considered to be of equal magnitude $$\begin{aligned} \frac{|E|}{M} &= O(v) & \frac{|k^0|}{M} = O(v) \\ \frac{|\mathbf p|}{M} &= O(v) & \frac{|\mathbf k|}{M} = O(v). \end{aligned}$$ The name is an adaptation from HBCHPT [@hbchpt-review], where this counting scheme is used. 2. Non-Relativistic (NR) scheme. In this scheme, the energy of a heavy particle is counted like its three-momentum squared $$\begin{aligned} \frac{|E|}{M} &= O(v^2) & \frac{|k^0|}{M} = O(v)\\ \frac{|\mathbf p|}{M} &= O(v) & \frac{|\mathbf k|}{M} = O(v). \end{aligned}$$ The name is derived from the fact that the lowest order effective Lagrangian is Galilei-invariant and thus represents a true non-relativistic theory. Perturbation Theory {#sec:perturbation-theory} ------------------- We have seen that the relativistic Green’s functions can be expanded in different ways. What does this mean for the effective theory? The effective Lagrangian contains space and time derivatives of the fields. In momentum space they become three-momentum and energy variables and the question of ordering arises already on the level of the Lagrangian. It is clear that the perturbation theory looks different for the different counting schemes. The resulting Green’s functions can then be identified with the different expansions of the relativistic Green’s functions. In section \[sec:M-expansion-of-L\], we have ordered the effective Lagrangian according to powers of $1/M$. It is useful to reorder it now. First of all, we should collect the terms with the same power of $e$ or, which is equivalent, the same number of light fields. Then we should assign the differential operators $\partial_t/M$ and $\nabla/M$ some power of the parameter $v$ introduced in the previous section according to one of the power counting schemes. The Lagrangian is then of the form $${{\mathcal L}_+}= \bar{{\mathcal L}_+}^0 + {\mathcal L}_{l}^0 + \sum_{\mu,\nu}^\infty \left( \frac{e}{2M} \right)^\mu{{\mathcal L}_+}^{(\mu,\nu)}.$$ Here, $\bar{{\mathcal L}_+}^0$ contains only the leading part of ${{\mathcal L}_+}^0$ in the parameter $v$. The term ${{\mathcal L}_+}^{(\mu,\nu)}$ contains $\mu$ light fields and $\nu$ powers of $v$. In the HM scheme we formally assign $$\begin{aligned} \frac{i\partial_t-M}{M} &= O(v) \\ \frac{\nabla}{M} &= O(v)\end{aligned}$$ irrespective of what field they act on. The interaction independent pieces are given by $$\begin{aligned} \bar{{\mathcal L}_+}^0 &\equiv {{\mathcal L}_{+,\text{HM}}^0}\doteq {{H}_+}^*{D_{+,\text{HM}}}{{H}_+}\\ {D_{+,\text{HM}}}&= i\partial_t - M \\ \sum_{m=2}^\infty{{\mathcal L}_+}^{(0,m)} &= {{H}_+}^*(M-\sqrt{M^2-\Delta}){{H}_+}.\end{aligned}$$ In the NR scheme we count time derivatives differently: $$\begin{aligned} \frac{i\partial_t-M}{M} &= O(v^2) \;\;\text{when acting on a heavy field} \\ \frac{i\partial_t}{M} &= O(v) \;\;\text{when acting on a light field} \\ \frac{\nabla}{M} &= O(v) \;\;\text{always}\end{aligned}$$ and $$\begin{aligned} \bar{{\mathcal L}_+}^0 &\equiv {{\mathcal L}_{+,\text{NR}}^0}\doteq {{H}_+}^*{D_{+,\text{NR}}}{{H}_+}\\ {D_{+,\text{NR}}}&= i\partial_t - M + \frac{\Delta}{2M} \\ \sum_{m=2}^\infty{{\mathcal L}_+}^{(0,m)} &= {{H}_+}^*(M - \frac{\Delta}{2M} -\sqrt{M^2-\Delta}){{H}_+}. \end{aligned}$$ The leading terms of the interaction Lagrangians ${{\mathcal L}_+}^{(\mu,\nu)}$ for both schemes are shown in table \[tab:Lmunu\]. ------------------------------------------------------------------------------------------------------------------------------------------ $(\mu,\nu)$ HM NR ------------- -------------------------------------------------------- ------------------------------------------------------------------- $(1,0)$ ${{H}_+}^*{l}{{H}_+}$ ${{H}_+}^*{l}{{H}_+}$ $(1,2)$ $\frac{1}{4M^2}{{H}_+}^*({l}\Delta+\Delta{l}){{H}_+}$ $\frac{1}{4M^2}{{H}_+}^*({l}\Delta+\Delta{l}){{H}_+}$ $(1,4)$ $\frac{1}{16M^4}{{H}_+}^*(\Delta{l}\Delta + $\frac{1}{16M^4}{{H}_+}^*(\Delta{l}\Delta + \frac{5}{2}{l}\Delta^2$ \frac{5}{2}{l}\Delta^2$ $+\frac{5}{2}\Delta^2{l}){{H}_+}$ $+\frac{5}{2}\Delta^2{l}){{H}_+}$ $(2,0)$ $\frac{1}{2M}{{H}_+}^*{l}^2{{H}_+}$ $\frac{1}{2M}{{H}_+}^*{l}^2{{H}_+}$ $(2,1)$ $-\frac{1}{4M^2}{{H}_+}^*{l}(i\partial_t-M){l}{{H}_+}$ $\frac{-1}{4M^2}{{H}_+}^*{l}(i\partial_t{l}){{H}_+}$ $(2,2)$ $\frac{1}{8M^3}{{H}_+}^*({l}^2\Delta+\Delta{l}^2]$ $\frac{1}{8M^3}{{H}_+}^*({l}^2\Delta+\Delta{l}^2 + 3{l}\Delta{l}$ $+ 3{l}\Delta{l}+ {l}[i\partial_t-M]^2{l}){{H}_+}$ $- {l}(\partial_t^2{l}) -\frac{1}{2M}{l}^2(i\partial_t - M)){{H}_+}$ $(3,0)$ $\frac{1}{4M^2}{{H}_+}^*{l}^3{{H}_+}$ $\frac{1}{4M^2}{{H}_+}^*{l}^3{{H}_+}$ ------------------------------------------------------------------------------------------------------------------------------------------ : The leading terms of the interaction Lagrangians ${{\mathcal L}_+}^{(\mu,\nu)}$. $\mu$ and $\nu$ denote the number of powers of $e$ and $v$, respectively.[]{data-label="tab:Lmunu"} In the following, we first consider a free field and discuss the form of the propagators to be used in perturbation theory. Then we formulate a power counting for Green’s functions to find out which vertices of the effective Lagrangian must be used to calculate them to some order in $v$. Finally, we state how Green’s functions can be calculated in a systematic way from the generating functional. ### Free Propagators The propagators to be used in perturbation theory are derived from the Lagrangians ${{\mathcal L}_{+,\text{HM}}^0}$ and ${{\mathcal L}_{+,\text{NR}}^0}$. In the notation of appendix \[app:notation\] $$\begin{aligned} {\langlex|{D_{+,\text{HM}}}^{-1}|y\rangle} &= {\Delta_+^{\text{HM}}}(x-y) = \int\frac{d^4p}{(2\pi)^4} \frac{e^{-ip(x-y)}}{E+i\epsilon} \\ {\langlex|{D_{+,\text{NR}}}^{-1}|y\rangle} &= {\Delta_+^{\text{NR}}}(x-y) = - \int\frac{d^4p}{(2\pi)^4} \frac{e^{-ip(x-y)}}{\frac{\mathbf p^2}{2M}-E-i\epsilon},\end{aligned}$$ where $E=p^0-M$. The operators in the Lagrangian ${{\mathcal L}_+}^{(0,\nu)}$ are considered to be corrections to these lowest order propagators. By resumming insertions of $\mathbf p^2/2M$ in the HM propagator, we obtain the propagator of the NR scheme. $$\begin{aligned} {\Delta_+^{\text{HM}}}(p)\left(1+\frac{\mathbf p^2}{2M}\frac{1}{E} + \left(\frac{\mathbf p^2}{2M}\frac{1}{E}\right)^2 + \dots\right) = {\Delta_+^{\text{NR}}}(p).\end{aligned}$$ Similarly, by including higher and higher corrections and resumming them, we recover the full propagator $$\Delta_+(p) = -\frac{1}{\omega(\mathbf p) - p^0 - i\epsilon}.$$ ### Naive Power Counting for Green’s Functions We would like to find a way how one can read off the power of $v$ to which a certain Graph contributes. Every Graph can be characterized by the following parameters --------------- ------------------------------------------------------------------- $E_{H}$ $\#$ of external heavy lines $I_{H}$ $\#$ of internal heavy lines $I_{l}$ $\#$ of internal light lines $N_{\mu,\nu}$ $\#$ of vertices with $\mu$ powers of $e$ and $\nu$ powers of $v$ $L$ $\#$ of loops. --------------- ------------------------------------------------------------------- In addition, let $P$ denote the power of $1/v$ of the heavy propagator. We have $P=1$ and $P=2$ in the HM and NR schemes, respectively. Excluding external lines, the total power $d$ of $v$ of the diagram is given by $$d = 4L - PI_{H}- 2I_{l}+ \sum_{\mu,\nu} \nu N_{\mu,\nu}.$$ Using the well known “topological” relations (the factor 2 in front of $N_{\mu,\nu}$ is due to the fact that at each vertex exactly two heavy lines meet) $$\begin{aligned} L &= I_{H}+ I_{l}+ 1 - \sum_{\mu,\nu} N_{\mu,\nu} \\ E_{H}&= \sum_{\mu,\nu}2N_{\mu,\nu} - 2I_{H},\end{aligned}$$ we get $$\label{eq:power-counting} d = 2(L+1) - \frac{2-P}{2} E_{H}+ \sum_{\mu,\nu} N_{\mu,\nu}(\nu-P).$$ This formula is certainly correct for $L=0$ because all factors of $v$ are explicit and there are no integrations over internal momenta. Loops are complicated functions of the external momenta and may produce additional factors of $v$ which can upset this naive power counting. We will briefly come back to this point below and consider only tree graphs for now. Remember that we always work to a fixed order in the fundamental coupling in $e$. Therefore, the sum $\sum_{\mu,\nu}\mu N_{\mu,\nu}$ must be the same for every graph contributing to some Green’s function. From eq.  we can see that the leading contribution is given by the graph with as few powers of $v$ as possible. Corrections can be systematically obtained by including vertices with more powers of $v$. ### Perturbation Series We are now in a position to formulate how a Green’s function can be calculated from the generating functional $$\begin{aligned} Z[j,j^*,J] &= \frac{1}{\mathcal Z}\int [dl][d{{H}_+}][d{{H}_+}^*] e^{iS_+ + \int j^*{{H}_+}+ {{H}_+}^* j + J{l}} \\ \mathcal Z &= \int [dl][d{{H}_+}][d{{H}_+}^*] e^{iS_+} \\ S_+ &= \int d^4x {{\mathcal L}_+}(x).\end{aligned}$$ The first step towards the perturbation theory is the separation of the action $S_+$ into a “free” part (which must be quadratic in the field) and an “interacting” part $$S_+ = S_+^0 + S_+^{\text{int}}.$$ This decomposition depends on the counting scheme and we set $$\begin{aligned} S_{+,\text{HM,NR}}^0 &= \int d^4x {\mathcal L}_{+,\text{HM,NR}}^0 \\ S_{+,\text{HM,NR}}^{\text{int}} &= S_+ - S_{+,\text{HM,NR}}^0.\end{aligned}$$ The Gaussian average of some functional $F$ of the fields ${{H}_+}$ and ${l}$ is denoted by $${\langle\langleF[{{H}_+},{{H}_+}^*,{l}]\rangle\rangle}^{\text{HM,NR}} \doteq \frac{\int [d{l}][d{{H}_+}][d{{H}_+}^*] F[{{H}_+},{{H}_+}^*,{l}] e^{iS_{+,\text{HM,NR}}^0}}{\int [d{l}][d{{H}_+}][d{{H}_+}^*]e^{iS_{+,\text{HM,NR}}^0}}.$$ In particular, the free propagators are given by $$\begin{aligned} i{\Delta_+^{\text{HM}}}(x-y) &= {\langle\langle{{H}_+}(x){{H}_+}^*(y)\rangle\rangle}^{\text{HM}} \\ i{\Delta_+^{\text{NR}}}(x-y) &= {\langle\langle{{H}_+}(x){{H}_+}^*(y)\rangle\rangle}^{\text{NR}} \\ \frac{1}{i}\Delta_m(x-y) &= {\langle\langle{l}(x){l}(y)\rangle\rangle}.\end{aligned}$$ The latter is the same in both schemes. In the notation set up in section \[sec:p-q-greens-functions\], the connected $n$-point functions are written as (we should put indices HM or NR here as well but we suppress them in order to simplify the notation) $$G_+^{(a,b)}(x,y,z) = {\langle\langle\hat{{H}_+}(x)\hat{{H}_+}^*(y)\hat{l}(z)e^{iS_+^{\text{int}}}\rangle\rangle}_c.$$ The perturbation series is obtained by expanding the exponential in powers of $v$ with the constraint that $\sum_{\mu,\nu}\mu N_{\mu,\nu}$ is fixed (see above). After the expansion we are left with Gaussian integrals which can be reduced to sums of products of propagators owing to the Wick theorem. Compton Scattering at Tree Level -------------------------------- Let us calculate the tree level truncated Green’s function $G_{+,tr}^{(1,2)}$ in the HM scheme to next-to-next-to leading order. Applying formula , we find the combinations of vertices that yield a specific power of $v$ displayed in table \[tab:compton-vertices\] $d$ Vertices ----- ----------------------------------------------------------- -1 $N_{1,0}=2$ 0 $N_{2,0}=1;\; N_{1,0}=2,N_{0,2}=1$ 1 $N_{2,1}=1;\; N_{1,0}=2,N_{0,2}=2;\; N_{1,2}=1,N_{1,0}=1$ : The combination of vertices that yield a certain power $d$ of $v$ for Compton scattering in the HM scheme.[]{data-label="tab:compton-vertices"} We can see, for example, that the leading term is of $O(1/v)$ (it is just the propagator ${\Delta_+^{\text{HM}}}$) and consists of two vertices of the Lagrangian ${{\mathcal L}_+}^{(1,0)}$. At $O(v^0)$, we can either use one vertex from ${{\mathcal L}_+}^{(2,0)}$ or two from ${{\mathcal L}_+}^{(1,0)}$ together with ${{\mathcal L}_+}^{(0,2)}$, which is an insertion of $\mathbf p^2/2M$. The result is $$\begin{aligned} \frac{1}{i}G_{+,tr}^{(1,2)} =&\; \frac{-1}{4M^2}\frac{1}{E_q + k_1^0} \left( 1 + \frac{(\mathbf q + \mathbf k_1)^2}{2M(E_q+k_1^0)} \right. \\ &\;+ \left. \frac{(\mathbf q + \mathbf k_1)^4}{4M^2(E_q+k_1^0)^2} - \frac{\mathbf q^2 + \mathbf p^2 + 2(\mathbf q + \mathbf k_1)^2}{4M^2} + O(v^3) \right) \\ &\;+ \frac{1}{8M^3}\left( 1 - \frac{E_q+k_1^0}{2M} + O(v^2) \right).\end{aligned}$$ According to the truncation rule given in eq. , we must multiply with $$\sqrt{2\omega(\mathbf q)}\sqrt{2\omega(\mathbf p)} = 2M\left( 1 + \frac{\mathbf q^2 + \mathbf p^2}{4M^2} + O(v^4) \right)$$ to compare with the truncated Greens function $G_{tr}^{(1,2)}$ of the relativistic theory. Comparing the result with eq. , we see that $$G_{tr}^{(1,2)} = \bar G_{+,tr}^{(1,2)}$$ is true to $O(e^2 v^2)$. The amplitude $B_s^{(2)}$ for Compton scattering obtained from $G_{+,tr}^{(1,2)}$ is therefore the same as the one in the relativistic theory with the same precision. Power Counting Beyond Tree-Level {#sec:beyond-tree} -------------------------------- Consider the contribution of the graph $$\includegraphics[scale=0.8]{figure10.ps}$$ to the self energy of the heavy particle, where the boxes represent vertices from ${{\mathcal L}_+}^{(1,2)}$, i.e. they have two powers of $v$. According to the formula given in eq. , this diagram is of $O(v^5)$ in the HM scheme and of $O(v^4)$ in NR. If $p$ is the momentum that flows through the diagram, the loop is a function that depends only on $E=p^0-M$ and $m$ in the HM scheme but on $E,m$ and $\mathbf p$ in the NR scheme. The integrals are of the form $$\begin{aligned} I_{\text{HM}} &= I_{\text{HM}}\left(\frac{E}{m}\right) \\ I_{\text{NR}} &= I_{\text{NR}}\left(\frac{E}{m},\frac{\mathbf p}{m},\frac{m}{M}\right). \end{aligned}$$ The argument of $I_{\text{HM}}$ is of $O(1)$ but the one of $I_{\text{NR}}$ contains a part that is of $O(v)$. Therefore, the loop destroys the naive power counting in the NR scheme[^7]: unless the integral does not really depend on $m/M$ by chance, it produces factors of $v$ which either raise or lower the naive power of $v$. The former would not be so bad but the latter is a disaster because one must expect that [*all*]{} loop graphs start contributing at lowest order. There is, however, a scenario, where this catastrophe is reduced to a mere inconvenience. If the terms that contribute to a lower order than the naive one are such that they can be absorbed in the coupling constants of the Lagrangian (i.e. polynomials in the energies and momenta), systematic perturbation theory is still possible, because only a finite number of graphs contribute to the “interesting” (non-polynomial) part of the Green’s function. The inconvenience is that whenever one pushes the calculation to the next higher order one has to re-match the effective coupling constants (either to the fundamental theory, if possible, or directly to experiment). It is believed that this is indeed what happens and was checked in an explicit one-loop calculation [@1-loop-renormalization]. In the HM scheme, the problem is absent[^8]. However, as mentioned in the introduction, this scheme is not suited for systems where two heavy particles can form a bound state because it leads to spurious infrared divergences, which vanish only upon a resummation of certain contributions (see for example [@eichten-hill1]). Summary and Outlook =================== In this work, we have investigated effective theories describing heavy and light scalar particles in the low–energy regime. First, we discussed the concept of the physical mass and of the matching condition for S–matrix elements in a general setting, and then proposed a matching procedure for off–shell Green’s functions, that leads – due to different notions of one–particle irreducibility in the original and effective theory – to a specific truncation prescription in the effective theory. We then investigated these matching conditions for a Yukawa interaction between two heavy (${H}$) and one light field (${l}$). First, we treated the light field as an external source and constructed two non-local Lagrangians that are equivalent to the full theory in the pure particle- and anti-particle sectors. Adding dynamics for the light field, we showed that the amplitudes and properly truncated Green’s functions of the effective Lagrangians indeed satisfy the proposed matching conditions to all orders in the coupling in the presence of a UV regulator. In order to arrive at a local Lagrangian, we discussed the $1/M$ expansion of tree-level scattering. We classified all physical processes by the number of heavy particles and anti-particles in the initial and final states, distinguishing - [*soft*]{} processes: initial and final states contain only heavy particles or only heavy anti particles, like $${H}{l}\rightarrow{H}{l}{l},$$ - [*semi-hard*]{} processes: both types of particles are present, but their number is separately conserved, like $${H}\bar{{H}}\rightarrow {H}\bar{{H}}{l}{l},$$ - [*hard*]{} processes: number of particles and anti particles is not conserved separately, e.g., $${H}\bar{{H}}\rightarrow {l}{l}.$$ Starting from the nonlocal Lagrangian, we then constructed the effective local Lagrangian for soft and semi–soft processes at low orders in the $1/M$ expansion. Hard processes play a special role in this setting: their $1/M$ expansion is difficult, because there is so much energy released that some light particles may become very hard, while others stay soft. Neither did we find a satisfactory treatment of these processes in the literature, nor can we offer one at this moment[^9]. Work on the problem is in progress. Extending the expansion to off-shell Green’s functions, we found that there is no natural way to count the energies of heavy particles relative to their momenta (being no longer related through the on-shell condition). We introduced a bookkeeping parameter $v$ and defined two possible counting schemes by assigning powers of it to energies and momenta of the particles and showed how – in a systematic expansion in the fundamental coupling and in the parameter $v$ – tree-level Green’s functions can be calculated. We checked the method in the case of Compton scattering. The final aim of this programme is the application of effective theories to the decay of bound states, like $\pi^+\pi^-\rightarrow \pi^0\pi^0$, and to relate these processes to the underlying theory of strong interactions. For this purpose, one needs to include hard processes in the framework, and to set up a consistent and systematic power counting in the scattering sector (including loops) as well as in the bound state calculation where Rayleigh–Schrödinger perturbation theory may be applied. Finally, one has to show how the effective Lagrangians describing QCD at low energies are incorporated in order to arrive at the above described aim. First steps in this direction are already done [@labelle-thesis; @labelle-lepage; @kinoshita-nio-1; @labelle-buckley; @kong-ravndall; @labelle-retardation] or will soon be completed [@energy-shift]. Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank Jürg Gasser for his advice, guidance and encouragement and for passing his insight into physics on to me. My thanks also go to Vito Antonelli for collaboration during his stay in Bern and to Akaki Rusetsky for interesting discussions and his hospitality during my brief stay in Dubna. Last but not least I would like to thank all the people of the institute who have contributed to this work in one way or another. I especially enjoyed the many discussions with Thomas Becher and Markus Leibundgut, who were always willing to stop working long enough to talk about life, physics and the world in general. Notation {#app:notation} ======== #### Metric We work in Minkowski space with a signature of $(1,-1,-1,-1)$. Three-vectors, denoted by boldface letters, are the three-dimensional parts of contravariant four-vectors $$x^\mu = \{x^0,x^1,x^2,x^3\} = \{x^0, \mathbf x\}$$ except for the three-dimensional gradient $$\mathbf\nabla = \{\partial_1, \partial_2, \partial_3\},$$ where $$\partial_\mu \equiv \frac{\partial}{\partial x^\mu}.$$ #### Fourier Transform The Fourier transform $f(p)$ of a function $f(x)$ is defined by $$f(x) = \int\frac{d^4p}{(2\pi)^4} e^{-ipx}f(p).$$ #### Green’s functions {#greens-functions} Let $\phi$ be a complex field and $x$ and $y$ denote sets $(x_1,x_2,\dots,x_n)$, $(y_1,y_2,\dots,y_n)$ of coordinates. We use the shorthand form $$\hat\phi(x) \doteq \phi(x_1)\phi(x_2)\dots\phi(x_n).$$ The vacuum expectation value of the time ordered product of fields is written as $$G (x,y) = {\langle0|T\hat\phi(x)\hat\phi^\dag(y)|0\rangle}.$$ Assuming translation invariance, the Fourier transform of $G$ is defined by $$(2\pi)^4\delta^4(P-Q)G (p,q) = \int d^4x d^4y e^{i\sum_{i=1}^n(p_ix_i-q_iy_i)}G(x,y),$$ where $P=\sum_{i=1}^n p_i$ and $Q=\sum_{i=1}^n q_i$. With this convention, the $p_i$ and $q_i$ denote the physical momenta of outgoing and incoming particles if we let the time components $x_i^0$ and $y_i^0$ tend to $+\infty$ and $-\infty$, respectively. In the case of a real scalar field $\varphi$, we define $$G(x) = {\langle0|T\hat\varphi(x)|0\rangle}$$ and ($K=\sum_{i=1}^n k_i$) $$(2\pi)^4\delta^4(K)G(k) = \int d^4x e^{i\sum_{i=1}^nk_ix_i}G(x).$$ Here, the momenta $k_i$ correspond to outgoing particles in the same sense as above. #### Operators Let $\mathbf O$ be an operator that acts in some Hilbert space $\mathcal H$ of functions defined in Minkowski space. In Dirac notation, the orthogonality and closure relations for the $x$ basis read $$\begin{aligned} {\langlex|y\rangle} &= \delta^4(x-y) \\ \int d^4x {|x\rangle}{\langlex|} &= \mathbf 1.\end{aligned}$$ The $x$ representations of $f\in\mathcal H$ and $\mathbf O$ are denoted by $$\begin{aligned} f(x) &= {\langlex|f\rangle} \\ O(x,y) &= {\langlex|\mathbf O|y\rangle}.\end{aligned}$$ Accordingly, the action of $\mathbf O$ on $f$ reads $$(\mathbf O f)(x) = \int d^4 y O(x,y)f(y).$$ A differential operator $\mathbf D$ has the representation $${\langlex|\mathbf D|y\rangle} = \delta^4(x-y)D_y$$ so that $$(\mathbf D f)(x) = D_x f(x).$$ For any translation invariant operator, i.e. ${\langlex|\mathbf O|y\rangle} = O(x-y)$, we have $$\Box_x O(x-y) = \Box_yO(x-y).$$ If $\mathbf D$ is an invariant differential operator (i.e. a function of $\Box$) and $\mathbf O$ translation invariant, one may check, using partial integration, that $$\label{eq:rot-inv-diff-op} (\mathbf D\mathbf O f)(x) = (\mathbf O\mathbf D f)(x).$$ Klein-Gordon Green’s Functions {#app:KG} ============================== A Green’s function $G(x)$ of the Klein-Gordon equation is defined by $$\label{eq:KG-gf-def} D_MG(x)\doteq (\Box + M^2)G(x) = \delta^4(x)$$ together with some boundary conditions. The solution that is a superposition of incoming plane waves for $x_0<0$ and of outgoing plane waves for $x_0>0$ is the Feynman propagator $$\begin{aligned} \label{eq:feynman-prop} \Delta_M(x) &= \int\frac{d^4p}{(2\pi)^4} \frac{e^{-ipx}}{M^2-p^2-i\epsilon} \notag \\ &= i\int \frac{d^3p}{(2\pi)^3 2\omega(\mathbf p)}e^{i\mathbf p\cdot\mathbf x} \left( \theta(x^0)e^{-i\omega(\mathbf p)x^0} + \theta(-x^0) e^{i\omega(\mathbf p)x^0}\right),\end{aligned}$$ where $\omega(\mathbf p) = \sqrt{M^2+\mathbf p^2}$. The Klein-Gordon operator $D_M$ can be decomposed into two first order differential operators $$\begin{aligned} \label{eq:def-D+-} D_M &=& D_+ D_- \nonumber \\ D_\pm &=& \pm i\partial_t - \sqrt{M^2-\Delta}.\end{aligned}$$ The operator $$\label{eq:def-d} d = (2\sqrt{M^2-\Delta})^{-\frac{1}{2}}$$ plays an important role in the construction of the non-relativistic Lagrangian. Its action on a function $f$ is defined through the Fourier representation $$(df)(x) = \int \frac{d^4p}{(2\pi)^4} \frac{f(p)}{\sqrt{2\omega(\mathbf p)}} e^{-ipx}.$$ The functions $$\begin{aligned} \label{eq:d-pm} \Delta_\pm(x) &= - \int \frac{d^4p}{(2\pi)^4} \frac{e^{-ipx}}{\omega(\mathbf p) \mp p^0 - i \epsilon} \notag \\ &= -i\theta(\pm x^0) \int\frac{d^3p}{(2\pi)^3} e^{\mp i\omega(\mathbf p)x^0 + i\mathbf p\cdot\mathbf x}\end{aligned}$$ are Green’s functions of $D_\pm$, i.e. $$D_\pm\Delta_\pm(x) = \delta^4(x)$$ and the boundary conditions are chosen such that $\Delta_+(x)=0$ for $x^0<0$ and $\Delta_-(x)=0$ for $x^0>0$. Comparing (\[eq:d-pm\]) with (\[eq:feynman-prop\]) we find $$\label{eq:fp-as-d_pm} \Delta_M(x) = -d^2(\Delta_+(x)+ \Delta_-(x)).$$ The Green’s functions can be viewed as the inverse of the corresponding differential operators. In the notation introduced in appendix \[app:notation\], we write $$\begin{aligned} {\langlex|D_M^{-1}|y\rangle} &= \Delta_M(x-y) \\ {\langlex|D_\pm^{-1}|y\rangle} &= \Delta_\pm(x-y).\end{aligned}$$ In operator notation, eq.  can be written in any of the forms (cf. eq. ) $$\begin{aligned} \label{eq:DM-as-Dpm} D_M^{-1} &=& -d^2(D_+^{-1} + D_-^{-1}) = -(D_+^{-1} + D_-^{-1})d^2 \notag \\ &=& -d(D_+^{-1} + D_-^{-1})d.\end{aligned}$$ Finally, with the convention of appendix \[app:notation\], the Fourier transforms are given by $$\begin{aligned} \Delta_M(p) &= \frac{1}{M^2-p^2-i\epsilon} \\ \Delta_\pm(p) &= -\frac{1}{\omega(\mathbf p)\mp p^0-i\epsilon}.\end{aligned}$$ Canonical Quantization of Free Fields {#app:canonical-quantisation} ===================================== Let us briefly recall the canonical quantization procedure for a complex scalar field with the Lagrangian $${\mathcal L}^0 = \partial_\mu{H}^*\partial^\mu{H}- M^2{H}^*{H}.$$ The conjugate field is defined by $$\pi(t,\mathbf x) = \frac{\partial {\mathcal L}^0}{\partial\dot{H}(t,\mathbf x)} = \dot{H}^*(t,\mathbf x)$$ where $\dot{H}(t,\mathbf x) = \partial_0{H}(t,\mathbf x)$. The only non-vanishing Poisson bracket is $$\label{eq:rel-poisson} \{{H}(t,\mathbf x), \pi(t, \mathbf y)\} = \delta^3(\mathbf x-\mathbf y)$$ and the most general solution of the equation of motion $$(\Box + M^2){H}(x) = 0$$ is a superposition of plane waves $${H}(x) = \int d\mu(p)\left( a(\mathbf p)e^{-ipx} + b^*(\mathbf p)e^{ipx} \right),$$ where the invariant measure is defined by $$d\mu(p) = \frac{d^3p}{(2\pi)^32p^0}$$ and the momentum is on the mass shell $$p^0 = \omega(\mathbf p) = \sqrt{M^2+\mathbf p^2}.$$ The factor $(2\pi)^3$ is conventional and is chosen for later convenience. Quantization is performed by replacing ${H}$ and $\pi$ by operators which satisfy the equal-time commutation relation $$[{H}(t,\mathbf x), \pi(t, \mathbf y)] = i\delta^3(\mathbf x-\mathbf y).$$ The coefficient functions $a$ and $b$ are also operators and obey $$[a(\mathbf p), a^\dag(\mathbf q)] = [b(\mathbf p), b^\dag(\mathbf q)] = 2\omega(\mathbf p)(2\pi)^3\delta^3(\mathbf p-\mathbf q).$$ The operators $a^\dag$ and $b^\dag$ can be shown to create one-particle states out of the vacuum $$\begin{aligned} {|p\rangle} &= a^\dag(\mathbf p){|0\rangle} \\ {|\bar p\rangle} &= b^\dag(\mathbf p){|0\rangle}.\end{aligned}$$ We shall refer to them as particle and anti-particle states, respectively. The vacuum contains by definition no particles and is defined by the conditions $$\begin{aligned} a(\mathbf p){|0\rangle} &= b(\mathbf p){|0\rangle} = 0 \notag \\ {\langle0|0\rangle} &= 1.\end{aligned}$$ With these conventions, the states are normalized by $${\langlep|q\rangle} = {\langle\bar p|\bar q\rangle} = 2\omega(\mathbf p)(2\pi)^3\delta^3(\mathbf p-\mathbf q).$$ Let us apply this formalism to the Lagrangians $${\mathcal L}_\pm^0 = {H}_\pm^*(\pm i\partial_t - \sqrt{M^2-\Delta}){H}_\pm.$$ The conjugate fields are $$\begin{aligned} \pi_\pm(t,\mathbf x) = \frac{\partial{\mathcal L}_\pm}{\partial \dot{H}_\pm} = \pm i{H}_\pm^*\end{aligned}$$ and the Poisson brackets are analogous to eq. \[eq:rel-poisson\]. The most general solutions of the equations of motion $$(\pm i\partial_t-\sqrt{M^2-\Delta}){H}_\pm(x) = 0$$ are $$\begin{aligned} {{H}_+}(x) &= \int d\bar\mu(p) a(\mathbf p) e^{-ipx} \\ {{H}_-}(x) &= \int d\bar\mu(p) b^*(\mathbf p) e^{ipx}\end{aligned}$$ with $p^0=\omega(\mathbf p)$. This time, we chose the measure to be $$d\bar\mu(p) = \frac{d^3p}{(2\pi)^3}.$$ Replacing the functions by operators, we find $$[a(\mathbf p), a^\dag(\mathbf q)] = [b(\mathbf p), b^\dag(\mathbf q)] = (2\pi)^3\delta^3(\mathbf p-\mathbf q).$$ They create and destroy particles in the same way as described above. The only difference is the normalization of these states, which now reads $${\langlep|q\rangle} = {\langle\bar p|\bar q\rangle} = (2\pi)^3\delta^3(\mathbf p-\mathbf q).$$ We have sacrificed the rule to label different objects by different symbols to simplify the notation. Note that the Fock spaces of the theories defined by ${\mathcal L}^0$ and ${{\mathcal L}_+}+{{\mathcal L}_-}$ are identical. Two-Point Functions {#app:2pf} =================== In section \[sec:transition-amplitudes\], we consider transition amplitudes in the two models $$\begin{aligned} \tag{\ref{eq:L-gener}} {\mathcal L}&= \partial_\mu{H}^*\partial^\mu{H}-M^2{H}^*{H}+ \bar{\mathcal L}^0 + {\mathcal L}^{\text{int}} \\ \tag{\ref{eq:Lp-gener}} {{\mathcal L}_+}&= {{H}_+}^*(i\partial_t-\sqrt{M^2-\Delta}){{H}_+}+ \bar{\mathcal L}^0 + {{\mathcal L}_+}^{\text{int}}\end{aligned}$$ and show how they can be matched. For this procedure to work, it is necessary that there exists an unambiguous definition of the physical mass of the heavy particle. We must therefore examine the properties of the two-point functions $$\begin{aligned} G(p) &= \int d^4x e^{ipx}{\langle0|T {H}(x){H}^\dag(0)|0\rangle} \\ \label{eq:Gp-def} G_+(p) &= \int d^4x e^{ipx}{\langle0|T {{H}_+}(x){{H}_+}^\dag(0)|0\rangle}.\end{aligned}$$ It is convenient to express them in terms of one-particle irreducible functions[^10] $\Sigma$, $\Sigma_+$ $$\begin{aligned} G(p) &= \frac{1}{i}\frac{1}{M^2-p^2 + i\Sigma(p^2) -i\epsilon} \\ \label{eq:Gp} G_+(p) &= \frac{1}{i}\frac{1}{\omega(\mathbf p)-p^0 + i\Sigma_+(p^0,\mathbf p^2) - i\epsilon},\end{aligned}$$ with $\omega(\mathbf p) = \sqrt{M^2+\mathbf p^2}$. In the absence of interactions, they reduce to the free propagators $$\begin{aligned} G(p) &= \frac{1}{i}\Delta_M(p) = \frac{1}{i}\frac{1}{M^2-p^2-i\epsilon} \\ G_+(p) &= i\Delta_+(p) = \frac{1}{i} \frac{1}{\omega(\mathbf p)- p^0 -i\epsilon}\end{aligned}$$ discussed in appendix \[app:KG\]. The physical mass ${M_{\text{p}}}$ is defined as the location of the pole of $G$ $$\label{eq:Mp-def} M_{\text p}^2 = M^2 + i\Sigma(M_{\text p}^2)$$ and we can write $$\label{eq:G2-phys} G(p) = \frac{1}{i}\frac{Z_{H}}{M_{\text p}^2-p^2-i\epsilon} + \text{regular},p^2\rightarrow{M_{\text{p}}}^2,$$ where the residue is given by $$\label{eq:Z-def} Z_{H}^{-1} = 1 - i\Sigma'(M_{\text p}^2).$$ Let us focus on the pole at $p^0={\omega_{\text{p}}}(\mathbf p) = \sqrt{{M_{\text{p}}}^2+\mathbf p^2}$ $$\label{eq:G-pole-p0} G(p) = \frac{1}{i}\frac{1}{2{\omega_{\text{p}}}(\mathbf p)}\frac{Z_{H}}{{\omega_{\text{p}}}(\mathbf p)-p^0 - i\epsilon} + \text{regular},p^0\rightarrow{\omega_{\text{p}}}(\mathbf p).$$ We may isolate it in a different way by first writing $G$ as $$\label{eq:G-in-S} G(p) = \frac{1}{i}\Delta_M(p)\left(1 + S(p)\frac{1}{i}\Delta_M(p)\right),$$ with $$S(p) = \frac{\Sigma(p^2)}{1+\Delta_M(p)i\Sigma(p^2)}.$$ The r.h.s. sums up products of propagators with insertions of $\Sigma$. This representation shows that the latter is really the 1-particle irreducible two-point function with respect to $\Delta_M$. The idea is to define a new irreducible function [*with respect to*]{} $\Delta_+$. It is clear that $\Sigma$ is still irreducible in this new sense. In appendix \[app:KG\] it is shown that $$\Delta_M(p) = -\frac{1}{2\omega(\mathbf p)}\left( \Delta_+(p) + \Delta_-(p) \right),$$ where $\Delta_-(p)=-1/(\omega(\mathbf p) + p^0)$, and we find that $S$ contains new irreducible functions, namely those obtained by connecting factors of $\Sigma$ with $\Delta_-$, which is considered to be irreducible. Therefore, $$\begin{aligned} \label{eq:Sp} \hat\Sigma_+(p^0,\mathbf p^2) &\doteq \frac{\Sigma(p^2)}{2\omega(\mathbf p)} + \frac{\Sigma(p^2)}{2\omega(\mathbf p)}i\Delta_-(p) \frac{\Sigma(p^2)}{2\omega(\mathbf p)} + \dots \notag \\ &= \frac{\Sigma(p^2)}{2\omega(\mathbf p)-\Delta_-(p) i\Sigma(p^2)},\end{aligned}$$ is the fundamental irreducible function with respect to $\Delta_+$. One may easily check that in terms of $\hat\Sigma_+$, $S$ can be written as $$S(p) = \frac{2\omega(\mathbf p)\hat \Sigma_+(p^0,\mathbf p^2)}{1-\Delta_+(p) i\hat \Sigma_+(p^0,\mathbf p^2)}.$$ Let us also define $$\begin{aligned} \hat G_+(p) &\doteq i\Delta_+(p)\left( 1 + \frac{S(p)}{2\omega(\mathbf p)} i\Delta_+(p) \right) \\ &= \frac{1}{i}\frac{1}{\omega(\mathbf p)-p^0 + i\hat \Sigma_+(p^0,\mathbf p^2) - i\epsilon}.\end{aligned}$$ The equation $$\label{eq:Mpp-def} {\omega_{\text{p}}}(\mathbf p) = \omega(\mathbf p) + i\hat\Sigma_+({\omega_{\text{p}}}(\mathbf p^2), \mathbf p),$$ which defines the location of the pole of $\hat G_+$, is equivalent to  and we can write $$\label{eq:Gp-pole} \hat G_+(p) = \frac{1}{i}\frac{\hat Z_+(\mathbf p^2)}{{\omega_{\text{p}}}(\mathbf p) - p^0 - i\epsilon} + \text{regular},p^0\rightarrow{\omega_{\text{p}}}(\mathbf p),$$ where $$\hat Z_+^{-1}(\mathbf p^2) = 1 - i\hat\Sigma_+'({\omega_{\text{p}}}(\mathbf p),\mathbf p^2)$$ and the prime refers to the derivative with respect to $p^0$. With a little algebra, we can cast eq.  into the form $$\begin{aligned} G(p) =&\; \left(1-\frac{i\hat\Sigma_+(p^0, \mathbf p^2)}{\omega(\mathbf p)+p^0} \right) \frac{\hat G_+(p)}{\omega(\mathbf p)+p^0} \notag \\ =&\; \frac{1}{i} \frac{1}{\omega(\mathbf p)+p^0} \left(1-\frac{i\hat \Sigma_+(p^0, \mathbf p^2)}{\omega(\mathbf p)+p^0} \right) \frac{\hat Z_+(\mathbf p^2)}{{\omega_{\text{p}}}(\mathbf p) - p^0 - i\epsilon} \notag \\ &\; + \text{regular},p^0\rightarrow{\omega_{\text{p}}}(\mathbf p),\end{aligned}$$ which is to be compared with eq. . The relation between the residues can be read off to be $$\label{eq:Z-Zp} \hat Z_+(\mathbf p^2) = \frac{(\omega(\mathbf p)+ {\omega_{\text{p}}}(\mathbf p))^2} {4{\omega_{\text{p}}}(\mathbf p)\omega(\mathbf p)}Z_{H}.$$ Note that the $\mathbf p$ dependence of $\hat Z_+$ is entirely due to loop corrections. At tree-level, where ${\omega_{\text{p}}}(\mathbf p)=\omega(\mathbf p)$, the residues are, of course, both equal to $1$. Let us now come to the original $G_+$ defined in eq. . From the previous analysis we find that if we match the irreducible function $\Sigma_+$ defined in eq.  to the full theory according to $$\Sigma_+(p^0,\mathbf p^2) = \hat\Sigma_+(p^0,\mathbf p^2),$$ we also have $G_+(p)=\hat G_+(p)$ and the physical mass defined through eq.  is the same as the one in the relativistic theory. The residues are related by eq. . Note that the matching is done off-shell. The only relevant objects for physical quantities are the location of the pole, defining the physical mass in terms of the parameters of the theory, and its residue, providing the effective normalization of the field. Any off-shell matching that does not change these properties is allowed. The construction presented here singles out one of these possibilities rather naturally. To use these results in a calculation of physical quantities, we must renormalize the theories so that they yield finite results when the regulator is removed. The necessary counter terms at one-loop order are determined in appendix \[app:1-loop-renormalization\]. The statements derived here can be verified explicitly to this order in perturbation theory. Reduction Formulae {#app:reduction} ================== The reduction formula gives the relationship between the residues of certain Green’s functions and physical scattering amplitudes. The underlying assumption is that particles involved in a scattering process behave like free particles long before and long after the collision. This is called the asymptotic condition and must be formulated carefully, see for example [@zuber; @bogoliubov]. We first give a review of the facts in a relativistic theory and then consider an effective theory which is not manifestly Lorentz invariant. Relativistic Theory {#relativistic-theory} ------------------- We consider the generic Lagrangian $$\begin{aligned} {\mathcal L}&= {\mathcal L}^0+ \bar{\mathcal L}^0 + {\mathcal L}^{\text{int}} \notag \\ {\mathcal L}^0 &= \partial_\mu{H}^*\partial^\mu{H}-M^2{H}^*{H}\end{aligned}$$ introduced in section \[sec:transition-amplitudes\]. In the notation of appendix \[app:notation\], connected Green’s functions of the heavy field are denoted by $$\label{eq:red-def-G} G^{(2n)}(x,y) = {\langle0|T\hat{H}(x)\hat{H}^\dag(y)|0\rangle}_c.$$ Canonical quantization of the free ${H}$ field leads to creation and annihilation operators of one particle states as described in appendix \[app:canonical-quantisation\]. The asymptotic condition says that the interacting field behaves like a free field in the remote past and future in the weak sense (only for matrix elements) $$\begin{aligned} {H}(x) &\stackrel{x^0\rightarrow -\infty}{\rightarrow} Z_{H}^\frac{1}{2} {{H}_{\text{in}}}(x) \\ {H}(x) &\stackrel{x^0\rightarrow +\infty}{\rightarrow} Z_{H}^\frac{1}{2} {{H}_{\text{out}}}(x)\end{aligned}$$ The fields ${{H}_{\text{in}}}$, ${{H}_{\text{out}}}$ have all the properties of free fields and Lorentz invariance implies that $Z_{H}$ is a constant, which is given by the residue of the full two-point function $$\begin{aligned} \label{eq:G-rel} G^{(2)}(p) &= \int d^4x e^{ipx}{\langle0|T{H}(x){H}^\dag(0)|0\rangle} = \frac{1}{i}\frac{1}{M^2-p^2+i\Sigma(p^2)} \notag \\ &= \frac{1}{i}\frac{Z_{H}}{M_{\text p}^2-p^2-i\epsilon} + \dots\end{aligned}$$ The physical mass ${M_{\text{p}}}$ and the residue $Z_{H}$ are defined through the one-particle irreducible function $\Sigma$ by (see also appendix \[app:2pf\]) $$\begin{aligned} \label{eq:Mp-rel} {M_{\text{p}}}^2 &= M^2+i\Sigma({M_{\text{p}}}^2) \\ \label{eq:Z-rel} Z_{H}^{-1} &= 1 - i\Sigma'({M_{\text{p}}}^2).\end{aligned}$$ We define in- and out states by $$\begin{aligned} \label{eq:in-state} {|p;\text{in}\rangle} &= a_{\text{in}}^\dag(\mathbf p){|0\rangle} \notag \\ {|p;\text{out}\rangle} &= a_{\text{out}}^\dag(\mathbf p){|0\rangle}\end{aligned}$$ and similar for ${|\bar p;\text{in}\rangle}$, ${|\bar p;\text{out}\rangle}$. In fact, the Hilbert spaces of in- and out states are identical and the scattering operator $S$ is the isomorphism that maps a state ${|i;\text{in}\rangle}$ into the space of out-states $${|i;\text{in}\rangle} = S{|i;\text{out}\rangle}.$$ Defining the $T$ operator by $$S = 1+iT,$$ the amplitude to find the final state ${|f;\text{out}\rangle}$ is given by $$\begin{aligned} \label{eq:Tfi-rel} {\langlef;\text{out}|i;\text{in}\rangle} &= {\langlef;\text{in}|i;\text{in}\rangle}+i{\langlef|T|i\rangle} \notag \\ &= {\langlef;\text{in}|i;\text{in}\rangle}+i(2\pi)^4\delta^4\left(P_f-P_i\right) T_{fi}, \end{aligned}$$ where we have also defined the $T$-matrix element $T_{fi}$. If none of the initial one-particle states are contained in the final state, the first term on the r.h.s. vanishes. Let’s consider a configuration where there are $n$ heavy particles in the initial and final states, i.e. $$\begin{aligned} {|i;\text{in}\rangle} &= {|q_1,\dots,q_n;\text{in}\rangle} \\ {|f;\text{out}\rangle} &= {|p_1,\dots,p_n;\text{out}\rangle}.\end{aligned}$$ Reducing the in- and out states as described, for example, in [@zuber] we find $$\begin{gathered} {\langlep_1,\dots,p_n;\text{in}|S-1|q_1,\dots,q_n;\text{in}\rangle}_c = \left(iZ_{H}^{-\frac{1}{2}}\right)^{2n}\int d^4xd^4ye^{i\sum_i(p_ix_i-q_iy_i)} \\ (\Box_{x_1}+{M_{\text{p}}}^2)\dots(\Box_{y_n}+{M_{\text{p}}}^2)G^{(2n)}(x,y),\end{gathered}$$ where the subscript $c$ indicates that disconnected contributions[^11] are not included. In terms of the truncated Green’s function, this reads $$\begin{gathered} {\langlep_1,\dots,p_n;\text{in}|S-1|q_1,\dots,q_n;\text{in}\rangle}_c = \\ (2\pi)^4\delta^4\left(P-Q\right) Z_{H}^n \left.G_{tr}^{(2n)}(p,q)\right|_{\text{on-shell}},\end{gathered}$$ where $P = \sum_i p_i, Q = \sum_i q_i$ and “on-shell” means $p_i^0=\omega_{\text{p}}(\mathbf p_i) = \sqrt{{M_{\text{p}}}^2+\mathbf p_i^2}$, $q_i^0=\omega_{\text{p}}(\mathbf q_i)$. Finally, we read off the expression for the $T$-matrix element for this process[^12] $$\label{eq:Tn->n} T_{n\rightarrow n} = \frac{1}{i}Z_{H}^n \left.G_{tr}^{(2n)}(p,q)\right|_{\text{on-shell}}.$$ Effective Theory ---------------- Now consider the Lagrangian $$\begin{aligned} {{\mathcal L}_+}&= {{\mathcal L}_+}^0 + \bar{\mathcal L}^0 + {{\mathcal L}_+}^{\text{int}} \notag \\ {\mathcal L}_\pm^0 = {H}_\pm^*D_\pm{H}_\pm\end{aligned}$$ introduced in sections \[sec:free-fields\] and \[sec:eff-particle-theory\]. The Green’s functions $G_+^{(2n)}$ are defined in analogy with eq. . Again, we start with the quantization of the free ${{H}_+}$ field as described in appendix \[app:canonical-quantisation\]. It is very important that, up to the normalization, the one-particle state ${|p\rangle}$ is the same as the one of the relativistic theory: it describes a free scalar particle with momentum $\mathbf p$ and energy $\sqrt{{M_{\text{p}}}^2+\mathbf p^2}$. Therefore, the Fock spaces obtained by applying particle creation operators to the vacuum are the same in both theories. Since the in and out states live in this Fock space, the stage is set for an effective theory that can generate the same transition amplitudes as a relativistic theory (cf. sections \[sec:eff-particle-theory\] and \[sec:amplitude-matching\]). Lorentz symmetry is not respected and the asymptotic condition for the interacting field reads $$\begin{aligned} \label{eq:H-asymp-+} {{H}_+}(x) &\stackrel{x^0\rightarrow -\infty}{\rightarrow} Z_+(\Delta)^\frac{1}{2}{{H}_{+,\text{in}}}(x) \\ \label{eq:H-asymp--} {{H}_+}(x) &\stackrel{x^0\rightarrow +\infty}{\rightarrow} Z_+(\Delta)^\frac{1}{2}{{H}_{+,\text{out}}}(x). \end{aligned}$$ The symbol $Z_+(\Delta)$ represents a rotation invariant differential operator. In momentum space, it becomes a function of $\mathbf p^2$ and, as in the previous section, we expect it to be the residue of the full two-point function, which, in terms of the irreducible part $\Sigma_+$, reads $$G_+^{(2)}(p) = \frac{1}{i}\frac{1}{\omega(\mathbf p)-p^0+i\Sigma_+(p^0,\mathbf p^2)-i\epsilon}.$$ It seems obvious to define the physical mass by the zero of the denominator $$\label{eq:Mp-eff} {\omega_{\text{p}}}(\mathbf p) = \sqrt{{M_{\text{p}}}^2+\mathbf p^2} = \omega(\mathbf p)+i\Sigma_+({\omega_{\text{p}}}(\mathbf p),\mathbf p^2).$$ However, for the most general rotation invariant Lagrangian, this would yield a momentum-dependent object ${M_{\text{p}}}$, which cannot serve as a mass parameter. In view of our goal, which is to reproduce the scattering amplitudes of a fully relativistic theory, we may impose a constraint on the interaction Lagrangian, leading to a momentum-independent parameter ${M_{\text{p}}}$, which can play the role of the physical mass of the particle. This constraint can be easily implemented in perturbation theory, where it results in a relation among the coupling constants of the theory (see also the discussion in appendix \[app:2pf\]). Assuming this is done, we can write $$\label{eq:G2-eff-phys} G_+^{(2)}(p) = \frac{1}{i}\frac{Z_+(\mathbf p^2)}{{\omega_{\text{p}}}(\mathbf p)-p^0-i\epsilon} + \dots$$ The residue $Z_+$ is given by (the prime refers to the derivative with respect to $p^0$), $$\label{eq:Z-eff} Z_+^{-1} = 1-i\Sigma_+'({\omega_{\text{p}}}(\mathbf p),\mathbf p^2).$$ In- and out states are defined in analogy to eq.  and all that was said about the $S$ and $T$ matrix in the previous section applies also here. The fact that ${{H}_+}$ can only destroy a particle in the in-state together with the hermiticity of the Lagrangian implies that the number of heavy particles in the initial and final states must be the same. The procedure of the reduction of in- and out-states can be applied to the effective theory without any problems. The result is $$\begin{gathered} {\langlep_1,\dots,p_n|S-1|q_1,\dots,q_n\rangle}_c = \\ i^{2n}\prod_{i=1}^nZ_+(\mathbf p_i^2)^{-\frac{1}{2}} Z_+(\mathbf q_i^2)^{-\frac{1}{2}} \int d^4xd^4ye^{i\sum_i(p_ix_i-q_iy_i)} \\ (\sqrt{{M_{\text{p}}}^2-\Delta_{x_1}}-i\partial_{x_1^0})\dots (\sqrt{{M_{\text{p}}}^2-\Delta_{y_n}}+i\partial_{y_n^0})G_+^{(2n)}(x,y)\end{gathered}$$ or, in terms of the truncated function, $$\begin{gathered} {\langlep_1,\dots,p_n|S-1|q_1,\dots,q_n\rangle}_c = (2\pi)^4\delta^4\left(P-Q\right) \\ \prod_{i=1}^nZ_+(\mathbf p_i^2)^\frac{1}{2} Z_+(\mathbf q_i^2)^\frac{1}{2} \left.G_{+,tr}^{(2n)}(p,q)\right|_{\text{on-shell}}.\end{gathered}$$ The notion of on-shell is the same as before and the $T$ matrix element is given by $$\label{eq:Tfi-eff} T_{n\rightarrow n}^+ = \frac{1}{i}\prod_{i=1}^nZ_+(\mathbf p_i^2)^\frac{1}{2} Z_+(\mathbf q_i^2)^\frac{1}{2} \left.G_{+,tr}^{(2n)}(p,q)\right|_{\text{on-shell}}.$$ Proof of eq.  {#app:T} ============= We prove the decomposition for $G_1$ in eq.  – the proof for the decompositions $G_{2,3,4}$ in that equation is very similar. Using $$A=D_++eB{{\; \; ,}\;}C=D_- + eB{\; \; ,}\nonumber$$ one has $$A-e^2BC^{-1}B=D_+ + eB-e^2BC^{-1}B{\; \; ,}\nonumber$$ and $$\begin{aligned} eB-e^2BC^{-1}B &= eBC^{-1}[C-eB]=eBC^{-1}D_- \notag \\ &=eB(1+eD_-^{-1}B)^{-1}{\; \; ,}\nonumber\end{aligned}$$ as a result of which $G_1$ becomes $$G_1 =\left[1+eD_+^{-1}B (1+eD_-^{-1}B)^{-1}\right]^{-1}D_+^{-1}{ \; .}\nonumber$$ With $$1+eD_+^{-1}B(1+eD_-^{-1}B)^{-1}= \left[1+e(D_+^{-1}+D_-^{-1})B\right] (1+eD_-^{-1}B)^{-1}{\; \; ,}\nonumber$$ we find $$G_1=(1+eD_-^{-1}B)\left[1+e(D_+^{-1}+D_-^{-1})B\right]^{-1}D_+^{-1}{ \; .}\nonumber$$ From $$G_1\doteq D_+^{-1}-D_+^{-1}T_{++}D_+^{-1}{\; \; ,}\nonumber$$ one has $$T_{++}=eB\frac{1}{1+(D_+^{-1}+D_-^{-1})eB}{ \; .}\nonumber$$ This agrees indeed with $$dTd=d\frac{1}{1-elD_M^{-1}}el d{ \; .}\nonumber$$ Determinants {#app:det} ============ Consider the generating functional of Green’s functions $$Z[j,j^*,J] = \frac{\int[d{H}][d{H}^*][d{l}]e^{i\int{\mathcal L}+ j^*{H}+ {H}^*j +J{l}}} {\int [d{H}][d{H}^*][d{l}]e^{i\int{\mathcal L}}}$$ for the toy model $$\begin{aligned} {\mathcal L}&= {\mathcal L}^0 + {\mathcal L}_{l}^0 + e{H}^*{H}{l}\notag \\ {\mathcal L}^0 &= \partial_\mu{H}^*\partial^\mu{H}-M^2{H}^*{H}\notag \\ {\mathcal L}_{l}^0 &= \frac{1}{2}\partial_\mu {l}\partial^\mu{l}+ \frac{m^2}{2}{l}^2\end{aligned}$$ studied in chapter \[chap:toy-model\]. Performing the integration over ${H}$ we get $$\label{eq:Z-hf-int} Z[j,j^*,J] = \frac{\int[d{l}](\det D_M^{-1}D_e)^{-1}e^{i\int \mathcal L_{l}^0 + j^*D_e^{-1}j+J{l}}}{\int [d{l}](\det D_M^{-1}D_e)^{-1}e^{i\int \mathcal L_{l}^0}}.$$ The operator $D_e$ was introduced in section \[sec:external-field\] $$D_e \doteq D_M-e{l}$$ and $D_M = \Box + M^2$. The factors of $\det D_M$ were added to eq.  for later convenience. In this appendix, we first show that $$\label{eq:det-identity} \det D_M^{-1}D_e = \det D_+^{-1}\mathcal D_+ D_-^{-1}C,$$ with the symbols $$\begin{aligned} \mathcal D_+ &= D_+ + eB - e^2BC^{-1}B \\ D_\pm &= \pm i\partial_t-\sqrt{M^2-\Delta} \\ C &= D_- + eB \\ B &= d{l}d\end{aligned}$$ introduced in section \[sec:external-field\]. This decomposition allows us to prove that Z may be written in the form $$\label{eq:Z-equiv} Z[j,j^*,J] = \frac{\int [d{l}](\det D_+^{-1}\mathcal D_+)^{-1}e^{i\int \mathcal L_{l}^0 + j^*D_e^{-1}j+J{l}}}{\int [d{l}](\det D_+^{-1}\mathcal D_+)^{-1}e^{i\int \mathcal L_{l}^0}}.$$ The determinants are ill-defined as long as we don’t specify how to deal with the UV divergences inherent in their definitions. In the following we use dimensional regularization and work in $D\neq 4$ dimensions to render all expressions finite. The statements derived here are then valid to all orders in perturbation theory. Actual renormalization to one loop is performed in appendix \[app:1-loop-renormalization\]. In the following, we use the propagators $\Delta_M$ and $\Delta_\pm$ defined in appendix \[app:KG\] as representations of the operators $D_M^{-1}$ and $D_\pm^{-1}$, respectively. Consider first the l.h.s. of eq.  $$\det D_M^{-1}D_e = \det(1-D_M^{-1}e{l}) = e^{\operatorname{Tr}\ln(1-D_M^{-1}e{l})}.$$ Expanding the logarithm we can write $$\begin{gathered} \label{eq:ln-det-De} \operatorname{Tr}\ln(1-D_M^{-1}e{l}) = -\sum_{n=1}^{\infty}\frac{1}{n}\operatorname{Tr}( D_M^{-1}e{l})^n \\ = -\sum_{n=1}^\infty \frac{1}{n} \int d^Dx_1\dots d^Dx_n \Delta_M(x_1-x_2)e{l}(x_2) \dots \Delta_M(x_n-x_1)e{l}(x_1).\end{gathered}$$ The $n^{\text th}$ term of this sum is a loop formed by connecting $n$ fields ${l}$ by as many propagators $\Delta_M$ (see figure \[fig:det-expansion\]). ![A graphical representation of the r.h.s. of eq. . The line stands for a propagator $\Delta_M$ and the cross for a light field ${l}$.[]{data-label="fig:det-expansion"}](figure11.ps){width="8cm"} Using the identity $C^{-1}eB = 1-C^{-1}D_-$, which follows directly from the definition of $C$, we can cast the r.h.s. of eq. into the form $$\det D_+^{-1}\mathcal D_+ D_-^{-1}C = \det\left(1+\left(D_+^{-1}+D_-^{-1}\right)eB\right).$$ Proceeding as before, we find $$\begin{gathered} \label{eq:ln-det-D-p} \operatorname{Tr}\ln\left(1+\left(D_+^{-1}+D_-^{-1}\right)ed{l}d\right) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \int d^Dx_1\dots d^Dx_n \\ d_{x_1}(\Delta_+(x_1-x_2)+\Delta_-(x_1-x_2))d_{x_2}e{l}(x_2) \\ \dots d_{x_n}(\Delta_+(x_n-x_1)+\Delta_-(x_n-x_1))d_{x_1}e{l}(x_1),\end{gathered}$$ where the subscript of $d$ indicates on which variable it acts. Due to the operator relation $$\label{eq:DM-decomp} D_M^{-1} = -d(D_+^{-1} + D_-^{-1})d$$ derived in appendix \[app:KG\] this is indeed equal to the l.h.s. of eq. . Formally, we can write eq. as $$\label{eq:det-identity-2} \det D_M^{-1}D_e = \det D_+^{-1}\mathcal D_+ \det D_-^{-1}C.$$ Taking the logarithm, we find to first order in $e$ the tadpole term $$\Delta_M^D(0)\int d^Dx {l}(x) = - \left(\left. d^2\Delta_+^D(x)\right|_{x=0} + \left.d^2\Delta_-^D(x)\right|_{x=0}\right)\int d^Dx{l}(x).$$ The explicit expressions of the terms on the r.h.s. are $$\left.d^2\Delta_\pm^D\right|_{x=0} = - \int \frac{d^D p}{(2\pi)^D}\frac{1}{2\omega(\mathbf p)(\omega(\mathbf p)\mp p^0 - i\epsilon)}.$$ In standard dimensional regularization, where one writes $d^Dp = dp^0d^{D-1}\mathbf p$ and integrates over $p^0$ separately, these are not well defined because the integrand falls off only like $1/p^0$ for large $p^0$. One may use what is called split dimensional regularization ([@split-d]) where one introduces two independent regulators $\sigma$ and $D$ according to $$d^D p = d^\sigma p^0 d^{D-\sigma}\mathbf p.$$ In this scheme we find $$\left. d^2\Delta_+^D(x)\right|_{x=0} + \left.d^2\Delta_-^D(x)\right|_{x=0} = -e^{i\sigma\frac{\pi}{2}} \frac{M^{D-2}}{(4\pi)^\frac{D}{2}} \Gamma(1-\frac{D}{2})$$ and one can check that this is indeed equal to $-\Delta_M(0)$, evaluated with the same prescription. This subtlety only occurs in the tadpole. Every other graph has an integrand that falls off at least like $1/(p^0)^2$ and split dimensional regularization coincides with the standard dimensional regularization. Eq.  is therefore true within this special regularization scheme. Let us give an intuitive understanding of this decomposition. A loop containing $n$ propagators can be written as a sum of $2^n$ terms by decomposing $\Delta_M$ into $\Delta_\pm$ as in eq. . One of these terms will exclusively contain anti-particle propagators $\Delta_-$ and all of these graphs are collected in the expression $\det D_-^{-1}C$. Therefore, eq. can be interpreted as the separation of the contribution of the pure anti-particle sector to loops formed by the heavy field. To prove eq. , we show that $\det D_-^{-1}C$ does not contribute to any Green’s functions contained in Z. The explicit expression for this determinant is $$\begin{split} \det D_-^{-1}C = \exp\Bigl\{\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \int d^Dx_1 \dots d^Dx_n \Delta_-(x_1-x_2)eB(x_2)\dots \Bigr. \\ \Bigl. \times \Delta_-(x_n-x_1)eB(x_1) \Bigr\}. \end{split}$$ Upon performance of the ${l}$-integration in eq. it will produce sub-graphs of the type shown in figure \[fig:det-C-contrib\]. ![A typical contribution of a loop formed exclusively with anti-particle propagators (solid lines). It is connected to the rest of the diagram only by propagators of the light field (dashed lines).[]{data-label="fig:det-C-contrib"}](figure12.ps){width="5cm"} Because $\Delta_-(p^0,\vec p^2)$ contains only one pole in $p^0$, all poles of the integrand of such a loop lie in the same half-plane. We can close the contour of the integration in the other half-plane and find that the entire integral vanishes, irrespective of the rest of the diagram that this loop is part of. Therefore, we can drop this determinant in the expression for $Z$ [*without changing any Green’s functions*]{}, which completes the proof of eq. . Let us return to the tadpole contribution discussed above. We may separate it by defining $$\label{eq:tadpole-rem-1} \delta \doteq (\det D_M^{-1}D_e)^{-1} e^{-e\Delta_M(0)\int d^Dx l(x)}$$ Furthermore, let us add a term to the Lagrangian $$\bar {\mathcal L}= {\mathcal L}- e\frac{1}{i}\Delta_M(0){l}.$$ The corresponding generating functional $$\label{eq:tadpole-rem-2} \bar Z[j,j^*,J] = \frac{\int [d{l}]\delta e^{i\int{\mathcal L}_{l}^0 + j^*D_e^{-1}j + Jl}} {\int [dl]\delta e^{i\int{\mathcal L}_{l}^0}}$$ is identical to $Z$, except that it does not contain any one-loop tadpole contributions. The additional term in the definition of $\bar{\mathcal L}$ can be viewed as a 1-loop counter term. We have thus shown that renormalization can be done in such a way that the tadpole is removed from any Green’s function (see also appendix \[app:1-loop-renormalization\]). 1-Loop Renormalization ====================== \[app:1-loop-renormalization\] Relativistic Theory {#relativistic-theory-1} ------------------- We consider the tree-level Lagrangian $${\mathcal L}= -{H}^*D_M{H}-\frac{1}{2}{l}D_m{l}+ e{H}^*{H}{l}+j^*{H}+ {H}^*j + J{l},$$ where $D_M=\Box+M^2$ and $D_m=\Box+m^2$. It is convenient to replace the complex field ${H}$ by two real fields ${\phi_1}$, ${\phi_2}$ and the source $j$ by two real sources $j_1$, $j_2$ through $$\begin{aligned} {H}&= \frac{1}{\sqrt{2}}({\phi_1}+ i{\phi_2}) \notag \\ j &= \frac{1}{\sqrt{2}}(j_1 + ij_2).\end{aligned}$$ Renaming ${l}\equiv{\phi_3}$, $J\equiv j_3$, we can collect the fields and sources in three-dimensional vectors $$\begin{aligned} {\phi^{\text{T}}}&= ({\phi_1}, {\phi_2}, {\phi_3}) \notag \\ {j^{\text{T}}}&= (j_1,j_2,j_3).\end{aligned}$$ The Lagrangian then reads $${\mathcal L}= -\frac{1}{2}{\phi^{\text{T}}}D_0\phi + \frac{e}{2}({\phi_1}^2+{\phi_2}^2){\phi_3}+ {j^{\text{T}}}\phi,$$ with $$D_0 = \begin{pmatrix} D_M & 0 & 0 \\ 0 & D_M & 0 \\ 0 & 0 & D_m \end{pmatrix}.$$ In units where $\hbar$ is explicit, the generating functional $W$ of connected Green’s functions is defined by $$e^{\frac{i}{\hbar}W[j;\hbar]} = \frac{1}{\mathcal Z} \int [d\phi] e^{\frac{1}{\hbar} S[j]},$$ where $$\begin{aligned} \mathcal Z &= \int [d\phi] e^{\frac{i}{\hbar} S[0]} \notag \\ S[j] &= \int d^dx {\mathcal L}(x;j).\end{aligned}$$ We use dimensional regularization to give a meaning to the path integral and want to construct the counter term Lagrangian $${\mathcal L}_{ct} = \hbar{\mathcal L}^{(1)} + O(\hbar^2)$$ that absorbs the divergences in $d=4$. The expansion of $W$ in powers of $\hbar$ is equivalent to an expansion in the number of loops, so that $W_0$ and $W_1$ defined by $$W[j;\hbar] = W_0[j] + \hbar W_1[j] + O(\hbar^2)$$ generate tree- and one-loop graphs, respectively. This expansion is obtained by writing $\phi$ as fluctuation around the solution $\bar\phi$ of the equations of motion $$\begin{aligned} D_M\bar{\phi_1}- e\bar{\phi_1}\bar{\phi_3}- j_1 &= 0 \notag \\ D_M\bar{\phi_2}- e\bar{\phi_2}\bar{\phi_3}- j_2 &= 0 \notag \\ D_m\bar{\phi_3}-\frac{e}{2}(\bar{\phi_1}^2 + \bar{\phi_2}^2)\bar{\phi_3}- j_3 &= 0.\end{aligned}$$ Setting $\phi = \bar\phi + \hbar^{1/2}\eta$ and keeping only terms of $O(\hbar)$ we find $$\begin{aligned} W_0 &= \int d^dx \bar{\mathcal L}(x) \\ \label{eq:W1} W_1 &= \frac{i}{2}\ln\frac{\det D}{\det D_0} + \int d^dx\bar{\mathcal L}^{(1)}(x),\end{aligned}$$ where $$D = D_0 -e \begin{pmatrix} \bar{\phi_3}& 0 & \bar{\phi_1}\\ 0 & \bar{\phi_3}& \bar{\phi_2}\\ \bar{\phi_1}& \bar{\phi_2}& 0 \end{pmatrix}$$ and barred quantities are evaluated at $\phi=\bar\phi$. Applying the heat kernel technique, the contributions to $W_1$ that diverge in $d=4$ can be isolated. The result is $$\begin{aligned} \label{eq:W1div} W_1 = &\; \frac{e^2}{2} \Delta_1 \int d^dx \left( \bar{\phi_1}^2(x) + \bar{\phi_2}^2(x) \right) + \frac{e^2}{2}\Delta_2 \int d^dx \bar{\phi_3}^2(x) \notag \\ & + e \Delta_3 \int d^dx\bar{\phi_3}(x) + \text{finite}(d\rightarrow 4),\end{aligned}$$ with $$\begin{aligned} \Delta_1 &= \frac{1}{2} \frac{\Gamma(-\omega)}{(4\pi)^{2+\omega}} \left(M^{2\omega}+m^{2\omega}\right) \\ \Delta_2 &= \frac{\Gamma(-\omega)}{(4\pi)^{2+\omega}} M^{2\omega} \\ \Delta_3 &= \frac{\Gamma(-1-\omega)}{(4\pi)^{2+\omega}} M^{2(\omega+1)}\end{aligned}$$ and $\omega=(d-4)/2$. We introduce the renormalization scale $\mu$ with the object $$\begin{aligned} \hat L &= \left(\frac{M}{\mu}\right)^{2\omega} \frac{\mu^{2\omega}}{32\pi^2} \frac{\Gamma(-1-\omega)}{(4\pi)^\omega} \notag \\ &= L(\mu) + \frac{\mu^{2\omega}}{32\pi^2} \left(\ln\frac{M^2}{\mu^2} -1 \right) + a(\omega,\frac{M}{\mu}) \\ L(\mu) &= \frac{\mu^{2\omega}}{32\pi^2}\left(\frac{1}{\omega} - \Gamma'(1) - \ln 4\pi \right).\end{aligned}$$ The function $a$ vanishes in the limit $\omega\rightarrow 0$ and is not needed explicitly. $\hat L$ is independent of $\mu$ $$\mu\frac{\partial}{\partial\mu} \hat L = 0$$ and so are $$\begin{aligned} \Delta_1 &= -2\left[ L(\mu) + \frac{\mu^{2\omega}}{32\pi^2}\left\{ \ln\frac{M^2}{\mu^2} + \frac{1}{2}\ln\frac{m^2}{M^2}\right\} + b(\omega,\frac{M}{\mu},\frac{m}{\mu}) \right] \\ \Delta_2 &= -2\left(L(\mu) + \frac{\mu^{2\omega}}{32\pi^2}\ln\frac{M^2}{\mu^2} + c(\omega,\frac{M}{\mu})\right) \\ \Delta_3 &= 2M^2\hat L\end{aligned}$$ Like $a$, the functions $b$ and $c$ vanish for $\omega\rightarrow 0$. In order to cancel these divergences, we need a counter term Lagrangian of the form $${\mathcal L}^{(1)} = -\frac{e^2}{2}c_1\left({\phi_1}^2+{\phi_2}^2\right) - \frac{e^2}{2}c_2{\phi_3}^2 - c_3 eM^2{\phi_3},$$ The dimensionless constants $c_n$ can be chosen to be independent of $\mu$ and in the $\overline{MS}$ scheme we set $$c_n = c_{n}^r(\mu,\omega) + \Gamma_n L(\mu).$$ The renormalized couplings $c_n^r$ are finite and depend on the scale according to the renormalization group equations $$\mu\frac{\partial}{\partial\mu} c_n^r(\mu,\omega) = -2\omega\Gamma_n L(\mu).$$ From eq.  we can read off $$\begin{aligned} \Gamma_{1,2} = -2.\end{aligned}$$ The term $\Delta_3$ plays a special role. In appendix \[app:det\] it was identified with the loop of the tadpole graph $$\includegraphics[scale=0.7]{figure13.ps}$$ which is simply the Fourier transform $\Delta_M(0)$ of the heavy propagator at zero momentum. In fact, we have $\Delta_3 = -i\Delta_M(0)$. Now, in that appendix it was shown that by adding the term $ie\Delta_M(0){l}$ to the Lagrangian, the tadpole is removed from all the Green’s functions (see eqns.  through ). We therefore chose $$c_3 = 2\hat L.$$ Physical quantities can be expressed in terms of the scale-independent and finite couplings $$\bar c_n = -c_n^r(\mu,0) + \frac{\Gamma_n}{32\pi^2}\ln\frac{M^2}{\mu^2}.$$ They are determined through the condition that the parameters $M$ and $m$ should coincide with the physical masses ${M_{\text{p}}}$ and $m_{\text{p}}$. The explicit expressions are not needed here. Finally, we may go back to the original fields and find that $${\mathcal L}({H},{H}^*,{l}) - c_1 e^2 {H}^*{H}- c_2\frac{e^2}{2}{l}^2 - c_3 eM^2{l}$$ gives finite results in $d=4$ at 1-loop level. Effective Non-local Theory -------------------------- In appendix \[app:det\] it is shown that the non-local Lagrangian $${\mathcal L}= {\mathcal L}_+ + j^*{{H}_+}+ {{H}_+}^* j + J{l}$$ constructed in section \[sec:p-a-sector\] contains the same loops as the full theory. Therefore, the only divergent graphs to one loop are the self-energies and the vacuum polarization of the light field. Consider first the two-point function of the light field. We know that it is identical in both theories (this is evident by comparing the expressions  and  of the generating functionals) and so must be the counter terms. By comparing the two-point functions of the heavy field in eqns. and we find that $-e^2c_1(d{{H}_+}^*)(d{{H}_+})$ is the counter term needed to render the self-energy of ${H}$ in the effective theory finite. The vacuum expectation value of the light field is given by $$v = \int d^4 x e^{ipx}{\langle0|{l}(x)|0\rangle} = (2\pi)^4\delta^4(p)\Delta_m(p) ie[d^2\Delta_+](0)$$ where as in the relativistic theory we have $$v = (2\pi)^4\delta^4(p)\Delta_m(p) ie\Delta_M(0).$$ The quantity $[d^2\Delta_+](0)$ is evaluated in split dimensional regularization discussed in appendix \[app:det\] $$\begin{aligned} [d^2\Delta_+](0) &= -\frac{1}{2}e^{i\sigma\frac{\pi}{2}}\Delta_3 \nonumber\\ &= -M^2 e^{i\sigma\frac{\pi}{2}}\hat L.\end{aligned}$$ The appropriate counter term is therefore $-\tilde c_3 eM^2l$ with $$\tilde c_3 \doteq e^{i\sigma\frac{\pi}{2}}\hat L.$$ Putting everything together, the effective Lagrangian that is finite at 1-loop is given by $${\mathcal L}_+ - c_1e^2 (d{{H}_+}^*)(d{{H}_+}) - c_2\frac{e^2}{2}{l}^2 - \tilde c_3 eM^2l.$$ [^1]: In positronium, $e^+$ and $e^-$ can, however, annihilate into photons. We don’t want to go into this rather subtle issue here and ignore this effect. To be save, we could consider a stable system, like $e^+\mu^-$ as is actual done in [@caswell-lepage], introducing another scale $m_\mu$. [^2]: In HQET and HBCHPT one introduces a four-velocity $v_\mu$ to write down the Lagrangian in different frames of reference and Lorentz invariance is replaced by “reparametrisation invariance” [@reparametrization-invariance] [^3]: Due to the convention of the Fourier transform given in appendix \[app:notation\] the momenta $k_i$ with $k_i^0=\sqrt{m_{\text{p}}^2+\mathbf k^2}$ correspond to outgoing light particles. The amplitude for processes with incoming light particles can be obtained by crossing [^4]: We omit all tadpole graphs in accordance with the 1-loop renormalization discussed in appendix \[app:1-loop-renormalization\] [^5]: In the CMS [^6]: In no particular frame of reference [^7]: This actually depends on the regularization prescription. If one uses a momentum space cutoff $M\alpha\ll\Lambda\ll M$, the loop starts contributing at the naive order see, for example, ref. [@kinoshita-nio-1]. In dimensional regularization, however, this is not true [^8]: As a side remark, it is interesting to note that this fact is the reason why HBCHPT was introduced to replace the original relativistic treatment of the pion-nucleon system [@gasser-sainio-svarc], which suffers from the same power counting problem. It was recently shown that a new regularization scheme restores power counting in the manifestly relativistic formulation [@becher-leutwyler] [^9]: To mention an example, we consider the decay of Ortho– or Parapositronium in the framework of nonrelativistic QED – it requires the inclusion of the hard processes $e^+e^-\rightarrow n\gamma$. In the literature, the problem is circumvented by use of a nonhermitean Lagrangian [@labelle-thesis]. While this may be useful as far as the calculational purpose is concerned, it is clear that there is room for improving this framework. [^10]: They can be obtained in perturbation theory from the Legendre transform of the generating functional of connected Green’s functions [^11]: There are no subsets of particles that do not interact [^12]: The $T_{n\rightarrow n}$ is not precisely the one defined in eq.  but only the contribution of the connected part
--- abstract: 'Models that are learned from real-world data are often biased because the data used to train them is biased. This can propagate systemic human biases that exist and ultimately lead to inequitable treatment of people, especially minorities. To characterize bias in learned classifiers, existing approaches rely on human oracles labeling real-world examples to identify the “blind spots” of the classifiers; these are ultimately limited due to the human labor required and the finite nature of existing image examples. We propose a simulation-based approach for interrogating classifiers using generative adversarial models in a systematic manner. We incorporate a progressive conditional generative model for synthesizing photo-realistic facial images and Bayesian Optimization for an efficient interrogation of independent facial image classification systems. We show how this approach can be used to efficiently characterize racial and gender biases in commercial systems.' author: - | Daniel McDuff, Shuang Ma, Yale Song and Ashish Kapoor\ Microsoft\ Redmond, WA, USA\ `{damcduff,v-mashua,yalesong,akapoor}@microsoft.com`\ bibliography: - 'references.bib' title: Characterizing Bias in Classifiers using Generative Models --- Introduction ============ Models that are learned from found data (e.g., data scraped from the Internet) are often biased because these data sources are biased [@torralba2011unbiased]. This can propagate systemic inequalities that exist in the real-world [@caliskan2017semantics] and ultimately lead to unfair treatment of people. A model may perform poorly on populations that are minorities within the training set and present higher risks to them. For example, there is evidence of lower precision in pedestrian detection systems for people with darker skin tones (higher on the Fitzpatrick ([-@fitzpatrick1988validity]) scale). This exposes one group to greater risk from self-driving/autonomous vehicles than another [@wilson2019predictive]. Other studies have revealed systematic biases in facial classification systems [@buolamwini2017gender; @buolamwini2018gender], with the error rate of gender classification up to seven times larger on women than men and poorer on people with darker skin tones. Another study found that face recognition systems misidentify people with darker skin tones, women, and younger people at higher error rates [@klare2012face]. To exacerbate the negative effects of inequitable performance, there is evidence that African Americans are subjected to higher rates of facial recognition searches [@garvie2016perpetual]. Commercial facial classification APIs are already deployed in consumer-facing systems and are being used by law enforcement. The combination of greater exposure to algorithms and a reduced precision in the results for certain demographic groups deserves urgent attention. ![image](Overview.PNG){width="0.75\linewidth"} Many learned models exhibit bias as training datasets are limited in size and diversity. Let us take several benchmark datasets as exemplars. Almost 50% of the people featured in the widely used MS-CELEB-1M dataset [@guo2016ms] are from North America (USA and Canada) and Western Europe (UK and Germany), and over 75% are men. The demographic make up of these countries is predominantly Caucasian/white.[^1] Another dataset of faces, IMDB-WIKI [@rothe2015dex], features 59.3% men and Americans are hugely over-represented (34.5%). Another systematic profile [@buolamwini2017gender] found that the IARPA Janus Benchmark A (IJB-A) [@klare2015pushing] contained only 7.80% of faces with skin types V or VI (on the Fitzpatrick skin type scale) and again featured over 75% males. Sampling from the dataset listed here indiscriminately leads to a large proportion of images of males with lighter skin tones and upon training an image classifer often results in a biased system. Creating balanced datasets is a non-trivial task. Sourcing naturalistic images of a large number of different people is challenging. Furthermore, no matter how large the dataset is, it may still be difficult to find images that are distributed evenly across different demographic groups. Attempts have been made to improve facial classification by including gender and racial diversity. In one example, by Ryu et al. [@ryu2017improving], results were improved by scraping images from the web and learning facial representations from a held-out dataset with a uniform distribution across race and gender intersections. Improving the performance of machine-learned classifiers is virtuous but there are other approaches to addressing concerns around bias. The concept of *fairness through awareness* [@dwork2012fairness] is the principle that in order to combat bias, we need to be aware of the biases and why they occur. In complex systems, such as deep neural networks, many of the “unknowns” are unknown and need to be identified [@lakkaraju2016discovering; @lakkaraju2017identifying]. Identifying and characterizing “unknowns” in a model requires a combination of *exploration* to identify regions of the model that contain failure modes and *exploitation* to sample frequently from these region in order to characterize performance. Identifying failure modes is similar to finding adversarial examples for image classifiers [@athalye2017synthesizing; @tramer2017space], a subject that is of increasing interest. One way of characterizing bias that holds promise is data simulation. Parameterized computer graphics simulators are one way of testing vision models [@veeravasarapu2015model; @veeravasarapu2015simulations; @veeravasarapu2016model; @vazquez2014virtual]. Generally, it has been proposed that graphics models be used for performance evaluation [@haralick1992performance]. Recently, McDuff et al. [-@mcduff2018identifying] illustrated how highly realistic simulations could be used to interrogate the performance of face detection systems. However, creating high fidelity 3D assets for simulating many different facial appearances (e.g., bone structures, facial attributes, skin tones etc.) is time consuming and expensive. Generative adversarial networks (GANs) [@GANs] are becoming increasingly popular for synthesizing data [@shrivastava2017learning] and present an alternative, or complement, to graphical simulations. Generative models are trained to match the target distribution. Thus, once trained, a generative model can flexibly generate a large amount of diverse samples without the need for pre-built 3D assets. They can also be used to generate images with a set of desired properties by conditioning the model during the training stage and thus enabling generation of new samples in a controllable way at test time. Thus, GANs have been used for synthesizing images of faces at different ages [@yang2017learning; @choi2018stargan] or genders [@dong2017unsupervised; @choi2018stargan]. However, unlike parameterized models, statistical models (such as GANs) are fallible and might also have errors themselves. For example, the model may produce an image of a man even when conditioned on a woman. To use such a model for characterizing the performance of an independent classifier it is important to first characterize the error in the image generation itself. In this paper, we propose to use a characterized state-of-the-art progressive conditional generative model in order to test existing computer vision classification systems for bias, as shown in Figure \[fig:intro\]. In particular, we train a progressive conditional generative network that allows us to create high-fidelity images of new faces with different appearances by exploiting the underlying manifold. We train the model on diverse image examples by sampling in a balanced manner from men and women from different countries. Then we characterize this generator using oracles (human judges) to identify any errors in the synthesis model. Using the conditioned synthesis model and a Bayesian search scheme we efficiently exploit and explore the parameterized space of faces, in order to find the failure cases of a set of existing commercial facial classification systems and identify biases. One advantage of this scheme is that we only need to train and characterize the performance of the generator once and can then evaluate many classifiers efficiently and systematically, with potentially many more variations of facial images than were used to train the generator. The contributions of this paper are: (1) to present an approach for conditionally generating synthetic face images based on a curated dataset of people from different nations, (2) to show how synthetic data can be used to efficiently identify limits in existing facial classification systems, and (3) to propose a Bayesian Optimization based sampling procedure to identify these limits more efficiently. We release the nationality data, model and code to accompany the image data used in this paper (see the supplementary material). Related Work ============ **Algorithmic Bias.** There is wide concern about the equitable nature of machine learned systems. Algorithmic bias can exist for several reasons and the discovery or characterization of biases is non-trivial [@hajian2016algorithmic]. Even if biases are not introduced maliciously they can result from explicit variables contained within a model or via variables that correlate with sensitive attributes - *indirect discrimination*. Ideally we would minimize algorithmic bias or discrimination as much as possible, or prevent it entirely. However, this is challenging: First, algorithms can be released by third-parties who may not be acting in the public’s best interest and not take the time or effort required to maximize the fairness of their models. Second, removing biases is technically challenging. For example, balancing a dataset and removing correlates with sensitive variables is very difficult, especially when learning algorithms are data hungry and the largest, accessible data sources (e.g., the Internet) are biased [@baeza2016data]. Tools are needed to help practitioners evaluate models, especially black-box models. Making algorithms more transparent and increasing accountability is another approach to increasing fairness [@dwork2012fairness; @lepri2018fair]. A significant study [@buolamwini2018gender] highlighted that facial classification systems were not as accurate on faces with darker skin tones and on females. This paper led to improvements in the models behind these APIs being made [@raji2019actionable]. This illustrates how characterization of model biases can be used to advance the quality of machine learned systems. Biases often result from unknowns within a system. Methods have been proposed to help address the discovery of unknowns in predictive models [@lakkaraju2016discovering; @lakkaraju2017identifying]. In their work the search-space is partitioned into groups which can be given interpretable descriptions. Then an explore-exploit strategy is used to navigate through these groups systematically based on the feedback from an oracle (e.g., a human labeler). Bansal and Weld proposed a new class of utility models that rewarded how well the discovered $\textit{unknown unknowns}$ help explain a sample distribution of expected queries [@bansal2018coverage]. Using human oracles is labor intensive and not scalable. We employ an explore-exploit strategy in our work, but rather than rely on a human oracle we use an image synthesis model. Using a conditioned model we provide a systematic way of interrogating a black box model by generating variations on the target images and repeatedly testing a classifier’s performance. We incentivize the search algorithm to explore the parameter space of faces but also reward it for identifying failures and interrogating these regions of the space more frequently. **Generative Adversarial Networks.** Deep generative adversarial networks has enabled considerable improvements in image generation [@goodfellow2014generative; @zhang2017stackgan; @xu2018attngan]. Conditional GANs [@cGAN] allow for the addition of conditional variables, such that generation can be performed in a “controllable” way. The conditioning variables can take different forms (e.g. specific attributes or a raw image [@choi2018stargan; @yan2016attribute2image].) For facial images, this has been applied to control the gender [@dong2017unsupervised], age [@yang2017learning; @choi2018stargan], hair color, skin tone and facial expressions [@choi2018stargan] of generated faces. This allows for a level of systematic simulation via manifolds in the space of faces. Increasing the resolution of images synthesized using GANs is the focus of considerable research. Higher quality output images have been achieved by decomposing the generation process into different stages. The LR-GAN [@Yang2017LRGANLR] decomposes the process by generating image foregrounds and backgrounds separately. StackGAN [@zhang2017stackgan] decomposes generation stages into several steps each with greater resolution. PG-GAN [@karras2018progressive] has shown impressive performance using a progressive training procedure starting from very low resolution (4$\times$4) and ending with high resolution images (1024$\times$1024). It can produce high fidelity images that are often tricky to distinguish from real photos. In this paper, we employ a progressive conditional generative adversarial model for creating photo-realistic image examples with controllable “gender” and “race”. These images are then used to interrogate independent image classification systems. We model the problem as a Gaussian process, sampling images from the model iteratively based on the performance of the classifiers, to efficiently discover blind-spots in the models. Empirically, we find that these examples can be used to identify biases in the image classification systems. Approach ======== We propose to use a generative model to synthesize face images and then apply Bayesian optimization to efficiently generate images that have the highest likelihood of breaking a target classifier. **Image Generation.** To generate photo-realistic face images in a controllable way, we propose to adopt a progressively growing conditional GAN [@karras2018progressive; @cGAN] architecture. This model is trained so as to condition the generator $G$ and discriminator $D$ on additional labels. The given condition $\theta$ could be any kinds of auxiliary information; here we use $\theta$ to specify both the race $r$ and gender $g$ of the subject in the image, i.e., $\theta=[r; g]$. During testing time, the trained $G$ should produce face images with the race and gender as specified by $\theta$. We curated a dataset $\{x; \theta\}$ (described below), where $x$ is a face image and $\theta$ indicates the race $r$ and gender $g$ labels of $x$. To train the conditional generator, the input of the generator is a combination of a condition $\theta$ and a prior noise input $p_z(z)$; $z$ is a 100-D vector sampled from a unit normal distribution and $\theta$ is a one-hot vector that represents a unique combination of (race, gender) conditions. We concatenate $z$ and $\theta$ as the input to our model. $G$’s objective is defined by: $${\mathcal{L}_{G}} = - {\mathbb{E}_{z,\theta} \big[\log D(G(z,\theta))\big]}$$ The design of the discriminator $D$ is inspired by Thekumparampil et al.’s ([-@NIPSRC-GAN]) Robust Conditional GAN model which proved successful at delivering robust results. We train $D$ on two objectives: to discriminate whether the synthesized image is real or fake, and to classify the synthesized image into the correct class (e.g., race and gender). The training objective for $D$ is defined by: $${\mathcal{L}_{D}} = -{\mathbb{E}\big[\log D(x)\big]} - {\mathbb{E}_{z,\theta}\big[\log D(G(z,\theta))\big]} - {\mathbb{E}_{z,\theta}\big[\log C(G(z,\theta))\big]}$$ where $C$ is an N-way classifier. Our full learning objective is: $$\mathcal{L}_{adv} = \mathop {\min }\limits_{G} \mathop {\max }\limits_{D} {\mathcal{L}_{G}} + {\mathcal{L}_{D}}$$ ![image](Network.png){width="\linewidth"} We train the generator progressively [@karras2018progressive] by increasing the image resolution by a power of two at each step, from 4$\times$4 pixels to 128$\times$128 pixels (see Figure \[fig:architechture\]). The real samples are downsampled into the corresponding resolution in each stage. The training code is included in supplementary material. **Bayesian Optimization.** Now that we have a systematically controllable image generation model, we propose to combine this with Bayesian Optimization [@brochu2010tutorial] to explore and exploit the space of parameters $\theta$ to find errors in the target classifier. We have $\theta$ as parameters that spawn an instance of a simulation $f(\theta)$ (e.g., a synthesized face image). This instance is fed into a target image classifier to check whether the system correctly identifies $f(\theta)$. Consequently, we can define a composite function $L_c = \mbox{\em{Loss}}(f(\theta))$, where $\mbox{\em{Loss}}$ is the classification loss and reflects if target classifier correctly handles the simulation instance generated when applying the parameters $\theta$. Carrying out Bayesian optimization with $L_c$ allows us to find $\theta$ that maximizes the loss, thus discovering parameters that are likely to break the classifier we are interrogating (i.e., *exploitation*). However, we are not interested in just one instance but sets of diverse examples that would lead to misclassification. Consequently, we carry out a sequence of Bayesian optimization tasks, where each subsequent run considers an adaptive objective function that is conditioned on examples that were discovered in the previous round. Formally, in each round of Bayesian Optimization we maximize: $$L = (1 - \alpha) L_c + \alpha \min_i{||\Theta_i - \theta||}.$$ The above composite function is a convex combination of the misclassification cost with a term that encourages discovering new solutions $\theta$ that are diverse from the set of previously found examples $\Theta_i$. Specifically, the second term is the minimum euclidean distance of $\theta$ from the existing set and a high value of that term indicates that the example being considered is diverse from rest of the set. Intuitively, this term encourages *exploration* and prioritizes sampling a diverse set of images. Figure \[fig:intro\] graphically describes such a composition. The sequence of Bayesian Optimizations find a diverse set of examples by first modeling the composite function $\textit{L}$ as a Gaussian Process (GP) [@rasmussen2004gaussian]. Modeling as a GP allows us to quantify uncertainty around the predictions, which in turn is used to efficiently explore the parameter space in order to identify the spots that satisfy the search criterion. In this work, we follow the recommendations in [@snoek2012practical], and model the composite function via a GP with a Radial Basis Function (RBF) kernel, and use Expected Improvement (EI) as an acquisition function. The code is included in supplementary material. **Data.** We use the MS-CELEB-1M [@guo2016ms] for our experimentation. This is a large image dataset containing 1M different people and approximately 100 million images. To identify the nationalities of the people in the dataset we used the Google Search API and pulled biographic text associated with each person featured in the dataset. We then used the NLTK library to extract nationality and gender information from the biographies. Many nations have heterogeneous national and/or ethnic compositions and assuming that sampling from them at random would give consistent appearances is not well founded. Characterizing these differences is difficult, but necessary if we are to understand biases in vision classifiers. The United Nations (UN) notes that the ethnic and/or national groups of the population are dependent upon individual national circumstances and terms such as “race” and “origin” have many connotations. There is no internationally accepted criteria. Therefore, care must be taken in how we use these labels to generate images of different appearances. To help address this we used demographic data provided by the UN that gives the national and/or ethnic statistics for each country and then only sampled from countries with more homogeneous demographics. We selected four regions that have predominant and similar racial appearance groups. These group are Black (darker skin tones, Sub-Saharan African appearance), South Asian (darker skin tone, Caucasian appearance), Northeast Asian (moderate skin tone, East Asian appearance) and White (light skin tone, Caucasian appearance) and sampled from a set of countries to obtain images for each. We sampled 5,000 images (2,500 men and 2,500 women) from each region prioritizing higher resolution images (256$\times$256) and then lower resolution images (128$\times$128). The original raw images selected for training and the corresponding race and gender labels are included in the supplementary material. The nationality and gender labels for the complete MS-CELEB-1M will also be released. Table \[tab:dataset\] shows the nations from which we sampled images and the corresponding appearance group. The number of people and images that were used in the final data are shown. It was not necessary to use all the images from every country to create a model for generating faces, and to obtain evenly distributed data over both gender and region we used this subset. Examples of the images produced by our trained model (described below) are also shown in the table. Higher resolution images can be found in the supplementary material. -- ----------- ----- ----- ------ ------ --- --- M W M W M W Nigerian 81 28 768 467 Kenya 11 5 91 49 S. Africa 136 102 1641 1984 **Total** 228 135 2500 2500 India 142 83 2108 2267 Sri Lanka 1 2 11 7 Pakistan 19 11 381 226 **Total** 162 96 2500 2500 Australia 175 121 2500 2500 **Total** 175 121 2500 2500 Japan 105 89 930 1421 China 105 46 789 447 S. Korea 29 12 464 251 Hong Kong 36 28 317 381 **Total** 275 175 2500 2500 -- ----------- ----- ----- ------ ------ --- --- : The number of people and images we sampled from (by country and gender) to train our generation model. Examples of generated faces for each race and gender. M = Men, W = Women.[]{data-label="tab:dataset"} Experiments and Results ======================= **Validation of Image Generation.** Statistical generative models such as GANs are not perfect and may not always generate images that reflect the conditioned variables. Therefore, it is important to validate the performance of the GAN that we used at producing images that represent the specified conditions (race and gender) reliably. We generated a uniform sample of 50 images, at 128$\times$128 resolution, from each race and gender (total 50x4x2 = 400 images) and recruited five participants on MTurk to label the gender of the face in each image and the quality of the image (see Table \[tab:dataset\] for example images). The quality of the image was labeled on a scale of 0 (no face is identifiable) to 5 (the face is indistinguishable from a photograph). Of 400 images, the gender of only seven (1.75%) images was classified differently by a majority of the labelers than the intended condition dictated. The mean quality rating of the images was 3.39 (SD=0.831) out of 5. There was no significant difference in quality between races or genders. In none of the images was a face considered unidentifiable. **Classifier Interrogation.** Numerous companies offer services for face detection and gender detection from images (Microsoft, IBM, Amazon, SightEngine, Kairos, etc.). We selected two of these commercial APIs (IBM and SightEngine) to interrogate in our experiments. These are exemplars and the specific APIs used here are not the focus of our paper. Each API accepts HTTP POST requests with URLs of the images or binary image data as a parameter within the request. If a face is detected they return JSON formatted data structures with the locations of the detected faces and a prediction of the gender of the face. Details of the APIs can be found in the supplementary material. We ran our sampling procedure for 400 iterations (i.e., we sampled 400 images at 128$\times$128 resolution) in each trial. Table \[tab:results\] shows the error rates (in %) for face and gender detection. Figure \[fig:average\_faces\] shows the mean face of images containing faces that were detected and not detected by the API. The skin tones illustrate that missed faces had darker skin tones and gender detection was considerably less accurate on people from NE Asia. We found men were more frequently misclassified as women than the other way around. API Task All Black S Asian NE Asian White Men Women ----- ------ ------ ---------- ---------- ---------- ------- ---------- ------- 8.05 **16.9** **7.63** 3.96 3.8 **11.3** 2.27 8.26 **9.00** 2.13 **20.0** 1.87 **15.8** 0.27 0.13 0.00 0.00 0.53 0.00 0.21 0.00 2.84 3.39 0.74 **5.85** 1.38 5.14 0.00 : Face detection and gender detection error rates (in percentage). SE = SightEngine. []{data-label="tab:results"} ![image](AverageFaces.pdf){width="\linewidth"} ![image](Exploitation_vs_Exploration.pdf){width="\linewidth"} Next, we compare two approaches for searching our space of simulated faces for face detection and gender detection failure cases. For these analyses we used the IBM API as the target image classifier. First, we randomly sample parameters for generating face configurations and second we use Bayesian optimization. Again, we ran the sampling for 400 iterations in each case. In the case of Bayesian optimization, the image generation was updated dependant on the success or failure of classification of the previous image. This allows us to use an explore-exploit strategy and navigate through the facial appearance space efficiently using the feedback from the automated “oracle”. Figure \[fig:tradeoff\](a) shows the sample efficiency of finding face detection and gender detection failures. Figure \[fig:tradeoff\](b) shows the how the percentage of the errors found varies with the value of $\alpha$ and for random sampling. Discussion ========== Bias in machine learning classifiers is problematic and often these biases may not be introduced intentionally. Regardless, biases can still propagate systemic inequalities that exist in the real-world. Yet, there are still few practical tools for helping researchers and developers mitigate bias and create well characterized classifiers. Adversarial training is a powerful tool for creating generative models that can produce highly realistic content. By using an adversarial training architecture we create a model that can be used to interrogate facial classification systems. We apply an optimal search algorithm that allows us to perform an efficient exploration of the space of faces to reveal biases from a smaller number of samples than via a brute force method. We tested this approach on face detection and gender detection tasks and interrogated commercial APIs to demonstrate its application. **Can our conditional GAN produce sufficiently high quality images to interrogate a classifier?** Our validation of our face GAN shows that the model is able to generate realistic face images that are reliably conditioned on race and gender. Human subjects showed very high agreement with the conditional labels for the generated images and the quality of the images were rated similarly across each race and gender. This suggests that our balanced data set and training procedure produced a model that can generate images reliably conditioned on race and gender and of suitably equivalent quality. Examples of the generated images can be seen in Table \[tab:dataset\] (high resolution images are available in the supplementary material). Very few of the images have very noticeable artifacts. **How do commercial APIs perform?** Both of the commercial APIs we tested failed at significantly higher rates on images of people from the African and South Asian groups (see Table \[tab:results\]). For the IBM system the face detection error rate was more than four times as high on African faces as White and NE Asian faces. The error rates on Black and South Asian faces were the highest, suggesting that skin tone is a key variable here. Gender detection error rates were also high for African faces but unlike face detection the performance was worst for NE Asian faces. These results suggest that gender detection performance is not only impacted by skin tone but also other characteristics of appearance. Perhaps facial hair, or lack thereof, in NE Asian photographs, men with “bangs” and make-up (see Figure \[fig:tradeoff\]c and supplementary material for examples of images that resulted in errors.) **Can we efficiently sample images to find errors?** Errors are typically sparse (many APIs have a global error rate of less than 10%) and therefore simply randomly sampling images in order to identify biases is far from efficient. Using our optimization scheme we are able to identify an equivalent number of errors in significantly fewer samples (see Figure \[fig:tradeoff\]a). The results show that we are able to identify almost 50% more failure cases using the Bayesian sampling scheme than without. In some senses our approach can be thought of as a way of efficiently identifying adversarial examples. **Trading off exploitation and exploration?** In our sampling procedure we have an explicit trade-off, using $\alpha$, between exploration of the underlying manifold of face images and exploitation to find the highest number of errors (see Figure \[fig:tradeoff\](b)). With little exploration there is a danger that the sampling will find a local minima and continue sampling from a single region. Our results show that with $\alpha$ equal 0.6 we maximize the number of errors found. This is empirical evidence that exploration and exploration are both important. Otherwise there is risk that one might miss regions that have frequent failure cases. As the parameter space of $\theta$ grows in dimensionality our sampling procedure will become even more favorable compared to naive methods. Conclusions =========== We have presented an approach applying a conditional progressive generative model for creating photo-realistic synthetic images that can be used to interrogate facial classifiers. We test commercial image classification application programming interfaces and find evidence of systematic biases in their performance. A Bayesian search algorithm allows for efficient search and characterization of these biases. Biases in vision-based systems are of wide concern especially as these system become widely deployed by industry and governments. Generative models are a practical tool that can be used to characterize the performance of these systems. We hope that this work can help increase the prevalence of rigorous benchmarking of commercial classifier in the future. [^1]: http://data.un.org/\[unstat\]
--- abstract: 'We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n/\hspace{-0.8mm}/\mathtt{SL}_2$. Our main result is that $\overline{\mathcal{M}}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler $(\mathbb{P}^1)^n$ quotients. Moreover, these morphisms factor through Hassett’s moduli space of weighted pointed rational curves, where the weight data comes from the GIT linearization data.' author: - 'Noah Giansiracusa, Matthew Simpson' title: 'GIT Compactifications of $\mathcal{M}_{0,n}$ from Conics' --- Introduction and main results ============================= Inspired by recent work of the second author [@Simp], we study a family of GIT quotients parametrizing $n$-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}\mathtt{SL}_2$. These latter quotients compactify the moduli space $\mathcal{M}_{0,n}$ of nonsingular $n$-pointed rational curves by allowing points to collide as long as their *weight* (a number assigned to each point when choosing a linearization for the group action) is not too much. For the GIT quotients that we investigate, denoted $\mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$, the compactification allows a certain number of points to overlap based on their weights, but if too many points collide then the nonsingular conic degenerates into a nodal conic. Up to isomorphism nonsingular and nodal conics are a $\mathbb{P}^1$ and a pair of intersecting $\mathbb{P}^1$s, respectively, so the spaces $\mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$ can be viewed as intermediate compactifications between $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}\mathtt{SL}_2$ and the well-known Deligne-Mumford-Knudsen compactification $\overline{\mathcal{M}}_{0,n}$ [@Knud]. The following theorem characterizes GIT stability for pointed conics, generalizing a result of the second author (Theorem 3.1.5 in [@Simp]) which describes stability in the special case of $S_n$ invariant weights. \[thm:stability\] Let $(\gamma,c_1,\ldots,c_n)$ specify an ample fractional line bundle on the space of $n$-pointed conics $\mathtt{Con}(n)\subset \mathbb{P}(\mathtt{Sym}^2(V^*))\times(\mathbb{P}(V))^n$, $V=\mathbb{C}^3$, linearized for the natural action of $\mathtt{SL}(V)$. If $c := c_1 +\cdots + c_n$ is the total point weight then: - all non-reduced conics are unstable - a nodal conic is semistable iff 1. the weight of marked points at any smooth point is $\le \frac{c+\gamma}{3}$ 2. the weight of marked points at the node is $\le c - 2(\frac{c+\gamma}{3})$, and 3. the weight on each component is $\le \frac{2c-\gamma}{3}$; equivalently, the weight on each component away from the node is $\ge \frac{c+\gamma}{3}$ - a nonsingular conic is semistable iff the weight at each point is $\le \mathtt{min}\{\frac{c+\gamma}{3},\frac{c}{2}\}$ In particular, if $\gamma > \frac{c}{2}$ then nodal conics are unstable. Stability is characterized by the corresponding inequalities being replaced by strict inequalities. A variation of GIT perspective will be useful in our investigations. When a reductive group $G$ acts on a variety the space of linearized fractional polarizations forms a cone called the *$G$-ample cone*, and inside it sits the *$G$-effective cone* which is defined as the set of linearizations for which the semistable locus is nonempty. The $G$-effective cone admits a finite wall and chamber decomposition such that on each open chamber the GIT quotient is constant and when a wall is crossed the quotient undergoes a birational modification (see [@Thad] and [@DH]). In some cases this cone admits a natural cross-section so that the (closure of) the space of linearizations can be identified with a certain polytope which we call the *linearization polytope*. For example, the GIT quotients $(\mathbb{P}^m)^n{/\hspace{-1.2mm}/}\mathtt{SL}_{m+1}$ parametrizing configurations of $n$ points in $m$-dimensional projective space have $\mathtt{SL}_{m+1}$-ample cone $\mathbb{Q}_{>0}^n$ because $\text{Pic}((\mathbb{P}^m)^n) \cong \mathbb{Z}^n$ and each line bundle admits a unique linearization. A vector $\vec{c}=(c_1,\ldots,c_n)\in\mathbb{Q}_{>0}^n$ assigns a positive rational weight to each point, and a configuration is semistable if and only if the total weight lying in any proper linear subspace $W\subset\mathbb{P}^m$ is at most $\frac{\text{dim }W+1}{m+1}\cdot \sum_1^n c_i$ (see, e.g., Example 3.3.21 in [@DH]). Multiplying $\vec{c}$ by a positive constant does not affect stability so one can use the normalization $\sum_1^n c_i = m+1$. The semistable locus is then non-empty precisely when $\text{max}\{c_i\} \le 1$ so the linearization polytope is a hypersimplex $$\Delta(m+1,n) = \{\vec{c}\in\mathbb{Q}^{n}~|~0\le c_i \le 1, \sum_{i=1}^n c_i = m+1\}$$ with walls of the form $\sum_{i\in I} c_i=k$ for $I\subset\{1,\ldots,n\}$ and $1 \le k \le m$. In particular, for points on the line ($m=1$) we have $\Delta(2,n)$ with walls $\sum c_i = 1$, and for points in the plane ($m=2$) we have $\Delta(3,n)$ with walls $\sum c_i =1$ and $\sum c_i = 2$. A consequence of Theorem \[thm:stability\] is that for the space of $n$-pointed conics the effective linearizations form a 1-parameter family of hypersimplices that interpolate these two cases. \[cor:chamber\] The $\mathtt{SL}_3$-effective cone for $\mathtt{Con}(n)$ induced from that of the ambient $\mathbb{P}^5\times(\mathbb{P}^2)^n$ is subdivided by the hyperplane $\gamma = \frac{c}{2}$ into two subcones: $\gamma \le \frac{c}{2}$ for which semistable nodal conics occur, and $\gamma > \frac{c}{2}$ for which singular conics are unstable. When normalizing with cross-sections $\gamma + c = 3$ on the former and $c=2$ on the latter the linearization polytopes for fixed $\gamma$ are $\Delta(3-\gamma,n)$ with walls $\sum c_i =1$ and $\sum c_i=2$ if $0\le\gamma\le 1$, and $\Delta(2,n)$ with walls $\sum c_i = 1$ if $\gamma \ge 1$. These cross-sections meet at the codimension 2 linear subspace $\gamma=1,c=2$ and only miss the ray $\gamma \ne 0$, $c=0$. If $\gamma > \frac{c}{2}$ then, as we discuss later, $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3 \cong (\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_{\vec{c}}~\mathtt{SL}_2$ so to get novel compactifications we can restrict $\gamma$ to the interval $[0,\frac{c}{2}]$. In this case only one normalization is necessary (namely $\gamma+c=3$) and the linearization polytope is $\Delta(3,n+1)$ with walls $\sum c_i=1$ and $\sum c_i = 2$ which are “vertical” in the sense that they are independent of $\gamma$. Another interesting family of compactifications is provided by Hassett’s moduli spaces of stable weighted pointed curves [@Hass]. Recall that for a weight vector $\vec{c}\in[0,1]^n$ the space $\overline{\mathcal{M}}_{0,\vec{c}}$ parametrizes nodal rational curves with marked points $p_i$ avoiding the nodes such that on any component $C$ we have $\sum_{p_i \in C} c_i + \delta_C > 2$, where $\delta_C$ is the number of nodes on $C$. In particular, if $c_i=1$ for $1\le i \le n$ then $\overline{\mathcal{M}}_{0,\vec{c}}\cong \overline{\mathcal{M}}_{0,n}$ so these spaces can also be viewed as intermediate compactifications of $\mathcal{M}_{0,n}\subset \overline{\mathcal{M}}_{0,n}$. The main result of this paper (generalizing the $S_n$ invariant result Theorem 3.2.6 in [@Simp]) is that these Hassett compactifications and our conic compactifications are related in the following manner. \[thm:contraction\] For any $(\gamma,\vec{c})\in\Delta(3,n+1)$ such that all entries of $\vec{c}$ are nonzero there is a birational contraction morphism $$\overline{\mathcal{M}}_{0,n} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3$$ which factors through $\overline{\mathcal{M}}_{0,\vec{c}}$. The idea is that a stable $n$-pointed rational curve becomes a stable weighted pointed curve by contracting all components which carry too little weight and then this can be further contracted to at most two components which are then embedded in the plane as a conic stable with respect to the GIT linearization corresponding to the Hassett weight data. This theorem should be thought of as an analogue of the result of Kapranov [@Kapr] that $\overline{\mathcal{M}}_{0,n}$ admits a morphism to every GIT quotient $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}\mathtt{SL}_2$. In fact, because $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3 \cong (\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_{\vec{c}}~\mathtt{SL}_2$ for $\gamma > 1$, this theorem when combined with Kapranov’s result shows that $\overline{\mathcal{M}}_{0,n}$ admits a morphism to every GIT quotient $\mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$ with linearization induced from the ambient product of projective spaces. The remaining sections of this paper are devoted to proving the results described in this introduction, except for the last section in which we explore some examples and further properties.\ **Acknowledgements**. This project was suggested by the second author’s thesis adviser, Brendan Hassett, and was guided by helpful discussions with him as well as the first author’s thesis advisers, Dan Abramovich and Danny Gillam. We would like to extend our thanks to them for sharing their time and ideas with us. Numerical criteria for stability of conics ========================================== The goal of this section is to prove Theorem \[thm:stability\] by applying the Hilbert-Mumford numerical criterion for stability. Let us briefly recall how this criterion works (see [@GIT] and [@News] for details). Assume that $G$ is a reductive group, $L$ is an ample linearized line bundle, and $X$ is a projective variety over $\mathbb{C}$. Then $L$ (or a suitable tensor power) induces a morphism $X \rightarrow \mathbb{P}^N$ to the projective space determined by its sections and the linearization gives an action of $G$ on these sections. In particular, if $\lambda$ is a 1-parameter subgroup, i.e. a homomorphism $\mathbb{G}_m \rightarrow G$, then there is an induced linear action of $\lambda$ on $\mathbb{A}^{N+1}$. This action can always be diagonalized: there is a basis $e_0,\ldots,e_N$ such that $\lambda(t)\cdot e_i = t^{r_i}e_i$ for some integers $r_i$ which are called the *weights* of the action. For any point $x\in X$ we look at its image in $\mathbb{P}^N$, choose a point $\widetilde{x}=\sum_{i=0}^N \widetilde{x}_i e_i$ in the affine cone $\mathbb{A}^{N+1}$ lying over it, and then acting on this by $\lambda$ gives an expression of the form $\sum_{i=0}^N t^{r_i}\widetilde{x}_ie_i$. We also refer to $r_i$ as the weight of the coordinate $\widetilde{x}_i$. The “numerical criterion” says that a point $x$ is semistable if and only if the minimum $\mu$ of the weights of its coordinates over all nonzero coordinates and all 1-parameter subgroups is $\le 0$, and $x$ is stable if and only if $\mu <0$. We now turn to the specifics of our problem. We are interested in the space $$\mathtt{Con}(n) \subset \mathbb{P}^5\times(\mathbb{P}^2)^n$$ of $n$-pointed conics, where $\mathbb{P}^5 = \mathbb{P}(\mathtt{Sym}^2((\mathbb{C}^3)^*))$. If we take coordinates $(x_l,y_l,z_l)$ on the $l^{\text{th}}$ copy of $\mathbb{P}^2$ and $a_{ijk}$ ($i+j+k=2$) on the $\mathbb{P}^5$ then $\mathtt{Con}(n)$ is cut out by the equations $\sum_{i+j+k=2}a_{ijk}x_l^iy_l^jz_l^k=0$ for $l=1,\ldots, n$. The group $\mathtt{SL}_3$ acts as automorphisms on $\mathbb{P}^2$, so it acts on $\mathbb{P}^5$ by sending a conic to its image under the corresponding projective motion. This induces an action on $\mathtt{Con}(n)$. Concretely, if $$A = \left(\begin{array}{ccc} a_{200} & \frac{1}{2}a_{110} & \frac{1}{2}a_{101} \\ \frac{1}{2}a_{110} & a_{020} & \frac{1}{2}a_{011} \\ \frac{1}{2}a_{101} & \frac{1}{2}a_{011} & a_{002} \end{array}\right), \vec{x_l} =\left(\begin{array}{c} x_l \\ y_l \\ z_l \end{array} \right)$$ are the symmetric bilinear form associated to the conic $a_{ijk}$ and coordinate vectors on $\mathbb{P}^2$, respectively, then the action is given by $$g\cdot (A,\vec{x}_1,\ldots,\vec{x}_n) = ((g^{-1})^tAg^{-1},g\vec{x}_1,\ldots,g\vec{x}_n)$$ and the incidence correspondence $\mathtt{Con}(n)$ is defined by $\vec{x}^t_lA\vec{x}_l=0$ for $l=1,\ldots,n$. The Segre embedding $\mathbb{P}^5\times(\mathbb{P}^2)^n \hookrightarrow \mathbb{P}^N$ gives $\mathtt{Con}(n)$ homogeneous coordinates of the form $(\cdots,a_{ijk}w_{I,J,K},\cdots)$, where $i+j+k=2$, $I\cup J \cup K = \{1,\ldots, n\}$ is a disjoint union, and $w_{I,J,K}$ denotes the product of $x$ coordinates indexed by $I$, $y$ coordinates by $J$, and $z$ coordinates by $K$, so that for example if $n=6$ then $w_{125,4,36}=x_1x_2z_3y_4x_5z_6$. Any 1-parameter subgroup of $\mathtt{SL}_3$ can be diagonalized to the form $$t\mapsto\left(\begin{array}{ccc}t^{r_1} & 0 & 0 \\0 & t^{r_2} & 0 \\0 & 0 & t^{r_3}\end{array}\right)$$ where $r_1 + r_2 + r_3 = 0$ and $r_1\le r_2 \le r_3$. Since we are interested only in stability, which is invariant under scaling all the exponents by a common factor, we can divide the $r_i$ by $r_3$ and hence assume the subgroup is of the form $$t \mapsto \left(\begin{array}{ccc}t^b & 0 & 0 \\0 & t^{-1-b} & 0 \\0 & 0 & t\end{array}\right)$$ for $-2\le b \le -\frac{1}{2}$. For each weight vector $(\gamma, c_1,\ldots,c_n)\in \mathbb{Q}_{> 0}^{n+1}$ we consider the ample line bundle on $\mathtt{Con}(n)$ defined by $$(\pi_0^*\mathcal{O}_{\mathbb{P}^5}(\gamma)\otimes \pi_1^*\mathcal{O}_{\mathbb{P}^2}(c_1)\otimes \cdots \otimes \pi_n^*\mathcal{O}_{\mathbb{P}^2}(c_n))|_{\mathtt{Con}(n)}$$ where $\pi_i$ are the obvious projections, and equip it with the unique $\mathtt{SL}_3$-linearization. For any index set $I\subset \{1,\ldots,n\}$ we write $c_I := \sum_{i\in I}c_i$; the total weight of points is denoted $c := \sum_1^n c_i$. With this setup we claim that the weight of $a_{ijk}w_{I,J,K}$ is $$\label{eqn:weight} \gamma(i(1-b) + j(b+2) - 2) + bc_I + (-1-b)c_J + c_K.$$ Indeed, $t\cdot x_l = t^{bc_l} x_l$ so we get a term $bc_I$ coming from the $x$ variables indexed by $I$; similarly, $t \cdot y_l = t^{(-1-b)c_l}y_l$ yields $(-1-b)c_J$ and $t\cdot z_l = t^{c_l}z_l$ yields $c_K$. In other words, $t\cdot w_{I,J,K} = t^{bc_I + (-1-b)c_J + c_K}w_{I,J,K}$. Now to find the weight for the action on the $a_{ijk}$ factor we see from the matrix product $$\left(\begin{array}{ccc}t^{-b} & 0 & 0 \\0 & t^{1+b} & 0 \\0 & 0 & t^{-1}\end{array}\right)\left(\begin{array}{ccc} a_{200} & \frac{1}{2}a_{110} & \frac{1}{2}a_{101} \\ \frac{1}{2}a_{110} & a_{020} & \frac{1}{2}a_{011} \\ \frac{1}{2}a_{101} & \frac{1}{2}a_{011} & a_{002} \end{array}\right)\left(\begin{array}{ccc}t^{-b} & 0 & 0 \\0 & t^{1+b} & 0 \\0 & 0 & t^{-1}\end{array}\right)$$ that $t\cdot a_{ijk} = t^{\gamma(i(-b) + j(1+b) - k(-1))}a_{ijk}$. But $k = 2-(i+j)$ so $i(-b)+j(1+b)-k(-1) = i(1-b) + j(2+b) - 2$, which explains Formula (\[eqn:weight\]). Therefore, a pointed conic is semistable iff the minimum $\mu$ of these weights over all possible nonzero choices of $a_{ijk}w_{I,J,K}$ and all $b\in [-2,-\frac{1}{2}]$ is $\le 0$, and it is stable iff $\mu < 0$. Recall that conics have exactly three isomorphism classes: non-reduced (a double line), nodal (a pair of intersecting lines), and non-singular (isomorphic to $\mathbb{P}^1$). We examine these three cases in turn. ### Non-reduced conics are unstable {#non-reduced-conics-are-unstable .unnumbered} The orbit of any non-reduced conic contains $x^2=0$ so to show all non-reduced conics are unstable it is enough to show this one is unstable. Here we are forced to have $i=2,j=0,k=0$. Moreover, since all points on $x^2=0$ have vanishing $x$-coordinate we must have $I=\varnothing$, so the weight is $\gamma(-2b) + (-1-b)(c_J) + c_K$. Setting $b=-2$ this yields $4\gamma + c_J + c_K$, which is certainly positive. ### Three conditions for (semi)stability of a nodal conic {#three-conditions-for-semistability-of-a-nodal-conic .unnumbered} Here we derive three necessary conditions a semistable conic must satisfy and then show they are in fact sufficient. For the nodal conic $xy=0$ we have $i=1,j=1,k=0$ so the weight is $\gamma + bc_I + (-1-b)c_J + c_K$. Setting $b=-2$ yields $\gamma -2c_I + c_J + c_K = \gamma + c - 3c_I$ which is minimized when $c_I$ is maximized, so $\mu$ is computed by having $I$ index all points with nonzero $x$ coordinate. The remaining weight $c-c_I$ must lie on the line $x=0$ and $\mu \le 0$ implies $c-c_I \le \frac{2c-\gamma}{3}$ so there is $\le \frac{2c-\gamma}{3}$ weight on the component $x=0$. Since the action of $\mathtt{SL}_3$ sends any nodal conic to $xy=0$ and either component can be sent to $x=0$ this shows that any semistable nodal conic has $\le\frac{2c-\gamma}{3}$ weight on either component, or equivalently, it has $\le c-\frac{2c-\gamma}{3}=\frac{c+\gamma}{3}$ weight on each component off the node. Setting $b=-\frac{1}{2}$ the weight of $xy=0$ becomes $\gamma - \frac{1}{2}c_I - \frac{1}{2}c_J + c_K = \gamma + c - \frac{3}{2}(c_I + c_J)$ so $\mu$ is computed by indexing all points with nonzero $x$ or $y$ coordinate with $I\cup J$. The only remaining points are at the node $x=y=0$ and $\mu \le 0$ implies $c-(c_I + c_J) \le \frac{c-2\gamma}{3}$ so a semistable nodal conic has $\le\frac{c-2\gamma}{3} = c-2(\frac{c+\gamma}{3})$ weight at the node. For the conic $xz=0$ we have $i=1,j=0,k=1$ so for $b=-\frac{1}{2}$ the weight is $-\frac{\gamma}{2} -\frac{c_I}{2} - \frac{c_J}{2} + c_K = c - \frac{\gamma}{2} - \frac{3}{2}(c_I + c_J)$ which means again that $\mu$ is computed by indexing all points away from $x=y=0$ with $I\cup J$. But now this point is smooth so because $\mu\le 0$ implies $c - (c_I + c_J) \le \frac{c+\gamma}{3}$ we see that the weight at any smooth point of a semistable conic is $\le \frac{c+\gamma}{3}$.\ We next show that a nodal conic satisfying these three conditions is semistable: $$\text{min}_{a_{ijk}w_{I,J,K}\ne 0}\{\gamma(i(1-b) + j(b+2) - 2)) + bc_I + (-1-b)c_J + c_K\}\le 0$$ for all $b\in[-2,-\frac{1}{2}]$. It is enough to show that for each such conic there is a single coordinate $a_{ijk}w_{I,J,K}\ne 0$ with weight $\le 0$ at both endpoints $b=-2,b=-\frac{1}{2}$, since all other values of $b$ are linearly interpolated from these. If we define $\text{wt}(x^iy^jz^{2-i-j}) := i(1-b) + j(b+2) - 2$ then it is easy to see that $$-2 \le b \le -1 \Rightarrow \text{wt}(x^2) \ge \text{wt}(xy) \ge \text{wt}(xz) \ge 0 \ge \text{wt}(y^2) \ge \text{wt}(yz) \ge \text{wt}(z^2),$$ $$-1 \le b \le -\frac{1}{2} \Rightarrow \text{wt}(x^2) \ge \text{wt}(xy) \ge \text{wt}(y^2) \ge 0 \ge \text{wt}(xz) \ge \text{wt}(yz) \ge \text{wt}(z^2).$$ *Case 1: $x=y=0$ is a node*. The conic cannot be the double line $x^2=0$ so it has a monomial term with weight $\le \text{wt}(xy)=1$ and hence the pointed conic has weight $\le \gamma + bc_I + (-1-b)c_J + c_K$. Let $I$ index all points off $x=0$ and $J$ all remaining points off $y=0$. Semistability at the boundary values $b=-2$ and $b=-\frac{1}{2}$ translates into the inequalities $c-c_I \le \frac{2c-\gamma}{3}$ and $c - (c_I + c_J) \le c - 2(\frac{c+\gamma}{3})$, respectively, but these are satisfied by the assumption that there is $\le \frac{c-2\gamma}{3}$ weight on any component and $\le c - 2(\frac{c+\gamma}{3})$ weight at the node. *Case 2: $x=y=0$ is a smooth point*. An easy computation with partial derivatives shows that for $x=y=0$ to be smooth the conic cannot consist only of the monomials $x^2,xy,y^2$, so there is a monomial with weight $\le \text{wt}(xz)=-1-b$ and hence the pointed conic has weight $\le \gamma(-1-b) + bc_I + (-1-b)c_J + c_K$. The boundary values $b=-2$ and $b=-\frac{1}{2}$ now translate to $c-c_I \le \frac{2c-\gamma}{3}$ and $c-(c_I + c_J) \le \frac{c+\gamma}{3}$, respectively, which can be satisfied simultaneously by choosing $I,J$ as before and noting that there is $\le \frac{c+\gamma}{3}$ weight at the smooth point $x=y=0$. ### Two conditions for (semi)stability of a nonsingular conic {#two-conditions-for-semistability-of-a-nonsingular-conic .unnumbered} Consider the nonsingular curve $x^2 + xy + xz + y^2$. For $b=-\frac{1}{2}$ the monomial of minimal weight is $xz$ so when computing $\mu$ we set $i=1,j=0$ and find, as before, that semistability implies $\text{min}\{c - (c_I + c_J)\} \le \frac{c+\gamma}{3}$ and hence the weight at $x=y=0$ is $\le \frac{c+\gamma}{3}$ so the weight at any point of a semistable nonsingular conic is $\le \frac{c+\gamma}{3}$. For $b=-1$ the minimal weight monomial is either $xz$ or $y^2$ (both have weight $0$) so the total weight is $-c_I + c_K = c -2c_I - c_J$. Semistability implies $\text{max}\{2c_I + c_J\} \ge c$. This maximum occurs when $I$ indexes all points off the line $x=0$ and $J$ indexes all remaining points off $y=0$. The line $x=0$ is tangent to our conic, intersecting it at the unique point $x=y=0$, so $I$ indexes all points away from $x=y=0$ and $J=\varnothing$. This means $c_J=0$ so $2c_I \ge c$, or equivalently $c-c_I \le \frac{c}{2}$, and the number $c-c_I$ here measures the weight at $x=y=0$, so any point of a semistable nonsingular conic has weight $\le \frac{c}{2}$. Together this implies semistable nonsingular conics have $\le \mathtt{min}\{\frac{c+\gamma}{3},\frac{c}{2}\}$ weight at any point.\ Conversely, to see that any nonsingular conic satisfying these two inequalities is semistable takes a little more work. First, we verify semistability for conics that are not tangent to the line $x=0$. If the conic is in the $\mathbb{C}$-linear span of $\{x^2,xy,xz,y^2\}$ then the intersection with the line $x=0$ is the single point $x=y=0$ so $x=0$ is a tangent line. Therefore, we can assume there is a monomial with weight $\le \text{wt}(yz)$ (irrespective of $b$) so the pointed conic has weight $\le \gamma b + bc_I + (1-b)c_J + c_K$. For $b=-2$ this is $\le 0$ iff $c-c_I \le 2(\frac{c + \gamma}{3})$, so by taking $I$ to index all points off $x=0$ this inequality is satisfied since the line $x=0$ intersects the conic in exactly two points, each of which has $\le \frac{c+\gamma}{3}$ weight. For $b=-\frac{1}{2}$ the weight is $\le -\frac{\gamma}{2} - \frac{c_I}{2} - \frac{c_J}{2} + c_K$ which is $\le 0$ iff $c-(c_I + c_J) \le \frac{c+\gamma}{3}$, so if $J$ indexes all remaining points off $x=0$ then the inequality is satisfied since the weight at the point $x=y=0$ is $\le \frac{c+\gamma}{3}$. For conics tangent to the line $x=0$ we need to subdivide the interval $[-2,-\frac{1}{2}]$ to show that there is a single coordinate with weight $\le 0$ when $b=-2$ and $b=-1$, whence by linear interpolation for all $b\in [-2,-1]$, and then separately that there is a coordinate with weight $\le 0$ for $b=-1$ and $b=-\frac{1}{2}$. A conic in the $\mathbb{C}$-linear span of $\{x^2,xy,xz\}$ contains the line $x=0$ and hence cannot be nonsingular, so there is a monomial of weight $\le\text{wt}(y^2)$ and thus the pointed conic has weight $\le \gamma(2b+2) + bc_I + (-1-b)c_J + c_K$. For $b=-2$ this is $\le 0$ iff $c-c_I \le 2(\frac{c+\gamma}{3})$, and this inequality is easily satisfied when $I$ indexes the points off $x=0$ (in fact tangency gives the stronger condition that $c-c_I$ measures the weight at the single point $x=y=0$ so it is $\le\frac{c+\gamma}{3}$). For $b=-1$ the weight is $\le -c_I +c_K=c-2c_I - c_J$ so setting $J=\varnothing$ it is enough to check that $c-c_I \le \frac{c}{2}$. The condition that $x=0$ intersects the conic in a unique point forces this inequality to be satisfied. Finally, to prove semistability for $b\in[-1,-\frac{1}{2}]$ we observe that all conics in the $\mathbb{C}$-linear span of $\{x^2,xy,y^2\}$ are singular so there must be a monomial of weight $\le \text{wt}(xz)$. If $b=-1$ then $\text{wt}(xz)=\text{wt}(y^2)$ and we have already seen such conics have weight $\le 0$. If $b=-\frac{1}{2}$ then to show the weight is $\le 0$ it is enough to show that $c-(c_I+c_J) \le \frac{c+\gamma}{3}$, which holds by assumption. ### Conclusion {#conclusion .unnumbered} By the numerical criterion, the inequalities characterizing semistability of a pointed conic give a characterization for stability if they are replaced by strict inequalities. Therefore the proof of Theorem \[thm:stability\] is complete except for the final remark that singular conics are unstable if $\gamma > \frac{c}{2}$. But this is immediate since $\gamma > \frac{c}{2} \Rightarrow c-2(\frac{c+\gamma}{3})<0$ so a semistable nodal conic with $\gamma > \frac{c}{2}$ would have to satisfy the impossible condition of having negative weight at the node. Walls and chambers ================== In this section we use Theorem \[thm:stability\] to discuss and prove Corollary \[cor:chamber\]. The ample line bundles on $\mathbb{P}^5\times (\mathbb{P}^2)^n$ restrict to give ample line bundles on $\mathtt{Con}(n)$ and each comes with a unique linearization for the action of $\mathtt{SL}_3$ so $\mathtt{Con}(n)$ inherits the $\mathtt{SL}_3$-ample cone $\mathbb{Q}^{n+1}_{>0}$. Because $\frac{c+\gamma}{3} \le \frac{c}{2} \Leftrightarrow \gamma \le \frac{c}{2}$ we see by looking at the weight allowed at a smooth point of a conic that for a linearization $(\gamma,c_1,\ldots,c_n)$ the semistable locus is empty when $\mathtt{max}\{c_i\} > \frac{c+\gamma}{3}$ for $\gamma \le \frac{c}{2}$ and when $\mathtt{max}\{c_i\} > \frac{c}{2}$ for $\gamma \ge \frac{c}{2}$. This suggests taking cross-sections of the $\mathtt{SL}_3$-ample cone to normalize in the following way: $$\begin{cases}c+\gamma = 3 \text{ if } \gamma \le \frac{c}{2} \cr c=2 \text{ if } \gamma \ge \frac{c}{2} \cr \end{cases} \label{eqn:normalization}$$ The intersection of these two hyperplanes in the space of linearizations is the locus where $c=2$ and $\gamma=1$, so it is useful to view $\gamma$ as a parameter in $[0,\infty)$ such that for $\gamma \ge 1$ the point weights satisfy $c=2$ and for $\gamma \le 1$ they satisfy $c = 3 - \gamma$. If $\gamma > 1$ then singular conics are all unstable so the only relevant stability conditions are those of a nonsingular conic, namely that the weight at any point is $\le 1$. Therefore, for any such $\gamma$ the $c_i$ satisfy $0 < c_i \le 1$ and $\sum_{i=1}^n c_i = 2$ so the linearization polytope is $\Delta(2,n)$. The walls are where strictly semistable points occur, namely $\sum_{i\in I} c_i = 1$ for $I\subset\{1,\ldots,n\}$. This is the same polytope and chamber decomposition as the GIT quotient parametrizing $n$ points on $\mathbb{P}^1$. In fact, because a nonsingular conic is isomorphic to $\mathbb{P}^1$ and the action on marked points of $\mathtt{SL}_3$ stabilizing such a conic is the same as that of $\mathtt{SL}_2$ on $\mathbb{P}^1$ with the same stability conditions there is an isomorphism $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3 \cong (\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_{\vec{c}~}\mathtt{SL}_2$ for any $\gamma > 1$. On the other hand, if $\gamma \le 1$ then the linearization polytope is $\Delta(3-\gamma,n)$ since $\mathtt{min}\{\frac{c+\gamma}{3},\frac{c}{2}\} = \frac{c+\gamma}{3} = 1$. The stability conditions for points on a nonsingular conic, for the weight on each component of a nodal conic, and for the weight at a smooth point of a nodal conic all introduce walls of the form $\sum_{i\in I}c_i = 1$ for $I\subset\{1,\ldots,n\}$. The only remaining condition, the weight at the node, introduces walls $\sum_{i\in I}c_i = 2$ because if a collection of points indexed by $I\subset\{1,\ldots,n\}$ has total weight $c - 2$ then the complementary subset $\{1,\ldots,n\}\setminus I$ has total weight $2$. If we let $\gamma$ vary in the interval $[0,1]$ then the semistable locus is nonempty as long as all entries of the weight vector are between 0 and 1, so the linearization polytope for this range is $\Delta(3,n+1)$ with walls $\sum_{i\in I}c_i = 1, \sum_{i\in I}c_i = 2$ that are independent of $\gamma$. This concludes the proof of Corollary \[cor:chamber\], but it is worthwhile discussing the polytopes that arise and how they fit together. (135,70)(0,0) (45,35)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (35,45)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (40,30)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (30,30)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (30,30)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (30,40)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (40,40)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (30,40)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (40,30)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (35,60)(1.54,-1.28)[20]{}[(0,0)(0.15,-0.13)[5]{}[(1,0)[0.15]{}]{}]{} (20,20)(0.7,1.86)[22]{}[(0,0)(0.12,0.31)[3]{}[(0,1)[0.31]{}]{}]{} (40,40)[(0,0)\[cc\][\*]{}]{} (35,35)[(1,0)[40]{}]{} (35,35)[(0,1)[35]{}]{} (10,10)(0.12,0.12)[208]{}[(1,0)[0.12]{}]{} (105,35)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (95,45)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (100,30)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (90,30)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (90,30)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (90,40)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (100,40)(1.43,1.43)[4]{}[(0,0)(0.12,0.12)[6]{}[(1,0)[0.12]{}]{}]{} (90,40)(1.82,0)[6]{}[(1,0)[0.91]{}]{} (100,30)(0,1.82)[6]{}[(0,1)[0.91]{}]{} (95,35)[(1,0)[40]{}]{} (95,35)[(0,1)[35]{}]{} (70,10)(0.12,0.12)[208]{}[(1,0)[0.12]{}]{} (84.02,24.38)(1.89,0.61)[18]{}[(0,0)(0.32,0.1)[3]{}[(1,0)[0.32]{}]{}]{} (84.02,24.2)(0.7,1.81)[16]{}[(0,0)(0.12,0.3)[3]{}[(0,1)[0.3]{}]{}]{} (94.82,52.32)(1.57,-1.17)[15]{}[(0,0)(0.16,-0.12)[5]{}[(1,0)[0.16]{}]{}]{} (90,39.82)(0.12,-0.12)[81]{}[(1,0)[0.12]{}]{} (100.09,30.09)(0.12,0.36)[41]{}[(0,1)[0.36]{}]{} (90,40.18)(0.38,0.12)[39]{}[(1,0)[0.38]{}]{} (20.36,20.36)(1.89,0.62)[24]{}[(0,0)(0.32,0.1)[3]{}[(1,0)[0.32]{}]{}]{} (40,55)(0,0) (0,0)(0.12,0.12)[125]{}[(1,0)[0.12]{}]{} (15,15)[(1,0)[25]{}]{} (15,15)[(0,1)[35]{}]{} (15,50)[(0,1)[0.12]{}]{} (15,55)[(0,0)\[cc\][$\gamma$]{}]{} (15,45)(0.12,-0.12)[83]{}[(1,0)[0.12]{}]{} (25,35)(0.12,0.12)[83]{}[(1,0)[0.12]{}]{} (15,45)[(1,0)[20]{}]{} (15,30)(0.12,-0.12)[83]{}[(1,0)[0.12]{}]{} (25,20)(0.12,0.12)[83]{}[(1,0)[0.12]{}]{} (15,30)[(1,0)[20]{}]{} (15,30)(0.12,-0.24)[83]{}[(0,-1)[0.24]{}]{} (25,10)(0.12,0.24)[83]{}[(0,1)[0.24]{}]{} (25,10)[(0,1)[10]{}]{} (25,20)[(0,1)[20]{}]{} (25,40)[(0,1)[0.12]{}]{} (35,30)[(0,1)[20]{}]{} (35,50)[(0,1)[0.12]{}]{} (10,30)[(0,0)\[cc\][1]{}]{} The hypersimplex $\Delta(k,n)$ is sometimes described as the convex hull of all sums of $k$ distinct unit coordinate vectors in $\mathbb{R}^n$. However, in our setting $k$ can be rational so the relevant definition is the intersection of the hypercube $[0,1]^n$ with a scaled copy of the standard coordinate simplex. For $n=3$ we can draw the scaled 2-simplex and see how it intersects the cube $[0,1]^3$. When $k=3$ (so $\gamma=0$) the intersection is a single point, namely $\Delta(3,3) = \{(1,1,1)\}$. As $k$ varies from 3 to 2 (so $\gamma$ goes from 0 to 1) the 2-simplex shrinks so that the cube cuts out a larger piece of it. The hypersimplex $\Delta(2,3)$ is a 2-simplex dual to the ambient 2-simplex. See Figure \[fig:delta3\]. Putting this together gives a picture of the space of normalized linearizations for $n=3$ (admittedly an uninteresting case from a GIT/moduli perspective, but a helpful illustration nonetheless). See Figure \[fig:linpoly3\]. The polytope obtained by restricting $\gamma$ to the interval $[0,1]$ is seen to be the tetrahedron which, as we discuss momentarily, is the hypersimplex $\Delta(3,4)$. This exemplifies the general result that the polytope of linearizations for $\gamma$ on this interval is $\Delta(3,n+1)$. (152.5,75.18)(0,0) (7.14,25.18)(1.75,-1.03)[21]{}[(0,0)(0.22,-0.13)[4]{}[(1,0)[0.22]{}]{}]{} (43.04,4.11)(1.7,1.07)[21]{}[(0,0)(0.21,0.13)[4]{}[(1,0)[0.21]{}]{}]{} (39.64,73.57)(1.25,-1.56)[31]{}[(0,0)(0.13,-0.16)[5]{}[(0,-1)[0.16]{}]{}]{} (7.14,25.18)(1.1,1.64)[30]{}[(0,0)(0.11,0.16)[5]{}[(0,1)[0.16]{}]{}]{} (39.82,73.21)(0.09,-2)[35]{}[(0,0)(0.05,-1)[1]{}[(0,-1)[1]{}]{}]{} (7.32,25.71)(1.98,0.01)[36]{}[(0,0)(0.99,0)[1]{}[(1,0)[0.99]{}]{}]{} (30,37.14)(0.12,-0.16)[118]{}[(0,-1)[0.16]{}]{} (43.93,17.5)(0.12,0.19)[110]{}[(0,1)[0.19]{}]{} (30.54,36.96)(4.49,0.12)[6]{}[(1,0)[4.49]{}]{} (30.54,37.5)(0.32,0.12)[31]{}[(1,0)[0.32]{}]{} (40.36,41.07)(0.63,-0.12)[27]{}[(1,0)[0.63]{}]{} (40.36,41.07)(0.12,-0.83)[28]{}[(0,-1)[0.83]{}]{} (99.05,25.99)(1.7,-1.06)[15]{}[(0,0)(0.21,-0.13)[4]{}[(1,0)[0.21]{}]{}]{} (123.73,10.59)(1.65,1.11)[15]{}[(0,0)(0.17,0.11)[5]{}[(1,0)[0.17]{}]{}]{} (121.4,61.35)(1.22,-1.61)[22]{}[(0,0)(0.12,-0.16)[5]{}[(0,-1)[0.16]{}]{}]{} (99.05,25.99)(1.09,1.72)[21]{}[(0,0)(0.11,0.17)[5]{}[(0,1)[0.17]{}]{}]{} (121.51,61.09)(0.09,-1.98)[26]{}[(0,0)(0.04,-0.99)[1]{}[(0,-1)[0.99]{}]{}]{} (99.17,26.38)(1.97,0.01)[25]{}[(0,0)(0.98,0)[1]{}[(1,0)[0.98]{}]{}]{} (6.96,19.29)[(0,0)\[cc\][(3,0,0,0)]{}]{} (46.25,75.18)[(0,0)\[cc\][(0,0,3,0)]{}]{} (51.07,2.32)[(0,0)\[cc\][(0,3,0,0)]{}]{} (80.18,20.71)[(0,0)\[cc\][(0,0,0,3)]{}]{} (23.57,33.75)[(0,0)\[cc\][(1,1,1,0)]{}]{} (34.29,44.11)[(0,0)\[cc\][(1,0,1,1)]{}]{} (63.57,34.46)[(0,0)\[cc\][(0,1,1,1)]{}]{} (49.46,15.36)[(0,0)\[cc\][(1,1,0,1)]{}]{} (97.76,21.02)[(0,0)\[cc\][(2,0,0,0)]{}]{} (127.32,63.75)[(0,0)\[cc\][(0,0,2,0)]{}]{} (128.75,8.04)[(0,0)\[cc\][(0,2,0,0)]{}]{} (152.5,22.68)[(0,0)\[cc\][(0,0,0,2)]{}]{} (103.39,45)[(0,0)\[cc\][(1,0,1,0)]{}]{} (142.86,44.11)[(0,0)\[cc\][(0,0,1,1)]{}]{} (141.96,15.54)[(0,0)\[cc\][(0,1,0,1)]{}]{} (108.93,14.11)[(0,0)\[cc\][(1,1,0,0)]{}]{} (111.25,18.39)(0.12,0.16)[95]{}[(0,1)[0.16]{}]{} (110.18,42.86)(0.17,-0.12)[74]{}[(1,0)[0.17]{}]{} (110.18,42.86)(0.12,-2.74)[9]{}[(0,-1)[2.74]{}]{} (122.86,33.93)(0.12,-0.13)[115]{}[(0,-1)[0.13]{}]{} (136.07,41.79)(0.12,-3.75)[6]{}[(0,-1)[3.75]{}]{} (122.68,33.75)(0.2,0.12)[67]{}[(1,0)[0.2]{}]{} (110.54,43.04)(0.12,-0.19)[88]{}[(0,-1)[0.19]{}]{} (121.25,26.07)(0.12,0.12)[124]{}[(0,1)[0.12]{}]{} (110.36,42.32)(3.67,-0.13)[7]{}[(1,0)[3.67]{}]{} (111.25,18.04)(8.45,0.12)[3]{}[(1,0)[8.45]{}]{} (121.43,26.07)(0.24,-0.12)[63]{}[(1,0)[0.24]{}]{} (111.43,18.04)(0.15,0.12)[67]{}[(1,0)[0.15]{}]{} For $n=4$ the intersection of the scaled simplex and hypercube can be described by looking at what happens on each face of the simplex, since this just repeats the situation of $n=3$. For $k=3$ on each face the only point of intersection is the center point of the face so $\Delta(3,4)$ is the convex hull of these four points, a 3-simplex dual to the ambient 3-simplex. For $k=2$ the intersection with each face of the 3-simplex is a 2-simplex dual to that face and the convex hull of these forms an octahedron. See Figure \[fig:delta4\]. For intermediate values $3 > k > 2$ the 3-simplex shrinks with respect to the hypercube so that $\Delta(k,4)$ is a truncated tetrahedron. Thus, the linearization polytope $\Delta(3,5)$ for $n=4$ is obtained by starting with a tetrahedron and slowly truncating as $\gamma$ increase from 0 to 1 until it is truncated all the way to an octahedron. To see why the union of the linearization polytopes $\Delta(3-\gamma,n)$ as $\gamma$ ranges in $[0,1]$ is $\Delta(3,n+1)$, note that slicing with the hyperplane $\gamma = 0$ yields the hypersimplex $\Delta(3,n)$ which is the convex hull of the $\binom{n}{3}$ points whose coordinates have a 0 in the first entry (for $\gamma$) and all other entries are 0 except for three entries with a 1. Slicing with $\gamma = 1$ gives the hypersimplex $\Delta(2,n)$ whose $\binom{n}{2}$ coordinates have a 1 in the first entry (for $\gamma$) and the remaining entries are all zero except for two entries with a 1. Collectively these vertices are the $\binom{n}{3}+\binom{n}{2} = \binom{n+1}{3}$ vertices of the hypersimplex $\Delta(3,n+1)$ whose coordinates are all vectors of length $n+1$ with three 1s and 0s elsewhere. Moduli stability and GIT stability ================================== In this section we prove Theorem \[thm:contraction\]. The first step is to normalize the GIT linearization vector $(\gamma,c_1,\ldots,c_n)$ so that it is compatible with the weight data in Hassett’s moduli spaces. Recall (see [@Hass]) that for $\mathcal{A} := \{a_1,\ldots,a_n\}\in\mathbb{Q}^n\cap [0,1]^n$ the moduli space $\overline{\mathcal{M}}_{0,\mathcal{A}}$ parametrizes $n$-pointed nodal rational curves for which the *log canonical divisor* $K + a_1p_1 + \cdots + a_np_n$ (i.e. the dualizing sheaf twisted by the marked points $p_i$ with weights $a_i$) is ample and such that the $p_i$ are smooth and $\sum_{i\in I}a_i \le 1$ if $\{p_i\}_{i\in I}$ simultaneously collide. In what follows we take the moduli weight vector $\mathcal{A}$ to be the GIT point weights $\vec{c}=(c_1,\ldots,c_n)$. Because Hassett requires $c_i\in[0,1]$, the point weights in the GIT linearizations should also be in this range—and fortunately this is precisely the condition imposed by the normalization (\[eqn:normalization\]) used to describe the polytope of linearizations. Since Kapranov’s work [@Kapr] addresses GIT quotients $(\mathbb{P}^1){/\hspace{-1.2mm}/}\mathtt{SL}_2$, and our quotients $\mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$ are isomorphic to these for $\gamma > \frac{c}{2}$, where $c:=c_1+\cdots+c_n$, we can restrict attention to the part of the $\mathtt{SL}_3$-ample cone described by $\gamma \le \frac{c}{2}$. By Corollary \[cor:chamber\] this subcone admits a uniform cross-section $c+\gamma=3$ in such a way that rays are in bijection with the polytope $\Delta(3,n+1)$ and the point weights satisfy $c_i\in[0,1]$ as required. For fixed $\gamma$ the vector $(\gamma,\vec{c})\in\Delta(3,n+1)$ satisfies $\vec{c}\in\Delta(3-\gamma,n)$ and this forces all curves parametrized by $\overline{\mathcal{M}}_{0,\vec{c}}$ to be *chains* of $\mathbb{P}^1$s. Indeed, the dualizing sheaf restricted to a component with one node has degree $2g-2+1=-1$ so for a curve to be stable it must have $> 1 = \frac{\gamma+c}{3} \ge \frac{c}{3}$ weight on each such component so there can be at most two of them. There is a reduction morphism $\overline{\mathcal{M}}_{0,n} \rightarrow \overline{\mathcal{M}}_{0,\vec{c}}$ ([@Hass], Theorem 4.1) which contracts a tree of $\mathbb{P}^1$s down to a chain of $\mathbb{P}^1$s. To prove Theorem \[thm:contraction\] it is enough to show that there is a morphism $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3$ when $(\gamma,\vec{c})$ lies in the interior of a GIT chamber. Indeed, on the face $\gamma=1$ it is easy to see that $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3 \cong (\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_{\vec{c}}~\mathtt{SL}_2$ (even though the universal curves are different), and Hassett’s space $\overline{\mathcal{M}}_{0,\vec{c}}$ is by definition $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_{\vec{c}}~\mathtt{SL}_2$ here since $c=2$, so we can assume $\gamma < 1$, or equivalently $c > 2$. Thus for a linearization $(\gamma,\vec{c})$ on any wall except those of the form $c_i = 0$ we can find a nearby linearization $(\gamma+\epsilon,\vec{c}-\epsilon)$ lying in an open GIT chamber. By general principles of variation of GIT ([@Thad], Theorem 2.3) any GIT quotient for linearization on a wall or boundary receives a morphism from a quotient for linearization in the interior of an adjacent chamber, so we may deduce the desired factorization for a linearization on a wall or boundary from that of one in an open chamber by considering the composition $$\overline{\mathcal{M}}_{0,n} \rightarrow \overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \overline{\mathcal{M}}_{0,\vec{c}-\epsilon} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma+\epsilon,\vec{c}-\epsilon)}\mathtt{SL}_3 \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3.$$ All the morphisms in this composition are clearly birational. For the remained of this section, therefore, we assume $(\gamma,\vec{c})\in\Delta(3,n+1)$ lies in an open GIT chamber. Let $\pi : \mathcal{C}_{0,\vec{c}} \rightarrow\overline{\mathcal{M}}_{0,\vec{c}}$ be the universal curve with universal sections $p_1,\ldots,p_n$. Recall ([@Hass], Proposition 5.4) that $\mathcal{C}_{0,\vec{c}} = \mathcal{M}_{0,\vec{c}\cup\{\epsilon\}}$ for sufficiently small $\epsilon > 0$. Intuitively, because no points have weight 1 the curve represented by a point in the moduli space is traced out by throwing in an additional marked point with such small weight that it can freely pass by all the other points. \[prop:linebundle\] The line bundle $L := \omega_\pi^{-1}$ on $\mathcal{C}_{0,\vec{c}}$ induces a morphism which embeds each stable nonsingular curve as a nonsingular conic and contracts the inner components of each singular stable curve and embeds the resulting curve as a nodal conic (see Figure \[fig:contract\]). Moreover, the conics obtained this way are all GIT stable with respect to the linearization $(\gamma,\vec{c})$ and every GIT stable conic comes from a moduli-stable curve in this manner. (140.89,57.5)(0,0) (2.14,52.5)(0.12,-0.24)[83]{}[(0,-1)[0.24]{}]{} (2.14,42.5)[(1,0)[30]{}]{} (17.14,32.5)(0.12,0.12)[208]{}[(1,0)[0.12]{}]{} (27.14,57.5)(0.18,-0.12)[167]{}[(1,0)[0.18]{}]{} (42.14,37.5)(0.36,0.12)[83]{}[(1,0)[0.36]{}]{} (62.14,37.5)(0.12,0.16)[125]{}[(0,1)[0.16]{}]{} (122.14,52.5)(0.12,-0.24)[83]{}[(0,-1)[0.24]{}]{} (122.14,37.5)(0.12,0.16)[125]{}[(0,1)[0.16]{}]{} (6.43,52.5)[(0,0)\[cc\][1]{}]{} (72.14,57.5)[(0,0)\[cc\][1]{}]{} (15.71,46.07)[(0,0)\[cc\][0]{}]{} (59.46,46.79)[(0,0)\[cc\][0]{}]{} (29.11,49.11)[(0,0)\[cc\][0]{}]{} (47.5,48.39)[(0,0)\[cc\][0]{}]{} (81.43,43.39)(88.86,48.47)(91.07,44.51) (91.07,44.51)(93.28,40.55)(97.5,43.75) (97.5,43.75)(102.31,47.71)(104.64,45.85) (104.64,45.85)(106.98,43.98)(106.96,43.39) (106.96,43.39)[(1,0)[6.79]{}]{} (113.75,43.39)[(1,0)[0.12]{}]{} (140.89,11.64)[(0,1)[0.3]{}]{} (140.88,11.34)(0.01,0.3)[1]{}[(0,1)[0.3]{}]{} (140.85,11.05)(0.03,0.3)[1]{}[(0,1)[0.3]{}]{} (140.81,10.75)(0.04,0.29)[1]{}[(0,1)[0.29]{}]{} (140.75,10.46)(0.06,0.29)[1]{}[(0,1)[0.29]{}]{} (140.68,10.17)(0.07,0.29)[1]{}[(0,1)[0.29]{}]{} (140.6,9.88)(0.08,0.29)[1]{}[(0,1)[0.29]{}]{} (140.5,9.59)(0.1,0.29)[1]{}[(0,1)[0.29]{}]{} (140.39,9.3)(0.11,0.29)[1]{}[(0,1)[0.29]{}]{} (140.27,9.01)(0.12,0.29)[1]{}[(0,1)[0.29]{}]{} (140.13,8.73)(0.14,0.28)[1]{}[(0,1)[0.28]{}]{} (139.98,8.45)(0.15,0.28)[1]{}[(0,1)[0.28]{}]{} (139.82,8.17)(0.16,0.28)[1]{}[(0,1)[0.28]{}]{} (139.64,7.89)(0.18,0.28)[1]{}[(0,1)[0.28]{}]{} (139.45,7.62)(0.09,0.14)[2]{}[(0,1)[0.14]{}]{} (139.25,7.35)(0.1,0.13)[2]{}[(0,1)[0.13]{}]{} (139.03,7.08)(0.11,0.13)[2]{}[(0,1)[0.13]{}]{} (138.81,6.82)(0.11,0.13)[2]{}[(0,1)[0.13]{}]{} (138.57,6.56)(0.12,0.13)[2]{}[(0,1)[0.13]{}]{} (138.31,6.31)(0.13,0.13)[2]{}[(0,1)[0.13]{}]{} (138.05,6.06)(0.13,0.13)[2]{}[(1,0)[0.13]{}]{} (137.77,5.81)(0.14,0.12)[2]{}[(1,0)[0.14]{}]{} (137.49,5.57)(0.14,0.12)[2]{}[(1,0)[0.14]{}]{} (137.19,5.33)(0.15,0.12)[2]{}[(1,0)[0.15]{}]{} (136.88,5.1)(0.15,0.12)[2]{}[(1,0)[0.15]{}]{} (136.56,4.88)(0.16,0.11)[2]{}[(1,0)[0.16]{}]{} (136.23,4.66)(0.17,0.11)[2]{}[(1,0)[0.17]{}]{} (135.89,4.44)(0.17,0.11)[2]{}[(1,0)[0.17]{}]{} (135.54,4.23)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (135.18,4.03)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (134.81,3.83)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (134.43,3.64)(0.19,0.1)[2]{}[(1,0)[0.19]{}]{} (134.04,3.46)(0.19,0.09)[2]{}[(1,0)[0.19]{}]{} (133.65,3.28)(0.4,0.18)[1]{}[(1,0)[0.4]{}]{} (133.24,3.11)(0.4,0.17)[1]{}[(1,0)[0.4]{}]{} (132.83,2.94)(0.41,0.16)[1]{}[(1,0)[0.41]{}]{} (132.41,2.79)(0.42,0.16)[1]{}[(1,0)[0.42]{}]{} (131.98,2.64)(0.43,0.15)[1]{}[(1,0)[0.43]{}]{} (131.55,2.49)(0.43,0.14)[1]{}[(1,0)[0.43]{}]{} (131.11,2.36)(0.44,0.14)[1]{}[(1,0)[0.44]{}]{} (130.66,2.23)(0.45,0.13)[1]{}[(1,0)[0.45]{}]{} (130.21,2.11)(0.45,0.12)[1]{}[(1,0)[0.45]{}]{} (129.75,2)(0.46,0.11)[1]{}[(1,0)[0.46]{}]{} (129.29,1.89)(0.46,0.11)[1]{}[(1,0)[0.46]{}]{} (128.82,1.79)(0.47,0.1)[1]{}[(1,0)[0.47]{}]{} (128.35,1.7)(0.47,0.09)[1]{}[(1,0)[0.47]{}]{} (127.87,1.62)(0.48,0.08)[1]{}[(1,0)[0.48]{}]{} (127.39,1.55)(0.48,0.07)[1]{}[(1,0)[0.48]{}]{} (126.91,1.48)(0.48,0.07)[1]{}[(1,0)[0.48]{}]{} (126.42,1.42)(0.49,0.06)[1]{}[(1,0)[0.49]{}]{} (125.93,1.38)(0.49,0.05)[1]{}[(1,0)[0.49]{}]{} (125.44,1.33)(0.49,0.04)[1]{}[(1,0)[0.49]{}]{} (124.95,1.3)(0.49,0.03)[1]{}[(1,0)[0.49]{}]{} (124.45,1.28)(0.49,0.02)[1]{}[(1,0)[0.49]{}]{} (123.96,1.26)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (123.46,1.25)(0.5,0.01)[1]{}[(1,0)[0.5]{}]{} (122.97,1.25)[(1,0)[0.5]{}]{} (122.47,1.26)(0.5,-0.01)[1]{}[(1,0)[0.5]{}]{} (121.98,1.28)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (121.48,1.3)(0.49,-0.02)[1]{}[(1,0)[0.49]{}]{} (120.99,1.33)(0.49,-0.03)[1]{}[(1,0)[0.49]{}]{} (120.5,1.38)(0.49,-0.04)[1]{}[(1,0)[0.49]{}]{} (120.01,1.42)(0.49,-0.05)[1]{}[(1,0)[0.49]{}]{} (119.52,1.48)(0.49,-0.06)[1]{}[(1,0)[0.49]{}]{} (119.04,1.55)(0.48,-0.07)[1]{}[(1,0)[0.48]{}]{} (118.56,1.62)(0.48,-0.07)[1]{}[(1,0)[0.48]{}]{} (118.08,1.7)(0.48,-0.08)[1]{}[(1,0)[0.48]{}]{} (117.61,1.79)(0.47,-0.09)[1]{}[(1,0)[0.47]{}]{} (117.14,1.89)(0.47,-0.1)[1]{}[(1,0)[0.47]{}]{} (116.68,2)(0.46,-0.11)[1]{}[(1,0)[0.46]{}]{} (116.22,2.11)(0.46,-0.11)[1]{}[(1,0)[0.46]{}]{} (115.77,2.23)(0.45,-0.12)[1]{}[(1,0)[0.45]{}]{} (115.32,2.36)(0.45,-0.13)[1]{}[(1,0)[0.45]{}]{} (114.88,2.49)(0.44,-0.14)[1]{}[(1,0)[0.44]{}]{} (114.45,2.64)(0.43,-0.14)[1]{}[(1,0)[0.43]{}]{} (114.02,2.79)(0.43,-0.15)[1]{}[(1,0)[0.43]{}]{} (113.6,2.94)(0.42,-0.16)[1]{}[(1,0)[0.42]{}]{} (113.19,3.11)(0.41,-0.16)[1]{}[(1,0)[0.41]{}]{} (112.78,3.28)(0.4,-0.17)[1]{}[(1,0)[0.4]{}]{} (112.39,3.46)(0.4,-0.18)[1]{}[(1,0)[0.4]{}]{} (112,3.64)(0.19,-0.09)[2]{}[(1,0)[0.19]{}]{} (111.62,3.83)(0.19,-0.1)[2]{}[(1,0)[0.19]{}]{} (111.25,4.03)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (110.89,4.23)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (110.54,4.44)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (110.2,4.66)(0.17,-0.11)[2]{}[(1,0)[0.17]{}]{} (109.87,4.88)(0.17,-0.11)[2]{}[(1,0)[0.17]{}]{} (109.55,5.1)(0.16,-0.11)[2]{}[(1,0)[0.16]{}]{} (109.24,5.33)(0.15,-0.12)[2]{}[(1,0)[0.15]{}]{} (108.94,5.57)(0.15,-0.12)[2]{}[(1,0)[0.15]{}]{} (108.65,5.81)(0.14,-0.12)[2]{}[(1,0)[0.14]{}]{} (108.38,6.06)(0.14,-0.12)[2]{}[(1,0)[0.14]{}]{} (108.11,6.31)(0.13,-0.13)[2]{}[(1,0)[0.13]{}]{} (107.86,6.56)(0.13,-0.13)[2]{}[(0,-1)[0.13]{}]{} (107.62,6.82)(0.12,-0.13)[2]{}[(0,-1)[0.13]{}]{} (107.4,7.08)(0.11,-0.13)[2]{}[(0,-1)[0.13]{}]{} (107.18,7.35)(0.11,-0.13)[2]{}[(0,-1)[0.13]{}]{} (106.98,7.62)(0.1,-0.13)[2]{}[(0,-1)[0.13]{}]{} (106.79,7.89)(0.09,-0.14)[2]{}[(0,-1)[0.14]{}]{} (106.61,8.17)(0.18,-0.28)[1]{}[(0,-1)[0.28]{}]{} (106.45,8.45)(0.16,-0.28)[1]{}[(0,-1)[0.28]{}]{} (106.3,8.73)(0.15,-0.28)[1]{}[(0,-1)[0.28]{}]{} (106.16,9.01)(0.14,-0.28)[1]{}[(0,-1)[0.28]{}]{} (106.04,9.3)(0.12,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.93,9.59)(0.11,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.83,9.88)(0.1,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.75,10.17)(0.08,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.68,10.46)(0.07,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.62,10.75)(0.06,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.58,11.05)(0.04,-0.29)[1]{}[(0,-1)[0.29]{}]{} (105.55,11.34)(0.03,-0.3)[1]{}[(0,-1)[0.3]{}]{} (105.54,11.64)(0.01,-0.3)[1]{}[(0,-1)[0.3]{}]{} (105.54,11.64)[(0,1)[0.3]{}]{} (105.54,11.93)(0.01,0.3)[1]{}[(0,1)[0.3]{}]{} (105.55,12.23)(0.03,0.3)[1]{}[(0,1)[0.3]{}]{} (105.58,12.52)(0.04,0.29)[1]{}[(0,1)[0.29]{}]{} (105.62,12.82)(0.06,0.29)[1]{}[(0,1)[0.29]{}]{} (105.68,13.11)(0.07,0.29)[1]{}[(0,1)[0.29]{}]{} (105.75,13.4)(0.08,0.29)[1]{}[(0,1)[0.29]{}]{} (105.83,13.7)(0.1,0.29)[1]{}[(0,1)[0.29]{}]{} (105.93,13.99)(0.11,0.29)[1]{}[(0,1)[0.29]{}]{} (106.04,14.27)(0.12,0.29)[1]{}[(0,1)[0.29]{}]{} (106.16,14.56)(0.14,0.28)[1]{}[(0,1)[0.28]{}]{} (106.3,14.84)(0.15,0.28)[1]{}[(0,1)[0.28]{}]{} (106.45,15.13)(0.16,0.28)[1]{}[(0,1)[0.28]{}]{} (106.61,15.4)(0.18,0.28)[1]{}[(0,1)[0.28]{}]{} (106.79,15.68)(0.09,0.14)[2]{}[(0,1)[0.14]{}]{} (106.98,15.95)(0.1,0.13)[2]{}[(0,1)[0.13]{}]{} (107.18,16.22)(0.11,0.13)[2]{}[(0,1)[0.13]{}]{} (107.4,16.49)(0.11,0.13)[2]{}[(0,1)[0.13]{}]{} (107.62,16.75)(0.12,0.13)[2]{}[(0,1)[0.13]{}]{} (107.86,17.01)(0.13,0.13)[2]{}[(0,1)[0.13]{}]{} (108.11,17.27)(0.13,0.13)[2]{}[(1,0)[0.13]{}]{} (108.38,17.52)(0.14,0.12)[2]{}[(1,0)[0.14]{}]{} (108.65,17.76)(0.14,0.12)[2]{}[(1,0)[0.14]{}]{} (108.94,18)(0.15,0.12)[2]{}[(1,0)[0.15]{}]{} (109.24,18.24)(0.15,0.12)[2]{}[(1,0)[0.15]{}]{} (109.55,18.47)(0.16,0.11)[2]{}[(1,0)[0.16]{}]{} (109.87,18.7)(0.17,0.11)[2]{}[(1,0)[0.17]{}]{} (110.2,18.92)(0.17,0.11)[2]{}[(1,0)[0.17]{}]{} (110.54,19.13)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (110.89,19.34)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (111.25,19.54)(0.18,0.1)[2]{}[(1,0)[0.18]{}]{} (111.62,19.74)(0.19,0.1)[2]{}[(1,0)[0.19]{}]{} (112,19.93)(0.19,0.09)[2]{}[(1,0)[0.19]{}]{} (112.39,20.11)(0.4,0.18)[1]{}[(1,0)[0.4]{}]{} (112.78,20.29)(0.4,0.17)[1]{}[(1,0)[0.4]{}]{} (113.19,20.46)(0.41,0.16)[1]{}[(1,0)[0.41]{}]{} (113.6,20.63)(0.42,0.16)[1]{}[(1,0)[0.42]{}]{} (114.02,20.78)(0.43,0.15)[1]{}[(1,0)[0.43]{}]{} (114.45,20.93)(0.43,0.14)[1]{}[(1,0)[0.43]{}]{} (114.88,21.08)(0.44,0.14)[1]{}[(1,0)[0.44]{}]{} (115.32,21.21)(0.45,0.13)[1]{}[(1,0)[0.45]{}]{} (115.77,21.34)(0.45,0.12)[1]{}[(1,0)[0.45]{}]{} (116.22,21.46)(0.46,0.11)[1]{}[(1,0)[0.46]{}]{} (116.68,21.58)(0.46,0.11)[1]{}[(1,0)[0.46]{}]{} (117.14,21.68)(0.47,0.1)[1]{}[(1,0)[0.47]{}]{} (117.61,21.78)(0.47,0.09)[1]{}[(1,0)[0.47]{}]{} (118.08,21.87)(0.48,0.08)[1]{}[(1,0)[0.48]{}]{} (118.56,21.95)(0.48,0.07)[1]{}[(1,0)[0.48]{}]{} (119.04,22.02)(0.48,0.07)[1]{}[(1,0)[0.48]{}]{} (119.52,22.09)(0.49,0.06)[1]{}[(1,0)[0.49]{}]{} (120.01,22.15)(0.49,0.05)[1]{}[(1,0)[0.49]{}]{} (120.5,22.2)(0.49,0.04)[1]{}[(1,0)[0.49]{}]{} (120.99,22.24)(0.49,0.03)[1]{}[(1,0)[0.49]{}]{} (121.48,22.27)(0.49,0.02)[1]{}[(1,0)[0.49]{}]{} (121.98,22.3)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (122.47,22.31)(0.5,0.01)[1]{}[(1,0)[0.5]{}]{} (122.97,22.32)[(1,0)[0.5]{}]{} (123.46,22.32)(0.5,-0.01)[1]{}[(1,0)[0.5]{}]{} (123.96,22.31)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (124.45,22.3)(0.49,-0.02)[1]{}[(1,0)[0.49]{}]{} (124.95,22.27)(0.49,-0.03)[1]{}[(1,0)[0.49]{}]{} (125.44,22.24)(0.49,-0.04)[1]{}[(1,0)[0.49]{}]{} (125.93,22.2)(0.49,-0.05)[1]{}[(1,0)[0.49]{}]{} (126.42,22.15)(0.49,-0.06)[1]{}[(1,0)[0.49]{}]{} (126.91,22.09)(0.48,-0.07)[1]{}[(1,0)[0.48]{}]{} (127.39,22.02)(0.48,-0.07)[1]{}[(1,0)[0.48]{}]{} (127.87,21.95)(0.48,-0.08)[1]{}[(1,0)[0.48]{}]{} (128.35,21.87)(0.47,-0.09)[1]{}[(1,0)[0.47]{}]{} (128.82,21.78)(0.47,-0.1)[1]{}[(1,0)[0.47]{}]{} (129.29,21.68)(0.46,-0.11)[1]{}[(1,0)[0.46]{}]{} (129.75,21.58)(0.46,-0.11)[1]{}[(1,0)[0.46]{}]{} (130.21,21.46)(0.45,-0.12)[1]{}[(1,0)[0.45]{}]{} (130.66,21.34)(0.45,-0.13)[1]{}[(1,0)[0.45]{}]{} (131.11,21.21)(0.44,-0.14)[1]{}[(1,0)[0.44]{}]{} (131.55,21.08)(0.43,-0.14)[1]{}[(1,0)[0.43]{}]{} (131.98,20.93)(0.43,-0.15)[1]{}[(1,0)[0.43]{}]{} (132.41,20.78)(0.42,-0.16)[1]{}[(1,0)[0.42]{}]{} (132.83,20.63)(0.41,-0.16)[1]{}[(1,0)[0.41]{}]{} (133.24,20.46)(0.4,-0.17)[1]{}[(1,0)[0.4]{}]{} (133.65,20.29)(0.4,-0.18)[1]{}[(1,0)[0.4]{}]{} (134.04,20.11)(0.19,-0.09)[2]{}[(1,0)[0.19]{}]{} (134.43,19.93)(0.19,-0.1)[2]{}[(1,0)[0.19]{}]{} (134.81,19.74)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (135.18,19.54)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (135.54,19.34)(0.18,-0.1)[2]{}[(1,0)[0.18]{}]{} (135.89,19.13)(0.17,-0.11)[2]{}[(1,0)[0.17]{}]{} (136.23,18.92)(0.17,-0.11)[2]{}[(1,0)[0.17]{}]{} (136.56,18.7)(0.16,-0.11)[2]{}[(1,0)[0.16]{}]{} (136.88,18.47)(0.15,-0.12)[2]{}[(1,0)[0.15]{}]{} (137.19,18.24)(0.15,-0.12)[2]{}[(1,0)[0.15]{}]{} (137.49,18)(0.14,-0.12)[2]{}[(1,0)[0.14]{}]{} (137.77,17.76)(0.14,-0.12)[2]{}[(1,0)[0.14]{}]{} (138.05,17.52)(0.13,-0.13)[2]{}[(1,0)[0.13]{}]{} (138.31,17.27)(0.13,-0.13)[2]{}[(0,-1)[0.13]{}]{} (138.57,17.01)(0.12,-0.13)[2]{}[(0,-1)[0.13]{}]{} (138.81,16.75)(0.11,-0.13)[2]{}[(0,-1)[0.13]{}]{} (139.03,16.49)(0.11,-0.13)[2]{}[(0,-1)[0.13]{}]{} (139.25,16.22)(0.1,-0.13)[2]{}[(0,-1)[0.13]{}]{} (139.45,15.95)(0.09,-0.14)[2]{}[(0,-1)[0.14]{}]{} (139.64,15.68)(0.18,-0.28)[1]{}[(0,-1)[0.28]{}]{} (139.82,15.4)(0.16,-0.28)[1]{}[(0,-1)[0.28]{}]{} (139.98,15.13)(0.15,-0.28)[1]{}[(0,-1)[0.28]{}]{} (140.13,14.84)(0.14,-0.28)[1]{}[(0,-1)[0.28]{}]{} (140.27,14.56)(0.12,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.39,14.27)(0.11,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.5,13.99)(0.1,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.6,13.7)(0.08,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.68,13.4)(0.07,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.75,13.11)(0.06,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.81,12.82)(0.04,-0.29)[1]{}[(0,-1)[0.29]{}]{} (140.85,12.52)(0.03,-0.3)[1]{}[(0,-1)[0.3]{}]{} (140.88,12.23)(0.01,-0.3)[1]{}[(0,-1)[0.3]{}]{} (23.57,10.71)[(1,0)[23.39]{}]{} (62.86,10.89)(70.29,15.97)(72.5,12.01) (72.5,12.01)(74.71,8.05)(78.93,11.25) (78.93,11.25)(83.74,15.21)(86.07,13.35) (86.07,13.35)(88.4,11.48)(88.39,10.89) (88.39,10.89)[(1,0)[6.79]{}]{} (95.18,10.89)[(1,0)[0.12]{}]{} (34.64,14.46)[(0,0)\[cc\][2]{}]{} We first observe that on each fiber $L$ has vanishing higher cohomology. Indeed, a fiber is a moduli-stable curve $C$ and $L|_C = \omega_C^{-1}$ so by Serre duality $h^1(L|_C) = h^0(\omega_C^{\otimes 2}) = 0$. Now $L$ has degree 1 on extremal components and degree 0 on inner components of a singular curve and it has degree 2 on a nonsingular curve (as is indicated in Figure \[fig:contract\]) so as long as $L$ is basepoint-free it induces a morphism which contracts inner components. By the vanishing cohomology observation it is enough to check that $L$ is relatively basepoint-free, i.e. that $L|_C$ is basepoint-free for each fiber $C$. But if $C\cong \mathbb{P}^1$ then $L|_C\cong\mathcal{O}_{\mathbb{P}^1}(2)$ so it is obvious and if $C$ is a chain of $\mathbb{P}^1$s then on each extremal component $C'\subset C$ we have $L|_C'\cong\mathcal{O}_{\mathbb{P}^1}(1)$ and on each inner component $C''\subset C$ we have $L|_{C''}\cong\mathcal{O}_{\mathbb{P}^1}$ so it is also clear: global sections are constant on the inner components and do not simultaneously vanish at any points of the extremal components. Since $\text{deg}(L|_C)=2$ the image of each curve under the morphism induced by $L$ has degree 2, and it is mapped to $\mathbb{P}^2$ since $\text{dim }\Gamma(C,L|_C) = 3$. Indeed, Riemann-Roch says $\chi(L|_C) = \text{deg}(L|_C) + 1 - p_a,$ but $p_a=0$ and $\chi(L|_C) = h^0(L|_C)$ since $h^1(L|_C)=0$. We have shown that $L$ maps each moduli-stable curve to a degree 2 curve in $\mathbb{P}^2$, i.e. a conic. To finish the proof it only remains to verify the claim about GIT stability. Recall that we have normalized so that $\frac{c+\gamma}{3}=1$. Thus, by Theorem \[thm:stability\], GIT stability for nodal conics is characterized by the following three conditions: 1. there is $< c - 2$ weight at the node 2. there is $< 1$ weight at any smooth point, and 3. there is $> 1$ weight on each component away from the node But these follow immediately from the fact that on the original moduli-stable curve 1. the extremal components each have $> 1$ weight leaving $< c - 2$ weight remaining on the inner components—which are precisely the components that get contracted to the node of the conic (recall Figure \[fig:contract\]) 2. the smooth points of the conic come from smooth points on the extremal components of the origin curve, so there is $< 1$ weight at any such point (since we are assuming $\vec{c}$ lies in the interior of $[0,1]^n$), and 3. there is $> 1$ weight on the extremal components, as we have already noted. A nonsingular rational curve gets embedded as a nonsingular GIT-stable conic because the only condition to check is that there is $<1$ at each point. This shows that the image under $L$ of any stable rational curve is a stable conic, but it is easy to see that any stable conic can be obtained this way. We can use Proposition \[prop:linebundle\] to complete the proof of Theorem \[thm:contraction\]. The line bundle $L$ maps $\mathcal{C}_{0,\vec{c}}$ with its universal sections to a flat family of pointed curves over $\overline{\mathcal{M}}_{0,\vec{c}}$ and it embeds this family in a $\mathbb{P}^2$-bundle. This $\mathbb{P}^2$-bundle has an associated principal $\mathtt{PGL}_3$-bundle $P$. Pulling back the $\mathbb{P}^2$-bundle along the structure morphism $P \rightarrow \overline{\mathcal{M}}_{0,\vec{c}}$ trivializes it so we get a flat family of pointed conics over $P$ embedded in the trivial $\mathbb{P}^2$-bundle over $P$. Now $\mathbb{P}^5$ is the Hilbert scheme of conics, and $\mathtt{Con}(n)$ is the Hilbert scheme of $n$-pointed conics, so by the universal property of Hilbert schemes this family over $P$ induces a morphism $P \rightarrow \mathtt{Con}(n)$. But this morphism factors through the stable locus $\mathtt{Con}(n)_{s}$ because by Proposition \[prop:linebundle\] the image under $L$ of a moduli-stable curve is GIT-stable. Since $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3$ is the categorical quotient of $\mathtt{Con}(n)_s$ by $\mathtt{SL}_3$ (recall that by assumption stability and semistability coincide) and $\mathtt{SL}_3$ acts through $\mathtt{PGL}_3$ via the canonical isogeny $\mathtt{SL}_3 \rightarrow \mathtt{PGL}_3$ we see that the composition $$P \rightarrow \mathtt{Con}(n)_{s} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{\vec{w}~}\mathtt{SL}_3$$ is $\mathtt{PGL}_3$-invariant so it must factor through the categorical quotient of $P$ by $\mathtt{PGL}_3$, which by the definition of principal bundle is $\overline{\mathcal{M}}_{0,\vec{c}}$. This means precisely that there is a morphism $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{\vec{w}~}\mathtt{SL}_3$, thus concluding the proof of Theorem \[thm:contraction\]. Examples, further properties, and applications ============================================== In this section we explore some properties and manifestations of the conic compactifications constructed in this paper. Semistable reduction -------------------- The morphism described in Theorem \[thm:contraction\] and its proof can be used to study semistable reduction in the spaces $\mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$. Any 1-parameter family of semistable configurations of $n$ points on a conic must have a semistable limit since the GIT quotient is proper. If the conics are nonsingular we can identify them with $\mathbb{P}^1$ and the limit may be computed by first finding the limit as a stable curve in $\overline{\mathcal{M}}_{0,n}$ and then looking at the image of this curve under the morphism $\overline{\mathcal{M}}_{0,n} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}\mathtt{SL}_3$. Figure \[fig:limit\] shows an example with $\gamma=\frac{1}{8},\vec{c}=(\frac{5}{8},\frac{5}{8},\frac{5}{8},\frac{5}{8},\frac{2}{8},\frac{1}{8})$. (101.43,66.61)(0,0) (21.07,53.74)[(0,1)[0.38]{}]{} (21.04,53.36)(0.03,0.38)[1]{}[(0,1)[0.38]{}]{} (20.99,52.98)(0.05,0.38)[1]{}[(0,1)[0.38]{}]{} (20.91,52.61)(0.08,0.38)[1]{}[(0,1)[0.38]{}]{} (20.81,52.23)(0.1,0.37)[1]{}[(0,1)[0.37]{}]{} (20.68,51.87)(0.13,0.37)[1]{}[(0,1)[0.37]{}]{} (20.53,51.5)(0.15,0.36)[1]{}[(0,1)[0.36]{}]{} (20.36,51.15)(0.18,0.36)[1]{}[(0,1)[0.36]{}]{} (20.16,50.8)(0.1,0.17)[2]{}[(0,1)[0.17]{}]{} (19.93,50.46)(0.11,0.17)[2]{}[(0,1)[0.17]{}]{} (19.69,50.13)(0.12,0.17)[2]{}[(0,1)[0.17]{}]{} (19.42,49.8)(0.13,0.16)[2]{}[(0,1)[0.16]{}]{} (19.13,49.49)(0.14,0.16)[2]{}[(0,1)[0.16]{}]{} (18.82,49.19)(0.1,0.1)[3]{}[(1,0)[0.1]{}]{} (18.49,48.9)(0.17,0.14)[2]{}[(1,0)[0.17]{}]{} (18.14,48.63)(0.17,0.14)[2]{}[(1,0)[0.17]{}]{} (17.77,48.36)(0.18,0.13)[2]{}[(1,0)[0.18]{}]{} (17.39,48.12)(0.19,0.12)[2]{}[(1,0)[0.19]{}]{} (16.99,47.88)(0.2,0.12)[2]{}[(1,0)[0.2]{}]{} (16.57,47.67)(0.21,0.11)[2]{}[(1,0)[0.21]{}]{} (16.14,47.47)(0.21,0.1)[2]{}[(1,0)[0.21]{}]{} (15.7,47.28)(0.22,0.09)[2]{}[(1,0)[0.22]{}]{} (15.25,47.11)(0.45,0.17)[1]{}[(1,0)[0.45]{}]{} (14.78,46.97)(0.47,0.15)[1]{}[(1,0)[0.47]{}]{} (14.3,46.83)(0.48,0.13)[1]{}[(1,0)[0.48]{}]{} (13.82,46.72)(0.48,0.11)[1]{}[(1,0)[0.48]{}]{} (13.33,46.62)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (12.83,46.55)(0.5,0.08)[1]{}[(1,0)[0.5]{}]{} (12.33,46.49)(0.5,0.06)[1]{}[(1,0)[0.5]{}]{} (11.83,46.45)(0.5,0.04)[1]{}[(1,0)[0.5]{}]{} (11.32,46.43)(0.51,0.02)[1]{}[(1,0)[0.51]{}]{} (10.82,46.43)[(1,0)[0.51]{}]{} (10.31,46.45)(0.51,-0.02)[1]{}[(1,0)[0.51]{}]{} (9.81,46.49)(0.5,-0.04)[1]{}[(1,0)[0.5]{}]{} (9.31,46.55)(0.5,-0.06)[1]{}[(1,0)[0.5]{}]{} (8.81,46.62)(0.5,-0.08)[1]{}[(1,0)[0.5]{}]{} (8.32,46.72)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (7.84,46.83)(0.48,-0.11)[1]{}[(1,0)[0.48]{}]{} (7.36,46.97)(0.48,-0.13)[1]{}[(1,0)[0.48]{}]{} (6.89,47.11)(0.47,-0.15)[1]{}[(1,0)[0.47]{}]{} (6.44,47.28)(0.45,-0.17)[1]{}[(1,0)[0.45]{}]{} (6,47.47)(0.22,-0.09)[2]{}[(1,0)[0.22]{}]{} (5.57,47.67)(0.21,-0.1)[2]{}[(1,0)[0.21]{}]{} (5.15,47.88)(0.21,-0.11)[2]{}[(1,0)[0.21]{}]{} (4.75,48.12)(0.2,-0.12)[2]{}[(1,0)[0.2]{}]{} (4.37,48.36)(0.19,-0.12)[2]{}[(1,0)[0.19]{}]{} (4,48.63)(0.18,-0.13)[2]{}[(1,0)[0.18]{}]{} (3.65,48.9)(0.17,-0.14)[2]{}[(1,0)[0.17]{}]{} (3.32,49.19)(0.17,-0.14)[2]{}[(1,0)[0.17]{}]{} (3.01,49.49)(0.1,-0.1)[3]{}[(1,0)[0.1]{}]{} (2.72,49.8)(0.14,-0.16)[2]{}[(0,-1)[0.16]{}]{} (2.45,50.13)(0.13,-0.16)[2]{}[(0,-1)[0.16]{}]{} (2.21,50.46)(0.12,-0.17)[2]{}[(0,-1)[0.17]{}]{} (1.98,50.8)(0.11,-0.17)[2]{}[(0,-1)[0.17]{}]{} (1.78,51.15)(0.1,-0.17)[2]{}[(0,-1)[0.17]{}]{} (1.61,51.5)(0.18,-0.36)[1]{}[(0,-1)[0.36]{}]{} (1.46,51.87)(0.15,-0.36)[1]{}[(0,-1)[0.36]{}]{} (1.33,52.23)(0.13,-0.37)[1]{}[(0,-1)[0.37]{}]{} (1.23,52.61)(0.1,-0.37)[1]{}[(0,-1)[0.37]{}]{} (1.15,52.98)(0.08,-0.38)[1]{}[(0,-1)[0.38]{}]{} (1.1,53.36)(0.05,-0.38)[1]{}[(0,-1)[0.38]{}]{} (1.07,53.74)(0.03,-0.38)[1]{}[(0,-1)[0.38]{}]{} (1.07,53.74)[(0,1)[0.38]{}]{} (1.07,54.12)(0.03,0.38)[1]{}[(0,1)[0.38]{}]{} (1.1,54.5)(0.05,0.38)[1]{}[(0,1)[0.38]{}]{} (1.15,54.88)(0.08,0.38)[1]{}[(0,1)[0.38]{}]{} (1.23,55.25)(0.1,0.37)[1]{}[(0,1)[0.37]{}]{} (1.33,55.63)(0.13,0.37)[1]{}[(0,1)[0.37]{}]{} (1.46,55.99)(0.15,0.36)[1]{}[(0,1)[0.36]{}]{} (1.61,56.36)(0.18,0.36)[1]{}[(0,1)[0.36]{}]{} (1.78,56.71)(0.1,0.17)[2]{}[(0,1)[0.17]{}]{} (1.98,57.06)(0.11,0.17)[2]{}[(0,1)[0.17]{}]{} (2.21,57.4)(0.12,0.17)[2]{}[(0,1)[0.17]{}]{} (2.45,57.73)(0.13,0.16)[2]{}[(0,1)[0.16]{}]{} (2.72,58.06)(0.14,0.16)[2]{}[(0,1)[0.16]{}]{} (3.01,58.37)(0.1,0.1)[3]{}[(1,0)[0.1]{}]{} (3.32,58.67)(0.17,0.14)[2]{}[(1,0)[0.17]{}]{} (3.65,58.96)(0.17,0.14)[2]{}[(1,0)[0.17]{}]{} (4,59.23)(0.18,0.13)[2]{}[(1,0)[0.18]{}]{} (4.37,59.5)(0.19,0.12)[2]{}[(1,0)[0.19]{}]{} (4.75,59.74)(0.2,0.12)[2]{}[(1,0)[0.2]{}]{} (5.15,59.98)(0.21,0.11)[2]{}[(1,0)[0.21]{}]{} (5.57,60.19)(0.21,0.1)[2]{}[(1,0)[0.21]{}]{} (6,60.39)(0.22,0.09)[2]{}[(1,0)[0.22]{}]{} (6.44,60.58)(0.45,0.17)[1]{}[(1,0)[0.45]{}]{} (6.89,60.75)(0.47,0.15)[1]{}[(1,0)[0.47]{}]{} (7.36,60.89)(0.48,0.13)[1]{}[(1,0)[0.48]{}]{} (7.84,61.03)(0.48,0.11)[1]{}[(1,0)[0.48]{}]{} (8.32,61.14)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (8.81,61.24)(0.5,0.08)[1]{}[(1,0)[0.5]{}]{} (9.31,61.31)(0.5,0.06)[1]{}[(1,0)[0.5]{}]{} (9.81,61.37)(0.5,0.04)[1]{}[(1,0)[0.5]{}]{} (10.31,61.41)(0.51,0.02)[1]{}[(1,0)[0.51]{}]{} (10.82,61.43)[(1,0)[0.51]{}]{} (11.32,61.43)(0.51,-0.02)[1]{}[(1,0)[0.51]{}]{} (11.83,61.41)(0.5,-0.04)[1]{}[(1,0)[0.5]{}]{} (12.33,61.37)(0.5,-0.06)[1]{}[(1,0)[0.5]{}]{} (12.83,61.31)(0.5,-0.08)[1]{}[(1,0)[0.5]{}]{} (13.33,61.24)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (13.82,61.14)(0.48,-0.11)[1]{}[(1,0)[0.48]{}]{} (14.3,61.03)(0.48,-0.13)[1]{}[(1,0)[0.48]{}]{} (14.78,60.89)(0.47,-0.15)[1]{}[(1,0)[0.47]{}]{} (15.25,60.75)(0.45,-0.17)[1]{}[(1,0)[0.45]{}]{} (15.7,60.58)(0.22,-0.09)[2]{}[(1,0)[0.22]{}]{} (16.14,60.39)(0.21,-0.1)[2]{}[(1,0)[0.21]{}]{} (16.57,60.19)(0.21,-0.11)[2]{}[(1,0)[0.21]{}]{} (16.99,59.98)(0.2,-0.12)[2]{}[(1,0)[0.2]{}]{} (17.39,59.74)(0.19,-0.12)[2]{}[(1,0)[0.19]{}]{} (17.77,59.5)(0.18,-0.13)[2]{}[(1,0)[0.18]{}]{} (18.14,59.23)(0.17,-0.14)[2]{}[(1,0)[0.17]{}]{} (18.49,58.96)(0.17,-0.14)[2]{}[(1,0)[0.17]{}]{} (18.82,58.67)(0.1,-0.1)[3]{}[(1,0)[0.1]{}]{} (19.13,58.37)(0.14,-0.16)[2]{}[(0,-1)[0.16]{}]{} (19.42,58.06)(0.13,-0.16)[2]{}[(0,-1)[0.16]{}]{} (19.69,57.73)(0.12,-0.17)[2]{}[(0,-1)[0.17]{}]{} (19.93,57.4)(0.11,-0.17)[2]{}[(0,-1)[0.17]{}]{} (20.16,57.06)(0.1,-0.17)[2]{}[(0,-1)[0.17]{}]{} (20.36,56.71)(0.18,-0.36)[1]{}[(0,-1)[0.36]{}]{} (20.53,56.36)(0.15,-0.36)[1]{}[(0,-1)[0.36]{}]{} (20.68,55.99)(0.13,-0.37)[1]{}[(0,-1)[0.37]{}]{} (20.81,55.63)(0.1,-0.37)[1]{}[(0,-1)[0.37]{}]{} (20.91,55.25)(0.08,-0.38)[1]{}[(0,-1)[0.38]{}]{} (20.99,54.88)(0.05,-0.38)[1]{}[(0,-1)[0.38]{}]{} (21.04,54.5)(0.03,-0.38)[1]{}[(0,-1)[0.38]{}]{} (30.36,53.39)[(1,0)[25]{}]{} (78.75,42.68)[(0,1)[20]{}]{} (71.43,48.21)[(1,0)[30]{}]{} (93.39,42.86)[(0,1)[20]{}]{} (26.25,25.89)(0.12,-0.12)[208]{}[(1,0)[0.12]{}]{} (36.25,0.89)(0.12,0.12)[208]{}[(1,0)[0.12]{}]{} (25.36,54.29)[(0,0)\[cc\][$\cong$]{}]{} (60.54,52.5)(62.56,54.64)(63.71,52.52) (63.71,52.52)(64.86,50.41)(66.61,52.14) (66.61,52.14)(68.6,54.35)(70.31,54.15) (70.31,54.15)(72.03,53.95)(72.14,53.75) (15.48,40.03)(18.42,39.91)(17.58,37.66) (17.58,37.66)(16.73,35.41)(19.19,35.22) (19.19,35.22)(22.16,35.17)(23.14,33.74) (23.14,33.74)(24.12,32.31)(24.04,32.09) (64.64,31.07)(0.12,0.13)[58]{}[(0,1)[0.13]{}]{} (64.64,31.07)[(-1,-1)[0.12]{}]{} (70.54,40)(0.12,-0.12)[19]{}[(0,-1)[0.12]{}]{} (71.61,55)(0.12,-0.16)[10]{}[(0,-1)[0.16]{}]{} (71.43,52.5)(0.12,0.12)[9]{}[(0,1)[0.12]{}]{} (24.46,31.96)(0.12,0.54)[3]{}[(0,1)[0.54]{}]{} (22.5,31.96)[(1,0)[1.96]{}]{} (5.18,59.64)[(0,0)\[cc\][\*]{}]{} (10.18,61.43)[(0,0)\[cc\][\*]{}]{} (3.21,49.29)[(0,0)\[cc\][\*]{}]{} (17.86,48.21)[(0,0)\[cc\][\*]{}]{} (7.68,46.96)[(0,0)\[cc\][\*]{}]{} (20.36,51.25)[(0,0)\[cc\][\*]{}]{} (33.39,53.21)[(0,0)\[cc\][\*]{}]{} (37.14,53.21)[(0,0)\[cc\][\*]{}]{} (41.79,53.21)[(0,0)\[cc\][\*]{}]{} (45.36,53.04)[(0,0)\[cc\][\*]{}]{} (50.36,53.04)[(0,0)\[cc\][\*]{}]{} (54.11,53.21)[(0,0)\[cc\][\*]{}]{} (78.39,57.14)[(0,0)\[cc\][\*]{}]{} (2.68,63.75)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (10.18,66.61)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (1.43,46.07)[(0,0)\[cc\][$\frac{1}{8}$]{}]{} (6.43,43.75)[(0,0)\[cc\][$\frac{2}{8}$]{}]{} (18.57,44.83)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (22.5,48.39)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (32.5,58.39)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (37.5,58.21)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (41.96,58.39)[(0,0)\[cc\][$\frac{1}{8}$]{}]{} (45.71,58.21)[(0,0)\[cc\][$\frac{2}{8}$]{}]{} (54.29,58.03)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (50.36,58.22)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (74.82,58.57)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (74.82,52.5)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (48.39,13.21)[(0,0)\[cc\][\*]{}]{} (78.57,52.68)[(0,0)\[cc\][\*]{}]{} (82.5,48.04)[(0,0)\[cc\][\*]{}]{} (88.21,47.86)[(0,0)\[cc\][\*]{}]{} (36.43,15.36)[(0,0)\[cc\][\*]{}]{} (93.21,57.5)[(0,0)\[cc\][\*]{}]{} (93.21,52.32)[(0,0)\[cc\][\*]{}]{} (32.5,19.64)[(0,0)\[cc\][\*]{}]{} (43.75,8.57)[(0,0)\[cc\][\*]{}]{} (53.04,17.68)[(0,0)\[cc\][\*]{}]{} (82.68,44.46)[(0,0)\[cc\][$\frac{1}{8}$]{}]{} (88.57,44.28)[(0,0)\[cc\][$\frac{2}{8}$]{}]{} (29.82,17.14)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (97.14,52.14)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (97.14,58.57)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (55,15.36)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (51.07,11.25)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (34.11,13.75)[(0,0)\[cc\][$\frac{5}{8}$]{}]{} (43.75,14.64)[(0,0)\[cc\][$\frac{3}{8}$]{}]{} (7.16,58.24)(9.4,57.58)(10.25,58.43) (10.25,58.43)(11.1,59.29)(11.06,59.47) (7.25,58.27)(0.26,0.12)[6]{}[(1,0)[0.26]{}]{} (7.36,58.03)(0.12,-0.25)[5]{}[(0,-1)[0.25]{}]{} (18.9,52.86)(16.6,52.5)(16.2,51.36) (16.2,51.36)(15.8,50.22)(15.92,50.07) (16.02,50)(0.27,0.13)[5]{}[(1,0)[0.27]{}]{} (14.86,50.77)(0.16,-0.12)[7]{}[(1,0)[0.16]{}]{} (32.85,50.66)(34.82,49.42)(35.87,50.01) (35.87,50.01)(36.92,50.61)(36.93,50.79) (35.48,51.31)(0.47,-0.13)[3]{}[(1,0)[0.47]{}]{} (37.02,50.95)(0.12,-0.69)[2]{}[(0,-1)[0.69]{}]{} (50.61,50.39)(52.59,49.15)(53.64,49.75) (53.64,49.75)(54.69,50.34)(54.7,50.53) (50.57,50.6)(1.48,0.11)[1]{}[(1,0)[1.48]{}]{} (50.5,50.24)(0.12,-0.69)[2]{}[(0,-1)[0.69]{}]{} Kontsevich-Boggi compactifications ---------------------------------- In [@Kont], Kontsevich described certain topological modifications of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ which for $g=0$ were given an algebraic and moduli-theoretic description by Boggi in [@Bogg]. In the latter paper a rational pointed curve $(C,p_1,\ldots,p_n)$ is called *$I$-stable*, where $I\subset \{1,\ldots,n\}$ is nonempty, if i) $C$ has at worst ordinary multiple points and the points $p_i$ with $i\in I$ avoid the singularities, ii) each component of $C$ has at least 3 special points (marked points or singularities) and at least 1 point indexed by $I$, and iii) any collection of points is allowed to collide as long as none of the points are indexed by $I$. Boggi proves there is a normal projective variety, call it $\overline{\mathcal{M}}_{0,I}$, representing over $\text{Spec }\mathbb{Z}$ the moduli-functor of $I$-stable rational curves. Moreover, he shows there is a sequence of birational morphisms $$\overline{\mathcal{M}}_{0,n} \rightarrow \overline{\mathcal{M}}_{0,[n]} \rightarrow \overline{\mathcal{M}}_{0,[n-1]} \rightarrow \cdots \rightarrow \overline{\mathcal{M}}_{0,[2]} \rightarrow \overline{\mathcal{M}}_{0,[1]} = \mathbb{P}^{n-3}$$ where $[j] := \{1,\ldots,j\}$. Note that each stable curve parametrized by $\overline{\mathcal{M}}_{0,[j]}$ has at most $j$ components since each component must have a point indexed by $[j]$ and such points cannot lie on two components simultaneously. The smallest Boggi space, $\overline{\mathcal{M}}_{0,[1]}$, can be constructed as an $\mathtt{SL}_2$ GIT quotient. Indeed, this space parametrizes configurations of $n$ points on $\mathbb{P}^1$ such that $p_1$ is distinct from the others and the remaining $n-1$ points are supported in at least 2 points to ensure there are at least 3 special points. Therefore, by taking a linearization such as $L=(1-\epsilon,\frac{1+\epsilon}{n-1},\ldots,\frac{1+\epsilon}{n-1})$ for small $\epsilon >0$ we get $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_L\mathtt{SL}_2 \cong \overline{\mathcal{M}}_{0,[1]}$. The next Boggi compactification, $\overline{\mathcal{M}}_{0,[2]}$, arises as an $\mathtt{SL}_3$ GIT quotient, namely $$\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3 \cong \overline{\mathcal{M}}_{0,[2]}, \text{ where } (\gamma,\vec{c})=(3\epsilon,1-\epsilon,1-\epsilon,\frac{1-\epsilon}{n-2},\ldots,\frac{1-\epsilon}{n-2}).$$ Indeed, a nonsingular $[2]$-stable curve is a configuration of $n$ points on $\mathbb{P}^1$ such that $p_1$ and $p_2$ do not overlap with any points but the remaining $n-2$ can overlap—and a singular $[2]$-stable curve is a nodal curve with two components each marked by one of the heavy points $p_i$ ($i=1,2$) distinct from the rest as well as at least one other point away from the node. This agrees with the GIT stability conditions prescribed by such a linearization. Contraction of $F$-curves ------------------------- The boundary of $\overline{\mathcal{M}}_{0,n}$ is stratified by topological type, and irreducible components of the 1-strata are called $F$-curves (or sometimes *vital* curves). The curves parametrized by an $F$-curve have one *spine* which is a component with 4 special points, and up to 4 *legs* which are chains of $\mathbb{P}^1$s attached to the spine at these points. The cross-ratio of the 4 points on the spine traces a $\mathbb{P}^1$ in $\overline{\mathcal{M}}_{0,n}$. $F$-curves are determined up to linear equivalence by a partition of $\{1,\ldots,n\}$ into 4 nonempty subsets—corresponding to the indices of the points on each leg. The *$F$-conjecture* is the statement that $F$-curves span the entire cone of curves, i.e. that every curve is linearly equivalent to a non-negative sum of $F$-curves. This is known for $n\le 7$ in general and $n \le 24$ when considering an $S_n$-invariant analogue (see, e.g., [@GKM]). Given an $F$-curve corresponding to $\{1,\ldots,n\}=N_1\cup N_2 \cup N_3 \cup N_4$ and a weight vector $\vec{c} = (c_1,\ldots,c_n)$ we write $x_k = \sum_{i\in N_k}c_i$, so that $x_k$ measures the total weight on the $k^{\text{th}}$ leg. We can assume without loss of generality $x_1 \le x_2 \le x_3 \le x_4$. It is easy to see that the morphism $\overline{\mathcal{M}}_{0,n} \rightarrow \overline{\mathcal{M}}_{0,\vec{c}}$ contracts precisely those $F$-curves which satisfy $x_1 + x_2 + x_3 \le 1$ since this condition is necessary and sufficient for the spine to get contracted. It is also not hard to see what remaining $F$-curves get contracted by the morphism $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3$, where $\gamma = 3 - \sum c_i$. \[F-curves\] The $F$-curves contracted by the morphism $\overline{\mathcal{M}}_{0,n} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma,\vec{c})}\mathtt{SL}_3$ for $0 < \gamma < 1$ are precisely those satisfying $x_1 + x_2 + x_3 \le 1$ or $x_3 > 1$. If $x_3 > 1$ then the two heaviest legs become the components of a nodal conic and the spine is contracted to the node. Conversely, if $x_3 \le 1$ but $x_1 + x_2 + x_3 > 1$ then the three lightest legs are each contracted to a point in Hassett’s space and then the spine remains un-contracted as one of possibly two components in the resulting conic. Inverse limits -------------- In [@Kapr] it is shown, using the general machinery of Chow quotients for torus actions and the Gelfand-MacPherson isomorphism $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}\mathtt{SL}_2 \cong \mathtt{Gr}(2,n){/\hspace{-1.2mm}/}(\mathbb{C}^*)^n$, that $\overline{\mathcal{M}}_{0,n}$ is the inverse limit of all GIT quotients $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}\mathtt{SL}_2$. Assuming the $F$-conjecture (see the previous subsection) we can recognize any of the Hassett spaces appearing in Theorem \[thm:contraction\] as (the normalization of) the inverse limit of an appropriate family of $\mathtt{SL}_3$ GIT quotients. Fix $\vec{c}\in (0,1]^n\cap\mathbb{Q}^n$ with $2 < c < 3$ and write $\vec{c'} \le \vec{c}$ if $c'_i \le c_i$ for $i=1,\ldots,n$. Assuming the $F$-conjecture, the normalization of the inverse limit of all quotients $\mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma',\vec{c'})}\mathtt{SL}_3$ such that $\vec{c'} \le \vec{c}$ and $2 < \gamma' = 3 - c'$ is $\overline{\mathcal{M}}_{0,\vec{c}}$. We do not need the full $F$-conjecture to prove this, only a weaker form which asserts that $F$-curves generate the cone of curves for Hassett spaces $\overline{\mathcal{M}}_{0,\vec{c}}$ satisfying $2 < c < 3$. In ([@Simp], Theorem 3.3.2) this “weak” $F$-conjecture is proven for symmetric weights in this range, so in such cases the above proposition is unconditional. By Theorem \[thm:contraction\] there is a birational morphism $\overline{\mathcal{M}}_{0,\vec{c'}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma',\vec{c'})}\mathtt{SL}_3$ for any $\vec{c'} \le \vec{c}$, so by composing with the reduction morphisms $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \overline{\mathcal{M}}_{0,\vec{c'}}$ we get a birational morphism from $\overline{\mathcal{M}}_{0,\vec{c'}}$ to any of the GIT quotients described in the proposition—and hence to their inverse limit $\mathcal{L}$. This inverse limit may not be normal, but since all the Hassett spaces are normal this induced morphism factors through the normalization $\widetilde{\mathcal{L}} \rightarrow \mathcal{L}$. We claim the morphism $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \widetilde{\mathcal{L}}$ is an isomorphism. If it were not then it would have to contract a curve, and because we are assuming the $F$-conjecture this means it would have to contract an $F$-curve. This $F$-curve corresponds to a partition $N_1\cup\cdots\cup N_4 = \{1,\ldots,n\}$ and as above we write $x_k = \sum_{i\in N_k} c_i$ and assume without loss of generality $x_1 \le \cdots \le x_4$. For this $F$-curve to be contracted by $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \widetilde{\mathcal{L}}$ it must not be contracted by $\overline{\mathcal{M}}_{0,n} \rightarrow \overline{\mathcal{M}}_{0,\vec{c}}$, so $x_1 + x_2 + x_3 > 1$. Now the morphisms $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma',\vec{c'})}\mathtt{SL}_3$ all factor through $\widetilde{\mathcal{L}}$ so they must *all* contract this same $F$-curve. By Lemma \[F-curves\] this means in particular, taking $\vec{c'}=\vec{c}$, that $x_3 > 1$ (and hence $x_2 < 1$ since $\sum_{k=1}^4 x_k < 3$). Choose $\vec{c'}\le\vec{c}$ such that $x'_3 = 1$, where $x'_k := \sum_{i\in N_k}c'_i$. We still have $x'_1 \le x'_2 \le x'_3 \le x'_4$ since $x'_2 = x_2 < 1 = x'_3 < x_3 \le x_4 = x'_4$, and we certainly have $x'_1+x'_2+x'_3 > 1$ since $x'_3 =1$ and $x'_1 > 0$, and of course $\sum_{k=1}^4 x'_k > 2$, so by Lemma \[F-curves\] this $F$-curve is *not* contracted by $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_{(\gamma',\vec{c'})}\mathtt{SL}_3$, a contradiction. Therefore $\overline{\mathcal{M}}_{0,\vec{c}} \rightarrow \widetilde{\mathcal{L}}$ must in fact be an isomorphism. GIT Cones --------- If $X$ is a projective variety acted upon by a reductive group $G$, and $L$ is any $G$-linearized ample line bundle on $X$, then the GIT quotient may be defined as $X{/\hspace{-1.2mm}/}_L G := \mathtt{Proj}(\oplus_{m\ge 0}\Gamma(X,L^{\otimes m})^G)$. This perspective makes it clear that the quotient comes equipped with a distinguished polarization, i.e. an ample line bundle $L'$. If $\phi : Y \rightarrow X{/\hspace{-1.2mm}/}_L G$ is any morphism then the pull-backed line bundle $\phi^*L'$ on $Y$ is nef. Understanding the nef cone of a variety is a crucial part of understanding its birational geometry, so Alexeev and Swinarski ([@AS]) used this GIT setup to study the nef cone of $\overline{\mathcal{M}}_{0,n}$. Specifically, each of the quotients $(\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_L\mathtt{SL}_2$ comes with a distinguished polarization $L'$ determined by $L\in\Delta(2,n)$ so pulling back along the Kapranov morphisms $\overline{\mathcal{M}}_{0,n} \rightarrow (\mathbb{P}^1)^n{/\hspace{-1.2mm}/}_L\mathtt{SL}_2$ produces a collection of nef line bundles on $\overline{\mathcal{M}}_{0,n}$ which Alexeev and Swinarski term the *GIT cone*. For $n=5$ this GIT cone coincides with the full nef cone; for $n \ge 6$ it is strictly smaller, though still a useful object. Theorem \[thm:contraction\] allows one to setup an analogous framework for $\mathtt{SL}_3$ quotients. That is, as one varies the linearization $L\in\Delta(3,n+1)$ one can pull back the distinguished polarization on $\mathtt{Con}(n){/\hspace{-1.2mm}/}_L\mathtt{SL}_3$ along the morphism $\overline{\mathcal{M}}_{0,n} \rightarrow \mathtt{Con}(n){/\hspace{-1.2mm}/}_L\mathtt{SL}_3$, thereby producing a collection of nef line bundles on $\overline{\mathcal{M}}_{0,n}$ which we term the *second order GIT cone*. The set of line bundles coming from linearizations on the face $\gamma = 1$ of $\Delta(3,n+1)$ coincide with the first order GIT cone defined in [@AS], so by allowing arbitrary $\gamma$ we except to find a strictly larger GIT cone. This has not been explored much yet; we leave it as an open area of study. [MMMMM]{} Alexeev, V. and D. 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Kontsevich, M. “Intersection theory on the moduli space of curves and the matrix Airy function.” *Comm. Math. Phys.* **147** (1992), 1–23. Knudsen, F. “Projectivity of the moduli space of stable curves, II.” *Math. Scand.* **52** (1983), 1225-1265. Mumford, D., Fogarty, J. and F.C. Kirwan. *Geometric Invariant Theory.* Third Edition. Springer, 1994. Newstead, P.E. *Introduction to Moduli Problems and Orbit Spaces.* Tata Institute, 1978. Simpson, M. “On Log Canonical Models of the Moduli Space of Stable Pointed Genus Zero Curves.” Ph.D dissertation, Rice University, 2008. Thaddeus, M. “Geometric invariant theory and flips.” *J. Amer. Math. Soc.* **9** no. 3 (1996), 691-723.
--- abstract: 'We analytically study the superradiant instability of charged massless scalar field in the background of D-dimensional Reissner-Nordström (RN) black hole caused by mirror-like boundary condition. By using the asymptotic matching method to solve the Klein-Gordon equation that govern the dynamics of scalar field, we have derived the expressions of complex parts of boxed quasinormal frequencies, and shown they are positive in the regime of superradiance. This indicates the charged scalar field is unstable in D-dimensional Reissner-Nordström (RN) black hole surrounded by mirror. However, the numerical work to calculate the boxed quasinormal frequencies in this system is still required in the future.' author: - Ran Li - Junkun Zhao - Yanming Zhang title: 'Superradiant instability of D-dimensional Reissner-Nordström black hole mirror system' --- [^1] Press and Teukolsky [@press] proposed to built *black hole bomb* in terms of the classical superradiance phenomenon [@zeldovich; @bardeen; @misner; @starobinsky] in 70s last century. When an impinging bosonic wave with frequency satisfying superradiant condition is scattered by event horizon of rotating black hole, the scattered wave will be amplified. If one places a reflecting mirror outside of the black hole, the amplified wave will be reflected into the black hole once again. So it is obvious that the bosonic wave will be bounced back and forth between event horizon and mirror. Meanwhile, the energy of the wave can become sufficiently large in this black hole mirror system until the reflecting mirror is destroyed. Recent studies of black hole bomb mechanism was initiated by Cardoso et. al. in [@cardoso2004bomb]. See also the Refs.[@Rosa; @Lee; @leejhep; @jgrosa; @hod2013prd; @hodbhb] for the recent studies on this topic. Black hole bomb mechanism can also be realized in the following two cases. In the first case, the mass term of bosonic field plays the role of reflecting mirror, for example, the systems investigated in [@kerrunstable; @detweiler; @strafuss; @dolan; @Hod; @hodPLB2012; @konoplyaPLB; @DiasPRD2006; @zhangw; @dolanprd2013]. The second case is to study bosonic field perturbation in black hole background with Dirichlet boundary condition at the asymptotic infinity. These background spacetimes include the rotating black holes in AdS spacetime [@cardoso2004ads; @cardoso2006prd; @KKZ; @aliev; @uchikata; @rlplb; @zhang], in Gödel universe [@knopolya; @rlepjc], and in the linear dilaton background [@clement; @randilaton]. In all these cases, superradiance will trigger the instabilities of rotating black holes plus bosonic field perturbation. For a charged scalar wave in the background of spherical symmetric charged black hole, the wave scattered by event horizon will also undergo superradiant process if the frequency of this impinging wave satisfying superradiant condition [@bekenstein]. However, it was proved by Hod in [@hodrnplb2012; @hodrnplb2013] that, for four dimensional RN black holes, the existence of a trapping potential well outside black hole and superradiant amplification of trapped modes cannot be satisfied simultaneously. This means that the four dimensional RN black holes are stable under the perturbations of massive charged scalar fields. Soon after, Degollado et. al. [@Degolladoprd; @Degollado] found that the same system can be made unstable by adding a mirror-like boundary condition like the case of the Kerr black hole. In [@liprd], we also shown that the mass term of scalar field in charged stringy black hole is never able to generate a potential well outside the event horizon to trap superradiant modes. This is to say that the charged stringy black hole is stable against massive charged scalar perturbation. In [@ranliprogress; @ranliplb], we have further studied the superradiant instability of massless scalar field in the background of charged stringy black hole due to a mirror-like boundary condition. However, it is still interesting to study the high dimensional cases. More recently, Wang and Herdeiro [@mengjieprd] studied the superradiant instability of a charged scalar field in D-dimensional RN Anti-de Sitter black hole. According to the mechanism of black hole bomb, this work can also be generalized to study the superradiant instabilities of D-dimensional RN black hole caused by the mass of scalar field and mirror’s boundary condition. In [@zhangw], the authors have discussed superradiant instability of extremal brane-world RN black hole against the charged massive scalar perturbation. Scalar field quasinormal modes in the dyadosphere spacetime of charged black hole are studied by using the third-order WKB approximation in [@chunyan]. Quasinormal Modes of Phantom Scalar Perturbation in RN Black Hole was studied in [@Pan]. Hawking Radiation of charged GHS black hole was also recently studied in [@GHSH]. In this paper, we will further study the system composed by D-dimensional RN black hole, charged massless scalar field, and reflecting mirror outside of the black hole. By using the asymptotic matching method to solve the Klein-Gordon equation that govern the dynamics of scalar field, we will derive the expressions of the complex parts of boxed quasinormal frequencies, and show the corresponding instability caused by the reflecting mirror. The D-dimensional RN black hole [@MP] is described by the line element $$\begin{aligned} &&ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_n^2\;,\\ &&f(r)=1-\frac{\mu}{r^{n-1}}+\frac{q^2}{r^{2(n-1)}}\;, \end{aligned}$$ where the parameter $n=D-2$ is introduced to parameterize the dimension of spacetime for later convenience. $d\Omega_n^2$ denotes the line element of the n-dimensional unit sphere, $$\begin{aligned} d\Omega_n^2=d\theta_{n-1}^2+\sin^2\theta_{n-1}(d\theta_{n-2}^2+\sin^2\theta_{n-2} (\cdots&&\nonumber\\ +\sin^2\theta_2(d\theta_1^2+\sin^2\theta_1 d\phi^2)\cdots))\;,&& \end{aligned}$$ where the ranges of azimuthal coordinates are given by $\phi\in[0,2\pi]$ and $\theta_i\in[0,\pi] (i=1,\cdots,n-1)$. The Maxwell gauge potential is given by $$\begin{aligned} A=-\sqrt{\frac{n}{2(n-1)}}\frac{q}{r^{n-1}}dt\;. \end{aligned}$$ The event horizon $r_+$ is determined as the largest root of the equation $f(r)=0$. The parameters $\mu$ and $q$ are related to the mass $M$ and the charge $Q$ of the black hole towards the following relations $$\begin{aligned} &&\mu=\frac{4M}{nV_n}\;,\nonumber\\ &&q^2=\frac{2Q^2}{n(n-1)}\;, \end{aligned}$$ where $V_n=2\pi^{(n+1)/2}/\Gamma((n+1)/2)$ is the volume of the $n-$dimensional sphere. We consider the charged massless scalar field perturbation in the background of D-dimensional RN black hole. The dynamics is then governed by the corresponding charged Klein-Gordon equation $$\begin{aligned} \frac{1}{\sqrt{-g}}D_\mu\left[\sqrt{-g}g^{\mu\nu}D_\nu\right]\Psi=0\;, \end{aligned}$$ where $D_\mu=\partial_\mu-ieA_\mu$ with $e$ being the charge of the scalar field. The equation can be decomposed by the separation of variables for $\Psi$ as follows $$\begin{aligned} \Psi(t,r,\theta_i,\phi)=e^{-i\omega t} R(r) Y_{l,n}(\theta_1,\cdots,\theta_{n-1},\phi)\;, \end{aligned}$$ where $Y_{l,n}$ denotes the hyperspherical harmonics on the $n$-sphere with $l$ being the angular momentum quantum number. By substituting the above decomposition of the scalar field into the charged Klein-Gordon equation, we can obtain the radial wave equation as following $$\begin{aligned} \frac{\Delta}{r^{n-2}}\frac{d}{dr}\left(\frac{\Delta}{r^{n-2}}\frac{dR}{dr}\right) +r^{2n}(\omega+eA_t)^2 R-\lambda \Delta R=0\;, \end{aligned}$$ where the introduced new function $\Delta$ is explicitly given by $$\begin{aligned} \Delta=r^{2(n-2)}-\mu r^{n-1} +q^2\;, \end{aligned}$$ and the separation constant $\lambda$ is given by $$\begin{aligned} \lambda=l(l+n-1)\;. \end{aligned}$$ Near the event horizon $r=r_+$, the scalar field with the ingoing boundary condition behaves as $$\begin{aligned} \Psi\sim e^{-i\omega t} e^{-i(\omega-e\Phi_H)r_+^n r_*}\;, \end{aligned}$$ where the tortoise coordinate $r_*$ is defined explicitly by $$\begin{aligned} \frac{dr_*}{dr}=\frac{r^{n-2}}{\Delta}\;, \end{aligned}$$ and $\Phi_H=-A_t(r_+)=\sqrt{\frac{n}{2(n-1)}}\frac{q}{r_+^{n-1}}$ is the the electric potential at the event horizon. One can notice that if $$\begin{aligned} \omega<e\Phi_H\;, \end{aligned}$$ the wave appears to be outgoing for an inertial observer at spatial infinity. This gives us the superradiant condition of scalar field in D-dimensional RN black hole. Since we are working with the positive frequency $\omega$, the superraidance will occur only for the positive charge $e$ of the scalar field. In fact, we will consider the black hole in a box, i.e. the D-dimensional RN black hole surrounded by a reflecting mirror. More precisely, we will impose the mirror’s boundary condition that the scalar field vanishes at the mirror’s location $r_m$, i.e. $$\begin{aligned} \Psi(r=r_m)=0\;. \end{aligned}$$ The complex frequencies satisfying the purely ingoing boundary at the black hole horizon and the mirror’s boundary condition are called the boxed quasinormal (BQN) frequencies [@cardoso2004bomb]. The scalar modes in the superradiant regime will bounce back and forth between event horizon and mirror. Meanwhile, the energy extracted from black hole by means of superradiance process will grow exponentially. This will cause the instability of the black hole mirror system. In the following, we will present an analytical calculations of BQN frequencies in a certain limit and show the instability in the superradiant regime caused by the mirror’s boundary condition. Now we will employ the matched asymptotic expansion method [@page; @unruh] to compute the unstable modes of a charged scalar field in this black hole mirror system. We shall assume that the Compton wavelength of the scalar particles is muck larger than the typical size of the black hole, i.e. $1/\omega\gg \mu$. With this assumption, we can divide the space outside the event horizon into two regions, namely, a near-region, $r-r_+\ll 1/\omega$, and a far-region, $r-r_+\gg \mu$. The approximated solution can be obtained by matching the near-region solution and the far-region solution in the overlapping region $\mu\ll r-r_+\ll 1/\omega$. At last, we can impose the mirror’s boundary condition to obtain the analytical expression of the unstable modes in this system. Firstly, let us focus on the near-region in the vicinity of the event horizon, $\omega(r-r_+)\ll 1$. It is convenient to introduce a new variable as $x=r^{n-1}$, the radial wave equation can be rewritten as $$\begin{aligned} (n-1)^2 \Delta\frac{d}{dx}\left(\Delta\frac{dR}{dx}\right) +x^{2n/(n-1)}(\omega+eA_t)^2 R\nonumber\\ -l(l+n-1) \Delta R=0\;. \end{aligned}$$ By taking the near-region limit, the radial wave function can be reduced to the form $$\begin{aligned} (n-1)^2 \Delta\frac{d}{dx}\left(\Delta\frac{dR}{dx}\right) +r_+^{2n}(\omega+eA_t(r_+))^2 R\nonumber\\ -l(l+n-1)\Delta R=0\;. \end{aligned}$$ Introducing another new coordinate variable $$\begin{aligned} z=\frac{x-x_+}{x-x_-}\;, \end{aligned}$$ with $x_\pm=r_\pm^{n-1}$, the near-region radial wave equation can be rewritten in the form of $$\begin{aligned} z\partial_z(z\partial_z R(z)) +\left[\varpi^2-\alpha\left(\alpha+1\right)\frac{z}{(1-z)^2}\right]R(z)=0\;, \end{aligned}$$ where the parameter $\alpha$ and $\varphi$ are given by $$\begin{aligned} &&\alpha=\frac{l}{n-1}\;,\\ &&\varpi=\frac{r_+^n\left(\omega-e\Phi_H\right)}{(n-1)(r_+^{n-1}-r_-^{n-1})} \;. \end{aligned}$$ Through defining $$\begin{aligned} R=z^{i\varpi}(1-z)^{\alpha+1}F(z)\;, \end{aligned}$$ the near-region radial wave equation becomes the standard hypergeometric equation $$\begin{aligned} z(1-z)\partial_z^2F(z)+[c-(1+a+b)]\partial_zF(z)-abF(z)=0\;, \end{aligned}$$ with the parameters $$\begin{aligned} a&=&\alpha+1+2i\varpi\;,\nonumber\\ b&=&\alpha+1\;,\nonumber\\ c&=&\alpha+2i\varpi\;. \end{aligned}$$ In the neighborhood of $z=0$, the general solution of the radial wave equation is then given in terms of the hypergeometric function [@handbook] $$\begin{aligned} R&=&Az^{-i\varpi}(1-z)^{\alpha+1}F(\alpha+1,\alpha+1-2i\varpi,1-2i\varpi,z) \nonumber\\ &&+Bz^{i\varpi}(1-z)^{\alpha+1}F(\alpha+1,\alpha+1+2i\varpi,1+2i\varpi,z)\;. \end{aligned}$$ It is obvious that the first term represents the ingoing wave at the horizon, while the second term represents the outgoing wave at the horizon. Because we are considering the classical superradiance process, the ingoing boundary condition at the horizon should be employed. Then we have to set $B=0$. The physical solution of the radial wave equation corresponding to the ingoing wave at the horizon is then given by $$\begin{aligned} R=Az^{-i\varpi}(1-z)^{\alpha+1}F(\alpha+1,\alpha+1-2i\varpi,1-2i\varpi,z)\;. \end{aligned}$$ In the far-region, $r-r_+\gg M$, the effects induced by the black hole can be neglected. One can take the limit $\mu\rightarrow0$ and $q\rightarrow0$ to simplify the radial wave equation in the far-region. In this case, we have $\Delta\simeq r^{2(n-1)}$. The radial wave equation reduces to the wave equation of a massless scalar field in the D-dimensional flat background $$\begin{aligned} \partial_r^2(r^{n/2}R(r))+\left[\omega^2-\beta(\beta+1)\frac{1}{r^2}\right](r^{n/2}R(r))=0\;, \end{aligned}$$ with the parameter $\beta=l+\frac{n}{2}-1$. This equation can be solved by the Bessel function, and the general solution is given by [@handbook] $$\begin{aligned} R=r^{1/2-n/2}\left[C_1 J_{\beta+1/2}(\omega r)+C_2 Y_{\beta+1/2}(\omega r)\right]\;, \end{aligned}$$ where $J_{\beta+1/2}$ and $Y_{\beta+1/2}$ are the first and the second kind of the Bessel functions respectively. In order to match the far-region solution with the near-region solution, we should study the large $r$ behavior of the near-region solution and the small $r$ behavior of the far-region solution. For the sake of this purpose, we can us the $z\rightarrow 1-z$ transformation law for the hypergeometric function [@handbook] $$\begin{aligned} F(a,b,c;z)&=&\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} F(a,b,a+b-c+1;1-z)\nonumber\\ &&+(1-z)^{c-a-b} \frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\nonumber\\ &&\times F(c-a,c-b,c-a-b+1;1-z)\;\;. \end{aligned}$$ By employing this formula and using the properties of hypergeometric function $F(a,b,c,0)=1$, we can get the large $r$ behavior of the near-region solution as $$\begin{aligned} \label{nearsolutionlarge} R&\sim& A\Gamma(1-2i\varpi)\left[\frac{(r_+^{n-1}-r_-^{n-1})^{-\alpha}\Gamma(2\alpha+1)} {\Gamma(\alpha+1)\Gamma(\alpha+1-2i\varpi)}r^{l}\right. \nonumber\\&&\left. +\frac{(r_+^{n-1}-r_-^{n-1})^{\alpha+1}\Gamma(-2\alpha-1)} {\Gamma(-\alpha)\Gamma(-\alpha-2i\varpi)}r^{-l-n+1}\right]\;. \end{aligned}$$ On the other hand, using the asymptotic formulas of the Bessel functions [@handbook], $J_\nu(z)=(z/2)^\nu/\Gamma(\nu+1)\;(z\ll 1)$ and $Y_\nu(z)= -\frac{1}{\pi}\Gamma(\nu)(z/2)^{-\nu}$, one can get the small $r$ behavior of the far-region solution as $$\begin{aligned} R&\sim& C_1 \frac{\left(\frac{\omega}{2}\right)^{l+\frac{n}{2}-\frac{1}{2}}} {\Gamma\left(l+\frac{n}{2}+\frac{1}{2}\right)}r^l\nonumber\\ &&-\frac{C_2}{\pi}\left(\frac{\omega}{2}\right)^{-l-\frac{n}{2}+\frac{1}{2}} \Gamma\left(l+\frac{n}{2}-\frac{1}{2}\right)r^{-l-n+1}\;. \end{aligned}$$ By comparing the large $r$ behavior of the near-region solution with the small $r$ behavior of the far-region solution, one can conclude that there exists the overlapping region $\mu\ll r-r_+\ll 1/\omega$ where the two solutions should match. This matching yields the relation $$\begin{aligned} \frac{C_2}{C_1}&=&-\frac{\pi(r_+^{n-1}-r_-^{n-1})^{2\alpha+1}}{\left(l+\frac{n}{2}-\frac{1}{2}\right) \Gamma^2\left(l+\frac{n}{2}-\frac{1}{2}\right)}\nonumber\\ &&\times \frac{\Gamma(\alpha+1)}{\Gamma(2\alpha+1)} \frac{\Gamma(-2\alpha-1)}{\Gamma(-\alpha)}\nonumber\\ &&\times\frac{\Gamma(\alpha+1-2i\varpi)}{\Gamma(-\alpha-2i\varpi)} \left(\frac{\omega}{2}\right)^{2l+n-1}\;, \end{aligned}$$ where we have used the property of Gamma function $\Gamma(x+1)=x\Gamma(x)$. Now we want to impose the mirror’s boundary condition to study the unstable BQN modes. We assume that the mirror is placed near the infinity at a radius $r=r_m$. The far-region radial solution should vanish when reflected by the mirror. This yields the extra condition between the amplitudes $C_1$ and $C_2$ of the far-region radial solution, which is given by $$\begin{aligned} \frac{C_2}{C_1}=-\frac{J_{\beta+1/2}(\omega r_m)}{Y_{\beta+1/2}(\omega r_m)}\;. \end{aligned}$$ This mirror condition together with the matching condition give us the following equation which determines the BQN frequencies of the scalar field in this black hole mirror system $$\begin{aligned} \frac{J_{\beta+1/2}(\omega r_m)}{Y_{\beta+1/2}(\omega r_m)}&=& \frac{\pi(r_+^{n-1}-r_-^{n-1})^{2\alpha+1}}{\left(l+\frac{n}{2}-\frac{1}{2}\right) \Gamma^2\left(l+\frac{n}{2}-\frac{1}{2}\right)}\nonumber\\ &&\times \frac{\Gamma(\alpha+1)}{\Gamma(2\alpha+1)} \frac{\Gamma(-2\alpha-1)}{\Gamma(-\alpha)}\nonumber\\ &&\times\frac{\Gamma(\alpha+1-2i\varpi)}{\Gamma(-\alpha-2i\varpi)} \left(\frac{\omega}{2}\right)^{2l+n-1}\;, \end{aligned}$$ For the very small $\omega$, the analytical solution of BQN frequencies can be found from the above relation. In this case, the right hand side of the above relation is very small and then can be set to be zero. This means that $$\begin{aligned} J_{\beta+1/2}(\omega r_m)=0\;. \end{aligned}$$ The real zeros of the Bessel functions were well studied. We shall label the $N$-th positive zero of the Bessel function $J_{\beta+1/2}$ as $j_{\beta+1/2,N}$. Then we can get $$\begin{aligned} \omega r_m=j_{\beta+1/2,N}\;. \end{aligned}$$ In the first approximation for BQN frequencies, the solution of the Eq.(33) has a small imaginary part, which can be written as $$\begin{aligned} \omega_{BQN}=\frac{j_{\beta+1/2,N}}{r_m}+i\delta\;, \end{aligned}$$ where the introduced imaginary part $\delta$ is small enough comparing the real part of BQN frequency. It can be considered as a correction to Eq.(35). For the small $\delta$, we can use the Taylor expansion of Bessel function $J_{\beta+1/2}(\omega r_m)=i\delta r_m J'_{\beta+1/2}(j_{\beta+1/2,N})$ to proceed. Then the equation (33) can be reduced to $$\begin{aligned} i\delta r_m \frac{J'_{\beta+1/2}(j_{\beta+1/2,N})}{Y_{\beta+1/2}(j_{\beta+1/2,N})} &=& \frac{\pi(r_+^{n-1}-r_-^{n-1})^{2\alpha+1}}{\left(l+\frac{n}{2}-\frac{1}{2}\right) \Gamma^2\left(l+\frac{n}{2}-\frac{1}{2}\right)}\nonumber\\ &&\times \frac{\Gamma(\alpha+1)}{\Gamma(2\alpha+1)} \frac{\Gamma(-2\alpha-1)}{\Gamma(-\alpha)}\nonumber\\ &&\times\frac{\Gamma(\alpha+1-2i\varpi)}{\Gamma(-\alpha-2i\varpi)} \left(\frac{\omega}{2}\right)^{2l+n-1}\;, \end{aligned}$$ In order to go further, we should simplify the Gamma functions on the right hand side of the above equation. One can note that the simplification of the Gamma function depends on the parameter $\alpha$. Now, we have to investigate the following two cases separately. Case A: $\alpha$ is an integer {#case-a-alpha-is-an-integer .unnumbered} ============================== Generally, we consider the case that the dimension $D$ of spacetime is greater than $4$. So we must have that $\alpha$ is an non-negative integer. In this case, by using the property of Gamma function $\Gamma(x+1)=x\Gamma(x)$, we have the following formulas $$\begin{aligned} &&\frac{\Gamma(-2\alpha-1)}{\Gamma(-\alpha)}=\frac{(-1)^{\alpha+1}\alpha!}{(2\alpha+1)!}\;,\nonumber\\ &&\frac{\Gamma(\alpha+1-2i\varpi)}{\Gamma(-\alpha-2i\varpi)} =(-1)^{\alpha+1}2i\varpi \prod_{k=1}^{\alpha}(k^2+4\varpi^2)\;. \end{aligned}$$ From these we can easily obtain the small imaginary part of the BQN frequencies as $$\begin{aligned} \delta=\gamma\frac{Y_{\beta+1/2}(j_{\beta+1/2,N})}{J'_{\beta+1/2}(j_{\beta+1/2,N})} \frac{j_{\beta+1/2,N}/r_m-e\Phi_H}{r_m^{2l+n}}\;, \end{aligned}$$ where $$\begin{aligned} \gamma=\frac{\pi(r_+^{n-1}-r_-^{n-1})^{2\alpha+1}}{\left(2l+n-1\right) \Gamma^2\left(l+\frac{n}{2}-\frac{1}{2}\right)} \frac{(\alpha!)^2}{(2\alpha)!(2\alpha+1)!}\nonumber\\ \times \prod_{k=1}^{\alpha}(k^2+4\varpi^2)\left(\frac{j_{\beta+1/2,N}}{2}\right)^{2l+n-1}\;. \end{aligned}$$ One should note that the prefactor of $Y_{\beta+1/2}(j_{\beta+1/2,N})/J'_{\beta+1/2}(j_{\beta+1/2,N})$ is always negative for our relevant range of $\omega$. Then, it is easy to see that, in the superradiance regime, $\textrm{Re}[\omega_{BQN}]-e\Phi_H<0$, the imaginary part of the complex BQN frequency $\delta>0$. The scalar field has the time dependence $e^{-i\omega t}=e^{-i \textrm{Re}[\omega] t}e^{\delta t}$, which implies the exponential amplification of superradiance modes. This indicates that, in this case, the scalar field with the frequency in the superradiant regime will undergo an instability in D-dimensional RN black hole surrounded by mirror. Case B: $\alpha$ is not an integer {#case-b-alpha-is-not-an-integer .unnumbered} ================================== In this case, by using the property of Gamma function $\Gamma(z)\Gamma(1-z)=\pi/\sin\pi z$, we have the following relations $$\begin{aligned} &&\frac{\Gamma(-2\alpha-1)}{\Gamma(-\alpha)}=-\frac{1}{2\cos\frac{\pi l}{n-1}} \frac{\Gamma\left(1+\frac{l}{n-1} \right)}{\Gamma\left(2+\frac{2l}{n-1} \right)}\;,\\ &&\frac{\Gamma(\alpha+1-2i\varpi)}{\Gamma(-\alpha-2i\varpi)} =-\left[\frac{1}{\pi}\sin\frac{\pi l}{n-1}+ 2i\varpi\cos\frac{\pi l}{n-1}\right]\nonumber\\ &&~~~~~~~~~~~~~~~~~~~~~~~~\times\Gamma^2\left(1+\frac{l}{n-1} \right)\;, \end{aligned}$$ where we have assumed that the real part of BQN frequency is very near the superradiant bound and expanded the term $\Gamma(-\alpha-2i\varpi)$ for very small $\varpi$. It should be noted that $\cos\frac{\pi l}{n-1}=0$ when $\frac{l}{n-1}=p+\frac{1}{2}$ with $p$ being a non-negative integer. In this special case, according to the discussions in Ref.[@mengjieprd] the asymptotic matching method fails. The numerical method must be employed to study this case. Here, we will just consider the case that $\frac{l}{n-1}\neq p+\frac{1}{2}$. Then, the real part of $\delta$ is explicitly given by $$\begin{aligned} \textrm{Re}[\delta]=\gamma'\frac{Y_{\beta+1/2}(j_{\beta+1/2,N})}{J'_{\beta+1/2}(j_{\beta+1/2,N})} \frac{j_{\beta+1/2,N}/r_m-e\Phi_H}{r_m^{2l+n}}\;, \end{aligned}$$ where $$\begin{aligned} \gamma'&=&\frac{2(n-1)\pi}{(2l+n-1)^2} \frac{\Gamma^4\left(1+\frac{l}{n-1} \right)} {\Gamma^2\left(1+\frac{2l}{n-1} \right)\Gamma^2\left(l+\frac{n}{2}-\frac{1}{2}\right)} \nonumber\\ &&\times (r_+^{n-1}-r_-^{n-1})^{1+\frac{2l}{n-1}} \left(\frac{j_{\beta+1/2,N}}{2}\right)^{2l+n-1}\;. \end{aligned}$$ So, it is easy to see that, in the superradiance regime, $\textrm{Re}[\omega_{BQN}]-e\Phi_H<0$, the imaginary part of the complex BQN frequency $\delta>0$. This means that, in this case, the BQN frequencies in the superradiant regime is unstable for the charged scalar field in the D-dimensional RN black hole with a mirror placed outside of the hole. In summary, we have studied the instability of the massless charged scalar field in the D-dimensional RN black hole mirror system. By imposing the mirror boundary condition, we have analytically calculated the expression of BQN frequencies. In the first case, where $\alpha=\frac{l}{n-1}$ is an integer, the result is very similar to the case of Kerr black hole [@cardoso2004bomb], which indicates the feature of black hole bomb. Especially, for the case of $n=2$, the result (39) can be reduced to the case of four dimensional RN black hole bomb [@Degolladoprd]. In the second case, where $\alpha$ is not an integer and $\alpha\neq p+\frac{1}{2}$ with $p$ being an integer, the result also indicates an instability of the black hole mirror system. 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--- abstract: 'We show a systematic construction for implementing general measurements on a single qubit, including both strong (or projection) and weak measurements. We mainly focus on linear optical qubits. The present approach is composed of simple and feasible elements, i.e., beam splitters, wave plates, and polarizing beam splitters. We show how the parameters characterizing the measurement operators are controlled by the linear optical elements. We also propose a method for the implementation of general measurements in solid-state qubits.' author: - Yukihiro Ota - Sahel Ashhab - Franco Nori title: 'Implementing general measurements on linear optical and solid-state qubits' --- Introduction ============ The description of measurement in quantum mechanics[@vonNeumann:1955; @Davies:1976; @Kraus:1983; @Peres:1993] is formulated as an operation to extract information from a quantum system. This operation causes a disturbance to the system. This means that the measurement accuracy (or the amount of information extracted from the system) is closely related to the back-action[@Braginsky;Khalili:1992; @Banaszek:2001; @Ozawa:2004; @Clerk;Schoelkopf:2010; @Wiseman;Milburn:2010]. This is a curious feature of measurement processes in quantum mechanics. Such a character is actively used in measurement-based methods for quantum engineering (e.g., Refs.[@Nakazato;Yuasa:2003; @Koashi;Ueda:1999; @Korotkov;Keane:2010; @Ashhab;Nori:2010; @Paraoanu:2011:EPL; @Paraoanu:2011:FP; @Ota;Nori]). A general (or weak) measurement is associated with a positive-operator valued measure (POVM)[@Davies:1976]. A POVM on a measured quantum system can be expressed as a projection-valued measure in an extended system including the target system and an ancillary (or probe) system, as seen in, e.g., Ref.[@Peres:1993]. Therefore, we may construct arbitrary measurements from von Neumann measurements on the extended system. However, this statement does not give a specific and simple recipe to design measurement operators. Hence, a systematic approach to realize this general idea in specific physical systems is highly desirable. A large number of experimental studies on general measurements have been performed. Huttner [*et al.*]{}[@Huttner;Gisin:1996] discriminated with two nonorthogonal states. Gillett [*et al.*]{}[@Gillett;White:2010] demonstrated quantum feedback control for a photonic polarization qubit. A polarizing beam splitter with a tunable reflection coefficient was used on the above-mentioned two experiments. Kwiat [*et al.*]{}[@Kwiat;Gisin:2001] implemented an entanglement concentration protocol with a partial-collapse measurement in a photonic qubit. Kim [*et al.*]{}[@Kim;Kim:2009] demonstrated a reversal operation of a weak measurement for a photonic qubit. The key idea in the two experiments is to use Brewster-angle glass plates. Katz [*et al.*]{}[@Katz;Korotkov:2008] performed a conditional recovery of a quantum state with a partial-collapse measurement in Josephson phase qubits. Iinuma [*et al.*]{}[@Iinuma;Hofmann:2011] studied the observation of a weak value. Kocsis [*et al.*]{}[@Kocsis;Steinberg:2011] examined a photon’s “trajectories” through a double-slit interferometer with weak measurements of the photon momentum. In this paper, we show a method to implement general measurements on a single qubit, including both von Neumann and weak measurements. We mainly focus on a linear optical qubit. Depending on the path degree of freedom in an interferometer, a polarization state is transformed by a non-projective positive operator. We develop the idea proposed in Ref.[@Iinuma;Hofmann:2011] and give a systematic prescription for designing various measurement operators. The present approach is composed of simple and basic linear optical elements, i.e., beam splitters, wave plates, and polarizing beam splitters. We show how the parameters characterizing the measurement operators (e.g., the measurement strength) can be tuned. Furthermore, we show a method for implementing general measurements on a solid-state qubit. The paper is organized as follows. In Sec.\[sec:general\], we show the basic idea for constructing general measurements on linear optical qubits. The notation used in this paper is explained, there. In Sec.\[sec:los\], we propose systematic approaches for implementing measurement operators. These method are simple and could be implementable in experiments. In addition, in Sec.\[sec:sss\], we study how a general measurement can be implemented in solid-state qubits. Section \[sec:summary\] is devoted to a summary of the ideas and the results. Setting {#sec:general} ======= Let us consider a linear optical system with $N$ input and $N$ output modes. The following arguments are also applicable to other physical systems such as neutron interferometers[@Sponar;Hasagawa:2010]. The annihilation (creation) operator in the $n$th input mode is defined as ${\hat{a}}_{{\rm in},n}$ (${\hat{a}}_{{\rm in},n}^{\dagger}$) with $n=1,\,2,\ldots,N$. The vacuum ${| 0\rangle}$ is defined as ${\hat{a}}_{{\rm in},n}{| 0\rangle}=0$. The bosonic canonical commutation relations are $ [{\hat{a}}_{{\rm in},m},{\hat{a}}_{{\rm in},n}^{\dagger}]=\delta_{nm} $ and $ [{\hat{a}}_{{\rm in},m},{\hat{a}}_{{\rm in},n}] = [{\hat{a}}_{{\rm in},m}^{\dagger},{\hat{a}}_{{\rm in},n}^{\dagger}] =0 $. Similarly, we define the annihilation (creation) operator in the $n$th output mode as ${\hat{a}}_{{\rm out},n}$ (${\hat{a}}_{{\rm out},n}^{\dagger}$). A single photon with arbitrary polarization enters the system from the first input mode. The polarization is described by a vector ${| \psi\rangle}\in\mathbb{C}^{2}$, with ${\langle \psi|\psi \rangle}=1$. We assume that the input state is a pure state $ {| \Psi\rangle} = {| \psi\rangle} \otimes {\hat{a}}_{{\rm in},1}^{\dagger}{| 0\rangle} $. In this setting, the photon polarization is measured, using the path degree of freedom in the interferometer as an ancillary qubit. We note that the arguments in this section can be straightforwardly extended to the case of a mixed initial state of polarization. We calculate the photon state at each output mode. The field operators at the output modes are related to the input modes via a linear canonical transformation[@vanHemmen:1980; @Umezawa:1993; @Loudon:2000; @Kok;Milbrun:2007] because the system is composed of linear optical elements (e.g., beam splitters). The polarization of the photon at the $n$th output mode is described by $ {| \phi_{n}\rangle} = {\langle 0|}{\hat{a}}_{{\rm out},n}{| \Psi\rangle} $. Thus, we find that in terms of the output mode operators the photon state is written as $ {| \Psi^{({\rm out})}\rangle} = \sum_{n=1}^{N} {| \phi_{n}\rangle} \otimes {\hat{a}}_{{\rm out},n}^{\dagger}{| 0\rangle} $. Since we assume that there is no photon loss, we have $ {\langle \Psi^{\rm out} |\Psi^{\rm out} \rangle}={\langle \Psi |\Psi \rangle} $. The linearity of the system indicates that there exist linear operators ${\hat{X}}_{n}$ on $\mathbb{C}^{2}$ such that $ {| \phi_{n}\rangle} = {\hat{X}}_{n}{| \psi\rangle} $. The normalization condition ${\langle \Psi|\Psi \rangle}=1$ implies that $ \sum_{n=1}^{N}{\hat{X}}_{n}^{\dagger}{\hat{X}}_{n} = {\hat{I}}_{2} $, where the identity operator on $\mathbb{C}^{2}$ is ${\hat{I}}_{2}$. Therefore, we find that the initial state ${| \psi\rangle}$ is transformed by a linear operator which is related with an element of a POVM. It is convenient for designing various kinds of POVM to add another linear optical system to some of the output modes in the interferometer before the photon detection. For example, let us set wave plates in each output mode. The wave plates in the $n$th mode describe a unitary gate $V_{n}$ given as the unitary part of ${\hat{X}}_{n}$. Using the right-polar decomposition[@Horn;Johnson:1985], we find that $ {\hat{X}}_{n} = {\hat{V}}_{n}^{\dagger}{\hat{M}}_{n} $, where $ {\hat{M}}_{n}=({\hat{X}}_{n}^{\dagger}{\hat{X}}_{n})^{1/2} $ is a positive operator. The role of the wave plates is to remove the effect of the unitary evolution ${\hat{V}}_{n}^{\dagger}$ from ${\hat{X}}_{n}$. Using the application of such “path-dependent” unitary operators, we find that $ {| \Psi^{(\rm out)}\rangle} \to \sum_{n=1}^{N} {| \psi_{n}\rangle} \otimes {\hat{a}}_{{\rm out},n}^{\dagger}{| 0\rangle} $, where $ {| \psi_{n}\rangle} = {\hat{V}}_{n}{| \phi_{n}\rangle} = {\hat{M}}_{n}{| \psi\rangle} $. The measurement operators ${\hat{M}}_{n}$ satisfy $ \sum_{n=1}^{N}{\hat{M}}_{n}^{\dagger}{\hat{M}}_{n}={\hat{I}}_{2} $. Thus, we have the measurements corresponding to the POVM $ \{{\hat{E}}_{n}\}_{n=1}^{N} $ with ${\hat{E}}_{n}={\hat{M}}_{n}^{\dagger}{\hat{M}}_{n}$. The positive operator ${\hat{M}}_{n}$ is the essential part (i.e., the back-action associated with the measurement) of non-unitarity of ${\hat{X}}_{n}$ and called minimally-disturbing measurement, following Ref.[@Wiseman;Milburn:2010]. Throughout this paper, we will focus on such minimally-disturbing measurements. Let us characterize the measurement operators ${\hat{M}}_{n}$. We can expand ${\hat{M}}_{n}$ as $ {\hat{M}}_{n} = \sum_{i=1}^{2} m_{n,i}{| m_{n,i}\rangle\langle m_{n,i}|} $, with the eigenvalues $m_{n,i}\,(\ge 0)$ and the associated eigenvectors ${| m_{n,i}\rangle}$. The eigenvalues are related to the measurement strength, while the eigenvectors are regarded as the measurement direction. Let us consider the case when $m_{n,1}=1$ and $m_{n,2}=0$, for example. We find that ${\hat{M}}_{n}$ is a projection operator (i.e., a sharp measurement) in the direction of $\vec{m}_{n,1}$, where $\vec{m}_{n,1}$ is the Bloch vector corresponding to ${| m_{n,1}\rangle\langle m_{n,1}|}$. The indistinguishability between the elements of a POVM is characterized by the Hilbert-Schmidt inner product $ {\mbox{Tr}\,{\hat{E}}}_{n}^{\dagger}{\hat{E}}_{m} $ for $n\neq m$. The elements of a projection-valued measure are distinguishable (i.e., the inner products are zero), for example. Now, we pose the question: How are these measurement features controlled by typical linear optical devices? We will answer this question in the next section. \[0.43\][![Schematic diagram of a proposal for implementing symmetric, arbitrary-strength two-outcome measurements on a single-photon’s polarization. This setup is essentially equal to the one in Ref.[@Iinuma;Hofmann:2011]. Three kinds of field operators (i.e., ${\hat{a}}_{{\rm in},n}$, ${\hat{a}}_{{\rm int},n}$, and ${\hat{a}}_{{\rm out},n}$) are needed for representing the path degrees of freedom. A single photon with arbitrary polarization (described by ${| \psi\rangle}$) enters a polarizing beam splitter (PBS). Next, the polarization is transformed by a wave plate (WP) in each arm. Then, the two arms are recombined in a beam splitter (BS). After performing unitary gates ${\hat{V}}_{1}$ and ${\hat{V}}_{2}$ for compensation purposes, the resultant polarization states are expressed in terms of the measurement operators ${\hat{M}}_{1}$ and ${\hat{M}}_{2}$ given in Eqs.(\[eq:def\_elm1\]) and (\[eq:def\_elm2\]).[]{data-label="fig:diagram1"}](fig1 "fig:")]{} \[0.45\][![(Color online) Measurement direction of the measurement operators (\[eq:def\_elm1\]) and (\[eq:def\_elm2\]) for the measurement strength $\varepsilon=0.9$ (red solid), $0.6$ (blue dashed), and $0.3$ (magenta dotted). The horizontal axis represents the reflection coefficient $r$ of the beam splitter, while the vertical axis is the polar angle $\theta$ of the measurement direction. All curves merge at $(r,\,\theta)=(1/\sqrt{2},\,\pi/2)$. This case corresponds to the use of a $50:50$ beam splitter in Fig.\[fig:diagram1\].[]{data-label="fig:rc_theta"}](fig2 "fig:")]{} \[0.46\][![Schematic diagrams of the construction for general measurements with (a) two outcomes and (b) $N$ outcomes ($N=4$). In (a), the interferometer for a symmetric arbitrary-strength two-outcome measurement followed by a beam splitter leads to general two-outcome measurements. The final box leading to ${\hat{M}}_{2}^{\prime}{| \psi\rangle}$ after the last beam splitter represents a unitary gate for compensation purposes. Depending on system’s parameters, this unitary gate becomes either a phase-shift gate or the identity operator (i.e., does nothing). In (b), the repeated application of general two-outcome measurements (GTOM) allows the construction of a POVM with multiple outcomes.[]{data-label="fig:diagram2"}](fig3 "fig:")]{} Design of general measurements on linear optical qubits {#sec:los} ======================================================= Symmetric arbitrary-strength two-outcome measurements {#subsec:sas} ----------------------------------------------------- Let us first define a special class of general measurements, which is the target operation in this subsection. We consider a symmetric two-outcome POVM in $\mathbb{C}^{2}$ composed of convex combinations of two orthogonal projection operators $$\begin{aligned} && {\hat{M}}_{1} = \sqrt{\frac{1 + \varepsilon}{2}} {| m_{+}\rangle\langle m_{+}|} + \sqrt{\frac{1 - \varepsilon}{2}} {| m_{-}\rangle\langle m_{-}|} , \label{eq:def_elm1}\\ && {\hat{M}}_{2} = \sqrt{\frac{1 - \varepsilon}{2}} {| m_{+}\rangle\langle m_{+}|} + \sqrt{\frac{1 + \varepsilon}{2}} {| m_{-}\rangle\langle m_{-}|} , \label{eq:def_elm2} \end{aligned}$$ with $0 \le \varepsilon \le 1$ and $ {\langle m_{+}|m_{-} \rangle}=0 $. Adjusting a real parameter $\varepsilon$ allows arbitrary measurement strength. We find that if the coefficient for ${| m_{+}\rangle\langle m_{+}|}$ increases, then the one for ${| m_{-}\rangle\langle m_{-}|}$ decreases, and vice versa. Thus, these measurement operators are symmetric or balanced with respect to the parameter $\varepsilon$. We call the measurements described by these linear operators symmetric arbitrary-strength two-outcome measurments (SASTOM) on $\mathbb{C}^{2}$. The disturbance induced by a SASTOM is considered to be minimum, as shown in Ref.[@Banaszek:2001]. Another important property of a SASTOM is $ {\mbox{Tr}\,{\hat{M}}}_{1}^{\dagger}{\hat{M}}_{1} = {\mbox{Tr}\,{\hat{M}}}_{2}^{\dagger}{\hat{M}}_{2} = 1 $. An application of a SASTOM to quantum protocols is shown in, e.g., Ref.[@Ota;Nori]. Now, we propose a systematic way to construct this measurement. Let us consider an interferometer with two input and two output modes, as shown in Fig.\[fig:diagram1\]. A single photon enters a polarizing beam splitter[@Meschede:2007] from the first input mode. Subsequently, its polarization at each arm is transformed by wave plates. Then, the two modes are recombined in a beam splitter. This system is essentially the same as in Ref.[@Iinuma;Hofmann:2011]. We reformulate the construction manner of a SASTOM in Ref.[@Iinuma;Hofmann:2011] and extend it to make more general types of measurements. Unlike the proposal in Ref.[@Iinuma;Hofmann:2011], our proposals use beam splitters with tunable reflection coefficients. Let us apply the basic idea in Sec.\[sec:general\] to this interferometer. We need three kinds of field operators for describing the path degrees of freedom, i.e., input, intermediate, and output field operators, as shown in Fig.\[fig:diagram1\]. We have an initial pure state $${| \Psi^{({\rm in})}\rangle} = {| \psi\rangle} \otimes {\hat{a}}_{{\rm in},1}^{\dagger}{| 0\rangle}, \quad {| \psi\rangle} =c_{\rm H}{| {\rm H}\rangle} + c_{\rm V}{| {\rm V}\rangle},$$ with the horizontal polarization state ${| \rm H\rangle}$ and the vertical polarization state ${| \rm V\rangle}$. The field operators ${\hat{a}}_{{\rm int},n}$ and ${\hat{a}}_{{\rm int},n}^{\dagger}$ at the output modes of the polarizing beam splitter are related to ${\hat{a}}_{{\rm in},n}$ and ${\hat{a}}_{{\rm in},n}^{\dagger}$ via $ {\hat{a}}_{{\rm int,H},1} = {\hat{a}}_{{\rm in,H},1} $, $ {\hat{a}}_{{\rm int,H},2} = {\hat{a}}_{{\rm in,H},2} $, $ {\hat{a}}_{{\rm int,V},1} = {\hat{a}}_{{\rm in,V},2} $, and $ {\hat{a}}_{{\rm int,V},2} = {\hat{a}}_{{\rm in,V},1} $[@Kok;Milbrun:2007], where $ {\hat{a}}_{{\rm in,H},1} = {| {\rm H}\rangle\langle {\rm H}|}\otimes {\hat{a}}_{{\rm in},1} $, etc. After the single photon passes through the polarizing beam splitter, we find that the photon state is written as $ {| \Psi^{({\rm int})}\rangle} = {| u_{1}\rangle}\otimes {\hat{a}}_{{\rm int,}1}^{\dagger}{| 0\rangle} + {| u_{2}\rangle}\otimes {\hat{a}}_{{\rm int,}2}^{\dagger}{| 0\rangle} $ with $ {| u_{1}\rangle} = c_{\rm H}{| {\rm H}\rangle} $ and $ {| u_{2}\rangle} = c_{\rm V}{| {\rm V}\rangle} $. We remark that the polarizing beam splitter produces entanglement between the polarization and the path degrees of freedom in the interferometer. Next, we perform unitary operations for the polarization on each of these intermediate modes with the wave plates. The use of half- and quarter-wave plates allows the construction of arbitrary elements of $\mbox{SU}(2)$ [@Simon;Mukunda:1990; @Bhadari;Dasgupta:1990]. Hence, using the path-dependent wave plates, we find that $${| \Psi^{(\rm int)}\rangle} \to \sum_{n=1}^{2}{\hat{U}}_{n}{| u_{n}\rangle} \otimes {\hat{a}}_{{\rm int,}n}^{\dagger}{| 0\rangle},$$ with ${\hat{U}}_{n}\in \mbox{SU}(2)$. The canonical transformation associated with the beam splitter[@Loudon:2000; @Kok;Milbrun:2007] is $ {\hat{a}}_{{\rm out},1} = r {\hat{a}}_{{\rm int,}1} + t {\hat{a}}_{{\rm int,}2} $ and $ {\hat{a}}_{{\rm out},2} = t {\hat{a}}_{{\rm int,}1} - r {\hat{a}}_{{\rm int,}2} $, where $r^{2}+t^{2}=1$ and $0\le r,t \le 1$. After the single photon passes through the beam splitter, we find that the photon state is expressed by $${| \Psi^{(\rm out)}\rangle} = \sum_{n=1}^{2} {| \phi_{n}\rangle} \otimes {\hat{a}}_{{\rm out},n}^{\dagger}{| 0\rangle},$$ where $ {| \phi_{1}\rangle} = r {\hat{U}}_{1}{| u_{1}\rangle} + t {\hat{U}}_{2}{| u_{2}\rangle} $ and $ {| \phi_{2}\rangle} = t {\hat{U}}_{1}{| u_{1}\rangle} - r {\hat{U}}_{2}{| u_{2}\rangle} $. We find that the linear operators satisfying $ {| \phi_{n}\rangle} = {\hat{X}}_{n}{| \psi\rangle} $ are $ {\hat{X}}_{1} = r{\hat{U}}_{1}{| {\rm H}\rangle\langle {\rm H}|} + t {\hat{U}}_{2}{| {\rm V}\rangle\langle {\rm V}|} $ and $ {\hat{X}}_{2} = t {\hat{U}}_{1}{| {\rm H}\rangle\langle {\rm H}|} - r {\hat{U}}_{2}{| {\rm V}\rangle\langle {\rm V}|} $. The positive operators associated with ${\hat{X}}_{1}$ and ${\hat{X}}_{2}$ are given, respectively, by Eqs.(\[eq:def\_elm1\]) and (\[eq:def\_elm2\]) with $$\varepsilon = \sqrt{ 1 - 4r^{2}t^{2} (1-|w|^{2}) }, \quad w = {\langle {\rm H} | {\hat{U}}_{1}^{\dagger}{\hat{U}}_{2} | {\rm V} \rangle}. \label{eq:def_eps}$$ The definition of $w$ implies that the choice of the unitary gates ${\hat{U}}_{1}$ and ${\hat{U}}_{2}$ is not unique for constructing the measurement operators ${\hat{M}}_{n}$. For example, one can set ${\hat{U}}_{1}$ as the identity operator and still obtain any desired value of $w$ by adjusting ${\hat{U}}_{2}$. Iinuma [*et al.*]{}[@Iinuma;Hofmann:2011] set ${\hat{U}}_{1}={\hat{U}}_{2}^{\dagger}$, which results in a real value for $w$ and thus imposes a constraint on the measurement operators that can be constructed. The range of $|w|$ is $ 0 \le |w| \le 1 $ since $w$ is an off-diagonal element of a unitary operator. The basis vectors ${| m_{+}\rangle}$ and ${| m_{-}\rangle}$ for $w \neq 0$ are $$\begin{aligned} && {| m_{+}\rangle} = \cos\frac{\theta}{2}{| {\rm H}\rangle} + e^{-i\phi/2} \sin\frac{\theta}{2}{| {\rm V}\rangle}, \label{eq:def_mplus}\\ && {| m_{-}\rangle} = -e^{i\phi/2}\sin\frac{\theta}{2}{| {\rm H}\rangle} +\cos\frac{\theta}{2} {| {\rm V}\rangle}, \label{eq:def_mminus}\end{aligned}$$ where $ \tan(\theta/2) = (t^{2} - r^{2} + \varepsilon)/2rt |w| $ and $ e^{i\phi/2}=w/|w| $. When $w=0$ and $r^{2} \ge t^{2}$ ($r^{2} < t^{2}$), we have $ {| m_{+}\rangle} = {| {\rm H}\rangle} $ and $ {| m_{-}\rangle} = {| {\rm V}\rangle} $ ($ {| m_{+}\rangle} = {| {\rm V}\rangle} $ and $ {| m_{-}\rangle} = -{| {\rm H}\rangle} $). The fact that $ [{\hat{X}}_{1}^{\dagger}{\hat{X}}_{1}, {\hat{X}}_{2}^{\dagger}{\hat{X}}_{2}]=0 $ indicates that ${\hat{M}}_{1}$ and ${\hat{M}}_{2}$ are simultaneously diagonalizable. The expression for ${\hat{V}}_{n}$ such that ${\hat{X}}_{n}={\hat{V}}_{n}^{\dagger}{\hat{M}}_{n}$ is calculated straightforwardly. After the applications of ${\hat{V}}_{n}$, we find that $${| \Psi^{({\rm out})}\rangle} \to \sum_{n=1}^{2} {\hat{M}}_{n}{| \psi\rangle} \otimes {\hat{a}}_{{\rm out},n}^{\dagger}{| 0\rangle}.$$ The tunable parameter $\varepsilon$ is the measurement strength and completely determines the indistinguishability between the POVM elements, $ {\mbox{Tr}\,{\hat{E}}}_{1}^{\dagger}{\hat{E}}_{2} = (1-\epsilon^{2})/2 $. Thus, we have shown that the interferometer drawn in Fig.\[fig:diagram1\] is a measurement apparatus for performing a SASTOM. Let us show how the SASTOM can be adjusted via the linear optical elements. The interferometer contains three independent control parameters: the reflection coefficient $r$, the modulus of $w$, and the phase of $w$. The latter two parameters are related to the wave plates. The measurement direction is characterized by the Bloch vector $ ( \sin\theta \cos\phi, \sin\theta \sin \phi, \cos\theta ) $. For the calculation of the Bloch vector the Pauli matrices are defined as $ \hat{\sigma}_{x} = {| {\rm H}\rangle\langle {\rm V}|} + {| {\rm V}\rangle\langle {\rm H}|} $, $ \hat{\sigma}_{y} = -i{| {\rm H}\rangle\langle {\rm V}|} + i{| {\rm V}\rangle\langle {\rm H}|} $, and $ \hat{\sigma}_{z} = {| {\rm H}\rangle\langle {\rm H}|} - {| {\rm V}\rangle\langle {\rm V}|} $. Since the azimuthal angle $\phi$ is equal to the phase of $w$, this quantity is controlled at will via phase-shift gates. The polar angle $\theta$ and the measurement strength $\varepsilon$ are functions of $r$ and $|w|$. We can find that $0 \le \theta \le \pi$ and $0 \le \varepsilon \le 1$ when changing $r$ and $|w|$ independently. We evaluate $\theta$ as a function of $r$ and $\varepsilon$. Figure \[fig:rc\_theta\] shows that we can take an arbitrary measurement direction for given $\varepsilon$. The comparison to the method in Ref.[@Iinuma;Hofmann:2011] is useful for understanding our proposal for a SASTOM. Let us consider the case when the beam splitter is a $50:50$ beam splitter ($r=1/2$) and the unitary operators for the wave plates are ${\hat{U}}_{1}=\exp(-i2\eta \hat{\sigma}_{y})$ and ${\hat{U}}_{2} = \exp(i2\eta \hat{\sigma}_{y})$. We find that $\varepsilon = |w|=|\sin(4\eta)|$, $ \theta = \frac{\pi}{2} $, and $\phi=0$. The measurement direction is fixed, and it is characterized by $ {| m_{+}\rangle} = ({| {\rm H}\rangle} + {| {\rm V}\rangle})/\sqrt{2} $ and $ {| m_{-}\rangle} = (-{| {\rm H}\rangle} + {| {\rm V}\rangle})/\sqrt{2} $. In fact, all curves in Fig.\[fig:rc\_theta\] merge at $ (r,\,\theta)=(1/\sqrt{2},\,\pi/2) $. Thus, the only tunable parameter in Ref.[@Iinuma;Hofmann:2011] is $|w|$. Two additional real parameters are necessary for controlling the measurement direction of a SASTOM on a single qubit. For this purpose, our proposal uses the tunable reflection coefficient in the beam splitter and the phase of $w$. Alternatively, the measurement direction can be controlled by using additional wave plates (i.e., a unitary operator) before the polarizing beam splitter. General two-outcome measurements -------------------------------- Various quantum protocols with measurement operators not expressed by Eqs.(\[eq:def\_elm1\]) and (\[eq:def\_elm2\]) have been proposed (e.g., Refs.[@Korotkov;Keane:2010; @Ashhab;Nori:2010]). In the remaining parts of this section, we extend the approach developed in Sec.\[subsec:sas\] for implementing such general measurements. Two generalization routes may exist. One involves increasing the number of parameters characterizing the two measurement operators ${\hat{M}}_{1}$ and ${\hat{M}}_{2}$. The other is to increase the number of outcomes. First, we examine the former. The eigenvalues of the measurement operators (\[eq:def\_elm1\]) and (\[eq:def\_elm2\]) are parametrized by $\varepsilon$. The measurement direction contains the two parameters $\theta$ and $\phi$. Thus, the number of parameters in the measurement operators is equal to a density matrix on $\mathbb{C}^{2}$. This point is also confirmed by the fact that $ {\mbox{Tr}\,{\hat{M}}}_{n}^{\dagger}{\hat{M}}_{n} = 1 $. Since the positive operator ${\hat{M}}_{n}^{\dagger}{\hat{M}}_{n}$ is a density matrix on $\mathbb{C}^{2}$, its square root ${\hat{M}}_{n}$ is characterized by three real parameters. We consider the interferometer shown in Fig.\[fig:diagram2\](a). The main difference with Fig.\[fig:diagram1\] is that the present system has an additional beam splitter with reflection coefficient $r^{\prime}$ and transmission coefficient $t^{\prime}$. Accordingly, we need four kinds of field operators for describing the path degrees of freedom. The calculations before the last beam splitter are the same as in Sec.\[subsec:sas\]. Namely, the first polarizing beam splitter creates entanglement between the polarization and the path degree of freedom. The subsequent wave plates transform photon’s polarization through unitary operators depending on the path degree of freedom. The corresponding unitary operator ${\hat{U}}_{n}$ is an arbitrary element of $\mbox{SU}(2)$, as seen in Sec.\[subsec:sas\]. Then, the two paths are recombined in the intermediate beam splitter with reflection coefficient $r$. At this point, the polarization state in the $n$th mode is described by ${\hat{X}}_{n}{| \psi\rangle}$. After the photon passes through this intermediate beam splitter, the unitary operators ${\hat{V}}_{1}$ and ${\hat{V}}_{2}$ are applied to the first and the second paths, respectively, in order to remove the unitary parts of ${\hat{X}}_{1}$ and ${\hat{X}}_{2}$. These unitary operators ${\hat{V}}_{1}$ and ${\hat{V}}_{2}$ are automatically determined when calculating the right-polar decompositions of ${\hat{X}}_{1}$ and ${\hat{X}}_{2}$. Thus, the resultant state in the $n$th mode becomes ${\hat{M}}_{n}{| \psi\rangle}$, as seen in Eqs.(\[eq:def\_elm1\]) and (\[eq:def\_elm2\]). As shown in Sec.\[subsec:sas\], in this step, we have three tunable parameters, i.e., $r$, the modulus of $w$, and the phase of $w$, where $ w = {\langle {\rm H}|{\hat{U}}_{1}^{\dagger}{\hat{U}}_{2}|{\rm V} \rangle} $. Let us now consider what happens in the additional part. We write the input (output) field operators in the last beam splitter as $ {\hat{a}}_{{\rm int},n}^{\prime} $ and $ {\hat{a}}_{{\rm int},n}^{\prime\,\dagger} $ ($ {\hat{a}}_{{\rm out},n} $ and $ {\hat{a}}_{{\rm out},n}^{\dagger} $). The related linear canonical transformations is $ {\hat{a}}_{{\rm out},1} = r^{\prime}{\hat{a}}_{{\rm int},1}^{\prime} + t^{\prime}{\hat{a}}_{{\rm int},2}^{\prime} $ and $ {\hat{a}}_{{\rm out},2} = t^{\prime}{\hat{a}}_{{\rm int},1}^{\prime} - r^{\prime}{\hat{a}}_{{\rm int},2}^{\prime} $ with $(r^{\prime})^{2} + (t^{\prime})^{2}=1$ and $0\le r^{\prime},t^{\prime} \le 1$. After the photon passes through the last beam splitter, its state is written as $ {| \Psi^{({\rm out})}\rangle} = \sum_{n=1}^{2} {\hat{X}}_{n}^{\prime}{| \psi\rangle} \otimes {\hat{a}}_{{\rm out},n}^{\dagger}{| 0\rangle} $, where $ {\hat{X}}_{1}^{\prime} = r^{\prime}{\hat{M}}_{1} + t^{\prime}{\hat{M}}_{2} $ and $ {\hat{X}}_{2}^{\prime} = t^{\prime}{\hat{M}}_{1} - r^{\prime}{\hat{M}}_{2} $. In other words, we find that $$\begin{aligned} && {\hat{X}}_{1}^{\prime} = \left( r^{\prime}\sqrt{\frac{1+\varepsilon}{2}} + t^{\prime}\sqrt{\frac{1-\varepsilon}{2}} \right) {| m_{+}\rangle\langle m_{+}|} + \left( r^{\prime}\sqrt{\frac{1-\varepsilon}{2}} + t^{\prime}\sqrt{\frac{1+\varepsilon}{2}} \right) {| m_{-}\rangle\langle m_{-}|}, \\ && {\hat{X}}_{2}^{\prime} = \left( t^{\prime}\sqrt{\frac{1+\varepsilon}{2}} - r^{\prime}\sqrt{\frac{1-\varepsilon}{2}} \right) {| m_{+}\rangle\langle m_{+}|} + \left( t^{\prime}\sqrt{\frac{1-\varepsilon}{2}} - r^{\prime}\sqrt{\frac{1+\varepsilon}{2}} \right) {| m_{-}\rangle\langle m_{-}|}, \end{aligned}$$ where $\varepsilon$, ${| m_{+}\rangle}$, and ${| m_{-}\rangle}$ are given in Eqs.(\[eq:def\_eps\]), (\[eq:def\_mplus\]), and (\[eq:def\_mminus\]), respectively. The positive-operator parts of ${\hat{X}}_{1}^{\prime}$ and ${\hat{X}}_{2}^{\prime}$ are, respectively, $$\begin{aligned} && {\hat{M}}_{1}^{\prime} = \sqrt{p} {| m_{+}\rangle\langle m_{+}|} + \sqrt{q} {| m_{-}\rangle\langle m_{-}|}, \label{eq:def_gwm1}\\ && {\hat{M}}_{2}^{\prime} = \sqrt{1-p} {| m_{+}\rangle\langle m_{+}|} + \sqrt{1-q} {| m_{-}\rangle\langle m_{-}|}.\label{eq:def_gwm2}\end{aligned}$$ The measurement direction is the same as the SASTOM in the previous subsection. This means that the basis vectors ${| m_{+}\rangle}$ and ${| m_{-}\rangle}$ do not depend on the parameter $r^{\prime}$. We remark that ${\hat{M}}_{1}^{\prime}={\hat{X}}_{1}^{\prime}$ since ${\hat{X}}_{1}^{\prime}$ is a positive operator. The measurement operator ${\hat{M}}_{2}^{\prime}$ is related with the linear operator ${\hat{X}}_{2}^{\prime}$ via a unitary operator, $ {\hat{M}}_{2}^{\prime} = {\hat{S}}{\hat{X}}_{2}^{\prime} $, with ${\hat{S}}^{\dagger}{\hat{S}}={\hat{S}}{\hat{S}}^{\dagger}={\hat{I}}_{2}$. With some algebra one can show that when $$\sqrt{1-\varepsilon^{2}} \le 2r^{\prime}t^{\prime},$$ the unitary operator ${\hat{S}}$ is a phase-shift gate (i.e., ${\hat{S}}= {| m_{+}\rangle\langle m_{+}|} - {| m_{-}\rangle\langle m_{-}|}$). Otherwise, ${\hat{S}}$ is equal to the identity operator, up to an overall phase. The measurement operators ${\hat{M}}_{1}^{\prime}$ and ${\hat{M}}_{2}^{\prime}$ are characterized by two independent positive parameters $p$ and $q$ ($0 \le p,q\le 1$). In contrast to a SASTOM, the trace of ${\hat{M}}_{n}^{\prime\,\dagger}{\hat{M}}_{n}^{\prime}$ is not fixed. We find that $ {\mbox{Tr}\,{\hat{M}}}_{1}^{\prime\,\dagger}{\hat{M}}_{1}^{\prime} = 1 + \Delta $ and $ {\mbox{Tr}\,{\hat{M}}}_{2}^{\prime\,\dagger}{\hat{M}}_{2}^{\prime} = 1 - \Delta $, with $ \Delta = p+q-1 $. The indistinguishability between the elements of the corresponding POVM is $ {\mbox{Tr}\,{\hat{E}}}_{1}^{\prime\,\dagger}{\hat{E}}_{2}^{\prime} = p(1-p)+q(1-q) $, where ${\hat{E}}_{n}^{\prime}={\hat{M}}_{n}^{\prime\,\dagger}{\hat{M}}_{n}^{\prime}$. We stress that all of the features in the measurement operators are tunable via the basic linear optical elements. We also remark that the action of Eqs.(\[eq:def\_gwm1\]) and (\[eq:def\_gwm2\]) can be obtained by a polarizing beam splitter with tunable reflection coefficients, which has been used for implementing general two-outcome measurements in optical setups[@Huttner;Gisin:1996; @Gillett;White:2010]. We obtain an important special case of general two-outcome measurements when either $p=1$ or $q=1$. Let us consider the case $q=1$, for example. This situation is realized when $ \varepsilon = 1 - 2(r^{\prime})^{2} $. We now find that $ {\hat{M}}_{1}^{\prime} = \sqrt{p}{| m_{+}\rangle\langle m_{+}|} + {| m_{-}\rangle\langle m_{-}|} $ and $ {\hat{M}}_{2}^{\prime} = \sqrt{1-p}{| m_{+}\rangle\langle m_{+}|} $. This is nothing but the partial-collapse measurement of Refs.[@Koashi;Ueda:1999; @Katz;Korotkov:2008]. One possible application of this type of measurements is the proposal by Korotkov and Keane [@Korotkov;Keane:2010] for removing the effects of decoherence. General multi-outcome measurements {#subsec:gwmmo} ---------------------------------- Next, we examine another generalization of the SASTOM on $\mathbb{C}^{2}$. The repeated application of general two-outcome measurements allows the construction of a POVM with multiple outcomes. Let us consider a system composed of $(N-1)$ detectors, each of which performs a general two-outcome measurement. Figure \[fig:diagram2\](b) shows the case $N=4$. At the $\ell$th detector, the first output mode corresponds to an outcome, while the second output mode is regarded as an input mode for the subsequent device. Thus, we find that the entire system has $N$ outcomes. This “branch structure” is one possible realization of a multi-outcome POVM. Different geometric arrangements of detectors and paths from Fig.\[fig:diagram2\](b) can lead to the same result, as seen in, e.g., Ref.[@Andersson;Oi:2008]. In this branch structure, the number of measurements performed changes from run to run, with an average number $N/2$. In the binary-tree structure[@Andersson;Oi:2008], the number of measurements performed is $\log_{2}N$. Now, let us show how a general multi-outcome measurement is implemented. Let us write the measurement operators in the $\ell$th apparatus as ${\hat{M}}_{1}^{(\ell)}$ and ${\hat{M}}_{2}^{(\ell)}$. Their expressions are given in Eqs.(\[eq:def\_gwm1\]) and (\[eq:def\_gwm2\]). The measurement operator corresponding to the $\ell$th outcome is written as ${\hat{K}}_{\ell}$ ($\ell=1,\,2,\ldots,N$) and is constructed recursively using $$\begin{aligned} && {\hat{K}}_{\ell} = {\hat{W}}_{\ell}{\hat{M}}_{1}^{(\ell)}{\hat{Y}}_{\ell} \quad (2\le \ell \le N-1), \label{eq:def_Nout}\\ && {\hat{K}}_{1} = {\hat{M}}_{1}^{(1)}, \quad {\hat{K}}_{N} = {\hat{W}}_{N}{\hat{Y}}_{N}. \label{eq:def_Nout1}\end{aligned}$$ The linear operator ${\hat{Y}}_{\ell}$ is defined as $ {\hat{Y}}_{\ell} = {\hat{M}}_{2}^{(\ell-1)}{\hat{Y}}_{\ell-1} $ ($\ell \ge 2$) with $ {\hat{Y}}_{1} = {\hat{I}}_{2} $. We remark that ${\hat{Y}}_{\ell}$ ($2\le \ell \le N-1$) is associated with the input mode of the $\ell$th apparatus, while ${\hat{Y}}_{N}$ is related to the $N$th outcome. The unitary operator ${\hat{W}}_{\ell}$ is determined by imposing that ${\hat{K}}_{\ell}$ is a positive operator. The identity $ \sum_{n=1}^{2} {\hat{M}}_{n}^{(\ell)\,\dagger} {\hat{M}}_{n}^{(\ell)} = {\hat{I}}_{2} $ leads to the relation $ {\hat{K}}_{\ell}^{\dagger}{\hat{K}}_{\ell} + {\hat{Y}}_{\ell+1}^{\dagger}{\hat{Y}}_{\ell+1} = {\hat{Y}}_{\ell}^{\dagger}{\hat{Y}}_{\ell} $. This relation indicates the conservation law of probability. Using this formula, we show that $ \sum_{\ell=1}^{N}{\hat{K}}_{\ell}^{\dagger}{\hat{K}}_{\ell} = {\hat{I}}_{2} $. A number of interesting quantum protocols with multi-outcome POVM’s have been proposed in the literature. Two of the present authors (SA and FN)[@Ashhab;Nori:2010] proposed a measurement-only quantum feedback control of a single qubit, for example. Their proposal involves a four-outcome POVM satisfying $ {\mbox{Tr}\,{\hat{K}}}_{\ell}^{\dagger}{\hat{K}}_{\ell} = 1/2 $ and $ {\mbox{Tr}\,{\hat{E}}}_{\ell}^{\dagger}{\hat{E}}_{\ell^{\prime}} = f(x) $ for $\ell \neq \ell^{\prime}$, with ${\hat{E}}_{\ell}={\hat{K}}_{\ell}^{\dagger}{\hat{K}}_{\ell}$, a continuous real function $f$, and a real parameter $x$ ($0 \le x \le 1$). We remark that $f$ does not depend on the subscripts $\ell$ and $\ell^{\prime}$, but is a function of $x$. The real variable $x$ can be understood as the measurement strength. An important property of this POVM is that the indistinguishability is unbiased between arbitrary pairs of elements of the POVM. This mutually-unbiased feature can lead to an interesing quantum control. The present procedure is applicable to the construction of the corresponding measurement operators since one can freely control the number of outcomes, the trace of ${\hat{K}}_{\ell}^{\dagger}{\hat{K}}_{\ell}$, and the indistinguishability. Solid-state qubits {#sec:sss} ================== Let us consider methods for implementing general weak measurements on solid-state qubits. In this paper, we focus on superconducting qubits[@You;Nori:2005; @Nakahara;Ohmi:2008; @Clarke;Wilhelm:2008; @You;Franco:2011]. Superconducting qubits have many advantages for quantum engineering. Their current experimental status[@Buluta;Nori:2011] indicates that various important quantum operations, especially controlled operations are implemented reliably. The demonstration of controlled-NOT and controlled-phase gates was reported in various types of superconducting qubits[@Buluta;Nori:2011; @Yamamoto;Tsai:2003; @Plantenberg;Mooij:2007; @DiCarlo;Schoelkopf:2009; @Neeley;Martinis:2010; @deGroot;Mooij:2010; @Chow;Steffen:2011]. Therefore, it is important for development of measurement-based quantum protocols to explore the systematic construction methods for general measurements in such interesting physical systems. Several theoretical studies on the implementation of general measurements in superconducting qubits have been reported in, e.g., Refs.[@Paraoanu:2011:EPL; @Paraoanu:2011:FP; @Korotkov;Jordan:2006; @Ashhab;Nori:2009]. Analogies with linear optical qubits are useful for designing measurement operators in superconducting qubits. Let us consider two superconducting qubits, one of which is the measured system, while the other is an ancillary system. The former corresponds to the polarization in the previous arguments, and the latter is regarded as the path degree of freedom in the interferometer setup. In the interferometer, the polarizing beam splitter plays a central role to create entanglement between the polarization and the path. This operation can be replaced with a controlled operation (e.g., a controlled-NOT gate) between the two superconducting qubits. Now, we show a method for implementing a SASTOM on superconducting qubits. We use the following notation. The quantum states of the measured qubit is expressed in terms of the basis vectors ${| +\rangle}$ and ${| -\rangle}$ with $ {\langle +|{-} \rangle}=0 $. The ancillary qubit is described by ${| 0\rangle}$ and ${| 1\rangle}$ with ${\langle 0|1 \rangle}=0$. First, we prepare an initial state in the total system $${| \Psi^{({\rm in})}\rangle} = {| \psi\rangle}\otimes (\alpha{| 0\rangle}+\beta{| 1\rangle}), \label{eq:in_sss}$$ with $\alpha^{2}+\beta^{2}=1$, $\alpha,\beta\in\mathbb{R}$, and $0 \le \alpha,\beta \le 1$. We denote an arbitrary state in the measured qubit as ${| \psi\rangle}$. The state preparation in the ancillary system can be achieved using single-qubit operations. Next, we apply the controlled-NOT gate $ {| +\rangle\langle +|}\otimes {\hat{I}}_{2} + {| {-}\rangle\langle {-}|}\otimes \hat{\tau}_{x} $, where $ \hat{\tau}_{x} = {| 0\rangle\langle 1|} + {| 1\rangle\langle 0|} $. The resultant state is $ {| \Psi^{(\rm out)}\rangle} = {\hat{M}}_{0}{| \psi\rangle}\otimes {| 0\rangle} + {\hat{M}}_{1}{| \psi\rangle}\otimes {| 1\rangle} $, where using $\beta=\sqrt{1-\alpha^{2}}$, $$\begin{aligned} && {\hat{M}}_{0} = \alpha {| +\rangle\langle +|} + \sqrt{1-\alpha^{2}} {| {-}\rangle\langle {-}|}, \\ && {\hat{M}}_{1} = \sqrt{1-\alpha^{2}} {| +\rangle\langle +|} + \alpha {| {-}\rangle\langle {-}|}. \end{aligned}$$ Therefore, by performing a projective measurement on the state of the ancillary qubit, we have a SASTOM on ${| \psi\rangle}$. The measurement direction can be changed using single-qubit gates on the measured system before the controlled-NOT gate. The use of a partial controlled-NOT gate leads to the implementation of general two-outcome measurements. Let us now write down the recipe using $ {\hat{U}}= {| +\rangle\langle +|} \otimes {\hat{I}}_{2} + {| {-}\rangle\langle {-}|} \otimes \exp(i \xi \hat{\tau}_{x}) $ and the initial state (\[eq:in\_sss\]). See, e.g., Ref.[@deGroot;Mooij:2011] for details of a theoretical proposal for performing ${\hat{U}}$. We find that $ {| \Psi^{({\rm out})}\rangle} = {\hat{U}}{| \Psi^{({\rm in})}\rangle} = {\hat{X}}_{0}{| \psi\rangle}\otimes {| 0\rangle} + {\hat{X}}_{1}{| \psi\rangle}\otimes {| 1\rangle} $, where $ {\hat{X}}_{0} = \alpha {| +\rangle\langle +|} + (\alpha\cos\xi + i\beta\sin\xi){| -\rangle\langle -|} $ and $ {\hat{X}}_{1} = \beta {| +\rangle\langle +|} + (i\alpha\sin\xi + \beta\cos\xi){| -\rangle\langle -|} $. Depending on the readout result of the ancillary qubit, a proper single-qubit operation on the measured qubit is performed. Then, we find that the state ${| \psi\rangle}$ is transformed by the positive operator part of ${\hat{X}}_{n}$. Using the right-polar decomposition, we obtain the positive operator parts of ${\hat{X}}_{0}$ and ${\hat{X}}_{1}$, respectively, $$\begin{aligned} && {\hat{M}}_{0} = \alpha {| +\rangle\langle +|} + \sqrt{1 - \alpha^{\prime\,2}} {| -\rangle\langle -|}, \\ && {\hat{M}}_{1} = \sqrt{1-\alpha^{2}} {| +\rangle\langle +|} + \alpha^{\prime}{| -\rangle\langle -|},\end{aligned}$$ where $ \alpha^{\prime} = \sqrt{[1 - (2\alpha^{2}-1)\cos(2\xi)]/2} $. General measurements with multiple outcomes can be implemented in a similar manner to that given in Sec.\[subsec:gwmmo\]. If we obtain the result $0$ in the ancillary qubit, we do nothing. A measurement operator \[i.e., ${\hat{K}}_{1}$ in Eq.(\[eq:def\_Nout1\])\] is applied to ${| \psi\rangle}$. Otherwise we perform a single-qubit operation on the measured qubit to change the measurement direction and prepare a new superposition state in the ancillary qubit. Then, we apply a partial controlled-NOT gate to the two qubits again. Depending on the readout results of the ancillary qubit, we either obtain one element in the desired POVM \[i.e., ${\hat{K}}_{2}$ in Eq.(\[eq:def\_Nout\])\] or continue to the next step. Repeating this procedure, we can obtain any POVM with multiple outcomes. Compared to linear optical qubits, the implementation of a general multi-outcome measurement in superconducting qubits has an advantage with respect to scalability. In linear optical qubits, it is necessary for the implementation of a general multi-outcome measurement to prepare all the optical elements corresponding to all the possible outcomes before the measurement. When the number of the outcomes is large, the setup become large and complicated. In addition, most of the elements in the measurement apparatus are irrelevant to the state in any single run. For example, if one obtains the outcome corresponding to ${\hat{K}}_{1}^{\dagger}{\hat{K}}_{1}$, the remaining parts of the measurement apparatus are not used. In superconducting qubits, the ancillary qubit can be used in the different steps of the measurement process. In contrast to linear optical setups, the total system is a two-qubit system even if the number of outcomes is large. Summary {#sec:summary} ======= We have proposed methods for implementing general measurements on a single qubit in linear optical and solid-state qubits. We focused on three types of general measurements on $\mathbb{C}^{2}$. The first type is the SASTOM described by Eqs.(\[eq:def\_elm1\]) and (\[eq:def\_elm2\]). Their associated POVM is regarded as a minimal extension of a projection-valued measure. The second one is the general two-outcome measurements described by Eqs.(\[eq:def\_gwm1\]) and (\[eq:def\_gwm2\]). This is the most general form of the measurements with two outcomes on $\mathbb{C}^{2}$. These two kinds of measurements have only two outcomes. 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--- abstract: 'We report the initial results of a survey for intracluster planetary nebulae in the Virgo Cluster. In two $16'' \times 16''$ fields, we identify 69 and 16 intracluster planetary nebula candidates, respectively. In a third $16'' \times 16''$ field near the central elliptical galaxy M87, we detect 75 planetary nebula candidates, of which a substantial fraction are intracluster in nature. By examining the number of the planetaries detected in each field and the shape of the planetary nebula luminosity function, we show that 1) the intracluster starlight of Virgo is distributed non-uniformly, and varies between subclumps A and B, 2) the Virgo Cluster core extends $\sim 3$ Mpc in front of M87, and thus is elongated along the line-of-sight, and 3) a minimum of 22% of Virgo’s stellar luminosity resides between the galaxies in our fields, and that the true number may be considerably larger. We also use our planetary nebula data to argue that the intracluster stars in Virgo are likely derived from a population that is of moderate age and metallicity.' author: - 'John J. Feldmeier and Robin Ciardullo' - 'George H. Jacoby' title: ' Intracluster Planetary Nebulae in the Virgo Cluster I. Initial Results' --- Introduction ============ The concept of intracluster starlight was first proposed by Zwicky (1951), when he claimed to detect excess light between the galaxies of the Coma cluster. Follow-up photographic searches for intracluster luminosity in Coma and other rich clusters (Welch & Sastry 1971; Melnick, White, & Hoessel 1977; Thuan & Kormendy 1977) produced mixed results, and it was not until the advent of CCDs that more precise estimates of the amount of intracluster starlight were made (cf. Guldheus 1989; Uson, Boughn, & Kuhn 1991; Víchez-Gómez, Pelló, & Sanahuja 1994; Bernstein 1995). All these studies suffer from a fundamental limitation: the extremely low surface brightness of the phenomenon. Typically, the surface brightness of intracluster light is less than 1% that of the sky, and measurements of this luminosity must contend with the problems presented by scattered light from bright objects and the contribution of discrete sources below the detection limit. Consequently, obtaining detailed information on the distribution, metallicity, and kinematics of intracluster stars through these types of measurements is extremely difficult, if not impossible. An alternative method for probing intracluster starlight is through the direct detection and measurement of the stars themselves. Recent observations have shown this to be possible. In their radial velocity survey of 19 planetary nebulae (PN) in the halo of the Virgo Cluster galaxy NGC 4406 (M86), Arnaboldi (1996) found 3 objects with $v > 1300$ ; these planetaries are undoubtably intracluster in origin. Similarly, Ferguson, Tanvir, & von Hippel (1998) detected Virgo’s intracluster component via a statistical excess of red star counts in a Hubble Space Telescope (HST) Virgo field over that in the Hubble Deep Field. Finally, intracluster stars have been unambiguously identified from the ground via planetary nebula surveys in Fornax (Theuns & Warren 1997) and Virgo (Méndez 1997; Ciardullo 1998). Motivated by these results, we have begun a large scale \[O III\] $\lambda 5007$ survey of intergalactic fields in Virgo, with the goal of mapping out the distribution and luminosity function of intracluster planetary nebulae (IPN). Depending on the efficiency of tidal stripping, Virgo’s intracluster component is predicted to contain anywhere from 10% to 70% of the cluster’s total stellar mass (Richstone & Malumuth 1983; Miller 1983). A survey of several square degrees of Virgo’s intergalactic space with a four meter class telescope should therefore detect several thousand PN, and shed light on both the physics of tidal-stripping and on the initial conditions of cluster formation. Here, we present the first results of our survey. Observations and Reductions =========================== On 1997 March 6-9 and 16, we imaged the Virgo Cluster through a 44 Å wide redshifted \[O III\] $\lambda 5007$ filter (central wavelength, $\lambda_c = 5027$ Å) and a 267 Å wide off-band filter ($\lambda_c = 5300$ Å) with the prime focus camera of the Kitt Peak 4-m telescope and the T2KB $2048 \times 2048$ Tektronix CCD. In this initial survey, three $16\arcmin \times 16\arcmin$ fields were chosen for study: one located at the isopleth center of Virgo subclump A approximately $52\arcmin$ north and west from M87, one $\sim 34\arcmin$ north and east of the giant elliptical NGC 4472 in subclump B, and one $\sim 13\parcmin 8$ north of M87, also in subclump A (For the definitions of the subclumps, see [@bts1]). The conditions during the bulk of the observations were variable in both seeing and transparency. However, at least one image of each field was taken on the photometric nights of 1997 Mar 8 and 9. These photometric images served to calibrate the remaining frames, and allowed us to put all our measurements on a standard system. The exact coordinates of the field centers, as well as a summary of the observations appear in Table 1. Figure 1 displays the locations of our three fields, along with the locations of other detections of Virgo’s intracluster stars: M86 (via PN spectroscopy; Arnaboldi 1996), M87 (via PN imaging; Ciardullo 1998), an HST Virgo field (via excess red star counts; Ferguson, Tanvir, & von Hippel 1998), and a blank Virgo field observed with the William Hershel Telescope (via PN imaging; Méndez 1997). Our survey technique was as described in Jacoby (1989) and Ciardullo, Jacoby, & Ford (1989b). Planetary nebula candidates were identified by “blinking” the sum of the on-band images against corresponding offband sums, and noting those point sources which were only visible in \[O III\]. This procedure netted 76 planetary nebula candidates in Field 1, 16 PN in Field 2, and 75 in Field 3. The magnitudes of the IPN were then measured relative to bright field stars via the IRAF version of DAOPHOT (Stetson 1987), and placed on a standard system by comparing large aperture measurements of field stars on the Mar 8 and 9 images with similar measurements of Stone (1977) standard stars. Finally, monochromatic fluxes for the PN were computed using the techniques outlined in Jacoby, Quigley, & Africano (1987), Jacoby (1989) and Ciardullo, Jacoby, & Ford (1989b). These were converted to $m_{5007}$ magnitudes using: $$m_{5007} = -2.5 \log F_{5007} - 13.74$$ where $F_{5007}$ is in units of ergs cm$^{-2}$ s$^{-1}$. We note here that our $m_{5007}$ magnitudes carry an additional uncertainty which is unique to IPN photometry. In order to compare the monochromatic flux of an emission line object with the flux of a continuum source (a standard star), the transmission of the filter at the wavelength of the emission line must be known relative to its total integrated transmission (cf. Jacoby, Quigley, & Africano 1987). For PN observations within other galaxies, this quantity is known (at least, in the mean) from the recessional velocity and velocity dispersion of the target galaxy. However, for our intergalactic PN survey, we do not know [*a priori*]{} what the kinematic properties of the target objects are, and hence we do not know the mean wavelength of their redshifted \[O III\] $\lambda 5007$ emission lines. In deriving our monochromatic \[O III\] $\lambda 5007$ magnitudes, we have assumed that the velocity dispersion of the intracluster PN follows that of the Virgo Cluster as a whole (cf. Binggeli, Sandage, & Tammann 1985), but this may not be true. Although the systematic error introduced by this uncertainty is small ($\sim 5\%$), the effect may be important for individual objects. Obtaining a Statistical Sample ------------------------------ Before we can compare the numbers and luminosity functions of IPN, we must first determine the photometric completeness limit in each field, and define a statistical sample of objects. Because our data were taken in mostly blank fields, our ability to detect IPN was not a strong function of position. We therefore used the results of Jacoby (1989) and Hui (1993) and equated our limiting magnitude for completeness with a signal-to-noise of 9. This is approximately the location where the PNLF (which should be exponentially increasing at the faint end) begins to turn down. The relative depth of each field derived in this way also agrees with that expected from measurements of the seeing and mean transparency on the individual images. In addition to excluding faint sources, we must make one other modification to the IPN luminosity function. Any galaxy that is within, or adjacent to our data frames, can potentially contaminate our IPN sample with PN still bound to their parent galaxies. Because PN are relatively rare objects, this source of contamination is unimportant for the small, low luminosity, dwarf galaxies that are scattered throughout the Virgo Cluster. However, PN in the halos of bright galaxies can be mistaken for intracluster objects, and must be taken into account. In Fields 1 and 2, there is only one contaminating object of any consequence, NGC 4425, a relatively bright ($0.2 L^*$) lenticular galaxy on the extreme western edge of Field 1. From numerous observations (examples include [@paper5]; [@rbc1]), PN very closely follow the spatial distribution of galaxy starlight. We therefore used surface photometry measurements to exclude seven of our PN candidates that fell within $2\parcmin 4$ of NGC 4425’s nucleus. From the $B$-band surface photometry of Bothun & Gregg (1990), this distance corresponds to approximately five effective radii or seven disk scale lengths. Without these objects, we have 69 and 16 IPN candidates in Fields 1 and 2, respectively, and a total of 44 and 9 objects in the photometrically complete samples. For Field 3, which has a total of 47 objects in its photometrically complete sample, the situation is more complex. Because the field is only $13\parcmin 8$ from the center of M87, some fraction of the objects in this region are probably bound to the galaxy. The angular distribution of the 75 planetary candidates does roughly follow the gradient defined by the M87 surface brightness measurements of Caon, Capaccioli, & Rampazzo (1990), and this does suggest an association with the galaxy. However, these data, by themselves, do not exclude the possibility that a large fraction of our PN are intracluster in origin. Indeed, since intracluster stars contribute luminosity just as galactic stars do, it is difficult to distinguish intracluster light from galactic light from surface photometry alone. The situation is further complicated by the recent discovery of an extremely large, highly elongated, low surface brightness halo which surrounds M87 and extends $\sim 15\arcmin$ on the sky (Weil, Bland-Hawthorn, & Malin 1997). This structure is so large, that it is probably not all bound to the galaxy. The complexity of the region makes it impossible to determine the precise number of galactic PN contaminating the intracluster counts of Field 3 using surface photometry. We will return to this issue in §3 below. We note that there is one other possible source of contamination to our intracluster PN sample. Any object that emits a large amount of flux in our redshifted $\lambda 5007$ filter, but is undetectable in the offband filter will be mistaken for a planetary nebula. Thus, distant objects, such as gas-rich starburst galaxies or quasars, could have enough flux in a redshifted emission line to be detected in our survey. In practice, however, this is extremely unlikely. The emission lines of importance are \[O II\] $\lambda 3727$ at $z \sim 0.35$ and Ly$\alpha$ at $z \sim 3.1$. From surveys of high-redshift quasars (Schmidt, Schneider, & Gunn 1995), we should expect much less than one $z = 3.1$ bright quasar in our three fields combined. Additionally, since our PN candidates have point-like point-spread-functions, those $z \sim 0.35$ galaxies with linear extents greater than $\sim 10$ kpc should have all been excluded on the basis of their angular size. These and similar arguments made by other authors (Theuns & Warren 1997; Méndez 1997), imply that contamination from background sources is probably not significant in our survey. The final luminosity functions for our 3 fields are plotted in Figure 2. For comparison, a sample of suspected intracluster PN’s from M87’s inner halo (Ciardullo 1998) are also plotted. The Distribution of Intracluster Stars ====================================== The most obvious feature displayed in Figure 2 is the dramatically different numbers of IPN in Fields 1 & 2. Although the survey depths of the three regions differ (due to differences in sky transparency and seeing), it is clear that the density of intracluster objects in Field 1 at the center of subclump A is much larger than that in Field 2. After accounting for the different depths, the PN density in Field 3, near M87, is smaller by a factor of at least two, and the number of PN in Field 2, which is near the edge of the 6 degree Virgo Cluster core in subclump B, is down by a factor of $\sim 4$. This behavior is quite intriguing. The fact that subclump B has fewer PN than subclump A can probably be attributed to cluster environment. It is well known that subclump B has fewer early-type galaxies than subclump A ([@bts1]). If ellipticals and intracluster stars have a related formation mechanism (galaxy interactions), then a direct correlation between galaxy type and stellar density in the intergalactic environment might be expected. What is harder to understand is the high IPN density in Field 1. Although galaxy isopleths place the Virgo Cluster center in Field 1 ([@bts1]), X-ray data clearly demonstrate that the true center of the cluster is at M87 (Böhringer 1994). The reason for the offset is a subcluster of galaxies associated with M86. Kinematic data (Binggeli, Popescu, & Tammann 1993) and ROSAT X-ray measurements (Böhringer 1994) both show that the center of Virgo subclump A is contaminated by a separate group of galaxies which is falling in from the far side of the cluster. This interpretation is confirmed via direct distance measurements using the planetary nebula luminosity function (Jacoby, Ciardullo, & Ford 1990) and surface brightness fluctuation method (Ciardullo, Jacoby, & Tonry 1993; Tonry 1997): both place M86 $\sim 0.3$ mag behind M87. At this distance, the PN associated with M86 and its surroundings should be beyond the completeness limit of our survey, and should not be contributing to our PN counts. The observed IPN density of Field 1 should therefore be smaller than that measured near M87, not greater. A second feature of Figure 2 is the slow fall-off of the bright-end of Field 3’s planetary nebula luminosity function (PNLF). Observations in $\sim 30$ elliptical, spiral, and irregular galaxies have demonstrated that a system of stars at a common distance will have a PNLF of the form: $$N(M) \propto e^{0.307M} \, [1 - e^{3(M^{*}-M)}]$$ where $M^* \approx -4.5$ (Jacoby 1992). [*In no isolated galaxy has the PNLF ever deviated from this form.*]{} However, in a cluster environment such as Virgo, the finite depth of the cluster can distort the PNLF, as PN at many different distances contribute to the observed luminosity function. Due to the lack of survey depth in Field 1, and small number of PN in Field 2, we cannot determine whether the PNLFs of these fields differ from the empirical function. However the PNLF of Field 3 is clearly distorted, in the manner identical to that found by Ciardullo (1998) in their survey of the envelope of M87. Figure 3 illustrates this effect in more detail. In the figure, we first compare the observed PNLF of Field 3 to the most likely empirical curve (solid line) as found via the method of maximum likelihood (Ciardullo 1989a). The fit is extremely poor, and a Kolmogorov-Smirnov test rejects the empirical law at the 93% confidence level. The reason for the poor fit is simple: the presence of bright, “overluminous” objects forces the maximum-likelihood technique to find solutions which severely overpredict the total number of bright PN in the field. There is only one plausible hypothesis for the distorted PNLF and the overabundance of bright PN — the presence of intracluster objects. An instrumental problem is ruled out, since the bright-end distortion has also been seen in data taken of M87’s envelope two years earlier with a different filter and under different observing conditions (Ciardullo 1998). Similarly, extreme changes in metallicity cannot explain the discrepancy. Unless the bulk of the intracluster stars are super metal-rich (\[O/H\] $\simgt 0.5$), abundance shifts can only decrease the luminosity of the \[O III\] $\lambda 5007$ line, not increase it (Dopita, Jacoby, & Vassiliadis 1992; Ciardullo & Jacoby 1992). Finally, neither population age nor the existence of dust is a likely scenario: the former requires an unreasonably young ($< 0.5$ Gyr) age for M87’s stellar envelope (Dopita, Jacoby, & Vassiliadis 1992; Méndez 1993; Han, Podsiadlowski, & Eggleton 1994), and the latter implies a significant gradient in foreground extinction between 2 and 7 effective radii from M87’s center. Thus, one is left with the hypothesis, first presented by Jacoby (1996) and Ciardullo (1998), that intracluster PN are responsible for the distorted PNLF. From our own study, and that of others (Arnaboldi 1995; Méndez 1997; Ferguson, Tanvir, & von Hippel 1998), it is clear that a significant fraction of intracluster stars exist in Virgo, and some of these stars should be positioned in front of M87 along the line-of-sight. The existence of foreground PN naturally explains the existence of “overluminous” PN, and is supported by the fact that the brightest PN in Field 3 are of comparable magnitude to the brightest objects found in the Ciardullo (1998) survey of M87. The distorted PNLF gives us a way to estimate a lower limit to the number of IPN in Field 3. To do this, we plot a dashed line in Figure 3, which represents the expected luminosity function of M87 planetaries using the observed PNLF distance modulus of the inner part of the galaxy ($m-M = 30.87$; Ciardullo 1998). To normalize this curve, we assume that [*all*]{} the PN at $m_{5007} = 26.9$ belong to the galaxy. As is illustrated, there is an excess of bright objects compared to that expected from M87 alone: these are objects at the bright end of the intracluster planetary nebula population. If we statistically subtract the M87 luminosity function from the Field 3 data, we arrive at the conclusion that at least $10 \pm 2$ planetaries with $m_{5007}$ brighter than 26.5 are intracluster in nature, where the uncertainty is solely due to the uncertainty in the normalization value. Note that this estimate is a bare minimum for the number of intracluster PN. For all reasonable planetary nebula luminosity functions, there are many more faint PN than bright PN. Thus, our assumption that all $m_{5007} = 26.9$ PN are galactic is clearly wrong. However, without some model for the distribution of intracluster stars, the shape of the intracluster PNLF cannot be determined. This makes it impossible to photometrically distinguish faint intracluster PN from PN that are bound to M87. For the moment, we therefore conservatively claim that at least $10 \pm 2$ intracluster planetaries are present in Field 3. In the future, it should be possible to refine this estimate with improved observations of the precise shape of the PNLF, and with the use of dynamical information obtained from PN radial velocity measurements. A final feature of Figure 2, and perhaps the most remarkable, deals with distance. Even a cursory inspection of Figure 2 shows that the IPN of Field 1 are significantly brighter than those of Field 2. If we fit the two distributions to the PNLF of equation (2) via the technique of maximum likelihood (Ciardullo 1989a), then the most likely distance to the PN of Field 1 is $11.8 \pm 0.7$ Mpc, while that for Field 2 is $14.7 \pm 1.5$ Mpc. The fact that Field 2 is more distant is not surprising, since both Yasuda, Fukugita, & Okamura (1997) and Federspiel, Tammann, & Sandage (1997) place the galaxies of subclump B $\sim 0.4$ mag behind those of subclump A. What is surprising is the relatively small distance to the IPN of Field 1. Most modern distance determinations, including the analysis of Cepheids in spirals by van den Bergh (1996) and the measurement of the planetary nebula luminosity function and surface brightness fluctuations in ellipticals (Jacoby, Ciardullo, & Ford 1990; Tonry 1997), place the core of the Virgo Cluster at a distance of between 14 and 17 Mpc. No modern measurement to Virgo gives a distance smaller than $\sim 14$ Mpc, and 11.8 Mpc is certainly not a reasonable value for the distance to the cluster. The reason for the $\sim 3$ Mpc discrepancy is that the PNLF law has an extremely sharp cutoff at the bright end of the luminosity function. As a result, our PN detections are severely biased towards objects on the front edge of the cluster, and our distance estimates carry the same bias. Our derived value of 11.8 Mpc therefore represents the distance to the front edge of the Virgo Cluster, not the distance to the galaxies in the Virgo Cluster core. Another way of looking at the data is to consider the brightest IPN in Fields 1 and 3. If we assume that these bright objects are indeed part of the Virgo Cluster and have an absolute magnitude near $M^*$, then their apparent magnitudes yield an immediate upper limit to the front edge of each field. The result is that the brightest IPN of Fields 1 and 3 have distances of no more than $11.7$ Mpc, they are $\sim 3$ Mpc in front of the cluster core, as defined by the original PNLF measurements of Jacoby, Ciardullo, & Ford (1990). The same conclusion was reached by Ciardullo (1998) using the sample of PN around M87’s inner halo. This distance is significantly larger than the size of the cluster projected on the sky: at a distance of $\sim 15$ Mpc, the classical 6 degree core of Virgo translates to a linear extent of only $\sim 1.5$ Mpc. Although it is unclear how our measurement of cluster depth quantitatively compares to the classical angular estimate of the core (Shapley & Ames 1926; see the discussion in de Vaucouleurs & de Vaucouleurs 1973), our data does suggest that Virgo is elongated along our line-of-sight, perhaps by as much as a factor of two. It is worth repeating here that the planetary nebula luminosity function is extremely insensitive to the details of its parent population, and those dependences that do exist cannot explain the bright apparent magnitudes seen in the IPN population. The problem is more fully discussed in Ciardullo (1998), but, in summary, there is no reasonable explanation for the existence of these bright \[O III\] $\lambda 5007$ sources other than that their location is in the foreground of the Virgo Cluster. We note that many authors have suggested that the Virgo Cluster has substantial depth (for example, see the galaxy measurements of Pierce & Tully 1988; Tonry, Ajhar, & Luppino 1990; Yasuda, Fukugita, & Okamura 1997), but this direct measurement is still quite surprising. The Stellar Population and Total Number of Intracluster Stars ============================================================= Renzini & Buzzoni (1986) have shown that bolometric-luminosity specific stellar evolutionary flux of non-star-forming stellar populations should be $\sim 2 \times 10^{-11}$ stars-yr$^{-1}$-$L_{\odot}^{-1}$, nearly independent of population age or initial mass function. If the lifetime of the planetary nebula stage is $\sim 25,000$ yr, then every stellar system should have $\alpha \sim 50 \times 10^{-8}$ PN-$L_{\odot}^{-1}$. Observations in elliptical galaxies and spiral bulges have shown that no galaxy has a value of $\alpha $ greater than this number, but $\alpha $ can be a up to a factor of five smaller ([@rbciau]). Nevertheless, the direct relation between number of planetary nebulae and parent system luminosity does provide us with a tool with which to estimate the number of stars in Virgo’s intergalactic environment. In order for us to estimate the density of intracluster stars in Virgo, we must first fit the observed PNLF with a model which represents the distribution of planetary nebulae along the line-of-sight. Such a model is a necessary step in the analysis: as seen above, Virgo’s intracluster component extends $\sim 0.5$ mag in front of its core, and thus our sample of PN is severely biased towards objects on the near side of the cluster. To investigate the effect of this bias, we considered two extreme models for the distribution of Virgo’s intracluster stars: a single component model, in which all the PN are at a common distance, and a radially symmetric model, in which the intracluster stars are distributed isotropically throughout a sphere of radius 3 Mpc. The symmetric model was then adjusted to deliver a “best fit” to the observed PNLF at the assumed cluster distance of 15 Mpc (Jacoby, Ciardullo, & Ford 1990). For the single component models, we adopted the distances derived in §3. The most likely value for the underlying population’s stellar luminosity was then calculated using the method of maximum likelihood (Ciardullo 1989a) and an assumed value of $\alpha_{2.5} = 20 \times 10^{-9}$, which is an average value for elliptical galaxies ($\alpha_{2.5}$ is the number of PN within 2.5 mag of $M^*$ per unit bolometric luminosity). Because of our limited knowledge of the luminosity function of Field 3, we used single component models only for this field, assuming that field contains 10 IPN within our completeness limit. In addition, because of the very large uncertainties in the above models, we also computed a single component model using $\alpha_{2.5} = 50 \times 10^{-9}$ PN-$L_{\odot}^{-1}$; this last model represents the minimum amount of intracluster starlight necessary to be consistent with our data. The results of our models are summarized in columns 1-4 in Table 2. The total amount of intracluster starlight found is quite large, at least $6.8 \times 10^{9} L\subsun$ in the 768 square arcminutes we surveyed, and probably much more. As expected, the density of intracluster stars varies significantly between the fields. For Fields 1 and 2, the choice of cluster model changes the amount of derived intracluster light dramatically. In the single component model, most of the PN can be on the near side of the cluster, where we can see objects relatively far down the luminosity function. In this scenario, the size of the IPN’s parent population is relatively small, as is the amount of intracluster light. In contrast, the symmetric model places a large number of stars on the back side of the cluster, where they contribute light, but do not populate the bright end of the PN luminosity function. The intracluster luminosity in this picture is correspondingly larger. How likely is it that the intracluster stars are distributed symmetrically in the cluster? There is strong evidence to suggest that neither the galaxies nor the intracluster stars are in virial equilibrium. Based on the distribution and kinematics of galaxies, Binggeli, Tammann, & Sandage (1987) and Bingelli, Popescu, & Tammann (1993) concluded that the core of Virgo exhibits a significant amount of substructure. Similarly, Ciardullo (1998) showed that the PNLF of intracluster stars near M87 is incompatible with that of a relaxed system. Although these analyses do not formally exclude all symmetric distributions, they do suggest that a symmetric distribution is unlikely. Although our limited amount of photometric data do not allow us to address the question of Virgo’s structure directly, we can gain some insight into the state of the intracluster stars by speculating about their likely origin. To do this, we first focus on the metallicity of the observed IPN. Many of the planetary nebulae detected in our survey are extremely bright: some are more than 0.6 mag brighter than $m^*$ at the center of the cluster. In §3, we interpreted this brightness in terms of distance, and were thus able to place the IPN $\sim 3$ Mpc in front of M87. This argument, however, assumes that $M^*$, the absolute magnitude of the PNLF cutoff, is well known. For most stellar populations, this is a good assumption, as evidenced by the agreement between PNLF distances and distances determined from other methods (cf. Jacoby 1992; Ciardullo, Jacoby, & Tonry 1993; Feldmeier, Ciardullo, & Jacoby 1997). However, in extremely metal poor systems, the decreased number of oxygen atoms present in a PN’s nebula does have an affect. Specifically, PN from a population whose metallicity is one-tenth solar are expected to have a value of $M^*$ that is fainter than the nominal value by more than 0.25 mag (Ciardullo & Jacoby 1992; Dopita, Jacoby, & Vassiliadis 1992; Richer 1994). This increase in $M^*$ translates directly into an error in distance. For example, if the intracluster environment of Virgo were filled with stars with one-tenth solar abundance, our derived distance to the front of the cluster would be overestimated by $\sim 11\%$, and our value for the distance between the front of the cluster and the Virgo core galaxies would be underestimated by almost 50%. Note, however, that we already measure a Virgo Cluster depth that is $\sim 2$ times that of its projected size; lowering the metallicity of the stars would only increase this ratio. It therefore seems likely that the PN detected in our survey are of moderate metallicity. Similarly, the mere fact that we do see a large number of IPN suggests that most of the intergalactic stars are not extremely old. In their \[O III\] $\lambda 5007$ survey of Galactic globular clusters, Jacoby (1997) found a factor of $\sim 4$ fewer PN than expected from stellar evolution theory. Jacoby attribute this small number to the extreme age of the stars. Observations in the Galaxy suggest that stars with turnoff masses of $\sim 0.8 M\subsun$ produce post asymptotic giant branch stars with extremely small cores, $M_c < 0.55 M\subsun$ (Weidemann & Koester 1983). Objects such as these evolve to the planetary nebula phase very slowly: so slowly, in fact, that their nebulae can diffuse into space before the stars become hot enough to produce a significant number of ionizing photons. If this scenario is correct, then the fact that we see large numbers of IPN indicates that the intracluster stars of Virgo cannot be as old as the globular cluster stars of the Galaxy. If the intracluster stars of Virgo are, indeed, of moderate age and metallicity, then they must have been stripped out of their parent galaxies at a relatively recent epoch. One likely way of doing this is through “galaxy harassment” whereby high-speed encounters between galaxies rip off long tails of matter which lead and follow the galaxy (Moore 1996). Although this tidal debris will eventually dissolve into the intracluster environment, it is possible that the increased energy of these stars may cause them to linger in the outer parts of the cluster for a long time. It is therefore possible that the intergalactic environment of Virgo is not a homogeneous region, but is instead clumpy, and filled with filaments. It may be that the bright PN present in Field 1 belong to one such structure that happens to be in the front side of the cluster. If this is correct, the symmetric model for Field 1 should be a gross overestimate of the true amount of intracluster starlight. If we assume that the intracluster stars come from an old stellar population, similar to that found in M87 with a $B$$-$$V$ color of 1.0 (de Vaucouleurs 1991), a bolometric correction of $-0.85$ (Jacoby, Ciardullo, & Ford 1990), and a mean distance of 15 Mpc, then the luminosities implied by our PN observations can be expressed in terms of $B$-band surface brightnesses. Although the values are position and model dependent (cf. Table 2, column 5), the result is quite interesting. Our PN observations imply that the surface brightness in the Virgo Cluster core varies between $B \sim 26$ and $B \sim 30$ mag per sq. arcsec. These values are in excellent agreement with other PN-derived surface brightness values in Virgo (Méndez 1997) and Fornax (Theuns & Warren 1997). They are, however, significantly larger than the value of $B \sim 31.2$ implied from red star counts on [*HST*]{} frames (Ferguson, Tanvir, & von Hippel 1998). One possible explanation for this discrepancy lies in the locations of the fields: the [*HST*]{} field is $50\arcmin$ east of M87, whereas Fields 1 and 2 are in denser regions of the cluster. If intracluster stars are concentrated towards the cores of the subclumps of Virgo, then the low number of red stars observed by [*HST*]{} may be attributable to the field’s location. Under this assumption, and using the same stellar population model used by Ferguson, Tanvir & von Hippel (1998), we would expect only $\sim 5$ IPN brighter than $m_{5007} = 27.0 $ in a $16\arcmin \times 16\arcmin$ field centered on the [*HST*]{} position. On the other hand, our fields may, indeed, be typical locations in the Virgo Cluster. As described above, the high galaxy density in Field 1 is due, in part, to objects on the far side of the cluster which do not contribute to the observed sample of planetaries. Similarly, the shape of the PNLF of Field 3 suggests that much of the intracluster luminosity in that region is not associated with the physical core of Virgo, but is only present in the field through projection. If our survey regions are typical of Virgo in general, then we can use our observations to estimate the total fraction of Virgo’s starlight which is between the galaxies. By fitting the surface distribution of Virgo galaxies in subclump A to a King model with core radius $1\pdegree 7$, Binggeli, Tammann, & Sandage (1987) derived a central luminosity density of the cluster of $1 \times 10^{11} L\subsun$ per square degree. If we scale this galactic luminosity density, which we denote as L$_{\it galaxies}$ (cf. Table 2, column 6), to the sizes and locations of Fields 1 and 3, we can directly determine the importance of intracluster starlight. Due to its irregularity, Binggeli, Tammann, & Sandage (1987) did not fit an analytical model to subclump B. However, the luminosity density in Field 2 is certainly not greater than that for Field 1. We therefore use Field 1’s value to set a lower limit on the fraction of intracluster starlight in Field 2. The results are given in column 7 of Table 2. Depending on the model and the field, the fraction of intracluster light varies from 12% to 88% of the cluster’s total luminosity. To set a lower limit to the relative importance of intracluster starlight, we average the results from our three fields, using the smallest fraction found for each field. We find an average fraction of 22%. Similarly, to find the upper limit to the intracluster fraction, we average the largest fractions determined. In this case, we find an average fraction of 61%. This range is in rough agreement with the results derived from other PN observations in Virgo (Méndez 1997) and Fornax (Theuns & Warren 1997). They also agree with the direct measurement of intracluster light in Coma (Bernstein 1995). We stress that these results are very uncertain, due to our lack of knowledge about the true distribution of intracluster stars, and the possible variation of $\alpha_{2.5}$. In addition, it is also probable that we have missed a small fraction of the IPN present in our fields due to the finite width of our interference filter. If the IPN velocity dispersion follows that of the galaxies, then $\sim 8\%$ of the IPN will be Doppler shifted out of the bandpass of our 44 Å full-width half-maximum filter. However, since the true kinematics of the IPN are unknown, this fraction cannot be determined precisely. Nevertheless, regardless of the particular model, a large fraction of intracluster stars are present in the Virgo Cluster. The large amount of intracluster starlight found by this and other studies places new constraints on models of the formation and evolution of galaxy clusters. If, as we have suggested, the intracluster stars are of moderate age and metallicity, they must not have been formed in the intracluster environment, but instead have been removed from their parent galaxies during encounters. Therefore, the number and distribution of intracluster stars could be a powerful tool for discovering the history of individual galaxy clusters. Furthermore, since intracluster stars are already free of the potential wells of galaxies, they may contribute a significant fraction of metals and dust to the intracluster medium. The large amount of intracluster stars is also an unrecognized source of baryonic matter that must be taken into account in studies of galaxy clusters. Though not enough to account for all of the dark matter, intracluster stars do increase the fraction of matter that is in baryonic form. This has potentially serious consequences for cosmological models. From their calculation of the baryon fraction in the Coma Cluster, White (1993) found that their derived value was too large for the universe to simultaneously have $\Omega_0 = 1$ and also be in accord with calculations of cosmic nucleosynthesis. This result, sometimes called the “baryon catastrophe,” has been confirmed in several other galaxy clusters (White & Fabian 1995; Ettori, Fabian, & White 1997). Although the exact amount of intracluster starlight is currently very uncertain, the presence of a large number of intracluster stars can only increase the baryon discrepancy already found. More observations will be necessary to establish how much intracluster starlight adds to the observed baryon fraction of galaxy clusters. Conclusion ========== We report the results of a search in three fields in the Virgo Cluster for intracluster planetary nebulae, and have detected a total of 95 intracluster candidates. From analysis of the numbers of the planetaries, we find that the amount of intracluster light in Virgo is large (at least 22% of the cluster’s total luminosity), distributed non-uniformly, and varies between subclump A and B. By using the planetary nebulae luminosity function, we derive an upper limit of $\sim 12$ Mpc for the distance to the front edge of the Virgo Cluster and use this to show that the cluster must be elongated along our line of sight. We also use the properties of planetary nebulae to suggest that the intracluster stars of Virgo have moderate age and metallicity. The large fraction of intracluster stars found has potentially serious consequences for models of cluster formation and evolution, and for cosmological models. Finally, we note that this survey included less than 0.2% of the traditional 6 degree core of the Virgo Cluster. Many more intracluster planetary nebulae wait to be discovered. We thank Allen Shafter for some additional off-band observations and Ed Carder at NOAO, for his measurements of our on-band filter so that we could begin our observations on time. We would also like to thank the referee, R. Corradi, for several suggestions that improved the quality of this paper. Figure 1 was extracted from the Digitized Sky Survey, which was produced at the Space Telescope Science Institute under U.S. Government grant NAGW-2166. 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Panagia (Cambridge: Cambridge Univ. Press), 197 Jacoby, G. H., Ciardullo, R., & Ford, H. C. 1990, , 356, 332 Jacoby, G. H., Ciardullo, R., Ford, H. C., & Booth, J. 1989, , 344, 70 Jacoby, G. H., Branch, D., Ciardullo, R., Davies, R. L., Harris, W. E., Pierce, M. J., Pritchet, C. J., Tonry, J. L., & Welch, D. L. 1992, , 104, 599 Jacoby, G. H., Morse, J. A., Fullton, L. K., Kwitter, K. B., & Henry, R. B. C. 1997, , 114, 2611 Jacoby, G. H., Quigley, R. J., & Africano, J. L. 1987, , 99, 672 Melnick, J., White, S. D. M., & Hoessel, J. 1977, , 180, 207 Méndez, R. H., Guerrero, M. A., Freeman, K. C., Arnaboldi, M., Kudritzki, R. P., Hopp, U., Capaccioli, M. & Ford, H. 1997, , 491, 23 Méndez, R. H., Kudritzki, R. P., Ciardullo, R., & Jacoby, G. H. 1993, , 275, 534 Miller, G. E. 1983, , 268, 495 Moore, B., Katz, N., Lake, G., Dressler, A., & Oemler, A. 1996, , 379, 613 Pierce, M. J., & Tully, R. B. 1988, , 330, 579 Renzini, A., & Buzzoni, A. 1986, in [Spectral Evolution of Galaxies,]{} ed. C. Chiosi, & A. Renzini (Dordrecht: Reidel), p. 195 Richer, M. G. 1994, Ph.D. Thesis, York University Richstone, D. O., & Malumuth, E. M. 1983, , 268, 30 Schmidt, M., Schneider, D. P., & Gunn, J. E. 1995, , 110, 68 Shapley, H., & Ames, A. 1926, Harvard Circ. No. 294 Stetson, P. B. 1987, , 99, 191 Stone, R. P. S. 1977, , 218, 767 Theuns, T., & Warren, S.J. 1997, , 284, L11 Thuan, T. X., & Kormendy, J. 1977, , 89, 466 Tonry, J. L., Ajhar, E. A., & Luppino, G. A. 1990, , 100, 1416 Tonry, J. L., Blakeslee, J. P., Ajhar, E. A., & Dressler, A. 1997, , 475, 399 Uson, J. M., Boughn, S. P., & Kuhn, J. R. 1991, , 369, 46 van den Bergh, S. 1996, , 108, 1091 Vílchez-Gómez, R., Pelló, R., & Sanahuja, B. 1994, , 283, 37 Weidemann, V., & Koester, D. 1983, , 121, 77 Weil, M. L., Bland-Hawthorn, J., & Malin, D. F. 1997, , 490, 664 Welch, G. A., & Sastry, G. N. 1971, , 169, l3 White, D. A., & Fabian A. C. 1995, , 273, 72 White, S. D. M., Navarro, J. F., Evrard, A. E., & Frenk, C. 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--- author: - 'P. A. James , J. O’Neill , N. S. Shane ,' bibliography: - 'refs.bib' date: 'Received ; accepted ' subtitle: 'VI. Star-forming companions of nearby field galaxies' title: 'The H$\alpha$ Galaxy Survey [^1]' --- Introduction ============ Satellite galaxies are of great importance in our understanding of the formation and morphological evolution of all types of galaxy. Minor mergers of disk and dwarf galaxies provide one possible process for forming or enlarging bulges [e.g. @hayn00; @kann04; @elic06], and the associated tidal perturbation is likely to distort and ultimately thicken disks [@quin86; @quin93]. Indeed, several minor mergers may suffice to convert a disk galaxy into an elliptical [@bour07; @kavi07]. Gas-rich dwarfs may provide gas reservoirs for rejuvenating early-type disk galaxies and thus prolonging or re-establishing star formation (SF) activity [@whit91; @hau07], whilst also changing the metallicity of the remaining gas component. Finally, accretion of dwarf galaxies may be very significant for the formation of the stellar haloes of galaxies [e.g. @sear78; @read06]. However, in order to quantify the typical effects of such mergers on disk galaxies, it is necessary to know what fraction actually have close companions, and how many of these contain significant gas reservoirs. These fractions, when combined with a typical time for the decay of orbits through dynamical friction, should ultimately enable an estimate to be made of the minor merger rate, and the resulting impact on the SF of the central galaxies. A starting-point for such an investigation is provided by the Milky Way, which has a significant number of satellite galaxies. Of these, most are very faint and low-surface brightness dwarf spheroidal and dwarf irregular galaxies; in terms of mass, and future evolutionary impact, by far the most dominant of the probable satellites are the Large and Small Magellanic Clouds (LMC and SMC henceforth). Since the Milky Way is often assumed to be a typical field galaxy, probably of type SBb - SBc, with a total luminosity close to the characteristic value L$^{\star}$ in the [@sche76] luminosity function, it might be natural to assume that Magellanic Cloud-like companions are also typical of the field galaxy population. Indeed, there are several well-studied bright satellites around local galaxies, such as M 32 and NGC 205, the early-type companions of M 31. However, no systematic search for Magellanic-type companions around a representative sample of field galaxies has yet been carried out, so it is not clear whether the Milky Way is typical or unusual in having two near neighbours of this type. Some studies of field galaxy companions have been presented in the literature. @zari97 searched for companions at least 2.2 mag fainter than the primary galaxy, within 1000 km s$^{-1}$ in velocity and 500 kpc in projected separation, and found 115 satellites around 69 primary galaxies. However, the allowed separations are very much greater than the separation of the Magellanic Clouds from the Milky Way, and @zari97 comment that some of the primaries have four or five companions, and these systems may thus be better regarded as galaxy groups. Only about 10% of the companions have projected separations less than 70 kpc, potentially comparable to the Magellanic Clouds. @noes01 found $\sim$30% of a sample of field dwarf galaxies to have companions within a projected separation of 100 kpc and a recession velocity difference of $\pm500$ km s$^{-1}$. @mado04, following a pioneering study by @both77, performed a search for companions around isolated elliptical galaxies. Within a 75 kpc projected radius they found 1.0$\pm$0.5 companions per elliptical galaxy, higher than the 0.12$\pm$0.42 companions per galaxy found by @both77. One requirement of any such study is for recession velocity information to remove line-of-sight pairs, which generally dominate the population of apparent pairs found in pure imaging studies, particularly for blue companions [@chen07]. A complicating factor in obtaining such data through spectroscopic surveys is the problem of fibre collisions with multi-object spectrographs, which means that close pairs are under-represented in many existing surveys. Here we undertake a search using an alternative method based on narrow-band [H$\alpha$]{} imaging of the areas around field galaxies in the local Universe. Specifically, we search for star forming companions of these galaxies, which will show up via [H$\alpha$]{} line emission, and as a result this study says nothing about non-star forming companions. Any companions detected in [H$\alpha$]{} are likely, given the $\sim$50 Angstrom width of [H$\alpha$]{} filters used, to be truly associated with central target galaxy. The velocity range for [H$\alpha$]{} to lie within the same narrow-band filter is $\pm$1000 km s$^{-1}$, which excludes projected companions with good efficiency, but the number of detected companions will be an upper limit as there will still be some line-of-sight projections within this range. Thus the radial distance range in which satellites could lie is 29 Mpc in depth for an assumed Hubble constant of 70 km s$^{-1}$Mpc$^{-1}$; as explained in section \[sec:galstats\], we search for satellites to a projected distance of 75kpc in the plane of the sky, giving a total search volume of $\sim$0.5 Mpc$^{3}$. Of course, any ‘true’ satellites will lie in a much smaller volume than this about the central galaxy. There is also the possibility of including very distant background galaxies, with other emission lines redshifted into the [H$\alpha$]{} filter bandpass. Whilst this technique will clearly detect only the line-emitting fraction of all satellite galaxies, it is important to note recent results demonstrating that this is equivalent to the entire gas-rich population in the equivalent mass range, i.e. that there are no or very few quiescent gas-rich galaxies. [@meur06] found that all 93 of the [H[i]{}]{}-selected galaxies in their sample were detected in [H$\alpha$]{}emission. @hain07, using SDSS data, found all of the $\sim600$ low-luminosity galaxies studied ($-18 < M_r < -16$) in the lowest-density environments to have [H$\alpha$]{} emission. The previous paper in this series [@paper5] found that all 117 late-type Sm and Im galaxies in the [H$\alpha$]{}GS sample are actively forming stars, and these correspond to more than 50% of the galaxies of these types satisfying the selection criteria of this sample. Thus possession of significant gas reservoirs and star formation (with corresponding [H$\alpha$]{} emission) appear to be synonymous. Data and methods {#sec:method} ================ We use imaging data from the [H$\alpha$]{} Galaxy Survey [@paper1], [H$\alpha$]{}GS henceforth. This survey contains data for 327 galaxies selected from the Uppsala General Catalogue of Galaxies [@nils73] (UGC) to have diameters between 17 and 60, measured recession velocities of less than 3000 km s$^{-1}$, major-to-minor axis ratios less than 4.0, and Hubble type later than S0a. The Virgo cluster core was excluded, so this is effectively a field galaxy sample, but otherwise it should be unbiased with respect to presence or absence of companion or satellite galaxies. The [H$\alpha$]{}GS data comprise broad-band $R$ imaging, and [H$\alpha$]{} in one of several narrow band filters, selected to match the recession velocity of the target UGC galaxy, obtained at the 1.0 metre Jacobus Kapteyn Telescope. The field of view of the JKT CCD camera was $\sim$ 11$\times$11 arcmin, but we conservatively use only the central 9.5$\times$9.5 arcmin area to avoid vignetting and cosmetic problems with the edges of the frames. The [H$\alpha$]{} images used here were continuum-subtracted, using either observations in an intermediate-width continuum filter, or scaled $R$-band exposures, and flux calibrated, as outlined in @paper1. All galaxy distances (from Virgo-infall corrected recession velocities) and $R$-band magnitudes for the central galaxies used in the present analysis are those listed in Table 3 of @paper1. However, for consistency with the analysis presented in @paper5 (hereafter referred to as paper V), the SF rates for central galaxies used here are multiplied by a factor 0.7, consistent with the assumption of a ‘Salpeter light’ stellar initial mass function, rather than the Salpeter function assumed by [@kenn98] and adopted in paper I. All [H$\alpha$]{}GS galaxies with distances greater than or equal to 20 Mpc were included in this analysis; this was a total of 119 central galaxies. Galaxies within this distance limit were excluded because of the small effective volume included within the CCD field of view. The method employed when searching for potential satellite galaxies was initially to scan the continuum-subtracted [H$\alpha$]{} images, looking for any apparent regions of emission that are detached from the main galaxy. Any potential sources found were then ‘blinked’ with the $R$-band image (which is aligned to sub-pixel accuracy) to check for, e.g., poorly-subtracted foreground stars, cosmetic defects or bright cosmic ray trails. The brightest of such spurious sources can be excluded from consideration if they are not present in the $R$-band image (since this filter includes the [H$\alpha$]{} emission), and they can also be excluded if not at least as extended as the stellar point spread function. In most of the frames studied, the situation was completely unambiguous, as no possible satellite galaxies were seen on the [H$\alpha$]{} frame, and almost all of the spurious objects were quickly identified as such. This left 32 cases where an apparently real emission-line source was identified somewhere within the CCD field of view. One possible source of incompleteness concerns those galaxies for which the [H$\alpha$]{} line from the central galaxy lies close to the edge of the bandpass of the narrow band filter used. There is significant overlap between the redshifted [H$\alpha$]{} filters that were available for use with the JKT, so this was not severe problem, but there is a small effect for those galaxies lying close to the ‘changeover’ recession velocity from one filter to the next. To quantify this, the filter throughputs were calculated for each galaxy recession velocity, and for velocities 300 km s$^{-1}$ on either side of this to represent the kinematic limits of likely satellite populations. The most severe bias found was for UGC 4260, where the galaxy itself lies just on the ‘wrong’ side of a changeover point; adding 300 km s$^{-1}$ to this velocity drops the filter throughput to 62% of the value at the galaxy recession velocity. There are 4 cases where this throughput drop at $\pm$300 km s$^{-1}$ is 69 or 70%, 19 cases from 71 to 75%, and 7 cases from 76 to 80%. For the remaining 88 out of 119 galaxies, the throughput across the entire likely satellite velocity range is greater than 80% of that at the galaxy recession velocity. The flux limit for our [H$\alpha$]{} images was calculated from faint sources in real images, including photon noise and systematic effects due to irregularities in the background ‘sky’ regions. A moderately extended source like our putative companions would be detected at 5$\sigma$ for an observed [H$\alpha$]{} plus [\[N[ii]{}\]]{} flux of 4.3$\times 10^{-18}$ W m$^{-2}$. This corresponds to a SF rate of 0.004 – 0.0076 M$_{\odot}$yr$^{-1}$ for a galaxy at a distance of 30 Mpc, with the exact value in the range depending on the extinction corrections adopted. The remaining, and distinctly problematic issue concerning the 32 putative satellites was to distinguish whether these sources were truly separate galaxies, or just outlying SF regions of the central galaxy. Conservatively, all 32 sources were included for further analysis, even where the appearance on the $R$-band image pointed towards the latter being the case, and resolution of this question was left to further analysis of the source properties in section \[sec:satclass\] below. This analysis was based on total [H$\alpha$]{} and $R$-band fluxes of the companions, which were measured using matched elliptical apertures. The [H$\alpha$]{} fluxes were then converted to SF rates using the conversion formula of @kenn98 scaled by the 0.7 factor mentioned above, but with internal extinction corrections based on companion object $M_R$-magnitudes using the methods of @helm04. Classification of putative satellite galaxies {#sec:satclass} ============================================= The first stage in separating the 32 [H$\alpha$]{}-emitting sources into true companions and outlying [H[ii]{}]{} regions was to examine their distributions of SF rate and $R$-band luminosity, as shown in Fig. \[fig:dsfvdrt\]. The range of SF rates is broadly consistent with those found in Magellanic dwarf galaxies, but in the $R$-band luminosity distribution there is already the hint of bimodality, with two components around $10^7$ and $10^{8.5}~L_{\odot}$. The Magellanic Clouds are included in this plot (crosses), with the SF rates having been calculated from the [H$\alpha$]{} data of @kenn86, using the same method of correcting for internal extinction as was applied to the SF rates of the putative satellite galaxies. Both Magellanic Clouds lie in the group of higher luminosity objects in Fig. \[fig:dsfvdrt\]. Figure \[fig:dgdvdRT\] shows the projected separation of the putative companion from the nucleus of the central galaxy, plotted against the companion $R$-band luminosity in solar units. This figure adds weight to the supposition that the candidate companions with lower luminosities in the $R$-band are [H[ii]{}]{} regions rather than separate galaxies, as they show a strong tendency to lie closer to the central galaxy than do the brighter sources. A further plot used to discriminate between [H[ii]{}]{} regions and satellite galaxies makes use of the SF timescale, a parameter explored extensively in paper V. The SF timescale is given by the total stellar mass of the galaxy divided by the current SF rate, such that a constant SF rate throughout a galaxy’s history would result in a SF timescale of a Hubble time. As explained in paper V, the total stellar mass is derived from $R$-band photometry, with an adopted $R$-band mass-to-light ratio for late-type dwarfs of 0.65, based on the models of [@bell01]. The timescale is increased by a factor of 1.67 to account for gas recycling of 40% of the mass of any new generation of stars [@vanz01]. As was concluded in paper V, Fig. \[fig:sftsat\] shows the field Sm and Im late-type galaxies from the [H$\alpha$]{}GS sample to scatter around a mean SF timescale of just under a Hubble time, indicating approximately constant SF activity in these galaxies. The crosses show the positions of the LMC and SMC within this distribution, where the Magellanic Cloud values are based on SF rates from the [H$\alpha$]{} measurements of [@kenn86]. Interestingly, both lie close to the centre of the range of SF timescales of the field galaxies, with values of $\sim$10–15 Gyr. The brightest, and hence highest mass, of the putative satellite galaxies also lie within the distribution of field galaxies shown in Fig. \[fig:sftsat\], and thus appear similar in their stellar masses and current SF activity to field Sm and Im galaxies. The one bright satellite which is displaced somewhat to a short SF timescale is the companion to UGC 4541, which appears tidally disturbed and may be undergoing an interaction-induced starburst or other nuclear activity. However, [*all*]{} of the putative companions with stellar masses below $\sim 5 \times 10^7~M_{\odot}$ have short SF timescales of $<$10$^{10}$ yr, and the average for these objects is $<$10$^9$ yr. This is most simply explained if these objects are SF regions in the outer disks of the central [H$\alpha$]{}GS galaxies, which would naturally have low SF timescales since they represent short-lived local enhancements in the SF rate of their host galaxies. This is confirmed by the position of known [H[ii]{}]{} regions from the disk regions of [H$\alpha$]{}GS galaxies, shown as stars in Fig. \[fig:sftsat\]. The existence of SF at a low level in the outer regions of disk galaxies has also been noted in UV imaging from the GALEX mission [@thil05; @gild05]. However, even though Figs. \[fig:dsfvdrt\] and \[fig:sftsat\] indicate that the 32 objects split into two groups as described in the previous paragraph, they are not completely conclusive. In particular, objects in the high luminosity/mass end of the distribution of probable outer [H[ii]{}]{} regions lie on the outskirts of the distribution of field Sm and Im galaxies in Fig. \[fig:sftsat\]. As a final test, the sizes of the objects were measured from our $R$-band and [H$\alpha$]{} images, and plotted in Figs. \[fig:Rsatsize\] and \[fig:Hasatsize\]. In both figures, the sizes were determined in a completely automated and objective fashion, using Full Width at Half Maximum (FWHM) values from the SExtractor package. The resulting $R$-band sizes are plotted against the distance to the central galaxy in Mpc, in Fig. \[fig:Rsatsize\]. This confirms that all 9 of the likely satellite galaxies are very extended in terms of their continuum emission, with FWHM sizes of 1.5 – 6.0 kpc. Many of the probable outer disk sources are too faint in the $R$-band to be picked up as separate sources by the SExtractor software; most of the remainder have $R$-band FWHM sizes between 0.2 and 1.0 kpc, identical to the range of sizes of disk [H[ii]{}]{} regions, also shown in Fig. \[fig:Rsatsize\] for comparison. However, two of the ‘outer disk’ objects do show significant extents in the $R$-band images. These are objects associated with UGC 3530 and UGC 4260, with $R$-band extents of 1.8 and 2.2 kpc respectively. Both objects are amongst the most massive of the 23 ‘outer disk’ objects, at just over 10$^7$ M$_{\odot}$, and have star-formation timescales of $\sim$1 and 2.5 Gyr respectively. Thus these may be satellite galaxies in a star-bursting phase, and we will keep them in our analysis as ‘possible’ satellites. Images of these objects are shown in Appendix A, with the putative satellites indicated by lines above and to one side of their location. The [H$\alpha$]{} sizes of objects, plotted on the y-axis in Fig. \[fig:Hasatsize\], show no significant dependence on object type. The SExtractor algorithm split the likely satellite galaxies into separate resolved [H[ii]{}]{} regions in several cases, and the resulting regions had FWHM sizes consistent with those of both the 23 outlying regions (all of which were detected in the [H$\alpha$]{} analysis), and a selection of disk [H[ii]{}]{} regions. The mean FWHM of 36 disk [H[ii]{}]{} regions was 0.55 kpc (standard deviation 0.19 kpc, standard error on the mean 0.03 kpc); for the 23 outlying regions, the mean was 0.55 kpc (s.d. 0.24 kpc, s.e. 0.05 kpc); and for the 9 probable satellites, which were detected as 17 [H[ii]{}]{} regions, the mean FWHM was 0.63 kpc (s.d. 0.29 kpc, s.e. 0.07 kpc). Henceforth, we consider only the 9 companion objects with $R$-band luminosities greater than $10^8$ L$_{\odot}$ to be ‘probable’ satellite galaxies, and the companions to UGC 3530 and UGC 4260 as ‘possible’ satellites. The remaining 21 objects are consistent with being outer disk [H[ii]{}]{} regions in terms of all the parameters considered here. However, they will be targetted for spectroscopic follow-up in future work (outlined in section \[sec:disc\] of this paper). [rlccccccccccc]{} UGC$_{\rm c}$ & Type$_{\rm c}$ & Dist & SFR$_{\rm c}$ & $\delta$SFR$_{\rm c}$ & $R_{\rm tot,c}$ & $\delta R_{\rm tot,c}$ & L$_{R,{\rm c}}$ & SFR$_{\rm s}$ & $\delta$SFR$_{\rm s}$ & L$_{R, {\rm s}}$ & $\delta$L$_{R, {\rm s}}$ & Sepn & & Mpc & M$_{\odot}$ yr$^{-1}$ & & mag & & L$_{\odot}$ & M$_{\odot}$ yr$^{-1}$ & & L$_{\odot}$ & & kpc 2603 & Im & 33.7 & 0.28 & 0.10 & 14.59 & 0.06 & 9.70(08) & 0.032 & 0.015 & 2.71(8) & 5.4(7) & 44.6\ 4273 & SBb & 35.4 & 1.92 & 0.36 & 11.84 & 0.04 & 1.36(10) & 0.018 & 0.004 & 5.81(8) & 3.0(7) & 35.0\ 4362 & S0a & 33.1 & 0.46 & 0.17 & 11.96 & 0.04 & 1.06(10) & 0.023 & 0.009 & 2.96(8) & 1.2(7) & 51.2\ 4469 & SBcd & 31.5 & 1.52 & 0.25 & 12.50 & 0.04 & 5.81(09) & 0.072 & 0.014 & 6.86(8) & 3.9(7) & 33.3\ 4541 & Sa & 31.4 & 0.21 & 0.06 & 11.37 & 0.04 & 1.63(10) & 0.478 & 0.197 & 9.21(8) & 5.0(7) & 21.7\ 4574 & SBb & 31.1 & 2.96 & 0.60 & 11.26 & 0.04 & 1.77(10) & 0.058 & 0.012 & 8.55(8) & 3.7(7) & 35.2\ 5688 & SBm & 29.2 & 0.44 & 0.08 & 13.57 & 0.05 & 1.87(09) & 0.097 & 0.021 & 4.81(8) & 5.2(7) & 16.5\ 6506 & SBd & 29.1 & 0.06 & 0.02 & 14.74 & 0.06 & 6.31(08) & 0.011 & 0.004 & 2.86(8) & 1.9(7) & 29.5\ 12788 & Sc & 32.8 & 1.54 & 0.30 & 12.51 & 0.04 & 6.25(09) & 0.004 & 0.001 & 1.16(8) & 7.6(6) & 9.5\ Properties of the 9 probable satellite galaxies {#sec:galprop} =============================================== The main data for the 9 probable satellite galaxies are presented in Table 1, which is organised as follows. Columns 1 and 2 give the UGC number and classification of the central galaxy, with the latter being taken from the NASA Extragalactic Database (NED henceforth). Column 3 gives the distance for the system, calculated from the central galaxy recession velocity, using a Virgo-infall corrected model [@paper1]. Columns 4 and 5 give the [H$\alpha$]{}-derived SF rate and error for the central galaxy, and columns 6 and 7 the $R$-band total magnitude and error. The corresponding $R$-band luminosity in solar units is given in column 8. Columns 9 and 10 give the SF rate and error for the satellite galaxy, calculated as described in section 2, and columns 11 and 12 the satellite galaxy $R$-band luminosity and error. The final column lists the projected separation in kpc between the centres of the central and satellite galaxies. This assumes that central and satellite galaxies are at the same distance, as listed in column 3. Images of the 9 galaxy systems are shown in Appendix B. In each case, the upper $R$-band image is a sufficently wide field to show both central and satellite galaxies, and again lines have been added to indicate the location of the satellites. The satellites are shown in more detail in the lower images, with the $R$-band image on the left and the continuum-subtracted [H$\alpha$]{} image on the right. In every case, even the upper image shows only a subset of the area searched for satellites. Our method of identifying companions as likely satellites relies on the detection of line emission in a narrow-band filter selected to include [H$\alpha$]{} from the central galaxy. This method is not foolproof, and it is possible that some of the putative satellites are background galaxies with shorter-wavelength lines redshifted into the same bandpass. One check for this is to search for previously catalogued recession velocities for the 9 satellites in the literature. The results of this search, using the NED “Search for Objects Near Object Name” facility, are given in Table 2. All bar one had names or identifiers in at least one catalogue or survey, listed in the second column of table 2; the angular separation between central and satellite galaxies is given in arcmin in column 3, and in kpc in column 4. Only 3 satellites had separate recession velocities; these are listed in column 5, with the central galaxy recession velocities being in the final column. All 3 satellite velocities lie within $\sim$100 km s$^{-1}$ of that of the central galaxy, which gives some confidence in our methods. Comments on individual galaxies:\ [**UGC 2603**]{} - the main galaxy is classified as an Im, so this can be considered a binary pair rather than a central plus satellite system. The satellite galaxy has substantially higher surface brightness in $R$-band continuum than the [H$\alpha$]{}GS galaxy, but fairly diffuse [H$\alpha$]{} emission.\ [**UGC 4273**]{} - the companion is one of the most weakly detected in [H$\alpha$]{} but this has an independent recession velocity listed in NED, which is very close to that of the central galaxy.\ [**UGC 4362**]{} - the companion is bright in the $R$-band image, and shows strong SF with clumpy, resolved [H[ii]{}]{} regions, but has not been previously catalogued according to NED. The companion is on the extreme edge of the CCD frame so the measured SF rate and $R$-band luminosities in Table 1 should be considered as lower limits, although the majority of the companion does appear to have been imaged, from inspection of sky survey images of this field.\ [**UGC 4469**]{} - the companion is clearly a Magellanic irregular, with clumpy [H$\alpha$]{} emission indicating several off-centre [H[ii]{}]{} regions. There is an independent recession velocity for the companion in NED, again very close to that of the the central galaxy.\ [**UGC 4541**]{} - this system appears to be undergoing significant tidal disturbance; unsurprisingly, the catalogued recession velocity of the companion is close to that of UGC 4541. The [H$\alpha$]{} morphology of the companion reveals a strong nuclear starburst and/or AGN activity; the strength of the [H$\alpha$]{} emission results in the short SF timescale found for this object in section 3.\ [**UGC 4574**]{} - the [H$\alpha$]{} image of the companion shows two clumps of SF, located at either end of the moderately elongated $R$-band light distribution.\ [**UGC 5688**]{} - the central galaxy has low $R$-band luminosity and surface brightness. The [H$\alpha$]{} image shows clumps of emission from both ends of the elongated $R$-band light distribution, like the companion of UGC 4574, resembling the pattern of SF seen in many galaxy bars.\ [**UGC 6506**]{} - this is a similar case to UGC 2603 and UGC 5688, with a low luminosity central galaxy, and this should probably be considered a binary system rather than as a central galaxy with a satellite.\ [**UGC 12788**]{} - the putative companion is projected on the outer disk of the central galaxy, but does not look like an HII region or part of a spiral arm. This is a possible case of tidal interaction, revealed in the distorted and strongly star-forming arm in UGC 12788, close to the putative companion. However, it should be noted that the object considered here to be a companion galaxy is listed in NED as ‘West [H[ii]{}]{} region in UGC 12788’.\ Two of the central/satellite systems studied by @zari97 are in the current sample. The first is NGC 5921 and and its companion, identified by @zari97 as NGC 5921:\[ZSF97\]b. This companion is clearly detected in our data, both in $R$-band and [H$\alpha$]{} emission, and indeed was independently selected as a putative satellite (it is the square at SF rate $=$0.012 M$_{\odot}$ yr$^{-1}$, $R$-band luminosity 4 $\times 10^6$ L$_{\odot}$ in Fig. \[fig:dsfvdrt\]). However, this region, along with one other similar region in NGC 5921 (not identified by @zari97), was considered highly likely to be an outer [H[ii]{}]{} region. The second of the systems listed by @zari97 is NGC 5962 and the companion they identify as NGC 5962:\[ZSF97\]b. The region containing the latter was searched in the present study, but the companion is present only as a very faint $R$-band source, with no detected [H$\alpha$]{} emission. The NED “Search for Objects Near Object Name” was also carried out for the 23 outer disk objects. None of these was found to a measured recession velocity, and the only two to have individual NED entries are both classified as [H[ii]{}]{} regions, one in UGC 9935/NGC 5964 [@brad06], and one in UGC 12343/NGC 7479 [@roza99]. [rlcccc]{} UGC$_{\rm c}$ & Satellite name & Separation & Separation & V$_{\rm rec}$ sat & V$_{\rm rec}$ cent & & (arcmin) & (kpc) & (km s$^{-1}$) & (km s$^{-1}$) 2603 & 2MASXJ03192345+8116238 & 4.4 & 43.1 & – & 2516\ 4273 & KUG0809+363 & 3.5 & 36.0 & 2483 & 2471\ 4362 & – & – & – & – & 2344\ 4469 & NGC 2406B & 3.6 & 33.0 & 2104 & 2078\ 4541 & CGCG 060-036 & 2.4 & 21.9 & 2115 & 2060\ 4574 & 2MASXJ08482381+7402176 & 3.9 & 35.3 & – & 2160\ 5688 & VV 294b & 1.9 & 16.1 & – & 1920\ 6506 & MAPS-NGP O$\_$319$\_$1199567 & 3.6 & 30.5 & – & 1580\ 12788 & UM 007 NED01 & 1.0 & 9.5 & – & 2956\ Statistics of star forming satellites {#sec:galstats} ===================================== The next stage of the analysis is to look at the numbers of star-forming satellites found, to put constraints on the overall abundance of such systems around field galaxies. Given that the regions around 119 [H$\alpha$]{}GS galaxies were searched, 9 satellites found seems a small number, given that we have two such systems around the Milky Way, but in order to make this comparison quantitative we need to correct for sources of incompleteness in our search method. As a starting point for this analysis, we need to confirm that our data and methods are sufficiently sensitive to detect ‘Magellanic Cloud like’ satellites. Figure \[fig:dsfvdmpc\] shows the [H$\alpha$]{}-derived SF rates for 32 regions identified in the current paper as possible satellites, with the ringed points identifying the 9 likely satellites. These rates are plotted against the distance in Mpc of the central galaxy in the system, and the horizontal lines show the SF rates of the Magellanic Clouds. This confirms that our H$\alpha$ technique is easily sensitive enough to detect star forming galaxies fainter than the Magellanic Clouds to the spatial limits of our survey, as objects with [H$\alpha$]{} luminosities significantly lower than that of the SMC are seen over the full range of distances studied. Surface brightness is also important in determining the detectability of galaxies; it is possible that Magellanic-type companions with quite high total SF rates could be missed if the line emission were very extended and hence of low surface brightness. This possibility is investigated in Fig. \[fig:dsbvdmpc\], which is similar to Fig. \[fig:dsfvdmpc\], but with the SF rate of each region divided by its area in kpc$^2$, measured from our [H$\alpha$]{} images. The same quantity for the Magellanic Clouds is again shown by the dashed lines. The latter values were derived from @kenn86, who give [H$\alpha$]{} flux measurements measured in large apertures comparable in kpc sizes to the apertures used to measure the [H$\alpha$]{} fluxes in our companion objects. Specifically, @kenn86 find 25% of the [H$\alpha$]{} flux from the LMC to come from a 30 arcmin aperture centred on 30 Doradus; this would be both spatially resolved at the distance of our satellite galaxies, and easily detectable in surface brightness as shown from the position of the upper dashed line in Fig. \[fig:dsbvdmpc\]. Similarly, the lower dashed line comes from a 16 arcmin aperture flux measurement on the SMC, which contains 17% of the total [H$\alpha$]{} emission and hence of the inferred SF activity in that galaxy. Thus we can again conclude that Magellanic-type galaxies should be easily detectable with our data and techniques. In order to calculate incompleteness corrections, it is necessary to decide on a definition of a close companion galaxy. For the present work, a ‘Magellanic-type’ companion is defined to be one that lies within a spherical region, centred on the UGC galaxy, with a volume twice as large as the volume that just contains both the Magellanic Clouds. This corresponds to a radial separation of less than $60 \times \sqrt[3] 2 = 75.2$kpc, where 60 kpc is our adopted distance to the SMC. However, for any given galaxy, it is not possible to see the full volume potentially occupied by satellites. For closer galaxies, the outlying parts of the volume are missed as they lie outside the CCD field of view. In addition, there is a ‘blank spot’ along the line-of-sight of the main galaxy, in all cases, as the [H$\alpha$]{} emission from any companion would be confused with that of the central galaxy, in our narrow-band images. Knowing the distance of each of the central galaxies, their optical major and minor axes, and the size of the imaged field, it is then possible to calculate the fraction of the spherical halo volume that is missed due to these two causes. This fraction was then assumed to be equal to the fraction of satellites that were missed, i.e. implicitly assuming that satellites were equally likely to be found anywhere within the spherical volume. In support of this assumption, several studies have found satellite distributions to be isotropic, for blue central galaxies such as those studied here [@yang06; @agus07; @azza07]. A programme was written to calculate the completeness fraction for all 119 of the [H$\alpha$]{}GS galaxies, and the results are summarised in Table \[tbl:satstats\]. This lists, as a function of Hubble $T$-type, the total numbers of galaxies observed (column 2); the sums of the observable fractions of the satellite volumes (column 3); the number of star-forming satellites found (column 4); and ratio of satellites found to observable volumes, i.e. column 4 divided by column 3 (listed in the final column). To summarise these results, the 119 potentially satellite-containing volumes were covered with a mean efficiency of just below 50%, resulting in an effective search of 53 volumes. This search yielded 9 star-forming satellites, or 0.17$^{+0.08}_{-0.06}$ satellite galaxies per full volume searched. This fraction applies whether all T-types included, or just the more luminous galaxies with $T$ types in the range 0 to 5 (bottom and penultimate lines of Table \[tbl:satstats\]). The errors quoted here and in the final column of Table \[tbl:satstats\] are 1-$\sigma$ limits derived from the tables of @gehr86. It could be argued that the companion to UGC 12788, which lies inside the projected disk of UGC 12788 but is detected due to its distinctive morphology, should be excluded from these statistics. It should also be noted that some of the detected satellites may actually lie outside the 75.2 kpc radius circle due to projection uncertainties. Thus the fractions listed in the final column of Table \[tbl:satstats\] could be considered as upper limits for the numbers of ‘Magellanic Cloud type’ companions as we have defined them here. However, we should also take into account to possibility that some of the objects classified as outlying [H[ii]{}]{} regions may in fact be satellite galaxies. If the two ‘possible’ objects identified from their $R$-band sizes in Fig. \[fig:Rsatsize\] are included, the overall fraction of satellites per central galaxy search increases from 0.17 to 0.21$^{+0.08}_{-0.06}$. If we very conservatively include all the outlying objects not positively identified elsewhere as being [H[ii]{}]{} regions, the total number increases to 30, for an overall fraction of satellites per galaxy searched of 0.57$^{+0.12}_{-0.10}$. [crrcc]{} $T$-type & N$_{\rm Gal}$ & N$_{\rm Corr}$ & N$_{\rm Sat}$ & N$_{\rm Sat}$/N$_{\rm Corr}$ 0 & 6 & 2.4 & 1 & 0.42$^{+0.97}_{-0.35}$\ 1 & 7 & 3.5 & 1 & 0.29$^{+0.67}_{-0.24}$\ 2 & 4 & 2.1 & 0 & 0.00$^{+0.88}_{-0.00}$\ 3 & 15 & 7.2 & 2 & 0.28$^{+0.37}_{-0.18}$\ 4 & 15 & 6.8 & 0 & 0.00$^{+0.27}_{-0.00}$\ 5 & 14 & 5.9 & 1 & 0.17$^{+0.39}_{-0.14}$\ 6 & 15 & 6.5 & 1 & 0.15$^{+0.35}_{-0.12}$\ 7 & 14 & 5.7 & 1 & 0.18$^{+0.41}_{-0.15}$\ 8 & 13 & 5.5 & 0 & 0.00$^{+0.33}_{-0.00}$\ 9 & 6 & 2.4 & 1 & 0.42$^{+0.97}_{-0.35}$\ 10 & 10 & 4.6 & 1 & 0.22$^{+0.51}_{-0.18}$\ 0-5 & 61 & 28.0 & 5 & 0.18$^{+0.12}_{-0.08}$\ 0-10 &119 & 52.7 & 9 & 0.17$^{+0.08}_{-0.06}$\ Correlation of satellite and central galaxy properties {#sec:galcorr} ====================================================== The search carried out in the present study has identified 9 probable satellite galaxies, with significant stellar masses, in close proximity (in projection) to their central galaxies. Given these properties, it is interesting to check whether the SF activity, either of the central galaxies or of the probable companions, is affected by the tidal forces between central and satellite galaxies. The overall question of the effect of environment on SF rates for the entire [H$\alpha$]{}GS sample is the subject of a future paper, so this analysis concerns only the subsample of galaxies with identified companions. Figure \[fig:gsfvdgd\] shows the [H$\alpha$]{}-derived SF rates of the central galaxies plotted against projected galaxy-companion separation in kpc. The ringed points indicate the 9 probable satellite systems as identified in section 3. This figure shows no correlation between central galaxy SF rate and projected separation, so there is no obvious effect of the presence of the satellites on the their central galaxies. Similarly, Fig. \[fig:dsfvdgd\] shows no correlation between satellite galaxy SF rate and projected distance from the central galaxy, at least for the 9 probable satellites. There is a significant difference between these probable satellites and the objects identified in section 3 as outlying [H[ii]{}]{} regions, but this is as expected under our preferred interpretation of these objects. Finally, Fig. \[fig:dsftvdgd\] shows companion object SF timescales, as defined in section 3, against projected separation between central and satellite galaxies. Again, the only clear result from this plot is the short SF timescales of the outlying [H[ii]{}]{} regions, already identified in Fig. \[fig:sftsat\]. For the 9 probable satellites, no correlation is found between SF timescale and projected separation from their central galaxies. The satellite galaxy with the shortest SF timescale is the companion to UGC 4541; this not the closest galaxy-satellite pair investigated here, but there does appear to be tidal distortion associated with this interaction. The closest pairing in projected separation is UGC 12788 and its companion; here the companion appears completely unaffected by tidal effects, in terms of both optical morphology and SF properties, so it is possible that there is a significant line-of-sight separation in this case. The second closest in apparent separation are UGC 5688 and companion; the latter has a SF timescale below 10 Gyr, which may show a modest enhancement in SF rate as a the result of tidal effects. However, UGC 5688 is a low luminosity galaxy of type SBm, and so tidal effects are likely to be minimal due to the low mass of the primary galaxy. Discussion and future work {#sec:disc} ========================== The major result of this paper is that actively star-forming satellite galaxies, with luminosities and SF rates within approximately an order of magnitude of those of the Magellanic Clouds, are fairly rare. Most galaxies in our survey of field spirals have no such satellites. Previous searches for satellites of all types [@zari97; @mado04] have found significantly more than this, typically one substantial satellite per central galaxy surveyed, although it is very difficult to make quantitative comparisons because of the varying depths and methods of the different surveys. However, it does appear likely that most satellites, even of isolated field galaxies, are of non-star forming types. Examples close to home are M 32 and NGC 205, early type companions of the Andromeda spiral M 31. Given that most isolated galaxies of Magellanic-Cloud like luminosities are actively forming stars (paper V), this would seem to imply the efficient truncation of star formation in satellites, with a timescale short compared to a Hubble time. This then raises interesting questions about the fate of the gas originally in the satellite dwarfs; is it consumed in a burst of star formation triggered by interaction with the central galaxy, or is a significant fraction of it expelled from the satellite, enabling it to be accreted onto the central galaxy? Evidence of this process in the Milky Way system may be provided by the Magellanic Stream. Given the importance of gas supply in disk galaxy evolution, it would clearly be useful to know the frequency of such gas-yielding interactions, and the timescales and gas masses involved. Observationally, there are 3 main requirements: to extend the sample size of star-forming satellite candidates; to discriminate definitively between true satellites and outlying parts of disks; and to determine the red-to-blue satellite fraction. The first can simply be done by further [H$\alpha$]{} imaging, preferably going rather deeper and over wider fields than the present data set. The second requires spectroscopy to determine velocity differences between central and putative satellite galaxies, and preferably velocity cubes (e.g. from [H$\alpha$]{} Fabry-Perot instruments or [H[i]{}]{} maps) to look for signs of interaction. Data enabling the identification of quiescent red-sequence satellites are problematic given the greater difficulty of measuring recession velocities for faint and possibly low-surface-brightness absorption line sources, but the situation is being improved with surveys using sensitive multi-fibre instruments. We are actively pursuing all of these approaches. One caveat on the current results concerns the conclusion that the 9 satellites are fairly similar to the Magellanic Clouds. This can be seen as unsurprising given that we used the similarity to the Magellanic Clouds, particularly in Figs. \[fig:dsfvdrt\] and \[fig:sftsat\], to argue for the 9 objects being true satellites. There is indeed circularity in this argument, and it is clearly important to include the provisionally rejected objects in follow-up spectroscopy. Given that the latter tend to be found at small projected separations from the central galaxies and have similar properties to [H[ii]{}]{} regions in Fig. \[fig:sftsat\], it is likely that this interpretation will be confirmed for most or all of them, but there may be some important objects lurking in this category. Conclusions {#sec:conc} =========== This study has identified 9 probable star-forming satellite galaxies with projected separations consistent with their being as close to their central galaxies as the Magellanic Clouds are to the Milky Way. Figure \[fig:dsfvdrt\] illustrates that the stellar luminosities and SF rates of the Magellanic Clouds are comparable to those of the 9 probable satellites found here. Overall, the satellite galaxies (including the Magellanic Clouds) are currently forming stars at a rate comparable to field Sm and Im galaxies in the [H$\alpha$]{}GS sample. The only evidence of a strong starburst is in the tidally-disturbed companion to UGC 4541. Considering the 9 probable satellites and the Magellanic Clouds together, the LMC and SMC are the brightest and 7th brightest in $R$-band luminosity, and the 2nd and 9th most rapidly star-forming. Thus the LMC is clearly a large satellite, whereas the SMC is close to or just below average amongst those found here. We find no cases of 2 satellites around any of the 119 central galaxies studied, so the Milky Way appears well-favoured in the number of large star-forming companions in its immediate neighbourhood. In this context, it is interesting to note that @puec07 have recently concluded that the MW is distinctive for its [*lack*]{} of merging activity over its history; the results found here indicate that any such deficiency is likely to be rectified in the future. The Jacobus Kapteyn Telescope was operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This research has made extensive use of 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. Jane O’Neill contributed to this work while on a summer placement supported by the Nuffield Science Bursary Scheme. PAJ thanks Chris Moss and Sue Percival for useful comments, and the referee is also thanked for many constructive suggestions. Images of the 2 possible satellites and their central galaxies ============================================================== ![ Upper images: UGC 3530 and satellite galaxy, showing the full field of view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower image: UGC 3530 and satellite galaxy, close-up view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u3530"}](9297a1a.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 3530 and satellite galaxy, showing the full field of view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower image: UGC 3530 and satellite galaxy, close-up view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u3530"}](9297a1b.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 3530 and satellite galaxy, showing the full field of view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower image: UGC 3530 and satellite galaxy, close-up view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u3530"}](9297a1c.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 3530 and satellite galaxy, showing the full field of view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower image: UGC 3530 and satellite galaxy, close-up view (image size 160$^{\prime \prime}$ by 160$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u3530"}](9297a1d.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 4260 and satellite galaxy, showing the full field of view (image size 560$^{\prime \prime}$ by 560$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower images: UGC 4260 and satellite galaxy, close-up view (image size 154$^{\prime \prime}$ by 154$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u4260"}](9297a2a.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 4260 and satellite galaxy, showing the full field of view (image size 560$^{\prime \prime}$ by 560$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower images: UGC 4260 and satellite galaxy, close-up view (image size 154$^{\prime \prime}$ by 154$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u4260"}](9297a2b.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 4260 and satellite galaxy, showing the full field of view (image size 560$^{\prime \prime}$ by 560$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower images: UGC 4260 and satellite galaxy, close-up view (image size 154$^{\prime \prime}$ by 154$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u4260"}](9297a2c.ps "fig:"){height="4.0cm" width="4.0cm"} ![ Upper images: UGC 4260 and satellite galaxy, showing the full field of view (image size 560$^{\prime \prime}$ by 560$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). Lower images: UGC 4260 and satellite galaxy, close-up view (image size 154$^{\prime \prime}$ by 154$^{\prime \prime}$); $R$-band (left) and [H$\alpha$]{} (right). []{data-label="fig:u4260"}](9297a2d.ps "fig:"){height="4.0cm" width="4.0cm"} Images of the 9 probable satellites and their central galaxies ============================================================== ![ Upper image: UGC 2603 and satellite galaxy (image size 343$^{\prime \prime}$ by 325$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 47$^{\prime \prime}$ by 47$^{\prime \prime}$). []{data-label="fig:u2603"}](9297b1a.ps "fig:"){height="4.0cm"}\ ![ Upper image: UGC 2603 and satellite galaxy (image size 343$^{\prime \prime}$ by 325$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 47$^{\prime \prime}$ by 47$^{\prime \prime}$). []{data-label="fig:u2603"}](9297b1b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 2603 and satellite galaxy (image size 343$^{\prime \prime}$ by 325$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 47$^{\prime \prime}$ by 47$^{\prime \prime}$). []{data-label="fig:u2603"}](9297b1c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4273 and satellite galaxy (image size 338$^{\prime \prime}$ by 335$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 56$^{\prime \prime}$ by 56$^{\prime \prime}$). []{data-label="fig:u4273"}](9297b2a.ps "fig:"){height="4.0cm"}\ ![ Upper image: UGC 4273 and satellite galaxy (image size 338$^{\prime \prime}$ by 335$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 56$^{\prime \prime}$ by 56$^{\prime \prime}$). []{data-label="fig:u4273"}](9297b2b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4273 and satellite galaxy (image size 338$^{\prime \prime}$ by 335$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 56$^{\prime \prime}$ by 56$^{\prime \prime}$). []{data-label="fig:u4273"}](9297b2c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4362 and satellite galaxy (image size 359$^{\prime \prime}$ by 177$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 44$^{\prime \prime}$ by 44$^{\prime \prime}$). []{data-label="fig:u4362"}](9297b3a.ps){height="4.0cm"} ![ Upper image: UGC 4362 and satellite galaxy (image size 359$^{\prime \prime}$ by 177$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 44$^{\prime \prime}$ by 44$^{\prime \prime}$). []{data-label="fig:u4362"}](9297b3b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4362 and satellite galaxy (image size 359$^{\prime \prime}$ by 177$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 44$^{\prime \prime}$ by 44$^{\prime \prime}$). []{data-label="fig:u4362"}](9297b3c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4469 and satellite galaxy (image size 248$^{\prime \prime}$ by 220$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u4469"}](9297b4a.ps){height="4.0cm"} ![ Upper image: UGC 4469 and satellite galaxy (image size 248$^{\prime \prime}$ by 220$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u4469"}](9297b4b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4469 and satellite galaxy (image size 248$^{\prime \prime}$ by 220$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u4469"}](9297b4c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4541 and satellite galaxy (image size 196$^{\prime \prime}$ by 161$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 78$^{\prime \prime}$ by 78$^{\prime \prime}$). []{data-label="fig:u4541"}](9297b5a.ps){height="4.0cm"} ![ Upper image: UGC 4541 and satellite galaxy (image size 196$^{\prime \prime}$ by 161$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 78$^{\prime \prime}$ by 78$^{\prime \prime}$). []{data-label="fig:u4541"}](9297b5b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4541 and satellite galaxy (image size 196$^{\prime \prime}$ by 161$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 78$^{\prime \prime}$ by 78$^{\prime \prime}$). []{data-label="fig:u4541"}](9297b5c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4574 and satellite galaxy (image size 318$^{\prime \prime}$ by 309$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 53$^{\prime \prime}$ by 53$^{\prime \prime}$). []{data-label="fig:u4574"}](9297b6a.ps){height="4.0cm"} ![ Upper image: UGC 4574 and satellite galaxy (image size 318$^{\prime \prime}$ by 309$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 53$^{\prime \prime}$ by 53$^{\prime \prime}$). []{data-label="fig:u4574"}](9297b6b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 4574 and satellite galaxy (image size 318$^{\prime \prime}$ by 309$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 53$^{\prime \prime}$ by 53$^{\prime \prime}$). []{data-label="fig:u4574"}](9297b6c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 5688 and satellite galaxy (image size 160$^{\prime \prime}$ by 154$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u5688"}](9297b7a.ps){height="4.0cm"} ![ Upper image: UGC 5688 and satellite galaxy (image size 160$^{\prime \prime}$ by 154$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u5688"}](9297b7b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 5688 and satellite galaxy (image size 160$^{\prime \prime}$ by 154$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 51$^{\prime \prime}$ by 51$^{\prime \prime}$). []{data-label="fig:u5688"}](9297b7c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 6506 and satellite galaxy (image size 212$^{\prime \prime}$ by 206$^{\prime \prime}$). The central part of this image suffered contamination from a band of scattered light in a diagonal strip between the two galaxies; this has been removed in the data reduction, along with some stars which lie in the centre of this field. The relative brightness and orientation of the two galaxies are not affected. Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 54$^{\prime \prime}$ by 54$^{\prime \prime}$). []{data-label="fig:u6506"}](9297b8a.ps){height="4.0cm"} ![ Upper image: UGC 6506 and satellite galaxy (image size 212$^{\prime \prime}$ by 206$^{\prime \prime}$). The central part of this image suffered contamination from a band of scattered light in a diagonal strip between the two galaxies; this has been removed in the data reduction, along with some stars which lie in the centre of this field. The relative brightness and orientation of the two galaxies are not affected. Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 54$^{\prime \prime}$ by 54$^{\prime \prime}$). []{data-label="fig:u6506"}](9297b8b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 6506 and satellite galaxy (image size 212$^{\prime \prime}$ by 206$^{\prime \prime}$). The central part of this image suffered contamination from a band of scattered light in a diagonal strip between the two galaxies; this has been removed in the data reduction, along with some stars which lie in the centre of this field. The relative brightness and orientation of the two galaxies are not affected. Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 54$^{\prime \prime}$ by 54$^{\prime \prime}$). []{data-label="fig:u6506"}](9297b8c.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 12788 and satellite galaxy (image size 113$^{\prime \prime}$ by 82$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 33$^{\prime \prime}$ by 33$^{\prime \prime}$). []{data-label="fig:u12788"}](9297b9a.ps){height="4.0cm"} ![ Upper image: UGC 12788 and satellite galaxy (image size 113$^{\prime \prime}$ by 82$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 33$^{\prime \prime}$ by 33$^{\prime \prime}$). []{data-label="fig:u12788"}](9297b9b.ps "fig:"){height="4.0cm"} ![ Upper image: UGC 12788 and satellite galaxy (image size 113$^{\prime \prime}$ by 82$^{\prime \prime}$). Lower images: Satellite galaxy in $R$-band (left) and continuum-subtracted [H$\alpha$]{} (right: image sizes 33$^{\prime \prime}$ by 33$^{\prime \prime}$). []{data-label="fig:u12788"}](9297b9c.ps "fig:"){height="4.0cm"} [^1]: Based on observations made with the Jacobus Kapteyn Telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias
--- author: - | Arun Jambulapati\ Stanford University\ `[email protected]` - | Yang P. Liu\ Stanford University\ `[email protected]` [^1] - | Aaron Sidford\ Stanford University\ `[email protected]` [^2] bibliography: - 'refs.bib' title: Parallel Reachability in Almost Linear Work and Square Root Depth --- [^1]: Research supported by the U.S. Department of Defense via an NDSEG fellowship. [^2]: Research supported by NSF CAREER Award CCF-1844855.
--- abstract: 'Contact-rich manipulation tasks in unstructured environments often require both haptic and visual feedback. It is non-trivial to manually design a robot controller that combines these modalities which have very different characteristics. While deep reinforcement learning has shown success in learning control policies for high-dimensional inputs, these algorithms are generally intractable to deploy on real robots due to sample complexity. In this work, we use self-supervision to learn a compact and multimodal representation of our sensory inputs, which can then be used to improve the sample efficiency of our policy learning. Evaluating our method on a peg insertion task, we show that it generalizes over varying geometries, configurations, and clearances, while being robust to external perturbations. We also systematically study different self-supervised learning objectives and representation learning architectures. Results are presented in simulation and on a physical robot.' author: - 'Michelle A. Lee, Yuke Zhu, Peter Zachares, Matthew Tan, Krishnan Srinivasan, Silvio Savarese, Li Fei-Fei, Animesh Garg, Jeannette Bohg [^1]' bibliography: - 'multimodal.bib' title: 'Making Sense of Vision and Touch: Learning Multimodal Representations for Contact-Rich Tasks' --- Deep Learning in Robotics and Automation, Perception for Grasping and Manipulation, Sensor Fusion, Sensor-based Control [^1]: ------------------------------------------------------------------------ Authors are with the Department of Computer Science, Stanford University. [@stanford.edu]{}. A. Garg is also at Nvidia, USA.
--- abstract: 'We present an HST/ACS weak gravitational lensing analysis of 13 massive high-redshift () galaxy clusters discovered in the South Pole Telescope (SPT) Sunyaev-Zel’dovich Survey. This study is part of a larger campaign that aims to robustly calibrate mass-observable scaling relations over a wide range in redshift to enable improved cosmological constraints from the SPT cluster sample. We introduce [new strategies]{} to ensure that systematics in the lensing analysis do not degrade constraints on cluster scaling relations significantly. First, we efficiently remove cluster members from the source sample by selecting very blue galaxies in colour. Our estimate of the source redshift distribution is based on CANDELS data, where we carefully mimic the source selection criteria of the cluster fields. We apply a statistical correction for systematic photometric redshift errors as derived from [*Hubble*]{} Ultra Deep Field data and verified through spatial cross-correlations. We account for the impact of lensing magnification on the source redshift distribution, finding that this is particularly relevant for shallower surveys. Finally, we account for biases in the mass modelling caused by miscentring and uncertainties in the [concentration–mass]{} relation using simulations. [In combination with temperature estimates from *Chandra* we constrain the normalisation of the mass–temperature scaling relation $\ln\left(E(z) M_\mathrm{500c}/10^{14}\mathrm{M}_\odot \right)=A+1.5\ln\left(kT/7.2\mathrm{keV}\right)$ to , consistent with self-similar redshift evolution when compared to lower redshift samples.]{} Additionally, the lensing data constrain the average concentration of the clusters to .' author: - | T. Schrabback[$^{\Bonn,\StanfordKIPAC,\StanfordPhysics}$]{}[^1], D. Applegate[$^{\Bonn,\KICPChicago}$]{}, J. P. Dietrich[$^{\Munich,\ExcellenceCluster}$]{}, H. Hoekstra[$^{\Leiden}$]{}, S. Bocquet[$^{\KICPChicago,\ANL,\Munich,\ExcellenceCluster}$]{}, A. H. Gonzalez[$^{\UFlorida}$]{}, A. von der Linden[$^{\StanfordKIPAC,\StanfordPhysics,\DARK,\StonyBrook}$]{}, M. McDonald[$^{\MIT}$]{}, C. B. Morrison[$^{\Bonn,\Washington}$]{}, S. F. Raihan[$^{\Bonn}$]{}, S. W. Allen[$^{\StanfordKIPAC,\StanfordPhysics,\SLAC}$]{}, M. Bayliss[$^{\Harvard,\CfA,\Colby}$]{}, B. A. Benson[$^{\FNAL,\AAUChicago,\KICPChicago}$]{}, L. E. Bleem[$^{\KICPChicago,\PhysicsUChicago,\ANL}$]{}, I. Chiu[$^{\Munich,\ExcellenceCluster,\ASIAA}$]{}, S. Desai[$^{\Munich,\ExcellenceCluster,\Hyderabad}$]{}, R. J. Foley[$^{\UCStCruz}$]{}, T. de Haan[$^{\McGill,\Berkeley}$]{}, F. W. High[$^{\KICPChicago,\AAUChicago}$]{}, S. Hilbert[$^{\Munich,\ExcellenceCluster}$]{}, A. B. Mantz[$^{\StanfordKIPAC,\StanfordPhysics}$]{}, R. Massey[$^{\Durham}$]{}, J. Mohr[$^{\Munich,\ExcellenceCluster,\MPE}$]{}, C. L. Reichardt[$^{\Melbourne}$]{}, A. Saro[$^{\Munich,\ExcellenceCluster}$]{}, P. Simon[$^{\Bonn}$]{}, C. Stern[$^{\Munich,\ExcellenceCluster}$]{}, C. W. Stubbs[$^{\Harvard,\CfA}$]{}, A. Zenteno[$^{\CTIO}$]{}\ \ bibliography: - 'oir.bib' title: 'Cluster Mass Calibration at High Redshift: HST Weak Lensing Analysis of 13 Distant Galaxy Clusters from the South Pole Telescope Sunyaev-Zel’dovich Survey' --- gravitational lensing: weak – cosmology: observations – galaxies: clusters: general Introduction {#sec:intro} ============ Constraints on the number density of clusters as a function of their mass and redshift probe the growth of structure in the Universe, therefore holding great promise to constrain cosmological models [e.g. @haiman01; @allen11; @weinberg13]. Previous studies using samples of at most a few hundred clusters have delivered some of the tightest cosmological constraints currently available on dark energy properties, theories of modified gravity, and the species-summed neutrino mass (e.g.@vikhlinin09; @rapetti09 [@rapetti13]; @schmidt09; @mantz10 [@mantz15]; @bocquet15a; @dehaan16). Recently, CMB experiments have begun to substantially increase the number of massive, high-redshift clusters found with well-characterised selection functions, detected via their Sunyaev-Zel’dovich [SZ, @sunyaev70; @sunyaev72] signature from inverse Compton scattering off the electrons in the hot cluster plasma [@hasselfield13; @bleem15; @planck15sz]. Upcoming experiments such as SPT-3G [@benson14] and eROSITA [@merloni12] are expected to soon provide samples of $10^4$–$10^5$ massive clusters with well-characterised selection functions, yielding a statistical constraining power that may mark the transition between “Stage III” and “Stage IV” dark energy constraints [see @albrecht06] from clusters if systematic uncertainties are well controlled. Cluster observables such as X-ray luminosity, SZ signal, or optical/NIR richness and luminosity have been shown to scale with mass [e.g. @reiprich02; @lin04a; @andersson11]. In order to adequately exploit the statistical constraining power of large cluster surveys, an accurate and precise calibration of the scaling relations between such mass proxies and mass is needed. Already for current surveys cosmological constraints are primarily limited by uncertainties in the calibration of mass–observable scaling relations [e.g. @rozo10; @sehgal11; @benson13; @vonderlinden14b; @mantz15; @planck15szconstraints]. It is therefore imperative to improve this calibration empirically. In this context our work focuses especially on calibrating mass–observable relations at high redshifts, which together with low-redshift measurements, provides constraints on their redshift evolution. Particularly for constraints on dark energy properties, which are primarily derived from the redshift evolution of the cluster mass function, it is critical to ensure that systematic errors in the evolution of mass–observable scaling relations do not mimic the signature of dark energy. Most previous cosmological cluster studies had to rely on priors for the redshift evolution derived from numerical cluster simulations [e.g. @vikhlinin09; @benson13; @dehaan16]. It is crucial to test the assumed models of cluster astrophysics in these simulations by comparing their predictions to observational constraints on the scaling relations [e.g. @lebrun14], and to shrink the uncertainties on the scaling relation parameters. Progress in the field critically requires improvements in the cluster mass calibration through large multi-wavelength follow-up campaigns. For example, high-resolution X-ray observations provide mass proxies with low intrinsic scatter, which can be used to constrain the relative masses of clusters [e.g. @vikhlinin09b; @reichert11; @andersson11]. On the other hand, weak gravitational lensing has been recognised as the most direct technique for the absolute calibration of the normalisation of cluster mass observable relations [@allen11; @hoekstra13; @applegate14; @mantz15]. The main observable is the weak lensing reduced shear, a tangential distortion caused by the projected tidal gravitational field of the foreground mass distribution. It is directly related to the differential projected cluster mass distribution, and can be estimated from the observed shapes of background galaxies [e.g. @bartelmann01; @schneider06]. To date, the majority of cluster weak lensing mass estimates have been obtained for lower redshift clusters () using ground-based observations [e.g. @high12; @israel12; @oguri12; @applegate14; @gruen14; @umetsu14; @hoekstra15; @ford15; @kettula15; @battaglia16; @lieu16; @vanuitert16; @okabe16; @simet17; @melchior17]. To constrain the evolution of cluster mass-observable scaling relations, these measurements need to be complimented with constraints for higher redshift clusters. Here, ground-based measurements suffer from low densities of sufficiently resolved background galaxies with robust shape measurements. This can be overcome using high-resolution [*Hubble Space Telescope*]{} (HST) images, where so far @jee11 present the only weak lensing constraints for the cluster mass calibration of a large sample of massive high-redshift () clusters, which were drawn from optically, NIR, and X-ray-selected samples. Interestingly, their results suggest a possible evolution in the $M_\mathrm{2500c}-T_\mathrm{X}$ scaling relation in comparison to self-similar extrapolations from low redshifts, with lower masses at the level. HST weak lensing measurements have also been used to constrain mass-observable scaling relations for lower [@leauthaud10] and intermediate mass clusters [@hoekstra11b]. This paper is part of a larger effort to obtain improved observational constraints on the calibration of cluster masses as function of redshift. Here we analyse new HST observations of 13 massive high-$z$ clusters detected by the South Pole Telescope [@carlstrom11] via the SZ effect. This constitutes the first high-$z$ sample of clusters with HST weak lensing observations which were drawn from a single, well-characterised survey selection function. As a major part of this paper, we carefully investigate and account for the relevant sources of systematic uncertainty in the weak lensing mass analysis, and discuss their relevance for future studies of larger samples. The primary technical challenges for weak lensing studies are accurate measurements of galaxy shapes from noisy data in the presence of instrumental distortions, and the need for an accurate knowledge of the source redshift distribution which enters through the geometric lensing efficiency. Within the weak lensing community substantial progress has been made on the former issue through the development of improved shape measurement algorithms tested using image simulations [e.g. @miller13; @hoekstra15; @bernstein16; @fenechconti17]. For the latter issue, previous studies have typically estimated the redshift distribution from photometric redshifts (photo-$z$s) given the incompleteness of spectroscopic redshift samples (spec-$z$s) at the relevant magnitudes, requiring that the photo-$z$-based estimates are sufficiently accurate. If sufficient wavelength coverage is available, photo-$z$s can be estimated directly for the weak lensing survey fields of interest [used in the cluster context e.g. by @leauthaud10; @applegate14; @ford15]. Otherwise, photo-$z$s can be used from external reference deep fields, requiring that statistically consistent and sufficiently representative galaxy populations are selected in both the survey and reference fields. For cluster weak lensing studies both approaches are complicated by the fact that the presence of a cluster means that the corresponding line-of-sight is over-dense at the cluster redshift, while both the default priors of photo-$z$ codes and the reference deep fields ought to be representative for the cosmic mean distribution. Previous studies employing reference fields have typically dealt with this issue by applying colour selections (“colour cuts”) that remove galaxies at the cluster redshift [e.g. @high12; @hoekstra12; @okabe16]. In case of incomplete removal the approach can be complemented by a statistical correction for the residual cluster member contamination if that can be estimated sufficiently well [e.g. @hoekstra15]. For cluster weak lensing studies a further complication arises when parametric models are fitted to the measured tangential reduced shear profiles, as issues such as miscentring [e.g. @johnston07; @george12] or uncertainties regarding assumed cluster concentrations can lead to non-negligible biases, introducing the need for calibrations using simulations [e.g. @becker11]. This paper is organised as follows: Sect.\[sec:sheartheory\] summarises relevant aspects of weak lensing theory. This is followed by a description of our cluster sample in Sect.\[sec:clusters\] and a description of the analysed data and image processing in Sect.\[sec:data\]. Sect.\[sec:shear\] details on the weak lensing shape measurements and a new test for signatures of potential residuals of charge-transfer inefficiency in the weak lensing catalogues. In Sect.\[sec:phot\] we describe in detail our approach to remove cluster galaxies via colour cuts and reliably estimate the source redshift distribution using data from the CANDELS fields. In Sect.\[sec:wlmasses\] we present our weak lensing shear profile analysis, mass reconstructions, and mass estimates, which we use in Sect.\[se:mtx\] to constrain the mass–temperature scaling relation. Finally, we discuss our [findings]{} in Sect.\[sec:discussion\] and conclude in Sect.\[sec:conclusions\]. Throughout this paper we assume a standard flat $\Lambda$CDM cosmology characterised by $\Omega_\mathrm{m}=0.3$, $\Omega_\Lambda=0.7$, and $H_0=70h_{70}$ km/s/Mpc with $h_{70}=1$, as approximately consistent with recent CMB constraints [@hinshaw13; @planck15cosmo]. For the computation of large-scale structure noise on the weak lensing estimates [and the concentration–mass relation according to [@diemer15]]{} we furthermore assume , , and . All magnitudes are in the AB system and are corrected for extinction according to @schlegel98. Summary of Relevant Weak Lensing Theory {#sec:sheartheory} ======================================= The images of distant background galaxies are distorted by the tidal gravitational field of a foreground mass concentration, see e.g. the reviews by @bartelmann01 [@schneider06], as well as @hoekstra13 in the context of galaxy clusters. In the weak lensing regime the size of a source is much smaller than the characteristic scale on which variations in the tidal field occur. In this case the lens mapping as function of observed position ${\boldsymbol{\theta}}$ can be described using the reduced shear $g({\boldsymbol{\theta}})$ and the convergence $\kappa({\boldsymbol{\theta}})=\Sigma({\boldsymbol{\theta}})/\Sigma_\mathrm{crit}$, which is the ratio of the surface mass density $\Sigma({\boldsymbol{\theta}})$ and the critical surface mass density $$\label{eqn:sigmacrit} \Sigma_{\mathrm{crit}} = \frac{c^2}{4\pi G}\frac{1}{D_{\mathrm{l}} \beta },$$ with the speed of light $c$, the gravitational constant $G$, and the geometric lensing efficiency $$\beta=\mathrm{max}\left[0,\frac{D_\mathrm{ls}}{D_\mathrm{s}}\right] \, ,$$ where $D_\mathrm{s}$, $D_\mathrm{l}$, and $D_\mathrm{ls}$ indicate the angular diameter distances to the source, to the lens, and between lens and source, respectively. The reduced shear $$g({\boldsymbol{\theta}})=\frac{\gamma({\boldsymbol{\theta}})}{1-\kappa({\boldsymbol{\theta}})}$$ describes the observable anisotropic shape distortion due to weak lensing. It is a two component quantity, conveniently written as a complex number $$g=g_1+\mathrm{i}g_2=|g|\mathrm{e}^{2\mathrm{i}\varphi} \,,$$ where $|g|$ constitutes the strength of the distortion and $\varphi$ its orientation with respect to the coordinate system. The reduced shear $g({\boldsymbol{\theta}})$ is a rescaled version of the unobservable shear $\gamma({\boldsymbol{\theta}})$, and can be estimated from the ensemble-averaged PSF-corrected ellipticities of background galaxies (see Sect.\[sec:shear\]), with the expectation value $$\langle\epsilon\rangle = g \, .$$ Due to noise from the intrinsic galaxy shape distribution and measurement noise we need to average the ellipticities of a large ensemble of galaxies $$\langle \epsilon_\alpha \rangle = \frac{\sum \epsilon_{\alpha,i} w_i}{\sum w_i}$$ to obtain useful constraints, where $\alpha\in\{1,2\}$ indicates the two ellipticity components and $i$ indicates galaxy $i$. The shape weights are included to improve the measurement signal-to-noise ratio, where $\sigma_{\epsilon,i}$ contains contributions both from the measurement noise and the intrinsic shape distribution (see Appendix \[app:shapes\_candels\], where we constrain both contributions empirically using CANDELS data). It is often useful to decompose the shear, reduced shear, and the ellipticity into their tangential components, e.g. $g_\mathrm{t}$, and cross components, e.g. $g_\times$, with respect to the centre of a mass distribution as $$\begin{aligned} g_\mathrm{t} & = & - g_1 \cos{2 \phi} - g_2 \sin{2\phi}\label{eq:gt} \\ g_\times & = & + g_1 \sin{2\phi} - g_2 \cos{2 \phi} \,, \label{eq:gx}\end{aligned}$$ where $\phi$ is the azimuthal angle with respect to the centre. The azimuthal average of the tangential shear $\gamma_\mathrm{t}$ at a radius $r$ around the centre of the mass distribution is linked to the mean convergence $\bar{\kappa}(<r)$ inside $r$ and $\bar{\kappa}(r)$ at $r$ via $$\langle \gamma_\mathrm{t} \rangle (r) = \bar{\kappa}(<r) - \bar{\kappa}(r)\, .$$ The weak lensing convergence and shear scale for an individual source galaxy at redshift with the geometric lensing efficiency $\beta(z_i)$, which is often conveniently written as $$\gamma=\beta_s(z_i)\gamma_\infty \, , \, \kappa=\beta_s(z_i)\kappa_\infty \,,$$ where $\kappa_\infty$ and $\gamma_\infty$ correspond to the values for a source at infinite redshift, and . In practise, we average the ellipticities of an ensemble of galaxies distributed in redshift, providing an estimate for $$\langle g \rangle=\left\langle \frac{\beta_s(z_i)\gamma_\infty}{1-\beta_s(z_i)\kappa_\infty} \right\rangle \,.$$ While one could in principle compute the exact model prediction for this from the source redshift distribution weighted by the lensing weights, a sufficiently accurate approximation is provided in @hoekstra00: $$g^\mathrm{model}\simeq \left[ 1+ \left( \frac{\langle\beta_s^2\rangle}{\langle\beta_s\rangle^2} -1 \right)\langle\beta_s\rangle\kappa_\infty^\mathrm{model} \right] \frac{\langle\beta_s\rangle\gamma_\infty^\mathrm{model}}{1-\langle\beta_s\rangle\kappa_\infty^\mathrm{model}} \label{eq:gbeta2corrected}$$ [see also @seitz97; @applegate14], where $$\langle\beta_s\rangle=\frac{\sum \beta_s(z_i) w_i}{\sum w_i} \, ,\, \langle\beta_s^2\rangle=\frac{\sum \beta_s^2(z_i) w_i}{\sum w_i}\,$$ need to be computed from the estimated source redshift distribution, taking the shape weights into account. When the signal of lenses at different redshifts is compared or stacked, it can be useful to conduct the analysis in terms of the differential surface mass density $$\Delta \Sigma(r)=\frac{\sum_{i} w_i \left(\epsilon_\mathrm{t}\Sigma_\mathrm{crit}\right)_i}{\sum_i w_i}\, \label{eq:deltaSigma}$$ to compensate for the redshift dependence of the signal, where the the summation is conducted over sources in a separation interval around $r$. Gravitational lensing leaves the surface brightness invariant. Accordingly, a relative change in the observed flux of a source due to lensing is solely given by the relative magnification of the source $$\mu = \frac{1}{(1-\kappa)^2-|\gamma|^2} \,. \label{eq:magnification_correct}$$ Together with the change in solid angle this also changes the observed density of background sources and their redshift distribution, as investigated in Sect.\[sec:magnification\_model\]. The Cluster Sample {#sec:clusters} ================== We study a total of 13 distant galaxy clusters detected by the SPT in the redshift range via the SZ effect; see Table \[tab:clusters\] for details and Fig.\[fi:zdist\_cluster\_surveys\] for a comparison of the cluster redshift distribution to recent large weak lensing cluster samples from the Canadian Cluster Comparison Project [CCCP; @hoekstra15], Weighing the Giants [WtG; @vonderlinden14], the Cluster Lensing And Supernova survey with Hubble [CLASH; @umetsu14], the Local Cluster Substructure Survey [LoCuSS; @okabe16], and the analysis of HST observations of X-ray, optically, and NIR selected high-redshift clusters by @jee11. The SPT clusters were observed in HST Cycles 18 and 19. At the time of the target selection, the SPT cluster follow-up campaign was still incomplete. From the clusters with measured spectroscopic redshifts prior to the corresponding cycle, we selected the most massive SPT-SZ clusters at for the Cycles 18 programme, and the most massive clusters at for the Cycle 19 programme. Nine clusters in our overall sample originate from the first 178 deg$^2$ of the sky surveyed by SPT [@vanderlinde10 hereafter ]. Using updated estimates of the SZ detection significance $\xi$ from the cluster catalogue for the full 2,500 deg$^2$ SPT-SZ survey [@bleem15 hereafter ], our selection of clusters from the sample includes all clusters from the first 178 deg$^2$ at with plus all clusters at with (see Table \[tab:clusters\]), except for SPT-CL$J$0540$-$5744 (). Additionally, our sample includes all clusters at from @williamson11 [henceforth ], who present a catalogue of the 26 most significant SZ cluster detections in the full 2500 deg$^2$ SPT survey region. This adds three clusters in addition to SPT-CL$J$2337$-$5942, which is part of both samples. Finally, with SPT-CL$J$2040$-$5725 a single further cluster is included from @reichardt13 [hereafter ], who present the cluster sample constructed from the first 720 deg$^2$ of the SPT cluster survey. In addition to the aforementioned sample papers, more detailed studies of individual clusters were published for SPT-CLJ0546$-$5345 [@brodwin10] and SPT-CLJ2106$-$5844 [@foley11]. Spectroscopic cluster redshift measurements are described in @ruel14 and [@bayliss16]. In Table \[tab:clusters\] we also list X-ray centroids as estimated from the available [*Chandra*]{} or XMM-Newton data [detailed in @andersson11; @benson13; @mcdonald13; @chiu16 see also Sect.\[se:mtx\]], and BCG positions from @chiu16. ![Comparison of the cluster redshift distribution of our sample with several recent independent studies, plus the larger high-redshift sample from @jee11, which includes a combination of optically, NIR, and X-ray-selected clusters. \[fi:zdist\_cluster\_surveys\]](cluster_zdist.png){width="0.99\columnwidth"} ---------------------- ------- ------- ------------- ---------------------- ---------------- ---------------- -------------- -------------- -------------------------------------------- --- Cluster name $z_l$ $\xi$ $M_\mathrm{500c,SZ}$ Sample SZ $\alpha$ SZ $\delta$ X-ray $\alpha$ X-ray $\delta$ BCG $\alpha$ BCG $\delta$ \[$10^{14} \mathrm{M}_\odot h_{70}^{-1}$\] SPT-CL$J$0000$-$5748 0.702 8.49 0.2499 $-57.8064$ 0.2518 $-57.8094$ 0.2502 $-57.8093$ $4.56\pm0.80$ SPT-CL$J$0102$-$4915 0.870 39.91 15.7294 $-49.2611$ 15.7350 $-49.2667$ 15.7407 $-49.2720$ $14.43\pm2.10$ SPT-CL$J$0533$-$5005 0.881 7.08 83.4009 $-50.0901$ 83.4018 $-50.0969$ 83.4144 $-50.0845$ $3.79\pm0.73$ SPT-CL$J$0546$-$5345 1.066 10.76 86.6525 $-53.7625$ 86.6532 $-53.7604$ 86.6569 $-53.7586$ $5.05\pm0.82$ SPT-CL$J$0559$-$5249 0.609 10.64 89.9251 $-52.8260$ 89.9357 $-52.8253$ 89.9301 $-52.8241$ $5.78\pm0.95$ SPT-CL$J$0615$-$5746 0.972 26.42 93.9650 $-57.7763$ 93.9652 $-57.7788$ 93.9656 $-57.7802$ $10.53\pm1.55$ SPT-CL$J$2040$-$5725 0.930 6.24 310.0573 $-57.4295$ 310.0631$^*$ $-57.4287$ 310.0552 $-57.4209$ $3.36\pm0.70$ SPT-CL$J$2106$-$5844 1.132 22.22 316.5206 $-58.7451$ 316.5174 $-58.7426$ 316.5192 $-58.7411$ $8.35\pm1.24$ SPT-CL$J$2331$-$5051 0.576 10.47 352.9608 $-50.8639$ 352.9610 $-50.8631$ 352.9631 $-50.8650$ $5.60\pm0.92$ SPT-CL$J$2337$-$5942 0.775 20.35 354.3523 $-59.7049$ 354.3516 $-59.7061$ 354.3650 $-59.7013$ $8.43\pm1.27$ , SPT-CL$J$2341$-$5119 1.003 12.49 355.2991 $-51.3281$ 355.3009 $-51.3285$ 355.3014 $-51.3291$ $5.59\pm0.89$ SPT-CL$J$2342$-$5411 1.075 8.18 355.6892 $-54.1856$ 355.6904 $-54.1838$ 355.6913 $-54.1848$ $3.93\pm0.70$ SPT-CL$J$2359$-$5009 0.775 6.68 359.9230 $-50.1649$ 359.9321 $-50.1697$ 359.9324 $-50.1722$ $3.60\pm0.71$ ---------------------- ------- ------- ------------- ---------------------- ---------------- ---------------- -------------- -------------- -------------------------------------------- --- [Note. — Basic data from @bleem15 and @chiu16 for the 13 clusters targeted in this weak lensing analysis. [*Column 1:*]{} Cluster designation. [*Column 2:*]{} Spectroscopic cluster redshift. [*Column 3:*]{} Peak signal-to-noise ratio of the SZ detection. [*Columns 4–9:*]{} Right ascension $\alpha$ and declination $\delta$ of the cluster centres used in the weak lensing analysis from the SZ peak, X-ray centroid, and BCG position. $^*$: X-ray centroid from XMM-Newton data, otherwise [*Chandra*]{} (see Sect.\[se:mtx\]). [*Column 10:*]{} Mass derived from the SZ-Signal. [*Column 11:*]{} SPT parent sample for HST follow-up selection.\ ]{} Data and data reduction {#sec:data} ======================= In this section we provide details on the data analysed in this study and their reduction. For the SPT clusters we make use of HST observations (Sect.\[se:hst\_observations\]) for shape and colour measurements, as well as VLT observations (Sect.\[se:data\_vlt\]) for colour measurements in the outer cluster regions. To optimise our weak lensing pipeline, and to be able to apply consistent selection criteria to photo-$z$ catalogues from @skelton14, we also process HST observations of the CANDELS fields (Sect.\[se:data\_hst\_candels\]). HST/ACS data {#sec:hstdata} ------------ ### SPT cluster observations {#se:hst_observations} We measure weak lensing galaxy shapes from high-resolution [*Hubble Space Telescope*]{} imaging obtained during Cycles 18 and 19 as part of programmes 12246 (PI: C. Stubbs) and 12477[^2] (PI: F. W. High), and observed between Sep 29, 2011 and Oct 24, 2012 under low sky background conditions. Each cluster was observed with a $2\times 2$ ACS/WFC mosaic in the F606W filter, where each tile consists of 4 dithered exposures of 480s, adding to a total exposure time of 1.92ks per tile. These mosaic observations allow us to probe the cluster weak lensing signal out to approximately the virial radius. Additionally, a single tile was observed with ACS in the F814W filter on the cluster centre (1.92ks). These data are included in our photometric analysis (Sect.\[sec:phot\]). For the weak lensing shape measurements we chose observations in the F606W filter as it is the most efficient ACS filter in terms of weak lensing galaxy source density [see, e.g. @schrabback07]. However note that our analysis in Appendix \[app:606vs814\] suggests that future programmes could benefit from mosaic observations in both F606W and F814W to simultaneously obtain robust shape measurements and colour estimates. In fact, a F814W ACS mosaic was obtained for one of the clusters in our sample, [[SPT-CL]{}]{}, through the independent HST programme 12757 (PI: Mazzotta), with observations conducted Jan 19–22, 2012. For the current analysis we include these additional data in the colour measurements but not the shape analysis. We denote magnitudes measured from the ACS F606W and F814W images as $V_{606}$ and $I_{814}$, respectively. By default these correspond to magnitudes measured in circular apertures with a diameter 07 unless explicitly stated differently. ### HST data reduction {#sec:reduction} For basic image reductions we largely employ the standard ACS calibration pipeline `CALACS`. The main exception is our use of the @massey2014 [ henceforth] algorithm for the correction of charge-transfer inefficiency (CTI). CTI constitutes an important systematic effect for HST weak lensing shape analyses if left uncorrected [e.g. @rhodes07; @schrabback2010 henceforth]. It is caused by radiation damage in space. The resulting CCD defects act as charge traps during the read-out process, introducing non-linear charge-trails behind objects in the parallel-transfer read-out direction. updated their time-dependent model of the charge trap densities by fitting charge trails behind hot pixels in CANDELS ACS/F606W imaging exposures of the COSMOS field [@grogin2011], which were obtained at a similar epoch as our cluster data (between Dec 06, 2011 and Apr 15, 2012). Given that we conduct the CTI correction using the code, we also have to CTI-correct the master dark frames using this pipeline. As further differences to standard `CALACS` processing we compute accurately normalised r.m.s. noise maps as detailed in and optimise the bad pixel mask, where we flag satellite trails and cosmic ray clusters, and unflag the removed CTI trails of hot pixels. The further data reduction for the individual ACS tiles closely follows , to which we refer the reader for details. As the first step, we carefully refine relative shifts and rotations between the exposures by matching the positions of compact objects. We then use `MultiDrizzle` [@koekemoer2003] for the cosmic ray removal and stacking, where we employ the `lanczos3` kernel at the native pixel scale 005 to minimise noise correlations while only introducing a low level of aliasing for ellipticity measurements [@jee2007]. The pipeline also generates correctly scaled r.m.s. noise maps for stacks that are used for the object detection. We conduct weak lensing shape measurements on these individual stacked ACS tiles (see Sect.\[sec:shear\]). For the joint photometric analysis with available VLT data (Sect.\[se:source\_sel\_scatter\] with details given in Appendix \[app:details\_fors2color\_scatter\]) we additionally generate stacks for the ACS mosaics. Here we iteratively align neighbouring tiles by first resampling them separately onto a common pixel grid, only stacking the exposures of the corresponding tile. We then use the differences between the positions of matched objects in the overlapping regions to compute shifts and rotations, in order to update the astrometry. ### CANDELS HST data {#se:data_hst_candels} When estimating the redshift distribution of our source sample (see Sect.\[sec:phot\]) we need to apply the same selection function (consisting of photometric, shape, and size cuts) to the galaxies in the CANDELS fields, which act as our reference sample. To be able to employ consistent weak lensing cuts, we reduce and analyse ACS imaging in the CANDELS fields with the same pipeline as the HST observations of the SPT clusters. This includes data from the CANDELS [@grogin2011 Proposal IDs 12440, 12064], GOODS [@giavalisco2004 Proposal IDs 9425, 9583], GEMS [@rix2004 Proposal ID 9500], and AEGIS [@davis2007 Proposal ID 10134] programmes. Here we perform a tile-wise analysis, always stacking exposures with good spatial overlap which add to approximately 1-orbit depth, roughly matching the depth of our cluster field data (see Appendix \[app:candels:data\] for additional information). We use these blank field data also as a calibration sample to derive an empirical weak lensing weighting scheme that is based on the measured ellipticity dispersion as function of logarithmic signal-to-noise ratio and employed in our cluster lensing analysis (see Appendix \[app:shapeweights\]). This analysis also provides updated constraints on the dispersion of the intrinsic galaxy ellipticities and allows us to compare the weak lensing performance of the ACS F606W and F814W filters, aiding the preparation of future weak lensing programmes (see Appendix \[app:606vs814\]). VLT/FORS2 data {#se:data_vlt} -------------- For our analysis we make use of VLT/FORS2 imaging of all of our targets taken as part of programmes 086.A-0741 (PI: Bazin), 088.A-0796 (PI: Bazin), 088.A-0889 (PI: Mohr), and 089.A-0824 (PI: Mohr) in the $I_\mathrm{BESS}$ pass-band, which we call $I_\mathrm{FORS2}$. The FORS2 focal plane is covered with two $2k\times 4k$ MIT CCDs. The data were taken with the standard resolution collimator in $2\times 2$ binning, providing imaging over a $6\farcm8 \times 6\farcm8$ field-of-view with a pixel scale of 025, matching the size of our ACS mosaics well. ---------------------- ------------------ ------------------ ----- ------------ ----------- Cluster name $t_\mathrm{exp}$ $I_\mathrm{lim}$ IQ bright cut faint cut SPT-CL$J$0000$-$5748 2.1ks 26.0 065 24.0–25.5 25.5–26.0 SPT-CL$J$0102$-$4915 2.1ks 25.8 075 24.0–25.0 25.0–25.5 SPT-CL$J$0533$-$5005 2.1ks 25.8 073 24.0–25.5 - SPT-CL$J$0546$-$5345 2.1ks 25.7 075 24.0–25.0 25.0–25.5 SPT-CL$J$0559$-$5249 1.9ks 25.6 065 24.0–25.0 25.0–25.5 SPT-CL$J$0615$-$5746 2.5ks 25.6 093 24.0–24.5 24.5–25.5 SPT-CL$J$2040$-$5725 2.9ks 25.7 070 24.0–25.0 25.0–25.5 SPT-CL$J$2106$-$5844 4.8ks 25.8 080 24.0–25.0 25.0–25.5 SPT-CL$J$2331$-$5051 2.4ks 25.9 083 24.0–25.5 25.5–26.0 SPT-CL$J$2337$-$5942 2.1ks 25.7 080 24.0–25.5 25.5–26.0 SPT-CL$J$2341$-$5119 2.1ks 25.8 080 24.0–25.5 25.5–26.0 SPT-CL$J$2342$-$5411 2.1ks 25.7 093 24.0–25.0 25.0–25.5 SPT-CL$J$2359$-$5009 2.1ks 25.9 068 24.0–25.5 25.5–26.0 ---------------------- ------------------ ------------------ ----- ------------ ----------- [Note. — Details of the analysed VLT/FORS2 imaging data. [*Column 1:*]{} Cluster designation. [*Column 2:*]{} Total co-added exposure time. [*Column 3:*]{} $5\sigma$-limiting magnitude computed for 15 apertures in the stack from the single pixel noise r.m.s. values of the contributing exposures. [*Column 4:*]{} Image Quality defined as $2\times \texttt{FLUX\_RADIUS}$ from `Source Extractor`. [*Column 5:*]{} $V_{606}$ magnitude range with low photometric colour scatter , for which the “bright” colour cut is applied (see Table \[tab:app:colourcuts\] in Appendix \[app:details\_fors2color\_scatter\]). [*Column 6:*]{} $V_{606}$ magnitude range with increased photometric colour scatter , for which the “faint” colour cut is applied (see Table \[tab:app:colourcuts\] in Appendix \[app:details\_fors2color\_scatter\]).\ ]{} We reduced the data using `theli` [@erben05; @schirmer13], applying bias and flat-field correction, relative photometric calibration, and sky background subtraction using `Source Extractor` [@bertin1996]. We use the object positions in the HST F606W image as astrometric reference for the distortion correction. For an initial absolute photometric calibration using the stars located in the central HST $I_{814}$ tile we employ the relation $$\label{eq:picklesoffset} I_\mathrm{FORS2} - I_{814} = -0.052 + 0.0095 (V_{606} - I_{814}) \,,$$ which was derived employing the @pickles98 stellar library. This relation is valid for and assumes total magnitudes for the computation of . We list total exposure times, limiting magnitudes, and delivered image quality for the co-added images in Table \[tab:vltdata\]. For further details on the data reduction see @chiu16, who also analyse observations obtained with FORS2 in the $B_\mathrm{HIGH}$ and $z_\mathrm{GUNN}$ pass-bands. In our analysis we do not include these additional bands. Our initial testing indicates that their inclusion would only yield a minor increase in the usable background galaxy source density given the depth of the different observations and typical colours of the dominant background source population. Weak lensing galaxy shapes {#sec:shear} ========================== Shape measurements {#sec:shapemeasure} ------------------ For the generation of weak lensing shape catalogues we employ the pipeline from , which was successfully used for cosmological weak lensing measurements that typically have more stringent requirements on the control of systematics than cluster weak lensing studies. We refer the reader to this publication for a more detailed pipeline description. Here we summarise the main steps and provide details on recent changes to our pipeline only. One of the main changes is the application of the pixel-based CTI correction from (Sect.\[sec:reduction\]), which is more accurate than the catalogue-level correction employed in . This change has become necessary as we analyse more recent ACS data with stronger CTI degradation. As the first step in the catalogue generation we use `Source Extractor` [@bertin1996] to detect objects in the F606W stacks and measure basic object properties. For the ellipticity measurement and correction for the point-spread function (PSF) we employ the KSB+ formalism [@kaiser1995; @luppino1997; @hoekstra1998] as implemented by @erben2001 with modifications from @schrabback07 and . We interpolate the spatially and temporally varying ACS PSF using a model derived from a principal component analysis of PSF variations in dense stellar fields. showed that the dominant contribution to ACS PSF ellipticity variations can be described with a single principal component (related to the HST focus position). This one-parameter PSF model is sufficiently well constrained by the $\sim 10-20$ high-$S/N$ stars available for PSF measurements in extragalactic ACS pointings. We obtain a PSF model for each contributing exposure based on stellar ellipticity and size measurements in the image prior to resampling (to minimise noise), from which we compute the combined model for the stack. For the current work we recalibrated this algorithm using archival ACS F606W stellar field observations taken after Servicing Mission 4. [We processed these data]{} with the same CTI correction method as our cluster field data. Following we select galaxies in terms of their half-light radius , where $r_\mathrm{h}^{*,\mathrm{max}}$ is the upper limit of the $0.25$ pixel wide stellar locus, and “pre-seeing” shear polarisability tensor with . Deviating from we exclude very extended galaxies with pixels, as they are poorly covered by the employed postage stamps. As done in we mask galaxies close to the image boundaries, large galaxies, or bright stars. introduced an empirical correction for noise bias in the ellipticity measurement as a function of the KSB signal-to-noise ratio from @erben2001. calibrated this correction using simulated images of ground-based weak lensing observations from STEP2 [@massey2007], and verified that the same correction robustly corrects simulated high-resolution ACS-like weak lensing data with less than 2% residual multiplicative ellipticity bias ($0.8\%$ on average). However, as recently shown by @hoekstra15, the STEP2 image simulations lack sources at the faint end, affecting the derived bias calibration [see also @hoekstra17]. Also, deviations in the assumed intrinsic galaxy shape distribution influence the noise-bias correction [e.g. @viola2014]. To minimise the impact of such uncertainties we apply a more conservative galaxy selection requiring from `Source Extractor`[^3]. To be conservative, we additionally double the systematic uncertainty for the shear calibration in the error-budget of our current cluster study (4%), which is comparable to the mean shear calibration correction of the galaxies passing our cuts (average factor 1.05). In the context of cluster weak lensing studies a relevant question is also if the image simulations probe the relevant range of shears sufficiently well. We expect that this is not a major concern for our study given that for all of our clusters within the radial range used for the mass constraints (see Sect.\[sec:wlmasses\]). For comparison, the basic KSB+ implementation used in our analysis was tested in @heymans06 using shears up to , where no indications were found for significant quadratic shear bias terms that would result in an inaccurate correction using our linear correction scheme. We apply the same shape measurement pipeline to the CANDELS data discussed in Sect.\[se:data\_hst\_candels\]. When mimicking our cluster field selection in these catalogues and assigning weights, we rescale the values prior to the cut to account for slight differences in depth. Hence, if a CANDELS tile is slightly shallower (deeper) compared to the cluster tile considered, we will apply a correspondingly slightly lower (higher) cut in the CANDELS tile to select consistent galaxy samples. On average the depth of our CANDELS stacks agrees well with the depth of the cluster field stacks (to 0.065 mag). Together with the fact that $\langle\beta\rangle$ depends only weakly on $V_{606}$ for our colour-selected sample at the faint end (see Sect.\[se:analysis\_in\_magbins\]), we therefore ignore second-order effects such as incompleteness differences between the CANDELS and cluster field catalogues. Test for residual CTI signatures in the ACS cluster data {#sec:ctitests} -------------------------------------------------------- CTI generates charge-trails behind objects dominantly in the parallel-transfer readout direction. For raw ACS images this corresponds to the $y$-direction, and this is approximately also the case for distortion-corrected images if `MultiDrizzle` is run using the native detector orientation. test the performance of their pixel-based CTI correction by averaging the PSF-corrected ellipticity estimates of galaxies in blank field CANDELS data. Images without CTI correction show a prominent alignment with the $y$-axis (), where the magnitude of the effect increases with the $y$-separation relative to the readout amplifiers. In contrast, this alignment is undetected if the correction is applied. ![Testing for residual CTI systematics in the cluster fields: [*Top:*]{} Illustration for the separation of the tangential and cross components of the ellipticity into components affected by CTI ($\epsilon_{\mathrm{t},1}$, $\epsilon_{\times,1}$), and those unaffected by CTI ($\epsilon_{\mathrm{t},2}$, $\epsilon_{\times,2}$). The [*middle*]{} ([*bottom*]{}) panel shows the difference in the tangential (cross) ellipticity component with respect to the cluster centre as estimated from the CTI-affected and the CTI-unaffected components. Here we combine the signal from all galaxies passing the shape cuts with in all cluster fields. The points are consistent with zero () suggesting that the CTI has been fully corrected within the statistical precision of the data. For comparison, the dotted curve shows the signal which would be measured from an uncorrected CTI saw-tooth ellipticity pattern with , where small wiggles are caused by the sampling at the galaxy positions and the masks applied. \[fi:ctitest\]](gtx_e1_e2_top.png){width="0.95\columnwidth"} ![Testing for residual CTI systematics in the cluster fields: [*Top:*]{} Illustration for the separation of the tangential and cross components of the ellipticity into components affected by CTI ($\epsilon_{\mathrm{t},1}$, $\epsilon_{\times,1}$), and those unaffected by CTI ($\epsilon_{\mathrm{t},2}$, $\epsilon_{\times,2}$). The [*middle*]{} ([*bottom*]{}) panel shows the difference in the tangential (cross) ellipticity component with respect to the cluster centre as estimated from the CTI-affected and the CTI-unaffected components. Here we combine the signal from all galaxies passing the shape cuts with in all cluster fields. The points are consistent with zero () suggesting that the CTI has been fully corrected within the statistical precision of the data. For comparison, the dotted curve shows the signal which would be measured from an uncorrected CTI saw-tooth ellipticity pattern with , where small wiggles are caused by the sampling at the galaxy positions and the masks applied. \[fi:ctitest\]](gtx_e1_e2.png){width="0.99\columnwidth"} We cannot apply the same test to our ACS data of the cluster fields given the presence of massive clusters, which are always located at the same position within the mosaics, and whose weak gravitational lensing shear would add to the saw-tooth CTI signature. However, we can make use of the fact that CTI primarily affects the $\epsilon_1$ ellipticity component (measured along the image axes) but not the $\epsilon_2$ ellipticity component (measured along the field diagonals). The tangential and cross components of the ellipticity with respect to the cluster centre $$\begin{aligned} \epsilon_\mathrm{t} & = & \epsilon_{\mathrm{t},1} + \epsilon_{\mathrm{t},2}\\ \epsilon_\times & = & \epsilon_{{\times},1} + \epsilon_{\times,2}\end{aligned}$$ (compare Equations\[eq:gt\] and \[eq:gx\]) receive contributions from both ellipticity components with $$\begin{aligned} \epsilon_{\mathrm{t},1} & = & - \epsilon_1 \cos{2 \phi}\\ \epsilon_{\mathrm{t},2} & = & - \epsilon_2 \sin{2\phi}\\ \epsilon_{\times,1} & = & + \epsilon_1 \sin{2\phi} \\ \epsilon_{\times,2} & = & - \epsilon_2 \cos{2 \phi} \,,\end{aligned}$$ see the sketch in the top panel of Fig.\[fi:ctitest\] for an illustration of these components. In our test we stack the signal from all clusters. Here we expect that any anisotropy in the reduced shear pattern due to cluster halo ellipticity will average out leading to an approximately circularly symmetric shear field. Accordingly, in the absence of residual systematics we expect that and are consistent with zero when averaged azimuthally. Fig.\[fi:ctitest\] shows that this is indeed the case for our data (), confirming the success of the CTI correction within the statistical precision of the data. For comparison, the dotted line in Fig.\[fi:ctitest\] shows the signal that would be caused by a typical uncorrected CTI ellipticity saw-tooth pattern with [^4]. Cluster member removal and estimation of the source redshift distribution {#sec:phot} ========================================================================= Robust weak lensing mass measurements require accurate knowledge of the mean geometric lensing efficiency $\langle \beta \rangle$ of the source sample and its variance $\langle \beta^2 \rangle$ (see Sect.\[sec:sheartheory\]). For a given cosmological model these depend only on the source redshift distribution and cluster redshift. Surveys with sufficiently deep imaging in sufficiently many bands can attempt to estimate the probability distribution of source redshifts directly via photo-$z$s [e.g. @applegate14]. However, such data are not available for our cluster fields. Hence, we have to rely on an estimate of the redshift distribution from external reference fields. Here we use photometric redshift estimates for the CANDELS fields from the 3D-HST team [@skelton14] as primary data set (see Sect.\[se:photo\_ref\_cats\]). Additionally, we use spectroscopic and grism redshift estimates for galaxies in the CANDELS fields, as well as much deeper data from the [*Hubble*]{} Ultra Deep field (HUDF) to investigate and statistically correct for systematic features in the CANDELS photo-$z$s (Sect.\[sec:correct\_sys\_photoz\]). Given that our cluster fields are over-dense at the cluster redshift we have to apply a colour selection that robustly removes galaxies at the cluster redshift both in the reference catalogue and our actual cluster field catalogues. Here we use colour estimates from the HST/ACS F606W and F814W images in the inner regions (“ACS-only” selection, Sect.\[se:photo\_color\_select\_acs\]), and we use VLT/FORS2 $I$-band imaging for the cluster outskirts (“ACS+FORS2” selection, Sect.\[se:source\_sel\_scatter\] with details given in Appendix \[app:details\_fors2color\_scatter\]). As discussed in Appendix \[app:why\_not\_boost\] we also explored a different analysis scheme which substitutes the colour selection with a statistical correction for cluster member contamination, but we found that we could not control the systematics of the correction to the needed level due to the limited radial range probed by the F606W images. We optimise the analysis by splitting the colour-selected sources into magnitude bins (Sect.\[se:analysis\_in\_magbins\]), investigate the influence of line-of-sight variations (Sect.\[sec:beta\_los\_variation\]), and account for weak lensing magnification (Sect.\[sec:magnification\_model\]). Sect.\[se:photo\_number\_density\_tests\] presents consistency checks for our analysis based on the source number density measured as function of magnitude and cluster-centric distance. CANDELS photometric redshift reference catalogues from 3D-HST {#se:photo_ref_cats} ------------------------------------------------------------- We make use of photometric redshift catalogues computed by the 3D-HST team [@brammer12; @skelton14 hereafter ] for the CANDELS fields [@grogin2011], which consist of five independent lines-of-sight (AEGIS, COSMOS, GOODS-North, GOODS-South, UDS). Hence, their combination efficiently suppresses the impact of sampling variance. All CANDELS field were observed by HST with ACS and WFC3, including ACS F606W and F814W[^5] imaging mosaics that have at least the depth of our cluster field observations [see @koekemoer11]. This includes observations from the CANDELS program [@grogin2011] and earlier projects [@giavalisco2004; @rix2004; @davis2007; @scoville07]. The catalogues are based on detections from combined HST/WFC3 NIR F125W+F140W+F160W images, and include photometric measurements from a total of 147 distinct imaging data sets from HST, [*Spitzer*]{}, and ground-based facilities with a broad wavelength coverage from ( data sets per field). compute photometric redshifts using <span style="font-variant:small-caps;">EAZY</span> [@brammer08], which fits the observed SED constraints of each object with a linear combination of galaxy templates. We have matched the catalogues with our F606W-detected shape catalogues of the CANDELS fields (see Sect.\[sec:shear\]). After applying weak lensing cuts, accounting for masks, and restricting the analysis to the overlap region of the ACS and WFC3 mosaics, we find that of the galaxies in the shape catalogues with have a direct match within 05 in the catalogues, showing that they are nearly complete within our employed magnitude range (see Appendix \[se:app:non\_matches\] for an investigation of the of non-matching galaxies which shows that they have a negligible impact). Source selection using ACS-only colours {#se:photo_color_select_acs} --------------------------------------- ![Measured $V_{606}-I_{814}$ colours as function of $V_{606}$ for galaxies in the field of SPT-CL$J$2337$-$5942 that pass our weak lensing shape cuts, and that are located within the central $I_{814}$ ACS tile. The blue lines indicate the region of blue galaxies that pass our colour selection. The cluster red sequence is clearly visible at . \[fig:v\_vi\]](v_vi.png){width="0.99\columnwidth"} ![$V_{606}-I_{814}$ colours of galaxies in the CANDELS fields as function of the peak photometric redshift $z_\mathrm{p}$ from . The colour coding splits the galaxies into our different magnitude bins. The horizontal lines mark our different colour cuts (dependent on cluster redshift and galaxy magnitude, see Sect.\[se:photo\_color\_select\_acs\]), while the vertical lines indicate the cluster redshift range (solid), as well as (dashed), at which cluster redshift the colour cuts change. The curves indicate synthetic $V_{606}-I_{814}$ colours of galaxy SED templates from @coe06. \[fig:z\_vi\]](z_vi.png){width="1.03\columnwidth"} In the inner cluster regions we apply a colour selection (indicated in Fig.\[fig:v\_vi\]) using our ACS F606W and F814W images, selecting only galaxies that are bluer than nearly all galaxies at the cluster redshift. This is illustrated in Fig.\[fig:z\_vi\], where we plot the <span style="font-variant:small-caps;">EAZY</span> peak photometric redshift $z_\mathrm{p}$ for the CANDELS galaxies as function of colour from (measured with the same 07 aperture diameter as employed for our ACS colour measurements). Figures \[fig:z\_vi\] and \[fig:zdist\_f814w\] illustrate that the selection of blue galaxies in colour in CANDELS is very effective in removing galaxies at our cluster redshifts, while it selects the majority of the background galaxies. The latter are high-redshift star-forming galaxies observed at rest-frame UV wavelength with very blue spectral slopes. In contrast, nearly all galaxies at the cluster redshifts show a redder colour, as they contain either the 4000Åbreak (early type galaxies, see the cluster red sequence in Fig.\[fig:v\_vi\]) or the Balmer break (late type galaxies) within the filter pair. We note that our approach rejects both red and blue cluster members. It is therefore more conservative and robust than redder colour cuts that some studies have used to remove red sequence cluster members only [e.g. @jee11]. Note that, in contrast, @okabe13 select only galaxies that are redder than the red sequence. This is a useful approach for the low-redshift clusters targeted in their study, but less effective for the high-redshift clusters studied here, as most of the background galaxies are blue at optical wavelengths (see Fig.\[fig:zdist\_f814w\]). Likewise, some studies of lower redshift clusters have used combinations of blue and red regions in colour space to minimise cluster member contamination [e.g. @medezinski10; @high12; @umetsu14]. [It is evident from Fig.\[fig:z\_vi\] that a selection of blue galaxies in $V-I$ colour is inefficient for clusters at low redshifts , as it would either require extremely blue cuts that drastically shrink the source sample, or lead to a larger residual contamination by galaxies at the cluster redshift. Similar results were found by @ziparo16, who conclude that optical observations alone are not sufficient to reduce the cluster member contamination below the per-cent level for blue source samples and clusters at . ]{} ![image](pofz_scheme_3_cut_0_3_stack_24_26_5.png){width="0.99\columnwidth"} ![image](pofz_scheme_3_cut_0_2_stack_24_26_5.png){width="0.99\columnwidth"} ![image](pofz_scheme_3_cut_0_3_stack_24_26_5_spec.png){width="0.99\columnwidth"} ![image](pofz_scheme_3_cut_0_2_stack_24_26_5_spec.png){width="0.99\columnwidth"} For clusters at we select source galaxies with . This maximises the background galaxy density while at the same time removing of the CANDELS galaxies at that pass the other weak lensing cuts, see the top left panel of Fig.\[fig:zdist\_f814w\]. For the higher redshift clusters we apply a more stringent cut which still yields a suppression of galaxies at , at the expense of a slightly lower source density (top right panel of Fig.\[fig:zdist\_f814w\]). When conducting the analysis for our cluster fields we apply slightly more conservative colour cuts that are bluer by 0.1mag for the faintest sources in our analysis, as they show the largest photometric scatter. As a result, we obtain a similar fraction of removed galaxies at the cluster redshifts when taking photometric scatter into account (see Sect.\[se:source\_sel\_scatter\] and Appendix \[app:scatter\]). In Fig.\[fig:z\_vi\] we also over-plot synthetic colours of redshifted SED templates for star forming galaxies employed in the Bayesian Photometric Redshift (`BPZ`) algorithm [@benitez00]. This includes the SB3 and SB2 star burst templates from @kinney96 as recalibrated by @benitez04. We additionally include a young star burst model (SSP 25Myr), which is one of the templates introduced by @coe06 into `BPZ` to improve photometric redshift estimates for very blue galaxies in the HUDF. The shown SED corresponds to a simple stellar population (SSP) model with an age of 25 Myr and metallicity [@bruzual03]. At the cluster redshifts, the colours of the SB3 and SB2 templates approximately describe the range of colours of typical blue cloud galaxies, which are well removed by our colour selection. In contrast, while the colour of the SSP 25 Myr model appears to be representative for a considerable fraction of the background galaxies, it approximately marks the location of the most extreme blue outliers at the cluster redshifts, which are not fully removed by our colour selection scheme. If the clusters contain a substantial fraction of such extremely blue galaxies, this might introduce some residual cluster member contamination in our lensing catalogue. We investigate this issue in Appendix \[se:test\_extremely\_blue\], concluding that such galaxies have a negligible impact for our analysis despite the physical over-density of galaxies in clusters. We also present empirical tests for residual contamination by cluster galaxies in Sect.\[se:photo\_number\_density\_tests\]. Statistical correction for systematic features in the photometric redshift distribution {#sec:correct_sys_photoz} --------------------------------------------------------------------------------------- ![image](zz_grismspec_1.png){width="0.68\columnwidth"} ![image](zz_BPZfixgrismspec_1.png){width="0.68\columnwidth"} ![image](zz_BPZfix_1.png){width="0.68\columnwidth"} We base our estimate of the source redshift distribution on the CANDELS photo-$z$ catalogues because of their high completeness at the depth of our SPT ACS observations (Sect.\[se:photo\_ref\_cats\]), allowing us to select galaxies that are representative for the galaxies used in our lensing analysis. However, it is important to realise that such photo-$z$ estimates may contain systematic features (e.g. catastrophic outliers) that can bias the inferred redshift distribution and accordingly the lensing results. As an example, the cosmological weak lensing analysis of COSMOS data by suggests that the majority of faint galaxies in the COSMOS-30 photometric redshift catalogue [@ilbert2009] that have a primary peak in their posterior redshift probability distribution $p(z)$ at low redshifts but also a secondary peak at high redshifts, are truly at high redshift. Likewise, the galaxy-galaxy lensing analysis of CFHTLenS data by @heymans12 indicates that a significant fraction of galaxies with an assigned photometric redshift are truly at high redshift. In the following subsections we exploit additional data sets to check the accuracy of the CANDELS photo-$z$s and implement a statistical correction for relevant systematic features. ### Tests and statistical correction based on HUDF data {#se:sub:catastrophic_outliers} The [*Hubble*]{} Ultra Deep Field (HUDF) is located within one of the CANDELS fields (GOODS-South). The very deep multi-wavelength observations conducted in the HUDF can therefore be used for cross-checks of the CANDELS photo-$z$s. As first data set we use a combination of high-fidelity spectroscopic redshifts (“spec-$z$s”, $z_\mathrm{s}$) compiled by @rafelski15[^6], and redshift estimates extracted by the 3D-HST team [@brammer12; @brammer13] from the combination of deep HST WFC3/IR slitless grism spectroscopy and very deep HST optical/NIR imaging. These “grism-$z$s” ($z_\mathrm{g}$) significantly enlarge the sample of high-$z$ () galaxies with high quality redshift estimates, where typical errors of the grism-$z$s are [@brammer12; @momcheva16]. We compare the CANDELS photo-$z$s to the HUDF $z_\mathrm{s/g}$ estimates in the left panel of Fig.\[fig:udf\_correction\]. The majority of the data points closely follow the diagonal, suggesting that the 3D-HST photo-$z$s are overall well calibrated as needed for unbiased estimates of the redshift distribution. However, we note the presence of two relevant systematic features: first, there are three catastrophic outliers that are at high , but are assigned a low . Second, there is an increased, asymmetric scatter at . Most notably, many galaxies with an assigned photometric redshift are actually at higher redshift. This is likely the result of redshift focusing effects [e.g. @wolf09] caused by the broad band HST filters. While this comparison allows us to identify these issues, the matched catalogue is insufficient to derive a robust statistical correction for our full photometric sample given the incompleteness of the sample. To overcome this limitation of incompleteness, we use deep photometric redshifts computed by @rafelski15 using HUDF data as a second comparison sample. Compared to the CANDELS photo-$z$s they benefit from much deeper HST optical [@beckwith06] and NIR imaging [@koekemoer13], and additionally incorporate new HST/UVIS Near UV imaging from the UVUDF project [@teplitz13] taken in the F225W, F275W, and F336W filters. These bands probe the Lyman break in the redshift range , which contains most of our weak lensing source galaxies. At these redshifts, the NIR imaging additionally probes the location of the Balmer/4000Å break. Hence, we expect that the resulting photo-$z$ should be highly robust against catastrophic outliers. We test this by comparing them to the $z_\mathrm{s/g}$ redshifts in the middle panel of Fig.\[fig:udf\_correction\]. Here we use the photo-$z$ estimates $z_\mathrm{BPZ}$ obtained by @rafelski15 using `BPZ` as it yields the highest robustness against catastrophic outliers in their analysis. Note that the comparison of $z_\mathrm{BPZ}$ and $z_\mathrm{s/g}$ suggests that $z_\mathrm{BPZ}$ slightly overestimates the redshifts for the colour-selected sample in the redshift intervals and , with median redshift offsets of 0.071 and 0.171, respectively. We have therefore subtracted these offsets in the corresponding redshift intervals, yielding $z_\mathrm{BPZ,fix}$, which is shown in Fig.\[fig:udf\_correction\]. As visible in the middle panel of Fig.\[fig:udf\_correction\], $z_\mathrm{BPZ,fix}$ correlates tightly with $z_\mathrm{s/g}$. In particular, the three catastrophic outliers from the left panel are now correctly placed at high redshifts. Likewise, the redshift focusing effects are basically removed. The remaining scatter with one moderate outlier has negligible impact on our results. For example, agrees to 0.4% between $z_\mathrm{BPZ,fix}$ and $z_\mathrm{s/g}$ for the matched catalogue and clusters at (we include this in the systematic error budget of Sect.\[se:zdist\_fix\_uncertainty\]). This suggests that $z_\mathrm{BPZ,fix}$ provides a sufficiently accurate approximation for the true redshift. Hence, we use $z_\mathrm{BPZ,fix}$ as a reference to obtain a statistical correction for the systematic features of the CANDELS photo-$z$s. We compare the 3D-HST photo-$z$s $z_\mathrm{p}$ in the HUDF to $z_\mathrm{BPZ,fix}$ in the right panel of Fig.\[fig:udf\_correction\], again showing the previously identified catastrophic outliers at and redshift focusing effects at , but now at the full depth of our photometric sample. The catastrophic outliers with are dominated by blue galaxies, for which 9 out of 12 galaxies appear to be truly at high redshifts. In order to implement a statistical correction for these outliers for the full CANDELS catalogue, we note the 12 redshift offsets . We bootstrap this empirically defined distribution to define the correction: for each CANDELS galaxy with and we add a randomly drawn offset to its $z_\mathrm{p}$. Likewise, we apply a statistical correction for the redshift focusing within the redshift range for galaxies with (which are most strongly affected, see Fig.\[fig:udf\_correction\]), again randomly sampling from the corresponding offsets in the HUDF. For the latter correction we split the galaxies into two magnitude ranges ( and ) given that the fainter galaxies appear to suffer from the redshift focusing effects more strongly. We show the resulting distribution of statistically corrected redshifts $z_\mathrm{f}$ as magenta dashed histograms in the top panels of Fig.\[fig:zdist\_f814w\]. As expected, it has a lower fraction of low-$z$ galaxies compared to the uncorrected $z_\mathrm{p}$ distribution, as well as a reduction of the redshift focusing peak at . Both effects are compensated by a higher fraction of high-$z$ galaxies, where we also note that the local minimum at , which likely results from the redshift focusing (see also Sect.\[se:zdist\_candels\_checks\]), is reduced. Averaged over our full cluster sample, and accounting for the magnitude-dependent effects explained in the following sections (e.g. shape weights), the application of this correction scheme leads to a 12% decrease of the resulting cluster masses. Of this, 10% originate from the correction for catastrophic outliers, and 2% from the correction for redshift focusing. ### Uncertainty of the statistical correction of the redshift distribution {#se:zdist_fix_uncertainty} The statistical correction of the redshift distribution explained in Sect.\[se:sub:catastrophic\_outliers\] has a non-negligible impact on our analysis. Therefore it is important to quantify its uncertainty. We consider a number of effects that affect the uncertainty: first, we estimate the statistical uncertainty originating from the limited size of the HUDF catalogue by generating bootstrapped versions of it, which are then used to generate the offset samples. This yields a small, 0.5% uncertainty regarding the average masses. Second, our correction scheme assumes that the relative effects seen in the HUDF are representative for the full CANDELS area. However, some previous studies suggest that the GOODS-South field, which contains the HUDF, could be somewhat under-dense at lower redshifts compared to the cosmic mean [e.g. @schrabback07; @hartlap09]. To obtain a worst case estimate of the impact this could have, we assume that the GOODS-South field could be under-dense at low redshifts by a factor 3 compared to the cosmic mean. Hence, we artificially boost the number of HUDF galaxies with that are truly at low-$z$ by a factor 3 for the generation of the offset pool. On average this leads to a 3% increase of the cluster masses. Third, we note that our correction for redshift focusing incorporates most but not all of the corresponding outliers in the right panel of Fig.\[fig:udf\_correction\]. We assume a conservative 50% relative uncertainty on the 2% correction, corresponding to an absolute 1% uncertainty. Adding all individual systematic uncertainties identified here and in Sect.\[se:sub:catastrophic\_outliers\] in quadrature yields a combined systematic uncertainty for the systematic corrections to the photometric redshifts of 3.3% in the average cluster mass. ### Consistency checks using spectroscopic and grism redshifts in the CANDELS fields {#se:zdist_candels_checks} In Sect.\[se:sub:catastrophic\_outliers\] we obtained a statistical correction for systematic features in the CANDELS photo-$z$s using very deep data available in the HUDF. Here we present cross-checks for this correction using the CANDELS redshift catalogue from @momcheva16, which combines a compilation of high fidelity spectroscopic redshifts from with redshift estimates derived from their joint analysis of slitless WFC3/NIR grism spectra from the 3D-HST project and the photometric catalogues. These grism data are shallower than those available in the HUDF (see Sect.\[se:sub:catastrophic\_outliers\]) but cover a much wider area. We restrict the use of these grism-$z$s to relatively bright galaxies (NIR magnitude ). These galaxies were individually inspected by the 3D-HST team, allowing us to select galaxies classified to have robust redshift estimates. For these relatively bright galaxies the continuum emission is comfortably detected in the grism data, yielding high-quality redshift estimates with a typical redshift error of [@momcheva16], which we can neglect compared to the photo-$z$ uncertainties. For the combined sample of galaxies with spec-$z$s and grism-$z$s we compare the colour-selected histogram of spec-$z$s/grism-$z$s ($z_\mathrm{s/g}$, using $z_\mathrm{s}$ in case both are available) to the histogram of their photo-$z$s in the bottom panels of Fig.\[fig:zdist\_f814w\]. Here we note two points: First, the spec-$z$s/grism-$z$s confirm that the colour selection indeed provides a very efficient removal of galaxies at our targeted cluster redshifts. Second, the high-$z$ galaxies are distributed in a relatively symmetric, unimodal peak that has a maximum at according to spec-$z$s/grism-$z$s. In contrast, the photo-$z$ histogram shows two slight peaks ( and ). This is consistent with the conclusion from Sect.\[se:sub:catastrophic\_outliers\] that the peaks in the photo-$z$ histogram of the full photometric sample (top panels of Fig.\[fig:zdist\_f814w\]) at these redshifts are a result of redshift focusing effects and not true large-scale structure peaks in the galaxy distribution. As a further cross-check we reconstruct the redshift distribution of the photometric sample by exploiting its spatial cross-correlation with the spec-$z$s/grism-$z$ sample, applying the technique developed by @newman08 [@schmidt13; @menard13]. Specifically, we use the implementation in <span style="font-variant:small-caps;">The-wiZZ</span>[^7] redshift recovery code [@morrison17]. We provide the details of this analysis in Appendix \[app:candels\_x\], showing that it independently confirms the presence of the catastrophic redshift outliers and redshift focusing effects. ### Limitations of the averaged posterior probability distribution {#se:posterior} Past weak lensing studies suggest that a better approximation of the true source redshift distribution may be given by the average photometric redshift posterior probability distribution $p(z)$ of all sources compared to a histogram of the best-fit (or peak) photometric redshifts [see e.g. @heymans12; @benjamin13; @bonnett15]. To test this we recompute the $p(z)$ using <span style="font-variant:small-caps;">EAZY</span> from the photometric catalogues, which is necessary as the $p(z)$ are not reported in the catalogues. As visible in Fig.\[fig:zdist\_f814w\], the redshift distribution inferred from the averaged $p(z)$ is relatively similar to the normalised histogram of the peak photometric redshifts $z_\mathrm{p}$. We note that the redshift focusing peak at and local minimum at are slightly less pronounced in the averaged $p(z)$, but they do not reach the level suggested by the corrected $z_\mathrm{f}$ histogram. More severely, the averaged $p(z)$ over predicts the fraction of low-$z$ galaxies compared to the $z_\mathrm{f}$ distribution similarly to the $z_\mathrm{p}$ histogram. We therefore conclude that the use of the averaged $p(z)$ instead of the $z_\mathrm{p}$ histogram is insufficient to account for the systematic features identified in Sect.\[se:sub:catastrophic\_outliers\]. ![ Normalised histogram of the statistically corrected photometric redshift estimates $z_\mathrm{f}$ for all galaxies in our CANDELS catalogues that pass the weak lensing cuts and the colour selection after adding noise to mimic the properties of the SPT-CL$J$0000$-$5748 data, both for the ACS+FORS2 (magenta dotted) and the ACS-only (black solid) selection. The vertical dotted line indicates the cluster redshift, at which both selections achieve an efficient suppression also in the presence of noise. \[fig:zdist\_noisy\_minimalistic\]](SPT-CLJ0000-5748_zf_histo.png){width="0.99\columnwidth"} Source selection in the presence of photometric scatter {#se:source_sel_scatter} ------------------------------------------------------- Outside the area of the central F814W ACS tile we only have single band F606W observations from HST. For the colour selection we therefore have to combine the F606W data with the VLT/FORS2 $I$-band imaging (see Sect.\[se:data\_vlt\]). We measure colours between these images as described in Appendix \[app:details\_fors2color\]. However, VLT/FORS2 $I$-band observations are not available in all CANDELS fields. We therefore need to accurately map the selection in the ACS+FORS2-based $V_\mathrm{606,con}-I_\mathrm{FORS2}$ colour to the colour available in CANDELS. We empirically obtain this mapping through the comparison of both colour estimates in the inner cluster regions, where both are available (see Appendix \[app:tiecolor\]). As described in Appendix \[app:scatter\] we add photometric scatter to the catalogues from the CANDELS fields to mimic the noise properties of the cluster fields for the colour selection. In particular, we apply an empirical model for the (non-Gaussian) scatter between the ACS-only and the ACS+FORS2 colours derived from the comparison of the colour measurements in the inner cluster regions. The ACS-only colour selection has higher signal-to-noise, allowing us to include galaxies with in the analysis. In contrast, the ACS+FORS2 colour selection is more noisy, which is why we have to employ shallower magnitude limits (dependent on the depth of the VLT data, see Table \[tab:vltdata\]) and more stringent colour cuts (see Table \[tab:app:colourcuts\] in Appendix \[app:details\_fors2color\_scatter\]). Fig.\[fig:zdist\_noisy\_minimalistic\] demonstrates that this approach leads to a robust removal of galaxies at the cluster redshift despite the presence of noise. Here we show the histogram of the statistically corrected redshift estimates $z_\mathrm{f}$ for the CANDELS galaxies passing the colour selection for SPT-CL$J$0000$-$5748 after application of the photometric scatter. Averaged over the cluster sample we find that 98.9% (98.1%) of the CANDELS galaxies with are removed by the ACS+FORS2 (ACS-only) colour selection scheme when the noise is taken into account. As shown in Appendix \[se:test\_extremely\_blue\] this translates into a negligible expected cluster member contamination in the weak lensing analysis. In addition, we will show in Sect.\[se:photo\_number\_density\_tests\] that the total source density and the source density profiles provide limits on the residual cluster member contamination, which are consistent with no contamination. Analysis in magnitude bins {#se:analysis_in_magbins} -------------------------- As shown in Fig.\[fig:beta\_beta2\_w\], $\langle\beta\rangle$ increases moderately within the magnitude range , which is due to a larger fraction of high-redshift galaxies passing the colour selection at fainter magnitudes. We only include galaxies with in our analysis as brighter galaxies contain only a low fraction of background sources. We split the source galaxies into subsets according to $V_{606}$ magnitude (0.5mag-wide bins) in order to optimise the $S/N$ of our measurement. This allows us to adequately weight the bins in the analysis not only accounting for the shape weight $w$, but also the geometric lensing efficiency. ![Analysis of SPT-CLJ0000$-$5748 as function of $V_{606}$ magnitude, where the solid (open) symbols correspond to the ACS-only (ACS+FORS2) analysis. [*Top:*]{} Mean weak lensing shape weight $w$ with error-bars indicating the dispersion from all selected galaxies in the magnitude bin. [*Bottom:*]{} $\langle\beta \rangle$ (circles) and $\langle\beta^2 \rangle$ (squares) with error-bars showing the dispersion of their estimates computed from all CANDELS sub-patches (see Sect.\[sec:beta\_los\_variation\]), thus indicating the expected line-of-sight variations for the field sizes of our cluster observations. \[fig:beta\_beta2\_w\]](SPT-CLJ0000-5748_mag_beta_w.png){width="1\columnwidth"} Accounting for line-of-sight variations {#sec:beta_los_variation} --------------------------------------- There is statistical uncertainty on how well we can estimate the cosmic mean $\langle \beta \rangle$ in a magnitude bin (given our lensing and colour selection) due to sampling variance and the finite sky-coverage of CANDELS. Furthermore, the actual redshift distribution along the line-of-sight to each of our clusters will be randomly sampled from this cosmic mean distribution, leading to additional statistical scatter, see e.g. @hoekstra11, who show that this is particularly relevant for high-$z$ clusters. To account for the statistical scatter in our weak lensing mass analysis (Sect.\[sec:wlmasses\]), we subdivide the CANDELS fields into sub-patches that match the size of our cluster field observations (single ACS tiles for the ACS-only colour selection and mosaics for the ACS+FORS2 selection) and compute $\langle \beta \rangle_i$ and $\langle \beta^2 \rangle_i$ from the redshift distribution of each sub-patch $i$. From the scatter of these quantities between all sub-patches we compute the resulting scatter in the mass constraints in Sect.\[sec:profiles\]. Furthermore, we need to investigate if the uncertainty on the estimate of the cosmic mean $\langle \beta \rangle$ due to the finite sky-coverage of CANDELS adds a significant systematic uncertainty in our error-budget. For this, we first compute the uncertainty on the mean $\langle \beta \rangle$ from the variance of the $\langle \beta \rangle_i$. Assuming all $N$ sub-patches were statistically independent, we find a very small relative uncertainty ($0.6\%$) for our lowest-redshift cluster SPT-CLJ2331$-$5051 at and $0.4\%$ ($1.1\%$) for our highest-redshift cluster SPT-CLJ2106$-$5844 at using the ACS-only (ACS+FORS2) colour selection combining all magnitude bins. However, due to large-scale structure the $\langle \beta \rangle_i$ within each CANDELS field will be correlated. A more conservative estimate can be obtained by computing $\langle \beta \rangle_i$ for each CANDELS field (without sub-patches) and estimating $\frac{\Delta \langle \beta \rangle}{\langle \beta \rangle}$ from the variation between the five fields[^8]. This yields (0.6%) for SPT-CLJ2331$-$5051 and $0.4\%$ ($1.0\%$) for SPT-CLJ2106$-$5844, again employing the ACS-only (ACS+FORS2) colour selection. This uncertainty is taken into account in our systematic error budget in Sect.\[se:systematic\_error\_budget\], but we note that it is very small compared to our statistical errors in all cases. Accounting for magnification {#sec:magnification_model} ---------------------------- ![image](magnification_ngal.png){width="1\columnwidth"} ![image](magnification_beta.png){width="1\columnwidth"} In addition to the shear, the weak lensing effect of the clusters magnifies background sources by a factor $\mu(z)$ given by Eq.(\[eq:magnification\_correct\]). This effectively alters the source redshift distribution, but this effect has typically been ignored in previous studies. For our analysis this has three effects: first, it increases the fluxes of sources by a factor $\mu(z)$, which may place them into brighter magnitude bins, thus increasing the total source density by including galaxies which are intrinsically too faint to be included. Second, it reduces the source sky area we observe, diluting the number density of sources by a factor $\mu(z)$. Finally, the magnification of object sizes may lead to the inclusion of some small galaxies which would otherwise be excluded by the lensing size cut. However, the large majority of our galaxies are well-resolved with HST, so we will ignore this latter effect (but it may be more relevant for data with lower image quality). We estimate the impact of the first and second effect from a colour-selected[^9] CANDELS catalogue (lensing is achromatic). Here we restrict the analysis to the deeper GOODS fields, initially including galaxies down to . For this part of the analysis we do not require a matching entry in our 1 orbit-depth shape catalogue in order to maximise the completeness at the faint end. We include the statistical correction for catastrophic redshift outliers and redshift focusing from Sect.\[se:sub:catastrophic\_outliers\], where we apply the same scheme also for one additional magnitude bin with . For each cluster redshift we compute $\beta(z_i)$ for each galaxy $i$ (using ) in the CANDELS catalogue and approximate the magnification as $$\mu(z)-1\simeq \frac{\beta(z)}{\beta_0}\left(\mu_0-1\right) \, , \label{eq:mu_approximated}$$ where $\mu_0$ indicates the magnification at an arbitrary fiducial $\beta_0$, for which we use close to the mean $\beta$ for our higher redshift clusters (compare Table \[tab:betastats\]). The scaling in Eq.(\[eq:mu\_approximated\]) is adequate in the weak lensing limit (, ), in which case Eq.(\[eq:magnification\_correct\]) simplifies to $$\mu(z)=\frac{1}{1-2\frac{\beta(z)}{\beta_0}\kappa_0+\left(\frac{\beta(z)}{\beta_0}\right)^2(\kappa_0^2-|\gamma_0|^2)}\simeq 1+2\frac{\beta(z)}{\beta_0}\kappa_0 \, , \label{eq:mu_expand}$$ where $\kappa_0$ and $\gamma_0$ are the convergence and shear for . In practice we find that the assumed linear scaling with $\beta$ in Eq.(\[eq:mu\_approximated\]) is sufficiently accurate for all of our clusters within the considered radial range of the tangential reduced shear profile fits (see Sect.\[sec:profiles\]). ---------------------- ----------------------- ------------------------- ------------------------------------------ ---------- ----------- Cluster $\langle\beta\rangle$ $\langle\beta^2\rangle$ $\sigma_{\langle \beta \rangle_i}/\langle \beta\rangle$ ACS-only ACS+FORS2 SPT-CL$J$0000$-$5748 0.466 0.243 0.053 18.2 7.2 SPT-CL$J$0102$-$4915 0.374 0.163 0.068 20.4 3.6 SPT-CL$J$0533$-$5005 0.368 0.159 0.062 19.7 5.4 SPT-CL$J$0546$-$5345 0.303 0.107 0.083 13.1 2.9 SPT-CL$J$0559$-$5249 0.505 0.288 0.064 18.2 4.0 SPT-CL$J$0615$-$5746 0.334 0.132 0.075 18.0 2.3 SPT-CL$J$2040$-$5725 0.344 0.141 0.077 16.2 3.5 SPT-CL$J$2106$-$5844 0.282 0.093 0.087 9.2 2.0 SPT-CL$J$2331$-$5051 0.522 0.308 0.059 16.2 8.3 SPT-CL$J$2337$-$5942 0.425 0.205 0.059 18.3 7.6 SPT-CL$J$2341$-$5119 0.320 0.122 0.067 19.1 9.3 SPT-CL$J$2342$-$5411 0.300 0.105 0.082 15.8 2.5 SPT-CL$J$2359$-$5009 0.423 0.204 0.055 16.6 8.7 ---------------------- ----------------------- ------------------------- ------------------------------------------ ---------- ----------- [Note. — [*Column 1:*]{} Cluster designation. [*Columns 2–4:*]{} $\langle\beta\rangle$, $\langle\beta^2\rangle$, and $\sigma_{\langle \beta \rangle_i}/\langle \beta\rangle$ averaged over both colour selection schemes and all magnitude bins that are included in the NFW fits according to their corresponding shape weight sum. [*Columns 5–6:*]{} Density of selected sources in the cluster fields for the ACS-only and the ACS+FORS2 colour selection schemes, respectively.\ ]{} For each galaxy in the CANDELS catalogue we compute $\mu(z_i)$ for a range of $\mu_0$. We then estimate the magnified magnitude $V_{606,i}^\mathrm{lensed}=V_{606,i}-2.5\log_{10}\mu(z_i)$ for each galaxy, and keep track of the reduced sky area through a weight $W_i=1/\mu(z_i)$. By binning in $V_{606,i}^\mathrm{lensed}$ we then compute the lensed number density $$n_\mathrm{gal}^\mathrm{lensed}=\sum_\mathrm{galaxies} W_i/\mathrm{area}$$ and the mean lensed geometric lensing efficiency $$\langle \beta \rangle^\mathrm{lensed}=\sum_\mathrm{galaxies} W_i \beta_i/\sum_\mathrm{galaxies} W_i \, ,$$ where the summations are over all galaxies with lensed magnitudes falling into the corresponding bin. In Fig.\[fig:magnification\] we plot the ratio of these quantities to their not-lensed counterparts and computed in $V_{606}$ bins with uniform weight[^10]. This shows that magnification has only a minor net effect at magnitudes , which contain a large fraction of our source galaxies. In contrast, it significantly boosts both quantities for brighter magnitudes . The net impact of magnification on high-$z$ cluster mass estimates therefore strongly depends on the depth of the observations. For illustration we also show the redshift distributions within three magnitude bins and their relative change after applying magnification with and in Fig.\[fig:magnification\_zdist\]. [Previous weak lensing magnification studies have made the simplifying assumptions that sources are located at a single redshift and that the source counts can be described as a power law. Under these assumptions the ratio of the lensed and non-lensed cumulative source densities above a magnitude $m_\mathrm{cut}$ $$\frac{n(<m_\mathrm{cut})}{n_0(<m_\mathrm{cut})}=\mu^{2.5s-1}$$ depends only on the magnification $\mu$ and the slope of the logarithmic cumulative number counts $$s= \frac{\mathrm{d}\log_{10}n(<m)}{\mathrm{d}m}$$ [e.g. @broadhurst95; @chiu16b], where slopes () lead to a net increase (decrease) of the counts. As an illustration we estimate from our colour () and shape-selected GOODS-South and GOODS-North catalogue, finding that it can approximately be described as $$s(V_{606})\simeq +0.88\pm 0.03 -(0.15\pm 0.02)(V_{606}-24)\,$$ for . Under these simplifying assumptions we therefore expect a significant boost in the source density at bright magnitudes () where the slope of the number counts is steep, and only a small boost towards fainter magnitudes (), where the slope of the number counts is shallower. This roughly agrees with the more accurate results shown in Fig.\[fig:magnification\], but there are noticeable differences, such as the slight net decrease in the source density at in Fig.\[fig:magnification\]. As our sources are not at a single redshift, the simplifying assumptions are clearly not met, which is why we base our analysis on the more accurate approach described above. ]{} When fitting the reduced cluster shear profiles with NFW models in Sect.\[sec:wlmasses\], we compute a $\mu(\langle\beta\rangle_j,r)$ profile for magnitude bin $j$ and a given mass from the NFW model predictions for both $\kappa(\langle\beta\rangle_j,r)$ and $\gamma(\langle\beta\rangle_j,r)$ according to Eq.(\[eq:magnification\_correct\]). Employing Eq.(\[eq:mu\_approximated\]) with we compute the corresponding $(\mu_0-1)(r)$ profile and obtain radius-dependent corrections $\langle\beta\rangle^\mathrm{lensed}_j/\langle\beta\rangle_j(r)$ and $n_{\mathrm{gal},j}^\mathrm{lensed}/n_{\mathrm{gal},j}(r)$ by interpolating our CANDELS-based estimates (Fig.\[fig:magnification\]) between the discrete $\mu_0$ values. The fact that we compute the magnification in the NFW prediction from both $\kappa$ and $\gamma$ is our primary motivation to conduct the interpolation in terms of $\mu_0$ and not $\kappa_0$. This provides a more accurate correction than if the shear contribution is ignored, even though we assume the linear scaling in $\beta$ in Eq.(\[eq:mu\_approximated\]) to simplify the CANDELS analysis. ![[*Top:*]{} Distribution of the statistically corrected photometric redshifts $z_\mathrm{f}$ for galaxies in the GOODS-South and GOODS-North catalogues located in the ACS+WFC3 area when applying our ACS-only colour selection. The different histograms correspond to three different $V_{606}$ magnitude bins. [*Bottom:*]{} Relative change in those redshift distributions after application of weak lensing magnification for a lens at with and . \[fig:magnification\_zdist\]](zdist_magnified.png){width="1.0\columnwidth"} On average the application of the correction for magnification-induced changes in the redshift distribution reduces our estimated cluster masses by 3%. This net impact is relatively small since the majority of our sources are at , requiring small corrections. Also, we exclude the cluster cores, where the correction is the largest (see Fig.\[fig:magnification\_shear\]), from our tangential shear profile fits (see Sect.\[sec:wlmasses\]). However, we emphasise that weak lensing studies of high-$z$ clusters using shallower data will be affected more strongly and should adequately model this effect. ![Correction factors to the reduced shear profile model of a $M_\mathrm{200c}=7\times 10^{14}\mathrm{M}_\odot$ galaxy cluster at due to the impact of magnification on the source redshift distribution (solid curves) and the finite width of the redshift distribution ($\langle \beta^2\rangle$, see Eq.\[eq:gbeta2corrected\], dotted). The different colours correspond to different bins in the $V_{606}$ aperture magnitude. \[fig:magnification\_shear\]](shear_correction_factors.png){width="1.0\columnwidth"} We note a subtle limitation of our modelling approach for magnification which results from our choice to conduct the analysis as function of aperture magnitude. Here we ignore the fact that the increase in size due to magnification will lead to a larger fraction of the flux being outside the fixed aperture radius than without magnification. As a test we also conducted the magnification analysis using aperture-corrected magnitudes from CANDELS[^11], finding similar models as in Fig.\[fig:magnification\] but shifted to brighter magnitudes, with reaching unity at . Given the very minor impact magnification has for our data compared to the statistical uncertainties, the described subtle limitation can safely be ignored for the current study. In the future this can be avoided by computing aperture corrections in the filter used for shape measurements both for CANDELS and the cluster fields. Number density consistency tests {#se:photo_number_density_tests} -------------------------------- The measurements of the total source density and its radial dependence can be used to test the cluster member removal and our procedure to consistently select galaxies in the cluster and CANDELS fields in the presence of noise (Sect.\[se:source\_sel\_scatter\]). When computing the source density we account for masks and apply an approximate correction[^12] for the impact of obscuration by cluster members [@simet15]. We also account for the impact of cluster magnification, employing the corresponding radius-dependent magnification model for each cluster from Sect.\[sec:magnification\_model\]. ### Total source density {#se:total_ngal} In Fig.\[fig:ngal\] we compare the average density of selected source galaxies in the cluster fields as function of $V_{606}$ to the corresponding average density in the CANDELS fields corrected for the expected influence of magnification given our best-fit NFW cluster mass models. There is very good agreement for the ACS+FORS2 selection and reasonable agreement for the ACS-only selection (error-bars are correlated because of large-scale structure). Fig.\[fig:ngal\] also visualises that the ACS-only analysis (with two ACS bands) provides a substantially higher average total density of selected sources in the cluster fields of 16.8 galaxies/arcmin$^2$ compared to 5.2 galaxies/arcmin$^2$ for the ACS+FORS2 colour selection (see Table \[tab:betastats\] for the total source densities in each field). This shows that either substantially deeper ground-based imaging or ACS-based colours for the full imaging area would be required for the colour selection in order to adequately exploit the full depth of the ACS shape catalogues. ![Selected source density $n_\mathrm{gal}$ as function of $V_{606}$ accounting for masks and averaged over all the cluster fields (small solid symbols) and the corresponding source density averaged over the CANDELS fields when mimicking the same selection and accounting for photometric scatter and magnification (large open symbols). Black squares show the ACS-only selection, while magenta hexagons correspond to the ACS+FORS2 selection. We include only galaxies located within the fit range of the shear profiles (see Sect.\[sec:profiles\]) to avoid limitations of the magnification correction at small radii. The error-bars show the uncertainty on the mean from the variation between the contributing cluster fields or the five CANDELS fields, respectively, assuming Gaussian scatter. They are correlated between magnitude bins due to large-scale structure. \[fig:ngal\]](ngal.png){width="1\columnwidth"} ### Source density profile As an additional cross-check for the removal of cluster galaxies and our magnification model we plot the radial source density profiles for the ACS-only and ACS+FORS2 selected samples in Fig.\[fig:ngal\_radius\], averaged over all clusters, as function of cluster-centric distance from the X-ray centre (nearly identical results are obtained when using the SZ peak location, see Sect.\[sec:wlmasses\]) in units of their corresponding as estimated from the SZ signal. Here we compare the case with and without applying the magnification correction. [ The difference is small given that the magnification is relatively weak for the majority of the clusters. Also, most of the source galaxies have faint magnitudes, where the net impact of magnification is small]{} (see Fig.\[fig:magnification\]). The net difference is strongest for the inner cluster regions where the magnification is strongest. [In the case of a complete cluster galaxy removal and an accurate correction for magnification we expect]{} to measure a number density that is consistent with being flat as function of radius. To test this, we perform a model comparison test using the cluster sample-averaged number density profiles, assuming that errors were independent and Gaussian distributed. Each radial bin used in the test is the average of at least three clusters, while uncertainties are determined by bootstrapping the cluster sample. With these measures, the $\chi^2$ statistic should be a crude yet useful approximation to the true uncertainty distribution, while allowing us to use analytic model quality of fit and comparison tests. As expected for adequate removal of cluster members and magnification correction, we find that the source density profiles are consistent with being flat. For the ACS-only case, the maximum likelihood constant number density model returns a with 4 degrees of freedom, while a $1/r$ inverse-r profile with two parameters, the contamination fraction $f_{500}$ at $r_\mathrm{500c}$ and the background number density [@hoekstra07], returns a with 3 degrees of freedom. Both are acceptable models at , [ where the improvement in $\chi^2$ is consistent with random according to an F-test (). The rather low $\chi^2$ values might be due to our assumption of independent errors between bins, which neglects the effects of large-scale structure. For]{} the ACS+FORS2 selection, the constant number density model returns with 8 degrees of freedom, while an exponential model [see Appendix \[app:why\_not\_boost\] and @applegate14], which is preferred over the inverse-r model in this case, returns with 7 degrees of freedom, again suggesting that a flat number density model is sufficient ( from the F-test). For a general test for the consistency of the combined number density profile being flat we allow for negative $f_{500}$ [in these fits]{}, which could for example be mimicked by an incorrect magnification correction. The maximum likelihood parameter value for the inverse-r contamination model fit to the ACS-only number density profile [indeed]{} peaks at slightly, [but not significantly]{} negative values, . Fig.\[fig:ngal\_radius\] shows the measured number density profiles and maximum-likelihood model fits for both selections. ![Density of sources $n_\mathrm{gal}$ for the ACS-only and ACS+FORS2 colour selections as function of cluster centric distance around the X-ray centre in units of , as estimated from the SZ signal. The profiles account for both masks and obscuration by cluster members. Solid (open) symbols include (do not include) the correction for magnification. The coloured regions indicate the one sigma constraints on the mean background density from the five CANDELS fields. Black solid and dotted lines show the maximum likelihood and 68% uncertainty range for a constant density model. The dashed line shows for the ACS-only selection the maximum likelihood contamination model following a $1/r$ functional form. \[fig:ngal\_radius\]](ngal_radius.png){width="1\columnwidth"} Weak lensing constraints and mass analysis {#sec:wlmasses} ========================================== We reconstruct the projected mass distribution in our cluster fields in Sect.\[sec:massmaps\] and constrain the cluster masses via fits to the tangential reduced shear profile in Sect.\[sec:profiles\]. In Sect.\[se:stack\_mc\] we compare the stacked shear profiles from all clusters for the different centres used in the analysis and investigate the consistency of the data with different [concentration–mass]{} relations. In Sect.\[sec:sims\] we detail on the simulations used to calibrate the mass estimates. We discuss the systematic error budget in Sect.\[se:systematic\_error\_budget\]. Mass maps {#sec:massmaps} --------- ![image](SPT-CL\sclusterdemo_mass_sn.jpg){height="7.4cm"} ![image](SPT-CL\sclusterdemo_mass_sn_sb.jpg){height="7.4cm"} The weak lensing shear and convergence are linked as they are both based on second-order derivatives of the lensing potential. Therefore, a reconstruction of the convergence field can be obtained from the shear field up to a constant [@kaiser93], which is the mass-sheet degeneracy [@schneider95]. Motivated by the different colour-selected source densities in the inner and outer regions of our clusters we employ a Wiener filter for the convergence reconstruction using an implementation described in @mcinnes09 and @simon09. This code estimates the convergence on a grid taking the spatial variation in the source number density into account; it applies more smoothing where the number density of sources is lower. The smoothing in the Wiener filtered map employs the shear two-point correlation function $\xi_+(\theta)$ [e.g. @schneider06] as a prior on the angular correlation of the convergence, which affects the degree of smoothing. For this, we measure $\xi_+(\theta)$ in the cluster fields and find that it is on average approximately described by the fitting function $\xi_+^\mathrm{fit}(\theta)=0.012(1+\theta/\mathrm{arcmin})^{-2}$. We fix the mass-sheet degeneracy by setting the average convergence inside each cluster field to zero. While this underestimates the overall convergence for our relatively small cluster fields, this is irrelevant as we use the reconstructions to study positional offsets and the signal-to-noise ($S/N$) ratio of relative mass distributions, but not to obtain quantitative mass constraints. To compute the $S/N$ mass maps we generate 500 noise maps for each cluster by randomising the ellipticity phases and repeating the mass reconstruction. We then define the $S/N$ mass map as the ratio of the reconstruction from the actual data and the r.m.s. image of the noise reconstructions. ---------------------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ----- Cluster $\alpha$ $\delta$ $\Delta\alpha$ $\Delta\delta$ $\Delta\alpha$ $\Delta\delta$ \[deg J2000\] \[deg J2000\] \[arcsec\] \[arcsec\] \[kpc\] \[kpc\] SPT-CL$J$0000$-$5748 0.25195 $ -57.80875 $ 1.9 2.2 14 16 5.7 SPT-CL$J$0102$-$4915 15.71743 $ -49.25458 $ 7.1 7.9 55 61 5.7 SPT-CL$J$0533$-$5005 83.39772 $ -50.09984 $ 10.2 8.0 79 62 3.0 SPT-CL$J$0546$-$5345 86.65396 $ -53.75861 $ 5.1 3.7 41 30 3.6 SPT-CL$J$0559$-$5249 89.92875 $ -52.82297 $ 4.3 3.6 29 24 5.0 SPT-CL$J$0615$-$5746 93.96562 $ -57.77979 $ 4.3 2.8 34 23 5.1 SPT-CL$J$2040$-$5725 310.06389 $ -57.42232 $ 5.0 5.1 40 40 3.1 SPT-CL$J$2106$-$5844 316.52210 $ -58.74336 $ 7.2 4.5 59 37 2.9 SPT-CL$J$2331$-$5051 352.96521 $ -50.86360 $ 1.8 2.3 12 15 5.1 SPT-CL$J$2337$-$5942 354.35384 $ -59.70819 $ 1.8 2.5 13 19 6.0 SPT-CL$J$2341$-$5119 355.30057 $ -51.33015 $ 5.3 5.4 42 44 3.3 SPT-CL$J$2342$-$5411 355.69305 $ -54.18043 $ 3.7 9.9 30 80 3.1 SPT-CL$J$2359$-$5009 359.93213 $ -50.16822 $ 4.7 4.8 35 35 5.2 ---------------------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ----- As an example, Fig.\[fig:massplotdemo\_map\] shows an overlay of the contours of the mass reconstruction (starting at $2\sigma$ in steps of $0.5\sigma$) for [SPT-CL]{} on a FORS2 $BIz$ colour image (left) as well as a cut-out of the ACS imaging (right). Here we also indicate the locations of the X-ray centroid, BCG, and SZ peak (see Table \[tab:clusters\]). The corresponding figures for the other clusters are shown in Appendix \[se:additional\_figs\]. For all clusters the weak lensing reconstruction shows a mass peak associated with the cluster, with a peak signal-to-noise ratio between $2.9\sigma$ and $6.0\sigma$. Typically, the mass reconstructions follow the distribution of red cluster galaxies well, especially for the clusters with . Of these clusters, [SPT-CL]{}and [SPT-CL]{} show relatively symmetric mass reconstructions consistent with more relaxed morphologies, while [SPT-CL]{}, [SPT-CL]{}, [SPT-CL]{}, [SPT-CL]{}, and [SPT-CL]{} show more elongated or perturbed morphologies. In particular, [SPT-CL]{} is known to be an extreme merger [@menanteau12], for which our mass reconstruction separates both main components well [see also the independent weak lensing analysis by @jee14]. ![Histograms of spatial offsets between the peak in the mass reconstruction signal-to-noise map and the indicated centres. \[fig:massoffset\_histo\]](offset_histo.png){width="1\columnwidth"} In the mass signal-to-noise maps we determine the position of the mass peak of the corresponding cluster by identifying the pixel with the highest $S/N$ within 90$^{\prime\prime}$ from the SZ peak location. We report these positions in Table \[tab:masspeaklocations\] along with estimates of their uncertainty and peak signal-to-noise $(S/N)_\mathrm{peak}$. The positional uncertainties are estimated by generating 500 bootstrap samples of the source catalogue for which we repeat the reconstruction and identification of the peak location. The average r.m.s. positional uncertainty (including both directions) for the full sample is 59 kpc. [@dietrich12] investigate the origin of offsets between peaks in weak lensing mass reconstructions and the projected position of the 3D centre (defined as the minimum of the potential) of cluster-scale dark matter haloes in the Millennium Simulation [@springel05; @hilbert09]. Without shape noise and smoothing applied in the mass reconstruction they find very small offsets: their analysis using sources at is most similar to the set-up of our study, yielding a 90th percentile offset of $5.6 h^{-1}$ kpc. Hence, projection effects and large-scale structure have a negligible impact for the measured offsets in typical observing scenarios. [@dietrich12] find that smoothing and shape noise increase the offsets substantially, where the addition of shape noise has the biggest impact unless unnecessarily large smoothing kernels are used. Our bootstrap analysis provides an estimate for the positional uncertainty due to shape noise. However, the analysis likely underestimates the true positional uncertainty with respect to the 3D cluster centre as it does not explicitly account for the impact of smoothing. Nevertheless, we can use the distribution of offsets between the peaks in the mass signal-to-noise maps and different proxies for the cluster centre, namely the X-ray centroid, SZ peak, and BCG position, to test if these are similarly good proxies for the true 3D cluster centre position. Fig.\[fig:massoffset\_histo\] shows a histogram of the corresponding offset distributions. We also summarise the average, r.m.s., and median of these offset distributions in Table \[tab:offsetsstats\], where errors indicate the dispersion of the corresponding values when bootstrapping the cluster sample. While the X-ray centroids yield the smallest average and median offsets, their r.m.s. offset is similar to the one for the SZ peaks. The distribution of offsets between the BCG locations and the mass signal-to-noise peaks has the largest r.m.s. offset, resulting from two outliers: the largest offset occurs for the merger [[SPT-CL]{}]{} (642 kpc), where the BCG is located in the South-Eastern component while the highest signal-to-noise peak in the mass reconstruction coincides with the North-Western cluster component, which also shows a strong concentration of galaxies but has a less bright BCG (see Fig.\[fig:massplotb\]). In contrast, both the SZ peak and the X-ray centroid are located between the two cluster cores and peaks of the mass reconstruction, resulting in smaller offsets. [[SPT-CL]{}]{} also shows a large (522 kpc) offset between the BCG and the mass signal-to-noise peak, while the latter is broadly consistent with the SZ peak, X-ray centroid, and strongest galaxy concentration. This could also be explained with a merger scenario, where a smaller component hosting a brighter BCG is falling into the main cluster. Centre average r.m.s. median ---------------- ---------------- ---------------- ------------------ SZ peak $ 137 \pm 21 $ $ 158 \pm 23 $ $ 100 \pm 13 $ X-ray centroid $ 105 \pm 30 $ $ 154 \pm 50 $ $ 64 \pm 44 $ BCG location $ 159 \pm 52 $ $ 249 \pm 72 $ $ 80 \pm 54 $ : Average, r.m.s., and median of the offsets \[kpc\] between the peaks in the mass reconstruction signal-to-noise maps and the SZ peak, X-ray centroid, and BCG location. \[tab:offsetsstats\] [Note. — Errors indicate the dispersion of the values when bootstrapping the cluster sample.]{} Individual shear profile analysis {#sec:profiles} --------------------------------- ![Tangential reduced shear profile (black solid circles) of [[SPT-CL]{}]{} centred on the X-ray centroid and obtained by combining the profiles of all contributing magnitude bins of the ACS-only plus the ACS+FORS2 selection (see Sect.\[sec:profiles\]). The curve shows the correspondingly-combined best-fitting NFW model prediction obtained by fitting the data in the range and using the @diemer15 [concentration–mass]{} relation. The grey open circles indicate the 45 degrees-rotated reduced cross-shear component, which is a test for systematics, shifted by for clarity. The corresponding plots of the other clusters are shown in Appendix \[se:additional\_figs\]. \[fig:massplotdemo\_profile\]](SPT-CL\sclusterdemo-a-c_gt.png){width="1\columnwidth"} We compute profiles of the tangential reduced shear (Eq.\[eq:gt\]) around the cluster centres in 14 linearly-spaced bins of transverse physical separation between 300kpc and 1.7Mpc (100kpc-wide bins), but note that we restrict the fit range to when deriving mass constraints. Smaller scales are more susceptible to the impact of miscentring, cluster substructure, uncertainties in the [concentration–mass]{} relation, and shear calibration, while larger scales suffer from an increasingly incomplete azimuthal coverage, where 1.5Mpc (1.3Mpc) equals the largest radius with full azimuthal coverage at the median (lowest) redshift of the targeted clusters. We repeat the analysis for the different proxies for the cluster centre (X-ray centroid, SZ peak, and BCG position), but regard the measurements using the X-ray centroids as our primary (default) results, given that they yield the smallest average and median offsets from the peaks in the mass signal-to-noise maps (Sect.\[sec:massmaps\]). We compute separate tangential reduced shear profiles for each magnitude bin and colour selection scheme, where we use the ACS-only selection in the inner cluster regions where both ACS bands are available, and the ACS+FORS2 selection in the outer cluster regions. Each magnitude bin for both colour selection schemes has a separate value for $\langle \beta \rangle$ and $\langle \beta^2 \rangle$ (see Sect.\[se:analysis\_in\_magbins\], Fig.\[fig:beta\_beta2\_w\], and average values reported in Table \[tab:betastats\]), which we correct for magnification as a function of cluster centric distance as described in Sect.\[sec:magnification\_model\]. We fit the profiles from all ACS-only magnitude bins plus those ACS+FORS2 bins that have sufficiently low photometric scatter (Sect.\[se:source\_sel\_scatter\] and Appendix \[se:scatter\_acsfors2\]) jointly with a reduced shear profile model (see Eq.\[eq:gbeta2corrected\]) according to @wright00, assuming spherical mass distributions that follow the NFW density profile [@navarro97]. Here we use a fixed [concentration–mass]{} ([$c(M)$]{}) relation, where we by default employ the [$c(M)$]{} relation from @diemer15, but also test the consistency of the data with other relations in Sect.\[se:stack\_mc\]. While the mass distributions in individual clusters may well deviate from an NFW profile, we account for the net impact on an ensemble of clusters in Sect.\[sec:sims\]. Due to the fixed [concentration–mass]{} relation we fit a one-parameter model to each cluster. Here we perform a $\chi^2$ minimisation using $M_{200\mathrm{c}}$ as free parameter, comparing the predicted reduced shear values to the measured values in each contributing bin in radius and magnitude. Given its dominance we employ a pure shape noise covariance matrix derived from our empirical weighting scheme (see Appendix \[app:shapes\_candels\]). In the fit we also allow for , which we model by switching the sign of the tangential reduced shear profile. For the calibration of scaling relations we make use of the full likelihood distribution (see Sect.\[se:mtx\]). In addition, we identify the maximum likelihood (minimum $\chi^2$) location and the $\Delta\chi^2=1$ points, which we report in Table \[tab:mass\], where conversions between over-density masses use the assumed [$c(M)$]{} relation. Note that the derived mass constraints are expected to be biased due to effects in the mass modelling such as miscentring, which will be addressed in Sect.\[sec:sims\]. The statistical errors in Table \[tab:mass\] include two additional minor noise sources. The first source is given by line-of-sight variations in the redshift distribution, which we estimate from the dispersion $\sigma_{\langle \beta \rangle_i}$ of the estimates $\langle \beta \rangle_i$ from the CANDELS sub-patches (see Sect.\[sec:beta\_los\_variation\]). In Table \[tab:betastats\] we report $\sigma_{\langle \beta \rangle_i}/\langle \beta\rangle$, which introduces an additional relative noise in the mass estimates of $\sigma_{M,z}/M\simeq 1.5\sigma_{\langle \beta \rangle_i}/\langle \beta\rangle$, where . Further statistical noise is added by projections of uncorrelated large-scale structure [@hoekstra01]. To estimate it we compute 500 random realisations of the cosmic shear field per cluster for our reference cosmology and the colour-selected source redshift distribution as detailed in Appendix B of @simon12, with the non-linear matter power spectrum estimated following @takahashi12[^13]. We add these cosmic shear field realisations to the measured shear field in the SPT cluster fields and repeat the cluster mass analysis for each realisation. The dispersion in the best-fit mass estimates then yields an estimate for the large-scale structure noise. We find that it amounts to of the statistical errors from shape noise. [Additional scatter between profile-fitted weak lensing mass estimates and halo masses defined via spherical overdensities is caused by halo triaxiality, variations in cluster density profiles, and correlated large-scale structure [e.g. @gruen15; @umetsu16]. This scatter typically amounts to for massive clusters [@becker11] and is not explicitly listed in Table \[tab:mass\]. Instead, we absorb it in the intrinsic scatter accounted for in the scaling relation analysis (see Sect.\[se:mtx\] and Dietrich et al. in prep.). ]{} =0.13cm ---------------------- -------------------------------------- --------------------- ------------------------------------- --------------------- -------------------------------------- --------------------- -------------------------------------- --------------------- -- -- -- Cluster $b_{200\mathrm{c}}$ $b_{500\mathrm{c}}$ $b_{200\mathrm{c}}$ $b_{500\mathrm{c}}$ SPT-CL$J$0000$-$5748 $6.2_{-2.4}^{+2.6} \pm 1.1 \pm 0.5$ 0.91 $4.2_{-1.6}^{+1.8} \pm 0.7 \pm 0.3$ 0.88 $6.5_{-2.5}^{+2.6} \pm 1.1 \pm 0.5$ 0.80 $4.5_{-1.7}^{+1.8} \pm 0.7 \pm 0.3$ 0.82 SPT-CL$J$0102$-$4915 $11.1_{-2.8}^{+2.9} \pm 1.2 \pm 1.1$ 0.86 $7.9_{-2.1}^{+2.2} \pm 0.9 \pm 0.8$ 0.88 $14.4_{-2.8}^{+2.8} \pm 1.2 \pm 1.5$ 0.79 $10.3_{-2.1}^{+2.1} \pm 0.9 \pm 1.1$ 0.79 SPT-CL$J$0533$-$5005 $4.3_{-2.4}^{+2.7} \pm 1.0 \pm 0.4$ 0.88 $2.9_{-1.6}^{+1.9} \pm 0.7 \pm 0.3$ 0.87 $2.4_{-1.8}^{+2.4} \pm 1.0 \pm 0.2$ 0.80 $1.6_{-1.2}^{+1.7} \pm 0.7 \pm 0.1$ 0.81 SPT-CL$J$0546$-$5345 $5.4_{-3.3}^{+3.7} \pm 1.1 \pm 0.7$ 0.86 $3.7_{-2.3}^{+2.6} \pm 0.8 \pm 0.5$ 0.85 $2.6_{-2.4}^{+3.5} \pm 1.1 \pm 0.3$ 0.72 $1.8_{-1.6}^{+2.4} \pm 0.8 \pm 0.2$ 0.73 SPT-CL$J$0559$-$5249 $8.0_{-2.9}^{+3.1} \pm 1.0 \pm 0.8$ 0.79 $5.4_{-2.0}^{+2.2} \pm 0.7 \pm 0.5$ 0.81 $4.7_{-2.5}^{+2.9} \pm 1.0 \pm 0.5$ 0.84 $3.2_{-1.7}^{+2.0} \pm 0.7 \pm 0.3$ 0.85 SPT-CL$J$0615$-$5746 $6.8_{-2.6}^{+2.9} \pm 1.0 \pm 0.8$ 0.88 $4.7_{-1.8}^{+2.0} \pm 0.7 \pm 0.5$ 0.85 $5.8_{-2.5}^{+2.8} \pm 1.0 \pm 0.7$ 0.82 $3.9_{-1.7}^{+1.9} \pm 0.7 \pm 0.4$ 0.80 SPT-CL$J$2040$-$5726 $2.1_{-1.9}^{+2.9} \pm 0.8 \pm 0.2$ 0.87 $1.4_{-1.3}^{+2.0} \pm 0.6 \pm 0.2$ 0.81 $2.1_{-2.0}^{+2.9} \pm 0.8 \pm 0.2$ 0.80 $1.4_{-1.3}^{+2.0} \pm 0.6 \pm 0.2$ 0.80 SPT-CL$J$2106$-$5844 $8.8_{-4.6}^{+5.0} \pm 1.5 \pm 1.1$ 0.85 $6.1_{-3.3}^{+3.7} \pm 1.1 \pm 0.8$ 0.86 $8.2_{-4.3}^{+5.0} \pm 1.5 \pm 1.1$ 0.81 $5.7_{-3.1}^{+3.6} \pm 1.1 \pm 0.7$ 0.78 SPT-CL$J$2331$-$5051 $3.8_{-2.1}^{+2.5} \pm 1.1 \pm 0.3$ 0.85 $2.6_{-1.4}^{+1.7} \pm 0.7 \pm 0.2$ 0.92 $4.0_{-2.1}^{+2.5} \pm 1.1 \pm 0.4$ 0.85 $2.7_{-1.4}^{+1.7} \pm 0.7 \pm 0.2$ 0.87 SPT-CL$J$2337$-$5942 $10.5_{-2.8}^{+2.9} \pm 1.3 \pm 0.9$ 0.88 $7.2_{-2.0}^{+2.1} \pm 0.9 \pm 0.7$ 0.91 $10.0_{-2.8}^{+2.9} \pm 1.3 \pm 0.9$ 0.82 $6.9_{-2.0}^{+2.1} \pm 0.9 \pm 0.6$ 0.83 SPT-CL$J$2341$-$5119 $2.4_{-1.9}^{+2.5} \pm 1.1 \pm 0.2$ 0.91 $1.6_{-1.3}^{+1.7} \pm 0.7 \pm 0.2$ 0.89 $2.3_{-1.8}^{+2.5} \pm 1.1 \pm 0.2$ 0.80 $1.5_{-1.2}^{+1.7} \pm 0.7 \pm 0.1$ 0.80 SPT-CL$J$2342$-$5411 $8.6_{-3.5}^{+3.8} \pm 1.4 \pm 1.1$ 0.87 $6.0_{-2.5}^{+2.8} \pm 1.0 \pm 0.7$ 0.84 $7.0_{-3.4}^{+3.7} \pm 1.4 \pm 0.9$ 0.79 $4.8_{-2.4}^{+2.7} \pm 1.0 \pm 0.6$ 0.81 SPT-CL$J$2359$-$5009 $5.0_{-2.6}^{+3.0} \pm 1.1 \pm 0.4$ 0.91 $3.4_{-1.8}^{+2.1} \pm 0.8 \pm 0.3$ 0.92 $5.7_{-2.5}^{+2.8} \pm 1.1 \pm 0.5$ 0.83 $3.9_{-1.7}^{+1.9} \pm 0.8 \pm 0.3$ 0.84 ---------------------- -------------------------------------- --------------------- ------------------------------------- --------------------- -------------------------------------- --------------------- -------------------------------------- --------------------- -- -- -- For visualisation we show profiles in Figs.\[fig:massplotdemo\_profile\] and \[fig:massplota\] to \[fig:massplotm\], where we have combined shear estimates from the different magnitude bins and colour selections for the analysis using the X-ray centroids as centres. Here we stack all profiles scaled to the same average $\langle \beta \rangle$ of all magnitude bins of the cluster as $$\langle g_\mathrm{t} \rangle_\mathrm{comb}(r_k)=\sum_{j \in \mathrm{mag \, bins}}\langle g_\mathrm{t} \rangle_j(r_k)\frac{\langle \beta \rangle}{\langle \beta \rangle_j}W_{j,k}/\sum_{j \in \mathrm{mag \, bins}} W_{j,k} \, ,$$ where $k$ indicates the radial bin, $j$ the magnitude bin and colour selection scheme, and $W_{j,k}=( {\langle \beta \rangle_j}/{\langle \beta \rangle})^{2}\sum w_i$ is the rescaled sum of the shape weights of the contributing galaxies. ![[[*Top*]{}:]{} Weighted average of the rescaled differential surface mass density profiles from all clusters. The circles, squares, crosses, and triangles show the signal measured around the X-ray centroids, the SZ peak positions, the BCG locations, and the weak lensing mass peaks, respectively. The circles showing the signal around the X-ray centroids are displayed at the correct radius, while the other symbols are shown with a horizontal offset for clarity. The curves show the correspondingly averaged best-fit model predictions for different fixed concentrations for the analysis employing the X-ray centres and using an extended fit range 300kpc to 1.5Mpc, which increases the sensitivity for constraints on the average concentration. [[*Bottom*]{}: Profile of the stacked reduced cross-shear component of all clusters measured with respect to their X-ray centres.]{} \[fig:shear\_profile\_stack\]](stack_gt.png "fig:"){width="1\columnwidth"} ![[[*Top*]{}:]{} Weighted average of the rescaled differential surface mass density profiles from all clusters. The circles, squares, crosses, and triangles show the signal measured around the X-ray centroids, the SZ peak positions, the BCG locations, and the weak lensing mass peaks, respectively. The circles showing the signal around the X-ray centroids are displayed at the correct radius, while the other symbols are shown with a horizontal offset for clarity. The curves show the correspondingly averaged best-fit model predictions for different fixed concentrations for the analysis employing the X-ray centres and using an extended fit range 300kpc to 1.5Mpc, which increases the sensitivity for constraints on the average concentration. [[*Bottom*]{}: Profile of the stacked reduced cross-shear component of all clusters measured with respect to their X-ray centres.]{} \[fig:shear\_profile\_stack\]](stack_gx.png "fig:"){width="0.95\columnwidth"} Stacked signal and constraints on the average cluster concentration {#se:stack_mc} ------------------------------------------------------------------- Miscentring reduces the shear signal at small radii. To test if our data show signs for this, we compare the stacked signal for the different centres ([top panel of]{} Fig.\[fig:shear\_profile\_stack\]). To stack the signal from clusters at different redshifts and lensing efficiencies we employ the differential surface mass density $\Delta \Sigma(r)$ (see Eq.\[eq:deltaSigma\]), where we compute $\Sigma_\mathrm{crit}$ using the $\langle\beta\rangle$ of the corresponding magnitude bin and colour selection scheme. Our clusters span a significant range in mass. Here we expect higher $\Delta \Sigma(r)$ profiles for the more massive clusters. Before stacking, we therefore scale them to approximately the same signal amplitude. For this we compute a theoretical NFW model for the differential surface mass density $\Delta\Sigma_\mathrm{model}$ for each cluster assuming its mass inferred from the SZ signal $M_{500\mathrm{c,SZ}}$ [@bleem15][^14] and a fixed , and then scale the cluster signal as $$\Delta \Sigma^*(r)=s \Delta \Sigma(r)\equiv \frac{\langle \Delta\Sigma_\mathrm{model}(800\mathrm{kpc})\rangle}{\Delta\Sigma_\mathrm{model}(800\mathrm{kpc})} \Delta \Sigma(r) \,.$$ We evaluate the theoretical model at an intermediate scale $r=800\mathrm{kpc}$, but note that the exact choice is not important as we are only interested in an approximate rescaling to optimise the weighting. We then compute the weighted average $$\langle \Delta \Sigma^*(r_j) \rangle = \sum_{i\in \mathrm{clusters}} \Delta \Sigma^*_i(r_j)\hat{W_{ij}}/ \sum_{i\in \mathrm{clusters}} \hat{W_{ij}} \,,$$ with $\hat{W_{ij}}=\left(s\sigma(\Delta \Sigma(r_j))\right)^{-2}$ and $\sigma(\Delta \Sigma(r_j))$ indicating the $1\sigma$ uncertainty of $\Delta \Sigma(r_j)$. The results are shown in [in the top panel of]{} Fig.\[fig:shear\_profile\_stack\]. We first note that the stacked profiles are fairly similar for the different centre definitions. This is also the case for an analysis centred on the peaks in the weak lensing mass reconstruction. Such an analysis should not suffer from miscentring, but is rather expected to deliver shear estimates that are biased high [see e.g. @dietrich12]. The similarity of the shear profiles suggests that, for the sample as a whole, miscentring appears to have relatively minor impact at the radial scales considered in our analysis. We also fit the reduced shear profiles of all clusters using models with different fixed concentrations. For three of these fixed concentrations and the analysis using the X-ray centres we show the averaged best-fit models from all clusters in Fig.\[fig:shear\_profile\_stack\], using the same scale factors and weights as used for the data. In these fits we use an extended fit range 300kpc to 1.5Mpc to increase the sensitivity of the data for constraints on the concentration, which are mostly derived from the change in the slope between small and large radii (compare Fig.\[fig:shear\_profile\_stack\]). Adding the $\chi^2$ from the individual clusters with equal weights we compute the total $\chi^2_\mathrm{tot}$ of the sample as function of the fixed concentration, allowing us to place constraints on the average concentration of the sample[^15] to using the X-ray centres (), using the SZ centres (), using the BCG centres (), and () when centring on the weak lensing mass peaks. We stress that the fitting was conducted for each cluster separately (see Sect.\[sec:profiles\]), and that the stacked signal shown in Fig.\[fig:shear\_profile\_stack\] is for illustrative purposes only. This is important given that the scaling is only approximate, while the individual analyses account for all effects (e.g. reduced shear). Due to miscentring the estimates using the X-ray, SZ, and BCG centres may be slightly biased low, while the estimate based on the mass peak centre is likely biased high. Given that all constraints are well consistent within the uncertainties, we conclude that miscentring has a negligible impact for the constraints on concentration at the current statistical precision. These estimates are consistent with predictions from recent numerical simulations. In particular, the [$c(M)$]{} relation from @diemer15, which corresponds to our default analysis, yields average concentrations (average 3.8) in our mass and redshift range, fully consistent with our constraints. Accordingly, it is not surprising that it provides similarly good fits to the data as the best-fit fixed concentrations, e.g. we obtain with the @diemer15 [$c(M)$]{} relation for the analysis using the X-ray centres. For comparison, the [$c(M)$]{} relation from @duffy08 yields lower average concentrations (average 2.7) in our mass and redshift range, which agrees with our constraints at the level only. [In the bottom panel of Fig.\[fig:shear\_profile\_stack\] we additionally show the stacked profile of the reduced cross-shear component of all clusters measured with respect to their X-ray centres (computed without rescaling). We find that it is consistent with zero, providing another consistency check for our analysis.]{} Calibration of the mass estimates with simulations and consistency checks in the data {#sec:sims} ------------------------------------------------------------------------------------- We have adopted a simplistic model for the mass distribution in clusters, namely a spherical NFW halo with a known centre and a concentration fixed by a [concentration–mass]{} relation. However, effects such as choosing an improper cluster centre (“miscentring”), variations in cluster density profiles, and noise bias in statistical estimators can introduce substantial biases in the mass constraints derived from fits of such a model to cluster weak lensing shear profiles [e.g. @becker11; @gruen15]. To estimate and correct for these biases in our analysis we apply our measurement procedure to a large sets of simulated cluster weak lensing data based on the Millennium XXL simulation [@angulo12] and the simulations created by @becker11 [henceforth ]. The details of this analysis will be presented in Applegate et al. (in prep.). Here we only summarise the most important points relevant to this analysis. ### Simulations The two simulations considered for our calibration differ in the redshifts of the available snapshots, in the cluster mass range, and the input cosmology. The difference in cosmology alters the [concentration–mass]{} relation in the simulation [e.g. @diemer15], but this is small compared to the range of [$c(M)$]{} relations we consider (Sect.\[se:shear\_profile\_compare\_mc\]). Likewise, the calibration does not depend on mass to a level that is important for this analysis when the full likelihood distribution of the mass constraints is used (see Applegate et al. in prep.). We find that the bias has some dependence on redshift and therefore interpolate between the two available snapshots that match our observations best. For the calibration of both $M_{200\mathrm{c}}$ and $M_{500\mathrm{c}}$ these are the snapshot of the simulation and the snapshot of the Millennium XXL simulation. Note that both simulations yield consistent bias calibrations at , where data are available from both simulations (see Applegate et al. in prep.). For the snapshot we include 788 haloes with , providing a good match to the SPT cluster mass range. Since the sample provided to us by is selected in , we are only able to measure the bias in at , above which we are still complete. For the MXXL snapshot we include the 2100 most massive haloes, corresponding to . For the calibration of weak lensing estimates for this sample is complete for , matching the mass range of the studied SPT clusters well (compare to Table \[tab:clusters\]). The generation of simulated shear fields from the underlying N-body simulations is described in BK11. In short, all particles within Mpc along the line-of-sight to each cluster are projected onto a common plane to produce a $\kappa$ map, from which a fast Fourier transform can compute the shear field on a regular grid. The procedure is similar for MXXL, except that particles within Mpc are used, and three orthogonal projection directions are employed. We create mock observations matching each cluster in our observed sample. We first select a profile centre location by randomly choosing an offset from the true cluster centre, which is defined as the position of the most-bound particle in the simulation, according to different probability distributions reflecting our assumptions on the miscentring distributions of SZ and X-ray centres (Sect.\[se:miscentring\_distributions\]). We then bin and azimuthally average the simulated reduced shear grid, matching the binning in the observed shear profile, and add Gaussian random noise to each bin matching the observed noise levels. We fit the cluster masses from these simulated weak lensing data as done for the real clusters, calculating scans of $\chi^2$ versus $M_{\mathrm{meas}}$. To obtain a bias calibration for the scaling relation analysis (see Sect.\[se:mtx\]) we model the ratio $M_{\mathrm{meas}}/M_{\mathrm{true}}$, where $M_{\mathrm{true}}$ denotes the corresponding halo mass, as a log-normal distribution. We associate the mean of the log-normal distribution as the inferred average bias and the width of the distribution as the intrinsic scatter from cluster triaxiality, substructure, and line-of-sight projections. We fit the log-normal distribution to the population of clusters in each snapshot, marginalising over the statistical uncertainty for each cluster (see Applegate et al. in prep). While we perform the analysis in bins of true halo mass to check for mass-dependence of the bias, we instead only use one all-encompassing mass bin to determine the bias for this analysis. We repeat the whole procedure for a number of miscentring distributions and [$c(M)$]{} relations. We list individual bias numbers for each cluster for the X-ray and SZ miscentring distributions and the @diemer15 [$c(M)$]{} relation in Table \[tab:mass\], and sample-averaged values for a number of configurations in Table \[tab:bias\]. We stress that the quoted bias numbers are adequate for quantitative analyses that take the full likelihood distribution of the mass constraints into account, as done in our scaling relation analysis presented in Sect.\[se:mtx\]. We correct the mass estimates as $$M_x^\mathrm{WL}=\frac{M_x^\mathrm{biased}}{b_x} \, .$$ As an approximation we also apply these bias correction factors to the maximum likelihood values and confidence intervals indicated in Figures \[fig:mass\_corrected\_comparison\_centres\] to \[fig:m\_tx\_sz\] in the following sections. However, note that the bias factors may differ at some level for the maximum likelihood estimates and the fits that use the full likelihood distribution due to differences in the impact of noise bias. We plan to investigate this issue further in Applegate et al. (in prep.). ### Miscentring distributions {#se:miscentring_distributions} For the SPT clusters we have proxies for the cluster centres, where we in particular use the X-ray centroids and SZ peaks for the mass analysis. These need to be related to the cluster centres defined by halo finding algorithms used to predict the cluster mass function from simulations. These offsets will typically lower the measured shear from the expected NFW signal at small radii [e.g. @johnston07; @george12]. To mimic this effect in the BK11 and MXXL N-body simulations, where we have neither mock SZ nor mock X-ray observations, we employ offset distributions derived from the Magneticum Pathfinder Simulation (@dolag16; see also @bocquet16), which is a large volume, high-resolution cosmological hydrodynamical simulation. It includes simulated SZ and X-ray observations, where we make use of SPT mock catalogues (@saro14; Gupta et al. in prep.) that include the full SPT cluster detection procedure. We find that the most relevant parameter regarding the centring uncertainty when using the SZ centres is the smoothing scale $\theta_\mathrm{c}$ used for the cluster detection [see @bleem15]. We therefore use the actual distribution of $\theta_\mathrm{c}$ values for our clusters from @bleem15 for the generation of the miscentring distribution. Miscentring [$c(M)$]{} rel. ------------- ----------------- ------ ------ None Diemer+15 0.95 0.96 X-ray-hydro Diemer+15 0.87 0.87 SZ-hydro Diemer+15 0.81 0.81 SZ-hydro 0.79 0.81 SZ-hydro 0.89 0.86 SZ-hydro 0.73 0.77 : Mass recovery bias factors for the analysis taking the full likelihood distribution into account, averaged over all of our clusters, for different miscentring distributions and [concentration–mass]{} relations. The statistical uncertainty of the bias correction ranges from $1.5\%$ for our lower redshift clusters to $2.5\%$ for our highest redshift clusters. \[tab:bias\] ### Impact and uncertainty of the miscentring correction {#se:impact_miscentring} Using the default [$c(M)$]{} relation [@diemer15] and comparing the analyses using the miscentring distributions from the hydrodynamical simulation to the case without miscentring, we estimate that miscentring on average introduces a moderate mass bias of when using the X-ray centres, and a more substantial bias of using the SZ centres (see Table \[tab:bias\]). The SZ measurements less accurately determine the cluster centre, which on-average increases the bias correction. This result is consistent with the smaller average offsets from the mass peaks found for the X-ray centres (Sect.\[sec:massmaps\]). ![Comparison of the bias-corrected weak lensing mass estimates using the X-ray versus the SZ centres. The high-mass outlier is the merger SPT-CL$J$0102$-$4915, for which the location of the SZ peak is closer to the centre between the two peaks of the mass reconstruction (see Fig.\[fig:massplotb\]), resulting in a higher mass estimate. \[fig:mass\_corrected\_comparison\_centres\]](mass200_mass_cor1_-1_0_0_5_3_-1_1_0_5_xraymag.png){width="0.9\columnwidth"} As a consistency check for the miscentring correction we compare the bias-corrected mass estimates using the X-ray and SZ centres in Fig.\[fig:mass\_corrected\_comparison\_centres\]. Their median ratio , with an uncertainty estimated by bootstrapping the clusters, is consistent with unity as expected in the case of accurate bias correction. We, however, note that the small sample size leads to a significant uncertainty of this median ratio, making it not a very stringent test for the accuracy of the bias correction. The accurate correction for mass modelling biases such as the one introduced by miscentring is an active field of research [e.g. @lsst12]. Our analysis using a miscentring distribution based on a hydrodynamical simulation is a step forward in this respect, but we acknowledge that it is still simplistic. In particular, it ignores that positional offsets are not always in a random direction. This is prominently demonstrated by the merger SPT-CL$J$0102$-$4915, for which the location of the SZ peak is closer to the centre between the two peaks of the mass reconstruction (see Fig.\[fig:massplotb\]), leading to an increased mass estimate (compare Fig.\[fig:mass\_corrected\_comparison\_centres\]). Due to this simplification in our current analysis, we conservatively assign a large uncertainty for the miscentring correction, which amounts to 50% of the correction, corresponding to a 4% uncertainty in mass when using the X-ray centres and 7% when using the SZ centres. Future analyses can reduce this uncertainty by simulating all observables including the weak lensing data from the same hydrodynamical simulation (see Sect.\[se:dis\_massmodel\]). ### Uncertainties in the [concentration–mass]{} relation {#se:shear_profile_compare_mc} For the case of SZ miscentring Table \[tab:bias\] lists average bias numbers for the [$c(M)$]{} relation from @diemer15, as well as fixed concentrations . Our bias correction procedure effectively maps the [$c(M)$]{} relation used for the fit to the observed [$c(M)$]{} relations in the simulations that are used for the bias correction (BK11, Millennium XXL). The remaining question is how well the [$c(M)$]{} relations in these simulations resemble the true average [$c(M)$]{} relation in the Universe, especially regarding the impact of baryons. @duffy10 show that the impact of baryon physics appears to have only a relatively minor () influence on the concentrations of very massive clusters. @deboni13 find similar numbers at low redshifts (for complete halo samples), and slightly stronger effects at (). Interpolating between the values in Table \[tab:bias\] we estimate that a 10–20% uncertainty on the concentration around leads to a systematic uncertainty for the constraints on , where we conservatively adopt the larger number in our systematic error budget (see Sect.\[se:systematic\_error\_budget\]). @deboni13 note that differences in the definition of the concentration can lead to shifts in the values measured from N-body simulations of up to 20%. This is not a concern for our analysis, as we directly estimate the calibration from the simulated weak lensing data, and therefore do not rely on concentration measurements in the simulations. ------------------------------------------------ ------------ -- ------------ -------- ------------------------------------ --------------------------- ------ -- Source rel. error rel. error Sect./ Improve via signal signal App. [**Shape measurements:**]{} Shear calibration 4% 1% \[sec:shear\] Image simulations [**Redshift distribution:**]{} $\langle\beta \rangle$ sys. photo-$z$ 2.2% 1.5% \[se:zdist\_fix\_uncertainty\] Improved priors + $p(z)$ $\langle\beta \rangle$ cosmic variance 1% 1% \[sec:beta\_los\_variation\] More reference fields $\langle\beta \rangle$ deblending 0.5% 0% \[se:app:non\_matches\] F606W-detected photo-$z$s $\langle\beta \rangle$ LCBG contamination 0.9% 0.5% \[se:test\_extremely\_blue\] Apply model [**Mass model:**]{} Miscentring for X-ray (SZ) centres \[se:miscentring\_distributions\] Hydro sims, weak lensing [$c(M)$]{} relation \[se:shear\_profile\_compare\_mc\] Hydro sims, weak lensing [**Total**]{} for X-ray (SZ) centres [**:**]{} ------------------------------------------------ ------------ -- ------------ -------- ------------------------------------ --------------------------- ------ -- Statistical precision versus systematic uncertainty {#se:systematic_error_budget} --------------------------------------------------- We summarize the identified sources of systematic uncertainty for our study in Table \[tab:sys\], pointing to their corresponding sections, and listing their associated relative uncertainties in the measured weak lensing signal and mass constraints. Combining all systematic error contributions in quadrature, we estimate an overall systematic mass uncertainty of 9% (11%) for the analysis using the X-ray (SZ) centres. This can be compared to the combined statistical mass signal-to-noise ratio of the sample, which we approximate as $$\left(S/N \right)_\mathrm{mass}^\mathrm{sample}=\sqrt{\sum_\mathrm{clusters}\left(M_{\mathrm{500c},i}/\Delta M_{\mathrm{500c},i}^\mathrm{stat.}\right)^2}\simeq 7.3\,,$$ which corresponds to a precision, ignoring the impact of intrinsic scatter, e.g. from cluster triaxiality. Accordingly, our total uncertainty is dominated by statistical measurement noise and not systematic uncertainties. For the analysis of larger future data sets with improved statistical precision it will be important to further reduce systematic uncertainties. When discussing the individual sources of systematic uncertainty we have already suggested strategies how their influence can be reduced in the future. The largest contributions to the systematic error budget currently come from the shear calibration, miscentring corrections, and uncertainties in the [$c(M)$]{} relation. All of these can be reduced with better simulations. For the latter two issues the weak lensing data can themselves provide information that help to reduce these uncertainties (see also Sect.\[se:dis\_massmodel\]). As a rough guess we expect that it should be possible to cut the systematic uncertainties associated with the mass modelling by half in the coming years with moderate effort (compare Table \[tab:sys\]), and note that some improved shape measurement techniques have already reached significantly higher accuracy [e.g. @bernstein16; @fenechconti17]. We further discuss the strategies to reduce systematic uncertainties in Sect.\[sec:discussion\]. Constraints on the $M$–$T_\mathrm{X}$ scaling relation {#se:mtx} ====================================================== In the self-similar model [e.g. @kaiser86] galaxy clusters form through the gravitational collapse of the most overdense regions in the early Universe. In this model the cluster baryons are heated through gravitational processes only, leading to predictions for cluster scaling relations. Deviations from self-similarity, e.g. regarding the slope of the X-ray luminosity–temperature relation [e.g. @arnaud99], suggest that non-gravitational effects, such as heating by active galactic nuclei or radiative cooling, provide non-negligible contributions to the energy budget of clusters. However, the redshift evolution of cluster X-ray observables appears to be consistent with self-similar predictions [e.g. @maughan06], suggesting that non-gravitational effects have a similar impact at low and high redshifts. If this “weak self-similarity” [e.g. @bower97] also applies to cluster masses, we expect a scaling between temperature and mass in the form $$\label{eqn:METx} M_x E(z) \propto T^\alpha \, ,$$ [e.g. @mathiesen01; @boehringer12], where $$E(z)=\frac{H(z)}{H_0}=\sqrt{\Omega_\mathrm{m}(1+z)^3+\Omega_\Lambda}$$ indicates the redshift dependence of the Hubble parameter, here assuming a flat $\Lambda$CDM cosmology, and corresponds to the self-similar prediction for the slope of the relation. The main constraints on cluster scaling relations from our sample will be presented in a forthcoming paper (Dietrich et al. in prep.) that combines our measurements with a complementary sample of clusters at lower redshifts with Magellan/Megacam observations and accounts for the SPT selection function, which is especially important when calibrating SZ scaling relations. However, here we already combine our measurements with core-excised *Chandra* X-ray temperature estimates $T_\mathrm{X}$ that are available for 12 clusters in our sample. Details of the specific measurements are provided in [@mcdonald13], with the analysis pipeline adapted based on [@vikhlinin06]. In short, *Chandra* ACIS-I data were reduced using <span style="font-variant:small-caps;">ciao</span> v4.7 and <span style="font-variant:small-caps;">caldb</span> v4.7.1. All exposures were initially filtered for flares, before applying the latest calibrations and determining the appropriate epoch-based blank-sky background. Point sources were identified via an automated wavelet decomposition technique [@vikhlinin98] and masked. Spectra were extracted in a core-excised region from $(0.15$–$1)\times r_{500c}$ [@mcdonald13] and fit over 0.5–10.0keV using a combination of an absorbed, optically-thin plasma (<span style="font-variant:small-caps;">phabs</span>$\times$<span style="font-variant:small-caps;">apec</span>), an absorbed hard background component (<span style="font-variant:small-caps;">phabs</span>$\times$<span style="font-variant:small-caps;">bremss</span>), and a soft background (<span style="font-variant:small-caps;">apec</span>), see [@mcdonald13] for details. Figures \[fig:m\_tx\] and \[fig:m\_tx\_sz\] show the bias-corrected $M_\mathrm{500c}^\mathrm{WL}E(z)$ using the @diemer15 [$c(M)$]{} relation as function of the core-excised $T_\mathrm{X}$ estimates (Table \[tab:tx\]) for the analyses centring on the X-ray centroids or SZ peaks, respectively. For comparison we show best-fit estimates for the scaling relation derived by @arnaud05 [based on their sample], @vikhlinin09b, and @mantz16 using clusters at lower and intermediate redshifts ($z \lesssim 0.6$). To obtain quantitative constraints on the scaling relation, we assume the functional form $$\ln \left(E(z) M_\mathrm{500c}/10^{14}\mathrm{M}_\odot \right)=A+\alpha \left[\ln \left(kT/7.2\mathrm{keV}\right)\right]\, ,$$ where the temperature pivot point roughly corresponds to the mean temperature of the sample. Our fitting method is based on the approach of [@kelly07], which incorporates measurement errors in the $x$- and $y$- coordinates and has been extended to include log-normal intrinsic scatter. The method has been generalized to use the exact likelihood from the lensing analysis, and a two-piece normal approximation to the X-ray likelihood [@applegate16]. For this analysis we use the lensing likelihood based on the dominant shape noise only and absorb the minor contributions from large-scale structure projections and line-of-sight variations in the redshift distribution (see Sect.\[sec:profiles\]) in the intrinsic scatter $\sigma_\mathrm{M}$. Cluster $T_\mathrm{X}$ \[keV\] ---------------------- ------------------------ SPT-CL$J$0000$-$5748 $6.7_{-1.6}^{+2.9}$ SPT-CL$J$0102$-$4915 $13.5_{-0.6}^{+0.5}$ SPT-CL$J$0533$-$5005 $4.6_{-1.7}^{+2.0}$ SPT-CL$J$0546$-$5345 $6.7_{-0.9}^{+1.4}$ SPT-CL$J$0559$-$5249 $6.1_{-0.6}^{+0.8}$ SPT-CL$J$0615$-$5746 $13.1_{-1.8}^{+1.1}$ SPT-CL$J$2106$-$5844 $8.7_{-0.7}^{+1.2}$ SPT-CL$J$2331$-$5051 $5.6_{-0.7}^{+1.4}$ SPT-CL$J$2337$-$5942 $7.0_{-0.9}^{+1.6}$ SPT-CL$J$2341$-$5119 $10.4_{-1.9}^{+2.5}$ SPT-CL$J$2342$-$5411 $4.0_{-0.8}^{+0.6}$ SPT-CL$J$2359$-$5009 $5.7_{-1.3}^{+1.2}$ : Core-excised [*Chandra*]{} X-ray temperatures used for our constraints on the $M$–$T_\mathrm{X}$ scaling relation. \[tab:tx\] ![ Core-excised X-ray temperatures measured in the range $(0.15$–$1)\times r_{500c}$ based on [*Chandra*]{} data versus $E(z) M_\mathrm{500c}^\mathrm{WL}$ from the weak lensing analysis using the X-ray centroids and assuming the [$c(M)$]{} relation from @diemer15. The solid black line shows our best-fit estimate of the scaling relation when assuming a fixed slope $\alpha=3/2$. The dotted lines correspond to normalisations that are lower or higher by $1\sigma$, combining the statistical and systematic uncertainties of our constraints. The dashed and dashed-dotted lines indicate best-fit estimates derived by @arnaud05 [@vikhlinin09b] and @mantz16. \[fig:m\_tx\]](tx_500_cen1_biascor1.png){width="1\columnwidth"} ![As Figure \[fig:m\_tx\], but employing the weak lensing results for the SZ centres. \[fig:m\_tx\_sz\]](tx_500_cen0_biascor1.png){width="1\columnwidth"} We fix the slope of the scaling relation to the self-similar prediction ($\alpha=3/2$) for the current analysis, given the limited sample size and mass range. [We then obtain constraints for our default analysis using the X-ray centres. When alternatively using the SZ peaks as centre for the weak lensing analysis we obtain consistent results .]{} In addition to these statistical uncertainties there is a 9% (11%) systematic uncertainty for the analysis using the X-ray (SZ) centres, directly propagating into the normalisation of the scaling relation (see Sect.\[se:systematic\_error\_budget\]). The obtained constraints are consistent with the aforementioned results from lower redshift samples when assuming self-similar redshift evolution [within ]{} (see Figures \[fig:m\_tx\] and \[fig:m\_tx\_sz\]). @jee11 present an HST weak lensing analysis for 27 galaxy clusters at , using a heterogeous sample that includes optically, NIR, and X-ray-selected clusters. [Their analysis suggests a possible evolution in the $M_\mathrm{2500c}$–$T_\mathrm{X}$ scaling relation in comparison to self-similar extrapolations from lower redshifts. For example, at their estimated scaling relation has a lower amplitude by (statistical uncertainty from @jee11 only) compared to the best-fit relation from\ @arnaud05. We do not find significant indications for a similar evolution for the $M_\mathrm{500c}$–$T_\mathrm{X}$ scaling relation, but note that our statistical uncertainties are significantly larger given our smaller sample size and more conservative radial fit range. There are various additional differences in the analyses, such as different samples for the calibration of the source redshift distribution, our more conservative removal of cluster galaxies, and our calibration of modelling biases on simulations, making the direct comparison difficult. Importantly, both studies use different overdensities for the scaling relation constraints[^16]. Furthermore, @jee11 use X-ray temperature estimates from the literature that typically do not exclude the core regions. Including the cores should, on average, reduce the temperatures in the presence of cool-core clusters. This would, however, aggravate the tension between the @jee11 results and the self-similar extrapolations from lower redshift]{} samples. Discussion {#sec:discussion} ========== In our analysis we have introduced a number of new aspects and systematic investigations for weak lensing studies of high-redshift clusters. Here we discuss their relevance also in the context of future weak lensing programmes. Our study using HST and VLT data provides a demonstration for future weak lensing science investigations that combine deep high-resolution space-based shape measurements, e.g. from *Euclid* [@laureijs11] or WFIRST [@spergel15], with deep photometry, e.g. from LSST [@lsst09]. The benefits and challenges of using faint blue galaxies for weak lensing ------------------------------------------------------------------------- For deep weak lensing surveys conducting shape measurements at optical wavelengths the majority of the high-redshift () sources are blue star forming galaxies observed at rest-frame UV wavelengths with blue observed optical colours (see the top left panel of Fig.\[fig:zdist\_f814w\]). These galaxies are useful as the source sample in weak lensing studies of high-redshift clusters both because of their high source density and high geometric lensing efficiency, but also because they can be readily distinguished from both blue and red cluster galaxies using optical colours (see Sect.\[se:photo\_color\_select\_acs\]). This enables a nearly complete removal of cluster galaxies from the weak lensing source sample, which is important both in order to minimise modelling uncertainties regarding cluster member contamination (Appendix \[app:why\_not\_boost\]), and to ensure that intrinsic alignments of galaxies within the targeted clusters cannot bias mass constraints [but note that this appears to be a negligible effect at the precision of current samples, see @sifon15]. To exploit these benefits, a number of challenges need to be overcome. Here we first stress that high signal-to-noise optical photometry is needed to robustly select these galaxies in colour space. In the case of our study a well-matched colour selection was possible in areas covered by ACS in both F606W and F814W. However, outside the F814W footprint we had to rely on the combination of F606W ACS imaging and VLT $I_\mathrm{FORS2}$ images, which, despite a good VLT integration time and seeing, delivered a density of usable sources that is only 32% of the density from the ACS-only $V_{606}-I_{814}$ selection (Sect.\[se:total\_ngal\]). This highlights that future weak lensing programmes and surveys should carefully tune the relative depth of their bands (regarding both red and blue filters) to maximise the science output of their data. While our analysis is based on simple colour cuts due to the limited data available in our cluster fields, we expect that similar conclusions apply for surveys that aim at computing individual photometric redshifts for the weak lensing source galaxies. Photometric redshift selections correspond to higher dimensional cuts in colour-colour space. However, depending on the survey characteristics, the large population of blue high-$z$ galaxies may only be detected in a few of the bluer optical pass bands, effectively reducing photo-$z$ cuts to a selection in a relatively small colour-colour space. As a result, individual photometric redshift estimates for faint blue galaxies have typically large uncertainties unless deep photometry is available over a very broad wavelength range (in particular including deep $u$-band and NIR observations). For cluster weak lensing studies noise in individual photometric redshifts is not a problem as long as cluster galaxies can be removed robustly and the overall source redshift distribution can be modelled accurately. Robust estimates of the source redshift distribution ---------------------------------------------------- We employ a statistically consistent selection of source galaxies matched in filter, magnitude, colour, and shape properties in our cluster fields and observations of the CANDELS fields. This allows us to estimate the average source redshift distribution and its statistical variation between lines-of-sights using the CANDELS data and apply this information for the cluster weak lensing analyses. At depths similar to our data, the CANDELS fields are currently among the extragalactic fields that are best studied both photometrically and spectroscopically. We have shown that they cover enough sky area to reduce the cosmic variance contribution to the uncertainty on the mean lensing efficiency at our cluster redshifts to the level (Sect.\[sec:beta\_los\_variation\]), which is much smaller than current statistical weak lensing uncertainties. Therefore, we expect that the CANDELS fields will remain to be an important calibration sample for estimates of the source redshift distribution in deep weak lensing data in the near future. As revealed by our comparison to HUDF data (Sect.\[se:sub:catastrophic\_outliers\]) and confirmed via spatial cross-correlations with spectroscopic/grism redshifts (Appendix\[app:candels\_x\]), the 3D-HST CANDELS photo-$z$s suffer from catastrophic redshift outliers (primarily galaxies at that are assigned a low photometric redshift ) and redshift focussing effects at . Together these would on average bias our mass estimates high by 12% if not accounted for. For our current study we have implemented an empirical correction for these systematics effects. We plan to investigate this issue and its causes in detail in a future paper (Raihan et al. in prep.). Given the high photometric quality, depth, and broad wavelength coverage of the CANDELS data, we speculate that some other current photometric redshift data sets might suffer from similar effects. This is supported by the weak lensing analyses of and @heymans12 as discussed in Sect.\[sec:correct\_sys\_photoz\]. We therefore expect that also other weak lensing programmes will have to implement similar correction schemes or improved photometric redshift algorithms, and apply these either to deep field data in case of colour cut analyses, or their survey data in case of individual photo-$z$ estimates. Surveys that obtain individual photo-$z$s can also attempt to identify and remove galaxies in problematic ranges at the cost of reduced sensitivity. We stress that the use of the average redshift posterior probability distribution instead of the peak photometric redshift estimates is not sufficient to cure the identified issue for the 3D-HST photo-$z$s (Sect.\[se:posterior\]). One route to calibrate photo-$z$s is via very deep spectroscopy for representative galaxy samples. At present, such spectroscopic samples are very incomplete at the depth of our analysis, which is why we resorted to the comparison of the CANDELS photo-$z$s to photometric redshifts for the HUDF [@rafelski15], which are based on deeper data and a broader wavelength coverage. We find that this is a viable approach at the precision of current and near-term high-$z$ cluster samples with weak lensing measurements, but it is likely not of sufficient accuracy for the calibration of very large future data sets. To prepare for the analyses of such data sets it is vital and timely to obtain larger spectroscopic calibration samples, including both highly complete deep samples for direct calibration, but also very large, potentially shallower and less complete samples [@newman15]. The later can be used to infer information on the redshift distribution via spatial cross-correlations [e.g. @newman08; @matthews10; @schmidt13; @rahman15; @rahman16], for which we provide one of the first practical applications in the context of weak lensing measurements [see Appendix \[app:candels\_x\] and @hildebrandt17]. As an important ingredient for our modelling of the redshift distribution we carefully matched the selection criteria and noise properties between our cluster field data and the CANDELS data to ensure that consistent galaxy populations are selected between both data sets (see Sect.\[se:source\_sel\_scatter\] and Appendix \[app:scatter\]). For the colours obtained from the combination of ACS F606W and VLT $I_\mathrm{FORS2}$ data we empirically estimated the net scatter distribution by comparing to the colours estimated in the inner cluster regions from ACS F606W and F814W data. We note that systematic effects such as residuals from the PSF homogenisation can add scatter which may well deviate from Poisson noise distributions that are often assumed, e.g. in photometric redshift codes. As we empirically sample from the actual scatter distribution such effects are automatically accounted for in our analysis. For future surveys that vary in data quality we recommend to obtain repeated imaging observations of spectroscopic reference fields that span the full range of varying observing conditions, in order to generate similar empirical models for the impact of the actual noise properties. Accounting for magnification {#se:dis:mag} ---------------------------- The impact of weak lensing magnification on the source redshift distribution has typically been ignored in past weak lensing studies. Our investigation of this effect in Sect.\[sec:magnification\_model\] indicates that the net effect is small for our study given the depth of our data. However, shallower programmes such as DES [@des05] or KiDS [@kuijken15], which aim to calibrate high-$z$ cluster masses by combining measurements from a large number of clusters, will need to carefully account for the resulting boost in the average lensing efficiency . For example, Fig.\[fig:magnification\_shear\] illustrates that the impact of magnification on the source redshift distribution has a larger impact on the reduced shear profile at brighter magnitudes than the typically applied correction for the finite width of the source redshift distribution. We point out that knowledge of the redshift distribution is needed at fainter magnitudes than the targeted depth limit of a survey in order to be able to compute the actual correction for the impact of magnification (Sect.\[sec:magnification\_model\]). Accordingly, it is necessary to obtain spectroscopic redshift samples for photo-$z$ calibration to greater depth than the targeted survey depth. The difference in depth depends on the maximum magnification that is considered, and therefore the magnitude limit, the cluster redshift and mass, as well as the considered fit range. We also note that it is important to take magnification into account when using the source density and the density profiles as validation tests for the cluster member removal (see Sect.\[se:photo\_number\_density\_tests\]). Programmes with ground-based resolution will also need to account for the change in source sizes due to magnification as function of redshift, cluster-centric distance, and mass, as shape cuts could otherwise introduce redshift- and mass-dependent selection biases. Shape measurement biases {#se:dis_shape} ------------------------ Currently the shear calibration uncertainty constitutes the largest individual contribution to the systematic error budget of our study (4% for the shear calibration corresponding to a 6% mass uncertainty). This is due to the fact that we base the calibration on simulations from the STEP project (Sect.\[sec:shapemeasure\]) which lack faint galaxies that influence the bias calibration [@hoekstra15] and do not probe shears as high as those used in our analysis. However, this source of systematics can easily be reduced through image simulations that resemble real galaxy populations and cluster-regime shears more accurately, and which can be generated recent tools such as <span style="font-variant:small-caps;">GalSim</span> [@rowe15]. We therefore expect that shear measurement biases in cluster weak lensing studies will soon be reduced to the levels reached in cosmic shear measurements [e.g. @fenechconti17]. Also see @hoekstra17, [whose results suggest]{} that the impact of the higher density of sources in cluster regions on shape measurement biases should be negligible for current data. In addition, additive shape measurement biases can be relevant for cluster weak lensing in particular for pointed follow-up programmes where the clusters are always centred at similar detector positions. An example for such a potential source of bias can be CTI residuals. However, through a new null test we have shown that our data show no significant CTI-like residuals within the current statistical uncertainty (Sect.\[sec:ctitests\]). Accounting for biases in the mass modelling {#se:dis_massmodel} ------------------------------------------- We have calibrated our mass estimates using reduced shear profile fits to simulated cluster weak lensing data from N-body simulations (see Sect.\[sec:sims\]). One important source for bias is miscentring of the reduced shear profile. As we do not know the location of the centre of the 3D cluster potential we have to rely on observable proxies for the cluster centre, leading to a suppression of the expected reduced shear signal at small radii. Based on the work from [@dietrich12] we expect that the peaks in the reconstructed weak lensing mass maps of the clusters (see Sect.\[sec:massmaps\]) should provide a tight tracer for the centre of the 3D cluster potential, but we do not use these centres for our mass constraints in our current analysis as they are expected to yield masses that are biased high. By studying the offset distributions between the mass peaks and the other proxies for the cluster centre we find that the X-ray centroids provide the smallest average offsets, closely followed by the SZ peak locations. Hence, they also provide good proxies for the cluster centre, which is why we employ them as centres for our mass constraints. To account for the expected remaining bias caused by miscentring, we randomly misplace the centre in the simulated weak lensing data based on offset distributions measured between the 3D cluster centre and the SZ peak location or X-ray centroid in hydrodynamical simulations (see Sect.\[se:miscentring\_distributions\]). Future studies could further advance this approach by simulating all observables including SZ, X-ray, and weak lensing data from the same hydrodynamical simulation, in order to also account for possible covariances between these observables. Our analysis of the prominent merger SPT-CL$J$0102$-$4915 demonstrates that such covariances exist, as both the X-ray centroid and SZ peak are located between the two peaks of the mass reconstruction (see Fig.\[fig:massplotb\]). Hence, the misplacement is not in a random direction. To validate the accuracy of the employed simulations, the measured offset distributions between the mass peaks and the different proxies for the centre can be compared between the real data and the simulations. This approach could be further expanded by explicitly accounting for the miscentring in the fitted reduced shear profile model (e.g. @johnston07 [@george12]; also see @koehlinger15 for the impact of miscentring in stacked Stage IV analyses). A further uncertainty for the mass constraints arises from uncertainties in the assumed [$c(M)$]{} relation. The applied calibration procedure essentially maps the measurements onto the [$c(M)$]{} relation of the simulation. Remaining uncertainties reflect our ability to simulate the true [$c(M)$]{} relation of the Universe, especially with respect to the impact of baryons. These uncertainties are expected to shrink with further advances in simulations, in particular thanks to the recent advent of large hydrodynamical simulations [e.g. @dolag16]. In addition, the weak lensing measurements themselves can be used to test if the inferred reduced shear profiles are consistent with the simulation-based priors on the [$c(M)$]{} relation, in particular if information from the inner reduced shear profiles is incorporated. Using the X-ray centroids our analysis yields a best-fitting fixed concentration for the sample of when including scales (Sect.\[se:stack\_mc\]). This is fully consistent with recent results for the [$c(M)$]{} relation from simulations [e.g. @diemer15], but higher than earlier results from @duffy08, [which, however, are based on a WMAP5 cosmology [@komatsu09] with lower $\Omega_\mathrm{m}$ and $\sigma_8$, reducing the resulting concentrations]{}. We note that future studies that aim to obtain tighter constraints on the concentration will have to account for the impact of miscentring and stronger shears in the inner cluster regions, which we could ignore for this part of our analysis given the statistical uncertainties. Conclusions {#sec:conclusions} =========== We have presented a weak gravitational lensing analysis of 13 high-redshift clusters from the SPT-SZ Survey, based on shape measurements in high resolution HST/ACS data and colour measurements that also incorporate VLT/FORS2 imaging. We have introduced new methods for the weak lensing analysis of high redshift clusters and carefully investigated the impact of systematic uncertainties as discussed in Sect.\[sec:discussion\] in the context of future programmes. In particular, we select blue galaxies in colour to achieve a nearly complete removal of cluster galaxies, while selecting most of the relevant source galaxies at (see Sect.\[se:photo\_color\_select\_acs\]). Carefully matching our selection criteria we estimate the source redshift distribution using data from CANDELS, where we apply a statistical correction for photometric redshift outliers. This correction is derived from the comparison to deep spectroscopic and photometric data from the HUDF (see Sect.\[sec:correct\_sys\_photoz\]), and checked using spatial cross-correlations (see Appendix \[app:candels\_x\]). We account for the impact of lensing magnification on the source redshift distribution, which we find is especially important for shallower surveys (see Sect.\[sec:magnification\_model\]). We also introduce a new test for residual contamination of galaxy shape estimates from charge-transfer inefficiency, which is in particular applicable for pointed cluster follow-up observations (see Sect.\[sec:ctitests\]). Finally, we account for biases in the mass modelling through simulations (see Sect.\[sec:sims\]). At present, our weak lensing mass constraints are limited by statistical uncertainties given the small cluster sample and the limited depth of the data for the colour selection in the cluster outskirts. For the current study the total systematic uncertainty on the cluster mass scale at high-$z$ is at the level, where the largest contributions come from the shear calibration and mass modelling. As discussed in Sect.\[se:systematic\_error\_budget\] we have identified strategies how this can be reduced to the level in the near future based on exisiting calibration data and improved simulations. This is particularly relevant for near-term studies using larger HST data sets. We have used our measurements to derive updated constraints on the scaling relation for massive high-$z$ clusters in combination with [*Chandra*]{} observations. Compared to scaling relations calibrated at lower redshifts we find no indication for a significant deviation from self-similar redshift evolution at our [current precision]{} (see Sect.\[se:mtx\]). Our measurements will additionally be used in companion papers to derive updated constraints on additional mass-observable scaling relations, where we also incorporate weak lensing measurements at lower redshifts from Magellan/Megacam (Dietrich et al. in prep.) and the Dark Energy Survey [DES, @des05 Stern et al. in prep.], and to derive improved cosmological constraints from the SPT-SZ cluster sample. We investigate the offset distributions between different proxies for the cluster centre and the weak lensing mass reconstruction, where we find that the X-ray centres provide the smallest average offsets (see Sect.\[sec:massmaps\]). Our analysis constrains the average concentration of the cluster sample to (Sect.\[se:stack\_mc\]) when using the X-ray centres and including information from smaller scales ($300\, \mathrm{kpc}<r<500\, \mathrm{kpc}$), which are excluded for the conservative mass constraints. With the advent of the next generation of deep cluster surveys such as SPT-3G [@benson14], the Dark Energy Survey [DES @des05], Hyper-Suprimecam [HSC @miyazaki12], eROSITA [@merloni12], and Advanced ACTPol [@henderson16] it will be vital to further tighten the weak lensing calibration of cluster masses in order to exploit these surveys for constraints on cosmology and cluster astrophysics. At low and intermediate redshifts, weak lensing surveys such as DES, HSC, and KiDS are expected to soon calibrate cluster masses at the few per cent level, especially if large numbers of clusters can be reliably selected down to lower masses and if their weak lensing signatures are combined statistically [e.g. @rozo11]. Such surveys will also provide some statistical weak lensing constraints for clusters out to [e.g. @vanuitert16], but it still needs to be demonstrated how reliably such measurements can be conducted from the ground as most of the distant background galaxies are poorly resolved. At such cluster redshifts HST is currently unique with its capabilities to measure robust individual cluster masses with good signal-to-noise. [Clusters at high redshifts and high masses are very rare. As a result,]{} stacking analyses of shallower wide-area surveys cannot compete in terms of precision for their mass calibration with a large HST program that obtains pointed follow-up observations for all of them. Our current study is an important pathfinder towards such a program. For comparison, stacked analyses tend to be more powerful for lower mass clusters, which are too numerous to be followed up individually. The combination of deep pointed follow-up for high-mass clusters and stacked shallower measurements for lower mass clusters is therefore particularly powerful for obtaining constraints on the slope of mass-observable scaling relations. In addition, good signal-to-noise ratios [for individual clusters,]{} as provided by deep pointed follow-up, are needed for constraints on intrinsic scatter. In the 2020s weak lensing Stage IV dark energy experiments such as *Euclid* [@laureijs11], LSST [@lsst09], and WFIRST [@spergel15] are expected to provide a precise calibration of cluster masses over a wide range in redshift [for a forecast for *Euclid* see @koehlinger15]. To reach their weak lensing science goals they will require highly accurate calibrations for the redshift distribution and shear estimation. Further efforts will be needed to fully exploit these calibrations and weak lensing data sets for cluster mass estimation. For example, the shear calibration needs to be extended towards stronger shear, and magnification has to be taken into account when estimating the source redshift distribution (Sect.\[se:dis:mag\]). We also stress that it will be vital to pair such observational studies with analyses of large sets of hydrodynamical simulations, in order to accurately calibrate the weak lensing mass estimates and account for covariances with other observables (see Sect.\[se:dis\_massmodel\]). LSST and *Euclid* will still have signficantly lower densities of high-redshift background source galaxies compared to HST observations. In order to extend the mass calibration for massive clusters out to very high redshifts (), large pointed HST and subsequently JWST programmes may therefore remain the most effective approach until similarly deep data become available from WFIRST. Acknowledgements {#acknowledgements .unnumbered} ================ This work is based on observations made with the NASA/ESA [*Hubble Space Telescope*]{}, using imaging data from the SPT follow-up GO programmes 12246 (PI: C. Stubbs) and 12477 (PI: F. W. High), as well as archival data from GO programmes 9425, 9500, 9583, 10134, 12064, 12440, and 12757, obtained via the data archive at the Space Telescope Science Institute, and catalogues based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) and the UVUDF Project (GO 12534, also based on data from GO programmes 9978, 10086, 11563, 12498). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. It is also based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programmes 086.A-0741, 088.A-0796, 088.A-0889, 089.A-0824. The scientific results reported in this article are based in part on observations made by the [*Chandra*]{} X-ray Observatory (ObsIDs 9332, 9333, 9334, 9335, 9336, 9345, 10851, 10864, 11738, 11739, 11741, 11742, 11748, 11799, 11859, 11864, 11870, 11997, 12001, 12002, 12014, 12091, 12180, 12189, 12258, 12264, 13116, 13117, 14017, 14018, 14022, 14023, 14349, 14350, 14351, 14437, 15572, 15574, 15579, 15582, 15588, 15589, 18241). It is a pleasure to thank Gabriel Brammer, Pieter van Dokkum, Mattia Fumagalli, Ivelina Momcheva, Rosalind Skelton, and the 3D-HST team for helpful discussions and for making their photometric and grism redshift catalogues available to us prior to public release. We thank Matthew Becker and Andrey Kravtsov for making results from their N-body simulation [@becker11] available to us, Raul Angulo for providing data from the Millennium XXL Simulation, and Klaus Dolag for providing access to data from the Magneticum Pathfinder Simulation. TS, DA, and FR acknowledge support from the German Federal Ministry of Economics and Technology (BMWi) provided through DLR under projects 50 OR 1210, 50 OR 1308, 50 OR 1407, and 50 OR 1610. TS also acknowledges support from NSF through grant AST-0444059-001 and SAO through grant GO0-11147A. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grant NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli. HH acknowledges support from NWO VIDI grant number 639.042.814 and ERC FP7 grant 279396. Work at Argonne National Laboratory was supported under U.S. Department of Energy contract DE-AC02-06CH11357. The Munich group acknowledges the support by the DFG Cluster of Excellence “Origin and Structure of the Universe” and the Transregio program TR33 “The Dark Universe”. R.J.F. is supported in part by fellowships from the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation. TdH is supported by a Miller Research Fellowship. PS acknowledges support by the European DUEL Research-Training Network (MRTN-CT-2006-036133) and by the Deutsche Forschungsgemeinschaft under the project SCHN 342/7-1. CBM acknowledges the support of the DFG under Emmy Noether grant Hi 1495/2-1. BB is supported by the Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy. CR acknowledges support from the Australian Research Council’s Discovery Projects scheme (DP150103208). The Dark Cosmology Centre is funded by the Danish National Research Foundation. Galaxy Ellipticity Dispersion and Shape Measurement Weights {#app:shapes_candels} =========================================================== ![image](erms_sn_cc.png){width="1\columnwidth"} ![image](erms_sn_nocc.png){width="1\columnwidth"} ![image](erms_mag_cc.png){width="1\columnwidth"} ![image](erms_mag_nocc.png){width="1\columnwidth"} ![image](erms_magauto_cc.png){width="1\columnwidth"} ![image](erms_magauto_nocc.png){width="1\columnwidth"} ![image](erms_redshift_bright.png){width="1\columnwidth"} ![image](erms_redshift_faint.png){width="1\columnwidth"} As explained in Section \[sec:shear\] we processed ACS observations in the CANDELS fields to be able to mimic our source selection in the photometric redshift reference catalogues. These blank field data also enable us to study the galaxy ellipticity distribution as detailed in this Appendix. On the one hand this allows us to optimise our weighting scheme for the current study. In addition, these estimates can be used to optimise future weak lensing observing programmes and forecast their performance. For the latter purpose we have studied shape estimates from both ACS standard lensing filters F606W and F814W. This also updates earlier results on the intrinsic ellipticity dispersion estimated by @leauthaud07 for F814W observations in the COSMOS Survey. Method ------ Our ellipticity measurements $\epsilon$ provide estimates for the reduced shear $g$. We model the measured dispersion of the galaxy ellipticity $\sigma_\epsilon$ with contributions from the intrinsic galaxy shapes $\sigma_\mathrm{int}$ and measurement noise $\sigma_\mathrm{m}$ as $$\label{eq:sigmae2} \sigma_\epsilon^2=\sigma_\mathrm{int}^2+\sigma_\mathrm{m}^2 \,.$$ The contribution from the cosmological shear in CANDELS is small compared to $\sigma_\epsilon$, and for the purpose of this study we regard it as part of $\sigma_\mathrm{int}$. To estimate $\sigma_\mathrm{m}$ we make use of the overlap region of neighbouring ACS tiles (that have similar noise properties), where we have two estimates (a,b) of the ellipticity of each galaxy with two independent realisations of the measurement noise for identical $\epsilon_\mathrm{int}$. After rotating the ellipticities to the same coordinate frame, the dispersion of their difference allows us to estimate $$\sigma_\mathrm{m}^2=\sigma_{\Delta\epsilon}^2/2\,,$$ from which we compute $\sigma_\mathrm{int}$ according to (\[eq:sigmae2\]). Generally, we quote r.m.s. ellipticity values [*per ellipticity component*]{}, where we compute the average from both components as $$\sigma_\epsilon^2= \left(\sigma_{\epsilon,1}^2+\sigma_{\epsilon,2}^2\right)/2\,.$$ Data {#app:candels:data} ---- For this analysis we generated and analysed tile-wise F606W and F814W stacks of 4 ACS exposures each. We include the initial AEGIS ACS F606W and F814W observations\ [@davis2007 Proposal ID 10134]. Similar to @schrabback07 we generate F606W stacks in GOODS-South and GOODS-North that always combine two epochs of the observations from @giavalisco2004 [Proposal IDs 9425, 9583]. In GOODS-South we also include F606W observations from GEMS [@rix2004 Proposal ID 9500], which provides some additional overlap with the S14 WFC3/IR-detected catalogues. Generally, we limit our analysis to the overlap region with the S14 catalogues to enable the colour selection and provide constraints as function of photometric redshift. For the COSMOS and UDS fields we use the F606W and F814W observations from CANDELS [@grogin2011 Proposal IDs 12440, 12064]. Here, the tile-wise F606W stacks have slightly shorter integration times of 1.3–1.7ks compared to our targeted $\sim 2$ks depth. For the constraints on the ellipticity dispersion we therefore include these observations only when studying the ellipticity dispersion as function of flux signal-to-noise ratio, where the impact of the shallower depth is minimal. Discussion {#discussion} ---------- We plot our estimates for the measured ellipticity dispersion $\sigma_\epsilon$, the intrinsic ellipticity dispersion $\sigma_\mathrm{int}$, and the measurement noise $\sigma_\mathrm{m}$ for both ACS filters in Figures \[fig:erms\_sn\] to \[fig:erms\_redshift\]. We investigate the dependencies on the logarithmic flux signal-to-noise ratio , defined via the ratio `FLUX_AUTO`/`FLUXERR_AUTO` from `SExtractor` in Fig.\[fig:erms\_sn\], on the aperture magnitude in Fig.\[fig:erms\_mag\], and on the auto magnitude from `SExtractor` in Fig.\[fig:erms\_magauto\], in all cases with (left panels) and without (right panels) applying our colour selection. As expected, the measurement noise $\sigma_\mathrm{m}$ increases steeply towards low signal-to-noise and fainter magnitudes. This is one of the reasons why $\sigma_\epsilon$ increases towards lower signal-to-noise and fainter magnitudes. Interestingly, we find that $\sigma_\mathrm{int}$ also increases towards fainter magnitudes. The analysis of COSMOS data by @leauthaud07 also hinted at this trend with magnitude, but these authors discussed that it might be an artefact from their simplified estimator of the measurement error. We expect that our estimate of the measurement noise from overlapping tiles is fairly robust, and therefore suggest that this indeed appears to be a real effect, showing that intrinsically fainter galaxies have a broader ellipticity distribution. As a function of the signal-to-noise ratio we largely observe the corresponding trend of an increasing $\sigma_\epsilon$ and $\sigma_\mathrm{int}$ towards lower , but note that our estimate for $\sigma_\mathrm{int}$ flattens at and eventually turns over to decreasing $\sigma_\mathrm{int}$. Using stacks of different depth we verified that this flattening is not intrinsic to the galaxies. Instead, we expect that the validity of Eq.\[eq:sigmae2\] breaks down for large $\sigma_\mathrm{m}$. In addition, selection effects may have some influence, e.g. the cuts applied in size and , as well as non-Gaussian tails in the measured ellipticity distribution at low signal-to-noise. Comparing the left and right panels in Figures\[fig:erms\_sn\] to \[fig:erms\_magauto\] we find that the application of our colour selection to remove cluster galaxies has only a relatively small impact on the ellipticity dispersion: Applying the colour selection (which preferentially selects blue high-$z$ background galaxies) increases $\sigma_\epsilon$ by () and $\sigma_\mathrm{int}$ by () at magnitudes in the F606W (F814W) filter. This can be compared to the dependence of the ellipticity dispersion on photometric redshift shown in Fig.\[fig:erms\_redshift\], where we split the sample into bright (left panel) and faint (right panel) galaxies. Over the broad redshift range covered by the HST data the redshift dependence appears to be relatively weak. Most notably, the faint galaxies show an increase in $\sigma_\epsilon$ and $\sigma_\mathrm{int}$ between redshift 0 and . In principle, one expects such a trend, as galaxies at higher redshifts are observed at bluer rest-frame wavelengths, with stronger light contributions from sites of star formation. However note that it is more challenging to robustly infer conclusions on the redshift dependence of the shape distribution, as this is more strongly affected by large-scale structure variations [compare e.g. @kannawadi15]. We therefore suggest to investigate these trends further in the future with larger data sets. Comparing the weak lensing efficiency of F606W and F814W {#app:606vs814} -------------------------------------------------------- In Figures\[fig:erms\_sn\] to \[fig:erms\_magauto\] $\sigma_\mathrm{int}$ is typically lower for the analysis of the F814W data than for the F606W images at a given signal-to-noise ratio or magnitude. However, when interpreting this one has to keep in mind that the bins do not contain identical sets of galaxies. To facilitate a fair direct comparison of the performance of both filters for weak lensing measurements we limit the analysis to the F606W and F814W AEGIS observations, which were taken under very similar conditions with similar exposure times. As a first test, we compare the ellipticity dispersions computed from those galaxies that have robust shape estimates and in [*both*]{} bands. Including the matched galaxies with we find that on average $\sigma_\mathrm{int}$ ($\sigma_\epsilon$) is lower for the F814W shape estimates by $0.022\pm0.003$ ($0.019\pm0.003$) compared to the F606W shapes when no colour selection is applied, and by $0.016\pm0.006$ ($0.009\pm0.004$) when blue galaxies are selected with . Hence, we find that intrinsic galaxy shapes are slightly rounder when observed in the redder filter, which reduces their weak lensing shape noise. However, the quantity that actually sets the effective noise level for weak lensing studies is the effective source density after colour selection, which we define as $$n_\mathrm{eff}=\sum_\mathrm{mag} n(\mathrm{mag}) \times \left(\frac{\sigma_\epsilon^\mathrm{ref}}{\sigma_\epsilon(\mathrm{mag})}\frac{\langle\beta\rangle(\mathrm{mag})}{\langle\beta\rangle^\mathrm{ref}}\right)^2 \,.$$ For a cluster at we find from the AEGIS data that $n_\mathrm{eff}$ is higher by a factor 1.28 (1.06) for F606W compared to F814W when applying (when not applying) the colour selection with . Hence, if only a single band is observed with HST, F606W is slightly more efficient for the shape measurements than F814W. However, given that the ratio between the estimates is close to unity, we expect that programmes which have observations in [*both*]{} F606W and F814W can achieve a higher effective source density when jointly estimating shapes from both bands. Our work has shown the necessity for depth-matched colours for the cluster member removal. Therefore, we suggest that future HST weak lensing programmes of clusters at should consider to split their observations between F606W and F814W to obtain both colour estimates and joint shape measurements from both bands. Fitting functions and shape weights {#app:shapeweights} ----------------------------------- We compute second-order polynomial interpolations for the ellipticity dispersions as a function of logarithmic signal-to-noise and magnitude within limits as $$y=a+b\hat{x}+c\hat{x}^2\,,$$ where . For our weak lensing analysis of SPT clusters we compute empirical shape weights for galaxy $i$ as $$\label{eq:shapeweights} w_i=\left[ \sigma_\epsilon^\mathrm{fit}\left(\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_{\mathrm{auto},i}\right) \right]^{-2}\,.$$ from the interpolation of $\sigma_\epsilon$ as function of the logarithmic signal-to-noise ratio for the colour-selected CANDELS galaxies. We plot the best-fit interpolations in Figures \[fig:erms\_sn\] to \[fig:erms\_magauto\] and list their polynomial coefficients in Table \[tab:sigmae\_fits\]. Band Colour $x$ $x_\mathrm{min}$ $x_\mathrm{max}$ $y$ ----------- ----------------------- ------------------------------------------------------------------- ------------------ ------------------ ----------------------- ----------- ------------ ------------ -- $I_{814}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\epsilon$ $0.36777$ $-0.18359$ $0.06843$ $I_{814}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{int}$ $0.27050$ $0.03504$ $-0.05252$ $I_{814}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{m}$ $0.26390$ $-0.42101$ $0.18058$ $I_{814}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\epsilon$ $0.22123$ $0.01644$ $0.00340$ $I_{814}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\mathrm{int}$ $0.21232$ $0.03411$ $-0.00402$ $I_{814}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\mathrm{m}$ $0.01480$ $-0.01211$ $0.01453$ $I_{814}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\epsilon$ $0.24301$ $0.01649$ $0.00201$ $I_{814}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\mathrm{int}$ $0.23712$ $0.02839$ $-0.00433$ $I_{814}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\mathrm{m}$ $0.01925$ $-0.00090$ $0.01200$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\epsilon$ $0.38420$ $-0.19190$ $0.05716$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{int}$ $0.28447$ $0.03555$ $-0.07253$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{m}$ $0.27431$ $-0.43743$ $0.18966$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\epsilon$ $0.22602$ $0.00757$ $0.00698$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\mathrm{int}$ $0.21238$ $0.02943$ $-0.00130$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26 $\sigma_\mathrm{m}$ $0.01583$ $-0.01478$ $0.01571$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\epsilon$ $0.23050$ $0.02525$ $0.00195$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\mathrm{int}$ $0.22288$ $0.03886$ $-0.00469$ $I_{814}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26 $\sigma_\mathrm{m}$ $0.01869$ $-0.00322$ $0.01307$ $V_{606}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\epsilon$ $0.38882$ $-0.16903$ $0.05008$ $V_{606}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{int}$ $0.29414$ $0.05089$ $-0.07555$ $V_{606}$ all $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{m}$ $0.27001$ $-0.44604$ $0.19850$ $V_{606}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\epsilon$ $0.22918$ $0.01439$ $0.00371$ $V_{606}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\mathrm{int}$ $0.21549$ $0.03276$ $-0.00186$ $V_{606}$ all $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\mathrm{m}$ $0.01564$ $-0.01605$ $0.01140$ $V_{606}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\epsilon$ $0.24435$ $0.01885$ $0.00208$ $V_{606}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\mathrm{int}$ $0.23647$ $0.03082$ $-0.00233$ $V_{606}$ all $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\mathrm{m}$ $0.01257$ $-0.00372$ $0.00912$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\epsilon$ $0.39491$ $-0.16019$ $0.04158$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{int}$ $0.29096$ $0.08216$ $-0.09812$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{log}_{10}(\mathrm{Flux}/\mathrm{Fluxerr})_\mathrm{auto}$ 0.75 2 $\sigma_\mathrm{m}$ $0.28751$ $-0.48200$ $0.22022$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\epsilon$ $0.24319$ $0.00763$ $0.00486$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\mathrm{int}$ $0.22404$ $0.03115$ $-0.00180$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{aper}$ 22.5 26.5 $\sigma_\mathrm{m}$ $0.01884$ $-0.01892$ $0.01190$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\epsilon$ $0.24607$ $0.01653$ $0.00295$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\mathrm{int}$ $0.23311$ $0.03313$ $-0.00234$ $V_{606}$ $V_{606}-I_{814}<0.3$ $\mathrm{Mag}_\mathrm{auto}$ 22.5 26.5 $\sigma_\mathrm{m}$ $0.01309$ $-0.00857$ $0.01044$ Non-matching galaxies in CANDELS {#se:app:non_matches} ================================ We have investigated the of non-matching galaxies between our CANDELS F606W shear catalogue and the S14 photo-$z$ catalogue (see Sect.\[se:photo\_ref\_cats\]) by visually inspecting a random subset. Most of the non-matching galaxies can be explained through differences in the object detection or deblending given the different detection bands (optical F606W vs. NIR F125W+F140W+F160W). For of the total galaxies centroid shifts prevent a match. These should not affect the source redshift distribution. For the S14 catalogue contains a single object which is associated with two deblended objects in our F606W shear catalogue. If such differences in the deblending would occur independent of redshift, there would be no net effect on the source redshift distribution. However, such differences might be more frequent for high-$z$ ($z\gtrsim 1$) galaxies, where the F606W images probe rest-frame UV wavelengths and mostly detect sites of star formation, while the IR imaging probes the stellar content of the galaxies. Finally, of the total galaxies show clear isolated galaxies in our F606W shear catalogue that are missing in the S14 NIR-detected catalogue, possibly because they are too faint and too blue. To obtain a rough estimate for the resulting uncertainty of these effects on our analysis, we assume a scenario where both the missing isolated galaxies () plus the excess half of the differently deblended galaxies () constitute an excess population of 100% blue () galaxies at high redshifts (). This scenario is pessimistic for the differently deblended galaxies as explained above (no impact if the effect is redshift independent). For the missing isolated galaxies the scenario is likely to be realistic, but we note that it would also overestimate the impact in case some of the galaxies are redder and removed by our colour selection. At our median cluster redshift the scenario leads to a [*relative*]{} increase in $\langle\beta\rangle$ by only +0.5%, thanks to our colour selection which already selects mostly galaxies. Cross-check for the redshift distribution using spatial cross-correlations {#app:candels_x} ========================================================================== ![image](z_cc_recovery_1_sample3.png){width="0.69\columnwidth"} ![image](z_cc_recovery_1_sample4.png){width="0.69\columnwidth"} ![image](z_cc_recovery_1_sample_added.png){width="0.69\columnwidth"} A number of studies have explored the use of spatial cross-correlation techniques to constrain source redshift distributions [e.g. @newman08; @matthews10; @benjamin13]. In particular, @newman08 [@matthews10; @schmidt13; @rahman15; @rahman16; @scottez16] aim at reconstructing the redshift distribution of a sample with an unknown redshift distribution (“photometric sample”) via its spatial cross-correlation with galaxies in redshift slices of an incomplete spectroscopic reference sample. The cross-correlation amplitude increases if a larger fraction of the photometric sample is located within the redshift range of the corresponding slice. As a result, information on the redshift distribution of the photometric sample can be inferred. When using photometric samples with a broad redshift distribution the accuracy of the method is limited by how well a potential redshift evolution of the relative galaxy bias between the populations can be accounted for [e.g. @rahman15]. However, the impact of this limitation can be reduced if the photometric sample can be split into sub-samples with relatively narrow individual redshift distributions, as suggested by [@schmidt13; @menard13] and applied to SDSS data in [@rahman16]. The CANDELS data are well suited to employ this technique, as considerable spectroscopic (or grism) redshift samples are available (Sect.\[se:zdist\_candels\_checks\]), and given that the 3D-HST photo-$z$s allow for a relatively clean subdivision into narrower redshift slice for most of the galaxies. We employ the <span style="font-variant:small-caps;">The-wiZZ</span>[^17] implementation [@morrison17] of the cross-correlation technique described in @schmidt13 [@menard13] to obtain an independent cross-check for our estimate of the colour-selected CANDELS redshift distribution. For this we use the combined sample of high-fidelity spectroscopic and high-quality grism redshifts (see Sect.\[se:zdist\_candels\_checks\]) as spectroscopic reference sample (without colour selection) and the colour-selected photo-$z$ sample as photometric sample, splitting galaxies into 25 linear bins in $z_\mathrm{s/g}$ or $z_\mathrm{p}$, respectively, between and . We compare the estimate for the redshift probability distribution obtained from the cross-correlation analysis using physical separations between 30 kpc and 300 kpc to the $z_\mathrm{p}$ and $z_\mathrm{f}$ histograms in Fig.\[fig:zdist\_cross\] using galaxies with and the actual shape weights from our CANDELS shear catalogue. The left and the middle panels of Fig.\[fig:zdist\_cross\] correspond to the subset of CANDELS galaxies for which we implemented statistical corrections (see Sect.\[sec:correct\_sys\_photoz\]) for catastrophic redshift outliers (, ) or redshift focusing (, ), respectively. In both cases we find that the redshift distribution inferred from the cross-correlation analysis is largely consistent with the statistically corrected distribution based on the HUDF analysis ($z_\mathrm{f}$), while it is clearly incompatible with the uncorrected distribution in the selected $z_\mathrm{p}$ ranges, providing an independent confirmation for the HUDF-based correction scheme. The right panel of Fig.\[fig:zdist\_cross\] shows the combined reconstruction for the full colour selected sample (). Consistent with the other panels the reconstruction describes the $z_\mathrm{f}$ histogram better than the $z_\mathrm{p}$ histogram, both at low redshifts () and around the broad peak at . The statistical error-bars shown in Fig.\[fig:zdist\_cross\] indicate the dispersion of the reconstruction when splitting the combined CANDELS data set into 10 subareas of equal area and obtaining 1000 bootstrap resamples of the subareas included in the analysis. We expect that this yields a good approximation for the statistical uncertainty for most of the redshift range of interest. However, at the highest redshifts () the spectroscopic samples become very small (compare Fig.\[fig:zdist\_f814w\]), likely introducing additional uncertainties that are not fully captured by the error-bars. This is also suggested by the large fluctuations of both the recovered and the error-bars between neighbouring high-$z$ bins. We note the substantially negative reconstructions at in the middle and right panel of Fig.\[fig:zdist\_cross\]. At these redshifts the full spectroscopic sample contains a large number of galaxies (no colour selection applied to the spectroscopic sample). We therefore expect that the error-bars are robust and that the negative estimates are indeed significant. We interpret these negative values as a spurious effect caused by our colour selection, which explicitly removes galaxies at these redshifts from the photometric sample. Therefore, the photometric sample is spatially underdense in regions that are physically overdense at these redshifts. In contrast, the spectroscopic sample is spatially over-represented in regions of physical overdensities at these redshifts. This results in a net anti-correlation between the samples and negative estimates. As a possible solution to this problem [@rahman15] suggest to homogenise the spatial density of the spectroscopic sample by removing galaxies in overdense regions. However, as the spectroscopic sample employed in our analysis (14,472 galaxies) is already much smaller than the sample employed by [@rahman15] (791,546 galaxies) we do not follow this approach. As an approximate solution for this systematic effect we instead set the values of the two bins in Fig.\[fig:zdist\_cross\] at to zero when computing $\langle\beta \rangle$ as described in the next paragraph. This is justified by multiple tests presented in this work that suggest that the residual contamination by galaxies at these redshifts should be very low and close to zero (Sections \[se:photo\_color\_select\_acs\], \[se:zdist\_candels\_checks\], \[se:source\_sel\_scatter\], \[se:photo\_number\_density\_tests\]). However, outside this redshift range we treat bins with negative as negative contributions in the computation of $\langle\beta \rangle$. This is needed in order to achieve unbiased results in the case of purely statistical scatter that has equal chance to be positive or negative. For a quantitative comparison of the distribution and the histograms shown in the right panel of Fig.\[fig:zdist\_cross\] we compute $\langle\beta \rangle$ for our median cluster redshift , dealing with negative as explained in the previous paragraph, and generally limiting the considered redshift range to to minimise the impact of the highest-$z$ data points for which the recovery suffers the strongest from the small spectroscopic sample. The resulting from the cross-correlation analysis (the error indicates the statistical scatter from the bootstrap resamples) is consistent with from the HUDF-corrected catalogues within $2\sigma$. We conclude that the cross-correlation analysis independently supports the results from the HUDF analysis, but note that the spectroscopic samples within the relatively small CANDELS areas are not yet sufficiently large to constrain the redshift distribution with very high precision. Details of the ACS+FORS2 colour measurements and the accounting for photometric scatter {#app:details_fors2color_scatter} ======================================================================================== ACS+FORS2 colour measurement {#app:details_fors2color} ---------------------------- To measure colours between the F606W and FORS2 I-band images images we convolve each mosaic F606W image with a Gaussian kernel such that the resulting PSF has the same `FLUX_RADIUS` measured by `Source Extractor` as the corresponding FORS2 I-band image (we empirically account for the impact of non-Gaussian VLT PSF profiles in Appendix \[app:tiecolor\]). For some of the FORS2 stacks we found small residual systematic offsets of object positions in some image regions with respect to their location in the corresponding ACS mosaic (typically ). To not bias the colour measurement, we therefore fit and subtract a smooth 5th-order 2D-polynomial interpolation of the measured positional offsets to the catalogue positions. We overlayed and visually inspected these corrections on all images to ensure that they are robust. We then measure object fluxes in circular apertures with diameter 15 both in the VLT and the convolved ACS image. We transform them into magnitudes, correct these for galactic extinction, and compute the colour estimate . ![image](SPT-CLJ0000-5748_coldiff_p.png){width="0.9\columnwidth"} ![image](SPT-CLJ0000-5748_coldiff_p_fix.png){width="0.9\columnwidth"} ![image](SPT-CLJ0546-5345_coldiff_p.png){width="0.9\columnwidth"} ![image](SPT-CLJ0546-5345_coldiff_p_fix.png){width="0.9\columnwidth"} Tying the ACS+FORS2 colours to the ACS-only colours {#app:tiecolor} --------------------------------------------------- We have ACS-based and ACS+FORS2-based colour estimates for the galaxies in the inner cluster regions. We use these galaxies to refine the calibration of the colours for all galaxies and tie them to the colour selection available in the 3D-HST CANDELS catalogues. The left panels of Fig.\[fig:scatter\] shows the difference of these colour estimates as function of for two example clusters. The top row corresponds to SPT-CL$J$0000$-$5748, which has one of the deepest and best-seeing FORS2 I-band stacks in our sample, resulting in relatively moderate photometric scatter. Here the analysis reveals a mag colour offset for bright galaxies. We expect that this offset is in part caused by the offset in Eq.(\[eq:picklesoffset\]). Further contributions might come from uncertainties in the $I_\mathrm{FORS2}$ zero-point calibration due to the small number of stars available for its determination, or inaccuracies in the PSF homogenisation. In comparison, the bottom row reveals a larger photometric scatter for SPT-CL$J$0546$-$5345, which has a shallower magnitude limit and worse image quality (see Table \[tab:vltdata\]). For such VLT data we typically detect a shift of the median colour difference (indicated through the open circles) at faint magnitudes towards negative values. In part this is caused by the asymmetric and biased scatter in logarithmic magnitude space. However, further effects could lead to a magnitude-dependent colour offset: for example, we acknowledge that our PSF homogenisation only ensures equal flux radii between the bands. However, residual differences in the actual PSF shapes might lead to slightly different fractions in the total PSF flux lost outside the aperture. This would lead to a magnitude-dependent colour offset given that fainter objects are typically less resolved. Understanding the exact combination of these effects for each cluster is not necessary given that we directly tie the colours to the colours empirically: To do so, we fit the median values of the colour offsets determined in 0.5mag-wide bins between with a second order polynomial in $V_{606}$ and subtract this model from all colour estimates in the cluster field to obtain (see Fig.\[fig:scatter\]). We only use relatively blue galaxies with to derive this fit. This is motivated by small differences in the effective filter curves of $I_\mathrm{FORS2}$ and $I_{814}$. In particular, $I_\mathrm{FORS2}$ cuts off transmission red-wards of , while $I_{814}$ has a transmission tail out to . Thus, we expect non-negligible colour differences for very red objects. Given that we generally apply fairly blue cuts in colour this is not a problem for our analysis. However, we exclude red galaxies when deriving the fit as they are over-represented compared to CANDELS in the cluster fields. Accounting for photometric scatter {#app:scatter} ---------------------------------- ### ACS-only colour selection {#sec:noise:acsonly} In the inner cluster regions covered by the F606W and F814W ACS images we include galaxies in the magnitude range . The brighter magnitude limit has been chosen as galaxies passing our colour selection at even brighter magnitudes are dominated by foreground galaxies. The fainter magnitude limit approximately matches the cut applied in the weak lensing shape analysis (see Sect.\[sec:shear\]). Our ACS images have typical $5\sigma$ limits for the adopted 07 apertures of and . Therefore, the faintest galaxies included at the colour cut (, ) still have fairly high photometric signal-to-noise and . Accordingly, photometric noise has only minor impact on the colour selection for these galaxies. Nonetheless, we account for it by adding random Gaussian scatter to the catalogues, which are typically based on deeper ACS mosaic stacks compared to the ones used for our shape analysis, prior to the colour selection, such that they have the same limiting magnitudes in $V_{606}$ and $I_{814}$ as our cluster field observations. Also, we apply a slightly bluer colour selection for the galaxies in the faintest magnitude bin (see Table \[tab:app:colourcuts\] and Sect.\[se:photo\_color\_select\_acs\]). ---------------- ----- ----- -------- -------- $z_\mathrm{l}$ bright faint $<1.01$ 0.3 0.2 0.2 0.0 $>1.01$ 0.2 0.1 0.1 $-0.1$ ---------------- ----- ----- -------- -------- [Note. — Colour cut limits applied in our analysis. [*Column 1:*]{} Cluster redshift range. [*Column 2:*]{} Colour-cut in ACS-only colour for galaxies with . [*Column 3:*]{} Colour-cut in ACS-only colour for galaxies with . [*Column 4:*]{} Colour cut in the ACS+FORS2 colour after tying it to the colour (see Appendix \[app:tiecolor\]), as employed for “bright cut” magnitude bins with low photometric scatter . [*Column 5:*]{} As column 4, but for the “faint cut” magnitude bins with increased photometric scatter .\ ]{} ### ACS+FORS2 colour selection {#se:scatter_acsfors2} The colour estimates that include the FORS2 data are more strongly affected by photometric scatter than the colours obtained from the high-resolution ACS data only (see Fig.\[fig:scatter\]). To ensure that we can still apply a consistent colour selection to the catalogues we do the following: First, we limit the analysis to relatively bright $V_{606}$ magnitudes, to ensure that the scatter is small enough to not compromise the exclusion of galaxies at the cluster redshift considerably. For this we compute the r.m.s. scatter $\sigma_{\Delta(V-I)}$ in the colour difference of blue galaxies () in 0.5mag-wide bins in $V_{606}$. For the ACS+FORS2 colour selection we only include magnitudes bins with scatter . Here we employ our standard (“bright”) colour cut for the magnitude bins with low scatter , and a more conservative (“faint”) colour cut for magnitude bins with slightly larger scatter , see columns 5 and 6 in Table \[tab:vltdata\] for the corresponding magnitude bins in each cluster and Table \[tab:app:colourcuts\] for the colour cuts as function of cluster redshift. Second, we add noise to the colour estimates in the CANDELS catalogue prior to the colour cut, similarly to our approach for the ACS-only selection. However, in contrast to Appendix\[sec:noise:acsonly\] we do not assume a Gaussian noise distribution here, but randomly sample the noise from the actual distribution of the colour differences shown in the right panels of Fig.\[fig:scatter\]. The motivation for not using a Gaussian approximation is given by the skewness in the distribution and presence of outliers. In practice, we again divide the galaxies into 0.5mag-wide bins in $V_{606}$. We further subdivide these galaxies into sub-bins according to their colour if sufficiently many galaxies are available to provide sub-bins containing at least 30 galaxies each. For each galaxy in the CANDELS catalogue we then identify the corresponding bin/sub-bin and randomly assign a colour difference drawn from this bin/sub-bin. Note that we introduce the further colour subdivision as red galaxies (which are later removed by the colour cut) show a lower scatter at a given $V_{606}$ magnitude[^18]. Limitations of a statistical correction for cluster member contamination {#app:why_not_boost} ======================================================================== Weak lensing studies that use wide-field imaging data and do not have sufficient colour information for a robust removal of cluster galaxies can attempt to statistically correct their shear profiles for the dilution effect of cluster members in the source samples [see e.g. @hoekstra15]. For this, they need to estimate the relative excess counts as function of cluster-centric distance, ideally accounting for the impact of masks, obscuration by cluster members, and magnification, and fit it with a model, typically in the form $$n_\mathrm{measure}(r)=\frac{n_\mathrm{bg}}{1-f(r)}\,,$$ and scale the shear profile as $$\langle g_\mathrm{t} \rangle^\mathrm{boosted}(r)=\langle g_\mathrm{t} \rangle(r) \frac{1}{1-f(r)}\,.$$ Here we consider two previously employed models for the projected density profiles of cluster galaxies, namely the projected singular isothermal sphere (SIS) model $$f(r)=f_{500} \frac{r_\mathrm{500c}}{r}$$ [e.g. @hoekstra07] and an exponential model $$f(r)=f_{500} e^{1-r/r_\mathrm{500c}}$$ [e.g. @applegate14], where $f_\mathrm{500}$ corresponds to the contamination at $r_\mathrm{500c}$. ![image](nocc_stacked_xray.png){width="0.99\columnwidth"} ![image](nocc_f500bins.png){width="0.99\columnwidth"} We do not use this approach for our HST analysis as the ACS mosaics are too small to derive a robust estimate of the background source density directly. To test this, we use our source catalogues without colour selection, estimate the mask- and obscuration-corrected source density profiles in magnitude bins, and fit them with both $f(r)$ profiles. Combining the analysis from all clusters we find that both profiles provide [acceptable]{} fits for most of the radial range covered by the ACS data. For example, when using only a single broad magnitude bin, the SIS model returns for 7 degrees of freedom, whereas the exponential model returns . [The SIS model is clearly a better fit at small radii (see the *left* panel of Fig.\[fig:ngal\_nocc\]), but the exponential profile is not ruled out at high significance.]{} Yet, the two models yield [uncomfortably]{} different contamination fractions [(shown in the *right* panel of Fig.\[fig:ngal\_nocc\] as function of $V_{606}$)]{}. As a test for the impact of these differences we artificially apply the two different boost correction schemes [(taking their magnitude dependencies into account)]{} to our colour-selected shear profiles and compare the resulting mass estimates. Here we find that the exponential model leads to mass estimates which are higher compared to those from the isothermal model by . As it is currently not clear what the correct functional form would be, we conclude that the application of such a contamination correction would introduce substantial systematic uncertainty. One could consider to reduce this uncertainty by using external blank fields to constrain the background source density. Using our colour-selected catalogues we have demonstrated that the careful matching of source selection criteria and noise properties between the cluster and reference fields, which would be required for such an approach, is in principle possible. However, instead of providing an important validation test as in our study, the information in the number density would then be used to correct the signal, assuming that all other related analyses steps were done correctly. Large-scale structure variations also introduce significant variations in the source densities between the five CANDELS fields. Without colour selection we find that they lead to uncertainties in the estimated mean background density of at to at . In addition to the increased systematic uncertainty, the use of a contamination correction also increases the statistical uncertainty compared to a robust colour selection that adequately removes cluster galaxies. First, the cluster members dilute the small-scale signal, which is the regime providing the highest signal-to-noise contribution for our analysis. Second, source density profiles are typically too noisy to measure the contamination for individual clusters. On the other hand, if an average contamination model is applied, extra scatter in the mass constraints is introduced. Impact of contamination by very blue cluster members {#se:test_extremely_blue} ==================================================== The tests presented in Sect.\[se:photo\_number\_density\_tests\] show no indication for a significant residual contamination by cluster members. However, our estimates from Sect.\[se:source\_sel\_scatter\] suggest that, in the presence of noise and averaged over our cluster sample, our ACS-only (ACS+FORS2) colour selection should leave a residual contamination of () of very blue field galaxies at the corresponding cluster redshifts. Whether or not this can introduce a residual excess contamination by cluster members depends on the relative properties of the galaxy distributions in the field and cluster environment. Luminous Compact Blue Galaxies [LCBGs, e.g. @koo94] represent an extreme star-bursting population of galaxies with very blue colours and compact sizes. Such galaxies were also identified in cluster environments [@koo97], making them the most relevant potential contaminant for our colour-selected weak lensing source sample. @crawfords11 [@crawfords14] and @crawfords16 identify and study LCBGs in five massive clusters at using a photometric preselection, Keck/DEIMOS spectroscopy, and HST morphological measurements. For the clusters in their sample @crawfords11 find that the number density enhancement of the cluster LCBG population compared to the LCBG field density is comparable to or lower than the corresponding enhancement of the total cluster population compared to the total field population. In addition, @crawfords16 find that the relevant properties of the cluster LCBGs (star-formation rate, dynamical mass, size, luminosity, and metallicity) are indistinguishable from the properties of field LCBGs at the same redshift. Accordingly, we can make the conservative assumption that the relative fraction of cluster members that pass our colour selection is equal to or lower than the fraction of field galaxies passing the selection for the ACS-only (ACS+FORS2) selection, accounting for noise (see Sect.\[se:source\_sel\_scatter\]). We then estimate the approximately expected average fraction of cluster galaxies in our colour-selected source sample at as $$\begin{aligned} f_\mathrm{500,expected}&=& f_\mathrm{pass,field} f_\mathrm{500,no-cc} \left(\frac{n_\mathrm{gal,cc}}{n_\mathrm{gal,no-cc}}\right)^{-1} \\ \nonumber &=&0.009\thinspace (0.008) \,,\end{aligned}$$ where indicates an estimate for the average contamination at based on a number density profile analysis when the colour selection is [*not*]{} applied (see [the right panel of Fig.\[fig:ngal\_nocc\], averaging the values for the more conservative exponential model according to the relative weight of the corresponding magnitude bin in the reduced shear profile fits]{}), and corresponds to the fraction of galaxies in the cluster fields passing the colour selection within the magnitude range of the ACS-only (ACS+FORS2) analysis. We do not attempt to model the radial distribution of the expected contaminating cluster galaxies, as LCBGs appear to follow a rather shell-like distribution with a depletion in the cluster core [@crawfords06]. Instead, we assume that $f_\mathrm{500,expected}$ provides a reasonable approximation for the typical contamination, which is likely conservative given that the average of our cluster sample [based on the lensing analysis and assuming the [concentration–mass]{} relation from @diemer15 see Sect.\[sec:wlmasses\]] is more representative for the inner ($500$ kpc) than the outer ($1.5$ Mpc) limit of the fit range for our default analysis. With these conservative assumptions, the relative bias for the average lensing efficiency caused by the expected cluster contamination is . Given that this is even smaller than the uncertainty on from line-of-sight variations between the CANDELS fields (Sect.\[sec:beta\_los\_variation\]), this bias could well be ignored [(we still include it in the systematic error budget in Table \[tab:sys\])]{}. But we note that future studies could attempt to model the contamination more accurately and apply a correction. Additional figures {#se:additional_figs} ================== Figures \[fig:massplota\] to \[fig:massplotm\] complement Fig.\[fig:massplotdemo\_map\] and Fig.\[fig:massplotdemo\_profile\], showing the corresponding results for the other clusters. In particular, the left and middle panels show the weak lensing mass reconstructions overlaid onto the corresponding VLT/FORS2 $BIz$ and central ACS colour images, as well as the locations of the different cluster centres used in our analysis. In the corresponding right panels we show the weak lensing shear profiles centred onto the X-ray centroids. ![image](SPT-CL\sclustera_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclustera_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclustera-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterb_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterb_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterb-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterc_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterc_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterc-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterd_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterd_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterd-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclustere_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclustere_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclustere-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterf_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterf_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterf-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterg_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterg_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterg-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterh_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterh_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterh-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusteri_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusteri_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusteri-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterk_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterk_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterk-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterl_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterl_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterl-a-c_gt.png){width="5.4cm"} ![image](SPT-CL\sclusterm_mass_sn.jpg){width="6.cm"} ![image](SPT-CL\sclusterm_mass_sn_sb.jpg){width="6.cm"} ![image](SPT-CL\sclusterm-a-c_gt.png){width="5.4cm"} Affiliations {#affiliations .unnumbered} ============ [$^{\Bonn}$]{} Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121, Bonn, Germany\ [$^{\StanfordKIPAC}$]{} Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305-4060, USA\ [$^{\StanfordPhysics}$]{} Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305-4060, USA\ [$^{\KICPChicago}$]{} Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637\ [$^{\Munich}$]{} Faculty of Physics, Ludwig-Maximilians University, Scheinerstr. 1, 81679 München, Germany\ [$^{\ExcellenceCluster}$]{} Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany\ [$^{\Leiden}$]{} Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2300 CA Leiden, The Netherlands\ [$^{\ANL}$]{} Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL, USA 60439\ [$^{\UFlorida}$]{} Department of Astronomy, University of Florida, Gainesville, FL 3261\ [$^{\DARK}$]{} Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark\ [$^{\StonyBrook}$]{} Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA\ [$^{\MIT}$]{} MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139\ [$^{\Washington}$]{} Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA\ [$^{\SLAC}$]{} SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\ [$^{\Harvard}$]{} Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138\ [$^{\CfA}$]{} Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138\ [$^{\Colby}$]{}[Department of Physics & Astronomy, Colby College, 5800 Mayflower Hill, Waterville, Maine 04901]{}\ [$^{\FNAL}$]{}[Fermi National Accelerator Laboratory, Batavia, IL 60510-0500, USA]{}\ [$^{\AAUChicago}$]{} Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637\ [$^{\PhysicsUChicago}$]{} Department of Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637\ [$^{\ASIAA}$]{} Academia Sinica Institute of Astronomy and Astrophysics (ASIAA) 11F of AS/NTU Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan\ [$^{\Hyderabad}$]{} Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India\ [$^{\UCStCruz}$]{} Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\ [$^{\McGill}$]{} Department of Physics, McGill University, 3600 Rue University, Montreal, Quebec H3A 2T8, Canada\ [$^{\Berkeley}$]{} Department of Physics, University of California, Berkeley, CA 94720\ [$^{\Durham}$]{} Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK\ [$^{\MPE}$]{} Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany\ [$^{\Melbourne}$]{} School of Physics, University of Melbourne, Parkville, VIC 3010, Australia\ [$^{\CTIO}$]{} Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile\ [^1]: E-mail: [email protected] [^2]: This program also includes observations of SPT-CLJ0205$-$5829 (). However, we do not include it in the current analysis given its high-redshift, which would require deeper $z$-band observations for the background selection (see Sect.\[sec:phot\]) than currently available. [^3]: This cut is more conservative than the cut from S10, which is based on the @erben2001 signal-to-noise ratio definition that includes a radial weak lensing weight function. approximately corresponds to for our typical source galaxies, but note that there is a significant scatter between both estimates due to the different radial weighting. [^4]: measure an average uncorrected CTI-induced galaxy ellipticity at of from CANDELS/COSMOS F606W images, which were observed at a similar epoch but have higher background levels than our data, and thus weaker CTI signals. [^5]: For the GOODS-North field we estimate the $I_{814}$ magnitudes from the flux measurements in the F775W and F850LP filters. When conducting selections or binning in $V_{606}$ based on the photometry we undo their correction for total magnitudes in order to employ aperture magnitudes that are consistent with our cluster field measurements. [^6]: @rafelski15 note that the object 10157 in their catalogue is problematic as it consists of a blend of two galaxies at different redshifts. We therefore exclude it from the spec-$z$/grism-$z$ sample used in our analysis. [^7]: Available at <https://github.com/morriscb/The-wiZZ> [^8]: Here we want to investigate how well we can estimate the cosmic mean redshift distribution from CANDELS, for which sub-patches are not needed. The sub-patches are needed to estimate the line-of-sight scatter in $\langle \beta \rangle$ between the different cluster fields, as discussed in the second paragraph of this subsection. [^9]: Here we account for the magnitude-dependence of our colour cut (see Table \[tab:app:colourcuts\] in Appendix \[app:details\_fors2color\_scatter\]), by basing it on the lensed magnitude. [^10]: When computing the [*relative*]{} impact of magnification on the number density and mean lensing efficiency we deliberately do not include the shape weights, as we would otherwise need to account for the increase in $S/N$ and thus $w$ due to the magnification. Since we perform the full analysis in magnitude bins, with very little variation in $w$ within a bin, our approach constitutes a very good approximation. [^11]: The 3D-HST CANDELS catalogue provides aperture magnitudes, which we can directly compare to our measurements, plus an aperture correction based on the $H$-band, which is however not available for our cluster fields. [^12]: Here we approximate the sky area blocked by a galaxy through the $N_\mathrm{pix}$ parameter from `Source Extractor`. @hoekstra15 present a more detailed treatment using image simulations, finding that obscuration by cluster members is a relatively minor effect for their analysis. Our cluster galaxies are at higher redshift and are thus more strongly dimmed, leading to an even smaller impact of obscuration by cluster members. Our pipeline automatically masks the image region around bright and very extended galaxies. With this applied we find that accounting for the sky area blocked by unmasked brighter galaxies via the $N_\mathrm{pix}$ parameter leads to $\lesssim 1$ per cent changes in the source density even for the faintest galaxies considered in our analysis. [^13]: This approach generates Gaussian random shear fields based on the matter power spectrum. Comparing the resulting scatter in cluster mass estimates, @hoekstra11 show that approaches using the shear power spectrum provide good approximations to more accurate estimates from a ray-tracing analysis through the Millennium Simulation [@springel05; @hilbert09]. [^14]: We weight according to the SZ mass and not the lensing-inferred mass. The latter is more noisy and would give higher weight to clusters for which the lensing mass estimate scatters up. [^15]: An alternative approach to constrain cluster concentrations from weak lensing data is to fit both mass and concentration simultaneously for each cluster. These individual constraints are however very weak due to shape noise, and they are strongly affected by large-scale structure projections [e.g. @hoekstra03]. [^16]: We do not report $M_\mathrm{2500c}$ masses as these are not available in the simulation, preventing us to compute accurate bias corrections for this overdensity. [^17]: <https://github.com/morriscb/The-wiZZ> [^18]: This is expected since receives roughly comparable scatter contributions from $V_{606,\mathrm{con}}$ and $I_\mathrm{FORS2}$, with a reduced scatter in $I_\mathrm{FORS2}$ for red objects.
--- abstract: 'We develop some design examples for approximating a target surface at the final rigidly folded state of a developable quadrilateral creased paper, which is folded with a 1-DOF rigid folding motion from the planar state. The final rigidly folded state is reached due to the clashing of panels. Now we can approximate some specific types of non-developable surfaces, but we do not yet fully understand how to approximate an arbitrary surface with a developable creased paper that has limited DOFs. Our designs might have applications in areas related to the formation of a shell structure from a planar region.' bibliography: - 'Rigid-Folding.bib' title: 'Approximating a Target Surface with 1-DOF Rigid Origami' --- Introduction ============ We discuss here the inverse problem of rigid origami, that is, to approximate a target surface by rigid origami — usually starting from a planar creased paper. This problem has been preliminarily discussed in the review [@callens_flat_2017]. Generally, we need to consider the following factors when designing a creased paper for approximation. 1. Which surface we can approximate. 2. The DOF (degree of freedom) during the rigid folding motion. 3. The utilization of materials. Experience shows that, it is hard to make the creased paper behave well in all aspects. Generally, as we reduce the possible DOFs during the rigid folding motion, less surfaces can be approximated. For example, [@demaine_origamizer:_2017] give a universal algorithm to fold a planar creased paper to any piecewise-polygon orientable 2-manifold, which can be used to approximate any orientable 2-manifold. Although to archive the “watertight” property, they add small additional features at the vertices and along the edges, this algorithm is practical and guarantees a minimum number of seams. This result is undoubtedly successful, but will have many DOFs during its rigid folding motion, and a large proportion of materials are used in the connections and “walls” hidden in the “tuck” side. On the other hand, [@dudte_programming_2016] and [@song_design_2017] use a flat-foldable creased paper. These algorithms guarantee one degree-of-freedom (1-DOF) during the rigid folding motion and high utilization of materials, but only possible for a cylinderical developable surface or a surface of revolution, otherwise the creased paper will not be rigid-foldable. If we triangulate this creased paper, we can approximate more surfaces but without being 1-DOF. In this article we will focus on a branch of the inverse problem, to design a 1-DOF rigid origami approximating some desired shapes. More formally, given a connected surface $S$ in $\mathbb{R}^3$, for any positive real number $\epsilon>0$, find a creased paper $(P,C)$, which is the union of a connected planar paper $P \subset \mathbb{R}^2$ and a straight-line crease pattern $C$ embedded on $P$, such that 1. $(P,C)$ is rigid-foldable to its final rigidly folded state $(P',C')$, where the rigid folding motion halts because some panels clash. 2. $(P,C)$ has one degree of freedom during the rigid folding motion. 3. the Hausdorff distance $d$ between $S$ and $P$ satisfies $d \le \epsilon$. More details of the terminologies used here are given in [@he_rigid_2018]. By using a family of developable quadrilateral creased papers that are rigid-foldable but not necessarily flat-foldable, we are able to give solutions for approximating some surfaces at the final rigidly folded state, but they cannot be completely arbitrary. Furthermore, the approximation problem naturally induces an optimization problem, that is, find the “best” creased paper that fits the extra presupposed requirements. We will discuss it at the end of the article. Choosing Design Example ======================= To our best knowledge, the constraints on 1-DOF rigid-foldability for a creased paper restrict the configurations it may have. Therefore it seems hard to approximate an arbitrary connected surface with 1-DOF rigid origami. Our idea is, from several types of 1-DOF developable and rigid-foldable quadrilateral creased papers we have known [@he_new_2018; @he_rigid_2018-1], we choose some of them as the *design examples*, and study the rigid folding motions of them. For each design example, the profile of its inner vertices can approximate certain types of surfaces. The more design examples we can use, the more surfaces we can approximate. In this section we will analyze two design examples mentioned in [@he_rigid_2018-1]. The first one is the developable case of the “parallel repeating” type, which is generated among rows of parallel inner creases (Figure \[fig: unit columns\]); and the second one is the developable case of the “orthodiagonal” type, which is generated among several parallel straight line segments (Figure \[fig: orthodiagonal creased paper\]). ![\[fig: unit columns\](a) is the developable case of the parallel repeating type. In each row there are three independent sector angles $\alpha_i, \beta_i, \gamma_i$, and $\delta_i=2\pi-\alpha_i-\beta_i-\gamma_i$. Column 1 is an example of general input sector angles, where a “basic unit” of this column is labelled by a dashed red cycle; columns 2, 3 and 4 are flat-foldable and “straight-line” input sector angles. (b) is a rigidly folded state of (a) plotted by Freeform Origami [@tachi_freeform_2010-1]. The mountain and valley creases are colored red and blue. The inner vertices of a column are co-planar (see Proposition \[prop co-planar\]).](unit_columns){width="1\linewidth"} Parallel Repeating Type ----------------------- Figure \[fig: unit columns\] shows the developable case of the parallel repeating type of rigid-foldable quadrilateral creased papers, which is generated among rows of parallel inner creases. In each column we can choose independent input sector angles $\alpha_i, \beta_i, \gamma_i$, and $\delta_i=2\pi-\alpha_i-\beta_i-\gamma_i$, then construct the rest of the creased paper with both these angles and the supplement of these angles. $\alpha_i, \beta_i, \gamma_i,\delta_i$ should not form a cross. Here the length of creases does not affect the rigid-foldability, but will affect the profile of inner vertices. Based on that we start to analyze which surface the parallel repeating type can approximate. \[prop co-planar\] The inner vertices on a column of the parallel repeating type are co-planar. Figure \[fig: degree-4 single-vertex creased paper\](a) shows a general column. We know $A_1A_2 \parallel A_3A_4$ at any rigidly folded state. Thus $A_1, A_2, A_3, A_4$ are coplanar. Because $A_iA_{i+1} \parallel A_{i+2}A_{i+3}$, this argument continues down the column, which means all the inner vertices are co-planar, and the angle between any two adjacent inner creases is $\xi$, as illustrated in Figure \[fig: degree-4 single-vertex creased paper\](b). Figure \[fig: degree-4 single-vertex creased paper\](c) demonstrates a rigidly folded state of this column. ![\[fig: degree-4 single-vertex creased paper\](a) A “basic unit” of a column of the parallel repeating type. The mountain and valley creases are colored red and blue. Note that the folding angles $\rho_1, \rho_2, \rho_3$ on corresponding inner creases are opposite. (b) The inner creases of a column with its independent parameters labelled. (c) A view of a column from a view point in the plane coincident with the inner vertices, plotted by Freeform Origami [@tachi_freeform_2010-1].](coplanar){width="1\linewidth"} To study the profile of inner vertices on a column of the parallel repeating type, we use A $x$-$y$ coordinate system is built on the plane mentioned in Proposition \[prop co-planar\]. We say the inner vertices $(x_i,y_i)$ of a column can *approximate* a given planar curve $f: I \rightarrow \mathbb{R}^2$ if $\forall \epsilon>0$, there exists a column whose inner vertices $(x_i,y_i)$ make the Hausdorff distance $d$ between the set $(x_i,y_i)$ and the curve $f$ satisfy $d \le \epsilon$. \[prop: approximation\] The inner vertices on a column of the parallel repeating type can only approximate a planar curve $f:I \ni t \rightarrow (x(t),y(t)) \in \mathbb{R}^2$ that satisfies the following condition: there exists a rotation $\theta \in [0, 2\pi)$ and a shear transformation of magnitude $\pi/2-\xi$, $\xi \in [0, \pi]$, s.t. after the affine transformation $f \rightarrow \overline{f}$ described below, $\overline{f}$ is monotone decreasing. $$\label{eq: affine} \left[ \begin{array}{c} \overline{x}(t)\\ \overline{y}(t) \end{array} \right]= \left[ \begin{array}{cc} 1 & -1 / \tan \xi \\ 0 & 1 / \sin \xi \end{array} \right] \left[ \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right] \left[ \begin{array}{c} x(t)\\ y(t) \end{array} \right]$$ We name the curve approximated by the inner vertices of a column as a *target curve*. Sufficiency: An example of approximation is shown in Figure \[fig: target curve\]. For a given curve $f$, after a rotation by $\theta$ and a shear transformation by $\pi/2-\xi$, as in equation , $f$ (black curve in (b)) is mapped to $\overline{f}$ (black curve in (c)), where the inner creases are parallel to the $\overline{x}$ and $\overline{y}$ axes. Then if $\overline{f}$ is monotonous, we can construct an approximation corresponding to the partition when we apply the Darboux sum to describe the Darboux-integrability (red line segments in (c)). Because $\overline{f}$ is monotonous, it is Darboux-integrable (if $\overline{f}(I)$ is unbounded, the limit of difference between lower and upper Darboux sum is zero when the partition is infinitesimally refined) and only has countable first-kind discontinuity points, so arbitrarily refining the partition will make the Hausdorff distance be arbitrarily small. Hence we can approximate $f$ by re-transforming the approximation in the $\overline{x}$-$\overline{y}$ coordinate system to the $x$-$y$ coordinate system (red line segments in (b)). Here the requirement of monotone decreasing makes the angle between adjacent inner creases $\xi$, not $\pi-\xi$. Necessity: If a curve $f$ can be approximated by the inner vertices of a column, we can always find corresponding $\theta$ and $\xi$ and do the affine transformation in equation . Because the approximation turns left and right alternatively, $\overline{f}$ must be monotonous. To make the angle between adjacent inner creases $\xi$, not $\pi-\xi$, $\overline{f}$ should be monotone decreasing. ![\[fig: target curve\](a) shows a target curve (coloured black) and the result of an approximation (coloured red). Here $f(t)=[t,t^2/4]^T$, $t \in [-2,2]$. (b) is the image of (a) under equation with $\theta=70^{\circ}$ and $\xi=60^{\circ}$. The image of the target curve and the approximation are coloured black and red respectively.](target_curve){width="1\linewidth"} The inner vertices of a column cannot approximate a closed planar curve. \[rem: comment\] We can require $f$ to be continuous and $I$ to be a closed interval in $\mathbb{R}$ for the following reason. If $f(I)$ is not connected, $f(I)$ is the union of countable disjoint closed intervals. Each such subset can be approximated independently because the rigid folding motions of disconnected creased papers are independent. Therefore we can require $f(I)$ to be connected, which means $\overline{f}(I)$ is continuous and $f(I)$ is continuous. Then we can re-parametrize $f(I)=f(I')$, where $I'$ is a closed interval in $\mathbb{R}$. Next we will analyze the rigid folding motion of a row of the parallel repeating type. \[prop datum curve\] The inner vertices on a row of the parallel repeating type can approximate a Darboux-integrable curve $\Gamma: J \rightarrow \mathbb{R}^3$ at its final rigidly folded state, named the *datum curve*. ![\[fig: approximation\](a) An example of the datum curve $\Gamma$ (coloured black) with its approximation by line segments (coloured orange). In this example, $n=9$, $\Gamma(u)=[u+\cos u, -2u^2, \sin u]^T$, $u \in [-1,0.5]$, $\rho_4=5\pi/6$. Here we partition $\Gamma$ uniformly in $u$. (b) Part of the creased paper. The inner creases of its final rigidly folded state (coloured orange) exactly form the approximation in (a). We label all partition points $A_k$, $k \in [0,n+1]$; folding angles $\rho_k$, $k \in [1,3n+1]$; sector angles $\alpha_k$, $k \in [1,4n]$. The planar pattern is viewed along the direction perpendicular to the plane.](approximation){width="1\linewidth"} An approximation can be generated following the steps mentioned below. 1. Partition $\Gamma$ (black curve in Figure \[fig: approximation\](a)) by $n+2$ points and connect adjacent partition points in sequence by line segments (orange line segments in Figure \[fig: approximation\](a)). Any given Hausdorff distance $\epsilon$ can be satisfied by choosing a sufficiently large $n$. 2. Assign a direction of $\Gamma$, label each partition point $A_i$ ($i \in [0,n+1]$) along this direction. From the coordinates of $A_i$, calculate $l_i= \|A_iA_{i+1}\|$ ($i \in [0,n]$), $\beta_i=\angle A_{i-1}A_iA_{i+1}$ ($i \in [1,n]$) and $\theta_i=\langle \triangle A_{i-1}A_iA_{i+1}, \triangle A_iA_{i+1}A_{i+2} \rangle$ ($i \in [1,n-1]$). $\theta_i \in [0,2\pi)$ is calculated by the rotation angle along the vector $\overrightarrow{A_iA_{i+1}}$. 3. Choose the first inner vertex on the left from column 3 in Figure \[fig: unit columns\], set $\rho_4$ as a specific angle and $\rho_2=\pi$ (see Figure \[fig: approximation\](b)), with a known $\beta_1$, solve the following equations: $$\label{eq: the first beta and theta} \begin{gathered} \rho_2=2\arccos\bigg(\dfrac{\cos \alpha_2 \cos \beta_1-\cos \alpha_1}{\sin \alpha_2 \sin \beta_1}\bigg) \\ \rho_4=2\arccos\bigg(\dfrac{\cos \alpha_1 \cos \beta_1-\cos \alpha_2}{\sin \alpha_1 \sin \beta_1}\bigg) \end{gathered}$$ we can obtain $\alpha_1$ and $\alpha_2$. 4. Then we continue to obtain other sector angles ($i \in [1,n-1]$) in this row, as shown in Figure \[fig: approximation\](b). Regarding $\alpha_{4i-3}$, $\alpha_{4i-2}$, $\alpha_{4i-1}$, $\alpha_{4i}$, $\beta_i$, $\beta_{i+1}$ and $\theta_i$ as known variables, there are four equations for $\alpha_{4i+1}$, $\alpha_{4i+2}$, $\alpha_{4i+3}$ and $\alpha_{4i+4}$. Two of them are related to $\beta_i$, $\beta_{i+1}$ and $\theta_i$. $$\begin{gathered} \label{eq: general beta and theta} \arccos\bigg(\dfrac{\cos \alpha_{4i-3} \cos \beta_i-\cos \alpha_{4i-2}}{\sin \alpha_{4i-3} \sin \beta_i}\bigg) \pm \arccos\bigg(\dfrac{\cos \alpha_{4i-1}-\cos \alpha_{4i} \cos \beta_i}{\sin \alpha_{4i} \sin \beta_i}\bigg) = \\ \arccos\bigg(\dfrac{\cos \alpha_{4i+2} \cos \beta_{i+1}-\cos \alpha_{4i+1}}{\sin \alpha_{4i+2} \sin \beta_{i+1}}\bigg) \pm \arccos\bigg(\dfrac{\cos \alpha_{4i+4}-\cos \alpha_{4i+3} \cos \beta_{i+1}}{\sin \alpha_{4i+3} \sin \beta_{i+1}}\bigg) \\ \shoveleft{\theta_i= \pm \arccos\bigg(\dfrac{\cos \alpha_{4i-2}-\cos \alpha_{4i-3} \cos \beta_i}{\sin \alpha_{4i-3} \sin \beta_i}\bigg)} \\ \pm \arccos\bigg(\dfrac{\cos \alpha_{4i+1}-\cos \alpha_{4i+2} \cos \beta_{i+1}}{\sin \alpha_{4i+2} \sin \beta_{i+1}}\bigg) \end{gathered}$$ Note that the $\pm$ depends on the rigid folding motion we choose and the magnitude of sector angles. These equations can be directly derived from spherical trigonometry. Another equation is $\alpha_{4i+1}+\alpha_{4i+2}+\alpha_{4i+3}+\alpha_{4i+4}=2\pi$, but in order to simplify the steps we suppose the other vertices in this row are from column 2 in Figure \[fig: unit columns\]. Hence there are two more equations: $$\alpha_{4i+1}+\alpha_{4i+3}=\pi \quad \alpha_{4i+2}+\alpha_{4i+4}=\pi$$ 5. With $l_i$ and $\alpha_{4i-3}$, $\alpha_{4i-2}$, $\alpha_{4i-1}$, $\alpha_{4i}$ ($i \in [1,n]$), draw the creased paper. \[rem: comment 2\] Generically, We can require $\Gamma$ to be continuous and $J$ to be a closed interval in $\mathbb{R}$ for the following reason. Similar to our analysis in Remark \[rem: comment\], $\Gamma(J)$ can be required as a connected and closed set. If $\Gamma(J)$ has no second-kind discontinuity points, $\Gamma(J)$ is continuous. Then we can re-parametrize $\Gamma(J)=\Gamma(J')$, where $J'$ is a closed interval in $\mathbb{R}$. Besides, $\Gamma$ can be a closed curve. Here the final rigidly folded state is not special, we make it final by designing the rigid folding motion to be halted by clashing of panels at the first column of inner vertices from the left. If we release this condition we can make the datum curve be approximated by an intermediate rigidly folded state. \[cor: s1-s1\] In step 4 of Proposition \[prop datum curve\], If we choose other vertices in the first row from column 3 in Figure \[fig: unit columns\] , equation will be simplified: $$\begin{gathered} \dfrac{\cos \alpha_{4i-3} \cos \beta_i-\cos \alpha_{4i-2}}{\sin \alpha_{4i-3} \sin \beta_i} =\dfrac{\cos \alpha_{4i+2} \cos \beta_{i+1}-\cos \alpha_{4i+1}}{\sin \alpha_{4i+2} \sin \beta_{i+1}} \\ \theta_1 = 0 ~~ \textrm{if} ~~ (\alpha_{4i-3}+\alpha_{4i-2}-\pi)(\alpha_{4i+1}+\alpha_{4i+2}-\pi)>0 \\ ~~~~ = \pi ~~ \textrm{if} ~~ (\alpha_{4i-3}+\alpha_{4i-2}-\pi)(\alpha_{4i+1}+\alpha_{4i+2}-\pi)<0 \end{gathered}$$ Now there is only one branch of rigid folding motion and only one equation for $\alpha_{4i+1}$ and $\alpha_{4i+2}$. However, it requires $\theta_1=0$ or $\pi$, which means $\Gamma$ should be locally planar in a discrete sense. More essentially, [@fuchs_more_1999] illustrate this in a continuous sense. From Propositions \[prop co-planar\], \[prop: approximation\] and \[prop datum curve\], we are now in a position of studying which surface the parallel repeating type can approximate. ![\[fig: approximation-2\](a) is the creased paper approximating the target surface described in Proposition \[prop: surface\]. The equation of the datum curve is given in the caption of Figure \[fig: approximation\]; $f_1(t)=[t,\exp(t)]$, $t \in [0,1]$. We choose $\theta=73^\circ$. The orange and green inner creases are to approximate the datum curve $\Gamma$ and the target curve $f_1$. The first column of inner vertices from the left approximates the datum curve is from column 3 in Figure \[fig: unit columns\], and the others are from column 2 in Figure \[fig: unit columns\]. The planar creased paper is viewed along the direction perpendicular to the plane. (b) is the final rigidly folded state of (a), where we approximate these two curves, plotted by Freeform Origami [@tachi_freeform_2010-1]. The rigid folding motion halts due to a clash at the column we approximate the target curve. The mountain and valley creases are colored red and blue.](approximation-2){width="0.98\linewidth"} \[prop: surface\] The parallel repeating type can approximate a surface $S: V \ni (u,t) \rightarrow \vec{r}(u,t) \in \mathbb{R}^3$ under a given Hausdorff distance $\epsilon$. The creased paper is generated by the following steps, and $S$ is described below. (see Figure \[fig: approximation-2\]) 1. Choose a datum curve $\Gamma: u \in J \rightarrow \mathbb{R}^3$ and approximate it in a row under a sufficiently small Hausdorff distance $\epsilon_1<\epsilon$, as described in Proposition \[prop datum curve\]. 2. Choose a target curve $f_1: t \in I \rightarrow \mathbb{R}^2$ and approximate it the first column from the left under a sufficiently small Hausdorff distance $\epsilon_2<\epsilon$, as described in Proposition \[prop: approximation\]. 3. In other columns, the shape of the target curve $f_{i+1} : I \ni t \rightarrow (x_{i+1}(t),y_{i+1}(t))$ ($i \in [1,n-1]$) is an affine transformation of $f_1 : I \ni t \rightarrow (x_1(t),y_1(t))$. $$\label{eq: affine 2} \left[ \begin{array}{c} x_{i+1}(t)\\ y_{i+1}(t) \end{array} \right]= A^{-1}(\xi_{i+1},\theta) \prod_{j=1}^{i} \left[ \begin{array}{cc} k_1 & 0 \\ 0 & k_2 \end{array} \right] A(\xi_1,\theta) \left[ \begin{array}{c} x_1(t)\\ y_1(t) \end{array} \right] \\$$ where, $$k_1= \frac{\sin \alpha_{4j-3}}{\sin \alpha_{4j+2}}, \quad k_2= \frac{\sin \alpha_{4j}}{\sin \alpha_{4j+3}}$$ or vice versa. This depends on whether the line segments start in the $\overline{x}$ or $\overline{y}$ direction as shown in Figure \[fig: target curve\](b). Additionally, $$\begin{gathered} A(\xi,\theta)= \left[ \begin{array}{cc} 1 & -1 / \tan \xi \\ 0 & 1 / \sin \xi \end{array} \right] \left[ \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right] \\ \cos \xi_{i+1}=\cos \alpha_{4i+2}\cos \alpha_{4i+3}+\dfrac{\sin \alpha_{4i+2}\sin \alpha_{4i+3}}{\sin \alpha_{4i-3}\sin \alpha_{4i}}(\cos \xi_i - \cos \alpha_{4i-3}\cos \alpha_{4i}) \\ \xi_1=|2\alpha_2-\pi| \end{gathered}$$ $A$ is the affine matrix we used in Proposition \[prop: approximation\], and $\theta$ is the rotation angle in the approximation of $f_1$. We then define the angle between the planes where $f_i$ and $f_{i+1}$ locate by $\phi_i$ ($i \in [1,n-1]$), which can be expressed by $$\label{eq: phi} \begin{gathered} \eta_1=\arccos\bigg(\dfrac{\cos \alpha_{4i}-\cos \alpha_{4i-3} \cos \xi_i}{\sin \alpha_{4i-3} \sin \xi_i}\bigg) \\ \eta_2=\arccos\bigg(\dfrac{\cos \alpha_{4i+3}-\cos \alpha_{4i+2} \cos \xi_{i+1}}{\sin \alpha_{4i+2} \sin \xi_{i+1}}\bigg) \\ \cos \phi_i=- \cos \eta_1 \cos \eta_2-\sin \eta_1\sin \eta_2 \cos (\alpha_{4i-3}+\alpha_{4i+2}) \end{gathered}$$ When the Hausdorff distance $\epsilon \rightarrow 0$, the surface $S$ can be expressed as: $$\label{eq: surface} \vec{r}(u,t)=\Gamma(u)+\widetilde{f}_u(t)$$ where $\widetilde{f}_u$ depends on $u$ (equation ) and locates on different planes determined by $\phi_i$ (equation ). \[cor first special\] If $\Gamma$ is piecewise-spiral, on each piece $\Gamma_j$, $\alpha_{4i+1}$, $\alpha_{4i+2}$, $\alpha_{4i+3}$ and $\alpha_{4i+4}$ ($i \in [1,n-1]$) will keep the same from the second column respectively. Then each piece $S_j$ induced by $\Gamma_j$ can be expressed analytically as $f_1$ scanning along $\Gamma$. Some examples are demonstrated in [@song_design_2017] when $\Gamma$ is circular. Besides, if $f_1$ is piecewise-linear, $f_u$ remains piecewise-linear. Orthodiagonal Type ------------------ The developable case of the orthodiagonal type of rigid-foldable quadrilateral creased papers is generated among several parallel line segments [@he_rigid_2018-1], as shown in Figure \[fig: orthodiagonal creased paper\]. For all $i,j$, The sector angles here should satisfy $$\label{eq: non-stitching} \dfrac{\tan \alpha_{ij}}{\tan \alpha_{ij+1}}=\dfrac{\tan \alpha_{i+1j}}{\tan \alpha_{i+1j+1}}$$ $i \ge 1$, $j \ge 0$. This equation guarantees each column and each row of inner vertices to be co-planar during the rigid folding motion. ![\[fig: orthodiagonal creased paper\](a) is an example of the developable case of the orthodiagonal type, where we label out the relations among the sector angles. Additionally, the sector angles should satisfy equation . (b) is a rigidly folded state of (a), plotted by Freeform Origami [@tachi_freeform_2010-1]. The mountain and valley creases are colored red and blue.](orthodiagonal){width="1\linewidth"} \[prop: datum curve 2\] The inner vertices on a column of the orthodiagonal type can approximate a Darboux-integrable curve $\Gamma: J \rightarrow \mathbb{R}^2$, named a *datum curve*. An approximation can be generated following the steps mentioned below. 1. Under a given Hausdorff distance $\epsilon$, partition $\Gamma$ (black curve in Figure \[fig: straight-line approximation\](a)) by $n+2$ points and connect adjacent partition points in sequence by line segments (orange line segments in Figure \[fig: straight-line approximation\](a)). To make the angle between adjacent inner creases not too close to $\pi$ we choose partition points on an $\epsilon$-tube of $\Gamma$ (purple curves in Figure \[fig: straight-line approximation\](a)). Note that here we use a method different from step 1 in Proposition \[prop datum curve\], and there are many techniques to approximate a Darboux-integrable curve by a series of line segments. 2. Assign a direction of $\Gamma$, label each partition point $A_i$ ($i \in [0,n+1]$) along this direction. From the coordinates of $A_i$, calculate $l_i= \|A_iA_{i+1}\|$ ($i \in [0,n]$) and $\beta_i=\angle A_{i-1}A_iA_{i+1}$ ($i \in [1,n]$). 3. Without loss of generality, we just consider the first column from the left. Calculate all the sector angles on the left $\alpha_{i0}$ ($i \in [1,n]$) from equation , which is to make the rigid folding motion halt at the left side of the first column. $$\label{eq: straight-line alpha 1} \alpha_{i0}=\dfrac{\pi \pm \beta_i}{2}$$ Note that the $\pm$ depends on whether the line segments (coloured orange in Figure \[fig: straight-line approximation\](a)) “turn left” or “turn right” along the direction assigned for $\Gamma$. Only one of $\alpha_{i1}$ can be randomly chosen. Suppose it is $\alpha_{11}$, which should satisfy equation to make sure the inner creases turn left or right properly, and the rigid folding motion halts at the left side. $$\label{eq: straight-line alpha 2} \begin{gathered} (\alpha_{11}-\pi/2)(\alpha_{10}-\pi/2)>0 \\ 0<|\alpha_{11}-\pi/2|<|\alpha_{10}-\pi/2| \end{gathered}$$ the other sector angles $\alpha_{i1}$ ($i \in [2,n]$) can be calculated from equation . 4. With $l_i$ and $\alpha_{i0}$, $\alpha_{i1}$ ($i \in [1,n]$), draw the creased paper. As shown in Remark \[rem: comment 2\], generically, we can require $\Gamma$ to be continuous and $J$ to be a closed interval in $\mathbb{R}$. Besides, $\Gamma$ can be a closed curve. Another point is the approximation of $\Gamma$ does not need to turn left or right alternately, which means the mountain-valley assignment in each row not necessarily change alternately. Then we will analyze which surface the orthodiagonal type can approximate. From equation , we know the sector angles in the first column and first row are independent, therefore we can also approximate a Darboux-integrable curve with the inner vertices in a row. The problem of such approximation is, we cannot write a clear expression for the curve approximated by other rows of inner vertices in this creased paper, which makes it hard to grasp the feature of the target surface. (In step 3 of Proposition \[prop: surface\], we express the curve approximated by other columns of inner vertices as an affine transformation of the target curve.) Based on that we set $\alpha_{1j+1}=\alpha_{1j}$ ($j \ge 1$), and the result in Proposition \[prop: approximation\] can be applied. For the orthodiagonal type, this simplification makes it possible to express the curve approximated by other rows of inner vertices as an affine transformation of the curve approximated by the first row of inner vertices. If we regard the surface approximated by such a simplified creased paper as a piece, the target surface can consist of these pieces stitched in the transverse direction, as shown in Proposition \[prop: approximation 2\]. ![\[fig: straight-line approximation\](a) An example of the datum curve $\Gamma$ (coloured black) with its approximation by line segments (coloured orange) generated from a $\epsilon$-tube of $\Gamma$ (coloured purple). In this example, $n=9$, $\Gamma(u)=[u, \sin u]^T$, $u \in [0,\pi]$, $\alpha_1=\pi/4+\alpha_2/2$. Here we partition $\Gamma$ uniformly in $u$. (b) is the creased paper approximating a piece described in Proposition \[prop: approximation 2\], and $f_1(t)=[t,t-\ln(t)]^T$, $t \in [0.5,1.5]$. We choose $\theta=30^\circ$. The orange and green inner creases are to approximate the datum curve $\Gamma$ and the target curve $f_1$. (c) is the final rigidly folded state of (b), where we approximate these two curves, plotted by Freeform Origami [@tachi_freeform_2010-1]. The rigid folding motion halts due to a clash at the column we approximate the datum curve. The mountain and valley creases are colored red and blue.](straight-line_approximation){width="1\linewidth"} \[prop: approximation 2\] The orthodiagonal type can approximate a surface $S$ which is stitched by countable pieces $S_k: V \ni (u,t) \rightarrow \vec{r}(u,t) \in \mathbb{R}^3$ in the transverse direction under a given Hausdorff distance $\epsilon$. Each piece is generated by the following steps. (see Figure \[fig: straight-line approximation\]) 1. Choose a datum curve $\Gamma: u \in J \rightarrow \mathbb{R}^2$ and approximate it in a column under a sufficiently small Hausdorff distance $\epsilon_1<\epsilon$, as described in Proposition \[prop: datum curve 2\]. 2. Choose a target curve $f_1: t \in I \rightarrow \mathbb{R}^2$ and approximate it in the first row from the top under a sufficiently small Hausdorff distance $\epsilon_2<\epsilon$, as described in Proposition \[prop: approximation\]. 3. Here $\alpha_{1j+1}=\alpha_{1j}$ ($j \ge 1$). With equation , calculate the other sector angles in the creased paper. In other columns, the shape of the target curve $f_i : I \ni t \rightarrow (x_i(t),y_i(t))$ ($i \in [2,n]$) is an affine transformation of $f_1 : I \ni t \rightarrow (x_1(t),y_1(t))$. $$\label{eq: straight-line target curves} \left[ \begin{array}{c} x_i(t)\\ y_i(t) \end{array} \right]= \frac{\sin \alpha_{i1}}{\sin \alpha_{11}} A^{-1}(\xi_i,\theta) A(\xi_1,\theta) \left[ \begin{array}{c} x_1(t)\\ y_1(t) \end{array} \right]$$ where, $$A(\xi,\theta)= \left[ \begin{array}{cc} 1 & -1 / \tan \xi \\ 0 & 1 / \sin \xi \end{array} \right] \left[ \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right]$$ is the affine matrix we used in Proposition \[prop: approximation\], and for $i \in [1,n]$ $$\cos \xi_{i}=\dfrac{4 \cos^2\alpha_{i1}}{1-\cos\beta_i}-1$$ $\theta$ is the rotation angle in the approximation of $f_1$. $f_i$ locates on a plane perpendicular to $\Gamma$. When the Hausdorff distance $\epsilon \rightarrow 0$, the piece $S_k$ can be expressed as: $$\vec{r}(u,t)=\Gamma(u)+\widetilde{f}_u(t)$$ where $\widetilde{f}_u$ depends on $u$ (equation ) and locates on the tangent plane of $\Gamma$ at $u$ that is also perpendicular to $\Gamma$. \[cor second special\] If $\Gamma$ is piecewise-circular, on each piece $\Gamma_j$, $\alpha_{i0}$ and $\alpha_{i1}$ ($i \in [1,n]$) will keep the same respectively. Then each piece $S_j$ induced by $\Gamma_j$ can be expressed analytically as $f_1$ scanning along $\Gamma_j$. Besides, if $f_1$ is piecewise-linear, $f_u$ remains piecewise-linear, then each piece induced by $f_1$ will become a cylinderical developable surface. Discussion ========== Comment on the Algorithms ------------------------- This article presents some initial results for the problem we have set. We recognize there are flaws and limitations in the algorithms, some of which are discussed here. 1. In Proposition \[prop: approximation\], for a given target curve $f$ and an angle $\xi$, the condition we give to find an appropriate $\theta$ is not convenient. For the examples in this article we try scattered $\theta \in [0,2\pi)$ to find possible approximations. 2. In Proposition \[prop datum curve\], the folding angle $\rho_4$ is a flexible parameter, that can be used to adjust the sector angles $\alpha_1$ and $\alpha_2$. Actually, $\rho_4$ is the magnitude of all the folding angles on every row of inner creases. If we set $\rho_4$ too close to $\pi$, $\alpha_1$ will increase, but the width of approximation will be too small in the longitudinal direction, which may not be suitable for application. If we set $\rho_4$ too far from $\pi$, the singularity of solutions will increase. For the approximation shown in Figures \[fig: approximation\](c) and \[fig: approximation\](d) we choose $\rho_4=5\pi/6$. 3. The algorithm in Proposition \[prop datum curve\] does not guarantee a solution. We need to assume the rigid folding motion and choose the signs that make equation most likely to be solved. If it happens that the parameters are on the singular points of equations and , there may be no solution. 4. Even if we have obtained all the sector angles, when we plot the creased paper as described in Propositions \[prop: surface\] and \[prop: approximation 2\], the inner creases in different columns may intersect at points other than vertices. We haven’t found a good way to control this. Sometimes scaling the datum or target curve will help. 5. In Proposition \[prop: datum curve 2\], the sector angle $\alpha_1$ is also a flexible parameter, which can be used to control the width of the approximation. There is no singularity in the algorithm proposed here. Comment on the Approximation ---------------------------- Apart from the algorithms, we want to mention a few points related to an approximation for a surface. 1. In Propositions \[prop: surface\] and \[prop: approximation 2\], when the Hausdorff distance $\epsilon \rightarrow 0$, we haven’t found a good way to express the target surface $S$ analytically, even though it is determined by the datum curve $\Gamma$, target curve $f_1$, and a folding angle $\rho_4$ (Proposition \[prop: surface\]) or a sector angle $\alpha_1$ (Proposition \[prop: approximation 2\]). We only give analytical expressions for special cases mentioned in Corollaries \[cor first special\] and \[cor second special\]. However, an analytical solution for how to approximate a curve is given in [@tachi_composite_2013], which might be helpful in solving this problem. 2. It is not necessary to make the rigid folding motion halt at the first column from the left. Such halting columns can be inserted to the creased paper arbitrarily. Besides, the two examples shown in sections 2.1 and 2.2 design the halting column differently, and there are many other possible techniques. 3. The utilization of materials of our design depends on the proportion of halting columns with respected to the whole creased paper, which is at a relatively high level. 4. If there are some holes on the target surfaces in Propositions \[prop: surface\] and \[prop: approximation 2\], we can follow the algorithms described in this article and generate the approximation by applying kirigami on the creased papers. 5. In this article we show that a series of developable surfaces can approximate a non-developable surface, which means the collection of all developable surfaces is not a closed set. 6. Another possible design example is a quadrilateral creased paper with no inner panel [@he_rigid_2018-1], which can approximate more surfaces but will have some big “cracks” almost running through the final rigidly folded state. We will include the discussion on it in a future article. Optimization ------------ Optimizations can be applied to the algorithms mentioned in this article, which will lead to future work. 1. In Propositions \[prop: approximation\], \[prop datum curve\] and \[prop: datum curve 2\], the piecewise-linear approximation do not need to be uniformly-spaced. Furthermore, the partition points are not necessarily on the curve. Given the number of partition points, there are some classic methods to approximate the datum and target curves, which will improve the accuracy of approximation. 2. There are some other criteria for an approximation in Propositions \[prop: approximation\], \[prop datum curve\] and \[prop: datum curve 2\] under a given Hausdorff distance $\epsilon$, such as minimizing the number of inner vertices, the total bending energy [@solomon_flexible_2012]; or restricting the minimum length of all the creases, the minimum of sector angles, etc, which will involve more complex calculations. Conclusion ========== We have shown that it is possible to approximate some types of non-developable surfaces with a 1-DOF rigid folding motion starting from a planar creased paper. This might have useful engineering applications, for instance as a way of forming a shell structure in 3-dimensional space. However, given an arbitrary surface, the methods that can be used to generate an approximation with a planar creased paper that has limited DOFs are not fully understood. Acknowledgment ============== We thank Tom Aldridge for some preliminary works on this topic, and Hanxiao Cui for helpful discussions on Proposition \[prop: approximation\]. This paper has been awarded the 7OSME Gabriella & Paul Rosenbaum Foundation Travel Award.
--- abstract: 'We show the merits of plasma enhanced atomic layer deposition (PEALD) of catalytic substrate for chemical vapour deposition (CVD) graphene growth. The high quality multilayer graphene (MLG) on molybdenum carbide ($MoC_{x}$) thin film exhibits excellent uniformity and layer homogeneity over a large area. Moreover, we demonstrate how to achieve control of graphene layers thickness and properties, by varying the specific catalytic film chemical and physical properties. The control of growth is not digital, but is broad ranged from few layer graphene to a graphitic film of $\sim{75}$ graphene layers grown on the respective ALD catalytic substrates. Characterisation of the MLG has been performed using Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), spectral ellipsometry (SE), and scanning low-energy electron microscopy (SLEEM). By varying MLG thickness in a uniform homogeneous way, we can tailor the desired MLG properties for different application needs. Furthermore, the PEALD process can be readily adapted to high volume manufacturing processes, and combined with existing production lines.' author: - Eldad Grady - 'W.M.M. Kessels' - 'Ageeth A. Bol' bibliography: - 'MLG.bib' title: Control of Graphene Layer Thickness Grown on Plasma Enhanced Atomic Layer Deposition of Molybdenum Carbide --- Introduction ============ Graphene, a two dimensional sheet of carbon atoms, has attracted great interest in research because of its unique properties. A vast field of applications could utilise graphene to realise the next generation technology [@el2016graphene; @bianco2018carbon]. Consequently, the ability to fabricate this material in a uniform scalable method and integrate into applications is the main challenge for both industry and academia. Various techniques have been explored to prepare graphene sheets, such as liquid phase exfoliation [@hernandez2008high], epitaxy on SiC [@sutter2009epitaxial] and chemical vapour deposition (CVD) [@mattevi2011review]. Among these, CVD is the most promising for industrial scale production of high quality graphene. Graphene can be grown by CVD on a various of catalytic metallic substrates, including Cu, Co, Ni to name a few. CVD growth on Cu allows for growth of mostly single layer graphene sheets, but Cu grain size induce a polycrystalline graphene growth with defects on the graphene grain boundaries. Cu foils are commonly used for growth, and graphene defects form along the Cu railing sites. Sputtered Cu thin film can mitigate Cu foil related defects, but the thin film has its limitations as well [@lee2015suppression; @tao2012synthesis]. Graphene growth is performed close to the melting point of Cu causing mass loss of Cu and subsequently holes in the film due to dewetting. The difference between the thermal expansion in the Cu itself and the substrate cause for strain which leads to a wrinkly graphene sheet. Ni is commonly used for growth of multilayer graphene (MLG), due to the high solubility of carbon in Ni. While MLG grown on Ni is typically not uniform due to island like growth around the Ni grain boundaries, significant advancements have been made [@zakar2011nucleation; @rameshan2018role]. Recently, a scalable and uniform growth of MLG has been demonstrated on Mo thin films deposited by physical vapour deposition (PVD) [@GRACHOVA20141501; @Ricciardella_2019]. Mo has the advantage of a higher melting point than Cu, which is vital to inhibit mass loss during growth. Furthermore, a similar thermal expansion coefficient to Si (4.8 and 2.6 $\mu{m}\cdot m^{-1}\cdot K^{-1}$ ) as oppose to Cu (17 $\mu{m}\cdot m^{-1}\cdot K^{-1}$), that reduces stress during growth and thus wrinkles in the graphene layers. Mo has also the advantage of being a clean room compatible material commonly used in IC manufacturing environment. However, PVD Mo thin films still suffer from from limitations related to the deposition technique. For the next generation applications, requirements for film uniformity, coverage and homogeneity are very strict, and PVD has reached its limitations in a high aspect ratio (HAR) topologies. As PVD utilises highly kinetic ion bombardment to deposit the film, it is not suitable for coating MLG if a stack layer is required, as it will severely damage the MLG. Finally, impurities in sputtered Mo will also contribute to defects in the grown MLG. In order to answer the vast demands of graphene applications, such as high current density and specific capacitance, and to tailor them to specific application’s needs, a control of graphene layers would be of high value. Extensive work on control of graphene layer numbers have been performed in recent years. Liu et. al. have shown that $H_2$ concentration and oxide nanoparticles influence the number of layers grown on Cu foil [@liu2015controllable]. However, the control was not uniform and varied within the graphene domain. Limbu et. al. [@limbu2018novel] have demonstrated layer control during hot fillament CVD growth on Cu (111) surface. However, the surface reaction growth is time dependent, and can take up to 200 mins to achieve few layer graphene. The first graphene adlayer is approximately 4 times longer than the first graphene layer, due to the lower adsorption energy of carbon on graphene than on Cu (111). In this work we use atomic layer deposition techniques to fabricate the catalytic substrate for CVD graphene growth. Atomic layer deposition (ALD) allows for sub nanometre thickness control suitable for the strictest application demands to date. Additionally, we are able to tune the film composition and properties with high accuracy and reproducibility [@grady2019tailored]. By employing ALD we gain an additional merit of research, that allows us to study the effects and correlations between the two in a manner that is not possible with other deposition techniques mentioned above. As a result, we are able to present a controlled graphene layer thickness with a standard CVD growth process, by means of a bulk induced layer growth. The MLG grown were on molybdenum carbide ($MoC_{x}$) substrate that we fabricated by plasma enhanced atomic layer deposition (PEALD) [@grady2019tailored]. We refer to these samples as ALD $MoC_{x}$ films. Thus, high quality MLG were grown on large area, with excellent uniformity and homogeneity in CVD graphene fabrication. In the characterisation section we compare graphene grown on various types of $MoC_{x}$ films. An overview of results is presented, depicting the characterisation and a comparison of the grown MLG. The technique is scalable for large volume manufacturing and is compatible with IC and sensors fabrication environment. Experimental methods ==================== $MoC_{x}$ thin films have been deposited by plasma enhanced atomic layer deposition (PEALD) at various temperatures and plasma conditions. PEALD was performed on 100 mm Si (100) wafers coated with 450 nm of thermally grown $SiO_2$. The depositions were performed in an Oxford instruments FlexAL2 ALD reactor, which is equipped with an inductively coupled remote RF plasma (ICP) source (13.56 MHz) with alumina dielectric tube. $MoC_{x}$ thin films have been deposited by PEALD at various temperatures and plasma conditions, with $MoC_{x}$ films varying from 15$\mu{m}$ to 30$\mu{m}$ in thickness. We define 4 generic types of $MoC_{x}$ films by their physical and chemical properties. Type A is defined as mostly amorphous film with low mass density and high carbon content (typically $C/Mo$ ratio $>0.9$). Type B has lower C/Mo ratio and higher mass density, amorphous film embedded with crystalline islands. Type C has high mass density ($\sim{9}$) low C/Mo ratio ($\sim{0.6}$) and is mostly amorphous. Type D films are highly crystalline cubic $\delta-MoC_{0.75}$, with high mass density. MLG was grown by low-pressure CVD (LPCVD) in a quartz tube (d=50mm, l=60cm) furnace with 3 heat zones set to 1100$^\circ$C. The typical base pressure when evacuated is $10^{-3}$ mbar. The furnace is set on cart wheels, to allow samples to be rapid annealed, as furnace temperature stabilises within 3.5 minutes after tube insertion. When moved away from the furnace, sample cooling down duration is typically 15 minutes. Carbon feedstock gas ($CH_4$) is fed along with Argon through a quartz inner tube of 5 mm in diameter to the sealed side of the outer tube. Graphene films have been grown using CVD with $CH_4/Ar$ gas flow at 1100$^\circ$C after carbon saturation at a lower temperature. More details can be found elsewhere [@grady_eldad_2019_3542084]. Characterisation of Graphene Films ================================== We studied the CVD grown MLG films, comparing graphene quality, uniformity and homogeneity. Characterisation of the MLG has been performed using a Reinshaw InVia 514 nm laser Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), spectral ellipsometry (SE) and scanning low-energy electron microscopy (SLEEM). The thermal stability of $MoC_{x}$ ALD film was very high, and no delamination or mass loss was noticed for all measured film thicknesses. The experiments presented here have been performed on $\sim{25}\,\mu{m}$ thick $MoC_{x}$ films. A comparative analysis has been performed in order to understand the correlation between the catalytic substrate properties and the subsequent graphene layers grown on top. Using PEALD $MoC_{x}$, we are able to control the characteristics $MoC_{x}$ film with excellent uniformity, by altering the $MoC_{x}$ film mass density and crystallinity as demonstrated in [@grady2019tailored]. Raman Spectroscopy ------------------ Raman spectroscopy is a non-intrusive measurement which indicates graphene quality by the characteristic bands of graphene at 1350 $cm^{-1}$ (D peak) 1580 $cm^{-1}$ (G peak) and 2700 $cm^{-1}$ (2D peak). In short, the D peak is related to disorder in the graphene lattice and considered to indicate defects in the graphene layers. The ratio between 2D and G peak indicates the nature of the graphene, with MLG ranges between 0.4-0.8 2D/G ratio, while single layer graphene (SLG) ratio is typically $>1$. The FWHM for SLG is 30$cm^{-1}$ and for MLG ranges between 60 - 120 $cm^{-1}$ . Raman mapping scans were also used to characterise a large area by scanning multiple adjacent spots. We have analysed graphene samples on a large scale with non- intrusive methods, to gain statistical data on uniformity, homogeneity and defects of our graphene samples. To the best of our knowledge, this is the first time graphene quality has been quantified statistically by analysing single point Raman diagnostics of characteristic graphene spectra to produce an overview of the entire scanned sheet attributes. We measured areas of 30x30 $\mu{m}^2$ with 1 $\mu{m}$ steps (laser spot resolution limit), to derive information on the MLG uniformity and defects variations. To do so, the 2D/G and D/G ratios were calculated from $\sim$1000 scans for each area, for statistical analysis. Variations in 2D/G are a good indicator for uniformity of MLG layers. Both variation and value of D/G ratios are an indicator to amount of defects and uniformity in MLG layers. We show the results both in form of colour maps to visualise uniformity variations and as density distribution functions, to quantify these variations. The covariance of each dataset indicates the uniformity and homogeneity of the graphene layers. As seen in figure \[fig:2D-G ALD\] the uniformity of MLG grown on $MoC_{x}$ ALD at 350$^{\circ}C$ is high, with typical ratio of MLG as well as FWHM of 2D peak. The narrow distribution function shows high MLG uniformity with 2D/G ratio at about 0.9 and variance of $1.5\cdot10^{-2}$. Similarly, MLG grown on $MoC_{x}$ ALD film fabricated at low deposition temperature exhibits high uniformity and low defects ratio \[see table \[fig:xps\]\]. As can be seen in figure \[fig:D-G ALD\], MLG grown on ALD film show lower average D/G peak ratios, indicating the overall high quality and uniformity, as confirmed by the low D/G covariance. X-ray photoelectron spectroscopy -------------------------------- The film composition was analysed by X-ray photoelectron spectroscopy (XPS) with a Thermo Scientific KA1066 spectrometer, using monochromatic Al $K\alpha$ x-rays with an energy of 1486.6 eV. The films were sputtered with $Ar^{+}$ ion gun prior to scans, in order to remove surface oxide and adventitious carbon. A continuous electron flood gun was employed during measurements to compensate for charging. XPS is surface-sensitive quantitative spectroscopic technique, that allows for analysis of a substrate surface chemical composition. We compare MLG grown on various types of $MoC_{x}$ films with C1s core level spectra and reference it to graphene grown on Cu thin film. MLG grown on ALD films of type A shows pure carbon presence at the surface (within error margins) and no trace of the catalytic substrate or oxygen. The peak position confirms a $sp^{2}$ carbon bonds of MLG. As can be shown in figure \[fig:XPS\_C1s\_ALD\], the MLG has high purity $sp^2$ bonding with main contributing peak located at 284.4 eV. Due to the interaction of the photoelectron with free electrons in the film, a corresponding plasmon loss feature is observed at $\sim{290.4}$eV signifying the graphene sheet conductivity. We can further examine the bulk of the material by etching with $Ar^+$ ion gun to perform depth profiling and analyse the elemental composition and the nature of the chemical bondings of each layer. A comparison of MLG grown on various types of $MoC_{x}$ ALD films is performed by etching 100 levels (2000 seconds). As can be shown in figure \[fig:XPS-controlofmlgthickness1\], the MLG varies in thickness between different type of catalytic substrates, as indicated by diminishing C1s peak intensities [@xu2010auger]. The diminished peak intensities are accompanied with the rise of Mo peak, which indicates a transition to the underlying catalytic substrate. The thinnest graphene sample was measured on type C $MoC_{x}$, with only 1 level with typical $sp^{2}$ carbon peak and $\sim{65} \,at.\%$ carbon on the surface, matching typical values measured for graphene grown on Cu thin film. The thickest MLG layers have shown a consistent $sp^{2}$ carbon peak for all 100 etched levels, with no indication of any other element peak. Carbon content remained stable around $\sim{98}\, at.\%$ as is shown in figure \[XPS-Thick\_film\]. Film Thickness Characterisation ------------------------------- ### ultra-low energy SEM/STEM of graphene The electron transmissivity at low energy (up to tens of eV) has proven a reliable tool for counting graphene layers as an alternative to Raman spectroscopy, providing an enhanced lateral resolution. Graphene layers exhibit contrasts connected with electron reflectivity fluctuations below 8 eV and in an additional band around 15 eV in the ultra-low-energy SEM. This phenomenon can also be employed for counting the graphene layers with a correlation of n-1 minima reflectivity for n graphene layers. Studies of graphene layers number and homogeneity were performed, as shown in figure \[fig:VASP\] shows simulation of the fluctuations as a function of graphene layers as was simulated in VASP. Figure \[fig:number-of-layers\] shows measured fluctuations on ALD $MoC_{x}$ MLG sample corresponding to 6 layers of graphene (5 minima). Measurements performed concluded high layer homogeneity across the sample. ### Spectral ellipsometry Film thickness and optical properties of the deposited films have been studied with a J.A. Woollam UV-spectroscopic ellipsometer (SE). Data was obtained within the range of 190 nm 990 nm, and refractive index (n) and extinction coefficient (k) were determined. Figure 7 shows an extract of the layers thickness and composition, as modelled in CompleteEASE software. Relevant selected oscillators were chosen to achieve optimal fitting of the data [@grady2019tailored].\ For a type A $MoC_{x}$ sample of $25\pm{1}$ nm thick film, we estimated an equivalent $25\pm{1}$ thick MLG film grown on top. The MLG thickness corresponds to approximately 75 graphene layers, as was confirmed with XPS depth profiling matching all etch levels to $sp^{2}$ carbon peak positions. Discussion and Conclusions ========================== We have demonstrated the effects of the catalytic substrate physical and chemical properties on the CVD grown MLG properties. The results presented have been collected over a time span of 18 months, with excellent reproducibility. While ALD reactor conditions may vary, the correlation between the ALD film and the CVD grown graphene remained identical. With the ability to precisely control PEALD $MoC_{x}$ composition, we can control the grown MLG film ontop. $MoC_{x}$ film with low density, low crystallinity and high carbon content (type A) are readily able to saturate and precipitate free carbon content required to form $sp^{2}$ carbon bonds on the surface, as oppose to crystalline $MoC_{x}$ films with high mass density (type D). Moreover, these $MoC_{x}$ films will yield MLG with significantly lower defects. $MoC_{x}$ mass density has shown to have a reverse proportion to MLG thickness, and a good correlation to defects ratio in MLG film, as Type C films have shown to inhibit graphene growth. The high crystalline type D film has shown to have high 2D/G ratio typical to FLG and a higher D/G ratio. The defects are likely occuring as the $MoC_{x}$ film transition from a cubic $MoC_{0.75}$ phase to orthorhombic phase during the high temperature MLG growth process [@grady_eldad_2019_3542084].Type A films with the lowest mass density and highest C/Mo ratio have produced the thickest MLG layers. A film of $\sim{25}$ nm $MoC_{x}$ produced over 75 graphene layers, as was estimated by UV-SE. Type C high denisty film with low C/Mo ratio yielded the lowest number of MLG layers. We could accurately determine 6 graphene layers using SLEEM measurements grown on ALD $MoC_{x}$. Nevertheless, we stipulate thinner FLG could be achieved, as the Raman spectrum of type D graphene suggests. More work needs to be done in fine tuning MLG thickness by a study of the critical regimes of plasma time exposures, and more parameters should be looked at, specifically substrate thickness and crystallinity. In addition, other catalytic substrates should be evaluated, both as homogeneous elements and alloys, for suitability [@guo2018atomic]. By profiling the chemical composition of 100 etch levels, we could confirm our stipulation of MLG thickness control, with proportional increase between graphene film thickness and initial $MoC_{x}$ film C/Mo ratio and inverse to mass density. This control can be achieved very accurately using PEALD film, which allows for reproducible and scalable fabrication. Furthermore, the catalytic substrate purity, homogeneity and uniformity plays an important role to the same properties of the graphene grown on top, namely MLG defects, homogeneity and uniformity. Therefore, a uniform homogeneous film is vital in order to optimise MLG quality in a uniform manner. In this regard, ALD catalytic substrates would have an advantage over other commercial deposition techniques to achieve an overall higher quality film. In addition, the outstanding step coverage makes ALD a tool of choice if 3D objects are of interest. By varying MLG thickness in a uniform homogeneous way, we can tailor the desired MLG properties for different application needs. Furthermore, the process is reproducible, and suitable for implementation in IC and sensors fabrication environments. The PEALD process can be readily adapted to high volume manufacturing processes, and combined with existing production lines. The MLG film grown on $MoC_{x}$ ALD film demonstrated here are of excellent quality and uniformity. The technique presented here has a significant advantage to other commercial CVD graphene fabrication techniques known thus far when scalability and compatibility to IC fabrication environment are the decisive factors. The rapid advancement in ALD tools and techniques such as role to role, batch production, and selective area PEALD promise a leap forward in advancing commercial high quality graphene production. These can be combined to achieve graphene devices with atomic resolution and mitigate current edge placement errors in manufacturing by a complete bottom up process. By utilising recent developments in the field of ALD, the path to next generation electronics based on high quality graphene is opened. Acknowledgements {#acknowledgements .unnumbered} ---------------- This research is supported by the Dutch Technology Foundation STW (project number 140930), which is part of the Netherlands Organization for Scientific Research (NWO), and partly funded by the Ministry of Economic Affairs as well as ASML and ZEISS. Eliska Mikmekova is acknowledged for SLEEM measurements and analysis of the data. E. Grady thanks Cristian Helvoirt, Janneke Zeegbregts, Jeroen van Gerwen and the lab technical staff for their support. Figures ======= [|&gt;m [1.5cm]{}|&gt;m [1cm]{}|&gt;m [1cm]{}|&gt;m [1cm]{}|&gt;m [1cm]{}|&gt;m [1.5cm]{}|&gt;m [1cm]{}|&gt;m [1.5cm]{}|]{} & & ------------------------------------------------------------------------ Temp.($^{\circ}$C) & \[C\] (at%)& \[O\] (at%)& \[Mo\] (at%) & 2D/G& 2D/G variance & D/G& D/G variance ------------------------------------------------------------------------ High temp. & 97.9 & $< d.l. $& $< d.l. $ &0.96 & $1.5\cdot10^{-2}$ & 0.19& $7\cdot10{-4}$ ------------------------------------------------------------------------ Low temp. & 98.5 & $< d.l. $ & $< d.l. $ & 0.91& $2\cdot10^{-2}$ & 0.31& $1.2\cdot10^{-3}$ [0.5]{} ![Raman spectra of MLG grown on 25 $\mu{m}$ thick $MoC_{x}$ ALD film. (a) Top: single Raman scan of the graphene. Bottom: Raman spectra of peaks ratio measured by a mapping scan of 30 × 30$\mu{m}^2$ area with 1$\mu{m}$ step resolution. (b) Left: 2D/G peak ratio of MLG film. Variation in colour is inversely proportional to MLG uniformity. (c) Right: D/G peak ratio of MLG shows low D/G peak ratio with high uniformity.[]{data-label="fig:Raman map"}](MLG_Raman_single_350C.png "fig:"){width="\textwidth"} [0.5]{} ![Raman spectra of MLG grown on 25 $\mu{m}$ thick $MoC_{x}$ ALD film. (a) Top: single Raman scan of the graphene. Bottom: Raman spectra of peaks ratio measured by a mapping scan of 30 × 30$\mu{m}^2$ area with 1$\mu{m}$ step resolution. (b) Left: 2D/G peak ratio of MLG film. Variation in colour is inversely proportional to MLG uniformity. (c) Right: D/G peak ratio of MLG shows low D/G peak ratio with high uniformity.[]{data-label="fig:Raman map"}](2D_G_peak_ratio_2h_800c_10min_gg.jpg "fig:"){width="\textwidth"} [0.5]{} ![Raman spectra of MLG grown on 25 $\mu{m}$ thick $MoC_{x}$ ALD film. (a) Top: single Raman scan of the graphene. Bottom: Raman spectra of peaks ratio measured by a mapping scan of 30 × 30$\mu{m}^2$ area with 1$\mu{m}$ step resolution. (b) Left: 2D/G peak ratio of MLG film. Variation in colour is inversely proportional to MLG uniformity. (c) Right: D/G peak ratio of MLG shows low D/G peak ratio with high uniformity.[]{data-label="fig:Raman map"}](30C_20s_2017_03_D_G_ratio_map.png "fig:"){width="\textwidth"} [0.5]{} ![Distribution function of MLG Raman peak ratios, based on $\sim{1000}$ scanning points per function. Narrower distribution is generally attributed to higher uniformity of MLG. A narrower distribution of the 2D/G ratio indicates higher homogeneity of the MLG.[]{data-label="fig:Ratio_distribution"}](20s_80s_100V_bias_comparison_2D_G_ratio_paper_new.png "fig:"){width="\textwidth"} [0.5]{} ![Distribution function of MLG Raman peak ratios, based on $\sim{1000}$ scanning points per function. Narrower distribution is generally attributed to higher uniformity of MLG. A narrower distribution of the 2D/G ratio indicates higher homogeneity of the MLG.[]{data-label="fig:Ratio_distribution"}](20s_80s_100V_bias_comparison_D_G_ratio_paper-new.png "fig:"){width="\textwidth"} [0.5]{} ![XPS spectra of C1s core level. (a) Left: XPS C 1s spectrum of single layer graphene (SLG) on Cu PVD thin film.(b) Right: C1s scan of MLG on ALD $MoC_{x}$. A typical asymmetric $sp^{2}$ peak at 284.4 eV indicate high purity graphene material. Plasmon loss features at $\sim{290.5}$ and $\sim{293}$ are caused by interaction of the photoelectron with free electrons present in the graphene sheet. No sign of carbidic metal from catalytic substrate. []{data-label="fig:XPS_C1s"}](Cu_C1s_peaks.png "fig:"){width="\textwidth"} [0.5]{} ![XPS spectra of C1s core level. (a) Left: XPS C 1s spectrum of single layer graphene (SLG) on Cu PVD thin film.(b) Right: C1s scan of MLG on ALD $MoC_{x}$. A typical asymmetric $sp^{2}$ peak at 284.4 eV indicate high purity graphene material. Plasmon loss features at $\sim{290.5}$ and $\sim{293}$ are caused by interaction of the photoelectron with free electrons present in the graphene sheet. No sign of carbidic metal from catalytic substrate. []{data-label="fig:XPS_C1s"}](ALD_C1s_peaks.jpg "fig:"){width="\textwidth"} [0.5]{} ![XPS depth profiling of MLG on ALD catalytic substrate. (a) High purity $sp^{2}$ carbon content is shown in top figure for the entire 100 etch levels (20s etch time) for MLG grown on type A substrate. No impurities are measured within the XPS error range ($\pm{2}\,at.\%$). (b) C 1s spectra of samples deposited at various conditions, demonstrating the control of MLG layer thickness. Type C substrate inhibits MLG growth, with merely 65% surface $sp^{2}$ carbon content, typical to $\leq{1}$ graphene layer.[]{data-label="fig:XPS-controlofmlgthickness1"}](MLG_350C_20s_pl_paper.png "fig:"){width="\textwidth"} [0.5]{} ![XPS depth profiling of MLG on ALD catalytic substrate. (a) High purity $sp^{2}$ carbon content is shown in top figure for the entire 100 etch levels (20s etch time) for MLG grown on type A substrate. No impurities are measured within the XPS error range ($\pm{2}\,at.\%$). (b) C 1s spectra of samples deposited at various conditions, demonstrating the control of MLG layer thickness. Type C substrate inhibits MLG growth, with merely 65% surface $sp^{2}$ carbon content, typical to $\leq{1}$ graphene layer.[]{data-label="fig:XPS-controlofmlgthickness1"}](control_of_mlg_thickness_1.png "fig:"){width="\textwidth"} [0.9]{} ![Count of graphene layers by scanning low energy electron microscopy. Measurements of electron reflectivity in MLG/$MoC_{x}$ ALD sample at two energy bands. Simulations and measurements performed by Eliska Mikmekova (The Czech Academy of Sciences)[]{data-label="fig:SLEEM"}](LEEM_DFT_sim_VASP.png "fig:"){width="\textwidth"} [0.5]{} ![Count of graphene layers by scanning low energy electron microscopy. Measurements of electron reflectivity in MLG/$MoC_{x}$ ALD sample at two energy bands. Simulations and measurements performed by Eliska Mikmekova (The Czech Academy of Sciences)[]{data-label="fig:SLEEM"}](LEEM_1.png "fig:"){width="\textwidth"} [0.5]{} ![Count of graphene layers by scanning low energy electron microscopy. Measurements of electron reflectivity in MLG/$MoC_{x}$ ALD sample at two energy bands. Simulations and measurements performed by Eliska Mikmekova (The Czech Academy of Sciences)[]{data-label="fig:SLEEM"}](LEEM_2.png "fig:"){width="\textwidth"} ![Overview of ALD $MoC_{x}$ temperature of deposition effects on graphene growth and properties. The effects of $MoC_{x}$ film on MLG are presented by comparison of D/G peaks as measurement with Raman mapping, % of graphene as determined by $sp^{2}$ peak positions in each etch layer, and % of carbon at the surface of the MLG that relates to the $sp^{2}$ peak position.[]{data-label="fig:overview-ald"}](graphene_lvls_Defects_after_gg.png){width="0.9\linewidth"}
Introduction ============ Since the discovery of high temperature superconductors, the origin of the pairing mechanism has been controversial. Numerous studies suggest that the normal state properties are anomalous and, as a consequence, above the critical temperature, ${ T_c}$, the strongly correlated character of the conduction electrons in the cuprates should be taken into account in any realistic study. In particular, the presence of short-range antiferromagnetic fluctuations have been invoked by several groups as the origin of the deviations from a conventional Fermi liquid behavior observed in transport measurements above $T_c$. Theories based on antiferromagnetic fluctuations quite clearly predict a superconducting condensate in the $d_{x^2-y^2}$ channel.[@AF-potential; @review] These studies are supported by experimental results for the bilayer cuprate ${\rm YBa_2Cu_3O_{7-\delta}}$, which are compatible with a highly anisotropic pairing state, likely a ${ d_{x^2-y^2}}$ singlet.[@gap-symmetry] Photoemission results for Bi-2212, which also have two $CuO_2$ layers per unit cell, support this scenario,[@ARPES] and thus superconductivity induced by AF correlations is a strong candidate to explain the pairing mechanism of the cuprates. The authors and collaborators have recently explored these ideas using simple models for quasiparticles (q.p.) considered as holes strongly dressed by AF fluctuations, and interacting through a nearest-neighbor (NN) density-density attraction, as suggested by the study of the two-holes bound-state of the $t-J$ model.[@afvh] The presence of a large accumulation of weight in the density of states, induced by the small bandwidth of the q.p. dispersion, enhances $T_c$ beyond what would be naively expected from the strength of the hole-hole attraction. These ideas were originally referred to as the “antiferromagnetic van Hove” (AFVH) scenario, but it is important to remark that the boost in $T_c$ does not arise exclusively from a van Hove singularity but mainly from the “flat” regions observed in the q.p. dispersion, effect which is induced by AF correlations.[@flat-theory; @other-flat] The purpose of this paper is (i) the study of the properties of holes and (ii) the extension of previous analysis for the symmetry of the superconducting order parameter to the case of lightly doped bilayers and 3D antiferromagnets. The effective Hamiltonian introduced in previous publications for one plane[@afvh] is defined on the 2D square lattice as $$H = \sum_{{\bf k}\alpha} \epsilon_{AF}({\bf k}) c^\dagger_{{\bf k}\alpha} c_{{\bf k}\alpha} - V \sum_{\langle {\bf ij} \rangle } n_{\bf i} n_{\bf j}, \eqno(1)$$ where $c_{{\bf k}\alpha}$ denotes a destruction operator of a q.p. with dispersion extracted from accurate numerical and analytical studies of one hole in the undoped $t-J$ model given by $\epsilon_{AF}({\bf k})/eV=0.165\cos k_x\cos k_y\, +\,0.0435(\cos 2k_x+\cos 2k_y)$, where $t=0.4eV$ and $J/t = 0.3$ are assumed.[@flat-theory] The quasiparticles move within the same sublattice to avoid distorting the AF background ($\alpha=A,B$ denotes the sublattice). This is correct in the limit of a small number of holes in a long-range ordered AF system, and it is expected to be a good approximation even with AF short-range order, as long as the AF correlations are strong. The dispersion $\epsilon_{AF}({\bf k})$ reflects a remarkable feature of the cuprates, observed in photoemission experiments, namely the presence of flat bands near $(0,\pi)$ $(\pi,0)$ in the standard square lattice notation. [@flat-theory; @other-flat] The parameter $V$ is the intersublattice density-density attractive coupling strength between holes also discussed in previous literature,[@afvh] with $n_{\bf i}$ the number operator. The rest of the notation is standard. The interaction in Eq.(1) should be considered as the dominant piece of a more general and extended AF-induced effective potential between holes which in the presence of long-range order corresponds to a sharp $\delta$-function centered at ${\bf Q}=(\pi,\pi)$ in momentum space, and it acquires a width as the AF correlation length $\xi_{AF}$ decreases. Even with $\xi_{AF}$ as small as a couple of lattice spacings, it has been shown that the NN interaction used in Eq.(1) remains robust.[@dos] Equivalently, the NN hole-hole attraction can be considered as arising from the “minimization of broken AF links” argument[@review] in the large $J/t$ limit. It is expected that Hamiltonian Eq.(1), which is certainly a very simplified version of the low energy behavior of the $t-J$ model, nevertheless captures the important features of holes moving in AF backgrounds at low doping, and temperatures smaller than $J$ where the quasiparticles are dominant. This is a regime difficult to study with numerical methods directly applied to the $t-J$ model. The analysis of Eq.(1) using standard BCS techniques, Exact Diagonalization approaches, and the Eliashberg equations, has been fairly successful in reproducing some experimental features of the cuprates.[@afvh; @naza1] For example, in this model superconductivity appears in the $d_{x^2-y^2}$ channel, as it seems to occur in experiments, with a critical temperature $T_c \sim 100 K$. In addition, the concept of “optimal doping” appears naturally due to the robust peak in the density of states (DOS) produced by the dispersion $\epsilon_{AF}({\bf k})$, which is crossed by the chemical potential as the hole density grows. In this respect, the theory has features very similar to those discussed before in “van Hove” theories for the cuprates.[@vh] Indeed, the quasiparticle lifetime is linear with energy at optimal doping. However, the flat regions in the dispersion and the associated accumulation of weight in the DOS are produced by strong correlations and thus they are stable against small perturbations, like impurities and extra couplings, effects that usually destroy weak logarithmic van Hove singularities. The specific heat jump, $2\Delta/T_c$ and other important BCS ratios are in good agreement with the experimental data.[@afvh] Although certainly more work is needed to show that these “real-space” pairing ideas presented in previous literature are a strong candidate to describe the cuprates, its quantitative success led us here to study geometries and couplings beyond those of the single layer with some confidence. Our goal is to report the trends observed when systems different from a single layer cuprate are analyzed. Here, the hole spectrum in the bilayer and 3D antiferromagnets is calculated with the SCBA which was shown to reproduce accurately numerical results for the 2D $t-J$ model,[@Rainbow] and it was successfully applied to the 2D $t-t^{\prime}-J$ model to compare its predictions with the photoemission spectra of ${\rm Sr_2CuO_2Cl_2}$.[@Nazarenko] Using the formalism previously described, a hole-hole attraction will be introduced producing a superconducting state. Here, the symmetry of this superconducting state will be analyzed and compare with other predictions for the same bilayer and 3D systems. Self Consistent Born Approximation. =================================== The Hamiltonian for spins and holes used in this section is defined as, $$H = -\sum_{\alpha} t_{\alpha} \sum_{ {\bf i}\sigma } ({\bar c}^{\dagger}_{{\bf i}\sigma} {\bar c}_{{\bf i}+{\bf e}_{\alpha}\sigma} +h.c.)$$ $$-\frac{1}{2} {\textstyle {\displaystyle\sum_{ \alpha\beta }}} t_{ \alpha\beta } \sum_{ {\bf i}\sigma } ( {\bar c}^{\dagger}_{{\bf i}\sigma} {\bar c}_{{\bf i}+{\bf e}_{\alpha}\pm{\bf e}_{\beta}\sigma} + h.c.)~~$$ $$+ \sum_{\alpha}J_{\alpha}\sum_{ {\bf i} } [ ( S^{z}_{{\bf i}} S^{z}_{{\bf i}+{\bf e}_{\alpha}} - {{1}\over{4}} n_{\bf i} n_{{\bf i}+{\bf e}_{\alpha}})~~$$ $$+ \frac{1}{2} \vartheta_{\alpha} ( S^{+}_{{\bf i}} S^{-}_{{\bf i}+{\bf e}_{\alpha}}+ S^{-}_{{\bf i}} S^{+}_{{\bf i}+{\bf e}_{\alpha}}) ], \eqno(2)$$ where ${\bf i}$ denote sites of a bilayer or simple 3D cubic cluster, ${\bf e}_{\alpha}$ is the unit vector in the $\alpha$-direction ($\alpha=x,y,z$), $t_{\alpha}$ and $J_{\alpha}$ correspond to the NN hopping amplitude and exchange coupling, respectively, in the direction $\alpha$, $t_{ \alpha\beta }$ is the next-nearest-neighbors (NNN) hopping in the directions defined by ${\bf e}_{\alpha} \pm {\bf e}_{\beta}$, the parameter $\vartheta_{\alpha} ( \epsilon [0,1] )$ represents a possible exchange anisotropy added for completeness, and the rest of the notation is standard. In the terms with NNN interactions, the summation is done with the condition $\alpha\neq\beta$, i.e. we only consider NNN hopping along the diagonals of the plaquettes. A generalization to include NNN hopping at distance of two lattice spacings along the main axes is straightforward. The constraint of no double occupancy is implemented in the kinetic energy by means of the standard definition for hole operators ${\bar c}_{{\bf i}\sigma}~=~c_{{\bf i}\sigma}(1-n_{{\bf i}-\sigma})$, with $n_{{\bf i}\sigma}=c^{\dagger}_{{\bf i}\sigma}c_{{\bf i}\sigma}$. Let us extend the SCBA analysis which has been applied before to single planes[@Rainbow] to the bilayer and 3D Hamiltonian Eq.(2). Following steps that by now are standard, first we redefine operators according to $S_{\bf j}^{\pm} \rightarrow S_{\bf j}^{\mp}$, $S_{\bf j}^{z} \rightarrow -S_{\bf j}^{z}$, $c_{{\bf j}\sigma}\rightarrow c_{{\bf j}-\sigma}$, where ${\bf j}$ denotes sites on the $B$ sublattice. Spins on the $A$ sublattice remain intact. With this procedure an AF state becomes ferromagnetic. Then, the usual linearized Holstein-Primakoff transformation is used. This transformation is defined in terms of Bose operators $a_{\bf i}$ and $a^{\dagger}_{{\bf i}}$, according to $S_{{\bf i}}^+ \approx a_{{\bf i}}\sqrt {2S}$, $S_{{\bf i}}^- \approx a^{\dagger}_{{\bf i}}\sqrt {2S}$, and $S_{{\bf i}}^z=S-a^{\dagger}_{{\bf i}}a_{{\bf i}}$. To handle the hole sector, the spin-charge decomposition $c_{{\bf i}\uparrow}=h^{\dagger}_{{\bf i}}$ and $c_{{\bf i}\downarrow} = h^{\dagger}_{{\bf i}}S_{{\bf i}}^+$ is introduced, where $h_{\bf i},h^{\dagger}_{\bf i}$ are operators corresponding to spinless holes. Finally, using periodic boundary conditions and after long but straightforward algebra, we can rewrite the resulting Hamiltonian in momentum space as $$H = E_0 + \sum_{\bf k}\epsilon_{\bf k}h^{\dagger}_{\bf k}h_{\bf k} + \sum_{\bf q}\omega_{\bf q}b^{\dagger}_{\bf q}b_{\bf q}$$ $$+ \frac{1}{\sqrt N}\sum_{{\bf k}{\bf q}}[ M_{{\bf k},{\bf q}}h^{\dagger}_{\bf k}h_{{\bf k}-{\bf q}}b_{\bf q} + h.c.], \eqno(3)$$ where the Bose operators $b(b^{\dagger})$ and $a(a^{\dagger})$ are connected by a Bogoliubov transformation,[@Rainbow] $N$ is the number of sites in the lattice, $\epsilon_{\bf k}=4(t_{xy}\cos k_x\cos k_y +t_{xz}\cos k_x\cos k_z+t_{yz}\cos k_y\cos k_z)$ is the bare spinless hole dispersion (which cancels in the absence of an explicit NNN hopping in the Hamiltonian), and $M_{{\bf k},{\bf q}} = u_{\bf q}\beta_{{\bf k}-{\bf q}}+ v_{\bf q}\beta_{\bf k}$ is the hole-magnon vertex. The magnon dispersion is given by $\omega_{\bf q}=\omega_0\nu_{\bf q}$, where $\omega_0=zSJ$, $J=\frac{1}{d} \sum_{\alpha}J_{\alpha}$, $S=\frac{1}{2}$ is the spin of the electrons in the original $t-J$ model language, $d$ is the number of dimensions (which will be taken equal to 3 in the rest of the paper), $z=2d$ is the coordination number, and we have used units where $\hbar=1$. The other quantities are defined as follows: $$\beta_{\bf k} = 2 \sum_{\alpha} t_{\alpha}\cos k_{\alpha},$$ $$u_{\bf q}=\sqrt \frac{1+\nu_{\bf q}}{2\nu_{\bf q}},\,\,\,\,\, v_{\bf q}=-sgn \kappa_{\bf q}\sqrt \frac{1-\nu_{\bf q}}{2\nu_{\bf q}},$$ $$\nu_{\bf q}=\sqrt {1-\kappa^{2}_{\bf q}},\,\,\,\,\, \kappa_{\bf q}= \frac{ 2S\sum_{{\alpha}} J_{\alpha}\vartheta_{\alpha}\cos q_{\alpha}} {\omega_0}. \eqno(4)$$ The ground state energy $E_0$ has the form: $$E_0=-dJN[S^2+S(1-\frac{1}{N}\sum_{\bf q}\nu_{\bf q})+\frac{1}{4}]. \eqno(5)$$ Care must be taken in order to apply Eqs.(3) and (4) to the bilayer. The small size in the z-direction (one lattice spacing) does not allow us to apply periodic boundary conditions (BC) to simulate bulk effects, as we do in the x- and y-directions. The z-axis BC should be open. However, for a bilayer open BC are equivalent to periodic BC but with parameters $J_{z}$, $t_{z}$, $t_{xz}$, and $t_{yz}$ reduced in half from their actual values. Remember also that $k_z$ can only be $0$ or $\pi$. Thus, for a bilayer assumed symmetric with respect to the $x\leftrightarrow y$ interchange in the planes, and spin symmetric ($\vartheta_{\alpha} =1$), two branches of the magnon spectrum are found, namely $$\omega_{\bf q}=SJ\sqrt {16-\tau^2_{\bf q}+\frac{2J_{\perp}}{J} (4\pm\tau_{\bf q})} \eqno(6)$$ where ${\bf q}$ is a 2D vector, $J=J_x = J_y$ is the in-plane coupling, $J_{\perp}=J_z$ is the interlayer coupling, and $\tau_{\bf q}=2(\cos q_{\bf x}+\cos q_{\bf y})$.[@Bonesteel]. With this information, we can find the quasiparticle hole spectrum in the AF background which is defined by the position of the peak with the lowest binding energy (assuming its intensity $Z$ is finite) in the spectral function $$A({\bf k},\omega)=-\frac{1}{\pi}Im\frac{1} {\omega-\epsilon_{\bf k}-\Sigma({\bf k}, \omega+i\delta)}, \eqno(7)$$ where the self-energy $\Sigma({\bf k},\omega)$ is calculated self-consistently from $$\Sigma({\bf k},\omega) =\frac{1}{N}\sum_{\bf q}\frac{M^{2}_{{\bf k},{\bf q}}} {\omega-\omega_{\bf q}-\epsilon_{{\bf k}-{\bf q}}-\Sigma({\bf k}-{\bf q}, \omega-\omega_{\bf q})}, \eqno(8)$$ which corresponds to the sum of magnon rainbow diagrams, as described in previous literature.[@Rainbow] Eq.(8) allow us to consider, at least in part, the dressing of the hole by spin excitations, and it has been shown that the predictions arising from this approach are in excellent agreement with Exact Diagonalization numerical results for 2D finite clusters.[@Rainbow] Thus, we are confident that the rainbow approximation may maintain its accuracy when applied to bilayer and 3D systems where numerical results are difficult to obtain. The solution of Eq.(8) is obtained numerically on finite but large clusters. While the SCBA formalism described here was setup for arbitrary values of the many couplings appearing in Hamiltonian Eq.(2), in the results reported in the following sections, we have neglected NNN hoppings in all directions and set the magnetic anisotropy parameter $\vartheta_{\alpha}$ to 1(spin isotropic system). We also assumed $xy$ symmetry in the planes, i.e. orthorhombic distortions are not considered here. Thus, in the rest of the paper we will use the following notations: $J$ and $t$ will refer to the in-plane exchange and hopping parameters, while $J_{\perp}$ and $t_{\perp}$ correspond to the inter-plane ones. The generality of Eqs.(2-8) would allow the interested reader to obtain SCBA results for arbitrary situations where spin or lattice anisotropies are important, or including NNN hopping amplitudes. Regarding the numerical values of the couplings, we have introduced relations between exchange and hopping amplitudes to simplify the multiparameter analysis. In particular, we have here assumed $\frac{J_{\perp}}{J}\,=\,(\frac{t_{\perp}}{t})^2$, which arises from the $t-J$ model when derived from the one band Hubbard model in strong coupling. In addition, we have fixed $t\,=\,0.4$ eV throughout the paper. Thus, as parameters we only have $J/t$ and $J_{\perp}/J$. The actual value of the inter-plane exchange is controversial. The two-magnon peak in Raman spectra experiments for ${\rm YBa_2Cu_3O_{6+x}}$ is consistent with an in-plane $J\sim$ 125 meV, with a negligible interplane exchange.[@Ratio] Recent measurements of the nuclear quadrupole and nuclear magnetic resonances on Cu-sites in ${\rm YBa_2Cu_3O_{15}}$ (alternating 1-2-3 and 1-2-4 structures) suggest a ratio $\frac{J_{\perp}}{J}\,\sim\,$0.3.[@Ratio1] However, a different value $\frac{J_{\perp}}{J}\,\sim\,$0.55 was found from infrared transmission and reflection measurements, after applying linear spin wave theory.[@Ratio2] A hopping amplitude ratio $\frac{t_{\perp}}{t}\,=\,$0.4 was reported on the basis of first-principles linear density approximation calculations,[@Andersen] corresponding to $\frac{J_{\perp}}{J}\,\sim\,$0.16 with our convention. In view of these discrepancies, in most of the results below the perpendicular exchange is consider a free parameter. Results ======= Bilayer. -------- Let us first discuss the one hole spectrum in the AF bilayer system using the exchange ratio $\frac{J_{\perp}}{J}\,=\,(\frac{t_{\perp}}{t})^2\,=\,0.16$ suggested by band structure calculations, $t=0.4$eV and $J\,=\,0.125$ eV, as found in some experiments ($J/t=0.3125$). The self-consistent calculation was performed on a $16 \times 16 \times 2$ cluster. In Fig.1 the spectrum is presented. It has two branches that correspond to the two possible momenta in the z-direction i.e. $0$ and $\pi$. These branches obey the relation $\epsilon_{+}({\bf k})\,=\, \epsilon_{-}({\bf k}+{\bf Q})$, where ${\bf Q}\,=\,(\pi, \pi)$ and ${\bf k}$ is the 2D momentum ($\epsilon_{\pm}({\bf k})$ here denote the even and odd branches of the spectrum). This relation is induced by the presence of antiferromagnetic long-range order in the system, which effectively reduces the size of the Brillouin zone. The minima of the branches are close to, but not exactly at, ${\bf k}=({\pi/2, \pi/2})$ i.e. where the minimum of the one-layer dispersion resides. This small effect is observed from fits of the data using trigonometric functions and it is barely noticeable in Fig.1, but as we increase $J_{\perp}$ the splitting of the minima away from ${\bf k}=({\pi/2, \pi/2})$ becomes larger, and at $J_{\perp}\,=\,J$ it is clearly visible. The bilayer one-hole spectrum can be accurately fit by NN, NNN and next to NNN hoppings amplitudes, namely $\epsilon_{\pm}({\bf k})\,= \pm\,0.01(\cos k_x+\cos k_y)\, +\,0.099\cos k_x\cos k_y\,+\,0.033(\cos 2k_x+\cos 2k_y)$ (eV) (see Fig.1). Note that each individual branch contains a NN amplitude which is nonzero. This is necessary since individually the even and odd branches are not invariant under a momentum shift in $\bf Q$. It should also be remarked that near $(\pi,0)$ there are “flat” dispersion regions, as observed in previous studies of the $t-J$ model.[@flat-theory; @other-flat] It has been argued that AF correlations induce these features in models of correlated electrons even when the $\xi_{AF}$ is only a couple of lattice spacings.[@moreo] Fig.1 shows that these features also survive the introduction of a realistic bilayer coupling. Our SCBA calculations indicate that the antiferromagnetic bilayer has quasiparticle excitations since a large peak appears at the bottom of the hole spectral function $A({\bf k},\omega)$. Typical spectral functions for the high symmetry points are presented in Fig.2. The quasiparticle peak carries a substantial percentage of the spectral weight in agreement with the calculations for a single layer.[@review] Note also that the closer the momentum to the bottom of the band, the stronger the quasiparticle peak. We observed a similar behavior even for interlayer exchanges as large as $\frac{J_{\perp}}{J}\,=\,3$. The evolution of the Fermi Surface (FS) as the SCBA hole dispersion is populated in the rigid band picture is shown in Fig.3. At very low hole density it starts with hole pockets around $({\pi/2, \pi/2})$ (Fig.3(a)) for the even and odd branches. The pockets are longer along the $(0,\pi)-(\pi,0)$ direction than along the main diagonal in the Brillouin zone as observed in similar studies for the single layer problem.[@Trugman; @flat-theory] As these pockets grow in size, the Fermi level eventually hits the saddle-points and the FS changes its topology (Fig.3(b) and (c)), becoming a large FS when the chemical potential is above the saddle points (Fig.3(d)). It should be remarked that after the hole pockets disappear, the FS acquire quasi-nesting features, which may lead to an enhancement of certain susceptibilities. The energy scale of the rapid change from hole pockets to a large FS is about 400K. This implies that a strong temperature dependence should be expected in transport measurements of a doped AF bilayer, similar to those observed in a single AF layer.[@Trugman] Actually the overall FS evolution obtained from the rigid band filling of the bilayer dispersion for $\frac{J_{\perp}}{J}\,=\,(\frac{t_{\perp}}{t})^2\,=\,0.16$ is qualitatively similar to results found for just one ${\rm CuO_2}$ plane using the same approximations,[@Trugman; @flat-theory] with the main difference being the presence of two branches. Identifying similar features in experiments could give us information about the strength of the bilayer coupling and antiferromagnetic correlations. 3D Cubic Lattice ---------------- In this subsection, the properties of a hole in a 3D antiferromagnetic environment will be analyzed. The SCBA quasiparticle hole dispersion for the case of an isotropic cubic lattice (i.e. $\frac{J_{\perp}}{J}=(\frac{t_{\perp}}{t})^2=$1) is shown in Fig.4. The numerical self-consistent calculation was carried out on an $8 \times 8 \times 8 $ cluster with parameters chosen as $t=0.4$eV and $J=0.16$eV, which gives $\frac{J}{t}\,=\,0.4$. The minimum of the dispersion is at $({\pi/2, \pi/2, \pi/2})$, which is natural based on the results reported for the one and two coupled planes. Our best fit of the numerical data corresponds to the following dispersion: $\epsilon({\bf k})\,= 0.082(\cos k_x\cos k_y+ \cos k_y\cos k_z+\cos k_x\cos k_z)\, +\,0.022(\cos 2k_x+\cos 2k_y+\cos 2k_z)$ (eV). As in the case of the one layer system, holes tend to move within the same sublattice to avoid distorting the AF background. It is interesting to analyze the dependence of the quasiparticle dispersion total bandwidth, $W$, with the ratio $\frac{J}{t}$ for the isotropic 3D AF. Here $W$ is defined as the difference in energy between the state with the highest energy, typically at $(0,0,0)$ and $(\pi,\pi,\pi)$, and the state with the lowest energy at $({\pi/2, \pi/2, \pi/2})$. Fig.5 shows $W$ as the ratio $\frac{J}{t}$ varies from 0.1 to 1. Results for a single layer on an $8 \times 8$ cluster are also shown for comparison. The overall behavior of the bandwidths is qualitatively the same in both cases, with $W$ for a 3D AF being slightly larger. In both cases $W/t$ grows approximately linearly for small $J/t$, and reaches saturation around $\frac{J}{t}\,\sim\,0.6$. In the linear regime, we find $W \sim 2.5J$ showing that the characteristic energy scale of the dispersion is $J$ rather than $t$, a well-known result obtained before in the context of holes in 2D antiferromagnets.[@review] It is interesting to observe that the increase in the dimensionality of the problem does not change dramatically the bandwidth of holes. As for planes and bilayers, the SCBA calculations show a strong quasiparticle peak in the hole spectral function $A({\bf k}, \omega)$ of an isotropic 3D AF. In Fig.6 results are shown at particular high symmetry points. The parameters are the same as those used to calculate the dispersion of Fig.4. We checked that the relative weight of the quasiparticle peak increases with the ratio $\frac{J}{t}$. The quasiparticle weight $Z$ is similar in 2D and 3D systems, at least at the bottom of the hole spectrum. Superconductivity. ================== If there is a source of attraction between the quasiparticles described in the previous sections, then the ground state of the system could become superconducting. In the underdoped regime, AF correlations are strong and, thus, it is natural to study pairing mediated by AF fluctuations assuming that the dispersion at half-filling is not drastically distorted when a finite but small hole density is studied. Since $\xi_{AF}$ is likely of the order of only a couple of lattice spacing at realistic dopings, a $real-space$ approach to pairing is more suitable than a picture where extended states (magnons) are interchanged between carriers ($k-space$ approach). While these two variations of AF-mediated pairing are quantitatively different, nevertheless it has been shown that they lead to the same symmetry for the superconducting order parameter (SCOP).[@afvh] Thus, independent of the discussion of real-space vs k-space approaches, we believe that it is safe to use the real-space pairing ideas of Ref.[@afvh] to study the symmetry of the SCOP for the case of a bilayer and 3D AF systems. As explained before, the simplified picture introduced in previous literature[@afvh] is to use the spectrum of a hole, calculated numerically or using the SCBA, as the dispersion of the quasiparticles which interact through a NN attraction regulated by a parameter $V$ (of the order of $J$). This interaction is motivated by AF-induced pairing since it can be shown that a potential $V({\bf q})$ in momentum space maximized at ${\bf q=Q}$ and smeared by a finite $\xi_{AF}$, induces a real-space potential which is dominated by a NN attraction. Previous estimations suggest a value $V=0.6J$ from the study of the two-holes bound state on a single layer.[@afvh] Here, the same value will be employed for the in-plane NN interaction, while for the interaction in the z-direction $V_{\perp}=0.6J_{\perp}$ will be used. In the absence of accurate Exact Diagonalization or Monte Carlo results for the dispersion of holes away from half-filling, a rigid-band filling of the SCBA quasiparticle spectra obtained at half-filling will be assumed in the BCS gap equation. Previous studies for single layer systems have shown that this approach provides a SCOP with a symmetry consistent with other more traditional methods, and in addition the actual values of $T_c$ and its density dependence are in qualitative agreement with experiments.[@afvh] Thus, it is natural to employ the same approach to analyze the SC properties of doped bilayers and 3D antiferromagnets. Superconductivity in a bilayer. ------------------------------- Using the bilayer SCBA hole dispersion and the hole-hole NN attraction described before, we have studied superconductivity within the BCS formalism. To calculate the SCBA hole dispersion, we fixed $\frac{J}{t}\,=\,0.3125$ and allow for the interlayer coupling to take the values $\frac{J_{\perp}}{J}\,=\,$0.16, 0.3, 0.5, 1, 2 and 3. Solving numerically the gap equation on large clusters, we observed that for the first three ratios $\frac{J_{\perp}}{J}$ the symmetry of the superconducting order parameter (SCOP) is $d_{x^2-y^2}$ for all the hole densities where superconductivity exists. This is the region analytically connected to the physics of one layer, where a similar condensate was found in previous papers. In Fig.7, we show $T_c$ vs the quasiparticle hole density for a bilayer with $\frac{J_{\perp}}{J}\,=\,$1. This dependence is also shown for the case of a single layer with the same $J/t$, for comparison. Both curves exhibit the “optimal doping” behavior and high values of $T_c$ of the order of 100K. The difference in $T_c$ between a bilayer and a single layer is small for practical purposes, except for the region of high densities where a bilayer changes the symmetry of the superconducting condensate into an “s-wave” state to be discussed below (note that a quasiparticle weight $Z$ smaller than one implies the existence of incoherent weight in the spectral function at energies larger than those of the q.p. band. Then, “high density” in this context does not imply a very small electronic density $\langle n \rangle$). Similar conclusions were obtained in Ref.[@Jose] where an Exact Diagonalization study of coupled layers was carried out to calculate the superconducting pairing correlations. No important enhancement of pairing correlations changing from a single $t-J$ layer to a bilayer was reported.[@Jose] According to the classification of Liu, Levin and Maly[@Levin] the $d_{x^2-y^2}$ state that we observed is labeled $(d,d)$, which means that a gap with the symmetry $d_{x^2-y^2}$ opens in both the bonding and antibonding Fermi surfaces (i.e. the Fermi surfaces corresponding to $k_z = 0$ and $\pi$ shown in Fig.1). This d-wave channel dominates up to values close to $J_{\perp}\,\sim\,J$. At the ratio $\frac{J_{\perp}}{J}\,=\,$1 the symmetry of the SC order parameter changes only at high densities (i.e. around 55% filling of the quasiparticle band as shown in Fig.7) to a mixture of $d_{3z^2-r^2}$ and $s_{x^2+y^2+z^2}$, with real amplitudes of the same sign. $s_{x^2+y^2+z^2}$ is the 3D analog of the extended s-wave function. The relative weight of the $d_{3z^2-r^2}$ component is dominant in the bilayer case. In general, the tetragonal point symmetry group consists of 10 irreducible representations among which only five correspond to singlet pairing. The functions ${3z^2-r^2}$, ${x^2+y^2+z^2}$ and also a $constant$ function have the same transformation properties with respect to the tetragonal symmetry operations and thus belong to the same representation $\Gamma_{1}^{+}$ of the tetragonal lattice symmetry group. Then, they mix together and they form the “s-wave” representation with respect to rotations in the planes, while $d_{x^2-y^2}$ belongs to $\Gamma_{3}^{+}$ and $d_{xy}$ to $\Gamma_{4}^{+}$ for a tetragonal system.[@Sigrist] Note that in an orthorhombic lattice $d_{3z^2-r^2}$, $s_{x^2+y^2+z^2}$ as well as $d_{x^2-y^2}$ correspond to the same representation $\Gamma_{1}$. A less trivial issue is how the symmetry evolves from a mixture of $d_{3z^2-r^2}$ and $s_{x^2+y^2+z^2}$ to the so-called $d_z$ state[@dz] (the $d_z$ state is a singlet along the link in the z-direction of the bilayer, i.e. it is “s-wave” with respect to $\pi/2$ rotations of the planes). This transition occurs as the ratio $\frac{J_{\perp}}{J}$ grows at a fixed density. The relative weight of $s_{x^2+y^2+z^2}$ in the mixture increases until both functions become equally weighted with the same signs of the weight coefficients in the limit of $J_{\perp} \gg J$. In this situation the mixture is exactly equivalent to the $d_z$ symmetry. In our calculations a coupling $\frac{J_{\perp}}{J}\,=\,$2 is enough to have almost $d_z$ symmetry for all densities, with a substantial increase of $T_c$ up to 150K. The mixture of $d_{3z^2-r^2}$ and $s_{x^2+y^2+z^2}$ can also be viewed as the mixture of $d_z$ and the extended in-plane s-wave $s_{x^2+y^2}$, or in terms of odd-even gaps $\Delta_{\pm}({\bf k})\,\propto\,\pm \alpha\,+\,\beta(\cos k_x+\cos k_y)$, where ${\bf k}$ is the in-plane momentum of the quasiparticle (the even and odd gaps differ by a constant corresponding to the inter-plane pairing amplitude). In a similar fashion, as the ratio $\frac{J_{\perp}}{J}$ increases, the relative weight of the in-plane extended s-wave decreases. There is a simple intuitive way to understand the evolution with $J_{\perp}/J$ of the symmetry of the superconducting order parameter found here.[@naza1] Consider the limit of Eq.(1) applied to a bilayer where the NN in-plane and inter-plane attractions dominate over the kinetic energy. In this limit, let us analyze the two quasiparticle problem and its bound state. The quasiparticles will be roughly at a distance of one lattice spacing (tight bound state). In the reference frame of one particle, the other can occupy any of the five NN sites (four in the same plane and one in the other plane). These sites are actually $linked$ by the dispersion in the SCBA since the five sites belong to the same sublattice. In other words, the wave function of one quasiparticle orbiting around the other can be found in this limit from a simple 5-site 1-particle problem as shown schematically in Fig.8, with the ratio of hopping amplitudes $t_1$ and $t_2$ simulating the effect of $J_{\perp}/J$. Solving this trivial problem we find that for $J_{\perp}/J <1$ (or $t_2/t_1 <1$) the lowest energy is obtained for a state with amplitudes $c,-c,c,-c,0$ $(c>0)$ at sites $1,2,3,4,5$, respectively, which is the $d_{x^2 - y^2}$ state. On the other hand, for $J_{\perp}/J >1$ (or $t_2/t_1 >1$) the ground state amplitudes have the same value (with its sign) in-plane, and a different value and sign out of the plane, as expected from the discussion above. Note also that these simple calculations illustrate the fact that a NN attraction and on-site repulsion is $not$ enough to have a tendency to form d-wave pairs in a $dilute$ gas. In the present case this potential must be supplemented by a particular hole dispersion that allows for the movement of carriers within the same sublattice. Comparison with other theories ------------------------------ In the literature, several types of interactions between quasiparticles in a bilayer have been proposed. Of particular interest for our discussion are those based on phenomenological magnetic susceptibilities in the presence of dynamical short-range AF order, both in and between the planes. Recently it has been suggested[@Mazin] that the SCOP in a bilayer can be of a particular s-wave type named $(s,-s)$ with opposite signs in the odd and even branches of the spectrum of carriers. The assumption made in Ref.[@Mazin] is that the in-plane magnetic susceptibility $\chi_{\|}({\bf q})$ has the opposite sign, but the same absolute value, as the inter-plane susceptibility $\chi_{\perp}({\bf q})$. In other words, $\chi_{\|}({\bf q})\,=\,-\chi_{\perp}({\bf q})\,=\, \chi_{0}({\bf q})$, where $\chi_{0}({\bf q})$ is the phenomenological susceptibility of a doped 2D antiferromagnet peaked at ${\bf q}\,=\,{\bf Q}\,=\,(\pi,\pi)$ introduced to fit NMR data.[@MMP] This assumption needs further justification since the analysis of experimental results provides an estimation 5 meV $<\,J_{\perp}\,<$ 20 meV[@Monien] for the interplane coupling, while $J\,\approx\,$120 meV. Thus, it is somewhat risky to assume that the spin correlations between planes have the same strength as in the planes. Then, it seems more natural to employ the RPA approximation for the susceptibility[@Monien] given by $\chi^{-1}_{\pm}({\bf q})\,=\,\chi^{-1}_{0}({\bf q})\mp J_{\perp}$, where $\chi_{\pm}({\bf q})$ are the even and odd branches of the bilayer susceptibility, respectively. In this case, $\chi_{\|}({\bf q})$ and $\chi_{\perp}({\bf q})$ are quite different in absolute value, i.e. $|\chi_{\|}({\bf q})|\,\gg\,|\chi_{\perp}({\bf q})|$ in order to satisfy the applicability of the RPA approximation. We have checked by a direct calculation, that this approach provides $d_{x^2-y^2}$ symmetry for the SCOP on both branches of the spectrum i.e. the so-called $(d,d)$ state. This is in agreement with Ref.[@Levin] where a similar phenomenological in-plane susceptibility derived from the neutron scattering data was used to construct the effective potential with an arbitrary relation between the in-plane and inter-plane coefficients. The above mentioned RPA model is certainly relevant to the more phenomenological approach used in this paper if one considers the expansion of $\chi_{0}({\bf q})$ in real space, assuming that the effective coupling between quasiparticles is proportional to $\chi_0$. For correlation lengths $\xi\sim2-3$ lattice spacings, the expansion consists of a strong on-site repulsion $U$, and also a relatively strong NN attraction $V$ in-plane, with a ratio $\frac{U}{|V|}\sim3$. The NNN interaction is repulsive and about one order of magnitude weaker than $U$. Other terms are negligibly small. Then, the phenomenological interaction used here and in previous literature[@afvh] is very closely related to the RPA interaction in the realistic regime $J_{\perp}\,\ll\,J$. To show this result, first let us expand the susceptibility as $$\chi{\pm}({\bf q})\,=\,\frac{1}{\chi^{-1}_{0}({\bf q})\pm J_{\perp}} \simeq \chi_{0}({\bf q})\mp J_{\perp}\chi^{2}_{0}({\bf q}). \eqno(9)$$ Using a 3D notation one can write $\chi({\bf q}, q_z) \simeq \chi_{0}({\bf q})-\cos (q_z) J_{\perp}\chi^{2}_{0}({\bf q})$, for the case of a bilayer with $q_z\,=\,$0 or $\pi$. The effective in-plane potential must contain an strong on-site repulsion. Consequently, $\chi_{0}({\bf q})$ can be written as $\chi_{0}({\bf q})\,=\, u+f({\bf q})$, where $u$ is a constant proportional to the on-site repulsion $U$, and the function $f({\bf q})$ satisfies the condition $\sum_{\bf q}f({\bf q})=0$. Hence, the susceptibility can be rewritten as $\chi({\bf q}, q_z,) \simeq u+f({\bf q})-\cos (q_z) J_{\perp}[u+f({\bf q})]^2$. Assuming that the ratio $\frac{f({\bf q})}{u}$ is small (i.e. the on-site repulsion is strong), we finally get: $$\chi({\bf q}, q_z) \simeq u+f({\bf q})- J_{\perp}u\,\cos (q_z) - 2J_{\perp}u\,\cos (q_z)f({\bf q}). \eqno(10)$$ The first three terms are accounted for in the calculation discussed in this paper both for a bilayer and a 3D antiferromagnet since, as it was argued above, the largest contribution to $f({\bf q})$ comes from the NN attraction. The last term in Eq.(10) is relatively small compared to the other three, but it is assumed to have a significant contribution in the models of Refs.[@Levin; @Mazin] explaining the difference between their results (i.e. $(s,-s)$) and ours (i.e. mixture of $d_{3z^2-r^2}$ and $s_{x^2+y^2+z^2}$). Nevertheless, both cases are “s-wave” in-plane and thus there is no qualitative difference in their transformation properties (they belong to the same representation). In addition, since $J_{\perp}/J \ll 1$, both are unlikely to be stable in the cuprates. Superconductivity in the isotropic 3D AF. ----------------------------------------- To calculate $T_c$ and the symmetry of the SCOP in the case of a three dimensional isotropic AF, we here use the SCBA hole dispersion obtained at $J/t\,=\,0.4$ and again the BCS gap equation. As expected,[@Miyake; @Scalapino] when $\frac{J_{\perp}}{J}\,=\,1$, the two channels $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ are degenerate, i.e. $T_c$ is the same for both and varies with the hole density as it is shown in Fig.9. The critical temperature in 3D is slightly smaller than for planes and bilayers. The “optimal doping” behavior is also present in 3D, but it is less pronounced than in 2D. When $\frac{J_{\perp}}{J}\,\neq\,1$, the system acquires tetragonal symmetry and the channels $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ are no longer degenerate. In fact, if $\frac{J_{\perp}}{J}\,<\,1$, the gap possesses $d_{x^2-y^2}$ character, while if $\frac{J_{\perp}}{J}\,>\,1$, it becomes again a mixture of $d_{3z^2-r^2}$ and $s_{x^2+y^2+z^2}$, as it was mentioned in the case of a bilayer. Again, as the ratio $\frac{J_{\perp}}{J}$ increases, the symmetry of the SCOP becomes $d_z$. All these results can also be obtained following a discussion similar to that of Fig.8 but now using six sites instead of five. Conclusions =========== In this paper, we have investigated the properties of bilayer and 3D antiferromagnets specially regarding the behavior of holes in such environments and the eventual formation of a superconducting state upon doping. The main assumption is that AF correlations remain of at least a few lattice spacings in range which, as a first approximation, allow us to use the dispersion of holes at half-filling and a density-density NN attraction between quasiparticles also generated by antiferromagnetism. 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--- abstract: 'We develop a general theory of “almost Hadamard matrices”. These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the $1$-norm on $O(N)$. Our study includes a detailed discussion of the circulant case ($H_{ij}=\gamma_{j-i}$) and of the two-entry case ($H_{ij}\in\{x,y\}$), with the construction of several families of examples, and some $1$-norm computations.' address: - 'T.B.: Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France. [[email protected]]{}' - 'I.N.: CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France. [[email protected]]{}' - 'K.Z.: Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland. [[email protected]]{}' author: - Teodor Banica - Ion Nechita - Karol Życzkowski title: 'Almost Hadamard matrices: general theory and examples' --- Introduction {#introduction .unnumbered} ============ An Hadamard matrix is a square matrix having $\pm 1$ entries, whose rows are pairwise orthogonal. The simplest example, appearing at $N=2$, is the Walsh matrix: $$H_2=\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}$$ At $N=3$ we cannot have examples, due to the orthogonality condition, which forces $N$ to be even. At $N=4$ now, we have several examples, for instance $H_4=H_2\otimes H_2$: $$H_4=\begin{pmatrix}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{pmatrix}$$ For higher values of $N$, the construction of Hadamard matrices is quite a tricky problem. First, by permuting rows and columns or by multiplying them by $-1$, we can always assume that the first 3 rows of our matrix look as follows: $$H=\begin{pmatrix} 1\ldots 1&1\ldots 1&1\ldots 1&1\ldots 1\\ 1\ldots 1&1\ldots 1&-1\ldots -1&-1\ldots -1\\ 1\ldots 1&-1\ldots -1&1\ldots 1&-1\ldots -1\\ \ldots&\ldots&\ldots&\ldots \end{pmatrix}$$ Now if we denote by $x,y,z,t$ the sizes of the 4 columns, the orthogonality conditions between the first 3 rows give $x=y=z=t$, so $N=x+y+z+t$ is a multiple of 4. A similar analysis with 4 rows instead of 3, or any other kind of abstract or concrete consideration doesn’t give any further restriction on $N$, and we have: [**Hadamard Conjecture (HC).**]{} [*There is at least one Hadamard matrix of size $N\times N$, for any $N\in 4\mathbb N$.*]{} This conjecture, going back to the 19th century, is probably one of the most beautiful statements in combinatorics, and in mathematics in general. The numeric verification so far goes up to $N=664$, see [@kta]. For a general presentation of the problem, see [@lgo]. At the level of concrete examples, the only ones which are simple to describe are the tensor powers of the Walsh matrix, having size $2^k$. For some other examples, see [@hor]. Yet another good problem, simple-looking as well, concerns the circulant case. Given a vector $\gamma\in(\pm 1)^N$, one can ask whether the matrix $H\in M_N(\pm 1)$ defined by $H_{ij}=\gamma_{j-i}$ is Hadamard or not. Here is a solution to the problem, appearing at $N=4$: $$K_4=\begin{pmatrix}-1&1&1&1\\ 1&-1&1&1\\ 1&1&-1&1\\ 1&1&1&-1\end{pmatrix}$$ More generally, any vector $\gamma\in(\pm 1)^4$ satisfying $\sum\gamma_i=\pm 1$ is a solution to the problem. The following conjecture, going back to [@rys], states that there are no other solutions: [**Circulant Hadamard Conjecture (CHC).**]{} [*There is no circulant Hadamard matrix of size $N\times N$, for any $N\neq 4$.*]{} The fact that such a simple-looking problem is still open might seem quite surprizing. If we denote by $S\subset\{1,\ldots,N\}$ the set of positions of the $-1$ entries of $\gamma$, the Hadamard matrix condition is simply $|S\cap(S+k)|=|S|-N/4$, for any $k\neq 0$, taken modulo $N$. Thus, the above conjecture simply states that at $N\neq 4$, such a set $S$ cannot exist! Summarizing, the Hadamard matrices are very easy to introduce, and they quickly lead to some difficult and interesting combinatorial problems. Regarding now the other motivations for studying such matrices, these are quite varied: 1. The Hadamard matrices were first studied by Sylvester [@syl], who was seemingly attracted by their plastic beauty: just replace the $\pm 1$ entries by black and white tiles, and admire the symmetries and dissymmetries of the resulting design! 2. More concretely now, the Hadamard matrices can be used for various coding purposes, and have several applications to engineering, and quantum physics. For instance the Walsh matrices $H_2^{\otimes k}$ are used in the Olivia MFSK radio protocol. Most applications of the Hadamard matrices, however, come from their generalizations. A “complex Hadamard” matrix is a matrix $H\in M_N(\mathbb C)$ all whose entries are on the unit circle, and whose rows are pairwise orthogonal. The basic example is $\widetilde{F}_N=\sqrt{N}F_N$, where $F_N$ is the matrix of the Fourier transform over $\mathbb Z_N$. That is, with $\omega=e^{2\pi i/N}$: $$\widetilde{F}_N=\begin{pmatrix} 1&1&1&\ldots&1\\ 1&\omega&\omega^2&\ldots&\omega^{N-1}\\ 1&\omega^2&\omega^4&\ldots&\omega^{2(N-1)}\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 1&\omega^{N-1}&\omega^{2(N-1)}&\ldots&\omega^{(N-1)^2} \end{pmatrix}$$ As a first observation, the existence of this matrix prevents the existence of a “complex version” of the HC. However, when trying to construct complex Hadamard matrices by using roots of unity of a given order, a wide, subtle, and quite poorly understood generalization of the HC problematics appears. See e.g. [@bbs], [@but], [@lle], [@lau]. As for the motivations and applications, these partly come from pure mathematics, cf. e.g. [@ba2], [@haa], [@jon], [@pop], and partly come from quantum physics, cf. e.g. [@bb+], [@tzy]. Let us go back now to the real case. Since the determinant of $N$ vectors is maximized when these vectors are chosen pairwise orthogonal, we have the following result: [**Theorem A.**]{} [*For a matrix $H\in M_N(\pm 1)$ we have $|\det H|\leq N^{N/2}$, with equality if and only if $H$ is Hadamard.*]{} This result, due to Hadamard himself [@had], has led to a number of interesting problems, and to the general development of the theory of Hadamard matrices. See [@hor]. As already mentioned, in order for an Hadamard matrix to exist, its size $N$ must be a multiple of $4$. For numbers of type $N=4n+k$ with $k=1,2,3$, several “real” generalizations of the Hadamard matrices have been constructed. The idea is usually to consider matrices $H\in M_N(\pm 1)$, whose rows are as orthogonal as they can be: [**Definition A.**]{} [*A “quasi-Hadamard” matrix is a square matrix $H\in M_N(\pm 1)$ which is as orthogonal as possible, e.g. which maximizes the quantity $|det H|$.*]{} This definition is of course a bit vague, but the main idea is there. For a detailed discussion of the different notions here, we refer to the articles [@aa+], [@kko], [@pso]. Yet another interpretation of the Hadamard matrices comes from the Cauchy-Schwarz inequality. Since for $U\in O(N)$ we have $||U||_2=\sqrt{N}$, we obtain: [**Theorem B.**]{} [*For a matrix $U\in O(N)$ we have $||U||_1\leq N\sqrt{N}$, with equality if and only if $H=\sqrt{N}U$ is Hadamard.*]{} This result, first pointed out in [@bc1], shows that the matrices of type $H=\sqrt{N}U$, with $U\in O(N)$ being a maximizer of the 1-norm on $O(N)$, can be thought of as being some kind of “analytic generalizations” of the Hadamard matrices. Note that such matrices exist for any $N$, in particular for $N=4n+k$ with $k=1,2,3$. As an example, the maximum of the 1-norm on $O(3)$ can be shown to be the number $5$, coming from the matrix $U_3=K_3/\sqrt{3}$ and its various conjugates, where: $$K_3=\frac{1}{\sqrt{3}}\begin{pmatrix}-1&2&2\\ 2&-1&2\\ 2&2&-1\end{pmatrix}$$ This result, proved in [@bc1] by using the Euler-Rodrigues formula, is actually something quite accidental. In general, the integration on $O(N)$ is quite a subtle business, and the maximum of the 1-norm is quite difficult to approach. See [@bc2], [@bsc], [@csn]. These integration problems make the above type of matrices quite hard to investigate. Instead of looking directly at them, we will rather enlarge the attention to the matrices of type $H=\sqrt{N}U$, where $U\in O(N)$ is a local maximizer of the 1-norm on $O(N)$. Indeed, according to the Hessian computations in [@bc1], these latter matrices are characterized by the fact that all their entries are nonzero, and $SU^t>0$, where $S_{ij}={\rm sgn}(U_{ij})$. Summarizing, Theorem B suggests the following definition: [**Definition B.**]{} [*An “almost Hadamard” matrix is a square matrix $H\in M_N(\mathbb R)$ such that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the $1$-norm on $O(N)$.*]{} The basic examples are of course the Hadamard matrices. We have as well the following $N\times N$ matrix, with $N\in\mathbb N$ arbitrary, which generalizes the above matrices $K_3,K_4$: $$K_N=\frac{1}{\sqrt{N}}\begin{pmatrix} 2-N&2&\ldots&2&2\\ 2&2-N&\ldots&\ldots&2\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 2&\ldots&\ldots&\ldots&\ldots\\ 2&2&\ldots&\ldots&2-N \end{pmatrix}$$ A lot of other interesting examples exist, as we will show in this paper. Here is for instance a remarkable matrix, having order $N\in 2\mathbb N+1$, and circulant structure: $$L_N=\frac{1}{\sqrt{N}} \begin{pmatrix} 1&-\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&\ldots&\cos^{-1}\frac{(N-1)\pi}{N}\\ \cos^{-1}\frac{(N-1)\pi}{N}&1&-\cos^{-1}\frac{\pi}{N}&\ldots&-\cos^{-1}\frac{(N-2)\pi}{N}\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ -\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&-\cos^{-1}\frac{3\pi}{N}&\ldots&1 \end{pmatrix}$$ Yet another series, with $N=q^2+q+1$, where $q=p^k$ is a prime power, comes from the adjacency matrix of the projective plane over $\mathbb F_q$. Here is for instance the matrix associated to the Fano plane ($q=2$), where $x=2-4\sqrt{2}$, $y=2+3\sqrt{2}$: $$I_7=\frac{1}{2\sqrt{7}}\begin{pmatrix} x&x&y&y&y&x&y\\ y&x&x&y&y&y&x\\ x&y&x&x&y&y&y\\ y&x&y&x&x&y&y\\ y&y&x&y&x&x&y\\ y&y&y&x&y&x&x\\ x&y&y&y&x&y&x \end{pmatrix}$$ The aim of the present paper is to provide a systematic study of such matrices, with the construction of a number of non-trivial examples, and with the development of some general theory as well. Our motivation comes from two kinds of problems: 1. The Hadamard matrix problematics. The world of Hadamard matrices is extremely rigid, and we think that our study of almost Hadamard matrices, where there is much more freedom, can help. As an example, there are several non-trivial classes of circulant almost Hadamard matrices at any $N\in\mathbb N$, and trying to understand them might end up in sheding some new light on the CHC. 2. Generalizations of Hadamard matrices. The Hadamard matrices have applications in a number of areas of physics and engineering, notably in coding theory, and in various branches of quantum physics. One problem, however, is that these matrices exist only at $N=4n$. At $N=4n+k$ with $k=1,2,3$ some generalizations would be needed, and we believe that our almost Hadamard matrices can help. The paper is organized as follows: 1 is a preliminary section, in 2-3 we investigate two special cases, namely the circulant case and the two-entry case, and 4 contains a list of examples. The final section, 5, contains a few concluding remarks. [**Acknowledgements.**]{} We would like to thank Guillaume Aubrun for several useful discussions. The work of T.B. was supported by the ANR grant “Granma”. I.N. acknowledges financial support from the ANR project OSvsQPI 2011 BS01 008 01 and from a CNRS PEPS grant. The work of K.Z. was supported by the grant N N202 261938, financed by the Polish Ministry of Science and Higher Education. Preliminaries ============= We consider in this paper various square matrices $M\in M_N(\mathbb C)$. The indices of our matrices will usually range in the set $\{0,1,\ldots,N-1\}$. We use the following special $N\times N$ matrices: 1. $1_N$ is the identity matrix. 2. $J_N$ is the “flat” matrix, having all entries equal to $1/N$. 3. $F_N$ is the Fourier matrix, given by $(F_N)_{ij}=\omega^{ij}/\sqrt{N}$, with $\omega=e^{2\pi i/N}$. We denote by $D$ the generic diagonal matrices, by $U$ the generic orthogonal or unitary matrices, and by $H$ the generic Hadamard matrices, and their generalizations. Our starting point is the following observation, from [@bc1]: For $U\in O(N)$ we have $||U||_1\leq N\sqrt{N}$, with equality if and only if $H=\sqrt{N}U$ is Hadamard. The first assertion follows from the Cauchy-Schwarz inequality: $$\sum_{ij}|U_{ij}|\leq\left(\sum_{ij}1^2\right)^{1/2}\left(\sum_{ij}|U_{ij}|^2\right)^{1/2}=N\sqrt{N}$$ For having equality the numbers $|U_{ij}|$ must be equal, and since the sum of squares of these numbers is $N$, we must have $|U_{ij}|=1/\sqrt{N}$, which gives the result. As already mentioned in the introduction, the study of maximizers of the 1-norm on $O(N)$ is a quite difficult task. However, we have here the following result, from [@bc1]: For $U\in O(N)$, the following are equivalent: 1. $U$ is a local maximizer of the $1$-norm on $O(N)$. 2. $U_{ij}\neq 0$, and $SU^t>0$, where $S_{ij}={\rm sgn}(U_{ij})$. As already mentioned, this result is from [@bc1]. Here is the idea of the proof: Let us first prove that if $U$ is a local maximizer of the 1-norm, then $U_{ij}\neq 0$. For this purpose, let $U_1,\ldots,U_N$ be the columns of $U$, and let us perform a rotation of $U_1,U_2$: $$\begin{pmatrix}U^t_1\\ U^t_2\end{pmatrix}=\begin{pmatrix} \cos t\cdot U_1-\sin t\cdot U_2\\ \sin t\cdot U_1+\cos t\cdot U_2 \end{pmatrix}$$ In order to compute the 1-norm, let us permute the columns of $U$, in such a way that the first two rows look as follows, with $X_k\neq 0$, $Y_k\neq 0$, $A_kC_k>0$, $B_kD_k<0$: $$\begin{pmatrix}U_1\\ U_2\end{pmatrix} =\begin{pmatrix} 0&0&Y&A&B\\ 0&X&0&C&D \end{pmatrix}$$ If we agree that the lower-case letters denote the 1-norms of the corresponding upper-case vectors, and we let $K=u_3+\ldots+u_N$, then for $t>0$ small we have: $$\begin{aligned} ||U^t||_1 &=&||\cos t\cdot U_1-\sin t\cdot U_2||_1+||\sin t\cdot U_1+\cos t\cdot U_2||_1+K\\ &=&(\cos t+\sin t)(x+y+a+d)+(\cos t-\sin t)(b+c)+K\end{aligned}$$ Now since $U$ locally maximizes the 1-norm on $O(N)$, the derivative of this quantity must be negative in the limit $t\to 0$. So, we obtain the following inequality: $$(x+y+a+d)-(b+c)\leq 0$$ Consider now the matrix obtained by interchanging $U_1,U_2$. Since this matrix must be as well a local maximizer of the 1-norm on $O(N)$, we obtain: $$(x+y+b+c)-(a+d)\leq 0$$ We deduce that $x+y=0$, so $x=y=0$, and the $0$ entries of $U_1,U_2$ must appear at the same positions. By permuting the rows of $U$ the same must hold for any two rows $U_i,U_j$. Now since $U$ cannot have zero columns, all its entries must be nonzero, as claimed. It remains to prove that if $U_{ij}\neq 0$, then $U$ is a local maximizer of $F(U)=||U||_1$ if and only if $SU^t>0$, where $S_{ij}={\rm sgn}(U_{ij})$. For this purpose, we differentiate $F$: $$dF=\sum_{ij}S_{ij}dU_{ij}$$ We know that $O(N)$ consists of the zeroes of the polynomials $A_{ij}=\sum_kU_{ik}U_{jk}-\delta_{ij}$. So, $U$ is a critical point of $F$ if and only if $dF\in span(dA_{ij})$. Now since $A_{ij}=A_{ji}$, this is the same as asking for a symmetric matrix $M$ such that $dF=\sum_{ij}M_{ij}dA_{ij}$. But: $$\sum_{ij}M_{ij}dA_{ij} =\sum_{ijk}M_{ij}(U_{ik}dU_{jk}+U_{jk}dU_{ik}) =2\sum_{lk}(MU)_{lk}dU_{lk}$$ Thus the critical point condition reads $S=2MU$, so the matrix $M=SU^t/2$ must be symmetric. Now the Hessian of $F$ applied to a vector $X=UY$, with $Y\in O(N)$, is: $$Hess(F)(X) =\frac{1}{2}Tr(X^t\cdot SU^t\cdot X) =\frac{1}{2}Tr(Y^t\cdot U^tS\cdot Y)$$ Thus the Hessian of $F$ is positive definite when $U^tS$ is positive definite, which is the same as saying that $U(U^tS)U^t=SU^t$ is positive definite, and we are done. The above result gives rise to the following definition: A square matrix $H\in M_N(\mathbb R^*)$ is called “almost Hadamard” if $U=H/\sqrt{N}$ is orthogonal, and the following equivalent conditions are satisfied: 1. $U$ is a local maximizer of the $1$-norm on $O(N)$. 2. $U_{ij}\neq 0$, and with $S_{ij}={\rm sgn}(U_{ij})$, we have $SU^t>0$. If so is the case, we call $H$ “optimal” if $U$ is a maximizer of the $1$-norm on $O(N)$. Let $J_N$ be the flat $N\times N$ matrix, having all the entries equal to $1/N$. Also, let us call “Hadamard equivalence”, or just “equivalence”, the equivalence relation on the $N\times N$ matrices coming from permuting the rows and columns, or multiplying them by $-1$. The class of almost Hadamard matrices has the following properties: 1. It contains all the Hadamard matrices. 2. It contains the matrix $K_N=\sqrt{N}(2J_N-1_N)$. 3. It is stable under equivalence, tensor products, and transposition. All the assertions are clear from definitions: \(1) This follows either from Proposition 1.2, or from the fact that $U=H/\sqrt{N}$ is orthogonal, and $SU^t=HU^t=\sqrt{N}1_N$ is positive. \(2) First, the matrix $U=K_N/\sqrt{N}$ is orthogonal, because it is symmetric, and: $$U^2=(2J_N-1_N)^2=4J_N^2-4J_N+1_N=1_N$$ Also, we have $S=NJ_N-21_N$, so the matrix $SU^t$ is indeed positive: $$SU^t=(NJ_N-21_N)(2J_N-1_N)=(N-2)J_N+2(1_N-J_N)$$ \(3) For a tensor product of almost Hadamard matrices $H=H'\otimes H''$ we have $U=U'\otimes U''$ and $S=S'\otimes S''$, so that $U$ is unitary and $SU^t$ is positive, as claimed. As for the assertions regarding equivalence and transposition, these are clear from definitions. Regarding now the optimal case, we have the following result, from [@bc1]: The optimal almost Hadamard matrices are as follows: 1. At any $N$ where HC holds, these are the $N\times N$ Hadamard matrices. 2. At $N=3$, these are precisely $K_3=\sqrt{3}(2J_3-1_3)$ and its conjugates. The assertion (1) is clear from Proposition 1.2. For (2) we must prove that for $U\in O(3)$ we have $||U||_1\leq 5$, with equality when $U$ is conjugate to $U_3=2J_3-1_3$. But here we can assume $U\in SO(3)$, and use the Euler-Rodrigues formula: $$U=\begin{pmatrix} x^2+y^2-z^2-t^2&2(yz-xt)&2(xz+yt)\\ 2(xt+yz)&x^2+z^2-y^2-t^2&2(zt-xy)\\ 2(yt-xz)&2(xy+zt)&x^2+t^2-y^2-z^2 \end{pmatrix}$$ Here $(x,y,z,t)\in S^3$ comes from the standard cover map $S^3\simeq SU(2)\to SO(3)$. Now by linearizing, we must prove that for any $(x,y,z,t)\in\mathbb R^4$ we have: $$||U||_1\leq 5(x^2+y^2+z^2+t^2)$$ The proof of this latter inequality is routine, and the equality situation turns to hold indeed exactly for the matrix $U_3=2J_3-1_3$ and its conjugates. See [@bc1]. Finally, let us mention that a version of Proposition 1.2 above, using the Hölder inequality, shows that the matrices of type $U=H/\sqrt{N}$ with $H\in M_N(\pm 1)$ Hadamard maximize the $p$-norm on $O(N)$ at $p\in [1,2)$, and minimize it at $p\in(2,\infty]$. See [@bc1]. Part of the above $p=1$ results extend to the general setting $p\in [1,\infty]-\{2\}$, and in particular to the exponents $p=4$ and $p=\infty$, which are of particular interest in connection with several quantum physics questions. This will be discussed in a forthcoming paper. The circulant case ================== In this section we study the almost Hadamard matrices which are circulant. We recall that a matrix $H\in M_N(\mathbb C)$ is called circulant if it is of the form: $$H= \begin{pmatrix} \gamma_0&\gamma_1&\ldots&\gamma_{N-1}\\ \gamma_{N-1}&\gamma_0&\ldots&\gamma_{N-2}\\ \ldots&\ldots&\ldots&\ldots\\ \gamma_1&\gamma_2&\ldots&\gamma_0 \end{pmatrix}$$ Let $F\in U(N)$ be the Fourier matrix, given by $F_{ij}=\omega^{ij}/\sqrt{N}$, where $\omega=e^{2\pi i/N}$. Given a vector $\alpha\in\mathbb C^n$, we associate to it the diagonal matrix $\alpha'=diag(\alpha_0,\ldots,\alpha_{N-1})$. We will make a heavy use of the following well-known result: For a matrix $H\in M_N(\mathbb C)$, the following are equivalent: 1. $H$ is circulant, i.e. $H_{ij}=\gamma_{j-i}$, for a certain vector $\gamma\in\mathbb C^N$. 2. $H$ is Fourier-diagonal, i.e. $H=FDF^*$, with $D\in M_N(\mathbb C)$ diagonal. In addition, if so is the case, then with $D=\sqrt{N}\alpha'$ we have $\gamma=F\alpha$. (1)$\implies$(2) The matrix $D=F^*HF$ is indeed diagonal, given by: $$D_{ij}=\frac{1}{N}\sum_{kl}\omega^{jl-ik}\gamma_{l-k}=\delta_{ij}\sum_r\omega^{jr}\gamma_r$$ (2)$\implies$(1) The matrix $H=FDF^*$ is indeed circulant, given by: $$H_{ij}=\sum_kF_{ik}D_{kk}\bar{F}_{jk}=\frac{1}{N}\sum_k\omega^{(i-j)k}D_{kk}$$ Finally, the last assertion is clear from the above formula of $H_{ij}$. Let us investigate now the circulant orthogonal matrices. We let the matrix indices $i,j$ vary modulo $N$. We denote by $\mathbb T$ the unit circle in the complex plane. For a matrix $U\in M_N(\mathbb C)$, the following are equivalent: 1. $U$ is orthogonal and circulant. 2. $U=F\alpha'F^*$ with $\alpha\in\mathbb T^N$ satisfying $\bar{\alpha}_i=\alpha_{-i}$ for any $i$. We will use many times the fact that given $\alpha\in\mathbb C^N$, the vector $\gamma=F\alpha$ is real if and only if $\bar{\alpha}_i=\alpha_{-i}$ for any $i$. This follows indeed from $\overline{F\alpha}=F\tilde{\alpha}$, with $\tilde{\alpha}_i=\bar{\alpha}_{-i}$. (1)$\implies$(2) Write $H_{ij}=\gamma_{j-i}$ with $\gamma\in\mathbb R^N$. By using Proposition 2.1 we obtain $H=FDF^*$ with $D=\sqrt{N}\alpha'$ and $\gamma=F\alpha$. Now since $U=F\alpha'F^*$ is unitary, so is $\alpha'$, so we must have $\alpha\in\mathbb T^N$. Finally, since $\gamma$ is real we have $\bar{\alpha}_i=\alpha_{-i}$, and we are done. (2)$\implies$(1) We know from Proposition 2.1 that $U$ is circulant. Also, from $\alpha\in\mathbb T^N$ we obtain that $\alpha'$ is unitary, and so must be $U$. Finally, since we have $\bar{\alpha}_i=\alpha_{-i}$, the vector $\gamma=F\alpha$ is real, and hence we have $U\in M_N(\mathbb R)$, which finishes the proof. Let us discuss now the almost Hadamard case. First, in the usual Hadamard case, the known examples and the corresponding $\alpha$-vectors are as follows: The known circulant Hadamard matrices, namely $$\pm\begin{pmatrix} -1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!-1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!-1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!-1 \end{pmatrix}, \pm\begin{pmatrix} 1\!\!&\!\!-1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!-1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!-1\\ -1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!1 \end{pmatrix}, \pm\begin{pmatrix} 1\!\!&\!\!1\!\!&\!\!-1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!-1\\ -1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!-1\!\!&\!\!1\!\!&\!\!1 \end{pmatrix}, \pm\begin{pmatrix} 1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!-1\\ -1\!\!&\!\!1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!-1\!\!&\!\!1\!\!&\!\!1\\ 1\!\!&\!\!1\!\!&\!\!-1\!\!&\!\!1 \end{pmatrix}$$ come from the vectors $\alpha=\pm(1,-1,-1,-1),\pm(1,-i,1,i),\pm(1,1,-1,1),\pm(1,i,1,-i)$. At $N=4$ the conjugate of the Fourier matrix is given by: $$F^*=\frac{1}{2}\begin{pmatrix} 1&1&1&1\\ 1&-i&-1&i\\ 1&-1&1&-1\\ 1&i&-1&-i \end{pmatrix}$$ Thus the vectors $\alpha=F^*\gamma$ are indeed those in the statement. We have the following “almost Hadamard” generalization of the above matrices: If $q^N=1$ then the vector $\alpha=\pm(1,-q,-q^2,\ldots,-q^{N-1})$ produces an almost Hadamard matrix, which is equivalent to $K_N=\sqrt{N}(2J_N-1_N)$. Observe first that these matrices generalize those in Proposition 2.3. Indeed, at $N=4$ the choices for $q$ are $1,i,-1,-i$, and this gives the above $\alpha$-vectors. Assume that the $\pm$ sign in the statement is $+$. With $q=\omega^r$, we have: $$\sqrt{N}\gamma_i=\sum_{k=0}^{N-1}\omega^{ik}\alpha_k=1-\sum_{k=1}^{N-1}\omega^{(i+r)k}=2-\sum_{k=0}^{N-1}\omega^{(i+r)k}=2-\delta_{i,-r}N$$ In terms of the standard long cycle $(C_N)_{ij}=\delta_{i+1,j}$, we obtain: $$H=\sqrt{N}(2J_N-C_N^{-r})$$ Thus $H$ is equivalent to $K_N$, and by Proposition 1.5, it is almost Hadamard. In general, the construction of circulant almost Hadamard matrices is quite a tricky problem. At the abstract level, we have the following technical result: A circulant matrix $H\in M_N(\mathbb R^*)$, written $H_{ij}=\gamma_{j-i}$, is almost Hadamard if and only if the following conditions are satisfied: 1. The vector $\alpha=F^*\gamma$ satisfies $\alpha\in\mathbb T^N$. 2. With $\varepsilon={\rm sgn}(\gamma)$, $\rho_i=\sum_r\varepsilon_r\gamma_{i+r}$ and $\nu=F^*\rho$, we have $\nu>0$. In addition, if so is the case, then $\bar{\alpha}_i=\alpha_{-i}$, $\rho_i=\rho_{-i}$ and $\nu_i=\nu_{-i}$ for any $i$. According to Definition 1.4 our matrix $H$ is almost Hadamard if any only if the matrix $U=H/\sqrt{N}$ is orthogonal and $SU^t>0$, where $S_{ij}={\rm sgn}(U_{ij})$. By Lemma 2.2 the orthogonality of $U$ is equivalent to the condition (1). Regarding now the condition $SU^t>0$, this is equivalent to $S^tU>0$. But, with $k=i-r$, we have: $$(S^tH)_{ij}=\sum_kS_{ki}H_{kj}=\sum_k\varepsilon_{i-k}\gamma_{j-k}=\sum_r\varepsilon_r\gamma_{j-i+r}=\rho_{j-i}$$ Thus $S^tU$ is circulant, with $\rho/\sqrt{N}$ as first row. From Proposition 2.1 we get $S^tU=FLF^*$ with $L=\nu'$ and $\nu=F^*\rho$, so $S^tU>0$ iff $\nu>0$, which is the condition (2). Finally, the assertions about $\alpha,\nu$ follow from the fact that $F\alpha,F\nu$ are real. As for the assertion about $\rho$, this follows from the fact that $S^tU$ is symmetric. For $N$ odd the following matrix is almost Hadamard, $$L_N=\frac{1}{\sqrt{N}} \begin{pmatrix} 1&-\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&\ldots&\cos^{-1}\frac{(N-1)\pi}{N}\\ \cos^{-1}\frac{(N-1)\pi}{N}&1&-\cos^{-1}\frac{\pi}{N}&\ldots&-\cos^{-1}\frac{(N-2)\pi}{N}\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ -\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&-\cos^{-1}\frac{3\pi}{N}&\ldots&1 \end{pmatrix}$$ and comes from an $\alpha$-vector having all entries equal to $1$ or $-1$. Write $N=2n+1$, and consider the following vector: $$\alpha_i=\begin{cases} (-1)^{n+i}&{\rm for }\ i=0,1,\ldots,n\\ (-1)^{n+i+1}&{\rm for}\ i=n+1,\ldots,2n \end{cases}$$ Let us first prove that $(L_N)_{ij}=\gamma_{j-i}$, where $\gamma=F\alpha$. With $\omega=e^{2\pi i/N}$ we have: $$\sqrt{N}\gamma_i=\sum_{j=0}^{2n}\omega^{ij}\alpha_j=\sum_{j=0}^n(-1)^{n+j}\omega^{ij}+\sum_{j=1}^n(-1)^{n+(N-j)+1}\omega^{i(N-j)}$$ Now since $N$ is odd, and since $\omega^N=1$, we obtain: $$\sqrt{N}\gamma_i=\sum_{j=0}^n(-1)^{n+j}\omega^{ij}+\sum_{j=1}^n(-1)^{n-j}\omega^{-ij}=\sum_{j=-n}^n(-1)^{n+j}\omega^{ij}$$ By computing the sum on the right, with $\xi=e^{\pi i/N}$ we get, as claimed: $$\sqrt{N}\gamma_i=\frac{2\omega^{-ni}}{1+\omega^i}=\frac{2\xi^{-2ni}}{1+\xi^{2i}}=\frac{2\xi^{-Ni}}{\xi^{-i}+\xi^i}=(-1)^i\cos^{-1}\frac{i\pi}{N}$$ In order to prove now that $L_N$ is almost Hadamard, we use Lemma 2.5. Since the sign vector is simply $\varepsilon=(-1)^n\alpha$, the vector $\rho_i=\sum_r\varepsilon_r\gamma_{i+r}$ is given by: $$\sqrt{N}\rho_i=(-1)^n\sum_{r=0}^{2n}\alpha_r\sum_{j=-n}^n(-1)^{n+j}\omega^{(i+r)j}=\sum_{j=-n}^n(-1)^j\omega^{ij}\sum_{r=0}^{2n}\alpha_r\omega^{rj}$$ Now since the last sum on the right is $(\sqrt{N}F\alpha)_j=\sqrt{N}\gamma_j$, we obtain: $$\rho_i=\sum_{j=-n}^n(-1)^j\omega^{ij}\gamma_j=\frac{1}{\sqrt{N}}\sum_{j=-n}^n(-1)^j\omega^{ij}\sum_{k=-n}^n(-1)^{n+k}\omega^{jk}$$ Thus we have the following formula: $$\rho_i=\frac{(-1)^n}{\sqrt{N}}\sum_{j=-n}^n\sum_{k=-n}^n(-1)^{j+k}\omega^{(i+k)j}$$ Let us compute now the vector $\nu=F^*\rho$. We have: $$\nu_l=\frac{1}{\sqrt{N}}\sum_{i=0}^{2n}\omega^{-il}\rho_i=\frac{(-1)^n}{N}\sum_{j=-n}^n\sum_{k=-n}^n(-1)^{j+k}\omega^{jk}\sum_{i=0}^{2n}\omega^{i(j-l)}$$ The sum on the right is $N\delta_{jl}$, with both $j,l$ taken modulo $N$, so it is equal to $N\delta_{jL}$, where $L=l$ for $l\leq n$, and $L=l-N$ for $l>n$. We get: $$\nu_l=(-1)^n\sum_{k=-n}^n(-1)^{L+k}\omega^{Lk}=(-1)^{n+L}\sum_{k=-n}^n(-w^L)^k$$ With $\xi=e^{\pi i/N}$, this gives the following formula: $$\nu_l=(-1)^{n+L}\frac{2(-\omega^L)^{-n}}{1+\omega^L}=(-1)^L\frac{2\omega^{-nL}}{1+\omega^L}$$ In terms of the variable $\xi=e^{\pi i/N}$, we obtain: $$\nu_l=(-1)^L\frac{2\xi^{-2nL}}{1+\xi^{2L}}=(-1)^L\frac{2\xi^{-NL}}{\xi^{-L}+\xi^L}=\cos^{-1}\frac{L\pi}{N}$$ Now since $L\in[-n,n]$, all the entries of $\nu$ are positive, and we are done. At the level of examples now, at $N=3$ we obtain the matrix $L_3=-K_3$. At $N=5$ we obtain a matrix having as entries 1 and $x=-\cos^{-1}\frac{\pi}{5}$, $y=\cos^{-1}\frac{2\pi}{5}$: $$L_5=\frac{1}{\sqrt{5}}\begin{pmatrix} 1&x&y&y&x\\ x&1&x&y&y\\ y&x&1&x&y\\ y&y&x&1&x\\ x&y&y&x&1 \end{pmatrix}$$ Let us look now more in detail at the vectors $\alpha\in\mathbb T^N$ appearing in Proposition 2.4 and in the proof of Theorem 2.6. In both cases we have $\alpha_i^2=\omega^{ri}$ for a certain $r\in\mathbb N$, and this might suggest that any circulant almost Hadamard matrix should come from a vector $\alpha\in\mathbb T^N$ having the property that $\alpha^2$ is formed by roots of unity in a progression. However, the rescaled adjacency matrix of the Fano plane, to be discussed in the next section, is circulant almost Hadamard, but does not have this property. The problem of finding the correct extension of the circulant Hadamard conjecture to the almost Hadamard matrix case is a quite subtle one, that we would like to raise here. The two-entry case ================== In this section we study the almost Hadamard matrices having only two entries, $H\in M_N(x,y)$, with $x,y\in\mathbb R$. As a first remark, the usual Hadamard matrices $H\in M_N(\pm 1)$ are of this form. However, when trying to build a combinatorial hierarchy of the two-entry almost Hadamard matrices, the usual Hadamard matrices stand on top, and there is not so much general theory that can be developed, as to cover them. We will therefore restrict attention to the following special type of matrices: An $(a,b,c)$ pattern is a matrix $M\in M_N(x,y)$, with $N=a+2b+c$, such that, in any two rows, the number of $x/y/x/y$ sitting below $x/x/y/y$ is $a/b/b/c$. In other words, given any two rows of our matrix, we are asking for the existence of a permutation of the columns such that these two rows become: $$\begin{matrix} x\ldots x&x\ldots x&y\ldots y&y\ldots y\\ \underbrace{x\ldots x}_a&\underbrace{y\ldots y}_b&\underbrace{x\ldots x}_b&\underbrace{y\ldots y}_c \end{matrix}$$ Oberve that the Hadamard matrices do not come in general from patterns. However, there are many interesting examples of patterns coming from block designs [@cdi], [@sti]: A $(v,k,\lambda)$ symmetric balanced incomplete block design is a collection $B$ of subsets of a set $X$, called blocks, with the following properties: 1. $|X|=|B|=v$. 2. Each block contains exactly $k$ points from $X$. 3. Each pair of distinct points is contained in exactly $\lambda$ blocks of $B$. The incidence matrix of a such block design is the $v\times v$ matrix defined by: $$M_{bx}=\begin{cases} 1&\text{if }x\in b\\ 0&\text{if }x\notin b \end{cases}$$ The connection between designs and patterns comes from: If $N=a+2b+c$ then the adjacency matrix of any $(N,a+b,a)$ symmetric balanced incomplete block design is an $(a,b,c)$ pattern. Indeed, let us replace the $0-1$ values in the adjacency matrix $M$ by abstract $x-y$ values. Then each row of $M$ contains $a+b$ copies of $x$ and $b+c$ copies of $y$, and since every pair of distinct blocks intersect in exactly $a$ points, cf. [@sti], we see that every pair of rows has exactly $a$ variables $x$ in matching positions, so that $M$ is an $(a,b,c)$ pattern. As a first example, consider the Fano plane. The sets $X,B$ of points and lines form a $(7,3,1)$ block design, corresponding to the following $(1,2,2)$ pattern: $$I_7=\begin{pmatrix} x&x&y&y&y&x&y\\ y&x&x&y&y&y&x\\ x&y&x&x&y&y&y\\ y&x&y&x&x&y&y\\ y&y&x&y&x&x&y\\ y&y&y&x&y&x&x\\ x&y&y&y&x&y&x \end{pmatrix}$$ Now remember that the Fano plane is the projective plane over $\mathbb F_2=\{0,1\}$. The same method works with $\mathbb F_2$ replaced by an arbitrary finite field $\mathbb F_q$, and we get: Assume that $q=p^k$ is a prime power. Then the point-line incidence matrix of the projective plane over $\mathbb F_q$ is a $(1,q,q^2-q)$ pattern. The sets $X,B$ of points and lines of the projective plane over $\mathbb F_q$ are known to form a $(q^2+q+1,q+1,1)$ block design, and this gives the result. There are many other interesting examples of symmetric balanced incomplete block designs, all giving rise to patterns, via Proposition 3.3. For instance the famous Paley biplane [@bro], pictured below, is a $(11,5,2)$ block design, and hence gives rise to a $(2,3,3)$ pattern. When assigning certain special values to the parameters $x,y$ we obtain a $11\times 11$ almost Hadamard matrix, that we believe to be optimal. See section 4 below. We consider now the problem of associating real values to the symbols $x,y$ in an $(a,b,c)$ pattern such that the resulting matrix $U(x,y)$ is orthogonal. Given $a,b,c\in\mathbb N$, there exists an orthogonal matrix having pattern $(a,b,c)$ iff $b^2\geq ac$. In this case the solutions are $U(x,y)$ and $-U(x,y)$, where $$x=-\frac{t}{\sqrt{b}(t+1)},\quad\quad y=\frac{1}{\sqrt{b}(t+1)}$$ where $t=(b\pm\sqrt{b^2-ac})/a$ can be any of the solutions of $at^2-2bt+c=0$. First, in order for $U$ to be orthogonal, the following conditions must be satisfied: $$ax^2+2bxy+cy^2=0,\quad (a+b)x^2+(b+c)y^2=1$$ The first condition, coming from the orthogonality of rows, tells us that $t=-x/y$ must be the variable in the statement. As for the second condition, this becomes: $$y^2=\frac{1}{(a+b)t^2+(b+c)}=\frac{1}{(at^2+c)+(bt^2+b)}=\frac{1}{2bt+bt^2+b}=\frac{1}{b(t+1)^2}$$ This gives the above formula of $y$, and hence the formula of $x=-ty$ as well. Let $U=U(x,y)$ be orthogonal, corresponding to an $(a,b,c)$ pattern. Then $H=\sqrt{N}U$ is almost Hadamard iff $(N(a-b)+2b)|x|+(N(c-b)+2b)|y|\geq 0$. We use the criterion in Definition 1.4 (2). So, let $S_{ij}={\rm sgn}(U_{ij})$. Since any row of $U$ consists of $a+b$ copies of $x$ and $b+c$ copies of $y$, we have: $$(SU^t)_{ii}=\sum_k{\rm sgn}(U_{ik})U_{ik}=(a+b)|x|+(b+c)|y|$$ Regarding now $(SU^t)_{ij}$ with $i\neq j$, we can assume in the computation that the $i$-th and $j$-th row of $U$ are exactly those pictured after Definition 3.1 above. Thus: $$\begin{aligned} (SU^t)_{ij} &=&\sum_k{\rm sgn}(U_{ik})U_{jk}\\ &=&a\,{\rm sgn}(x)x+b\,{\rm sgn}(x)y+b\,{\rm sgn}(y)x+c\,{\rm sgn}(y)y\\ &=&a|x|-b|y|-b|x|+c|y|\\ &=&(a-b)|x|+(c-b)|y|\end{aligned}$$ We obtain the following formula for the matrix $SU^t$ itself: $$\begin{aligned} SU^t &=&2b(|x|+|y|)1_N+((a-b)|x|+(c-b)|y|)NJ_N\\ &=&2b(|x|+|y|)(1_N-J_N)+((N(a-b)+2b)|x|+(N(c-b)+2b)|y|))J_N\end{aligned}$$ Now since the matrices $1_N-J_N,J_N$ are orthogonal projections, we have $SU^t>0$ if and only if the coefficients of these matrices in the above expression are both positive. Since the coefficient of $1_N-J_N$ is clearly positive, the condition left is: $$(N(a-b)+2b)|x|+(N(c-b)+2b)|y|\geq 0$$ So, we have obtained the condition in the statement, and we are done. Assume that $a,b,c\in\mathbb N$ satisfy $c\geq a$ and $b(b-1)=ac$, and consider the $(a,b,c)$ pattern $U=U(x,y)$, where: $$x=\frac{a+(1-a-b)\sqrt{b}}{Na},\quad y=\frac{b+(a+b)\sqrt{b}}{Nb}$$ Then $H=\sqrt{N}U$ is an almost Hadamard matrix. We have $b^2-ac=b$, so Lemma 2.5 applies, and shows that with $t=(b-\sqrt{b})/a$ we have an orthogonal matrix $U=U(x,y)$, where: $$x=-\frac{t}{\sqrt{b}(t+1)},\quad y=\frac{1}{\sqrt{b}(t+1)}$$ In order to compute these variables, we use the following formula: $$(a+b)^2-b=a^2+b^2+2ab-b=a^2+2ab+ac=Na$$ This gives indeed the formula of $y$ in the statement: $$y=\frac{a}{(a+b)\sqrt{b}-b}=\frac{(a+b)\sqrt{b}+b}{Nb}$$ As for the formula of $x$, we can obtain it as follows: $$x=-ty=\frac{(\sqrt{b}-b)((a+b)\sqrt{b}+b)}{Nab}=\frac{a+(1-a-b)\sqrt{b}}{Na}$$ Let us compute now the quantity appearing in Lemma 3.6. We have: $$\begin{aligned} N(a-b)+2b &=&(a+2b+c)(a-b)+2b\\ &=&a^2+ab-2b^2+ac-bc+2b\\ &=&a^2+ab-ac-bc\\ &=&(a-c)(a+b)\end{aligned}$$ Similarly, $N(c-b)+2b=(c-a)(c+b)$, so the quantity in Lemma 3.6 is $Ky$, with: $$\begin{aligned} K &=&(a-c)(a+b)t+(c-a)(c+b)\\ &=&(c-a)(c+b-(a+b)t)\\ &=&\frac{c-a}{a}(ac+ab-(a+b)(b-\sqrt{b}))\\ &=&\frac{c-a}{a}((ac-b^2)+(a+b)\sqrt{b})\\ &=&\frac{c-a}{a}((a+b)\sqrt{b}-b)\end{aligned}$$ Since this quantity is positive, Lemma 3.6 applies and gives the result. Assume that $q=p^k$ is a prime power. Then the matrix $I_N\in M_N(x,y)$, where $N=q^2+q+1$ and $$x=\frac{1-q\sqrt{q}}{\sqrt{N}},\quad y=\frac{q+(q+1)\sqrt{q}}{q\sqrt{N}}$$ having $(1,q,q^2-q)$ pattern coming from the point-line incidence of the projective plane over $\mathbb F_q$ is an almost Hadamard matrix. Indeed, the conditions $c\geq a$ and $b(b-1)=ac$ needed in Proposition 3.7 are satisfied, and the variables constructed there are $x'=x/\sqrt{N}$ and $y'=y/\sqrt{N}$. There are of course many other interesting examples of two-entry almost Hadamard matrices, all worth investigating in detail, but we will stop here. Indeed, the main purpose of the reminder of this paper is to provide a list of almost Hadamard matrices which are “as optimal as possible”, at $N=2,3,\ldots,13$, and in order to establish this list, we will just need the incidence matrices $I_N$, plus the matrix coming from the Paley biplane. Let us mention however two more important aspects of the general theory: 1. The series $I_N$ is the particular case of a 2-parameter series $I_N^{(d)}$. Indeed, associated to $q=p^k$ and $d\in\mathbb N$ is a certain $([d+2]_q,[d+1]_q,[d]_q)$ block design coming from $\mathbb F_q$, where $[e]_q=(q^e-1)/(q-1)$, cf. [@cdi], [@sti]. Thus by Proposition 3.7 we obtain an almost Hadamard matrix $I_N^{(d)}$, having pattern $(\frac{q^d-1}{q-1},q^d,q^d(q-1))$. 2. Trying to find block designs and patterns leads to the following chess problem: consider a $N\times N$ chessboard, take $NM$ rooks with $M\leq N/2$, fix an integer $K\leq M$, and try to place all the rooks on the board such that: (a) there are exactly $M$ rooks on each row and each column of the board, and (b) for any pair of rows or columns, there are exactly $K$ pairs of mutually attacking rooks. Finally, let us mention that there are many questions raised by the almost Hadamard matrices, at the quantum algebraic level. For instance the symmetries of Hadamard matrices are known to be described by quantum permutations [@ba2], and it would be interesting to have a similar result for the almost Hadamard matrices. This might probably bring some new ideas on the “homogeneous implies quantum homogeneous” question, raised in [@ba1], and having connections with the finite projective planes. Also, an interesting link between mutually unbiased bases, complex Hadamard matrices and affine planes was emphasized in [@ber], but its relation with our present investigations is not known yet. List of examples ================ In this section we present a list of examples of almost Hadamard matrices, for small values of $N$. Since we are mainly interested in the optimal case, our examples will be chosen to be “as optimal as possible”, i.e. will be chosen as to have big $1$-norms. So, let us first compute the 1-norms for the examples that we have. In what follows $L_N$ is the matrix found in Theorem 2.6, and $I_N$ is the matrix found in Theorem 3.8. The $1$-norms of the basic examples of almost Hadamard matrices are: 1. Hadamard case: $||H/\sqrt{N}||_1=N\sqrt{N}$, for any $H\in M_N(\pm 1)$ Hadamard. 2. Basic series case: $||K_N/\sqrt{N}||_1=3N-4$, where $K_N=\sqrt{N}(2J_N-1_N)$. 3. Circulant series case: $||L_N/\sqrt{N}||_1=\frac{2}{\pi}N\log N+O(N)$. 4. Incidence series case: $||I_N/\sqrt{N}||_1=(q^2-q-1)+2q(q+1)\sqrt{q}$. The first two assertions are clear. For the third one, with $N=2n+1$ we have: $$||L_N/\sqrt{N}||_1=2\sum_{i=0}^n\cos^{-1}\frac{i\pi}{N}+O(N)=2\sum_{i=0}^n\sin^{-1}\frac{(2i+1)\pi}{2N}+O(N)$$ Now by using $\sin x\sim x$ and $\sum_{i=1}^k1/i=\log k+O(1)$ we obtain, as claimed: $$||L_N/\sqrt{N}||_1=\frac{4N}{\pi}\sum_{i=0}^n\frac{1}{2i+1}+O(N)=\frac{2N}{\pi}\log N+O(N)$$ As for the last assertion, let first $U$ be the matrix in Lemma 3.6. We have: $$||U||_1=N((a+b)|x|+(b+c)|y|)$$ In the particular case of the orthogonal matrices in Proposition 3.7, we get: $$||U||_1=(c-a)+\frac{(a+b)(2a+2b-2)}{a}\sqrt{b}$$ Now with $a=1$, $b=q$, $c=q^2-q$, this gives the formula in the statement. Observe that at $N$ big the above matrices $K_N,L_N,I_N$ are far from being optimal. In fact, with $N\to\infty$, the corresponding orthogonal matrices don’t even match the average of the 1-norm on $O(N)$, which, according to [@bc1], is $\sim cN\sqrt{N}$, with $c=0.797..$ With these ingredients in hand, let us discuss now the various examples: At $N=2$ we have the Walsh matrix $H_2$, which is of course optimal. At $N=3$ we have the almost Hadamard matrix $K_3$, which is optimal: $$K_3=\frac{1}{\sqrt{3}}\begin{pmatrix}-1&2&2\\ 2&-1&2\\ 2&2&-1\end{pmatrix}$$ At $N=4$ we have the Hadamard matrix $H_4\sim K_4$, once again optimal. At $N=5$ we have the matrix $K_5$, that we believe to be optimal as well: $$K_5=\frac{1}{\sqrt{5}}\begin{pmatrix}-3&2&2&2&2\\ 2&-3&2&2&2\\ 2&2&-3&2&2\\ 2&2&2&-3&2\\ 2&2&2&2&-3\end{pmatrix}$$ At $N=6$ it is plausible that the optimal AHM is simply $K_3\otimes H_2$: $$K_3\otimes H_2=\frac{1}{\sqrt{3}}\begin{pmatrix}-1&2&2&-1&2&2\\ 2&-1&2&2&-1&2\\ 2&2&-1&2&2&-1\\ -1&2&2&1&-2&-2\\ 2&-1&2&-2&1&-2\\ 2&2&-1&-2&-2&1\end{pmatrix}$$ At $N=7$ we have the incidence matrix of the Fano plane ($x=2-4\sqrt{2}$, $y=2+3\sqrt{2}$): $$I_7=\frac{1}{2\sqrt{7}}\begin{pmatrix} x&x&y&y&y&x&y\\ y&x&x&y&y&y&x\\ x&y&x&x&y&y&y\\ y&x&y&x&x&y&y\\ y&y&x&y&x&x&y\\ y&y&y&x&y&x&x\\ x&y&y&y&x&y&x \end{pmatrix}$$ At $N=8$ we have the third Walsh matrix $H_8=H_2\otimes H_4$, of course optimal. At $N=9$ we just have the matrix $K_3\otimes K_3$, which can be shown not to be optimal. At $N=10$ we believe that the matrix $K_5\otimes H_2$ is optimal: $$K_5\otimes H_2=\frac{1}{\sqrt{5}}\begin{pmatrix} -3&2&2&2&2&-3&2&2&2&2\\ 2&-3&2&2&2&2&-3&2&2&2\\ 2&2&-3&2&2&2&2&-3&2&2\\ 2&2&2&-3&2&2&2&2&-3&2\\ 2&2&2&2&-3&2&2&2&2&-3\\ -3&2&2&2&2&3&-2&-2&-2&-2\\ 2&-3&2&2&2&-2&3&-2&-2&-2\\ 2&2&-3&2&2&-2&-2&3&-2&-2\\ 2&2&2&-3&2&-2&-2&-2&3&-2\\ 2&2&2&2&-3&-2&-2&-2&-2&3 \end{pmatrix}$$ At $N=11$ we have the matrix of the Paley biplane ($x=6-12\sqrt{3}$, $y=6+10\sqrt{3}$): $$P_{11}=\frac{1}{6\sqrt{11}}\left(\begin{array}{cccccccccccccc} y& x& y& x& x& x& y& y& y& x& y\\ y& y& x& y& x& x& x& y& y& y& x\\ x& y& y& x& y& x& x& x& y& y& y\\ y& x& y& y& x& y& x& x& x& y& y\\ y& y& x& y& y& x& y& x& x& x& y\\ y& y& y& x& y& y& x& y& x& x& x\\ x& y& y& y& x& y& y& x& y& x& x\\ x& x& y& y& y& x& y& y& x& y& x\\ x& x& x& y& y& y& x& y& y& x& y\\ y& x& x& x& y& y& y& x& y& y& x\\ x& y& x& x& x& y& y& y& x& y& y \end{array}\right)$$ At $N=12$ we have the Sylvester Hadamard matrix $S_{12}$, of course optimal. At $N=13$ we have the incidence matrix of $P(\mathbb F_3)$ ($x=3-9\sqrt{3}$, $y=3+4\sqrt{3}$): $$I_{13}=\frac{1}{3\sqrt{13}}\left(\begin{array}{cccccccccccccc} x&x&x&x&y&y&y&y&y&y&y&y&y\\ x&y&y&y&x&x&x&y&y&y&y&y&y\\ x&y&y&y&y&y&y&x&x&x&y&y&y\\ x&y&y&y&y&y&y&y&y&y&x&x&x\\ y&x&y&y&y&y&x&y&x&y&y&y&x\\ y&x&y&y&y&x&y&y&y&x&x&y&y\\ y&x&y&y&x&y&y&x&y&y&y&x&y\\ y&y&x&y&y&y&x&x&y&y&x&y&y\\ y&y&x&y&x&y&y&y&y&x&y&y&x\\ y&y&x&y&y&x&y&y&x&y&y&x&y\\ y&y&y&x&y&x&y&x&y&y&y&y&x\\ y&y&y&x&y&y&x&y&y&x&y&x&y\\ y&y&y&x&x&y&y&y&x&y&x&y&y \end{array}\right)$$ The norms of the above matrices can be computed by using the various formulae in Theorem 4.1 and its proof, and the results are summarized in Table 1. $N$ matrix 1-norm (formula) 1-norm (numeric) $N\sqrt{N}$ remarks ----- ------------------ ------------------ ------------------ ------------- ---------- -- 2 $H_2$ $2\sqrt{2}$ 2.828 2.828 Hadamard 3 $K_3$ 5 5.000 5.196 optimal 4 $K_4$ 8 8.000 8.000 Hadamard 5 $K_5$ 11 11.000 11.118 6 $K_3\otimes H_2$ $10\sqrt{2}$8 14.142 14.697 7 $I_7$ $1+12\sqrt{2}$ 17.971 18.520 8 $H_8$ $16\sqrt{2}$ 22.627 22.627 Hadamard 9 ? – $>26.513$ 27.000 10 $K_5\otimes H_2$ $22\sqrt{2}$ 31.113 31.623 11 $P_{11}$ $1+20\sqrt{3}$ 35.641 36.483 12 $S_{12}$ $24\sqrt{3}$ 41.569 41.569 Hadamard 13 $I_{13}$ $5+24\sqrt{3}$ 46.569 46.872 : Almost Hadamard matrices $H\in M_N(\mathbb R)$, chosen as for the corresponding orthogonal matrices $U=H/\sqrt{N}$ to have big 1-norm. All matrices are believed to be optimal. The lower bound for the maximum of the 1-norm on $O(9)$, which is not optimal, was obtained by numerical simulation.[]{data-label="table1"} Conclusion ========== We have seen in this paper that the Hadamard matrices are quite nicely generalized by the almost Hadamard matrices (AHM), which exist at any given order $N\in\mathbb N$. Our study of these matrices, which was for the most of algebraic nature, turns to be related to several interesting combinatorial problems, notably to the Circulant Hadamard Conjecture. We believe that the AHM can be used as well in connection with several problems in quantum physics, in a way somehow similar to the way the complex Hadamard matrices (CHM) are used. Indeed, since the CHM exist as well at any given order $N\in\mathbb N$, these matrices proved to be useful in several branches of quantum physics. For instance in quantum optics they are sometimes called the “Zeilinger matrices”, as they can be applied to design symmetric linear multiports, used to split the beam into $N$ parts of the same intensity and to analyze interference effect [@rz+], [@jsz]. In the theory of quantum information one uses quantum Hadamard matrices to construct mutually unbiased bases (MUB) [@iva], [@de+] to design teleportation and dense coding schemes. As shown in the seminal work of Werner [@wer] these two problems are in fact equivalent and also equivalent to construction of unitary depolarisers and maximally entangled bases [@wgc]. Although quantum mechanics in a natural way relays on a complex Hilbert space it is often convenient to study a simplified problem and restrict attention to the subset of real quantum states only. Such an approach can be useful in theoretical investigations of quantum entanglement [@cfr] or also in experimental studies on engineering of quantum states, as creating of a real state by an orthogonal rotation usually requires less effort than construction of an arbitrary complex state. There exists therefore a natural motivation to ask similar problems concerning e.g. unbiased bases and teleportation schemes in the real setup. For instance, it is known that for any $N$ there exist $\leq N/2+1$ real MUB and for most dimensions their actual number is not larger than 3, cf. [@bs+], while for any prime $N$ there exist $N+1$ complex MUB. Note that the real MUB and Hadamard matrices are closely related to several combinatorial problems [@hko], [@lmo], [@mrw]. In the case the maximal number of real MUB does not exist one can search for an optimal set of real bases which are approximately unbiased. 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--- abstract: 'Thermodynamic metric usually works only for those black holes with more than one conserved charge, therefore the Schwarzschild black hole was excluded. In this letter, we compute and compare different versions of [*offshell*]{} thermodynamic metric for the Schwarzschild-like black hole by introducing a new degree of freedom. This new degree of freedom could be the running Newton constant, a cutoff scale for regular black hole, a noncommutative deformation, or the deformed parameter in the nonextensive Tsallis-R[è]{}nyi entropy. The [*onshell*]{} metric of the deformed Schwarzschild solution would correspond to the submanifold by gauge fixing of this additional degree of freedom. In particular, the thermal Ricci scalar for the Schwarzschild black hole, though different for various deformation, could be obtained by switching off the deformation.' author: - 'Wen-Yu Wen' title: Offshell thermodynamic metrics of the Schwarzschild black hole --- [^1] Introduction ============ The thermodynamic metric has been studied for decades. First concept was brought by Weinhold [@Weinhold:1975], who regarded the Hessian of internal energy as a mretic of quilibrium states. Then there was the Ruppeiner metric [@Ruppeiner:1979] given by the negative Hessian of entropy with respect to conserved charges. It was shown that the Weinhold and Ruppeiner are conformally equivalent to each other [@Salamon:1984]. Quevedo suggested a Legendre invaraint form of metric in the formalism of geometrothermodynamics[@Quevedo:2006xk; @Quevedo:2007ws] and Hendi et al [@Hendi:2015rja] proposed another new metric such that divergence of Ricci scalar coincides with phase transition points. For the black hole with single charge, such as the Schwarzschild black hole, the charge has to be its ADM mass $M$ and represents the internal energy of the system. According to the first law of thermodynamics, its mass can be expressed as a function of the Bekenstein-Hawking entropy $S$, say $M=M(S)$, which is a monotonically increasing function of single varaible and $S$ could be seemed as a reparametrization of a one-dimensional curve for $M\ge 0$. The thermodynamic metric, of course, is trivial for such a one-dimensional space. However, a theory of quantum gravity is expected to generate a ultraviolate (UV) scale, denoting $\Lambda_X$, to prevent the catastrophe of infinite temperature predicted by the Schwarzschild metric at its final stage. Although satisfactory theory of quantum gravity is still unavailable, it may be still possible to construct an effective theory consistent with the General Relativiry (GR) but with the desired UV scale. This effective theory should correspond to a modified Schwarzschild metric and one may assign a one-parameter deformation function $M=M_X(S)$, where deformation parameter $X$ is determined by the desired UV scale $\Lambda_X$. If we regard $X$ to be a new degree of freedom, meaning that scale $\Lambda_X$ is allowed to run freely, we obtain an energy function of two variables instead, say $M=M(S,X)$. In this way, we may have a nontrivial two-dimensional thermodynamic metric and the modified Schwarzschild-like solution is just the submanifold after a specific $X$ is chosen. As an analogy to the gauge theory, one may regard this two-dimensional thermodynamic metric offshell and the onshell effective theory is obtained by gauge fixing $X$. In this letter we will focus on four different effective theories: the Schwarzschild solution with a running Newton’s gravitational constant [@Falls:2012nd], the regular Schwarzschild-like black hole with a UV cutoff [@Hayward:2005gi], the noncommutative geometry inspried Schwarzschild black hole [@Nicolini:2005vd], and the Schwarzschild black hole with a nonextensive Tsallis entropy[@Biro:2013cra]. This paper is organized as follows: in the section II, we studied various thermodynamic metric for the Schwarzschild black hole with a running Newton’s gravitational constant $G$. We found it give vanishing Ricci scalar if one simply regards $G$ as a free parameter. In stead, if one regarded the UV scale on which $G$ depends as the new degree of freedom, then nontrivial Ricci scalar can be obtained. In the section III, we studied various thermodynamic metric for the regular Schwarzschild-like black hole and regarded the UV length scale as the free parameter. Zeros of heat capacities and poles in the Ricci scalars in each thermodynamic metric were computed and compared. In the limit of ordinary Schwarzschild black hole, most Ricci scalars are found inversely proportional to the entropy to some power. In the section IV, we studied various thermodynamic metric for the noncommutative geometry inspired Schwarzschild black hole. We computed and compared the zeros and spikes in the heat capacities with the poles and spikes in the Ricci scalars. The heat capacities bahave like the ordinary Schwarzschild black hole for large entropy, while the Ricci scalar was found inversely proportional to the noncommutative parameter at the extremal limit where the Hawking temperature vanishes. In the section V, we studied various thermodynamic metric for the Schwarzschild black hole but with the Tsallis-R[è]{}nyi entropy. While the thermal Ricci scalars in the Ruppeiner and Weinhold behaved similarly as those in the running Newton constant, they behaved very differently in the HPEM metric. We summarize our result in the section VI. Shcwarzschild black hole with running Newton constant ===================================================== We first regard the Newton’s gravitational constant $G$ as a free parameter and the Bekenstein-Hawking entropy as a function of both $G$ and mass $M$, that is $$\label{eqn:entropy_function} S(G,M) = \frac{4\pi G M^2}{c^4}.$$ The idea of time-varying $G$ was started long ago by Dirac to explain the coincidence between two large dimensionless numbers of ${\cal O}(10^{39})$ [@Dirac:1937]. It appeared later in the Weinberg’s asymptotic saftey for quantum gravity, which is based on the assumption that there exists a nontrivial fixed point for the renormalization group flow of $G$ [@Weinberg:1979]. Now we compute the Ruppeiner metric, which is given by he negative Hessians of the entropy function (\[eqn:entropy\_function\]): $$\begin{aligned} ds^2_R =&& -S_{MM}dMdM - 2S_{MG}dMdG -S_{GG}dGdG\nonumber\\ =&& -\frac{8\pi G}{c^4} dM^2 -\frac{16\pi M}{c^4} dMdG,\end{aligned}$$ where $S_{XY}$ denotes the differential $\partial_X\partial_Y S$. It can be shown that Ruppeiner metric is flat. Now we rewrite the ADM mass as a function of entropy and Newton constant, that is $$M(S,G)=\frac{c^2}{2}\sqrt{\frac{S}{\pi G}},$$ which is identified as the internal energy of the black hole. The Hessians of the internal energy gives rise to the Weinhold metric: $$\begin{aligned} ds^2_W =&& M_{SS}dSdS + 2M_{SG}dSdG + M_{GG}dGdG\nonumber\\ =&& -\frac{c^2}{8\sqrt{\pi}}\frac{1}{\sqrt{S^3G}}dS^2 -\frac{c^2}{4\sqrt{\pi}}\frac{1}{\sqrt{SG^3}} dSdG + \frac{3c^2}{8\sqrt{\pi}}\sqrt{\frac{S}{G^5}}dG^2, \end{aligned}$$ where $M_{XY}$ denotes the differential $\partial_X\partial_Y M$. It can be shown that the Weinhold metric is also flat. Now we move to the Quevedo metric, which is given by[@Quevedo:2008xn] $$\begin{aligned} ds^2_Q =&& \Omega (-M_{SS}dSdS + M_{GG}dGdG) \nonumber\\ =&& \Omega (\frac{c^2}{8\sqrt{\pi}}\frac{1}{\sqrt{S^3G}}dS^2 + \frac{3c^2}{8\sqrt{\pi}}\sqrt{\frac{S}{G^5}}dG^2)\end{aligned}$$ with some conformal factor $\Omega$. In the Quevedo metric of case I, one has the choice $$\Omega_I = SM_S + GM_G = \frac{c^2}{4\sqrt{\pi}}\sqrt{\frac{S}{G}}+(-\frac{c^2}{4\sqrt{\pi}}\sqrt{\frac{S}{G}})=0$$ or $$\Omega_{II} = SM_S = \frac{c^2}{4\sqrt{\pi}}\sqrt{\frac{S}{G}}$$ for the case II. We remark that there could exist infinite number of Legendre-invariant metrics and the above choices were picked up for simplicity. Another new metric proposed by Hendi et al [@Hendi:2015rja], denoting HPEM metric, claimed to remove the unwanted poles in the thermal Ricci scalar. This corresponds to the choice of conformal factor $\Omega_N$: $$\Omega_N = S\frac{M_S}{M_{GG}^3} = \frac{128\pi}{27c^4}\frac{G^7}{S}$$ However, all Quevedo and HPEM metric are flat and have vanishing Ricci scalar. This seems to imply that there is no thermodynamic interaction. To have a nontrivial Ricci scalar, we assume a toy model for running Newton constant[@Falls:2012nd]: $$\frac{1}{G(k)} = \frac{1}{k} + \frac{1}{G_{\infty}},$$ where $k$ is the energy scale. The finite $G_{\infty}$ implies a vanishing Newton constant at IR and asymptotically UV safe theory of gravity. Now we regard energy scale $k$, instead of $G$, as a free parameter and calculate various thermodynamic metric. That is, we consider the entropy function $S=S(M,G(k))=S(M,k)$ for the Ruppeiner metric. The Ricci scalar for the conformally flat Ruppeiner metric is found to be $$\label{eqn:Ricci_regular} R_R = -\frac{c^4}{16\pi G_{\infty} M^2},$$ which is independent of $k$. The Ricci scalar for the Weinhold metric reads $$\label{eqn:Ricci_Weinhold_regular} R_W = -\frac{\sqrt{\pi}}{c^2G_{\infty}}\sqrt{\frac{G(k)^3}{S}},$$ which vanishes at IR and approaches constant at UV. The heat capacity for constant $k$ and constant $S$ are computed respectively $$C_k = -2S, \qquad C_S = -\frac{2k(k+G_\infty)}{4k+3G_\infty}.$$ We remark that the heat capacity $C_k$ is explicitly independent on the scale $k$ and its negative value, which signals the instability, agrees with the conventional Schwarzschild black hole with constant $G$. We plot them in the figure \[fig:Weinhold\_runG\]. The Ricci scalars for both Ruppeiner and Weinhold metric remain negative for finite $S$ and $k$. It becomes divergent as $M\to 0$ and vanished at IR limit ($M\to \infty$ or $k\to 0$). ![\[fig:Weinhold\_runG\](Left) Ricci scalar $R_W$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_W$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$.](Weinhold_runG_k.eps "fig:"){width="48.00000%"} ![\[fig:Weinhold\_runG\](Left) Ricci scalar $R_W$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_W$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$.](Weinhold_runG_S.eps "fig:"){width="48.00000%"} While the Quevedo metric of case I turns out to be flat, the Quevedo metric of case II has a running Ricci scalar: $$R_{\Omega_{II}}=-\frac{64\pi G_\infty k^2}{c^4 S (4k+3G_\infty )^2}$$ which takes the following form at UV limit $$\lim_{k\to \infty}R_{\Omega_{II}} = -\frac{4\pi G_\infty}{c^4 S}.$$ We remark that both $R_R$ and $R_{\Omega_{II}}$ at UV limit are inversely proportional to $C_k$ if area law is true for all scales. At last, the Ricci scalar for the HPEM metric reads $$R_{N} = -\frac{c^2G_\infty S (72k^2+116kG_\infty+33 G_\infty^2)}{16\sqrt{\pi}k^6 (k+G_\infty)^4}\sqrt{\frac{G(k)}{S}},$$ which vanishes at UV limit and scales as $-|C_k|^{1/2}$ for fixed $k$. We plot them in the figure \[fig:Quevedo\_runG\] and \[fig:HPEM\_runG\] ![\[fig:Quevedo\_runG\](Left) Ricci scalar $R_{\Omega_{II}}$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_{\Omega_{II}}$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$.](Ricci_QII_runG_k.eps "fig:"){width="48.00000%"} ![\[fig:Quevedo\_runG\](Left) Ricci scalar $R_{\Omega_{II}}$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_{\Omega_{II}}$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$.](Ricci_QII_runG_S.eps "fig:"){width="48.00000%"} ![\[fig:HPEM\_runG\](Left) Ricci scalar $R_N$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_N$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$. ](Ricci_HEPM_runG.eps "fig:"){width="48.00000%"} ![\[fig:HPEM\_runG\](Left) Ricci scalar $R_N$ and heat capacities $C_{k}$ and $C_S$ for various $k$ at fixed $S=1$. (Right) Ricci scalar $R_N$ and heat capacity $C_{k}$ and $C_S$ for various $S$ at fixed $k=1$. ](Ricci_HEPM_runG_S.eps "fig:"){width="48.00000%"} Regular Schwarzschild-like black hole ===================================== Now we inspect various thermodynamic metric for a regular black hole which has the Schwarzschild metric from outset but a singularity-free interior. It was first proposed by Hayward[@Hayward:2005gi] and recently served as a toy model to answer Hawking’s resolution to the firewall and information paradox[@Frolov:2014jva]. The metric reads: $$g_{tt} = g^{rr} = 1-\frac{2GMr^2}{r^3+2\l^2GM},$$ which behaves like the ordinary Schwarzschild black hole at large distance but a de Sitter space for $r \ll r_+$. Now regarding $\l$ to be a new degree of freedom, we have the internal energy $$M(S,\l)=\frac{\sqrt{S^3}}{2\sqrt{\pi}(S-\pi \l^2)}.$$ The Hessian of $M$ gives the Weinhold metric: $$ds^2_W = -\frac{S^2-6\pi\l^2 S-3\pi^2\l^4}{8\sqrt{\pi S}(S-\pi\l^2)^3} dM^2 - \frac{\sqrt{\pi S^3}\l (S+3\pi\l^2)}{S(S-\pi\l^2)^3} dM d\l + \frac{\sqrt{\pi S^3}(S+3\pi\l^2)}{(S-\pi\l^2)^3} d\l^2.$$ which is conformally flat with Ricci scalar $$R_W= \frac{2\sqrt{\pi}(S^2+6\pi\l^2S-15\pi^2\l^4)}{\sqrt{S}(S+3\pi\l^2)(S-3\pi\l^2)^2}$$ and the heat capacities evaluated at fixed $\l$ or fixed $S$ respectaively read $$C_{\l} = -\frac{2S(S-\pi\l^2)(S-3\pi\l^2)}{S^2-6\pi\l^2 S-3\pi^2\l^4},\qquad C_S = -\l\frac{S-\pi\l^2}{S+3\pi\l^2}.$$ We remark that a pole of $R_W$, $S=3\pi\l^2$, coincides with a zero of $C_{\l}$ at the extremal limit $r_+=\sqrt{3}\l$. Furthermore, the ratio $\gamma \equiv C_S/C_{\l}$ has the same poles as that of Ricci scalar and removes the unphysical zero at $S=\pi \l^2$. The heat capacity $C_{\l} \to -2S$ at the limit $S \gg \l^2$, as expected from the Schwarzschild black hole. Now we study the Quevedo metric. The conformal factors for the case I and II are $$\Omega_I = \frac{\sqrt{S^3}(S+\pi\l^2)}{4\sqrt{\pi}(S-\pi\l^2)^2},\qquad \Omega_{II} = \frac{\sqrt{S^3}(S-3\pi\l^2)}{4\sqrt{\pi}(S-\pi\l^2)^2}$$ respectively. The corresponding Ricci scalar for case I is lengthy to be skipped here, but its denominator reads $$\text{demoninator}( R_{Q_I})=S^2(S+\pi\l^2)^3(S+3\pi\l^2)^2(S^2-6\pi\l^2 S-3\pi^2\l^4)^2.$$ Although there is no pole which agrees with the extremal limit, it poccesses the same positive pole as that of $C_{\l}$, i.e. $S=(3+2\sqrt{3})\pi\l^2$. The denominator of Ricci scalar for case II reads $$\text{demoninator}( R_{Q_{II}} )= S^2(S-3\pi\l^2)(S+3\pi\l^2)(S^2-6\pi\l^2-3\pi^2\l^4)^2$$ and the desired pole, $S=3\pi\l^2$, is one of them. As to the HPEM metric, the denominator for its Ricci scalar reads $$\text{demoninator}( R_N )= (S-3\pi\l^2)(S-\pi\l^2)^4(S^2-6\pi\l^2-3\pi^2\l^4)^2,$$ while it keeps the desired pole but replaces the pole $S=0$ by $S=\pi\l^2$ instead. We remark that at the Schwarzschild limit $\l \to 0$, the Ricci scalar for different metrics read $$R_W = 2\sqrt{\frac{\pi}{S}},\quad R_{Q_I} = 0,\quad R_{Q_{II}} = \frac{32\pi}{S}, \quad R_N =200\sqrt{\frac{\pi^5}{S^5}}.$$ We remark that beside the case I of Quevedo metric, Ricci scales are inversely proportional to $S$ to some power as $\l \to 0$. We plot those Ricci scalars with heat capacities in the figure \[fig:Ricci\_regular\] and \[fig:Ricci\_Q\_regular\]. The nonvanishing Ricci scalar implies that strength of thermodynamic interaction increases as the black hole evaporates. The divergance at the final stage suggests a phase transition where new degrees of freedom in the quantum gravity would play an important role. ![\[fig:Ricci\_regular\] (Left) Ricci scalar $R_W$ for the Weinhold metric and heat capacities $C_{\l}$ and $C_S$. Here we set $\l=1$ for all simulation. (Right) Ricci scalar $R_N$ for the HPEM metric and heat capacity $C_{\l}$ and $C_S$.](Weinhold_Ricci.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_regular\] (Left) Ricci scalar $R_W$ for the Weinhold metric and heat capacities $C_{\l}$ and $C_S$. Here we set $\l=1$ for all simulation. (Right) Ricci scalar $R_N$ for the HPEM metric and heat capacity $C_{\l}$ and $C_S$.](New_regular.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_Q\_regular\] (Left) Ricci scalar $R_{\Omega_I}$ for the Quevedo model I and heat capacities $C_{\l}$ and $C_S$. (Right) Ricci scalar $R_{\Omega_{II}}$ for the Quevedo model II and heat capacities $C_{\l}$ and $C_S$. ](Quevedo_I_regular.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_Q\_regular\] (Left) Ricci scalar $R_{\Omega_I}$ for the Quevedo model I and heat capacities $C_{\l}$ and $C_S$. (Right) Ricci scalar $R_{\Omega_{II}}$ for the Quevedo model II and heat capacities $C_{\l}$ and $C_S$. ](Quevedo_II_regular.eps "fig:"){width="48.00000%"} Noncommutative geometry inspired Schwarzschild black hole ========================================================= In the noncommutative geometry, a smear source was proposed to resolve the singularity at the center of Schwarzschild black hole[@Nicolini:2005vd]: $$M(\sqrt{\frac{S}{\pi}},\theta) = \frac{\sqrt{S}}{4\gamma(\frac{3}{2},\frac{S}{4\pi\theta})},$$ where the imcomplete gamma function $\gamma(a,x)=\int_0^x{t^{a-1}e^{-t}dt}$. ![\[fig:Capacity\_NC\] (Left) Heat capacity $C_\theta$ for fixed $\theta=0.01$ The inset plot shows a zero at small $S$. (Right) Heat capacity $C_S$ for different $S$. ](Heat_Capacity_NC.eps "fig:"){width="48.00000%"} ![\[fig:Capacity\_NC\] (Left) Heat capacity $C_\theta$ for fixed $\theta=0.01$ The inset plot shows a zero at small $S$. (Right) Heat capacity $C_S$ for different $S$. ](Heat_Capacity1_NC.eps "fig:"){width="48.00000%"} Now regarding the noncommutative parameter $\theta$ as a free varaible, one can compute the lengthy Weinhold metric, wchih is flat. Heat capacities $C_\theta$ at fixed $\theta$ and $C_S$ at fixed $S$ are plotted in the figure \[fig:Capacity\_NC\]. We remark that $C_\theta$ has a zero at small $S$ and follows by a spike. It approaches the line $C_\theta = -2S$ for large $S$, as expected for the Schwarzschild black hole. On the other hand, $C_S$ monotoneously decreases with $S$. Both models in the Quevedo metric are all conformally flat. We plot their Ricci scalars together with heat capicities in the figure \[fig:Ricci\_NC\_Q1\] and figure \[fig:Ricci\_NC\_Q2\]. We remark that the pole of $R_{\Omega_I}$ matches with the spike of $C_\theta$, but their zeros at small $S$ do not agree. On the other hand, the spike of $R_{\Omega_{II}}$ agrees with the zero of $C_\theta$ at small $S$. The Riccis scalar of HPEM metric $R_N$ is potted in the figure \[fig:Ricci\_NC\_HPEM\] and the zero and spike of $C_\theta$ agrees with the spike and pole of $R_N$. It is not straightforward to take $\theta \to 0$ limit for the thermal Ricci scalar. However, if the above onshell limit is taken with the extremal condition $S=9\pi \theta$, it is found that $$R \sim -\frac{C}{\theta},$$ where $C\simeq 0.72$ for the model I of Quevedo metric and $C\simeq 1.28\times 10^5$ for the model II. Since the extremal limit can be regarded as a kind of vacuum (ground state) of noncommutative space for its zero temperature, it is entertaining to suggest the divergence of Ricci scalar at $\theta \to 0$ limit implies a phase transition from the noncommutative space to the commutative space. ![\[fig:Ricci\_NC\_Q1\] (Left) Ricci scalar for Quevedo metric model I of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for Quevedo metric model I of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_Q1_NC.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_NC\_Q1\] (Left) Ricci scalar for Quevedo metric model I of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for Quevedo metric model I of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_Q1zoom_NC.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_NC\_Q2\] (Left) Ricci scalar for Quevedo metric model II of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for Quevedo metric model II of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_Q2_NC.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_NC\_Q2\] (Left) Ricci scalar for Quevedo metric model II of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for Quevedo metric model II of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_Q2zoom_NC.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_NC\_HPEM\] (Left) Ricci scalar for HPEM metric of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for HPEM metric of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_new_NC.eps "fig:"){width="48.00000%"} ![\[fig:Ricci\_NC\_HPEM\] (Left) Ricci scalar for HPEM metric of NC black hole and heat capacity for constant $\theta$. (Right) Ricci scalar for HPEM metric of NC black hole and heat capacity for constant $\theta$ at small $S$. ](Ricci_newclose_NC.eps "fig:"){width="48.00000%"} Shcwarzschild black hole with Tsallis-R[è]{}nyi entropy ======================================================= Bir[ó]{} and Czinner [@Biro:2013cra] regarded the Hawking-Bekenstein entropy as the non-extensive Tsallis or R[è]{}nyi entropy. The Tsallis entropy has been devised to fit the power-law tailed spectra, and the Schwarzschild black hole becomes thermally stable at a fixed temperature in a similar way as that in the anti-de Sitter space[@Biro:2013cra; @Czinner:2015eyk]. Here the R[è]{}nyi entropy reads, $$\label{eqn:Renyi_entropy} S_a(M) = \frac{1}{a}\ln(1+4\pi a M^2),$$ where the Bekenstein-Hawking entropy is obtained at the Schwarzschild limit $a\to 0$. Now we regard the entropy as a function of both mass and $a$, say $S(M,a)=S_a(M)$. The Ruppeiner metric is conformally flat. The Ricci scalar expands around the Schwarzschild limit as: $$R_R = \frac{45}{256\pi M^2} + \frac{171}{1280}a - \frac{549 \pi}{5120} M^2 a^2 +\cdots.$$ The Weinhold metric can be obtained from inverting (\[eqn:Renyi\_entropy\]) and the Ricci scalar expands around the Schwarzschild limit as: $$R_W = -\frac{19}{16}\sqrt{\frac{\pi}{S}} + \frac{123}{160}\sqrt{\pi S} a - \frac{3721}{15360}\sqrt{\pi S^3}a^2 + \cdots.$$ We remark that the leading terms in $R_R$ and $R_W$ behave like those in (\[eqn:Ricci\_regular\]) and (\[eqn:Ricci\_Weinhold\_regular\]) but with different coefficients. We plot the Weinhold Ricci scalar and heat capacities against entropy and parameter $a$ respectively in the figure \[fig:Renyi\_Weinhold\]. ![\[fig:Renyi\_Weinhold\] (Left) Ricci scalar for Weinhold metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. (Right) Ricci scalar for Weinhold metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $S=1$. ](Renyi_Weinhold.eps "fig:"){width="48.00000%"} ![\[fig:Renyi\_Weinhold\] (Left) Ricci scalar for Weinhold metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. (Right) Ricci scalar for Weinhold metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $S=1$. ](Renyi_Weinhold_S.eps "fig:"){width="48.00000%"} In the figure \[fig:Renyi\_Quevedo\], we plot the Ricci scalars for Quevedo metric and HPEM metric. The divergence of both Ricci scalars agree with the heat capacity $C_a$. At the Schwarzschild limit, their Ricci scalars behave as follows: $$R_{\Omega_I}= \frac{3456\pi}{25 S},\qquad R_{\Omega_{II}} = \frac{352\pi}{5S}, \qquad R_{N} = \frac{965}{27648}\sqrt{\frac{S^{13}}{\pi}}.$$ ![\[fig:Renyi\_Quevedo\] (Left) Ricci scalar for Quevedo metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. (Right) Ricci scalar for HPEM metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. ](Renyi_Quevedo.eps "fig:"){width="48.00000%"} ![\[fig:Renyi\_Quevedo\] (Left) Ricci scalar for Quevedo metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. (Right) Ricci scalar for HPEM metric for black hole with R[è]{}nyi entropy and heat capacity for constant $a$ and constant $S$ at fixed $a=1$. ](Renyi_HPEM.eps "fig:"){width="48.00000%"} Discussion ========== In this letter, we study the offshell thermal metric of Schwarzschild black hole by introducing a new degree of freedom, which could be the running Newton constant, a cutoff scale for regular black hole, a noncommutative deformation, or the deformed parameter in the nonextensive Tsallis-R[è]{}nyi entropy. The onshell thermal Ricci scalar of original Schwarzschild black hole is obtained by guage fixing this freedom. In the table \[table:summary\], we summarize our result for different metric and various deformation. In many cases, the Ricci scalar blows up at the final stage of evaporation where $S\to 0$. This calls for a theory of quantum gravity to resolve the divergence. For those effective theories with deformation, the Ricci scalar in the Quevedo or HPEM metric usually has a pole at finite entropy, which agrees with either a zero or pole in the heat capacity. This implies the existence of a phase transition at some UV scale due to emerged degrees of freedom in quantum effects. Theories Ruppeiner Weinhold Quevedo I Quevedo II HPEM ----------------------- ----------- ------------- ----------- ------------ ------------ Running G[^2] $-S^{-1}$ $-S^{-1/2}$ $0$ $-S^{-1}$ $0$ Regular BH -[^3] $S^{-1/2}$ $0$ $S^{-1}$ $S^{-5/2}$ Noncommutative BH[^4] - $-S^{-1}$ $-S^{-1}$ $-S^{-1}$ $-S^{-1}$ R[è]{}nyi entropy $S^{-1}$ $-S^{-1/2}$ $S^{-1}$ $S^{-1}$ $S^{13/2}$ : \[table:summary\][*Onshell*]{} thermal Ricci scalar at the Schwarzschild limit. Here we assume the Bekenstein-Hawking area law is correct. The author is grateful to the package Riemannian Geometry & Tensor Calculus and the software Mathematica. 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--- abstract: 'The cross section for inclusive prompt photon production with polarized hadron beams is calculated at order $\alpha\alpha_s^2$ using the phase space slicing or analytic/Monte Carlo method. Isolation cuts are placed on the photon and the results are compared to a previous fully analytic calculation. Numerical results for the isolated cross section are presented for for $\vec{p}\vec{p}\rightarrow \gamma+X$ at RHIC center-of-mass energies with plausible isolation parameters using the most modern polarized parton densities evolved in next-to-leading order QCD. The perturbative stability of the asymmetries and scale dependence of the results are briefly discussed.' address: 'High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439' author: - 'L. E. Gordon' title: 'Isolated Photons at Hadron Colliders at $O(\alpha\alpha_s^2)$(II): Spin Dependent Case' --- Introduction ============ Prompt photon production is at the top of the list of the most important processes to be studied at the BNL Relativistic Heavy Ion Collider (RHIC) which is expected to start taking data in the next few years. It is well established as one of the main processes which is sensitive to the polarized gluon density of the proton $\Delta G$. This is mainly because the cross section is dominated by the quark-gluon scattering process $qg\rightarrow\gamma+X$ which contributes at leading order (LO). Another factor in the importance of this process is that photons are in principle experimentally more simple to detect, and hence the kinematic variables can be more accurately determined as compared to high transverse momentum ($p_T$) jets, for example. Unfortunately, in practice this is not always the case. At collider energies experimentalists are forced to impose isolation cuts on the photon in order to accurately detect it from among the copious hadronic debris produced simultaneously in the high energy collisions. In effect what is detected in many cases is a photon in a jet, where the jet energy is restricted by the isolation cuts. The isolation procedure has proved difficult to implement in a theoretically consistent way especially on the component of the cross section where the photon is produced by bremsstrahlung off a final state parton, the fragmentation component. This happens to be the component of the cross section most affected by the isolation procedure since the photon is always accompanied by a jet in this case. These difficulties have restricted the usefulness of prompt photon production for extracting information on gluon distributions in the unpolarized case since they highlight the fact that isolation is not yet theoretically fully understood. In fact there is at present some controversy pertaining to the use of the conventional factorization theorem in the infrared regions of phase space for the fragmentation contribution in NLO [@bergerqui1; @auretal] when isolation cuts are imposed. Since in spite of these difficulties prompt photon production has proved useful for helping to constrain the unpolarized gluon distributions, it has been suggested that it might also be useful in the case of the polarized gluon distribution $\Delta G$ [@bergerqiu2], about which little or no experimental information is presently available. The spin structure of nucleons has been a topic of much activity ever since the European Muon Collaboration (EMC) first published results of a measurement of the first moment of the spin-dependent proton structure function $g_1^p(x,Q^2)$ [@emc], obtained from polarized deep inelastic scattering experiments. The results were in disagreement with the Ellis-Jaffe sum rule which is based on the naive parton model [@hai-yang] and suggested that much less of the nucleon spin was carried by quarks than would be expected from this model. Since that time much progress has been made on the understanding of the problem on the theoretical front and new more precise data have also become available, which on average reduces somewhat the discrepancy with parton model expectations [@smc] although still leaving a very significant one. Most of the progress that can be made from purely theoretical investigations alone has now been achieved and therefore activity is at present more focussed on obtaining more precise experimental information on the spin structure of the nucleons. Deep inelastic scattering processes do not directly probe the gluon structure of hadrons since photons do not couple directly to gluons, and hence only limited further information on $\Delta G$ can be obtained from this source via processes such as heavy quark or two jet production. One must therefore look at other scattering processes which involve a direct gluon coupling. Prompt photon production is one such process, but others have also been suggested and some have been calculated in NLO [@nlo1; @nlo2]. In this context, inclusive prompt photon production with polarized beam and target, $\vec{p}\vec{p}\rightarrow \gamma +X$, was first examined in LO [@bergerqiu2] and shown to be sensitive to $\Delta G$. Sizeable asymmetries were also predicted, indicating a sensitivity to polarization effects. The NLO corrections were calculated in [@contogouris] and [@gorvogel], numerical estimates were also presented and it was established that the LO results were perturbatively stable. In [@contogouris] and [@gorvogel] the phase space integrations were carried out analytically, hence isolation restrictions could not be imposed. Furthermore, at that time only LO parton distributions were available so a fully consistent NLO analysis could not be performed. Recently, due to the calculation of the spin dependent splitting functions at NLO [@mertig], new polarized parton distributions evolved fully in NLO QCD, which take into account all recent data have become available [@grsv; @gs; @forte; @ramsey]. In these analyses different assumptions are made about both the size and shape of $\Delta G$ which is hardly constrained by the available data. This is reflected in the fact that in general they each give more than one parametrization of the polarized parton densities each having a different input for $\Delta G$. In [@gorvogel1] we recently provided estimates for the polarized non-isolated prompt photon cross section at energies relevant for the proposed HERA-$\vec{\rm N}$ collider, using these new polarized parton distributions. In this paper this analysis is extended the case of the RHIC collider with the inclusion of isolation restrictions on the photon. In this case the calculation is carried out using the Monte Carlo method. Recently [@conto2] the analytic calculation of [@contogouris] was updated to include the use of NLO structure functions. In this calculation the authors chose to completely ignore the fragmentation contributions, and of course isolation effects are not included since the calculation was done analytically. In [@gordon] the results of the analytic and Monte Carlo methods of calculating both the inclusive and isolated cross sections were compared for both the polarized and unpolarized cases and exact agreement was found as expected for the inclusive case. For the isolated case agreement was found over a wide range of the isolation parameters for centrally produced photons. As expected there are regions where the analytic method breaks down. It gives results in disagreement with the Monte Carlo method for very large ($R\geq 1$) or very small ($R\leq 0.1$) isolation cone sizes, or at rapidities away from the central regions. It is therefore useful to calculate the polarized cross section using the more robust and flexible Monte Carlo method, particularly since the values of the isolation parameters necessary for RHIC have not yet been decided. If it turns out that they are chosen outside the range where the analytic calculation is valid then it will be useful to have the Monte Carlo calculation. The details of the calculation can be found in [@nlo2] where it was applied to prompt photon plus jet production. In the unpolarized case the cross section was first calculated in [@baeretal] using the Monte Carlo method, and this calculation is thus the second calculation in this case. A new feature in the present calculation is that the various subprocess contributions are kept separate which allows one to tell how much of the cross section is due to $qg$ or $q\bar{q}$ scattering, for example. This information could be useful if one is interested in the sensitivity of the cross section to the gluon distribution. For the polarized case, the present calculation is the first using this method. The only major drawback of the present calculation is that the fragmentation contributions can only be calculated in LO since the matrix elements for the NLO case have not yet been calculated. In this study an attempt is made to assess the importance of these contributions at RHIC center-of-mass (cms) energies. Since RHIC is expected to run at different cms energies between $50$ and $500$ GeV, it is possible that the fragmentation contributions may not be numerically important at the lower energies, thereby reducing the need to impose isolation cuts on the photon, or at least reducing the theoretical uncertainty from incomplete calculations of these contributions. The rest of this paper is as follows; in section II a brief theoretical background to the calculations is given in order to make the paper as self contained as possible. In section III numerical results are presented for the polarized and unpolarized cross sections at RHIC, and in section IV the conclusions are given. Isolated Prompt Photons ======================= In this section a brief description of the ingredients used in the calculation of the inclusive and isolated prompt photon cross sections is given in order to make the paper as self contained as possible. More details of the calculation can be found in refs.[@nlo2; @gordon]. Only the polarized case is discussed explicitly but all the arguments are valid in the unpolarized case with the replacements discussed at the end of the section. Contributions to the prompt photon cross section are usually separated into two classes in both LO and NLO. There are the so-called direct processes, $ab\rightarrow \gamma c$ in LO and $ab\rightarrow \gamma c d$, in NLO, $a,b,c$ and $d$ referring to partons, where the photon is produced directly in the hard scattering. In addition there are the fragmentation contributions where the photon is produced via bremsstrahlung off a final state quark or gluon, $ab\rightarrow c d (e)$ followed by $c->\gamma + X$ for instance. Experimentally, a prompt photon is considered isolated if inside a cone of radius $R$ centered on the photon the hadronic energy is less than $\epsilon E_\gamma$, where $E_\gamma$ is the photon energy and $\epsilon$ is the energy resolution parameter, typically $\epsilon \sim 0.1$. The radius of the circle defined by the isolation cone is given in the pseudo-rapidity $\eta$ and azimuthal angle $\phi$-plane by $R=\sqrt{(\Delta \eta)^2 + (\Delta \phi)^2}$. In the case of a small cone the parameter used is the half angle of the cone, $\delta$, where $\delta \approx R$ for small rapidities of the photon. The exact relation is $R=\delta/{\cosh \eta}$. The LO Case ----------- In LO, $O(\alpha \alpha_s)$, the direct subprocesses contributing to the cross section are $$\begin{aligned} qg&\rightarrow& \gamma q \nonumber \\ q\bar{q}&\rightarrow& \gamma g.\end{aligned}$$ In addition there are the fragmentation processes $$\begin{aligned} qg &\rightarrow& q g \nonumber \\ qq &\rightarrow& q q \nonumber \\ qq' &\rightarrow& q q' \nonumber \\ q\bar{q} &\rightarrow& q \bar{q} \nonumber \\ qg &\rightarrow& q g \nonumber \\ q\bar{q} &\rightarrow& g g \nonumber \\ gg &\rightarrow& g g \nonumber \\ gg &\rightarrow& q \bar{q} \end{aligned}$$ where one of the final state partons fragments to produce the photon, i.e., $q (g)\rightarrow \gamma + X$. In the direct processes in LO, the photon is always isolated since it must always balance the transverse momentum $p_T$ of the other final state parton and is thus always in the opposite hemisphere. In this case the differential cross section is given by $$E_\gamma\frac{d\Delta\sigma_{dir}^{LO}}{d^3p_\gamma}=\frac{1}{\pi S}\sum_{i,j} \int^V_{V W}\frac{dv}{1-v}\int^1_{VW/v}\Delta f^i_1(x_1,M^2)\Delta f^j_2(x_2,M^2) \frac{1}{v}\frac {d\Delta\hat{\sigma}_{ij\rightarrow\gamma}}{dv}\delta(1-w)$$ where $S=(P_1+P_2)^2$, $V=1+T/S$, $W=-U/(T+S)$, $v=1+\hat{t}/\hat{s}$, $w=-\hat{u}/(\hat{t}+\hat{s})$, $\hat{s}=x_1 x_2 S$, and $T=(P_1-P_\gamma)^2$ and $U=(P_2-P_\gamma)^2$. As usual the Mandelstam variables are defined in the upper case for the hadron-hadron system and in lower case in the parton-parton system. $P_1$ and $P_2$ are the momenta of the incoming hadrons and $f^i_1(x_1,M^2)$ and $f^j_2(x_2,M^2)$ represent the respective probabilities of finding parton $i$ and $j$ in hadrons $1$ and $2$ with momentum fractions $x_1$ and $x_2$ at scale $M^2$. For the fragmentation processes, the photon is always produced nearly collinearly to the fragmenting parton and an isolation cut must be placed on the cross section to remove the remnants of the fragmenting parton if it has more energy than $\epsilon E_{\gamma}$. In this case this restriction is quite easy to implement. The [*inclusive*]{} differential cross section is given by $$\begin{aligned} E_\gamma\frac{d\Delta\sigma_{frag}^{incl}}{d^3p_\gamma}&=&\frac{1}{\pi S} \sum_{i,j,l}\int^1_{1-V+VW}\frac{dz}{z^2}\int^{1-(1-V)/z}_{V W/z}\frac{dv}{1-v}\int^1_{VW/vz}\frac{dw}{w}\Delta f^i_1(x_1,M^2)\Delta f^j_2(x_2,M^2) \nonumber \\ &\times &\frac{1}{v} \frac{d\Delta\hat{\sigma}_{ij\rightarrow l}}{dv}\delta(1-w)D^\gamma_l(z,M_f^2),\end{aligned}$$ where $D_{\gamma/l}(z,M_f^2)$ represents the probability that the parton labelled $l$ fragments to a photon with a momentum fraction $z$ of its own momentum at scale $M_f^2$ (note that $D_{\gamma/l}(z,M_f^2)$ is the usual unpolarized fragmentation function, since the final state is not polarized). This is the non-perturbative fragmentation function which must be obtained from experiment at some scale and evolved to $M_f^2$ using the usual evolution equations. This means that in order to obtain the isolated cross section we simply have to cut on the variable $z$. If isolation is defined in the usual way by only accepting events with hadronic energy less than fraction $\epsilon$ in a cone of radius $R=\sqrt{(\Delta \phi)^2+(\Delta\eta)^2}$ drawn in the pseudo-rapidity azimuthal angle plane around the photon, then the hadronic remnants of the fragmenting parton will always automatically be inside the cone with the photon, for suitable choices of $M_f$, and the isolated cross section is given by the equation $$\begin{aligned} E_\gamma\frac{d\Delta\sigma_{frag}^{isol}}{d^3p_\gamma}&=&\frac{1}{\pi S} \sum_{i,j,l} \int^1_{Max[z_{min},1/(1+\epsilon)]}\frac{dz}{z^2}\int^{1-(1-V)/z}_{V W/z}\frac{dv}{1-v}\int^1_{VW/vz}\frac{dw}{w}\nonumber \\ &\times & \Delta f^i_1(x_1,M^2) \Delta f^j_2(x_2,M^2)\frac{1}{v}\frac {d\Delta\hat{\sigma}_{ij\rightarrow l}}{dv}\delta(1-w)D^\gamma_l(z,M_f^2),\end{aligned}$$ where $z_{min}=1-V+V W$. It is also suggested that the fragmentation scale should be replaced by $(R M_f)$ or $(\delta M_f)$ in order to ensure that all fragmentation remnants are radiated inside the cone [@bergerqiu4], but this argument not universally accepted. It was shown in [@gorvogel3] that the choice is numerically irrelevant, since the dependence of the cross section on the fragmentation scale is negligible after isolation in NLO. The NLO Case ------------ ### The Non-Fragmentation Contribution In NLO order, $O(\alpha\alpha_s^2)$, there are virtual corrections to the LO non-fragmentation processes of eq.(2.1), as well as the further three-body processes: \[eq:1\] $$\begin{aligned} g &+q \rightarrow g + q + \gamma\label{eq:11}\\ g &+ g \rightarrow q +\bar{q} + \gamma\label{eq:12}\\ q &+ \bar{q} \rightarrow g + g + \gamma\label{eq:13}\\ q &+ q \rightarrow q + q + \gamma\label{eq:14}\\ \bar{q} &+ q \rightarrow \bar{q} + q + \gamma\label{eq:15}\\ q &+ \bar{q} \rightarrow q' + \bar{q}' + \gamma\label{eq:16}\\ q &+ q' \rightarrow q + q' + \gamma\label{eq:17}\end{aligned}$$ In principle the the fragmentation processes of eq.(2.2) should now be calculated to $O(\alpha_s^3)$ and convoluted with the NLO photon fragmentation functions whose leading behaviour is $O(\alpha/\alpha_s)$, but the hard subprocess matrix elements are not yet available in the polarized case, hence, in both the polarized and unpolarized cases, we include the leading order contributions to these processes only. Numerically the fragmentation processes are not as significant except at low $p_T$ after isolation cuts are implemented, but for a theoretically consistent calculation they should nevertheless be included as they help to reduce scale dependences, and as was demonstrated in ref.[@reya] they also help to improve the agreement between theory and experiment in the low $p_T^\gamma$ region. The direct contribution to the inclusive cross section is given by $$\begin{aligned} E_\gamma\frac{d\Delta\sigma^{incl}_{dir}}{d^3p_\gamma}&=&\frac{1}{\pi S} \sum_{i,j}\int^V_{V W}\frac{dv}{1-v}\int^1_{VW/v}\frac{dw}{w}\Delta f^i_1(x_1,M^2)\Delta f^j_2(x_2,M^2) \nonumber \\ &\times&\left[ \frac{1}{v}\frac {d\Delta\hat{\sigma}_{ij\rightarrow\gamma}}{dv}\delta(1-w)+ \frac{\alpha_s(\mu^2)}{2\pi} \Delta K_{ij\rightarrow\gamma}(\hat{s},v,w,\mu^2,M^2,M_f^2)\right],\end{aligned}$$ where $\Delta K_{ij\rightarrow\gamma}(\hat{s},v,w,\mu^2,M^2,M_f^2)$ represents the higher corrections to the hard subprocess cross sections calculated in [@gorvogel] and $\mu$ is the renormalization scale. In [@gorvogel3] the isolated cross section is first written as the inclusive cross section minus a subtraction piece, along the lines suggested in [@bergerqiu4]: $$E_\gamma\frac{d\Delta\sigma^{isol}_{dir}}{d^3p_\gamma}= E_\gamma\frac{d\Delta\sigma^{incl}_{dir}}{d^3p_\gamma}- E_\gamma\frac{d\Delta\sigma^{sub}_{dir}}{d^3p_\gamma},$$ $E_\gamma\frac{d\Delta \sigma^{sub}_{dir}}{d^3p_\gamma}$ being the cross section for producing a prompt photon with energy $E_\gamma$ which is accompanied by more hadronic energy than $\epsilon E_\gamma$ inside the cone. The question is then how to calculate the subtraction piece. In [@gorvogel3] it is calculated by an approximate analytic method for a small cone of half angle $\delta$ as well as for a cone of radius $R$ as defined above using Monte Carlo integration methods. The complete details of the calculation can be found in ref.[@gorvogel3]. The final form for the subtraction piece assuming a small cone of half angle $\delta$ is given by $$E_\gamma\frac{d^3\Delta\sigma^{sub}_{dir}}{d^3p_\gamma}= A\ln \delta +B +C\delta^2\ln\epsilon,$$ where $A,B$ and $C$ are functions of the kinematic variables of the photon and $\epsilon$. A detailed study was then made of the difference between the analytic and numerical Monte Carlo [*subtraction*]{} pieces for various values of the isolation parameters $\epsilon$ and $\delta$ at $\sqrt{S}=1$ TeV. It was found that the small cone approximation was within $10\%$ of the Monte Carlo results for the subtraction piece except for very large values of $\epsilon$ and $\delta$, greater than $0.25$ and $0.8$ respectively. This translated into a [*very small*]{} error for the [*full*]{} isolated cross section even for large values of the parameters, as the subtraction piece is numerically much smaller than the inclusive piece. The Monte Carlo method of calculation differs from the method outlined above in some important ways. The phase space is only integrated over analytically in those regions where soft collinear singularities occur. These are cancelled or subtracted in the usual way leaving the rest of the phase space to be integrated over numerically. The flexibility of the method lies in the fact that any infrared safe experimental cuts can be imposed on the phase space by imposing restrictions on the regions which are integrated over numerically by Monte Carlo methods. Thus it is straightforward to impose isolation cuts on the photon in this case without making any further approximations. In ref.[@gordon] a detailed comparison of the results of the analytic and Monte Carlo methods was made for both the inclusive and isolated cross sections, and agreement was found. A similar comparison was made for the polarized case with the same results, but since the comparisons follow along the exact same lines as that presented in [@gordon] with similar results the details will not be repeated in this paper. Polarized vs Unpolarized Cases ------------------------------ When the initial hadrons are longitudinally polarized, all the usual formulas used in the spin averaged case can be taken over, expect that now the hard subprocess cross sections and the parton distributions must be replaced by the corresponding spin dependent versions. For example the polarized hard subprocess matrix elements in LO used in eqs.(2.3-2.5) and (2.7) were defined by $$\frac{d\Delta\hat{\sigma}}{dv}=\frac{1}{2}\left[\frac{d\hat{\sigma}(++)}{dv}- \frac{d\hat{\sigma}(+-)}{dv}\right],$$ where $+, -$ denote the helicities of the initial partons. The usual spin averaged versions are defined by $$\frac{d\hat{\sigma}}{dv}=\frac{1}{2}\left[\frac{d\hat{\sigma}(++)}{dv}+ \frac{d\hat{\sigma}(+-)}{dv}\right].$$ LO matrix elements for the direct and fragmentation processes have been presented in many places (see eg. [@gorvogel1]) and the NLO ones integrated analytically over phase are given in the appendix of [@gorvogel]. The unintegrated three-body matrix elements are collected in the appendix of [@nlo2]. The polarized parton distributions are similarly defined by $$\Delta f_a^i(x,M^2)=f^i_{a,+}(x,M^2)-f^i_{a,-}(x,M^2),$$ where $f^i_{a,\pm}(x,M^2)$ is the distribution of parton type $i$ with positive (+) or negative (-) helicity in hadron $a$, whereas the usual unpolarized ones are given by $$f_a^i(x,M^2)=f^i_{a,+}(x,M^2)+f^i_{a,-}(x,M^2).$$ Numerical Results ================= In this section predictions for the isolated prompt photon cross section for polarized proton-proton collision at RHIC energies are investigated. The fragmentation contribution is always estimated with LO matrix elements for both the polarized and unpolarized cases, although NLO structure and fragmentation functions are used throughout. The renormalization, factorization, and fragmentation scales are always set to a common value $\mu = p_T^{\gamma}$ unless otherwise stated. The fragmentation functions evolved in NLO from ref.[@vogtfrag] are used throughout. There are various parametrizations of the polarized proton densities at NLO on the market [@grsv; @gs; @forte]. In ref.[@gs] three different sets are parametrized (the GS sets), all fitting the DIS data, but due to the freedom in fixing the various flavour of quark densities as well as the gluon densities, the actual distributions differ. In this paper the three GS distributions (GSA, GSB and GSC) are used and the predictions using them compared. For the unpolarized cross sections, the CTEQ4M [@CTEQ4] distributions are used throughout. The NLO expression for $\alpha_s$ is always used and four quark flavors are assumed although no contribution from initial charm quark scattering is included in the calculations. The value of $\Lambda$ used is chosen to correspond with the unpolarized parton parametrization used. Isolated Prompt Photons at RHIC ------------------------------- RHIC is expected to run at center of mass energies between $\sqrt{s}=50$ and $500$ GeV. At the lower energies the fragmentation contribution to prompt photon production is expected to be much less. This has two important consequences. First, since the matrix elements for the fragmentation contribution in the polarized case are still unknown the estimates using LO matrix elements should be more reliable, and secondly, there will be less need to place isolation restrictions on the cross section at lower energies, thereby avoiding all the attendant uncertainties. In Figs.1a and 1b the isolated and non-isolated cross sections for prompt photon production are compared at $50$ and $500$ GeV at an average rapidity $y=0$. The GSA polarized distributions is used and the isolation parameters used are $R=1.0$ and $\epsilon=2\;{\rm GeV}/p_T^{\gamma}$. As expected to $\sqrt{s}=50$ GeV the effect of isolation on the cross section is negligible, whereas at $500$ GeV it is more significant. The total rates are also substantial enough to be measured out to $p_T^{\gamma}=15-20$ GeV at $\sqrt{s}=50$ GeV and $50-60$ GeV at $\sqrt{s}=500$ GeV. In figs.1c and 1d the ratio $\sigma^{frag}/\sigma^{full}$, where $\sigma^{frag}$ is the fragmentation contribution to the cross section and $\sigma^{full}$ is the sum of direct and fragmentation contributions, are plotted vs $p_T^{\gamma}$. This is done for both the isolated and non-isolated cases. The ratio is typically less than $15\%$ for the unpolarized and less than $10\%$ for the polarized case at $\sqrt{s}=50$ GeV, and falls with increasing $p_T^{\gamma}$. Isolation cuts have the effect of reducing the ratio only slightly at this energy. At $\sqrt{s}=500$ GeV the situation is very different. For the range of $p_T^{\gamma}$ shown, before isolation, the fragmentation contribution makes up to around $50\%$ of the cross section at low $p_T^{\gamma}$ and is still significant at medium values of $p_T^{\gamma}$. Once isolation cuts are imposed the fragmentation contribution falls dramatically as one might expect. The fragmentation contributions seem to be more important for the polarized case before isolation, but after isolation the ratio is similar for both the polarized and unpolarized cases. It turn out that this effect is mostly due to the interplay between the various subprocess contributions and depends significantly on the choice of parton distributions made. In figs.2a and 2b predictions for the polarized isolated cross section are compared for the GSA, GSB and GSC parametrizations of the parton distributions at $\sqrt{s}=50$ and $500$ GeV respectively at average rapidity $y=0$. The corresponding asymmetries, defined as the ratio of the polarized to the unpolarized cross section are plotted in figs.2c and 2d. Larger asymmetries are preferred as they indicate that the cross section is sensitive to polarization effects. The GSA and GSB distributions give rather similar predictions at both cms energies in the low $p_T^{\gamma}$ region but tend to diverge as $p_T^{\gamma}$ increases, whereas the GSC predictions are very different at all $p_T^{\gamma}$ values. In both cases GSC predicts negative cross sections for part of the $p_T^{\gamma}$ range covered. This is because in this case $\Delta G$ is negative over part of the $x$-range at input [@gs]. The asymmetry plots reflect the differences between the three parametrizations. There is a wide spread in the three curves of figs.2c and 2d as $p_T^{\gamma}$ increases. The differences should be experimentally distinguishable. In fig.2d the solid line is the predicted asymmetry for the non-isolated cross section using the GSA parametrization. It is very similar to the isolated prediction, although the corresponding cross sections are very different in magnitude. This indicates that the predictions for the asymmetries do not depend very much on whether the photon is isolated or not although the actual sizes of the cross sections are significantly affected at higher cms energies. Figs.3a and 3b show the rapidity distributions of the isolated cross section for the various parametrizations at a fixed value of $p_T^{\gamma}=10$ GeV at $\sqrt{s}=50$ and $500$ GeV respectively. The unpolarized cross section is also shown. Again the three parametrizations of the polarized proton distributions give distinguishable results. This is reflected in the asymmetry plots in figs.3c and 3d which show differences in both shapes and sizes. Changing the factorization/renormalization scales do not have a substantial effect on the asymmetries although the individual cross sections can increase by as much as $50\%$ if $\mu^2=(p_T^{\gamma}/2)^2$ is used. Including fragmentation contributions fully at NLO would likely reduce this scale sensitivity and thus improve the reliability of the predictions for the cross sections. Conclusions =========== The cross section for prompt photon production was presented at RHIC cms energies using polarized proton densities evolved in NLO QCD for the first time. The calculation was performed using the Monte Carlo method and compared to a previous calculation using purely analytic methods and agreement was found. The Monte Carlo method used allowed the inclusion of isolation cuts on the direct component of the cross section in NLO without any further approximations. The fragmentation contribution was estimated in LO where isolation is trivial to implement. The cross section was studied at two cms energies, $\sqrt{s}=50$ and $500$ GeV, typical for the RHIC collider. At $\sqrt{s}=50$ GeV, the fragmentation contributions which can only be estimated in LO were found be small and isolation hardly change the predictions. At $\sqrt{s}=500$ GeV, fragmentation contributes up to $50\%$ of the cross section before isolation, and thus imposition of the isolation cuts substantially reduced the cross section. This means that the predictions presented here for the cross sections are more likely to be reliable at lower cms energies. It turned out that the asymmetries were hardly affected by isolation since both the polarized and unpolarized cross sections are similarly affected and the effect cancelled out in the ratio. A similar effect was found when the factorization/renormalization scales were varied. Thus the asymmetries are the most stable predictions of this calculation, and it can be anticipated that even the inclusion of higher order corrections to the fragmentation contributions are unlikely to change them very much. The three parametrizations for the polarized proton densities gave distinguishable results, particularly at higher $p_T^{\gamma}$ values. This suggests that they should also be distinguishable in the experiment and that, as expected, prompt photon production will definitely prove useful in determining the size of $\Delta G$. Acknowledgments =============== This work was supported by the US Department of Energy, Division of High Energy Physics, Contract number W-31-109-ENG-38. I would like to thank E. L. Berger and G. P. Ramsey for reading the manuscript. E. L. Berger, X. Guo and J.-W. Qiu, Phys. Rev. Lett. [**76**]{}, 2234, (1996); Phys. Rev. [**D 54**]{}, 5470, (1996). P. Aurenche et. al., ENSLAPP-A-595/96, LPTHEORSAY 96-40, hep-ph/9606287. E. L. Berger and J. 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E. Gordon and W. Vogelsang Phys. Rev. [**D50**]{}, 1901 (1993). E. L. Berger and J. Qiu, Phys. Lett. [**B 248**]{}, 371 (1990); Phys. Rev. [**D 44**]{}, 2002 (1991). M. Glück, E. Reya and A. Vogt, Phys. Rev. [**D48**]{}, 116 (1993). H. Lai et al., CTEQ Collaboration, MSUHEP-60426,CTEQ-604, hep-ph/9606399. [FIGURE CAPTIONS]{} [\[\]]{}[ ]{} \(a) The inclusive and isolated differential cross sections at $\sqrt{s}=50$ GeV and rapidity averaged between $-0.5\leq y\leq 0.5$ plotted vs photon $p_T$ for the polarized and unpolarized cases. (b0 same as (a) at $\sqrt{s}=500$ GeV. (c) ratios of the fragmentation contribution estimated with LO matrix elements to the full (direct plus fragmentation) of the polarized and unpolarized cross sections both before and after isolation cuts are implemented at $\sqrt{s}=50$ GeV. (d) same as (a) at $\sqrt{s}=500$ GeV. Polarized differential cross section at average rapidity $y=0$ plotted vs photon $p_T$ at $\sqrt{s}=50$ GeV as predicted using the GSA, GSB and GSC parametrizations of the polarized proton distributions. (b) same as (a) at $\sqrt{s}=500$ GeV. (c) asymmetry plots vs photon $p_T$ for the cross sections given in (a) using the CTEQ4M proton distributions for the unpolarized cross section. (d) same as (c) but for the differential cross sections plotted in (b). The solid curve is for the non-isolated cross section predicted using the GSA polarized distributions. Rapidity distributions for the polarized and unpolarized differential cross sections at $p_T=10$ GeV and $\sqrt{s}=50$ GeV. (b) same as (a) at $\sqrt{s}=500$ GeV. (c) and (d) asymmetry plots for the distributions plotted in (a) and (b) respectively.
--- abstract: | We find that the fraction of early-type galaxies in poor groups (containing from 4 to 10 members) is a weakly increasing function of the number of the group members and is about two times higher than in a sample of isolated galaxies. We also find that the group velocity dispersion increases weakly with the fraction of early-type galaxies. Early-type galaxies in poor groups are brighter in the near-infrared with respect to isolated ones by $\Delta M_{K}\sim 0\fmag75$, and to a lesser degree also in the blue ($\Delta M_{B} \sim 0\fmag5$). We also find early-type galaxies in groups to be redder than those in the field. These findings suggest that the formation history for early-type galaxies in overdense regions is different from that of in underdense regions, and that their formation in groups is triggered by merging processes. author: - 'Hrant M. Tovmassian, Manolis Plionis Heinz Andernach' title: Morphological and Luminosity Content of Poor Galaxy Groups --- Introduction ============ It is known that most galaxies in the Universe occur in small groups (cf. Geller & Huchra 1983; Tully 1987; Nolthenius & White 1987; Fukugita et al. 1998). In the dense and relatively low-velocity dispersion group environments one expects frequent galaxy interactions. Indeed, Tran et al. (2001) showed that a certain fraction of galaxies in evolved, X-ray luminous groups are significantly asymmetric which is evidence of galaxy interactions. N-body simulations of isolated groups indicate that dynamical friction should also play an important role in the evolution of groups (Bode, Cohn, & Lugger 1993; Bode et al. 1994; Athanassoula, Makino, & Bosma 1997). In such a scenario the groups are not in a dynamical equilibrium because of the high frequency of galaxy interactions. Interactions and merging of galaxies has been shown to play a major role in the evolution of galaxy morphology (eg. Toomre & Toomre 1972; Schweizer & Seitzer 1992). The general belief is that early-type (E/S0) galaxies are formed by merging of spiral galaxies (Barnes & Hernquist 1992; Mihos 1995). As a result of multiple merging, a massive, central galaxy can be formed. Obviously this phenomenon must be especially rapid in compact groups. For this reason a lot of efforts have been devoted to the study of dynamical evolution and morphological content of compact groups (e.g. Hickson & Mendes de Oliveira 1992, Mendes de Oliveira & Hickson 1994; Tovmassian 2001, 2002; Kelm & Focardi 2004; Lee et al. 2004; Coziol, Brinks, & Bravo-Alfaro 2004). Contrary to expectation, Shimada et al. (2000) found no statistical difference in the frequency of occurrence of emission-line galaxies between the Hickson compact groups (Hickson 1982; Hickson et al. 1992) and the field. However, Kelm & Focardi (2004) found that compact groups identified in the Updated Zwicky catalogue of galaxies contain a higher fraction of early-type galaxies with respect to the field. On the other hand, Colbert, Mulchaey, & Zabludoff (2001) have shown that early-type galaxies in the field appear to have more shells and tidal features than those in groups, a fact that they attribute to merging events that occur also in the field, and whose signatures probably survive due to isolation of these galaxies.These results seem to be confirmed by Marcum, Aars, & Fanelli (2004) although they have also identified a small fraction of isolated early-type galaxies that show no evidence of a merger history, and thus appear to be passively evolving primordial galaxies. Recently we have used the Ramella et al. (2002) UZC-SSRS2 group catalogue, based on the Updated Zwicky Catalogue (UZC; Falco et al. 1999) and the Southern Sky Redshift Survey (SSRS2; da Costa et al. 1998) to show that poor groups have a prolate spheroidal shape configuration with a mean intrinsic axial ratio ($\beta$) of $\langle \beta\rangle \approx0.3$ and standard deviation $\sigma_{\beta}\approx 0.15$ (Plionis, Basilakos, & Tovmassian 2004, \[hereafter PBT04\]). As interaction are more likely in such elongated systems, poor groups are good laboratories to test galaxy formation theories and study their evolution. In this paper we compare the morphological and luminosity content (in the K and B bands) of the group galaxy members with those of isolated galaxies. Observational Data ================== We have used the poor groups of the UZC-SSRS2 catalogue (USGC, Ramella et al. 2002) as defined in PBT04, i.e., groups with galaxy membership $n_m$ (“richness“ in what follows) in the range $4\le n_m\le 10$ and radial velocity $cz \le 5500$ km $s^{-1}$, since within this velocity limit the groups appear to have constant space density and therefore they are assumed to constitute a roughly volume-limited sample. As in PBT04, we divide groups in different richness classes. All candidate fake groups (see PBT04) were excluded from our analysis. Furthermore we excluded those of our $n_m=4$ groups which Focardi & Kelm (2002) identified as triplets (due to possible projection contamination), i.e., USGC U033, U039, U066, U070, U076, and U127 (which correspond to FK 12, 14, 26, 27, 29 and 35). As a starting point for our analysis we use 169 groups containing in total 932 galaxies, out of which 58, 59, 35 and 17 groups have a richness ($n_m$) of 4, 5-6, 7-8 and 9-10 galaxies, respectively. For comparison we used the catalogue of Isolated galaxies (IGs) compiled by Karachentseva, Lebedev & Shcherbanovskij (1986). This catalogue is widely used as a source of isolated galaxies (e.g. Marcum et al. 2004; Varela et al. 2004). It contains 1051 entries, among which there are 329 galaxies with $cz \le 5500$ km $s^{-1}$. Note that this galaxy sample has the same magnitude limit as our group galaxies ($m_{lim}\sim 15.5$). Two of the IGs, PGC 008220 and PGC 059971, were found in USGC groups and were deleted from the IG list. Galaxy morphological types for both poor group galaxies (GGs) and isolated ones were taken from the NED (http://nedwww.ipac.caltech.edu). Since the galaxies we are dealing with are nearby and sufficiently bright ($m_{\rm lim}\simeq 15.5$), we expect that the NED morphological classification into the two broad categories (early and late types) is very accurate. We deduce absolute magnitudes in the $K$ band for two families of galaxies: (a) E/S0 galaxies and (b) spiral galaxies (Sa and later, including irregulars) for both the considered USGC group members and the IGs. We used the 2[MASS]{} $K_{\rm total}$ magnitudes (see Jarrett et al. 2000 and http://www.ipac.caltech.edu/2mass), corrected for the extinction in our Galaxy according to Schlegel, Finkbeiner, & Davis (1998) as given in NED. The quoted 2[MASS]{} average magnitude 1$\sigma$ uncertainty is $\sim$0.04 for $m_k<11$ and $\sim$0.09 for $m_k\ge 11$. The absolute $B$ magnitudes for all studied galaxies were extracted from LEDA (Paturel et al. 1997, http://leda.univ-lyon1.fr) and their quoted individual 1$\sigma$ uncertainties are quite large, the average of which is estimated to be $\sim 0.3$. Note that in order to determine absolute magnitudes the radial velocities of groups or individual galaxies were corrected for the peculiar velocity of the local group and a local velocity field that contains a Virgo-centric infall component and a bulk flow given by the expectations of linear theory (see Branchini, Plionis, & Sciama 1996), assuming a Hubble constant of $H_0=70$ km s$^{-1}$ Mpc$^{-1}$. We excluded from our analysis groups and isolated galaxies that have $cz \le 1000$ km s$^{-1}$, as their radial velocities may be significantly contaminated by peculiar velocities. In order to investigate the mutual completeness of the GG and IG samples we employed the non-parametric Kolmogorov-Smirnov (KS) test and found that their redshift distributions may be considered (at a ${\raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept\hbox{$>$}\ }}$25% level) as having been drawn from the same parent population. This also implies that there is no systematic difference in their limiting magnitudes. The fraction of early-type galaxies =================================== We determine the mean fraction of E/S0 galaxies, $f_{E/S0} (\equiv n(E/S0)/n_m$), over our group sample, as well as in the sample of IGs. We use only those groups of which all members have known morphological type, in total 102 groups spanning the whole original sample richness range and which contain in total 583 galaxies (38, 27, 27 and 10 groups with $n_m=4$, 5-6, 7-8 and 9-10 members, respectively). For the sample of 329 IGs we find $f_{E/S0}\sim 0.15$ while for the group galaxies we obtain a significantly higher mean fraction of $f_{E/S0}\sim 0.23$. This result is in accordance with the “morphology - density” relation of Dressler (1981) Kuntschner et al. (2002) also noticed that early-type galaxies are rare in low-density environments. The higher relative number of E/S0 galaxies in groups favors the view that they are formed by the interaction and merging of spiral galaxies. The number of such galaxy encounters is expected to be even higher in high density environments, ie., in groups with large $n_m$. To test this we estimate the mean fraction of early-type galaxies as a function of increasing group richness ($n_m$) and find: $f_{E/S0}\sim$ 0.21, 0.19, 0.29 and 0.26 for groups with $n_m=4$, 5-6, 7-8, 9-10 members, respectively, suggesting a weak dependence of $f_{E/S0}$ on the group richness. However, it seems surprising that the relative number of E/S0’s and possibly their formation rate depends only weakly on the group richness and thus on the group mass. We investigate this issue further by correlating $f_{E/S0}$ with the velocity dispersion, $\sigma_v$, of the corresponding groups (Fig. 1), and we find that there is indeed a significant although weak correlation with the group velocity dispersion increasing with $f_{E/S0}$ (in agreement with Zabludoff 2000). The Spearman rank correlation coefficient is 0.2 and the probability of random correlation is 0.045. Meanwhile, such a correlation is not seen for groups of a given richness class, which confirms that the effect is due to an increase of the velocity dispersion with the group mass. The true underlying correlation should be even stronger than the observed one since the dependence of the measured group velocity dispersion on group orientation can weaken it. This effect results from the fact that elongated prolate-like groups orientated roughly along the line of sight will appear to have a higher velocity dispersion while when seen orthogonal to the line of sight they will appear to have smaller velocity dispersions. Another parameter that should play a significant role in the quantification of the rate of galaxy encounters in groups is the group crossing time, $\tau_c$, defined by: $\tau_{c}=\overline{a}/\overline{\sigma}_v \;,$ with $\overline{a}$ the median true group major axis and $\overline{\sigma_v}$ the median true (deprojected) group velocity dispersion. The probability of an encounter in a group with a given number of members should be higher in groups with smaller $\tau_c$. However, effective interactions that could alter galaxy morphologies should happen in environments of relatively low velocity dispersion which implies a relatively large $\tau_c$ for a given group size. Note also that in high $\sigma_v$ systems that are also extremely elongated, effective galaxy interactions may happen at the apogees of the spatial configuration where the relative velocities of galaxies will be minimal and where the member galaxies will tend to accumulate. Clearly, the dynamics of galaxy mergers in non-spherical structures is a multi-parameter problem and not easily quantified. We attempt to estimate the mean crossing time for groups of different richness ($n_m$) by taking into account the fact that they are prolate-like (Plionis et al. 2004) and that their member galaxies will probably have mostly radial orbits (moving along the major axis of the prolate spheroid). The [*true*]{} median $\sigma_v$ is estimated using the quasi-spherical groups i.e. those with axial ratio close to unity, since due to orientation effects, the quasi-spherical groups should typically be those elongated groups seen end-on and therefore their velocity dispersion is closer to the true value. We find $\overline{\sigma}_v\simeq 170$ km/sec and 360 km/sec for the $n_m=4$ and $n_m=9,10$ groups respectively. In order to estimate the true size of groups we rely on a Monte-Carlo simulation method, the details of which will be presented elsewhere (Plionis & Tovmassian 2004, in preparation). It is based on searching for the intrinsic group major axis distribution that once folded through the intrinsic group axial ratio distribution (determined in Plionis et al. 2004) and random orientations with respect to the line of sight, will produce the projected Monte-Carlo group shape parameters (axial ratio, minor and major axis) distributions which are in agreement with the corresponding observed ones. We find $\overline{a}_{3D}\simeq 1.5\pm 0.3$ and $2\pm 0.5 \; h_{70}^{-1}$ Mpc, respectively for the $n_m=4$ and $n_m=9,10$ groups and therefore the group crossing times are: $\tau_{c}\sim 8.6$ and 5.8 Gyrs, respectively. To test the effect of the different values of $\tau_c$ on the galaxy encounter and merging rate, disentangling the possible effect of the different group orientations with respect to the line of sight, we determined the mean $f_{E/S0}$ in the most elongated chain-like groups (ie., those which are preferentially seen perpendicular to the line of sight) for two group categories: those with $\tau_c$ higher and smaller than the mean value of the corresponding group subsample. There are 36 groups with axial ratio $q<0.3$ (all of them with $n_m\le 6$) and with known morphology of all their members. We find $f_{E/S0} \simeq 0.25\pm0.055$ and $0.165\pm0.045$ for groups with $\tau_c$ smaller and larger than the corresponding group mean value, respectively. Although the difference is not very significant it does imply, as expected, that the crossing time is an important parameter in the quantification of the rate of galaxy interactions. Absolute magnitudes and colours of galaxies. ============================================ If E/S0 group galaxies are formed as the result of galaxy merging, one should expect them to be more luminous than isolated E/S0’s. We estimated the mean $\langle M_{K}\rangle$ and $\langle M_{B}\rangle$ absolute magnitudes of E/S0 and spiral galaxies (Sa and later) in our original group sample ($N=169$) and isolated galaxies (see Table 1). The frequency distribution of the absolute magnitudes of the group and isolated early-type galaxies is shown in Fig. 2. We also derived the absolute magnitude difference ($\Delta M$) between GGs and IGs and found $\Delta M_K \approx0\fmag75$ and $\Delta M_B \approx0\fmag5$ for the E/S0’s and $\Delta M_K \approx0\fmag05$ and $\Delta M_B \approx0\fmag02$ for the spirals (see Table 1). The KS test shows that the GG and IG $K$-band absolute magnitude distributions of early-type galaxies are significantly different, with the probability ${\cal P}\simeq 0.05$ of being drawn from the same population. The corresponding $M_B$ distributions do not show a significant difference. Furthermore, no difference in absolute magnitude (neither for $M_K$ nor for $M_B$) is observed between group and isolated spirals. The KS test shows that their distributions have a probability of being drawn from the same population of ${\cal P}\simeq0.7$ and $0.95$ respectively for the $K$ and $B$ bands. We have also found that the values $\langle M_{B}\rangle$ and $\langle M_{K}\rangle$ of E/S0 galaxies do not depend on the group richness. We conclude that indeed there is strong evidence that the group E/S0 galaxies are brighter, especially in $K$ with respect to the IGs of either type, E/S0 or spiral. The fact that absolute magnitudes of the group E/S0 galaxies are on average brighter than those of isolated galaxies by $\approx0\fmag75$ is consistent with the idea that E/S0 group galaxies are formed by merging of two spirals of about the same luminosity. The latter is in general agreement with the environmental dependence of the IR luminosity function found by Balogh et al. (2001). Finally, we compare the $M_{B}-M_{K}$ colours of the considered subsamples. We find that the mean colour of the E/S0 galaxies in groups is redder by about $0\fmag30$ than that of the IGs, and the corresponding colour distributions are significantly different, having a KS probability of consistency of only ${\cal P}\simeq 0.006$. A redder colour of the group E/S0’s may be due to shed of gas during merging. As expected, we find that both group and isolated spiral galaxies are bluer on average in comparison to the group and isolated E/S0 galaxies (see Fig. 3 and Table 1). Fig. 3 also shows the presence of a few relatively blue galaxies ($M_{B}-M_{K}\le 1$) among the group E/S0’s for which weak starburst processes, induced by interactions, could be responsible (Zepf, Whitmore, & Levison 1991). Conclusions =========== We have found that the mean fraction of early-type galaxies, $f_{E/S0}$, in poor USGC groups of galaxies (with $1000<cz<5500$ km s$^{-1}$ and galaxy members $4\le n_m\le 10$) is $\sim 0.23 \pm 0.04$, which is $\sim 65\%$ higher than that of the field galaxies. This fraction is a weakly increasing function of group richness and thus of group mass. This trend is also confirmed from the statistically significant, although weak, increase of $f_{E/S0}$ with group velocity dispersion. We have also found that the mean near-infrared absolute magnitude of E/S0 galaxies in our groups ($\langle M_K \rangle\approx-23\fmag4$) does not depend on the number of the group members and is $\sim 0\fmag75$ brighter than that of isolated E/S0 galaxies, while the mean blue absolute magnitude, $\langle M_B\rangle$, of E/S0 galaxies in these groups is brighter by $0\fmag5$ than the isolated ones. No such differences are found in the corresponding spiral galaxy samples. We conclude that E/S0 galaxies in groups may be the result of merging of two galaxies of about the same luminosity. The $M_B-M_K$ colours of the group E/S0 galaxies are redder, by about $0\fmag30$ on average, than those of isolated E/S0 galaxies. These results are in agreement with the paradigm in which the early-type galaxies, in relatively dense environments, are formed by merging. The redder $M_B-M_K$ colours of the group E/S0 galaxies in comparison to the isolated ones shows that the formation processes of the former somehow differs from that of isolated galaxies. An open question is still how the isolated E/S0 galaxies are formed. Various authors have found signs of disturbed morphologies in field E/S0’s (eg. Colbert et al. 2001; Kuntschner et al. 2002; Marcum et al. 2004). Although the bright isolated E/S0 galaxies may also be remnants of merged groups or pairs of galaxies, the isolated faint E/S0’s probably are not (eg. Marcum et al. 2004); their origin may be primordial and thus different from those in groups. 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E. Hibbard, M. P. Rupen, & J.H. van Gorkom, San Francisco: Astronomical Society of the Pacific, p. 547 Zepf, S. E., Whitmore, B. C., & Levison, H. F. 1991, ApJ, 383, 524 ---- ------------------------ ----------------------- ----------------- ------------------------ ----------------------- ----------------- $\langle M_K \rangle$ $\langle M_B \rangle$ $\Delta M$ $\langle $\langle M_B \rangle$ $\Delta M$ M_K\rangle$ GG $-23.42\pm 1.39$ (183) $-19.92\pm1.17$ (194) -3.47$\pm 0.76$ $-22.51\pm1.51$ (398) $-19.47\pm1.36$ (500) -2.89$\pm 0.89$ IG $-22.68\pm 1.64$ (43) $-19.44\pm1.32$ (48) -3.16$\pm 0.72$ $-22.54\pm 1.46$ (205) $-19.44\pm1.40$ (269) -2.82$\pm 0.86$ ---- ------------------------ ----------------------- ----------------- ------------------------ ----------------------- ----------------- : The mean absolute magnitudes $\langle M_K \rangle$ and $\langle M_B \rangle$ of E/S0 and of spiral galaxies (Sa and later) in groups and in the field. In parenthesis the number of corresponding galaxies is given.
--- abstract: 'We review models of chondrite component transport in the gaseous protoplanetary disk. Refractory inclusions were likely transported by turbulent diffusion and possible early disk expansion, and required low turbulence for their subsequent preservation in the disk, possibly in a dead zone. Chondrules were produced locally but did not necessarily accrete shortly after formation. Water may have been enhanced in the inner disk because of inward drift of solids from further out, but likely not by more than a factor of a few. Incomplete condensation in chondrites may be due to slow reaction kinetics during temperature decrease. While carbonaceous chondrite compositions might be reproduced in a “two-component” picture (Anders 1964), such components would not correspond to simple petrographic constituents, although part of the refractory element fractionations in chondrites may be due to the inward drift of refractory inclusions. Overall, considerations of chondrite component transport alone favor an earlier formation for carbonaceous chondrites relative to their noncarbonaceous counterparts, but independent objections have yet to be resolved.' address: 'Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St Georges Street, Toronto, ON M5S 3H8, Canada' author: - Emmanuel Jacquet bibliography: - 'bibliography.bib' title: 'Transport of solids in protoplanetary disks: Comparing meteorites and astrophysical models' --- Introduction ============ With the accelerating pace of exoplanet detections, the protoplanetary disk phase of stellar evolution enjoys considerable interest. Thanks to increasing computational power, theorists can test mechanisms for disk transport [@Turneretal2014] and planet formation [@YoudinKenyon2013]. Observations of present-day protoplanetary disks [@WilliamsCieza2011] probe the disk mass, size, structure, chemical species and solids [@Nattaetal2007]. However, even with the Atacama Large Millimeter/Submillimeter Array, it will remain challenging to resolve scales below a few AUs and probe the optically thick midplane of the inner disks where planet formation should occur. To understand the evolution of solids in disks, we must turn our attention to constraints provided closer to us by our own solar system in the form of primitive meteorites, or *chondrites*. Indeed, chondrites date back to the protoplanetary disk phase of the solar system, 4.57 Ga ago, and with more than 40,000 specimens classified to date, not to mention samples returned from comet Wild 2 [e.g. @Zolenskyetal2008Wild2] or asteroid Itokawa [@Nakamuraetal2011], they offer a considerable wealth of petrographic, chemical, and isotopic data at all examination scales. Yet chondrites arrive in our laboratories without geological context. While orbit determinations consistently assign their parent bodies to the asteroid main belt—with the exception of micrometeorites [e.g. @EngrandMaurette1998] and perhaps some carbonaceous chondrites [@Gounelleetal2008] possibly derived from further out—, exactly where and when they originally accreted is largely unknown. Still, chondrites exhibit considerable compositional variations, and space and time were obviously important dimensions behind them. In fact, each individual chondrite is a mixture of components (chondrules, refractory inclusions, etc.) formed in different locations, epochs and environments in the disk [@BrearleyJones1998; @Krotetal2009]. This is evidence for considerable transport in the disk. In order to place the meteoritical record in context, the relevant transport processes have to be understood, and as such meteorites are sensors of the dynamics of protoplanetary disks. Our purpose here is to review transport mechanisms of chondrite components before accretion. An earlier review on particle-gas dynamics was given by @CuzziWeidenschilling2006 and @Boss2012 reviewed transport and mixing from the perspective of isotopic heterogeneity. The formation *per se* of chondrite components is essentially beyond our scope but the reader may be referred to recent reviews by @Krotetal2009 and @Aleon2010. @Wood2005 and @Chambers2006 proposed syntheses on the origin of chondrite types from cosmochemical and astrophysical viewpoints, respectively. Here, the discussion will be organized around meteoritical constraints as follows: In section \[Background\], we provide background on chondrites and the basic physics of the protoplanetary disk before embarking in section \[Transport\] on an examination of transport constraints from specific chondrite components. We then review the interpretation of fractionation trends exhibited by chondrites as wholes (section \[Fractionation\]) in light of which we will discuss the chronological and/or spatial ordering of chondrite groups (section \[Chondrites in space and time\]). Background {#Background} ========== Chondrites: a brief presentation -------------------------------- Chondrites are assemblages of various mm- and sub-mm-sized solids native to the protoplanetary disk. Oldest among them are the *refractory inclusions* [@MacPherson2005; @Krotetal2004], further divided in calcium-aluminum-rich inclusions (CAI) and (less refractory) amoeboid olivine aggregates (AOA), which presumably originated by high-temperature gas-solid condensation, 4568 Ma ago [@BouvierWadhwa2010; @Connellyetal2012; @Kitaetal2013], although many have since experienced melting. More abundant than those are *chondrules*, silicate spheroids 1-4 Ma younger than refractory inclusions [@KitaUshikubo2012; @Connellyetal2012], likely formed by melting of isotopically and chemically diverse precursor material. The nature of the melting events remains however elusive, with “nebular” (e.g. shock waves) and “planetary” (e.g. collisions) environments still being considered [@Boss1996; @Deschetal2012]. *Metal* and *sulfide* grains also occur, either inside or outside chondrules [@Campbelletal2005]. All these components are set in a fine-grained *matrix*, a complex mixture of presolar grains, nebular condensates and/or smoke condensed during chondrule-forming events [@Brearley1996]. While all chondrites roughly exhibit solar abundances for nonvolatile elements [@PalmeJones2005], with CI chondrites providing the best match, they are petrographically, chemically and isotopically diverse, and 14 discrete *chemical groups*, each believed to represent a distinct parent body (or a family of similar ones), have hitherto been recognized. To first order, one may partition these groups in two *super-clans* [@Kallemeynetal1996; @Warren2011b], namely the *carbonaceous chondrites* (with the CI, CM, CO, CV, CK, CR, CB, CH groups), and the *non-carbonaceous chondrites*, which comprise the enstatite (EH, EL), ordinary (H, L, LL) and Rumuruti (R) chondrites. Carbonaceous chondrites are more “primitive” in the sense that they have a higher abundance of refractory inclusions and matrix, a solar Mg/Si ratio, and an $^{16}$O-rich oxygen isotopic composition closer to that of the Sun [@McKeeganetal2011]. Non-carbonaceous chondrites, though poorer in refractory elements, are more depleted in volatile elements, have subsolar Mg/Si ratios and a more terrestrial isotopic composition for many elements [e.g. @Trinquieretal2009]. What these differences mean and how some may relate to the transport of chondrite components is one of the main focuses of this review. Dynamics of the early solar system {#Disks} ---------------------------------- The exact structure of our protoplanetary disk remains very conjectural. If we mentally add gas to a smoothed version of the current planetary system to restore solar abundances, we obtain a density profile known as the “Minimum Mass Solar Nebula” (MMSN; @Hayashi1981) with an integrated mass $\sim$0.01 M$_\odot$ (1 M$_\odot$ $\equiv$ 1 solar mass). While this agrees with disk masses estimated for most T Tauri stars [@WilliamsCieza2011], it may be one order of magnitude below the original disk mass at the cessation of infall from the parent molecular cloud [e.g. @YangCiesla2012]. The MMSN, though a useful reference, ignores the extensive redistribution and losses of gas and solids occurring in disks which funnel gas onto the central stars at observed rates of $10^{-8\pm 1}\:\mathrm{M_\odot/a}$ [@WilliamsCieza2011]. What drives the evolution of gas disks? Since the molecular viscosity is far too small to account for the $\sim$1-10 Ma lifetime of protoplanetary disks [@WilliamsCieza2011], disk theorists generally rely on turbulence which, in a *rough, large-scale* sense, may mimic the effects of an enhanced viscosity $$\label{nu} \nu=\alpha \frac{c_s^2}{\Omega},$$ with $c_s$ the (isothermal) sound speed, $\Omega$ the Keplerian angular velocity and $\alpha$ the dimensionless “turbulence parameter” (see e.g. @BalbusPapaloizou1999), for which values around $10^{-2}$ are inferred from observations [@Armitage2011]. The exact source of this turbulence is still contentious. As yet, the leading candidates are *gravitational instabilities* [@Durisenetal2007] and the *magneto-rotational instability* (MRI; @BalbusHawley1998). While the former would be important in the earliest epochs where the disk is massive enough, the latter essentially only requires the gas ionization fraction to be above a small threshold, and may operate at all times. However, this threshold may not be attained over a considerable range of heliocentric distances, yielding a *dead zone* of low turbulence, unless other instabilities are at work (see @Turneretal2014). Eventually, after the disk mass has significantly dropped, photoevaporation due to the central star (and/or close neighbours) should completely clear the gas [@Armitage2011]. Except in late stages of the disk where photophoresis may become important [@WurmKrauss2006], the dynamics of small solids are primarily dictated by *gas drag*. The stopping time, for a spherical particle of radius $a$ smaller than the molecular mean free path, is [@Weidenschilling1977]: $$\label{tstop} \tau=\sqrt{\frac{\pi}{8}}\frac{\rho_sa}{\rho c_s},$$ with $\rho_s$ and $\rho$ the solid and gas densities, respectively. For instance, at 3 AU in a MMSN, a 0.3 mm radius chondrule has $\tau = 2$ h at the midplane, much shorter than the orbital period. Hence, to zeroth order, chondrite components should follow the gas, but they cannot have *exactly* the same velocity because they do not “feel” the pressure gradient acceleration experienced by it. There is thus a systematic drift velocity of the solids relative to the gas in the direction of larger pressures, that is, toward both the midplane and the Sun. The radial drift is given by: $$\begin{aligned} v_{\rm drift,R}=\frac{\tau}{\rho}\frac{\partial P}{\partial R}=0.004\:\mathrm{m/s}\frac{\partial\mathrm{ln}P}{\partial\mathrm{ln}R}\left(\frac{\rho_sa}{1\:\rm kg/m^2}\right)\left(\frac{10^{-6}\:\rm kg/m^3}{\rho}\right)\nonumber\\ \left(\frac{1\:\rm AU}{R}\right)\left(\frac{T}{300\:K}\right)^{1/2}\end{aligned}$$ with $P$ and $T$ the gas pressure and temperature and $R$ the heliocentric distance. While this drift is generally smaller than the turbulent velocity fluctuations of the gas ($\sim\sqrt{\alpha}c_s$), it may become important in the long run as these average out. A measure of this importance is the “gas-grain decoupling parameter” [@Cuzzietal1996; @Jacquetetal2012S]: $$\begin{aligned} \label{S} S & \equiv & \frac{\Omega\tau}{\alpha} \\ &=& 0.1\left(\frac{\rho_sa}{1\:\rm kg/m^2}\right)\left(\frac{10^{-8}\:\rm M_\odot/a}{\dot{M}}\right)\left(\frac{R}{1\:\rm AU}\right)^{3/2}\left(\frac{T}{300\:\rm K}\right),\nonumber\end{aligned}$$ where the last equality is for a steady disk of mass accretion rate $\dot{M}$. For $S\ll 1$, the particles are tightly coupled to the gas, while for $S\gtrsim 1$, they settle to the midplane (with a concentration factor $\sim\sqrt{S}$) and drift radially sunward faster than the gas. Over time, grains collide and coagulate, as evidenced by detection, in protoplanetary disks, of mm-sized solids [@Nattaetal2007], much larger than typical interstellar grains ($\sim 0.1\:\mu$m). Growth to meter size is frustrated by bouncing/fragmentation at high collision speeds and by increased radial drift which would remove them from the disk within centuries [@Weidenschilling1977; @Braueretal2007; @Birnstieletal2010]. Settling to the midplane might help self-gravity of the solids to intervene but is limited by gas turbulence, so that other mechanisms such as turbulent concentration or streaming instabilities may have to bridge the gap [@CuzziWeidenschilling2006; @YoudinKenyon2013]. Here, we will mostly restrict attention to sub-mm/cm-sized bodies since chondrite composition was established at the agglomeration of such components. Transport of chondrite components {#Transport} ================================= With these fundamentals in mind, we now turn to constraints provided by specific chondrite components, and the astrophysical processes which may satisfy them. Refractory inclusions {#CAI} --------------------- The oldest solids of the solar system, the refractory inclusions, are estimated to have formed at $\sim$1400-1800 K [@Grossman2010]. Such high temperatures, presumably due to small-scale dissipation of turbulence, did not obtain at heliocentric distances $>$ 1 AU for more than a few $10^5$ years after disk formation [e.g. @YangCiesla2012]. This is consistent with the old age of CAIs and AOAs; yet to account for their presence in chondrites and even in comets [e.g. @Simonetal2008Inti], outward transport is in order. It would also account for the abundance of crystalline silicates in comets [e.g. @Zolenskyetal2008Wild2]. @Shuetal1996 proposed that solids processed at $R<0.1$ AU were entrained by stellar winds and fell back onto the disk further out [@Hu2010]. @Deschetal2010 however criticized the role of this “X-wind” in processing chondrite components e.g. regarding the very survival of solids (see also @Cuzzietal2005 on the formation timescales of Wark-Lovering rims) so that one may have to seek transport within the disk itself. In a turbulent disk, velocity fluctuations may send high-temperature material outward [e.g. @Gail2001; @BockeleeMorvanetal2002; @Cuzzietal2003; @Boss2004; @Ciesla2010; @HughesArmitage2010]. The envisioned random-walk paths may explain complex thermal histories recorded by some refractory inclusions [@Ciesla2011radial; @Bossetal2012]. The efficiency of this turbulent diffusion against the mean inward flows depends sensitively on the Schmidt number Sc$_R\equiv\nu/D_R$, with $D_R$ the turbulent diffusivity, although many studies have simply equated it to 1. Sc$_R<1$ seems required for efficient outward diffusion [@ClarkePringle1988; @PavlyuchenkovDullemond2007; @HughesArmitage2010]. While MRI-driven turbulence would likely not satisfy this requirement [@Johansenetal2006], this is expected from hydrodynamical turbulence [@Prinn1990], but save for a few laboratory experiments, of uncertain relevance to protoplanetary disks [e.g. @Launder1976; @Lathropetal1992], empirical evidence is largely wanting. Turbulent diffusion may have been supplemented by *outward* advection flows early in the disk evolution. Indeed, the disk may have been initially compact ($\lesssim$ 10 AU in radius), and its ensuing expansion would have begun in the condensation region of refractory inclusions [@Jacquetetal2011a; @YangCiesla2012]. Outward transport would then have been efficient for the earliest generation of CAIs, hence perhaps their narrow *observed* age range [@Ciesla2010; @YangCiesla2012]. Outward flows have been proposed to persist around the disk midplane, even if the vertically integrated flow is inward, and ease the outward transport of inner disk material at later stages [@Ciesla2007; @HughesArmitage2010]. This so-called meridional circulation arises when turbulence is modeled as a viscosity in a very literal sense [e.g. @TakeuchiLin2002]. However, while the gross properties of turbulent disks may be obtained with this ansatz, there is no first principle reason that the resulting two-dimensional flow structure should also hold true, and in fact meridional circulation has not been observed in numerical simulations of MRI-driven turbulence by @Fromangetal2011 or @Flocketal2011. At any rate, @Jacquet2013 showed that even if meridional circulation existed, it would not, because of the inward flows in the upper layers, make a significant difference in terms of net radial transport compared to 1D models, given current uncertainties in turbulence parameters. Not only were refractory inclusions transported in the disk, they were preserved there quite efficiently, for the CAI fraction in some carbonaceous chondrites is comparable to what *in situ* condensation out of a solar gas would have produced ($\sim$6 %; @Grossman2010). However, the inward gas flows and grain-gas radial drift have long been expected to remove them from the chondrite-forming regions within a few $10^5$ years—especially for the large type B CAIs in CV chondrites—, even though the drift slows down closer to the Sun [@Laibeetal2012]. Indeed turbulent diffusion calculations by @Cuzzietal2003 and @Ciesla2010 underpredicted CAI abundances by 1-2 orders of magnitude—unless the “CAI factory” was enriched in condensible matter, which, if not very carbon-rich, would however yield too oxidizing conditions [@Jacquetetal2011a]. In simulations starting with compact disks, however, @YangCiesla2012 achieved retention of refractory inclusions for $>$2 Ma, presumably because many of them were sent far from the Sun (10-100 AU), their disk remained quite massive ($\gtrsim 0.1\:\rm M_\odot$) even after a few Ma, and $\alpha$ was relatively low ($10^{-3}$; see also equation 8 of @Jacquetetal2011a). In fact, low turbulence levels $\alpha\lesssim 10^{-4}$ could *alone* account for the preservation of refractory inclusions, assuming outward transport was accomplished somehow before, and low turbulence is exactly what is generically expected from the dead zone picture [@Jacquetetal2011a]. Indeed, the dead zone, which would have emerged *after* an initially turbulent phase conducive to extensive transport, would slow down gas accretion toward the Sun, and by forcing gas to accumulate there, would also reduce the stopping time and thence the drift of refractory inclusions. Importantly, efficient diffusion as expected in the early disk would rapidly homogenize any short-lived radionuclide like $^{26}$Al [e.g. @Bossetal2012] and hence validate its use as a chronometer. Same would not hold, however, if such isotopes were injected into the disk after the formation of a dead zone. Chondrules {#Chondrules} ---------- The low turbulence levels invoked above for the preservation of refractory inclusions would as well account for the few-Ma age range of chondrules measured in single meteorites [@KitaUshikubo2012; @Connellyetal2012], for chondrules and refractory inclusions have comparable sizes. However, @AlexanderEbel2012 argued that turbulent mixing would homogenize chondrule populations over chondrite-forming regions within a few $10^5$ years, at variance with the distinctive chondrule populations of the different chemical groups [@Jones2012]. They thus suggested that the Al-Mg ages—although broadly corroborated by Pb-Pb dating—were perturbed (but see @KitaUshikubo2012) and that the data are consistent with chondrule formation immediately preceding chondrite accretion. From an astrophysical standpoint, this may be a premature conclusion, though. The diffusion length after a time $t$ is $$\sqrt{2D_R t} =1\:\mathrm{AU}\left(\frac{t}{1\:\rm Ma}\right)^{1/2}\mathrm{Sc}_R^{-1/2}\left(\frac{\alpha}{10^{-4}}\right)^{1/2}\left(\frac{T}{300\:\rm K}\right)^{1/2}\left(\frac{R}{1\:\rm AU}\right)^{3/4}.$$ So whether this exceeds the separation between different chondrule-forming regions depends, among other things, on the exact values of $\alpha$ and on where the chondrule- and chondrite-forming regions actually were in the disk. These locales may have been quite distinct from the present-day position of chondrite parent bodies, in the asteroid main belt, especially if their orbits were significantly reshuffled e.g. during a “Grand Tack” [@Walshetal2011]. Moreover, this calculation ignores the barrier that gas drag-induced drift may have posed to outward mixing (if $S>1$). In fact, if mixing had been as efficient as to homogenize the chondrite-forming region in $\lesssim 1$ Ma timescales, bulk chemical fractionations observed across chondrite groups would be difficult to understand (see section \[Fractionation\]). Another constraint that chondrule transport must satisfy is *chondrule-matrix complementarity* [@Hussetal2005]. This is the observation, for carbonaceous chondrites, that while the bulk rocks have solar Mg/Si ratios (or other interelement ratios, see @HezelPalme2010), this does not hold for chondrules or matrix taken individually. Complementarity, if confirmed (but see @Zandaetal2012), requires that chondrules and matrix be genetically related (unlike, e.g., a X-wind scenario), but also that chondrules and dust from a given chondrule-forming region did not drift apart until accretion. While a particular chondrule and a particular dust grain would quickly separate barring immediate accretion, this would not be true of the populations of chondrules and dust grains *as wholes* which would remain spatially indistinguishable for some time due to turbulence. In fact, in the regime $S<1$ (for chondrules), @Jacquetetal2012S showed that this overlap would continue over their whole drift timescale, so that accretion at any time would yield complementarity. Complementarity would not be compromised by mixing between products of several chondrule-forming events provided transport from each of these sources was likewise unbiased as to the chondrule/dust ratio [@Cuzzietal2005; @Jacquetetal2012S]. Chondrule transport in the disk is thus still compatible with observations, although a link between *chondrule* and *chondrite* formation cannot be ruled out. Metal and sulfide grains {#Metal} ------------------------ Chondrites, and especially non-carbonaceous ones—plus the very metal-rich CHs and CBs, although their genesis likely was very anomalous [e.g. @Krotetal2005]—, have undergone metal/silicate fractionation prior to accretion [@LarimerWasson1988; @Wood2005]. While early workers invoked some separation of metal grains directly condensed from the hot solar nebula, e.g. because of ferromagnetically enhanced coagulation (@HarrisTozer1967; see also a review by @Kerridge1977), there is little evidence of *isolated* pristine nebular metal condensates in meteorites, although such grains are found *enclosed* in refractory inclusions [@Weisbergetal2004; @Schwanderetal2013]. Actually, chondrite metal grains mostly seem to be byproducts of chondrule-forming events [@Campbelletal2005]. Metal/silicate fractionation may have arisen locally by aerodynamic sorting [e.g. @Zandaetal2006], e.g. because of differential radial and/or vertical drift, or turbulent concentration. Indeed, in both ordinary [@Kuebleretal1999; @NettlesMcSween2006] and enstatite [@Schneideretal1998] chondrites, metal and sulfide grains have a somewhat lower $\rho_sa$ than chondrules on average, but are closest to aerodynamic equivalence with them for their most Fe-rich varieties (H and EH, respectively) which have the smaller chondrules. Therefore, metal/sulfide grains and small chondrules could have been segregated together. Alternatively, metal/silicate fractionation might reflect varying contributions of debris of differentiated planetesimals predating chondrule formation (e.g. @SandersScott2012; but see @FischerGoeddeetal2010). Matrix grains ------------- The grains of chondritic matrices are typically sub-$\mu$m-sized [@PontoppidanBrearley2010], likely too small to show any decoupling relative to the nebular gas prior to agglomeration as fluffy aggregates or fine-grained rims around chondrules [e.g. @MetzlerBischoff1996]. For $\alpha=10^{-3}$, surface densities below 1 kg/m$^2$ (2 orders of magnitude lower than the MMSN at 30 AU) would be required to see any effect on radial drift (i.e. $S\gtrsim 1$). It is then surprising that silicate and sulfide grains in chondritic porous interplanetary dust particles are aerodynamically equivalent [@Wozniakiewiczetal2012]; if non-coincidental, it could indicate very low densities in the outer disk when these (likely comet-derived) objects formed. While no differential drift of dust grains is expected anyway in the inner disk, presolar grains show some variations across chondrite groups, e.g. the proportions of type X SiC grains [@Zinner2003] or the carriers of $^{54}$Cr anomalies [e.g. @Trinquieretal2009]. These may be due to non uniform injection or thermal processing of these grains [@Trinquieretal2009] and their persistence (at a few tenths of the anomalies of refractory inclusions) suggest limited turbulence levels, as similarly inferred above for refractory inclusions and chondrules. As the high-temperature events which produced the latter would have destroyed presolar grains, their very survival indicates that these events were localized, allowing subsequent mixing between processed and unprocessed matter. Water ----- Water makes up about half of condensable matter in a solar mix [@Lodders2003]. While chondrites that survive atmospheric entry are mostly dry, hydrated silicates, mostly found in carbonaceous chondrites, testify to aqueous alteration on their parent body [@Brearley2003]. Also, water may have been partly responsible for the high oxygen fugacities recorded by many chondrules [e.g. @Schraderetal2013], which require 10-1000-fold enhancements over solar abundances. Water was likely $^{16}$O-poor [@Sakamotoetal2007], and possibly responsible for the variations of the oxygen isotopic composition of the inner solar system, from $^{16}$O-rich signatures of CAIs to $^{16}$O-poor, “planetary” ones (but see @Krotetal2010; see also @Yurimotoetal2008 for an overview of oxygen isotopic data). In protoplanetary disks, water condenses as ice beyond the “snow line” ($\sim$170 K). Ice and intermingled silicates would drift inward and enrich the inner disk inside the snow line [@StepinskiValaegas1997; @CuzziZahnle2004]. In the popular CO self-shielding scenario, whether in the parental molecular cloud [@YurimotoKuramoto2004] or in the disk [@Lyonsetal2009], as $^{16}$O-poor water may be most efficiently produced and/or preserved at large heliocentric distances, this would account for its addition to inner solar system material, although detailed calculations of O isotopic evolution in disks have yet to be published (it remains in particular to be seen whether the existence of *both* $^{16}$O-rich and -poor reservoirs already during CAI formation as recorded by some reversely zoned melilite grains [@Parketal2012] can be reproduced). Because of the finite supply of water in the outer disk (and/or the “bouncing barrier” to grain growth which would limit drift), the enrichment would be limited to a factor of a few [@CieslaCuzzi2006; @HughesArmitage2012], insufficient to account for FeO contents in chondrules. Settling to the midplane might further enhance the (dust $\pm$ ice)/gas ratio to the desired levels, depending on the enhancement due to radial drift, but would require very low turbulence [@CuzziWeidenschilling2006]. Turbulent concentration is yet another possibility [@Cuzzietal2001]. For efficient ice accretion beyond the snow line [e.g. @StevensonLunine1988], diffusion may later *deplete* the inner disk in water, perhaps accounting for the (reduced) enstatite chondrites [@Paseketal2005], although replenishment from further out would limit this to a factor of a few [@CieslaCuzzi2006]. Another important constraint on water is the D/H ratio which, for carbonaceous chondrites appears systematically lower than most comets [@Alexanderetal2012]. This may require efficient outward diffusion (i.e. low Schmidt number) of D-poor water from the warm inner disk (@JacquetRobert2013; see also @Yangetal2013), consistent with the requirement of efficient outward transport of high-temperature minerals (@BockeleeMorvanetal2002; see section \[CAI\]). Fractionation trends in chondrites {#Fractionation} ================================== We have investigated above the constraints given by individual petrographic components of chondrites on their transport in the protoplanetary disk. On a more integrated perspective, such redistribution of material may have caused the compositional variations exhibited by the different chemical groups of chondrites. We have already mentioned metal/silicate fractionation (section \[Metal\]); here, we focus on lithophile element fractionations and their possible dynamical interpretations. With respect to solar abundances, the most striking pattern exhibited by bulk chondrite chemistry (except CIs) is the depletion in volatile elements, increasing with decreasing nominal condensation temperature [@Palmeetal1988]. How did this *incomplete condensation* come about? @Yin2005 suggested that it was inherited from the interstellar medium, but isotope systems involving elements of different volatilities (e.g. Rb-Sr) yield whole-rock isochrons consistent with the age of the solar system [@PalmeJones2005], indicating, along with the very existence of undepleted CI chondrites native to the solar system, that the depletion arose in the disk itself. @Cassen1996 reproduced some of the elemental trends by assuming that the chondrite parent bodies started to form while the disk was hot and massive, but this is inconsistent with more recent evidence that chondrites accreted after a few Ma [@Ciesla2008]. Timescale considerations also exclude the suggestions by @WassonChou1974 of gas-solid separation by settling, radial drift (both of which require $S>1$ which would obtain late in the disk history [@Jacquetetal2012S]), or gas photoevaporation. The last plausible alternative may then be a slowing of reaction kinetics upon decrease of temperature—which are anyway required to explain the preservation of CAIs in the first place [@Ciesla2008]. Whether the temperature changes witnessed by individual condensates (over 10-1000 years in simulations by @Bossetal2012 and @Taillifetetal2013) would be sufficiently rapid to incur such an effect has yet to be investigated. Whatever process caused incomplete condensation, it did not operate to the same degree in all regions and epochs of the disk, and undoubtedly, there has been mixing by diffusion and differential drift between these different reservoirs. For example, CM chondrites are enriched in refractory lithophile elements but moderately volatile elements exhibit a plateau at about half the CI chondritic value, suggesting a 50 % admixture of CI-like material to an otherwise smoothly volatile-depleted material [@Cassen1996]. @Anders1964 proposed that the composition of chondrites resulted from varying proportions of an unfractionated CI chondritic and a high-temperature component. @Zandaetal2006 recently developed this *two-component model* by identifying these components with petrographic constituents such as CAIs, chondrules and matrix, whose proportions would have varied independently accross the different chondrite groups (this may be called the “strong” two-component model). It is however questionable whether these petrographic components were dynamically independent from each other, in particular for carbonaceous chondrites. In the regime $S<1$, which would hold for the first few Ma, there would be indeed little decoupling between these. Observational evidence for coherence between chondrite components is provided by (i) matrix-chondrule complementarity (see section \[Chondrules\]) and (ii) the *subsolar* Al/Si ratios of *CAI-subtracted* carbonaceous chondrites [@Hezeletal2008], contrary to a simple picture of CAI addition to a CI chondritic material, suggesting a genetic link between at least some of the CAIs and their host chondrite [@Jacquetetal2012S]. Also, the distinctiveness of chondrules in different chondrite groups [@Jones2012] excludes that a single chondrule population was distributed throughout the disk. Thus, while carbonaceous chondrite bulk compositions might conceivably be modeled in a simple two-component picture [@Zandaetal2012], with higher high-temperature fractions presumably representing earlier times and/or shorter heliocentric distances, such chemical components would likely have no straightforward petrographic manifestation. Non-carbonaceous chondrites, to which we now turn attention, are depleted in refractory lithophile elements relative to CIs, but this trend does not actually simply complement the enrichment exhibited by carbonaceous chondrites, as it is accompanied by a decrease in the Mg/Si ratios (roughly uniformly solar for carbonaceous chondrites; @LarimerWasson1988refractory). Another process must be at play. @LarimerWasson1988refractory proposed a loss of a refractory olivine-rich material, possibly AOAs [see also @Ruzickaetal2012AOA]. This could be accomplished by inward drift to the Sun (in the regime $S>1$) provided that this component was present in grains systematically coarser than the other [@Jacquetetal2012S]—at least before chondrule formation. @Hutchison2002 proposed instead the addition of low Mg/Si material to CI composition to account for non-carbonaceous chondrite composition, which may be implemented in a X-wind model but would be subject to the drawbacks of such scenarios [@Deschetal2010]. In the case of enstatite chondrites, which have the lowest Mg/Si ratios of chondrite groups, @Lehneretal2013 proposed that sulfidation of silicates may have led to evaporative loss of Mg. The concentration of chondrules, e.g. due to preferential settling [@Jacquetetal2012S] or turbulent concentration [@Cuzzietal2001], might explain the volatile-depleted composition of non-carbonaceous chondrites relative to their carbonaceous counterparts. Chondrites in space and time {#Chondrites in space and time} ============================ In this final section, we would like to return to our original question—how the different chondrite groups may be ordered in space and time in the early protoplanetary disk. It is widely assumed that chondrite groups represent different heliocentric distances of formation, with enstatite chondrites closest to the Sun, followed by ordinary, Rumuruti and carbonaceous chondrites [e.g. @RubinWasson1995; @Wood2005; @Warren2011b]. Certainly, spectroscopic observations—and the sample return mission to S(IV) asteroid Itokawa [@Nakamuraetal2011]—suggest that enstatite, ordinary and carbonaceous chondrites are associated with E, S, and C-type asteroids, respectively, and these do exhibit this radial sequence [@Burbineetal2008], although with wide overlap (e.g. @Usuietal2013 find that most of the *large* E-type asteroids actually lie in the *middle* of the asteroid belt). *Ab initio* rationalization of this trend as a purely spatial effect is however problematic. It is e.g. no longer possible to ascribe the implied increase in oxidation state with heliocentric distance to a temperature decrease as in the classic “hot solar nebula” picture, not only because high temperatures would not have prevailed long, but also because ferroan olivine in chondrites is difficult to ascribe to nebular condensation [@Grossmanetal2012]; and in fact, *chondrules* in carbonaceous chondrites are *more reduced* than their ordinary chondrite counterparts. The nonmonotonic trend in isotopic ratios of oxygen or other elements [@Warren2011b] is also difficult to ascribe to episodic infall [@RubinWasson1995] as infall would long have ceased. Could time of formation have then played a role? We have seen in the previous section that carbonaceous chondrites were enriched in refractory elements (in particular in CAIs) compared to non-carbonaceous chondrites. Regardless of the details of the fractionation mechanisms, if, from the above, they did not form closer to the Sun, it seems unavoidable that they formed earlier, as proposed by @Cuzzietal2003 (specifically for CV chondrites with their large type B CAIs, similar to @Wood2005) and @Chambers2006. @Jacquetetal2012S showed that the retention of CAIs required, along with other properties, that $S<1$ for carbonaceous chondrites, and which would also plead in favor of an earlier epoch, as $S$ tends to increase with time and heliocentric distance (see equation (\[S\])). The fact that noncarbonaceous chondrite parent bodies seem on average closer to the Sun may be due to inward drift which tended to increasingly concentrate solids in the inner regions. Then, the $^{16}$O-poorer composition of non-carbonaceous chondrites may be ascribed to a later, more advanced stage of influx of $^{16}$O-poor water from the outer disk if the self-shielding picture holds. While modelling of chondrite component transport thus suggests that carbonaceous chondrites accreted earlier than non-carbonaceous chondrites, other points of view are allowed by other lines of evidence. One is that chondrules in CO and LL chondrites exhibit a similar range of Al-Mg ages ($\sim 1-3$ Ma after CAIs, with younger ages in CRs and EHs [@KitaUshikubo2012; @Guanetal2006]). While not strictly contradicting a difference in chondrite *accretion* time, this could indeed suggest that time was not an important factor. Also, non-carbonaceous chondrites have generally been more thermally metamorphosed on their parent body than carbonaceous chondrites [@Hussetal2006]. If the heating is ascribed to $^{26}$Al decay, which should decrease over time, this would on the contrary suggest that carbonaceous chondrites accreted *later* than non-carbonaceous chondrites (see @GrimmMcSween1993), unless the higher water content of the former or some difference in the structure or size of the parent bodies was responsible (@Chambers2006; see also @ElkinsTantonetal2011). It thus appears that there are cogent arguments for the three possible chronological orderings of carbonaceous and noncarbonaceous chondrites (with the former either older, younger or contemporaneous with the latter). Obviously, however, two of these reasonings have to give, but it may still be dicey to decide which with any authority. Resolution of this critical issue in the interpretation of the meteoritical record will await further advances on the transport of chondrite components, as reviewed here, but also on their formation models as well as the thermal and collisional evolution of the chondrite parent bodies. Acknowledgments {#acknowledgments .unnumbered} =============== Reviews by Prof. Fred Ciesla, Jeffrey Cuzzi and Alexander Krot were greatly appreciated. I am grateful to the “Meteoritics and Solar System History” reading group at the University of Toronto for wide-ranging discussions. This review is dedicated to the memory of Guillaume Barlet (1985-2014), a friend and tireless promoter of synergies between astronomers and cosmochemists.
--- abstract: 'We compute the average luminosity of X-ray flares as a function of time, for a sample of 10 long-duration gamma-ray burst afterglows. The mean luminosity, averaged over a timescale longer than the duration of the individual flares, declines as a power-law in time with index $\sim-1.5$. We elaborate on the properties of the central engine that can produce such a decline. Assuming that the engine is an accreting compact object, and for a standard conversion factor between accretion rate and jet luminosity, the switch between a neutrino-cooled thin disk and a non-cooled thick disk takes place at the transition from the prompt to the flaring phase. We discuss the implications of this coincidence under different scenarios for the powering of the GRB outflow. We also show that the interaction of the outflow with the envelope of the progenitor star cannot produce flares out of a continuous relativistic flow, and conclude that it is the dynamics of the disk or the jet-launching mechanism that generates an intrinsically unsteady outflow on timescales much longer than the dynamical timescale of the system. This is consistent with the fact that X-ray flares are observed in short-duration GRBs as well as in long-duration ones.' author: - | Davide Lazzati, Rosalba Perna, Mitchell C. Begelman\ JILA, University of Colorado, 440 UCB, Boulder, CO 80309-0440, USA title: 'X-ray flares, neutrino cooled disks, and the dynamics of late accretion in GRB engines' --- gamma-ray: bursts Introduction ============ The [*Swift*]{} mission, with its ability to localize gamma-ray bursts (GRBs) in real time, has revolutionized our understanding of these phenomena in many ways. One of the most interesting discoveries is that the light curve of the X-ray afterglow displays a large diversity of behaviors (Nousek et al. 2006), rather than being a relatively featureless power-law. The X-ray afterglow sets in as a rapidly fading source at the end of the prompt emission (Tagliaferri et al. 2005). This early phase is understood as the radiation of the prompt phase reaching the observer from off-axis angles $\theta\gg\Gamma^{-1}$ (Kumar & Panaitescu 2000; Kumar et al. 2006; Lazzati & Begelman 2006; Zhang et al. 2007). The steep decay phase is usually followed by a flat component, whose origin is still highly debated (Granot & Kumar 2006; Uhm & Beloborodov 2007; Genet et al. 2007; Ghisellini et al. 2007). Finally, at random times between a few hundred seconds up to several tens of thousands of seconds after the onset, the X-ray afterglow displays sudden rebrightenings, known as X-ray flares (Burrows et al. 2005; Falcone et al. 2006, 2007; Chincarini et al. 2007, hereafter C07). X-ray flares could be due to a variety of causes, either related to external shock activity or to the inner engine itself. It can be shown that any mechanism related to the external shock would produce flares with a characteristically long timescale (Lazzati et al. 2002; Lazzati & Perna 2007). Most of the observed flares have fast rise and decay times (C07; Kocevski, Butler & Bloom 2007) and must therefore be related to activity of the central engine at times comparable to those at which the flare is observed. For this reason, they are of great importance for our understanding of the mechanism that powers the GRB outflows. They are potentially unique laboratories to investigate the properties of relativistic outflows from compact objects over a broad range of luminosities. As we will see in the following, the isotropic equivalent luminosity of the outflow ranges from $\sim10^{53}$ erg/s during the prompt phase to $\sim 10^{47}$ erg/s during flares at late times. We study a sample of GRBs with X-ray flares that have been observed by the Swift X-ray Telescope (XRT) for a sufficiently long time and for which redshift information is available. We compute the average energy output from the inner engine as a function of the time elapsed since the GRB explosion, and we compare it to several mechanisms for energy extraction from a magnetar or a black hole. This letter is organized as follows: in § 2 we discuss the sample and the procedure used to derive the cumulative light curve, in § 3 we discuss different mechanisms that could power the flares, in § 4 we discuss the role of the progenitor star in shaping the late time outflow, and in § 5 we discuss our results, their limitations and their implications. Data analysis and results ========================= We selected from the tables in Falcone et al. (2007; hereafter F07) the sub-sample of bursts with X-ray flares for which redshift information is available. The bursts and some of their key properties are listed in Tab. \[tab:sample\]. The sample consists of 10 GRBs for a total of 24 flares. All the bursts have XRT observations up to comoving time $t\sim3\times10^4$ s. The flare isotropic equivalent energy was computed from the fluence ${\cal F}$ in Tab. 6 of F07. The average flare light curve between the times $t_1$ and $t_2$ was computed as: $$\langle L \rangle_{t_1,t_2} = \frac{1}{n_{\rm{GRB}}}\sum_{i=1}^{n_f} L_i\,\delta{t}_{i,1,2} \qquad ,$$ where $L_i$ is the average luminosity of the $i^{th}$ flare during its active time, $n_{\rm{GRB}}=10$ is the total number of GRBs considered, $n_f=24$ is the total number of flares and $0\le\delta{t}_{i,1,2}\le1$ is the fraction of time in the interval $(t_1,t_2)$ during which the $i^{\rm{th}}$ flare is active. This prescription is equivalent to the assumption that the flares have a square shape. Even though this is a very poor approximation to the real shape of the flares, its effect is negligible when many flares are averaged to compute the light curve. We tested both a Gaussian shape (C07) and a triangular shape, and we obtained consistent results within the errors. The computation of the errors in the average light curve is nontrivial, since the main contribution is not the uncertainty in each fluence measurement, but rather the uncertainty in the number of flares that are observed on average during a given time interval. For this reason we assume that the uncertainty of each fluence is as large as the measurement itself and we compute the errors on the average light curve as: $$\sigma_{\langle L \rangle_{t_1,t_2}} = \frac{1}{n_{\rm{GRB}}}\left[\sum_{i=1}^{n_f} \left(L_i\,\delta{t}_{i,1,2}\right)^2\right]^{1/2}.$$ This prescription ensures that if the average flux at a certain time is dominated by a single bright flare, the error is large. Small errors occur only in time intervals with a large number of flares. GRB $z$ $N_{\rm{Flares}}$ $T_{\max}$ (s) --------- -------- ------------------- ----------------- 050724 0.258 3 $3.4\times10^5$ 050730 3.967 4 $9.9\times10^4$ 050802 1.71 1 $3.0\times10^5$ 050814 5.3 2 $1.1\times10^5$ 050820a 2.612 1 $1.1\times10^6$ 050904 6.29 7 $4.3\times10^4$ 050908 3.344 2 $2.8\times10^4$ 051016b 0.9364 1 $6.9\times10^5$ 060115 3.53 1 $7.9\times10^4$ 060124 2.296 2 $6.3\times10^5$ : [Properties of the bursts selected for the analysis]{} \[tab:sample\] The resulting light curve is shown in Fig. \[fig:z\], where $\langle L \rangle_{t_1,t_2}$ is shown versus time ($t=(t_2-t_1)/2$). The solid dark points with error bars show the results assuming a Band model spectrum for the flares (Band et al. 1993; F07), while the white lozenges show the results of the power-law spectral model (F07). We model the average light-curve as a power-law, excluding the first point (at $t\sim40$ s) since it is still contaminated by the prompt emission in many cases. We find that the average light curve is very well described by a power-law with index ($1\sigma$ errors): $$\alpha=-1.5 \pm 0.16 \,. \label{eq:slope}$$ The fit has $\chi^2/$d.o.f.$=0.2$, a small value that is not surprising given the very conservative assumptions on the uncertainties. The formal error of 0.16 is, as a consequence, very conservative. In order to check the result, we computed the average light curve also as the derivative of the cumulative energy produced as a function of time since the GRB onset, finding analogous results. Averaging many bursts together allowed us to increase the signal-to-noise ratio of the flare light curve. We now examine whether the resulting slope can be recovered in individual GRBs that display a large number of flares. When we compute the flare light curve for an individual event, the knowledge of the redshift is not necessary and we can select the GRBs from a larger pool of events. There are three GRBs in the F07 catalog with six or more flares. One is GRB 050904, which is in our sample and has 7 flares. Another is GRB 051117a, which has 7 flares according to F07. However, the seven flares of GRB 051117a overlap one another and therefore we discard this event from our analysis. Finally, GRB 050803 has six flares. Figure \[fig:050803\] shows the flare light curve, in observed time and flux, for these two bursts. Power-law fits were performed for these cases, yielding $\alpha=-1\pm0.4$ and $\alpha=-1\pm0.3$ for GRB 050803 and GRB 050904, respectively. Even though the uncertainties are larger, as expected given the smaller statistics, the individual cases confirm that flares produce more luminosity at early times than at late times, their light curve is consistent with a power-law, and the index of the power-law is consistent with the cumulative curve of Fig. \[fig:z\]. GRB 050904 has seven flares and is also part of our main sample. Since it has so many flares, we checked that the results for the average flare energy are not dependent on the presence of GRB 050904 in the sample. Reproducing Fig. \[fig:z\] without the seven flares of GRB 050904 yields similar results, with a slope $\alpha=-1.8 \pm 0.3$, in agreement with the result of Eq. \[eq:slope\]. Powering modes ============== Even though collimated outflows originating from compact objects accreting matter from a disk are ubiquitous in the universe (e.g., Ferrari 1998; Blandford 2001), the actual mechanism that produces the jet and determines its properties (luminosity, entropy, opening angle, structure, and magnetization) remains elusive. In addition, we see jets in other types of objects, such as pulsars, where mechanisms other than accretion could play a role in the jet production (but see Blackman & Perna 2004). In the case of GRBs, several mechanisms have been proposed to produce the relativistic jet: conversion of internal energy into bulk motion with hydrodynamic collimation (Cavallo & Rees 1978; Lazzati & Begelman 2005), energy deposition from neutrinos, energy released from a rapidly spinning, newly born magnetar (Usov 1992), and magnetic collimation and acceleration (Vlahakis & Königl 2001). It has proven so far extremely challenging to prune some of these possibilities, since the GRB radiation is produced far away from the place where the outflow is accelerated, and we have not been able to connect the properties of the radiation (light curves and spectra) to the (magneto)-hydrodynamical properties of the plasma producing it. Let us first consider a system made by a newly formed stellar mass black hole accreting matter at a high rate from a disk. During the prompt GRB phase, such a disk is so hot and dense that neutrino losses provide an effective cooling mechanism (Popham, Woosley & Fryer 1999; Narayan, Piran & Kumar 2001; Chen & Beloborodov 2007). In this phase, the jet may be powered by neutrino annihilation, even though magnetic effects could play a dominant role. In the latter case, the outflow may either originate on the disk surface, as material escapes along low inclination magnetic field lines (e.g., Levinson 2006), or by Blandford-Znajek mechanisms (e.g., McKinney & Gammie 2004). 1D steady-state calculations of Chen and Beloborodov (2007) showed that, for a rotating BH, the switch from a neutrino-cooled to an advective disk takes place at an accretion rate of $10^{-3}-10^{-2} M_\odot/$s, depending on the viscosity prescription used. In all cases, the switch-off of the neutrino cooling is expected to be associated to a change in the GRB outflow characteristics, since the magnetic field that can be anchored to a dense, high pressure, thin disk is expected to be stronger than the one anchored to a lower-density, lower-pressure, thick inflow. Figure \[fig:mdot\] shows the average flare luminosities in units of the accretion rate in solar masses per second, where we assumed that the GRB is beamed into 1 per cent of the sky, that the efficiency of converting the accretion rate into jet luminosity is 0.1 per cent, and that these two parameters are constant. The two parameters are rather standard (e.g., Chen & Beloborodov 2007; Podsiadlowski et al. 2004). Changing them by a factor up to ten would not affect the main conclusion since the prompt and flare luminosities would scale in the same way. The average luminosity of the prompt phase of the 10 GRBs is also shown in the figure with a thick solid line. Such accretion rates are compared to the one at which neutrino cooling is switching on/off (Chen & Beloborodov 2007). This transition lies suggestively at the boundary between the prompt phase and the flaring phase. We argue that this could be an indication of the fact that the prompt phase of GRBs, the most luminous one, is powered by accretion onto a black hole in the form of a geometrically thin – neutrino cooled – disk. As the accretion rate drops under the critical value, the fast accretion is switched off and the prompt phase ends. The disk swells under the effect of internal pressure and a new accretion geometry, with a thick disk, sets in. This gives rise to the flares that we observe during the afterglow phase. In the case of a magnetically driven jet, the change in the outflow dynamics would be brought about by the fact that a thick disk has a lower pressure and therefore can anchor a magnetic field with lower intensity. It would also change the pitch angle of low-inclination magnetic field lines. In the case of a neutrino powered outflow, the switch off of the neutrino cooling would shut off completely the outflow. It seems therefore that the observations of late-time, low-luminosity flares provide evidence against the powering of GRB outflows exclusively by neutrino annihilation. The material necessary to provide the accretion at late stages can be provided, in long-duration gamma-ray bursts, by the fallback of material that did not reach the escape velocity in the stellar explosion (Chevalier 1989; MacFadyen, Woosley & Heger 2001; Zhang & Woosley 2008). Alternatively, especially in the case of short GRBs, the natural evolution of an accretion disk can supply the accretion rate necessary to explain the observations. Consider an accretion disk that forms in a short timescale (comparable to or smaller than the duration of the GRB prompt phase) and is left to evolve without any sizable mass input thereafter. The accretion rate depends on the assumptions made on the nature of viscosity. The case of a thin disk has been studied thoroughly. Frank, King & Raine (2002) report the exact solution for the case of a thin disk with constant viscosity. They show that, after several tens of viscous time-scales, the accretion rate onto the central object approaches the asymptotic form $\dot{m}\propto{}t^{-1.25}$. Cannizzo, Lee & Goodman (1990) performed numerical simulations for the same initial conditions using the prescriptions of $\alpha$-viscosity (Shakura & Sunyaev 1973). They find that the accretion rate at late times scales with time as $\dot{m}\propto{}t^{-1.2}$, independently of the value of $\alpha_\nu$ assumed. The case of a thick disk needs to be studied in detail (Lazzati & Begelman in preparation). The comparison of the observed flare luminosity to the theoretical rates of late-time accretion implies that there is an almost linear relation between accretion and the luminosity of the outflow. In our favored scenario, where the outflow is powered by magnetic processes, this relation would be caused by the decay of the magnetic field as the disk becomes less massive and dense. The reason why we observe such a linear relation is not entirely clear and deserves further investigation. An alternative to the accretion disk–BH system is a rapidly spinning magnetar that powers the outflow as a consequence of spin-down (Thompson, Chang & Quataert 2004; Bucciantini et al. 2008). This system can provide late-time energy either through neutrino emission or through dipole radiation. The neutrino emission decays exponentially and cannot provide the power required (Thompson et al. 2004). In the simple vacuum dipole scenario the decay of the late-time energy deposition is too steep ($L\propto{}t^{-2}$). However, the interaction of the field with the stellar material can create shallower slopes consistent with the observations. In this scenario, the fact that the neutrino cooling switch coincides with the transition from the prompt to the flaring phase is purely coincidental. In addition, it is not clear how the continuous luminosity produced by the spin-down of the magnetar can be converted into a flaring source. From steady state to impulsive ============================== Even though the average flare curve of Fig. \[fig:z\] is featureless, we ought to keep in mind that flares are episodic events. The fact that an accretion disk or a spinning magnetar can provide energy at late time does not ensure that the energy will be released intermittently. It is not surprising to observe variability of relativistic sources on timescales comparable to the dynamical time of the system. However, in the case of flares, the variability timescale is many orders of magnitude longer than the dynamical one for a solar mass black hole, and it grows approximately linearly with time (C07). It is more likely that this variability is associated to a viscous time-scale rather than to the dynamical time scale. We must therefore seek either a mechanism that can release energy episodically from the inner engine, or a way to transform a continuous output of energy into a highly variable one as it propagates from the engine to the radiation zone. The propagation of the jet through the cold stellar progenitor material provides in principle a way by which a continuous outflow can be converted into a succession of individual fireballs. Morsony et al. (2007) showed that even if a continuous jet is injected into the core of a massive star, the ensuing light curve is highly variable. Consider an engine releasing an outflow with continuous but decreasing luminosity. Inside the star the jet is in pressure equilibrium with a high pressure cocoon of material (Lazzati & Begelman 2005). As the jet luminosity decreases, the cocoon becomes over-pressured and squeezes the jet. At the same time, the cocoon pressure decreases due to the fact that the cocoon material is escaping from the surface of the star. If, at any point in time, the cocoon pressure overcomes the stagnation pressure of the jet, the jet would be choked and the flow of energy interrupted. The stagnation pressure of a relativistic outflow with Lorentz factor $\Gamma$ and cross section $\Sigma$ is given by $p_{\rm{stag}}=L_j\,\Gamma^2/(4c\Sigma)$, where $L_j$ is the luminosity of the jet. Following Lazzati & Begelman (2005), the cocoon pressure can be written as: $$p_{\rm{cocoon}} = \left(\frac{L_{\{j,0\}}\rho_\star} {3\,r_\star\,t_{\rm{br}}}\right)^{1/2}\, e^{-\frac{c\,t}{\sqrt{3}\,r_\star}} \label{eq:pcocoon}$$ where $L_{\{j,0\}}$ is the average jet luminosity before breakout, $\rho_\star$ is the average stellar progenitor density and $r_\star$ its radius, and $t_{\rm{br}}$ is the breakout time. A condition for the stagnation of the jet can be obtained by comparing the stagnation pressure with Eq. \[eq:pcocoon\]. It can be seen that the cocoon pressure cannot reach the jet stagnation pressure for any reasonable parameter set. We conclude therefore that the instability giving rise to the flaring behavior of the late time activity of the inner engine has to be intrinsic to the jet release process or to its transition from the non-relativistic to the relativistic stage, and cannot be brought about by the propagation of the relativistic outflow in the star. Discussion and Conclusions ========================== We have computed the average energy released in the form of X-ray flares overlaid on the power-law decay of the afterglow of long-duration gamma-ray bursts. A sample of 10 long duration GRBs from the catalog of F07 with redshift measurements was used for the analysis. We conclude that, on average, the late-time energy release approximately follows the power-law scaling $L\propto{}t^{-1.5}$. Our analysis is possibly affected by several biases. First, the definition of a flare is fraught with uncertainty, as the comparison of the C07 and the F07 catalogs easily reveals. We here adopt the definition of F07 and refer to that paper for a discussion of their selection criteria. Second, the measurement of the slope of the power-law depends on our capability to select flares. Low brightness flares could be lost in the early phases when the afterglow can easily outshine them. On the other hand, very short duration flares could be missed at late times when the observations are not continuous in time. It appears that the first bias is most serious (since no very short duration flare was ever detected in the late phases) and that the slope is possibly underestimated. We do not believe this should affect our conclusions, since almost all the GRBs we considered have an early flare and so the contribution of shallow flares would be minimal. One important effect could, however, create a systematic overestimation of the slope. The late time flares may be less beamed than the early ones, and therefore their isotropic equivalent luminosities would appear smaller than the one of early flares due, in part, to geometric end not intrinsic effects. According to simulations, however, most of the opening angle evolution takes place at very early times (Morsony et al. 2007), as long as the injection opening angle does not evolve. The energy to power the flares may come from accretion or from the spinning down of a magnetar. In the first case, we were able to estimate the accretion rate required to power the prompt emission as well as the flaring phase. Interestingly, during the prompt phase the accretion rate is so large that neutrino emission cools the flow and accretion takes place in the form of a thin disk. At accretion rates of about 0.001 to 0.01 solar masses per year, the neutrino luminosity drops and the accretion disk becomes thick. We find that the switch-off of the neutrino cooling, and the transition from a thin to a thick disk, take place between the prompt and the flaring phase. Indeed, the prompt emission spikes and the late-time flares exhibit some differences. While the prompt spikes do not show any evolution in their duration or peak luminosity, the flares become longer and shallower with time. This property naturally arises in a disk that fragments and accretes the various blobs of material on their viscous timescales (Perna, Armitage & Zhang 2006). In addition, flares are very episodic and the engine is “off" most of the time at late stages, while during the prompt phase the engine is “on" most of the time. We propose that the differences are due to the switch from a thin disk to a thick disk configuration. A more detailed analysis of the thick disk accretion dynamics is on-going (Lazzati & Begelman in preparation). An alternative scenario is that of a spinning-down pulsar. In this case, the fact that the neutrino switch coincides with the transition from the prompt to the flare phase would be a pure coincidence. More observations and a better understanding of both engine models are needed before definitive conclusions can be drawn. 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--- abstract: 'Use of the effective Lagrangian incorporating both the scalar and pseudoscalar mesons gives a possibility to calculate the $\pi\pi$-scattering lengths without attraction of the ChPT theory.' --- [On the $\pi\pi$ - scattering lengths in the theory with effective Lagrangian]{} [**E.P. Shabalin [^1]**]{}\ Institute of Theoretical and Experimental Physics; National Research Center Kurchatov Institute, Bol’shaya Cheremushkinskaya ul., 25, Moscow, 117218 Russia. Introduction ============ The question about the numerical quantities of the $\pi\pi$ -scattering lengths engages the minds of theoreticians and the experimentalists so far. Nowdays, a number of the observed events of $\pi\pi$ - scattering increased many times, that gives a possibility to verify the different approaches to work out a theory of the low-energy $\pi\pi$ - interaction. It is clear, that an approach based on the use of the Lagrangian scheme looks as more convenient for deep understanding of the theorems concerning the soft pions. Just in this scheme, it is possible to obtain the precise results of the current algebra even on the “trees” level, that is, on the level of the diagrams without loops \[1\]. In this paper, this peculiarity of the Lagrangian scheme will be used for the calculation of the $\pi\pi$ scattering lengths. And besides, the conformity of the properties of the QCD objects and objects of the real world will be ensured also. In QCD, the spinless flavoured objects are $\bar q_R t^a q_L$ and their Hermitian conjugate. These objects are formed by opposite-parity components. For this property to be reproduced in the Lagrangian of real particles, it must be expressed in terms of the matrix $$U = (\sigma_a + i\pi_a)t_a, \label{1}$$ where $t_0 = \sqrt{1/3} $ , $t_{1....8} = \sqrt{1/2}_{1,....8}$ and $\sigma_a$ and $\pi_a$ are nonets of, respectively, scalar and pseudoscalar mesons. This idea is not new and was used in $ [2] $ and $ [3 ]$, but we go over directly to the final form of the Lagrangian containing all forms of breakdown of chiral symmetry \[4\], \[5\]: $$\begin{array}{ll} L = \frac{1}{2}Tr(\partial U_{\mu}\partial U^+ _{\mu}) - cTr(UU^+ - A^2t^2_0)^2 - c\xi( Tr(UU^+ - A^2t^2_0))^2 + \nonumber \\ \frac{F_{\pi}}{2\sqrt{2}}Tr{(M(U +U^+)} + \Delta L^{U(1)}_{PS}. \end{array} \label{2}$$ These forms permit to express all properties of $\sigma$ - mesons using the properties of $\pi$ - mesons and the parameters $R=F_K/F_{\pi}$ and $\xi$. The values of them can be found from the data on the decays $K,\pi \to \mu \nu$ and identification of the scalar $\sigma_{\pi}$ - meson with the resonance $a_0(980)$ \[4\], \[5\], \[6\]. In a theory with the broken $U(3)_L\otimes U(3)_R$ chiral symmetry there are two isosinglet $\sigma$-particles having nonzero vacuum expectation values $<\sigma>$. As a result, the Lagrangian(2) contains the vertices $<\pi\pi|\sigma>$. Then, the set of the pole diagrams with the intermediate $\sigma$ - meson appears. The amplitude of the $\pi\pi \to \pi\pi$ - scattering acquires the form: $$\begin{array}{ll} T_{\sigma} = <\pi_k(p'_1)\pi_l(p'_2)|\pi_i(p_1)\pi_j(p_2)> = A_{\sigma}\delta_{ij}\delta_{kl} + B_{\sigma} \delta_{ik}\delta_{jl} + C_{\sigma} \delta_{il}\delta_{jk}, \label{3} \end {array}$$ where $$\begin{array}{ll|} A_{\sigma} = (s-\mu^2)\sum_{n=1,2} \frac{G_n}{m^2_{\sigma_n}-s}, \quad B_{\sigma} = (t-\mu^2 )\sum_{n=1,2} \frac{G_n}{m^2_{\sigma_n}-t},\nonumber \\ ~~ \nonumber \\ C_{\sigma} = (u-\mu^2)\sum_{n=1,2} \frac{G_n}{m^2_{\sigma_n} - u}, \quad \mu \equiv m_{\pi}, \end{array} \label{4}$$ and where $$\begin{array}{ll} s = (p_1 + p_2)^2, \quad t = (p_1 - p'_1)^2, \quad u = (p_1 - p'_2)^2, \quad G_n=\frac{g^2_n}{m^2_{\sigma_n} - \mu^2}. \end{array} \label{5}$$ In the theory specified by the Lagrangian (2) , the following relation holds: $$\frac{G_1}{m^2_{\sigma_1} - \mu^2} +\frac{G_2}{m^2_{\sigma_2} - \mu^2}=\frac{1}{F^2_{\pi} }, \quad F_{\pi} = 93 \:{ MeV}. \label{6}$$ For the case of fixed total isospin of a system of initial pions, the expressions for the amplitudes are given in \[7\]: $$T^{(0)} = 3A + B + C. \label{7}$$ $$T^{(1)} =B-C \label{8}$$ $$T^{(2)} = B +C. \label{9}$$ The decomposition of the isotopic amplitudes into amplitudes corresponding to fixed values of the orbital angular momentum is given by $$T^{(I)} = 32\pi\sum_{l=0}^{\infty} (2l+1)t^{(I)}_l(s)P_l(\cos \theta). \label{10}$$ It follows from (10) that the partial-wave amplituda $t^{(I)}$ is: $$t^{(I)}_l(s) = \frac{1}{64\pi} \int_{-1}^{1} T^{(I)}P_I(\cos\theta) d \cos\theta. \label{11}$$ The scattering lengths arise from the expansion $$t^I_l(s)m^{-1}_{\pi} = q^{2l}[a^{(I)}_l + b^{(I)}_l q^2 + O(q^4)], \qquad q^2=\frac{s}{4}-\mu^2. \label{12}$$ To calculate $a^{(0)}_0$ and $a^{(2)}_0$, we make use of the relations (3,7,9,11). The scattering length $a^{(1)}_1$ will be considered later, since besides the $\sigma$ - mesons, the $\rho$-mesons also contribute into this scattering length. We begin calculations from $a^{(2)_0}$, because this scattering length, according to (9) and (7),enters into $a^{(0)}_0$. The scattering length $a^{(2)}_0$ ================================= According to (9) $$T^{(2)}_{\sigma} = \sum_{n=1,2} G_n\left(\frac{t - \mu^2}{m^2_{\sigma_n} - t} +\frac{u - \mu^2}{m^2_{\sigma_n} -u} \right). \label{13}$$ On the threshold $t$ and $u$ are equal to zero. Using the expressions for masses and coupling constants of the $\sigma_n$ - mesons \[5\] and the last numerical data on their values \[6\], we obtain: $$T^{(2)}_{\sigma} = - \left(\frac{2G_1\mu^2}{m^2_{\sigma_1|}} +\frac{2G_2\mu^2}{m^2_{\sigma_2}} \right) = - 4.3171. \label{14}$$ According to (11) and (12) $$a^{(2)}_0 = - 0.04294m^{-1}_{\pi}. \label{15}$$ This result is in agreement with the result of analysis of the $K^{\pm} \to \pi^{+}\pi^{-}e^{\pm}\nu$ decay, based on the statistics of 1.13 million decays \[8\]: $$a^{(2)}_0 = (- 0.0432 \pm 0.0086_{stat} \pm 0.0034_{syst} \pm 0.0028_{th})m^{-1}_{\pi} \label{16}$$ The scattering length $a^{(0)}_0$ ================================== In accordance with (7) , the amplitude of $S$-wave with the isospin 0 looks as $$T^{(0)} = \sum_{n=1,2} G_n \left( 3\frac{s - \mu^2}{m^2_{\sigma_n} - s} \right) +T^{(2)}_{\sigma}. \label{17}$$ At the threshold, the first addendum in (17) performs into $$9\mu^2 \sum_{n=1,2} G_n(m^2_{\sigma_n} - 4\mu^2)^{-1} = 23.3335. \label{18}$$ Adding the result (14) for $T^{(2)}_{\sigma}$, we come to $$T^{(0)}_{\sigma} = 19.0164. \label{19}$$ And the final result for $a^{(0)}_0$ is: $$(a^{(0)}_0)_{\sigma} = \frac{19.0164}{32\pi m_{\pi}} = 0.18916m^{-1}_{\pi}. \label{20}$$ This value agrees with the experimental one: $$(a^{(0)}_0)_{exp} = (0.197\pm 0.010)m^{-1}_{\pi} \label{21}$$ obtained from the analysis of all data near threshold of the reaction $\pi N \to \pi\pi N$ . The data obtained for $a^{(0)} _0$ by the collaboration E865 \[10\] depend from the Models used at analysis. In particular, in the Model A $a^{(0)}_0 = (0.184 \pm 0.010)m^{-1}_{\pi}$, in the Model B $a^{(0)}_0= (0.179 \pm 0.033)m^{-1}_{\pi}$ and in the Model C $a^{(0)}_0=(0.213 \pm 0.013)m^{(-1)}_{\pi}$. [^2] However, these results appear after taking into account the isospin corrections, also depending from the Models. Our result (20) does not demand any additional model corrections. The scattering length $a^{(1)}_1$ ================================= In the used by us theory, besides the $\sigma$ - mesons, the intermediate $\rho$ - mesons also give a contribution into $a^{(1)}_1$. In the present paper, a nature of the $\rho$ - mesons will be not associated with the vector quark current, as it was considered usually, but their nature will be associated with the divergence of vector quark current \[11 - 15\]. A contibution of the $\sigma$ - mesons into isovector amplitude is: $$T^{(1)}_{\sigma} = \sum_{n=1,2}G_n \left[\frac{t - \mu^2}{m^2_{\sigma_n} - t} - \frac{u - \mu^2}{m^2_{\sigma_n} -u} \right]. \label{22}$$ According to the relations (11) and (12), the part of scattering length $a^{(1)}_1$ produced by $\sigma$ - mesons is: $$(a^{(1)}_1)_{\sigma} = \frac{1}{24\pi m_{\pi}} \left[ \frac{g^2_1}{m^4_{\sigma_1}} + \frac{g^2_2}{m^4_{\sigma_2}} \right]. \label{23}$$ Using the numerical values of $m_{\sigma_{1,2}}$ and $g_{1,2}$, we find: $$(a^{(1)}_1)_{\sigma} = 0.02744m^{(-3)}_{\pi}. \label{24}$$ The part of the isovector amplitude produced by the intermediate $\rho$ - mesons looks like the relation (3), but with the differnt A, B and C \[15\]. Namely, $$A_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(u)\frac{(s-t)u}{M^2_{\rho} -u} + g^2(t)\frac{(s-u)t}{M^2_{\rho} -t} \right). \label{25}$$ $$B_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(s)\frac{(t-u )s}{M^2_{\rho} -s} + g^2(u)\frac{(t-s)u}{M^2_{\rho} -u} \right). \label{26}$$ $$C_{\rho} = \frac{1}{M^2_{\rho}} \left(g^2(s)\frac{(u-t)s}{M^2_{\rho} -s} + g^2(t)\frac{(u-s)t}{M^2_{\rho} -t} \right). \label{27}$$ In accordance with (8) $$T^{(1)}_{\rho} = B_{\rho} - C_{\rho}. \label{28}$$ As the explicit form of $g(x=s,t,u)$ is not specified by our theory, we are forced to resort to the phenomenological relation elaborated in \[15\]: $$g(x) = g_{\rho} exp \left(0.7855 \left[\frac{x}{2M^2_{\rho}} - \left(\frac{x}{2m^2_{\rho}}\right)^2 \right] \right), \qquad x \le M^2_{\rho}. \label{29}$$ As we are interesting in behavior of the partial wave $t^{(1)}_1(s)$ near threshold, we obtain the following result: $$(t^{(1)}_1)^{threshold}_{\rho} = \frac{4\mu^2}{3\pi M^2_{\rho}} \left(\frac{2g^2(4\mu^2)}{M^2_{\rho} - 4\mu^2} +\frac{g^2(0)}{M^2_{\rho}} \right) \label{30}$$ Using the experimental value of $g(M^2_{\rho})$=5.9764 and the formula (29), we find: $$g(0) = 4.9108, \qquad g(4\mu^2) =5.5520. \label{31}$$ The part of the scatterinag length $a^{(1)}_1$ produced by the intermediate $\rho$ - mesons turns out to be: $$(a^{(1)}_1)_{\rho} = 0.005918m^{(-3)}_{\pi}. \label{32}$$ Together with the part (24) produced by the $\sigma$ -mesons we get: $$(a^{(1)}_1)^{total} =0.03336m^{-3}_{\pi}. \label{33}$$ Conclusion ========== The requirement of conformity between the properties of the QCD objects and objects of the real world gives rise to necessity of existance of the scalar mesons, that, as it turned out, play the principal role in the low-energy $\pi\pi$ - interactions. Being the chiral partners of the pseudoscalar mesons, the scalar mesons possess quite definite properties, ascribed by the structure of the Lagrangian (2). Their masses and coupling constants depend only on two parameters:$ R = F_{K}/F_{\pi}$ and $\xi$, the values of which are determined experimentally. The found by us scattering lengths are: $$a^{(0)}_0=0.18916m^{-1}_{\pi}, \; a^{(2)}_0=-0.04294m^{-1}_{\pi}, \;a^{(1)}_1=0.03336m^{(-3)}_{\pi}. \label{33}$$ And they did not require to attract the special models, concerning these scattering lengths. In our theory, the current-algebra prediction \[16\]: $$\frac{2a^{(0)}_0 - 5a^{(2)}_0}{18\mu^2{\pi}a^{(1)}_1} = 1 \label{34}$$ is satisfied to within 1.24%. [99]{} 1. V. de Alfaro, S.Fubini, G. Furlan, and C. Rossetti, Currents in Hadron Physics (North-Holland, Essevier, Amsterdam,1973)\ 2. M. Levy, Nuovo Cimento A [**52**]{}, 23 (1976); G. Cicongna, Phys. Rev. D [**1**]{}, 1786 (1970); J. Schechter and Y. Ueda, Phys. Rev. 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--- author: - 'Xinglong Zhang,  Wei Jiang,  Zhizhong Li, and Xin Xu[^1][^2][^3][^4][^5]' bibliography: - 'IEEEabrv.bib' - 'ref.bib' title: 'A Dual-level Model Predictive Control Scheme for Multi-timescale Dynamical Systems' --- [Shell : A Dual-level Model Predictive Control Scheme for Multi-timescale Dynamical Systems]{} Introduction ============ Many industrial processes are characterized by separable fast-slow dynamics, which can be called “multi-timescale dynamic systems". In a multi-timescale dynamic system, for a given constant input signal, some of the output variables reach their steady-state values fast while the other ones may experience a longer transient time period, see for instance [@naidu2002singular; @kokotovic1999singular; @mishchenko1994asymptotic]. A widely-acceptable approach for the control of such systems consists in resorting to hierarchical control synthesis that possibly relies on singular perturbation theory (see the book [@naidu1988singular]), where time-scale separation technique is adopted to define regulators at different control frequencies so as to guarantee the stability and performance of the dynamics associated with the each channel. As another example, there might be systems that their dynamics are not separable but must be controlled in the same way with a multi-rate control setting, see for instance the control of a Boiler Turbine (BT) system considered in [@RV2008]. [In this case, usually, the crucial controlled variables of the considered system must be adjusted in a faster rate to meet the control performance requirement, while other outputs can be controlled more smoothly in a slower time scale.]{} Model predictive control (MPC) is an advanced process control technique, widely used in industrial processes, such as chemical plants and smart grids, see  [@forbes2015model; @qin2003survey; @li2017nonlinear; @jin2018improved; @roshany2013kalman], in robotics, see [@zhou2017real; @santoso2018robust], in urban traffics, see [@zhou2017two], and in computing data center, see [@fang2017thermal]. In MPC, the control problem is reformulated as an optimization one, that has to be solved on-line iteratively. This allows to explicitly consider the control, state and output constraints in the control problem. At any generic time instant $k$, an finite horizon optimization problem must be solved to compute the optimal control sequence. Only the first element is applied to the system, and the state and output variables are updated. The optimization problem is then repeated at the next instant $k+1$. This is the so called “receding horizon" strategy. As an alternative, in the context of MPC, “shrinking horizon" strategy can also be used. The resultant algorithm is usually called “shrinking horizon MPC", in which, the on-line optimization problem is still solved recursively, however the prediction horizon reduces with the time to go. This re-optimization nature paves the way for endpoint tracking objective and for disturbance/uncertainty effect reduction. see [@picasso2016hierarchical; @grune2013benefit]. In the framework of MPC, many solutions have been developed based on time-scale separation technique for systems characterized by open-loop separable dynamics. Among them, solutions on theoretical developments can be found in [@chen2011model; @chen2012composite; @ellis2013economic] for nonlinear singularly perturbed systems, in [@wogrin2010mpc; @ma2017slow; @miao2015fast] for linear systems, and in [@niu2009two] for input/output models, while the results focusing on the application aspect are reported in  [@brdys2008hierarchical; @ohshima1994multirate; @van2009time; @sanila2015simultaneous]. However, most of the aforementioned works are tailored for systems with clearly different dynamics due to their dependencies on singular perturbation theory, so not suitable for the multirate control of systems with non-separable open-loop dynamics.\ Motivated by this, the scope of this work concerns designing dual-level algorithms based on MPC for linear multi-timescale dynamical systems subject to input and state constraints, such that the resulting closed-loop systems exhibit separable dynamic behaviors. A novel dual-level MPC (D-MPC) algorithm is initially presented. At the high level of the control structure, a slow time scale associated with $N$ period of the basic time scale is adopted to define a stabilizing MPC problem with respect to the sampled version of the original system so as to ensure convergence at the considered time scale. At the low level, a shrinking horizon MPC is designed in the basic time scale to refine the computed control actions in order to derive satisfactory short-term transient associated with the closed-loop fast dynamics and to verify endpoint state constraint computed from the high level. In doing so, the recursive feasibility and stability properties of the closed-loop system can be guaranteed. Moreover, an incremental form of D-MPC (i.e., *incremental* D-MPC) is proposed with an emphasis on the controller modification at the high level, including specifically an integral action on the “slow" outputs and, a prior explicit design of the reference trajectory of the “fast" outputs relying on an auxiliary optimization variable to be optimized at the slow time scale. Also, the recursive feasibility and the convergence properties of the *incremental* D-MPC closed-loop control systems are proven. The control of a BT system is considered as a case study. The simulation results with the proposed approaches, including their comparisons with a traditional decentralized PID controller and a multirate MPC controller, are reported to witness the potentiality of the proposed algorithms in imposing different closed-loop dynamics and to show the advantages of producing satisfactory control performance in this respect. Differently from the aforementioned works, see such as [@chen2011model; @chen2012composite; @wogrin2010mpc; @niu2009two; @van2009time], the proposed D-MPC and *incremental* D-MPC do not rely on singular perturbation theory, so suitable not only for singularly perturbed systems but also for the ones that do not exhibit strictly separable dynamics. [Note that, a similar problem has been addressed in [@zhang2018], however the control scheme described in this paper shows a significant improvement for the following reasons: i) The algorithm in [@zhang2018] is proposed for system described by impulse responses, with a special focus on the viewpoint of application, while this paper presents the novel solutions on the theoretical developments based on a state-space formulation, with verified closed-loop recursive feasibility and stability. ii) Due to the usage of impulse response representation, the concerned system in [@zhang2018] is assumed to be strictly stable. In this case, for examples that have poles on the unitary disk, a stable feedback control law must be designed primarily (see the Section \[sec:example\]). Whereas in this paper this restriction is slightly relaxed, i.e., the considered system has no poles larger than 1. iii) The algorithm described in  [@zhang2018] only permits the input associated with the slow dynamics to be manipulated in the slow control channel. This could lead to control performance degradation especially for systems that are strongly coupled. To solve this problem, herein, the corresponding “slow" control variable is allowed to be furtherly refined in the fast control channel according to a properly designed performance index.]{} The rest of the paper is organized as follows: Section 2 presents the problem description and the proposed control structure. The MPC problems at the high and low levels of the D-MPC are introduced in Section 3, while the *incremental* D-MPC algorithm is described in Section 4. Simulation example concerning the BT control is studied in Section 5, while some conclusions are drawn in Section 6. Proofs of the theoretical results are given in the Appendix A.\ **Notation:** for a given set of variables $z_{i}\in{\mathbb{R}}^{q_{i}}$, $i=1,2,\dots,M$, we define $(z_{1}, z_{2}, \cdots, z_{M})=[\,z_{1}^{\top}\ z_{2}^{\top}\ \cdots\ z_{M}^{\top}\,]^{\top}\in{\mathbb{R}}^{q}$, where $q= \sum_{i=1}^{M}q_{i}$. We use $\mathbb{C}$ to denote the set of the complex plane. Given a matrix $P$, we use the symbol $P^{\top}$ to denote its transpose. We use $\|x\|_Q^2$ to represent $x^{\top}Qx$. We use $\mathbb{N}$ and $\mathbb{N}_+$ to denote the set of non-negative and positive integers respectively. Given two sets $A$ and $B$, we denote $A\times B$ as the Cartesian product. Given the signal $r$, we denote $\overrightarrow{v}(k:k+N-1)$ the sequence $r (k)\ldots r(k+N-1)$, where $k$ is the discrete time index, $N$ is a positive integer. Problem formulation =================== The system to be controlled is described by a discrete-time linear system consisting of two interacting subsystems expressed as \[Eqn:sigma\_s\_f\] $$\begin{aligned} &\Sigma_s:\ \left\{\begin{array}{l} x_s(h+1)=A_{ss}x_s(h)+A_{sf}x_f(h)+B_{ss}u_s(h)+B_{sf}u_f(h)\\ [0.2cm]y_s(h)=C_{ss}x_s(h), \end{array} \right.\qquad\label{Eqn:sigma_s}\\ &\Sigma_f:\ \left\{\begin{array}{l} x_f(h+1)=A_{fs}x_s(h)+A_{ff}x_f(h)+B_{fs}u_s(h)+B_{ff}u_f(h)\\ [0.2cm]y_f(h)=C_{ff}x_f(h), \end{array} \right.\qquad\label{Eqn:sigma_f} \end{aligned}$$ where $u_s\in \mathbb{R}^{m_s}$, $x_s\in \mathbb{R}^{n_s}$, $y_s\in \mathbb{R}^{p_s}$ are the input, state, and output variables belonged to $\Sigma_s$, while $u_f\in \mathbb{R}^{m_f}$, $x_f\in \mathbb{R}^{n_f}$, $y_f\in \mathbb{R}^{p_f}$ are the ones associated with $\Sigma_f$, $h$ is a basic discrete-time scale index, the matrices $A_{\ast}$, $B_{\ast}$ (where $\ast$ is $sf$ or $fs$ in turn) represent the couplings between $\Sigma_s$ and $\Sigma_f$ through the state and input variables respectively. Similar to [@zhang2018], in this paper, models and are assumed to satisfy at least one of the following scenarios: - $\Sigma_s$ is characterized by a slower dynamics in contrast to $\Sigma_f$ in the sense that some of the triples $(u_f,\, x_f,\, y_f)$ reach their final steady-state values fast while the other ones, i.e. $(u_s,\, x_s,\, y_s)$ may have begun their main dynamic motions, see the examples in [@naidu2002singular; @kokotovic1999singular; @mishchenko1994asymptotic]; - even if the dynamics of $\Sigma_s$ and $\Sigma_f$ might not be strictly separable, however they must be controlled in a multi-rate fashion, e.g., the triples $(u_f,\, x_f,\, y_f)$ must react promptly to respond to operation (reference) variations while the triples $(u_s,\, x_s,\, y_s)$ can be controlled in a smoother fashion, see for instance [@RV2008]. Combining , , the overall system is written as $$\label{Eqn:CL} \Sigma:\ \left\{\begin{array}{l} x(h+1)=Ax(h)+Bu(h)\\ [0.2cm]y(h)=Cx(h), \end{array}\right. \qquad$$ where $u=(u_s,\ u_f)\in\mathbb{R}^m$, $m=m_s+m_f$, $x=(x_s,\, x_f)\in\mathbb{R}^n$, $n=n_s+n_f$, $y=(y_s,\, y_f)\in \mathbb{R}^{p}$, $p=p_s+p_f$. The diagonal blocks of the collective state transition matrix $A$ and input matrix $B$ are $A_{ss}$, $A_{ff}$ and $B_{ss}$, $B_{ff}$ respectively; whereas their non-diagonal blocks correspond to the coupling terms of the state and input variables between $\Sigma_s$ and $\Sigma_f$. The collective output matrix is $C=\text{diag}(C_{ss},\ C_{ff})$.\ The following assumption is assumed to hold: (1) $A$ is stable, i.e., all the eigenvalues of $A$ are in the unitary disk $\mathcal{D}=\{z\in \mathbb{C}||z|\leq 1\}$; (2) \[assum1:nozero\] $m= p$, $m_s=p_s$, and the system has no invariant zeros in 1, i.e., $\text{det}(\Phi)\neq 0$ where $$\Phi=\begin{bmatrix} I-A&-B\\C&0 \end{bmatrix}.$$ \[assump:A\] The control objectives to be achieved are introduced here. - **Output tracking**: for a given reference value $y_{r}=(y_{s,r},\,y_{f,r})$, we aim to drive \[eq:output\_r\] $$\begin{aligned} y_s(h)\rightarrow y_{s,r},\label{Eqn:y_s-obje}\\ y_f(h)\rightarrow y_{f,r}\label{Eqn:y_f-obje} \end{aligned}$$ - **Input and state constraints**: enforce the input and state constraints of the type \[Eqn:out-in-con\] $$\begin{aligned} u_s(h)\in&\mathcal{U}_s,\label{eq:u_s_constraint}\\ u_f(h)\in& \mathcal{U}_f,\label{eq:u_f_constraint}\\ {x}_s(h)\in&\mathcal{X}_s,\label{eq:y_s_constraint}\\ x_f(h)\in&\mathcal{X}_f \label{eq:y_f_constraint} \end{aligned}$$ where $\mathcal{X}_s$, $\mathcal{X}_f$, $\mathcal{U}_s$, $\mathcal{U}_f$ are convex sets. Thanks to Assumption \[assump:A\]., from , it is possible to compute the steady-state input and state, i.e., $u_r=(u_{s,r},\, u_{f,r})$, $x_r=(x_{s,r},\, x_{f,r})$ such that $y_r=Cx_r$ and $x_r=Ax_r+Bu_r$. It is assumed that the sets $\mathcal{U}=\mathcal{U}_s\times \mathcal{U}_f$, $\mathcal{X}=\mathcal{X}_s\times \mathcal{X}_f$ contain $u_r$, $x_r$ in their interiors respectively. In principle, a centralized MPC problem with respect to $\Sigma$ can be solved so as to achieve the above objectives. However, the resulting control performance might be hampered in the aforementioned scenarios due to the conflicting requirements of the sampling period and prediction horizon for $\Sigma_s$ and $\Sigma_f$ respectively. For instance, the control of the associated dynamics $\Sigma_f$ needs a higher input update frequency to ensure the short-term dynamic behaviors, while a larger prediction horizon might be expected for $\Sigma_s$ to guarantee the feasibility and stability of the adopted algorithm in the long term.\ For this reason, a dual-level MPC (D-MPC) control scheme is initially proposed in this paper to fulfill the aforementioned control objectives. At the high level, a slow time scale $k$ associated with $N$ ($N\in\mathbb{N}$) period of the basic time scale $h$ is adopted to define a stabilizing MPC problem with respect to the sampled version of $\Sigma$ penalizing the deviation between the output and its setpoint. The computed values of the control actions at this level (i.e., $u_f^{\scriptscriptstyle[N]}(k)$, $u_s^{\scriptscriptstyle[N]}(k)$) are held constant within the long sampling time interval $[kN,kN+N)$, i.e., $\bar u_f(h)=u_f^{\scriptscriptstyle[N]}(k)$, $\bar u_s(h)=u_s^{\scriptscriptstyle[N]}(k)$ for all $h\in [kN,kN+N)$. At the low level, a shrinking horizon MPC is designed at the basic time scale to refine control actions with additional corrections (i.e., $\delta u_f(h)$, $\delta u_s(h)$) in order to derive satisfactory short-term transient associated with the closed-loop fast dynamics and to verify endpoint state constraint computed from the high level. The overall control actions of the D-MPC regulator are described by \[eq:input\_contributions\] $$\begin{aligned} u_s(h)&=\bar u_s(h)+\delta u_s(h),\label{Eqn:slow-in}\\ u_f(h)&=\bar u_f(h)+\delta u_f(h)\label{Eqn:fast-in} \end{aligned}$$ where - the control actions $\bar u_s(h)$ and $\bar u_f(h)$ will be computed by solving a MPC problem according to receding horizon principle in the slow time scale to fulfill objective and , ; - the corrections $\delta u_s(h)$ and $\delta u_f(h)$ will be defined by a shrinking horizon MPC regulator running in the basic time scale to fulfill objective and to enforce constraints –. Moreover, with the objective of further improving the control behavior of the fast and slow controlled variables, the *incremental* D-MPC algorithm is also proposed (see Section \[sec:iD-MPC\]), in which the MPC at the high level is modified. To be specific, this version includes at the high level an integral action on the controlled variable $y_s$ and, a prior explicit design of the output trajectory of $y_f$ relying on an auxiliary optimization variable to be optimized at the slow time scale with the objective to enforce $y_f$ to steer to the reference value or its neighbor promptly. A brief diagram of the proposed approaches is displayed in Fig. \[fig:control\_scheme\]. ![A brief diagram of the proposed control scheme: HMPC (LMPC) stands for the MPC at the higher (lower) level, *i*HMPC represents the *incremental* HMPC, while ZOH is the zero order holder.[]{data-label="fig:control_scheme"}](control_scheme){width="1\columnwidth"} D-MPC algorithm =============== In this section, the D-MPC algorithm consisting of a stabilizing MPC at the high level and a shrinking horizon MPC at the low level is devised. Stabilizing MPC at the high level --------------------------------- In order to design the high-level regulator in the slow time scale, first define the time index $k\in \mathbb{N}$ associated with a fixed positive integer $N$ so that $h=kN$ and denote by $u_{\ast}^{\scriptscriptstyle[N]}$, $x_{\ast}^{\scriptscriptstyle[N]}$, and $y_{\ast}^{\scriptscriptstyle[N]}$ the samplings of $u_{\ast}$, $x_{\ast}$, and $y_{\ast}$ (where $\ast$ is $s$ or $f$, in turn) and by $u^{\scriptscriptstyle[N]}=(u_s^{\scriptscriptstyle[N]},\,u_f^{\scriptscriptstyle[N]})$, $x^{\scriptscriptstyle[N]}=(x_s^{\scriptscriptstyle[N]},\,x_f^{\scriptscriptstyle[N]})$ , and $y^{\scriptscriptstyle[N]}=(y_s^{\scriptscriptstyle[N]},\,y_f^{\scriptscriptstyle[N]})$ the samplings of the input, state, and output variables corresponding to the time scale $k$. Hence, the sampled system of with $N$ period is given as $$\label{Eqn:CL_k} \Sigma^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{l} x^{\scriptscriptstyle[N]}(k+1)=A^{\scriptscriptstyle[N]}x^{\scriptscriptstyle[N]}(k)+B^{\scriptscriptstyle[N]}u^{\scriptscriptstyle[N]}(k)\\ [0.2cm]y^{\scriptscriptstyle[N]}(k)=Cx^{\scriptscriptstyle[N]}(k), \end{array}\right. \qquad$$ where $A^{\scriptscriptstyle[N]}=A^{N}$, $B^{\scriptscriptstyle[N]}=\sum_{j=0}^{N-1}A^{N-j-1}B$. Notice that, from and , if $x(kN)=x^{\scriptscriptstyle[N]}(k)$ and the control $u(h)=u^{\scriptscriptstyle[N]}(k)\ \forall h\in[kN,kN+N)$, it holds that $x(kN+N)=x^{\scriptscriptstyle[N]}(k+1)$ and $y(kN+N)=y^{\scriptscriptstyle[N]}(k+1)$.\ The following proposition can be stated for $\Sigma^{[N]}$: \[prop:obser-con\] The pair $(A^{\scriptscriptstyle[N]},\,C)$ is detectable if $(A,\,C)$ is detectable. Also, the following assumption about $\Sigma^{[N]}$ is assumed to be holding: The pair $(A^{\scriptscriptstyle[N]},\,B^{\scriptscriptstyle[N]})$ is stabilizable. \[assum:stab\] Note that, slightly different from the detectability condition in Proposition \[prop:obser-con\], to meet the stabilizability requirement of $\Sigma^{\scriptscriptstyle[N]}$, we require Assumption \[assum:stab\] to be verified a posteriori once the sampling period $N$ is chosen. This is due to the fact that, starting from the stabilizability of $(A,\,B)$, there is no guarantee the resultant sampled pair $(A^{\scriptscriptstyle[N]},\,B^{\scriptscriptstyle[N]})$ is also stabilizable. A simple example to illustrate this point is as follows: consider a SISO system described by $x(h+1)=-x(h)+u(h)$, which is obviously stabilizable. However, the $N=2$ period sampled version $x^{\scriptscriptstyle[2]}(k+1)=(-1)^2x^{\scriptscriptstyle[2]}(k)+(-1+1)u^{\scriptscriptstyle[2]}(k)=x^{\scriptscriptstyle[2]}(k)$ is not stabilizable. With , it is now possible to state the MPC problem at the high level. At each slow time-step $k$ we solve an optimization problem according to receding horizon principle as follows: $$\begin{array}{cc} \min & {J_{{\rm\scriptscriptstyle H}}}\\ {\scriptstyle \overrightarrow{u^{\scriptscriptstyle[N]}}{(k:k+N_{\rm\scriptscriptstyle H}-1)}}\\ \end{array}\label{Eqn:HLoptimiz}$$ where $$\begin{array}{cl} J_{{\rm\scriptscriptstyle H}}=&\sum_{i=0}^{N_{\rm\scriptscriptstyle H}-1}\big(\|y^{\scriptscriptstyle[N]}(k+i)-y_r\|_{Q_{\rm\scriptscriptstyle H}}^{2}+\|u^{\scriptscriptstyle[N]}(k+i)-u_r\|_{R_{\rm\scriptscriptstyle H}}^{2}\big)\\ & +\|x^{\scriptscriptstyle[N]}(k+N_{\rm\scriptscriptstyle H})-x_r\|_{P_{\rm\scriptscriptstyle H}}^{2}, \end{array}\label{Eqn:HL_cost}$$ and where $N_{\rm\scriptscriptstyle H}>0$ is the adopted prediction horizon. The parameters ${Q_{\rm\scriptscriptstyle H}}\in \mathbb{R}^{p\times p}$, ${R_{\rm\scriptscriptstyle H}}\in \mathbb{R}^{m\times m}$ are positive definite and symmetric weighting matrices, while ${P_{\rm\scriptscriptstyle H}}\in \mathbb{R}^{n\times n}$ is computed as the solution to the Lyapunov equation described by $$\label{Eqn:lyap} F_{\rm\scriptscriptstyle H}^{\top} P_{\rm\scriptscriptstyle H} F_{\rm\scriptscriptstyle H}-P_{\rm\scriptscriptstyle H}=-(C^{\top}Q_{\rm\scriptscriptstyle H}C+K_{\rm\scriptscriptstyle H}^{\top}R_{\rm\scriptscriptstyle H} K_{\rm\scriptscriptstyle H})$$ where matrix $F_{\rm\scriptscriptstyle H}=A^{\scriptscriptstyle[N]}+B^{\scriptscriptstyle[N]}K_{\rm\scriptscriptstyle H}$ is Schur stable and $K_{\rm\scriptscriptstyle H}$ is a stabilizing gain matrix.\ The optimization problem is performed under the following constraints: (1) the dynamics ; (2) the input and state constraints $$\begin{array}{ccc} x^{\scriptscriptstyle[N]}(k+i)\in \mathcal{X}\\ u^{\scriptscriptstyle[N]}(k+i)\in \mathcal{U}; \end{array}$$ (3) the terminal state constraint $$x^{\scriptscriptstyle[N]}(k+N_{\rm\scriptscriptstyle H})\in \mathcal{X}_{\rm\scriptscriptstyle F}.$$ Thanks to Assumption \[assum:stab\], the set $\mathcal{X}_{\rm\scriptscriptstyle F}$ is chosen as a positively invariant set for system  controlled with the stabilizing control law $u^{\scriptscriptstyle[N]}(k)=K_{\rm\scriptscriptstyle H} (x^{\scriptscriptstyle[N]}(k)-x_r)+u_r$, satisfying $K_{\rm\scriptscriptstyle H}(\mathcal{X}_{\rm\scriptscriptstyle F}\ominus x_r) \subseteq \mathcal{U}\ominus u_r$. Let $\overrightarrow{u^{\scriptscriptstyle[N]}}{(k:k+N_{\rm\scriptscriptstyle H}-1|k)}$ be the optimal solution to optimization . Only the first element $u^{\scriptscriptstyle[N]}(k|k)$ is applied at time instant $k$, then the values of $x^{\scriptscriptstyle[N]}(k+1|k)$ and $y^{\scriptscriptstyle[N]}(k+1|k)$ are updated and the optimization is repeated at the next time instant $k+1$ according to receding horizon principle.\ Shrinking horizon MPC at the low level -------------------------------------- Assume now to be at a specific fast time instant $h=kN$ such that the high-level optimization problem with cost  has been successfully solved. Thus the computed values of the input $u^{\scriptscriptstyle[N]}(k)=(u_s^{\scriptscriptstyle[N]}(k),\,u_f^{\scriptscriptstyle[N]}(k))$ and the one-step ahead state prediction $x^{\scriptscriptstyle[N]}(k+1|k)$ are available. Let us focus on the output performance in the fast time scale within the interval $h\in[kN,\,kN+N)$. Denoting by $\tilde{y}(h)=(\tilde y_s(h),\,\tilde y_f(h))$ the output resulting from  with $u(h)=u^{\scriptscriptstyle[N]}(\lfloor{h/N}\rfloor)$, the component $\tilde y_f(h)$ may expect undesired transient due to the use of the long sampling period at the high level. For this reason, at the low level the overall control action associated with $y_f$ is refined as \[Eqn: u\_redefine\] $$\begin{aligned} \label{Eqn: u_f_redefine} u_f(h)=\bar u_f(h)+\delta u_f(h) \end{aligned}$$ where $\bar u_f(h)=u_f^{\scriptscriptstyle[N]}(\lfloor{h/N}\rfloor)$, $\delta u_f$ is computed at the low level by a properly defined optimization problem, see .\ Since $\delta u_f(h)$ could influence the value of $y_s(h)$ in the fast time scale due to possible nonzero coupling terms from $\Sigma_f$ to $\Sigma_s$ (e.g. $A_{sf}$, $B_{sf}$), it is also convenient to allow a further control freedom of $u_s$ leading to the correction as follows: $$\begin{aligned} \label{Eqn: u_s_redefine} u_s(h)=\bar u_s(h)+\delta u_s(h) \end{aligned}$$ where $\bar u_s(h)=u_s^{\scriptscriptstyle[N]}(\lfloor{h/N}\rfloor)$, $\delta u_s$ is another decision variable at the low level.\ In view of , we write dynamics  in the form: $$\label{Eqn:CL_final} \Sigma:\ \left\{\begin{array}{l} x(h+1)=Ax(h)+B\bar u(h)+B\delta u(h)\\ [0.2cm]y(h)=Cx(h) \end{array}\right. \qquad$$ where $\bar u=(\bar u_s,\ \bar u_f)$, $\delta u=(\delta u_s,\ \delta u_f)$. Accordingly, at any fast time instant $h=kN+t$, letting $\overrightarrow{\delta{u}}(h:(k+1)N-1)=\big[\, \delta u(h|h)\ \cdots\ \delta u((k+1)N-1|h)\,\big]\in({\mathbb R}^{m})^{N-t}$, a shrinking horizon MPC problem can be solved at the low level: $$\label{Eqn:LLoptimiz} \begin{array}{ccl} \min& {J_{\rm\scriptscriptstyle L}}\\ {\scriptstyle\overrightarrow{\delta {u}}{\scriptscriptstyle(h:(k+1)N-1)}}& \end{array}$$ where $$\label{Eqn:LLoptimiz_cost} J_{\rm\scriptscriptstyle L}=\sum_{j=0}^{N-t-1}\|y(h+j|h)-\tilde y^\ast(h)\|_{Q}^2+\| \delta u(h+j|h)\|_{R}^2 $$ and where $\tilde y^\ast(h)=(\tilde y_s(h),\ \tilde y_f(kN+N))$, for $h\in[kN,kN+N)$.\ The optimization problem  is performed under the following constraints: (1) the dynamics ; (2) the state and input constraints \[Eqn:LLoptimiz1\_con\] $$\begin{aligned} x(h+j|h)&\in\mathcal{X},\ \forall\,j=0,\dots,N-t-1,\\ u^{\scriptscriptstyle[N]}(\lfloor{h/N}\rfloor)+\delta u(h+j|h)&\in{\mathcal U},\ \forall\,j=0,\dots,N-t-1; \end{aligned}$$ (3) the state terminal constraint $$\label{LLoptimiz_term_con} x(kN+N|h)= x^{\scriptscriptstyle[N]}(k+1|k)$$ Thanks to the , at the high level the state in the next slow time instant $x^{\scriptscriptstyle[N]}(k+1)$ can be recovered by the predicted value $x^{\scriptscriptstyle[N]}(k+1|k)$, i.e., $x^{\scriptscriptstyle[N]}(k+1)=x^{\scriptscriptstyle[N]}(k+1|k)$. The rationale of choosing signal $\tilde y^\ast(h)$ as the reference for the low level lies in the fact that $y_f(h)$ is expected to react promptly to respond to $\tilde y_f(kN+N)$, while $y_s(h)$ can be controlled to follow the smooth trajectory $\tilde y_s(h)$ generated from the high level. It is highlighted that the structure of the proposed approach is different from that of the cascade ones, see for instance [@picasso2010mpc], for the reason that in the cascade algorithm, the computed input from the high level is considered as the output reference to be tracked at the low level, while the proposed schemes utilize the control action computed from the high level, i.e., $\bar u(h)$, to generate the possible reference profile with model  in an open-loop fashion. Summary of the D-MPC algorithm ------------------------------ In summary, the main steps for the on-line implementation of the D-MPC algorithm is given in Algorithm \[Atm:1\]. initialization\ The following theoretical results can be stated: \[theorem\] Under Assumption \[assump:A\], if the initial condition is such that $x^{\scriptscriptstyle[N]}(0)=x(0)$ and  is feasible at $k=0$, then the following results can be stated for the proposed D-MPC control algorithm - The feasibility can be guaranteed: - for the high-level problem at all slow time instant $k>0$; - for the low-level problem at all fast time instant $h\geq 0$. - The slow-time scale system $\Sigma^{\scriptscriptstyle[N]}$ enjoys the convergence property, i.e., $\lim_{k\to+\infty} (u^{\scriptscriptstyle[N]}(k),\, x^{\scriptscriptstyle[N]}(k),\, y^{\scriptscriptstyle[N]}(k))=(u_r,\, x_r,\, y_r)$. - Moreover, for the low-level problem , it holds that $\lim_{h\to+\infty} \delta u(h)=0$. Finally, $\lim_{h\to+\infty}(u(h),\, x(h),\, y(h))=(u_r,\, x_r,\, y_r)$. Note that, the terminal constraint  plays a crucial role for the closed-loop properties of the D-MPC due to the fact that it guarantees $x^{\scriptscriptstyle[N]}(k+1)=x^{\scriptscriptstyle[N]}(k+1|k)$. However, since the proposed D-MPC control structure is an upper-bottom one, the computed value of $x_f^{\scriptscriptstyle[N]}(k)$ at the high level influences the control performance at the low level due to . For this reason, the state $x_f(h)$ associated with $y_f(h)$ in the basic time scale might not converge to its nominal value faster than $x_s(h)$ especially for systems that exhibit nonseparable open-loop dynamics.This problem can be properly coped with in the framework of the *incremental* D-MPC whose details will be given in the following section. *Incremental* D-MPC algorithm {#sec:iD-MPC} ============================= In this section, we design an *incremental* D-MPC algorithm that includes at the high level an explicit design of the trajectory of “fast" output $y_f^{\scriptscriptstyle[N]}$ such that it reaches the reference value $y_{f,r}$ or its neighbour promptly in the slow time scale, also an integral action so as to improve robustness in case of model uncetainties, e.g., due to modelling errors. Compared with D-MPC, this version requires a major modification on the high-level MPC formulation, meanwhile preserves the previous design of the fast shrinking horizon MPC at the low level.\ Design of the incremental D-MPC ------------------------------- In the following we mainly focus on the redesign of the MPC regulator at the high level. As a further attention is paid at this level to design the trajectory of $y_f^{\scriptscriptstyle[N]}$, it is convenient to partition the sampled system as the one with the structure similar to . To proceed, according to the structures of $A$ and $B$ (see ), we first rewrite the matrices $A^{\scriptscriptstyle[N]}$, $B^{\scriptscriptstyle[N]}$ into the following forms $$A^{\scriptscriptstyle[N]}=\begin{bmatrix}A_{ss}^{\scriptscriptstyle[N]}&A_{sf}^{\scriptscriptstyle[N]}\\A_{fs}^{\scriptscriptstyle[N]}&A_{ff}^{\scriptscriptstyle[N]}\end{bmatrix},\ \ B^{\scriptscriptstyle[N]}=\begin{bmatrix}B_{ss}^{\scriptscriptstyle[N]}&B_{sf}^{\scriptscriptstyle[N]}\\B_{fs}^{\scriptscriptstyle[N]}&B_{ff}^{\scriptscriptstyle[N]}\end{bmatrix},$$ where $A_{ss}^{\scriptscriptstyle[N]}\in\mathbb{R}^{n_s\times n_s}$, $B_{ss}^{\scriptscriptstyle[N]}\in\mathbb{R}^{n_s\times m_s}$. In view of this, finally system  can be partitioned as two interacting subsystems represented by $\Sigma_s^{\scriptscriptstyle[N]}$ and $\Sigma_f^{\scriptscriptstyle[N]}$ described as follows: \[Eqn:sigma\_s\_f-k-im\] $$\begin{aligned} &\Sigma_s^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{ll} x_s^{\scriptscriptstyle[N]}(k+1)=&A_{ss}^{\scriptscriptstyle[N]}x_s^{\scriptscriptstyle[N]}(k)+A_{sf}^{\scriptscriptstyle[N]}x_f^{\scriptscriptstyle[N]}(k)+B_{ss}^{\scriptscriptstyle[N]}u_s^{\scriptscriptstyle[N]}(k)\\&+B_{sf}^{\scriptscriptstyle[N]}u_f^{\scriptscriptstyle[N]}(k)\\ [0.2cm]y_s^{\scriptscriptstyle[N]}(k)=&C_{ss}x_s^{\scriptscriptstyle[N]}(k), \end{array} \right.\qquad\label{Eqn:sigma_s-k}\\ &\Sigma_f^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{ll} x_f^{\scriptscriptstyle[N]}(k+1)=&A_{fs}^{\scriptscriptstyle[N]}x_s^{\scriptscriptstyle[N]}(k)+A_{ff}^{\scriptscriptstyle[N]}x_f^{\scriptscriptstyle[N]}(k)+B_{fs}^{\scriptscriptstyle[N]}u_s^{\scriptscriptstyle[N]}(k)\\ &+B_{ff}^{\scriptscriptstyle[N]}u_f^{\scriptscriptstyle[N]}(k)\\ [0.2cm]y_f^{\scriptscriptstyle[N]}(k)=&C_{ff}x_f^{\scriptscriptstyle[N]}(k), \end{array} \right.\qquad\label{Eqn:sigma_f-k} \end{aligned}$$ The following assumption about $\Sigma_f^{\scriptscriptstyle[N]}$ is assumed to hold: \[assum:cb-fullrank\] Matrix $C_{ff}B_{ff}^{\scriptscriptstyle[N]}$ is full rank. [With , with the goal of guaranteeing satisfactory control performance related to $y_f$ in the basic time scale, it is convenient to enforce that all the future predictions $y_f^{\scriptscriptstyle[N]}(k),\,\forall k>0$ associated with $\Sigma_f^{\scriptscriptstyle[N]}$ are equal to the reference value $y_{f,r}$ at the high level. In this way, the real output $y_f$ resulting from the controller  will reach the reference $y_{f,r}$ in only one slow-time step.]{} However, this restriction might cause infeasibility issue in the cases where the reference $y_{f,r}$ is far from its initial value $y_f(0)$, and/or constraints on the control increments are enforced. For this reason, instead of imposing $y_f^{\scriptscriptstyle[N]}(k)=y_{f,r}\ \forall k>0$, we enforce the following relation $$\label{Eqn:yf-gov} y_f^{\scriptscriptstyle[N]}(k)=\tilde y_{f,r}\ \forall k>0,$$ where $$\tilde y_{f,r}=y_f(0)+\alpha(k)(y_{f,r}-y_f(0))$$ and where $\alpha(k)$ is defined as an optimization variable that its value is restricted by $0\leq\alpha(k)\leq1$ and reaches $1$ in finite time steps, i.e., $$\label{Eqn:alpha} \left\{\begin{array}{ll} \alpha(k)=0,& k=0 \\ 0\leq\alpha(k)\leq 1,& k\in[1,\ N_{\alpha})\\ \alpha(k)=1,& k\geq N_{\alpha} \end{array}\right.$$ where $N_{\alpha}$ is a positive integer. Thanks to Assumption \[assum:cb-fullrank\], from , imposing  is equivalent to considering the constraint as follows: $$\label{Eqn:uf-yf-con-k-im} \begin{array}{l} u_f^{\scriptscriptstyle[N]}(k)=\\ (C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}\big( \tilde y_{f,r}-C_{ff}(A_{fs}^{\scriptscriptstyle[N]}x_s^{\scriptscriptstyle[N]}(k)+A_{ff}^{\scriptscriptstyle[N]}x_f^{\scriptscriptstyle[N]}(k)+B_{fs}^{\scriptscriptstyle[N]}u_s^{\scriptscriptstyle[N]}(k))\big) \end{array}$$ Under constraint , the time steps required for $y_f^{\scriptscriptstyle[N]}$ being converging to its reference value $y_{f,r}$ can be defined via properly tuning parameter $N_{\alpha}$. With a slightly abuse of notation, we denote $\tilde{\Sigma}^{\scriptscriptstyle[N]}$ the system from  by substituting $u_f^{\scriptscriptstyle[N]}(k)$ with , that is $$\label{Eqn:sigma_s-us-k-im} \tilde{\Sigma}^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{l} x^{\scriptscriptstyle[N]}(k+1)=\tilde A^{\scriptscriptstyle[N]}x^{\scriptscriptstyle[N]}(k)+\tilde B_{s}^{\scriptscriptstyle[N]} u_s^{\scriptscriptstyle[N]}(k)+\tilde B_{f}^{\scriptscriptstyle[N]}\tilde y_{f,r}\\ [0.2cm]y_s^{\scriptscriptstyle[N]}(k)=\tilde C_{s}x^{\scriptscriptstyle[N]}(k), \end{array} \right.\qquad$$ where $\tilde A^{\scriptscriptstyle[N]}=\begin{bmatrix}\tilde A_{ss}^{\scriptscriptstyle[N]}&\tilde A_{sf}^{\scriptscriptstyle[N]}\\ \tilde A_{fs}^{\scriptscriptstyle[N]}&\tilde A_{ff}^{\scriptscriptstyle[N]}\end{bmatrix}$, $\tilde B_s^{\scriptscriptstyle[N]}=\begin{bmatrix}\tilde B_{ss}^{\scriptscriptstyle[N]}\\\tilde B_{fs}^{\scriptscriptstyle[N]}\end{bmatrix}$, $\tilde B_f^{\scriptscriptstyle[N]}=\begin{bmatrix}\tilde B_{sf}^{\scriptscriptstyle[N]}\\\tilde B_{ff}^{\scriptscriptstyle[N]}\end{bmatrix}$, $\tilde C_s=\begin{bmatrix}C_{ss}\\0\end{bmatrix}^{\top}$, and where $$\begin{aligned} &\tilde A_{ss}^{\scriptscriptstyle[N]}=A_{ss}^{\scriptscriptstyle[N]}-B_{sf}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}A_{fs}^{\scriptscriptstyle[N]}\\ &\tilde A_{sf}^{\scriptscriptstyle[N]}=A_{sf}^{\scriptscriptstyle[N]}-B_{sf}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}A_{ff}^{\scriptscriptstyle[N]}\\ &\tilde A_{fs}^{\scriptscriptstyle[N]}=A_{fs}^{\scriptscriptstyle[N]}-B_{ff}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}A_{fs}^{\scriptscriptstyle[N]}\\ &\tilde A_{ff}^{\scriptscriptstyle[N]}=A_{ff}^{\scriptscriptstyle[N]}-B_{ff}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}A_{ff}^{\scriptscriptstyle[N]}\\ &\tilde B_{ss}^{\scriptscriptstyle[N]}=B_{ss}^{\scriptscriptstyle[N]}-B_{sf}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}B_{fs}^{\scriptscriptstyle[N]}\\ &\tilde B_{fs}^{\scriptscriptstyle[N]}=B_{fs}^{\scriptscriptstyle[N]}-B_{ff}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}B_{fs}^{\scriptscriptstyle[N]}\\ &\tilde B_{sf}^{\scriptscriptstyle[N]}=B_{sf}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}\\ &\tilde B_{ff}^{\scriptscriptstyle[N]}=B_{ff}^{\scriptscriptstyle[N]}(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}.\end{aligned}$$ \[assum:tilde A-stable\] The integer $N$ is such that $\tilde A^{\scriptscriptstyle[N]}$ is stable. [ With , it is reasonable to define a corresponding MPC problem at the high level similar to . We highlight that, with this formulation, the closed-loop performance might be hampered due to possible model uncertainties. Also note that, in view of the definition of $\alpha(k)$, the value of $\tilde y_{f,r}$ will reach $y_{f,r}$ as long as the time index $k\geq N_{\alpha}$. For these reasons, model  is reformulated in the corresponding *incremental form*. In doing so, the effects by slow (or constant) disturbance can be alleviated (or cancelled) and the dependency on $y_{f,r}$ will be disappeared for $k\geq N_{\alpha}$. With the model in velocity form, it is possible to define a MPC problem including an integral action with the goal of output offset-free tracking control.]{} [In doing so, the closed-loop system is capable to compensate for constant or slow disturbances, see [@betti].]{} To this end, denoting $\Delta x^{\scriptscriptstyle[N]}(k)=x^{\scriptscriptstyle[N]}(k)-x^{\scriptscriptstyle[N]}(k-1)$, $\Delta u_s^{\scriptscriptstyle[N]}(k)=u_s^{\scriptscriptstyle[N]}(k)-u_s^{\scriptscriptstyle[N]}(k-1)$, $\Delta \alpha(k)=\alpha(k)-\alpha(k-1)$, $\bar x^{\scriptscriptstyle[N]}(k)=(y_s^{\scriptscriptstyle[N]}(k),\ \Delta x^{\scriptscriptstyle[N]}(k))$, from  we compute $$\label{Eqn:sigma_s-dus-k} \bar{\Sigma}^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{rl} \bar x^{\scriptscriptstyle[N]}(k+1)=&\bar A^{\scriptscriptstyle[N]}\bar x^{\scriptscriptstyle[N]}(k)+\bar B_{s}^{\scriptscriptstyle[N]} \Delta u_s^{\scriptscriptstyle[N]}(k)+\\ &\bar B_{f}^{\scriptscriptstyle[N]}\Delta \alpha(k)(y_{f,r}-y_f(0))\\ \alpha(k)=&\alpha(k-1)+\Delta\alpha(k)\\ [0.2cm]y_s^{\scriptscriptstyle[N]}(k)=&\bar C\bar x^{\scriptscriptstyle[N]}(k), \end{array} \right.\qquad$$ where $\bar A^{\scriptscriptstyle[N]}=\begin{bmatrix}I& \tilde C_s\tilde A^{\scriptscriptstyle[N]}\\0&\tilde A^{\scriptscriptstyle[N]}\end{bmatrix}$, $\bar B_s^{\scriptscriptstyle[N]}=\begin{bmatrix}\tilde C_s\tilde B_s^{\scriptscriptstyle[N]}\\ \tilde B_s^{\scriptscriptstyle[N]}\end{bmatrix}$, $\bar B_f^{\scriptscriptstyle[N]}=\begin{bmatrix}\tilde C_s\tilde B_f^{\scriptscriptstyle[N]}\\ \tilde B_f^{\scriptscriptstyle[N]}\end{bmatrix}$, $\bar C=\begin{bmatrix}I&0\end{bmatrix}$. \[prop1\] The pair $(\bar A^{\scriptscriptstyle[N]},\ \bar B_s^{\scriptscriptstyle[N]})$ is stabilizable if and only if $$\bullet\ \ \text{rank}\big(\begin{bmatrix} \tilde C_s\tilde A^{\scriptscriptstyle[N]}&\tilde C_s\tilde B_s^{\scriptscriptstyle[N]}\\\tilde A^{\scriptscriptstyle[N]}-I&\tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{\top} \big)= n+p_s,$$ $$\bullet\ \ \text{rank}\big(\begin{bmatrix} 2I&0\\ \tilde C_s\tilde A^{\scriptscriptstyle[N]}&\tilde A^{\scriptscriptstyle[N]}+I\\\tilde C_s\tilde B_s^{\scriptscriptstyle[N]}&\tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{\top}\big)= n+p_s.$$ Under proposition \[prop1\], it is possible to find a gain matrix $\bar K_{s,\rm\scriptscriptstyle H}$ such that $\bar F_{s,\rm\scriptscriptstyle H}=\bar A^{\scriptscriptstyle[N]}+\bar B^{\scriptscriptstyle[N]}\bar K_{s,\rm\scriptscriptstyle H}$ is Schur stable. Note that, the original constraints on $ x_s^{\scriptscriptstyle[N]}(k)$, $\bar u_s^{\scriptscriptstyle[N]}(k)$, and $\bar u_f^{\scriptscriptstyle[N]}(k)$ are not compatible with model . We are going to show that, along the same line described in [@betti], it is possible to represent the control and state variables by the state and input variables in the velocity form, that is \[Eqn:con-velocity\] $$\begin{aligned} & x_s^{\scriptscriptstyle[N]}(k)=\Gamma_{xs}\bar x^{\scriptscriptstyle[N]}(k+1)\\ & u_s^{\scriptscriptstyle[N]}(k)=\Gamma_{us}\bar x^{\scriptscriptstyle[N]}(k+1),\label{Eqn:us-con-velocity}\\ u_f^{\scriptscriptstyle[N]}(k)=&(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}\alpha(k) y_{f,r}-\Gamma_{uf}\bar x^{\scriptscriptstyle[N]}(k+1)\label{Eqn:uf-con-velocity} \end{aligned}$$ where $\Gamma_{uf}= (C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1}C_{ff}\begin{bmatrix}A_{fs}^{\scriptscriptstyle[N]}&A^{\scriptscriptstyle[N]}_{ff}&B_{fs}^{\scriptscriptstyle[N]} \end{bmatrix}\Gamma$, $\Gamma_{xs}=\begin{bmatrix} I_{n_s}&0 \end{bmatrix}\Gamma$, $\Gamma_{us}=\begin{bmatrix} 0&I_{m_s} \end{bmatrix}\Gamma$, and where $\Gamma=\begin{bmatrix} \tilde C_s\tilde A^{\scriptscriptstyle[N]}&\bar C_s\tilde B_s^{\scriptscriptstyle[N]}\\\tilde A^{\scriptscriptstyle[N]}-I_{n}&\tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{-1}$.\ In view of , , , , the constraints to be considered at the high level are as follows: $$\label{Eqn:con-velocity-all} \begin{array}{c} \begin{bmatrix} \Gamma_{xs}\\\Gamma_{us}\\-\Gamma_{uf} \end{bmatrix}\bar x^{\scriptscriptstyle[N]}(k+1)+\begin{bmatrix} 0\\0\\(C_{ff}B_{ff}^{\scriptscriptstyle[N]})^{-1} \end{bmatrix}\alpha(k) y_{f,r}=\\ (y_s^{\scriptscriptstyle[N]}(k+1), u_s^{\scriptscriptstyle[N]}(k), u_f^{\scriptscriptstyle[N]}(k)) \in \mathcal{X}_s\times\mathcal{U}_s\times\mathcal{U}_f \end{array}$$ Based on   and , now it is possible to state the *incremental* MPC problem at the high level. At each slow time-step $k$ we solve an optimization problem according to receding horizon principle as follows: $$\begin{array}{cc} \min & {\bar J_{{\rm\scriptscriptstyle H}}}\\ {\scriptstyle \overrightarrow{\Delta u_s^{\scriptscriptstyle[N]}}{(k:k+\bar N_{\rm\scriptscriptstyle H}-1)}}\\ \end{array}\label{Eqn:HLoptimiz-im}$$ where $$\begin{array}{cll} \bar J_{{\rm\scriptscriptstyle H}}=&\sum_{i=0}^{\bar N_{\rm\scriptscriptstyle H}-1}\|\bar x^{\scriptscriptstyle[N]}(k+i)-\bar C^{\top} y_{s,r}\|_{\bar Q_{s,\rm\scriptscriptstyle H}}^{2}+\|\Delta u_s^{\scriptscriptstyle[N]}(k+i)\|_{\bar R_{s,\rm\scriptscriptstyle H}}^{2}+\\ &\gamma(\alpha(k+i)-1)^2+ \|\bar x^{\scriptscriptstyle[N]}(k+\bar N_{\rm\scriptscriptstyle H})-\bar C^{\top}y_{s,r}\|_{\bar P_{\rm\scriptscriptstyle H}}^{2}. \end{array}\label{Eqn:HL_cost-im}$$ $\gamma$ is a positive scalar, $\bar N_{\rm\scriptscriptstyle H}>N_{\alpha}$ is the adopted prediction horizon. The positive definite and symmetric weighting matrices ${\bar Q_{s,\rm\scriptscriptstyle H}}\in \mathbb{R}^{(n+p_s)\times (n+p_s)}$, ${\bar R_{s,\rm\scriptscriptstyle H}}\in\mathbb{R}^{m_s\times m_s}$ are free design parameters, while ${\bar P_{\rm\scriptscriptstyle H}}$ is computed as the solution to the Lyapunov equation $$\label{Eqn:lyap-im} \bar F_{s,\rm\scriptscriptstyle H}^{\top} \bar P_{\rm\scriptscriptstyle H} \bar F_{s,\rm\scriptscriptstyle H}-\bar P_{\rm\scriptscriptstyle H}=-(Q_{s,\rm\scriptscriptstyle H}+\bar K_{s,\rm\scriptscriptstyle H}^{\top} \bar R_{s,\rm\scriptscriptstyle H} \bar K_{s,\rm\scriptscriptstyle H})$$ The optimization problem  is performed under the following constraints: (1) dynamics , constraint  and ; (2) the state terminal constraint $$\begin{array}{ccc} \bar x^{\scriptscriptstyle[N]}(k+\bar N_{\rm\scriptscriptstyle H})\in \bar{\mathcal{X}}_{\rm\scriptscriptstyle F},\\ \end{array}$$ where the set $\bar{\mathcal{X}}_{\rm\scriptscriptstyle F}$ is a positively invariant set for the nominal system of , i.e., $$\label{Eqn:sigma_s-dus-k-nominal} \Delta \hat{\Sigma}^{\scriptscriptstyle[N]}:\ \left\{\begin{array}{l} \hat {\bar x}^{\scriptscriptstyle[N]}(k+1)=\bar A^{\scriptscriptstyle[N]}\hat {\bar x}^{\scriptscriptstyle[N]}(k)+\bar B_{s}^{\scriptscriptstyle[N]} \Delta u_s^{\scriptscriptstyle[N]}(k)\\ [0.2cm]\hat y_s^{\scriptscriptstyle[N]}(k)=\bar C\hat{\bar x}^{\scriptscriptstyle[N]}(k), \end{array} \right.\qquad$$ that is controlled with the stabilizing control law $\Delta u_s^{\scriptscriptstyle[N]}(k)=\bar K_{s,\rm\scriptscriptstyle H} (\hat{\bar x}^{\scriptscriptstyle[N]}(k)-\bar C^{\top}y_{s,r})$ such that $\bar F_{s,\rm\scriptscriptstyle H}\bar{\mathcal{X}}_{\rm\scriptscriptstyle F}\subseteq \bar{\mathcal{X}}_{\rm\scriptscriptstyle F}$ under constraint . Let $\overrightarrow{\Delta u_s^{\scriptscriptstyle[N]}}{(k:k+\bar N_{\rm\scriptscriptstyle H}-1|k)}$ be the optimal solution to optimization . Only the first element $\Delta u_s^{\scriptscriptstyle[N]}(k|k)$ is applied at time instant $k$, then the values of $\bar x^{\scriptscriptstyle[N]}(k+1|k)$, $y_s^{\scriptscriptstyle[N]}(k+1|k)$ are updated. The real input $u_s^{\scriptscriptstyle[N]}(k)$ at time instant $k$ is given by $u_s^{\scriptscriptstyle[N]}(k)=u_s^{\scriptscriptstyle[N]}(k-1)+\Delta u_s^{\scriptscriptstyle[N]}(k|k)$, also from  we can compute the value of $u_f^{\scriptscriptstyle[N]}(k)$. At this time instant, the state $x^{\scriptscriptstyle[N]}(k+1|k)$ is available by applying $u^{\scriptscriptstyle[N]}(k)=(u_s^{\scriptscriptstyle[N]}(k),u_f^{\scriptscriptstyle[N]}(k))$ to . With the above available information, the fast MPC problem described in the previous section, i.e., with cost  is solved recursively in the fast interval $[kN,\ kN+N)$ according to shrinking horizon principle. Then the optimization problem  with cost  is repeated at $k+1$ according to receding horizon principle. Summary of the incremental D-MPC algorithm ------------------------------------------ To better clarify the requirements for the implementation of the *incremental* D-MPC algorithm and its difference with the D-MPC algorithm, the main steps for the on-line implementation is given in Algorithm \[Atm:2\]. initialization with $N_{\alpha}=1$\ The following theoretical results are stated: \[theorem-im\] Under Assumptions \[assump:A\]–\[assum:tilde A-stable\], if the initial condition is such that $x^{\scriptscriptstyle[N]}(0)=x(0)$ and $N_{\alpha}$ is reachable by Algorithm \[Atm:2\] such that is feasible at $k=0$, then the following results can be stated for the *incremental* D-MPC algorithm (i.e., high-level problem  with cost  and low-level problem  with ): - The feasibility can be guaranteed: - for the high-level problem at all slow time instant $k>0$; - for the low-level problem at all fast time instant $h\geq 0$. - The slow-time scale system enjoys the convergence property, i.e., $\lim_{k\to+\infty} (\bar x^{\scriptscriptstyle[N]}(k),\, \Delta u_s^{\scriptscriptstyle[N]}(k))=(\bar C^{\top} y_{s,r},\, 0)$. Consequently, $\lim_{k\to+\infty} (x^{\scriptscriptstyle[N]}(k),\, u^{\scriptscriptstyle[N]}(k))=(x_r,\, u_r)$. - For the low-level problem it holds that $\lim_{h\to+\infty} \delta u(h)=0$. Finally, $\lim_{h\to+\infty} (u(h),\, x(h),\, y(h))=(u_r,\, x_r,\, y_r)$. Simulation example {#sec:example} ================== The proposed D-MPC and *incremental* D-MPC algorithms are used for the control of a BT system including the comparisons of their control performances with a traditional decentralized PID controller and a multi-rate MPC. Description of the BT system and its linearized model ----------------------------------------------------- A $160$ MV BT system in [@aastrom1987dynamic] is considered and its dynamic diagram is presented in Fig. \[fig:boiler-turbine\]. The input variables applied to the boiler are the fuel flow $q_f$ ($\rm kg/s$) and feedwater flow $q_w$ ($\rm kg/s$), while the controlled variables of the boiler are drum pressure $P$ ($\text{ kg/cm}^2$) and water level $L$ ($\text{m}$). The control and controlled variables of the turbine are the steam control $q_s$ ($\rm kg/s$) and the electrical power output $Q$ ($\text{MV}$). Typically, the goal of BT control is to regulate the electrical power to meet the load demand profile meanwhile to minimize the variations of internal variables such as water level and drum pressure within their safe sets. Moreover, drum pressure must also be controlled properly in the operation range to respond to possible turbine speed changes caused by load demand variations. Many works have been addressed at this point that focus on deriving satisfactory closed-loop control performance of electrical power, see e.g. [@chen2013multi; @wang2017improved; @li2005new]. In this scenario, the control related to the output variables such as electrical power and drum pressure is a major issue that must be tackled properly to respond to frequent load demand variations, while the water level can be adjusted smoothly under its constraint with the possibility to follow its desired value. This makes it reasonable in this case to apply the proposed dual-level control algorithms. The continuous nonlinear dynamic model described in [@aastrom1987dynamic] is given as $$\label{Eqn:non-noiler} \begin{array}{l} \dot \rho=(141q_w-(1.1q_s-0.19)P)/85\\ \dot P=-0.0018q_sP^{9/8}+0.9q_f-0.15q_w\\ \dot Q=(0.073q_s-0.016)P^{9/8}-0.1Q \end{array}$$ where $\rho$ is the fluid density $(\rm kg/cm^3)$. The control variables are limited by $0\leq q_f,q_w,q_s\leq 1$ and their rate constraints are also considered, i.e., $-0.007\leq\dot q_f\leq 0.007$, $-2\leq \dot q_s\leq 0.2$, $-0.05\leq \dot q_w\leq 0.05$. The water level relies on a static nonlinear mapping from the state and input variables in . For simplicity, instead of water level, the fluid density is selected as a controlled variable. The considered operation point is $(\rho_r,\,P_r,\, Q_r)=(513.6,\break\,129.6,\,105.8)$, $(q_{w,r},\,q_{f,r},\, q_{s,r})=(0.663,\,0.505,\,0.828)$. The continuous linearized model at this operation point is computed and described by $$\label{Eqn:CL_C} \left\{\begin{array}{l} \dot x=A_cx+B_cu\\ [0.2cm]y=C_cx, \end{array}\right. \qquad$$ where $$A_c= \begin{bmatrix} 0&-0.008&0\\0&-0.003&0\\0&0.092&-0.1 \end{bmatrix},\ \ \ \ B_c= \begin{bmatrix} 1.66&0&-1.68\\-0.15&0.9&-0.43\\0&0&17.4\\ \end{bmatrix},$$ $C=I$, the state and output variables are $y=x=( \rho-\rho_r,\,P-P_r,\, Q-Q_r)$, while the input variables are $u=(q_w-q_{w,r},\, q_{f}-q_{f,r},\, q_s-q_{s,r})$.[ The unitary step response of is presented in Figure \[fig:boiler-turbine\_step\], which displays that the system outputs are strongly coupled, and the dynamics is not strictly separable.]{} ![Diagram of the BT dynamics.[]{data-label="fig:boiler-turbine"}](boiler-turbine){width="0.9\columnwidth"} ![Unitary impulse response of the BT dynamics.[]{data-label="fig:boiler-turbine_step"}](step){width="0.9\columnwidth"} ![Control variables of the BT plant: red continuous lines (blue dot-dashed lines) represent the inputs computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones computed with the decentralized PIDs.[]{data-label="fig:input"}](input_pid){width="0.9\columnwidth"} ![Outputs of the BT plant: red continuous lines (blue dot-dashed lines) represent the output variables computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones obtained with the decentralized PIDs.[]{data-label="fig:state-output"}](output_pid){width="0.9\columnwidth"} Devising the D-MPC and incremental D-MPC regulators --------------------------------------------------- In order to implement the proposed dual-level control algorithms, the system’s continuous-time model has been sampled with $\Delta t=1$s to obtain the discrete-time counterpart in the fast time scale. The resulting system has been rewritten to derive model , where the input, state, and output variables associated with $\Sigma_s$ to be controlled smoothly are chosen as $u_s=q_w-q_{w,r}$, $x_s=\rho-\rho_r$, and $y_s=x_s$ while the corresponding ones associated with $\Sigma_f$ to be controlled in a prompt fashion are $u_f=(q_{f}-q_{f,r},\, q_s-q_{s,r})$, $x_f=(P-P_r,\, Q-Q_r)$, and $y_f=x_f$. The resulting model has been re-sampled with $N=20$ to obtain and  to be used at the high level.\ ### Design of the D-MPC regulator - The high-level stabilizing MPC with cost has been implemented with $Q_{\rm\scriptscriptstyle H}=I$ and $R_{\rm\scriptscriptstyle H}=\text{diag}(2,\ 20,\ 20)$, and prediction horizon $N_{\rm\scriptscriptstyle H}=20$. The control gain matrix $K_{\rm\scriptscriptstyle H}$ is selected by solving an infinite horizon LQ problem. The terminal penalty is computed according to and the solution is $$P_{\rm\scriptscriptstyle H}=\begin{bmatrix} 10.34&2.54&0.65\\ 2.54&59.19&4.73\\ 0.65&4.73&8.01 \end{bmatrix}.$$ The terminal set has been chosen according to the algorithm described in [@hu2008model], i.e., $\mathcal{X}_{\rm\scriptscriptstyle F}=\{x|(x-x_r)^TP_{\rm\scriptscriptstyle H}(x-x_r)\leq0.221$. - The low-level shrinking horizon MPC with cost has been designed with $Q=I$ and $R=\text{diag}(1,\, 1,\, 10)$. ### Design of the incremental D-MPC regulator - The high-level stabilizing MPC with cost has been implemented with $N_{\alpha}=2$ (see Algorithm \[Atm:2\]), $\bar Q_{\rm\scriptscriptstyle H}=I$ and $\bar R_{\rm\scriptscriptstyle H}=\text{diag}(2,\, 20,\, 20)$, and prediction horizon $\bar N_{\rm\scriptscriptstyle H}=20$. The control gain matrix $\bar K_{\rm\scriptscriptstyle S, H}$ is selected by solving the corresponding infinite horizon LQ problem. The terminal penalty is computed according to and the solution is $$\bar P_{\rm\scriptscriptstyle H}=\begin{bmatrix} 5.89&-1.39&-0.0001&-0.0002\\ -1.39&7.2&-0.0002&0.0003\\ -0.0001&-0.0002&5&0\\ -0.0002&0.0003&0&5 \end{bmatrix}.$$ The terminal set has been chosen according to the algorithm described in [@hu2008model], i.e., $\bar{\mathcal{X}}_{\rm\scriptscriptstyle F}=\{x|(x-x_r)^T\bar P_{\rm\scriptscriptstyle H}(x-x_r)\leq=0.269$. - The low-level shrinking horizon MPC with cost has been designed with $Q=I$ and $R=\text{diag}(1,\, 1,\, 10)$. ### Simulation results The proposed dual-level control algorithms have been applied to the linear BT system by solving an output reference tracking problem in the basic time scale. The output set-point $y_r=(10,1,-2)$ is initially considered; while at time $t=400$ $s$, due to load variation, the reference value has been reset according to the new load profile, i.e., $y_r=(5,2,4)$. The dual-level control algorithms have been implemented from null initial conditions. In the following, the simulation results has been reported including the comparisons with a traditional decentralized PID controller and the multirate MPC described in [@zhang2018].\ ### Comparison with the traditional decentralized PID controller {#comparison-with-the-traditional-decentralized-pid-controller .unnumbered} The proposed algorithms have been firstly compared with decentralized continuous PID controller. The decnetralized PIDs, one for each input/output pair, have been designed with all the selected tuning parameters listed in Tab. \[tab:Tab\_com\]. 0.2cm The simulation results have been reported in Fig. \[fig:input\]-\[fig:state-output\], which show that, after an initial transient, inputs and outputs return to their nominal values, until the change of the reference occurs when the dual-level and decentralized PID control systems properly reacts to bring the input and output variables to their new steady-state values. Note that the closed-loop dynamics of the three approaches associated with input/output pair $(u_s,y_s)$ are comparable, while the corresponding control performances of the input/output pair $(u_f,y_f)$ are significantly different. To be specific, the pairs $(u_f,y_f)$ with the proposed D-MPC and *incremental* D-MPC algorithms react promptly to reference variations that the corresponding $y_f$ can be recovered in about $10$ $sec$ and $80$ $sec$ respectively; while the one with the PIDs experiences a longer transient period (that is almost $350$ $sec$). This reveals that, compared with the PIDs, the proposed D-MPC and *incremental* D-MPC show strong points in this respect. Also, the control system with the *incremental* D-MPC reacts slightly faster to reference variations than the one with the D-MPC especially for the control pair $(u_f(2),y_f(2))$. ### Comparison with the multirate MPC in [@zhang2018] {#comparison-with-the-multirate-mpc-in .unnumbered} The proposed approaches have also been compared with the multirate MPC described in [@zhang2018]. Note that, due to the usage of finite impulse response representation, the model used for the multirate MPC must be strictly stable. However, the considered system in this paper has a pole on the unitary circle, see . In order to implement the multirate MPC successfully, a feedback compensator $u_s=ky_s+v$ has been used firstly, where $v$ is defined as an auxiliary control variable, and the feedback gain $k$ is chosen as $k=-0.005$. For fair comparison, the design parameters $Q_s$ and $R_s$ have been selected coincident with the proposed MPC algorithms, i.e., $Q_s=\text{diag}\{1,2,\cdots,2,20,\cdots,20\}$, $R_s=2$. All the comparative simulation experiments have been implemented within Yalmip toolbox installed in MATLAB environment, see [@Lofberg2004], in a Laptop with Intel 8 Core i5-4200U 2.30 GHz running Windows 10 operating system. [The average values of the computational time for all the approaches are listed in Table \[tab:Tab\_com\]. The average computational times of the proposed algorithms are slightly greater than that of the multirate MPC. This result is acceptable as the state constraints are considered at the both levels and the terminal state constraints are included in  and .]{} 0.2cm The corresponding simulation results have been reported in Fig. \[fig:input\_q5\]-\[fig:state-output\_q5\]. These results show that, after an initial transient, inputs and outputs with the proposed MPC algorithms return to their nominal values, while the pair $(u_s,y_s)$ with the multirate MPC does not converge to their nominal values. Moreover, when the change of the reference occurs, the input and output variables steer to their new steady-state values, except for the output $y_s$ computed with the multirate MPC, whose value is far from the new steady-state one. According to the above analysis and as shown in Fig. \[fig:input\_q5\]-\[fig:state-output\_q5\], the control performances of the pairs $(u_f,y_f)$ with the three approaches are comparable. However, comapred to that with the proposed algorithms, the pair $(u_s,y_s)$ with the multirate MPC experiences a longer transient period and even does not converge to the reference value. This is possibly due to the strong interactions from $u_f$ to $y_s$ and to the lack of compensation term $\delta u_s$ in the fast timescale of the multirate MPC. Therefore, for fully comparison and analysis, we have also repeated the simulation with a larger penalty value associated with $y_s$ for multirate MPC, i.e., $Q_s(1,1)=100$. Similarly, the penalty value associated with $y_s$ of the proposed approaches are changed corrspondingly with $Q_{\rm\scriptscriptstyle H}(1,1)=\bar Q_{\rm\scriptscriptstyle H}(1,1)=100$. The repeated simulation results of the three approaches are presented in Fig. \[fig:input\_q500\]-\[fig:state-output\_q500\]. These results reveals that, compared with the previous case, the tracking performance of the pair $(u_s,y_s)$ with multirate MPC is better, but the residual amplitude of the static tracking error is still evident. For a numerical comparison, the cumulative square tracking errors $J_s=\sum_{i=1}^{N_{sim}}\|y_s(i)-y_{s,r}\|^2$, and $J_f=\sum_{i=1}^{N_{sim}}\|y_f(i)-y_{f,r}\|^2$, along the simulation steps from $1$ to $N_{sim}=800$ are collected for all the three approaches and listed in Tab. \[tab:Tab\_J\_com\], which show that, the proposed algorithms enjoy smaller cumulative square tracking errors than the multirate MPC. Interestingly, the tracking error corresponding to the fast control channel with the *incremental* D-MPC is better than that with the D-MPC at the price of a slightly larger tracking error in the slow channel. 0.2cm ![Control variables of the BT plant with $Q_s(1,1)=1$: red continuous lines (blue dot-dashed lines) represent the inputs computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones computed with the multirate MPC.[]{data-label="fig:input_q5"}](input_q5){width="0.9\columnwidth"} ![Outputs of the BT plant with $Q_s(1,1)=1$: red continuous lines (blue dot-dashed lines) represent the output variables computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones obtained with the multirate MPC.[]{data-label="fig:state-output_q5"}](output_q5){width="0.9\columnwidth"} ![Control variables of the BT plant with $Q_s(1,1)=100$: red continuous lines (blue dot-dashed lines) represent the inputs computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones computed with the multirate MPC.[]{data-label="fig:input_q500"}](input_q500){width="0.9\columnwidth"} ![Outputs of the BT plant with $Q_s(1,1)=100$: red continuous lines (blue dot-dashed lines) represent the output variables computed with the D-MPC (*incremental* D-MPC) approach, while black dashed lines are the ones obtained with the multirate MPC.[]{data-label="fig:state-output_q500"}](output_q500){width="0.9\columnwidth"} Conclusion ========== In this paper, two dual-level MPC control algorithms have been proposed for linear multi-timescale systems subject to input and output constraints. The proposed MPC algorithms rely on clearly time separation, so allow to deal with control problems in different channels. In view of their main properties, the proposed algorithms are, based on the solution based on MPC with dual-level structure, suitable not only to cope with control of singularly perturbed systems but also to impose different closed-loop dynamical performance for systems with nonseparable openloop dynamics. Their recursive feasibility and convergence properties have been proven under mild assumptions. The simulation results concerning the use of the proposed approaches for the control of a BT system including their comparisons with a traditional decentralized PID regulator and a multirate MPC are reported, which show that the proposed algorithms are both effective in generating significantly different closed-loop control behaviors to the output variables. In this respect, the proposed D-MPC has shown an advantageous feature with respect to the PIDs and the multirate MPC, while the *incremental* D-MPC enjoys better tracking performance than the D-MPC in the fast channel at the price of a slightly larger tracking error in the slow channel.\ Acknowledgment ============== The first author wish to thank Prof. Riccardo Scattolini and Prof. Marcello Farina from Politecnico di Milano, for their fruitful discussions. Proof of Proposition \[prop:obser-con\] {#sec:prop-con-proof} --------------------------------------- According to PBH detectability rank test, the pair $(A,\,C)$ is detectable if and only if $\text{rank}(\begin{bmatrix} \lambda I-A&C \end{bmatrix})=n$, $\forall\,\lambda \in \mathbb{C}$ and $|\lambda| \geq 1$. An equivalent form to this condition is that $v=0$ is the unique solution to the following linear equations $$\label{Eqn:con-cond} \left\{ \begin{array}{c} Av=\lambda v\\ Cv=0, \end{array}\right.$$ $\forall\,\lambda\in \mathbb{C}$ and $|\lambda|\geq 1$. From , $v=0$ is the unique solution to $\lambda^{i-1}Av=\lambda^i v,\, Cv=0,\,\forall\, i\in\mathbb{N}_+$, which is $A^iv=\lambda^iv,\, Cv=0,\,\forall\, i\in\mathbb{N}_+$. In view of this, recalling that $(A,\,C)$ is detectable, it holds that $v=0$ is the only solution to $$\left\{ \begin{array}{c} A^{\scriptscriptstyle[N]}v=\mu v\\ Cv=0, \end{array}\right.$$ where $\mu=\lambda^{N}$, which implies $(A^{\rm \scriptscriptstyle[N]},\, C)$ is observable for all the modes that their poles $|\lambda|\geq 1$. Hence, Proposition \[prop:obser-con\] holds. $\square$ Proof of Theorem \[theorem\] {#sec:theo-proof} ---------------------------- ### Recursive feasibility of the D-MPC (i.e., high-level problem  and low-level problem ) As the problem is assumed to be feasible at time $k=0$, with resorting to Mathematical Induction technique, one can prove the closed-loop recursive feasibility by verifying that if is feasible at any time $k$, then - the low-level problem is feasible at any fast time $h\in[kN,kN+N)$; - also the high-level problem is feasible at the subsequent slow time instant $k+1$. First, we show that condition *(i)* can be verified. To proceed, recalling that the high-level problem is feasible at time $k$, from initial condition $x(0)=x^{\scriptscriptstyle[N]}(0)$, it holds that $x(kN)=x^{\scriptscriptstyle[N]}(k)$. In view of this and of , it is easy to see that the null input sequence $\overrightarrow{{\delta {u}}}{(kN:(k+1)N-1)}= 0$ is a feasible solution to problem at time $h=kN$. In other words, the feasibility of the low-level problem is guaranteed at the fast time instant $h=kN$. Based on this, we assume problem is feasible at any time $h\in[kN,kN+N)$ and let $\big[\, \delta u(h|h)\ \cdots\ \delta u((k+1)N-1|h)\,\big]$ be its optimal solution. Only the first element $\delta u(h|h)$ is applied at the current time $h$ and the remaining control sequence $\big[\, \delta u(h+1|h)\ \cdots\ \delta u_f((k+1)N-1|h)\,\big]$ is a feasible choice at the subsequent time $h+1$ according to shrinking horizon principle. Hence, is feasible for any fast time $h\in[kN,kN+N)$.\ As for *(ii)*, assume that at any slow time instant $k$ the optimal control sequence of can be found, i.e., $\overrightarrow {u^{\scriptscriptstyle[N],o}}(k:k+N_{\rm\scriptscriptstyle H}-1|k)=\big(u^{\scriptscriptstyle[N],o}(k|k),\cdots,u^{\scriptscriptstyle[N],o}(k+N_{\rm\scriptscriptstyle H}-1|k)\big)$ such that $x^{\scriptscriptstyle[N]}(k+N_{\rm\scriptscriptstyle H}|k)\in \mathcal{X}_{\rm\scriptscriptstyle F}$. In view of the definition of $\mathcal{X}_{\rm\scriptscriptstyle F}$ and of the gain matrix $K_{\rm\scriptscriptstyle H}$, and thanks to constraint , the input sequence $\overrightarrow {u^{\scriptscriptstyle[N],s}}(k+1:k+N_{\rm\scriptscriptstyle H}|k+1)=\big(u^{\scriptscriptstyle[N],o}(k+1|k),\cdots,u^{\scriptscriptstyle[N],o}(k+N_{\rm\scriptscriptstyle H}-1|k),K_{\rm\scriptscriptstyle H}(x^{\scriptscriptstyle[N]}(k+N_{\rm\scriptscriptstyle H}|k)-x_r)+u_r\big)$ is a feasible choice at the next time instant $k+1$, such that the terminal constraint $x(k+N_{\rm\scriptscriptstyle H}+1|k+1)\in \mathcal{X}_{\rm\scriptscriptstyle F}$ is verified. Hence, the recursive feasibility of follows. ### Convergence of the high-level problem  Denote by $J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))$ the optimal cost associated with $\overrightarrow {u^{\scriptscriptstyle[N],o}}(k:k+N_{\rm\scriptscriptstyle H}-1|k)$ at time $k$ and by $J_{\rm\scriptscriptstyle H}^s(x^{\scriptscriptstyle[N]}(k+1|k))$ the suboptimal cost associated with $\overrightarrow {u^{\scriptscriptstyle[N],s}}(k+1:k+N_{\rm\scriptscriptstyle H}|k+1)$ at time $k+1$. It is possible to write $$\label{Eqn:cost_decre1} \begin{array}{lll} &J_{\rm\scriptscriptstyle H}^s(x^{\scriptscriptstyle[N]}(k+1|k))-J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))=\\&=-(\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}})+\\ &\ \ \ \|y^{\scriptscriptstyle[N]}(k+N|k)-y_r\|_{Q_{\rm\scriptscriptstyle H}}^2+ \|K_{\rm\scriptscriptstyle H}(x^{\scriptscriptstyle[N]}(k+N_{\rm\scriptscriptstyle H}|k)-x_r)\|_{R_{\rm\scriptscriptstyle H}}^2+\\ &\ \ \ \|F_{\rm\scriptscriptstyle H}(x^{\scriptscriptstyle[N]}(k+N|k)-x_r)\|^2_{P_{\rm\scriptscriptstyle H}}-\|x^{\scriptscriptstyle[N]}(k+N|k)-x_r\|^2_{P_{\rm\scriptscriptstyle H}}\\ &=-(\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}})\\ &\ \ \ +\|x^{\scriptscriptstyle[N]}(k+N|k)-x_r\|^2_{ F_{\rm\scriptscriptstyle H}^{\top} P_{\rm\scriptscriptstyle H} F_{\rm\scriptscriptstyle H}-P_{\rm\scriptscriptstyle H}+C^{\top}Q_{\rm\scriptscriptstyle H}C+K_{\rm\scriptscriptstyle H}^{\top}R_{\rm\scriptscriptstyle H} K_{\rm\scriptscriptstyle H}=0} \end{array}$$ In view of and from , one has $$\begin{array}{l} J_{\rm\scriptscriptstyle H}^s(x^{\scriptscriptstyle[N]}(k+1|k))-J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))=\\ -(\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}}). \end{array}$$ Recalling the fact that $J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k+1|k))\leq J_{\rm\scriptscriptstyle H}^s(x^{\scriptscriptstyle[N]}(k+1|k))$, then $$\label{Eqn:cost_monotonic1} \begin{array}{l} J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k+1|k))-J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))\leq\\ -(\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}}), \end{array}$$ which implies that $J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k+1|k))-J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))$ converges to zero. Moreover, from , one has $J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k))-J_{\rm\scriptscriptstyle H}^o(x^{\scriptscriptstyle[N]}(k+1|k))\geq\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}}$, then $\|y^{\scriptscriptstyle[N]}(k)-y_r\|^2_{Q_{\rm\scriptscriptstyle H}}+\|u^{\scriptscriptstyle[N],o}(k|k)-u_r\|^2_{R_{\rm\scriptscriptstyle H}}\rightarrow 0$. Recalling the definitions of $Q_{\rm\scriptscriptstyle H}$ and $R_{\rm\scriptscriptstyle H}$, one has $\lim_{k\to+\infty} y^{\scriptscriptstyle[N]}(k)=y_r$ and $\lim_{k\to+\infty} u^{\scriptscriptstyle[N]}(k)=u_r$. In view of Proposition \[prop:obser-con\], consequently $\lim_{k\to+\infty} x^{\scriptscriptstyle[N]}(k)=x_r$. ### Convergence of the low-level problem  Assume that the high-level system variables have reached their reference values, i.e., $u^{\scriptscriptstyle[N]}(k)\equiv u_r$ $x^{\scriptscriptstyle[N]}(k)\equiv x_r$, $y^{\scriptscriptstyle[N]}(k)\equiv y_r$. Define $\delta x(k)=x(kN)-x_r$ and $\delta y(k)=y(kN)-y_r$. Along the same line as described in [@picasso2016hierarchical], in view of dynamics at time instant $h=kN$, the low-level dynamics at the slow time scale is defined $$\label{Eqn:CL_delta_k} \left\{\begin{array}{l} \delta x(k+1)=A^N\delta x(k)+w(k)\\ [0.2cm]\delta y(k)=C\delta x(k), \end{array}\right. \qquad$$ where $w(k)=\sum_{j=0}^{N-1}A^{N-j-1}B\delta u(kN+j)$. Since $\delta x(k)=0,\, \forall k\geq 0$ (due to ), it holds that $w(k)=0$. In view of the cost function at the low level, the null sequence $\overrightarrow{\delta u}(h:(k+1)N-1)=0$ solves the problem , which implies that $\lim_{h\to+\infty} \delta u(h)=0$ and $\lim_{h\to+\infty} u(h)=u_r$. Finally, $\lim_{h\to+\infty} y(h)=y_r$ and$\lim_{h\to+\infty} x(h)=x_r$. $\square$ Proof of proposition \[prop1\] ------------------------------ According to PBH stabilizability rank test, the pair $(\bar A^{\scriptscriptstyle[N]},\,\bar B_s^{\scriptscriptstyle[N]})$ is stabilizable if and only if $\text{rank}(\begin{bmatrix} \lambda I-\bar A^{\scriptscriptstyle[N]}&\bar B_s^{\scriptscriptstyle[N]} \end{bmatrix})=n+p_s$, for $\lambda \in \mathbb{C}$ and $|\lambda| \geq 1$. An equivalent form to this condition is that $v=0$ is the unique solution to the following linear equations $$\label{Eqn:PBH_test} \left\{\begin{array}{l} (\bar A^{\scriptscriptstyle[N]})^{\top}v=\lambda v\\ [0.2cm] (\bar B_s^{\scriptscriptstyle[N]})^{\top}v=0, \end{array}\right.\qquad$$ where $\lambda\in \mathbb{C}$ and $|\lambda|\geq 1$.\ In view of , it is possible to write  in the form $$\label{Eqn:PBH_lambda1} \begin{bmatrix} I-\lambda I& 0\\\tilde C_s\tilde A^{\scriptscriptstyle[N]}&\tilde A^{\scriptscriptstyle[N]}-\lambda I\\ \tilde C_s\tilde B_s^{\scriptscriptstyle[N]}& \tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{\top}v=0$$ Since $\tilde A^{\scriptscriptstyle[N]}$ is stable by Assumption \[assum:tilde A-stable\], it is obvious to see that for $|\lambda|> 1$, $v=0$ is the unique solution to .\ For $\lambda=1$, $v=0$ is the unique solution to  if and only if $$\text{rank}\big(\begin{bmatrix} \tilde C_s\tilde A^{\scriptscriptstyle[N]}&\tilde A^{\scriptscriptstyle[N]}-I\\\tilde C_s\tilde B_s^{\scriptscriptstyle[N]}&\tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{\top}\big)=n+p_s.$$ As for $\lambda=-1$, $v=0$ is the unique solution to  if and only if $$\text{rank}\big(\begin{bmatrix} 2I&0\\ \tilde C_s\tilde A^{\scriptscriptstyle[N]}&\tilde A^{\scriptscriptstyle[N]}+I\\\tilde C_s\tilde B_s^{\scriptscriptstyle[N]}&\tilde B_s^{\scriptscriptstyle[N]} \end{bmatrix}^{\top}\big)=n+p_s.$$ $\square$ Proof of Theorem \[theorem-im\] ------------------------------- ### Recursive feasibility of the incremental D-MPC (i.e., high-level problem  and low-level problem ) As $N_{\alpha}$ is assumed to be reachable by Algorithm  \[Atm:2\] such that is feasible at time $k=0$, along the same line of Section \[sec:theo-proof\], the closed-loop system is recursively feasible as long as the following statement is verified: if the high-level problem is feasible at any time $k$, then - the low-level problem is feasible at any fast time $h\in[kN,kN+N)$; - also the high-level problem is feasible at the subsequent slow time instant $k+1$. The proof of condition *(i)* is similar to Section \[sec:theo-proof\]. As for *(ii)*, assume that at time instant $k$ the optimal control sequence is found, i.e., $\overrightarrow {\Delta u_s^{\scriptscriptstyle[N],o}}(k:k+\bar N_{\rm\scriptscriptstyle H}-1|k)=\big(\Delta u_s^{\scriptscriptstyle[N],o}(k|k),\cdots,\Delta u_s^{\scriptscriptstyle[N],o}(k+\bar N_{\rm\scriptscriptstyle H}-1|k)\big)$ such that $\bar x^{\scriptscriptstyle[N]}(k+\bar N_{\rm\scriptscriptstyle H}|k)\in \bar{\mathcal{X}}_{\rm\scriptscriptstyle F}$. Noting the fact that $\bar N_{\rm\scriptscriptstyle H}\geq N_{\alpha}$, one has $ \alpha(k+\bar N_{\rm\scriptscriptstyle H})=1\, \forall k\geq 0$. In view of this, the input sequence $\overrightarrow {\Delta u_s^{\scriptscriptstyle[N],s}}(k+1:k+\bar N_{\rm\scriptscriptstyle H}|k+1)=\big(\Delta u_s^{\scriptscriptstyle[N],o}(k+1|k),\cdots,\break\Delta u_s^{\scriptscriptstyle[N],o}(k+\bar N_{\rm\scriptscriptstyle H}-1|k),\bar K_{\rm\scriptscriptstyle H}(\bar x^{\scriptscriptstyle[N]}(k+\bar N_{\rm\scriptscriptstyle H}|k)-\bar C^{\top}y_{s,r})\big)$ is a feasible choice at the next time instant $k+1$ such that $\bar x(k+\bar N_{\rm\scriptscriptstyle H}+1|k+1)\in \bar{\mathcal{X}}_{\rm\scriptscriptstyle F}$ can also be verified. Hence, the recursive feasibility of  follows. ### Convergence of the incremental D-MPC In view of  and recalling the feasibility result of , along the same line of Section \[sec:theo-proof\], one can compute $$\label{Eqn:cost_monotonic1-im} \begin{array}{l} \bar J_{\rm\scriptscriptstyle H}^o(\bar x^{\scriptscriptstyle[N]}(k+1|k))-\bar J_{\rm\scriptscriptstyle H}^o(\bar x^{\scriptscriptstyle[N]}(k))\leq\\ -(\|\bar x_s^{\scriptscriptstyle[N]}(k)-\bar C^{\top} y_{s,r}\|^2_{Q_{s,\rm\scriptscriptstyle H}}+\|\Delta u_s^{\scriptscriptstyle[N],o}(k|k)\|^2_{R_{s,\rm\scriptscriptstyle H}}) \end{array}$$ where $\bar J_{\rm\scriptscriptstyle H}^o$ is the optimal cost. implies that $\bar J_{\rm\scriptscriptstyle H}^o(\bar x^{\scriptscriptstyle[N]}(k+1|k))-\bar J_{\rm\scriptscriptstyle H}^o(\bar x^{\scriptscriptstyle[N]}(k))$ converges to zero. Consequently, it holds that $\|\bar x_s^{\scriptscriptstyle[N]}(k)-\bar C^{\top} y_{s,r}\|^2_{\bar Q_{\rm\scriptscriptstyle H}}+\|\Delta u_s^{\scriptscriptstyle[N],o}(k|k)\|^2_{\bar R_{\rm\scriptscriptstyle H}}\rightarrow 0$ as well. Recalling the definitions of $\bar Q_{\rm\scriptscriptstyle H}$ and $\bar R_{\rm\scriptscriptstyle H}$, it holds that $\lim_{k\to+\infty} \bar x_s^{\scriptscriptstyle[N]}(k)=\bar C^{\top} y_{s,r}$ and $\lim_{k\to+\infty} \Delta u_s^{\scriptscriptstyle[N]}(k)=0$. Consequently, one has $\lim_{k\to+\infty} y^{\scriptscriptstyle[N]}(k)=y_r$, $\lim_{k\to+\infty} u_s^{\scriptscriptstyle[N]}(k)=const$. In view of Proposition \[prop:obser-con\], it promptly follows that, $\lim_{k\to+\infty} x^{\scriptscriptstyle[N]}(k)=x_r$, $\lim_{k\to+\infty} u_s^{\scriptscriptstyle[N]}(k)=u_{s,r}$. The arguments for the results $\lim_{h\to+\infty} \delta u(h)=0$, $\lim_{h\to+\infty} y(h)=y_r$, $\lim_{h\to+\infty} x(h)=x_r$, and $\lim_{h\to+\infty} u(h)=u_r$ are similar to Section \[sec:theo-proof\]. $\square$ [^1]: Xinglong Zhang,Wei Jiang, and Xin Xu are with the College of Intelligence Science, National University of Defense Technology, Changsha 410073, China. email: ([email protected], xuxin\[email protected]) [^2]: Xinglong Zhang was with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan 20133, Italy. [^3]: Wei Jiang was with the Research Institute for National Defense Engineering of Academy of Military Science, Luoyang, 471300, P.R. China e-mail: ([email protected]). [^4]: Zhizhong Li is with the State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University, Nanjing, Jiangsu 210007, China e-mail: ([email protected]). [^5]: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.
--- abstract: 'The optical properties of (001)-oriented NbP single crystals have been studied in a wide spectral range from 6 meV to 3 eV from room temperature down to 10 K. The itinerant carriers lead to a Drude-like contribution to the optical response; we can further identify two pronounced phonon modes and interband transitions starting already at rather low frequencies. By comparing our experimental findings to the calculated interband optical conductivity, we can assign the features observed in the measured conductivity to certain interband transitions. In particular, we find that transitions between the electronic bands spilt by spin-orbit coupling dominate the interband conductivity of NbP below 100 meV. At low temperatures, the momentum-relaxing scattering rate of the itinerant carriers in NbP is very small, leading to macroscopic characteristic length scales of the momentum relaxation of approximately 0.5 $\mu$m.' author: - David Neubauer - Alexander Yaresko - Weiwu Li - Anja Löhle - Ralph Hübner - | \ Micha B. Schilling - Chandra Shekhar - Claudia Felser - Martin Dressel - 'Artem V. Pronin' date: 'March 26, 2018' title: Optical conductivity of the Weyl semimetal NbP --- Introduction ============ NbP is a nonmagnetic non-centrosymmetric Weyl semimetal (WSM) with extremely large magnetoresistance and ultrahigh carrier mobility [@Shekhar2015; @Wang2016]. These extraordinary transport properties are believed to be caused by quasiparticles in chiral Weyl bands. According to band-structure calculations NbP possesses 24 Weyl nodes, i.e. twelve pairs of the nodes with opposite chiralities [@Huang2014; @Weng2015; @Sun2015; @Lee2015]. The nodes are “leftovers” of nodal rings, which are gapped by spin-orbit coupling (SOC) everywhere in the Brillouin zone (BZ), except of these special points [@Huang2014; @Weng2015; @Lee2015; @Ahn2015]. The nodes can be divided in two groups, commonly dubbed as W1 (8 nodes) and W2 (16 nodes). Most recent band-structure calculations agree well on the energy position of the W1 nodes: 56 - 57 meV below the Fermi level $E_{F}$ [@Klotz2016; @Wu2017; @Grassano2018]; the position of the W2 nodes is specified less accurately, ranging from 5 [@Klotz2016; @Wu2017] to 26 meV [@Grassano2018] above $E_F$. Furthermore, the W2 cones could be strongly tilted along a low-symmetric direction in BZ [@Wu2017], realizing thus a type-II WSM state [@Soluyanov2015]. The free-carrier dynamics of NbP has attracted considerable attention due to the possibility of hydrodynamic election behavior. Recently, a so-called axial-gravitational anomaly, relevant in the hydrodynamic regime [@Lucas2016], was reported in NbP [@Gooth2017NbP]. In turn, studying the hydrodynamic behavior of electrons is very interesting because the conduction of viscous electron fluids can be extremely high, exceeding the fundamental ballistic limit [@Gurzhi1963; @Gurzhi1968; @Guo2017]. Dirac materials with highly mobile electrons are prime candidates for realizing this super-ballistic conductivity. It was recently reported that electrons flow in a hydrodynamic fashion in clean samples of graphene [@Bandurin2016; @Crossno2016] and WP$_{2}$ [@Gooth2017WP2]; and the higher-than-ballistic conduction was found through graphene constrictions [@Kumar2017]. Similar to all WSM, the physical properties of NbP are determined by their low-energy electron dynamics, including both, free-carrier response and interband transitions [@Wehling2014]. Infrared optical methods enable direct access to this dynamics. For example, the interband optical response of a single isotropic three-dimensional Weyl band, being expressed in terms of the real part of the complex conductivity, should follow a linear frequency dependence with the pre-factor given by the band Fermi velocity $v_F$ [@Hosur2012; @Bacsi2013; @Ashby2014]: $$\sigma(\omega) = \frac{e^2} {12 h} \frac{\omega} {v_F} \quad . \label{simple}$$ Here, electron-hole symmetry is assumed, and the complex conductivity is $\hat{\sigma}(\omega) = \sigma(\omega) - {\rm i}\omega(\varepsilon(\omega)-1)/4\pi$, with $\varepsilon(\omega)$ the real part of the dielectric function. For $N_\text{W}$ identical Weyl bands, the right side of Eq. (\[simple\]) should be multiplied by $N_\text{W}$. Such a linear behavior of the optical conductivity ($\sigma\propto\omega$) has indeed been observed in a number of well-established and proposed three-dimensional Weyl/Dirac-semimetal systems [@Timusk2013; @Chen2015; @Sushkov2015; @Xu2016; @Neubauer2016; @Ueda2016; @Kimura2017; @Huett2018]. In the hydrodynamic regime, the rates for momentum-relaxing scattering $\Gamma_{\textrm{mr}} = 1/\tau_{\textrm{mr}}$ and momentum-conserving scattering $\Gamma_{\textrm{mc}} = 1/\tau_{\textrm{mc}}$ of the itinerant carriers differ appreciably: $\Gamma_{\textrm{mc}} \gg \Gamma_{\textrm{mr}}$ [@Gurzhi1963; @Gurzhi1968; @Scaffidi2017; @Lucas2017]. The momentum-relaxing scattering manifests itself in the optical-conductivity spectra [@Dressel2002], thus one can directly determine the corresponding scattering time $\tau_{\textrm{mr}}$ from $\sigma(\omega)$. For this paper, we have measured and analyzed both, interband and itinerant-carrier, conductivity of NbP. We show that the low-energy interband conductivity of NbP is dominated by transitions between the bands with parallel dispersions that are split by spin-orbit coupling. These excitations, as well as the Drude response of the itinerant carriers, completely mask the linear-in-frequency $\sigma(\omega)$ due to the three-dimensional chiral Weyl bands. At somewhat higher frequencies (1400 - 2000 cm$^{-1}$, 175 - 250 meV), $\sigma(\omega)$ becomes roughly linear. Our calculations demonstrate that this linearity stems from the fact that all electronic bands, which are involved in the transitions with relevant energies, are roughly linear. In addition, we find that at low temperatures the itinerant carriers in at least one of the conduction channels possess an extremely long momentum-relaxing scattering time; the characteristic length scale of momentum relaxation, $\ell_{\textrm{mr}} = v_F\tau_{\textrm{mr}}$, is then basically macroscopic, supporting that the hydrodynamic regime can be realized in NbP. Samples Preparation, Experimental and Computational Details =========================================================== Single crystals of NbP were synthesized according to the description reported in Refs. [@Martin1990; @Shekhar2015]: a polycrystalline NbP powder was synthesized in a direct reaction of pure niobium and red phosphorus; the single NbP crystals were grown from the powder via vapor-transport reaction with iodine. ![Temperature-dependent (001)-plane dc resistivity of the NbP sample used in the optical measurements.[]{data-label="fig:res"}](rho.eps){width="0.9\columnwidth"} The electrical resistivity, $\rho_{dc} (T)$, was measured as a function of temperature within the (001)-plane in four-contact geometry. The experiments were performed on a small piece, cut from the specimen used for the optical investigations. The results of the dc measurements are plotted in Fig. \[fig:res\]. A clear metallic behavior with linear-in-temperature resistivity is observed down to approximately 100 K. Below, $\rho_{dc} (T)$ levels off and approaches the residual resistivity $\rho_0$ = $0.55~\mu\Omega$cm; the resistivity ratio is $\rho$($300$ K)/$\rho_0 = 40$. These values are well comparable with the ones reported in literature [@Shekhar2015; @Wang2016; @Zhang2015; @Wang2015c]. Note, NbP exhibits an extremely high mobility independent on the residual resistivity ratio [@Wang2015c]. The normal-incidence optical reflectivity $R(\nu,T)$ was measured on the (001)-surfaces of a large ($2~{\rm mm} \times 2~{\rm mm}$ in lateral dimensions) single crystal from room temperature down to $T=10$ K covering a wide frequency range from $\nu = \omega/(2\pi c)= 50$ to 12000 cm$^{-1}$. The temperature-dependent experiments were supplemented by room-temperature reflectivity measurements up to 25000 cm$^{-1}$. In the far-infrared spectral range below 700 cm$^{-1}$, a Bruker IFS 113v Fourier-transform spectrometer was employed with *in situ* gold coating of the sample surface for reference measurements. At higher frequencies, we used a Bruker Hyperion infrared microscope attached to a Bruker Vertex 80v spectrometer. Here, freshly evaporated gold mirrors (below 12000 cm$^{-1}$) and protected silver (above 12000 cm$^{-1}$) served as reference. For the Kramers-Kronig analysis [@Dressel2002] we involved the x-ray atomic scattering functions for high-frequency extrapolations [@Tanner2015]. From recent optical investigations of materials with highly mobile carriers, it is known [@Schilling2017Yb; @Schilling2017Zr] that the commonly applied Hagen-Rubens extrapolation to zero frequency is not adequate: the very narrow zero-frequency component present in the spectra corresponds to a scattering rate comparable to (or even below) our lowest measurement frequency, $\nu_{\rm min} \approx 50$ cm$^{-1}$. Thus, we first fitted the spectra with a set of Lorentzians (similar fitting procedures can be utilized as a substitute of the Kramers-Kronig analysis [@Kuzmenko2005; @Chanda2014]) and then we used the results of these fits between $\nu = 0$ and $\nu_{\rm min}$ as zero-frequency extrapolations for subsequent Kramers-Kronig transformations. We note that our optical measurements probe the bulk material properties, as the penetration depth exceeds 20 nm for any measurement frequency. We performed band structure calculations within the local density approximation (LDA) based on the crystal structure of NbP determined by experiments [@XGEH+96]; we employed the linear muffin-tin orbital method [@And75] as implemented in the relativistic PY LMTO computer code. Some details of the implementation can be found in Ref. . The Perdew-Wang parametrization [@PW92] was used for the exchange-correlation potential. SOC was added to the LMTO Hamiltonian in the variational step. BZ integrations were done using the improved tetrahedron method [@BJA94]. Dipole matrix elements for interband optical transitions were calculated on a $96 \times 96 \times 96$ $k$-mesh using LMTO wave functions. As was shown in Ref., it is necessary to use sufficiently dense meshes in order to resolve transitions between the SOC-split bands. The real part of the optical conductivity was calculated by the tetrahedron method. Results and Analysis ==================== Fig. \[ref\_sig\_eps\] displays the overall reflectivity $R(\nu)$, the real part of the conductivity $\sigma(\nu)$, and the dielectric constant $\varepsilon(\nu)$ of NbP for different temperatures. For frequencies higher than 5000 cm$^{-1}$, the optical properties are basically independent on temperature. In the spectra we can identify the signatures of: (i) phonons, (ii) itinerant-carrier (intraband) absorption, and (iii) interband transitions. Below we discus these spectral features separately. ![(a) Optical reflectivity, real parts of the (b) optical conductivity and (c) dielectric permittivity of NbP at selected temperatures between $T=10$ and 300 K; note the logarithmic frequency scale. The inset (d) shows a simple fit of the low-energy $\sigma(\nu)$ at $T=10$ K by the sum of two Drude terms (narrow and broad) and two Lorentzians centered at 250 and 500 cm$^{-1}$, which mimic the interband transitions. The inset (e) displays the phonon modes in $\sigma(\nu)$ on an enlarged frequency scale. The dashed red line corresponds to a fit of the features at $T=10$ K by two narrow Lorentzians centered at 336 and 370 cm$^{-1}$, respectively.[]{data-label="ref_sig_eps"}](R_sig_eps.eps){width="0.8\columnwidth"} in TaAs, for example, such a Fano resonance reportedly signals a strong coupling between phonons and electronic transitions \[50\]). Phonons ------- In the far-infrared range, two sharp phonon peaks can be seen in Fig. \[ref\_sig\_eps\] at 336 and 370 cm$^{-1}$. Due to their symmetric shape, they can be nicely fitted with Lorentzians, as demonstrated in panel (e). This is in contrast to the Fano-shape phonon resonances observed in TaAs, where the asymmetric line shape reportedly signals a strong coupling between phonons and electronic transitions [@FanoTaAs]. In NbP, four infrared-active phonons are expected [@Chang2016]; however, there is no full consensus on the calculated frequencies [@Chang2016; @Liu2016]. The phonon positions observed in our spectra agree very well with the calculations from Ref. [@Liu2016] as well as with the Raman data presented there (in non-centrosymmetric structures, same phonon modes can be both, infrared and Raman, active). Thus, following Ref. [@Liu2016], we assign the observed features at 336 and 370 cm$^{-1}$ to those lattice vibrations that mainly involve the light P atoms. The other two infrared-active phonon modes are apparently too weak to be resolved on the electronic background. Itinerant charge carriers ------------------------- At the lowest frequencies, NbP exhibits an optical response typical for metals, i.e. the itinerant carriers dominate: the reflectivity approaches unity, $\varepsilon(\omega)$ is negative and diverges as $\omega \rightarrow 0$, $\sigma(\omega)$ exhibits a narrow zero-frequency peak, as seen in Fig. \[ref\_sig\_eps\] (a-c). Panel (d) clearly shows a shoulder on this peak that can be fitted by a Lorentzian; it will be discussed in the next section. For fitting the optical spectra at our lowest frequencies, we use the measured dc-conductivity values as the zero-frequency limit of the Drude term [@Dressel2002], and we allow both, the plasma frequency and the scattering rate, to vary freely. The best fit to the data is then obtained with a momentum-relaxing scattering rate, $\gamma_{\textrm{mr}} = 1/(2\pi c \tau_{\textrm{mr}})$, as low as 4.5 cm$^{-1}$ at $T=10$ K. Thus, $\tau_{\textrm{mr}} (10 \textrm{K}) = 1.2$ ps and the momentum-relaxation length, $\ell_{\textrm{mr}} = v_\textrm{F}\tau_\textrm{mr}$, becomes as long as 0.2 to 0.6 $\mu$m. Here we utilized the lower, $1.5\times 10^{5}$ m/s, and, respectively, the upper, $4.8 \times 10^{5}$ m/s, boundaries for the (001)-plane Fermi velocity $v_\textrm{F}$ obtained from experiment [@Shekhar2015; @Wang2016] and theory [@Lee2015; @Ahn2015]. At elevated temperatures, $\gamma_{\textrm{mr}}$ rises, reaching 35 cm$^{-1}$ at $T=300$ K. This corresponds to $\ell_{\textrm{mr}}$ of 20 to 70 nm. Since the typical sample size for ballistic transport measurements is around 0.1 to 1 $\mu$m [@Kumar2017; @Datta1995], one can actually realize ballistic conduction in NbP at low temperatures. For the hydrodynamic electron behavior discussed in the introduction, the energy-relaxing momentum-conserving electron-electron scattering must happen on time scales shorter than $\tau_{\rm{mr}}$ [@Gurzhi1963; @Gurzhi1968; @Scaffidi2017; @Lucas2017]. Because the found $\tau_{\textrm{mr}}$ is rather large, the hydrodynamic behavior and, eventually, the super-ballistic electron flows may be possible in NbP. Less mobile carriers from parabolic bands [@Shekhar2015; @Sun2015; @Ahn2015; @Klotz2016; @Wu2017], however, can obscure the observation of this behavior. Thus, a proper adjustment of the Fermi level is crucial, as can be achieved, e.g., by Gd doping [@Niemann2017]. Fingerprints of these low-mobility carriers can be identified in the optical conductivity of undoped NbP as another, very broad and relatively weak, Drude band. Such two-channel optical conductivity has been recently observed, for instance, in YbPtBi [@Schilling2017Yb]. In the case of NbP, this broad Drude term is present in the far-infrared spectral range as a basically frequency-independent pedestal in $\sigma(\omega)$, clearly seen in Fig. \[ref\_sig\_eps\](d). ![\[fig:sigma\]Interband optical conductivity of NbP calculated with (red line) and without (blue line) SOC and the total experimental NbP conductivity at 10 K (black line). Intraband (Drude) contributions to the conductivity are not included in the computations. Inset shows same sets of data on a broader photon energy scale (0 – 3 eV). All spectra are for the (001)-plane response.](Fig_sigma_le_inset){width="\columnwidth"} Interband transitions: optical experiments and calculations based on the electronic band structure -------------------------------------------------------------------------------------------------- Calculations of the interband optical conductivity based on calculated electronic band structure are very useful, but seem to be rather challenging in (topological) semimetals. A survey of the available literature reveals only a qualitative match between the calculated optical conductivity and experimental results [@Grassano2018; @Kimura2017; @Frenzel2017; @Chaudhuri2017]. In the most interesting low-energy part of the spectrum (less than a few hundred meV), a reasonable agreement is particularly hard to achieve [@Kimura2017; @Chaudhuri2017]. In the case of NbP, we reach a fairly good qualitative match between our calculations of the optical conductivity and experimental spectra even at low energies; this allows us to identify the origin of various spectral features observed in $\sigma(\omega)$. In Fig. \[fig:sigma\] we compare the experimental low-temperature optical conductivity to the interband (001)-plane conductivity calculated with and without SOC. Note, that the itinerant-carriers contributions have not been subtracted from the experimental conductivity. The calculated spectra qualitatively agree with the experiment: the inset in Fig. \[fig:sigma\] illustrates that the peaks and deeps in the calculated $\sigma(\omega)$ reasonably coincide with those in the experimental data. The calculated spectra contain more fine structures than the experimental $\sigma(\omega)$, as no broadening was applied to the computed spectra in order to simulate finite life-time effects. The experimental $\sigma(\omega)$ in general falls above the calculated interband conductivity. Partly, this can be due to the broad Drude contribution present in the experimental spectra. Above an energy of 0.2 eV, the effect of SOC on the theoretical $\sigma(\omega)$ is negligible; but for smaller energies the spectra computed with and without taking SOC into account differ significantly. The interband contribution to the conductivity calculated without SOC increases smoothly when raising the photon energy to 0.1 eV. When SOC is included, however, two sharp peaks appear around 30 and 65 meV (corresponding to $\sim 250$ and 500 cm$^{-1}$). The latter matches very well the shoulder we observed on the narrow Drude term, as shown in Fig. \[ref\_sig\_eps\](d). The former feature can be directly associated with the bump observed at all temperatures at around 500 cm$^{-1}$ in the measured spectra plotted in Fig. \[ref\_sig\_eps\](b). The fact that these two peaks appear only in those calculations including SOC indicates that they must be related to the transitions between the SOC-split bands. Our conclusion gets support when decomposing the calculated $\sigma(\omega)$ into contributions coming from transitions between different pairs of bands crossing $E_{\rm F}$. When SOC is neglected, two doubly degenerate bands with predominant Nb $d$ character cross on a $k_{x/y}=0$ mirror plane and form one electron and one hole Fermi surface with crescent-shaped cross sections by this plane [@Lee2015]. This degeneracy is lifted when SOC is accounted for; thus, four non-degenerate bands, numbered 19 to 22, now cross $E_{\rm F}$ as shown in Figs. \[fig:transitions\] and \[fig:fs\] (at every given **k** point, the bands are numbered with increasing energy). The band structure of NbP, calculated along selected lines in the BZ \[cf. Fig.\[fig:fs\](b)\], and the allowed transitions between different bands are shown in Figs. \[fig:transitions\](a,b). Here, the thickness of the vertical lines connecting occupied initial and unoccupied final states is proportional to the probability of the interband transition at a given $\mathbf{k}$ point. In panel (c) of Fig. \[fig:transitions\], we plot the various contributions to the total interband conductivity from the individual interband transitions $19 \to 20$, $21 \to 22$, $19 \to 22$, and $20 \to 21$. The contributions of the two remaining transitions, $19 \to 21$, $20 \to 22$, are much smaller. Thus, the total low-energy conductivity $\sigma(\omega)$ is basically the sum of the four contributions shown in Fig. \[fig:transitions\](c). The pronounced narrow peak at 30 meV is solely formed by transitions between the bands 21 and 22, while the 65 meV mode is due to the $19 \to 20$ transitions; i.e. both features stem from transitions between SOC-split bands. These transitions are only allowed in a small volume of the $\mathbf{k}$-space, where one of the two SOC-split bands is occupied, while the other one is empty (see the red and blue lines in Fig. \[fig:transitions\](a)), i.e., between the nested crescents in Fig. \[fig:fs\](a). Within this volume, the SOC-split bands are almost parallel to each other. This ensures the appearance of strong and narrow peaks in the joint density of states for the corresponding interband transitions. The peak positions are determined by the average band splitting, which, in turn, is of the order of the SOC strength of the Nb $d$ states, $\xi_{d} \approx 85$ meV. Finally, since the dipole matrix elements for these transitions are rather large, the two peaks dominate the low-energy interband conductivity. ![image](Fig_tr_wp){width="\textwidth"} Besides these two peaks, there are also contributions to the optical conductivity with a smooth $\omega$ dependence. These contributions originate in transitions between the touching bands 20 and 21 \[green lines in Fig. \[fig:transitions\](b,c)\] and between the bands 19 and 22, which are separated by a finite gap everywhere in the BZ \[magenta lines\]. Accordingly, $\sigma_{20\to21}(\omega)$ starts at zero energy, while $\sigma_{19\to22}(\omega)$ at 0.1 eV. Both conductivity contributions, $\sigma_{20\to21}$ and $\sigma_{19\to22}$, increase, when the photon energy rises from 0 to 0.4 eV. Both contributions, $\sigma_{19\to22}$ and $\sigma_{19\to22}$, as well as the total calculated $\sigma(\omega)$ exhibit sharp kinks (Van Hove singularities [@Yu2010]) at 0.4 eV, which are related to the transitions between flat parallel bands near the N point, see Fig. \[fig:transitions\](b). The experimental $\sigma(\omega)$ demonstrates such a kink at somewhat lower energy, 0.27 eV (Fig. \[fig:sigma\]); still, we find this match reasonable. In the vicinity of a Weyl point, the optical conductivity is expected to be proportional to frequency, Eq. (\[simple\]). The two sets of Weyl points in NbP, W1 and W2, are formed by touching points of the bands 20 and 21. In agreement with previous results [@Lee2015; @Klotz2016; @Wu2017; @Grassano2018], our LMTO calculations yield the W1 points approximately 50 meV below $E_F$. Consequently, their contribution to the conductivity cannot start at zero frequency. On the other hand, the energy of the W2 points in the present calculations is very close to $E_F$ and, thus, transitions near W2 may provide a linearly vanishing $\sigma(\omega)$ as $\omega \rightarrow 0$. To verify this behavior, we calculate the contribution to $\sigma_{20\to 21}(\omega)$ from a $\mathbf{k}$ volume with a radius of $\sim 0.05 \frac{2\pi}{a}$ ($a$ is the in-plane lattice constant) around the averaged position of a pair of W2 points. The contribution indeed shows a linear $\omega$ dependence as $\omega \rightarrow 0$, see Fig.\[fig:transitions\](d). The smooth kink at $\sim 15$ meV, marked with an arrow, corresponds to the merging point of the chiral Weyl bands [@Yu2010; @Tabert2016]. We should note that in experiments, this linear interband optical conductivity at low $\omega$ is completely masked by the itinerant carriers and by the strong peaks due to the transitions between the SOC-split bands, as discussed above. Although no linear-in-frequency $\sigma(\omega)$ due to the transitions within the chiral Weyl bands can be seen in NbP at low frequencies, both, experimental and computed, $\sigma(\omega)$ demonstrate a sort of linear increase with $\omega$ at higher frequencies: 180 to 250 meV for the experimental and up to 360 meV for the calculated optical conductivity, see Fig. \[fig:sigma\]. As apparent from our calculations, the linearity just reflects the fact that all the electronic bands, involved in the transitions with corresponding energies, are roughly linear (but not parallel to each other), see Fig. \[fig:transitions\](b). ![\[fig:fs\](a) Fermi surface cross sections by $k_y=0$ plane. Small black circles illustrate integration volumes around Weyl points. (b) BZ of NbP. Weyl points from the W1 set are situated near the S point, while the points from the W2 set are close to the N–M line.](Fig_fs){width="0.9\columnwidth"} Based on the comparison between the calculated and the experimental conductivity, we can assign the observed spectral features to different absorption mechanisms. Fig. \[sigma010K\] schematically summarizes these assignments. Before we conclude, we would like to emphasize the importance of the transitions between the SOC-split bands. So far, the strong influence of these transitions on the low-energy conductivity of WSMs has not been fully appreciated. Using a modified Dirac Hamiltonian [@Burkov2011], Tabert and Carbotte [@Tabert2016] calculated the optical conductivity for a four-band model, relevant for many WSMs. In this model, the band structure consists of four isotropic non-degenerate three-dimensional bands, two of which cross and the other two are gapped. The band structure is mirror symmetric in energy with respect to the Weyl nodes and the Weyl cones are anisotropic in $\mathbf{k}$-space at low energies. This model definitely grasps the main features of the band structure of many WSMs, including those from the TaAs family; but – apart from neglecting possible band anisotropy – it does not take into account the transitions between the SOC-split bands: these transitions are considered forbidden in the model, while in the real WSMs they might play an important role, as we have shown for NbP. ![Low-frequency portion of the real part of NbP optical conductivity at 10 K and assignment of the observed features to different absorption mechanisms, as discussed in the course of this paper.[]{data-label="sigma010K"}](sigma_010K.eps){width="0.9\columnwidth"} Finally, we would like to note that TaAs and TaP demonstrate sharp absorption peaks at frequencies, which compare well to those of the transitions between the SOC-split bands in NbP, cf. Fig. 3 from Ref. [@Kimura2017]. Previously, these features have been assigned to transitions between the merging (saddle) points of the Weyl bands, even though, e.g., in TaP this assignment is at odds with the absence of chiral carriers [@Arnold2016]. We suggest to reconsider this assignment. Conclusions =========== We measure and analyze the interband and itinerant-carrier optical conductivity of NbP. From the electronic band structure, we calculate the interband optical conductivity and decompose it into contributions from the transitions between different bands. By comparing these contributions to the spectral features in the experimental conductivity, we assign the observed features to certain interband transitions. We argue that the low-energy (below 100 meV) interband conductivity is dominated by transitions between almost parallel bands, split by spin orbit coupling. Hence, these transitions manifest themselves as relatively sharp peaks centered at 30 and 65 meV. These peaks and the low-energy itinerant-carrier conductivity (Drude-like) conceal the linear-in-frequency contribution to $\sigma(\omega)$ from the transitions within the chiral Weyl bands. Our calculations demonstrate that the nearly linear in $\omega$ conductivity at around 200 – 300 meV is naturally explained by the fact that all electronic bands, involved in the transitions with such frequencies, possess approximately linear dispersion relations. We also identify two optical phonons and assign them to the vibrations, which mostly involve P atoms. Finally, we find that the carriers in one of the conduction channels possess extremely low momentum-relaxing scattering rates at low temperatures, leading to macroscopic characteristic length scales of momentum relaxation of about 0.5 $\mu$m at $T=10$ K. Acknowledgements ================ We thank G. Untereiner and S. Prill-Diemer for experimental support, and U. S. Pracht and H. B. Zhang for fruitful discussions. 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--- abstract: 'We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians.' author: - | Kentaro Nagao\ RIMS, Kyoto University\ Kyoto 606-8502, Japan title: 'Donaldson-Thomas theory and cluster algebras' --- Introduction {#introduction .unnumbered} ============ Donaldson-Thomas invariants ([@thomas-dt; @mnop]) are defined as the topological Euler characteristics (more precisely, the weighted Euler characteristics weighted by Behrend function [@behrend-dt]) of the moduli spaces of sheaves on a Calabi-Yau $3$-fold (more generally, the moduli spaces of objects in a $3$-Calabi-Yau category [@szendroi-ncdt; @joyce-4; @ks; @joyce-song]). Dominic Joyce introduced the motivic Hall algebra for an Abelian category in his study of generalized Donaldson-Thomas invariants ([@joyce-2]). One of the important results is that for a $3$-Calabi-Yau category there exists a Poisson algebra homomorphism, so called the [*integration map*]{}, from the motivic Hall algebra to a power series ring ([@joyce-2; @joyce-song; @bridgeland-hall]) The integration map is given by taking the (weighted) Euler characteristic of an element in the motivic Hall algebra. Due to the integration map, we get the following powerful method in Donaldson-Thomas theory for $3$-Calabi-Yau categories, which originates with Reineke’s computation of the Betti numbers of the spaces of stable quiver representations ([@reineke-HN]): > Starting from a simple categorical statement, provide an identity in the motivic Hall algebra. Pushing it out by the integration map, we get a power series identity for the generating functions of Donaldson-Thomas invariants. The aim of this paper is to provide - Theorem \[thm\_trans\] (Theorem \[thm\_01\]) : a transformation formula of the noncommutative Donaldson-Thomas invariants, and - Theorem \[thm\_CC\] (Theorem \[thm\_02\]) and the results in §\[\] : its application to the theory of cluster algebras using this method. Transformation formula of ncDT invariants {#transformation-formula-of-ncdt-invariants .unnumbered} ----------------------------------------- Let $Q$ be a quiver and $W$ be a potential. In this paper, we always assume that - the quiver has the vertex set $I=\{1,\ldots,n\}$, - the quiver has no loops and oriented $2$-cycles, and - the potential is [*finite*]{}, i.e. a [finite]{} linear combination of oriented cycles. Let $J=J_{Q,W}$ be the (non-complete) Jacobi algebra. We have a $3$-Calabi-Yau triangulated category (the derived category of Ginzburg’s dg algebra) with a t-structure whose core ${{\mathcal{A}}}$ is the module category of the Jacobi algebra. It was proposed by B. Szendroi ([@szendroi-ncdt]) to study Donaldson-Thomas theory for the Abelian category ${{\mathcal{A}}}\simeq {{\mathrm{mod}}}J$ ([*non-commutative Donaldson-Thomas theory*]{}). For a vertex $i\in I$, let $P_i$ denote the projective indecomposable $J$-module corresponding to the vertex $i$. For a dimension vector $\mathbf{v}\in ({{\mathbf{Z}}}_{\geq 0})^{I}$, let ${{\mathrm{Hilb}}}_J(i;\mathbf{v})$ be the moduli scheme which parametrizes elements in $V\in {\mathrm{mod}\hspace{1pt}J}$ equipped with a surjection from $P_i$ such that $[V]=\mathbf{v}$: $${{\mathrm{Hilb}}}_J(i;\mathbf{v}):=\{P_i\twoheadrightarrow V\mid V\in {\mathcal{A}}, [V]=\mathrm{v} \}.$$ The (Euler characteristic version of the) non-commutative Donaldson-Thomas invariant is defined by $${{\mathrm{DT}}}_{J,+}(i;\mathbf{v})=e_+({{{\mathrm{Hilb}}}_J(i;\mathbf{v})}):= e({{{\mathrm{Hilb}}}_J(i;\mathbf{v})})$$ where $e(\bullet)$ denote the topological Euler characteristic. In the context of this paper, we will also deal with the invariant $${{\mathrm{DT}}}_{J,-}(i;\mathbf{v})=e_-({{{\mathrm{Hilb}}}_J(i;\mathbf{v})})$$ where $e_-(\bullet)$ denote the weighted Euler characteristic weighted by the Behrend function (Definition \[defn\_ncdt\]). For a vertex $k$, we assume that the mutation $\mu_k(Q,W)$ is well-defined. Due to the result by Keller and Yang, $(Q,W)$ and $\mu_k(Q,W)$ provide the same derived category with different t-structures ([@dong-keller; @keller-completion]). Kontsevich and Soibelman ([@ks]) observed that the cluster transformation appears in the transformation formula of non-commutative Donaldson-Thomas invariants under a mutation. In this paper, generalizing their observation, we provide a transformation formula of the non-commutative Donaldson-Thomas invariants under a composition of mutations. We put $$\begin{array}{c} T_{Q,\pm}:=\mathbb{C}\left[(y_{1,\pm})^{\pm 1},\ldots ,(y_{n,\pm})^{\pm 1}\right],\quad T^\vee_{Q,\pm}:=\mathbb{C}\left[(x_{1,\pm})^{\pm 1},\ldots ,(x_{n,\pm})^{\pm 1}\right],\vspace{2mm}\\ \mathbb{T}_{Q,\pm}:=T^\vee_{Q,\pm}\otimes_{\mathbb{C}}T_{Q,\pm} \end{array}$$ They are called the semiclassical limits of [*quantum torus*]{}, [*quantum dual torus*]{} and [*quantum double torus*]{}[^1] respectively. They are taken as the group algebra of the lattices $M_Q$, $L_Q$ and $M_Q\oplus L_Q$ which are related to the Grothendieck group of the derived category (§\[subsec\_Gro\]). Since we have derived equivalences between $(Q,W)$ and $\mu_\mathbf{k}(Q,W)$, we have isomorphisms of the corresponding tori. We identify them by these isomorphisms. We take a certain completion $\widehat{\mathbb{T}}_{Q,\pm}$ of $\mathbb{T}_{Q,\pm}$ (§\[subsub\_auto\]). We define the generating function of the Donaldson-Thomas invariants by $${{\mathcal{Z}}}_{J,\pm}^{i}:= \sum_{\mathbf{v}}{{\mathrm{DT}}}_{J,\pm}(i;\mathbf{v})\cdot y^\mathbf{v}_\pm$$ where $y^\mathbf{v}_\pm:=\prod (y_{i,\pm})^{v_i}$. Using the generating functions, we define algebra automorphisms ${{\mathcal{DT}}}_{J,\pm}$ of $\widehat{\mathbb{T}}_{Q,\pm}$ by $${{\mathcal{DT}}}_{J,\pm}(x_{i,\pm}):=x_{i,\pm}\cdot {{\mathcal{Z}}}_{J,\pm}^{i},\quad {{\mathcal{DT}}}_{J,\pm}(y_{i,\pm}):=y_{i,\pm}\cdot \prod_j({{\mathcal{Z}}}_{J,\pm}^{j})^{\bar{Q}(j,i)}$$ where $$\bar{Q}(j,i):=Q(i,j)-Q(j,i),\quad Q(i,j)=\sharp \{\text{arrows from $i$ to $j$ in $Q$}\}.$$ For a sequence of vertices $\mathbf{k}=(k_1,\ldots,k_l)\in I^l$, let $\mu_\mathbf{k}(Q,W)$ denote the new QP $\mu_{k_l}(\cdots \mu_{k_1}(Q,W)\cdots)$ and $J_\mathbf{k}$ denote the Jacobi algebra associated to $\mu_\mathbf{k}(Q,W)$. Then we have two isomorphisms ${{\mathcal{DT}}}_{J,\pm}$ and ${{\mathcal{DT}}}_{J_{\mathbf{k}},\pm}$ of the torus [^2]. Our transformation formula of DT invariants is given as the relation of these isomorphisms. In §\[subsec\_grass\], we construct a $J$-module $R_{{\mathbf{k}},i}$ and define the [*quiver Grassmannian*]{} which parametrizes quotient modules of $R_{{\mathbf{k}},i}$ : $$\mathrm{Grass}(\mathbf{k};i,\mathbf{v}):= \{R_{{\mathbf{k}},i}\twoheadrightarrow V\mid V\in {\mathcal{A}},\ [V]=\mathbf{v}\}.$$ The formula is described in terms of (weighted) Euler characteristics of the quiver Grassmannians. \[thm\_01\] Assume that the $(Q,W)$ is successively f-mutatable with respect to the sequence ${\mathbf{k}}$ (see §\[subsub\_pmutation\] for the details of the assumption). Then we have the following “commutative diagram” [^3]: $$\xymatrix{ \widehat{\mathbb{T}}_{Q_{\mathbf{k}},\pm} \ar[rr]^{{{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\pm}} \ar[d]_{{{\mathcal{DT}}}_{\hspace{-1mm}J_{\mathbf{k}},\pm}} & & \widehat{\mathbb{T}}_{Q,\pm} \ar[d]^{{{\mathcal{DT}}}_{\hspace{-1mm}J,\pm}}\\ \widehat{\mathbb{T}}_{Q_{\mathbf{k}},\pm} \ar[rr]_{{{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\pm}} & & \widehat{\mathbb{T}}_{Q,\pm}. }$$ The morphism ${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\pm}$ is given by $$\begin{aligned} {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\pm}(x_{{\mathbf{k}},i,\pm}) &=x_{{\mathbf{k}},i,\pm}\cdot \Biggl(\sum_\mathbf{v}e_\pm\Bigr(\mathrm{Grass}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}_\pm^{-\mathbf{v}}\Biggr),\\ {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\pm}(y_{{\mathbf{k}},i,\pm}) &=y_{{\mathbf{k}},i,\pm}\cdot \prod_j\Biggl(\sum_\mathbf{v}e_\pm\Bigr(\mathrm{Grass}(\mathbf{k};j,\mathbf{v})\Bigr)\cdot\mathbf{y}_\pm^{-\mathbf{v}}\Biggr)^{\bar{Q}(j,i)}.\end{aligned}$$ where $x_{{\mathbf{k}},i,\pm}$ and $y_{{\mathbf{k}},i,\pm}$ are generators of ${\mathbb{T}}_{Q_{\mathbf{k}},\pm}$[^4]. The morphism ${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\pm}$ is given by $$\label{eq_Sigma} {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\pm} :=\Sigma\circ {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\pm}\circ \Sigma$$ where $\Sigma$ is the involution of the tori given by $$\Sigma(x_{i,\pm})=(x_{i,\pm})^{-1},\quad \Sigma(y_{i,\pm})=(y_{i,\pm})^{-1}.$$ If we take a sequence ${\mathbf{k}}=(k)$ of length $1$, then we have $$R_{(k),i}=\begin{cases} 0 & i\neq k,\\ s_k & i=k. \end{cases}$$ Hence we have $$\label{eq_preCT} {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{(k)}[-1],\pm}(x_{(k),i,\pm})= \begin{cases} x_{(k),i,\pm} & i\neq k,\\ x_{(k),k,\pm}(1+(y_{k,\pm})^{-1}) & i=k. \end{cases}$$ This recovers the results in [@ks pp143]. Composition of cluster transformations {#composition-of-cluster-transformations .unnumbered} -------------------------------------- Cluster algebras were introduced by Fomin and Zelevinsky ([@fomin-zelevinsky1]) in their study of dual canonical bases and total positivity in semi-simple groups. Although the initial aim has not been established, it has been discovered that the theory of cluster algebras has many links with a wide range of mathematics (see [@keller-survay §1.1] and the references there). Since a cluster transformation helps us to understand the whole structure in an inductive way, study of compositions of cluster transformations is important. A [*seed*]{} is a pair $(Q\mid\underline{u})$, where 1. $Q$ is a quiver without loops and oriented $2$-cycles, and 2. ${\underline{u}}=(u_1,\ldots,u_n)$ is a free generating set of the field ${{{\mathbb{C}}}}(x_1,\ldots,x_n)$. For a vertex $k\in I$, the [*mutation*]{} $\mu_k(Q\mid{\underline{u}})$ of $(Q\mid{\underline{u}})$ at $k$ is the seed $(\mu_kQ\mid{\underline{u}}^{{{\mathrm{new}}}})$, where $\mu_kQ$ is the mutation of the quiver (§\[qmutation\]) and ${\underline{u}}^{{{\mathrm{new}}}}$ is obtained from ${\underline{u}}$ by replacing $u_k$ with $$\label{eq_cluster} u_k^{{{\mathrm{new}}}}=u_k^{-1}\Biggl(\prod_i (u_k)^{Q(i,k)}+\prod_i (u_k)^{Q(k,i)}\Biggr)$$ This is called the [*cluster transformation*]{}. Given a quiver $Q$, we call $(Q\mid\underline{x})=(Q,(x_1,\ldots,x_n))$ an [*initial seed*]{}. For a sequence of vertices $\mathbf{k}=(k_1,\ldots,k_l)\in I^l$ and a vertex $i\in I$, we define rational functions $FZ_{\mathbf{k},i}(\underline{x})$ by $$\mu_{k_l}(\cdots(\mu_2(\mu_1(Q\mid\underline{x}))\cdots) =(Q_{\mathbf{k}}\mid(FZ_{\mathbf{k},i}(\underline{x}))).$$ In the case of a quiver of finite type, Caldero and Chapoton ([@caldero-chapoton]) described a composition of cluster transformations in terms of quiver Grassmannians of the original quiver. This result is generalized by many people (see the references in [@plamondon] for example). Finally, Derksen-Weyman-Zelevinsky and Plamondon ([@DWZ2; @plamondon]) provided the Caldero-Chapoton type formula for an arbitrary quiver without loops and oriented $2$-cycles. In this paper, we provide an alternative proof of the Caldero-Chapoton type formula under the assumption that there is a potential $W$ such that the QP $(Q,W)$ is successively f-mutatable with respect to the sequence ${\mathbf{k}}$ (§\[subsub\_pmutation\]). We identify ${{{\mathbb{C}}}}(x_1,\ldots,x_n)$ with the fractional field of $T_{Q,+}$. We will omit “$+$” in the notations. \[thm\_02\] We have $$\label{eq_CC} \mathrm{FZ}_{\mathbf{k},i}(\underline{x})= x_{{\mathbf{k}},i}\cdot \Biggl(\sum_\mathbf{v}e\Bigr(\mathrm{Grass}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}^{-\mathbf{v}}\Biggr).$$ where $(\underline{y})^{-\mathbf{v}}=\prod_j (y_j)^{-v_j}$ and $y_j=\prod_i(x_i)^{\bar{Q}(i,j)}$. Application to cluster algebras {#application-to-cluster-algebras .unnumbered} ------------------------------- In [@DWZ2; @plamondon], they prove six conjectures given in [@fomin-zelevinsky4] for cluster algebras associated to quivers [^5]. In §\[subsec\_F\_and\_g\] and §\[subsec\_g\_to\_F\] we give alternative proofs for them under the assumption that the quiver with principal framing is successively f-mutatable. [^6]. Let $Q{^{{{\mathrm{pf}}}}}$ be the following quiver: [vertices]{} : $I\sqcup I^*$ where $I^*=\{1^*,\ldots,n^*\}$, [arrows]{} : $\{\text{arrows in $Q$}\}\sqcup \{i^*\to i\mid i\in I\}$. This is called the quiver with the principal framing associated to $Q$. Let us use $\{X_i\}$ and $\{Y_i\}$ for generators of the tori associated to $Q{^{{{\mathrm{pf}}}}}$. - The $F$-polynomial associated to $(Q,W)$, ${\mathbf{k}}$ and $i$ is the following : $$F_{{\mathbf{k}},i}(\underline{y}):={FZ}{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}(\underline{X})|_{X_i=1,X_{i^*}=y_i}.$$ - The $g$-vector $g_{{\mathbf{k}},i}\in M_Q$ associated to $(Q,W)$, ${\mathbf{k}}$ and $i$ is the element which is characterized by the following identity : $$\mathrm{FZ}_{\mathbf{k},i}(\underline{x})= \mathbf{x}^{g_{{\mathbf{k}},i}} \cdot{{\mathrm{F}}}_{{\mathbf{k}},i}(\underline{y}^{-1})$$ where the last term is given by substituting $y_i^{-1}$ to $y_i$. It is $y_i^{-1}$ in our notation what is denoted by $y_i$ in Fomin-Zelevinsky’s notation. We use this notation since $y_i$ corresponds to the simple module in our notation. The potential $W$ of $Q$ can be taken as a potential of $Q{^{{{\mathrm{pf}}}}}$. We [*assume*]{} that $(Q{^{{{\mathrm{pf}}}}},W)$ is successively f-mutatable with respect to the sequence ${\mathbf{k}}$. We will apply an argument similar to the one in §\[sec\_proof\], for $(Q{^{{{\mathrm{pf}}}}},W)$. Then we get descriptions of $g$-vectors and $F$-polynomials in terms of the $3$-Calabi-Yau category : cluster algebra DT theory ----------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------- $y$-variable $y_i$ formal variable corresponding to the simple module $s_i$ $x$-variable $x_i$ formal variables corresponding to the projective module $P_i$ (or $\Gamma_i$) $F$-polynomial generating function of the Euler characteristics of the quiver Grassmannians $g$-vector $\phi_{{\mathbf{k}}}^{-1}([\Gamma_{{\mathbf{k}},i}])\in M_Q=K_0({{\mathrm{per}}}\Gamma)\simeq {{{\mathbb{Z}}}}^I$ ${}^tg$-vector $\phi_{{\mathbf{k}}}([s_i])\in L_{Q_{\mathbf{k}}}\simeq {{{\mathbb{Z}}}}^I$ $c$-vector $\phi_{{\mathbf{k}}}^{-1}([s_{{\mathbf{k}},i}])\in L_{Q}\simeq {{{\mathbb{Z}}}}^I$ sign coherence of ${}^tg$-vectors $s_i\in {{\mathcal{T}}}_{\mathbf{k}}\subset {{\mathcal{A}}}_{\mathbf{k}}[1]$ or $s_i\in {{\mathcal{F}}}_{\mathbf{k}}\subset {{\mathcal{A}}}_{\mathbf{k}}$ sign coherence of $c$-vectors $s_{{\mathbf{k}},i}\in {{\mathcal{T}}}_{\mathbf{k}}[-1]\subset {{\mathcal{A}}}[-1]$ or $s_{{\mathbf{k}},i}\in {{\mathcal{F}}}_{\mathbf{k}}\subset {{\mathcal{A}}}$ $g$-vectors determine $F$-vectors Bridgeland stability on walls Contents {#contents .unnumbered} -------- From §\[sec\_1\] to §\[subsec\_stab\], we study some categorical properties of the $3$-dimensional Calabi-Yau category associated to a quiver with a potential. The statements of our main results appear in §\[sec\_statement\]. We prove the theorems using motivic Hall algebra, on which we give a brief review in §\[sec\_MH\]. For the proof, first, we show in §\[subsec\_eq\_in\_MH\] some identities on the motivic Hall algebra using the results from §\[sec\_1\] to §\[subsec\_stab\]. They are translated in §\[subsec\_auto\] into the main results via the integration map. Finally, we study quivers with principal framings to provide alternative proofs for the six conjectures given in [@fomin-zelevinsky4] (§\[sec\_F\_and\_g\]). Comments {#comments .unnumbered} -------- \(1) Throughout this paper, we assume that all the potentials are finite. As we mentioned, from the view points of applications to cluster algebras, we would like to remove the assumption. If we take an infinite potential, then the moduli spaces will not be schemes (or stacks) but formal schemes (or stacks). Once we construct a theory of the motivic Hall algebra in the formal setting, we can apply all the arguments in this paper. \(2) A typical example of a finite potential is a potential associated to a triangulated surface [@QP-surface]. We will apply the results in this paper for a triangulated surface in [@DT_surface]. \(3) It is expected that there is a refinement of the DT theory, which is called the [*motivic DT theory*]{} ([@ks; @behrend_bryan_szendroi]). Wall-crossing phenomena of the motivic DT theory has been studied in [@ks; @WC_motivic]. We hope to study quantum cluster algebras from the view point of motivic DT theory in the future. Acknowledgement {#acknowledgement .unnumbered} --------------- I would like to express my gratitude for all of the following mathematicians; Bernhard Keller who patiently explained many things about the cluster categories and the cluster algebras, indicated many stupid mistakes in the very preliminary version of this paper; Tom Bridgeland who showed me the preliminary version of his paper [@bridgeland-hall] and gave me a lot of helpful comments and encouragement. In particular, the proof of Theorem \[thm\_tilting\] is due to him; Pierre-Guy Plamondon who kindly explained the results in his PhD thesis [@plamondon]; Hiraku Nakajima who explained me his results in [@nakajima-cluster] and encouraged me to promote the result of [@ks]; Bernard Leclerc who recommended me to give alternative proofs for the conjectures in [@fomin-zelevinsky4]; Andrei Zelevinsky who gave me some comments on the preliminary version of this paper. The first version of this paper was written while I have been visiting the University of Oxford. I am grateful to Dominic Joyce for the invitation and to the Mathematical Institute for hospitality. The author is supported by the Grant-in-Aid for Research Activity Start-up (No. 22840023) and for Scientific Research (S) (No. 22224001). Preliminary {#sec_0} =========== QP, dga and Jacobi algebra -------------------------- A quiver with a potential (QP, in short) is a pair $(Q,W)$ of a quiver $Q$ and a potential $W$, a linear combination of oriented cycles. We say that $W$ (or $(Q,W)$) is [*finite*]{} when $W$ is a [finite]{} linear combination of oriented cycles. In this paper, we always assume that a QP is finite. First, we define the derivation of the potential. For an arrow $a$ and a oriented cycle $a_1\cdots a_l$, we put $$\partial_a(a_1\cdots a_l) := \sum_i\delta_{a,a_i}a_{i+1}\cdots a_la_1\cdots a_{i-1}.$$ For an arrow $a$ and a potential $W$, we define the derivation $\partial_aW$ by the linear combination of the derivations of the oriented cycles. For a QP $(Q,W)$, we define Ginzburg’s differential graded algebra $\Gamma=\Gamma_{Q,W}$. As a graded algebra, $\Gamma_{Q,W}$ is given by the path algebra $\mathbb{C}\hat{Q}$ of the following graded quiver $\hat{Q}$. The vertex set of $\hat{Q}$ is the same as $Q$ and the arrow set is the union of the following three sets : - arrows in $Q$ (degree $0$), - opposite arrow $a^*$ for each arrow $a$ in $Q$ (degree $-1$), - loop $t_i$ at $i$ for each vertex $i$ in $Q$ (degree $-2$). We define the differential $d=d_W$ of degree $1$ on the path algebra $\mathbb{C}\hat{Q}$ as follows: - $da=0$ for any arrow $a$ in $Q$, - $d(a^*) = \partial_aW$ for any arrow $a$ in $Q$,and - $d(t_i) = e_i\left(\prod_{a}[a,a^*]\right)e_i$ for any vertex $i$ in $Q$. <!-- --> - The differential graded algebra $\Gamma_{Q,W}=(\mathbb{C}\hat{Q},d_W)$ is called the [*Ginzburg differential graded algebra*]{} (dga, in short). - The algebra $J=J_{Q,W}:=H^0\Gamma_{Q,W}$ is called the [*Jacobi algebra*]{}. The Jacobi algebra can be described as the quiver with the relations : $$J_{Q,W}=\mathbb{C}Q/\langle\partial_a W;a\in Q_1\rangle.$$ Quiver mutation {#qmutation} --------------- In this paper, we always assume that a quiver has - the vertex set $I=\{1,\ldots,n\}$, and - no loops and oriented $2$-cycles. For vertices $i$ and $j\in I$, we put $$Q(i,j)=\sharp\{\text{arrows from $i$ to $j$}\},\quad \bar{Q}(i,j)=Q(i,j)-Q(j,i).$$ Note that the quiver $Q$ is determined by the matrix $\bar{Q}(i,j)$ under the assumption above. For the vertex $k$, we define the new quiver $\mu_kQ$ as follows : - First, we define a new quiver $\mu_k^{{\mathrm{pre}}}{Q}$ as follows : - For any subquiver $u\overset{\alpha}{\to} k\overset{\beta}{\to}v$, we associate a new arrow $[\beta\alpha]\colon u \to v$. - replace any arrow $a$ incident to the vertex $k$ with an opposite arrow $a^*$. - Remove all oriented cycles of length $2$ in $\mu_k^{{\mathrm{pre}}}{Q}$. QP mutation ----------- ### Reduced part of a potential Let $\widehat{\mathbb{C}Q}$ be the completion of $\mathbb{C}Q$ with respect to path lengths. A potential of $Q$ is an element in $\widehat{{{\mathbf{C}}}Q}$ which is described as a linear combination of oriented cycles in $Q$. We identify two potentials which are related via rotations of oriented cycles. A potential is said to be finite if it is an element in $\mathbb{C}Q$. Two QP $(Q,W)$ and $(Q',W')$ are said to be right equivalent, which is denoted by $(Q,W)\sim (Q',W')$, if there exists an algebra isomorphism $\psi$ between $\widehat{\mathbb{C}Q}$ and $\widehat{\mathbb{C}Q'}$ so that $\psi(W)=W'$. Two finite QP $(Q,W)$ and $(Q',W')$ are said to be right f-equivalent, which is denoted by $(Q,W)\overset{{{\mathrm{fin.}}}}{\sim} (Q',W')$, if there exists an algebra isomorphism $\psi$ between ${\mathbb{C}Q}$ and ${\mathbb{C}Q'}$ so that $\psi(W)=W'$. A potential is said to be reduced if it has no oriented cycles of length less than $3$, and said to be trivial if its Jacobi algebra is trivial. For quivers $Q$ and $Q'$ with the same vertex set, let $Q\cup Q'$ denote the quiver given by taking union the arrow sets. For QPs $(Q,W)$ and $(Q',W')$ with the same vertex set, we take $W$ and $W'$ as potentials of $Q\cup Q'$ and let $(Q,W)\oplus(Q',W')$ denote the new QP $(Q\cup Q',W+W')$.. For any QP $(Q,W)$, we have a right equivalence $$(Q,W) \sim (Q,W)_{{\mathrm{red}}} \oplus (Q,W)_{{\mathrm{triv}}}$$ with reduced $W_{{\mathrm{red}}}$ and trivial ${W_{{\mathrm{triv}}}}$ ([@quiver-with-potentials Lemma 4.6]). Moreover, $(Q_{{\mathrm{red}}},W_{{\mathrm{red}}})$ and $(Q_{{\mathrm{triv}}},W_{{\mathrm{triv}}})$ are determined uniquely up to right equivalences. We call $(Q_{{\mathrm{red}}},W_{{\mathrm{red}}})$ as the reduced part of $(Q,W)$. A finite QP $(Q,W)$ is said to be f-reducible if we have a right f-equivalence $$(Q,W) \overset{{{\mathrm{fin.}}}}{\simeq} (Q_{{\mathrm{red}}},W_{{\mathrm{red}}}) \oplus (Q_{{\mathrm{triv}}},W_{{\mathrm{triv}}})$$ with finite reduced $(Q_{{\mathrm{red}}},W_{{\mathrm{red}}})$ and finite trivial $(Q_{{\mathrm{triv}}},W_{{\mathrm{triv}}})$. ### Potential mutation {#subsub_pmutation} For a QP $(Q,W)$ and a vertex $k$, we define the potential $\mu_k^{{\mathrm{pre}}}{W}$ of the quiver $\mu_k^{{\mathrm{pre}}}Q$ by $$\mu_k^{{\mathrm{pre}}}{W}:=[W]+\Delta$$ where - $[W]$ is the potential which is obtained from $W$ by replacing all the composition $u\overset{\alpha}{\to} k\overset{\beta}{\to}v$ with $[\beta\alpha]$, and - $\Delta:=\sum \alpha^*\beta^*[\beta\alpha]$. The mutation $\mu_k(Q,W)$ of the QP $(Q,W)$ at $k$ is the reduced part $(\mu_k^{{\mathrm{pre}}}{Q},\mu_k^{{\mathrm{pre}}}{W})_{{\mathrm{red}}}$ of $(\mu_k^{{\mathrm{pre}}}{Q},\mu_k^{{\mathrm{pre}}}{W})$. - We say that a QP $(Q,W)$ is mutatable at $k$ if the underlying quiver of $\mu_k(Q,W)$ is $\mu_kQ$, the mutation of the quiver defined in §\[qmutation\]. - We say that a finite QP $(Q,W)$ is f-mutatable at $k$ if it is mutatable and $(\mu_k^{{\mathrm{pre}}}{Q},\mu_k^{{\mathrm{pre}}}{W})$ is f-reducible. Let ${\mathbf{k}}=(k_1,\ldots,k_l)$ be a sequence of vertices. A finite QP $(Q,W)$ is said to be successively f-mutatable with respect to the sequence ${\mathbf{k}}$ if $$\mu_{k_{s-1}}(\cdots (\mu_{k_1}(Q,W))\cdots)$$ is f-mutatable at $k_s$. Derived categories {#sec_1} ================== Categories {#subsubsec_111} ---------- For a QP $(Q,W)$, we have the following triangulated categories : [${\mathcal{D}\Gamma}$ : ]{}the derived category of right dg-modules over Ginzburg dga $\Gamma$, [${\mathrm{per}\Gamma}$ : ]{}the smallest full subcategory of ${\mathcal{D}\Gamma}$ containing $\Gamma$ and closed under extensions, shifts and direct summands, [${\mathcal{D}^{\mathrm{fd}}\Gamma}$ : ]{}the full subcategory of ${\mathcal{D}\Gamma}$ consisting of dg-modules with finite dimensional cohomologies. The triangulated categories ${\mathcal{D}\Gamma}$ and ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ have the canonical t-structures whose cores are [${\mathrm{Mod}\hspace{1pt}J}$ : ]{}the category of finitely generated right modules over the (non-complete) Jacobi algebra, and [${\mathrm{mod}\hspace{1pt}J}$ : ]{}the full subcategory of ${\mathrm{Mod}\hspace{1pt}J}$ consisting of finite dimensional modules respectively. For a vertex $i\in I$, we have the following objects: [$s_i$ : ]{}the simple $J$-module, [$\Gamma_i:=e_i\Gamma$ : ]{}the $\Gamma$-module , which is a direct summand of $\Gamma$, and [$P_i:=H^0_{{\mathrm{Mod}\hspace{1pt}J}}(\Gamma_i)$ : ]{}the projective indecomposable $J$-module. Here $e_i$ is the idempotent. Grothendieck groups {#subsec_Gro} ------------------- We put $M=M_Q:=K_0({\mathrm{per}\Gamma})$ and $L=L_Q:=\mathbb{Z}^{I}$, where $L$ is taken as the target of the map $$K_0({\mathcal{D}^{\mathrm{fd}}\Gamma})\to \mathbb{Z}^{I}=L$$ defined by $[E]\mapsto \underline{\mathrm{dim}}(E)$. With a slight abuse of notations, we will write $[E]\in L$ instead of $\underline{\mathrm{dim}}(E)$. We put $M_{{{{\mathbb{R}}}}}=M_{Q,{{{\mathbb{R}}}}}:=M_Q\otimes {{{\mathbb{R}}}}$ and $L_{{{\mathbb{R}}}}=L_{Q,{{{\mathbb{R}}}}}:=L_Q\otimes {{{\mathbb{R}}}}$. Let $\chi$ denote the Euler pairing $L\times L\to {{{\mathbb{Z}}}}$ given by $$\chi([E],[F])=\sum_i(-1)^i\dim\mathrm{Hom}(E,F[i]).$$ We put $\mathbf{w}_i:=[\Gamma_i]$ and $\mathbf{v}_i:=[s_i]$. The set $\{\mathbf{w}_i\}$ forms a basis of $M$ and the set $\{\mathbf{v}_i\}$ forms a basis of $L$. We extend $\chi$ on $L\otimes M$ by $$\chi(\mathbf{w}_i,\mathbf{v}_j):=\delta_{i,j},\quad \chi(\mathbf{w},\mathbf{w'})=0$$ for any $\mathbf{w}, \mathbf{w}'\in M$. This gives $M_{Q,{{{\mathbb{R}}}}}\simeq (L_{Q,{{{\mathbb{R}}}}})^*$. Tori {#subsub_torus} ---- Let $\sigma$ be a sign; $\sigma=\pm$. We define $\mathrm{T}^\vee_{Q,\sigma}$, $\mathrm{T}_{Q,\sigma}$ and $\mathbb{T}_{Q,\sigma}$ by $$\begin{array}{c} \mathrm{T}_{Q,\sigma}^\vee:=\bigoplus_{\mathbf{w}\in M}{{{\mathbb{C}}}}\cdot \mathbf{x}^{\mathbf{w}}_{\sigma},\quad \mathrm{T}_{Q,\sigma}:=\bigoplus_{\mathbf{v}\in L}{{{\mathbb{C}}}}\cdot \mathbf{y}^{\mathbf{v}}_{\sigma},\vspace{5pt}\\ \mathbb{T}_{Q,\sigma}:=\mathrm{T}_{Q,\sigma}^\vee\otimes\mathrm{T}_{Q,\sigma}, \end{array}$$ with the following products: $$\mathbf{x}^{\mathbf{w}}_{\sigma}\cdot \mathbf{x}_{\sigma}^{\mathbf{w'}}=\mathbf{x}^{\mathbf{w+w'}}_{\sigma}, \quad \mathbf{y}^{\mathbf{v}}_{\sigma}\cdot \mathbf{y}^{\mathbf{v'}}_{\sigma}=\sigma^{\chi(\mathbf{v},\mathbf{v'})}\mathbf{y}_{\sigma}^{\mathbf{v+v'}}, \quad \mathbf{x}^{\mathbf{w}}_{\sigma} \cdot \mathbf{y}^{\mathbf{v}}_{\sigma} = \mathbf{y}^{\mathbf{v}}_{\sigma} \cdot \mathbf{x}^{\mathbf{w}}_{\sigma}$$ where we identify $\sigma$ with $\pm 1$. We put $x_{i,\sigma}:=\mathbf{x}^{[\Gamma_i]}_\sigma$ and $y_{i,\sigma}:=\mathbf{y}^{[s_i]}_\sigma$, then we have $$\begin{array}{c} \mathrm{T}_{Q,\sigma}^\vee={{{\mathbb{C}}}}[x_{1,\sigma}^\pm,\ldots,x_{n,\sigma}^\pm],\quad \mathrm{T}_{Q,\sigma}={{{\mathbb{C}}}}[y_{1,\sigma}^\pm,\ldots,y_{n,\sigma}^\pm],\vspace{5pt}\\ \mathbb{T}_{Q,\sigma}={{{\mathbb{C}}}}[x_{1,\sigma}^\pm,\ldots,x_{n,\sigma}^\pm,y_{1,\sigma}^\pm,\ldots,y_{n,\sigma}^\pm]. \end{array}$$ They are called the semiclassical limits of [*quantum torus*]{}, [*quantum dual torus*]{} and [*quantum double torus*]{} respectively [^7]. We define the surjective algebra homomorphism $\pi_\sigma\colon \mathbb{T}_{Q,\sigma}\twoheadrightarrow \mathrm{T}_{Q,\sigma}^\vee$ by $$\label{eq_pi} x_{i,\sigma} \otimes 1 \longmapsto x_{i,\sigma},\quad 1 \otimes y_{i,\sigma} \longmapsto \mathbf{x}_\sigma^{[s_i]}.$$ The kernel of $\pi_\sigma$ is generated by $\{(\mathbf{x}_\sigma^{[s_i]}\otimes 1) - (1\otimes y_{i,\sigma})\mid i \in I\}$. We sometimes identify an element in ${{{\mathrm{T}}}}_{Q,\sigma}$ with its image in ${{{\mathrm{T}}}}_{Q,\sigma}^\vee$ under the composition $${{{\mathrm{T}}}}_{Q,\sigma} \hookrightarrow \mathbb{T}_{Q,\sigma} \overset{\pi_\sigma}{\twoheadrightarrow}{{{\mathrm{T}}}}_{Q,\sigma}^\vee.$$ Let $\Sigma$ denote the automorphism of the tori given by $$\Sigma(\mathbf{x}_\sigma^\mathbf{w})=\mathbf{x}_\sigma^{-\mathbf{w}},\quad \Sigma(\mathbf{y}_\sigma^\mathbf{v})=\mathbf{y}_\sigma^{-\mathbf{v}}.$$ Mutation and derived equivalence -------------------------------- ### Derived equivalence {#subsub_derived_equiv} Let $(Q,W)$ be a finite QP which is f-mutatable at a vertex $k$. Let $\mu_k\Gamma$ be the Ginzburg dga associated to the mutation $\mu_k(Q,W)$. \[thm\_KY\] There exist equivalences of triangulated categories $$\Phi_{k,+},\Phi_{k,-}\ \colon {\mathcal{D}\Gamma}\overset{\sim}{\longrightarrow} \mathcal{D}(\mu_k\Gamma)$$ such that - $\Phi_{k,\pm}^{-1}(\Gamma_i')=\Gamma_i$ for $i\neq k$, and - $\Phi_{k,+}^{-1}(\Gamma_k')$ and $\Phi_{k,-}^{-1}(\Gamma_k')$ are involved in the following triangles : $$\begin{array}{ccccccc} \Phi_{k,+}^{-1}(\Gamma'_k)[-1]&\to& \bigoplus_{j}\Gamma_{j}^{\oplus Q(k,j)}&\to& \Gamma_k&\to &\Phi_{k,+}^{-1}(\Gamma'_k),\vspace{1.5mm} \\ \Phi_{k,-}^{-1}(\Gamma'_k)&\to& \Gamma_k&\to & \bigoplus_{j}\Gamma_{j}^{\oplus Q(j,k)} &\to& \Phi_{k,-}^{-1}(\Gamma'_k)[1] \end{array}$$ where $\Gamma_j'$ is the direct summand of $\mu_k\Gamma$. Moreover, $\Phi_{k,\pm}$ restricts to equivalences from ${\mathrm{per}\Gamma}$ to $\mathrm{per}(\mu_k\Gamma)$ and from ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ to $\mathcal{D}^{\mathrm{fd}}(\mu_k\Gamma)$. It is $\Phi_{k,-}^{-1}$ that is studied in [@dong-keller Theorem 3.2]. The equivalences induce isomorphisms $$\phi_{k,\pm}\colon M_Q \overset{\sim}{\longrightarrow} M_{\mu_kQ}$$ and $$\phi_{k,\pm}\colon L_Q \overset{\sim}{\longrightarrow} L_{\mu_kQ}.$$ By the triangles in Theorem \[thm\_KY\] we have $$\label{eq_yy'} \begin{array}{l} \phi_{k,+}^{-1}([\Gamma_i'])=\begin{cases} [\Gamma_i] & i\neq k, \\ -[\Gamma_k]+\sum_j Q(k,j)[\Gamma_j] & i=k, \end{cases}\\ \phi_{k,-}^{-1}([\Gamma_i'])=\begin{cases} [\Gamma_i] & i\neq k, \\ -[\Gamma_k]+\sum_j Q(j,k)[\Gamma_j] & i=k \end{cases} \end{array}$$ in $M_Q$. Since $\phi_{k,\pm}$ preserves $\chi$ we have $$\label{eq_phi_for_s} \begin{array}{l} \phi_{k,+}^{-1}([s_i'])= \begin{cases} [s_i]+ Q(k,i)[s_k] & i\neq k, \\ -[s_k] & i=k, \end{cases}\\ \phi_{k,-}^{-1}([s_i'])=\begin{cases} [s_i]+Q(i,k)[s_k] & i\neq k, \\ -[s_k] & i=k \end{cases} \end{array}$$ in $L_Q$. Note that $\phi_{k,\pm}$ also induce isomorphisms between $\mathrm{T}_{Q,\sigma}$ and $\mathrm{T}_{\mu_kQ,\sigma}$. We sometimes identify them with each other and write simply $\mathrm{T}_{\sigma}$ since we do not want to specify a choice of a quiver. Tilting of t-structures {#sec_2} ======================= Torsion pair and tilting ------------------------ Let ${{\mathcal{D}}}$ be a triangulated category and ${{\mathcal{A}}}$ be the core of a t-structure. A pair $({{\mathcal{T}}},{{\mathcal{F}}})$ of full subcategories of ${{\mathcal{A}}}$ is called a [*torsion pair*]{} if the following conditions are satisfied : - for any $T\in{{\mathcal{T}}}$ and any $F\in{{\mathcal{F}}}$, we have ${{{\mathrm{Hom}}}}(T,F)=0$, - for any $X\in {{\mathcal{A}}}$, there exists an exact sequence $$0\to T\to X\to F\to 0$$ with $T\in{{\mathcal{T}}}$ and $F\in{{\mathcal{F}}}$. We sometimes illustrate the torsion pair as in Figure \[fig\_TP\]. In the figure, we have no non-trivial morphism from an object on left to an object on right. Given a torsion pair $({{\mathcal{T}}},{{\mathcal{F}}})$, let ${{\mathcal{D}}}_{\leq -1}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}$ denote the full subcategory of ${{\mathcal{D}}}$ consisting of objects $E$ which satisfy $$H^i_{{{\mathcal{A}}}}(E) \begin{cases} \in {{\mathcal{T}}}& i=0,\\ = 0 & i \geq 1, \end{cases}$$ and let ${{\mathcal{D}}}_{\geq 0}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}$ denote the full subcategory of ${{\mathcal{D}}}$ consisting of objects $E$ which satisfy $$H^i_{{{\mathcal{A}}}}(E) \begin{cases} \in {{\mathcal{F}}} & i=0,\\ = 0 & i \leq -1. \end{cases}$$ Then the pair of full subcategories $$\left({{\mathcal{D}}}_{\leq -1}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])},{{\mathcal{D}}}_{\geq 0}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}\right)$$ gives a t-structure of ${{\mathcal{D}}}$ (see Figure \[fig\_TILTING\]). Let $${{\mathcal{A}}}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}:= {{\mathcal{D}}}_{\leq -1}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}[-1] \cap {{\mathcal{D}}}_{\geq 0}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}$$ be the heart of the t-structure. That is, ${{\mathcal{A}}}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}$ is the full subcategory of ${{\mathcal{D}}}$ consisting of objects $E$ which satisfy $$H^i_{{{\mathcal{A}}}}(E) \begin{cases} \in {{\mathcal{F}}} & i=0,\\ \in {{\mathcal{T}}} & i=1,\\ = 0 & i \neq 0,1. \end{cases}$$ Composition of tilting ---------------------- Let $({{\mathcal{T}}},{{\mathcal{F}}})$ be a torsion pair of ${{\mathcal{A}}}$ and we put ${{\mathcal{A}}}':={{\mathcal{A}}}^{({{\mathcal{F}}},{{\mathcal{T}}}[-1])}$. Let $({{\mathcal{T}}}',{{\mathcal{F}}}')$ be a torsion pair of ${{\mathcal{A}}}'$ such that ${{\mathcal{T}}}'\subset {{\mathcal{F}}}$. We put ${{\mathcal{A}}}''=({{\mathcal{A}}}')^{({{\mathcal{F}}}',{{\mathcal{T}}}'[-1])}$. Let ${{\mathcal{T}}}''$ denote the full subcategory of ${{\mathcal{A}}}$ consisting of $X$ with $F_X\in {{\mathcal{T}}}'$ where $F_X\in {{\mathcal{F}}}$ is the quotient object of $X$ associated to the exact sequence (TP2) for $({{\mathcal{T}}},{{\mathcal{F}}})$. Let ${{\mathcal{F}}}''$ denote the full subcategory of ${{\mathcal{F}}}$ consisting of $Y$ with $T_Y=0$ where $T_Y\in {{\mathcal{T}}}'$ is the subobject of $Y$ associated to the exact sequence (TP2) for $({{\mathcal{T}}}',{{\mathcal{F}}}')$. (See Figure \[fig\_lem23\].) We can easily verify the following lemma. \[lem\_23\] The pair of the full subcategories $({{\mathcal{T}}}'',{{\mathcal{F}}}'')$ gives a torsion pair of ${{\mathcal{A}}}$ and $${{\mathcal{A}}}'' = {{\mathcal{A}}}^{({{\mathcal{F}}}'',{{\mathcal{T}}}''[-1])} $$ On the other hand, assume that $({{\mathcal{T}}}',{{\mathcal{F}}}')$ is a torsion pair of ${{\mathcal{A}}}'$ such that ${{\mathcal{F}}}'\subset {{\mathcal{T}}}[-1]$. Let ${{\mathcal{F}}}''$ denote the full subcategory of ${{\mathcal{A}}}$ consisting of $Y$ with $T_Y\in {{\mathcal{F}}}'[1]$ where $T_Y\in {{\mathcal{T}}}$ is the subobject of $Y$ associated to the exact sequence (TP2) for $({{\mathcal{T}}},{{\mathcal{F}}})$. Let ${{\mathcal{T}}}''$ denote the full subcategory of ${{\mathcal{T}}}$ consisting of $X$ with $F_X=0$ where $F_X\in {{\mathcal{F}}}'[1]$ is the quotient object of $X$ associated to the exact sequence (TP2) for $({{\mathcal{T}}}'[1],{{\mathcal{F}}}'[1])$ (see Figure \[fig\_lem24\]). We put ${{\mathcal{A}}}'':=({{\mathcal{A}}}')^{({{\mathcal{F}}}'[1],{{\mathcal{T}}}')}$ We can also verify the following lemma. \[lem\_24\] The pair of the full subcategories $({{\mathcal{T}}}'',{{\mathcal{F}}}'')$ gives a torsion pair of ${{\mathcal{A}}}$ and $${{\mathcal{A}}}'':={{\mathcal{A}}}^{({{\mathcal{F}}}'',{{\mathcal{T}}}''[-1])}.$$ Mutation and tilting {#sec_} -------------------- Let ${{\mathcal{S}}}_k$ be the full subcategory consisting of $J_{Q,W}$-modules supported on the vertex $k$. We put $$\begin{aligned} ({{\mathcal{S}}}_k)^{\bot}&:= \{ E\in {{\mathrm{Mod}}}J_{Q,W}\mid {{\mathrm{Hom}}}(s_k,E)=0\},\\ {}^{\bot}({{\mathcal{S}}}_k)&:= \{ E\in {{\mathrm{Mod}}}J_{Q,W}\mid {{\mathrm{Hom}}}(E,s_k)=0 \}.\end{aligned}$$ Then both $({{\mathcal{S}}}_k,({{\mathcal{S}}}_k)^{\bot})$ and $({}^{\bot}({{\mathcal{S}}}_k),{{\mathcal{S}}}_k)$ give torsion pairs of ${{\mathrm{Mod}}}J_{Q,W}$. It is shown in [@dong-keller Corollary 5.5] that the derived equivalences associated to a mutation are given by tilting with respect to these torsion pairs : $$\begin{aligned} \Phi_{k,+}^{-1}\left({{\mathrm{Mod}}}J_{\mu_k(Q,W)}\right) & = ({{\mathrm{Mod}}}J_{Q,W})^{\left(({{\mathcal{S}}}_k)^{\bot},{{\mathcal{S}}}_k[-1]\right)},\label{eq_tilting+}\\ \Phi_{k,-}^{-1}\left({{\mathrm{Mod}}}J_{\mu_k(Q,W)}\right) & = ({{\mathrm{Mod}}}J_{Q,W})^{\left({{\mathcal{S}}}_k[1],{}^\bot({{\mathcal{S}}}_k)\right)}.\label{eq_tilting-}\end{aligned}$$ Composition of mutations and tilting {#subsec_comp} ------------------------------------ The proof of the following theorem is due to Tom Bridgeland. We put $\bar{{{\mathcal{A}}}}:={{\mathrm{Mod}}}J_{Q,W}$. \[thm\_tilting\] There exists a unique sequence ${\varepsilon}(1),\ldots, {\varepsilon}(l)$ of signs which satisfies the following conditions; We put $$\Phi_{{\mathbf{k}}}:= \Phi_{k_l,{\varepsilon}(l)}\circ\cdots\circ\Phi_{k_1,{\varepsilon}(1)}\colon {{\mathcal{D}}}\Gamma\overset{\sim}{\longrightarrow} {{\mathcal{D}}}\Gamma_{\mu_{\mathbf{k}}(Q,W)}$$ and $$\bar{{{\mathcal{A}}}}_{{\mathbf{k}}}:=\Phi_{{\mathbf{k}}}^{-1}({{\mathrm{Mod}}}J_{\mu_{\mathbf{k}}(Q,W)}).$$ Then - there exists a torsion pair $(\bar{{{\mathcal{T}}}}_{{\mathbf{k}}},\bar{{{\mathcal{F}}}}_{{\mathbf{k}}})$ of $\bar{{{\mathcal{A}}}}$ such that $$\bar{{{\mathcal{A}}}}^{(\bar{{{\mathcal{F}}}}_{{\mathbf{k}}},\bar{{{\mathcal{T}}}}_{{\mathbf{k}}}[-1])}=\bar{{{\mathcal{A}}}}_{{\mathbf{k}}}. $$ - $\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{F}}}}_{{\mathbf{k}}}$ or $\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{T}}}}_{{\mathbf{k}}}[-1]$ for any $i\in Q_0$ where $s_{{\mathbf{k}},i}$ is the simple $J_{\mu_{{\mathbf{k}}}(Q,W)}$-module. We prove the claim by induction with respect to the length $l$ of the sequence. First of all, $(A_1)$ is hold if we take ${\varepsilon}(1)=+$ by . : Since $(\bar{{{\mathcal{F}}}}_{{\mathbf{k}}},\bar{{{\mathcal{T}}}}_{{\mathbf{k}}}[-1])$ give a torsion pair for $\bar{{{\mathcal{A}}}}_{{\mathbf{k}}}$, an exact sequence is associated to $\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})$. Because $\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})$ is simple in $\bar{{{\mathcal{A}}}}_{{\mathbf{k}}}$, we have $$\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{F}}}}_{{\mathbf{k}}}$$ or $$\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{T}}}}_{{\mathbf{k}}}[-1].$$ : We define ${\varepsilon}(l)$ by $${\varepsilon}(l)= \begin{cases} + & \text{if }\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{F}}}}_{{\mathbf{k}}},\\ - & \text{if }\Phi_{{\mathbf{k}}}^{-1}(s_{{\mathbf{k}},i})\in \bar{{{\mathcal{T}}}}_{{\mathbf{k}}}[-1].\\ \end{cases}$$ Then the claim follows Lemma \[lem\_23\] and Lemma \[lem\_24\]. A similar statement has been shown in [@plamondon Theorem 2.15]. For $1\leq r\leq l$, let ${\mathbf{k}}^{(r)}$ denote the truncated sequence $(k_1,\ldots,k_r)$. For $i\in I$, we define $s^{(r)}_{i}\in \bar{{{\mathcal{A}}}}$ $$s^{(r)}_{i}:= \begin{cases} \Phi_{{\mathbf{k}}^{(r)}}^{-1}(s_{{\mathbf{k}}^{(r)},i}) & \text{if } \Phi_{{\mathbf{k}}^{(r)}}^{-1}(s_{{\mathbf{k}}^{(r)},i})\in \bar{{{\mathcal{F}}}}_{{\mathbf{k}}^{(r)}},\\ \Phi_{{\mathbf{k}}^{(r)}}^{-1}(s_{{\mathbf{k}}^{(r)},i})[1] & \text{if } \Phi_{{\mathbf{k}}^{(r)}}^{-1}(s_{{\mathbf{k}}^{(r)},i})\in \bar{{{\mathcal{T}}}}_{{\mathbf{k}}^{(r)}}[-1].\\ \end{cases}$$ We put $s^{(r)}:=s^{(r)}_{k_r}$. The canonical t-structure of ${\mathcal{D}\Gamma}$ induces a t-structure of ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ whose core is ${{\mathcal{A}}}:={{\mathrm{mod}}}J_{Q,W}$. Since $s^{(r)}\in {{\mathcal{A}}}$ for any $r$, we have $\bar{{{\mathcal{T}}}}_{\mathbf{k}}\in {{\mathcal{A}}}$. We put $${{\mathcal{A}}}_{\mathbf{k}}:=\Phi_{{\mathbf{k}}}^{-1}({{\mathrm{mod}}}J_{\mu_{\mathbf{k}}(Q,W)}),\quad {{\mathcal{T}}}_{\mathbf{k}}:=\bar{{{\mathcal{T}}}}_{\mathbf{k}},\quad {{\mathcal{F}}}_{\mathbf{k}}:=\bar{{{\mathcal{F}}}}_{\mathbf{k}}\cap{{\mathcal{A}}}.$$ Then we can verify the following : The pair of the full subcategories $({{\mathcal{T}}}_{\mathbf{k}},{{\mathcal{F}}}_{\mathbf{k}})$ gives a torsion pair of ${{\mathcal{A}}}$ and $${{\mathcal{A}}}_{\mathbf{k}}= {{\mathcal{A}}}^{({{\mathcal{F}}}_{\mathbf{k}},{{\mathcal{T}}}_{\mathbf{k}}[-1])}. $$ Stability condition on ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ {#subsec_stab} ========================================================== In this section, we study the space of stability conditions on ${\mathcal{D}^{\mathrm{fd}}\Gamma}$. For a subcategory ${{\mathcal{C}}}\subset {\mathcal{D}^{\mathrm{fd}}\Gamma}$, let $C_{{{\mathcal{C}}}}\subset L_{{{\mathbb{R}}}}$ be the minimal cone containing all the classes of elements in ${{\mathcal{C}}}$ and we define its dual cone $C_{{{\mathcal{C}}}}^*$ by $$C_{{{\mathcal{C}}}}^*:=\bigl\{\theta\in (L_{{{\mathbb{R}}}})^*=M_{{{\mathbb{R}}}}\mid \langle\theta,\mathbf{v}\rangle>0\text{ for any }\mathbf{v}\in C_{{\mathcal{C}}}\bigr\}.$$ Throughout this section, we fix an element $\delta\in {{\mathcal{C}}}^*_{\mathcal{A}}$. For $\theta\in (L_{{{\mathbb{R}}}})^*=M_{{{\mathbb{R}}}}$, let $$Z_\theta\colon L\to {{{\mathbb{C}}}}$$ denote the group homomorphism given by $Z_\theta:=\langle-\delta+\sqrt{-1}\theta,\bullet\rangle$. Embedding of $M_{{{\mathbb{R}}}}$ {#subsec_embed} --------------------------------- If $\theta\in C_{\mathcal{A}}^*$, the pair $\zeta(\theta):=({{\mathcal{A}}},Z_\theta)$ gives a Bridgeland’s stability condition on ${\mathcal{D}^{\mathrm{fd}}\Gamma}$. This gives an embedding $$\zeta \colon C_{\mathcal{A}}^* \hookrightarrow \mathrm{Stab}({\mathcal{D}^{\mathrm{fd}}\Gamma})$$ where the right hand side is the space of Bridgeland stability conditions on ${\mathcal{D}^{\mathrm{fd}}\Gamma}$. We will extend this to an embedding of $(L_{{{\mathbb{R}}}})^*=M_{{{\mathbb{R}}}}$. For two real numbers $t$ and $\phi$, we define $t^*\phi\in {{{\mathbb{R}}}}$ so that $${{\mathrm{tan}}}((t^*\phi)\pi)={{\mathrm{tan}}}(\phi\pi)+t, \quad 0^*\phi=\phi$$ and so that the map $(t,\phi)\mapsto t^*\phi$ is continuous. For $\theta\in C_{\mathcal{A}}^*$, let ${{\mathcal{P}}}_\theta$ the slicing of ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ corresponding to the stability condition $\zeta(\theta)$ ([@bridgeland-stability Definition 5.1 and Proposition 5.3]). That is, ${{\mathcal{P}}}_\theta(\phi)$ is the full subcategory of semistable objects with phase $\phi\in {{{\mathbb{R}}}}$ with respect to the stability condition $\zeta(\theta)$. We define the slicing $t^*{{\mathcal{P}}}_\theta$ by $$t^*{{\mathcal{P}}}_\theta(\phi):={{\mathcal{P}}}_\theta(t^*\phi).$$ Then the pair $(t^*{{\mathcal{P}}}_\theta,Z_{\theta-t\delta})$ gives a stability condition (Figure \[fig\_stability\]). We define the map $$\zeta \colon (L_{{{\mathbb{R}}}})^*=M_{{{\mathbb{R}}}}\to \mathrm{Stab}({\mathcal{D}^{\mathrm{fd}}\Gamma}).$$ by $${\zeta}(\theta-t\delta):=(t^*{{\mathcal{P}}}_\theta,Z_{\theta-t\delta})$$ for any $\theta\in C_{\mathcal{A}}^*$ and $t\in \mathbb{R}$. We can verify that this is well-defined and injective. T-structures {#subsec_Tstr} ------------ In this subsection, we describe the t-structures corresponding to some stability conditions in $\zeta(M_{{{\mathbb{R}}}})$. For a stability condition $\zeta$, let $\mathcal{A}_\zeta$ denote the core of the t-structure corresponding to $\zeta$, i.e. the full subcategory of objects whose HN factors have phases in $[0,1)$. \[prop\_27\] For $\theta\in C_{{\mathcal{A}}_{\mathbf{k}}}^*$, we have $\mathcal{A}_{\zeta(\theta)}=\mathcal{A}_{{\mathbf{k}}}$. We will prove by induction with respect to the length $l$ of the sequence ${\mathbf{k}}$. For $1\leq r\leq l$ we put ${{\mathcal{A}}}^{(r)}:={{\mathcal{A}}}_{{\mathbf{k}}^{(r)}}$, where ${\mathbf{k}}^{(r)}$ is the truncated sequence. For $i\in I$, let $W^{(r-1)}_{i}$ denote the hyperplane which is perpendicular to $s^{(r)}_{i}$: $$W^{(r-1)}_{i}:= \Bigl\{\theta\,\Big|\, \bigl\langle\theta,[s^{(r)}_{i}]\bigr\rangle = 0 \Bigr\}.$$ Note that the boundary of $C_{{\mathcal{A}}^{{(r-1)}}}^*$ is contained in the union of $W^{(r-1)}_{i}$’s. Assume that $\mathcal{A}_{\zeta(\theta)}={\mathcal{A}}^{(r-1)}$ for $\theta\in C_{{\mathcal{A}}^{(r-1)}}^*$. Take $\theta'\in C_{{\mathcal{A}}_{(r)}}^*$ which is sufficiently close to the hyperplane $W^{(r-1)}_{k_r}=W^{(r)}_{k_r}$ and which is sufficiently far from the other hyperplanes. It is enough to show that ${\mathcal{A}}_{\zeta(\theta')}={\mathcal{A}}^{(r)}$. In the case of ${\varepsilon}(r)=+$, we have $${{\mathrm{Re}}}Z_{\theta}(s(r)),{{\mathrm{Re}}}Z_{\theta'}(s(r))<0,\quad {{\mathrm{Im}}}Z_{\theta}(s(r))>0,\quad {{\mathrm{Im}}}Z_{\theta'}(s(r))<0 $$ (see Figure \[fig\_WC\]). So the core ${\mathcal{A}}_{\zeta(\theta')}$ is given by tilting the core ${\mathcal{A}}_{\zeta(\theta)}$ with respect to the torsion pair $$\left({{\mathcal{S}}}{(r)},({{\mathcal{S}}}{(r)})^\bot\right)$$ where $${{\mathcal{S}}}{(r)}:=\bigl\{\bigl(s^{(r)}\bigr)^{\oplus n}\mid n\geq 0\bigr\}.$$ By , we get ${\mathcal{A}}_{\zeta(\theta')}={\mathcal{A}}^{(r)}$. We can see in the case of ${\varepsilon}(r)=-$ in the same way. Assume we have ${C}_{\mathbf{k}}^*={C}_{{\mathbf{k}}'}^*$. Then, the equivalence $\Phi_{{\mathbf{k}}'}\circ\Phi_{\mathbf{k}}^{-1}$ induces an equivalence from ${{\mathrm{mod}}}J_{{\mathbf{k}}}$ to ${{\mathrm{mod}}}J_{{\mathbf{k}}'}$. Moreover, there is a unique permutation $\kappa\in \mathfrak{S}_I$ of $I$ such that $$\Phi_{{\mathbf{k}}'}\circ\Phi_{\mathbf{k}}^{-1}(s_{{\mathbf{k}},i})=s_{{\mathbf{k}}',\kappa(i)} $$ The equivalence for ${{\mathrm{mod}}}J_{{\mathbf{k}}}$ is a consequence of Proposition \[prop\_27\]. The permutation is induced by the description of the boundary of the chamber. Statements {#sec_statement} ========== Quiver Grassmannian {#subsec_grass} ------------------- Let $\Gamma_{{\mathbf{k}},i}$ denote the direct summand of $\Gamma_{\mathbf{k}}$ and $P_{{\mathbf{k}},i}$ denote the projective indecomposable $J_{\mathbf{k}}$-module. We put $$R_{{\mathbf{k}},i}:=H^{1}_{\bar{{\mathcal{A}}}}(\Phi_{{\mathbf{k}}}^{-1}(\Gamma_{{\mathbf{k}},i}))=H^{1}_{\bar{{\mathcal{A}}}}(\Phi_{{\mathbf{k}}}^{-1}(P_{{\mathbf{k}},i}))\in {{\mathcal{T}}}_{\mathbf{k}}\subset {\mathcal{A}}.$$ For $\mathbf{v}\in L$, let $\mathrm{Grass}(\mathbf{k};i,\mathbf{v})$ be the moduli scheme which parametrizes elements $V$ in ${\mathcal{A}}$ equipped with surjections from $R_{{\mathbf{k}},i}$ such that $[V]=\mathbf{v}$ $$\mathrm{Grass}(\mathbf{k};i,\mathbf{v}):=\{R_{{\mathbf{k}},i}\twoheadrightarrow V\mid V\in {\mathcal{A}},\ [V]=\mathbf{v}\}.$$ We call $\mathrm{Grass}(\mathbf{k};i,\mathbf{v})$ as a [*quiver Grassmannian*]{}. We can construct the moduli scheme as a GIT quotient . Let ${{\mathcal{M}}}_{\mathcal{A}}$ be the moduli stack of objects in ${\mathcal{A}}$ and $\nu_{{{\mathcal{M}}}_{\mathcal{A}}}$ be the Behrend function on it ([@behrend-dt]). We define $$\begin{aligned} e_+(\mathrm{Grass}(\mathbf{k};i,\mathbf{v}))&:=e(\mathrm{Grass}(\mathbf{k};i,\mathbf{v})),\\ e_-(\mathrm{Grass}(\mathbf{k};i,\mathbf{v}))&:= \int_{\mathrm{Grass}(\mathbf{k};i,\mathbf{v})}\pi^*(\nu_{{{\mathcal{M}}}_{\mathcal{A}}})\cdot{{\mathrm{d}}}e\\ &=\sum_{n\in {{{\mathbb{Z}}}}} n\cdot e(\pi^*(\nu_{{{\mathcal{M}}}_{\mathcal{A}}})^{-1}(n)),\end{aligned}$$ where $\pi\colon \mathrm{Grass}(\mathbf{k};i,\mathbf{v})\to {{\mathcal{M}}}_{\mathcal{A}}$ is the forgetful morphism and $e(-)$ represents the topological Euler characteristics. On non-commutative Donaldson-Thomas invariants {#subsec_ncdt} ---------------------------------------------- ### Non-commutative Donaldson-Thomas invariants {#subsub_ncdt} For a vertex $i\in I$ and an element $\mathbf{v}\in L$, let ${{\mathrm{Hilb}}}_J(i;\mathbf{v})$ be the moduli scheme which parametrizes elements in $V\in {\mathrm{Mod}\hspace{1pt}J}$ equipped with a surjection from $P_i$ such that $[V]=\mathbf{v}$: $${{\mathrm{Hilb}}}_J(i;\mathbf{v}):=\{P_i\twoheadrightarrow V\mid V\in {\mathcal{A}}, [V]=\mathrm{v} \}.$$ \[defn\_ncdt\] We define invariants by $$\begin{aligned} &{{\mathrm{DT}}}_{J,+}(i;\mathbf{v}):=e({{{\mathrm{Hilb}}}_J(i;\mathbf{v})}),\\ &{{\mathrm{DT}}}_{J,-}(i;\mathbf{v}):= \int_{{{\mathrm{Hilb}}}_J(i;\mathbf{v})}\pi^*(\nu_{{{\mathcal{M}}}_{\mathcal{A}}})\cdot {{\mathrm{d}}}e=\sum_{n\in Z}n\cdot e(\pi^*(\nu_{{{\mathcal{M}}}_{\mathcal{A}}})^{-1}(n)),\end{aligned}$$ where $\pi\colon {{{\mathrm{Hilb}}}_J(i;\mathbf{v})}\to {{\mathcal{M}}}_{\mathcal{A}}$ is the forgetful morphism. The non-commutative Donaldson-Thomas invariants in [@szendroi-ncdt] are defined using the Behrend function on ${{\mathrm{Hilb}}}_J(i;\mathbf{v})$ $${{\mathrm{DT}}}_{J,{{\mathrm{Sze}}}}(i;\mathbf{v}):= \int_{{{\mathrm{Hilb}}}_J(i;\mathbf{v})}\nu_{{{\mathrm{Hilb}}}_J(i;\mathbf{v})}\cdot {{\mathrm{d}}}e:=\sum_{n\in Z}n\cdot e(\nu_{{{\mathrm{Hilb}}}_J(i;\mathbf{v})}^{-1}(n)).$$ We have $\pi^*(\nu_{{{\mathcal{M}}}_{\mathcal{A}}})=(-1)^{v_i}\cdot\nu_{{{\mathrm{Hilb}}}_J(i;\mathbf{v})}$ and $${{\mathrm{DT}}}_{J,{{\mathrm{Sze}}}}(i;\mathbf{v})=(-1)^{v_i}\cdot{{\mathrm{DT}}}_{J,-}(i;\mathbf{v}).$$ We define generating functions by $${{\mathcal{Z}}}_{J,\sigma}^{i}:=\sum_{\mathbf{v}}{{\mathrm{DT}}}_{J,\sigma}(i;\mathbf{v})\cdot \mathbf{y}^{\mathbf{v}}_\sigma. $$ ### Torus automorphism via ncDT invariants {#subsub_auto} For a full subcategory ${{\mathcal{C}}}\subset{\mathcal{D}^{\mathrm{fd}}\Gamma}$, we define ${\widehat{{{\mathrm{T}}}}}_{{{\mathcal{C}}},\sigma}$ and ${\widehat{\mathbb{T}}}_{{{\mathcal{C}}},\sigma}$ by $$\begin{aligned} &{\widehat{{{\mathrm{T}}}}}_{{{\mathcal{C}}},\sigma}:=\biggl(\,\prod_{\mathbf{v}\in L\cap C_{{\mathcal{C}}}} {{{\mathbb{C}}}}\cdot \mathbf{y}_\sigma^\mathbf{v}\biggr) \oplus \biggl(\,\bigoplus_{\mathbf{v}\in L-C_{{\mathcal{C}}}} {{{\mathbb{C}}}}\cdot \mathbf{y}_\sigma^\mathbf{v}\biggr),\\ &{\widehat{\mathbb{T}}}_{{{\mathcal{C}}},\sigma}:={{{\mathrm{T}}}}_{{{\mathcal{C}}},\sigma}^\vee\otimes {\widehat{{{\mathrm{T}}}}}_{{{\mathcal{C}}},\sigma}\end{aligned}$$ where $C_{{\mathcal{C}}}\subset L_{{{\mathbb{R}}}}$ is the minimal cone which contains all the classes of elements in ${{\mathcal{C}}}$. They are called [*the completions with respect to ${{\mathcal{C}}}$*]{}. If ${{\mathcal{C}}}$ is a subcategory of a core of a t-structure, then the products extend to these completions. Moreover, if ${{\mathcal{C'}}}\subset{{\mathcal{C}}}$ then the completions with respect to ${{\mathcal{C'}}}$ give subalgebras of the ones with respect to ${{\mathcal{C}}}$. If an automorphism of the completion with respect to ${{\mathcal{C}}}'$ lifts to the one for ${{\mathcal{C}}}$, we use the same symbol as the original automorphism for the lift. Note that ${{\mathcal{Z}}}_{J,\sigma}^{i}$ gives an elements in ${\widehat{{{\mathrm{T}}}}}_{{\mathcal{A}},\sigma}$. We define torus automorphisms $${{\mathcal{DT}}}_{J,\sigma}\colon {\widehat{\mathbb{T}}}_{{\mathcal{A}},\sigma} \overset{\sim}{{\longrightarrow}} {\widehat{\mathbb{T}}}_{{\mathcal{A}},\sigma} $$ by $${{\mathcal{DT}}}_{J,\sigma}(x_{i,\sigma}):=x_{i,\sigma}\cdot {{\mathcal{Z}}}_{J,\sigma}^{i},\quad {{\mathcal{DT}}}_{J,\sigma}(y_{i,\sigma}):=y_{i,\sigma}\cdot \prod_j({{\mathcal{Z}}}_{J,\sigma}^{j})^{\bar{Q}(j,i)}.$$ ### Transformation formula of ncDT invariants {#subsub_trans} We define torus automorphisms $${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}\colon {\mathbb{T}}_{\sigma} \overset{\sim}{{\longrightarrow}} {\mathbb{T}}_{\sigma}$$ by $$\begin{aligned} {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}(x_{{\mathbf{k}},i,\sigma}) &:=x_{{\mathbf{k}},i,\sigma}\cdot \Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Grass}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{-\mathbf{v}}\Biggr),\\ {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}(y_{{\mathbf{k}},i,\sigma}) &:=y_{{\mathbf{k}},i,\sigma}\cdot \prod_j\Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Grass}(\mathbf{k};j,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{-\mathbf{v}}\Biggr)^{\bar{Q}(j,i)}.\end{aligned}$$ They lift to the completions with respect to ${\mathcal{A}}_{\mathbf{k}}$. We also define ${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma}$ by $${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma} :=\Sigma\circ {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}\circ \Sigma $$ which lift to the completions with respect to ${\mathcal{A}}$. See for the definition of $\Sigma$. \[thm\_trans\] The composition $${{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma}^{-1}\circ {{\mathcal{DT}}}_{J,\sigma}$$ preserves ${\widehat{\mathbb{T}}}_{{{\mathcal{F}}}_{\mathbf{k}},\sigma}$ and lifts to the automorphism of ${\widehat{\mathbb{T}}}_{{\mathcal{A}}_{\mathbf{k}},\sigma}$. Moreover, we have the following identity of automorphisms of ${\widehat{\mathbb{T}}}_{{\mathcal{A}}_{\mathbf{k}},\sigma}$ $${{\mathcal{DT}}}_{J_{\mathbf{k}},\sigma} = {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma}^{-1}\circ {{\mathcal{DT}}}_{J,\sigma}\circ {{\mathrm{Ad}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}.$$ Caldero-Chapoton formula ------------------------ In this subsection, we put $\sigma=+$ and use notations without “+”. We identify ${{{\mathbb{C}}}}(x_1,\ldots,x_n)$ with the fractional field of $T_Q$. \[thm\_CC\] We have $$\label{eq_CC} \mathrm{FZ}_{{\mathbf{k}},i}(\underline{x})= x_{{\mathbf{k}},i}\cdot \Biggl(\sum_\mathbf{v}e\Bigr(\mathrm{Grass}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}^{-\mathbf{v}}\Biggr).$$ where $(\underline{y})^{-\mathbf{v}}=\prod_j (y_j)^{-v_j}$ and $y_j=\prod_i(x_i)^{\bar{Q}(i,j)}$ \[ex\_cluster\] If we take a sequence ${\mathbf{k}}=(k)$ of length $1$, then we have $$R_{(k),i}=\begin{cases} 0 & i\neq k,\\ s_k & i=k. \end{cases}$$ Hence we have $$\label{eq_preCT} FZ_{i,k}(\underline{x})= \begin{cases} x_i' & i\neq k,\\ x_k'(1+(y_k)^{-1}) & i=k, \end{cases}$$ where $y_k=\prod_j (x_j)^{\bar{Q}(j,k)}$, $x_i':=\mathbf{x}^{[\Gamma'_i]}$ and $\Gamma'_i$ is the direct summand of $\mu_k\Gamma$. Note that by we have $$x_i'=x_i\ (i\neq k),\quad x_k'=(x_k)^{-1}\prod_j (x_j)^{Q(j,k)},\quad$$ Substituting these for , we get the cluster transformation . Review: Motivic Hall algebra {#sec_MH} ============================ Motivic Hall algebra and its limit ---------------------------------- ### Relative Grothendieck ring of stacks For an algebraic stack ${\mathcal{S}}$, let $\mathrm{St}/{\mathcal{S}}$ denote the category whose objects are finite type stacks ${\mathcal{X}}$ over ${{{\mathbb{C}}}}$ equipped with a morphism to ${\mathcal{S}}$. A morphism of stacks $f\colon \to Y$ is said to be a [*geometric bijection*]{} if it is representable and the induced functor on groupoids of ${{{\mathbb{C}}}}$-valued points $$f({{{\mathbb{C}}}})\colon X({{{\mathbb{C}}}}) \to Y({{{\mathbb{C}}}})$$ is an equivalence of categories ([@bridgeland-hall Definition 3.1]). A morphism of stacks $f\colon \to Y$ is said to be a [*Zariski fibration*]{} if its pullback to any scheme is a Zariski fibration of schemes ([@bridgeland-hall Definition 3.3]). We define $K(\mathrm{St}/{\mathcal{S}})$ by the free Abelian group spanned by isomorphism classes of $\mathrm{St}/{\mathcal{S}}$ modulo the following relations: 1. $[{\mathcal{X}}_1\sqcup {\mathcal{X}}_2\overset{f_1\sqcup f_2}{{\longrightarrow}}{\mathcal{S}}]= [{\mathcal{X}}_1\overset{f_1}{{\longrightarrow}}{\mathcal{S}}]+ [{\mathcal{X}}_2\overset{f_2}{{\longrightarrow}}{\mathcal{S}}]$, 2. $[{\mathcal{X}}_1\overset{f_1}{{\longrightarrow}}{\mathcal{S}}]=[{\mathcal{X}}_2\overset{f_2}{{\longrightarrow}}{\mathcal{S}}]$ if there is a geometric bijection $g\colon{\mathcal{X}}_1\to{\mathcal{X}}_2$ with $f_1=f_2\circ g$, 3. $[{\mathcal{X}}_1\overset{f_1}{{\longrightarrow}}{\mathcal{S}}]=[{\mathcal{X}}_2\overset{f_2}{{\longrightarrow}}{\mathcal{S}}]$ if there is a factorisations $f_i=g\circ h_i$ such that $h_i\colon {\mathcal{X}}_i\to {\mathcal{Y}}$ are Zariski fibrations with the same fibres ([@bridgeland-hall Definition 3.6]). We call $K(\mathrm{St}/{\mathcal{S}})$ as the [*relative Grothendieck ring of stacks over ${\mathcal{S}}$*]{}. A morphism of stacks $\psi\colon\mathcal{T}\to\mathcal{S}$ induces a map $$\psi_*\colon K({{\mathrm{St}}}/{\mathcal{T}})\to K({{\mathrm{St}}}/{\mathcal{S}})$$ sending $[g\colon {\mathcal{Y}}\to{\mathcal{T}}]$ to $[\psi \circ g\colon {\mathcal{X}}\to{\mathcal{S}}]$. If $\psi$ is of finite type it also induces a map $$\psi^*\colon K({{\mathrm{St}}}/{\mathcal{S}})\to K({{\mathrm{St}}}/{\mathcal{T}})$$ sending $[f\colon {\mathcal{X}}\to{\mathcal{S}}]$ to the map $[g\colon {\mathcal{Y}}\to{\mathcal{T}}]$ in the following Cartesian diagram: $$\xymatrix{ \mathcal{Y} \ar[d] \ar[r]^g \ar@{}[dr]|\square & {\mathcal{T}}\ar[d]^{\psi} & \\ \mathcal{X} \ar[r]_f & {\mathcal{S}}\ar@{}[r]_. & }$$ ### Motivic Hall algebra Let ${\mathcal{M}_{{\mathcal{A}}}}$ be the moduli stack of all objects in ${\mathcal{A}}=\mathrm{mod}J$ and ${\mathcal{M}^{(2)}_{{\mathcal{A}}}}$ be the moduli stack of all exact sequences in ${\mathcal{A}}$. Let $p_\varepsilon\colon {\mathcal{M}^{(2)}_{{\mathcal{A}}}}\to{\mathcal{M}_{{\mathcal{A}}}}$ ($\varepsilon=1,2,3$) be the morphism given by taking $\varepsilon$-th terms of exact sequences. Then, $p_2$ is of finite type. Using the diagram $$\xymatrix{ {\mathcal{M}^{(2)}_{{\mathcal{A}}}}\ar[d]_{p_1\times p_3} \ar[r]^{p_2} & {\mathcal{M}_{{\mathcal{A}}}}\\ {\mathcal{M}_{{\mathcal{A}}}}\times{\mathcal{M}_{{\mathcal{A}}}}& }$$ we define a product $*$ on $K(\mathrm{St}/{\mathcal{M}_{{\mathcal{A}}}})$ by $$\label{eq_hall} *:=(p_2)_*(p_1\times p_3)^*\colon K(\mathrm{St}/{\mathcal{M}_{{\mathcal{A}}}})\otimes K(\mathrm{St}/{\mathcal{M}_{{\mathcal{A}}}})\to K(\mathrm{St}/{\mathcal{M}_{{\mathcal{A}}}}).$$ We put $\mathrm{MH}({\mathcal{A}}):=K(\mathrm{St}/{\mathcal{M}_{{\mathcal{A}}}})$. The algebra $(\mathrm{MH}({\mathcal{A}}),*)$ is called the [*motivic Hall algebra*]{} of the Abelian category ${\mathcal{A}}$. The motivic Hall algebra $(\mathrm{MH}({\mathcal{A}}),*)$ is associative. The $3$-Calabi-Yau property of the category ${\mathcal{A}}$ is not necessary for this theorem. ### Semi-classical limit of the motivic Hall algebra Let ${{\mathrm{MH}}}_0({\mathcal{A}})\subset{{\mathrm{MH}}}({\mathcal{A}})$ be the $K({{\mathrm{Var}}}/{{{\mathbb{C}}}})[\mathbb{L}^{-1}]$-submodule generated by classes $$[X\overset{f}{{\longrightarrow}}{{\mathcal{M}}}_{{\mathcal{A}}}]$$ with $X$ a variety. - $\mathrm{MH}_0({\mathcal{A}})\subset \mathrm{MH}({\mathcal{A}})$ is a subring. - The product induced on the quotient $$\mathrm{MH}_{{{\mathrm{sc}}}}({\mathcal{A}}):=\mathrm{MH}_{0}({\mathcal{A}})/(\mathbb{L}-1)\mathrm{MH}_{0}({\mathcal{A}})$$ is commutative $K({{\mathrm{Var}}}/{{{\mathbb{C}}}})$-algebra. We define a Poisson bracket $\{-,-\}$ on $\mathrm{MH}_{{{\mathrm{sc}}}}({\mathcal{A}})$ by $$\{f,g\}=\frac{f*g-g*f}{\mathbb{L}-1}\quad \mathrm{mod}(\mathbb{L}-1).$$ ### Completion of the motivic Hall algebra Note that the moduli stack has the canonical decomposition $${\mathcal{M}_{{\mathcal{A}}}}=\bigsqcup_{\mathbf{v}\in C_{{\mathcal{A}}}\cap L} \mathcal{M}_{{\mathcal{A}}}(\mathbf{v}).$$ We put $${\widehat{{{\mathrm{MH}}}}}({\mathcal{A}}):=\prod_{\mathbf{v}\in C_{{\mathcal{A}}}\cap L} K({{\mathrm{St}}}/{\mathcal{M}_{{\mathcal{A}}}}(\mathbf{v})),$$ then the $*$-product canonically extends to ${\widehat{{{\mathrm{MH}}}}}({\mathcal{A}})$. Let ${{\mathcal{C}}}\subset{{\mathcal{A}}}$ be an extension closed full subcategory. Assume that the moduli stack ${{\mathcal{M}}}_{{{\mathcal{C}}}}\subset{{\mathcal{M}}}_{{\mathcal{A}}}$ of objects in ${{\mathcal{C}}}$ is algebraic. Let $\mathrm{MH}({{\mathcal{C}}})$ denote the subalgebra consisting of the elements $[f\colon {{\mathcal{X}}}\to{{\mathcal{M}}}_{{\mathcal{A}}}]$ such that $f$ factors through ${{\mathcal{M}}}_{{{\mathcal{C}}}}\subset{{\mathcal{M}}}_{{\mathcal{A}}}$. We put $${\widehat{{{\mathrm{MH}}}}}({{\mathcal{C}}}):=\prod_{\mathbf{v}\in C_{{\mathcal{C}}}\cap L}\Bigl({{{\mathrm{MH}}}}({{\mathcal{C}}})\cap K({{\mathrm{St}}}/{\mathcal{M}_{{\mathcal{A}}}}(\mathbf{v}))\Bigr).$$ We define ${\widehat{{{\mathrm{MH}}}}}_{0}({\mathcal{A}})$, ${\widehat{{{\mathrm{MH}}}}}_{{{\mathrm{sc}}}}({\mathcal{A}})$ ${\widehat{{{\mathrm{MH}}}}}_{0}({{\mathcal{C}}})$ and ${\widehat{{{\mathrm{MH}}}}}_{{{\mathrm{sc}}}}({{\mathcal{C}}})$ in the same way. Quantum torus and integration map --------------------------------- ### Quantum torus and semi-classical limit The [*quantum torus*]{}, [*dual quantum torus*]{} and [*double quantum torus*]{} for ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ is the ${{{\mathbb{C}}}}(t)$-vector spaces $$\mathrm{QT}:=\sum_{\mathbf{v}\in L}{{{\mathbb{C}}}}(t)\cdot \mathbf{y}^{\mathbf{v}}, \quad \mathrm{QT}^\vee:=\sum_{\mathbf{w}\in M}{{{\mathbb{C}}}}(t)\cdot \mathbf{x}^{\mathbf{w}}, \quad \mathrm{Q}\mathbb{T}:=\mathrm{QT}^\vee\otimes\mathrm{QT}$$ with the following product structure: $$\mathbf{y}^\mathbf{v}\cdot\mathbf{y}^{\mathbf{v}'}=t^{\chi(\mathbf{v}',\mathbf{v})}\mathbf{y}^{\mathbf{v}+\mathbf{v}'},\quad \mathbf{x}^\mathbf{w}\cdot\mathbf{x}^{\mathbf{w}'}=\mathbf{x}^{\mathbf{w}+\mathbf{w}'},\quad y_i\cdot x_j=t^{\delta_{i,j}}x_j\cdot y_i $$ where $y_i:=\mathbf{y}^{[s_i]}$ and $x_j:=\mathbf{x}^{[P_i]}$. We define the surjective algebra homomorphism $\pi\colon {{{\mathrm{Q}}}\mathbb{T}}\twoheadrightarrow \mathrm{QT}^\vee$ by $$x_i \otimes 1 \longmapsto x_i,\quad 1 \otimes y_i \longmapsto \mathbf{x}^{[s_i]}.$$ The kernel of $\pi$ is generated by $\{(\mathbf{x}^{[s_i]}\otimes 1) - (1\otimes y_i)\mid i \in I\}$. Let ${{{\mathrm{QT}}}}_{0}$, ${{{\mathrm{QT}}}}_{0}^\vee$ and ${{{\mathrm{Q}}}\mathbb{T}}_{0}$ denote the ${{{\mathbb{C}}}}[t^\pm]$-subalgebra generated by $\mathbf{y}^\mathbf{v}$’s and $\mathbf{x}^\mathbf{w}$’s. We put $${{{\mathrm{QT}}}}_{{{\mathrm{sc}}},\pm}:={{{\mathrm{QT}}}}_0/(t\pm 1){{{\mathrm{QT}}}}_0$$ and define ${{{\mathrm{QT}}}}_{{{\mathrm{sc}}},\pm}^\vee$ and ${{{\mathrm{Q}}}\mathbb{T}}_{{{\mathrm{sc}}},\pm}^\vee$ in the same way. Let $\{-,-\}$ denote the Poisson bracket on these quotients. ### Integration map \[thm\_integral\] There is a unique $L$-graded linear map $$I_\pm\colon \mathrm{MH}_{{{\mathrm{sc}}}}({\mathcal{A}})\to \mathrm{QT}_{{{\mathrm{sc}}},\pm}$$ such that if $X$ is a variety with a map $f\colon X\to{\mathcal{M}_{{\mathcal{A}}}}$ factoring through ${\mathcal{M}_{{\mathcal{A}}}}(\mathbf{v})\subset {\mathcal{M}_{{\mathcal{A}}}}$ then $$\begin{aligned} I_+\bigl([f\colon X\to{\mathcal{M}_{{\mathcal{A}}}}]\bigr)&:=e(X)\cdot \mathbf{y}^{\mathbf{v}},\\ I_-\bigl([f\colon X\to{\mathcal{M}_{{\mathcal{A}}}}]\bigr)&:=\biggl(\,\sum_{n\in{{{\mathbb{Z}}}}}n\cdot e\bigl((f^*\nu_{{{\mathcal{M}}}_{\mathcal{A}}})^{-1}(n)\bigr)\biggr)\cdot \mathbf{y}^{\mathbf{v}}\end{aligned}$$ Moreover, $I_\pm$ is a Poisson algebra homomorphism. \[conj\_ks\] There exists[^8] an $L$-graded $\Lambda$-algebra homomorphism $$I_{{{\mathrm{KS}}}}\colon \mathrm{MH}({\mathcal{A}})\to \mathrm{QT}({\mathcal{A}})$$ defined by taking “motivic invariants”. Since it is $L$-graded, the homomorphism $I$ (and $I_{{{\mathrm{KS}}}}$ if it exists) extends to the completion. We use the same symbol for the extended homomorphism. Absence of poles {#subsec_absence} ---------------- Let ${{\mathcal{C}}}$ be one of the categories ${\mathcal{A}}$, ${\mathcal{A}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}[-1]$, ${{\mathcal{T}}}_{\mathbf{k}}^\bot$ and ${{\mathcal{S}}}(r)$. As we showed in §\[subsec\_Tstr\], we have a Bridgeland’s stability condition $(Z,{{\mathcal{P}}})$ on ${\mathcal{D}^{\mathrm{fd}}\Gamma}$ such that $${{\mathcal{P}}}((0,1])={\mathcal{A}},\quad {{\mathcal{P}}}((0,\phi])={{\mathcal{C}}}$$ for some $0<\phi\leq 1$. By the results in [@joyce-1], we get the algebraic moduli stacks ${{\mathcal{M}}}_{{\mathcal{C}}}$ of objects in ${{\mathcal{C}}}$. We put $$\begin{aligned} \label{eq_ep} {\varepsilon}_{{{\mathcal{C}}}}:=\log(1+{{\mathcal{M}}}_{{{\mathcal{C}}}}):=\sum_{l\geq 1}\frac{(-1)^l}{l}{{\mathcal{M}}}_{{{\mathcal{C}}}}*\cdots*{{\mathcal{M}}}_{{{\mathcal{C}}}}\in {\widehat{{{\mathrm{MH}}}}}({{\mathcal{C}}})\end{aligned}$$ and ${\tilde{\varepsilon}}_{{{\mathcal{C}}}}:=(\mathbb{L}-1){\varepsilon}_{{{\mathcal{C}}}}\in {\widehat{{{\mathrm{MH}}}}}({{\mathcal{C}}})$. Then we have $$\label{eq_exp} {{\mathcal{M}}}_{{{\mathcal{C}}}}=\exp({\varepsilon}_{{{\mathcal{C}}}}):=\sum_{l\geq 1}\frac{1}{l!}\,{\varepsilon}_{{{\mathcal{C}}}}*\cdots*{\varepsilon}_{{{\mathcal{C}}}}.$$ \[thm\_pole\] ${\tilde{\varepsilon}_{{{\mathcal{C}}}_{\mathbf{k}}}}\in {\widehat{{{\mathrm{MH}}}}}_{0}({{\mathcal{C}}})$. We put $${\hat{\varepsilon}}_{{\mathcal{C}}}:={\tilde{\varepsilon}}_{{\mathcal{C}}}|_{\mathbb{L}-1}\in \widehat{{{\mathrm{MH}}}}_{{{\mathrm{sc}}}}.$$ Proof {#sec_proof} ===== Hall algebra identities {#subsec_eq_in_MH} ----------------------- Throughout this subsection. let ${{\mathcal{C}}}$ be one of the categories ${\mathcal{A}}$, ${\mathcal{A}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}[-1]$, ${{\mathcal{T}}}_{\mathbf{k}}^\bot$ and ${{\mathcal{S}}}_{(r)}$. Let ${{\mathcal{M}}}_{{\mathcal{C}}}$ be the moduli stack of objects in ${{\mathcal{C}}}$. For an element $P\in{\mathrm{per}\Gamma}$ we define the following moduli stacks: $$\mathfrak{Hom}(P,{{{\mathcal{C}}}}) := \{(f,E)\mid E\in {{{\mathcal{C}}}}, f\in {{{\mathrm{Hom}}}}(P,E)\}.$$ \[prop\_mhsi\] $${{\mathrm{Hilb}}}_{J}(i)={{\mathfrak{Hom}}}(P_i,{{{\mathcal{A}}}})*{{\mathcal{M}}}_{\mathcal{A}}^{-1}\quad \text{\textup{(motivic Hilbert scheme identity)}}.$$ \[prop\_mtpi\] $${{\mathcal{M}}}_{{\mathcal{A}}}={{\mathcal{M}}}_{\mathcal{T}_{{\mathbf{k}}}}*{{\mathcal{M}}}_{\mathcal{T}_{{\mathbf{k}}}^\bot},\quad {{\mathcal{M}}}_{{\mathcal{A}}_{\mathbf{k}}}={{\mathcal{M}}}_{\mathcal{T}_{{\mathbf{k}}}^\bot}*{{\mathcal{M}}}_{\mathcal{T}_{{\mathbf{k}}}[-1]}\quad \text{\textup{(motivic torsion pair identity)}}.$$ For an element $R\in {\mathcal{A}}$, let ${{\mathrm{Grass}}}(R,{{\mathcal{A}}})$ be the moduli stack of elements in ${\mathcal{A}}$ equipped with surjections from $R$: $${{\mathrm{Grass}}}(R,{{\mathcal{A}}}) := \{(f,E)\mid E\in {{\mathcal{A}}}, f\in {{{\mathrm{Hom}}}}(R,E), \text{$f$\,:\,surjective}\}.$$ \[prop\_grass\] $$\label{eq_grass} {{\mathfrak{Hom}}}(R,{{\mathcal{A}}})= {{\mathfrak{Grass}}}(R,{{\mathcal{A}}})* {{\mathcal{M}}}_{{\mathcal{A}}}^{-1}.$$ We can prove in the same way as the motivic Hilbert scheme identities. \[prop\_hom2\] $$\label{eq_hom2} {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}^\bot_{\mathbf{k}}} = {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{\mathcal{A}}).$$ As in [@bridgeland-dtpt §4], a ${{{\mathbb{C}}}}$-valued point of ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}^\bot_{\mathbf{k}}} $ is represented by a diagram $$\xymatrix{ & P_{{\mathbf{k}},i}[1]\ar[d] & & & \\ 0\ar[r] & Y\ar[r] & X\ar[r] & Z\ar[r] & 0 }$$ with $F\in {{\mathcal{T}}}_{\mathbf{k}}$ and $Z \in {{\mathcal{T}}}{}^\bot_{\mathbf{k}}$. By composing the morphisms in the diagram, we get a family of morphism $P_{{\mathbf{k}},i}[1]\to X$ on ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}^\bot_{\mathbf{k}}} $, which induces a morphism from ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}^\bot_{\mathbf{k}}} $ to the moduli stack ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{\mathcal{A}})$. Since ${{{\mathrm{Hom}}}}(P_{{\mathbf{k}},i}[1],Z)={{{\mathrm{Hom}}}}(P_{{\mathbf{k}},i}[1],Z[-1])=0$, we have $${{{\mathrm{Hom}}}}(P_{{\mathbf{k}},i}[1],X)={{{\mathrm{Hom}}}}(P_{{\mathbf{k}},i}[1],Y).$$ The axiom of the torsion pair and the equation above provide an equivalence of ${{{\mathbb{C}}}}$-valued points induced by the morphism from ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}^\bot_{\mathbf{k}}} $ to ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{\mathcal{A}})$. The following lemma is clear: $$\label{eq_hom3} {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{\mathcal{A}})={{\mathfrak{Hom}}}(R_{{\mathbf{k}},i},{\mathcal{A}}).$$ The following equation plays a principal role in this paper: \[prop\_hom4\] We have $$\label{eq_hom4} {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}^{-1}= {{\mathrm{Grass}}}(R_{{\mathbf{k}},i},{\mathcal{A}})$$ . In particular, ${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}^{-1}\in {{{\mathrm{MH}}}}_0$. $$\begin{aligned} {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})*{{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}^{-1} \ \ =\ &\ {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{\mathbf{k}}})* {{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}{}^\bot}* {{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}^\bot}^{-1}* {{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}^{-1}\\ \overset{\eqref{eq_hom2}}{=} &\ {{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{\mathcal{A}}})*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\\ \overset{\eqref{eq_hom3}}{=} &\ {{\mathfrak{Hom}}}(R_{{\mathbf{k}},i},{{\mathcal{A}}})*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\\ \overset{\eqref{eq_grass}}{=} &\ {{\mathrm{Grass}}}(R_{{\mathbf{k}},i},{{\mathcal{A}}}[-1]).\end{aligned}$$ \[prop\_mfi\] We have $${{\mathcal{M}}}_{\mathcal{T}_{{\mathbf{k}}}}=\bigl({{\mathcal{M}}}_{\mathcal{S}{(1)}}\bigr)^{{\varepsilon}(1)}*\cdots *\bigl({{\mathcal{M}}}_{\mathcal{S}{(l)}}\bigr)^{{\varepsilon}(l)}\quad ({{\mathrm{motivic\ factorization\ identity}}}).$$ We can prove in the same way as the torsion pair identities. For $\mathbf{w}\in M$, we define $${{\mathcal{M}}}_{{{\mathcal{C}}}}[\mathbf{w}]:= \sum_{\mathbf{v}}\mathbb{L}^{\chi(\mathbf{w},\mathbf{v})}\cdot{{\mathcal{M}}}_{{{\mathcal{C}}}}(\mathbf{v}).$$ We put $\mathbf{w}_{{\mathbf{k}},i}:=[\Gamma_{{\mathbf{k}},i}]$. $${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i},{{{\mathcal{C}}}}) = {{\mathcal{M}}}_{{{\mathcal{C}}}}[\mathbf{w_{{\mathbf{k}},i}}]$$ We can realize ${{\mathcal{M}}}_{{{\mathcal{C}}}}({{{\mathbf{v}}}})$ as a quotient stack $[{\mathcal{X}}/GL({{{\mathbf{v}}}})]$, where $GL({{{\mathbf{v}}}})$ is a direct product of ${{\mathrm{GL}}}(v_i)$’s. Note that ${{\mathfrak{Hom}}}(\Gamma_i,{{{\mathcal{C}}}})({{{\mathbf{v}}}})$ is a vector bundle of rank $v_i$ on ${{\mathcal{M}}}_{{{\mathcal{C}}}}({{{\mathbf{v}}}})$, whose pull-back on ${\mathcal{X}}$ is trivial. Since $GL({{{\mathbf{v}}}})$ is special, ${{\mathfrak{Hom}}}(\Gamma_i,{{{\mathcal{C}}}})({{{\mathbf{v}}}})$ is Zariski locally trivial. \[cor\_hom\_factor\] $${{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{T}}}_{{\mathbf{k}}}})=\bigl({{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{S}}}(1)})\bigr)^{{\varepsilon}(1)}*\cdots *\bigl({{\mathfrak{Hom}}}(P_{{\mathbf{k}},i}[1],{{{\mathcal{S}}}(l)})\bigr)^{{\varepsilon}(l)}.$$ Idea {#subsec_idea} ---- The purpose of this subsection is to show the idea of the proof. In this subsection we [*assume*]{} Conjecture \[conj\_ks\] is true, since it would make the argument clearer. The actual proof starts from the next subsection, which is independent from Conjecture \[conj\_ks\]. We define the torus automorphism $$\begin{array}{ccccc} \widehat{{{\mathrm{q\text{-}Ad}}}}_{{\mathcal{A}}}:=\mathrm{Ad}_{I_{{{\mathrm{KS}}}}({{\mathcal{M}}}_{{{\mathcal{A}}}})} & \colon & {\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}({{\mathcal{A}}}) & \overset{\sim}{\longrightarrow} & {\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}({{\mathcal{A}}})\\ && \bullet & \longmapsto & I_{{{\mathrm{KS}}}}({{\mathcal{M}}}_{{\mathcal{A}}})\times \bullet \times I_{{{\mathrm{KS}}}}({{\mathcal{M}}}_{{\mathcal{A}}})^{-1}. \end{array}$$ - We have $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{A}}}}(x_i)= x_i\cdot I_{{{\mathrm{KS}}}}\Bigl({{{\mathrm{Hom}}}}(P_i,{{\mathcal{A}}})*{{\mathcal{M}}}_{{{\mathcal{A}}}}^{-1}\Bigr).$$ - We have $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{\mathcal{A}}}(x_i)= x_i\cdot I_{{{\mathrm{KS}}}}({{\mathrm{Hilb}}}_J(i)).$$ In particular, the non-commutative Donaldson-Thomas invariants for $J$ are encoded in the torus automorphism $\widehat{{{\mathrm{q\text{-}Ad}}}}_{{\mathcal{A}}}$ Note that we have $$\label{eq_commutator} {{\mathcal{E}}}\cdot x_i=x_i\cdot {{\mathcal{E}}}|_{y_j=y_j\cdot t^{2\delta_{ij}}}$$ for ${{\mathcal{E}}}\in{\widehat{{{\mathrm{QT}}}}}({\mathcal{A}})$, where ${{\mathcal{E}}}|_{y_j=y_j\cdot t^{2\delta_{ij}}}$ is given by substituting $y_j\cdot t^{2\delta_{ij}}$ for $y_j$. We call this as [*the commutator identity*]{}. The first claim is a consequence of the commutator identity. The second one follows from the “motivic Hilbert scheme identity” (Proposition \[prop\_mhsi\]). We define the torus automorphism $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{C}}}}:=\mathrm{Ad}_{I_{{{\mathrm{KS}}}}({{\mathcal{M}}}_{{{\mathcal{C}}}})}\colon{\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}({{\mathcal{C}}})\overset{\sim}{\longrightarrow}{\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}({{\mathcal{C}}})$$ in the same way. We have the factorization identities $$\begin{aligned} &\widehat{{{\mathrm{q\text{-}Ad}}}}_{{\mathcal{A}}}= \widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}}\circ \widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}^\bot},\label{eq_q_factor1}\\ &\widehat{{{\mathrm{q\text{-}Ad}}}}_{{\mathcal{A}}_{\mathbf{k}}}= \widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}{}^\bot}\circ \widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]},\label{eq_q_factor2}. $$ In particular, $\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}}$ and $\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]}$ provide a transformation formula of non-commutative Donaldson-Thomas invariants between $J$ and $J_{\mathbf{k}}$. They are consequences of the “motivic torsion pair identity” (Proposition \[prop\_mtpi\]). - We have $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]}(x_{{\mathbf{k}},i})= x_{{\mathbf{k}},i}\cdot I_{{{\mathrm{KS}}}}\Bigl({{{\mathrm{Hom}}}}(P_{{\mathbf{k}},i},{{\mathcal{T}}}_{\mathbf{k}}[-1])*{{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]}^{-1}\Bigr).$$ - The torus automorphism can be described in terms of quiver Grassmannians: $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]}(x_{{\mathbf{k}},i})= x_{{\mathbf{k}},i}\cdot I_{{{\mathrm{KS}}}}\bigl({{\mathrm{Grass}}}(R_{{\mathbf{k}},i}[-1],{\mathcal{A}}[-1])\bigr).$$ In particular, the transformation formula of the non-commutative Donaldson-Thomas invariants can be described in terms of quiver Grassmannians. The first claim follows by the commutator identity. The second one follows by the “motivic quiver Grassmannian identity” (Proposition ). We have the factorization identity $$\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1]}= \Bigl(\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{S}}}(1)[-1]}\Bigr)^{{\varepsilon}(1)}\circ\cdots\circ \Bigr(\widehat{{{\mathrm{q\text{-}Ad}}}}_{{{\mathcal{S}}}(l)[-1]}\Bigr)^{{\varepsilon}(l)}. $$ This is a consequences of the “motivic factorization identity” (Proposition \[prop\_mfi\]). Proof {#subsec_auto} ----- ### Definition of the automorphism {#subsec_531} Let ${{\mathcal{C}}}$ be one of the categories ${\mathcal{A}}$, ${\mathcal{A}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}[-1]$, ${{\mathcal{T}}}_{\mathbf{k}}^\bot$ and ${{\mathcal{S}}}(r)$. Recall that we have ${\hat{\varepsilon}}_{{\mathcal{C}}}\in \widehat{{{\mathrm{MH}}}}_0({{\mathcal{C}}})$. $$\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}},\sigma} := \exp\Bigl(\mathrm{ad}_{I_\sigma\bigl({\hat{\varepsilon}}_{{\mathcal{C}}}\bigr)}\Bigr)\colon {\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}_{{{\mathrm{sc}}},\sigma}({{\mathcal{C}}})\overset{\sim}{{\longrightarrow}}{\widehat{{{{\mathrm{Q}}}\mathbb{T}}}}_{{{\mathrm{sc}}},\sigma}({{\mathcal{C}}})$$ We will prove Theorem \[thm\_520\], \[thm\_521\], \[thm\_factor1\] and \[thm\_factor2\] which induce all the results in §\[sec\_statement\]. ### Infinitesimal commutator identity For ${\varepsilon}=\sum{\varepsilon}(\mathbf{v})\in {\widehat{{{\mathrm{MH}}}}}({\mathcal{A}})$ and ${\hat{\varepsilon}}=\sum{\hat{\varepsilon}}(\mathbf{v})\in {\widehat{{{\mathrm{MH}}}}}_{{{\mathrm{sc}}}}({\mathcal{A}})$ we define ${\varepsilon}[\mathbf{w}_{i}]\in {\widehat{{{\mathrm{MH}}}}}({\mathcal{A}})$ and ${\hat{\varepsilon}}\{\mathbf{w}_i\}\in {\widehat{{{\mathrm{MH}}}}}_{{{\mathrm{sc}}}}({\mathcal{A}})$ by $${\varepsilon}[\mathbf{w}_i]:=\sum \mathbb{L}^{\chi([\mathbf{w}_i],\mathbf{v})}\cdot {\varepsilon}(\mathbf{v}),\quad {\hat{\varepsilon}}\{\mathbf{w}_i\}:=\sum {\chi(\mathbf{w}_i,\mathbf{v})}\cdot {\hat{\varepsilon}}(\mathbf{v})$$ respectively. Then we have the following: We have ${\varepsilon}_{{\mathcal{A}}}[\mathbf{w}_i]-{\varepsilon}_{{\mathcal{A}}}\in {\widehat{{{\mathrm{MH}}}}}_{0}({{\mathcal{A}}})$ and $$({\varepsilon}_{{\mathcal{A}}}[\mathbf{w}_i]-{\varepsilon}_{{\mathcal{A}}})|_{\mathbb{L}=1}={\hat{\varepsilon}}_{{\mathcal{A}}}\{\mathbf{w}_i\}.$$ (See §\[subsec\_absence\] for the definitions.) We put $${{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}:=\sum_j\frac{(-1)^jp!}{j!(p-j)!}{\varepsilon}_{{\mathcal{A}}}[\mathbf{w}_i]^{*(p-j)}*{\varepsilon}_{{\mathcal{A}}}^{*(j)}\in {\widehat{{{\mathrm{MH}}}}}({\mathcal{A}}).$$ ${{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\in \widehat{{{\mathrm{MH}}}}_{0}({\mathcal{A}})$. Since we have $${{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p+1\rangle}:= ({\varepsilon}_{\mathcal{A}}[\mathbf{w}_i]-{\varepsilon}_{\mathcal{A}})\cdot {{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle} + \frac{1}{\mathbb{L}-1}\Bigl[{\tilde{\varepsilon}}_{\mathcal{A}},{{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\Bigr],$$ the claim follows by induction. $$\label{eq_517} {{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p+1\rangle}\big|_{\mathbb{L}=1}= {\hat{\varepsilon}}_{\mathcal{A}}\{\mathbf{w}_i\}\cdot {{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1} + \Bigl\{{\hat{\varepsilon}}_{\mathcal{A}},{{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\Bigr\}.$$ \[prop\_commutator\] $$\begin{aligned} \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}},\sigma}(x_{i,\sigma})&=x_{i,\sigma}\cdot I_\sigma\Bigl(\bigl({{\mathcal{M}}}_{{\mathcal{A}}}[\mathbf{w}_i]*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\bigr)\big|_{\mathbb{L}=1}\Bigr),\\ \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}},\sigma}(y_{i,\sigma})&=y_{i,\sigma}\cdot \prod_{j}I_\sigma\Bigl(\bigl({{\mathcal{M}}}_{{\mathcal{A}}}[\mathbf{w}_j]*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\bigr)\big|_{\mathbb{L}=1}\Bigr)^{\bar{Q}(j,i)}.\end{aligned}$$ We define ${E}_{i,{\mathcal{A}}}^{\langle p\rangle}\in {\widehat{{{\mathrm{QT}}}}}_{{{\mathrm{sc}}},\sigma}({\mathcal{A}})$ by $$\bigl(\{I_\sigma({\hat{\varepsilon}}_{\mathcal{A}}),-\}\bigr)^p(x_{i,\sigma})={E}_{i,{\mathcal{A}}}^{\langle p\rangle}\cdot x_{i,\sigma}.$$ By , it is suffice to show that ${E}_{i,{\mathcal{A}}}^{\langle p\rangle}=I_\sigma\Bigl({{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\Bigr)$. Since we have $$\begin{aligned} &\Bigl\{I_\sigma({\hat{\varepsilon}}_{\mathcal{A}}),I_\sigma\bigl({{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\bigr)\cdot x_{i,\sigma}\Bigr\}\\ &\ =\ \Bigl(I_\sigma({\hat{\varepsilon}}_{\mathcal{A}}\{\mathbf{w}_i\})\cdot I_\sigma\bigl({{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\bigr) +\Bigl\{I_\sigma({\hat{\varepsilon}}_{\mathcal{A}}), I_\sigma\bigl({{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\bigr) \Bigr\}\Bigr)\cdot x_{i,\sigma}\\ &\ =\ I_\sigma\biggl({\hat{\varepsilon}}_{\mathcal{A}}\{\mathbf{w}_i\}\cdot {{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1} + \Bigl\{{\hat{\varepsilon}}_{\mathcal{A}}, {{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1} \Bigr\}\biggr)\cdot x_{i,\sigma}\\ &\overset{\eqref{eq_517}}{=}I_\sigma\bigl({{\mathcal{E}}}_{i,{\mathcal{A}}}^{\langle p\rangle}\big|_{\mathbb{L}=1}\bigr)\cdot x_{i,\sigma},\end{aligned}$$ the first equation follows by induction. The second one follows since we have $${{\mathcal{M}}}_{{\mathcal{A}}}[\mathbf{w}+\mathbf{w}']*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1} = \bigl({{\mathcal{M}}}_{{\mathcal{A}}}[\mathbf{w}]*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\bigr)[\mathbf{w}']* \bigl({{\mathcal{M}}}_{{\mathcal{A}}}[\mathbf{w}']*{{\mathcal{M}}}_{{\mathcal{A}}}^{-1}\bigr).$$ Similarly we have the following: \[prop\_commutator2\] Let ${{\mathcal{C}}}$ be one of ${\mathcal{A}}_{\mathbf{k}}$, ${{\mathcal{T}}}_{\mathbf{k}}$ and ${{\mathcal{S}}}{(r)}$. Then we have $$\begin{aligned} \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}},\sigma}(x_{{\mathbf{k}},i,\sigma})&=x_{{\mathbf{k}},i,\sigma}\cdot I_\sigma\Bigl(\bigl({{\mathcal{M}}}_{{{\mathcal{C}}}}[\mathbf{w}_{{\mathbf{k}},i}]*{{\mathcal{M}}}_{{{\mathcal{C}}}}^{-1}\bigr)\big|_{\mathbb{L}=1}\Bigr).\\ \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}},\sigma}(y_{{\mathbf{k}},i,\sigma})&=y_{{\mathbf{k}},i,\sigma}\cdot \prod_j I_\sigma\Bigl(\bigl({{\mathcal{M}}}_{{{\mathcal{C}}}}[\mathbf{w}_{{\mathbf{k}},i}]*{{\mathcal{M}}}_{{{\mathcal{C}}}}^{-1}\bigr)\big|_{\mathbb{L}=1}\Bigr)^{\bar{Q}_{\mathbf{k}}(j,i)}.\end{aligned}$$ ### Hilbert/Grassmann in the automorphisms \[thm\_520\] $$\begin{aligned} \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}},\sigma}(x_{i,\sigma})&= x_{i,\sigma}\cdot \Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Hilb_{J}}(i,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{\mathbf{v}}\Biggr),\\ \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}},\sigma}(y_{i,\sigma})&= y_{i,\sigma}\cdot \prod_j\Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Hilb_{J}}(j,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{\mathbf{v}}\Biggr)^{\bar{Q}(j,i)}.\end{aligned}$$ This is a consequence of the motivic Hilbert scheme identity (Proposition \[prop\_mhsi\]) and Proposition \[prop\_commutator\]. \[thm\_521\] $$\begin{aligned} \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}(x_{{\mathbf{k}},i,\sigma})&= x_{{\mathbf{k}},i,\sigma}\cdot \Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Grass}({\mathbf{k}},i,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{-\mathbf{v}}\Biggr),\\ \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}(y_{{\mathbf{k}},i,\sigma})&= y_{{\mathbf{k}},i,\sigma}\cdot \prod_j\Biggl(\sum_\mathbf{v}e_\sigma\Bigr(\mathrm{Grass}({\mathbf{k}},j,\mathbf{v})\Bigr)\cdot\mathbf{y}_\sigma^{-\mathbf{v}}\Biggr)^{\bar{Q}(j,i)}.\end{aligned}$$ This is a consequence of the motivic quiver Grassmannian identity (Proposition \[prop\_hom4\]) and Proposition \[prop\_commutator2\]. The automorphism $\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}$ preserves ${{{\mathrm{Q}}}\mathbb{T}}_{{{\mathrm{sc}}},\sigma}$ and induces an automorphism of ${{{\mathrm{QT}}}}_{{{\mathrm{sc}}},\sigma}$. The first half is clear from Theorem \[thm\_521\] and the second half follows since $\mathrm{ad}$ preserves the kernel of the map given in . Let ${{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}$ denote the automorphism on ${{{\mathrm{QT}}}}_{{{\mathrm{sc}}},\sigma}$ induced by $\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}}_{\mathbf{k}},\sigma}$. We put $$x_{(r),i,\sigma}:=\mathbf{x}_\sigma^{[\Gamma_{(r),i}]},\quad y_{(r),i,\sigma}:=\mathbf{y}_\sigma^{[\sigma_{(r),i}]}.$$ Then we have $${{\mathrm{Ad}}}_{{{\mathcal{S}}}(k),\sigma}(x_{(r),i,\sigma})= \begin{cases} x_{(r),i,\sigma} & i\neq k,\\ x_{(r),k,\sigma}(1+(y_{(r-1),k,\sigma})^{-1}) & i=k. \end{cases}$$ This gives the cluster transformation for the quiver $Q_{(r-1)}$ see Example \[ex\_cluster\]. ### Factorization identity \[lem\_factor\] For $X\in {\widehat{{{\mathrm{MH}}}}}_{0}({{\mathcal{C}}})$ we have $${{\mathrm{Ad}}}_{{{\mathcal{C}}},\sigma}(I_\sigma(X|_{\mathbb{L}=1}))=I_\sigma\Bigl(\bigl({{\mathcal{M}}}_{{{\mathcal{C}}}}*X*{{\mathcal{M}}}_{{{\mathcal{C}}}}^{-1}\bigr)|_{\mathbb{L}=1}\Bigr)$$ Note that we have $$\begin{aligned} {{\mathcal{M}}}_{{{\mathcal{C}}}}*{X}*{{\mathcal{M}}}_{{{\mathcal{C}}}}^{-1}|_{\mathbb{L}=1} &= \bigl(\exp([{\varepsilon}_{{{\mathcal{C}}}},-])({X})\bigr)|_{\mathbb{L}=1}\\ &= \Bigl(\exp\Bigl(\frac{1}{\mathbb{L}-1}[{\tilde{\varepsilon}}_{{{\mathcal{C}}}},-]\Bigr)({X})\Bigr)|_{\mathbb{L}=1}\\ &= \exp(\{{\hat{\varepsilon}}_{{{\mathcal{C}}}},-\})({X}|_{\mathbb{L}=1}).\end{aligned}$$ Then the claim follows since $I_\sigma$ respects the Poisson bracket. \[thm\_factor1\] $$\begin{aligned} \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}},\sigma}= \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma}\circ \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}^\bot,\sigma},\quad \widehat{{{\mathrm{Ad}}}}_{{\mathcal{A}}_{\mathbf{k}},\sigma}= \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}^\bot,\sigma}\circ \widehat{{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}.\end{aligned}$$ \[thm\_factor2\] $${{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}}[-1],\sigma}= \Bigl({{{\mathrm{Ad}}}}_{{{\mathcal{S}}}(1)[-1],\sigma}\Bigr)^{{\varepsilon}(1)} \circ\cdots\circ \Bigl({{{\mathrm{Ad}}}}_{{{\mathcal{S}}}(l)[-1],\sigma}\Bigr)^{{\varepsilon}(l)}.$$ We will show the proof of $$\label{eq_factorization} {{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma}= \Bigl({{{\mathrm{Ad}}}}_{{{\mathcal{S}}}(1),\sigma}\Bigr)^{{\varepsilon}(1)} \circ\cdots\circ \Bigl({{{\mathrm{Ad}}}}_{{{\mathcal{S}}}(l),\sigma}\Bigr)^{{\varepsilon}(l)}$$ which is equivalent to Theorem \[thm\_factor2\] (we can prove Theorem \[thm\_factor1\] in the same way). We put $\delta(r):={{\mathcal{M}}}_{{{\mathcal{C}}}(r)}$. First, we can see the following identity by induction with respect to $r$ using Proposition \[prop\_commutator\] and Lemma \[lem\_factor\] : $$\begin{aligned} &\Bigl(\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}}(r),\sigma}\Bigr)^{{\varepsilon}(r)} \circ\cdots\circ \Bigl(\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}}(l),\sigma}\Bigr)^{{\varepsilon}(l)}(\mathbf{x}_\sigma^\mathbf{w})=\\ &\mathbf{x}_\sigma^\mathbf{w}\cdot \prod_{r'=r}^{l} I_\sigma\Bigl(\delta_{(r)}^{{\varepsilon}(r)}*\cdots *\delta_{(r'-1)}^{{\varepsilon}(r'-1)}*(\delta_{(r')}[\mathbf{w}])^{{\varepsilon}(r')}*\delta_{(r')}^{-{\varepsilon}(r')}*\cdots *\delta_{(r)}^{-{\varepsilon}(r)}\Big|_{\mathbb{L}=1}\Bigr)\end{aligned}$$ Then we have $$\begin{aligned} &\Bigl(\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}}(1),\sigma}\Bigr)^{{\varepsilon}(1)} \circ\cdots\circ \Bigl(\widehat{{{\mathrm{Ad}}}}_{{{\mathcal{C}}}(l),\sigma}\Bigr)^{{\varepsilon}(l)}(\mathbf{x}_\sigma^\mathbf{w})\\ & = \mathbf{x}_\sigma^\mathbf{w}\cdot \prod_{r'=1}^{l} I_\sigma\Bigl(\delta_{(1)}^{{\varepsilon}(r)}*\cdots *\delta_{(r'-1)}^{{\varepsilon}(r'-1)}*(\delta_{(r')}[\mathbf{w}])^{{\varepsilon}(r')}*\delta_{(r')}^{-{\varepsilon}(r')}*\cdots *\delta_{(1)}^{-{\varepsilon}(1)}\Big|_{\mathbb{L}=1}\Bigr)\\ & = \mathbf{x}_\sigma^\mathbf{w}\cdot I_\sigma\Bigl((\delta_{(1)}[\mathbf{w}])^{{\varepsilon}(1)}*\cdots *(\delta_{(l)}[\mathbf{w}])^{{\varepsilon}(l)}*\delta_{(l)}^{-{\varepsilon}(l)}*\cdots *\delta_{(1)}^{-{\varepsilon}(1)}\Big|_{\mathbb{L}=1}\Bigr)\\ &= \mathbf{x}_\sigma^\mathbf{w}\cdot I_\sigma\Bigl({{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}[\mathbf{w}_{{\mathbf{k}},i}]*{{\mathcal{M}}}_{{{\mathcal{T}}}_{\mathbf{k}}}^{-1}\big|_{\mathbb{L}=1}\Bigr)\\ &= {{{\mathrm{Ad}}}}_{{{\mathcal{T}}}_{\mathbf{k}},\sigma} (\mathbf{x}_\sigma^\mathbf{w}).\end{aligned}$$ Here we use Proposition \[prop\_mfi\] and Corollary \[cor\_hom\_factor\] for the second equation and Proposition \[prop\_commutator\] for the last one. Applications for cluster algebras {#sec_F_and_g} ================================= Quiver with the principal framing --------------------------------- Let $Q{^{{{\mathrm{pf}}}}}$ be the following quiver: [vertices]{} : $I\sqcup I^*$ where $I^*=\{1^*,\ldots,n^*\}$, [arrows]{} : $\{\text{arrows in $Q$}\}\sqcup \{i^*\to i\mid i\in I\}$. This is called the quiver with the principal framing associated to $Q$. A potential $W$ of $Q$ can be taken as a potential of $Q{^{{{\mathrm{pf}}}}}$. In the rest of this paper, we assume that $(Q{^{{{\mathrm{pf}}}}},W)$ is successively f-mutatable with respect to a sequence ${\mathbf{k}}$. We apply Theorem \[thm\_tilting\] for $(Q{^{{{\mathrm{pf}}}}},W)$ and ${\mathbf{k}}$ to get the sequence ${\varepsilon}'(1),\ldots,{\varepsilon}'(l)$. We put $$\Phi{^{{{\mathrm{pf}}}}}_{{\mathbf{k}}}:=\Phi{^{{{\mathrm{pf}}}}}_{k_l,{\varepsilon}'(l)}\circ\cdots\circ\Phi{^{{{\mathrm{pf}}}}}_{k_l,{\varepsilon}'(1)}\colon \bar{{{\mathcal{D}}}}{^{{{\mathrm{pf}}}}}\overset{\sim}{\longrightarrow} \bar{{{\mathcal{D}}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}. $$ Let $\phi{^{{{\mathrm{pf}}}}}_{{\mathbf{k}}}$ denote the homomorphism induced on the lattices $L_{Q{^{{{\mathrm{pf}}}}}}$ or $M_{Q{^{{{\mathrm{pf}}}}}}$. Let ${\varepsilon}(1),\ldots,{\varepsilon}(l)$ be the sequence associated to the QP $(Q,W)$ and the sequence ${\mathbf{k}}$. \[prop\_pf\] - ${\varepsilon}(r)={\varepsilon}'(r)$ for any $r$. - $\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}([s_{i^*}])=[s_{{\mathbf{k}},i^*}]$ for any $i^*\in I^*$. - $\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}(L_Q)\subset L_{Q_{\mathbf{k}}}$ and $\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}|_{L_Q}\colon L_Q\to L_Q$ coincides with $\phi_{\mathbf{k}}$. - $\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}(M_Q)\subset M_{Q_{\mathbf{k}}}$ and $\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}|_{M_Q}\colon M_Q\to M_Q$ coincides with $\phi_{\mathbf{k}}$. We prove all the claims together by induction with respect to the length of the sequence ${\mathbf{k}}$. Assume the claim holds for the sequence ${\mathbf{k}}^-=(k_1,\ldots,k_{l-1})$. Note that $\varepsilon (l)$ (resp. $\varepsilon' (l)$) is determined by the condition $$\varepsilon(l)\times (\phi_{{\mathbf{k}}^-})^{-1}([s_{{{\mathbf{k}}^-},k_l}])\in C_{{{\mathcal{A}}}}\subset L_Q\quad \text{(resp. $\varepsilon'(l)\times (\phi_{{\mathbf{k}}^-}{^{{{\mathrm{pf}}}}})^{-1}([s_{{{\mathbf{k}}^-},k_l}])\in C_{{{\mathcal{A}}}{^{{{\mathrm{pf}}}}}}\subset L_{Q{^{{{\mathrm{pf}}}}}}$)}.$$ By the induction assumption (3), we get $\varepsilon (l)=\varepsilon' (l)$. We assume $\varepsilon (l+1)=+$ (for the case of $\varepsilon (l+1)=-$, we can see in the same way). Then we have $$\begin{aligned} Q_{{\mathbf{k}}'}(k_{l+1},i^*) & = \chi([s_{{{\mathbf{k}}'},k_{l+1}}],[s_{{{\mathbf{k}}'},i^*}])\\ & = \chi((\phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}})^{-1}[s_{{{\mathbf{k}}'},k_{l+1}}],(\phi_{{\mathbf{k}}'}{^{{{\mathrm{pf}}}}})^{-1}[s_{{{\mathbf{k}}'},i^*}])\\ & = \chi((\phi_{{\mathbf{k}}'})^{-1}[s_{{{\mathbf{k}}'},k_{l+1}}],[s_{i^*}])\\ & \geq 0\end{aligned}$$ for any $i^*$. Then the claims follow and . \[rem\_c\] [For a sequence ${\mathbf{k}}$ and a vertex $i\in I$, the vector $(Q_{{\mathbf{k}}}(i,j^*))_{j^*\in I^+}$ is so called the [*$c$-vector*]{}. Now we see that the $c$-vector is given by $(\phi_{{\mathbf{k}}'})^{-1}[s_{{\mathbf{k}},i}]$.]{} We have the following triangulated categories : [$\bar{{{\mathcal{D}}}}^{{{\mathrm{pf}}}}$]{} : $={{\mathcal{D}}}\Gamma_{(Q^{{\mathrm{pf}}},W)}$, [${{{\mathcal{D}}}}^{{{\mathrm{pf}}}}$]{} : $={{\mathcal{D}}}^{{\mathrm{fd}}}\Gamma_{(Q^{{\mathrm{pf}}},W)}$, [$\bar{{{\mathcal{D}}}}'$]{} : the full subcategory of $\bar{{{\mathcal{D}}}}^{{{\mathrm{pf}}}}$ consisting of objects whose cohomologies are supported on $I$, [${{\mathcal{D}}}'$]{} : $={{{\mathcal{D}}}}^{{{\mathrm{pf}}}}\cap \bar{{{\mathcal{D}}}}'$ The canonical t-structure of $\bar{{{\mathcal{D}}}}^{{{\mathrm{pf}}}}$ induces t-structures of ${{{\mathcal{D}}}}^{{{\mathrm{pf}}}}$, $\bar{{{\mathcal{D}}}}'$ and ${{\mathcal{D}}}'$. Let $\bar{{{\mathcal{A}}}}^{{{\mathrm{pf}}}}$, ${{{\mathcal{A}}}}^{{{\mathrm{pf}}}}$, $\bar{{{\mathcal{A}}}}'$ and ${{\mathcal{A}}}'$ denote the cores of t-structures. \[lem\_82\] - $\Phi_{\mathbf{k}}{^{{{\mathrm{pf}}}}}(s_{i^*})=s_{{\mathbf{k}},i^*}$ for any $i^*\in I^*$. - $\bar{{{\mathcal{T}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}\subset {{\mathcal{A}}}'$. The first claim follows Lemma \[prop\_pf\] (2). For (2), let $s^{{{\mathrm{pf}}},(r)}$ be the spherical object in ${{\mathcal{A}}}{^{{{\mathrm{pf}}}}}$ defined in the same way as $s^{(r)}$ (see §\[subsec\_comp\]). By Lemma \[prop\_pf\] (3), we have $[s^{{{\mathrm{pf}}},(r)}]\in L_Q$. Hence we have $s^{{{\mathrm{pf}}},(r)}\in {{\mathcal{A}}}'$ and so $\bar{{{\mathcal{T}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}\subset {{\mathcal{A}}}'$. We put $$\bar{{{\mathcal{A}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}:=(\Phi{^{{{\mathrm{pf}}}}}_{{\mathbf{k}}})^{-1}({{\mathrm{Mod}}}J_{\mu_{\mathbf{k}}(Q{^{{{\mathrm{pf}}}}},W)})$$ and $$\bar{{{\mathcal{A}}}}'_{\mathbf{k}}:= \bar{{{\mathcal{A}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}\cap \bar{{{\mathcal{D}}}}',\quad {{{\mathcal{A}}}}'_{\mathbf{k}}:= \bar{{{\mathcal{A}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}\cap {{{\mathcal{D}}}}'.$$ Then $\bar{{{\mathcal{A}}}}'_{\mathbf{k}}$ (resp. ${{{\mathcal{A}}}}'_{\mathbf{k}}$) coincides with the full subcategory of $\bar{{{\mathcal{A}}}}_{\mathbf{k}}{^{{{\mathrm{pf}}}}}$ consisting of objects supported on $I$ (with finite dimensional cohomologies). We set $\bar{{{\mathcal{T}}}}'_{\mathbf{k}}:=\bar{{{\mathcal{T}}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}$ (resp. ${{{\mathcal{T}}}}'_{\mathbf{k}}:=\bar{{{\mathcal{T}}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}$) and $$\bar{{{\mathcal{F}}}}'_{\mathbf{k}}:= \bar{{{\mathcal{F}}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}\cap \bar{{{\mathcal{A}}}}',\quad \text{(resp. ${{{\mathcal{F}}}}'_{\mathbf{k}}:= \bar{{{\mathcal{F}}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}\cap {{{\mathcal{A}}}}'$)}.$$ Then, $(\bar{{{\mathcal{T}}}}'_{\mathbf{k}},\bar{{{\mathcal{F}}}}'_{\mathbf{k}})$ (resp. $({{{\mathcal{T}}}}'_{\mathbf{k}},{{{\mathcal{F}}}}'_{\mathbf{k}})$) gives a torsion pair of $\bar{{{\mathcal{A}}}}'$ (resp. ${{{\mathcal{A}}}}'$) and the tilted t-structure coincides with $\bar{{{\mathcal{A}}}}'_{\mathbf{k}}$ (resp. ${{{\mathcal{A}}}}'_{\mathbf{k}}$). CC formula for $(Q,W)$ and $(Q{^{{{\mathrm{pf}}}}},W)$ ------------------------------------------------------ We put $$R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}:=H^{1}_{\bar{{\mathcal{A}}}{^{{{\mathrm{pf}}}}}}((\Phi{^{{{\mathrm{pf}}}}}_{{\mathbf{k}}})^{-1}(P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}))$$ and $$\mathrm{Grass}{^{{{\mathrm{pf}}}}}(\mathbf{k};i,\mathbf{v}) :=\{R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\twoheadrightarrow V\mid V\in {\mathcal{A}}',\ [V]=\mathbf{v}\}.$$ Applying Theorem \[thm\_CC\] for $(Q{^{{{\mathrm{pf}}}}},W)$ we get $$\mathrm{FZ}{^{{{\mathrm{pf}}}}}_{i,\mathbf{k}}(\underline{X})= X_{{\mathbf{k}},i}\cdot \Biggl(\sum_\mathbf{v}e\Bigr(\mathrm{Grass}{^{{{\mathrm{pf}}}}}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{Y}^{-\mathbf{v}}\Biggr)$$ where $Y_j=X_{j^*}^{-1}\cdot \prod_i(X_i)^{\bar{Q}(i,j)}$. Then we have $$\label{eq_F} F_{{\mathbf{k}},i}(\underline{y}):={FZ}{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}(\underline{X})|_{X_i=1,X_{i^*}=y_i}=\sum_\mathbf{v}e\Bigr(\mathrm{Grass}{^{{{\mathrm{pf}}}}}(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}^{\mathbf{v}}.$$ On the other hand, we put $$R'_{{\mathbf{k}},i}:=H^{1}_{\bar{{\mathcal{A}}}'}((\Phi{^{{{\mathrm{pf}}}}}_{{\mathbf{k}}})^{-1}(P_{{\mathbf{k}},i}))$$ and $$\mathrm{Grass}'(\mathbf{k};i,\mathbf{v}) :=\{R'_{{\mathbf{k}},i}\twoheadrightarrow V\mid V\in {\mathcal{A}}',\ [V]=\mathbf{v}\}.$$ We will apply the same arguments as in §\[sec\_proof\] for ${{\mathcal{D}}}'\Gamma$. Let ${{\mathfrak{Hom}}}_{{{\mathcal{D}}}'\Gamma}(P_{{\mathbf{k}},i},{{\mathcal{T}}}_{\mathbf{k}})$ be the moduli stack which parametrizes homomorphisms in ${{\mathcal{D}}}'\Gamma$ from $P_{{\mathbf{k}},i}$ to elements in ${{\mathcal{T}}}{^{{{\mathrm{pf}}}}}_{\mathbf{k}}$. We can verify all the lemmas and propositions in §\[subsec\_idea\] and §\[subsec\_auto\] if we replace $\mathrm{Grass}(\mathbf{k};i,\mathbf{v})$ with $\mathrm{Grass}'(\mathbf{k};i,\mathbf{v})$. As a consequence, we get the following modification of the Caldero-Chapoton type formula for $(Q,W)$ : $$\label{eq_mCC} \mathrm{FZ}_{i,\mathbf{k}}(\underline{x})= x_{{\mathbf{k}},i}\cdot \Biggl(\sum_\mathbf{v}e\Bigr(\mathrm{Grass}'(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}^{-\mathbf{v}}\Biggr).$$ where $(\underline{y})^{-\mathbf{v}}=\prod_j (y_j)^{-v_j}$ and $y_j=\prod_i(x_i)^{\bar{Q}(i,j)}$. \[prop\_gr\] $$R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}=R'_{{\mathbf{k}},i}$$ Since we have no non-trivial morphism from $${{\mathrm{ker}}}\left( P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\twoheadrightarrow R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i} \right)$$ to $R'_{{\mathbf{k}},i}$, the composition $$P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\twoheadrightarrow P_{{\mathbf{k}},i}\twoheadrightarrow R'_{{\mathbf{k}},i}$$ factors through $P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\twoheadrightarrow R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}$ : $$\xymatrix{ P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\ar@{->>}[r]\ar@{->>}[d] & R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\ar@{-->>}[d]\\ P_{{\mathbf{k}},i}\ar@{->>}[r] & R'_{{\mathbf{k}},i}. }$$ On the other hand, since $R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}$ is supported on $I$ the surjection $P_{{\mathbf{k}},i}{^{{{\mathrm{pf}}}}}\twoheadrightarrow R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}$ factors through $P_{{\mathbf{k}},i}$. By the same reason, this map factors through $P_{{\mathbf{k}},i}\twoheadrightarrow R'_{{\mathbf{k}},i}$. $$\xymatrix{ P{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\ar@{->>}[r]\ar@{->>}[d] & R{^{{{\mathrm{pf}}}}}_{{\mathbf{k}},i}\\ P_{{\mathbf{k}},i}\ar@{->>}[r]\ar@{-->>}[ru] & R'_{{\mathbf{k}},i}\ar@{-->>}[u] }.$$ These two morphisms are the inverse of each other. $F$-polynomials and $g$-vectors {#subsec_F_and_g} ------------------------------- Combining , and Proposition \[prop\_gr\], we get $$\label{eq_Fg} \mathrm{FZ}_{i,\mathbf{k}}(\underline{x})= x_{{\mathbf{k}},i}\cdot F_{{\mathbf{k}},i}(\underline{y}^{-1}).$$ Finally, we have the following description of $F$-polynomials and $g$-vectors : \[thm\_Fg\] - $$F_{{\mathbf{k}},i}(\underline{y})= \sum_\mathbf{v}e\Bigr(\mathrm{Grass}'(\mathbf{k};i,\mathbf{v})\Bigr)\cdot\mathbf{y}^{\mathbf{v}}$$ - $$g_{{\mathbf{k}},i} = \phi_{{\mathbf{k}}}^{-1}([\Gamma_{{\mathbf{k}},i}])\in M_Q.\label{eq_g}$$ Since $L_{J}\otimes {{{\mathbb{R}}}}$ and $M_{J}\otimes {{{\mathbb{R}}}}$ are dual to each other via $\chi$ and $\phi_{\mathbf{k}}$ preserves $\chi$, we get the following description of the ${{}^{{{\mathrm{t}}}}\hspace{-2pt} g}$-vector. \[cor\_619\] $${{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{{\mathbf{k}},i} = \phi_{{\mathbf{k}}}([s_i]) \in L_{Q_{\mathbf{k}}}.$$ The $g$-vector can be viewed as a tropical counterpart of the $x$-variable, while the $c$-vector can be viewed as a tropical counterpart of the $y$-variable. The duality between the $g$- and the $c$-vectors is called toropical duality in [@nakanishi-zelevinsky] [^9]. From our view point, the $x$-variable corresponds to the “projective" $\Gamma_i$ and the $y$-variable corresponds to the simple $s_i$, and the toropical duality is a consequence of the duality between $\{\Gamma_i\}$ and $\{s_i\}$, ### Conjectures on $F$-polynomials The following claims follow directly from the description in Theorem \[thm\_Fg\]. Each polynomial $F_{{\mathbf{k}},i}(\underline{y})$ has constant term $1$. Each polynomial $F_{{\mathbf{k}},i}(\underline{y})$ has a unique monomial of maximal degree. Furthermore, this monomial has coefficient $1$, and it is divisible by all the other occurring monomials. ### Conjectures on $g$-vectors For any sequence ${\mathbf{k}}$, the vectors $\{g_{{\mathbf{k}},i}\}_{i\in I}$ form a ${{{\mathbb{Z}}}}$-basis of the lattice ${{{\mathbb{Z}}}}^n$. This is clear from Theorem \[thm\_Fg\] (2). For any sequence ${\mathbf{k}}$ and a vertex $i\in I$, the components of the vector ${{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{{\mathbf{k}},i}$ are either all non-negative, or all non-positive. In the same way as Theorem \[thm\_tilting\], we can see that $\Phi_{\mathbf{k}}(s_i)\in {{\mathcal{A}}}_{\mathbf{k}}$ or $\Phi_{\mathbf{k}}(s_i)\in {{\mathcal{A}}}_{\mathbf{k}}[1]$. Then, the claim is a consequence Corollary \[cor\_619\]. For a sequence ${\mathbf{k}}=(k_0,\ldots,k_l)$, we take a new sequence $${\mathbf{k}}^\circ:=(k_1,\ldots,k_l).$$ Then we have $${{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{{\mathbf{k}},i^\circ}=\begin{cases} -{{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{i,{{\mathbf{k}}}} & i=k_0, \\ {{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{i,{{\mathbf{k}}}} + Q(i,k_0)\cdot {{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{k_0,{{\mathbf{k}}}} & i\neq k_0,\ {\varepsilon}(0)=-,\\ {{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{i,{{\mathbf{k}}}} + Q(k_0,i)\cdot {{}^{{{\mathrm{t}}}}\hspace{-2pt} g}_{k_0,{{\mathbf{k}}}} & i\neq k_0,\ {\varepsilon}(0)=+. \end{cases}$$ This is a consequence of and Corollary \[cor\_619\]. $g$-vectors determine $F$-polynomials {#subsec_g_to_F} ------------------------------------- We define $$\zeta' \colon M_\mathbb{R} \to \mathrm{Stab}({{\mathcal{D}}}'\Gamma).$$ in the same way as §\[subsec\_embed\]. For $\theta\in M_\mathbb{R}$, let ${{\mathcal{A}}}'_\theta$ denote the core of the t-structure corresponding to $\zeta'(\theta)$ and $({{\mathcal{T}}}'_\theta,{{\mathcal{F}}}'_\theta)$ be the corresponding torsion pair. For $\theta\in C_{{\mathcal{A}}_{\mathbf{k}}}^*$, we have $\mathcal{A}_{\zeta(\theta)}=\mathcal{A}'_{{\mathbf{k}}}$. Suppose we have $$\sum_{i\in J}a_i\cdot g_{{\mathbf{k}},i}= \sum_{i\in J'}a'_i\cdot g_{{\mathbf{k}},i'}$$ for some nonempty subsets $J,J'\subset I$ and some positive real numbers $a_i$ and $a'_i$. Then there is a bijection $\kappa\colon J\to J'$ such that for every $i\in J$ we have $$a_i=a'_{\kappa(i)},\quad g_{{\mathbf{k}},i}=g_{\kappa(i),{\mathbf{k}}'},\quad F_{{\mathbf{k}},i}=F_{\kappa(i),{\mathbf{k}}'}.$$ In particular, $F_{{\mathbf{k}},i}$ is determined by $g_{{\mathbf{k}},i}$. By Theorem \[thm\_Fg\], $g_{{\mathbf{k}},i}$ is primitive and $$g_{{\mathbf{k}},i}\in \bigcap_{j\neq i}W_{{\mathbf{k}},j}\cap \overline{C_{\mathbf{k}}^*}$$ where $$W_{{\mathbf{k}},j}:=\{\theta\in M_\mathbb{R}\mid \langle \theta,[s_{{\mathbf{k}},i}]\rangle=0\}.$$ Then we have $${{\mathrm{Int}}}\left(\bigcap_{j\notin J}W_{{\mathbf{k}},j}\cap \overline{C_{\mathbf{k}}^*}\right) = \left\{\sum_{i\in J}a_i\cdot g_{{\mathbf{k}},i}\mid a_i>0\right\}.$$ The bijection $\kappa\colon J\to J'$ and $$a_i=a'_{\kappa(i)},\quad g_{{\mathbf{k}},i}=g_{{\mathbf{k}}',\kappa(i)}$$ follow from this description. Let ${{\mathcal{C}}}_{{\mathbf{k}},J}$ be the full subcategory of $J_{\mathbf{k}}$-modules supported on $$\{i\mid i\neq J,{\varepsilon}({\mathbf{k}},i)=+\}.$$ Then we have $${\mathcal{A}}'_\theta={\mathcal{A}}'_{\mathbf{k}}({{\mathcal{C}}}_{{\mathbf{k}},J}[1],{}^\bot{{\mathcal{C}}}_{{\mathbf{k}},J})\footnote{The category ${\mathcal{A}}{^{{{\mathrm{pf}}}}}_\theta$ is not a module category in general.}.$$ We define $$\mathrm{Ad}{^{{{\mathrm{pf}}}}}_{{{\mathcal{T}}}'_{\theta}}, \mathrm{Ad}{^{{{\mathrm{pf}}}}}_{{{\mathcal{C}}}_{{\mathbf{k}},J}}\colon {{{\mathrm{QT}}}}{^{{{\mathrm{pf}}}}}_{{{\mathrm{sc}}}}\overset{\sim}{{\longrightarrow}} {{{\mathrm{QT}}}}{^{{{\mathrm{pf}}}}}_{{{\mathrm{sc}}}}$$ in the same way in §\[subsec\_531\], then we have $$\mathrm{Ad}{^{{{\mathrm{pf}}}}}_{{{\mathcal{T}}}'_{\theta}}\circ\mathrm{Ad}{^{{{\mathrm{pf}}}}}_{{{\mathcal{C}}}_{{\mathbf{k}},J}}=\mathrm{Ad}{^{{{\mathrm{pf}}}}}_{{{\mathcal{T}}}'_{{\mathbf{k}}}}$$ as Theorem \[thm\_factor2\]. 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T. Nakanishi, *Periodicities in cluster algebras and dilogarithm identities*, arXiv:1006.0632. T. Nakanishi and A. Zelevinsky, *On tropical dualities in cluster algebras*, arXiv:1101.3736. P-G. Plamondon, *Cluster characters for cluster categories with infinite-dimensional morphism spaces*, arXiv:1002.4956. M. Reineke, *The [Harder-Narasimhan]{} system in quantum groups and cohomology of quiver moduli*, Invent. Math. **152** (2003), no. 2, 349–368. B. Szendroi, *Non-commutative [Donaldson-Thomas]{} invariants and the conifold*, Geom. Topol. **12(2)** (2008), 1171–1202. R. P. Thomas, *A holomorphic [Casson]{} invariant for [Calabi-Yau]{} $3$-folds, and bundles on [$K3$]{} fibrations*, J. Differential Geom. **54** (2000), no. 2, 367–438. [^1]: Since ${{\mathrm{Spec}}}$ of them are algebraic tori, we call them tori with a slight abuse. [^2]: To be precise, they are isomorphisms of different completions. See Theorem \[thm\_trans\] for the precise statement. [^3]: This diagram is not rigorous in that the compositions of the maps are not well-defined. See Theorem \[thm\_trans\] for the precise statement. [^4]: The variables $x_{{\mathbf{k}},i,\pm}$ and $y_{{\mathbf{k}},i,\pm}$ on the left hand side of the equations does make sense since we have identified the two tori ${\mathbb{T}}_{Q_{\mathbf{k}},\pm}$ and ${\mathbb{T}}_{Q,\pm}$. [^5]: Cluster algebras are associated not only with quivers without loops and oriented 2-cycles (equivalently, with skew-symmetric integer matrices) but also with [*skew-symmetrizable*]{} matrices, [^6]: From the view points of applications to cluster algebras, the finite assumption is too strong. In this sense, our result on the Fomin-Zelevinsky conjectures is weaker than ones in [@DWZ2; @plamondon]. [^7]: Since ${{\mathrm{Spec}}}$ of them are algebraic tori, we call them tori with a slight abuse. [^8]: They construct $I_{{{\mathrm{KS}}}}$ using [*motivic Milnor fiber*]{} and give a proof of the conjecture modulo certain expected formula on motivic Milnor fibers. [^9]: The duality has proved in [@nakanishi-periodicity] for skewsymmetric matrices. For skewsymmetrizable matrices, it is still a conjecture.
--- abstract: 'Within a genuinely gauge invariant approach recently developed for the computation of the cosmological backreaction, we study, in a cosmological inflationary context and with respect to various observers, the impact of [*scalar*]{} fluctuations on the space-time dynamics in the long wavelength limit. We stress that such a quantum backreaction effect is evaluated in a truly gauge independent way using a set of effective equations which describe the dynamics of the averaged geometry. In particular we show under what conditions the free falling (geodetic) observers do not experience any scalar-induced backreaction in the effective Hubble rate and fluid equation of state.' author: - 'F. Finelli' - 'G. Marozzi' - 'G. P. Vacca' - 'G. Venturi' title: 'Backreaction during inflation: a physical gauge invariant formulation' --- [*Introduction*]{}. The computation of backreaction effects induced by cosmological fluctuations in an inflationary era  [@MAB1; @ABM2], has been the subject of controversial analysis [@Uc; @AW; @GB; @LU; @FMVV_II]. Such a task has been plagued by fundamental ambiguities in constructing perturbatively gauge invariant (GI) observables  [@Uc] and average quantities. The basic fact that the averaging procedure does not commute with the non linear evolution of Einstein equations [@GE] was first exploited to study the effective dynamics of the averaged geometry for a dust universe [@Buchert] (see [@BO] for a recent application in the context of inhomogeneity driven inflation). Gauge invariance of averaged quantities has recently been addressed in a novel context which introduces a GI but observer dependent averaging prescription [@GMV1] (see [@ULC] for a recent application of such a prescription to the analysis of the present Hubble rate) whereas the effective equations for the averaged geometry [@Buchert] have been generalized in a covariant and GI form in [@GMV2]. Taking advantage of these recent results and having in mind the backreaction of quantum fluctuations during inflation, we devote this Letter to describing, for the first time in this context, an analysis of the GI effective equations which nevertheless depend on the different observers intrinsically used in the GI construction. [*Gauge Invariant Backreaction*]{}. We start by illustrating how, following a recent proposal [@GMV1; @GMV2], one may define observables, of a non local nature and constructed with quantum averages, which obey GI dynamical equations. Specifically what has been investigated is how to give a classical or quantum GI average of a scalar $S(x)$, for a classical field or a composite quantum operator, assumed to be renormalized, respectively. In such an approach the fundamental point is the choice of a hypersurface, which defines a class of observers, with respect to which the average is done. In particular a hypersurface $\Sigma_{A_0}$ is defined, using another scalar field $A(x)$ with a timelike gradient, through the constraint $A(x)=A_0$, where $A_0$ is a constant. Both the scalars $S$ and $A$ are not GI but one may construct an average which does have the desired property. We simply consider here a spatially unbounded $\Sigma_{A_0}$ which is reasonable in an inflationary context. Eventually one might constrain it to be bounded in the spatial direction by using other scalar fields having a spacelike gradient, if available. In investigations such as the actual dynamics of the universe, one may imagine more suitable choices such as past light cone regions, which however make the problem extremely more complicated. Once the GI definition is given, the computation of the averaged quantity can be done in any gauge (coordinate frame). We can define the quantum (classical, see notation of [@GMV1]) averaging prescription of a scalar quantity $S(x)$ as a functional of $A(x)$ which can be reduced, in the (barred) coordinate system $\bar{x}^{\mu}=(\bar{t}, \vec{x})$ where the scalar $A$ is homogeneous, to the simpler form [@GMV1; @GMV2] S \_[A\_0]{}=[   (t\_0, ) ]{}, \[media\] where we have called $t_0$ the time $\bar{t}$ when $\bar{A}(\bar{x})=A^{(0)}(\bar{t})=A_0$. The quantity $\sqrt{|\overline{\gamma}(t_0, {\vec{x}})|}$ is the square root of the determinant of the induced three dimensional metric on $\Sigma_{A_0}$. Note that in a general covariant definition the average is defined on a spacetime region where the distribution $n^\mu\nabla_\mu\theta(A(x)\!-\!A_0)$ has support. The vector $n^\mu$ defines the associated observer by $n^\mu= -Z_A^{-1/2}\partial^\mu A$, $Z_A=-\partial^\mu A\partial_\mu A$ and $\theta$ is the Heaviside step function. Following the results of [@GMV1; @GMV2] one can consider an effective scale factor $a_{eff}$ which describes the dynamics of a perfect fluid-dominated early universe as $a_{eff}=\langle\sqrt{|\bar{\gamma}|}\, \rangle ^{1/3}$ (where we have chosen $A^{(0)}(t)=t$ to have standard results at the homogeneous level [@Mar]), and obtain a quantum gauge invariant version of the effective cosmological equations for the averaged geometry: ( )\^2 &=& \_[A\_0]{} - \_[A\_0]{} - + \_[A\_0]{} = \_[A\_0]{}\^2, \[EQB1\]\ - &=& \_[A\_0]{}- \_[A\_0]{} + \_[A\_0]{} - + \_[A\_0]{} \[EQB2\] where ${\cal R}_s$ is a generalization of the intrinsic scalar curvature [@GMV2], $\Theta=\nabla_\mu n^\mu$ the expansion scalar and $\sigma^2$ the shear scalar with respect to the observer. We then define $$\begin{aligned} \varepsilon &=& \rho - (\rho+p)\left(1- (u^\mu n_\mu)^2 \right)\; , \ \\ \pi &=& p - \frac13 (\rho+p) \left(1- (u^\mu n_\mu)^2\right)\;,\end{aligned}$$ with $u_\mu$ the 4-velocity comoving with the perfect fluid and $\rho$ and $p$ are, respectively, the (scalar) energy density and pressure in the fluid’s rest frame. In the inflationary scenario we consider the inflaton field as the fluid. Moreover we define the effective observer dependent energy density $\rho_{eff A}$ by writing the r.h.s. of Eq. as $(8\pi G/3) \rho_{eff A}$ while the effective pressure $p_{eff A}$ is obtained by rewriting the r.h.s. of Eq. as $(4\pi G/3) (\rho_{eff A}+3\, p_{eff A})$. In order to deal with the metric components in any specific frame we employ the standard decomposition of the metric in terms of scalar, transverse vector ($B_i$,$\chi_i$) and traceless transverse tensor ($h_{ij}$) fluctuations up to the second order around a homogeneous FLRW zero order space-time g\_[00]{}&=& -1-2 -2 \^[(2)]{}, g\_[i0]{}=-[a2]{}(\_[,i]{}+B\_i ) -(\^[(2)]{}\_[,i]{}+B\^[(2)]{}\_i)\ g\_[ij]{} &=& a\^2 \[GeneralGauge\] where $D_{ij}=\partial_i\partial_j-\delta_{ij}\nabla^2/3$ and for notational simplicity we have removed an upper script for first order quantities. The Einstein equations connect those fluctuations with the matter ones. In particular the inflaton field is written to second order as $\Phi(x)=\phi(t)+\varphi(x)+\varphi^{(2)}(x)$. These general perturbed expressions can be gauge fixed. Let us recall some common gauge fixing (of the scalar and vector part): the synchronous gauge (SG) is defined by $g_{00}=-1$ and $g_{i0}=0$, the uniform field gauge (UFG) apart from setting $\Phi(x)=\phi(t)$ must be supplemented by other conditions (we consider $g_{i0}=0$), finally the uniform curvature gauge (UCG) is defined by $g_{ij}=a^2\left[\delta_{ij}+\frac{1}{2} \left(h_{ij}+h^{(2)}_{ij}\right)\right]$. In the following we shall take the long wavelength (LW) limit as our approximation and consider the cosmological backreaction with respect to different observers: \(a) the geodetic, or free falling, observers which are associated with a scalar field homogeneous in the SG [@Mar]. We consider such observers to be the most interesting ones from a physical point of view. \(b) the observers associated with a scalar homogeneous in the UFG. We shall show that up to second order in perturbation theory they are equivalent to free falling observers. \(c) Let us also briefly comment on the possibility of defining an observer which measures an unperturbed e-fold number $N$ in the LW limit, i.e. an unperturbed effective expansion factor. Consistently one finds, for such an observer which can be associated with a scalar homogeneous in the UCG, identically zero backreaction effects in Eqs. and . [*Geodetic Observers*]{}. We start by defining a free falling observer whose kinematics is determined by the equation $t_\mu=v^\nu \nabla_\nu v_\mu=0$ for its velocity $v_\mu$, which can be determined in any reference frame from the corresponding metric. In our analysis we keep contributions up to second order in the fluctuations: $v_\mu=v_\mu^{(0)}+ v_\mu^{(1)} +v_\mu^{(2)}$. In the SG one has $v_\mu=(-1,\vec{0})$. Our first task is to define the scalar field $A(x)$ associated with this observer. Such a scalar field should give $n_\mu=v_\mu$ and appears to be the one homogeneous up to second order in the SG (see [@Mar] for the complete description). For later use, let us exhibit the general condition for a scalar field $A(x)$ to be associated with free falling observers at first order. In this case $t_\mu$ should be zero up to the first order. The zero order condition is trivially satisfied for any scalar. At first order the $\mu=0$ condition is always satisfied while the $\mu=i$ condition gives ()-=0. \[Cond\_Geodesic\] As is easy to check, the l.h.s. of this condition is GI since the vector $t_\mu^{(0)}$ is identically zero. In general, using the coordinate transformations up to second order [@MetAll] x\^|[x]{}\^=x\^+\^\_[(1)]{}+ ( \^\_[(1)]{}\_\^\_[(1)]{}+\^\_[(2)]{}), \[CoordTrasf\] we define the espressions associated with Eq. , going from a general coordinate system to the barred one. Our strategy is therefore the following: we shall construct the observables and study Eqs. (\[EQB1\],\[EQB2\]) using Eq.(\[media\]). The results are by definition GI, this allows us to use the results for the dynamics of the inflaton and metric fluctuations, which satisfy the Einstein equations, up to second order, in any frame convenient for the calculations (we shall use the results computed in the UCG [@FMVV_II]). In the LW limit we have $$\begin{aligned} & & \bar{\Theta}=3H-3 H \bar{\alpha}-3\dot{\bar{\psi}} +\frac{9}{2} H \bar{\alpha}^2+3 \bar{\alpha} \dot{\bar{\psi}} -6\bar{\psi}\dot{\bar {\psi}} \nonumber \\ & & \,\,\,\,\,\,\,\,\,\,\,\,\,-3 H \bar{\alpha}^{(2)}- 3 \dot{\bar{\psi}}^{(2)}-\frac{1}{8}h_{ij}\dot{h}^{ij} \label{Theta} \\ & & \!\!\!\!\!- \partial_\mu \bar{A} \partial^\mu \bar{A}= 1- 2 \bar{\alpha}+4 \bar{\alpha}^2- 2 \bar{\alpha}^{(2)} \label{partA}\end{aligned}$$ and for the measure in the spatial section =a\^3 (1-3 |+|\^2 - h\^[ij]{}h\_[ij]{} -3|\^[(2)]{}). \[detgamma\] Let us note that we shall neglect in our computations the dependence on the tensor fluctuations hereafter (see however [@FMVV_GW] for the backreaction of tensor fluctuations in de Sitter space-time). Inserting Eqs.(\[Theta\], \[partA\], \[detgamma\]) in Eq.(\[EQB1\]) one obtains the simple expression ( )\^2 = H\^2 \[EQ1simpl\_2\] Let us turn our attention to the SG observers. The coordinate transformations needed to go to the SG are characterized by \_[(1)]{}\^0= \^t dt’ , \_[(2)]{}\^0= - \^t dt’ + \^t dt’ ( 2 \^[(2)]{} - \^2 ) , where we neglect a non dynamical constant contribution. Using Eq.(\[CoordTrasf\]) we have $$\begin{aligned} & &\bar{\alpha}=0\,\,\,\,,\,\,\,\,\bar{\psi}=\psi+H \int^t dt' \alpha\,\,\,\,,\,\,\,\,\bar{\varphi}=\varphi -\dot{\phi} \int^t dt' \alpha \nonumber \\ & & \bar{\psi}^{(2)}=\psi^{(2)}-H \alpha \int^t dt' \alpha-\frac{1}{2} \left(\dot{H}+2 H^2\right) \left[\int^t dt' \alpha\right]^2 \nonumber \\ & &-\left(2 H \psi+\dot{\psi}\right)\int^t dt' \alpha+ \frac{H}{2} \int^t dt' \left(2 \alpha^{(2)}-\alpha^2\right) \nonumber\end{aligned}$$ In order to evaluate the backreaction Eq. (\[EQB1\]), as said, we choose to perform the calculation in the UCG. In this gauge we need the solution to the equations of motion for the inflaton and the metric. To first order, one has = , \[Eq\_usefull1\] where $M_{pl}^{-2}=8\pi G$, and considering only the LW limit one gets =f() \^t dt’ =-f() \^t dt’ = . \[Eq\_usefull2\] These lead to $\bar{\varphi}=0$, $\bar{\alpha}=0$ and $\bar{\psi}=\frac{H}{\dot{\phi}} \varphi$ where the last term, which we insert in , is constant in such a limit. To second order we restrict to the particular case of a non self-interacting massive inflaton field [@FMVV_II]. In such a case, in the LW limit and at the leading order in the slow-roll approximation, one has $$\begin{aligned} \langle \alpha^{(2)} \rangle = \frac{1}{M_{pl}^2}\,\epsilon \, \langle \varphi^2 \rangle \,, \nonumber\end{aligned}$$ where $\epsilon=-\dot H/H^2$. We then obtain \^[(2)]{}= H [O]{}(\^2) . Inserting the various results into Eq. , we find ( )\^2=H\^2, \[reseq1\] where, for a massive chaotic model [@FMVV_II] in the LW limit and $H_i=H(t_i)\gg H$, one has - \~ a \[phiquad\] (see [@starall] for a generic single field inflationary scenario). Therefore there is no [*leading*]{} backreaction in the slow-roll parameter $\epsilon$ on the effective Hubble factor induced by scalar fluctuations. Provided the coefficient of $\langle \varphi^2 \rangle$ in Eq.  does not turn out to be zero, the quantum backreaction has the chance of appearing in the next-to-leading order, with a secular term related to the infrared growth of inflaton fluctuations. On the other hand such a growth gives a negligible effect whenever the quantity in is much smaller than $\epsilon^{-2}$. One can proceed in a similar way with the analysis of the second backreaction equation (\[EQB2\]) by evaluating directly the expressions. However we give here a more general result valid for any observer and slow-roll inflationary models. If for the effective Hubble factor one finds ( )\^2=H\^2, then, from the consistency between the effective equations for the averaged geometry, one obtains - = --H\^2-H\^2 and it is easy to see that the effective equation of state is unperturbed up to the leading non trivial order, i.e. w\_[effA]{}==-1+ +[O]{} (\^[n+1]{} ) . For the SG observer one has $n=2$ and we obtain the same condition as before to have no appreciable scalar backreaction effects. Non negligible effects could appear at the end of inflation ($H\sim m$) for initial conditions such that $H(t_i)\sim (m^2 M_{pl})^{1/3}$, which is an energy scale much smaller than the Planck one but is however associated with an extremely long inflationary era. Indeed such values give a typical number of e-folds of the order of ${\cal O}(10^4)$, for $M_{pl}=10^5m$, and correspond to the case where non-linear corrections become really important [@starall; @BPT]. In this case other computational techniques are required. [*UFG Observer*]{}. Let us introduce the observers associated with a scalar homogeneous in the UFG. Such an observer always sees as homogeneous the inflaton field $\Phi$ along its whole evolution during the inflationary regime. It is easy, following [@Mar], to identify the scalar associated to this observer as A(x)=A\^[(0)]{}+ + \^[(2)]{} + ( - )\^[2]{} . In this way the condition (\[Cond\_Geodesic\]) to have geodesic observers to first order becomes ()-=0. \[Cond\_Geodesic\_UFG\] Such a condition is trivially satisfied in the LW limit (as can be seen in the UCG using Eqs.(\[Eq\_usefull1\],\[Eq\_usefull2\])). To see if this equivalence is valid also to second order one should study the condition $t_\mu^{(2)}=0$. The quantity $t_\mu^{(2)}$ is GI in this case since $t_\mu^{(0)}=t_\mu^{(1)}=0$, and the condition can be studied in any gauge. Choosing the barred gauge it becomes $\bar{\alpha}^{(2)}=0$ and is satisfied in the LW limit. As a consequence the UFG observers are physically equivalent to the free falling ones and experience the same backreaction, as can be explicity verified by calculations. Let us emphasize that this property is not valid in general for all observers in the LW limit. Let us, for example, consider the observer define by the scalar homogeneous in the longitudinal gauge ($\beta=E=0$). This is defined to first order by A(x)=A\^[(0)]{}+\^[(0)]{} and the condition (\[Cond\_Geodesic\]) is not verified as can be easily seen. Such observer is not in geodesic motion and it may see a backreaction effect, even to leading order in the slow-roll parameter, which is in general different with respect to that experienced by a free falling observer. This case will be studied elsewhere. [*Conclusions*]{}. We have applied for the first time the gauge invariant observer dependent approach introduced in [@GMV1; @GMV2] to analyze backreaction effects induced by long wavelength scalar fluctuations in the cosmological early universe during an inflationary era. We have seen how for geodetic observers the backreaction of long wavelength scalar fluctuations does not appear to leading order in the slow-roll approximation for a $m^2 \phi^2$ chaotic inflationary model. Moreover we have shown under what conditions the backreation is negligible or not in the next-to-leading order. In the same long wavelength limit the tensorial contribution also disappear. This is a physical result which is derived in a truly gauge invariant way. Modes with shorter wavelengths, which typically behave less classically, may be a source of backreaction seen by physical observers, through all the terms, present in the equations, which have been neglected here. 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--- abstract: 'We analyse the darkweb and find its structure is unusual. For example, $ \sim 87 \%$ of darkweb sites *never* link to another site. To call the darkweb a “web” is thus a misnomer – it’s better described as a set of largely isolated dark silos. As we show through a detailed comparison to the World Wide Web (www), this siloed structure is highly dissimilar to other social networks and indicates the social behavior of darkweb users is much different to that of www users. We show a generalized preferential attachment model can partially explain the strange topology of the darkweb, but an understanding of the anomalous behavior of its users remains out of reach. Our results are relevant to network scientists, social scientists, and other researchers interested in the social interactions of large numbers of agents.' author: - 'Kevin P. O’Keeffe' - Virgil Griffith Yang Xu - Paolo Santi - Carlo Ratti title: 'The darkweb: a social network anomaly' --- Introduction ============ Studies of the World Wide Web (www) have had much impact. An understanding of its topology have let us better understand how its information is stored and can be retrieved [@albert1999diameter; @barabasi2000scale]. Insight into its paradoxical resilience [@albert2000error] have allowed us to design more fault tolerant networks [@wang2007fault], and the universal dynamics of it growth patterns [@barabasi1999emergence] have shed light on the behaviors of many others classes of networks [@barabasi2003scale]. And perhaps most importantly, being one of the earliest studied networks[^1] [@albert1999diameter; @barabasi2000scale], the www helped give rise to the flame of attention that network science enjoys today. This paper is about the www’s shady twin: the darkweb. Though the darkweb is similar to the www, both being navigated through a web browser, a key difference the two is that the identities of darkweb users are hidden – that’s what makes it ‘dark’. This gives the darkweb an infamous air of mystery, which, along with the sparse academic attention it has received [@everton2012disrupting; @darknet], makes it ripe for analysis. And beyond satisfying curiosity, its reasonable to think studies of the darkweb could have as much applied impact as the studies of the www. Insight about the structure or dynamics of the darkweb, for instance, could potentially allow policy-makers to better police its more sinister aspects. There are many natural questions to ask about the darkweb. Does it have the same topology as the www, and therefore hold and distribute information in the same way? Is it resilient to attack? Social questions can also be posed. Results from psychology and game theory show people behave more socially when they are watched [@bradley2018does; @kristofferson2014nature] or have reputations to uphold [@nowak2011supercooperators]. Do darkweb users, with their masked faces, therefore behave more selfishly than www users, whose faces are bare? Here we address some of these questions by performing a graph theoretical analysis of the darkweb (we define exactly what we mean by the darkweb below) and comparing it to the www. We find the topologies of the darkweb and the www are starkly different – in fact, the darkweb is much different to many social networks – and conjecture this is due to their users’ differing social behaviour. We hope our work stimulates further research on the darkweb’s structure as well as the social dynamics of its users. Data Collection =============== There is no single definition of the darkweb. It is sometimes loosely defined as “anything seedy on the Internet”, but in this work we define the darkweb as all domains underneath the “.onion” psuedo-top-level-domain[@rfc7686] (which is sometimes called the onionweb). Specifically, we mean the subset of the web where websites are identified not by a human-readable hostname (e.g., yahoo.com) or by a IP number (e.g., 206.190.36.45), but by a randomly generated 16-character address (specifically, a hash fingerprint). Each website can be accessed via its hash, but it is very difficult to learn the IP number of a website from its hash – this is what makes the web ‘dark’; without an IP number, it is exceedingly difficult to trace the geographical origin of a communication. Crawling the darkweb is not much harder than crawling the regular web. In our case, we crawled the darkweb through the popular tor2web proxy onion.link. Onion.link does all of the interfacing with Tor, and one can crawl all darkweb pages without a login simply by setting a standard crawler to specifically crawl the domain `*.onion.link`. Darkweb pages are written in the same HTML language as the regular web which means we could crawl onion.link using the commercial service scrapinghub.com. Starting from two popular lists of darkweb sites,[^2] we accessed each page and crawled all linked pages using breadth-first search. Most analyses of the www are at the page-level, where each node is an individual URL. One could adopt this model for the darkweb. It would be a natural choice for engineering motivated research question, such as studying crawlability. But the page-level model is not natural for socially motivated research questions, which we are interested in in this study, because the page-level graph is influenced more by the various choices of content management system, rather than by social dynamics. So we instead follow the modeling choice in [@lehmberg2014graph] and *aggregate by second-level domain* (for the onionweb the second-level domain is equivalent to [@lehmberg2014graph]’s “pay-level domain”). This means that links within a second-level domain are ignored as socially irrelevant self-connections. In this formulation, each vertex in the darkweb is a *domain* and every directed edge from $u \rightarrow v$ means there exists a page within domain $u$ linking to a page within domain $v$. The weight of the edge from $u \rightarrow v$ is the number of pages on domain $u$ linking to pages on domain $v$. The Tor Project Inc. (the creators and custodians of the darkweb) state there are $\sim \!\! 60,000$ distinct, active .onion addresses [@tormetrics]. In our analysis, however, we found merely [$7,178$]{}active .onion domains. We attribute this high-discrepancy to various messaging services—particularly TorChat [@wiki:torchat], Tor Messenger [@wiki:tormessenger], and Ricochet [@wiki:ricochet]. In all of these services, each user is identified by a unique .onion domain. The darkweb has a notoriously high attrition rate; its sites regularly appear and disappear. This complicates our analysis because it creates dead links, links to pages which have been removed. We do not want the dead links in our datasets so we collected responsive sites only; if we discover a link to a page on domain $v$, but domain $v$ could not be reached after $>\!10$ attempts across November 2016–February 2017, we delete node $v$ and all edges to node $v$. Before pruning nonresponding domains, our constructed graph had 13,117 nodes and 39,283 edges. After pruning, it has [$7,178$]{}nodes and [$25,104$]{}edges (55% and 64% respectively). The pruned graph is the one used in the rest of this paper, which from now on we call “ the darkweb graph”. We note that the darkweb as defined above is different from the network described in [@darknet]. There, the network represents volunteer-operated nodes that could be connected in the sense that they could appear as consecutive nodes in a path through the Tor network. This is completely different from our network. Graph-theoretic Results ======================= Table \[fig:summary1\] reports summary statistics of the darkweb and www. In what follows, we discuss these and other statistics. Measure www [@lehmberg2014graph] darkweb --------------------------- -------------------------- -------------- **Num nodes** 43M [$7,178$]{} **Num edges** 623M [$25,104$]{} **Prop. connected pairs** $\sim\!\! 42\%$ 8.11% **Average SPL** [$4.27$]{} [$4.35$]{} **Edges per node** 14.5 3.50 **Network diameter**\* 48 5 **Harmonic diameter** $\sim\!\!9.55$ 232.49 : Summarized network level properties between the www and the darkweb. Asterisk for the entries requiring conversion to an undirected graph.[]{data-label="fig:summary1"} Degree distribution ------------------- We begin with statistics on degree distributions, reported in Figure \[fig:degree\_and\_pr\]. Panels (a) and (b) show the in and out degree distributions of the darkweb resemble power laws[^3], just like the www, but with one crucial difference: the location of the $y$-intercept. In (a) we see $\sim\! 30\%$ of domains have *exactly one* incoming link ($k_{in} = 1$), with $62\%$ come from one of the five largest out-degree hubs. In (b), we see a whopping of sites *do not link to any other site* ($k_{out} = 0$)! – these are the dark silos that we mentioned in the abstract. These findings tell us the darkweb is a sparse hub-and-spoke place. The bulk of its sites live in seclusion, not wanting to connect with the outside world. Panel (c) confirms this picture by showing the vast majority of pages have low pagerank (and so are isolated). Panel (d) shows that when a site does link to another, $32\%$ of the time it’s only a single page linking out. As we show in the next section, this siloed structure is not shared by many social networks. \ Bow-tie decomposition --------------------- A useful way to describe directed graphs is via *bow-tie decomposition*, where the nodes are divided into six disjoint parts [@broder2000graph]: 1. <span style="font-variant:small-caps;">CORE</span> — Also called the “Largest Strongly Connected Component”. It is defined as the largest subset of nodes such that there exists a directed path (in both directions, from $u \rightarrow \cdots \rightarrow v$ as well as $v \rightarrow \cdots \rightarrow u$) between every pair of nodes in the set. 2. <span style="font-variant:small-caps;">IN</span> — The set of nodes, excluding those in the <span style="font-variant:small-caps;">CORE</span>, that are ancestors of a <span style="font-variant:small-caps;">CORE</span> node. 3. <span style="font-variant:small-caps;">OUT</span> — The set of nodes, excluding those in the CORE, that are descendants of a <span style="font-variant:small-caps;">CORE</span> node. 4. <span style="font-variant:small-caps;">TUBES</span> — The set of nodes, excluding those in the <span style="font-variant:small-caps;">CORE</span>, <span style="font-variant:small-caps;">IN</span>, and <span style="font-variant:small-caps;">OUT</span>, who have an ancestor in <span style="font-variant:small-caps;">IN</span> as well as a descendant in <span style="font-variant:small-caps;">OUT</span>. 5. <span style="font-variant:small-caps;">TENDRILS</span> — Nodes that have an ancestor in <span style="font-variant:small-caps;">IN</span> but do not have a descendant in <span style="font-variant:small-caps;">OUT</span>. Also, nodes that have a descendant in OUT but do not have an ancestor in <span style="font-variant:small-caps;">IN</span>. 6. <span style="font-variant:small-caps;">DISCONNECTED</span> — Everything else. Figure \[fig:bowtie\] compares the bowtie decompositions of the darkweb and www. The www numbers are taken from [@Serrano2007; @lehmberg2014graph; @meusel2015graph]. We chose these works because of the size of their crawls and the rigor of their analyses. Notice the www has each of one the 6 components of the bow-tie decomposition, whereas the darkweb only has a <span style="font-variant:small-caps;">CORE</span> and an <span style="font-variant:small-caps;">OUT</span> component. Moreover, the OUT component of the darkweb contains $\sim 96\%$ of the mass (these are the dark silos), leaving the CORE with just $\sim 4 \%$. This is unusual for a social network; most have large COREs. The www’s CORE has $> 50 \%$ of the mass, while the cores of Orkut, YouTube, Flickr [@mislove2007measurement] and Twitter [@garcia2017understanding] are even larger. In this sense, the darkweb is a social network anomaly. -------------- ------- -------- ------ -------- CORE 22.3M 51.94% 297 4.14% IN 3.3M 7.65% 0 0.0% OUT 13.3M 30.98% 6881 95.86% TUBES 17k .04% 0 0.0% TENDRILS 514k 1.2% 0 0.0% DISCONNECTED 3.5M 8.2% 0 0.0% -------------- ------- -------- ------ -------- Diameter analysis ----------------- Next we examine the darkweb’s internal connectivity by computing the shortest-path-length (SPL) between nodes. Figure \[fig:spl\](a) and (b) shows the SPL’s for the www and darkweb. For all pairs of nodes $\{u,v\}$ in the darkweb, only 8.11% are connected by a directed path from $u \rightarrow \cdots \rightarrow v$ or $v \rightarrow \cdots \rightarrow u$. This is drastically lower than the $\sim\!\!43.42\%$ found in the www [@lehmberg2014graph]. We again attribute this to the low out-degree per \[fig:degree\_and\_pr\]. Of the connected pairs, the darkweb’s average shortest path length is [$4.35$]{}compared to the [$4.27$]{}in the world-wide-web [@lehmberg2014graph]. It’s surprising to see a graph as small as the darkweb have a higher mean SPL than the entire world-wide-web, and is a testament to how sparse the darkweb graph really is. Figure \[fig:spl\](c) plots the distribution of SPLs for the [$297$]{}nodes of the <span style="font-variant:small-caps;">CORE</span>. To our surprise, the mean SPL within the <span style="font-variant:small-caps;">CORE</span> is [$3.97$]{}, only $9\%$ less than the entire darkweb. From this we conclude the <span style="font-variant:small-caps;">CORE</span> is not densely interconnected. Robustness and Fragility ------------------------ Does the peculiar structure of the darkweb make it resilient to attack? Figure \[fig:knockouts\] shows it does not. As seen, the entire network (WCC) as well as the <span style="font-variant:small-caps;">CORE</span> quickly disintegrates under node removal. In Figures \[fig:knockouts\](a) and (b) we see the familiar resistance to random failure yoked with fragility to targeted attacks, in keeping with the findings of [@bollobas2004robustness]. Figures \[fig:knockouts\](b) shows that, unlike the www [@lehmberg2014graph], the WCC is *more susceptible to high in-degree deletions than the <span style="font-variant:small-caps;">CORE</span>*. This elaborates the view from Figures \[fig:knockouts\](c) that the <span style="font-variant:small-caps;">CORE</span> is – in addition to not being strongly interconnected – also not any kind of high in-degree nexus. Figures \[fig:knockouts\](c) and (d) show the breakdown when removing central nodes. In (c), the CORE is largely unaffected by low centrality deletions. In Figure \[fig:knockouts\](c) we see that although the <span style="font-variant:small-caps;">CORE</span> isn’t disproportionately held together by high in-degree nodes, it is dominated by very central nodes. Comparing Figures \[fig:knockouts\](b) and (f) we see the <span style="font-variant:small-caps;">CORE</span> relative to the entire network consists of more high-pagerank nodes than high in-degree nodes. This implies <span style="font-variant:small-caps;">CORE</span> nodes are not created by their high-indegree (\[fig:dw\_removing\_indgree\_highest\]), but by their high centrality, amply corroborated by Figures \[fig:knockouts\](c) and (d). Likewise, Figures \[fig:knockouts\](e) recapitulates \[fig:dw\_removing\_indgree\_lowest\], that nodes with especially low in-degree or centrality are, unsurprisingly, not in the <span style="font-variant:small-caps;">CORE</span>. \ Reciprocal Connections ---------------------- The authors of [@Serrano2007] stress the importance of reciprocal connections in maintaining the www’s graph properties. We compute two of their measures. First, we compute [@Serrano2007]’s measure $\frac{\langle k_{in}k_{out} \rangle }{ \langle k_{in} \rangle \langle k_{out} \rangle } = \frac{ \mathbb{E}[ k_{in} k_{out} ] }{ \mathbb{E}[ k_{in} ] \mathbb{E}[ k_{out} ] }$, to quantify in-degree and out-degree’s deviation from independence. For the darkweb, we arrive at $\frac{\langle k_{in}k_{out} \rangle}{\langle k_{in} \rangle \langle k_{out} \rangle}=3.70$. This is in the middle of the road of prior estimates of the www, and means that the out-degree and in-degree are positively correlated. For greater clarity, we also plot the average out-degree as a function of the in-degree, given as, $$\langle k_{out}(k_{in}) \rangle = \frac{1}{N_{k_{in}}} \sum_{i \in \Upsilon(k_{in})} k_{out,i} \; , \label{eq:serrano}$$ which is simply “For all nodes of a given in-degree, what is the mean out-degree?”. The results are depicted in Figure \[fig:outdeg\_as\_func\_of\_indeg\]. In short, in the darkweb there’s no obvious pattern to the relationship between in-degree and out-degree. ![image](scatter-plot-kin-kout.pdf){height="3cm"}![image](scatter-plot-kin-mean-kout.pdf){height="3cm"} ![image](dw2dw_vs_dw2www.pdf){height="3cm"} Measure www [@lehmberg2014graph] darkweb --------------------------- -------------------------- -------------- **Num nodes** 43M [$7,178$]{} **Num edges** 623M [$25,104$]{} **Prop. connected pairs** $\sim\!\! 42\%$ 8.11% **Average SPL** [$4.27$]{} [$4.35$]{} **Edges per node** 14.5 3.50 **Network diameter**\* 48 5 **Harmonic diameter** $\sim\!\!9.55$ 232.49 : Summarized network level properties between the www and the darkweb. Asterisk for the entries requiring conversion to an undirected graph.[]{data-label="fig:summary"} Network growth model -------------------- Are standard network growth models able to capture the topology of the darkweb? Here we show a generalized preferential attachment model approximately can. In regular preferential attachment, nodes of only one type are added sequentially to a network. Here, we generalize this procedure to two types of nodes: *pages* and *portals*. Page nodes model the nodes in the darkweb which do not link to anyone (and so have $k_{in} = 0$). Portals model the rest of the nodes, which act as ‘portals’ to the page nodes. The dynamics of the ‘page-portal’ model are defined as follows. At $t = 0$ $N_0$ portals are arranged in a given network. At each time step after $t>0$, a new page is added to the network and $m$ different portals are chosen to link to $m$ end nodes (these end nodes can be either pages or portals). The portal which does the linking is chosen according to preferential attachment, that is, chosen with probability $\propto (1 + k_{out}^{\beta}$), where $k_{out}$ is the out degree and $\beta$ is a free parameter. The end node chosen with probability $\propto (1 + k_{in}^{\beta})$. Notice this scheme means that only portals can link, which means page nodes forever remain with $k_{in} = 0$, as desired. The model has three free parameters $m, \beta , N_0$. We chose $N_0 = 397$ so as to match the number of portals in our datasets (defined as all nodes with $k_{in} > 0$) and ran the model for $6242$ timesteps i.e added $6242$ page nodes) so that there were $7178$ nodes at the end of the process which again matched the dataset. Figure \[alphas\] shows the page-portal model with parameters $(m,\beta) = (4, 2)$ approximates the darkweb, the in and out degree distributions of the model approximately mimicking the data. We report exponents of best fit, found using the powerlaw package python. But keep in mind that as mentioned earlier, there are not enough orders of magnitude in the data for the estimates of the exponents to be reliable; thus the page and portal model is intended as a first step in modeling the darkweb. The values were $(\alpha_{in}, \alpha_{out})_{data} = (3.09, 2.10)$ and $(\alpha_{in}, \alpha_{out})_{model} = (4.3 \pm 1.3 , 2.4 \pm 0.2 )$, where the model figures report means of 10 realizations and the error is given as half the range. The fitting was performed using the python powerlaw package. $\alpha_{in}$ was much more difficult to measure than $\alpha_{out}$. As much as half of realizations led to estimates $ \alpha_{in} > 50$, which we interpret as the failure of the fitting to converge, and so were discarded. ![image](alphas.pdf){width=".75\linewidth"} Conclusion ========== Our primary finding is that the darkweb is a social network anomaly. Its light CORE and massive OUT components distinguishes it from other popular social networks. In fact, calling it a ‘web’ is a connectivity misnomer; it is more accurate to view it as a set of dark silos – a place of isolation, entirely distinct from the well connected world of the www and other social network. What causes the darkweb to be so isolated? We see two possible explanations: - **The technological explanation.** In the darkweb, sites go up and go down all the time. Why bother linking if there’s little chance that the destination will still exist? - **The social explanation.** People who create sites on the darkweb are cut from a different social cloth than those who create sites on the www (or at least when using the darkweb, these people behave differently) To test the technological explanation, we performed a second crawl collecting instances of darkweb sites linking to the www and compared the relative rates of outbound linking in Figure \[fig:outdeg\_as\_func\_of\_indeg\](c). There are essentially equal rates of outbound linking to the www as well as the darkweb which tells us (i) the low outbound linking is not due to the impermanence of onion sites and (ii) if onion sites got drastically more stable, we would still see very low rates of linking. Taken together, these indicate the technological explanation is likely untrue. Thus, the social explanation is likely the cause of the darkweb’s anomalous topology. Rigorously testing the social hypothesis is however beyond the scope of this work. Although, in a sense we have taken a first step in this direction by generalizing preferential attachment which itself can be viewed as model of social behavior; it is a model of trust: highly linked nodes are perceived as ‘trustworthy’ sources of information, and so receive a disproportionate number of links; the rich get richer. The isolated silos of the darkweb, however, indicate trust does *not* play a role in the dynamics governing its evolution. Rather, one might say it indicates that *distrust* does. The passive pages of the page and portal model (which recall, do not link to anybody through the dynamics, and are in that sense passive) were a crude way to incorporate this effect. But a more principled behavioral model (i.e. one consistent with known results from psychology) is needed, which were were unable to develop. We hope psychology-fluent researchers will take up this task in future work. Future work could also study the temporal aspects of the darkweb. Is the topology we have found stationary? For example, in the work most closely related to ours [@darknet], it was found that the resilience of the studied ‘darknet’ evolved over time (as discussed in the Data Collection section, our darkweb graph is much different to the darknet in [@darknet]). It would be interesting to see if the resilience of our darkweb graph behaves like this too. ![image](dw2dw_vs_dw2www.pdf){width="70.00000%"} Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank all members of the MIT Senseable City Lab consortium. 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Mislove, Alan and Marcon, Massimiliano and Gummadi, Krishna P and Druschel, Peter and Bhattacharjee, Bobby (2007) Measurement and analysis of online social networks Garcia, David and Mavrodiev, Pavlin and Casati, Daniele and Schweitzer, Frank (2017) Understanding popularity, reputation, and social influence in the twitter society (2007) Measurement and analysis of online social networks Bollob[á]{}s B, Riordan O (2004) Robustness and vulnerability of scale-free random graphs. Internet Mathematics 1: 1–35. Biryukov A, Pustogarov I, Weinmann RP (2013) Trawling for tor hidden services: Detection, measurement, deanonymization. In: Security and Privacy (SP), 2013 IEEE Symposium on. IEEE, pp. 80–94. [^1]: alongside smallworld networks [@watts1998collective]. [^2]: `http://directoryvi6plzm.onion` and `https://ahmia.fi/onions/` [^3]: Unfortunately however, we were able to confirm the power laws quantitatively. 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--- abstract: 'Large-scale quantum networks promise to enable secure communication, distributed quantum computing, enhanced sensing and fundamental tests of quantum mechanics through the distribution of entanglement across nodes [@kimble_quantum_2008; @broadbent_universal_2009; @jiang_quantum_2009; @ekert_ultimate_2014; @gottesman_longer-baseline_2012; @nickerson_freely_2014; @komar_quantum_2014]. Moving beyond current two-node networks [@Hucul2015; @Hensen2015; @Kalb2017; @Reiserer2015; @Hofmann72; @northup2014quantum] requires the rate of entanglement generation between nodes to exceed their decoherence rates. Beyond this critical threshold, intrinsically probabilistic entangling protocols can be subsumed into a powerful building block that deterministically provides remote entangled links at pre-specified times. Here we surpass this threshold using diamond spin qubit nodes separated by 2 metres. We realise a fully heralded single-photon entanglement protocol that achieves entangling rates up to 39 Hz, three orders of magnitude higher than previously demonstrated two-photon protocols on this platform [@Pfaff2014]. At the same time, we suppress the decoherence rate of remote entangled states to 5 Hz by dynamical decoupling. By combining these results with efficient charge-state control and mitigation of spectral diffusion, we are able to deterministically deliver a fresh remote state with average entanglement fidelity exceeding 0.5 at every clock cycle of $\sim$100 ms without any pre- or post-selection. These results demonstrate a key building block for extended quantum networks and open the door to entanglement distribution across multiple remote nodes.' author: - 'Peter C. Humphreys' - Norbert Kalb - 'Jaco P. J. Morits' - 'Raymond N. Schouten' - 'Raymond F. L. Vermeulen' - 'Daniel J. Twitchen' - Matthew Markham - Ronald Hanson bibliography: - 'bib\_library.bib' title: Deterministic delivery of remote entanglement on a quantum network --- The power of future quantum networks will derive from entanglement that is shared between the network nodes. Two critical parameters for the performance of such networks are the entanglement generation rate $r_\text{ent}$ between nodes and the entangled-state decoherence rate $r_\text{dec}$. Their ratio, that we term the quantum link efficiency $\eta_{\text{\,link}} = r_\text{ent} / r_\text{dec}$ [@Monroe2014; @Hucul2015], quantifies how effectively entangled states can be preserved over the timescales necessary to generate them. Alternatively, from a complementary perspective, the link efficiency determines the average number of entangled states that can be created within one entangled state lifetime. Achieving a link efficiency of unity therefore represents a critical threshold beyond which entanglement can be generated faster than it is lost. Exceeding this threshold is central to allowing multiple entangled links to be created and maintained simultaneously, as required for the distribution of many-body quantum states across a network [@Monroe2014; @nickerson_freely_2014]. Consider an elementary entanglement delivery protocol that delivers states at pre-determined times. This can be achieved by making multiple attempts to generate entanglement, and then protecting successfully generated entangled states from decoherence until the required delivery time (Fig. \[fig:QuantumCapacity\]a, steps 1, 2 & 3). If we try to generate entanglement for a period $t_\text{ent}$, the cumulative probability of success will be $p_\text{succ} = 1-e^{- r_\text{ent} t_\text{ent}}$. For a given $p_\text{succ}$, the average fidelity $F_\text{succ}$ of the successfully generated states is solely determined by the quantum link efficiency $\eta_{\text{\,link}}$ (Supplementary Information (SI) section I). We plot $F_\text{succ}$ versus $p_\text{succ}$ for several values of $\eta_{\text{\,link}}$ in Fig \[fig:QuantumCapacity\]b. ![image](Fig1/Fig1_QuantumCapacity){width="14.0cm"} This protocol allows entangled states to be delivered at specified times, but with a finite probability of success. By delivering an unentangled state (state fidelity $F_\text{unent} \leq \frac{1}{2}$) in cycles in which all entanglement generation attempts failed, the protocol can be cast into a fully deterministic black-box (Fig \[fig:QuantumCapacity\]a, step 4). The states output from such a black-box will have a fidelity with a Bell state of $$F_\text{det} = p_\text{succ} F_\text{succ} + (1 - p_\text{succ}) F_\text{unent}.$$ The maximum achievable fidelity $F_\text{det}^{\text{max}}$ of this deterministic state delivery protocol, found by optimising $p_\text{succ}$, is also only determined by the quantum link efficiency $\eta_{\text{\,link}}$. For $F_\text{unent} = \frac{1}{4}$ (fully mixed state), we find (see Fig \[fig:QuantumCapacity\]c): $$F_\text{det}^{\text{max}} = \frac{1}{4} \left(1+3 {\eta_\text{\,link}}^{\frac{1}{1-\eta_{\text{\,link}}}}\right).$$ Beyond the threshold $\eta_{\text{\,link}} \gtrsim 0.83$, there exists a combination of $p_\text{succ}$ and $F_\text{succ}$ high enough to compensate for cycles in which entanglement is not heralded, allowing for the deterministic delivery of states that are on-average entangled ($F_\text{det}^{\text{max}}\geq\frac{1}{2}$). Demonstrating deterministic entanglement delivery therefore presents a critical benchmark of a network’s performance, certifying that the network quantum link efficiency is of order unity or higher. Furthermore, the ability to specify in advance the time at which entangled states are delivered may assist in designing multi-step quantum information tasks such as entanglement routing [@2017arXiv170807142P; @2016arXiv161005238S]. To date, this threshold has remained out of reach for solid-state quantum networks. Quantum dots have demonstrated kHz entanglement rates $r_\text{ent}$, but tens of MHz decoherence rates $r_\text{dec}$ limit their achieved quantum link efficiencies to ${\eta_{\text{\,link}}\sim10^{-4}}$ [@Stockill2017; @Delteil2016]. Nitrogen vacancy (NV) centres, point-defects in diamond with a long-lived electron spin and bright optical transitions, have demonstrated entanglement rates $r_\text{ent}$ of tens of mHz [@Kalb2017; @Pfaff2014] and, in separate experiments, decoherence rates $r_\text{dec}$ of order 1 Hz [@bar2013solid], which would together give $\eta_{\text{\,link}} \sim10^{-2}$. Here we achieve $\eta_{\text{\,link}}$ well in excess of unity by realising an alternative entanglement protocol for NV centres in which we directly use the state heralded by the detection of a single photon (Fig. \[fig:SingleClickEntResults\]) [@Cabrillo_1999; @Minar2008]. The rate for such single-photon protocols scales linearly with losses, which, in comparison with previously used two-photon-mediated protocols [@Pfaff2014; @Hensen2015], provides a dramatic advantage in typical remote entanglement settings. Recent experiments have highlighted the potential of such single-photon protocols by generating local entanglement [@Casabone2013; @Sipahigil2016], and remote entanglement in post-selection [@Stockill2017; @Delteil2016]. By realising a single-photon protocol in a fully heralded fashion and protecting entanglement through dynamical decoupling, we achieve the deterministic delivery of remote entangled states on a $\sim$10 Hz clock. Our experiment employs NV centres residing in independently operated cryostat setups separated by 2 metres (SI section II). We use qubits formed by two of the NV centre ground-state spin sub-levels ($\ket{\uparrow} \equiv \ket{m_s = 0}, \ket{\downarrow} \equiv \ket{m_s=-1}$). Single-photon entanglement generation (Fig. \[fig:SingleClickEntResults\]a) proceeds by first initialising each node in $\ket{\uparrow}$ by optical pumping [@Robledo2011], followed by a coherent rotation using a microwave pulse [@Fuchs1520] to create the state $$\ket{NV} = \sqrt{\alpha} \ket{\uparrow} + \sqrt{1-\alpha} \ket{\downarrow}.$$ We then apply resonant laser light to selectively excite the ‘bright’ state $\ket{\uparrow}$ to an excited state, which rapidly decays radiatively back to the ground state by emitting a single photon. This entangles the spin state of the NV with the presence $\ket{1}$ or absence $\ket{0}$ of a photon in the emitted optical mode: $$\ket{NV, \text{optical mode}} = \sqrt{\alpha} \ket{\uparrow} \ket{1} + \sqrt{1-\alpha} \ket{\downarrow} \ket{0}.$$ Emitted photons are transmitted to a central station at which a beamsplitter is used to remove their which-path information. Successful detection of a photon at this station indicates that at least one of the NVs is in the bright state $\ket{\uparrow}$ and therefore heralds the creation of a spin-spin entangled state. However, given the detection of one photon, the conditional probability that the other NV is also in the state $\ket{\uparrow}$, but the photon it emitted was lost, is given by $p =\alpha$ (in the limit that the photon detection efficiency $p_\text{det} \ll 1$). This degrades the heralded state from a maximally-entangled Bell state $\ket{\psi} = \frac{1}{\sqrt{2}} (\ket{\uparrow \downarrow} + \ket{\downarrow \uparrow})$ to $$\rho_{NV,NV} = (1-\alpha) \ket{\psi}\!\bra{\psi} + \alpha \ket{\uparrow \uparrow}\!\bra{\uparrow \uparrow}.\label{eqn:mixed}$$ The probability of successfully heralding entanglement is given by $ 2 \, p_\text{det} \alpha$. The state fidelity $F = 1-\alpha$ can therefore be directly traded off against the entanglement rate. The corresponding success probability of a two-photon protocol is given by $\frac{1}{2} p_\text{det}^2$; for a given acceptable infidelity $\alpha$, single-photon protocols will thus provide a rate increase of $4 \, \alpha / p_\text{det}$. For example, for our system’s ${p_\text{det}\sim4\times10^{-4}}$, if a 10% infidelity is acceptable, the rate can be increased by three orders of magnitude over two-photon protocols. The primary challenge in implementing single-photon entanglement is that the resulting entangled state depends on the optical phase acquired by the laser pulses used to create spin-photon entanglement at each node, as well as the phase acquired by the emitted single photons as they propagate (Fig \[fig:SingleClickEntResults\]b). The experimental setup therefore acts as an interferometer from the point at which the optical pulses are split to the point at which the emitted optical modes interfere. For a total optical phase difference of $\Delta \theta$, the entangled state created is given by $$\ket{\psi_{0/1}(\Delta \theta)} = \ket{\uparrow \downarrow} \pm e^{\mathrm{i} \Delta \theta} \ket{\downarrow \uparrow},$$ where $0/1$ (with corresponding $\pm$ phase factor) denotes which detector at the central station detected the incident photon. This optical phase difference must be known in order to ensure that entangled states are available for further use. We overcome this entangled-state phase sensitivity by interleaving periods of optical-phase stabilisation with our entanglement generation. During phase stabilisation we input bright laser light at the same frequency as the NV excitation light and detect the light reflected from the diamond substrate using the same detectors that are used to herald entanglement. The measured optical phase, estimated from the detected counts, is used to adjust the phase back to our desired value using a piezoelectric fibre stretcher. We achieve an average steady-state phase stability of $14.3(3)^\circ$, limited by the mechanical oscillations of the optical elements in our experimental setup (SI section V). To demonstrate the controlled generation of entangled states, we run the single-photon entangling protocol with a bright-state population of $\alpha = 0.1$. After entanglement is heralded, we apply basis rotations and single-shot state readout [@Robledo2011] at each node to measure $\avg{\sigma_i^A \sigma_j^B}$ correlations between the nodes, where the standard Pauli matrices will be referred to here in the shorthand $\sigma_{X},\sigma_{Y},\sigma_{Z} = X,Y,Z$. We observe strong correlations both for $\avg{XX}$ and $\avg{YY}$, and, when sweeping the readout basis for node A, oscillations of these coherences as expected from the desired entangled state (Fig. \[fig:SingleClickEntResults\]c, left panel). In combination with the measured $\avg{ZZ}$ correlations (Fig. \[fig:SingleClickEntResults\]c, right panel), this unambiguously proves the establishment of entanglement between our nodes. We explore the tradeoff between the entangled state fidelity and the entanglement rate by measuring $\avg{XX}$, $\avg{YY}$ and $\avg{ZZ}$ correlations for a range of different initial bright-state populations $\alpha$. Using these correlations, we calculate the fidelity of the heralded state to the desired maximally entangled Bell state for each value of $\alpha$ (Fig. \[fig:SingleClickEntResults\]d), along with the measured success rate (Fig. \[fig:SingleClickEntResults\]e). As predicted, the fidelity increases with decreasing $\alpha$ as the weight of the unentangled state $\ket{\uparrow \uparrow}\!\bra{\uparrow \uparrow}$ diminishes (Eqn. \[eqn:mixed\]). For small $\alpha$, the fidelity saturates because the detector dark-count rate becomes comparable to the detection rate. Choosing $\alpha$ to maximise fidelity, we find that our protocol allows us to generate entanglement with a fidelity of $0.81(2)$ at a rate of $r_\text{ent} = 6$ Hz (for $\alpha = 0.05$). Alternatively, by trading the entanglement fidelity for rate, we can generate entanglement at $r_\text{ent} = 39$ Hz with an associated fidelity of $0.60(2)$ ($\alpha = 0.3$). This represents a two orders of magnitude increase in the entangling rate over all previous NV experiments [@Kalb2017] and a three orders of magnitude increase in rates over two-photon protocols under the same conditions [@Pfaff2014]. ![**Coherence protection of remote entangled states.** **a**, Dynamical decoupling protects the state of the NV spins from quasi-static environmental noise. Applying $N$ pulses allows us to dynamically decouple the NV state for a time $2 N t$. **b**, Fidelity with the initial state for dynamical decoupling of the single-qubit state $\ket{\uparrow} + \ket{\downarrow}$ at our two NV nodes. Solid lines show exponential fits with coherence times of $290(20)$ ms and $680(70)$ ms for nodes A and B respectively. **c**, Dynamical decoupling of entangled states created using the single-photon entanglement protocol for bright-state populations $\alpha =0.12$ and $\alpha =0.2$. Solid lines show the predictions of our model based on the coherence times measured in (b), from which the effective entangled state coherence time is expected to be $\tau = 200(10)$ ms.[]{data-label="fig:SuccRateAndDD"}](Fig3/Fig3_SuccRateAndDD.pdf){width="8.8cm"} Compared to the maximum theoretical fidelity for $\alpha = 0.05$ of 0.95, the states we generate have a 3% reduction in fidelity due to residual photon distinguishability, 4% from double excitation, 3% from detector dark counts, and 2% from optical-phase uncertainty (SI sections V, VI & VII). In order to reach a sufficient link efficiency $\eta_{\text{\,link}}$ to allow for deterministic entanglement delivery, the single-photon protocol must be combined with robust protection of the generated remote entangled states. To achieve this, we carefully shielded our NVs from external noise sources including residual laser light and microwave amplifier noise, leaving as the dominant noise the slowly-fluctuating magnetic field induced by the surrounding nuclear spin bath. We mitigate this quasi-static noise by implementing dynamical decoupling with XY8 pulse sequences (Fig. \[fig:SuccRateAndDD\]a, SI section VIII). The fixed delay between microwave pulses in these sequences is optimised for each node [@2017abobeih]. Varying the number of decoupling pulses allows us to protect the spins for different durations. This dynamical decoupling extends the coherence time of Node A and B from a $T_2^*$ of $\sim5 \, \mu$s to $290(20)$ ms and $680(70)$ ms respectively, as shown in Fig. \[fig:SuccRateAndDD\]b. The difference in coherence times for the two nodes is attributed to differing nuclear spin environments and microwave pulse fidelities. ![image](Fig4/Fig4_DeterministicEntGenerationExpm){width="14.0cm"} To investigate the preservation of remote entangled states, we incorporate dynamical decoupling for varying time durations after successful single-photon entanglement generation (Fig. \[fig:SuccRateAndDD\]c). We find an entangled state coherence time of $200(10)$ ms (decoherence rate $r_\text{dec}$ of 5.0(3) Hz). The observed entangled-state fidelities closely match the predictions of our model, which is solely based on independently determined parameters (SI section III). In particular, the decoherence of the remote entangled state is fully explained by the combination of the individual decoherence rates of the individual nodes. The combination of dynamical decoupling with the single-photon entanglement protocol achieves a quantum link efficiency of ${\eta_{\text{\,link}}\sim8}$ (comparable to the published state-of-the-art in ion traps, ${\eta_{\text{\,link}}\sim5}$ [@Hucul2015]), pushing the NV-based platform well beyond the critical threshold of $\eta_{\text{\,link}} \gtrsim 0.83$. We capitalise on these innovations to design a deterministic entanglement delivery protocol that guarantees the delivery of entangled states at specified intervals, without any post-selection of results or pre-selection based on the nodes being in appropriate conditions (Fig. \[fig:DeterministicEntGenerationExpm\]a). Phase stabilisation occurs at the start of each cycle, after which there is a preset period before an entangled state must be delivered. This window must therefore include all NV state checks (necessary to mitigate spectral diffusion via feedback control and verify the charge-state and resonance conditions [@Hensen2015]), entanglement generation attempts and dynamical decoupling necessary to deliver an entangled state (further details are given in SI section II). We run our deterministic entanglement delivery protocol at two values of $\alpha$ (0.2 & 0.12) and for delivery rates ranging from 7-12 Hz. We divide the experiment into runs of 1500 cycles (i.e. 1500 deterministic state deliveries), for a total data set of 42000 cycles. We first confirm that heralded entanglement occurs with the expected probabilities (Fig. \[fig:DeterministicEntGenerationExpm\]a) by determining the fraction of cycles in which entanglement is heralded, in which no entangling attempts succeed, and in which entanglement attempts do not occur at all as the NV state check never succeeds. In order to establish reliable and useful quantum networks, it is important that entangled states can be delivered with high confidence over long periods. The nodes must therefore not be offline due, for example, to uncompensated drifts in the resonant frequencies of the optical transitions. We therefore do not stop the experiment from running once it starts and include any such offline cycles in our datasets. Their negligible contribution (0.8% of cycles) confirms the high robustness of our experimental platform and the effectiveness of our NV frequency and charge-state control (SI sections II & IV). For each value of $\alpha$ and for each pre-set delivery interval, we determine the average fidelity of the deterministically delivered states by measuring their $\avg{XX}$, $\avg{YY}$ and $\avg{ZZ}$ correlations (Fig. \[fig:DeterministicEntGenerationExpm\]b). We find that for $\alpha = 0.2$ and a rate of 9.9 Hz, we are able to create states with a fidelity of $0.56(1)$, proving successful deterministic entanglement delivery. During cycles in which entanglement is not successfully heralded, the spin states are nonetheless delivered and readout. In these case, we deliver the state that the NVs are left in after a failed entanglement attempt, which has a low fidelity with the desired Bell state (e.g. $F_\text{unent} = 0.04$ for $\alpha = 0.2$). While this stringent test highlights the robust nature of our protocol, we could instead deliver a mixed state ($F_\text{unent}=\frac{1}{4}$) or a classically-correlated state ($F_\text{unent} = \frac{1}{2}$) when a successful event is not heralded. The resulting fidelities for our experimental data if classically-correlated states were delivered are also plotted in Fig. \[fig:DeterministicEntGenerationExpm\]b (grey circles). In this case we would be able to deliver entangled states deterministically with fidelities of $0.62(1)$ at a rate of 9.9 Hz. The deterministic entanglement delivery between remote NV centres demonstrated here is enabled by a quantum link efficiency exceeding unity. Straightforward modifications to our experiment are expected to further increase our quantum link efficiency. Refinements to the classical experimental control will allow us to reduce the entanglement attempt duration from 5.5 $\mu$s to below 2 $\mu$s, which would more than double the entangling rate. Furthermore, the entangled state coherence time could be significantly improved by exploiting long-lived nuclear spin quantum memories [@Maurer2012; @Kalb2017; @yang2016high]. We anticipate that this will allow for link efficiencies in excess of 100 in the near term. Further improvements to the photon detection efficiency (including enhancement of zero-phonon line emission) [@Riedel_2017; @2017arXiv171101704W] would lead to an additional increase of at least an order of magnitude. In combination with recent progress on robust storage of quantum states during remote entangling operations [@Reiserer2016; @Kalb2017], the techniques reported here reveal a direct path to the creation of many-body quantum states distributed over multiple quantum network nodes. Moreover, given the demonstrated potential for phase stabilisation in optical fibre over tens of kilometre distances [@Minar2008], our results open up the prospect of entanglement-based quantum networks at metropolitan scales. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Suzanne van Dam, Mohamed Abobeih, Tim Taminiau, Filip Rozpdek, Kenneth Goodenough and Stephanie Wehner for helpful discussions. We acknowledge support from the Netherlands Organisation for Scientific Research (NWO) through a VICI grant and the European Research Council through a Starting Grant and a Synergy Grant. Correspondence and requests for materials should be addressed to R.H. ([email protected]).
--- abstract: 'We study the kinetics of chiral phase transitions in quark matter. We discuss the phase diagram of this system in both a microscopic framework (using the Nambu-Jona-Lasinio model) and a phenomenological framework (using a Landau free energy). Then, we study the far-from-equilibrium coarsening dynamics subsequent to a quench from the chirally-symmetric phase to the massive quark phase. Depending on the nature of the quench, the system evolves via either [*spinodal decomposition*]{} or [*nucleation and growth*]{}. The morphology of the ordering system is characterized using the order-parameter correlation function, structure factor, domain growth laws, etc.' address: - 'School of Physical Sciences, Jawaharlal Nehru University, New Delhi, 110067, India' - 'Theory Division, Physical Research Laboratory, Navrangpura, Ahmedabad – 380009, India.' author: - Awaneesh Singh - Sanjay Puri - Hiranmaya Mishra title: Domain Growth in Chiral Phase Transitions --- [Chiral symmetry breaking, Ginzburg-Landau expansion, TDGL equation, domain growth, quenching, dynamical scaling.]{} Introduction {#intro} ============ The nature of the [*quantum chromodynamics*]{} (QCD) phase diagram as a function of temperature ($T$) and baryon chemical potential ($\mu$) has been studied extensively over the last few years [@rischke]. For $\mu=0$, finite-temperature calculations have been complemented by first-principle approaches like lattice QCD simulations [@karsch]. However, for $\mu \ne 0$, the lattice simulations are limited to small values of $\mu$ [@latmu]. In QCD with two massless quarks, the chiral phase transition is expected to be second-order at zero baryon densities. In nature, the light quarks are not exactly massless and the sharp second-order transition is replaced by a smooth crossover. This picture is consistent with lattice QCD simulations with a transition temperature $T_c\sim 140 - 190$ MeV [@Tclattice]. On the other hand, calculations based on different effective models show that the transition becomes first-order at large $\mu$ and small $T$ [@mk98]. This means that the phase diagram will have a [*tricritical point*]{} (TCP), where the first-order chiral transition becomes second-order (for vanishing quark masses) or ends (for non-vanishing quark masses). The location of the TCP ($\mu_{\rm tcp}$, $T_{\rm tcp}$) in the phase diagram and its signature has been been under intense investigation, both theoretically and experimentally [@gavaigupta; @misha]. Heavy-ion collision experiments at high energies produce hot and dense strongly-interacting matter, and provide the opportunity to explore the phase diagram of QCD. The high-$T$ and low-$\mu$ region has been explored by recent experiments in the [*Relativistic Heavy Ion Collider*]{} (RHIC). This region will also be studied by planned experiments in the [*Large Hadron Collider*]{} (LHC). Further, future heavy-ion collision experiments like the [*Beam Energy Scan*]{} at RHIC, [*FAIR*]{} in GSI, and [*NICA*]{} in Dubna plan to explore the high-baryon-density region of the phase diagram, particularly around the TCP [@cpod]. The experiments at the RHIC provide clear signals that nuclear matter undergoes a phase transition to partonic phases at sufficiently large values of the energy density. However, the nature and kinetics of this transition remains an open question. We note here that lattice QCD assumes that the system is in equilibrium, whereas heavy-ion experiments are essentially nonequilibrium processes. Therefore, information about which equilibrium phase has the lowest free energy is not sufficient to discuss the properties of the system. One also has to understand the kinetic processes which drive the phase transition, and the properties of the nonequilibrium structures that the system must go through to reach equilibrium. In this paper, we study the kinetics of chiral transitions in quark matter. We focus on far-from-equilibrium kinetics, subsequent to a quench from the disordered phase (with zero quark condensate) to the ordered phase. This rapid quench renders the disordered system thermodynamically unstable. The evolution to the new equilibrium state is characterized by spatio-temporal pattern formation, with the emergence and growth of domains of the preferred phases. This nonlinear evolution is usually referred to as [*phase ordering dynamics*]{} or [*coarsening*]{} or [*domain growth*]{} [@aj94; @pw09]. Previous studies of ordering dynamics in quark matter, which we review shortly, have primarily focused upon early-time kinetics and the growth of initial fluctuations. The present paper is complementary to these studies. We investigate the universal properties (e.g., growth laws, scaling of correlation and structure functions, bubble dynamics, etc.) in the late stages of chiral kinetics. These properties are robust functions of the evolution dynamics, and only depend upon general features, e.g., scalar vs. vector order parameters, defect structures, conservation laws which govern dynamics, relevance of hydrodynamic effects, etc. Our results in this paper are obtained using a [*time-dependent Ginzburg-Landau*]{} (TDGL) model, which is derived from the [*Nambu-Jona-Lasinio*]{} (NJL) model with two light flavors [@klevansky]. However, we expect that our results apply to a much larger class of systems belonging to the same [*dynamical universality class*]{}. To place our work in the proper context, we provide an overview of studies of dynamical properties of quark matter. These focus on either (a) [*critical dynamics*]{}, i.e., time-dependent behavior in the vicinity of the critical points, or (b) [*far-from-equilibrium dynamics*]{}, which was explained above. In studies of critical dynamics, much interest has focused on the TCP. The [*static universality class*]{} of the QCD transition (for non-vanishing quark masses) is believed to be that of the $d=3$ Ising model, but there is debate regarding the dynamical universality class. For example, the TCP dynamics was argued [@bednikov] to be in the class of [*Model C*]{} in the Hohenberg-Halperin classification scheme [@hohenrev]. Essentially, the argument was that critical dynamics is described by a nonconserved order parameter (the quark condensate $\langle\bar\psi\psi\rangle$), in conjunction with conserved quantities like the baryon number density. If one includes the mode coupling between the quark condensate and the baryon density, the appropriate universality class is [*Model H*]{} with a different dynamical exponent [@son]. However, it was also argued that reversible couplings can play a crucial role in QCD critical dynamics, which may then differ from that of [*Model H*]{} [@koide]. In related work, Koide and Maruyama [@tomoinpa] have derived a linear Langevin equation for the chiral order parameter. This is obtained by applying the [*Mori projection operator technique*]{} to the NJL model. These authors study the solution of the Langevin equation, and investigate [*critical slowing down*]{} in the vicinity of the TCP. Let us next turn our attention to studies of far-from-equilibrium dynamics in quark matter. Sasaki et al. [@sasaki] have emphasized the importance of [*spinodal decomposition*]{} in understanding the chiral and deconfinement transitions in heavy-ion collisions. Their study was based on a mean-field approximation to the NJL model, as well as a phenomenological Landau theory. Sasaki et al. discussed fluctuations of the baryon number density as possible signatures of nonequilibrium transitions. However, they did not study the corresponding evolution dynamics. Scavenius et al. [@dumitru] have investigated the possibility of [*nucleation*]{} vs. [*spinodal decomposition*]{} in an effective field theory derived from the [*nonlinear sigma model*]{}. Again, these authors have not investigated time-dependent properties, which are of primary interest to us in this paper. An important study of evolution dynamics in [*quark-gluon plasma*]{} (QGP) is due to Fraga and Krein [@fragaplb], who modeled the relaxation to equilibrium via a phenomenological Langevin equation. (We follow a similar approach in this paper.) This Langevin equation can be derived from a microscopic field-theoretic model of kinetics of the chiral order parameter [@gr94; @dr98]. Fraga and Krein studied the early-time dynamics of spinodal decomposition in this model both analytically and numerically, and focused upon the effect of dissipation on the spinodal instability. In recent work, Bessa et al. [@fraga] studied bubble nucleation kinetics in chiral transitions, and the dependence of the nucleation rate on various parameters. Skokov and Voskresensky [@skokovdima] have also studied the kinetics of first-order phase transitions in nuclear systems and QGP. Starting from the equations of non-ideal non-relativistic hydrodynamics (i.e., Navier-Stokes equation, continuity and transport equations), they derived TDGL equations for the coupled order parameters. These TDGL equations were studied numerically and analytically in the vicinity of the critical point. Skokov-Voskresensky focus upon the evolution of density fluctuations in the metastable and unstable regions of the phase diagram, and the growth kinetics of seeds. They clarify the role of viscosity in the ordering kinetics. Finally, we mention the recent work of Randrup [@jr09], who has studied the fluid dynamics of relativistic nuclear collisions. The corresponding evolution equations reflect the conservation of baryon charge, momentum and energy. Randrup studied the amplification of spinodal fluctuations and the evolution of the [*real-space correlation function*]{} and the [*momentum-space structure factor*]{}. (We will study the scaling of these quantities in Sec. \[kct\] of this paper.) Randrup’s work mostly focused upon the evolution in the linearized regime, where there is an exponential growth of initial fluctuations. This paper is complementary to Refs. [@fragaplb; @fraga; @skokovdima; @jr09], and investigates the late stages of phase-separation kinetics in quark matter. The system is described by nonlinear evolution equations in this regime: the exponential growth of initial fluctuations is saturated by the nonlinearity. We study the coarsening dynamics from disordered initial conditions, and the scaling properties of emergent morphologies. We consider an initially disordered system which is quenched to the symmetry-broken phase through either the second-order line (relevant for high $T$ and small $\mu$) or the first-order line (relevant for small $T$ and large $\mu$) in the NJL phase diagram. We study domain growth for both types of quenching, and highlight quantitative features of the coarsening morphology. We also study the evolution kinetics of single droplets, and the dependence of the front velocity on system parameters. This paper is organized as follows. In Sec. \[pdct\], we discuss the equilibrium phase diagram of the two-flavor NJL model using a variational approach. Here, we describe chiral symmetry breaking as a vacuum realignment with quark-antiquark condensates. As we shall see, this method also captures some extra contributions proportional to 1/$N_c$ (where $N_c$ is the number of colors), as compared to mean-field theory. In Sec. \[pdct\], we will also discuss the corresponding Landau description of chiral transitions. In Sec. \[kct\], we introduce the TDGL equation which describes the evolution of the chiral order parameter, and use it to study the kinetics of chiral transitions. As mentioned earlier, we focus on pattern formation in the late-stage dynamics, which is characterized by scaling of the evolution morphologies, and the corresponding domain growth laws. Finally, we end this paper with a summary and discussion in Sec. \[summary\]. Our investigation has several novel features from the perspective of both QCD and domain growth studies. These can be highlighted as follows. First, we demonstrate a quantitative mapping between the phase diagrams of the NJL model as an effective model of QCD at low energy and the $\psi^6$-Landau potential. This mapping enables us to identify the relevant time-scales and length-scales in chiral transition kinetics. Second, we clarify the quantitative features of the coarsening morphology, e.g., correlation functions, growth laws, etc., in chiral transitions. These universal features are independent of system and model details, and can be measured in experiments on quark-gluon plasma. Third, the chiral transition provides a natural context to study ordering dynamics in the $\psi^6$-potential, which has received little attention. To date, most studies of domain growth have focused on the $\psi^4$-potential, which has a much simpler phase diagram. Finally, as we will discuss elsewhere, chiral dynamics also provides a framework to study the effect of inertial terms in phase ordering kinetics. Studies of domain growth have almost entirely focused on dissipative overdamped dynamics [@aj94; @pw09]. Phase Diagram for Chiral Transitions {#pdct} ===================================== An Ansatz for the Ground State {#gs} ------------------------------- For the consideration of chiral symmetry breaking, we denote the perturbative vacuum state with chiral symmetry as $|0\rangle$. We then assume a specific vacuum realignment which breaks chiral symmetry because of interactions. Let us first note the quark-field operator expansion in momentum space [@hmnj]: $$\begin{aligned} \psi (\vec{x}) &\equiv& \frac{1}{(2\pi)^{3/2}}\int \!d\vec{k}\; e^{i\vec{k}\cdot\vec{x}} \tilde\psi(\vec{k}) \nonumber\\ &=&\frac{1}{(2\pi)^{3/2}}\int\! d\vec{k} e^{i\vec{k}\cdot \vec{x}} \big[U_0(\vec{k})q^0_I(\vec{k})+V_0(-\vec{k})\tilde q^0_I(-\vec{k} )\big] , \label{psiexp}\end{aligned}$$ where $$\begin{aligned} U_0(\vec{k})=\left(\begin{array}{c}\cos \left(\phi^0/2 \right) \\ \vec{\sigma} \cdot \hat{k} \sin \left(\phi^0/2 \right) \end{array}\right),\;\; V_0(-\vec{k} )= \left(\begin{array}{c} - \vec{\sigma} \cdot \hat{k} \sin \left(\phi^0/2 \right) \\ \cos \left(\phi^0/2 \right) \end{array}\right). \label{uv0}\end{aligned}$$ The superscript $0$ indicates that $q_I^0$ and $\tilde q_I^0$ are two-component operators which annihilate or create quanta, and act upon the chiral vacuum $|0\rangle$. We have suppressed here the color and flavor indices of the quark-field operators. The function $\phi^0(\vec{k})$ in the spinors of Eq. (\[uv0\]) is obtained as $\cot{\phi_i^0}(\vec{k}) = m_i/k$ for free massive fermion fields, $i$ being the flavor index. For massless fields, $\phi^0(\vec{k})=\pi/2$. We now consider vacuum destabilization leading to chiral symmetry breaking [@hmnj], described by $$|{\rm vac} \rangle={\cal U}_Q|0\rangle, \label{u0}$$ where $${\cal U}_Q=\exp\left[ \int d\vec{k}~ q_I^{0i}(\vec{k})^\dagger(\vec{\sigma}\cdot\vec{k})h_i(\vec{k})\tilde q_I^{0i} (-\vec{k})-\mathrm{h.c.}\right]. \label{uq}$$ Here, $h_i(\vec{k})$ is a real function of $|\vec{k}|~(=k)$ which describes vacuum realignment for quarks of a given flavor $i$. We shall take the condensate function $h_i(\vec{k})$ to be the same ($h_i = h$) for $u$ and $d$ quarks. Clearly, a nontrivial $h_i(\vec{k})$ will break chiral symmetry. A sum over the three colors and three flavors is understood in the exponent of ${\cal U}_Q$ in Eq. (\[uq\]). Finally, to include the effect of temperature and density, we write down the state at nonzero temperature and chemical potential $|\Omega(\beta,\mu)\rangle$, where $\beta = 1/T$. This is done through a thermal Bogoliubov transformation of the state $|\Omega\rangle$, using [*thermo-field dynamics*]{} (TFD) [@tfd; @amph4]. We then have $$|\Omega(\beta,\mu)\rangle={\cal U}_{\beta,\mu}|\Omega\rangle={\cal U}_{\beta,\mu}{\cal U}_Q |0\rangle, \label{ubt}$$ where ${\cal U}_{\beta,\mu}$ is $${\cal U}_{\beta,\mu}=e^{{\cal B}^{\dagger}(\beta,\mu)-{\cal B}(\beta,\mu)}. \label{ubm}$$ Here, $${\cal B}^\dagger(\beta,\mu)=\int d\vec{k}~ \Big [ q_I^\prime (\vec{k})^\dagger \theta_-(\vec{k}, \beta,\mu) \underline q_I^{\prime} (\vec{k})^\dagger + \tilde q_I^\prime (\vec{k}) \theta_+(\vec{k}, \beta,\mu) \underline { \tilde q}_I^{\prime} (\vec{k})\Big ]. \label{bth}$$ In Eq. (\[bth\]), the ansatz functions $\theta_{\pm}(\vec{k},\beta,\mu)$ will be related to quark and anti-quark distributions. The underlined operators are defined in the extended Hilbert space associated with thermal doubling in the TFD method. In Eq. (\[bth\]), we have suppressed the color and flavor indices on the quarks and the functions $\theta_\pm (\vec{k},\beta,\mu)$. The ansatz functions $h(\vec{k})$, $\theta_\pm(\vec{k},\beta,\mu)$ will be determined by minimizing the thermodynamic potential in the next subsection. Minimization of Thermodynamic Potential and Gap Equations {#gap} --------------------------------------------------------- We next consider the NJL model, which is based on relativistic fermions interacting through local current-current couplings. It is assumed that gluonic degrees of freedom can be frozen into point-like effective interactions between the quarks. We shall confine ourselves to the two-flavor case only, with the Hamiltonian $${\cal H} = \sum_{i,a}\psi^{ia \dagger}\left(-i\vec{\alpha}\cdot\vec{\nabla} + \gamma^0 m_i \right)\psi^{ia} -G\left[(\bar\psi\psi)^2-(\bar\psi\gamma^5 \tau \psi)^2\right]. \label{ham}$$ Here, $m_i$ is the current quark mass. We take this to be the same ($m_i=m$) for both $u$ and $d$ quarks. The parameter $G$ denotes the quark-quark interaction strength. Further, $\tau$ is the Pauli matrix acting in flavor space. The quark operator $\psi$ has two indices $i$ and $a$, denoting the flavor and color indices, respectively. The point interaction produces short-distance singularities and, to regulate the integrals, we restrict the phase space to lie inside the sphere $k< \Lambda$, the ultraviolet cut-off in the NJL model. We next obtain the expectation values of various operators for the variational ansatz state in Eq. (\[ubt\]). One can calculate these using the fact that the state in Eq. (\[ubt\]) arises from successive Bogoliubov transformations. These expressions will then be used to calculate the thermal expectation value of the Hamiltonian, and to compute the thermodynamic potential. With $\tilde\psi(\vec{k})$ as defined in Eq. (\[psiexp\]), we evaluate the expectation values: $$\langle \Omega(\beta,\mu) |\tilde\psi_\alpha^{ia}(\vec{k})\tilde\psi ^{jb}_\beta(\vec{k}')^{\dagger} |\Omega(\beta,\mu)\rangle =\delta^{ij}\delta^{ab} \Lambda_{+\alpha\beta}^{ia}(\vec{k},\beta,\mu)\delta(\vec{k}-\vec{k}'), \label{psipsidb}$$ and $$\langle \Omega(\beta,\mu) |\tilde\psi_\beta^{ia\dagger}(\vec{k})\tilde\psi_\alpha^{jb}(\vec{k}') |\Omega(\beta,\mu)\rangle =\delta^{ij}\delta^{ab} \Lambda_{-\alpha\beta}^{ia}(\vec{k},\beta,\mu)\delta(\vec{k}-\vec{k}'). \label{psidpsib}$$ Here, $$\begin{aligned} \Lambda_\pm^{ia}(\vec{k},\beta,\mu) &=&\frac{1}{2}\big[1\mp(\sin^2\theta_-- \sin^2\theta_+)\pm \big(\gamma^0\cos\phi_i+\nonumber\\ &&\vec{\alpha}\cdot{\hat{k}}\sin \phi_i \big)\big(1-\sin^2\theta_--\sin^2\theta_+\big)\big]. \label{prpb}\end{aligned}$$ In Eq. (\[prpb\]), we have introduced the notation $\phi_i(\vec{k})=\phi_i^0(\vec{k})-2 h_i(\vec{k})$ in favor of the condensate function $h(\vec{k})$, which will later prove suitable for variation of the thermodynamic potential. Using Eqs. (\[psipsidb\])-(\[prpb\]), we can evaluate the expectation value of the NJL Hamiltonian in Eq. (\[ham\]) as $$\begin{aligned} \epsilon&=&\langle\Omega(\beta,\mu)|{\cal H}|\Omega(\beta,\mu)\rangle\nonumber\\ &=&-\frac{2N_c N_F}{(2\pi)^3}\int d\vec{k} ~ \left[m\cos\phi(\vec{k})+k\sin\phi(\vec{k})\right]\times \left(1-\sin^2\theta_ -- \sin^2\theta_+\right)\nonumber\\ && -G\left[\left(1+\frac{1}{4N_c}\right)\rho_s^2-\frac{1}{2N_c}\rho_v^2\right], \label{energy}\end{aligned}$$ where $N_F$ is the number of flavors. In Eq. (\[energy\]), we have set $m_i \equiv m$ and $\phi_i(\vec{k}) \equiv \phi(\vec{k})$. We have also defined the condensates. The scalar condensate is $$\rho_s=\langle \bar\psi\psi\rangle=-\frac{2N_c N_F}{(2\pi)^3} \int d\vec{k}\cos\phi(\vec{k})\left(1-\sin^2\theta_--\sin^2\theta_+\right) , \label{rhos}$$ and the expectation value of the number density is $$\rho_v=\langle \psi^\dagger\psi\rangle=\frac{2N_c N_F}{(2\pi)^3} \int d\vec{k}~\left(\sin^2\theta_--\sin^2\theta_+\right) . \label{rhov}$$ The thermodynamic grand potential is then given by $$\Omega=\epsilon-\mu \rho_v-\frac{1}{\beta}s, \label{omega}$$ where $s$ is the entropy density for the quarks. We have the expression [@tfd] $$\begin{aligned} s & = & -\frac{2N_c N_F}{(2\pi)^3}\int d\vec{k}~\Big[\sin^2{\theta_-}\ln \left(\sin^2{\theta_-}\right)+\cos^2{\theta_-}\ln\left(\cos^2{\theta_-}\right) \nonumber \\ && \hspace{3cm} + \sin^2{\theta_+}\ln\left(\sin^2{\theta_+}\right) +\cos^2{\theta_+}\ln \left(\cos^2{\theta_+}\right)\Big ]. \label{ent}\end{aligned}$$ Now, if we extremize $\Omega$ with respect to $h(\vec{k})$, or equivalently with respect to the function $\phi(\vec{k})$, we obtain $$\cot \phi(\vec{k}) = \frac{M}{k}, \label{tan2h}$$ where $M = m - 2g\rho_s$ with $g=G(1+1/4N_c)$. Substituting this in Eq. (\[rhos\]), we have the mass gap equation for the quarks as $$M = m + 2g \frac{2N_c N_F}{(2\pi)^3}\int d\vec{k}~\frac{M}{\sqrt{k^2 +M^2}} (1-\sin^2\theta_--\sin^2\theta_+). \label{mgap}$$ Similarly, minimization of the thermodynamic potential with respect to the thermal functions $\theta_{\pm}(\vec{k})$ gives $$\sin^2\theta_\pm = \frac{1}{\exp(\beta\omega_\pm)+1}, \label{them}$$ where $\omega_\pm=\sqrt{k^2+M^2}\pm\nu\equiv\epsilon(\vec{k})\pm\nu$. Here, $\nu$ is the interaction-dependent chemical potential given as $$\nu=\mu-\frac{G}{N_c}\rho_v. \label{nu}$$ In Eq. (\[omega\]), we substitute the expression for the condensate function from Eq. (\[tan2h\]), and the distribution functions from Eq. (\[them\]), to obtain $$\begin{aligned} \Omega(M,\beta,\mu)&=&-\frac{12}{(2\pi)^3\beta}\int d\vec{k}~\left\{\ln\left[1+\exp(-\beta\omega_-)\right]+\ln\left[1+\exp(-\beta\omega_+)\right] \right\} \nonumber\\ &&-\frac{12}{(2\pi)^3}\int d\vec{k}~ \sqrt{k^2+M^2}+g\rho_s^2 -\frac{G}{6}\rho_v^2, \label{omegam}\end{aligned}$$ where we have written down the expression for $N_c=3$ and $N_F=2$. From Eq. (\[omegam\]), we subtract the potential of the non-condensed state at $T=0$ and $\mu=0$ to obtain $$\begin{aligned} \tilde\Omega(M,\beta,\mu)&=&\Omega(M,\beta,\mu)-\Omega_0(m,\beta=\infty,\mu=0)\nonumber\\ &=&-\frac{12}{(2\pi)^3\beta}\int d\vec{k}~\left\{\ln\left[1+\exp(-\beta\omega_-)\right]+\ln\left[1+\exp(-\beta\omega_+)\right]\right\} \nonumber\\ &&-\frac{12}{(2\pi)^3}\int d\vec{k}~ \left(\sqrt{k^2+M^2}-\sqrt{k^2+m^2}\right)\nonumber\\ &&+g\rho_s^2-g\rho_{s0}^2 -\frac{G}{2N_c}\rho_v^2. \label{tomega}\end{aligned}$$ Here, $\rho_{s0}$ is the scalar density at zero temperature and quark mass $m$: $$\rho_{s0}=-\frac{12}{(2\pi)^3}\int d\vec{k}\frac{m}{\sqrt{k^2+m^2}}. \label{rho}$$ Landau Theory for Chiral Transitions {#glt} ------------------------------------ In the mean-field approximation and near the chiral transition line, the thermodynamic potential obtained above can also be described by Landau theory. Let us focus on the case with zero current quark mass. We consider the potential in Eq. (\[tomega\]) with $m=0$, and terms of order $N_c^{-1}$ being neglected (i.e., $N_c \rightarrow \infty$): $$\begin{aligned} \tilde\Omega(M,\beta,\mu) =& \;-\frac{12}{(2\pi)^3\beta}\int \! d\vec{k}\;\Big\{ \ln\left[1+e^{-\beta\left( \sqrt{k^2+M^2} - \mu \right)}\right] \notag\\ & \qquad\qquad\qquad\quad + \ln\left[1+e^{-\beta\left( \sqrt{k^2+M^2} + \mu \right)}\right] \Big\} \notag\\ &\; -\frac{12}{(2\pi)^3}\int \! d\vec{k} \; \left(\sqrt{k^2+M^2}-k\right)+ \frac{M^2}{4G}. \label{tomega1}\end{aligned}$$ To compute this potential numerically, we set the three-momentum ultraviolet cut-off $\Lambda=653.3$ MeV, and the four-fermion coupling $G=5.0163\times 10^{-6}$ MeV$^{-2}$ [@askawa]. With these values, the constituent quark mass at $\mu =0$ and $T=0$ is $M \simeq 312$ MeV. The variation of $M$ with $\mu$ at $T=0$ is shown in Fig. \[fig1\](a). For $\mu < \mu_1(T=0) \simeq 326.321$ MeV, the quark masses stay at their vacuum values. A first-order transition takes place at $\mu = \mu_1$, and the masses of $u$ and $d$ quarks drop from their vacuum values to zero. In Fig. \[fig1\](b), we show the $T$-dependence of $M$ at $\mu=0$. Chiral symmetry is restored for quarks at $T\simeq190$ MeV. In this case, the transition is second-order: this is reflected in the smooth variation of the mass, which is proportional to the order parameter $\langle\bar\psi\psi\rangle$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Variation of the mass gap $M$ (proportional to the scalar order parameter $\langle\bar\psi\psi\rangle$) with quark chemical potential $\mu$ at $T=0$. (b) Variation of $M$ with $T$ at $\mu = 0$.[]{data-label="fig1"}](fig1a.eps "fig:"){width="45.00000%"} ![(a) Variation of the mass gap $M$ (proportional to the scalar order parameter $\langle\bar\psi\psi\rangle$) with quark chemical potential $\mu$ at $T=0$. (b) Variation of $M$ with $T$ at $\mu = 0$.[]{data-label="fig1"}](fig1b.eps "fig:"){width="45.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In Fig. \[fig2\](a), we show the phase diagram resulting from Eq. (\[tomega1\]) for the chiral transition in the ($\mu, T$)-plane. The solid line is the critical line, and corresponds to the chiral phase transition, which can be either first-order or second-order. The first-order line I (at high $\mu$ and low $T$) meets the second-order line II (at low $\mu$ and high $T$) in a tricritical point ($\mu_\text{tcp}$, $T_\text{tcp}$) $\simeq$ ($282.58$, $78.0$) MeV. A first-order transition is characterized by the existence of metastable phases, e.g., supersaturated vapor. The masses corresponding to these metastable phases are local minima of the potential, but have higher free energy than the stable phase. The limit of metastability is denoted by the dot-dashed lines ($S_1$ and $S_2$) in Fig. \[fig2\](a) – these are referred to as [*spinodal*]{} lines. Before proceeding, we should stress that the phase diagram in Fig. \[fig2\](a) only considers homogeneous chiral condensates. However, recent calculations by Carignano et al. [@cnb10], Sadizkowski and Broniowski [@sadiz], and Nakano and Tatsumi [@nt05] show the existence of [*inhomogeneous*]{} chiral-symmetry-breaking phases in the NJL model, e.g., domain-wall solitons, chiral density waves, chiral spirals. In that case, the first-order line (and the associated spinodal lines) in Fig. \[fig2\](a) may be replaced by second-order transitions between inhomogeneous phases. In this paper, we confine ourselves to the kinetics of phase transitions between the homogeneous phases in Fig. \[fig2\](a). However, it is also of great interest to study the ordering dynamics from (say) a homogeneous phase to an inhomogeneous phase, or between different inhomogeneous phases. For example, there have been some studies of ordering to a lamellar (striped) phase in [*Rayleigh-Benard convection*]{}, described by the [*Swift-Hohenberg equation*]{} [@evg92; @cm95; @cb98; @gk10]. Another important system with ordering to inhomogeneous phases is that of [*phase-separating diblock copolymers*]{} [@bo90; @qw96; @rh01]. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Phase diagram of the Nambu-Jona-Lasinio (NJL) model in the ($\mu, T$)-plane for zero current quark mass. A line of first-order transitions (I) meets a line of second-order transitions (II) at the tricritical point (tcp). We have $(\mu_\text{tcp}, T_\text{tcp}) \simeq (282.58, 78)$ MeV. The dot-dashed lines $S_1$ and $S_2$ denote the spinodals or metastability limits for the first-order transitions. The open symbols denote 4 combinations of $\left(\mu, T\right)$ with $T=10$ MeV, chosen to represent qualitatively different shapes of the NJL potential. The asterisk and cross denote quench parameters for the simulations described in Sec. \[case1\] and Sec. \[case2\], respectively. (b) Plot of $\tilde\Omega\left(M, \beta, \mu \right) - \tilde\Omega\left(0, \beta, \mu \right)$ vs. $M$ from Eq. (\[tomega1\]). For a particular $(\mu, T)$-value, we denote the free energy by the same open symbol as in (a). The solid lines superposed on the potentials denote the Landau potential in Eq. (\[p6\]) with $a$ from Eq. (\[coff\]), and $b$, $d$ being fit parameters (see Table \[tab\]).[]{data-label="fig2"}](fig2a.eps "fig:"){width="44.00000%"} ![(a) Phase diagram of the Nambu-Jona-Lasinio (NJL) model in the ($\mu, T$)-plane for zero current quark mass. A line of first-order transitions (I) meets a line of second-order transitions (II) at the tricritical point (tcp). We have $(\mu_\text{tcp}, T_\text{tcp}) \simeq (282.58, 78)$ MeV. The dot-dashed lines $S_1$ and $S_2$ denote the spinodals or metastability limits for the first-order transitions. The open symbols denote 4 combinations of $\left(\mu, T\right)$ with $T=10$ MeV, chosen to represent qualitatively different shapes of the NJL potential. The asterisk and cross denote quench parameters for the simulations described in Sec. \[case1\] and Sec. \[case2\], respectively. (b) Plot of $\tilde\Omega\left(M, \beta, \mu \right) - \tilde\Omega\left(0, \beta, \mu \right)$ vs. $M$ from Eq. (\[tomega1\]). For a particular $(\mu, T)$-value, we denote the free energy by the same open symbol as in (a). The solid lines superposed on the potentials denote the Landau potential in Eq. (\[p6\]) with $a$ from Eq. (\[coff\]), and $b$, $d$ being fit parameters (see Table \[tab\]).[]{data-label="fig2"}](fig2b.eps "fig:"){width="46.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Close to the phase boundary, the thermodynamic potential (which is even in $M$) may be expanded as a Landau potential in the order parameter $M$: $$\tilde\Omega\left(M \right)= \tilde\Omega\left(0 \right) + \frac{a}{2}M^2 + \frac{b}{4}M^4 + \frac{d}{6}M^6 + O(M^8) \equiv f\left(M \right), \label{p6}$$ correct upto logarithmic factors [@sasaki; @iwasaki]. In the following, we consider the expansion of $\tilde\Omega\left(M \right)$ upto the $M^6$-term. This will be sufficient to recover the phase diagram in Fig. \[fig2\](a), as we see shortly. The first two coefficients \[$\tilde\Omega(0)$ and $a$\] in Eq. (\[p6\]) can be obtained by comparison with Eq. (\[tomega1\]) as $$\begin{aligned} \tilde\Omega(0) =&\;-\dfrac{6}{\pi^2\beta}\displaystyle\int_0^\Lambda \!\!\! dk\,\, k^2 \left\lbrace \ln\left[1+e^{-\beta(k-\mu)}\right] + \ln\left[1+e^{-\beta(k+\mu)}\right]\right\rbrace, \nonumber \\ a =& \; \dfrac{1}{2G} - \dfrac{3\Lambda^2}{\pi^2} + \dfrac{6}{\pi^2}\displaystyle\int_0^\Lambda \!\!\! dk\,\,k\left[ \dfrac{1}{1+e^{\beta(k-\mu)}} + \dfrac{1}{1+e^{\beta(k+\mu)}}\right]. \label{coff}\end{aligned}$$ We treat the higher coefficients in Eq. (\[p6\]) ($b$ and $d$) as phenomenological parameters. These are obtained by fitting $\tilde\Omega\left(M \right)$ in Eq. (\[p6\]) to the integral expression for $\tilde\Omega$ in Eq. (\[tomega1\]). There are two free parameters in the microscopic theory ($\mu$ and $T$), so we consider the $M^6$-potential with fitting parameters $b$ and $d$. For stability, we require $d>0$. In Fig. \[fig2\](b), we plot $\tilde\Omega\left(M \right)-\tilde\Omega\left(0\right)$ vs. $M$ from the integral expression in Eq. (\[tomega1\]). We show plots for 4 values of $\left(\mu, T\right)$ as marked in Fig. \[fig2\](a). These are chosen to represent qualitatively different shapes of the potential. The solid lines superposed on the data sets in Fig. \[fig2\](b) correspond to the Landau potential in Eq. (\[p6\]) with $a$ from Eq. (\[coff\]), and $b$, $d$ being fit parameters. The values of these parameters in dimensionless units are provided in Table \[tab\]. $(\mu,T)$ (MeV) $a/\Lambda^2$ $b$ $d\Lambda^{2}$ $\lambda=|a|d/|b|^2$ ----------------- ------------------------ -------- ---------------- ---------------------- (311.00,10) -1.306$\times 10^{-3}$ 0.092 0.439 0.067 (321.75,10) 3.539$\times 10^{-3}$ -0.101 0.402 0.140 (328.00,10) 6.431$\times 10^{-3}$ -0.111 0.396 0.206 (335.00,10) 9.736$\times 10^{-3}$ -0.101 0.265 0.255 : The coefficients ($a,b,d$) of the Landau potential in Eq. (\[p6\]) for 4 different values of $\mu$ at $T=10$ MeV. These parameters are specified in dimensionless units of $\Lambda^2$, $\Lambda^0$ and $\Lambda^{-2}$ respectively, where $\Lambda=653.3$ MeV. The dimensionless quantity $\lambda=|a|d/|b|^2$ will be useful in our discussion of the dynamics in Sec. \[kct\].[]{data-label="tab"} ![Phase diagram for the Landau free energy in Eq. (\[p6\]) in the \[$b/(d\Lambda^2), a/(d\Lambda^4)$\]-plane. A line of first-order transitions (I) meets a line of second-order transitions (II) at the tricritical point (tcp), which is located at the origin. The equation for I is $a_c = 3|b|^2/(16d)$, and that for II is $a_c = 0$. The dashed lines denote the spinodals $S_1$ and $S_2$, with equations $a_{S_1}=0$ and $a_{S_2} = |b|^2/(4d)$. The typical forms of the Landau potential in various regions are shown in the figure. The open symbols denote the ($\mu,T$)-values marked by the same symbols in Fig. \[fig2\](a). The cross denotes the point where we quench the system for $b<0$. The asterisk in Fig. \[fig2\] (a) corresponds to $(a/\Lambda^2,b,d\Lambda^2) = (-1.591 \times 10^{-2}, 8.985 \times 10^{-2}, 7.083 \times 10^{-2})$ or $b/(d\Lambda^2) = 1.269$, $a/(d\Lambda^4) = -0.225$. We do not mark this point in the figure as it results in a loss of clarity.[]{data-label="fig3"}](fig3.eps){width="75.00000%"} The order parameter values which extremize the Landau potential are given by the gap equation: $$f'\left(M \right)= aM + bM^3 + dM^5=0. \label{ge1}$$ The solutions of Eq. (\[ge1\]) are $$\begin{aligned} M &=& 0, \nonumber \\ M^2 &=& M_{\pm}^2=\dfrac{-b\pm \sqrt{b^2 -4ad}}{2d}. \label{ges}\end{aligned}$$ The phase diagram for the Landau potential in $[b/(d\Lambda^2), a/(d\Lambda^4)]$-space is shown in Fig. \[fig3\]. For $b>0$, the transition is second-order, as for the $M^4$-potential. The stationary points are $M=0$ (for $a>0$) or $M=0$, $\pm M_+$ (for $a<0$). For $a<0$, the preferred equilibrium state is the one with massive quarks. For $b<0$, the solutions of the gap equation are $$\begin{aligned} M &=& 0, \quad a>|b|^2/(4d), \nonumber \\ M &=& 0, \pm{M_+}, \pm{M_-}, \quad |b|^2/(4d)>a>0, \nonumber \\ M &=& 0, \pm{M_+}, \quad a<0. \label{mm1}\end{aligned}$$ As $a$ is reduced from large values, 5 roots appear at $a=|b|^2/(4d)$. However, this does not correspond to a phase transition. On further reduction of $a$, a first-order transition occurs at $a_c =3|b|^2/(16d)$. The order parameter jumps discontinuously from $M=0$ to $M=\pm M_+$, where $M_+ = [3|b|/(4d)]^{1/2}$. The tricritical point is located at $b_\text{tcp}=0$, $a_\text{tcp}=0$ \[cf. Fig. \[fig2\](a)\]. The 4 combinations of $\left(\mu, T\right)$-values marked in Fig. \[fig2\](a) are identified using the same symbols in Fig. \[fig3\]. Kinetics of Chiral Transitions {#kct} ============================== Dynamical Equation ------------------ Let us next study time-dependent problems in the context of the NJL or Landau free energy. Consider the dynamical environment of a heavy-ion collision. As long as the evolution is slow compared to the typical re-equilibration time, the order parameter field will be in local equilibrium. We consider a system which is rendered thermodynamically unstable by a rapid quench from the massless phase to the massive phase in Figs. \[fig2\](a) or \[fig3\]. In the context of Fig. \[fig3\], this corresponds to (say) quenching from $a>a_c(b)$ to $a<a_c(b)$ at a fixed value of $b$. (Of course, we can consider a variety of different quenches.) The unstable massless state (with $M \simeq 0$) evolves via the emergence and growth of domains rich in the preferred massive phase (with $M=\pm M_+$). There has been intense research interest in this far-from-equilibrium evolution [@aj94; @pw09]. Most problems in this area traditionally arise from materials science and metallurgy. However, equally fascinating problems are associated with the kinetics of phase transitions in high-energy physics or cosmology [@tk76; @tk07; @kp08]. In this paper, we focus on domain growth in QCD transitions, modeled by the $M^6$-free energy in Eq. (\[p6\]). The coarsening system is inhomogeneous, so we include a surface-tension term in the Landau free energy: $$\begin{aligned} \Omega\left[M\right] =& \; \int \! d\vec{r} \left[f\left(M\right) + \frac{K}{2} \left(\vec{\nabla}M\right)^2\right] \notag \\ =& \; \int \! d\vec{r} \left[\frac{a}{2}M^2 + \frac{b}{4}M^4 + \frac{d}{6}M^6 + \frac{K}{2} \left(\vec{\nabla}M\right)^2\right]. \label{Om}\end{aligned}$$ In Eq. (\[Om\]), $\Omega\left[M\right]$ is a functional of the spatially-dependent order parameter field $M(\vec{r})$, and $K$ measures the energy cost of spatial inhomogeneities, i.e., surface tension. The evolution of the system is described by the [*time-dependent Ginzburg-Landau*]{} (TDGL) equation: $$\begin{aligned} \frac{\partial}{\partial t}M\left(\vec{r},t\right)= -\Gamma \frac{\delta \Omega\left[M \right]}{ \delta M}+\theta\left(\vec{r},t\right), \label{ke}\end{aligned}$$ which models the over-damped (relaxational) dynamics of $M\left(\vec{r},t\right)$ to the minimum of $\Omega\left[M\right]$, i.e., the system is damped towards the equilibrium configuration [@hohenrev]. In Eq. (\[ke\]), $\Gamma$ denotes the inverse damping coefficient. The noise term $\theta(\vec{r},t)$ is taken to be Gaussian and white, and satisfies the fluctuation-dissipation relation [@hohenrev]: $$\begin{aligned} \left\langle \theta\left(\vec{r},t\right) \right\rangle =& \; 0,\notag \\ \left\langle \theta(\vec{r'},t')\theta(\vec{r''},t'') \right\rangle =& \; 2\Gamma T\delta(\vec{r'}-\vec{r''})\delta\left(t'-t''\right). \label{fdr}\end{aligned}$$ In Eq. (\[fdr\]), the angular brackets denote an averaging over different noise realizations. Replacing the potential from Eq. (\[Om\]) in Eq. (\[ke\]), we obtain $$\begin{aligned} \frac{\partial}{\partial t}M\left(\vec{r},t\right)= -\Gamma\left(aM + bM^3 + dM^5\right) + \Gamma K\nabla^2 M + \theta\left(\vec{r},t\right). \label{ke1}\end{aligned}$$ We use the natural scales of order parameter, space and time to introduce dimensionless variables: $$\begin{aligned} M &=& M_0 M', \quad M_0=\sqrt{|a|/|b|} , \nonumber \\ \vec{r} &=& \xi \vec{r'}, \quad \xi=\sqrt{K/|a|} , \nonumber \\ t &=& t_0 t', \quad t_0=(\Gamma|a|)^{-1} , \nonumber \\ \theta &=& (\Gamma |a|^{3/2}T^{1/2}/|b|^{1/2})~\theta' . \label{scale}\end{aligned}$$ Dropping primes, we obtain the dimensionless TDGL equation: $$\begin{aligned} \frac{\partial}{\partial t}M\left(\vec{r},t\right)= -\mathrm{sgn}\left(a\right)M - \mathrm{sgn}\left(b\right)M^3 - \lambda M^5 + \nabla^2 M +\theta\left(\vec{r},t\right), \label{ke2}\end{aligned}$$ where $\mathrm{sgn}(x)=x/|x|$ and $\lambda = |a|d/|b|^2 >0$. The values of $\lambda$ corresponding to $T=10$ MeV and $\mu =311,321.75,328,335$ (in MeV) are specified in Table \[tab\]. The dimensionless noise term obeys the fluctuation-dissipation relation: $$\begin{aligned} \left\langle \theta\left(\vec{r},t\right) \right\rangle =& \; 0,\notag \\ \left\langle \theta(\vec{r'},t')\theta(\vec{r''},t'') \right\rangle =& \; 2 \epsilon\; \delta(\vec{r'}-\vec{r''})\delta\left(t'-t''\right), \notag \\ \epsilon =& \frac{T |b|}{|a|^{1/2}K^{3/2}}. \label{fdr1}\end{aligned}$$ Our results in this paper are presented in dimensionless units of space and time. To obtain the corresponding physical units, one has to multiply by the appropriate dimensional length-scale $\xi$ and time-scale $t_0$. For this, we need to estimate the strength of the interfacial energy $K$, and the inverse damping coefficient $\Gamma$. The surface tension can be calculated as $\sigma = \sqrt{K}(|a|^{3/2}/|b|)\int dz~(dM/dz)^2$. For quark matter, $\sigma$ is poorly known and varies from 10-100 MeV/$\text{fm}^2$ at small temperatures [@hc93] – we take $\sigma \simeq 50$ MeV/$\text{fm}^2$. For $T=10$ MeV and $\mu = 321.75$ MeV, we then estimate $\xi = \sqrt{K/|a|} \simeq 2.8$ fm. Similarly, we set $\Gamma^{-1} \sim 2T/s$, where $s$ is a quantity of order 1 [@fragaplb; @kk92]. This leads to $t_0 = (\Gamma |a|)^{-1} \simeq 2.6$ fm/$s$. We study the phase-transition kinetics for two different quench possibilities. The first case corresponds to high $T$ and low baryon density ($\mu$), where the quenching is done through the second-order line (II) in Fig. \[fig2\](a) or Fig. \[fig3\]. The corresponding parameter values are $(\mu, T) = (231.6, 85)$ MeV \[marked by an asterisk in Fig. \[fig2\](a)\]; or $(a/\Lambda^2,b,d\Lambda^2) = (-1.591 \times 10^{-2}, 8.985 \times 10^{-2}, 7.083 \times 10^{-2})$ with $\lambda = |a|d/|b|^2 = 0.14$. The second case corresponds to low $T$ and high baryon density ($\mu$), where the chiral dynamics can probe the metastable region of the phase diagram. This can be achieved by shallow quenching through the first-order line (I) in Fig. \[fig2\](a) or Fig. \[fig3\], i.e., quenching to the region between I and S$_1$. This case is studied using parameter values $(\mu, T) = (321.75, 10)$ MeV; or $(a/\Lambda^2, b, d\Lambda^2) = ( 3.53885 \times 10^{-3}, -0.1005344, 0.4015734)$ with $\lambda = 0.14$. These points are marked by a cross in the phase diagrams of Figs. \[fig2\](a) and \[fig3\]. Quench through Second-order Line: Spinodal Decomposition {#case1} -------------------------------------------------------- Let us first focus on the ordering dynamics for quenches through the second-order line ($b > 0$) in Fig. \[fig3\]. For $b>0$, the chiral transition occurs when $a<0$. The quenched system is spontaneously unstable and evolves via [*spinodal decomposition*]{} [@aj94; @pw09]. The relevant TDGL equation is $$\begin{aligned} \frac{\partial}{\partial t}M\left(\vec{r},t\right)= M - M^3 - \lambda M^5 + \nabla^2 M +\theta\left(\vec{r},t\right), \label{ke3}\end{aligned}$$ with the dimensionless potential $$\begin{aligned} f\left(M \right)= -\frac{1}{2}M^2 + \frac{1}{4}M^4 + \frac{\lambda}{6}M^6. \label{dp6}\end{aligned}$$ The free-energy minima for this potential are $$\begin{aligned} M=\pm M_+ = \pm \left(\frac{-1+\sqrt{1+4\lambda}}{2\lambda} \right)^{1/2}. \label{min_dp6}\end{aligned}$$ We solve Eq. (\[ke3\]) with $\lambda = 0.14$ numerically using an Euler-discretization scheme. We implement this on a $d=3$ lattice of size $N^3~(N=256)$, with periodic boundary conditions in all directions. For numerical stability, the discretization mesh sizes must obey the condition $$\begin{aligned} \Delta t< \frac{2\Delta x^2}{4d + \alpha_1 \Delta x^2}, \label{stc1}\end{aligned}$$ where $\alpha_1 = 4+(1-\sqrt{1+4\lambda})/\lambda$. This condition is obtained from a linear stability analysis of Eq. (\[ke3\]) by requiring numerical stability of fluctuations about the stable fixed points in Eq. (\[min\_dp6\]) [@yp87; @red88]. For all results shown in this paper, we used the mesh sizes $\Delta x = 1.0$ and $\Delta t = 0.1$. We have confirmed that this spatial mesh size is sufficiently small to resolve the interface region, i.e., the boundary between domains with order parameter $-M_+$ and $M_+$. Further, we use an isotropic approximation to the Laplacian term $\nabla^2M$: $$\begin{aligned} \nabla^2M\left(\vec{r}, t\right) = \frac{1}{\Delta x^2} \left[\frac{1}{2} \displaystyle\sum_{\text{nn}}M + \frac{1}{4}\displaystyle\sum_{\text{nnn}}M - 6 M\left(\vec{r}, t\right)\right], \label{lap}\end{aligned}$$ which couples each cell to its 6 nearest neighbors (nn) and 12 next-nearest neighbors (nnn). Finally, the thermal noise $\theta(\vec{r}, t)$ is mimicked by uniformly-distributed random numbers between $[-A_n, A_n]$. We set $A_n = 0.5$, corresponding to $\epsilon = A_n^2{\left(\Delta x \right)}^d\Delta t/3 = 0.008$ in Eq. (\[fdr1\]). This noise amplitude is adequate to initiate the growth process from a metastable state on reasonable time-scales [@sp02], as we will see shortly. However, the asymptotic behavior of domain growth in both the unstable and metastable cases is insensitive to the noise term [@yp87; @po88]. ![Kinetics of chiral transition after a temperature quench through the second-order line (II) in Figs. \[fig2\](a) or \[fig3\]. The $d$=3 snapshots on the left show the interfaces ($M=0$) at $t=20, 50, 100$ (in units of $t_0$). The defects (interfaces) are kinks between domains of the massive phase with $M\simeq +M_+$ or $M\simeq -M_+$. The snapshots were obtained by numerically solving the TDGL Eq. (\[ke3\]) with $\lambda=0.14$, as described in the text. The frames on the right show a cross-section of the snapshots at $z=N/2$.[]{data-label="fig4"}](fig4.eps){width="82.00000%"} In Fig. \[fig4\], we show the evolution of Eq. (\[ke3\]) with a disordered initial condition, which consisted of small-amplitude random fluctuations about the massless phase $M=0$. The system rapidly evolves via spinodal decomposition into domains of the massive phase with $M\simeq +M_+$ and $M\simeq -M_+$. The coarsening is driven by interfacial defects, which are shown in Fig. \[fig4\]. These domains have a characteristic length scale $L(t)$, which grows with time. The growth process in Fig. \[fig4\] is analogous to coarsening dynamics in the TDGL equation with an $M^4$-potential [@aj94; @pw09], i.e., Eq. (\[ke3\]) with $\lambda = 0$, which describes coarsening in a ferromagnet subsequent to a temperature quench from $T>T_c$ to $T<T_c$. Coarsening in the ferromagnet is driven by kinks, with the equilibrium profile $M_s(z)= \tanh(\pm z/\sqrt{2})$. We can use the dynamics of kinks to obtain a good understanding of this evolution. The domain scale obeys the Allen-Cahn (AC) growth law, $L(t) \sim t^{1/2}$. Typically, the interface velocity $v\sim dL/dt \sim 1/L$, where $L^{-1}$ measures the local curvature of the interface. This yields the AC law. The same growth law has also been obtained via a closed-time-path formalism of relativistic finite-temperature field theory applied to the NJL model [@das]. The pattern morphology in Fig. \[fig4\] is statistically self-similar in time with $L(t)$ setting the scale. The morphology is studied experimentally using the [*order-parameter correlation function*]{} [@aj94; @pw09]: $$\begin{aligned} C\left(\vec{r},t\right) = \frac{1}{V}\!\! \int \!\!d\vec{R}\left[ \left\langle M(\vec{R},t)M(\vec{R}+\vec{r},t)\right\rangle - \left\langle M(\vec{R},t)\right\rangle \left\langle M(\vec{R}+\vec{r},t)\right\rangle\right], \label{cf} \end{aligned}$$ or its Fourier transform, the *structure factor*: $$\begin{aligned} S(\vec{k},t) = \int \!\! d\vec{r}\;e^{i\vec{k}\cdot\vec{r}}C\left(\vec{r},t\right). \label{sf}\end{aligned}$$ In Eq. (\[cf\]), $V$ denotes the volume, and the angular brackets denote an averaging over independent evolutions. As the system is translationally invariant and isotropic, $C(\vec{r},t)$ and $S(\vec{k},t)$ depend only on the vector magnitudes $r$ and $k$. The existence of a characteristic size $L(t)$ results in the [*dynamical scaling*]{} of $C(\vec{r},t)$ and $S(\vec{k},t)$: $$\begin{aligned} C(\vec{r},t) &=& g(r/L), \label{cscale} \\ S(\vec{k},t) &=& L^d f(kL), \label{dscale}\end{aligned}$$ where $g(x)$ and $f(p)$ are scaling functions which are independent of time. Let us demonstrate dynamical scaling for the spinodal decomposition illustrated in Fig. \[fig4\]. The statistical results presented in this paper correspond to the $d=2$ case, and are obtained as an average over 10 independent runs with $4096^2$ lattices. In Fig. \[fig5\](a), we plot the scaled correlation function \[$C\left(\vec{r},t\right)$ vs. $r/L$\] for 4 different times during the evolution. The length scale $L$ is obtained as the distance over which the correlation function decays to half its maximum value \[$C(r,t)=1$ at $r =0$\]. The data sets collapse onto a single master curve, confirming the scaling form in Eq. (\[cscale\]). The solid line in Fig. \[fig5\](a) denotes the analytical result due to Ohta et al. (OJK) [@ojk82], who studied ordering dynamics in a ferromagnet. The magnet is also described by a scalar order parameter, i.e., magnetization. The OJK function is $$\begin{aligned} C\left(\vec{r},t\right) =& \; \frac{2}{\pi}\sin^{-1}\left(e^{-r^2/L^2}\right). \label{ojk}\end{aligned}$$ (The corresponding result for the case with vector order parameter has been obtained by Bray and Puri [@bp91].) Our correlation-function data is in excellent agreement with the OJK function, showing that chiral spinodal decomposition is analogous to domain growth in a ferromagnet. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Dynamical scaling of the correlation function \[$C(r,t)$ vs. $r/L$\] for chiral spinodal decomposition at four different times. The length scale $L(t)$ is defined as the distance over which $C(r,t)$ decays to half its maximum value. The different data sets collapse onto a single master curve. The statistical data shown in this figure is obtained on $4096^2$ lattices as an average over 10 independent runs, and the correlation function is spherically averaged. The solid line denotes the OJK function in Eq. (\[ojk\]) [@ojk82]. (b) Dynamical scaling of the structure factor \[$L^{-2} S(k,t)$ vs. $kL$\] for the same times as in (a). The large-$k$ region (tail) of the structure factor obeys the Porod law [@porod], $S(k,t) \sim k^{-3}$ for $k \rightarrow \infty$, which results from scattering off kink defects.[]{data-label="fig5"}](fig5a.eps "fig:"){width="44.00000%"} ![(a) Dynamical scaling of the correlation function \[$C(r,t)$ vs. $r/L$\] for chiral spinodal decomposition at four different times. The length scale $L(t)$ is defined as the distance over which $C(r,t)$ decays to half its maximum value. The different data sets collapse onto a single master curve. The statistical data shown in this figure is obtained on $4096^2$ lattices as an average over 10 independent runs, and the correlation function is spherically averaged. The solid line denotes the OJK function in Eq. (\[ojk\]) [@ojk82]. (b) Dynamical scaling of the structure factor \[$L^{-2} S(k,t)$ vs. $kL$\] for the same times as in (a). The large-$k$ region (tail) of the structure factor obeys the Porod law [@porod], $S(k,t) \sim k^{-3}$ for $k \rightarrow \infty$, which results from scattering off kink defects.[]{data-label="fig5"}](fig5b.eps "fig:"){width="45.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Time-dependence of the domain size, $L(t)$ vs. $t$, for chiral spinodal decomposition (see Fig. \[fig5\]). The coarsening domains obey the Allen-Cahn (AC) growth law, $L(t)\sim t^{1/2}$.[]{data-label="fig6"}](fig6.eps){width="47.00000%"} In Fig. \[fig5\](b), we plot the scaled structure factor \[$L^{-2}S(\vec{k},t)$ vs. $kL$\] for the same times as in Fig. \[fig5\](a). Again, the data sets collapse neatly onto a single master curve, confirming the scaling form in Eq. (\[dscale\]). The scaling function is in excellent agreement with the corresponding OJK function. Notice that the tail of the structure factor shows the [*Porod law*]{} [@porod], $S(k,t) \sim k^{-(d+1)}$ for $k \rightarrow \infty$, which results from scattering off sharp interfaces. For an $n$-component vector order parameter, the system shows the [*generalized Porod law*]{} [@bp91], $S(k,t) \sim k^{-(d+n)}$ for $k \rightarrow \infty$. The scalar order parameter studied here corresponds to $n=1$ and the relevant defects are interfaces or kinks. Higher-order defects arise for vector fields, e.g., vortices or vortex strings ($n=2$), monopoles ($n=3$), etc. In Fig. \[fig6\], we plot $L(t)$ vs. $t$ on a log-log scale. The length scale data is consistent with the AC growth law, $L(t) \sim t^{1/2}$. As stated earlier, this law has also been obtained from finite-temperature field theory applied to the NJL model [@das]. Shallow Quench through First-order Line: Nucleation and Growth {#case2} -------------------------------------------------------------- Let us next consider shallow quenches from the massless state through the first-order line ($b<0$) in Fig. \[fig3\]. The chirally-symmetric phase is now metastable, and evolves to the stable massive phase via the [*nucleation and growth*]{} of droplets. From Fig. \[fig3\], we see that the first-order chiral transition occurs for $a<a_c=3|b|^2/(16d)$. (In terms of dimensionless variables, the transition occurs for $\lambda<\lambda_c=3/16$.) We consider quenches from $a>a_c$ (with $M=0$) to $a<a_c$. If we quench to $a<0$, the free energy has a double-well structure, as shown in Fig. \[fig3\]. Again, the ordering dynamics is analogous to that for the ferromagnet. We have confirmed (results not shown here) that the domain growth scenario is similar to that shown in Figs. \[fig4\]-\[fig6\]. Subsequently, we focus only on quenches to $0<a<a_c$ or $0<\lambda<\lambda_c$. The appropriate dimensionless TDGL equation is $$\begin{aligned} \frac{\partial}{\partial t}M\left(\vec{r},t\right)=\; -M + M^3 - \lambda M^5 + \nabla^2 M +\theta\left(\vec{r},t\right) , \label{ke4}\end{aligned}$$ with the potential $$\begin{aligned} f\left(M \right)= \frac{1}{2}M^2 - \frac{1}{4}M^4 + \frac{\lambda}{6}M^6 . \label{tp6}\end{aligned}$$ The free-energy extrema are located at $$\begin{aligned} M=\;0, \; \pm M_+,\; \pm M_- ,\end{aligned}$$ where $$\begin{aligned} M_+ =& \; \left(\frac{1+\sqrt{1-4\lambda}}{2\lambda} \right)^{1/2}, \notag \\ M_- =& \; \left(\frac{1-\sqrt{1-4\lambda}}{2\lambda} \right)^{1/2}.\end{aligned}$$ The extrema at $M=0, \pm M_+$ are local minima with $f(\pm M_+)<f(0)=0$ for $\lambda<\lambda_c$. ### Bubble Growth and Static Kinks Before we study the ordering dynamics of Eq. (\[ke4\]), it is useful to understand the traveling-wave and static solutions. After all, the phase transition is driven by the dynamics of kinks and anti-kinks (eventually 1-$d$ in nature). For the case with $b>0$ and $a<0$ discussed in Sec. \[case1\], the kinks have tanh-profiles with small corrections due to the $M^6$-term in the potential. We consider the deterministic version of Eq. (\[ke4\]) in $d$=1: $$\begin{aligned} \frac{\partial}{\partial t}M\left(z,t\right)= \: -M + M^3 - \lambda M^5 + \frac{\partial^2 M}{\partial z^2}. \label{dke4}\end{aligned}$$ We focus on traveling-wave solutions of this equation, $M\left(z,t\right) \equiv M\left(z-vt\right) \equiv M\left(\eta\right)$ with velocity $v>0$. This reduces Eq. (\[dke4\]) to the ordinary differential equation: $$\begin{aligned} \frac{d^2 M}{d\eta^2} + v\frac{d M}{d\eta} - M + M^3 -\lambda M^5 = 0. \label{ode}\end{aligned}$$ Equation (\[ode\]) is equivalent to a 2-$d$ dynamical system: $$\begin{aligned} \frac{d M}{d\eta} =& \; y, \notag \\ \frac{d y}{d\eta} =& \; M - M^3 + \lambda M^5 -vy. \label{ode1}\end{aligned}$$ To obtain the kink solutions of this system, we undertake a phase-plane analysis. The [*phase portrait*]{} will enable us to identify kink solutions of Eq. (\[dke4\]). The relevant fixed points (FPs) are $(M,y)= (0,0)$, $(\pm M_-,0)$, $(\pm M_+,0)$. We consider small fluctuations about these FPs: $$\begin{aligned} M =& \; M_0 + \phi, \quad (M_0 =0, \; \pm M_-, \; \pm M_+), \notag \\ y =& \; 0+y.\end{aligned}$$ We can linearize Eq. (\[ode1\]) about these FPs to obtain $$\begin{aligned} \frac{d \phi}{d\eta} =& \; y, \notag \\ \frac{d y}{d\eta} =& \; \left(1-3M_0^2+5\lambda M_0^4 \right)\phi -vy \equiv a\phi - vy. \label{ode2}\end{aligned}$$ The eigenvalues ($\lambda_e$) which determine the growth or decay of these small fluctuations are determined from $$\begin{aligned} \left|\begin {array}{cc} -\lambda_e & 1\\ \noalign{\medskip} a & -v-\lambda_e \end {array}\right| = 0, \label{array}\end{aligned}$$ or $$\begin{aligned} \lambda_{e\pm} = \frac{-v \pm \sqrt{v^2 + 4a}}{2}. \label{ev}\end{aligned}$$ We can combine this information to obtain the phase portrait of the system in Eq. (\[ode1\]). In Fig. \[fig7\], we show phase portraits for $\lambda=0.14$ ($<\lambda_c \simeq 0.1875$), which is the parameter value for most simulations presented in this subsection. Figure \[fig7\](a) corresponds to the case with $v=0$ (static solution). The saddle connections from $-M_+ \rightarrow +M_+$ and $+M_+ \rightarrow -M_+$ correspond to static kink solutions. These can be obtained by integrating $$\begin{aligned} \frac{d M_s}{dz} = \pm \sqrt{2}\left[\frac{1}{2}\left(M_s^2 - M_+^2 \right) - \frac{1}{4}\left(M_s^4 - M_+^4 \right) + \frac{\lambda}{6}\left(M_s^6 - M_+^6 \right)\right]^{1/2}. \label{kink1}\end{aligned}$$ The static kink profiles for several values of $\lambda < \lambda_c$ are shown in Fig. \[fig8\]. ![Phase portraits of the dynamical system in Eq. (\[ode1\]) with $\lambda = 0.14$. (a) Case with $v = 0$. The saddle connections from $-M_+ \rightarrow +M_+$ and $+M_+ \rightarrow -M_+$ correspond to static kink solutions. (b) Case with $v=v_s = 0.503$, corresponding to the appearance of saddle connections from $-M_+ \rightarrow 0$ and $+M_+ \rightarrow 0$. These correspond to kinks traveling with velocity $v_s > 0$.[]{data-label="fig7"}](fig7a.eps "fig:"){width="70.00000%"}\ ![Phase portraits of the dynamical system in Eq. (\[ode1\]) with $\lambda = 0.14$. (a) Case with $v = 0$. The saddle connections from $-M_+ \rightarrow +M_+$ and $+M_+ \rightarrow -M_+$ correspond to static kink solutions. (b) Case with $v=v_s = 0.503$, corresponding to the appearance of saddle connections from $-M_+ \rightarrow 0$ and $+M_+ \rightarrow 0$. These correspond to kinks traveling with velocity $v_s > 0$.[]{data-label="fig7"}](fig7b.eps "fig:"){width="70.00000%"}\ ![Spatial dependence of the order parameter \[$M_s(z)$ vs. $z$\] for static kink profiles of Eq. (\[dke4\]) with different $\lambda$-values ($< \lambda_c$).[]{data-label="fig8"}](fig8.eps){width="60.00000%"} In Fig. \[fig7\](b), we show the phase portrait for $v = v_s$, where $v_s$ corresponds to the appearance of saddle connections from $-M_+ \rightarrow 0$ and $+M_+ \rightarrow 0$. These correspond to kinks traveling with velocity $v_s >0$, as shown in Fig. \[fig7\](b). So far, our analysis has been done for the case with $v>0$, but it is straightforward to extend it to the case with $v<0$. In the latter case, the portrait in Fig. \[fig7\](b) is inverted, and the saddle connections (kinks) are from $0 \rightarrow -M_+$ and $0 \rightarrow +M_+$. In Fig. \[fig9\], we show the growth of a bubble (droplet) of the massive phase ($M=+M_+$) in the background of the metastable phase ($M=0$). These snapshots are obtained by solving Eq. (\[ke4\]) with $\lambda=0.14$ and $\theta = 0$. We start with an initial configuration of a $d=2$ bubble of radius $R_0>R_c$ such that $$\begin{aligned} M(r) &=& M_+, \quad r<R_0 , \nonumber \\ M(r) &=& 0, \quad r>R_0 .\end{aligned}$$ Here, $R_c(\lambda)$ is the critical size of the droplet, which diverges as $\lambda \rightarrow \lambda_c^-$. In Fig. \[fig10\](a), we plot the radius of the droplet \[$R(t) - R_0$\] vs. $t$. These curves are linear, showing that the bubble of the massive phase grows at a constant velocity $v_B$. In Fig. \[fig10\](b), we plot $v_B$ vs. $\lambda$. The quantity $(\lambda -\lambda_c)$ measures the degree of undercooling. Our numerical data is in good agreement with $v_s$, which is obtained from the phase-plane analysis \[cf. Fig. \[fig7\](b)\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Growth of a bubble or droplet of the massive phase ($M = +M_+$) in a background of the metastable massless phase ($M=0$). We show the boundary of the droplet at three different times. The innermost circle corresponds to the droplet at time $t=60$. (b) Variation of order parameter along the horizontal cross-section marked in (a).[]{data-label="fig9"}](fig9a.eps "fig:"){width="41.00000%"} ![(a) Growth of a bubble or droplet of the massive phase ($M = +M_+$) in a background of the metastable massless phase ($M=0$). We show the boundary of the droplet at three different times. The innermost circle corresponds to the droplet at time $t=60$. (b) Variation of order parameter along the horizontal cross-section marked in (a).[]{data-label="fig9"}](fig9b.eps "fig:"){width="44.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- It is also useful to study the limit $\lambda=\lambda_c$, where there are three coexisting solutions of the free energy in Eq. (\[tp6\]), viz., $M=0,\; \pm M_+$. The phase portrait for $v=0$ (static kinks) is shown in Fig. \[fig11\]. Now, there are saddle connections from $\pm M_+ \rightarrow 0$ and $0 \rightarrow \pm M_+$. The corresponding kink profiles are obtained as solutions of $$\begin{aligned} \frac{d M_s}{dz} = \pm \sqrt{2}\left(\frac{1}{2}M_s^2 - \frac{1}{4}M_s^4 + \frac{\lambda_c}{6}M_s^6\right)^{1/2}, \label{kink2}\end{aligned}$$ supplemented with appropriate boundary conditions. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![(a) Linear growth of a single bubble with time \[$R(t)-R_0$ vs. $t$\] for 4 different values of $\lambda$. (b) Plot of the bubble growth velocity $v_B$ vs. $\lambda$. The circles denote our numerical data, while the solid line is obtained from the phase-plane analysis (see Fig. \[fig7\]).[]{data-label="fig10"}](fig10a.eps "fig:"){width="45.00000%"} ![(a) Linear growth of a single bubble with time \[$R(t)-R_0$ vs. $t$\] for 4 different values of $\lambda$. (b) Plot of the bubble growth velocity $v_B$ vs. $\lambda$. The circles denote our numerical data, while the solid line is obtained from the phase-plane analysis (see Fig. \[fig7\]).[]{data-label="fig10"}](fig10b.eps "fig:"){width="41.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Phase portrait of the system in Eq. (\[ode1\]) with $\lambda = \lambda_c = 0.1875$ and $v=0$. Now, there are saddle connections (or static kinks) from $\pm M_+ \rightarrow 0$ and $0 \rightarrow \pm M_+$.[]{data-label="fig11"}](fig11.eps "fig:"){width="70.00000%"}\ ### Chiral Transition Kinetics Next, we consider the evolution from a disordered initial condition. We implemented an Euler-discretized version of the TDGL Eq. (\[ke4\]) on a $d=2$ lattice of size $N^2$. In this case, the mesh sizes must obey the numerical stability condition: $$\begin{aligned} \Delta t< \frac{2\Delta x^2}{4d + \alpha_2 \Delta x^2}, \label{stc2}\end{aligned}$$ where $\alpha_2 = -4 + (1+\sqrt{1-4\lambda})/\lambda$. We used the mesh sizes $\Delta x=1.0$ and $\Delta t=0.1$. The other numerical details are the same as in Sec. \[case1\]. ![Kinetics of chiral transition after a temperature quench through the first-order line (I) in Figs. \[fig2\](a) or \[fig3\]. The $d=2$ snapshots on the left show regions with $M=0$ at $t=200$, $400$, $4000$ (in units of $t_0$). They were obtained by numerically solving Eq. (\[ke4\]) with $\lambda=0.14$. The frames on the right show the variation of the order parameter along a diagonal cross-section ($y=x$) of the snapshots.[]{data-label="fig12"}](fig12.eps){width="90.00000%"} Recall the phase diagram in Fig. \[fig3\]. We now focus on the region $b<0$, and consider quenches from high values of $a$ (massless phase) to $0<a<a_c$ or $0<\lambda<\lambda_c$. The massless phase is a metastable state of the potential. The chiral transition proceeds via the nucleation and growth of droplets of the preferred phase ($M=\pm M_+$). This nucleation results from large fluctuations in the initial condition which seed bubbles, or thermal fluctuations during the evolution. This should be contrasted with the evolution in Fig. \[fig4\], where the massless phase is spontaneously unstable and the system evolves via spinodal decomposition. In Fig. \[fig12\], we show the nucleation and growth process which characterizes evolution. At early times ($t=200$), the system is covered with the massless phase, with only small bubbles of the massive phase. The bubbles grow with time (see Fig. \[fig9\]) and coalesce into domains ($t=400$). The coarsening of these domains is analogous to that in Fig. \[fig4\] – in the late stages of the transition, there is no memory of the nucleation which enabled growth in the early stages. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![(a) Scaling plot of the correlation function \[$C(r,t)$ vs. $r/L$\] for the evolution shown in Fig. \[fig12\]. The deviation of data sets from a master curve at early times reflect the morphological differences between the “nucleation and growth” and “domain coarsening” regimes. At later times, the data sets collapse onto a master curve. The solid line denotes the OJK result in Eq. (\[ojk\]). (b) Scaling plot of the structure factor \[$L^{-2} S(k,t)$ vs. $kL$\] for the same times as in (a). At late times, the tail of the structure factor shows the Porod law, $S(k,t) \sim k^{-3}$ for $k \rightarrow \infty$.[]{data-label="fig13"}](fig13a.eps "fig:"){width="44.00000%"} ![(a) Scaling plot of the correlation function \[$C(r,t)$ vs. $r/L$\] for the evolution shown in Fig. \[fig12\]. The deviation of data sets from a master curve at early times reflect the morphological differences between the “nucleation and growth” and “domain coarsening” regimes. At later times, the data sets collapse onto a master curve. The solid line denotes the OJK result in Eq. (\[ojk\]). (b) Scaling plot of the structure factor \[$L^{-2} S(k,t)$ vs. $kL$\] for the same times as in (a). At late times, the tail of the structure factor shows the Porod law, $S(k,t) \sim k^{-3}$ for $k \rightarrow \infty$.[]{data-label="fig13"}](fig13b.eps "fig:"){width="46.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ In Fig. \[fig13\](a), we plot the scaled correlation function \[$C(\vec{r},t)$ vs. $r/L$\] for the evolution in Fig. \[fig12\]. (Our statistical data is obtained as an average over 10 independent runs with $4096^2$ lattices.) The morphological differences between the “nucleation and growth” ($t=200,400$) and “domain coarsening” ($t=2000,4000$) regimes is reflected in the crossover of the scaling function. At later times, we recover dynamical scaling and the master function is in excellent agreement with the OJK function. Thus, the late-stage morphology in Fig. \[fig12\] is analogous to that for spinodal decomposition. In Fig. \[fig13\](b), we plot the scaled structure factor \[$L^{-2}S(\vec{k},t)$ vs. $kL$\] at the same times as in Fig. \[fig13\](a). As expected, the structure factor also violates dynamical scaling in the crossover regime. In Fig. \[fig14\](a), we plot the domain size $L(t)$ vs. $t$ for $\lambda = 0.12, 0.14, 0.15$ on a log-log scale. In contrast with Fig. \[fig6\], there is almost no growth in the early stages when droplets are being nucleated. The growth process begins once nucleation is over, with the onset being faster for lower $\lambda$ (or higher degree of undercooling). In Fig. \[fig14\](b), we plot $L(t)-L_0$ vs. $t-t_0$ on a log-log scale. Here, $L_0~(\simeq 10$) is the initial length scale and $t_0$ is the onset time for the different values of $\lambda$ in Fig. \[fig14\](a). We see that the asymptotic regime is again described by the AC growth law, $L(t)\sim (t-t_0)^{1/2}$. Let us note here that, converting to physical time scales (i.e., multiplying the dimensionless time $t$ by $t_0$), the time to reach equilibrium seems to be very large compared to the typical life-time $\tau$ of the fire-ball produced in a heavy-ion collision ($\tau\sim 10$ fm for the RHIC). Let us recall that $t_0 \simeq 2.6$ fm and is proportional to $\Gamma^{-1}$. Larger dissipation (larger $\Gamma^{-1}$) will make the equilibration time much larger. Thus, in a heavy-ion collision experiment, the system may linger in the QGP phase longer during the fire-ball expansion, even when the temperature has already decreased below the critical temperature. Thus, the critical temperature calculated within equilibrium thermodynamic models can be higher than the value that shows up in the growth of fluctuations in experiments. However, such conclusions depend upon our estimate of $\Gamma$. In principle, $\Gamma$ can be calculated within the model as has been attempted in Ref. [@tomoinpa] using the Mori projection operator method. However, such a calculation is beyond the scope of the present work and we have used the estimate of $\Gamma$ in Refs. [@fragaplb; @kk92]. Summary and Discussion {#summary} ====================== Let us conclude this paper with a summary and discussion of the results presented here. We have studied the kinetics of chiral phase transitions in quark matter. At the microscopic level, these transitions are described by the Nambu-Jona-Lasinio (NJL) model. In the NJL model with zero current quark mass, there can be either first-order (I) or second-order (II) transitions between a massless quark phase and a massive quark phase. The lines I and II meet in a tricritical point. At the coarse-grained level, chiral transitions can be modeled by a Landau potential with an $M^6$-functional. We have shown that there is a quantitative agreement between the NJL free energy as a function of $(T,\mu)$ and the Landau potential with an appropriate choice of parameters. Near the phase boundary, we can identify the coefficients of the Landau energy for different values of ($T,\mu$). However, we often consider parameter values far from the phase boundary, and it is more appropriate to interpret the Landau coefficients as phenomenological quantities. We studied the kinetics of chiral transitions from the massless (disordered) phase to the massive (ordered) phase, resulting from a sudden quench in parameters. We model the kinetics using the [*time-dependent Ginzburg-Landau*]{} (TDGL) equation, which describes the overdamped relaxation of the order parameter field (scalar condensate density) to the minimum of the corresponding Ginzburg-Landau (GL) free-energy functional. We consider quenches through both the first-order and second-order lines in the phase diagram. There have been some earlier studies of the TDGL equation in this context, as discussed in the introductory Sec. \[intro\]. However, these have primarily discussed the growth of initial fluctuations in the framework of a linearized theory. On the other hand, this paper focuses on the late stages of pattern formation where nonlinearities in the TDGL equation play an important role. For quenches through II, the chirally-symmetric phase is spontaneously unstable and evolves into the broken-symmetry phase via [*spinodal decomposition*]{}. The evolution morphologies show self-similarity and dynamical scaling, and can be quantitatively characterized by the [*order-parameter correlation function*]{} or its Fourier transform, the [*structure factor*]{}. The domains of the massive phase grow as $L(t) \sim t^{1/2}$. For deep quenches through I, the above scenario applies again. However, for shallow quenches, the chirally-symmetric phase is metastable. Then, the system evolves via the [*nucleation and growth*]{} of bubbles or droplets of the preferred massive phase. In this case, the early-stage dynamics is dominated by the appearance of bubbles. The growth and merger of these bubbles results in late-stage domain growth which is morphologically similar to that for spinodal decomposition. The correlation function, structure factor and growth law show a crossover, having different functional forms in the nucleation and coarsening regimes. Before concluding, it is important to discuss the relevance of these results for QCD phenomenology and experiments. In the context of heavy-ion collisions, we make the following observations. Within the uncertainities regarding values of dimensional quantities for quark matter (e.g., surface tension, dissipation), it is not clear whether the system equilibrates completely within the life-time of the fireball. If the system is nearly-equilibrated, the features of the coarsening morphology will be similar for quenches through both first- and second-order lines in the phase diagram. However, if the equilibration time-scale is much larger than the fireball life-time, the morphology is very different for quenches through the first-order line, with the system evolving through nucleation of bubbles. Consequences of such a first order transition have potential relevance since they imply the existence of a critical end point (CEP) in the QCD phase diagram. As a matter of fact, experimental studies of such signatures may be more convenient than directly searching for the CEP via critical fluctuations. The latter approach has not provided conclusive evidence of the existence of a CEP, presumably due to the smallness of the critical region. We also stress that relating our results to heavy-ion collision experiments requires information about the source size apart from its life-time. In this context, two-particle momentum correlations (i.e., the [*Hanbury-Brown-Twiss*]{} or HBT effect in heavy-ion collisions) could be relevant. In such two particle correlations, the inverse width of the correlation function in the “out” direction measures the life-time of the source, whereas the same in the “side” direction measures the transverse size of the source [@rischkegyulassy]. 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--- abstract: 'We study a system of a transition metal dichalcogenide (TMD) monolayer placed in an optical resonator, where strong light-matter coupling between excitons and photons is achieved. We present quantitative theory of the nonlinear optical response for exciton-polaritons for the case of doped TMD monolayer, and analyze in detail two sources of nonlinearity. The first nonlinear response contribution stems from the Coulomb exchange interaction between excitons. The second contribution comes from the reduction of Rabi splitting that originates from phase space filling at increased exciton concentration and the composite nature of excitons. We demonstrate that both nonlinear contributions are enhanced in the presence of free electrons. As free electron concentration can be routinely controlled by an externally applied gate voltage, this opens a way of electrical tuning of the nonlinear optical response.' author: - 'V. Shahnazaryan' - 'V. K. Kozin' - 'I. A. Shelykh' - 'I. V. Iorsh' - 'O. Kyriienko' title: Tunable optical nonlinearity for TMD polaritons dressed by a Fermi sea --- Introduction ============ Exciton-polaritons represent a robust platform for nonlinear optics. The hybridization of excitons with optical cavity photons leads to formation of ultralight interacting quasiparticles—polaritons—that enable polariton lasing [@Kasprzak2006; @Balili2007; @Schneider2013; @Ballarini2017] and emergent polariton fluid behavior [@Amo2009; @Amo2011; @Tercas2014; @CarusottoCiutiRev]. For conventional quantum well (QW) nanostructures in III-V and II-VI semiconductors various nonlinear effects were studied, including solitons [@Sich2012; @Hivet2012; @Chana2015; @Opala2018], vortices [@Lagoudakis2008; @Tosi2012; @Boulier2015; @Kwon2019; @Caputo2019], multistability [@Gippius2007; @Cerna2013; @Gavrilov2013; @Kyriienko2014; @Klaas2017; @Tan2018], and nontrivial polariton lattice dynamics [@Askitopoulos2013; @Ohadi2017; @Sigurdsson2017; @Gao2018; @Sigurdsson2019; @Mietki2018; @Kyriienko2019a]. The combination of such prominent properties gives rise to various for polariton-based nonlinear optical integrated devices [@ShelykhReview; @Amo2010; @Liew2010; @Ohadi2015; @Dreismann2016; @Askitopoulos2018; @Opala2019]. In these systems the nonlinear response originates from the Coulomb-based exciton-exciton scattering [@Ciuti1998; @Tassone1999; @Glazov2009; @Vladimirova2010; @Brichkin2011; @Estrecho2019; @Levinsen2019], typically observed at macroscopic mode occupations. For high quality samples prerequisite signatures of a quantum nonlinear behavior were recently observed [@Munoz-Matunano2019; @Delteil2019], thanks to outstanding fabrication advances. The limitations for QW-based platform come from low operation temperatures, relatively small light-matter coupling ($\sim 4$ meV per QW), and complex growth techniques [@DengRev]. Recent advances in the field of optically-active two-dimensional (2D) materials have largely increased capabilities of polaritonics [@Schneider2018; @Mortensen2020]. In this case excitons are hosted by monolayers of transition metal dichalcogenide (TMD) materials—atom-thick nanostructures with a direct optical bandgap [@Mak2010; @Chernikov2014; @Miwa2015; @Steinleitner2017; @Rostami2015; @Schwarz2014] and excellent optical properties [@Wang2018; @Wurstbauer2017]. To date light-matter coupling for TMD excitons was observed in various configurations, including optical microcavities [@LiuMenon2015; @Dufferwiel2015; @Dufferwiel2017; @Sidler2017; @Emmanuele2019], Tamm plasmon structures [@Lundt2017], photonic crystals [@Zhang2018; @Kravtsov2020], surface plasmons [@Kleemann2017; @Goncalves2018; @Geisler2019] and nanoantennas [@Antosiewicz2014; @Stuhrenberg2018]. Due to relatively large electron/hole masses and reduced screening, the TMD exciton binding energy ranges in hundreds of meVs, and multicharge bound complexes (trions [@Mak2013; @Ross2013; @Singh2016; @Courtade2017; @Lundt2018], biexcitons [@You2015]) can be observed. Importantly, small exciton volume leads to large Rabi frequency, and excitonic optical response dominates already at room temperature [@Wang2018]. Further list of exceptional properties of TMD monolayers includes strong spin-orbit interaction and valley-dependent physics [@Wang2017a; @Manca2017; @Lundt2019], peculiar exciton transport properties [@Kulig2018; @Zipfel2020], and strong dependence on dielectric properties for observed physical effects [@Huser2013; @Latini2015; @Shahnazaryan2019]. For doped and gated TMD samples bandgap renormalization was shown [@Chernikov2015a; @Chernikov2015; @Withers2015; @Raja2017], opening the route to engineering of material properties. Finally, studies of trapped excitons and strain-induced lattices has shown efficient defect-based single photon emission from two-dimensional materials [@Kumar2016; @Branny2017; @Carmen2017; @Flatten2018]. The natural next step of TMD polaritons is utilization of nonlinear response. This so far has proven to be a nontrivial task, as the very same large binding leads to reduction of the exciton-exciton scattering cross-section evidenced theoretically [@Shahnazaryan2017] and experimentally [@Barachati2018]. However, the situation changes drastically once large light-matter coupling is achieved and a TMD monolayer is doped with free carriers. First, the deviation of excitonic statistics from ideal bosons [@CombescotReview; @Combescot2006] leads to the nonlinear Rabi splitting behavior[@Brichkin2011; @Daskalakis2014; @Yagafarov2020; @Betzold2020]—optical saturation—that in the case of strongly-coupled TMD polaritons was shown to add a significant contribution [@Emmanuele2019]. Second, the presence of the free electron gas (Fermi sea) strongly modifies the optical response of TMD monolayers. It depends on the density of the electron gas, leading to several characteristic regimes [@Chang2018; @Shiau2017]. At low free electron concentrations a sharp additional peak appears that is typically attributed to charged exciton complexes (trions), being bound states of two electrons and one hole [@Mak2013; @Courtade2017; @Emmanuele2019]. At high free electron gas concentrations the broad spectral peak was observed and attributed to an exciton polaron-polariton—correlated state of a exciton dressed by the Fermi sea [@Efimkin17; @Sidler2017; @Ravets2018; @Tan2020]. In each case the enhancement of the nonlinear response was reported [@Emmanuele2019; @Tan2020; @Kyriienko2019]. One should note, however, that besides formation of the additional peak, corresponding to the appearance of new quasiparticles, the presence of free electrons shall modify substantially the optical response of the exciton mode itself. This is especially pronounced in the case of intermediate electron densities, where excitons are spectrally separated from other modes. In the current paper we aim to describe this intermediate regime, showing its significant impact on nonlinear optical properties. In particular, we account that Fermi sea strongly contributes to the screening of Coulomb potential and the onset of additional correlations stemming from the Pauli exclusion principle. These effects change both light matter coupling and exciton-exciton interactions, thus resulting in renormalization of the nonlinearity strength. Our theory shows that changing the free electron density, as routinely realized in gated TMD samples, gives a tool for controlling optical nonlinearity of TMD polaritons. The paper is organized as follows. In Section II we present a theory for an exciton in a TMD monolayer in the presence of Fermi sea. It accounts for both screening of the Coulomb potential and the Pauli blocking effect. We study the modification of excitonic wavefunctions and the dependence of exciton binding energy and Bohr radius. In Section III we calculate exciton-exciton interaction potentials in the presence of an electron gas. We demonstrate that screening and the Pauli blocking play opposite roles, the former increasing and the latter decreasing effective exciton-exciton interaction constant. We find that this counterintuitive effect of the Pauli blocking comes from the mixing of exciton ground and excited states. As a result of the competition of the two mechanisms we report overall increase of the interaction constant with electron density. In Section IV we analyze the impact of free electrons on nonlinear reduction of Rabi splitting in the system, and show that nonlinearity increases with electron density. Section V summarizes the findings. 2D excitons in the presence of a Fermi sea ========================================== ![Sketch of the system. (a) Absolute value of an excitonic wavefunction in a TMD monolayer that is strongly modified by the free electron gas in the conduction band. (b) Sketch of the excitation process where the Pauli principle excludes occupied states in the conduction band preventing the exciton formation. []{data-label="fig:sketch"}](Fig0.pdf){width="0.95\linewidth"} We study a transition metal dichalcogenide monolayer where optical response is strongly dominated by tightly-bound neutral Wannier excitons. Considering a doped monolayer, we account for the presence of the Fermi sea formed by the excessive charge. This can lead to the modification of an optical response in several different ways, and dominant contributions depend on the free electron density $n$ and correspondingly the location of the Fermi level $E_F$. At low electron concentrations charged excitons—trions—are formed. In TMD monolayers these three-particle bound states have binding energy $E^T_b$ being much smaller than an exciton binding energy $E^X_b$. Therefore, in the low-density regime $E_F\ll E^T_b$ trion- and exciton-based response is spectrally well-separated. However, properties of excitons in doped monolayers are modified by an electron gas through screening and Pauli blocking (Fig. \[fig:sketch\]). In the high density regime where $E_F\sim E^T_b$ strong many-body correlations between excitons and electrons become important, and the system is described in terms of exciton-polarons [@Efimkin17] (dressed exciton-electron quasiparticles). For instance, in MoS$_2$ monolayer with $E^T_b = 18$ meV this corresponds to the concentration of excess carriers $n\sim 10^{12}.. 10^{13}$ cm$^{-2}$. In the present work we focus on the low- and intermediate-density regime, where the exciton-based optical response is modified by the Fermi sea through the exciton wavefunction and energy renormalization. To account for electrons we solve the Wannier equation for the eigenenergy $E_X$ and the momentum-space exciton wavefunction $C_{\mathbf{p}}$ that reads[@Efimkin17] $$\label{eq:Wannier} \bigg(\frac{\hbar^2 k^2}{2 \mu}+\Sigma_g\bigg)C_{\mathbf{k}}-\sum_{\mathbf{k'}} B_{\mathbf{k}}V_{\mathbf{k}-\mathbf{k'}}B_{\mathbf{k'}}C_{\mathbf{k'}}=E_X C_{\mathbf{k}},$$ where the exciton binding energy $E^X_b = |E_X-\Sigma_g|$ accounts for the band gap renormalization $\Sigma_g$ caused by excessive charge carriers. In Eq.  $\mu=m m_v/(m+m_v)$ is an exciton reduced mass, and $m$ and $m_v$ stand for the conduction and valence band effective mass, respectively. $B_{\mathbf{k}}=[1-n_F(E^c_{\mathbf{k}})]^{1/2}$ is the Pauli blocking factor that excludes filled electronic states from the space available for exciton formation, $E^c_{\mathbf{k}}$ denotes an energy dispersion for the conduction band, and $n_F$ is a Fermi-Dirac distribution. To account for the effects of screening caused by the excess charge carriers [@Pol2DEG; @Glazov2018] and the atomic thickness of the material [@KeldyshRytova], we consider the screened interaction potential $$V_{\mathbf{k}}=\frac{2\pi e^2}{(4\pi\varepsilon_0\kappa)[k+\rho_0 k^2+k_{sc}(k)]},$$ where $\varepsilon_0$ is the vacuum permittivity, $\rho_0$ is a screening parameter associated to the intrinsic polarizability of the two-dimensional layer, $k_{sc}(k)=-2\pi e^2\Pi(k)/(4\pi\varepsilon_0\kappa)$ is the screening momentum, and $\kappa$ denotes a dielectric constant of the surrounding media. We use the static polarization operator of two-dimensional electron gas [@Pol2DEG] $\Pi(k)=-m/(\pi\hbar^2)[1-\Theta(k-2k_F)(1-4k_F^2/k^2)^{1/2}]$, where the Fermi wavevector is $k_F=\sqrt{2 m E_F}/\hbar$. $\Sigma_g$ accounts for the bandgap renormalization by carriers due to screening and phase space filling effects, and reads $$\Sigma_g=-\sum_{\mathbf{k}}V_\mathbf{k}n_F(E^c_\mathbf{k})-\sum_{\mathbf{k}}(V^0_\mathbf{k}-V_\mathbf{k})n_F(E^v_\mathbf{k}),$$ where $E^c_\mathbf{k}=\hbar^2 \mathbf{k}^2/2m-E_F$ and $E^v_\mathbf{k}=\hbar^2 \mathbf{k}^2/2m_v-E_g-E_F$ denote the energies of conduction and valence bands, $E_g$ is the non-screened bandgap width and $V^0_\mathbf{k}=2\pi e^2/[4\pi\varepsilon_0\kappa (k+\rho_0 k^2)]$. For the sake of simplicity we neglect the renormalization of electron masses in conduction and valence bands, and retain only the bandgap renormalization $\Sigma_g$. The rotational symmetry of the potential $V_{\mathbf{k}}$ allows one to write the wavefunction in the form $$C_{\mathbf{k}}=C^{n,m_z}(k)\frac{e^{i m_z \theta}}{\sqrt{2\pi}}$$ where we use polar coordinates $\mathbf{k}=(k,\theta)$. Then the Wannier equation for $C^{n,m_z}(k)$ reads $$\begin{aligned} \label{eq: exciton_Schrodinger} &E_X C^{n,m_z}(k)= \bigg(\frac{\hbar^2k^2}{2\mu}+\Sigma_g\bigg)C^{n,m_z}(k)+\\ &\int_0^{\infty} \frac{k'dk'}{(2\pi)^2} B(k)V_1(k,k')B(k')C^{n,m_z}(k')\nonumber\end{aligned}$$ where $V_1(k,k')$ is $$\begin{aligned} &V_1(k,k')= \nonumber \\ &-\int_0^{2\pi} d\theta V(\sqrt{k^2+k^{'2}-2kk'\cos{\theta}})e^{i m_z\theta}.\end{aligned}$$ We solve Eq.  numerically for a monolayer of transition metal dichalcogenide. The structure parameters vary a lot throughout the literature and depend on the choice of both TMD monolayer material and its surrounding. Here we set the screening length to $r_0=4$ nm and the bandgap to $E_g=2.6$ eV, which are typical for MoS$_2$ layer [@Stier2018; @Chernikov2014; @Qiu2013]. We also fix equal effective masses, $m_v=m$. As a reference, we choose the case of freestanding monolayer ($\kappa=1$), and set $m=0.35m_0$ ($m_0$ is a free electron mass), being typically the case for TMD monolayers [@Larentis2018]. We consider only the exciton ground-state, so that we set $m_z=0$. The results of calculations are shown in Fig. \[fig:WF\]. In Fig. \[fig:WF\](a) we present the momentum space distribution for the excitonic wavefunction. We observe that the increase of free electron gas density leads to the strong modification of the wavefunction as compared to standard two-dimensional hydrogen-like wavefunction that has a form $\propto \left[ 1+ (\lambda k)^2 \right]^{-3/2}$. The quenching of low-momenta region stems from the Pauli blocking effect. In order to extract the relative contributions of Pauli blocking factor and the screening of interaction induced by electron gas, we simulate Eq.  in the regimes when one of the factors is effectively turned off. At low density the impact of both effects is small, and the wavefunctions plotted for this two cases nearly coincide (not shown). At relatively high density of the free electron gas the momentum space wavefunction is shown in Fig. \[fig:WF\](c) (thick blue curve). We see that the additional screening leads to re-scaling of wavefunction \[Fig. \[fig:WF\](c), green curve\], while the Pauli blocking is responsible for the suppression of low momenta region \[Fig. \[fig:WF\](c), red curve\]. Plots in Fig. \[fig:WF\](b, d) illustrate the electron density dependence of the exciton binding energy and Bohr radius. The latter is defined as an average electron-hole separation $a_B(n) =\langle \psi_{n}(r) |r|\psi_{n}(r) \rangle$, where $\psi_{n}(r)$ is the exciton wavefunction in the real space, and we highlight that it depends on the free electron gas density $n$. The growth of the electron concentration leads to stronger interaction screening, which results in weaker binding of excitons. In the absence of screening the Pauli blocking factor becomes essential for larger values of electron density, leading to reduction of binding energy and corresponding increase for the exciton Bohr radius. ![ (a) Exciton wavefunction in the momentum space shown for different electron gas density. The shift of wavefunction maximum is caused by the increase of Fermi energy and the corresponding wavevector. (b) Exciton binding energy as a function of free electron concentration. Here the green curve corresponds to the absence of Pauli blocking, the red curve corresponds to the absence of interaction screening by electron gas, and the blue solid curve accounts both effects. (c) The impact of screening and Pauli blocking factors on exciton wavefunction at high density of free electron gas. Colors are the same as in panels (b). Notably, screening by the free electron gas leads to rescaling of hydrogen-like wave function (green curve), whereas the Pauli blocking determines the modified shape of the wavefunction (red curve). (d) Bohr radius shown as a function of free electron concentration. Labelling is the same as in (b). []{data-label="fig:WF"}](Fig1.pdf){width="0.98\linewidth"} Exciton-exciton interaction =========================== Next, we study the exciton-exciton interaction processes for TMD monolayers that originate from Coulomb interaction of electrons and holes. We use the standard scattering theory approach [@Tassone1999; @Ciuti1998; @Glazov2009] and exploit the calculated exciton wavefunction to account for the presence of the electron gas. First, we note that the direct interaction is suppressed due to the electron-hole equal effective masses, $m_v=m$ [@Ciuti1998]. Hence, the total interaction constant $g_{\mathrm{tot}}$ is determined by the electron and hole exchange terms, which are identical due to equal effective masses. Thus, $g_{\mathrm{tot}}=2g^e_{\mathrm{exch}}$, with $g^e_{\mathrm{exch}}$ denoting the electron exchange interaction constant. The latter reads [@Glazov2009] $$\begin{aligned} g^e_{\mathrm{exch}} (Q) =\frac{2}{A}\sum_{\mathbf{k},\mathbf{q} } &V_{\mathbf{q}} C_{ \mathbf{k}-\mathbf{q}/2 } C_{ \mathbf{k}-(\mathbf{Q}-\mathbf{q})/2 } C_{\mathbf{k}+\mathbf{q}/2 } \notag \\ & \left[ C_{ \mathbf{k}+(\mathbf{Q}-\mathbf{q})/2 } - C_{ \mathbf{k}+(\mathbf{Q}+\mathbf{q})/2 } \right],\end{aligned}$$ where $\mathbf{Q}$ is an exchanged momentum, and $A$ is the normalization area. ![Exciton-exciton exchange interaction. (a) The dependence of interaction rate on transfer momenta at different densities of free electron gas. While the interaction maxima demonstrates a moderate and non-monotonous shift with the increase of the free electron gas density, the shape for transfer momenta dependence is nearly unaltered. (b) The maxima of interaction rate as a function of exciton density. The inset illustrates zoomed-in region with the non-monotonous dependence. (c) The impact of screening and Pauli blocking factors on momentum dependence of exciton-exciton interaction at high density of free electron gas. Here the green curve corresponds to the absence of Pauli blocking, the red curve to the absence of interaction screening by electron gas, and blue solid curve accounts both effects. (d) The influence of screening and Pauli blocking factors on the maxima of exchange interaction vs the density of free electrons. Colors are the same as in panel (c). []{data-label="fig:exchscr"}](Fig2.pdf){width="0.98\linewidth"} In Fig. \[fig:exchscr\](a) we present the dependence of the electron exchange interaction constant as a function of exchange momenta. The maximum of interaction constant demonstrates a moderate increase with increasing $n$, and the shape of exchange momenta dependence is generally unchanged. Fig. \[fig:exchscr\](b) presents the dependence of interaction maxima on the electron density. Particularly one can see that the dependence is non-monotonous, with the local minima appearing at moderate electron densities. The latter stems from complex interplay between multiple factors, discussed below. In order to understand the origin of this non-trivial dependence of interaction on the electron gas density, we perform calculations (i) in the absence of Pauli blocking factor, and (ii) in the effective absence of screening. The corresponding dependence on exchange momenta is shown in Fig. \[fig:exchscr\](c) for $n=5\cdot 10^{11}$ cm$^{-2}$ density of free electrons. We observe that the interaction has its highest value when both effects are accounted (blue thick curve). In the absence of Pauli blocking the screening leads to the slight decrease of interaction maxima \[case (i), green curve\]. In turn the Pauli blocking leads to significant reduction \[case (ii), red curve\]. In Fig. \[fig:exchscr\](d) we plot the maxima of interaction versus the density of the electron gas. We observe that screening leads to the monotonous enhancement of interaction \[case (i), green curve\], and the Pauli blocking leads to its monotonous reduction \[case (ii), red curve\]. The enhancement of the interaction coefficient due to the interaction screening \[case (i)\] is caused by the enhancement of exciton Bohr radius that dominates the weakening of interaction potential. The origin of reduction due to the Pauli blocking \[case (ii)\] stems from the fact that the Pauli blocking leads to mixing of exciton ground and excited states. In its turn, it was shown earlier, that the interaction between excited exciton states is of attractive nature [@Shahnazaryan2016; @Shahnazaryan2017], explaining the overall decrease of repulsive interaction between excitons. Here we find that the exciton wave function in the presence of Pauli blocking can be expanded in terms of $1s$, $2s$, $3s$ exciton states, and the calculation of exciton-exciton interaction in terms of such functions agrees well with the one calculated in the presence of Pauli blocking (see Fig. \[fig:expansion\]). The details of the calculation are shown in appendix \[app:A\]. The presence of both Pauli blocking and screening leads to a complicated dependence on the density of free electron gas, with regions dominated by the reduction stemming from Pauli blocking and the enhancement arising from screening, as depicted in Fig. \[fig:exchscr\](d)\]. We further analyze the dependence of exciton-exciton interaction on material and substrate parameters. The results are shown in Fig. \[fig:exchpar\]. We observe that the interaction constant demonstrates non-monotonous dependence with local minima at intermediate density regardless the structure parameters. The latter means that the observed effect is of general character and does not depend qualitatively on the material choice. It is remarkable that the growth of Bohr radius due to increase of dielectric constant is nearly compensated by the corresponding reduction of interaction potential, leading to overall weak dependence of exciton-exciton rate on the dielectric properties of surrounding media \[cf. blue and black curves in Fig. \[fig:exchpar\](a)\]. On the other hand, the smaller effective mass of electrons leads to larger Bohr radius, resulting in enhancement of exchange interaction, as the increase of Bohr radius here is not compensated by corresponding reduction of interaction potential \[cf. blue and green curves in Fig. \[fig:exchpar\](a)\]. ![(a) The maximum of exciton-exciton exchange interaction and (b) exciton Bohr radius shown as a function of the free electron gas density for different material parameters. Colors in both panels correspond to parameters shown in panel (b). The increase of the dielectric constant $\kappa$ for surrounding media leads to the growth of exciton Bohr radius, which is nearly compensated by the reduction of interaction potential between excitons. Instead, the reduction of effective mass leads to the increase of Bohr radius, which is not compensated by change in interaction potential. This shows that for the fixed effective mass the interaction rate has weak dependence on the dielectric environment properties.[]{data-label="fig:exchpar"}](Fig3.pdf){width="0.98\linewidth"} Saturation effects and quenching of the Rabi splitting ====================================================== ![(a) Rabi splitting as a function of the free electron density in the weak excitation regime. (b) Light-matter coupling as a function of exciton density shown for different electron gas concentration. (c) Nonlinear polariton interaction rate and its contributions shown as a function of the free electron density. (d) Energy of the lower polariton branch relative to the reference value $E^0_{\mathrm{LP}} (n_0)$ as a function of the exciton density at different $n$. Here the solid curves correspond to Eq. , and the dashed curves to Eq. . []{data-label="fig:Rabi"}](Fig4.pdf){width="0.98\linewidth"} We proceed with the discussion of impact of free electron gas on the coupling between exciton and cavity photon modes. For a TMD monolayer put in a microcavity this corresponds to the electron density-dependent Rabi frequency. It can be expressed as $$\label{eq:coupling} \Omega_0 (n) = \sqrt{ \frac{E_C}{\kappa \varepsilon_0 L_C} } |\psi_n (0)| d_{cv},$$ where $E_C$ is the cavity resonance energy, $L_C=\pi \hbar c /(\sqrt{\kappa}E_C)$ is the cavity length, and $c$ is the speed of light. Here $\psi_n (0)$ is the real space exciton wavefunction at the origin that depends on the free electron gas density. Finally, $d_{cv}$ denotes the dipole matrix element for the optical interband transition. We consider the case of the optical cavity being resonant to the exciton transition in the absence of electron gas, $E_C=E_X^0(0)$. The exciton transition energy in the presence of electron gas reads $E_X^0 (n)=E_g +\Sigma_g (n)- E_b(n)$, where we recall that $E_b=|E_X -\Sigma_g|$ is the exciton binding energy. It should be noted, that the position of excitonic transition varies very slowly with the increase of $n$, as the reduction of binding energy is largely compensated by the corresponding bandgap renormalization. The latter is in good agreement with experimental [@Chernikov2015; @Ugeda2014; @Lin2014] and theoretical evidence [@Shahnazaryan2019]. The value of the dipole matrix element of interband transition is set to $d_{cv}=7$ D, leading to Rabi splitting of $\sim 30$ meV, in agreement with existing experimental results [@Dufferwiel2017]. Here we assume that the dependence of the interband transition matrix element on the density of free electron gas is negligible. Hence, the density of electron gas affects on the efficiency light-matter coupling only via the exciton wave function \[see Eq. \]. The dependence of light-matter coupling $\Omega_0(n)$ on the density of free electron gas is presented in Fig. \[fig:Rabi\](a). The reduction of coupling with the increase of electron density stems from the impact of Pauli blocking, and the detailed analysis is presented in Appendix \[app:B\]. Next we study the nonlinear part of light-matter interaction that is represented by optical saturation coming from the phase space filling. Recently it was shown to provide a significant nonlinear response contribution for TMD polaritons [@Emmanuele2019; @Kyriienko2019]. Together with nonlinear exciton-exciton interaction, the optical saturation effect leads to the energy blueshift for the lower polariton mode, coming from the renormalization of Rabi splitting. It depends on the density of excitons $n_X$ and the excitonic wavefunction. The generalized Rabi frequency can be written as [@Yagafarov2020] $$\begin{aligned} \label{eq:Omega_N} \Omega (n_X,n) \approx \Omega_0 (n) \sqrt{1 - 2 s(n) n_X} ,\end{aligned}$$ where the saturation factor $$s(n)=\frac{\sum_{\mathbf{k}} |C_{\mathbf{k}}|^2 C_{\mathbf{k}}}{\sum_{\mathbf{k'}} C_{\mathbf{k'}}^*}$$ accounts for the phase space filling arising from multiple exchange diagrams. In particular, in the case of effectively hydrogenic wavefunctions, this yields $s^{\mathrm{hyd}}={8\pi a_B^2}/7$, meaning that the larger Bohr radius provides larger nonlinearity. Here, however, the presence of Pauli blocking leads to a moderate dependence of the saturation factor on the density of free electron gas, discussed in Appendix \[app:B\]. As stated in Eq. , for growing density of excitons the Rabi splitting effectively shrinks. The corresponding dependence is illustrated in Fig. \[fig:Rabi\] (b) for various values of free electron gas density. The energy of lower polariton branch reads as $$\begin{aligned} \label{eq:E_LP} E_{\mathrm{LP}} (n_X,n) &= \frac{1}{2} \bigg[ E_C + E_X (n,n_X) \notag \\ & -\sqrt{\left[E_C-E_X (n,n_X)\right]^2 +\Omega^2(n_X,n)} \bigg],\end{aligned}$$ where the exciton energy is $E_X(n,n_X)=E_X^0(n)+ g_{\mathrm{tot}(n)} n_X /2$. Introducing the detuning between cavity and exciton modes as $\Delta(n)=E_C-E_X(n)$, and taking the limit of low exciton density, the energy for the lower polariton mode reads $$\label{eq:E_LPe} E_{\mathrm{LP}} (n_X, n) \approx E_{\mathrm{LP}}^0 (n)+ g_{\mathrm{eff}}(n) n_X,$$ which consists of the linear part equal to $$\label{eq:E_LP0} E_{\mathrm{LP}}^0 (n)= E_C- \frac{\Delta+\sqrt{\Delta^2+\Omega_0^2} }{2},$$ and nonlinear blueshift $g_{\mathrm{eff}}(n) n_X$. Here $g_{\mathrm{eff}}(n)$ is an effective polariton nonlinearity coefficient that is a sum of Coulomb-based interaction and saturative nonlinearity contributions, $$\begin{aligned} g_{\mathrm{eff}}(n) &= \left( 1+ \frac{\Delta}{\sqrt{\Omega^2_0 +\Delta^2} } \right) \frac{g_{\mathrm{tot}} }{4} + \frac{\Omega^2_0 }{2\sqrt{\Omega^2_0 +\Delta^2} }s \notag \\ &=:g^C_{\mathrm{eff}}(n)+g^s_{\mathrm{eff}}(n). \label{eq:g_eff}\end{aligned}$$ For the compactness we omitted the electron density dependence of the quantities $\Delta$, $\Omega_0$, $s$ etc. Polaritonic nonlinearity coefficient and its parts are shown in the Fig. \[fig:Rabi\](c) as a function of free electron gas density. Notably we observe that while the saturation coefficient $s$ increases more than the Coulomb interaction rate, the electron density dependence of its pre-factor diminishes its enhancement, making it nearly flat (dotted curve). Instead, the increase of Coulomb nonlinearity is further boosted by the corresponding growth of its pre-factor (the dashed curve). In Fig. \[fig:Rabi\](d) the dependence of the lower polariton energy on the exciton density is shown, where we plot its nonlinear contribution (as compared to $E^0_{\mathrm{LP}} (n_0)$ with $n_0=10^7$ cm$^{-2}$). At fixed density of free electrons the increase of the exciton density leads to both reduction of light-matter interaction, and the blueshift of the exciton energy. In each this leads to the blueshift of the lower polariton energy. With the increase of free electron gas density both the exciton-exciton interaction rate $g^e_{\mathrm{exch}}$ and the saturation factor $s$ are enhanced, leading to the corresponding growth of the nonlinear optical response. It should be mentioned that relation $E_{\mathrm{LP}} (n_X, n) \approx E_{\mathrm{LP}}^0 (n)+ g_{\mathrm{eff}}(n) n_X$ is valid in the moderate excitation regime $n_X \leq 10^{12}$ cm$^{-2}$, as for higher intensities the quadratic terms $\propto n_X^2$ become relevant. Conclusions =========== In this paper we analyzed the behavior of exciton polaritons in a TMD monolayer in the presence of a gas of free electrons. We revealed that the Fermi sea has a strong effect on the nonlinear optical response of the system. We found that the role of free electrons is twofold. First, doping leads to screening of the Coulomb interaction, and results in the increase of exciton Bohr radius and simultaneously the reduction of exciton-exciton interaction potential. Our calculations show that the overall impact of the screening leads to the enhancement of exciton-exciton interaction coefficient. Second, due to the Pauli exclusion principle, the presence of the free electrons also dramatically modifies the structure of the excitonic wave functions, suppressing the contribution of the harmonics corresponding to small electron wavevectors. Surprisingly the impact of the Pauli blocking factor leads to the reduction of exciton-exciton interaction. We found that the latter can be attributed to mixing of exciton ground and excited states, caused by the Pauli blocking factor. It is known that the interaction between excited exciton states is of attractive type, which explains the reduction of exciton-exciton repulsive interaction caused by Pauli principle. Finally, we showed that the combined impact of interaction screening and Pauli blocking leads to the non-monotonous dependence of the exciton-exciton interaction constant as a function of free electron gas density. The presence of Fermi sea substantially modifies also the statistics-based renormalization of the Rabi splitting at high exciton densities, which gives another contribution to the enhancement of the optical nonlinearity. It is important to note that both Coulomb nonlinearity and saturation-based nonlinearity generally grow with the increase of the free electron gas density. As the latter can be easily controlled by application of the external gate voltage, our findings pave the way to accessible and experimental friendly tuning of the degree of optical nonlinearity in TMD based samples. acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to D. Efimkin for valuable discussions. This work was supported by the Megagrant 14.Y26.31.0015 of the Ministry of Education and Science of Russian Federation, and ITMO Fellowship and Professorship Program. Numerical calculations of nonlinearities were supported by the Russian Science Foundation (grant No. 19-72-00171). V.K.K., I.A.S. and O.K. acknowledge support from Icelandic Science Foundation project “Hybrid polaritonics”.\ Expansion of an exciton wavefunction in terms of basis functions {#app:A} ================================================================ ![ (a) The real space dependence of exciton wave function in the presence of Pauli blocking at density of free electrons $n=5 \cdot 10^{11}$ cm$^{-2}$ (red solid curve), and the exciton ground and excited states wavefunctions in its absence. Here we neglect the impact of the interaction screening by free electrons. (b) Real and (c) momentum space dependence of wave function (red solid curve) and its expansion in terms of basis functions (black dashed curve). The inset in panel (c) demonstrates the contribution of excited states versus the density of free electron gas. (d) The maxima of exciton-exciton interactions as a function of free electron density. The red dots correspond to calculation using the actual wave functions, and the black dots is calculated using the wavefunctions expanded in terms of basis functions. The mismatch in the values is attributed to the imperfection of the fitting procedure. []{data-label="fig:expansion"}](FigS1.pdf){width="0.95\linewidth"} We analyze the impact of Pauli blocking factor on the excitonic wavefunction that results in the reduction of the exciton-exciton interaction for increasing electron gas density. First, we simulate Eq.  of the main text in the absence of both interaction screening and the Pauli blocking. We find wavefunctions for ground and excited exciton states ($1s$, $2s$, $3s$). Their real space distributions are shown in Fig. \[fig:expansion\](a). Next, for each value of $n$ we expand calculated wavefunctions in the presence of Pauli blocking (but non screening) in terms of bare basis functions. This procedure yields $$\begin{aligned} \label{eq:expansion} \psi_n(r) = a_1 (n) \psi_{1s} (r) +a_2 (n) \psi_{2s} (r) +a_3 (n) \psi_{3s} (r),\end{aligned}$$ where coefficients $a_i(n)$ are found from $$\begin{aligned} & \psi_n(0) =a_1 (n) \psi_{1s} (0) +a_2 (n) \psi_{2s} (0) +a_3 (n) \psi_{3s} (0), \notag \\ & \psi_n(r_1) =a_1 (n) \psi_{1s} (r_1) +a_2 (n) \psi_{2s} (r_1) +a_3 (n) \psi_{3s} (r_1), \notag \\ & 1=|a_1(n)|^2+|a_2(n)|^2+|a_3(n)|^2,\end{aligned}$$ and $r_1$ corresponds to the first root of $\psi_n$. The results of fitting for large density of the electron gas are shown in Fig. \[fig:expansion\](b, c). We find that the fit is nearly exact at small $r$, but strongly deviates at large distances (not shown). Correspondingly, for the small momenta there is a strong deviation, while for larger values there is a good agreement. Finally, in Fig. \[fig:expansion\](d) we provide the comparison of the exciton-exciton interaction coefficient calculated from exact wave functions (red dots), and the wave functions defined by the . Evidently, with the increase of electron gas density Pauli blocking leads to larger contribution of excited states \[see the inset in Fig. \[fig:expansion\](c)\], which interact attractively [@Shahnazaryan2016; @Shahnazaryan2017], resulting in corresponding reduction of the ground state exciton-exciton repulsive interaction. The dependence of Rabi splitting saturation rate on the free electron gas density {#app:B} ================================================================================= ![ (a) Light-matter coupling as a function of the free electron gas density at low excitation regime. Here green curves correspond to the absence of Pauli blocking, red curves correspond to the absence of interaction screening, and blue curves account both effects. (b), (c), (d) The saturation factor $s(n)$ (solid curves) and its hydrogen-like estimate $s^{\mathrm{hyd}} (n)$ (dashed curves) versus the free electron gas density. Panel (b) corresponds to the presence of both Pauli blocking and the interaction screening; panel (c) stands for the absence of the interaction screening; panel (d) illustrates the absence of Pauli blocking. []{data-label="fig:saturation"}](FigS2.pdf){width="0.95\linewidth"} In Fig. \[fig:saturation\](a) we present light-matter coupling at small exciton densities, $n_X a_B^2 \ll 1$, as a function of free electron gas density $n$. In the absence of the Pauli blocking the screening of Coulomb interaction leads to weaker binding of excitons, so that the wavefunction is less concentrated around the origin. The latter results in the quenching of light-matter coupling \[Fig. \[fig:saturation\](a), green curve\]. On the contrary, in the absence of screening the Pauli blocking leads to mixing with excited exciton states, leading to the increase of wavefunction amplitude at the origin \[see Fig. \[fig:expansion\](a)\]. This results in the corresponding enhancement of light-matter coupling \[Fig. \[fig:saturation\](a), red curve\]. Yet, the impact of screening is much stronger, so that the interplay of this counteracting effects leads to overall reduction of light-matter coupling with the increase of free electron gas density \[Fig. \[fig:saturation\](a), blue curve\]. We further analyze the Rabi splitting saturation factor $s(n)$. Fig. \[fig:saturation\](b) illustrates its dependence on the density of free electron gas. We observe a moderate enhancement of the saturation rate with the growth of the electron gas density. On the other hand, the estimate of saturation factor for the hydrogen-like exciton $s^{\mathrm{hyd}} (n)$ grows much faster \[dashed curve in Fig. \[fig:saturation\](b)\]. To get a better insight, we study the impacts of the screening of interaction and Pauli blocking separately. In Fig. \[fig:saturation\](c) we present the free electron gas density dependence of saturation factor and its estimate in the absence of interaction screening. As the density increases, the saturation factor moderately reduces, while its estimate enhances. This discrepancy with the hydrogen-based estimate stems from the Pauli blocking, which leads to emergence of strongly non-hydrogenic wavefunctions. Fig. \[fig:saturation\](d) illustrates the case of the absence of Pauli blocking and the presence of interaction screening. Here both the saturation factor and its estimate increase nearly on equal footing, indicating that in the absence of Pauli blocking effect the hydrogen-like model is valid up to a constant. In total, the interplay of these two counteracting impacts results in the moderate enhancement of saturation efficiency, as depicted in Fig. \[fig:saturation\](b). 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--- abstract: 'The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors’ earlier work ([@ShSh14-4]), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.' address: - 'Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849' - 'Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA' author: - Wenxian Shen - Zhongwei Shen title: 'Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media' --- Introduction ============ In the present paper, we study transition fronts of nonlocal dispersal equations of the form $$\label{main-eqn} u_{t}=J\ast u-u+f(t,u),\quad (t,x)\in\R\times\R,$$ where the dispersal kernel $J$ satisfies - [*$J\not\equiv0$, $J\in C^{1}(\R)$, $J(x)=J(-x)\geq0$ for all $x\in\R$, $\int_{\R}J(x)dx=1$ and $$\int_{\R}J(x)e^{\la x}dx<\infty,\quad\int_{\R}|J'(x)|e^{\la x}dx<\infty\quad\text{for any}\,\,\, \la\in\R.$$* ]{} We assume that $f$ is a Fisher-KPP or monostable type nonlinearity, that is, $f$ satisfies - *$f:\R\times[0,1]\ra\R$ is $C^{1}$ and $C^{2}$ in its first variable and second variable, respectively, and satisfies the following conditions:* - $f(t,0)=f(t,1)=0$ and $f(t,u)\geq0$ for all $(t,u)\in\R\times(0,1)$; - there exists $\theta_{1}\in(0,1)$ such that $f_{u}(t,u)\leq0$ for all $u\in[\theta_{1},1]$; - $\sup_{(t,u)\in\R\times[0,1]}\max\{|f_{t}(t,u)|,|f_{u}(t,u)|,|f_{uu}(t,u)|\}<\infty$; - there exists a $C^{1}$ function $g:[0,1]\to\R$ satisfying - $g(0)=g(1)=0<g(u)$ for $u\in(0,1)$, - $g_{u}(0)=1$, $g_{u}(1)\geq-1$, $g_{u}$ is decreasing and $\int_{0}^{1}\frac{u-g(u)}{u^{2}}du<\infty$ such that $$a(t)g(u)\leq f(t,u)\leq a(t)u,\quad (t,u)\in\R\times[0,1],$$ where $a(t):=f_{u}(t,0)$ satisfies $a_{\inf}:=\inf_{t\in\R}a(t)>0$. The decaying assumptions on $J$ are used implicitly in some places of the present paper. For example, to ensure Proposition \[prop-regularity\] below, we need $\la\geq r$ (see the proof of [@ShSh14-4 Theorem 1.3]). Here is a typical example of the function $f$, i.e., $f(t,u)=a(t)u(1-u)$. Equation with $J$ and $f$ satisfying (H1) and (H2) arises in, for example, population dynamics modeling the evolution of species. In which case, the unknown function $u(t,x)$ would be the normalized population density, the operator $v\mapsto J\ast v-v$ models nonlocal diffusion and $f(t,u)$ is the logistic-type growth rate. For equation , solutions of particular interest are global-in-time front-like solutions, more precisely, transition fronts, since they play the key role in describing extinction and persistence of the species. Recall that a global-in-time solution $u(t,x)$ of is called a (right-moving) *transition front* (connecting $0$ and $1$) in the sense of Berestycki-Hamel (see [@BeHa07; @BeHa12], also see [@Sh04; @Sh11]) if $u(t,x)\in(0,1)$ for all $(t,x)\in\R\times\R$ and there is a function $X:\R\to\R$, called *interface location function*, such that $$\lim_{x\to-\infty}u(t,x+X(t))=1\,\,\text{and}\,\,\lim_{x\to\infty}u(t,x+X(t))=0\,\,\text{uniformly in}\,\,t\in\R.$$ The interface location function $X(t)$ tells the position of the transition front as time $t$ elapses, while the uniform-in-$t$ limits shows the *bounded interface width*, that is, $$\forall\,\,0<\ep_{1}\leq\ep_{2}<1,\quad\sup_{t\in\R}{\rm diam}\{x\in\R|\ep_{1}\leq u(t,x)\leq\ep_{2}\}<\infty.$$ Thus, transition fronts are proper generalizations of traveling waves in homogeneous media and periodic (or pulsating) traveling waves in periodic media. Notice if $\xi(t)$ is a bounded function, then $X(t)+\xi(t)$ is also an interface location function. Therefore, interface location functions are not unique. But, it is not hard to check that the difference of two interface location functions is a bounded function. Hence, interface location functions are unique up to addition by bounded functions. We see that transition fronts can be defined in the same way for more general equations, say, $$\label{eqn-general} u_{t}=J\ast u-u+f(t,x,u).$$ Many results have been obtained for when $f(t,x,u)$ is of monostable or Fisher-KPP type in various special cases. For example, when $f(t,x,u)=f(u)$, traveling waves and minimal speeds have been studied in [@CaCh04; @CoDu05; @CoDu07; @Sc80]. When $f(t,x,u)=f(x,u)$ is periodic in $x$ or $f(t,x,u)$ is periodic in both $t$ and $x$, spreading properties and periodic traveling waves have been studied in [@CDM13; @RaShZh; @ShZh10; @ShZh12-1; @ShZh12-2]. When $f(t,x,u)=f(x,u)$, while principal eigenvalue, positive solution and long-time behavior of solutions was studied in [@BCV14], transition fronts were shown to exist in [@LiZl14]. In the present paper, we study transition fronts of when $f(t,x,u)=f(t,u)$, that is, . To state our results, we define for $\ka>0$ $$\label{speed-function-intro} c^{\ka}(t)=\frac{\int_{\R}J(y)e^{\ka y}dy-1}{\ka}t+\frac{1}{\ka}\int_{0}^{t}a(s)ds,\quad t\in\R.$$ We see that if $a(t)\equiv a>0$ is a constant function, then $c^{\ka}(t)$ is nothing but the front location function of traveling waves with speed $\frac{\int_{\R}J(y)e^{\ka y}dy-1+a}{\ka}$ of a homogeneous nonlocal Fisher-KPP equation (see e.g. [@CaCh04; @CoDu07; @Sc80]). Note that since $\inf_{t\in\R}a(t)>0$, $c^{\ka}(t)$ is increasing, and since $\sup_{(t,u)\in\R\times[0,1]}|f_{t}(t,u)|<\infty$, $c^{\ka}(t)$ is at most linear. Indeed, if $a_{\inf}=\inf_{t\in\R}a(t)$ and $a_{\sup}=\sup_{t\in\R}a(t)$, then $$c^{\ka}(t)\in\frac{\int_{\R}J(y)e^{\ka y}dy-1}{\ka}t+\bigg[\frac{a_{\inf}}{\ka}t,\frac{a_{\sup}}{\ka}t\bigg],\quad t\in\R.$$ This function will serve as the interface location function as in the definition of transition fronts. Our first result concerning the existence, regularity and decaying properties of transition fronts of is stated in the following theorem. \[thm-tf\] Assume (H1) and (H2). There exists $\ka_{0}>0$ such that for any $\ka\in(0,\ka_{0}]$, there is a transition front $u^{\ka}(t,x)$ of with interface location function $X^{\ka}(t)=c^{\ka}(t)$ and satisfying the following properties - $u^{\ka}(t,x)$ is decreasing in $x$ for any $t\in\R$; - there holds $\lim_{x\to\infty}\frac{u^{\ka}(t,x+X^{\ka}(t))}{e^{-\ka x}}=1$ uniformly in $t\in\R$; - $u^{\ka}(t,x)$ is continuously differentiable in $x$ for any $t\in\R$ and satisfies $$\sup_{(t,x)\in\R\times\R}\frac{|u^{\ka}_{x}(t,x)|}{u^{\ka}(t,x)}<\infty;$$ - there holds $\lim_{x\to-\infty}u^{\ka}_{x}(t,x+X^{\ka}(t))=0$ uniformly in $t\in\R$; - there holds $\lim_{x\to\infty}\frac{u_{x}^{\ka}(t,x+X^{\ka}(t))}{u^{\ka}(t,x+X^{\ka}(t))}=-\ka$ uniformly in $t\in\R$. Note that the continuity of a transition front $u(t,x)$ in the space variable $x$ is not assumed in the definition of transition fronts. But the space regularity of transition fronts plays an important role in the study of other important properties such as stability and uniqueness of transition fronts. In the random dispersal case, the space regularity of transition fronts follows from parabolic Schauder estimates, while, thanks to the lack of space regularity for the nonlocal dispersal equations (that is, the semigroup generated by the nonlocal dispersal operator has no regularizing effect), a transition front of a nonlocal dispersal equation may not be regular in space. We refer the reader to [@BaFiReWa97] for the existence of discontinuous traveling waves of $u_{t}=J\ast u-u+f_{B}(u)$, where $f_{B}$ is a balanced bistable nonlinearity. In [@ShSh14-4], we established some very general results on the space regularity of transition fronts of nonlocal dispersal equations. Among others, we proved in [@ShSh14-4] the following proposition. \[prop-regularity\] Assume (H1) and (H2). Let $w(t,x)$ be an arbitrary transition front of satisfying $$\label{harnack-type-intro} w(t,x)\leq Ce^{r|x-y|}w(t,y),\quad (t,x,y)\in\R\times\R\times\R$$ for some $C>0$ and $r>0$. Then, $w(t,x)$ is continuously differentiable in $x$ for any $t\in\R$ and satisfies $\sup_{(t,x)\in\R\times\R}\frac{|w_{x}(t,x)|}{w(t,x)}<\infty$. At this point, we mention that the regularity of pulsating fronts for nonlocal KPP equations in the space periodic case was treated in [@CDM13] (see also [@ShZh12-2]). We remark that Theorem \[thm-tf\]$\rm(iii)$ follows directly from Proposition \[prop-regularity\]. We point out that the existence of transition fronts in Theorem \[thm-tf\] is proven constructively via the construction of appropriate sub- and super-solutions. The $\ka_0$ in Theorem \[thm-tf\] is obtained in the construction of sub-solutions (see Proposition \[prop-subsol\]) and satisfies that $\ka_{0}<\inf_{t\in\R}\ka_{0}(t)$ (see ), where $\ka_{0}(t)>0$ is such that $$\frac{\int_{\R}J(y)e^{\ka_{0}(t) y}dy-1+a(t)}{\ka_{0}(t)}=\min_{\ka>0}\frac{\int_{\R}J(y)e^{\ka y}dy-1+a(t)}{\ka}.$$ The $\ka_0$ in Theorem \[thm-tf\] may be small and hence the set of transition fronts obtained in Theorem \[thm-tf\] may only contain those which move sufficiently fast to the right. In [@RaShZh], the authors proved the existence of periodic traveling waves in the time periodic case $f(t+T,u)=f(t,u)$ also constructively via the construction of appropriate sub- and super-solutions. The $\ka_0$ obtained in [@RaShZh] is given by $$\label{k0-eq} \frac{\int_{\R} J(y)e^{\ka_{0} y}dy-1+\hat a}{\ka_{0}}=\min_{\ka>0}\frac{\int_{\R} J(y)e^{\ka y}dy-1+\hat a}{\ka},$$ where $\hat a=\frac{1}{T}\int_0^T a(t)dt$. The $\ka_0$ in for the time periodic case is optimal and the value in is nothing but the minimal speed of periodic traveling waves. But the method to construct sub-solutions in the time periodic case adopted in [@RaShZh] is difficult to be applied to the general time dependent case. We adopt in the present paper a method based on an idea from [@Zl12] (also see [@LiZl14; @TaZhZl14]), which is different from that in [@RaShZh] and allows us to apply it to the general time dependent case, but does not enable us to obtain the optimal value of $\ka_0$. It would be interesting to determine the optimal value for $\ka_{0}$ (see Subsection \[subsec-some-remarks\] for some remarks). We also remark that if $a(t)=f_{u}(t,0)$ is uniquely ergodic, that is, the hull of $a(t)$ is compact and the dynamical system generated by the shift operators on the hull of $a(t)$ is uniquely ergodic, then the limit $\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}a(s)ds$ exists, and hence, the asymptotic speed $$\lim_{t\to\infty}\frac{X^{\ka}(t)}{t}=\lim_{t\to\infty}\frac{c^{\ka}(t)}{t}=\frac{\int_{\R}J(y)e^{\ka y}dy-1}{\ka}+\frac{1}{\ka}\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}a(s)ds$$ exists. Since interface location functions are unique up to addition by bounded functions as mentioned before, asymptotic speed (if exists) is independent of the choice of interface location functions. Note that asymptotic speed hardly exists in general. In the presence of space regularity, i.e., Theorem \[thm-tf\]$\rm(iii)$, we then move to the study of the asymptotic stability of $u^{\ka}(t,x)$. To do so, we further assume the uniform exponential stability of the constant solution $1$, that is, - [*There exists $\theta_{1}\in(0,1)$ and $\beta_{1}>0$ such that $f_{u}(t,u)\leq-\beta_{1}$ for all $(t,u)\in\R\times[\theta_{1},1]$.* ]{} Note that (H3) improves the corresponding assumption in (H2). From classical semigroup theory and comparison principle, we know that for any $u_{0}\in C^{b}_{\rm unif}(\R)$, the space of real-valued, bounded and uniformly continuous functions on $\R$, the solution $u(t,x;t_{0},u_{0})$ of with initial data $u(t_{0},\cdot;t_{0},u_{0})=u_{0}$ exists globally in the space $C^{b}_{\rm unif}(\R)$ and is unique. We then show \[thm-asympt-stability\] Assume (H1)-(H3). Let $\ka_{0}$ be as in Theorem \[thm-tf\]. For $\ka\in(0,\ka_{0}]$, let $u^{\ka}(t,x)$ be the transition front in Theorem \[thm-tf\]. Let $u_{0}:\R\to[0,1]$ be uniformly continuous and satisfy $$\liminf_{x\to-\infty}u_{0}(x)>0\quad\text{and}\quad\lim_{x\to\infty}\frac{u_{0}(x)}{u^{\ka}(t_{0},x)}=1$$ for some $t_{0}\in\R$. Then, there holds the limit $$\label{stability-intro} \lim_{t\to\infty}\sup_{x\in\R}\bigg|\frac{u(t,x;t_{0},u_{0})}{u^{\ka}(t,x)}-1\bigg|=0.$$ A generalization of Theorem \[thm-asympt-stability\] is given in Corollary \[cor-stability-general\], where we show that if $u_{0}$ is as in Theorem \[thm-asympt-stability\], but with $\lim_{x\to\infty}\frac{u_{0}(x)}{u^{\ka}(t_{0},x)}=1$ replaced by $\lim_{x\to\infty}\frac{u_{0}(x)}{u^{\ka}(t_{0},x)}=\la$ for some $\la>0$, then there holds $\lim_{t\to\infty}\sup_{x\in\R}\big|\frac{u(t,x;t_{0},u_{0})}{u^{\ka}(t,x-\frac{\ln\la}{\ka})}-1\big|=0$. More generally, if $\lim_{x\to\infty}\frac{u_{0}(x)}{e^{-\ka x}}=\tilde{\la}>0$, then, using Theorem \[thm-tf\]$\rm(ii)$ and the facts that $X^{\ka}(t)$ is continuous, increasing and satisfies $\lim_{t\to\pm\infty}X^{\ka}(t)=\pm\infty$, there exists a unique $\tilde{t}_{0}\in\R$ such that the limit $\lim_{x\to\infty}\frac{u_{0}(x)}{u^{\ka}(\tilde{t}_{0},x)}=1$ exists, which leads to the asymptotic dynamics of $u(t,x;\tilde{t}_{0},u_{0})$ as in with $t_{0}$ replaced by $\tilde{t}_{0}$. See Corollary \[cor-stability-more-general\] for more details. In particular, if $a(t)$ is uniquely ergodic, then $u(t,x;\tilde{t}_{0},u_{0})$ has asymptotic spreading properties in the following sense: if $\lim_{t\to\infty}\frac{X^{\ka}(t)}{t}=c^{\ka}_{*}$, then for any $\ep>0$ there holds $$\lim_{t\to\infty}\inf_{x\leq (c_{*}^{\ka}-\ep)t}u(t,x;\tilde{t}_{0},u_{0})=1\quad\text{and}\quad\lim_{t\to\infty}\inf_{x\geq (c_{*}^{\ka}+\ep)t}u(t,x;\tilde{t}_{0},u_{0})=0.$$ We remark that results as in Theorem \[thm-tf\] and Theorem \[thm-asympt-stability\] can also be established for the following reaction-diffusion equation $$\label{eqn-rd} u_{t}=u_{xx}+f(t,u),\quad (t,x)\in\R\times\R.$$ In particular, we have Assume $\rm(H2)$ and $\rm(H3)$. There exists $\ka_{0}>0$ such that for any $\ka\in(0,\ka_{0}]$, there is a transition front $u^{\ka}(t,x)$ of with interface location function $$X^{\ka}(t)=\ka t+\frac{1}{\ka}\int_{0}^{t}a(s)ds,\quad t\in\R$$ and satisfying the following properties - $u^{\ka}_{x}(t,x)<0$ for all $(t,x)\in\R\times\R$; - the limits $\lim_{x\to\infty}\frac{u^{\ka}(t,x+X^{\ka}(t))}{e^{-\ka x}}=1$ and $\lim_{x\to\infty}\frac{u_{x}^{\ka}(t,x+X^{\ka}(t))}{u^{\ka}(t,x+X^{\ka}(t))}=-\ka$ hold and are uniform in $t\in\R$; - let $u_{0}:\R\to[0,1]$ be uniformly continuous and satisfy $\liminf_{x\to-\infty}u_{0}(x)>0$ and $\lim_{x\to\infty}\frac{u_{0}(x)}{e^{-\ka x}}=\la$ for some $\la>0$; then, there exists $t_{0}\in\R$ such that $$\lim_{t\to\infty}\sup_{x\in\R}\bigg|\frac{u(t,x;t_{0},u_{0})}{u^{\ka}(t,x)}-1\bigg|=0.$$ It should be pointed out that transition fronts of were established in [@NaRo12], while no result concerning the stability exists in the literature. In the case $f(t,u)$ being uniquely ergodic in $t$, the existence, stability and uniqueness of transition fronts of were studied in [@Sh11]. Finally, we remark that while transition fronts of reaction-diffusion equations and nonlocal equations of Fisher-KPP type in general time or space heterogeneous media have been studied (see e.g. [@LiZl14; @NaRo12; @NRRZ12; @TaZhZl14; @Zl12]), there exists no result in the literature concerning the corresponding discrete equations in general time or space heterogeneous media, i.e., $$\label{discrete-time} \dot{u}_{i}=u_{i+1}-2u_{i}+u_{i-1}+f(t,u_{i}),\quad t\in\R,\,\,i\in\Z,$$ or $$\label{discrete-space} \dot{u}_{i}=u_{i+1}-2u_{i}+u_{i-1}+f(i,u_{i}),\quad t\in\R,\,\,i\in\Z,$$ where $f(t,u)$ and $f(i,u)$ are of Fisher-KPP type. Such discrete equations also arises naturally in applications (see e.g. [@ShSw90]), and hence, it is of great importance to study them. We refer the reader to [@CFG06; @ChGu02; @ChGu03; @FGS02; @HuZi94; @WuZo97; @ZHH93] and references therein for works of or in homogeneous media, i.e., $f(t,u)=f(u)$ or $f(i,u)=f(u)$, and to [@GuHa06; @GuWu09] for works of in periodic media, i.e., $f(i,u)=f(i+L,u)$ for some $L\in\Z$. The rest of the paper is organized as follows. In Section \[sec-sub-super-sol\], we construct appropriate global-in-time sub- and super-solutions of for the use to construct transition fronts. In Section \[sec-construct-tf\], we construct transition fronts and prove Theorem \[thm-tf\]. In Section \[sec-stability\], we study the stability of transition fronts constructed in Theorem \[thm-tf\] and prove Theorem \[thm-asympt-stability\]. Construction of sub- and super-solutions {#sec-sub-super-sol} ======================================== In this section, we construct appropriate global-in-time sub- and super-solutions of the equation . Throughout this section, we assume (H1) and (H2). Construction of the super-solution ---------------------------------- For $\ka>0$, let $$\label{speed-function} c^{\ka}(t)=\frac{\int_{\R}J(y)e^{\ka y}dy-1}{\ka}t+\frac{1}{\ka}\int_{0}^{t}a(s)ds,\quad t\in\R.$$ For $\ka>0$, define $$\label{supsol} \phi^{\ka}(t,x)=e^{-\ka(x-c^{\ka}(t))},\quad (t,x)\in\R\times\R.$$ \[prop-subsol\] For any $\ka>0$, $\phi^{\ka}(t,x)$ is a super-solution of . We check that $\phi^{\ka}(t,x)$ solves $\phi_{t}=J\ast\phi-\phi+a(t)\phi$. The proposition then follows from $f(t,u)\leq a(t)u$ for $u\geq0$ by (H2) (note it is safe to extend $f(t,u)$ to $u\in(1,\infty)$ in this way). Construction of the sub-solution -------------------------------- To construct the sub-solution, we borrow an idea from [@Zl12] (also see [@LiZl14; @TaZhZl14]). Let us consider the homogeneous reaction-diffusion equation $$\label{eqn-homo-g} u_{t}=u_{xx}+g(u),$$ where $g$, given in (H2), is of Fisher-KPP type with $g_{u}(0)=1$. It is known (see e.g. [@Uch78]) that for any $\al\in(0,1)$, traveling wave of the form $\phi_{\al}(x-c_{\al}t)$ with $\phi_{\al}(-\infty)=1$ and $\phi_{\al}(\infty)=0$ exists and is unique up to space translation, where $c_{\al}=\al+\frac{1}{\al}$. We assume, without loss of generality, that $\lim_{x\to\infty}e^{\al x}\phi_{\al}(x)=1$. Moreover, for any $\al\in(0,1)$, the exponential function $e^{-\al(x-c_{\al}t)}$ is a traveling wave of the linearization of at $0$, that is, $$\label{eqn-homo-g-linear} u_{t}=u_{xx}+u.$$ For $\al\in(0,1)$, we define a function $h_{\al}:[0,\infty)\to[0,1)$ by setting $$h_{\al}(s)=\begin{cases} 0,\quad&s=0,\\ \phi_{\al}(-\al^{-1}\ln s),\quad&s>0. \end{cases}$$ Note that $h_{\al}(e^{-\al x})=\phi_{\al}(x)$, that is, $h_{\al}$ takes the profile of traveling waves of with speed $c_{\al}$ to that of with the same speed. Since $\phi_{\al}$ solves $\phi_{\al}''+c_{\al}\phi_{\al}'+g(\phi_{\al})=0$, we check that $h_{\al}$ satisfies $$\label{eqn-h} \al^{2}s^{2}h_{\al}''(s)-sh_{\al}'(s)+g(h_{\al}(s))=0.$$ \[the-function-h\] Let $\al\in(0,1)$. Then, - $h_{\al}$ is increasing and satisfies $\lim_{s\to\infty}h_{\al}(s)=1$, $h_{\al}'(0)=1$ and $h_{\al}''(s)<0$ for $s>0$; - let $\beta=2+\frac{1}{\al^{2}}$ and $\rho_{\al}(x)=-h_{\al}''(e^{-x})$ for $x\in\R$; then there holds $$0<\rho_{\al}(x)\leq e^{\beta|x-y|}\rho_{\al}(y),\quad x,y\in\R.$$ See [@Zl12] for $\rm(i)$ and [@LiZl14 Lemma 3.1] for $\rm(ii)$. Set $\al=\frac{3}{4}$ (it is nothing special). We write $h(s):=h_{\frac{3}{4}}(s)$ and $\rho(x):=\rho_{\frac{3}{4}}(x)$. For $\ka>0$, define $$\psi^{\ka}(t,x)=h(\phi^{\ka}(t,x)),\quad (t,x)\in\R\times\R,$$ where $\phi^{\ka}(t,x)=e^{-\ka(x-c^{\ka}(t))}$ is given in . Then, \[prop-subsol\] There exists $\ka_{0}>0$ such that for any $\ka\in(0,\ka_{0}]$, $\psi^{\ka}(t,x)$ is a sub-solution of $u_{t}=J\ast u-u+a(t)g(u)$; in particular, it is a sub-solution of . Since $\phi^{\ka}(t,x)$ satisfies the equation $\phi^{\ka}_{t}=J\ast\phi^{\ka}-\phi^{\ka}+a(t)\phi^{\ka}$, we readily check that $N[\psi^{\ka}]:=\psi^{\ka}_{t}-[J\ast\psi^{\ka}-\psi^{\ka}]$ satisfies $$N[\psi^{\ka}]=a(t)\phi^{\ka}h'(\phi^{\ka})+h'(\phi^{\ka})[J\ast\phi^{\ka}-\phi^{\ka}]-\int_{\R}J(y)[h(\phi^{\ka}(t,x-y))-h(\phi^{\ka}(t,x))]dy.$$ By the second-order Taylor expansion, i.e., $$\begin{split} h(\phi^{\ka}(t,x-y))-h(\phi^{\ka}(t,x))&=h'(\phi^{\ka}(t,x))[\phi^{\ka}(t,x-y)-\phi^{\ka}(t,x)]\\ &\quad+\frac{h''(\zeta_{x,y,t})}{2}[\phi^{\ka}(t,x-y)-\phi^{\ka}(t,x)]^{2} \end{split}$$ for some $\zeta_{x,y,t}$ between $\phi^{\ka}(t,x-y)$ and $\phi^{\ka}(t,x)$, we see $$N[\psi^{\ka}]=a(t)\phi^{\ka}h'(\phi^{\ka})-\frac{1}{2}\int_{\R}J(y)h''(\zeta_{x,y,t})[\phi^{\ka}(t,x-y)-\phi^{\ka}(t,x)]^{2}dy.$$ For the integral in the above equality, we first see from the monotonicity of $\ln$ and the explicit expression of $\phi^{\ka}(t,x)$ that $$|\ln\zeta_{x,y,t}-\ln\phi^{\ka}(t,x)|\leq|\ln\phi^{\ka}(t,x-y)-\ln\phi^{\ka}(t,x)|\leq\ka|y|.$$ It then follows from Lemma \[the-function-h\]$\rm(ii)$ with $\beta=2+\frac{1}{(3/4)^{2}}<4$ that $$-h''(\zeta_{x,y,t})=\rho(-\ln\zeta_{x,y,t})\leq e^{4|\ln\zeta_{x,y,t}-\ln\phi^{\ka}(t,x)|}\rho(-\ln\phi^{\ka}(t,x))\leq e^{4\ka|y|}[-h''(\phi^{\ka}(t,x))].$$ Hence, $$\begin{split} &-\frac{1}{2}\int_{\R}J(y)h''(\zeta_{x,y,t})[\phi^{\ka}(t,x-y)-\phi^{\ka}(t,x)]^{2}dy\\ &\quad\quad\leq-\frac{1}{2}h''(\phi^{\ka}(t,x))\int_{\R}J(y)e^{4\ka|y|}[\phi^{\ka}(t,x-y)-\phi^{\ka}(t,x)]^{2}dy\\ &\quad\quad=-\frac{1}{2}h''(\phi^{\ka}(t,x))[\phi^{\ka}(t,x)]^{2}\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy, \end{split}$$ which leads to $$\label{an-equality-678543} N[\psi^{\ka}]\leq a(t)\phi^{\ka}(t,x)h'(\phi^{\ka}(t,x))-\frac{1}{2}h''(\phi^{\ka}(t,x))[\phi^{\ka}(t,x)]^{2}\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy.$$ Since $\frac{d}{d\ka}[e^{\ka y}-1]^{2}=2(e^{\ka y}-1)e^{\ka y}y\geq0$, the function $\ka\mapsto\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy$ is increasing on $(0,\infty)$. Moreover, we have $$\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy\to\begin{cases} 0,\quad& \ka\to0,\\ \infty,\quad&\ka\to\infty. \end{cases}$$ Thus, due to $\inf_{t\in\R}a(t)>0$ by (H2), there is a unique $\ka_{0}>0$ such that $$\label{parameter-ka-bound} \frac{1}{2}\int_{\R}J(y)e^{4\ka_{0}|y|}[e^{\ka_{0} y}-1]^{2}dy=\bigg(\frac{3}{4}\bigg)^{2}\inf_{t\in\R}a(t),$$ which implies $\frac{1}{2}\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy\leq\big(\frac{3}{4}\big)^{2}a(t)$ for $t\in\R$ and $\ka\in(0,\ka_{0}]$. It then follows from that $$N[\psi^{\ka}]\leq a(t)\bigg[\phi^{\ka}(t,x)h'(\phi^{\ka}(t,x))-\bigg(\frac{3}{4}\bigg)^{2}h''(\phi^{\ka}(t,x))[\phi^{\ka}(t,x)]^{2}\bigg]=a(t)g(\psi^{\ka}(t,x)),$$ where we used with $s=\phi^{\ka}(t,x)$ and $\al=\frac{3}{4}$. The proposition then follows from $a(t)g(u)\leq f(t,u)$ for $u\in[0,1]$ by (H2). Some remarks {#subsec-some-remarks} ------------ We make some remarks about the sub- and super-solutions constructed in the above two subsections. \(i) From Lemma \[the-function-h\]$\rm(i)$, there holds $h_{\al}(s)\leq s$ for all $s\geq0$. This, in particular, implies that $\psi^{\ka}(t,x)=h(\phi^{\ka}(t,x))\leq\phi^{\ka}(t,x)$ for all $(t,x)\in\R\times\R$ (actually, the strict inequality holds). This order relation between $\psi^{\ka}(t,x)$ and $\phi^{\ka}(t,x)$ is important in establishing various properties of approximating solutions, which will be studied in the next section, Section \[sec-construct-tf\]. Moreover, $\psi^{\ka}(t,x)$ and $\phi^{\ka}(t,x)$ propagate to the right with the same speed $\dot{c}^{\ka}(t)$. This fact says that any global-in-time solution of between $\psi^{\ka}(t,x)$ and $\min\{1,\phi^{\ka}(t,x)\}$ is a transition front. \(ii) In constructing the sub-solution $\psi^{\ka}(t,x)$, we restrict $\ka$ to take values in $(0,\ka_{0}]$, where $\ka_{0}>0$ is given by . If we let $\ka_{0}(t)>0$ be the one minimizing $\dot{c}^{\ka}(t)$, that is, $$\label{minimizer-148} \dot{c}^{\ka_{0}(t)}(t)=\min_{\ka>0}\dot{c}^{\ka}(t).$$ then we have $$\label{an-relation-parameter-ka} \ka_{0}<\inf_{t\in\R}\ka_{0}(t).$$ In fact, notice the critical points for $\dot{c}^{\ka}(t)$, i.e., points satisfying $\frac{d}{d\ka}\dot{c}^{\ka}(t)=0$, satisfies $$\ka\int_{\R}J(y)ye^{\ka y}dy-\int_{\R}J(y)e^{\ka y}dy+1=a(t).$$ Setting $$q_{1}(\ka)=\ka\int_{\R}J(y)ye^{\ka y}dy-\int_{\R}J(y)e^{\ka y}dy+1,$$ we see that $q_{1}(0)=0$ and $q_{1}'(\ka)=\ka\int_{\R}J(y)y^{2}e^{\ka y}dy>0$ for $\ka>0$. Setting $$q_{2}(\ka)=\frac{1}{2}\bigg(\frac{4}{3}\bigg)^{2}\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]^{2}dy,$$ we see that $q_{2}(0)=0$ and $$\begin{split} q_{2}'(\ka)&=\bigg(\frac{4}{3}\bigg)^{2}\int_{\R}J(y)2|y|e^{4\ka|y|}[e^{\ka y}-1]^{2}dy+\bigg(\frac{4}{3}\bigg)^{2}\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]e^{\ka y}ydy\\ &\geq\int_{\R}J(y)e^{4\ka|y|}[e^{\ka y}-1]e^{\ka y}ydy=\int_{\R}J(y)e^{4\ka|y|}e^{\ka y_{*}}\ka ye^{\ka y}ydy\quad(\text{where}\,\,y_{*}\in[0,y])\\ &=\ka\int_{\R}J(y)e^{2\ka|y|}y^{2}e^{\ka(|y|+y_{*})}e^{\ka(|y|+y)}dy\geq\ka\int_{\R}J(y)e^{2\ka|y|}y^{2}dy>q_{1}'(\ka),\quad\ka>0. \end{split}$$ This simply means that $q_{2}(\ka)>q_{1}(\ka)$ for $\ka>0$. Hence, $q_{1}(\ka_{0})<q_{2}(\ka_{0})=\inf_{t\in\R}a(t)$. Since $q_{1}(\ka)$ is continuous and increasing, we conclude . We will use in the proof of Lemma \[lem-asymptotic-at-infty\], which is the key to the stability of transition fronts. \(iii) A possible way to enlarge the value of $\ka_{0}$ is to make $\al$ change (we have fixed $\al=\frac{3}{4}$ in the above analysis). But, it seems not enough to push $\ka_{0}$ arbitrary close to $\inf_{t\in\R}\ka_{0}(t)$. Also, $\inf_{t\in\R}\ka_{0}(t)$ is hardly the optimal value for $\ka_{0}$, since in the periodic case, $\ka_0$ is exactly such that $$\frac{\int_{\R}J(y)e^{\ka_0 y}dy -1+\hat a}{\ka_0}=\min_{\ka >0} \frac{\int_{\R}J(y)e^{\ka y}dy -1+\hat a}{\ka}$$ where $\hat a=\frac{1}{T}\int_0^T a(t)dt$ and $T$ is the period (see e.g. [@RaShZh]). Construction of transition fronts {#sec-construct-tf} ================================= In this section, we construct transition fronts and study their space regularity and decaying properties, that is, we are going to prove Theorem \[thm-tf\]. Throughout this section, we assume (H1) and (H2). Fix any $\ka\in(0,\ka_{0}]$, where $\ka_{0}>0$ is given in Proposition \[prop-subsol\]. For this fixed $\ka$, we write $c(t)=c^{\ka}(t)$, $\phi(t,x)=\phi^{\ka}(t,x)$ and $\psi(t,x)=\psi^{\ka}(t,x)$. Again, $h(s)=h_{\frac{3}{4}}(s)$. The proof of the existence of transition fronts is constructive. We first construct approximating solutions. For $n\geq1$, let $u^{n}(t,x)$, $t\geq-n$ be the unique solution of $$\label{eqn-approximating-sol} \begin{cases} u_{t}=J\ast u-u+f(t,u),\quad t>-n,\\ u(-n,x)=\psi(-n,x)=h(\phi(-n,x)). \end{cases}$$ We prove some basic properties of $u^{n}(t,x)$ in the following \[lem-harnack-type\] The following statements hold: - for any $(t,x)\in[-n,\infty)\times\R$ and $n\geq1$, there holds $$0<\psi(t,x)\leq u^{n}(t,x)\leq\tilde{\phi}(t,x):=\min\{1,\phi(t,x)\};$$ - for any $(t,x,y)\in[-n,\infty)\times\R\times\R$ and $n\geq1$, there holds $$u^{n}(t,x)\leq\frac{e^{\ka|x-y|}}{h(1)}u^{n}(t,y);$$ - if $n>m\geq1$, then $u^n(t,x)\ge u^m(t,x)$ for all $(t,x)\in[-m,\infty)\times\R$. $\rm(i)$ By Lemma \[the-function-h\]$\rm(i)$, we have $h(s)\leq s$ for all $s\geq0$, which implies that $u(-n,x)\leq\phi(-n,x)$. The result then follows from comparison principle. $\rm(ii)$ Using the expression $\phi(t,x)=e^{-\ka(x-c(t))}$, we readily check $\tilde{\phi}(t,x)\leq e^{\ka|x-y|}\tilde{\phi}(t,y)$. Moreover, since $h(0)=0$ and $h''(s)<0$ for all $s>0$, we find $h(1)s\leq h(s)$ for $s\in[0,1]$. In particular, by the monotonicity of $h$, we have $$h(1)\tilde{\phi}(t,x)\leq h(\tilde{\phi}(t,x))\leq h(\phi(t,x))=\psi(t,x)\leq u^{n}(t,x),$$ which implies that $$u^{n}(t,x)\leq\tilde{\phi}(t,x)\leq e^{\ka|x-y|}\tilde{\phi}(t,y)\leq\frac{e^{\ka|x-y|}}{h(1)}u^{n}(t,y).$$ $\rm(iii)$ Let $n>m\geq1$. Since $\psi(t,x)$ is a sub-solution of and $\psi(-n,\cdot)=u^{n}(-n,\cdot)$, comparison principle yields $\psi(t,\cdot)\leq u^{n}(t,\cdot)$ for all $t\geq-n$. In particular, at time $t=-m$, we have $u^{n}(-m,\cdot)\geq\psi(-m,\cdot)=u^{m}(-m,\cdot)$. Again, by comparison principle, we arrive at the result. The sequence $\{u^{n}(t,x)\}$ is the approximating solutions. However, we can not conclude immediately the convergence of $\{u^{n}(t,x)\}$ to a global-in-time solution of due to the lack of regularity. Following the arguments in [@LiZl14], we can show the uniform Lipschitz continuity of $\{u^{n}(t,x)\}$ in $x$, which of course ensures the convergence. Here, we take a different approach, which is based on the monotonicity of $\{u^{n}(t,x)\}$ as in Lemma \[lem-harnack-type\]$\rm(iii)$. Now, we prove Theorem \[thm-tf\]. We first construct transition fronts. By Lemma \[lem-harnack-type\]$\rm(iii)$, for any fixed $(t,x)\in\R\times\R$, there exists $n_{0}=n_{0}(t)\geq1$ such that the sequence $\{u^{n}(t,x)\}_{n\geq n_{0}}$ is nondecreasing. Since it is clearly between $0$ and $1$, the limit $\lim_{n\to\infty}u^{n}(t,x)$ exists and equals to some number in $[0,1]$. Hence, there exists a function $u:\R\times\R\to[0,1]$ such that $$\label{locally-uniform-convergence-to-tf} \lim_{n\to\infty}u^{n}(t,x)=u(t,x)\quad\text{pointwise in}\quad(t,x)\in\R\times\R.$$ Moreover, since $u^{n}(t,x)$ satisfies $u^{n}_{t}=J\ast u^{n}-u^{n}+f(t,u^{n})$, we have that for any $t>-n$ and $x\in\R$, $$\label{existence-eq1} u^n(t,x)=u^n(0,x)+\int_0^t \int_{\R}J(y-x)u^n(s,y)dyds-\int_0^t u^n(s,x)ds+\int_0^t f(s,u^n(s,x))ds.$$ Passing to the limit $n\to\infty$ in , we conclude from the dominated convergence theorem that for any $t\in\R$ and $x\in\R$, $$\label{existence-eq2} u(t,x)=u(0,x)+\int_0^t \int_{\R}J(y-x)u(s,y)dyds-\int_0^t u(s,x)ds+\int_0^t f(s,u(s,x))ds.$$ From which, we conclude that $u(t,x)$ is differentiable in $t$ and satisfies . To see $u(t,x)$ is a transition front of , we notice $\psi(t,x)\leq u(t,x)\leq\phi(t,x)$ by Lemma \[lem-harnack-type\]$\rm(i)$ and . Taking $X(t)=c(t)$, we find $$\label{expo-decaying-tf} h(e^{-\ka x})=\psi(t,x+X(t))\leq u(t,x+X(t))\leq\phi(t,x+X(t))=e^{-\ka x},$$ which implies that $u(t,x)$ is a transition front of . We now prove the properties $\rm(i)$-$\rm(v)$ listed in the statement of Theorem \[thm-tf\]. $\rm(i)$ Since $\{u^{n}(t,x)\}_{n\geq1}$ are decreasing in $x$, $u(t,x)$ is nonincreasing in $x$. Since $u(t,x)$ is global-in-time, it is decreasing in $x$ due to comparison principle (see e.g. [@ShSh14 Proposition A.1(iii)]). $\rm(ii)$ By , we have $\frac{h(e^{-\ka x})}{e^{-\ka x}}\leq\frac{u(t,x+X(t))}{e^{-\ka x}}\leq1$. Since $\lim_{x\to\infty}\frac{h(e^{-\ka x})}{e^{-\ka x}}=h'(0)=1$ due to Lemma \[the-function-h\]$\rm(i)$, we find that the limit $\lim_{x\to\infty}\frac{u(t,x+X(t))}{e^{-\ka x}}=1$ exists and is uniform in $t\in\R$. We also have $$\label{asymptotic-9564} \lim_{x\to\infty}\frac{u(t,x+X(t))}{h(e^{-\ka x})}=1\quad\text{uniformly in}\quad t\in\R.$$ $\rm(iii)$ By and Lemma \[lem-harnack-type\]$\rm(ii)$, we have $$\label{harnack-type-tf} u(t,x)\leq\frac{e^{\ka|x-y|}}{h(1)}u(t,y), \quad(t,x,y)\in\R\times\R\times\R.$$ This verifies . The result then follows from Proposition \[prop-regularity\]. $\rm(iv)$ By $\rm(iii)$, $v(t,x):=u_{x}(t,x)$ exists and satisfies the equation $$v_{t}=a(t,x)v+b(t,x),$$ where $$a(t,x)=-1+f_{u}(t,u(t,x))\quad\text{and}\quad b(t,x)=\int_{\R}J'(y)u(t,x-y)dy.$$ We see that for any $t\in\R$ and $T>0$, there holds $$\label{aux-eqn-eqn} v(t,x)=e^{\int_{t-T}^{t}a(s,x)ds}v(t-T,x)+\int_{t-T}^{t}e^{\int_{\tau}^{t}a(s,x)ds}b(\tau,x)d\tau.$$ We are going to show $v(t,x+X(t))\to0$ as $x\to-\infty$ uniformly in $t\in\R$. To do so, we let $\ep>0$, and choose $T=T(\ep)>0$ and $L=L(\ep)>0$ such that $$Ce^{-T}\leq\frac{\ep}{3}\quad\text{and}\quad\int_{\R\backslash[-L,L]}|J'(x)|dx\leq\frac{\ep}{3},$$ where $C:=\sup_{(t,x)\in\R\times\R}|u_{x}(t,x)|<\infty$ by $\rm(iv)$. Notice such a $L$ exists due to (H1). For $\theta_{1}$ as in (H2), let $X_{\theta_{1}}(t)$ be the interface location function at $\theta_{1}$, i.e., $u(t,X_{\theta_{1}}(t))=\theta_{1}$ for all $t\in\R$. It is well-defined due to the monotonicity of $u(t,x)$ in $x$. Since $u(t,x)$ is a transition front, we have $\sup_{t\in\R}|X_{\theta_{1}}(t)-X(t)|<\infty$. Setting $$C_{T}=\Big[\sup_{t\in\R}\dot{X}(t)\Big]T+\sup_{t\in\R}|X_{\theta_{1}}(t)-X(t)|,$$ we readily check that if $x-X(t)\leq-C_{T}$, then $u(s,x)\geq\theta_{1}$ for all $s\in[t-T,t]$. As a result, $$a(s,x)=-1+f_{u}(s,u(s,x))\leq-1,\quad s\in[t-T,t],\quad x-X(t)\leq-C_{T}.$$ It then follows from that for $x-X(t)\leq-C_{T}$ there holds $$\label{aux-eqn-eqn-1} |v(t,x)|\leq Ce^{-T}+\int_{t-T}^{t}e^{-(t-\tau)}|b(\tau,x)|d\tau\leq\frac{\ep}{3}+\int_{t-T}^{t}e^{-(t-\tau)}|b(\tau,x)|d\tau.$$ For $b(\tau,x)$, we have $$\label{aux-eqn-eqn-2} \begin{split} |b(\tau,x)|&\leq\int_{\R\backslash[-L,L]}|J'(x)|dx+\bigg|\int_{-L}^{L}J'(y)u(\tau,x-y)dy\bigg|\\ &\leq\frac{\ep}{3}+\bigg|\int_{-L}^{L}J'(y)u(\tau,x-y)dy\bigg|. \end{split}$$ Since $J'$ is odd and $u(\tau,x+X(\tau))\to1$ as $x\to-\infty$ uniformly in $\tau\in\R$, we find some $C_{T,L}\geq C_{T}$ such that if $x-X(t)\leq-C_{T,L}$, then $$\bigg|\int_{-L}^{L}J'(y)u(\tau,x-y)dy\bigg|\leq\frac{\ep}{3},\quad\tau\in[t-T,t].$$ We then deduce from and that for $x-X(t)\leq-C_{T,L}$ there holds $$|v(t,x)|\leq\frac{\ep}{3}+\frac{2\ep}{3}\int_{t-T}^{t}e^{-(t-\tau)}d\tau\leq\ep.$$ Observing the above analysis is uniform in $t\in\R$, we find the limit. $\rm(v)$ By $\rm(ii)$, we have $\lim_{x\to\infty}\frac{u(t,x+X(t))}{e^{-\ka x}}=1$ uniformly in $t\in\R$. It follows that for any $\ep\in(0,1]$, there exists $M(\ep)>0$ such that $$u(t,x+X(t))\geq(1-\ep^{2})e^{-\ka x},\quad x\geq M(\ep),\quad t\in\R.$$ By Taylor expansion and , we find for $x\geq M(\ep)$ and $t\in\R$, there exists $\ep_{*}=\ep_{*}(t,x)\in(0,\ep)$ such that $$\begin{split} u_{x}(t,x+\ep_{*}+X(t))&=\frac{u(t,x+\ep+X(t))-u(t,x+X(t))}{\ep}\\ &\leq\frac{e^{-\ka(x+\ep)}-e^{-\ka x}+\ep^{2}e^{-\ka x}}{\ep}=e^{-\ka x}\bigg(\frac{e^{-\ka\ep}-1}{\ep}+\ep\bigg), \end{split}$$ which leads to $\frac{u_{x}(t,x+\ep_{*}+X(t))}{e^{-\ka(x+\ep_{*})}}\leq e^{\ka\ep}\big(\frac{e^{-\ka\ep}-1}{\ep}+\ep\big)$. In particular, for all $x\geq M(\ep)+1$ and $t\in\R$, $$\label{estimate-one-side-1} \frac{u_{x}(t,x+X(t))}{e^{-\ka x}}\leq e^{\ka\ep}\bigg(\frac{e^{-\ka\ep}-1}{\ep}+\ep\bigg).$$ Similarly, using $\lim_{x\to\infty}\frac{u(t,x+X(t))}{h(e^{-\ka x})}=1$ uniformly in $t\in\R$ by , we can show that for any $\ep\in(0,1]$, there exists $\tilde{M}(\ep)>0$ such that for $x\geq\tilde{M}(\ep)+1$ and $t\in\R$ there holds $$\label{estimate-one-side-2} \frac{u_{x}(t,x+X(t))}{e^{-\ka x}}\geq(1+\ep)\frac{e^{-\ka \ep}-1}{\ep}-\ep(1+\ep).$$ Since $\lim_{\ep\to0}\frac{e^{-\ka \ep}-1}{\ep}=-\ka$, and imply that the limit $\lim_{x\to\infty}\frac{u_{x}(t,x+X(t))}{e^{-\ka x}}=-\ka$ holds and is uniform in $t\in\R$. The result then follows from $\rm(ii)$. Stability of transition fronts {#sec-stability} ============================== Let $\ka_{0}$ be as in Proposition \[prop-subsol\] and fix $\ka\in(0,\ka_{0}]$. Let $u(t,x)=u^{\ka}(t,x)$ be the transition front constructed in Theorem \[thm-tf\] with interface location function $X(t)=c^{\ka}(t)$. We study the asymptotic stability of $u(t,x)$. Throughout this section, we assume (H1)-(H3). Sub- and super-solutions ------------------------ In this subsection, we construct appropriate sub- and super-solutions. We prove \[prop-sub-sup-solution\] Let $\ep_{*}\in(0,1-\theta_{1})$ and $\om_{*}=\beta_{1}$, where $\theta_{1}\in(0,1)$ and $\beta_{1}>0$ are as in (H3). Then, for any $\ep\in(0,\ep_{*}]$ and $\om\in(0,\om_{*}]$, there exists $C=C(\ep_{*},\om)>0$ such that for any $\xi\in\R$ $$\begin{split} u^{-}(t,x)&=(1-\ep e^{-\om(t-\tau)})u(t,x+\xi-C\ep e^{-\om(t-\tau)}),\quad t\geq\tau\quad\text{and}\\ u^{+}(t,x)&=(1+\ep e^{-\om(t-\tau)})u(t,x+\xi+C\ep e^{-\om(t-\tau)}),\quad t\geq\tau \end{split}$$ are sub-solution and sup-solution of , respectively. To prove Proposition \[prop-sub-sup-solution\], we need the uniform steepness of $u(t,x)$ given in the following \[lem-uniform-steepness\] For any $M>0$, there holds $\sup_{t\in\R}\sup_{|x-X(t)|\leq M}u_{x}(t,x)<0$. It can be proven along the same line as that of [@ShSh14-3 Theorem 3.1]. So we omit it. Now, we prove Proposition \[prop-sub-sup-solution\] Here, we only show that $u^{-}(t,x)$ is a sub-solution; $u^{+}(t,x)$ being a sup-solution can be proven along the same line. Note that by the space homogeneity of the equation , we may assume, without loss of generality, that $\xi=0$. Hence, we assume $$u^{-}=u^{-}(t,x)=(1-\ep e^{-\om(t-\tau)})u(t,x-C\ep e^{-\om(t-\tau)}),\quad t\geq\tau,$$ where $\ep\in(0,\ep_{*}]$, $\om\in(0,\om_{*}]$ and $C>0$ (to be chosen). With $N[u^{-}]:=u^{-}_{t}-[J\ast u^{-}-u^{-}]-f(t,u^{-})$, we compute $$\begin{split} N[u^{-}]&=\ep e^{-\om(t-\tau)}[\om u+C\om(1-\ep e^{-\om(t-\tau)})u_{x}-f(t,u)+f_{u}(t,u_{*})u]\\ &\leq\ep e^{-\om(t-\tau)}[\om u+C\om(1-\ep_{*})u_{x}+f_{u}(t,u_{*})u], \end{split}$$ where $u=u(t,x-C\ep e^{-\om(t-\tau)})$, $u_{x}=u_{x}(t,x-C\ep e^{-\om(t-\tau)})$ and $u_{*}\in[u^{-},u]$. By Theorem \[thm-tf\]$\rm(v)$, we find some $M_{0}>0$ such that $$\frac{u_{x}(t,x)}{u(t,x)}\leq-\frac{\ka}{2},\quad x\geq X(t)+M_{0},\quad t\in\R.$$ Let $M=M(\ep_{*})\geq M_{0}$ be such that $$u(t,x)\geq\frac{\theta_{1}}{1-\ep_{*}},\quad x\leq X(t)-M,\quad t\in\R.$$ By Lemma \[lem-uniform-steepness\], we have $C_{M}:=\sup_{t\in\R}\sup_{|x-X(t)|\leq M}u_{x}(t,x)<0$. Then, we choose $C=C(\ep_{*},\om)>0$ such that $$\begin{cases} \om+C\om(1-\ep_{*})C_{M}+\sup_{(t,u)\in\R\times[0,1]}|f_{u}(t,u)|\leq0,\\ \om-\frac{C\om(1-\ep_{*})\ka}{2}+\sup_{(t,u)\in\R\times[0,1]}|f_{u}(t,u)|\leq0. \end{cases}$$ Now, we consider three cases. If $x-C\ep e^{-\om(t-\tau)}-X(t)\leq-M$, we have $u\geq\frac{\theta_{1}}{1-\ep_{*}}$, and then, $u^{-}\geq(1-\ep e^{-\om(t-\tau)})\frac{\theta_{1}}{1-\ep_{*}}\geq\theta_{1}$. Thus, $f_{u}(t,u_{*})\leq-\beta_{1}$. Therefore, $\om\leq\beta_{1}$ leads to $$N[u^{-}]\leq\ep e^{-\om(t-\tau)}(\om+f_{u}(t,u_{*}))u\leq0.$$ If $x-C\ep e^{-\om(t-\tau)}-X(t)\in[-M,M]$, we have $u_{x}\leq C_{M}<0$. Hence, the choice of $C$ gives $$N[u^{-}]\leq\ep e^{-\om(t-\tau)}\Big[\om+C\om(1-\ep_{*})C_{M}+\sup_{(t,u)\in\R\times[0,1]}|f_{u}(t,u)|\Big]\leq0.$$ If $x-C\ep e^{-\om(t-\tau)}-X(t)\geq M$, we have $\frac{u_{x}}{u}\leq-\frac{\ka}{2}$. Then, the choice of $C$ leads to $$\begin{split} N[u^{-}]&\leq \ep e^{-\om(t-\tau)}u\bigg[\om +C\om(1-\ep_{*})\frac{u_{x}}{u}+f_{u}(t,u_{*})\bigg]\\ &\leq\ep e^{-\om(t-\tau)}u\bigg[\om-\frac{C\om(1-\ep_{*})\ka}{2}+f_{u}(t,u_{*})\bigg]\leq0. \end{split}$$ Hence, we have show $N[u^{-}]\leq0$, that is, $u^{-}$ is a sub-solution. Proof of Theorem \[thm-asympt-stability\] ----------------------------------------- This whole subsection is devoted to the proof of Theorem \[thm-asympt-stability\]. Let $u_{0}$ be as in the statement of Theorem \[thm-asympt-stability\] and $u(t,x;t_{0}):=u(t,x;t_{0},u_{0})$ be the solution of with initial data $u(t_{0},\cdot;t_{0})=u_{0}$. We are going to show $\lim_{t\to\infty}\sup_{x\in\R}\big|\frac{u(t,x;t_{0})}{u(t,x)}-1\big|=0$, which is true if we can show $$\label{estimate-limsup} \limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x)}\leq1$$ and $$\label{estimate-liminf} \liminf_{t\to\infty}\inf_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x)}\geq1.$$ Here, we only prove ; the proof of $\eqref{estimate-liminf}$ can be done along the same line except one that we comment after the proof of . To show , we consider the set $$\Xi=\bigg\{\xi\geq0\bigg|\limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi)}\leq1\bigg\}.$$ We claim that $\Xi\neq\emptyset$. In fact, since $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t_{0},x)}=1$, we have for any fixed $\ep_{*}\in(0,1-\theta_{1})$, $u_{0}(x)\leq(1+\ep_{*})u(t_{0},x)$ for all $x\gg1$. Then, we can find a large $\xi_{*}>0$ so that $$\label{initial-estimate-95678} u_{0}(x)\leq(1+\ep_{*})u(t_{0},x-\xi_{*}+C\ep_{*}), \quad x\in\R, \vspace{-0.04in}$$ where $C=C(\ep_{*},\om_{*})$ is as in Proposition \[prop-sub-sup-solution\]. Applying Proposition \[prop-sub-sup-solution\], we conclude $$u(t,x;t_{0})\leq(1+\ep_{*}e^{-\om_{*}(t-t_{0})})u(t,x-\xi_{*}+C\ep_{*}e^{-\om_{*}(t-t_{0})}),\quad x\in\R \vspace{-0.04in}$$ for all $t\geq t_{0}$. This and the monotonicity of $u(t,x)$ yield $\limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi_{*})}\leq1$, that is, $\xi_{*}\in\Xi$. Clearly, $\xi\in\Xi$ for all $\xi>\xi_{*}$. Set $\xi_{\inf}:=\inf\{\xi|\xi\in\Xi\}\geq0$. We claim that $\xi_{\inf}\in\Xi$. Indeed, let $\{\xi_{n}\}\subset\Xi$ and $\xi_{n}\to\xi_{\inf}$ as $n\to\infty$. We see $$\frac{u(t,x;t_{0})}{u(t,x-\xi_{\inf})}=\frac{u(t,x;t_{0})}{u(t,x-\xi_{n})}\bigg[1+\frac{u(t,x-\xi_{n})-u(t,x-\xi_{\inf})}{u(t,x-\xi_{\inf})}\bigg],$$ $$\frac{u(t,x-\xi_{n})-u(t,x-\xi_{\inf})}{u(t,x-\xi_{\inf})}=\frac{u_{x}(t,x-\xi_{n}^{*})}{u(t,x-\xi_{\inf})}(\xi_{n}-\xi_{\inf}),\,\,\text{where}\,\,\xi_{n}^{*}\in[\xi_{\inf},\xi_{n}]$$ and $$\frac{u_{x}(t,x-\xi_{n}^{*})}{u(t,x-\xi_{\inf})}\leq\bigg[\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}\bigg]\frac{u(t,x-\xi_{n}^{*})}{u(t,x-\xi_{\inf})}\leq\bigg[\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}\bigg]\frac{e^{\ka|\xi_{n}^{*}-\xi_{\inf}|}}{h(1)},$$ where we used . Setting $C_{0}:=\frac{1}{h(1)}\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}$, we then have $$\frac{u(t,x;t_{0})}{u(t,x-\xi_{\inf})}\leq\frac{u(t,x;t_{0})}{u(t,x-\xi_{n})}\big[1+C_{0}e^{\ka|\xi_{n}-\xi_{\inf}|}(\xi_{n}-\xi_{\inf})\big],$$ which leads to $\limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi_{\inf})}\leq1$, that is, $\xi_{\inf}\in\Xi$. It remains to show that $$\label{shift-is-zero} \xi_{\inf}=0.$$ To do so, we need the following crucial result concerning the exact asymptotic of $u(t,x;t_{0})$ as $x\to\infty$. \[lem-asymptotic-at-infty\] For any $\eta>0$, there exists $M=M(\eta)\gg1$ such that $$\label{sharp-asymptotics-key} u(t,x+X(t)+\eta)\leq u(t,x+X(t);t_{0})\leq u(t,x+X(t)-\eta),\quad x\geq M$$ for all $t\geq t_{0}$. Equivalently, $$\label{sharp-asymptotics-key-equiv} \lim_{x\to\infty}\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}=1\quad\text{uniformly in}\quad t\geq t_{0}$$ To show , we actually only need the upper bound in ; the lower bound in and the limit are needed for proving . The proof of Lemma \[lem-asymptotic-at-infty\] is postponed to Subsection \[subsec-proof-lemm-1\] Now, for contradiction, suppose is false, that is, $\xi_{\inf}>0$. We are going to find a number in $\Xi$, but smaller than $\xi_{\inf}$. To this end, let $\ep\in(0,\ep_{*})$ be sufficiently small, where $\ep_{*}\in(0,1-\theta_{1})$ is as in Proposition \[prop-sub-sup-solution\]. Since $\xi_{\inf}\in\Xi$, i.e., $\limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi_{\inf})}\leq1$, for any $\ep_{1}>0$, we can find some $T=T(\ep_{1})$ such that $\frac{u(T,x;t_{0})}{u(T,x-\xi_{\inf})}<1+\ep_{1}$ for all $x\in\R$, which implies $$\label{initial-condition-84294284284240539} u(T,x;t_{0})< u(T,x-\xi_{\inf})+\ep_{1},\quad x\in\R.$$ In particular, $$\label{estimate-1359-1} u(T,x+X(T);t_{0})<u(T,x+X(T)-\xi_{\inf})+\ep_{1},\quad x\in\R.$$ Let $C=C(\ep_{*},\om_{*})$ be as in Proposition \[prop-sub-sup-solution\]. We claim there exists $M_{1}>0$ such that $$\label{estimate-1359-2} u(t,x+X(t))\leq (1+\ep)u(t,x+X(t)+3\ep C),\quad x\leq -M_{1},\quad t\in\R.$$ Indeed, we have from Taylor expansion $$\begin{split} &(1+\ep)u(t,x+X(t)+3\ep C)-u(t,x+X(t))\\ &\quad\quad=3\ep Cu_{x}(t,x+X(t)+x_{*})+\ep u(t,x+X(t)+3\ep C)\geq0, \end{split}$$ where we used $u_{x}(t,x+X(t)+x_{*})\to0$ uniformly in $t\in\R$ as $x\to-\infty$ by Theorem \[thm-tf\]$\rm(iv)$ and $u(t,x+X(t)+3\ep C)\to1$ uniformly in $t\in\R$ as $x\to-\infty$. Now, applying with $t=T$, we deduce from that $$\label{estimate-1359-2-1} u(T,x+X(T);t_{0})\leq(1+\ep)u(T,x+X(T)-\xi_{\inf}+3\ep C)+\ep_{1},\quad x\leq-M_{1}.$$ Note that the inequality in is strict, which implies that if we choose $\ep$ so small that $3\ep C\ll1$, we will have $$u(T,x+X(T);t_{0})\leq u(T,x+X(T)-\xi_{\inf}+3\ep C)+\ep_{1},\quad x\in[-M_{1},M],$$ where $M=M(\frac{\xi_{\inf}}{2})\gg1$ is as in Lemma \[lem-asymptotic-at-infty\]. This, together with , give $$\label{estimate-1359-3} u(T,x+X(T);t_{0})\leq(1+\ep)u(T,x+X(T)-\xi_{\inf}+3\ep C)+\ep_{1},\quad x\leq M.$$ Since $\inf_{t\in\R}\inf_{x\leq M}u(t,x+X(t)-\xi_{\inf}+1)>0$, we can take $\ep_{1}$ so small that $$\ep_{1}\leq\ep\inf_{t\in\R}\inf_{x\leq M}u(t,x+X(t)-\xi_{\inf}+1),$$ and then conclude from that $$\label{estimate-1359-4} u(T,x+X(T);t_{0})\leq(1+2\ep)u(T,x+X(T)-\xi_{\inf}+3\ep C),\quad x\leq M.$$ Using Lemma \[lem-asymptotic-at-infty\], we see that $$\label{estimate-1359-5} u(T,x+X(T);t_{0})\leq u(T,x+X(T)-\frac{\xi_{\inf}}{2}),\quad x\geq M.$$ If $\ep$ is so small that $3\ep C\leq\frac{\xi_{\inf}}{2}$, we have from that $$u(T,x+X(T);t_{0})\leq u(T,x+X(T)-\xi_{\inf}+3\ep C),\quad x\geq M.$$ This, together with , give $$\label{estimate-1359-6} \begin{split} u(T,x+X(T);t_{0})&\leq(1+2\ep)u(T,x+X(T)-\xi_{\inf}+3\ep C)\\ &=(1+2\ep)u(T,x+X(T)-\xi_{\inf}+\ep C+2\ep C),\quad x\in\R. \end{split}$$ We now apply Proposition \[prop-sub-sup-solution\] to to conclude that $$\begin{split} u(t,x+X(t);t_{0})&\leq(1+2\ep e^{-\om_{*}(t-T)})\\ &\quad\times u(t,x+X(t)-\xi_{\inf}+\ep C+2\ep Ce^{-\om_{*}(t-T)}),\quad x\in\R \end{split}$$ for all $t\geq T$. From which and the monotonicity of $u(t,x)$ in $x$, we deduce $$\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t)-\xi_{\inf}+\ep C)}\leq(1+2\ep e^{-\om_{*}(t-T)}),\quad x\in\R$$ for all $t\geq T$, which leads to $\limsup_{t\to\infty}\sup_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi_{\inf}+\ep C)}\leq1$, that is, $\xi_{\inf}-\ep C\in\Xi$. It contradicts to the minimality of $\xi_{\inf}$. Hence, holds, i.e., $\xi_{\inf}=0$. Now, we comment on the proof of . Define $$\tilde{\Xi}=\bigg\{\xi\geq0\bigg|\liminf_{t\to\infty}\inf_{x\in\R}\frac{u(t,x;t_{0})}{u(t,x-\xi)}\geq1\bigg\}.$$ To show $\tilde{\Xi}\neq\emptyset$, as in , it seems that we have $$(1-\ep_{*})u(t_{0},x+\xi_{*}-C\ep_{*})\leq u_{0}(x),\quad x\in\R$$ for some $\ep_{*}\in(0,1-\theta_{1})$ and $\xi_{*}<0$, but it is wrong if $\limsup_{x\to-\infty}u_{0}(x)<\theta_{1}$. To overcome this, we only need to wait for a period of time, say $\tilde{T}=\tilde{T}(u_{0})>0$, so that $\liminf_{x\to-\infty}u(t_{0}+\tilde{T},x;t_{0})>\theta_{1}$. Then, at $t_{0}+\tilde{T}$, we still have limit $\lim_{x\to\infty}\frac{u(t_{0}+\tilde{T},x;t_{0})}{u(t_{0}+\tilde{T},x)}=1$ due to Lemma \[lem-asymptotic-at-infty\], which ensures $$(1-\ep_{*})u(t_{0}+\tilde{T},x+\xi_{*}-C\ep_{*})\leq u(t_{0}+\tilde{T},x;t_{0}), \quad x\in\R$$ for some $\ep_{*}\in(0,1-\theta_{1})$ and $\xi_{*}<0$. Applying Proposition \[prop-sub-sup-solution\], we then see that $\tilde{\Xi}\neq\emptyset$. The rest of the proof follows along the same line. This completes the proof. Proof of Lemma \[lem-asymptotic-at-infty\] {#subsec-proof-lemm-1} ------------------------------------------ Recall $u(t,x)=u^{\ka}(t,x)$ and $X(t)=c^{\ka}(t)$ for some fixed $\ka\in(0,\ka_{0}]$. Fix $\ka_{*}\in(\ka_{0},\inf_{t\in\R}\ka_{0}(t)]$, where $\ka_{0}(t)$ is given in . We prove the lemma within two steps. [**Step 1.**]{} We prove . Fix any $\eta>0$. We first prove the upper bound for $u(t,x;t_{0})$. Since $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t_{0},x)}=1$ by assumption and $\lim_{x\to\infty}\frac{u(t_{0},x+X(t_{0}))}{e^{-\ka x}}=1$ by Theorem \[thm-tf\]$\rm(ii)$, we can find some $\de=\de(\eta)\gg1$ such that $$u_{0}(x+\eta)\leq e^{-\ka(x-X(t_{0}))}+\de e^{-\ka_{*}(x-X(t_{0}))},\quad x\in\R.$$ Recall that $e^{-\ka(x-X(t))}$ is a solution of $u_{t}=J\ast u-u+a(t)u$. Setting $$v(t,x)=e^{-\ka(x-X(t))}+\de e^{-\ka_{*}(x-X(t))},\quad x\in\R,\quad t\geq t_{0},$$ we readily check that $$v_{t}-[J\ast v-v]-a(t)v=\ka_{*}\bigg[\frac{\int_{\R}J(y)e^{\ka y}dy-1+a(t)}{\ka}-\frac{\int_{\R}J(y)e^{\ka_{*}y}dy-1+a(t)}{\ka_{*}}\bigg].$$ Since the function $\zeta\mapsto\frac{\int_{\R}J(y)e^{\zeta y}dy-1+a(t)}{\zeta}$ is convex in $\zeta$ and decreasing on $(0,\ka_{0}(t))$, and $\ka\leq\ka_{0}<\ka_{*}\leq\inf_{t\in\R}\ka_{0}(t)$, we have $$\frac{\int_{\R}J(y)e^{\ka y}dy-1+a(t)}{\ka}\geq\frac{\int_{\R}J(y)e^{\ka_{*}y}dy-1+a(t)}{\ka_{*}},\quad t\in\R.$$ Hence, $v_{t}-[J\ast v-v]-a(t)v\geq0$. In particular, $v(t,x)$ is a super-solution of . It then follows from comparison principle and the space homogeneity of that $u(t,x+\eta;t_{0})\leq v(t,x)$ for all $x\in\R$ and $t\geq t_{0}$, that is, $$\frac{u(t,x+X(t);t_{0})}{e^{-\ka(x-\eta)}}\leq1+\de e^{-(\ka_{*}-\ka)(x-\eta)},\quad x\in\R$$ for $t\geq t_{0}$, which, together with $\lim_{x\to\infty}\frac{u(t,x+X(t))}{e^{-\ka x}}=1$ by Theorem \[thm-tf\]$\rm(ii)$, lead to $$\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t)-\eta)}\leq(1+\de e^{-(\ka_{*}-\ka)(x-\eta)})(1+\zeta(x-\eta)),\quad x\in\R$$ for $t\geq t_{0}$, where $\zeta(x)\geq0$ satisfies $\zeta(x)\to0$ as $x\to\infty$. It then follows that $$\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t)-2\eta)}\leq\frac{u(t,x+X(t)-\eta)}{u(t,x+X(t)-2\eta)}(1+\de e^{-(\ka_{*}-\ka)(x-\eta)})(1+\zeta(x-y)).$$ Since $$\begin{split} \frac{u(t,x+X(t)-\eta)}{u(t,x+X(t)-2\eta)}&=\frac{u(t,x+X(t)-\eta)}{e^{-\ka(x-\eta)}}\frac{e^{-\ka(x-\eta)}}{e^{-\ka(x-2\eta)}}\frac{e^{-\ka(x-2\eta)}}{u(t,x+X(t)-2\eta)}\\ &\to e^{-\ka\eta}<1\quad\text{as}\quad x\to\infty, \end{split}$$ and $\eta>0$ is arbitrary, the upper bound for $u(t,x;t_{0})$ follows. We now prove the lower bound for $u(t,x;t_{0})$. Since $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t_{0},x)}=1$ by assumption and $\lim_{x\to\infty}\frac{u(t_{0},x)}{h(e^{-\ka(x-X(t_{0}))})}=1$ by , we can find some $\de=\de(\ep)\gg1$ such that $$u_{0}(x-\eta)\geq h(e^{-\ka(x-X(t_{0}))})-\de e^{-\ka_{*}(x-X(t_{0}))},\quad x\in\R.$$ Setting $$v_{1}(t,x)=h(e^{-\ka(x-X(t))})\quad\text{and}\quad v_{2}(t,x)=e^{-\ka_{*}(x-X(t))}$$ for $x\in\R$ and $t\geq t_{0}$. By Proposition \[prop-subsol\], $v_{1}(t,x)$ is a sub-solution of $u_{t}=J\ast u-u+a(t)g(u)$, i.e., $(v_{1})_{t}\leq J\ast v_{1}-v_{1}+a(t)g(v_{1})$. Arguing as in the proof of the upper bound above, we see that $$\begin{split} &(v_{2})_{t}-[J\ast v_{2}-v_{2}]-a(t)v_{2}\\ &\quad\quad=\ka_{*}\bigg[\frac{\int_{\R}J(y)e^{\ka y}dy-1+a(t)}{\ka}-\frac{\int_{\R}J(y)e^{\ka_{*}y}dy-1+a(t)}{\ka_{*}}\bigg]\geq0. \end{split}$$ It then follows that $v(t,x)=v_{1}(t,x)-\de v_{2}(t,x)$ satisfies $$\begin{split} v_{t}-[J\ast v-v]&=\big((v_{1})_{t}-[J\ast v_{1}-v_{1}]\big)-\de\big((v_{2})_{t}-[J\ast v_{2}-v_{2}]\big)\\ &\leq a(t)g(v_{1})-\de a(t)v_{2}\leq a(t)g(v), \end{split}$$ where we used $g(v_{1})-g(v)=\de v_{2}g_{u}(v_{*})\leq\de v_{2}$, since $g_{u}\leq1$ (note that it’s safe to extend $g$ to $(-\infty,0)$ so that $g_{u}\leq1$ on this interval). Since $g(u)\leq f(t,u)$, we find $v_{t}\leq J\ast v-v+f(t,u)$. Then, by comparison principle, $u(t,x-\eta;t_{0})\geq v(t,x)$ for all $x\in\R$ and $t\geq t_{0}$, that is, $$\frac{u(t,x+X(t);t_{0})}{h(e^{-\ka(x+\eta)})}\geq1-\de\frac{e^{-\ka_{*}(x+\ep)}}{h(e^{-\ka(x+\eta)})},\quad x\in\R$$ for $t\geq t_{0}$, which, together with , lead to $$\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t)+\eta)}\geq\bigg(1-\de\frac{e^{-\ka_{*}(x+\eta)}}{h(e^{-\ka(x+\eta)})}\bigg)(1-\zeta(x+\eta)),\quad x\in\R$$ for all $t\geq t_{0}$, where $\zeta(x)$ satisfies $\zeta(x)\to0$ as $x\to\infty$. Since $\ka_{*}>\ka$ and $\lim_{x\to\infty}\frac{e^{-\ka x}}{h(e^{-\ka x})}=1$, we have $\lim_{x\to\infty}\frac{e^{-\ka_{*}x}}{h(e^{-\ka x})}=0$. Then, a similar argument as in the proof of the upper bound gives the lower bound for $u(t,x;t_{0})$. This finishes the proof of . [**Step 2.**]{} We prove the equivalence between and . Suppose first that holds. Then, $$\frac{u(t,x+X(t)+\eta)}{u(t,x+X(t))}\leq\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}\leq\frac{u(t,x+X(t)-\eta)}{u(t,x+X(t))},\,\, x\geq M,\,\, t\geq t_{0}.$$ We see that $$\begin{split} \frac{u(t,x+X(t)-\eta)}{u(t,x+X(t))}&=1-\frac{u_{x}(t,x+X(t)-\eta_{*})}{u(t,x+X(t)-\eta_{*})}\frac{u(t,x+X(t)-\eta_{*})}{u(t,x+X(t))}\eta\\ &\leq1+\bigg[\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}\bigg]\frac{e^{\ka|\eta_{*}|}}{h(1)}\eta, \end{split}$$ where $\eta_{*}\in[0,\eta]$ and we used . Similarly, we argue $$\frac{u(t,x+X(t)+\eta)}{u(t,x+X(t))}\geq1-\bigg[\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}\bigg]\frac{e^{\ka|\eta_{*}|}}{h(1)}\eta.$$ Setting $\tilde{C}=\frac{1}{h(1)}\big[\sup_{(t,x)\in\R\times\R}\frac{|u_{x}(t,x)|}{u(t,x)}\big]$, we obtain $$1-\tilde{C}e^{\ka|\eta_{*}|}\eta\leq\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}\leq1+\tilde{C}e^{\ka|\eta_{*}|}\eta,\quad x\geq M,\quad t\geq t_{0}.$$ Setting $x\to\infty$, we find $$1-\tilde{C}e^{\ka|\eta_{*}|}\eta\leq\liminf_{x\to\infty}\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}\leq\limsup_{x\to\infty}\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}\leq1+\tilde{C}e^{\ka|\eta_{*}|}\eta,\quad t\geq t_{0},$$ which leads to since $\eta>0$ is arbitrary. Conversely, suppose holds. We only show the upper bound for $u(t,x;t_{0})$ in ; the lower bound can be verified similarly. By , for any small $\eta>0$, there exists $M(\eta)>0$ such that $$\frac{u(t,x+X(t);t_{0})}{u(t,x+X(t))}\leq\frac{1}{1-\frac{\ka}{8}\eta e^{-\ka\eta}},\quad x\geq M(\eta),\quad t\geq t_{0}.$$ We claim that $$\label{claim-equiv} \begin{cases} \exists M_{1}(\eta)>0\,\,\text{s.t.}\,\, u(t,x+X(t))\leq(1-\frac{\ka}{8}\eta e^{-\ka\eta})u(t,x+X(t)-\eta)\\ \quad\quad \text{for all}\,\, x\geq M_{1}(\eta)\,\,\text{and}\,\, t\geq t_{0}. \end{cases}$$ Clearly, the result then follows from . Hence, it remains to show . We see that $$\begin{split} &\frac{u(t,x+X(t))}{u(t,x+X(t)-\eta)}\\ &\quad\quad=1-\bigg[\frac{-u_{x}(t,x+X(t)-\eta_{*})}{u(t,x+X(t)-\eta_{*})}\frac{u(t,x+X(t)-\eta_{*})}{e^{-\ka(x-\eta_{*})}}\frac{e^{-\ka(x-\eta_{*})}}{e^{-\ka(x-\eta)}}\frac{e^{-\ka(x-\eta)}}{u(t,x+X(t)-\eta)}\eta\bigg], \end{split}$$ where $\eta_{*}\in[0,\eta]$. Clearly, $\frac{e^{-\ka(x-\eta_{*})}}{e^{-\ka(x-\eta)}}=e^{-\ka(\eta-\eta_{*})}\geq e^{-\ka\eta}$. Moreover, by Theorem \[thm-tf\], we can find some $M_{1}(\eta)>0$ such that for all $x\geq M_{1}(\eta)$ and $t\geq t_{0}$, there hold the following $$\frac{-u_{x}(t,x+X(t)-\eta_{*})}{u(t,x+X(t)-\eta_{*})}\geq\frac{\ka}{2},\quad\frac{u(t,x+X(t)-\eta_{*})}{e^{-\ka(x-\eta_{*})}}\geq\frac{1}{2}\quad\text{and}\quad\frac{e^{-\ka(x-\eta)}}{u(t,x+X(t)-\eta)}\geq\frac{1}{2}.$$ It follows that $\frac{u(t,x+X(t))}{u(t,x+X(t)-\eta)}\leq1-\frac{\ka}{8}\eta e^{-\ka\eta}$ for all $x\geq M_{1}(\eta)$ and $t\geq t_{0}$, that is, is true, and the proof is complete. Stability: general cases {#subsec-stability-general} ------------------------ We prove the following generalizations of Theorem \[thm-asympt-stability\]. \[cor-stability-general\] Assume (H1)-(H3). Let $\ka_{0}$ be as in Theorem \[thm-tf\]. For $\ka\in(0,\ka_{0}]$, let $u^{\ka}(t,x)$ be the transition front in Theorem \[thm-tf\]. Let $u_{0}:\R\to[0,1]$ be uniformly continuous and satisfy $\liminf_{x\to-\infty}u_{0}>0$ and $\lim_{x\to\infty}\frac{u_{0}(x)}{u^{\ka}(t_{0},x)}=\la$ for some $t_{0}\in\R$ and $\la\in(0,\infty)$. Then, there holds the limit $$\lim_{t\to\infty}\sup_{x\in\R}\bigg|\frac{u(t,x;t_{0},u_{0})}{u^{\ka}(t,x-\frac{\ln\la}{\ka})}-1\bigg|=0.$$ Write $u(t,x)=u^{\ka}(t,x)$. To apply Theorem \[thm-asympt-stability\], we only need to show $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t_{0},x-\frac{\ln\la}{\ka})}=1$, which follows from $$\frac{u_{0}(x)}{u(t_{0},x-\frac{\ln\la}{\ka})}=\frac{u_{0}(x)}{u(t_{0},x)}\frac{u(t_{0},x)}{e^{-\ka(x-X(t_{0}))}}\frac{e^{-\ka(x-X(t_{0}))}}{e^{-\ka(x-X(t_{0})-\frac{\ln\la}{\ka})}}\frac{e^{-\ka(x-X(t_{0})-\frac{\ln\la}{\ka})}}{u(t_{0},x-\frac{\ln\la}{\ka})},$$ and $$\lim_{x\to\infty}\frac{u(t_{0},x)}{e^{-\ka(x-X(t_{0}))}}=1=\lim_{x\to\infty}\frac{e^{-\ka(x-X(t_{0})-\frac{\ln\la}{\ka})}}{u(t_{0},x-\frac{\ln\la}{\ka})}$$ thanks to Theorem \[thm-tf\](ii). \[cor-stability-more-general\] Assume (H1)-(H3). Let $\ka_{0}$ be as in Theorem \[thm-tf\]. For $\ka\in(0,\ka_{0}]$, let $u^{\ka}(t,x)$ be the transition front in Theorem \[thm-tf\]. Let $u_{0}:\R\to[0,1]$ be uniformly continuous and satisfy $\liminf_{x\to-\infty}u_{0}>0$ and $\lim_{x\to\infty}\frac{u_{0}(x)}{e^{-\ka x}}=\la$ for some $\la>0$. Then, there exists $t_{0}\in\R$ such that $$\lim_{t\to\infty}\sup_{x\in\R}\bigg|\frac{u(t,x;t_{0},u_{0})}{u^{\ka}(t,x)}-1\bigg|=0.$$ Write $u(t,x)=u^{\ka}(t,x)$. By assumption, we see that for any $t\in\R$, there holds the limit $\lim_{x\to\infty}\frac{u_{0}(x)}{e^{-\ka(x-X(t))}}=\la e^{-\ka X(t)}$. Then, by Theorem \[thm-tf\](ii), for any $t\in\R$, we have the limit $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t,x)}=\la e^{-\ka X(t)}$. Since $X(t)$ is continuous, increasing and satisfies $\lim_{t\to\pm\infty}X(t)=\pm\infty$, there exists a unique $t_{0}\in\R$ such that $\la e^{-\ka X(t_{0})}=1$, which implies that $\lim_{x\to\infty}\frac{u_{0}(x)}{u(t_{0},x)}=1$. We then apply Theorem \[thm-asympt-stability\] to conclude the result. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank anonymous referees for carefully reading the manuscript and providing useful suggestions. [10]{} P. W. Bates, P. C. Fife, X. Ren and X. 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--- abstract: 'We report the field-orientation dependent specific heat of the spin-triplet superconductor Sr$_2$RuO$_4$ under the magnetic field aligned parallel to the RuO$_2$ planes with high accuracy. Below about 0.3 K, striking 4-fold oscillations of the density of states reflecting the superconducting gap structure have been resolved for the first time. We also obtained strong evidence of multi-band superconductivity and concluded that the superconducting gap in the active band, responsible for the superconducting instability, is modulated with a minimum along the \[100\] direction.' author: - 'K. Deguchi' - 'Z. Q. Mao' - 'H. Yaguchi' - 'Y. Maeno' title: | Gap Structure of the Spin-Triplet Superconductor Sr$_2$RuO$_4$\ Determined from the Field-Orientation Dependence of Specific Heat --- Since the discovery of its superconductivity [@discovery], the layered ruthenate Sr$_2$RuO$_4$ has attracted a keen interest in the physics community [@Review]. The superconductivity of Sr$_2$RuO$_4$ has pronounced unconventional features such as: the invariance of the spin susceptibility across its superconducting (SC) transition temperature $T_{\rm c}$ [@NMR; @polarised], appearance of spontaneous internal field [@muSR], evidence for two-component order parameter [@small] and absence of a Hebel-Slichter peak [@NMR-HS]. These features are coherently understood in terms of spin-triplet superconductivity with the vector order parameter $\bm{d}(\bm{k})=\hat {\bm{z}}{\it \Delta}_{0}(k_x + ik_y)$, representing the spin state $S_z = 0$ and the orbital wave function with $L_z = +1$, called a chiral [*p*]{}-wave state. The above vector order parameter leads to the gap ${\it \Delta}(\bm{k})={\it \Delta}_{0}(k_x^2 + k_y^2)^{1/2}$, which is isotropic because of the quasi-two dimensionality of the Fermi surface consisting of three cylindrical sheets [@dHvA]. However, a number of experimental results [@Cp; @NQR; @penetration; @kappa; @USA] revealed the power-law temperature dependence of quasiparticle (QP) excitations, which suggest lines of nodes or node-like structures in the SC gap. There have been many theoretical attempts (anisotropic [*p*]{}-wave or [*f*]{}-wave states) to resolve this controversy [@Gamma; @AlphaBeta; @Miyake; @dxy; @dx2-y2; @Horizontal]. Although all these models suggest a substantial gap anisotropy, magnetothermal conductivity measurements with the applied field rotated within the RuO$_2$ plane down to 0.35 K revealed little anisotropy [@oscillationT; @oscillationI]. To explain those experimental facts as well as the mechanism of the spin-triplet superconductivity, several theories [@HorizontalODS; @VerticalODS], taking the orbital dependent superconductivity (ODS) into account [@Agterberg], have been proposed. In these models, there are active and passive bands to the superconductivity: the SC instability originates from the active band with a large gap amplitude; pair hopping across active to passive bands leads to a small gap in the passive bands. The gap structure with horizontal lines of nodes [@HorizontalODS] or strong in-plane anisotropy [@VerticalODS] in the passive bands was proposed. In order to identify the mechanism of the spin-triplet superconductivity, the determination of the gap structure in the active band is currently of prime importance. The field-orientation dependent specific heat is a direct measure of the QP density of states (DOS) and thus a powerful probe of the SC gap structure [@Vekhter; @WonO; @Miranovic; @Park]. In this Letter, we report high precision experiments of the specific heat as a function of the angle between the crystallographic axes and the magnetic field $\bm{H}$ within the RuO$_2$ plane. We reveal that the SC state of Sr$_2$RuO$_4$ has a band-dependent gap and that the gap of its active SC band has strong in-plane anisotropy with a minimum along the \[100\] direction, as illustrated in Fig. \[fig:3D\]. -0.3cm ![\[fig:3D\] Left: Electronic specific heat divided by temperature $C_{\rm e}/T$ for $\bm{H} \parallel [100]$, as a function of field strength and temperature. A contour plot is shown on the bottom $H-T$ plane, with the same color scale as the 3D plot. Right: Superconducting gap structure for the active band $\gamma$ deduced from the present study, corresponding to $\bm{d}(\bm{k})=\hat {\bm{z}}{\it \Delta}_{0}({\rm sin}ak_x + i{\rm sin}ak_y)$.](Fig1.eps){width="8.6cm"} -0.5cm -0.1cm Single crystals of Sr$_2$RuO$_4$ were grown by a floating-zone method in an infrared image furnace [@growth]. After specific-heat measurements on two crystals to confirm the reproducibility of salient characteristics such as a double SC transition [@Double], the sample with $T_{\rm c} = 1.48$ K, close to the estimated value for an impurity and defect free specimen ($T_{\rm c0}=1.50$ K) [@impurity], was chosen for detailed study. This crystal was cut and cleaved from the single crystalline rod, to a size of $2.8 \times 4.8 \;{\rm mm}^{2}$ in the $ab$-plane and $0.50 \;{\rm mm}$ along the $c$-axis. The side of the crystal was intentionally misaligned from the \[110\] axis by $16^{\circ}$. The field-orientation dependence of the specific heat was measured by a relaxation method with a dilution refrigerator. Since a slight field misalignment causes 2-fold anisotropy of the specific heat due to the large $H_{\rm c2}$ anisotropy $(H_{{\rm c2}\parallel ab}/H_{{\rm c2}\parallel c}\approx 20)$ [@Double], the rotation of the field $\bm{H}$ within the RuO$_2$ plane with high accuracy is very important. For this experiment, we built a measurement system consisting of two orthogonally arranged SC magnets [@VectorMag] to control the polar angle of the field $\bm{H}$. The two SC magnets are installed in a dewar seating on a mechanical rotating stage to control the azimuthal angle. With the dilution refrigerator fixed, we can rotate the field $\bm{H}$ continuously within the RuO$_2$ plane with a misalignment no greater than $0.01^{\circ}$ from the plane. The electronic specific heat $C_{\rm e}$ under the in-plane magnetic fields was obtained after subtraction of the phonon contribution with a Debye temperature of 410 K. The left panel of Fig. \[fig:3D\] shows $C_{\rm e}/T$ for the \[100\] field direction, as a function of field and temperature. The figure is constructed from data involving 13 temperature-sweeps and 11 field-sweeps. At low temperatures in zero field, power-law temperature dependence of $C_{\rm e}/T \propto T$ was observed, corresponding to the QPs excited from the line nodes or node-like structure in the gap. Now we focus on the field dependence of $C_{\rm e}/T$ at low temperature shown in Fig. \[fig:3D\] and Fig. \[fig:CHT\] (a). $C_{\rm e}/T$ increases sharply up to about 0.15 T and then gradually for higher fields. This unusual shoulder is naturally explained by the presence of two kinds of gaps [@Cp]. On the basis of the different orbital characters of the three Fermi surfaces ($\alpha, \beta,$ and $\gamma$) [@dHvA], the gap amplitudes ${\it \Delta}_{\alpha \beta }$ and ${\it \Delta}_{\gamma}$ are expected to be significantly different [@Agterberg]. The normalized DOS of those bands are $\frac{N_{\alpha \beta }}{N_{\rm total}} = 0.43$ and $\frac{N_{\gamma }}{N_{\rm total}} = 0.57$ [@Review]. Since the position of the shoulder in $C_{\rm e}/T$ corresponds well with the partial DOS of the $\alpha$ and $\beta$ bands, we conclude that the active band which has a robust SC gap in fields is the $\gamma$-band, mainly derived from the in-plane $d_{xy}$ orbital of Ru $4d$ electrons. Figures \[fig:CHT\] (a) and (b) show the field and temperature dependence of $C_{\rm e}/T$ under the in-plane magnetic fields $\bm{H} \parallel [100]$ and $\bm{H} \parallel [110]$ and indicate the existence of a slight in-plane anisotropy. In the mixed state, the QP energy spectrum is affected by the Doppler shift $\delta {\it \omega} = \hbar \bm{k} \cdot \bm{v}_s$, where $\bm{v}_s$ is the superfluid velocity around the vortices and $\hbar \bm{k}$ is the QP momentum. This energy shift gives rise to a finite DOS at the Fermi level in the case of $\delta {\it \omega}\geq {\it \Delta}(\bm{k})$ [@Volovik]. Since $\bm{v}_s \perp \bm{H}$, $\delta {\it \omega} = 0$ for $\bm{k} \parallel \bm{H}$. Thus the generation of nodal QPs is suppressed for $\bm{H} \parallel$ [*nodal*]{} directions and yields minima in $C_{\rm e}/T$ [@Vekhter; @WonO; @Miranovic]. ![\[fig:CHT\] (a), (b) Field and temperature dependence of $C_{\rm e}/T$ of Sr$_2$RuO$_4$ in magnetic fields parallel to the \[100\] (open circle) and \[110\] (closed circle) directions.](Fig2.eps){width="8cm"} -0.4cm -0.4cm -0.3cm ![\[fig:OSC\]The in-plane field-orientation dependence of normalized 4-fold component of the specific heat at several fields and temperatures. Only 1.45 T data is reduced to 1/3. The solid lines are fits with $f_{4}(\phi)$ given in the text.](Fig3.eps){width="7.2cm"} -0.5cm Figure \[fig:OSC\] shows the field-orientation dependence of the specific heat. The absence of a 2-fold oscillatory component in the raw data guarantees that the in-plane field alignment is accurate during the azimuthal-angle rotation. Thus $C_{\rm e}(T,H,\phi) $ can be decomposed into $\phi$-independent and 4-fold oscillatory terms, where the in-plane azimuthal field angle $\phi$ is defined from the \[100\] direction: $C_{\rm e}(T,H,\phi) = C_{\rm 0}(T,H) + C_{\rm 4}(T,H,\phi) $. $C_{\rm 4}(T,H,\phi)/C_{\rm N}$ is the normalized angular variation term, where $C_{\rm N}$ is the electronic specific heat in the normal state: $C_{\rm N} = \gamma_{\rm N}T$ with $\gamma_{\rm N} = 37.8$ mJ/K$^{2}$mol. There is no discernible angular variation in the normal state ($\mu_{0}H = 1.7$ T $> \mu_{0}H_{\rm c2}$); possibilities of angular variation originating from experimental setup or other extrinsic contributions are excluded. For fields near $H_{\rm c2}$ ($1.2$ T $\leq \mu_{0}H \leq 1.45$ T), a sinusoidal 4-fold angular variation is observed : $C_{\rm 4}(\phi) \propto f_{\rm 4}(\phi)=-{\rm cos}4\phi$. This is consistent with the in-plane sinusoidal anisotropy of $H_{\rm c2}$ with the maximum in the \[110\] direction [@Mao; @oscillationI]: $C_{{\rm 4}} = \frac{H_{{\rm c2} \parallel [110]}-H_{{\rm c2} \parallel [100]}}{H_{{\rm c2} \parallel [110]}+H_{{\rm c2} \parallel [100]}}\frac{{\rm d}C_{\rm e}}{{\rm d}H}H{\rm cos}4\phi$. Since $H_{\rm c2}$ decreases with increasing $T$, the oscillation amplitude at 1.3 T increases strongly at 0.51 K. For $\mu_{0}H < 1.2$ T, however, a [*non-sinusoidal*]{} 4-fold angular variation approximated as $C_{\rm 4}(\phi) \propto f_{\rm 4}(\phi)=2|{\rm sin}2\phi| -1$ is observed. Importantly, a phase inversion in $C_{\rm 4}(\phi)$ occurs across about $\mu_{0}H = 1.2$ T: $C_{\rm 4}(\phi)$ takes minima at $\phi = \frac{\pi}{2}n$ ($\phi = \frac{\pi}{4} + \frac{\pi}{2}n$, $n$: integer) for $\mu_{0}H < 1.2$ T ($\mu_{0}H \geq 1.2$ T), and thus the angular variation for $\mu_{0}H < 1.2$ T cannot be due to the in-plane $H_{\rm c2}$ anisotropy. Therefore we conclude that the [*non-sinusoidal*]{} 4-fold oscillations originate from the SC gap structure. This result does not contradict the previous measurements of the magnetothermal conductivity down to 0.35 K  [@oscillationT; @oscillationI], which reported little in-plane anisotropy, because these clear oscillations emerge only at lower $T$ $(T/T_{\rm c}\leq 0.2)$. For the field range $0.15$ T $< \mu_{0}H < 1.2$ T, where the QPs in the active band $\gamma$ are the dominant source of in-plane anisotropy, we first deduce the existence of a node or gap minimum along the \[100\] direction, because $C_{\rm e}$ takes a minimum. In addition, we found that the 4-fold oscillations have a [*non-sinusoidal*]{} form, approximated as $C_{\rm 4}(\phi) \propto 2|{\rm sin}2\phi| -1$, since cusp-like features are clearly seen at the minima ($\phi = \frac{\pi}{2}n$). Strong $k_z$ dependence of the gap function would enhance the QP excitations even if $\bm{H}$ is parallel to the nodal direction, so that the cusp-like features would have been strongly suppressed [@Vekhter]. -0.7cm \# $\bm{d}(\bm{k})$ direction of node or ${\it \Delta}_{\rm min}$ Ref. ---- ----------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------- --------------- 1 $\hat {\bm{z}}{\it \Delta}_{0}({\rm sin}ak_x + i{\rm sin}ak_y)$ \[100\] tiny gap [@Miyake] 2 $\hat {\bm{z}}{\it \Delta}_{0}k_xk_y(k_x + ik_y)$ \[100\] nodes [@dxy] 3 $\hat {\bm{z}}{\it \Delta}_{0}(k_x^2-k_y^2)(k_x + ik_y)$ \[110\] nodes [@dx2-y2] 4 $\left\{\begin{array}{cc} \hat {\bm{z}}{\it \Delta}_{0}(k_x + ik_y){\rm cos}ck_z\\\hat {\bm{z}}{\it \Delta}_{0}k_z(k_x + ik_y)^2\end{array}\right.$ horizontal nodes [@Horizontal] : \[tab:table1\]The classified order parameters with the typical gap structures for Sr$_2$RuO$_4$. Most of the proposed gap structures can be classified into four groups as summarized in Table \[tab:table1\]. \#1 and \#2 provide the direction of the gap minima consistent with our observation. To distinguish between \#1 with gap minima and \#2 with nodes, we examine the specific heat jump ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c}$ at $T_{\rm c}$ in zero field. The jump originates mainly from the active band with large ${\it \Delta}$ because of ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c} \propto \partial {\it \Delta}^2/\partial T|_{T_{\rm c}}$. We estimate the contribution of ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c}$ from the active band for the gap structures \#1 and \#2: ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c} = (1.22$ to $1.07) \times 0.57 = 0.70$ to $0.61$ with the gap minimum (${\it \Delta}_{\rm min}/{\it \Delta}_{\rm max} = 1/2$ to $1/4$) [@Miyake], while ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c} = 0.75 \times 0.57 = 0.42$ with the line nodes [@dxy]. From the experimental result ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c} = 0.73$ in Fig. \[fig:3D\] and the estimated additional contribution from the passive bands ${\it \Delta}C_{\rm e}/\gamma_{\rm N}T_{\rm c} \sim 0.04$ [@VerticalODS], $\bm{d}(\bm{k})=\hat {\bm{z}}{\it \Delta}_{0}({\rm sin}ak_x + i{\rm sin}ak_y)$ with the gap minimum is promising for the active band. To facilitate a comparison with theories, although they are presently available only for line-node gaps [@WonO; @Miranovic], we decomposed $C_{\rm e}$ into two parts: $C_{\rm 0}(T,0)$ due to the thermally excited QPs and ${\it \Delta}C_{\rm 0}(T,H)$ due to the field induced QPs, consisting of an isotropic component and a 4-fold anisotropic component $A_{\rm 4}(T,H)$: $C_{\rm e}(T,H,\phi) = C_{\rm 0}(T,0) + {\it \Delta}C_{\rm 0}(T,H)[1 + A_{\rm 4}(T,H)f_{\rm 4}(\phi)],$ $A_{\rm 4}(T,H)f_{\rm 4}(\phi) = \frac{C_{\rm 4}(T,H,\phi)}{{\it \Delta}C_{\rm 0}(T,H)},$ where $f_{\rm 4}$ was defined previously for low- and high-field ranges. Figures \[fig:C4HT\] (a) and (b) show the field and temperature dependence of $A_{\rm 4}$. The field dependence of $A_{\rm 4}$ with a maximum of 4% anisotropy at 0.31 K shows a monotone decrease from the delocalized-QP dominant region at low fields to the $H_{\rm c2}$-anisotropy dominant region at high fields. The temperature dependence of $A_{\rm 4}$ with 3% anisotropy at 0.9 T shows a smooth decrease with increasing temperature. These results are in semi-quantitative agreement with recent theories [@WonO; @Miranovic] which predict 4 to 1.5% anisotropy from gap structures with vertical line nodes. -0.2cm ![\[fig:C4HT\](a), (b) Field and temperature dependence of the 4-fold anisotropy $A_{\rm 4}$ in the specific heat. The points are evaluated from the fitting to the oscillatory data in Fig. \[fig:OSC\], while the lines from the difference in $C_{\rm e}$ between $\bm{H} \parallel [110]$ and $\bm{H} \parallel [100]$ in Fig. \[fig:CHT\]. Two methods yield consistent results.](Fig4.eps){width="8cm"} -0.5cm In contrast to the theoretical prediction [@Miranovic], however, at combined low fields ($\mu_{0}H \leq 0.15$ T) and low temperatures ($T \leq 0.3$ K) where QPs on both the active and passive bands are important for the anisotropy, $A_{\rm 4}$ rapidly decreases. This steep reduction of $A_{\rm 4}$ is primarily attributable to the non-zero gap minima ${\it \Delta}_{\rm min}$ of the active $\gamma$ band (\#1). In fact, at the lowest temperatures the 4-fold oscillations are suppressed below a threshold field of about 0.3 T, which should correspond to $\delta {\it \omega} = {\it \Delta}_{\rm min}$. However, even at 0.15 T below the threshold, increasing temperature leads to an increase in $A_{\rm 4}$ only up to about 0.25 K (Fig. \[fig:C4HT\] (b)). This behavior is at least in part explained by including contributions of both field and temperature in the QP excitations, but a more quantitative theoretical analysis is needed to examine this possibility. Let us finally discuss the roles of the passive bands in the oscillations in $C_{\rm e}$. While the present experimental study has resolved the directions of gap minima in the $\gamma$ band, there still remain two types of possibilities for the passive bands $\alpha$ and $\beta$ to account for the power-law QP excitations at low temperatures: (A) horizontal line nodes (\#4 in Table \[tab:table1\]) [@HorizontalODS] and (B) vertical gap minima along the \[110\] directions [@VerticalODS]. The gaps (A) in the passive bands will not contribute to any oscillatory component whether they are fully developed or filled with QPs induced by $H$ and/or $T$. Thus in this case the rather complex $H$ and $T$ dependence of $A_{\rm 4}$ needs to be accounted for solely by the gap structure of the active band. On the other hand, the gaps (B) will give rise to 4-fold oscillations which are out of phase with those originating from the active band, so that the oscillations will be additionally suppressed. Since Fig. 4 (b) shows that $A_{\rm 4}$ at 0.15 T decreases steeply with decreasing temperature in the temperature range where QP excitations strongly reflect the gap structure of the passive bands, the observed steep reduction of $A_{\rm 4}$ may be a consequence of an additional compensation by the passive bands. In conclusion, we have for the first time revealed the in-plane anisotropy in the SC gap of the spin-triplet superconductor Sr$_2$RuO$_4$, from the field-orientation dependence of the specific heat at low temperatures. We identified the multi-band superconductivity with the active band $\gamma$, which has a modulated SC gap with a minimum along the \[100\] direction with little interlayer dispersion. This gap structure is in good correspondence with the $p$-wave order parameter $\bm{d}(\bm{k})=\hat {\bm{z}}{\it \Delta}_{0}({\rm sin}ak_x + i{\rm sin}ak_y)$. 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--- abstract: 'We prove that a graph $C^*$-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in $K$-theory. We prove that a similar classification also holds for a graph $C^*$-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph $C^*$-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph $C^*$-algebra is stable.' address: - | Department of Mathematical Sciences\ University of Copenhagen\ Universitetsparken 5\ 2100 Copenhagen Ø\ Denmark - | Department of Mathematics\ University of Houston\ Houston, TX 77204-3008\ USA author: - Søren Eilers - Mark Tomforde title: 'On the classification of nonsimple graph $C^*$-algebras' --- Introduction ============ The classification program for $C^*$-algebras has for the most part progressed independently for the classes of infinite and finite $C^*$-algebras. Great strides have been made in this program for each of these classes. In the finite case, Elliott’s Theorem classifies all AF-algebras up to stable isomorphism by the ordered $K_0$-group. In the infinite case, there are a number of results for purely infinite $C^*$-algebras. The Kirchberg-Phillips Theorem classifies certain simple purely infinite $C^*$-algebras up to stable isomorphism by the $K_0$-group together with the $K_1$-group. For nonsimple purely infinite $C^*$-algebras many partial results have been obtained: Rørdam has shown that certain purely infinite $C^*$-algebras containing exactly one proper nontrivial ideal are classified up to stable isomorphism by the associated six-term exact sequence of $K$-groups [@Ror:ceccsteskt], Restorff has shown that nonsimple Cuntz-Krieger algebras satisfying Condition (II) are classified up to stable isomorphism by their filtrated $K$-theory [@gr:cckasi Theorem 4.2], and Meyer and Nest have shown that certain purely infinite $C^*$-algebras with a linear ideal lattice are classified up to stable isomorphism by their filtrated $K$-theory [@mn:cotpfkt Theorem 4.14]. However, in all of these situations the nonsimple $C^*$-algebras that are classified have the property that they are either AF-algebras or purely infinite, and consequently all of their ideals and quotients are of the same type. Recently, the first named author with Restorff and Ruiz have provided a framework for classifying nonsimple $C^*$-algebras that are not necessarily AF-algebras or purely infinite $C^*$-algebras. In particular, these authors have shown in [@segrer:cecc] that certain extensions of classifiable $C^*$-algebras may be classified up to stable isomorphism by their associated six-term exact sequence in $K$-theory. This has allowed for the classification of certain nonsimple $C^*$-algebras in which there are ideals and quotients of mixed type (some finite and some infinite). In this paper we consider the classification of nonsimple graph $C^*$-algebras. Simple graph $C^*$-algebras are known to be either AF-algebras or purely infinite algebras, and thus are classified by their $K$-groups according to either Elliott’s Theorem or the Kirchberg-Phillips Theorem. Therefore, we begin by considering nonsimple graph $C^*$-algebras with exactly one proper nontrivial ideal. These $C^*$-algebras will be extensions of simple $C^*$-algebras that are AF or purely infinite by other simple $C^*$-algebras that are AF or purely infinite — with mixing of the types allowed. These nonsimple graph $C^*$-algebras are similar to the extensions considered in [@segrer:cecc], however, the results of [@segrer:cecc] do not apply directly. Instead, we must do a fair bit of work, using the techniques from the theory of graph $C^*$-algebras, to show that the machinery of [@segrer:cecc] can be used to classify these extensions; it is verifying the requirement of **fullness** that is most difficult in this context. Ultimately, however, we are able to show that a graph $C^*$-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by the six-term exact sequence in $K$-theory of the corresponding extension. Additionally, we are able to show that a graph $C^*$-algebra with a largest proper ideal that is an AF-algebra is also classified up to stable isomorphism by the six-term exact sequence in $K$-theory of the corresponding extension. It is also worthwhile to note that the extensions of graph $C^*$-algebras classified in this paper constitute a very large class. Every AF-algebra is stably isomorphic to a graph $C^*$-algebra, and every Kirchberg algebra with free $K_1$-group is stably isomorphic to a a graph $C^*$-algebra. Thus the extensions we consider comprise a wide variety of extensions of AF-algebras (respectively, purely infinite algebras) by purely infinite algebras (respectively, AF-algebras). While there is little hope to generalize the methods in [@segrer:cecc] to general (even finite) ideal lattices in a context covering all graph $C^*$-algebras, the classifications we obtain in this paper suggest that a complete classification of graph $C^*$-algebras generalizing the Cuntz-Krieger case solved in [@gr:cckasi] may be possible by other methods. Such a result may involve generalizing the work of Boyle and Huang to graph $C^*$-algebras and mimicking the approach used by Restorff; or perhaps it may be accomplished by generalizing Kirchberg’s isomorphism theorem to allow for subquotients which are $AF$-algebras, and then in this special case (probably using the global vanishing of one connecting map of $K$-theory) overcoming the difficulties of projective dimension of the invariants exposed by Meyer and Nest. Neither of these approaches seem within immediate reach, but both appear plausible to be successful at some future stage. It is also an open problem, and possibly much less difficult, to establish isomorphism directly between unital graph $C^*$-algebras by keeping track of the class of the unit in the $K_0$-groups. We mention that the methods of [@grer:rccconiII] do not seem to generalize to this setting. This paper is organized as follows. In §\[not-conv-sec\] we establish notation and conventions for graph $C^*$-algebras and extensions. In §\[prelims-sec\] we derive a number of preliminary results for graph $C^*$-algebras with the goal of applying the methods of [@segrer:cecc]. In §\[class-sec\] we use our results from §\[prelims-sec\] and the results of [@segrer:cecc] to prove our two main theorems: In Theorem \[clas\] we show that if $A$ is a graph $C^*$-algebra with exactly one proper nontrivial ideal $I$, then $A$ is classified up to stable isomorphism by the six-term exact sequence in $K$-theory coming from the extension $0 \to I \to A \to A/I \to 0$. In Theorem \[clasimp\] we show that if $A$ is a graph $C^*$-algebra with a largest proper ideal $I$ that is an AF-algebra, then $A$ is classified up to stable isomorphism by the six-term exact sequence in $K$-theory coming from the extension $0 \to I \to A \to A/I \to 0$. In §\[ex-sec\] we consider a variety of examples, and also use our results to classify the stable isomorphism classes of the $C^*$-algebras of all graphs having exactly two vertices and satisfying Condition (K). (Be aware that although these graphs have only two vertices, the graphs are allowed to contain a finite or countably infinite number of edges.) We find that even for this small collection of graphs, the associated $C^*$-algebras fall into a variety of stable isomorphism classes, and there are quite a few cases to consider. We conclude in §\[stab-ideals-sec\] by proving that if $A$ is a graph $C^*$-algebra that is not a nonunital AF-algebra, and if $A$ contains a unique proper nontrivial ideal $I$, then $I$ is stable. Notation and conventions {#not-conv-sec} ======================== We establish some basic facts and notation for graph $C^*$-algebras and extensions. Notation and conventions for graph $C^*$-algebra ------------------------------------------------ A (directed) graph $E=(E^0, E^1, r, s)$ consists of a countable set $E^0$ of vertices, a countable set $E^1$ of edges, and maps $r,s: E^1 \rightarrow E^0$ identifying the range and source of each edge. A vertex $v \in E^0$ is called a *sink* if $|s^{-1}(v)|=0$, and $v$ is called an *infinite emitter* if $|s^{-1}(v)|=\infty$. A graph $E$ is said to be *row-finite* if it has no infinite emitters. If $v$ is either a sink or an infinite emitter, then we call $v$ a *singular vertex*. We write $E^0_\textnormal{sing}$ for the set of singular vertices. Vertices that are not singular vertices are called *regular vertices* and we write $E^0_\textnormal{reg}$ for the set of regular vertices. For any graph $E$, the *vertex matrix* is the $E^0 \times E^0$ matrix $A_E$ with $A_e(v,w) := | \{ e \in E^1 : s(e)= v \text{ and } r(e)=w \} |$. Note that the entries of $A_E$ are elements of $\{0, 1, 2, \ldots \} \cup \{ \infty \}$. If $E$ is a graph, a *Cuntz-Krieger $E$-family* is a set of mutually orthogonal projections $\{p_v : v \in E^0\}$ and a set of partial isometries $\{s_e : e \in E^1\}$ with orthogonal ranges which satisfy the *Cuntz-Krieger relations*: 1. $s_e^* s_e = p_{r(e)}$ for every $e \in E^1$; 2. $s_e s_e^* \leq p_{s(e)}$ for every $e \in E^1$; 3. $p_v = \sum_{s(e)=v} s_e s_e^*$ for every $v \in E^0$ that is not a singular vertex. The *graph algebra $C^*(E)$* is defined to be the $C^*$-algebra generated by a universal Cuntz-Krieger $E$-family. A *path* in $E$ is a sequence of edges $\alpha = \alpha_1 \alpha_2 \ldots \alpha_n$ with $r(\alpha_i) = s(\alpha_{i+1})$ for $1 \leq i < n$, and we say that $\alpha$ has length $|\alpha| = n$. We let $E^n$ denote the set of all paths of length $n$, and we let $E^* := \bigcup_{n=0}^\infty E^n$ denote the set of finite paths in $G$. Note that vertices are considered paths of length zero. The maps $r,s$ extend to $E^*$, and for $v,w \in E^0$ we write $v \geq w$ if there exists a path $\alpha \in E^*$ with $s(\alpha)=v$ and $r(\alpha) = w$. Also for a path $\alpha := \alpha_1 \ldots \alpha_n$ we define $s_\alpha := s_{\alpha_1} \ldots s_{\alpha_n}$, and for a vertex $v \in E^0$ we let $s_v := p_v$. It is a consequence of the Cuntz-Krieger relations that $C^*(E) = \overline{\textrm{span}} \{ s_\alpha s_\beta^* : \alpha, \beta \in E^* \text{ and } r(\alpha) = r(\beta)\}$. We say that a path $\alpha := \alpha_1 \ldots \alpha_n$ of length $1$ or greater is a *cycle* if $r(\alpha)=s(\alpha)$, and we call the vertex $s(\alpha)=r(\alpha)$ the *base point* of the cycle. A cycle is said to be *simple* if $s(\alpha_i) \neq s(\alpha_1)$ for all $1 < i \leq n$. The following is an important condition in the theory of graph $C^*$-algebras. $\text{ }$ **Condition (K)**: No vertex in $E$ is the base point of exactly one simple cycle; that is, every vertex is either the base point of no cycles or at least two simple cycles. $\text{ }$ For any graph $E$ a subset $H \subseteq E^0$ is *hereditary* if whenever $v, w \in E^0$ with $v \in H$ and $v \geq w$, then $w \in H$. A hereditary subset $H$ is *saturated* if whenever $v \in E^0_\textnormal{reg}$ with $r(s^{-1}(v)) \subseteq H$, then $v \in H$. For any saturated hereditary subset $H$, the *breaking vertices* corresponding to $H$ are the elements of the set $$B_H := \{ v \in E^0 : |s^{-1}(v)| = \infty \text{ and } 0 < |s^{-1}(v) \cap r^{-1}(E^0 \setminus H)| < \infty \}.$$ An *admissible pair* $(H, S)$ consists of a saturated hereditary subset $H$ and a subset $S \subseteq B_H$. For a fixed graph $E$ we order the collection of admissible pairs for $E$ by defining $(H,S) \leq (H',S')$ if and only if $H \subseteq H'$ and $S \subseteq H' \cup S'$. For any admissible pair $(H,S)$ we define $$I_{(H,S)} := \text{the ideal in $C^*(E)$ generated by $\{p_v : v \in H\} \cup \{p_{v_0}^H : v_0 \in S\}$},$$ where $p_{v_0}^H$ is the *gap projection* defined by $$p_{v_0}^H := p_{v_0} - \sum_{{s(e) = v_0} \atop {r(e) \notin H}} s_e s_e^*.$$ Note that the definition of $B_H$ ensures that the sum on the right is finite. For any graph $E$ there is a canonical gauge action $\gamma : \mathbb{T} \to \operatorname{Aut} C^*(E)$ with the property that for any $z \in \mathbb{T}$ we have $\gamma_z (p_v) = p_v$ for all $v \in E^0$ and $\gamma_z(s_e) = zs_e$ for all $e \in E^1$. We say that an ideal $I \triangleleft C^*(E)$ is *gauge invariant* if $\gamma_z(I) \subseteq I$ for all $z \in \mathbb{T}$. There is a bijective correspondence between the lattice of admissible pairs of $E$ and the lattice of gauge-invariant ideals of $C^*(E)$ given by $(H, S) \mapsto I_{(H,S)}$ [@bhrs:iccig Theorem 3.6]. When $E$ satisfies Condition (K), all ideals of $C^*(E)$ are gauge invariant [@mt:sgcg Theorem 2.24] and the map $(H, S) \mapsto I_{(H,S)}$ is onto the lattice of ideals of $C^*(E)$. When $B_H = \emptyset$, we write $I_H$ in place of $I_{(H, \emptyset)}$ and observe that $I_H$ equals the ideal generated by $\{ p_v : v \in H \}$. Note that if $E$ is row-finite, then $B_H$ is empty for every saturated hereditary subset $H$. Notation and conventions for extensions --------------------------------------- All ideals in $C^*$-algebras will be considered to be closed two-sided ideals. An element $a$ of a $C^*$-algebra $A$ (respectively, a subset $S \subseteq A$) is said to be *full* if $a$ (respectively, $S$) is not contained in any proper ideal of $A$. A map $\phi : A \to B$ is *full* if ${\operatorname{im }}\phi$ is full in $B$. If $A$ and $B$ are $C^*$-algebras, an *extension* of $A$ by $B$ consists of a $C^*$-algebra $E$ and a short exact sequence $$\xymatrix{{\mathfrak{e}}: & 0 \ar[r] & B \ar[r]^\alpha & E \ar[r]^\beta & A \ar[r] & 0.}$$ We say that the extension ${\mathfrak{e}}$ is *essential* if $\alpha(B)$ is an essential ideal of $E$, and we say that the extension ${\mathfrak{e}}$ is *unital* if $E$ is a unital $C^*$-algebra. For any extension there exist unique $*$-homomorphisms $\eta_{\mathfrak{e}}: E \to \mathcal{M} (B)$ and $\tau_{\mathfrak{e}}: A \to \mathcal{Q}(B) := \mathcal{M}(B)/B$ which make the diagram $$\xymatrix{ 0 \ar[r] & B \ar[r]^\alpha \ar@{=}[d] & E \ar[r]^\beta \ar[d]_{\eta_{\mathfrak{e}}} & A \ar[r] \ar[d]_{\tau_{\mathfrak{e}}} & 0 \\ 0 \ar[r] & B \ar[r]^i & \mathcal{M}(B) \ar[r]^\pi & \mathcal{Q}(B) \ar[r] & 0}$$ commute. The $*$-homomorphism $\tau_{\mathfrak{e}}$ is called the *Busby invariant* of the extension, and the extension is essential if and only if $\tau_{\mathfrak{e}}$ is injective. An extension ${\mathfrak{e}}$ is *full* if the associated Busby invariant $\tau_{\mathfrak{e}}$ has the property that $\tau_{\mathfrak{e}}(a)$ is full in $\mathcal{Q}(A)$ for every $a \in A \setminus \{ 0 \}$. For an extension ${\mathfrak{e}}$, we let ${K_\textbf{six}}({\mathfrak{e}})$ denote the cyclic six-term exact sequence of $K$-groups $$\xymatrix{ {K_0(B)}\ar[r]&{K_0(E)}\ar[r]&{K_0(A)}\ar[d]\\ {K_1(A)}\ar[u]&{K_1(E)}\ar[l]&{K_1(B)}\ar[l]}$$ where $K_0(B)$, $K_0(E)$, and $K_0(A)$ are viewed as (pre-)ordered groups. Given two extensions $$\xymatrix{{\mathfrak{e}}_1: & 0 \ar[r] & B_1 \ar[r]^{\alpha_1} & E_1 \ar[r]^{\beta_1} & A_1 \ar[r] & 0 & \\ {\mathfrak{e}}_2: & 0 \ar[r] & B_2 \ar[r]^{\alpha_2} & E_2 \ar[r]^{\beta_2} & A_2 \ar[r] & 0 & }$$ we say ${K_\textbf{six}}({\mathfrak{e}}_1)$ *is isomorphic to* ${K_\textbf{six}}({\mathfrak{e}}_2)$, written ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$, if there exist isomorphisms $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$ making the following diagram commute $$\xymatrix{ K_0(B_1) \ar[rr] \ar[rd]_\alpha & & K_0(E_1) \ar[rr] \ar[d]_\beta & & K_0(A_1) \ar[ddd] \ar[dl]^\gamma \\ & {K_0(B_2)}\ar[r]&{K_0(E_2)}\ar[r]&{K_0(A_2)}\ar[d] & \\ & {K_1(A_2)}\ar[u]&{K_1(E_2)}\ar[l]&{K_1(B_2)}\ar[l] & \\ K_1(A_1) \ar[uuu] \ar[ru]^\zeta & & K_1(E_1) \ar[ll] \ar[u]^\epsilon & & K_1(B_1) \ar[ll] \ar[ul]_\delta}$$ and where $\alpha$, $\beta$, and $\gamma$ are isomorphisms of (pre-)ordered groups. Preliminary graph $C^*$-algebra results {#prelims-sec} ======================================= In this section we develop a few results for graph $C^*$-algebras in order to apply the methods of [@segrer:cecc] in §\[class-sec\]. However, several of these results are interesting in their own right. \[E-structure-lem\] If $E$ is a graph such that $C^*(E)$ contains a unique proper nontrivial ideal $I$, then the following six conditions are satisfied: 1. $E$ satisfies Condition (K), 2. $E$ contains exactly three saturated hereditary subsets $\{ \emptyset, H, E^0 \}$, 3. $E$ contains no breaking vertices; i.e., $B_H = \emptyset$, 4. $I$ is a gauge invariant ideal and $I_H = I$, 5. If $X$ is a nonempty hereditary subset of $E$, then $X \cap H \neq \emptyset$, and 6. $E$ has at most one sink, and if $v$ is a sink of $E$ then $v \in H$. Suppose that $E$ does not satisfy Condition (K). Then by [@mt:sgcg Proposition 1.17] there exists a saturated hereditary subset $H \subseteq E^0$ such that $E \setminus H$ contains a cycle $\alpha = e_1 \ldots e_n$ with no exits. The set $X = \{ s(e_i) \}_{i=1}^n$ is a hereditary subset of $E \setminus H$, and $I_X$ is an ideal in $C^*(E \setminus H)$ Morita equivalent to $M_n (C(\mathbb{T}))$ (see [@bhrs:iccig Proposition 3.4] and [@rae:ga Example 2.14] for details). Thus $I_X$, and hence $C^*(E\setminus H)$, contains a countably infinite number of ideals. Since $C^*(E \setminus H) \cong C^*(E) / I_{(H,B_H)}$ [@bhrs:iccig Proposition 3.4], this implies that $C^*(E)$ has a countably infinite number of ideals. Hence if $C^*(E)$ has a finite number of ideals, $E$ satisfies Condition (K). Because $E$ satisfies Condition (K), it follows from [@ddmt:ckegc Theorem 3.5] that the ideals of $C^*(E)$ are in one-to-one correspondence with the pairs $(H, S)$ where $H$ is saturated hereditary, and $S \subseteq B_H$ is a subset of the breaking vertices of $H$. Since $E$ contains a unique proper nontrivial ideal, it follows that $E$ contains a unique saturated hereditary subset $H$ not equal to $E^0$ or $\emptyset$, and that there are no breaking vertices; i.e., $B_H = \emptyset$. It must also be the case that $I= I_H$. Moreover, since $E$ satisfies Condition (K), [@bhrs:iccig Corollary 3.8] shows that all ideals of $C^*(E)$ are gauge-invariant. In addition, suppose $X$ is a hereditary subset with $X \cap H = \emptyset$. Since $H$ is hereditary, none of the vertices in $H$ can reach $X$, and thus the saturation $\overline{X}$ contains no vertices of $H$, and $\overline{X} \cap H = \emptyset$. But then $\overline{X}$ is a saturated hereditary subset of $E$ that does not contain the vertices of $H$, and hence must be equal to $\emptyset$. Thus if $X$ is a nonempty hereditary subset of $E$, then $X \cap H \neq \emptyset$. Finally, suppose $v$ is a sink of $E$. Consider the hereditary subset $X := \{ v \}$. From the previous paragraph it follows that $X \cap H \neq \emptyset$ and hence $v \in H$. In addition, there cannot be a second sink in $E$, for if $v'$ is a sink, then $X := \{ v \}$ and $Y := \{ v' \}$ are distinct hereditary sets. Since $v$ cannot reach $v'$, we see that $v$ is not in the saturation $\overline{Y}$. Similarly, since $v'$ cannot reach $v$, we have that $v'$ is not in the saturation $\overline{X}$. Thus $\overline{X}$ and $\overline{Y}$ are distinct saturated hereditary subsets of $E$ that are proper and nontrivial, which is a contradiction. It follows that there is at most one sink in $E$. \[cases-remark\] According to [@kdjhhsw:srga Lemma 1.3] and [@bhrs:iccig Corollary 3.5] the $C^*$-algebras $I$ and $A/I$ are in this case also graph algebras. As they are necessarily simple, they must be either Kirchberg algebras or AF-algebras. We will denote the four cases thus occurring as follows $ $ Case $I$ $A/I$ ----------------------------- ----------- ----------- ${\mathbf{[11]}}$ AF AF ${\mathbf{[1\infty]}}$ AF Kirchberg ${\mathbf{[\infty 1]}}$ Kirchberg AF ${\mathbf{[\infty\infty]}}$ Kirchberg Kirchberg $ $ Let $A$ be a $C^*$-algebra. A proper ideal $I \triangleleft A$ is a *largest proper ideal* of $A$ if whenever $J \triangleleft A$, then either $J \subseteq I$ or $J=A$. Observe that a largest proper ideal is always an essential ideal. Also note that if $A$ is a $C^*$-algebra with a unique proper nontrivial ideal $I$, then $I$ is a largest proper ideal; and if $A$ is a simple $C^*$-algebra then $\{ 0 \}$ is a largest proper ideal. \[largest-g-i\] Let $E$ be a graph, and suppose that $I$ is a largest proper ideal of $C^*(E)$. Then $I$ is gauge invariant and $I = I_{(H, B_H)}$ for some saturated hereditary subset $H$ of $E^0$. Furthermore, if $K$ is any saturated hereditary subset of $E$, then either $K \subseteq H$ or $K = E^0$. Let $\gamma$ denote the canonical gauge action of ${\mathbb{T}}$ on $C^*(E)$. For any $z \in {\mathbb{T}}$ we have that $\gamma_z(I)$ is a proper ideal of $C^*(E)$. Since $I$ is a largest proper ideal of $C^*(E)$, it follows that $\gamma_z(I) \subseteq I$. A similar argument shows that $\gamma_{z^{-1}}(I) \subseteq I$. Thus $\gamma_z(I) = I$ and $I$ is gauge invariant. It follows from [@bhrs:iccig Theorem 3.6] that $I = I_{(H,S)}$ for some saturated hereditary subset $H$ of $E^0$ and some subset $S \subseteq B_H$. Because $I$ is a largest proper ideal, it follows that $S = B_H$, and hence $I = I_{(H,B_H)}$. Furthermore, if $K$ is a saturated hereditary subset, then either $I_{(K,B_K)} \subseteq I_{(H,B_H)}$ or $I_{(K,B_K)} = C^*(E)$. Hence either $K \subseteq H$ or $K = E^0$. \[cycle-exists-lem\] Let $E$ be a graph and suppose that $I$ is a largest proper ideal of $C^*(E)$ with the property that $C^*(E) / I$ is purely infinite. Then $I = I_{(H, B_H)}$ for some saturated hereditary subset $H$ of $E^0$, and there exists a cycle $\gamma$ in $E \setminus H$ and an edge $f \in E^1$ with $s(f) = s(\gamma)$ and $r(f) \in H$. Furthermore, if $x \in E^0$, then $x \geq s(\gamma)$ if and only if $x \in E^0 \setminus H$. Lemma \[largest-g-i\] shows that $I = I_{(H, B_H)}$ for some saturated hereditary subset $H$ of $E^0$. It follows from [@bhrs:iccig Corollary 3.5] that $C^*(E) / I_{(H, B_H)} \cong C^*(E \setminus H)$, where $E \setminus H$ is the subgraph of $E$ with $(E \setminus H)^0 := E^0 \setminus H$ and $(E \setminus H)^1 := E^1 \setminus r^{-1}(H)$. Since $C^*(E \setminus H)$ is purely infinite, it follows from [@ddmt:cag Corollary 2.14] that $E \setminus H$ contains a cycle $\alpha$. Define $K := \{ x \in E^0 : x \ngeq s(\alpha) \}$. Then $K$ is saturated hereditary, $H \subseteq K$, and $K \neq E^0$. Hence $I_{(H, B_H)} \subseteq I_{(K, B_K)} \neq C^*(E)$, and the fact that $I_{(H, B_H)}$ is a largest proper ideal implies that $I_{(H, B_H)} = I_{(K, B_K)}$ so that $H = K$. Hence for $x \in E^0$ we have $x \geq s(\alpha)$ if and only if $x \in E^0 \setminus H$. Consider the set $J := \{ x \in E^0 : s(\alpha) \geq x \}$. Then $J$ is a hereditary subset and we let $\overline{J}$ denote its saturation. Since $I_{(H,B_H)}$ is a largest proper ideal, it follows that either $\overline{J} \subseteq H$ or $\overline{J} = E^0$. Since $s(\alpha) \in \overline{J} \setminus H$, we must have $\overline{J} = E^0$. Choose any element $w \in H$. Since $w \in \overline{J}$ it follows that there exists $v \in J$ with $w \geq v$. But since $w \geq v$ and $H$ is hereditary, it follows that $v \in H$. Hence $v \in J \cap H$, and there is a path from $s(\alpha)$ to a vertex in $H$. Choose a path $\mu = \mu_1 \mu_2 \ldots \mu_n$ with $s(\mu) = s(\alpha)$, $r(\mu_{n-1}) \notin H$, and $r(\mu_n) \in H$. Since $r(\mu_{n-1}) \notin H$ the previous paragraph shows that there exists a path $\nu$ with $s(\nu) = r(\mu_{n-1})$ and $r(\nu) = s(\alpha)$. Let $\gamma := \nu \mu_1 \ldots \mu_{n-1}$ and let $f:= \mu_n$. Then $\gamma$ is a cycle in $E \setminus H$ and $f$ is an edge with $s(f) = s(\gamma)$ and $r(f) \in H$. Furthermore, since $s(\alpha)$ is a vertex on the cycle $\gamma$, we see that for any $x \in E^0$ we have $x \geq s(\gamma)$ if and only if $x \geq s(\alpha)$. It follows from the previous paragraph that if $x \in E^0$, then $x \geq s(\gamma)$ if and only if $x \in E^0 \setminus H$. Note that the conclusion of the above lemma does not hold if $I$ is a maximal proper ideal that is not a largest proper ideal. For example, if $E$ is the graph $ $ $$\xymatrix{ v & w \ar[l] \ar[r] & x \ar@(ul,ur)[] \ar@(dl,dr)[] }$$ $ $ and $H= \{v\}$ then $I := I_H$ is a maximal ideal that is AF, and $C^*(E) / I_H \cong M_2 (\mathcal{O}_2)$ is purely infinite. However, there is no edge from the base point of a cycle to $H = \{ v \}$. We say that two projections $p,q \in A$ are *equivalent*, written $p \sim q$, if there exists an element $v \in A$ with $p=vv^*$ and $q=v^*v$. We write $p \lesssim q$ to mean that $p$ is equivalent to a subprojection of $q$; that is, there exists $v \in A$ such that $p = vv^*$ and $v^*v \leq q$. Note that $p \lesssim q$ and $q \lesssim p$ does not imply that $p \sim q$ (unless $A$ is finite). If $e \in G^1$ then we see that $p_{r(e)} = s_e^*s_e$ and $s_es_e^* \leq p_{s(e)}$. Therefore $p_{r(e)} \lesssim p_{s(e)}$. More generally we see that $v \geq w$ implies $p_w \lesssim p_v$. \[stab-equiv\] Let $A$ be a $C^*$-algebra with an increasing countable approximate unit $\{ p_n \}_{n=1}^\infty$ consisting of projections. Then the following are equivalent. 1. $A$ is stable. 2. For every projection $p \in A$ there exists a projection $q \in A$ such that $p \sim q$ and $p \perp q$. 3. For all $n \in {\mathbb{N}}$ there exists $m > n$ such that $p_n \lesssim p_m-p_n$ The equivalence of and is shown in [@jhmr:sc Theorem 3.3]. The equivalence of and is shown in [@jh:piscgds Lemma 2.1]. \[sum-equiv-projs\] Let $A$ be a $C^*$-algebra. Suppose $p_1, p_2, \ldots, p_n$ are mutually orthogonal projections in $A$, and $q_1, q_2, \ldots, q_n$ are mutually orthogonal projections in $A$ with $p_i \sim q_i$ for $1 \leq i \leq n$. Then $\sum_{i=1}^n p_i \sim \sum_{i=1}^n q_i$. Since $p_i \sim q_i$ there exists $v_i \in A$ such that $v_i^*v_i = p_i$ and $v_iv_i^* = q_i$. Thus for $i \neq j$ we have $v_j^* v_i = v_j^*v_jv_j^* v_iv_i^*v_i = v_j^* q_j q_i v_i = 0$ and $v_i v_j^* = v_iv_i^*v_i v_j^* v_j v_j^* = v_i p_i p_j v_j^* = 0$. Hence $(\sum_{i=1}^n v_i)^* \sum_{i=1}^n v_i = \sum_{i=1}^n v_i^*v_i = \sum_{i=1}^n p_i$ and $\sum_{i=1}^n v_i ( \sum_{i=1}^n v_i)^* = \sum_{i=1}^n v_iv_i^* = \sum_{i=1}^n q_i$. Thus $\sum_{i=1}^n p_i \sim \sum_{i=1}^n q_i$. \[getcorner\] Let $E$ be a graph with no breaking vertices, and suppose that $I$ is a largest proper ideal of $C^*(E)$ and such that $C^*(E)/I$ is purely infinite and $I$ is AF. Then there exists a projection $p \in C^*(E)$ such that $pC^*(E)p$ is a full corner of $C^*(E)$ and $pIp$ is stable. Lemma \[cycle-exists-lem\] implies that $I = I_{(H,B_H)}$ for some saturated hereditary subset $H$ of $E^0$, and there exists a cycle $\gamma$ in $E \setminus H$ and an edge $f \in E^1$ with $s(f) = s(\gamma)$ and $r(f) \in H$; and furthermore, if $x \in E^0$, then $x \geq s(\gamma)$ if and only if $x \in E^0 \setminus H$. Since $E$ has no breaking vertices, we have that $B_H = \emptyset$ so that $I_{(H,B_H)}$ is the ideal generated by $\{p_v : v \in H \}$ and we may write $I_{(H,B_H)}$ as $I_H$. Let $v = s(f) = s(\gamma)$ and let $w = r(f)$. Define $p := p_v + p_w$. Suppose $J \triangleleft C^*(E)$ and $pC^*(E)p \subseteq J$. Since $v \notin H$ we see that $p_v \notin I$ and hence $p_v \in pC^*(E)p \setminus I \subseteq J \setminus I$. Thus $J \nsubseteq I$ and the fact that $I$ is a largest proper ideal implies that $J = C^*(E)$. Hence $pC^*(E)p$ is a full corner of $C^*(E)$. In addition, since there are no breaking vertices $$\begin{aligned} pIp &= pI_Hp \\ &= p \left( \overline{\operatorname{span}} \{ s_\alpha s_\beta^* : r(\alpha) = r(\beta) \in H \} \right)p \\ &= \overline{\operatorname{span}} \{ ps_\alpha s_\beta^*p : r(\alpha) = r(\beta) \in H \} \\ &= \overline{\operatorname{span}} \{ s_\alpha s_\beta^* : r(\alpha) = r(\beta) \in H \text{ and } s(\alpha), s(\beta) \in \{v, w \} \}.\end{aligned}$$ Let $S := \{ \alpha \in E^* : s(\alpha) = v \text{ and } r(\alpha) = w \}$. Since $S$ is a countable set we may list the elements of $S$ and write $S = \{ \alpha_1, \alpha_2, \alpha_3, \ldots \}$. Define $p_0 := p_w$ and $p_n := p_w + \sum_{k=1}^n s_{\alpha_k} s_{\alpha_k}^*$ for $n \in {\mathbb{N}}$. We will show that for $\mu, \nu \in S$ we have $$\label{S-eqn-displayed} s_\mu^* s_\nu := \begin{cases} p_{r(\mu)} & \text{ if $\mu = \nu$} \\ 0 & \text{ otherwise.} \end{cases}$$ First suppose that $s_\mu^* s_\nu \neq 0$. Then one of $\mu$ and $\nu$ must extend the other. Suppose $\mu$ extends $\nu$. Then $\mu = \nu \lambda$ for some $\lambda \in E^*$. Thus $s(\lambda) = r(\nu) = w$ and $r(\lambda) = r(\mu) = w$. However, $I_H$ is an AF-algebra, and $C^*(E_H)$ is strongly Morita equivalent to $I_H$ [@bhrs:iccig Proposition 3.4], so $C^*(E_H)$ is an AF-algebra. Thus $E_H$ contains no cycles. Since $\lambda$ is a path in $E_H$ with $s(\lambda) = r(\lambda) = w$, and since $E_H$ contains no cycles, we may conclude that $\lambda = w$. Thus $\mu = \nu$. A similar argument works when $\nu$ extends $\mu$. Hence the equation in holds. It follows that the elements of the set $\{ s_\alpha s_\alpha^* : \alpha \in S \} \cup \{p_w \}$ are mutually orthogonal projections, and hence $\{ p_n \}_{n=0}^\infty$ is an sequence of increasing projections. Next we shall show that $\{ p_n \}_{n=0}^\infty$ is an approximate unit for $pIp$. Given $s_\alpha s_\beta^*$ with $r(\alpha) = r(\beta) \in H$ and $s(\alpha), s(\beta) \in \{v, w \}$, we consider two cases. <span style="font-variant:small-caps;">Case I:</span> $s(\alpha) = w$. Then for any $\alpha_k \in S$ we see that $(s_{\alpha_k} s_{\alpha_k}^*) s_\alpha s_\beta^* = s_{\alpha_k} s_{\alpha_k}^* p_w s_\alpha s_\beta^* = 0$. In addition, $p_w (s_\alpha s_\beta^*) = s_\alpha s_\beta^*$. Thus $\lim_{n \to \infty} p_n s_\alpha s_\beta^* = s_\alpha s_\beta^*$. <span style="font-variant:small-caps;">Case II:</span> $s(\alpha) = v$. Then $\alpha = \alpha_j \lambda$ for some $\alpha_j \in S$ and some $\lambda \in E_H^*$ with $s(\lambda) = w$. We have $p_w (s_\alpha s_\beta) = 0$, and also implies that $$(s_{\alpha_k} s_{\alpha_k}^*) s_\alpha s_\beta^* = s_{\alpha_k} s_{\alpha_k}^* s_{\alpha_j} s_\lambda s_\beta^* = \begin{cases} s_\alpha s_\beta^* & \text{ $k = j$} \\ 0 & \text{ $k \neq j$.} \end{cases}$$ Thus $\lim_{n \to \infty} p_n s_\alpha s_\beta^* = s_\alpha s_\beta^*$. The above two cases imply that $\lim_{n \to \infty} p_n x = x$ for any $x \in \operatorname{span} \{ s_\alpha s_\beta^* : r(\alpha) = r(\beta) \in H \text{ and } s(\alpha), s(\beta) \in \{v, w \} \}$. Furthermore, an $\epsilon / 3$-argument shows that $\lim_{n \to \infty} p_n x = x$ for any $x \in pI_Hp = \overline{\operatorname{span}} \{ s_\alpha s_\beta^* : r(\alpha) = r(\beta) \in H \text{ and } s(\alpha), s(\beta) \in \{v, w \} \}$. A similar argument shows that $\lim_{n \to \infty} x p_n = x$ for any $x \in pI_Hp$. Thus $\{ p_n \}_{n=1}^\infty$ is an approximate unit for $pI_Hp$. We shall now show that $pI_H p$ is stable. For each $n \in {\mathbb{N}}$ define $$\lambda^n := \underbrace{ \gamma \gamma \ldots \gamma}_\text{$n$ times} f.$$ For any $k, n \in {\mathbb{N}}$ we have $$s_{\lambda^n} s_{\lambda^n}^* \sim s_{\lambda^n}^* s_{\lambda^n} = p_{r(\lambda^n)} = p_w = s_{\alpha_k}^* s_{\alpha_k} \sim s_{\alpha_k} s_{\alpha_k}^*.$$ For any $n \in {\mathbb{N}}$ choose $q$ large enough that $|\lambda^q| \geq |\alpha_k|$ for all $1 \leq k \leq n$. Then for all $j \in {\mathbb{N}}$ we see that $\lambda^{q+j} \in S$ and $\lambda^{q+j} \neq \alpha_k$ for all $1 \leq k \leq n$. Thus for any $1 \leq k \leq n$ we have $$s_{\alpha_k} s_{\alpha_k}^* \sim s_{\alpha_k}^* s_{\alpha_k} = p_{r(\alpha_k)} = p_w = p_{r(\lambda^{q+k})} = s_{\lambda^{q+k}}^* s_{\lambda^{q+k}} \sim s_{\lambda^{q+k}} s_{\lambda^{q+k}}^*$$ and $$p_w = s_{\lambda^q}^* s_{\lambda^q} \sim s_{\lambda^q} s_{\lambda^q}^*.$$ It follows from Lemma \[sum-equiv-projs\] that $$p_n = p_w + \sum_{k=1}^n s_{\alpha_k} s_{\alpha_k}^* \lesssim \sum_{k=0}^n s_{\lambda^{q+k}} s_{\lambda^{q+k}}^* \lesssim p_m - p_n$$ where $m$ is chosen large enough that $\lambda^{q+k} \in \{ \alpha_1, \alpha_2, \ldots, \alpha_m \}$ for all $0 \leq k \leq n$. Lemma \[stab-equiv\] shows that $pI_Hp$ is stable. Classification {#class-sec} ============== In this section we state and prove our main results. We apply the methods of [@segrer:cecc] to classify certain extensions of graph $C^*$-algebras in terms of their six-term exact sequences of $K$-groups. To do this we will need to discuss classes of $C^*$-algebras satisfying various properties. We give definitions of these properties here, and obtain a lemma that is a consequence of [@segrer:cecc Theorem 3.10]. \[Class-4-props-def\] We will be interested in classes $\mathcal{C}$ of separable nuclear unital simple $C ^*$-algebras in the bootstrap category $\mathcal{N}$ satisfying the following properties: - Any element of $\mathcal{C}$ is either purely infinite or stably finite. - $\mathcal{C}$ is closed under tensoring with $\textrm{M}_n$, where $\textrm{M}_n$ is the $C^*$-algebra of $n$ by $n$ matrices over $\mathcal{C}$. - If $A$ is in $\mathcal{C}$, then any unital hereditary $C^*$-subalgebra of $A$ is in $\mathcal{C}$. - For all $A$ and $B$ in $\mathcal{C}$ and for all $x$ in $KK(A, B)$ which induce an isomorphism from $(K^+_*(A), [1_A])$ to $(K^+_* (A), [1_B ])$, there exists a $*$-isomorphism $\alpha : A \to B$ such that $KK (\alpha) = x$. If $B$ is a separable stable $C^*$-algebra, then we say that $B$ has the *corona factorization property* if every full projection in $\mathcal{M} (B)$ is Murray-von Neumann equivalent to $1_{\mathcal{M}(B)}$. \[ERR-lem\] Let $\mathcal{C}_I$ and $\mathcal{C}_Q$ be classes of unital nuclear separable simple $C^*$-algebras in the bootstrap category $\mathcal{N}$ satisfying the properties of Definition \[Class-4-props-def\]. Let $A_1$ and $A_2$ be in $\mathcal{C}_Q$ and let $B_1$ and $B_2$ be in $\mathcal{C}_I$ with $B_1 \otimes {\mathbb{K}}$ and $B_2 \otimes {\mathbb{K}}$ satisfying the corona factorization property. Let $$\xymatrix{ {{\mathfrak{e}}_1:}&{0}\ar[r]&{B_1 \otimes {\mathbb{K}}}\ar[r]&{E_1}\ar[r]&{A_1}\ar[r]&{0} & }$$ $$\xymatrix{ {{\mathfrak{e}}_2:}&{0}\ar[r]&{B_2 \otimes {\mathbb{K}}}\ar[r]&{E_2}\ar[r]&{A_2}\ar[r]&{0} & }$$ be **essential** and **unital** extensions. If ${K_\textbf{six}}{}{}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$, then $E_1 \otimes {\mathbb{K}}\cong E_2 \otimes {\mathbb{K}}$. Tensoring the extension ${\mathfrak{e}}_1$ by ${\mathbb{K}}$ we obtain a short exact sequence ${\mathfrak{e}}_1'$ and and vertical maps $$\xymatrix{ {{\mathfrak{e}}_1:}&{0}\ar[r]&{B_1 \otimes {\mathbb{K}}}\ar[r]\ar@{^{(}->}[d]&{E_1}\ar[r]\ar@{^{(}->}[d]&{A_1}\ar@{^{(}->}[d]\ar[r] &{0}\\ {{\mathfrak{e}}_1':}&{0}\ar[r]&{(B_1 \otimes {\mathbb{K}}) \otimes {\mathbb{K}}}\ar[r]&{E_1 \otimes {\mathbb{K}}}\ar[r]&{A_1 \otimes {\mathbb{K}}}\ar[r]&{0}}$$ from ${\mathfrak{e}}_1$ into ${\mathfrak{e}}'_1$ that are full inclusions. These full inclusions induce isomorphisms of $K$-groups and hence we have that ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_1')$. In addition, since ${\mathfrak{e}}_1$ is essential, $B_1 \otimes {\mathbb{K}}$ is an essential ideal in $E_1$, and the Rieffel correspondence between the strongly Morita equivalent $C^*$-algebras $E_1$ and $E_1 \otimes {\mathbb{K}}$ implies that $(B_1 \otimes {\mathbb{K}}) \otimes {\mathbb{K}}$ is an essential ideal in $E_1 \otimes {\mathbb{K}}$, so that ${\mathfrak{e}}_1'$ is an essential extension. Furthermore, since $B_1 \otimes {\mathbb{K}}$ is stable and ${\mathfrak{e}}_1$ is essential and full [@segrer:cecc Proposition 1.5], it follows from [@segrer:cecc Proposition 1.6] that ${\mathfrak{e}}_1'$ is full. Moreover, since ${\mathbb{K}}\otimes {\mathbb{K}}\cong {\mathbb{K}}$, we may rewrite ${\mathfrak{e}}_1'$ as $$\xymatrix{ {{\mathfrak{e}}_1':}&{0}\ar[r]&{B_1 \otimes {\mathbb{K}}}\ar[r]&{E_1 \otimes {\mathbb{K}}}\ar[r]&{A_1 \otimes {\mathbb{K}}}\ar[r]&{0}.}$$ By a similar argument, there is an essential and full extension $$\xymatrix{ {{\mathfrak{e}}_2':}&{0}\ar[r]&{B_2 \otimes {\mathbb{K}}}\ar[r]&{E_2 \otimes {\mathbb{K}}}\ar[r]&{A_2 \otimes {\mathbb{K}}}\ar[r]&{0}}$$ such that ${K_\textbf{six}}({\mathfrak{e}}_2') \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. Thus ${K_\textbf{six}}({\mathfrak{e}}_1') \cong {K_\textbf{six}}({\mathfrak{e}}_2')$, and [@segrer:cecc Theorem 3.10] implies that $E_1 \otimes {\mathbb{K}}\cong E_2 \otimes {\mathbb{K}}$. \[full-inclusions-lem\] Let $A$ be a $C^*$-algebra and let $I$ be a largest proper ideal of $A$. If $p \in A$ is a full projection, then the inclusion map $pIp \hookrightarrow I$ and the inclusion map $pAp/pIp \hookrightarrow A/I$ are both full inclusions. Since $p$ is a full projection, we see that $A$ is Morita equivalent to $pAp$ and the Rieffel correspondence between ideals takes the form $J \mapsto pJp$. If $J$ is an ideal of $I$ with $pIp \subseteq J$, then by compressing by $p$ we obtain $pIp \subseteq pJp$. Since the Rieffel correspondence is a bijection, this implies that $I \subseteq J$, and because $J$ is an ideal contained in $I$, we get that $I = J$. Hence $pIp \hookrightarrow I$ is a full inclusion. Furthermore, because $I$ is a largest proper ideal of $A$, we know that $A/I$ is simple and thus $pAp/pIp \hookrightarrow A/I$ is a full inclusion. \[clas\] If $A$ is a graph $C^*$-algebra with exactly one proper nontrivial ideal $I$, then $A$ classified up to stable isomorphism by the six-term exact sequence $$\xymatrix{ {K_0(I)}\ar[r]&{K_0(A)}\ar[r]&{K_0(A/I)}\ar[d]\\ {K_1(A/I)}\ar[u]&{K_1(A)}\ar[l]&{K_1(I)}\ar[l]}$$ with all $K_0$-groups considered as ordered groups. In other words, if $A$ is a graph $C^*$-algebras with precisely one proper nontrivial ideal $I$, if $A'$ is a graph $C^*$-algebras with precisely one proper nontrivial ideal $I'$, and if $$\xymatrix{ {{\mathfrak{e}}_1:}&{0}\ar[r]& \, I \, \ar[r]&{\, A \,}\ar[r]&{\, A/I \, }\ar[r]&{0} & }$$ $$\xymatrix{ {{\mathfrak{e}}_2:}&{0}\ar[r]&I'\ar[r]&{A'}\ar[r]&{A'/I'}\ar[r]&{0} & }$$ are the associated extensions, then $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$ if and only if ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. Moreover, in cases ${\mathbf{[1\infty]}}$, ${\mathbf{[\infty 1]}}$, and ${\mathbf{[\infty\infty]}}$, the order structure on $K_0(A)$ may be removed from the invariant leaving it still complete. And in case ${\mathbf{[11]}}$, the ordered group $K_0(A)$ is a complete invariant in its own right. It is straightforward to show that $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$ implies that ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. Thus we need only establish the converse. To do this, we begin by assuming that ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. We define ${\mathcal{C}_{API}}$ as the union of the class of unital simple and separable $AF$-algebras and the class of simple, nuclear, unital, and separable purely infinite -algebras in the bootstrap category. The category ${\mathcal{C}_{API}}$ meets all the requirements in the list in Definition \[Class-4-props-def\]: We clearly have that each algebra in ${\mathcal{C}_{API}}$ is either purely infinite or stably finite, and that ${\mathcal{C}_{API}}$ is closed under passing to matrices and unital hereditary subalgebras. We also need to prove that the Elliott invariant is complete for ${\mathcal{C}_{API}}$, and this follows by the classification results of Elliott [@gae:cilssfa Theorem 4.3] and Kirchberg-Phillips (see [@kir:cpickt Theorem C] and [@phi:ctnpisc §4.2]) after noting that the classes are obviously distinguishable by the nature of the positive cone in $K_0$. Finally, as recorded in [@segrer:cecc Theorem 3.9], the stabilizations of the $C^*$-algebras in ${\mathcal{C}_{API}}$ all have the corona factorization property according to [@dkpwn:cfpaue Theorem 5.2] and [@pwn:cfp Proposition 2.1]. It follows from [@kdjhhsw:srga Lemma 1.3] and [@bhrs:iccig Corollary 3.5] that $I$ and $A/I$ are simple graph $C^*$-algebras and thus each of $I$ and $A/I$ is either an AF-algebra or a purely infinite algebra [@ddmt:cag Remark 2.16]. Similarly for $I'$ and $A'/I'$. Since ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$, we see that $K_0(I) \cong K_0(I')$ and $K_0(A/I) \cong K_0(A'/I')$ as ordered groups. By considering the positive cone in these groups, we may conclude that $I$ and $I'$ are either both purely infinite or both AF-algebras, and also $A/I$ and $A'/I'$ are either both purely infinite or both AF-algebras. Thus $A$ and $A'$ both fall into one of the four cases described in Remark \[cases-remark\]. **Cases ${\mathbf{[\infty\infty]}}$, ${\mathbf{[\infty 1]}}$**\ Write $A = C^*(E)$ for some graph $E$. Since $I$ is a largest proper ideal, Lemma \[largest-g-i\] implies that $I = I_{(H,B_H)}$ for some saturated hereditary subset $H \subsetneq E^0$. If we let $v \in E^0 \setminus H$, and define $p :=p_v$, then $p \notin I$. Since $I$ is a largest proper ideal in $A$, this implies that the projection $p$ is full. Thus we obtain a full hereditary subalgebra $pAp$, and as noted in Lemma \[full-inclusions-lem\] we have that all vertical maps in $$\xymatrix{ {{\mathfrak{e}}'_1:}&{0}\ar[r]&{pIp}\ar[r]\ar@{^{(}->}[d]&{pAp}\ar[r]\ar@{^{(}->}[d]&{pAp/pIp}\ar@{^{(}->}[d]\ar[r] &{0}\\ {{\mathfrak{e}}_1 :}&{0}\ar[r]&{I}\ar[r]&{A}\ar[r]&{A/I}\ar[r]&{0}}$$ are full inclusions. It follows that all of the above maps induce isomorphisms on the $K$-groups and ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}'_1)$. In addition, since $pIp$ is nonunital and purely infinite, the ideal $pIp$ is stable (by Zhang’s dichotomy) and we may write $pIp \cong B_1 \otimes {\mathbb{K}}$ for a suitably chosen $B_1 \in {\mathcal{C}_{API}}$. We now let $E_1 := pAp$ and $A_1 := pAp / pIp$. With this notation, ${\mathfrak{e}}'_1$ takes the form $$\xymatrix{ {{\mathfrak{e}}_1':}&{0}\ar[r]&{B_1 \otimes {\mathbb{K}}}\ar[r]&{E_1}\ar[r]&{A_1}\ar[r]&{0} }$$ with $B_1$ and $A_1$ unital $C^*$-algebras in ${\mathcal{C}_{API}}$. Furthermore, ${\mathfrak{e}}_1'$ is an essential extension because the ideal $I$ is a largest proper ideal in $A$, and thus also the ideal $pIp \cong B_1 \otimes {\mathbb{K}}$ is a largest proper ideal in $A_1 \otimes {\mathbb{K}}$, which implies that $pIp \cong B_1 \otimes {\mathbb{K}}$ is an essential ideal. By a similar argument, we may find an extension $$\xymatrix{ {{\mathfrak{e}}_2':}&{0}\ar[r]&{B_2 \otimes {\mathbb{K}}}\ar[r]&{E_2}\ar[r]&A_2 \ar[r]&{0}}$$ with $E_2 := qA'q$ for a full projection $q \in A'$, the $C^*$-algebras $B_2$ and $A_2$ in ${\mathcal{C}_{API}}$ with $B_2 \otimes {\mathbb{K}}$ satisfying the corona factorization property, and ${\mathfrak{e}}_2'$ an essential and full extension with ${K_\textbf{six}}({\mathfrak{e}}_2') \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. It follows from Lemma \[ERR-lem\] that $E_1 \otimes {\mathbb{K}}\cong E_2 \otimes {\mathbb{K}}$, or equivalently, that $pAp \otimes {\mathbb{K}}\cong qA'q \otimes {\mathbb{K}}$. Furthermore, because $pAp$ is a full corner of $A$, and $qAq$ is a full corner of $A'$, we obtain that $pAp \otimes {\mathbb{K}}\cong A \otimes {\mathbb{K}}$ and $qA'q \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$. It follows that $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$. We also observe that in this case the order structure on $K_0(A)$ is a redundant part of the invariant. **Case ${\mathbf{[1\infty]}}$**\ As seen in Lemma \[E-structure-lem\], we may write $A=C^*(E)$ where $E$ has no breaking vertices. By Proposition \[getcorner\] there exists a projection $p \in A$ such that $pAp$ is a full corner inside $A$, and $pIp$ is stable. Moreover, since $I$ is an AF-algebra by hypothesis and $pIp$ is a hereditary subalgebra of $I$, it follows from [@gae:admic Theorem 3.1] that $pIp$ is an AF-algebra. Hence we may choose a unital AF-algebra $B_1$ with $pIp \cong B_1 \otimes {\mathbb{K}}$. The extension $$\xymatrix{ {{\mathfrak{e}}_1':}&{0}\ar[r]&{B_2 \otimes {\mathbb{K}}}\ar[r]&{pAp}\ar[r]&{pAp/pIp}\ar[r] &{0}}$$ is essential. In addition, an argument as in Cases ${\mathbf{[\infty\infty]}}$, ${\mathbf{[\infty 1]}}$ shows that ${K_\textbf{six}}({\mathfrak{e}}_1') \cong {K_\textbf{six}}({\mathfrak{e}}_1)$. We may perform a similar argument for $A'$, and arguing as in Cases ${\mathbf{[\infty\infty]}}$, ${\mathbf{[\infty 1]}}$ and applying Lemma \[ERR-lem\] we obtain that $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$. Again, the order structure on $K_0(A)$ is a redundant part of the invariant. **Case ${\mathbf{[11]}}$**\ Since $I$ and $A/I$ are AF-algebras, it follows from a result of Brown that $A$ is an AF-algebra [@bro:eaplp] (or see [@ege:dca §9.9] for a detailed proof). Similarly, $A'$ is an AF-algebra. It follows from Elliott’s Theorem that $K_0(A)$ order isomorphic to $K_0(A')$ implies that $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$. Moreover, in this case the ordered group $K_0(A)$ is a complete invariant. Joint work in progress by Ruiz and the first named author provides information about the necessity of using order on the $K_0$-groups of ${K_\textbf{six}}(-)$. There are examples of pairs of non-isomorphic stable AF-algebras $A$ and $A'$ with exactly one ideal $I$ and $I'$ such that ${K_\textbf{six}}({\mathfrak{e}})\simeq {K_\textbf{six}}({\mathfrak{e}}')$ with group isomorphisms which are positive at $K_0(I)$ and $K_0(A/I)$ but not at $K_0(A)$. By [@kst:rafagaelua Corollary 4.8], $A$ and $A'$ may be realized as graph $C^*$-algebras. In the other cases, one may prove that any isomorphism between ${K_\textbf{six}}({\mathfrak{e}})$ and ${K_\textbf{six}}({\mathfrak{e}}')$ will automatically be positive at $K_0(A)$ if it is positive at $K_0(I)$ and $K_0(A/I)$. Thus it is possible that any isomorphism of our reduced invariant lifts to a $*$-isomorphism in the ${\mathbf{[\infty\infty]}}$, ${\mathbf{[\infty 1]}}$, and ${\mathbf{[1\infty]}}$ cases, but this has only been established in the ${\mathbf{[\infty\infty]}}$ case, cf.  [@eilres]. Our proof of Theorem \[clas\] can be modified slightly to give us an additional result. \[clasimp\] If $A$ is a the $C^*$-algebra of a graph satisfying Condition (K), and if $A$ has a largest proper ideal $I$ such that $I$ is an AF-algebra, then $A$ is classified up to stable isomorphism by the six-term exact sequence $$\xymatrix{ {K_0(I)}\ar[r]&{K_0(A)}\ar[r]&{K_0(A/I)}\ar[d]\\ {K_1(A/I)}\ar[u]&{K_1(A)}\ar[l]&{K_1(I)}\ar[l]}$$ with $K_0(I)$ considered as an ordered group. In other words, if $A$ is the $C^*$-algebra of a graph satisfying Condition (K) with a largest proper ideal $I$ that is an AF-algebra, if $A'$ is the $C^*$-algebra of a graph satisfying Condition (K) with a largest proper ideal $I'$ that is an AF-algebra, and if $$\xymatrix{ {{\mathfrak{e}}_1:}&{0}\ar[r]& \, I \, \ar[r]&{\, A \,}\ar[r]&{\, A/I \, }\ar[r]&{0} & }$$ $$\xymatrix{ {{\mathfrak{e}}_2:}&{0}\ar[r]&I'\ar[r]&{A'}\ar[r]&{A'/I'}\ar[r]&{0} & }$$ are the associated extensions, then $A \otimes {\mathbb{K}}\cong A' \otimes {\mathbb{K}}$ if and only if ${K_\textbf{six}}({\mathfrak{e}}_1) \cong {K_\textbf{six}}({\mathfrak{e}}_2)$. To begin, using the desingularization of [@ddmt:cag] we may find a row-finite graph $F$ such that $C^*(F)$ is stably isomorphic to $A$. Since $C^*(F)$ is Morita equivalent to $A$, the $C^*$-algebra $C^*(F)$ has a largest proper ideal that is an AF-algebra, and the associated six-term exact sequence of $K$-groups is isomorphic to ${K_\textbf{six}}({\mathfrak{e}}_1)$. Hence we may replace $A$ by $C^*(F)$ for the purposes of the proof. Likewise for $A'$. Thus we may, without loss of generality, assume that $A$ and $A'$ are $C^*$-algebras of row-finite graphs, and in particular that $A$ and $A'$ are $C^*$-algebras of graphs with no breaking vertices. To obtain the result, we simply argue as in Case ${\mathbf{[1\infty]}}$ of the proof of Theorem \[clas\], using [@segrer:cecc Theorem 3.13] in place of [@segrer:cecc Theorem 3.10], and noting that Proposition \[getcorner\] applies since the graphs have no breaking vertices. Examples {#ex-sec} ======== To illustrate our methods we give a complete classification, up to stable isomorphism, of all $C^*$-algebras of graphs with two vertices that have precisely one proper nontrivial ideal. Combined with other results, this allows us to give a complete classification of all $C^*$-algebras of graphs satisfying Condition (K) with exactly two vertices. If $E$ is a graph with two vertices, and if $C^*(E)$ has exactly one proper ideal, then $E$ must have exactly one proper nonempty saturated hereditary subset with no breaking vertices. This occurs precisely when the vertex matrix of $E$ has the form $$\begin{bmatrix}a&b\\0&d\end{bmatrix}$$ where $a,d\in\{0,2,3,\dots,\infty\}$ and $b\in\{1,2,3,\dots,\infty\}$ with the extra conditions $$a=0 \ \Longrightarrow \ b = \infty \qquad \text{ and } \qquad b = \infty \implies (a= 0 \text{ or } a = \infty),$$ Computing $K$-groups using [@ddmt:ckegc], we see that in all of these cases the $K_1$-groups of $C^*(E)$, the unique proper nontrivial ideal $I$, and the quotient $C^*(E)/I$ all vanish. Thus the six-term exact sequence becomes $0 \to K_0(I) \to K_0(C^*(E)) \to K_0(C^*(E)/I) \to 0$, and using [@ddmt:ckegc] to compute the $K_0$-groups and the induced maps we obtain the following cases. $ $ [|c|c|c||c|c|]{}$a$&$d$&$b$&$\ \ K_0(I) \to K_0(C^*(E)) \to K_0(C^*(E)/I) \ \ $&Case\ 0&0&$\infty$& ------------------------------------------------------------------------ ${\mathbb{Z}_{++}}\to{\mathbb{Z}}\oplus {\mathbb{Z}}\to{\mathbb{Z}_{++}}$&${\mathbf{[11]}}$\ 0&$n$&$\infty$& ------------------------------------------------------------------------ ${\mathbb{Z}}_{d-1}\to{\mathbb{Z}}_{d-1}\oplus {\mathbb{Z}}\to{\mathbb{Z}_{++}}$&${\mathbf{[\infty 1]}}$\ 0&$\infty$&$\infty$& ------------------------------------------------------------------------ ${{\mathbb{Z}}_{\pm}}\to{\mathbb{Z}}\oplus {\mathbb{Z}}\to{\mathbb{Z}_{++}}$&${\mathbf{[\infty 1]}}$\ $n$&0&$1,n$& ------------------------------------------------------------------------ ${\mathbb{Z}_{++}}\to \operatorname{coker}({\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}) \to {\mathbb{Z}}_{a-1}$&${\mathbf{[1\infty]}}$\ $n$&$n$&$1,n$& ------------------------------------------------------------------------ ${\mathbb{Z}}_{d-1}\to \operatorname{coker}({\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]})\to {\mathbb{Z}}_{a-1}$&${\mathbf{[\infty\infty]}}$\ $n$&$\infty$&$1,n$& ------------------------------------------------------------------------ ${{\mathbb{Z}}_{\pm}}\to \operatorname{coker}({\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}) \to {\mathbb{Z}}_{a-1}$&${\mathbf{[\infty\infty]}}$\ $\infty$&0&$1,n,\infty$& ------------------------------------------------------------------------ ${\mathbb{Z}_{++}}\to{\mathbb{Z}}\oplus {\mathbb{Z}}\to{{\mathbb{Z}}_{\pm}}$&${\mathbf{[1\infty]}}$\ $\infty$&$n$&$1,n,\infty$& ------------------------------------------------------------------------ ${\mathbb{Z}}_{d-1}\to{\mathbb{Z}}_{d-1}\oplus{\mathbb{Z}}\to{{\mathbb{Z}}_{\pm}}$&${\mathbf{[\infty\infty]}}$\ $\infty$&$\infty$&$1,n,\infty$& ------------------------------------------------------------------------ ${{\mathbb{Z}}_{\pm}}\to{\mathbb{Z}}\oplus {\mathbb{Z}}\to{{\mathbb{Z}}_{\pm}}$&${\mathbf{[\infty\infty]}}$\ $ $ where “$n$” indicates an integer $\geq 2$, “${\mathbb{Z}_{++}}$” indicates a copy of ${\mathbb{Z}}$ ordered with ${\mathbb{Z}}_+={\mathbb{N}}$ and “${{\mathbb{Z}}_{\pm}}$" indicates a copy of ${\mathbb{Z}}$ ordered with ${\mathbb{Z}}_+={\mathbb{Z}}$. In addition, in all cases we have written the middle group in such a way that the map from $K_0(I)$ to $K_0(C^*(E))$ is $[x] \mapsto [(x,0)]$, and the map from $K_0(C^*(E))$ to $K_0(C^*(E)/I)$ is $[(x,y)] \mapsto [y]$. Note that in all but the first case, the order structure of the middle $K_0$-groups is irrelevant and need not be computed. \[classthmtbt\]Let $E$ and $E'$ be graphs each with two vertices such that $C^*(E)$ and $C^*(E')$ each have exactly one proper nontrivial ideal, and write the vertex matrix of $E$ as ${\left[\begin{smallmatrix}a&b\\#3&d\end{smallmatrix}\right]}$ and the vertex matrix of $E'$ as ${\left[\begin{smallmatrix}a'&b'\\#3&d'\end{smallmatrix}\right]}$. Then $$C^*(E)\otimes{\mathbb{K}}\cong C^*(E')\otimes{\mathbb{K}}$$ if and only if the following three conditions hold: 1. $a=a'$ 2. $d=d'$ 3. If $a\in\{2,\dots\}$ then 1. If $d\in\{0,\infty\}$ then $[b] = [z] [b']$ in ${\mathbb{Z}}_{a-1}$ for a unit $[z] \in {\mathbb{Z}}_{a-1}$ 2. If $d\in\{2,\dots\}$ then $[z_1][b] = [z_2][b']$ in ${\mathbb{Z}}_{\gcd{(a-1, d-1)}}$ for a unit $[z_1] \in {\mathbb{Z}}_{d-1}$ and a unit $[z_2] \in {\mathbb{Z}}_{a-1}$. Suppose $C^*(E)\otimes{\mathbb{K}}\cong C^*(E')\otimes{\mathbb{K}}$. Then $K_0(I) \cong K_0(I')$ as ordered groups and $K_0(C^*(E)/I) \cong K_0(C^*(E')/I')$ as ordered groups. From a consideration of the invariants in the above table, this implies that $a = a'$, $d = d'$, and the invariants for $C^*(E)$ and $C^*(E')$ both fall into the same case (i.e. the same row) of the table. Thus we need only consider the two cases described in (3)(a) and (3)(b). $ $ <span style="font-variant:small-caps;">Case i:</span> $a \in \{2, \ldots \}$ and $d \in \{ 0, \infty \}$. In this case there are isomorphisms $\alpha$, $\beta$, and $\gamma$ such that $$\xymatrix{ 0 \ar[r] & {\mathbb{Z}}\ar[r] \ar[d]_\alpha & {\operatorname{coker }}({\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}) \ar[r] \ar[d]_\beta & {\mathbb{Z}}_{a-1} \ar[r] \ar[d]_\gamma & 0 \\ 0 \ar[r] & {\mathbb{Z}}\ar[r] & {\operatorname{coker }}({\left[\begin{smallmatrix}b'\\#2\end{smallmatrix}\right]}) \ar[r] & {\mathbb{Z}}_{a-1} \ar[r] & 0}$$ commutes. Since the only automorphisms on ${\mathbb{Z}}$ are $\pm \operatorname{Id}$, we have that $\alpha(x) = \pm x$. Also, since the only automorphisms on ${\mathbb{Z}}_{a-1}$ are multiplication by a unit, $\gamma ([x]) = [z] [x]$ for some unit $[z] \in {\mathbb{Z}}_{a-1}$. By the commutativity of the left square $\beta([1,0]) = [(\pm1, 0)]$. Also, by the commutativity of the right square, $\beta([0,1]) = ([y, z])$ for some $y \in {\mathbb{Z}}$. It follows from the ${\mathbb{Z}}$-linearity of $\beta$ that $\beta[(r,s)] = [(\pm r + sy, sz)]$, so $\beta$ is equal to left multiplication by the matrix ${\left[\begin{smallmatrix}\pm 1&y\\#3&z\end{smallmatrix}\right]}$. We must have $\beta [(b, a-1)] = [(0,0)]$, and thus $[(\pm b + (a-1)y, (a-1)z)] = [(0,0)]$ in ${\operatorname{coker }}({\left[\begin{smallmatrix}b'\\#2\end{smallmatrix}\right]})$. Hence $\pm b + (a-1)y = b't$ and $(a-1)z = (a-1)t$ for some $t \in {\mathbb{Z}}$. It follows that $z = t$ and $\pm b + (a-1)y = b'z$, so $b \equiv \pm z \mod (a-1)$. Since $[\pm z]$ is a unit for ${\mathbb{Z}}_{a-1}$ it follows that $[b] = [z] [b']$ in ${\mathbb{Z}}_{a-1}$ for a unit $[z] \in {\mathbb{Z}}_{a-1}$. Thus the condition in (a) holds. $ $ <span style="font-variant:small-caps;">Case ii:</span> $a \in \{2, \ldots \}$ and $d \in \{2, \ldots \}$. In this case there are isomorphisms $\alpha$, $\beta$, and $\gamma$ such that $$\xymatrix{ 0 \ar[r] & {\mathbb{Z}}_{d-1} \ar[r] \ar[d]_\alpha & {\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]}) \ar[r] \ar[d]_\beta & {\mathbb{Z}}_{a-1} \ar[r] \ar[d]_\gamma & 0 \\ 0 \ar[r] & {\mathbb{Z}}_{d-1} \ar[r] & {\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b'\\#3&a-1\end{smallmatrix}\right]}) \ar[r] & {\mathbb{Z}}_{a-1} \ar[r] & 0}$$ commutes. Since the only automorphisms on ${\mathbb{Z}}_{d-1}$ are multiplication by a unit, we have that $\alpha([x]) = [z_1] [x]$ for some unit $[z_1] \in {\mathbb{Z}}_{d-1}$. Likewise, $\gamma([x]) = [z_2][x]$ for some unit $[z_2] \in {\mathbb{Z}}_{a-1}$. By the commutativity of the left square $\beta([1,0]) = [(z_1, 0)]$. Also, by the commutativity of the right square, $\beta([0,1]) = ([y, z_2])$ for some $y \in {\mathbb{Z}}$. It follows from the ${\mathbb{Z}}$-linearity of $\beta$ that $\beta[(r,s)] = [(z_1r + ys, z_2s)]$, so $\beta$ is equal to left multiplication by the matrix ${\left[\begin{smallmatrix}z_1&y\\#3&z_2\end{smallmatrix}\right]}$. Since ${\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]} {\left[\begin{smallmatrix}0\\#2\end{smallmatrix}\right]} = {\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}$, we must have $\beta [(b, a-1)] = [(0,0)]$, and thus $[(z_1 b + y(a-1), z_2(a-1))] = [(0,0)]$ in ${\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b'\\#3&a-1\end{smallmatrix}\right]})$. Hence $z_1 b + y(a-1) = (d-1)s + b't$ and $z_2(a-1) = (a-1)t$ for some $s, t \in {\mathbb{Z}}$. It follows that $z_2 = t$ and $z_1 b + y(a-1) = (d-1)s + b'z_2$. Writing $(d-1)s - y(a-1) = k \gcd{(a-1, d-1)}$ we obtain $z_1b - z_2b' = k \gcd{(a-1, d-1)}$ so that $z_1 b \equiv z_2 b' \mod \gcd{(a-1, d-1)}$ and $[z_1][b] = [z_2][b']$ in ${\mathbb{Z}}_{\gcd{(a-1, d-1)}}$. Thus the condition in (b) holds. $ $ For the converse, we assume that the conditions in (1)–(3) hold. Consider the following three cases. $ $ <span style="font-variant:small-caps;">Case I:</span> $a = 0$ or $a = \infty$. In this case, by considering the invariants listed in the above table, we see that we may use the identity maps for the three vertical isomorphisms to obtain a commutative diagram. Thus the six-term exact sequences are isomorphic, and it follows from Theorem \[clas\] that $C^*(E)\otimes{\mathbb{K}}\cong C^*(E')\otimes{\mathbb{K}}$. $ $ <span style="font-variant:small-caps;">Case II:</span> $a \in \{2, \ldots \}$ and $[b] = [z] [b']$ in ${\mathbb{Z}}_{a-1}$ for a unit $[z] \in {\mathbb{Z}}_{a-1}$. Then $b \cong zb' \mod (a-1)$. Hence $zb' - b = (a-1)y$ for some $y \in {\mathbb{Z}}$. Consider ${\left[\begin{smallmatrix}1&y\\#3&z\end{smallmatrix}\right]} : {\mathbb{Z}}\oplus {\mathbb{Z}}\to {\mathbb{Z}}\oplus {\mathbb{Z}}$. It is straightforward to check that this matrix takes ${\operatorname{im }}{\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}$ into ${\operatorname{im }}{\left[\begin{smallmatrix}b'\\#2\end{smallmatrix}\right]}$. Thus multiplication by this matrix induces a map $\beta : {\operatorname{coker }}({\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}) \to {\operatorname{coker }}({\left[\begin{smallmatrix}b'\\#2\end{smallmatrix}\right]})$. In addition, if we let $\alpha = \operatorname{Id}$ and let $\gamma$ be multiplication by $[z]$, then it is straightforward to verify that the diagram $$\xymatrix{ 0 \ar[r] & {\mathbb{Z}}\ar[r] \ar[d]_\alpha & {\operatorname{coker }}({\left[\begin{smallmatrix}b\\#2\end{smallmatrix}\right]}) \ar[r] \ar[d]_\beta & {\mathbb{Z}}_{a-1} \ar[r] \ar[d]_\gamma & 0 \\ 0 \ar[r] & {\mathbb{Z}}\ar[r] & {\operatorname{coker }}({\left[\begin{smallmatrix}b'\\#2\end{smallmatrix}\right]}) \ar[r] & {\mathbb{Z}}_{a-1} \ar[r] & 0}$$ commutes. Since $\alpha$ and $\gamma$ are isomorphisms, an application of the five lemma implies that $\beta$ is an isomorphism. It follows from Theorem \[clas\] that $C^*(E)\otimes{\mathbb{K}}\cong C^*(E')\otimes{\mathbb{K}}$. $ $ <span style="font-variant:small-caps;">Case III:</span> Suppose that $[z_1][b] = [z_2][b']$ in ${\mathbb{Z}}_{\gcd{(a-1, d-1)}}$ for a unit $[z_1] \in {\mathbb{Z}}_{d-1}$ and a unit $[z_2] \in {\mathbb{Z}}_{a-1}$. Then $z_1b - z_2b' = k \gcd{(a-1, d-1)}$ for some $k \in {\mathbb{Z}}$. Furthermore, we may write $k \gcd{(a-1, d-1)} = s(d-1) -y(a-1)$ for some $s,y \in {\mathbb{Z}}$. Consider ${\left[\begin{smallmatrix}z_1&y\\#3&z_2\end{smallmatrix}\right]} : {\mathbb{Z}}\oplus {\mathbb{Z}}\to {\mathbb{Z}}\oplus {\mathbb{Z}}$. It is straightforward to check that this matrix takes ${\operatorname{im }}{\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]}$ into ${\operatorname{im }}{\left[\begin{smallmatrix}d-1&b'\\#3&a-1\end{smallmatrix}\right]}$. Thus multiplication by this matrix induces a map $\beta : {\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]}) \to {\operatorname{coker }}({\left[\begin{smallmatrix}d-1\\#2\end{smallmatrix}\right]}{0}{a-1})$. In addition, if we let $\alpha$ be multiplication by $[z_1]$ and and let $\gamma$ be multiplication by $[z_2]$, then it is straightforward to verify that the diagram $$\xymatrix{ 0 \ar[r] & {\mathbb{Z}}\ar[r] \ar[d]_\alpha & {\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b\\#3&a-1\end{smallmatrix}\right]}) \ar[r] \ar[d]_\beta & {\mathbb{Z}}_{a-1} \ar[r] \ar[d]_\gamma & 0 \\ 0 \ar[r] & {\mathbb{Z}}\ar[r] & {\operatorname{coker }}({\left[\begin{smallmatrix}d-1&b'\\#3&a-1\end{smallmatrix}\right]}) \ar[r] & {\mathbb{Z}}_{a-1} \ar[r] & 0}$$ commutes. Since $\alpha$ and $\gamma$ are isomorphisms, an application of the five lemma implies that $\beta$ is an isomorphism. It follows from Theorem \[clas\] that $C^*(E)\otimes{\mathbb{K}}\cong C^*(E')\otimes{\mathbb{K}}$. \[fourex\] Consider the three graphs $ $ $$\xymatrix{ \ar@(u, ur)[] \ar@(u, ul)[] \ar@(d, dr)[] \ar@(d, dl)[] \bullet \ar[rr] & & \bullet} \qquad \qquad \xymatrix{\ar@(u, ur)[] \ar@(u, ul)[] \ar@(d, dr)[] \ar@(d, dl)[] \bullet \ar@/^/[rr] \ar@/_/[rr] & & \bullet} \qquad \qquad \xymatrix{\ar@(u, ur)[] \ar@(u, ul)[] \ar@(d, dr)[] \ar@(d, dl)[] \bullet \ar[rr] \ar@/^0.9pc/[rr] \ar@/_0.9pc/[rr] & & \bullet }$$ $ $ $ $ which all have graph $C^*$-algebras with precisely one proper nontrivial ideal. By Theorem \[classthmtbt\] the $C^*$-algebras of the two first graphs are stably isomorphic to each other, but not to the $C^*$-algebra of the third graph. We mention that with existing technology it would be very difficult to see directly that the $C^*$-algebras of the two first graphs in Example \[fourex\] are stably isomorphic. One approach would be to form the *stabilized graphs* (see [@tom:socatg §4]) and then attempt to transform one graph to the other through operations that preserve the stable isomorphism class of the associated $C^*$-algebra (e.g., in/outsplittings, delays). However, even in this concrete example it is unclear what sequence of operations would accomplish this and we speculate that it would not be possible using the types of operations mentioned above and their inverses. In addition, the second author’s Ph.D. thesis (see [@Tom-thesis] and the resulting papers [@rtw:ctcgws], [@tom:ecefcgws], and [@tom:cefga]) deals with extensions of graph $C^*$-algebras and shows that under certain circumstances two essential one-sink extensions of a fixed graph $G$ have stably isomorphic $C^*$-algebras if they determine the same class in $\operatorname{Ext} C^*(G)$ [@tom:ecefcgws Theorem 4.1]. In Example \[fourex\], the three displayed graphs are all essential one-sink extensions of the graph with one vertex and four edges, whose $C^*$-algebra is $\mathcal{O}_4$. We also have that $\operatorname{Ext} \mathcal{O}_4 \cong {\mathbb{Z}}_3$, and the first two graphs in Example \[fourex\] determine the classes $[1]$ and $[2]$ in ${\mathbb{Z}}_3$, respectively. Consequently, we cannot apply [@tom:ecefcgws Theorem 4.1], and we see that the methods of this paper have applications to situations not covered by [@tom:ecefcgws Theorem 4.1]. (As an aside, we mention that the second author has conjectured that if $G$ is a finite graph with no sinks or sources, if $C^*(G)$ is simple, and if $E_1$ and $E_2$ are one-sink extensions of $G$, then $C^*(E_1)$ is stably isomorphic to $C^*(E_2)$ if and only if there exists an automorphism on $\operatorname{Ext} C^*(G)$ taking the class of the extension determined by $E_1$ to the class of the extension determined by $E_2$. We see that Example \[fourex\] is consistent with this conjecture since there is an automorphism of ${\mathbb{Z}}_3$ taking $[1]$ to $[2]$.) $ $ Using the Kirchberg-Phillips Classification Theorem and our results in Theorem \[classthmtbt\] we are able to give a complete classification of the stable isomorphism classes of $C^*$-algebras of graphs satisfying Condition (K) with exactly two vertices. We state this result in the following theorem. As one can see, there are a variety of cases and possible ideal structures for these stable isomorphism classes. \[class-two-vertices\] Let $E$ and $E'$ be graphs satisfying Condition (K) that each have exactly two vertices. Let $A_E$ and $A_{E'}$ be the vertex matrices of $E$ and $E'$, respectively, and order the vertices of each so that $c \leq b$ and $c' \leq b'$. Then $C^*(E) \otimes {\mathbb{K}}\cong C^*(E') \otimes {\mathbb{K}}$ if and only if one of the following five cases occurs. - $A_E = {\left[\begin{smallmatrix}a&b\\#3&d\end{smallmatrix}\right]}$ and $A_E = {\left[\begin{smallmatrix}a'&b'\\#3&d'\end{smallmatrix}\right]}$ with $$\quad (b \neq 0 \text{ and } c \neq 0) \quad \text{ or } \quad (a = 0, 0 < b < \infty, c= 0, \text{and } d \geq 2)$$ and $$\qquad (b' \neq 0 \text{ and } c' \neq 0) \quad \text{ or } \quad (a' = 0, 0 < b' < \infty, c'= 0, \text{and } d' \geq 2)$$ and if $B_E$ is the $E^0 \times E^0_\textnormal{reg}$ submatrix of $A_E^t-I$ and $B_{E'}$ is the $(E')^0 \times (E')^0_\textnormal{reg}$ submatrix of $A_{E'}^t-I$, then $${\operatorname{coker }}(B_E : {\mathbb{Z}}^{E^0_\textnormal{reg}} \to Z^{E^0}) \cong {\operatorname{coker }}(B_{E'} : {\mathbb{Z}}^{(E')^0_\textnormal{reg}} \to Z^{(E')^0})$$ and $$\ker (B_E : {\mathbb{Z}}^{E^0_\textnormal{reg}} \to Z^{E^0}) \cong \ker (B_{E'} : {\mathbb{Z}}^{(E')^0_\textnormal{reg}} \to Z^{(E')^0}).$$ In this case $C^*(E)$ and $C^*(E')$ are purely infinite and simple. - $A_E = {\left[\begin{smallmatrix}0&b\\#3&0\end{smallmatrix}\right]}$ and $A_{E'} = {\left[\begin{smallmatrix}0&b'\\#3&0\end{smallmatrix}\right]}$ with $0 < b < \infty$ and $0 < b' < \infty$. In this case $C^*(E) \cong M_{b+1}({\mathbb{C}})$ and $C^*(E') \cong M_{b'+1}({\mathbb{C}})$, so that both $C^*$-algebras are simple and finite-dimensional. - $A_E = {\left[\begin{smallmatrix}a&b\\#3&d\end{smallmatrix}\right]}$ and $A_{E'} = {\left[\begin{smallmatrix}a'&b'\\#3&d'\end{smallmatrix}\right]}$ with $b \neq 0$ and $b' \neq 0$, $a=0 \ \Longrightarrow \ b = \infty \qquad \text{ and } \qquad b = \infty \implies (a= 0 \text{ or } a = \infty),$ and $a'=0 \ \Longrightarrow \ b' = \infty \qquad \text{ and } \qquad b' = \infty \implies (a'= 0 \text{ or } a' = \infty),$ and the conditions (1)–(3) of Theorem \[classthmtbt\] hold. In this case $C^*(E)$ and $C^*(E')$ each have exactly one proper nontrivial ideal and have ideal structure of the form $$\xymatrix{ A \ar@{-}[d] \\ \ar@{-}[d] I \\ \ \{ 0 \}. }$$ - $A_E = {\left[\begin{smallmatrix}a&\infty\\#3&d\end{smallmatrix}\right]}$ and $A_{E'} = {\left[\begin{smallmatrix}a'&\infty\\#3&d'\end{smallmatrix}\right]}$ with $a \in \{ 2, 3, \ldots \}$ and $a' \in \{2, 3, \ldots \}$, and with $a = a'$ and $d = d'$. In this case $C^*(E)$ and $C^*(E')$ each have exactly two proper nontrivial ideals and have ideal structure of the form $$\xymatrix{ A \ar@{-}[d] \\ \ar@{-}[d] I \\ \ar@{-}[d] J \\ \ \{ 0 \}. }$$ - $A_E = {\left[\begin{smallmatrix}a&0\\#3&d\end{smallmatrix}\right]}$ and $A_{E'} = {\left[\begin{smallmatrix}a'&0\\#3&d'\end{smallmatrix}\right]}$ with $$(a = a' \text{ and } d = d') \quad \text{ or } \quad (a = d' \text{ and } d = a').$$ In this case $C^*(E) \cong C^*(E') \cong I \oplus J$, where $I := \begin{cases} \mathcal{O}_a & \text{ if $a \geq 2$} \\ {\mathbb{C}}& \text{ if $a =0$} \end{cases}$ and $J := \begin{cases} \mathcal{O}_d & \text{ if $d \geq 2$} \\ {\mathbb{C}}& \text{ if $d =0$} \end{cases}$, and each $C^*$-algebra has exactly two proper nontrivial ideals and ideal structure of the form $$\xymatrix{ & A \ar@{-}[dr] \ar@{-}[dl] & \\ I \ar@{-}[dr] & & J \ar@{-}[dl] \\ & \ \{ 0 \}. & }$$ We are not able to classify $C^*$-algebras of graphs with exactly two vertices that do not satisfy Condition (K). For example if $E$ and $E'$ are graphs with vertex matrices $A_E = {\left[\begin{smallmatrix}1&b\\#3&1\end{smallmatrix}\right]}$ and $A_{E'} = {\left[\begin{smallmatrix}1&b'\\#3&1\end{smallmatrix}\right]}$, then $C^*(E)$ and $C^*(E')$ each have uncountably many ideals, and are extensions of $C({\mathbb{T}})$ by $C({\mathbb{T}}\otimes {\mathbb{K}})$. Using existing techniques, it is unclear when $C^*(E)$ and $C^*(E')$ will be stably isomorphic. We conclude this section with an example showing an application of Theorem \[clasimp\] to $C^*$-algebras with multiple proper ideals. Consider the two graphs $$\qquad \xymatrix{ & & & v \\ E & \ar@(ur, ul)[] \ar@(dr, dl)[] \ar[rru] \ar[rrd] x & & \\ & & & w } \qquad \qquad \qquad \qquad\xymatrix{ & & & v' \\ E' & \ar@(ur, ul)[] \ar@(dr, dl)[] \ar@/^/[rru] \ar@/_/[rru] \ar@/^/[rrd] \ar@/_/[rrd] x' & & \\ & & & w' }$$ The ideal $I := I_{\{v,w\}}$ in $C^*(E)$ is a largest proper ideal that is an AF-algebra, and the six-term exact sequence corresponding to $$0 \to I \to C^*(E) \to C^*(E)/I \to 0$$ is $$0 \to {\mathbb{Z}}\oplus {\mathbb{Z}}\to {\operatorname{coker }}( \left[ \begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix} \right] ) \to 0$$ where the middle map is $[(x,y)] \mapsto [(x,y,0)]$. Likewise, the ideal $I' := I_{\{v',w'\}}$ in $C^*(E')$ is a largest proper ideal that is an AF-algebra, and the six-term exact sequence corresponding to $$0 \to I' \to C^*(E') \to C^*(E')/I' \to 0$$ is $$0 \to {\mathbb{Z}}\oplus {\mathbb{Z}}\to {\operatorname{coker }}( \left[ \begin{smallmatrix} 2 \\ 2 \\ 1 \end{smallmatrix} \right] ) \to 0$$ where the middle map is $[(x,y)] \mapsto [(x,y,0)]$. If we define $\beta : {\operatorname{coker }}( \left[ \begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix} \right] ) \to {\operatorname{coker }}( \left[ \begin{smallmatrix} 2 \\ 2 \\ 1 \end{smallmatrix} \right] )$ by $\beta[(x,y,z)] = [(x+z, y+z, z)]$, then we see that the diagram $$\xymatrix{ 0 \ar[r] & {\mathbb{Z}}\oplus {\mathbb{Z}}\ar[d]_{\operatorname{Id}} \ar[r] & {\operatorname{coker }}{(\left[\begin{smallmatrix}1\\1 \\1\end{smallmatrix}\right])} \ar[d]_{\beta} \ar[r] & 0 \\ 0 \ar[r] & {\mathbb{Z}}\oplus {\mathbb{Z}}\ar[r] &{\operatorname{coker }}{(\left[\begin{smallmatrix}2\\2 \\1\end{smallmatrix}\right])} \ar[r] & 0 }$$ commutes. An application of the five lemma shows that $\beta$ is an isomorphism. It follows from Theorem \[clasimp\] that $C^*(E) \otimes {\mathbb{K}}\cong C^*(E') \otimes {\mathbb{K}}$. In the examples above, both connecting maps in the six-term exact sequences vanish. Since all $C^*$-algebras considered (and, more generally, all graph $C^*$-algebras satisfying Condition (K)) have real rank zero, the exponential map $\partial : K_0(A/I) \to K_1(I)$ is always zero. However, the index map $\partial : K_1(A/I) \to K_0(I)$ does not necessarily vanish and may carry important information. In forthcoming work, the authors and Carlsen explain how to compute this map for graph $C^*$-algebras. Stability of ideals {#stab-ideals-sec} =================== In this section we prove that if $A$ is a graph $C^*$-algebra that is not an AF-algebra, and if $A$ contains a unique proper nontrivial ideal $I$ , then $I$ is stable. If $v$ is a vertex in a graph $E$ we define $$L(v) := \{w \in E^0 : \text{ there is a path from $w$ to $v$} \}.$$ We say that $v$ is *left infinite* if $L(v)$ contains infinitely many elements. If $E = (E^0, E^1, r, s )$ is a graph, then a *graph trace* on $E$ is a function $g : E^0 \rightarrow [0,\infty)$ with the following two properties: 1. \[g-t-1\] For any $v \in G^0$ with $0 < |s^{-1}(v)| < \infty$ we have $g(v) = \sum_{s(e) = v} g(r(e))$. 2. \[g-t-2\] For any infinite emitter $v \in G^0$ and any finite set of edges $e_1, \ldots, e_n \in s^{-1}(v)$ we have $g(v) \geq \sum_{i=1}^n g(r(e_i))$. We define the *norm* of a graph trace $g$ to be the (possibly infinite) quantity $\| g \| := \sum_{v \in E^0} g(v)$, and we say a graph trace $g$ is *bounded* if $\| g \| < \infty$. \[stable-l-i-lem\] Let $E$ be a graph such that $C^*(E)$ is simple. If there exists $v \in E^0$ such that $v$ is left infinite, then $C^*(E)$ is stable. Since $C^*(E)$ is simple, it follows from [@ddmt:cag Corollary 2.15] that $E$ is cofinal. Therefore, the vertex $v$ can reach every cycle in $E$, and any vertex that is on a cycle in $E$ is left infinite. In addition, if $g : E^0 \to [0,\infty)$ is a bounded graph trace on $E$, then since $v$ is left infinite, it follows that $g(v) = 0$. Furthermore, it follows from [@mt:sgcg Lemma 3.7] that $$H := \{ w \in E^0 : g(w) = 0 \}$$ is a saturated hereditary subset of vertices. Since $C^*(E)$ is simple, it follows from [@ddmt:cag Theorem 3.5] that the only saturated hereditary subsets of $E$ are $E^0$ and $\emptyset$. Because $v \in H$, we have that $H \neq \emptyset$ and hence $H = E^0$, which implies that $g \equiv 0$. Since we have shown that every vertex on a cycle in $E$ is left infinite, and that there are no nonzero bounded graph traces on $E$, it follows from [@mt:sgcg Theorem 3.2(d)] that $C^*(E)$ is stable. \[stable-or-AF-ideal\] Let $E$ be a graph such that $C^*(E)$ contains a unique proper nontrivial ideal $I$, and let $\{ E^0, H, \emptyset \}$ be the saturated hereditary subsets of $E$. Then there are two possibilities: 1. The ideal $I$ is stable; or 2. The graph $C^*$-algebra $C^*(E)$ is a nonunital AF-algebra, and $H$ is infinite. By Lemma \[E-structure-lem\], we see that $E$ contains a unique saturated hereditary subset $H$ not equal to either $E^0$ or $\emptyset$, and also $I = I_H$. In addition, it follows from [@kdjhhsw:srga Lemma 1.6] that $I_H$ is isomorphic to the graph $C^*$-algebra $C^*({}_HE_\emptyset)$, where ${}_HE_\emptyset$ is the graph described in [@kdjhhsw:srga Definition 1.4]. In particular, if we let $$F_H := \{ \alpha \in E^* : s(\alpha) \notin H, r(\alpha) \in H, \text{ and } r(\alpha_i) \notin H \text{ for $i < | \alpha |$} \}$$ then $${}_HE_\emptyset^0 := H \cup F_H \quad \text{ and } \quad {}_HE_\emptyset^1 := \{e \in E^1 : s(e) \in H \} \cup \{ \overline{\alpha} : \alpha \in F_H \}$$ where $s(\overline{\alpha}) = \alpha$, $r(\overline{\alpha}) = r(\alpha)$, and the range and source of the other edges is the same as in $E$. Note that since $I$ is the unique proper nontrivial ideal in $C^*(E)$, we have that $I \cong C^*({}_HE_\emptyset)$ is simple. Consider three cases. <span style="font-variant:small-caps;">Case I:</span> $H$ is finite. Choose a vertex $v \in E^0 \setminus H$. By Lemma \[E-structure-lem\] $v$ is not a sink in $E$, and thus there exists an edges $e_1 \in E^1$ with $s(e_1) = v$ and $r(e_1) \notin H$. Continuing inductively, we may produce an infinite path $e_1e_2e_3 \ldots$ with $r(e_i) \notin H$ for all $i$. (Note that the vertices of this infinite path need not be distinct.) We shall show that for each $i$ there is a path from $r(e_i)$ to a vertex in $H$. Fix $i$, and let $$X := \{ w \in E^0 : \text{ there is a path from $r(e_i)$ to $w$} \}.$$ Then $X$ is a nonempty hereditary subset, and by Lemma \[E-structure-lem\] it follows that $X \cap H \neq \emptyset$. Thus there is a path from $r(e_i)$ to a vertex in $H$. Since this is true for all $i$, it must be the case that $F_H$ is infinite. In the graph ${}_HE_\emptyset$ there is an edge from each element of $F_H$ to an element in $H$. Since $H$ is finite, this implies that there is a vertex in $H \subseteq {}_HE_\emptyset^0$ that is reached by infinitely many vertices, and hence is left infinite. It follows from Lemma \[stable-l-i-lem\] that $I \cong C^*({}_HE_\emptyset)$ is stable. Thus we are in the situation described in (1). <span style="font-variant:small-caps;">Case II:</span> $H$ is infinite, and $E$ contains a cycle. Let $\alpha = \alpha_1\ldots \alpha_n$ be a cycle in $E$. Since $H$ is hereditary, the vertices of $\alpha$ must either all lie outside of $H$ or all lie inside of $H$. If the vertices all lie in $H$, then the graph ${}_HE_\emptyset$ contains a cycle, and since $C^*({}_HE_\emptyset)$ is simple, the dichotomy for simple graph $C^*$-algebras [@ddmt:cag Remark 2.16] implies that $C^*({}_HE_\emptyset)$ is purely infinite. Since $H$ is infinite, it follows that ${}_HE_\emptyset^0$ is infinite and $C^*({}_HE_\emptyset)$ is nonunital. Because $C^*({}_HE_\emptyset)$ is a simple, separable, purely infinite, and nonunital $C^*$-algebra, Zhang’s Theorem [@sz:dpmamc] implies that $I \cong C^*({}_HE_\emptyset)$ is stable. Thus we are in the situation described in (1). If the vertices of $\alpha$ all lie outside $H$, then the set $$X := \{ w \in E^0 : \text{ there is a path from $r(\alpha_n)$ to $w$} \}$$ is a nonempty hereditary set. It follows from Lemma \[E-structure-lem\] that $X \cap H \neq \emptyset$. Thus there exists a vertex $v \in H$ and a path $\beta$ from $r(\alpha_n)$ to $v$ with $r(\beta_i) \notin H$ for $i < | \beta |$. Consequently there are infinitely many paths in $F_H$ that end at $v$ (viz.  $\beta, \alpha\beta, \alpha \alpha \beta, \alpha\alpha\alpha\beta, \ldots$). Hence there are infinitely many vertices in ${}_HE_\emptyset$ that can reach $v$, and $v$ is a left infinite vertex in ${}_HE_\emptyset$. It follows from Lemma \[stable-l-i-lem\] that $I \cong C^*({}_HE_\emptyset)$ is stable. Thus we are in the situation described in (1). <span style="font-variant:small-caps;">Case III:</span> $H$ is infinite, and $E$ does not contain a cycle. Since $E$ does not contain a cycle, it follows from [@ddmt:cag Corollary 2.13] that $C^*(E)$ is an AF-algebra. In addition, since $H$ is infinite it follows that $E^0$ is infinite and $C^*(E)$ is nonunital. Thus we are in the situation described in (2). \[usenext\] If $E$ is a graph with a finite number of vertices and such that $C^*(E)$ contains a unique proper nontrivial ideal $I$, then $I$ is stable. Furthermore, if $\{E^0, H, \emptyset \}$ are the saturated hereditary subsets of $E$, then $C^*(E_H)$ is a unital $C^*$-algebra and $I \cong C^*(E_H) \otimes \mathcal{K}$. 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--- abstract: 'We discuss the possible signatures in the electroweak symmetry breaking sector by new strong dynamics at future hadron colliders such as the Tevatron upgrade, the LHC and VLHC, and $e^+e^-$ linear colliders. Examples include a heavy Higgs-like scalar resonance, a heavy Technicolor-like vector resonance and pseudo-Goldstone states, non-resonance signatures via enhanced gauge-boson scattering and fermion compositeness.' author: - '[**Conveners:**]{} Timothy L. Barklow, R. Sekhar Chivukula, Joel Goldstein, Tao Han' - '[**Working group members:**]{} D. Dominici, S. Godfrey, R. Harris, H.-J. He, P. Kalyniak, W. Kilian, K. Lynch, T. Liu, J. Mizukoshi, M. Narain, T. Ohl, M. Popovic, A. Skuja, T. Tait, W. Walkowiak, J. Womersley' bibliography: - 'p1p3\_strong\_dynamics.bib' title: | Electroweak Symmetry Breaking by Strong Dynamics\ and the Collider Phenomenology[^1] --- Introduction ============ Particle physics is on the verge of major discovery. General arguments indicate that new physics in the electroweak symmetry breaking sector must show up below the scale of 1 TeV. The experiments at the Tevatron and next generation high energy colliders such as the LHC and a TeV $e^+e^-$ linear collider will fully explore the new physics at the electroweak scale. In theories of dynamical electroweak symmetry breaking, the electroweak interactions are broken to electromagnetism by the vacuum expectation value of a fermion bilinear. These theories may thereby avoid the introduction of fundamental scalar particles, of which we have no examples in nature thus far. Prominent examples include Technicolor, topcolor, and related models. If the new dynamical scale is somewhat higher than 1 TeV, then the low energy effects or the early signature at collider experiments may be anomalous gauge boson interactions, enhanced $WW$ scattering signals, or contact 4-fermion interactions. In this report, we first briefly introduce the dynamical electroweak symmetry breaking models and parameterization of the anomalous couplings. We then summarize the collider sensitivities to probe the new dynamics at future $e^+e^-$ linear colliders in Sec. II, and at hadron colliders in Sec. III. Technicolor ----------- The earliest models[@Weinberg; @Susskind] of dynamical electroweak symmetry breaking[@Chivukula:1998if; @Simmons:2001zt] include a new non-Abelian gauge theory (“Technicolor”) and additional massless fermions (“technifermions”) which feel this new force. The global chiral symmetry of the fermions is spontaneously broken by the formation of a technifermion condensate, just as the approximate chiral $SU(2) \times SU(2)$ symmetry in QCD is broken down to $SU(2)$ isospin by the formation of a quark condensate. If the quantum numbers of the technifermions are chosen correctly ([*e.g.*]{} by choosing technifermions in the fundamental representation of an SU$(N)$ Technicolor gauge group, with the left-handed technifermions being weak doublets and the right-handed ones weak singlets) this condensate can break the electroweak interactions down to electromagnetism. The breaking of the global chiral symmetries implies the existence of Goldstone bosons, the “technipions” ($\pi_T$). Through the Higgs mechanism, three of the Goldstone bosons become the longitudinal components of the $W$ and $Z$, and the weak gauge bosons acquire a mass proportional to the technipion decay constant (the analog of $f_\pi$ in QCD). The quantum numbers and masses of any remaining technipions are model dependent. There may be technipions which are colored (octets and triplets) as well as those carrying electroweak quantum numbers, and some color-singlet technipions are too light[@Eichten:1979ah; @tpnumbers] unless additional sources of chiral-symmetry breaking are introduced. The next lightest Technicolor resonances are expected to be the analogs of the vector mesons in QCD. The technivector mesons can also have color and electroweak quantum numbers and, for a theory with a small number of technifermions, are expected to have a mass in the TeV range[@Dimopoulos:1981yf]. While Technicolor chiral symmetry breaking can give mass to the $W$ and $Z$ particles, additional interactions must be introduced to produce the masses of the standard model fermions. The most thoroughly studied mechanism for this invokes “extended Technicolor” (ETC) gauge interactions[@Eichten:1979ah; @Dimopoulos:1979es]. In ETC, Technicolor, color and flavor are embedded into a larger gauge group which is broken to Technicolor and color at an energy scale of 100s to 1000s of TeV. The massive gauge bosons associated with this breaking mediate transitions between quarks/leptons and technifermions, giving rise to the couplings necessary to produce fermion masses. The ETC gauge bosons also mediate transitions among technifermions themselves, leading to interactions which can explicitly break unwanted chiral symmetries and raise the masses of any light technipions. The ETC interactions connecting technifermions to quarks/leptons also mediate technipion decays to ordinary fermion pairs. Since these interactions are responsible for fermion masses, one generally expects technipions to decay to the heaviest fermions kinematically allowed (though this need not hold in all models). In addition to quark masses, ETC interactions must also give rise to quark mixing. One expects, therefore, that there are ETC interactions coupling quarks of the same charge from different generations. A stringent limit on these flavor-changing neutral current interactions comes from $K^0$–$\overline K^0$ mixing[@Eichten:1979ah]. These force the scale of ETC breaking and the corresponding ETC gauge boson masses to be in the 100-1000 TeV range (at least insofar as ETC interactions of first two generations are concerned). To obtain quark and technipion masses that are large enough then requires an enhancement of the technifermion condensate over that expected naively by scaling from QCD. Such an enhancement can occur if the Technicolor gauge coupling runs very slowly, or “walks”[@walking]. Many technifermions typically are needed to make the TC coupling walk, implying that the Technicolor scale and, in particular, the technivector mesons may be much lighter than 1 TeV[@Chivukula:1998if; @Eichten:1996dx]. It should also be noted that there is no reliable calculation of electroweak parameters in a walking Technicolor theory, and the values of precisely measured electroweak quantities[@langacker] cannot directly be used to constrain the models. In existing colliders, technivector mesons are dominantly produced when an off-shell standard model gauge-boson “resonates” into a technivector meson with the same quantum numbers[@ehlq]. The technivector mesons may then decay, in analogy with $\rho\to \pi\pi$, to pairs of technipions. However, in walking Technicolor the technipion masses may be increased to the point that the decay of a technirho to pairs of technipions is kinematically forbidden[@Eichten:1996dx]. In this case the decay to a technipion and a longitudinally polarized weak boson (an “eaten” Goldstone boson) may be preferred, and the technivector meson would be very narrow. Alternatively, the technivector may also decay, in analogy with the decay $\rho\to\pi \gamma$, to a technipion plus a photon, gluon, or transversely polarized weak gauge boson. Finally, in analogy with the decay $\rho \to e^+ e^-$, the technivector meson may resonate back to an off-shell gluon or electroweak gauge boson, leading to a decay into a pair of leptons, quarks, or gluons. Top Condensate and Related Models --------------------------------- The top quark is much heavier than other fermions and must be more strongly coupled to the symmetry-breaking sector. It is natural to consider whether some or all of electroweak-symmetry breaking is due to a condensate of top quarks[@Chivukula:1998if; @topcondense]. Top-quark condensation alone, without additional fermions, seems to produce a top-quark mass larger[@Bardeen:1990ds] than observed experimentally, and is therefore not favored. Topcolor assisted Technicolor[@Hill:1995hp] combines Technicolor and top-condensation. In addition to Technicolor, which provides the bulk of electroweak symmetry breaking, top-condensation and the top quark mass arise predominantly from “topcolor,” a new QCD-like interaction which couples strongly to the third generation of quarks. An additional, strong, U(1) interaction (giving rise to a topcolor $Z'$) precludes the formation of a $b$-quark condensate. The top-quark seesaw model of electroweak symmetry breaking[@Dobrescu:1998nm] is a variant of the original top-condensate idea which reconciles top-condensation with a lighter top-quark mass. Such a model can easily be consistent with precision electroweak tests, either because the spectrum includes a light composite Higgs[@Chivukula:1998wd; @Chivukula:2001er] or because additional interactions allow for a heavier Higgs[@Collins:1999rz; @He:2001fz]. Such theories may arise naturally from gauge fields propagating in compact extra spatial dimensions[@dobrescu]. A variant of topcolor-assisted Technicolor is flavor-universal, in which the topcolor SU(3) gauge bosons, called colorons, couple equally to all quarks[@equala; @equalb]. Flavor-universal versions of the seesaw model[@colorons; @universalseesaw] incorporating a gauged flavor symmetry are also possible. In these models [*all*]{}left-handed quarks (and possibly leptons as well) participate in electroweak symmetry-breaking condensates with separate (one for each flavor) right-handed weak singlets, and the different fermion masses arise by adjusting the parameters which control the mixing of each fermion with the corresponding condensate. A prediction of these flavor-universal models, is the existence of new heavy gauge bosons, coupling to color or flavor, at relatively low mass scales. A mass limit of between 0.8 and 3.5 TeV is set[@Bertram:1998wf] depending on the coloron-gluon mixing angle. Precision electroweak measurements constrain[@Burdman:1999us] the masses of these new gauge bosons to be greater than 1–3 TeV in a variety of models, for strong couplings. Enhanced gauge-boson couplings and fermion compositeness -------------------------------------------------------- If the new strong dynamics scale is somewhat higher than that accessible to the next generation of colliders, the expected signature would be enhanced gauge-boson self-interactions conventionally parameterized by the “anomalous couplings”[@anom:1987; @dawson; @herrero; @wu:1993], and the fermion contact interactions the so-called “fermion compositeness”[@comp] at a scale $\Lambda$. Although the current LEP and Tevatron experiments have put stringent bounds on the anomalous gauge-boson self-interactions, the anticipated size of those couplings due to new strong dynamics may be of order $v^2/\Lambda^2\sim 1/16\pi^2 < 10^{-3}$, smaller than the current bounds. Experiments at future colliders will reach sensitivity to this level. In particular, high energy scattering of longitudinal gauge-bosons $W_L,Z_L$ as the electroweak Goldstone bosons should be the most direct probe to the electroweak symmetry breaking sector. General arguments such as unitarity [@strong:1977; @strong:1985] indicate that new physics associated with the electroweak symmetry breaking must show up in some form at the scale of TeV, which can be accessible most likely only at higher energy colliders of next generation. Regarding the fermion compositeness, higher sensitivity will be reached at higher energies due to the energy-dependent nature of the dimension 6-operators [@comp]. In the next two sections, we will summarize the studies of the above physics scenarios at future colliders. Strong Dynamics at $e^+e^-$ Linear Colliders ============================================ An $\ee$ linear collider with $\sqrt{s}=0.5-1.5$ TeV and a luminosity of $500-1000$ pb$^{-1}$ can be a very effective probe of strong electroweak symmetry breaking. Production mechanisms and backgrounds are limited to electroweak processes, so that signal and background cross sections can be calculated exactly. The initial state is well defined not only in terms of four-momentum, but also in terms of electron (and possibly positron) helicity. Also, complete final state helicity analyses are possible, due to the fact that most if not all of the final state kinematic variables can be reconstructed. In this section we review the $\ee$ collider phenomenology of strong $\ww$ interactions which appear when there is no light Higgs particle with large couplings to vector gauge bosons. Detection of directly produced narrow-width spinless particles such as technipions [@Casalbuoni:1998fs] and top-pions [@Yue:2000xa] is straightforward up to the kinematic limit, and will not be discussed further. $\ee\ra\nu \bar{\nu}\ww$,   $\nu \bar{\nu}ZZ,\ \ \ww Z,\ \ ZZZ, \ \ \nu \bar{\nu} t\bar{t}$ -------------------------------------------------------------------------------------------- The first step in studying the reaction $\ee\ra\nu \bar{\nu}\ww$ is to separate the scattering of a pair of longitudinally polarized $W$’s, denoted by $\wwl$, from transversely polarized $W$’s and background such as $\ee\ra\ee\ww$ and $e^- \bar{\nu}W^+Z$. Studies have shown that simple cuts[@Barger:1995cn] can be used to achieve this separation in $\ee\ra\nu \bar{\nu}\ww$, $\nu \bar{\nu}ZZ$ at $\sqrt{s}=1000$ GeV, and that the signals[@Boos:1998gw; @Boos:1999kj] are comparable to those obtained at the LHC[@barger:1990; @bagger:1994; @Chanowitz:1994zh; @bagger:1995]. Furthermore, by analyzing the gauge boson production and decay angles it is possible to use these reactions to measure chiral Lagrangian parameters with an accuracy greater than that which can be achieved at the LHC [@Chierici:2001ar]. The chiral Lagrangian parameters associated with quartic gauge boson couplings can also be measured with the triple gauge boson production processes $\ee\ra\ww Z$ and $\ee\ra ZZZ$ [@Barger:1988sq; @Barger:1989fd; @Han:1998ht]. These measurements complement the $\ww$ fusion measurements, and they will play a crucial role in multi-parameter chiral Lagrangian analyses. The reaction $\ee\ra \nu \bar{\nu} t\bar{t}$ provides unique access to $\ww\ra t\bar t$ since this process is overwhelmed by the background $gg\ra t\bar{t}$ at the LHC. Techniques similar to those employed to isolate $\wwl\ra \ww, ZZ$ can be used to measure the enhancement in $\wwl\ra t\bar{t}$ production[@Barklow:1996ti; @Larios:2000xj; @RuizMorales:1999kz; @Han:2000ic]. Even in the absence of a resonance it will be possible to establish a clear signal. The ratio $S/\sqrt{B}$ is expected to be 12 for a linear collider with $\sqrt{s}=1$ TeV, 1000 fb$^{-1}$ and $80\%/0\%$ electron/positron beam polarization, increasing to 22 for the same luminosity and beam polarization at $\sqrt{s}=1.5$ TeV. $\ee\ra\ww$ ----------- Strong gauge boson interactions induce anomalous triple gauge couplings (TGC’s) at tree-level[@anom:1987; @wu:1993; @dawson; @herrero]: $$\begin{aligned} \kappa_\gamma &=& 1+\frac{e^2}{32\pi^2s_w^2}\bigl(L_{9L}+L_{9R}\bigr) \nonumber \\ \kappa_Z &=& 1+\frac{e^2}{32\pi^2s_w^2} \bigl(L_{9L}-\frac{s_w^2}{c_w^2}L_{9R}\bigr) \nonumber \\ g_1^Z &=& 1+\frac{e^2}{32\pi^2s_w^2}\frac{L_{9L}}{c_w^2} \ \nonumber .\end{aligned}$$ where $\kappa_\gamma$, $ \kappa_Z$, and $g_1^Z$ are TGC’s, $s_w^2=\sin^2\theta_w$, $c_w^2=\cos^2\theta_w$, and $L_{9L}$ and $L_{9R}$ are chiral Lagrangian parameters[@Bagger:1993vu]. Assuming QCD values for $L_{9L}$ and $L_{9R}$, $\kappa_\gamma$ is shifted by $\dkg \sim -3\times 10^{-3}$. ------------------ ------ ------ ------ ------ TGC Re Im Re Im $g^\gamma_1$ 15.5 18.9 12.8 12.5 $\kappa_\gamma$  3.5  9.8  1.2  4.9 $\lambda_\gamma$  5.4  4.1  2.0  1.4 $g^Z_1$ 14.1 15.6 11.0 10.7 $\kappa_Z$  3.8  8.1  1.4  4.2 $\lambda_Z$  4.5  3.5  1.7  1.2 ------------------ ------ ------ ------ ------ : Expected errors for the real and imaginary parts of CP-conserving TGCs assuming $\sqrt{s}=500$ GeV, ${\cal L}=500$ fb$^{-1}$ and $\sqrt{s}=1000$ GeV, ${\cal L}=1000$ fb$^{-1}$. The results are for one-parameter fits in which all other TGCs are kept fixed at their SM values.[]{data-label="tab:cp-conserving"} Table \[tab:cp-conserving\] contains the estimates of the TGC precision that can be obtained at $\sqrt{s}=500$ and 1000 GeV for the CP-conserving couplings $g^V_1$, $\kappa_V$, and $\lambda_V$ [@Abe:2001wn]. These estimates are derived from one-parameter fits in which all other TGC parameters are kept fixed at their tree-level SM values. The $4\times 10^{-4}$ precision for the TGCs $\kappa_\gamma$ and $\kappa_Z$ at $\sqrt{s}=500$ GeV can be interpreted as a precision of $0.26$ for the chiral Lagrangian parameters $L_{9L}$ and $L_{9R}$. Assuming naive dimensional analysis[@Manohar:1984md] such a measurement would provide a $8\sigma$ ($5\sigma$) signal for $L_{9L}$ and $L_{9R}$ if the strong symmetry breaking energy scale were 3 TeV (4 TeV). When $\ww$ scattering becomes strong the amplitude for $\ee\ra\wwl$ develops a complex form factor $F_T$ in analogy with the pion form factor in $\ee\ra\pi^+\pi^-$[@Peskin:1984xw; @Iddir:1990xn]. To evaluate the size of this effect the following expression for $F_T$ can be used: $$F_T = \exp\bigl[{1\over \pi} \int_0^\infty ds'\delta(s',M_\rho,\Gamma_\rho) \{ {1\over s'-s-i\epsilon}-{1\over s'}\} \bigr]$$ where $$\delta(s,M_\rho,\Gamma_\rho) = {1\over 96\pi} {s\over v^2} + {3\pi\over 8} \left[ \ \tanh ( { s-M_\rho^2 \over M_\rho\Gamma_\rho } )+1\right] \ .$$ Here $M_\rho,\Gamma_\rho$ are the mass and width respectively of a vector resonance in $\wwl$ scattering. The term $$\delta(s) = {1\over 96\pi} {s\over v^2}$$ is the Low Energy Theorem (LET) amplitude for $\wwl$ scattering at energies below a resonance. Below the resonance, the real part of $F_T$ is proportional to $L_{9L}+L_{9R}$ and can therefore be interpreted as a TGC. The imaginary part, however, is a distinct new effect. The expected $95\%$ confidence level limits for $F_T$ for $\sqrt{s}=500$ GeV and a luminosity of 500 $fb^{-1}$ are shown in Figure \[fig:fteight\], along with the predicted values of $F_T$ for various masses $M_\rho$ of a vector resonance in $\wwl$ scattering. The signal significances obtained by combining the results for $\ee\ra\nu \bar{\nu}\ww$, $\nu \bar{\nu}ZZ$[@Barger:1995cn; @Boos:1998gw] with the $F_T$ analysis of $\ww$ [@Barklow:2000ci] are displayed in Fig. \[fig:strong\_lc\_lhc\] along with the results expected from the LHC[@unknown:1999fr]. At all values of the center-of-mass energy a linear collider provides a larger direct strong symmetry breaking signal than the LHC for vector resonance masses of 1200, 1600 and 2500 GeV. Only when the vector resonance disappears altogether (the LET case in the lower right-hand plot in Fig. \[fig:strong\_lc\_lhc\] ) does the direct strong symmetry breaking signal from the $\sqrt{s}=500$ GeV linear collider drop below the LHC signal. At higher $\ee$ center-of-mass energies the linear collider signal exceeds the LHC signal. ![Direct strong symmetry breaking signal significance in $\sigma$’s for various masses $M_\rho$ of a vector resonance in $\wwl$ scattering. The numbers below the “LC” labels refer to the center-of-mass energy of the linear collider in GeV. The luminosity of the LHC is assumed to be 300 $fb^{-1}$, while the luminosities of the linear colliders are assumed to be 500, 1000, and 1000 $fb^{-1}$ for $\sqrt{s}$=500, 1000, and 1500 GeV respectively. The lower right hand plot “LET” refers to the case where no vector resonance exists at any mass in strong $\wwl$ scattering.[]{data-label="fig:strong_lc_lhc"}](p1p3_strong_dynamics_strong_lc_lhc_4x4.eps){height="17cm"} Strong Dynamics at Hadron Colliders =================================== Hadron colliders offer exciting possibilities for searches for new particles and other signs of new strong dynamics and compositeness. High luminosity $pp$ and $p\overline{p}$ machines should copiously produce proposed strongly-coupled resonances including technihadrons and excited quarks. They also probe contact interactions and vector boson scattering at extremely high energy scales. In this section we describe the expected physics reach of hadron colliders that exist (the Tevatron), are under construction (the LHC) and are being designed (the VLHC). The Tevatron ------------ The Tevatron at Fermilab has taken approximately 100 pb$^{-1}$ of $p\overline{p}$ collision data at $\sqrt{s}=1.8$ TeV (Run I). In March 2001 Run II began, with an increased energy ($\sqrt{s} = 1.96$ TeV) and a planned integrated luminosity of 2 fb$^{-1}$ (Run IIa), followed by extended high luminosity running for a total in excess of 15 fb$^{-1}$ per experiment. In Tables \[p1p3harrisa\]–\[p1p3harrisd\] announced results from Run I are tabulated along with extrapolations to RunIIa and a possible 30 fb$^{-1}$ complete RunII. ---------------------------------------- ----------------------------------- ------------------------------------------ ------------------------------ -------------------------- Run I (100 pb$^{-1}$) Run IIa (2 fb$^{-1}$) Run II (30 fb$^{-1}$) (GeV at 95% CL) (GeV at 95% CL) (GeV at 95% CL) $\rho_{T1}\rightarrow W \pi_T$ $\rightarrow l \nu b\overline{b}$ $170<M_\rho<200$[@p1p3h1] $160<M_\rho<240$[@p1p3h2] $M\rho<350-450$[@p1p3h3] (for $M_\pi\approx M\rho/2$) (for $M_\pi\approx M\rho/2$) $\omega_{T1}\rightarrow\gamma\pi_T$ $\rightarrow\gamma b\overline{b}$ $240<M_\omega<310$ ($M_\pi=120$) - - $140<M_\omega<290$ ($M_\pi=60$)[@p1p3h4] $\rho_{T1},\omega_{T1}$ $\rightarrow e^+e^-$ $M<225$[@p1p3h5] $M<410$[@p1p3h6] - $\rho_{T8}\rightarrow qq,gg$ $\rightarrow jj$ $260<M<480$[@p1p3h7] $M<770$[@p1p3h9] $M<900$[@p1p3h9] ($M_\pi>M_\rho/2$) $\rightarrow b\overline{b}$ $350<M<440$[@p1p3h8] $\rho_{T8}\rightarrow\pi_{LQ}\pi_{LQ}$ $\rightarrow b\nu b\nu$ $M<600$[@p1p3h10] $M<850$[@p1p3h12] - ($M_\pi<M_\rho/2$) $\rightarrow c\nu c\nu$ $M<510$[@p1p3h10] - - $\rightarrow b\tau b\tau$ $M<470$[@p1p3h11] - - ---------------------------------------- ----------------------------------- ------------------------------------------ ------------------------------ -------------------------- : **Sensitivity to Technicolor at the Tevatron**[]{data-label="p1p3harrisa"} ----------------------------------------------------- ------------ ----------------------- ---------------------------- ---------------------------- Channel Width Run I (100 pb$^{-1}$) Run IIa (2 fb$^{-1}$) Run II (30 fb$^{-1}$) $\Gamma/M$ (GeV at 95% CL) (GeV for $5\sigma$ signal) (GeV for $5\sigma$ signal) 0.3 $280<M<670$ $M<950$ $M<1200$ $g_T\rightarrow b\overline{b}$ 0.5 $340<M<640$[@p1p3h21] $M<860$[@p1p3h22] $M<1100$[@p1p3h22] 0.7 $375<M<560$ $M<770$ $M<1000$ 0.3 - $M<1110$ $M<1400$ $g_T\rightarrow t\overline{t}\rightarrow l\nu+jets$ 0.5 - $M<1040$[@p1p3h23] $M<1350$[@p1p3h23] 0.7 - $M<970$ $M<1290$ 0.3 - $M<1000$ $M<1200$ $g_T\rightarrow t\overline{t}\rightarrow 6\ jets$ 0.5 - $M<900$[@p1p3h24] $M<1130$[@p1p3h24] 0.7 - $M<800$ $M<1100$ ----------------------------------------------------- ------------ ----------------------- ---------------------------- ---------------------------- : **Sensitivity to Topgluons at the Tevatron**[]{data-label="p1p3harrisb"} --------------------------------------------------------------------- ----------------- ----------------------- ---------------------------- ---------------------------- Channel Width Run I (100 pb$^{-1}$) Run IIa (2 fb$^{-1}$) Run II (30 fb$^{-1}$) $\Gamma/M$ (GeV at $95\%$ CL) (GeV for $5\sigma$ signal) (GeV for $5\sigma$ signal) $Z'$ Model I[^2] $\rightarrow t\overline{t}\rightarrow l\nu + jets$ 0.02 - - $M<830$[@p1p3h24] 0.04 - - $M<670$ $Z'$ Model II $\rightarrow t\overline{t}\rightarrow l\nu + jets$ 0.02 - $M<720$[@p1p3h24] $M<980$[@p1p3h24] 0.04 - $M<950$ $M<1200$ $Z'$ Model III $\rightarrow t\overline{t}\rightarrow l\nu + jets$ 0.02 - $M<600$[@p1p3h24] $M<910$[@p1p3h24] 0.04 - $M<800$ $M<1000$ 0.012 $M<480$ - - $Z'$ Model IV $\rightarrow t\overline{t}\rightarrow l\nu + jets$ 0.02 $M<650$[@p1p3h32] $M<980$[@p1p3h24] $M<1200$[@p1p3h24] 0.04 $M<780$ $M<1100$ $M<1300$ (GeV at 95% CL) (GeV at 95% CL) $b\overline{b}h_b\rightarrow b\overline{b} b\overline{b}$ - $M<270$[^3] $M<380$ --------------------------------------------------------------------- ----------------- ----------------------- ---------------------------- ---------------------------- : **Sensitivity to Topcolor $Z'$ and $h_b$ at the Tevatron**[]{data-label="p1p3harrisc"} ------------------------------------------ ---------------------------- ---------------------------- ------------------------- Channel Run I (100 pb$^{-1}$) Run IIa (2 fb$^{-1}$) Run II (30 fb$^{-1}$) (TeV at 95% CL) (TeV at 95% CL) (TeV at 95% CL) $\Lambda^\pm(qq\rightarrow qq)$ 2.7[@p1p3h41] - - 2.4 - - $\Lambda^\pm(qq\rightarrow ee)$ 3.3[@p1p3h42] 6.5[@p1p3h43] 14[@p1p3h43] 4.2 10 2 0 $\Lambda^\pm(qq\rightarrow\mu\mu)$ 2.9[@p1p3h44] - - 4.2 - - $\Lambda^\pm(qq\rightarrow\gamma\gamma)$ - 0.75[@p1p3h43; @p1p3h45] 0.9[@p1p3h43; @p1p3h45] - 0.71 - $q^*\rightarrow q\gamma,qW$ 0.54[^4][@p1p3h46] 0.91[@p1p3h9] 1.18[@p1p3h9] (TeV for $5\sigma$ signal) (TeV for $5\sigma$ signal) $q^*\rightarrow qg$ 0.76[^5] 0.94[@p1p3h49] 1.1[@p1p3h49] ------------------------------------------ ---------------------------- ---------------------------- ------------------------- : **Sensitivity to Compositeness at the Tevatron.** In each channel, $\Lambda^+$ is the upper entry and $\Lambda^-$ the lower.[]{data-label="p1p3harrisd"} The LHC ------- Despite the challenge at hadron colliders in the search for new strong dynamics at the TeV scale, much theoretical work has been performed at the LHC[@barger:1990; @bagger:1994; @Chanowitz:1994zh; @bagger:1995]. Many studies of strong EWSB at ATLAS and CMS have been performed and summarized in several places[@atlastdr; @p1p3ianh; @p1p3yellow; @New]. An expected “low luminosity” period will collect 30 fb$^{-1}$ of data at $\sqrt{s}=14$ TeV over the first three years of operation, and will be followed by a similar “high luminosity” period collecting up to 300 fb$^{-1}$. High luminosity running (up to $10^{34}$ cm$^{-2}$s$^{-1}$) presents many experimental challenges with an average of 20 collisions per beam crossing, degrading tracking and electron identification capabilities particularly in the forward region. As an example of a Technicolor resonance search, ATLAS have considered the production of 500 GeV technirho in a multiscale Technicolor model and its signal in the channel $\rho_T^\pm\rightarrow WZ\rightarrow l^\pm\nu l^+l^-$ [@atlastdr]. This study assumes the 30 fb$^{-1}$ of low luminosity data and hence the full lepton ID and tracking capabilities of the detector. The expected signal significance is strongly dependent on the input model parameters: a narrow resonance ($\Gamma_{\rho_T} = 1.1$ GeV) which is not allowed to decay to $\pi_T\pi_T$ ($m_{\pi_T}>m_{\rho_T}/2$) could have $S/\sqrt{B}\approx 80$; but for $\Gamma_{\rho_T}=110$ GeV and $m_{\pi_T}=110$ GeV this would drop to an indiscernable $S/\sqrt{B}\approx0.3$. The masses of observable resonances at the LHC are expected to be 5-10$\times$ those at the Tevatron. A $Z'$ with couplings similar to those of the Standard Model $Z$ should be observable up to $m_{Z'}\approx5$ TeV and direct observation of excited quarks of $m_{q^*}\approx 6$ TeV is possible[@p1p3h49]. The reach for compositeness scales is similarly enhanced, with 300 fb$^{-1}$ of dijet data being sensitive to $\Lambda\approx 40$ TeV. A further possibility at the LHC is that as $\sqrt{\hat{s}}$ begins to exceed 1 TeV, strong interaction effects in $WW$ scattering could become detectable. If jets can be reliably tagged in the forward region at high luminosities, a signal should be observable with the full 300 fb$^{-1}$. The Super-LHC ------------- There has been some discussion of upgrading the LHC in luminosity and energy after the 300 fb$^{-1}$ run is complete. A possible (though unlikely) doubling of the energy has been considered along with a tenfold increase in instantaneous luminosity. Since the LHC detectors were not designed for these conditions only jet and muon information is likely to be usefull. Such an upgrade could double the reach for a $Z'$ ($m_{Z'}\approx 10$ TeV) and compositeness ($\Lambda\approx 80$ TeV), and significantly increase the sensitivity for excited quarks ($m_{q^*}\approx 9$ TeV) and the scale of $WW$ scattering available ($\sqrt{\hat{s}}\approx1.5$ TeV, assuming that forward jet tagging is still possible). Unfortunately, most of these gains come from the energy increase, which is less plausible than a simple luminosity upgrade. The VLHC -------- A staged 40-175 TeV $p\overline{p}$ collider operating at luminosities comparable to the LHC (1-2$\times10^{34}$ cm$^{-2}$s$^{-1}$) has been proposed[@vlhctdr]. Studies of such a machine’s physics reach are in progress (see also the E4 Working Group report), but the direct reach for excited quark resonances is expected to be $m_{q^*}\approx 25$ TeV for 10 fb$^{-1}$ at $\sqrt{s}=100$ TeV[@p1p3h49], and $WW$ scattering could be probed at the scale of $2-3$ TeV. New signatures could become detectable at such high center-of-mass energies. For example, in topcolor models, direct $\chi$ pair production and subsequent decays $\chi\rightarrow ht\rightarrow t\bar{t}t$ could occur[@He:2001fz], with a $6t$ final state. Such a heavy state may only be copiously produced. The cross section for this process with $m_\chi=1$ TeV would be $\sim 10$ pb, as shown in Fig. [\[fig:timt\]]{}. ![$\chi$-pair production in top-color models at high energy hadron colliders leading to 6-top events. []{data-label="fig:timt"}](p1p3_strong_dynamics_ppcc.eps){height="10cm"} In interactions with $\sqrt{\hat{s}}\gg \Lambda_{TC}$, it is possible (in analogy with QCD) that asymptotically free techniquarks could be produced that subsequently hadronize into technijets consisting of weak vector bosons and technihadrons. A technijet would manifest itself as an extremely massive but significantly boosted (and hence not necessarily wide) jet in a VLHC detector. The production rate for such a process can be significant: For $m_{Q_T}=400$ GeV with $\sqrt{s}=100$ TeV the dijet differential cross section for technijets exceeds that for $t\overline{t}$ for dijet masses $>900$ GeV. Exploration of technijets could provide the ultimate determination of the TC dynamics. As shown in Fig. \[fig:tao\], a representative techni-quark may decay subsequently into multiple jets and the separation between any two jets may be small enough so that experimental signature would be a very massive (but not too fat) jet. ![Maximum separation in $\Delta R$ distribution among the jets from a heavy techni-quark decay. []{data-label="fig:tao"}](p1p3_strong_dynamics_deltar.ps){height="8.5cm"} We would like to thank the participants in the Strong Electroweak Symmetry Breaking Working group for the contribution during the Snowmass workshop, on which the materials summarized in this report are based. T.L.B. was supported by Department of Energy contract DE-AC03-76SF00515. R.S.C. was supported in part by a DOE grant DE-FG02-91ER40676. T.H. was supported in part by a DOE grant No. DE-FG02-95ER40896 and in part by the Wisconsin Alumni Research Foundation. [^1]: This is the report of the Strong Electroweak Symmetry Breaking Working Group at 2001 Snowmass summer studies. [^2]: $Z'$ models described in [@p1p3h31] [^3]: Using $y_b/y_b^{SM}=72$ in Fig 8b of [@p1p3h34] [^4]: 25 pb$^{-1}$ [^5]: DØ $q^*$ search (Bertram) combined with [@p1p3h48]
--- abstract: 'In three dimensional spacetime with negative cosmology constant, the general relativity can be written as two copies of SO$(2,1)$ Chern-Simons theory. On a manifold with boundary the Chern-Simons theory induces a conformal field theory–WZW theory on the boundary. In this paper, it is show that with suitable boundary condition for BTZ black hole, the WZW theory can reduce to a massless scalar field on the horizon.' author: - Jingbo Wang - 'Chao-Guang Huang' bibliography: - 'cft2.bib' title: The Conformal Field Theory on the Horizon of BTZ Black Hole --- Introduction ============ In three dimensional spacetime, the general relativity become simplified since it has no local degrees of freedom [@carbook1]. Indeed the theory is equivalent to Chern-Simons theory with suitable gauge group [@at1; @witten1]. It is a surprise that the black hole solution can exist when theory has negative cosmology constant $\Lambda<0$. This black hole-so called BTZ black hole [@btz1] can have an arbitrary high entropy which is difficult to understand since the theory has no local degrees of freedom. This mystery can be understood if one starts from the Chern-Simons formula. It is a standard result that on a manifold with boundary the Chern-Simons theory induces a Wess-Zumino-Witten (WZW) theory on the boundary which is a conformal field theory. Carlip use this WZW theory to explain the entropy of the BTZ black hole [@carlip1]. Later it was shown that, for the boundary at conformal infinity rather than the horizon, the Chern-Simons theory reduces to a Liouville theory on the boundary [@chp1; @rs1]. This Liouville theory has the right central charge to give the entropy of BTZ black hole if one use the Cardy formula [@cardy1; @cardy2]. For a review along this line, see Ref.[@carlip2]. There are other conformal field theories which start from Brown and Henneaux’s seminal work [@bh1]. They observed that the asymptotic symmetry group of $AdS_3$ is generated by two copies of Virasoro algebra, which correspond to a conformal field theory. This result can be seen as pioneering work of $AdS_3/CFT_2$ [@kra1]. Based on this result, the entropy of BTZ black hole can be calculated [@str1; @bss1], which matches the Bekenstein-Hawking formula. But most of those conformal field theories are taken to be at conformal infinity (although with exceptions, such as [@carlip1; @mp1; @mit1]). A physical more appealing location should be the horizon of black hole. In this paper, we consider the field theory just on the horizon. Starting from the Chern-Simons theory, with suitable boundary condition, it is shown that the WZW theory reduces to a chiral massless scalar field on the horizon. So on the BTZ horizon, there are two chiral massless scalar field since the 3D general relativity contain two copied of Chern-Simons theories. The paper is organized as follows. In section II, we summary the relation between gravity, Chern-Simons theory and the WZW theory. In section III, the BTZ black hole is considered. With suitable boundary condition, the boundary WZW theory reduces to a chiral massless scalar field theory. Section IV is the conclusion. Gravity, Chern-Simons theory and WZW theory =========================================== As first shown in Ref.[@at1], $(2+1)-$dimensional general relativity can be written as a Chern-Simons theory. For the case of negative cosmology constant $\Lambda=-1/L^2$, one can define two SO$(2,1)$ connection 1-form $$\label{1} A^{(\pm)a}=\omega^a\pm \frac{1}{L} e^a,$$where $e^a$ and $\omega^a$ are the co-triad and spin connection 1-form respectively. Then up to boundary term, the first order action of gravity can be rewritten as $$\label{2}\begin{split} I_{GR}[e,\omega]=\frac{1}{8\pi G}\int e^a \wedge ({\rm d}\omega_a+\frac{1}{2}\epsilon_{abc}\omega^b \wedge \omega^c)-\frac{1}{6L^2}\epsilon_{abc}e^a\wedge e^b \wedge e^c\\=I_{CS}[A^{(+)}]-I_{CS}[A^{(-)}], \end{split}$$ where $A^{(\pm)}=A^{(\pm)a}T_a$ are SO$(2,1)$ gauge potential, and the Chern-Simons action is $$\label{3} I_{CS}[A]=\frac{k}{4\pi}\int Tr\{A\wedge {\rm d}A+\frac{2}{3}A\wedge A \wedge A\},$$ with $$\label{4} k=\frac{L}{4G}.$$ Similarly, the CS equation $$\label{5} F^\pm={\rm d}A^{(\pm)}+A^{(\pm)} \wedge A^{(\pm)}=0$$is equivalent to the requirement that the connection is torsion-free and the metric has constant negative curvature. The equation implies that the potential $A$ can be locally written as $$\label{5a} A=g^{-1}{\rm d}g.$$ When the manifold has a boundary, a boundary term must be added. Assume the boundary has topology $\partial M=R\times S^1$. The usual boundary term is $$\label{6} I_{bd}=\frac{k}{4\pi}\int_{\partial M}Tr A_u A_{\tilde{u}},$$where $u$ and $\tilde{u}$ are two coordinates on the boundary. The boundary condition is chosen to be $$\label{7} \delta A_u|_{\partial M}=0,\quad or \quad \delta A_{\tilde{u}}|_{\partial M}=0$$ depend on the condition. With the boundary term, the total action, $I_{CS}[A]+I_{bd}[A]$, is not gauge-invariant under the gauge transformation $$\label{8} \bar{A}=g^{-1}Ag+g^{-1}{\rm d}g.$$To restore the gauge-invariant, the Wess-Zumino-Witten term is introduced for the first boundary condition[@ogu1; @carlipbd1]: $$\label{9} I^+_{WZW}[g^{-1},A_u]=\frac{1}{4\pi}\int_{\partial M}Tr(g^{-1}\partial_u g g^{-1}\partial_{\tilde{u}}g+2g^{-1}\partial_{\tilde{u}}gA_u)+\frac{1}{12\pi}\int_M Tr(g^{-1}{\rm d}g)^3,$$ which is chiral WZW action for a field $g$ coupled to a background gauge potential $A_u$. With the WZW term, the full action is gauge-invariant $$\label{10} (I_{CS}+I_{bd})[\bar{A}]+kI^+_{WZW}[e^{-1},\bar{A}]=(I_{CS}+I_{bd})[A]+kI^+_{WZW}[g^{-1},A].$$ Thus, the gauge transformation $g$ become dynamical at the boundary, and are described by the WZW action which is a conformal field theory. Those ‘would-be gauge degrees of freedom’[@carlipwb1] are present because the gauge invariant is broken at the boundary. The boundary action on the horizon of BTZ black hole ==================================================== In the previous section, the boundary of manifold can be arbitrary. If the horizon of BTZ black hole is considered, more reduction can be made due to the special property of the horizon. The BTZ black hole ------------------ To study the physics at horizon, it is more suitable to use advanced Eddington coordinate. The metric of BTZ black hole can be written as $$\label{11} ds^2=-N^2 dv^2+2 dv dr+r^2 (d\varphi+N^\varphi dv)^2.$$ Choose the following co-triads [@awd1] $$\label{12} l_a=-\frac{1}{2}N^2 dv+dr,\quad n_a=-dv,\quad m_a=r N^\varphi dv+r d\varphi,$$ which gives the following connection: $$\label{13} A^{-(\pm)}=-(N^\varphi\mp \frac{1}{L})dr-\frac{N^2}{2} d(\varphi\pm\frac{v}{L}),\quad A^{+(\pm)}=-d(\varphi\pm\frac{v}{L}),\quad A^{2(\pm)}=r (N^\varphi\pm\frac{1}{L}) d(\varphi\pm\frac{v}{L}),$$ where $A^{\pm}=(A^0\pm A^1)/\sqrt{2}$. Define new variables which are useful later, $$\label{14} u=\varphi-\frac{v}{L}, \quad \tilde{u}=\varphi+\frac{v}{L}.$$ A crucial property of the connection is that, on the whole manifold, one has $$\label{15} A^{(+)}_u\equiv 0,\quad A^{(-)}_{\tilde{u}}\equiv 0.$$ Since the topology of the space-section is cylinder, which is non-trivial, the vacuum Chern-Simons equation $F=0$ will be solved by non-periodic group element [@rs1] $$\label{16} A=Q^{-1} {\rm d} Q.$$ For a general SO$(2,1)$ group element $Q(\tilde{u},u,r)$, using the Gauss decomposition, it can be written as $$\label{17} Q=\left( \begin{array}{cc} 1 & \frac{1}{\sqrt{2}}x_1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{cc} e^{-\Psi_1/2} & 0 \\ 0 & e^{-\Psi_1/2} \\ \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{\sqrt{2}}y_1 & 1 \\ \end{array} \right).$$ Within this parameter, the WZW action is [@chp1] $$\label{18}\begin{split} kI_{WZW}=\frac{k}{4\pi}\int_{\partial M}du d\tilde{u}\frac{1}{2}(\partial_u \Psi \partial_{\tilde{u}} \Psi-e^\Psi (\partial_u x \partial_{\tilde{u}} y+\partial_u y \partial_{\tilde{u}} x)). \end{split}$$ Gauge transformation -------------------- Now we consider the gauge transformation (\[8\]) with group element $g_1$ for the $A^{(+)}$. In following we omit the superscript ${(+)}$. To preserve the boundary condition $$\label{19} \delta A_u|_{\partial M}=0,$$the gauge transformation should be $g_1=g_1(r,\tilde{u})$. But it is not enough. This boundary condition can’t tell us whether we are dealing with a black hole or not, so more restricted boundary conditions are need. Near the horizon, a small parameter $\epsilon=r-r_+$ can be defined, and $N^2\approx 2 \kappa \epsilon$, thus $$\label{20} A^-_{\tilde{u}}\approx-\kappa \epsilon.$$ Since this condition reflect the property of the horizon, we want the gauge transform to keep this property, thus $$\label{21} \bar{A}^-_{\tilde{u}}\sim O(\epsilon)=C_1 \epsilon.$$ Assume the gauge transformation is given by SO$(2,1)$ group element $$\label{22} g_1(x_1,y_1,\Psi_1)=\left( \begin{array}{cc} 1 & \frac{1}{\sqrt{2}}x_1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{cc} e^{-\Psi_1/2} & 0 \\ 0 & e^{-\Psi_1/2} \\ \end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{\sqrt{2}}y_1 & 1 \\ \end{array} \right),$$ under the gauge transformation (\[8\]), $$\label{23} \bar{A}^-=e^{\Psi_1} (A^--A^2 x_1+A^+ x_1^2/2+dx_1),$$ since $A^2, A^+$ are both finite at horizon, to keep the boundary condition (\[21\]), one need $$\label{24} x_1(r,\tilde{u})=\epsilon h(\tilde{u}),$$ where $h(\tilde{u})$ is a finite function at horizon. And also $\Psi_1(r,\tilde{u})$ is finite at horizon. The other component transforms into $$\label{25}\begin{split} \bar{A}^2=A^2 (1-e^{\Psi_1} y_1 x_1)-A^+ x_1(1-e^{\Psi_1} y_1 x_1/2)+A^- e^{\Psi_1} y_1 +d \Psi_1+e^{\Psi_1} y_1d x_1,\\ \bar{A}^+=A^+ e^{-\Psi_1}(1-e^{\Psi_1} y_1 x_1/2)^2+A^2 y_1(1-e^{\Psi_1} y_1 x_1/2)+A^- e^{\Psi_1} y_1^2/2+y_1 d\Psi_1+dy_1 +y_1^2e^{\Psi_1}dx_1/2. \end{split}$$ Those components are required to be finite at the horizon, so gives $$\label{26} y_1(r_+)=finite, \quad \Psi_1(r=r_+)=finite.$$ So the second term in the action (\[18\]) vanish $$\label{27} 2 e^{\Psi_1} \partial_a x_1 \partial_b y_1 \backsim \epsilon= 0$$ on the horizon. The final action on the horizon is $$\label{28}\begin{split} kI_{WZW}=\frac{k}{4\pi}\int_{\partial M}du d\tilde{u}\frac{1}{2}\partial_u \Psi_1 \partial_{\tilde{u}} \Psi_1\\ =\frac{k}{4\pi L}\int_{\partial M}d\varphi dv [(\partial_v \Psi_1)^2- L^2 (\partial_{\varphi} \Psi_1)^2], \end{split}$$ with $\Psi_1$ depend only on $\tilde{u}=\varphi+\frac{v}{L}$. So it is a chiral massless scalar field. The similar results can be get for the $A^{(-)}$, which gives another chiral massless scalar field $\Psi_2$ depending only on $u$. Conclusion ========== In this paper, the field theory on the horizon of BTZ black hole is investigated. Starting from the Chern-Simons formula, one get a chiral WZW theory on any boundary. Restrict to the horizon, this WZW theory reduces further to a chiral massless scalar field theory. Since the general relativity is equivalent to two copies of CS theory, the final theory on the horizon is two chiral massless scalar field theory with opposite chirality. Compared with the conformal field theories on the conformal boundary, the massless scalar field theory-which is also a conformal field theory–is more revelent to black hole physics. It is just on the horizon. But the central charge of this theory is $c=1$ [@cft1], which is too small to account the entropy of the BTZ black hole if one use the Cardy formula. The conformal symmetry here is different with that appears in Carlip’s effective description of the black hole entropy in arbitrary dimension [@carlip4]. As noticed in [@carlip3], the symmetry of this paper is on the $``\varphi-v$ cylinder“, while the symmetry of [@carlip4] is on the $``r-v$ plane”. In the previous work [@wmz; @wang1; @wh1; @wh2; @wh3; @hw1; @hw2], it was shown that the boundary degrees of freedom can also be described by a BF theory. Since both the BF theory and the massless scalar field theory are on the horizon, the relation between those two theories need further investigated. This work is supported by the NSFC (Grant No. 11690022 and No. 11647064).
--- abstract: 'Vortex fiber nulling (VFN) is a method that may enable the detection and characterization of exoplanets at small angular separations (0.5-2 $\lambda/D$) with ground- and space-based telescopes. Since the field of view is within the inner working angle of most coronagraphs, nulling accesses non-transiting planets that are otherwise too close to their star for spectral characterization by other means, thereby significantly increasing the number of known exoplanets available for direct spectroscopy in the near-infrared. Furthermore, VFN targets planets on closer-in orbits which tend to have more favorable planet-to-star flux ratios in reflected light. Here, we present the theory and applications of VFN, show that the optical performance is approximately equivalent for a variety of implementations and aperture shapes, and discuss the trade-offs between throughput and engineering requirements using numerical simulations. We compare vector and scalar approaches and, finally, show that beam shaping optics may be used to significantly improve the throughput for planet light. Based on theoretical performance, we estimate the number of known planets and theoretical exoEarths accessible with a VFN instrument linked to a high-resolution spectrograph on the future Thirty Meter Telescope.' author: - | Garreth Ruane, Daniel Echeverri, Nemanja Jovanovic, Dimitri Mawet, Eugene Serabyn J. Kent Wallace, Jason Wang, and Natasha Batalha\ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,\ Pasadena, CA 91109, USA\ Department of Astronomy, California Institute of Technology, 1200 E. California Blvd.,\ Pasadena, CA 91125, USA\ Department of Astronomy and Astrophysics, University of California Santa Cruz,\ Santa Cruz, CA 95064, USA bibliography: - 'Library.bib' title: '**Vortex fiber nulling for exoplanet observations: conceptual design, theoretical performance, and initial scientific yield predictions** ' --- INTRODUCTION {#sec:intro} ============ Nulling interferometry is used to reduce the intensity of starlight prior to detection while allowing for the detection and spectral characterization of exoplanets with acceptable losses in planet signal, resulting in a significant improvement to the raw signal-to-noise ratio ($S/N$). Ronald N. Bracewell originally proposed using an interferometer to null starlight for the purpose of exoplanet detection in 1978[@Bracewell1978]. Bracewell’s interferometer used two symmetric apertures to create an elongated interference pattern, or fringes, which could be rotated in time to modulate the planet signal, while the stellar leakage remained constant[@Bracewell1979]. Building on this concept, Haguenauer and Serabyn (2006)[@Haguenauer2006] showed that a single-mode optical fiber placed in the image plane improved the rejection of starlight in the Bracewell configuration. An alternate interferometric concept proposed by Swartzlander (2001)[@Swartzlander2001] used a phase mask in the pupil plane, rather than individual sub-apertures, to create an optical vortex in the image plane with a central null and a rotationally-symmetric bright fringe. Vortex fiber nulling (VFN)[@Ruane2018_VFN; @Echeverri2019_VFN] combines elements from all of these concepts to form a simple, single-aperture, broadband fiber nulling interferometer with rotationally-symmetric throughput for planet light. In scenarios where an exoplanet is barely resolved from its host star, VFN selectively suppresses the starlight while light from the planet may be efficiently routed to a diffraction-limited spectrograph via a single-mode optical fiber. The improved $S/N$ afforded by VFN may enable astronomers to detect the planet and potentially characterize its atmosphere by reducing the integration time required to do so by orders of magnitude. Although the optimal spectral resolution for this purpose remains an open question, with sufficiently high spectral resolution, cross-correlation techniques offer the potential to characterize the atmosphere of exoplanets in reflected light with modest $S/N$ per spectral channel[@SparksFord2002; @Riaud2007; @Snellen2015; @Wang2017]. When combined with such techniques, VFN is well suited for follow up observations of planets that were previously detected by radial velocity or transit methods, such as the Earth-sized planets in the habitable zone of nearby M stars: Proxima Centauri b[@ProxCen] and Ross 128 b[@Bonfils2017]. The two key optical components for VFN are a vortex phase mask and a single-mode fiber (SMF). Vortex phase masks are also utilized in vortex coronagraphs[@Mawet2005; @Foo2005], which are currently operating on several ground-based telescopes[@Mawet2010b; @Serabyn2010; @Serabyn2017; @Mawet2017; @Ruane2017; @Ruane2019_RDI] and are a leading design for future space telescopes with coronagraphs[@Ruane2018_JATIS]. VFN-capable instruments will take advantage of more than a decade of technological development to improve the quality and bandwidth of vortex phase masks at visible and infrared wavelengths for the purpose of high-contrast imaging[@Delacroix2013; @SerabynTDEM1; @SerabynTDEM2; @Serabyn2019]. Coronagraph instruments are also starting to take advantage of SMFs for improving the rejection of stellar speckles in cases where the planet is spatially resolved from the star[@Mawet2017_HDCII; @Llop2019]. Thus, the optics required for VFN are readily available and may be commonplace in future adaptive optics (AO) instruments designed for exoplanet science. While multi-aperture fiber nulling has been demonstrated on sky using the 200-inch Hale Telescope at Palomar Observatory[@Serabyn2019_PFN] and preparation is underway for the first on-sky demonstration of VFN as part of the Keck Planet Imager and Characterizer (KPIC) instrument[@Mawet2016_KPIC; @Mawet2017_KPIC; @Echeverri2019b_VFN], in the following we present conceptual design trades, theoretical performance, and estimated scientific yield of a VFN mode operating in the near-infrared on the future Thirty Meter Telescope (TMT) Multi-Object Diffraction-limited High-resolution Infrared Spectrograph (MODHIS) instrument[@Mawet2019_whitepaper]. Specifically, we will show that it is feasible to detect $>$10 known planets with integration times of tens of hours and potentially search for rocky planets in the habitable zones of nearby stars. ![(a) A schematic optical layout for an AO-fed fiber injection unit for exoplanet spectroscopy. VFN may be implemented in a number of ways, including configurations with the vortex mask (b) in a focal plane and (c) in a pupil. DM: Deformable mirror. OAP: Off-axis parabola. FP: Focal plane. PP: Pupil plane. SMF: Single-mode fiber. []{data-label="fig:opticallayouts"}](layouts.pdf){width="\linewidth"} VORTEX FIBER NULLING CONCEPT ============================ A vortex fiber nuller is formed by introducing a vortex mask within an AO-fed fiber injection unit (FIU), which forms a link between the telescope’s AO system and a spectrograph. Figure \[fig:opticallayouts\]a shows a representative optical layout of such an instrument. Light from astronomical sources couples into the SMF with coupling efficiency given by the square-magnitude of the so-called overlap integral: $$\eta=\left|\int E(\mathbf{r}) \Psi(\mathbf{r})dA\right|^2, \label{eqn:couplingeff}$$ where $\mathbf{r}=(r,\theta)$ are the polar coordinates in the plane of the SMF entrance face, $E(\mathbf{r})$ is the complex field, and $\Psi(\mathbf{r})$ is the fiber mode profile[@Shaklan1988; @Jeunhomme1989]. A fiber nuller[@Haguenauer2006; @Martin2017] creates a condition where the stellar field is orthogonal to $\Psi(\mathbf{r})$ and thus does not couple into the SMF, while light from planets within a certain range of angular separations couples into the SMF with reasonable efficiency. ![(a) The fundamental mode of a single-mode step-index fiber. (b) The nominal complex point spread function (PSF) of a telescope couples into a SMF, though with imperfect efficiency due to the mismatch between the mode and incident electric field profile. (c-d) However, a PSF containing an optical vortex at its center is orthogonal to the fiber mode and therefore does not couple into the SMF. (c) and (d) show cases with charges $l=1$ and $l=2$, respectively.[]{data-label="fig:fibermodes"}](fibermodes.pdf){width="\linewidth"} The principle of VFN is that the stellar beam is made into an optical vortex. For example, the field may take the form: $E(\mathbf{r})=f(r)\exp(il\theta)$, where $f(r)$ is the radial component of the field amplitude and $l$ is an integer known as the charge. The fundamental mode of a single-mode step-index fiber is also rotationally symmetric and approximately Gaussian. In this case, the overlap integral is separable and the polar term is $$\int_0^{2\pi} \exp(il\theta)d\theta = \left\{ \begin{array}{ll} 2\pi & l=0 \\ 0 & l\ne0 \\ \end{array} \right.. \label{eqn:overlap_az}$$ Thus, for $l\ne0$, the stellar field is orthogonal to the SMF mode and is therefore rejected by the SMF. Figure \[fig:fibermodes\] shows the fiber mode along with example stellar fields with $l=0$, which couples into the SMF, and with $l=1$ and $l=2$, which do not couple. While the rotationally symmetric amplitude profile is used here as a convenient example, Eqn. \[eqn:couplingeff\] computes to zero for a variety of symmetries for the complex fields produced in telescopes and SMF modes. In theory, the optical vortex can be introduced anywhere along the stellar beam path. Figs. \[fig:opticallayouts\]b,c show two examples where the phase would be singular along chief ray in an otherwise perfect system. In Fig. \[fig:opticallayouts\]b, the layout is similar to that of a vortex coronagraph[@Mawet2005; @Foo2005], which has the vortex mask in an intermediate focal plane and a so-called Lyot stop in the downstream pupil. On the other hand, Fig. \[fig:opticallayouts\]c has the vortex mask in a pupil plane. We will show in the next section that these arrangements perform similarly in terms of starlight suppression and the coupling of planet light. For context, an AO-fed FIU has several potential observing configurations (see Fig. \[fig:obs\_scenarios\]). Most commonly, the starlight is coupled directly into the SMF with high efficiency for stellar spectroscopy (Fig. \[fig:obs\_scenarios\]a). For resolved exoplanet spectroscopy (Fig. \[fig:obs\_scenarios\]b), the planet is aligned to the optical axis and the star is imaged at an angular separation $>\lambda/D$, where $\lambda$ is the wavelength and $D$ is the telescope diameter. In this case, an apodizer or coronagraph may be introduced to reduce the diffracted starlight at the position of the planet. VFN is advantageous when the star and planet of interest are separated by approximately $\lambda/D$. As shown in Fig. \[fig:obs\_scenarios\]c, the stellar beam containing the optical vortex is centered on the SMF and nulled, while the planet is slightly off-axis and therefore partially couples into the fiber. ![Observing configurations of an AO-fed FIU. (a) For stellar spectroscopy, the star is coupled directly into the SMF. (b) For resolved exoplanet spectroscopy, the planet is aligned to the optical axis and the stellar image appears at an angular separation $>\lambda/D$. (c) VFN is used when the star and planet are separated by $\sim\lambda/D$. The stellar beam containing the optical vortex (see inset) is centered on the SMF and nulled, while the planet is slightly off-axis and partially couples into the SMF. The inset shows the phase of the complex field of the stellar PSF.[]{data-label="fig:obs_scenarios"}](observingscenarios.pdf){width="0.9\linewidth"} THEORETICAL PERFORMANCE AND OPTICAL DESIGN TRADES ================================================= In this section, we present the simulated performance of example VFN instrument layouts on different telescopes, including their sensitivity to a variety of errors. We also discuss the pros and cons of pupil- and focal-plane VFN implementations as well as vector versus scalar vortex masks. Finally, we describe ways to use beam shaping optics in order to improve coupling efficiency. Throughput ---------- We define the throughput of a VFN instrument as the fraction of light from a distant point source that enters the SMF, normalized by the total energy. In the following, we ignore imperfect reflectance/transmission of the focusing and collimating optics, Fresnel losses at the ends of the SMF, and propagation losses within the SMF. The throughput is computed using Eqn. \[eqn:couplingeff\] and depends on the angular separation of the point source from the optical axis, $\alpha$. The throughput and coupling efficiency are equal if there is no Lyot stop in the system. The term “null depth" refers to the throughput for the star when the starlight is intentionally nulled. Figure \[fig:throughput\_grid\] shows the throughput for four example aperture shapes, with focal- and pupil-plane VFN configurations (Figs. \[fig:opticallayouts\]b,c), as well as charge $l=1$ and $l=2$ vortex masks. The aperture shapes (see Fig. \[fig:pupils\]) correspond to a circular aperture, the Keck telescopes, the Thirty Meter Telescope (TMT), and the Giant Magellan Telescope (GMT). We have omitted the Lyot stop for the sake of simplicity. Each case in Fig. \[fig:throughput\_grid\] confirms that the coupling efficiency, $\eta$, is zero where $\alpha=0$ (i.e. the starlight is nulled) and that off-axis sources partially couple into the fiber. The VFN performance is relatively insensitive to the shape of the aperture (Fig. \[fig:throughput\_grid\], columns) or whether the vortex mask is situated in the pupil or focal plane (Fig. \[fig:throughput\_grid\], rows), which differs considerably from the trade-offs typically encountered for coronagraphs[@Ruane2018_FALCO]. The maximum coupling efficiency, $\eta_\text{peak}$, is approximately 20% for $l=1$ and 10% for $\l=2$. The peaks occur at $\alpha_\mathrm{peak}=0.9~\lambda/D$ and $\alpha_\mathrm{peak}=1.3~\lambda/D$, respectively. See Table \[tab:optparams\] for a more complete list of important parameters for each case. Figure \[fig:etamaps\_pupils\] shows maps of $\eta$ versus the two-dimensional position of the source, demonstrating that the fiber collects light from a donut shaped region around the star in all cases. This highlights a key advantage of the VFN approach: light from planets is collected regardless of the planet’s position angle as projected on the sky, which is unknown for planets detected via the radial velocity or transit techniques. However, these techniques do constrain the separation of the planet from the star as a function of time, which is likely sufficient information to observe the planet with VFN. ![Optimal coupling efficiency, $\eta$, for a point source versus its angular separation from the optical axis, $\alpha$. These curves are calculated at the central wavelength assuming the optimal value of $q$ (see Table \[tab:optparams\]). Results are shown with the vortex mask placed in (a)-(d) the pupil and (e)-(h) the focal plane. The columns correspond to the pupil shapes shown in Fig. \[fig:pupils\].[]{data-label="fig:throughput_grid"}](throughput_grid.png){width="0.75\linewidth"} ![Pupil shapes for (a) a circular aperture, (b) the Keck Telescopes, (c) the Thirty Meter Telescope (TMT), and (d) the Giant Magellan Telescope (GMT). []{data-label="fig:pupils"}](pupils.png){width="0.8\linewidth"} ![Maps of $\eta$ versus angular offset in two dimensions over a 6$\times$6 $\lambda/D$ square for each pupil (columns) and charge (rows). Figure \[fig:throughput\_grid\] shows a horizontal line profile starting from the origin as indicated by the dashed yellow line. The throughput on non-circular apertures (e.g. GMT) has an weak modulation in the azimuthal direction. []{data-label="fig:etamaps_pupils"}](etamaps.png){width="0.8\linewidth"} ------- ----- ------------------ -------------------------------------- -------------------------- ------------------ -------------------------------------- -------------------------- Pupil $l$ $q_\mathrm{opt}$ $\alpha_\mathrm{peak}$ ($\lambda/D$) $\eta_\mathrm{peak}$ (%) $q_\mathrm{opt}$ $\alpha_\mathrm{peak}$ ($\lambda/D$) $\eta_\mathrm{peak}$ (%) 0 1.4 0.0 82 - - - Circ 1 1.4 0.85 19 2.5 0.84 20 2 1.3 1.3 10 3.6 1.3 10 0 1.5 0.0 67 - - - Keck 1 1.4 0.90 18 2.6 0.90 19 2 1.4 1.5 11 3.8 1.5 11 0 1.4 0.0 76 - - - TMT 1 1.4 0.87 19 2.5 0.87 20 2 1.3 1.4 9.6 3.6 1.4 9.9 0 1.5 0.0 62 - - - GMT 1 1.4 0.88 15 2.6 0.87 16 2 1.3 1.5 10 3.6 1.4 10 ------- ----- ------------------ -------------------------------------- -------------------------- ------------------ -------------------------------------- -------------------------- : Optimal parameters for pupil-plane and focal-plane VFN. $l$ is the charge of the vortex mask, $q_\mathrm{opt}$ is the value of $q=\mathrm{MFD}/(\lambda\,F\#)$ that maximizes the peak coupling efficiency (i.e. $\eta_\mathrm{peak}$), $\alpha_\mathrm{peak}$ is the angular separation at $\eta_\mathrm{peak}$, and $D$ is the circumscribed pupil diameter. []{data-label="tab:optparams"} Layout considerations --------------------- While the previous section demonstrated that there is little performance difference between pupil-plane and focal-plane layouts, there are several other pros and cons of each design that should be taken into consideration. **F\# compatibility.** An advantage of the pupil-plane layout is that the optimal focal ratio ($F\# = f/D$, where $f$ is the focal length) matches that of the standard FIU observing modes (stellar or direct exoplanet spectroscopy; Fig. \[fig:obs\_scenarios\]a,b). Specifically, when coupling starlight in an SMF, the optimal value for the ratio $q=\mathrm{MFD}/(\lambda\,F\#)$, where MFD is the mode field diameter of the SMF, is $\sim$1.4. Table \[tab:optparams\] gives the optimal ratios, $q_\mathrm{opt}$, for each design simulated above. Whereas $q_\mathrm{opt}$ remains between 1.3 and 1.5 for the pupil-plane version for all aperture shapes and vortex charges, the focal plane version needs $q=2.5$ and $q=3.6$, for charge 1 and 2 VFN modes, respectively. Practically, this means the final focusing optics in the FIU would be the same for standard modes (Fig. \[fig:obs\_scenarios\]a,b) and pupil-plane VFN (Fig. \[fig:obs\_scenarios\]c), while focal-plane VFN would need lenses with different focal lengths. With a fixed MFD and $F\#$, the differences in $q_\text{opt}$ for each $l$ value in the pupil-plane case shifts the wavelength of maximal coupling a small amount, which tends to have a minor impact on the integrated coupling efficiency across typical astronomical passbands. **Beam and defect size.** Another advantage of the pupil-plane layout is that the central singularity of the vortex masks may be obscured by the shadow of the secondary mirror on ground-based telescopes. For instance, the image of the Keck pupil in the KPIC instrument is 12 mm and the central obscuration is 4 mm in diameter. By comparison, the focal plane version requires the central defect on the vortex mask to be on order of microns. On the other hand, since the beam at the vortex mask is $\sim$200$\times$ larger in the pupil-plane case, the transmitted wavefront error must be minimized over a larger area of the mask. **On-sky operations.** Pupil-plane VFN may also simplify on-sky observing procedures compared to focal-plane VFN. For pupil-plane VFN, the observer only needs to align the image of the star to the fiber, while in the focal-plane version the star must be accurately aligned to both the center of the vortex mask and the fiber. These practical trade-offs have led our team to adopt the pupil-plane version for current laboratory testing[@Echeverri2019_VFN] and future plans for on-sky deployment[@Echeverri2019b_VFN], despite the fact that our first VFN paper[@Ruane2018_VFN] concentrated on the focal-plane approach. The remainder of this paper pertains to the pupil-plane version, but most of the theoretical performance properties also apply in the focal-plane case. Sensitivity of the null depth to tip/tilt jitter and finite size of the star ---------------------------------------------------------------------------- ![Theoretical coupling efficiency of (a) a point source as a function of angular offset from the optical axis and the fraction of starlight that couples into the fiber versus (b) tip/tilt jitter and (c) the size of the star. These values are calculated via a numerical simulations of the optical system with tip/tilt errors introduced in a Monte Carlo fashion. []{data-label="fig:ttsens"}](etas_vs_offset_tmt.png){height="0.32\linewidth"} A deep null with VFN requires precise control of tip/tilt errors. Figure \[fig:ttsens\]a shows the coupling efficiency of a point source, $\eta$, versus its angular separation from the optical axis, $\alpha$, for charges $l=1$ and $l=2$. For $\alpha\ll\lambda/D$, the coupling efficiency follows a simple power law: $\eta\propto\alpha^{2l}$. Therefore, while reducing the charge provides better throughput for sources closer to the star, higher values of the charge are much more robust to small tip/tilt errors. Thus, the requirements for residual tip/tilt jitter after AO correction depends on the value of the charge (see Fig. \[fig:ttsens\]b). Assuming uncorrected errors are normally-distributed and time-averaged over a single exposure, the fraction of starlight that couples into the SMF (also refereed to here as the “null depth") is approximately $$\eta_s\approx \left(\sigma \frac{D}{\lambda}\right)^{2l}, \label{eqn:null_jitter}$$ where $\sigma$ is the standard deviation of the tip/tilt error distribution in radians. For instance, to achieve $\eta_s=10^{-4}$, jitter needs to be less than 0.1 $\lambda/D$ RMS for $l=2$, which is routinely achieved by current ground-based AO systems. However, for the same null depth with $l=1$, the tip/tilt jitter requirement becomes roughly ten times smaller, which is challenging to achieve in the near-infrared with current instrumentation. To simulate a partially resolved star, we model it in a similar fashion, but as a disc of uniform emission. Figure \[fig:ttsens\]c shows the resulting null depth as a function of the angular size of the star with respect to the angular resolution of the telescope, $\lambda/D$. Although the $\eta$ values appear to be much smaller than the case shown for normally-distributed tip/tilt jitter, this is partially because the stellar size is specified by its full diameter rather than the standard deviation of the uniform distribution. Sensitivity of the null depth to low order aberrations ------------------------------------------------------ The performance of a VFN instrument is sensitive to some, but not all, low-order aberrations. Here, we describe the wavefront as a linear combination of Zernike polynomials. For small wavefront aberrations, $W(r,\theta)$, the field in the pupil is given by: $$E_p(r,\theta) = \exp(iW(r,\theta))\approx 1 + iW(r,\theta) = 1 + i\sum_{n,m}c_n^m R_n^m(r)\exp(im\theta), \label{eqn:loworderaber}$$ where $c_n^m$ are coefficients and $R_n^m(\rho)$ are the radial Zernike polynomials[@BornWolf]. The field in the image plane due to $W(r,\theta)$ takes a similar form, but with a different set radial functions described by Bessel functions. Centering the vortex mask on the beam modifies the polar component associated with each term in the overlap integral to $$\int_0^{2\pi} \exp(il\theta)\exp(im\theta)d\theta = \left\{ \begin{array}{ll} 2\pi & |l|=|m| \\ 0 & |l|\ne |m| \\ \end{array} \right.. \label{eqn:nullcondition}$$ Thus, the null depth will only be nonzero if $|m|=|l|$ and, under the first-order approximation, the charge 1 and 2 cases are most sensitive to coma ($m=\pm1$) and astigmatism ($m=\pm2$) orders, respectively. To illustrate this, Fig. \[fig:loworder\] shows the sensitivity of the null depth to each Zernike term. For the $l=1$ case, the null depth due to coma ($m=\pm1$) modes has second-order order dependence on wavefront error. For the $l=2$ case, coma ($m=\pm1$) modes have a 4th order dependence due to higher order terms in Eqn. \[eqn:loworderaber\], while astigmatism ($m=\pm2$) terms have a second-order order dependence. In each case, the wavefront errors in only a few Zernike modes (where $|m|=|l|$) tend to limit the nulling performance. ![Sensitivity to low-order aberrations for charges (a) $l=1$ and (b) $l=2$ assuming the star is a point source and the pupil is circular. The modes that appear in (a) have azimuthal index $m=\pm1$ (i.e. tip, tilt, and all orders of coma) and the null depth follows second-order power law. The same modes appear in (b) with a fourth-order power law, while the null depth has a second-order sensitivity to modes with azimuthal index $m=\pm2$ (i.e. all orders of astigmatism). Terms with $m \ne \pm1$ or $m \ne \pm2$ are omitted because they do not contribute to the null depth in either case.[]{data-label="fig:loworder"}](all_etas_vs_aber_charge1_noll2ind.png){height="0.32\linewidth"} ![Sensitivity to misalignment of the vortex mask for (a) a circular pupil, (b) the Keck telescope, and (c) the TMT.[]{data-label="fig:maskmisalignment"}](etas_vs_pupilmaskoffset_circ.png){height="0.32\linewidth"} Sensitivity to mask misalignment -------------------------------- All of the previous calculations assumed that the vortex mask was perfectly centered on the pupil. However, the null depth degrades when the vortex mask is misaligned in the transverse direction. Figure \[fig:maskmisalignment\] shows how the null depth degrades as the mask is offset by a fraction of the pupil diameter for three pupil shapes. For $l=1$, the effect is similar for each pupil, where the null depth follows a well behaved second-order power law. However, for $l=2$, the null depth follows a fourth-order power law in the case of a circular aperture, but deviates from this behavior when the pupil is centrally obscured, as in the case of the Keck telescope, TMT, and GMT. We find that the null depth does not degrade significantly until the mask moves by approximately the radius of the central obscuration. For instance, for the Keck telescope, the diameter of the central obscuration is 24% of the telescope outer diameter and, thus, a very low null depth is maintained until the mask is offset by $\sim$12% of the pupil diameter (see Fig. \[fig:maskmisalignment\]b). Mask designs ------------ There are several varieties of vortex masks in the literature; most types fall into two categories: *vectorial* and *scalar* masks. ### Vector vortex fiber nuller Vector phase masks may be manufactured via a variety of methods: liquid crystal (LC) [@Marrucci2006; @Mawet2009; @Mawet2010a; @Mawet2010b; @Serabyn2019], subwavelength gratings[@Biener2002; @Mawet2005b; @Niv2007], and photonic crystals[@Murakami2013]. Each impart conjugate phase patterns to the incident circular polarization states, while also inverting the two states. The phase pattern is set by the local orientation of the fast axis, i.e. the phase shift is $\Phi=\pm 2\chi(x,y)$, where $\chi(x,y)$ is the fast-axis angle. Thus, a vector vortex mask has $\chi=l\theta/2$ and Jones matrix in the circular polarization basis, $\mathbf{M}_\circlearrowright$, which acts on an incoming beam as follows: $$\left[ \begin{matrix} U^\prime_R \left(x,y\right) \\ U^\prime_L \left(x,y\right) \\ \end{matrix} \right]=\mathbf{M}_\circlearrowright \left(x,y\right) \left[ \begin{matrix} U_R \left(x,y\right) \\ U_L \left(x,y\right) \\ \end{matrix} \right] =\left[\begin{matrix} 0 & e^{il\theta} \\ e^{-il\theta} & 0 \\ \end{matrix} \right] \left[ \begin{matrix} U_R \left(x,y\right) \\ U_L \left(x,y\right) \\ \end{matrix} \right],$$ where $U_R \left(x,y\right)$ and $U_L \left(x,y\right)$ are the right- and left-handed circularly polarized field components in the $(x,y)$ plane. However, the half-wave condition is not achieved at all wavelengths simultaneously, which causes an additional leakage term, which may be represented by: $$\mathbf{M}_\circlearrowright =c_V \left[\begin{matrix} 0 & e^{il\theta} \\ e^{-il\theta} & 0 \\ \end{matrix} \right] + c_L \left[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],$$ where $c_V$ and $c_L$ are wavelength-dependent coefficients, $|c_L|^2\approx\epsilon(\lambda)^2/4$, and $\epsilon(\lambda)$ is the retardance error[@Mawet2010a]. The null depth is given by $\eta_s=|c_L|^2\eta_0$, where $|c_L|^2$ is fraction of light leaked through the vector vortex with $l=0$ and $\eta_0$ is the coupling efficiency for the $l=0$ mode (see Table \[tab:optparams\]). For $|c_L|^2<10^{-4}$, the retardance error requirement is $\sim1^\circ$, which may be achieved over substantial bandwidths using multi-layer LC[@Pancharatnam1955; @Komanduri2013]. ### Scalar vortex fiber nuller Scalar vortex approaches using spiral phase plates[@Swartzlander2008; @Ruane2019_scalarVC] or dispersion-compensated holograms[@Errmann2013; @Ruane2014], though appearing earlier in the literature[@Heckenberg1992; @Tidwell1993; @Beijersbergen1994], have not received as much attention in the context of exoplanet detection and characterization methods. However, there are several viable methods for implementing a scalar vortex fiber nuller. Perhaps the simplest scalar vortex mask is a spiral phase plate, which has transmittance of the form $t(\lambda)=\exp\left(il(\lambda)\theta\right)$, where $l(\lambda)$ is the effective charge as a function of wavelength, which takes on non-integer values: $$l(\lambda) = l_0\frac{\lambda_0}{\lambda}\left(\frac{n(\lambda)-1}{n(\lambda_0)-1}\right),$$ where $l_0$ is the the charge at the central wavelength $\lambda_0$ and $n(\lambda)$ is the refractive index of the material. A spiral phase plate with a wavelength-independent refractive index or a spiral phase mirror[@Campbell2012] would have a wavelength-dependent charge profile given by $l(\lambda)=l_0\lambda_0/\lambda$. For the sake of simplicity, we simulate the spiral phase mirror case below. Figure \[fig:etamap\_scalar\_vs\_wvl\] shows maps of $\eta$ versus angular offset for a scalar VFN using a spiral phase mirror with charge $l_0$ at a central wavelength of 2.2 $\mu m$. Although a true null only occurs at the central wavelength for a fiber centered on the “yellow cross," a null exists at other wavelengths with a wavelength-dependent offset from the origin. Therefore, a null may be created at any given wavelength by shifting the image of the star with respect to the fiber by the equal and opposite amount. In the charge 1 case, the null appears to follow a straight line as a function of wavelength, whereas in the charge 2 case, the null breaks up into two charge 1 nulls that move in opposite directions. At least in the charge 1 case, the wavelength-dependent offset of the null (see Fig. \[fig:scalaroffsets\]a) can be mitigated to first order by inserting a wedge of dielectric material. Since a peak-to-valley phase shift, $\Delta\Phi$, of one wave corresponds to a $\lambda/D$ offset in the image plane, the wedge must apply $$\Delta\Phi = 2 \pi p \left( \frac{\lambda-\lambda_0}{\lambda} \right) + c,$$ where $p=1.1$ (i.e. the slope of the linear fit in Fig. \[fig:scalaroffsets\]a) and $c$ is a constant representing a global offset in the image plane. A prism with refractive index $n$ and wedge angle $\beta$, applies $$\Delta\Phi = \frac{2 \pi}{\lambda}(n-1)d\tan(\beta),$$ where $d$ is the diameter of the beam. We find the wavelength-independent solution by setting $c=2\pi p$ (i.e. the broadband null will occur $p\;\lambda_0/D$ from the optical axis) and manufacture the prism such that $$d\tan(\beta) = \frac{p\;\lambda_0}{n-1}.$$ Figure \[fig:scalaroffsets\]b shows $>$300$\times$ improvement in the null depth after introducing a prism with $n=1.4$ and $d\tan(\beta)=6~\mu m$. In this example, the phase discontinuity occurs along the negative $x$-axis in the pupil, the offset moves along the positive $y$-axis with increasing wavelength, and therefore the wedge must compensate for dispersion in the $y$-direction (i.e. perpendicular to the phase discontinuity). ![Two-dimensional maps of $\log(\eta)$ versus angular separation for a scalar VFN for (top row) $l_0=1$ and (bottom row) $l_0=2$ at $\lambda_0=2.2~\mu m$. By comparison, a vector vortex is designed to give charge $\pm l_0$ at all wavelengths. []{data-label="fig:etamap_scalar_vs_wvl"}](etamaps_scalar_vs_wvl.png){height="0.32\linewidth"} ![Correction of the wavelength-dependent offsets of an $l=1$ scalar VFN by adding a wedge of dielectric. (a) The offsets seen in the $\eta$-maps in the top row of Fig. \[fig:etamap\_scalar\_vs\_wvl\]. (b) Comparison of null depth before and after adding a wedge with $n=1.4$ and $d\tan(\beta)=6~\mu m$.[]{data-label="fig:scalaroffsets"}](scalaroffsets_circ.png "fig:"){height="0.32\linewidth"} ![Correction of the wavelength-dependent offsets of an $l=1$ scalar VFN by adding a wedge of dielectric. (a) The offsets seen in the $\eta$-maps in the top row of Fig. \[fig:etamap\_scalar\_vs\_wvl\]. (b) Comparison of null depth before and after adding a wedge with $n=1.4$ and $d\tan(\beta)=6~\mu m$.[]{data-label="fig:scalaroffsets"}](scalar_eta_vs_wvl_circ.png "fig:"){height="0.32\linewidth"} Our team is actively exploring alternate approaches for implementing scalar VFN, including imprinting the vortex pattern directly on the deformable mirror, segmented primary mirror[@Tyson2008], or the tip of the SMF[@Vayalamkuzhi2016]. All of these methods are viable as long as the wavelength-dependent tilt can be accurately removed by the prism, or similar optic. In practice, the atmospheric dispersion compensator (ADC) may take the place of the prism, thereby potentially eliminating the need to add any new optic to existing AO-fed FIU instruments in order to implement VFN. Future work will explore the trade-offs between such methods. Beam shaping: a potential pathway towards improving planet coupling efficiency ------------------------------------------------------------------------------ We are also exploring methods to improve the VFN throughput. Beam shaping optics have been previously demonstrated to improve coupling efficiency of light from a large telescope into SMFs[@Jovanovic2017]. Building upon this work, we designed phase-induced amplitude-apodization (PIAA) lenses[@Guyon2005] to remap the collimated beam into a quasi-Gaussian beam in order to increase the coupling efficiency for the planet light in a VFN instrument. Figure \[fig:piaa\] shows two example PIAA lens pairs that are designed for circular (Fig. \[fig:piaa\]a) and annular (Fig. \[fig:piaa\]b) input beams. The latter is based on the dimensions of the Keck pupil. While each are designed to maximize the coupling efficiency in the $l=0$ case, they also provide significant throughput improvement in $l=1$ and $l=2$ VFN modes (see Fig. \[fig:piaa\]c). Specifically, for a circular pupil, the peak coupling efficiency increases by 20%, 30%, and 40% in the $l=0$, $l=1$, and $l=2$ cases, respectively, while for the Keck pupil, the coupling increases by approximately 30% for all three cases. Using PIAA lenses for beam shaping along with VFN may not require additional engineering in practice since PIAA lenses are already likely to be a key component of AO-fed FIUs for stellar spectroscopy and direct exoplanet spectroscopy where the target of interest is aligned to the optical axis (as in Fig. \[fig:obs\_scenarios\]). A potential drawback of the PIAA lenses is that they have an extremely narrow field of view (1-2 $\lambda/D$) within which wavefront aberrations are small enough to allow high coupling efficiency and, therefore, they are not applicable for simultaneous spectroscopy of multiple objects. In that case, image slicing methods may be necessary to achieve high efficiency using a pair of PIAA lenses and SMF for each object. ![Beam shaping with phase-induced amplitude-apodization (PIAA) lenses. (a)-(b) Ray trace through PIAA lenses designed for (a) a circular beam and (b) an annular beam. (c) The coupling efficiency as a function of angular separation of the planet from the optical axis, $\alpha$, (solid lines) with and (dashed lines) without the PIAA lenses as well as (blue lines) $l=1$ and (red lines) $l=2$ VFN modes. The coupling efficiencies shown are for a circular pupil, but we achieve similar gains for a large range of telescope pupils.[]{data-label="fig:piaa"}](piaa_raytrace.pdf "fig:"){height="0.33\linewidth"} ![Beam shaping with phase-induced amplitude-apodization (PIAA) lenses. (a)-(b) Ray trace through PIAA lenses designed for (a) a circular beam and (b) an annular beam. (c) The coupling efficiency as a function of angular separation of the planet from the optical axis, $\alpha$, (solid lines) with and (dashed lines) without the PIAA lenses as well as (blue lines) $l=1$ and (red lines) $l=2$ VFN modes. The coupling efficiencies shown are for a circular pupil, but we achieve similar gains for a large range of telescope pupils.[]{data-label="fig:piaa"}](piaa_thpt_circ.png "fig:"){height="0.32\linewidth"} THE POTENTIAL SCIENTIFIC YIELD FOR VFN ON TMT ============================================= The capability to probe very small angular separations may naturally lead to high scientific yield because the occurrence rate of planets is potentially much greater for closer-in orbits than direct imagers are currently able to observe (10-100 au). In fact, RV and direct imaging surveys suggest that there may be a peak in the occurrence rate of giant planets within 1-10 au[@Nielsen2019], which may be accessible in the near-infrared with a nuller on TMT beyond distances of 100 pc. VFN can therefore be used to detect and characterize young (1-10 Myr) giant planets in several nearby star forming regions and young moving groups[@Bowler2016]. VFN can also target smaller planets whose occurrence rate increases to almost one per star for periods less than 50 days and radii 0.5-4 $R_\oplus$ [@Dressing2013]. Though, the latter will lead to considerably longer integration time, $\tau$, which scales as $r_p^{-4}$ when observing a planet of radius $r_p$ in reflected light[@Ruane2018_VFN]. ![Integration time estimates for achieving the goal $S/N$ per spectral channel at a spectral resolution of $R$=100,000 in $J$ band for (a) known exoplanets and (b) theoretical exoEarths orbiting all nearby stars. We set the goal $S/N$ per spectral channel to 1 and 0.1 in each case, respectively. The blue circles indicate cases where $l=1$ provides a shorter integration time and likewise the red diamonds are cases where $l=2$ is more favorable. In this scenario, $l=2$ tends to be beneficial when the planets angular separation from the host star is $>\lambda_0/D$=8.6 mas, where $\lambda_0=1.25~\mu$m.[]{data-label="fig:tints"}](tau_TMT_J_R100000_spie_a.pdf "fig:"){height="0.33\linewidth"} ![Integration time estimates for achieving the goal $S/N$ per spectral channel at a spectral resolution of $R$=100,000 in $J$ band for (a) known exoplanets and (b) theoretical exoEarths orbiting all nearby stars. We set the goal $S/N$ per spectral channel to 1 and 0.1 in each case, respectively. The blue circles indicate cases where $l=1$ provides a shorter integration time and likewise the red diamonds are cases where $l=2$ is more favorable. In this scenario, $l=2$ tends to be beneficial when the planets angular separation from the host star is $>\lambda_0/D$=8.6 mas, where $\lambda_0=1.25~\mu$m.[]{data-label="fig:tints"}](tau_TMT_J_R100000_spie_b.pdf "fig:"){height="0.33\linewidth"} Possibly the most immediate application of VFN on TMT will be to independently confirm the existence of previously detected planets using other techniques, such as the RV or transit methods. To determine the potential number of feasible targets, we computed the $\tau$ needed detect all of the planets listed in the NASA Exoplanet Archive (accessed Dec 2018) at $S/N=1$ per spectral channel. To compute the planet-to-star flux ratio, we used the open-source atmospheric modeling package `PICASO`[@Batalha2019] to generate both the reflected light and thermal emission of the planet, assuming a single layer cloud deck in the atmosphere. Our current cloud model yielded an albedo of $\sim~0.2$ in $J$-band, but we note that the choice of cloud model can change reflected light flux by factors of several. The stellar size is calculated using an empirical radius-magnitude relationship for late-type stars[@Mann2015]. The telescope and instrument are assumed to have collecting area of 655 m$^2$, spectral resolution $R= 100,000$, transmission of 30%, detector quantum efficiency of 90%, no other detector noise, RMS tip-tilt jitter = $10^{-2}~\lambda/D$, an ideal vector vortex mask, and no PIAA optics. These calculations only take into account photon noise due to leaked starlight (i.e the dominant noise source) with the appropriate VFN throughput and expected leakage due to the estimated stellar size in addition to the assumed tip/tilt jitter. For each target, we determine the value of $l$ that gives the shortest integration times for these assumptions. Figure \[fig:tints\]a shows the integration times in $J$ band. The number of targets for which $S/N=1$ is reached in $\tau<$50 hr is 23, 17, and 23 in $J$, $H$, and $K$ bands, respectively. The targets with the shortest theoretical integration times are *tau Boo b* and *ups And b*. While the calculated $\tau$ for most planets is lower for $l=1$ under these assumptions, some cases prefer the $l=2$ mask, which are mostly scenarios where the planet separation is $>\lambda/D$; an example of such a case is *55 Cnc b* in $J$ band, which requires $\tau=2$ hr. A number of planets with relatively short integration time are within the conventional definition of the inner working angle (roughly 0.4 $\lambda/D$), demonstrating that VFN is beneficial even at separations of relatively low throughput (i.e. less than half of its maximum). VFN may also be a pathway towards detecting and characterizing Earth-sized planets in the habitable zone around nearby, cool stars. Figure \[fig:tints\]b shows the estimated number of such stars around which $S/N=0.1$ could be achieved on a theoretical 1 $R_\oplus$ planet at the center of the habitable zone (the input catalog is freely available at Olivier Guyon’s webpage[@guyonwebpage]). We otherwise apply equivalent assumptions as listed above. The number of targets where $\tau<$50 hr is 23, 18, and 10 in $J$, $H$, and $K$ bands, respectively. Since the prime targets tend to be nearby stars, the stellar angular sizes are larger with respect to $\lambda/D$ and therefore $l=2$ is more often favorable for this science case. One of the best targets is *Wolf 359*, which can be detected at $J$, $H$, and $K$ bands with integration times of $\sim$3 hr. While using $l=2$ in $J$ band leads to a shorter $\tau$ for *Wolf 359*, $l=1$ would be preferred for $H$ and $K$ bands where the planet-star separation is too small for $l=2$. CONCLUSIONS =========== VFN is a promising method for detecting and characterizing exoplanets with large-aperture ground- and space-based telescopes. Here, we showed that the small inner working angle provides access to close-in planets with separations from their host star on the order of $\lambda/D$. Increasing the charge of the vortex mask provides a way to reduce sensitivity to tip/tilt and mask misalignments at the expense of increasing the inner working angle. We demonstrated that (a) the theoretical performance of VFN is largely insensitive to the shape of the telescope aperture and some low order aberrations, (b) both vector and scalar implementations are possible, and (c) beam shaping can improve the throughput. Moreover, we showed that VFN is not sensitive to the planet’s position angle, which implies that the RV or transit methods provide sufficient information for follow up observations using VFN. We expect to be able to detect $>$10 known planets with integration times on the order of tens of hours using VFN on an AO-fed FIU on TMT, such as the MODHIS instrument[@Mawet2019_whitepaper], and possibly even enable the detection Earth-sized planets in the habitable zone of cool stars. This work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with NASA under the Exoplanet Exploration Program.
--- abstract: | [ *We consider magnetic fields generated by homogeneous isotropic and parity invariant turbulent flows. We show that simple scaling laws for dynamo threshold, magnetic energy and Ohmic dissipation can be obtained depending on the value of the magnetic Prandtl number.*]{} 0.5 [ : dynamo ; turbulence ; magnetic field]{} address: 'LPS, CNRS UMR 8550, ENS 24 rue Lhomond 75005 Paris France' author: - 'Stéphan Fauve, François Pétrélis' title: - Scaling laws of turbulent dynamos - Comportements asymptotiques des dynamos turbulentes --- Version française abrégée {#version-française-abrégée .unnumbered} ========================= Il est à présent admis que les champs magnétiques des étoiles voire même des galaxies sont engendrés par l’écoulement de fluides conducteurs de l’électricité [@Zeldovitch; @Widrow; @Brandenburg]. Ceux-ci impliquent des nombres de Reynolds cinétique, $Re$, et magnétique, $R_m$, très élevés ($Re = VL/\nu$, $R_m = \mu_0 \sigma VL$, où $V$ est l’écart-type des fluctuations de vitesse, $L$, l’échelle intégrale de l’écoulement, $\nu$, la viscosité cinématique du fluide, $\sigma$, sa conductivité électrique et $\mu_0$, la perméabilité magnétique). Aucune expérience de laboratoire ou simulation numérique directe des équations de la magnétohydrodynamique, ne permet l’ étude du problème dans des régimes de paramètres, $Re$ et $R_m$, d’intérêt astrophysique. Il est donc utile de considérer des hypothèses plausibles afin de pousser plus loin l’analyse dimensionnelle qui, à partir des paramètres $V$, $L$, $\nu$, $\sigma$, $\mu_0$ et de la densité du fluide $\rho$, prédit pour le seuil de l’effet dynamo et la densité moyenne d’énergie magnétique, $B^2 / 2\mu_0$, saturée non linéairement au-delà du seuil, $$R_m^{c} = f(Re), \label{thresholda}$$ $$\frac{B^2}{\mu_0} = \rho V^2 \,g(R_m, Re). \label{saturationa}$$ Dans le cas d’un écoulement turbulent homogène isotrope, donc de vitesse moyenne nulle, et invariant par symétrie plane, donc sans hélicité, les résultats des simulations numériques les plus performantes réalisées à ce jour montrent que $R_m^{c}$ augmente continuellement en fonction de $Re$ [@Schekochihin]. Schekochihin et al. proposent que deux scénarios extrêmes, schématisées dans la figure 1, seront susceptibles d’être observés lorsque les ordinateurs auront acquis la puissance requise pour effectuer des calculs à $Re$ plus élevé: (i) une saturation $R_m^{c} \rightarrow \rm{constante}$, ou alors (ii) une croissance de la forme $R_m^{c} \propto Re$. D’autres simulations numériques directes, réalisées avec des écoulements turbulents possédant un champ de vitesse moyen de géométrie fixée ${\overline{\bf v} (\bf { r})} \neq 0$, semblent suivre le scénario (i) [@PontyLaval]. Lorsque le nombre de Prandtl magnétique, $P_m = R_m / Re = \mu_0 \sigma \nu$, est faible, $P_m \ll 1$, l’échelle de dissipation Joule du champ magnétique, $l_{\sigma} = L R_m^{-3/4}$, est grande par rapport à l’échelle de Kolmogorov $l_K = L Re^{-3/4}$. Le champ magnétique se développe donc à une échelle suffisamment grande pour ne pas être affecté par la viscosité cinématique. Cette hypothèse, couramment effectuée en turbulence, permet de conclure en faveur du scénario (i). En effet, si $\nu$ n’est pas pris en compte, l’analyse dimensionnelle impose $R_m^{c} \rightarrow \rm{constante}$. Il n’est donc pas surprenant que les modélisations numériques des grandes échelles, qui ne résolvent pas les échelles dissipatives, donnent ce résultat. Le scénario (i) sera donc toujours observé à $P_m$ suffisamment faible sous réserve bien sûr que l’on ait dynamo. Il est cependant utile d’analyser le scénario (ii) d’autant plus que, comme nous pouvons le remarquer, il correspond à la prédiction faite par Batchelor en 1950 [@Batchelor]. En se basant sur une analogie entre l’équation de l’induction et celle de la vorticité, Batchelor avait estimé que le seuil d’une dynamo engendrée par un écoulement turbulent devait correspondre à $P_m$ d’ordre unité, soit $R_m^{c} \propto Re$. Même si nous savons aujourd’hui que l’analyse de Batchelor est discutable, il est intéressant de déterminer sous quelle hypothèse minimale sa prédiction est correcte. Supposons donc que nous nous limitions aux modes instables de champ magnétique, suffisamment localisés au sein de l’écoulement afin de ne pas être affectés par les conditions aux limites. Il est alors possible de ne pas prendre en compte l’échelle spatiale $L$, et l’analyse dimensionnelle impose pour le seuil, $P_m = \rm{constante}$, soit le scénario (ii) $R_m^{c} \propto Re$. Les scénarios considérés ci-dessus conduisent aussi à des prédictions différentes pour la densité d’énergie magnétique engendrée par effet dynamo. Le scenario (i) qui consiste à ne pas prendre en compte $\nu$ revient à négliger la dépendance en $Re$ de $g(R_m, Re)$ dans (\[saturationa\]). Au voisinage du seuil, $V$ est déterminé par $R_m^{c} \sim \mu_0 \sigma V L$ et $g(Rm) \propto R_m - R_m^{c}$ dans le cas d’une bifurcation supercritique. Il en résulte [@Petrelis01] $$B^2 \propto \frac{ \rho} {\mu_0 (\sigma L)^2} (R_m-R_m^{c}). \label{saturationseuila}$$ Loin du seuil pour $P_m \ll 1$, $Re \gg R_m \gg R_m^{c}$, on peut supposer que $B$ ne dépend plus de $\sigma$ à condition que le champ magnétique se développe à une échelle plus grande que $l_{\sigma}$. Il en résulte alors l’équipartition entre énergie magnétique et cinétique, $B^2/\mu_0 \propto \rho V^2$, tel que supposé initialement par Biermann et Schlüter [@Biermann]. Un résultat complètement différent est obtenu dans le scénario (ii). Il convient de considérer les paramètres du problème sous la forme équivalente, $B$, $\epsilon = V^3/L$, $L$, $\nu$, $\sigma$, $\mu_0$ et $\rho$. En effet, le champ magnétique à petite échelle est alimenté par la puissance par unité de masse $\epsilon$ qui cascade depuis l’échelle intégrale, et il est donc important de conserver ce paramètre même si l’on ne prend pas en compte explicitement $L$. L’analyse dimensionnelle conduit alors à $$\frac{B^2}{\mu_0} = \rho \sqrt{\nu \epsilon} \, h(P_m) = \frac{\rho V^2}{\sqrt{Re}} \, h(P_m), \label{saturationbatchelora}$$ qui, pour $P_m \sim 1$, n’est autre que le résultat obtenu par Batchelor en supposant que la saturation correspond à l’équipartition entre l’énergie magnétique et l’énergie cinétique à l’échelle de Kolmogorov. Revenons au cas $P_m \ll 1$ qui correspond aux écoulements de métaux liquides et plasmas à l’origine du champ magnétique des planètes et des étoiles ($P_m < 10^{-5}$). Dans ce cas, le champ magnétique se développe à des échelles a priori comprises entre $L$ et $l_{\sigma}$ avec $l_{\sigma} \gg l_K$ et il en résulte que $R_m^{c}$ ne dépend pas de variations de $P_m$ (ou de $Re$) et que $B^2/\mu_0 = \rho V^2 g(R_m)$ (scénario (i)). Intéressons nous à la puissance dissipée par effet Joule par une telle dynamo. Il faut à cet effet déterminer à quelles échelles se développe le champ magnétique. Utilisons pour cela un argument à la Kolmogorov en supposant que dans la zone inertielle, c’est à dire pour les nombres d’onde $k$ tels que $k l_{\sigma} \ll 1 \ll kL$, la puissance spectrale $\vert \hat B(k) \vert^2$ est indépendante de $L$, $\sigma$ and $\nu$. Il en résulte $$\vert \hat B \vert^2 \propto \mu_0 \rho \, \epsilon^{\frac{2}{3}} \, k^{-\frac{5}{3}}. \label{kolmogorova}$$ Ceci n’est pas la seule possibilité parmi les nombreuses prédictions relatives au spectre de la turbulence magnétohydrodynamique, mais dans le cas présent, c’est probablement la plus simple. L’intégration sur $k$ redonne l’équipartition $B^2/\mu_0 \propto \rho V^2$. La contribution dominante à l’effet Joule provient de l’échelle $l_{\sigma}$. Nous obtenons $$\frac{{\bf j}^2}{\sigma} = \frac{1}{\sigma} \int \vert \hat j \vert^2 \, dk \propto \frac{1}{\mu_0^2 \sigma} \int k^2 \vert \hat B \vert^2 \, dk \propto \frac{\rho}{\mu_0 \sigma} \, \epsilon^{\frac{2}{3}} \, l_{\sigma}^{-\frac{4}{3}} \propto {\rho} \frac{V^3}{L}, \label{joulea}$$ où ${\bf j}$ est le vecteur densité de courant. Nous constatons donc que la dissipation Joule est du même ordre que la puissance totale disponible. Remarquons qu’il en serait de même pour une dynamo de Batchelor suivant le scénario (ii) pour $P_m \sim 1$, car bien que la densité d’énergie soit plus faible, l’échelle caractéristique du champ magnétique l’est également. Introduction {#Magnetic fields generated by turbulent flows} ============ It is now believed that magnetic fields of stars and possibly galaxies are generated by the motion of electrically conducting fluids through the dynamo process [@Zeldovitch; @Widrow; @Brandenburg]. These flows involve huge kinetic, $Re$, and magnetic, $R_m$, Reynolds numbers ($Re = VL/\nu$, $R_m = \mu_0 \sigma VL$, where $V$ is the $rms$ velocity amplitude, $L$ is the integral length scale, $\nu$ is the kinematic viscosity of the fluid, $\sigma$ is its electrical conductivity and $\mu_0$ is the magnetic permeability). No laboratory experiments, neither direct numerical simulations are possible in the range of $Re$ and $R_m$ involved in astrophysical flows. It is thus interesting to try to guess scaling laws for the magnetic field using some simple hypothesis. We consider here the minimum set of parameters, $V$, $L$, $\nu$, $\mu_0$, $\sigma$ and $\rho$, the fluid density. We note that discarding global rotation makes our results certainly invalid for many astrophysical objects but not all of them. Rotation is indeed not assumed important for the galaxies which do not display a large scale coherent magnetic field [@Zeldovitch; @Widrow; @Brandenburg]. Calling $B$ its $rms$ value, dimensional analysis gives $$R_m^{c} = f(Re), \label{threshold}$$ for the dynamo threshold, and $$\frac{B^2}{\mu_0} = \rho V^2 \, g(R_m, Re), \label{saturation}$$ for the mean magnetic energy density in the nonlinearly saturated regime. Our aim is to determine $f$ and $g$ in various regions of the parameter space $(R_m, Re)$, assuming that turbulence is homogeneous, isotropic and parity invariant (thus with no mean flow and no mean magnetic field generation through an alpha effect). As already mentioned, this may look like an academic exercise compared to most natural dynamos. It is however not more academic that the concept of homogeneous and isotropic turbulence with respect to real turbulent flows. We thus expect that our simple arguments may shed some light on open problems concerning the effect of turbulence on the dynamo threshold and on the dynamic equilibrium between magnetic and kinetic energy. The dependence of the dynamo threshold $R_m^{c} = f(Re)$ in the limit of large $Re$ is still an open problem, even in the case of a homogeneous isotropic and parity invariant turbulent flow. Note that parity invariance prevents the generation of a large scale magnetic field via an alpha effect type mechanism and isotropy implies zero mean flow. Recent direct numerical simulations show that $R_m^{c}$ keeps increasing with $Re$ at the highest possible resolution without any indication of a possible saturation [@Schekochihin]. Schekochihin et al. thus propose that two limit scenarios, sketched in figure 1, could be observed when computers will be able to reach higher $Re$: (i) saturation, $R_m^{c} \rightarrow \rm{constant}$, or (ii) increasing threshold in the form $R_m^{c} \propto Re$. ![Dependence of the dynamo threshold $R_{m}^c$ as a function of the Reynolds number $Re$. Scenario (i): $R_m^c$ tends to a constant. Scenario (ii): $R_m^c$ is proportional to $Re$. []{data-label="fig1"}](dynscalfig.eps){width=".75\textwidth"} A lot of work has been performed on the determination of $R_m^{c}$ as a function of $Re$ for turbulent dynamos in the limit of large $Re$ (or small $P_m$). We recall that (ii) has been proposed by Batchelor in one of the first papers on turbulent dynamos [@Batchelor]. A lot of analytical studies have been also performed, mostly following Kazantsev’s model [@Kazantsev] in order to show that purely turbulent flows can generate a magnetic field. Kazantsev considered a random homegeneous and isotropic velocity field, $\delta$-correlated in time and with a wave number spectrum of the form $k^{-p}$. He showed that for $p$ large enough, generation of a homogeneous isotropic magnetic field with zero mean value, takes place. This is a nice model but its validity is questionable for realistic turbulent flows. However, Kazantsev’s model has been extrapolated to large $Re$. Various predictions, $R_m^{c} \propto Re$ [@Novikov], $R_m^{c} \rightarrow \rm{constant} \approx 400$ for velocity spectra with $3/2 < p < 3$ and no dynamo otherwise [@Rogachevskii], or dynamo for all possible slopes of the velocity spectrum $1 < p < 3$ [@Boldyrev] have been found. These discrepancies show that extrapolation of Kazantsev’s model to realistic turbulence cannot be rigorous. The calculation is possible only in the case of a $\delta$-correlated velocity field in time, and $\delta (t-t')$, which has the dimension of the inverse of time, should then be replaced by a finite eddy turn-over time in order to describe large $Re$ effects. As already noticed, its choice is crucial to determine the behavior of $R_m^{c}$ versus $Re$. A different problem about turbulent dynamos has been considered more recently. It concerns the effect of turbulent fluctuations on a dynamo generated by a mean flow. The problem is to estimate to which extent the dynamo threshold computed as if the mean flow were acting alone, is shifted by turbulent fluctuations. This question has been addressed only recently [@Fauve] and should not be confused with dynamo generated by random flows with zero mean. It has been shown that weak turbulent fluctuations do not shift the dynamo threshold of the mean flow at first order. In addition, in the case of small scale fluctuations, there is no shift at second order either, if the fluctuations have no helicity. This explains why the observed dynamo threshold in Karlsruhe and Riga experiments [@KarlsruheRiga] has been found in good agreement with the one computed as if the mean flow were acting alone, i.e. neglecting turbulent fluctuations. Recent direct numerical simulations have shown that in the presence of a prescribed mean flow, ${\overline{\bf v} (\bf { r})} \neq 0$, $R_m^{c}$ increases with $Re$ at moderate $Re$ but then seems to saturate at larger $Re$, thus following scenario (i). For the same flows, numerical modeling of large scales, large eddy simulations (LES) for instance, gives $R_m^{c} \sim \rm{constant}$ [@PontyLaval]. This last result follows from dimensional consideration as explained below, and has been also obtained for homogeneous isotropic turbulent non helical flows for which EDQNM closures have predicted $R_m^{c} \approx 30$ [@Leorat]. Turbulent dynamo threshold {#dynamo threshold} ========================== When the magnetic Prandtl number, $P_m = R_m / Re = \mu_0 \sigma \nu$, is small, $P_m \ll 1$, the Ohmic dissipative scale, $l_{\sigma} = L R_m^{-3/4}$ is much larger than the Kolmogorov $l_K = L Re^{-3/4}$. Thus, if there is dynamo action, the magnetic field grows at scales much larger than $l_K $ and does not depend on kinematic viscosity. This hypothesis is currently made for large scale quantities in turbulence and if correct, scenario (i) should be followed. If $\nu$ is discarded, $R_m^{c} = \rm{constant}$ indeed follows from dimensional analysis. It is thus not surprising that numerical models that do not resolve viscous scales, all gives this result, although the value of the constant seems to be strongly dependent on the flow geometry and on the model. We conclude that if dynamo action is observed for $P_m \ll 1$, the dynamo threshold is $$R_m^{c} \rightarrow \rm{constant} \; \rm{when} \; Re \rightarrow \infty. \label{thresholdsmallPm}$$ However, we emphasize that no clear-cut demonstration of dynamo action by homogeneous isotropic and parity invariant turbulence exists for $P_m \ll 1$. Experimental demonstrations as well as direct numerical simulations all involve a mean flow and analytical methods extrapolated to $P_m \ll 1$ are questionable. It may be instructive at this stage to recall the study on turbulent dynamos made more than half a century ago by Batchelor [@Batchelor]. Using a questionable analogy between the induction and the vorticity equations, he claimed that the dynamo threshold corresponds to $P_m = 1$, i.e. $R_m^{c} \propto Re$, using our choice of dimensionless parameters (scenario (ii)). It is now often claimed that Batchelor’s criterion $P_m > 1$ for the growth of magnetic energy in turbulent flows is incorrect. However, the weaker criterion $P_m > \rm{constant}$ (scenario (ii)) has not yet been invalidated by direct numerical simulations or by an experimental demonstration without mean flow. It is thus of interest to determine the minimal hypothesis for which Batchelor’s predictions for dynamo onset is obtained using dimensional arguments. To wit, assume that the dynamo eigenmodes develop at small scales such that the threshold does not depend on the integral scale $L$. Then, discarding $L$ in our set of parameters, dimensional analysis gives at once $P_m = P_m^{c} = \rm{constant}$ for the dynamo threshold, i. e. $$R_m^{c} \propto Re. \label{thresholdbatchelor}$$ It has been sometimes claimed that a non zero mean flow is necessary to get a dynamo following scenario (i). However, we note that even for a slow dynamo, i.e., growing on a diffusive time scale, the largest scales look stationary for a dynamo mode at wave length $l_{\sigma}$. For Kolmogorov turbulence, we indeed have, $\mu_0 \sigma l_{\sigma}^2/(L/V) \propto R_m^{-1/2} \ll 1$. This remains true for a $k^{-p}$ spectrum for $p<3$. Mean magnetic energy density {#Mean magnetic energy density} ============================ Dimensional arguments can be also used to determine scaling laws for the mean magnetic energy density. For $P_m \ll 1$ (scenario (i)), discarding $\nu$ gives $$\frac{B^2}{\mu_0} = \rho V^2 \, g_0(R_m), \label{saturationsmallPm}$$ where $g_0$ is an arbitrary function. Close to threshold, the $rms$ velocity $V$ is given by $\mu_0 \sigma VL \sim R_m^{c}$. In the case of a supercritical bifurcation, $g_0(R_m) \propto R_m - R_m^{c}$, and we obtain [@Petrelis01] $$B^2 \propto \frac{\rho}{\mu_0 (\sigma L)^2} \, (R_m - R_m^{c}). \label{saturationsmallPmthreshold}$$ Far from threshold, $Re \gg R_m \gg R_m^{c}$, one could assume that $B$ no longer depends on $\sigma$ provided that the magnetic field mostly grows at scales larger than $l_{\sigma}$. We then obtain equipartition between magnetic and kinetic energy densities, $$B^2/\mu_0 \propto \rho V^2, \label{equipartition}$$ as assumed by Biermann and Schlüter [@Biermann]. A completely different result is obtained in scenario (ii). Let us first recall that according to Batchelor’s analogy between magnetic field and vorticity [@Batchelor], the magnetic field should be generated mostly at the Kolmogorov scale, $l_K = L Re^{-3/4}$, where the velocity gradients are the strongest. He then assumed that saturation of the magnetic field takes place for $\langle B^2 \rangle /\mu_0 \propto \rho v_K^2 = \rho V^2/\sqrt{Re}$, where $v_K$ is the velocity increment at the Kolmogorov scale, $v_K^2 = \sqrt{\nu \epsilon}$. $\epsilon = V^3/L$ is the power per unit mass, cascading from $L$ to $l_K$ in the Kolmogorov description of turbulence. This can be easily understood. $\epsilon = V^3/L$ being the power per unit mass available to feed the dynamo, it may be a wise choice to keep it, instead of $V$ in our set of parameters, thus becoming $B$, $\rho$, $\epsilon$, $L$, $\nu$, $\mu_0$ and $\sigma$. Then, if we consider dynamo modes that do not depend on $L$, we obtain at once $$\frac{B^2}{\mu_0} = \rho \sqrt{\nu \epsilon} \, h (P_m) = \frac{\rho V^2}{\sqrt{Re}} \, h (P_m) \label{saturationbatchelor}$$ for saturation, where $h(P_m)$ is an arbitrary function of $P_m$. Close to dynamo threshold, $P_m \approx P_m^{c}$, we have $h(P_m) \propto P_m - P_m^{c}$ if the bifurcation is supercritical. Only the prefactor $\rho V^2 / \sqrt{Re}$ of (\[saturationbatchelor\]) is the kinetic energy at Kolmogorov scale, that was assumed to be in equipartition with magnetic energy in Batchelor’s prediction. This class of dynamos being small scale ones, it is not surprising that the inertial range of turbulence screens the magnetic field from the influence of integral size, thus $L$ can be forgotten. We emphasize that a necessary condition for Batchelor’s scenario is that the magnetic field can grow below the Kolmogorov scale, i.e. its dissipative length $l_{\sigma}$ should be smaller than $l_K$, thus $P_m > 1$. There is obviously a strong discrepancy between (\[equipartition\]) and (\[saturationbatchelor\]). The prefactors in these two laws are the upper and lower limits of a continuous family of scalings that are obtained by balancing the magnetic energy with the kinetic energy at one particular length scale within the Kolmogorov spectrum. It is not known if one of them is selected by turbulent dynamos. Ohmic losses ============ Ohmic losses due to currents generated by dynamo action give a lower bound to the power required to feed a dynamo. In order to evaluate them, it is crucial to know at which scales the magnetic field grows. Assuming that a dynamo is generated in the case $P_m \ll 1$ (scenario (i)), we want to give a possible guess for the power spectrum $\vert \hat B \vert^2$ of the magnetic field as a function of the wave number $k$ and the parameters $\rho$, $\epsilon$, $L$, $\nu$, $\mu_0$ and $\sigma$. Far from threshold, $Re \gg R_m \gg R_m^{c}$, the dissipative lengths are such that $l_K \ll l_{\sigma} \ll L$. For $k$ in the inertial range, i.e. $k l_{\sigma} \ll 1 \ll kL$, we may use a Kolmogorov type argument and discard $L$, $\sigma$ and $\nu$. Then, only one dimensionless parameter is left, and not too surprisingly, we get $$\vert \hat B \vert^2 \propto \mu_0 \rho \, \epsilon^{\frac{2}{3}} \, k^{-\frac{5}{3}}. \label{kolmogorov}$$ This is only one possibility among many others proposed for MHD turbulent spectra within the inertial range, but it is the simplest. Integrating over $k$ obviously gives the equipartition law (\[equipartition\]) for the magnetic energy. It is now interesting to evaluate Ohmic dissipation. Its dominant part comes from the current density at scale $l_{\sigma}$. We have $$\frac{{\bf j}^2}{\sigma} = \frac{1}{\sigma} \int \vert \hat j \vert^2 \, dk \propto \frac{1}{\mu_0^2 \sigma} \int k^2 \vert \hat B \vert^2 \, dk \propto \frac{\rho}{\mu_0 \sigma} \, \epsilon^{\frac{2}{3}} \, l_{\sigma}^{-\frac{4}{3}} \propto {\rho} \frac{V^3}{L}. \label{joule}$$ We thus find that Ohmic dissipation is proportional to the total available power which corresponds to some kind of optimum scaling law for Ohmic dissipation. Although, this does not give any indication that this regime is achieved, we note that the above scaling corresponds to the one found empirically from a set of numerical models [@christensen]. Their approximate fit, $(B^2/\mu_0)/(j^2/\sigma) \propto L/V$, indeed results from equations (\[kolmogorov\], \[joule\]). [00]{} Ya. B. Zeldovich , A. A. Ruzmaikin and D. D. 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--- abstract: 'A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is proven.' address: | Department of Mathematics and Statistics\ University of New Mexico\ Albuquerque, New Mexico, 87131-0001 author: - 'D. Vassilev' title: The monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds --- Introduction ============ The purpose of this note is to show the monotonicity of the entropy type energy for the heat equation on a compact quaternionic contact manifold inspired by the corresponding Riemannian fact related to Perelman’s entropy formula for the heat equation on a static Riemannian manifold, see [@Ni04]. More recently a similar quantity was considered in the CR case [@CW10]. Our goal is to give a relatively simple proof of the monotonicity, more in line with the Riemannian case, by resolving directly the difficulties arising in the sub-Riemannian setting. In Section \[s:CR\] we include a proof of the result of [@CW10] in the CR case from our point of view. To state the problem, let $M$ be a quaternionic contact manifold, henceforth abbreviated to qc, and $u$ be a smooth *positive* solution to the quaternionic contact heat equation $$\label{e:heat} \frac{\partial}{\partial t}u=\Delta u.$$ Hereafter, $\triangle u=tr^g(\nabla^2 u)$ is the negative sub-Laplacian with the trace taken with respect to an orthonormal basis of the horizontal $4n$-dimensional space. Associated to such a solution are the (Nash like) entropy $$\label{e:N definition} \mathcal{N}(t)=\int_M u\ln u {\, Vol_{\eta}}$$ and entropy energy functional $$\label{enf} \mathcal{E}(t)=\int_M |\nabla f|^2 u \, Vol_{\eta},$$ where, as usual, $f=-\ln u$ and $\, Vol_{\eta}$ is the naturally associated volume form on $M$, see and also [@IMV Chapter 8]. Exactly as in the Riemannian case, we have that the entropy is decreasing (i.e., non-increasing) because of the formula $${\frac{d}{dt}}\mathcal{N}=-\mathcal{E}(t).$$ Our goal is the computation of the second derivative of the entropy. In order to state the result we consider the Ricci type tensor $$\label{e:Lichnerowicz tensor} \mathcal{L}(X,X)\overset{def}{=}2 Sg(X,X)+\alpha_n T^0(X,X) +\beta_n U(X,X)\geq 4g(X,X),$$ where $X$ is any vector from the horizontal distribution, $\alpha_n=\frac {2(2n+3)}{2n+1}, \quad \beta_n=\frac {4(2n-1)(n+2)}{(2n+1)(n-1)}$, and $T^0$ and $U$ are certain invariant components of the torsion, see Subsection \[ss:prelim\]. In addition, following [@IPV2], we define the $P-$form of a fixed smooth function $f$ on $M$ by the following equation $$\begin{gathered} \label{e:P form} P_f(X) =\sum_{b=1}^{4n}\nabla ^{3}f(X,e_{b},e_{b})+\sum_{t=1}^{3}\sum_{b=1}^{4n}\nabla ^{3}f(I_{t}X,e_{b},I_{t}e_{b}) \\ -4nSdf(X)+4nT^{0}(X,\nabla f)-\frac{8n(n-2)}{n-1}U(X,\nabla f),\end{gathered}$$ which in the case $n=1$ is defined by formally dropping the last term. The $P-$function of $f$ is the function $P_f(\nabla f)$. The $C-$operator of $M$ is the 4-th order differential operator $$f\mapsto Cf =-\nabla^* P_f=\sum_{a=1}^{4n}(\nabla_{e_a} P_f)\,(e_a).$$In many respects the $C-$operator plays a role similar to the Paneitz operator in CR geometry. We say that the $P-$function of $f$ is non-negative if $$\int_M f\cdot Cf \, Vol_{\eta}= -\int_M P_f(\nabla f)\, Vol_{\eta} \geq 0.$$If the above holds for any $f\in \mathcal{C}^\infty_o\,(M)$ we say that the $C-$operator is [non-negative]{}, $\mathcal{C}\geq 0$. We are ready to state our first result. \[p:energy ineq\] Let $M$ be a compact QC manifold of dimension $4n+3$. If $u=e^{-f}$ is a positive solution to heat equation , then we have $$\frac {2n+1}{4n}\mathcal{E}'(t)=-\int_M \left[ |(\nabla^2f)_{0}|^2+\frac {2n+1}{2}\mathcal{L}(\nabla f,\nabla f) +\frac {1}{16n}| \nabla f|^4 \right]u\, Vol_{\eta} +\frac {3}{n}\int_M P_F(\nabla F)\, Vol_{\eta},$$ where $u=F^2$ ($f=-2\ln F$) and $(\nabla^2f)_{0}$ is the traceless part of horizontal Hessian of $f$. Several important properties of the C-operator were found in [@IPV2], most notable of which is the fact that the $C-$operator is non-negative for $n>1$. In dimension seven, $n=1$, the condition of non-negativity of the $C-$operator is non-trivial. However, [@IPV2] showed that on a 7-dimensional compact qc-Einstein manifold with positive qc-scalar curvature the $P-$function of an [eigenfunction]{} of the sub-Laplacian is non-negative. In particular, this property holds on any 3-Sasakian manifold. Clearly, these facts together with Proposition \[p:energy ineq\] imply the following theorem. \[t:monotone energy\] Let $M$ be a compact QC manifold of dimension $4n+3 $ of non-negative Ricci type tensor $\mathcal{L}(X,X) \geq 0$. In the case $n=1$ assume, in addition, that the $C-$operator is non-negative. If $u=e^{-f} $ is a positive solution to heat equation then the energy is monotone decreasing (i.e., non-increasing). The proof of Proposition \[p:energy ineq\] follows one of L. Ni’s arguments [@Ni04] in the Riemannian case, thus it relies on Bochner’s formula. More precisely, after Ni’s initial step, in order to handle the extra terms in Bochner’s formula, we will follow the presentation of [@IV14] where this was done for the qc Lichnerowicz type lower eigenvalue bound under positive Ricci type tensor, see [@IPV1; @IPV2] for the original result. In the qc case, similar to the CR case, the Bochner formula has additional hard to control terms, which include the $P$-function of $f$. In our case, since the integrals are with respect to the measure $u\, Vol_{\eta}$, rather than ${\, Vol_{\eta}}$ as in the Lichnerowicz type estimate, some new estimates are needed. The key is the following proposition which can be considered as an estimates from above of the integral of the $P$-function of $f$ with respect to the measure $u\, Vol_{\eta}$ when the $C-$operator is non-negative. \[p:Paneitz estimate\] Let $(M,\eta)$ be a compact QC manifold of dimension $4n+3$. If $u=e^{-f}$ is a positive solution to heat equation , then with $f=-2\ln F$ we have the identity $$\label{e:Paneitz estimate} \int_M P_f(\nabla f) u\, Vol_{\eta}=\frac 14\int_M |\nabla f|^4 u\, Vol_{\eta} +4\int_M P_F(\nabla F)\, Vol_{\eta}.$$ In the last section of the paper we apply the same method in the case of a strictly pseudoconvex pseudohermitian manifold and prove the following Proposition. \[p:CR energy ineq\] Let $M$ be a compact strictly pseudoconvex pseudohermitian CR manifold of dimension $2n+1$. If $u=e^{-f}$ is a positive solution to the heat equation , then we have $$\frac {n+1}{2n}\mathcal{E}'(t)=-\int_M \left[ |(\nabla^2f)_{0}|^2+\frac {2n+1}{2}\mathcal{L}(\nabla f,\nabla f) +\frac {1}{8n}| \nabla f|^4 \right]u\, Vol_{\eta} -\frac {6}{n}\int_M F\, \mathcal{C}(F){\, Vol_{\eta}},$$ where $u=F^2$, $(\nabla^2f)_{0}$ is the traceless part of horizontal Hessian of $f$ and $\mathcal{C}$ is the CR-Paneitz operator of $M$. We refer to Section \[s:CR\] for the relevant notation and definitions. As a consequence of Proposition \[p:CR energy ineq\] we recover the monotonicity of the entropy energy shown previously in [@CW10]. We note that one of my motivations to consider the problem was the application of the CR version of the monotonicity of the entropy like energy [@CW10 Lemma 3.3] in obtaining (non-optimal) estimate on the bottom of the $L^2$ spectrum of the CR sub-Laplacian. However, the proof of [@CW10 Corollary 1.9 and Section 6] is not fully justified since [@CW10 Lemma3.3] is proved for a compact manifold. It should be noted that a proof of S-Y Cheng’s type (even non-optimal) estimate in a sub-Riemannian setting, such as CR or qc-manifold, is an interesting problem in particular because of the lack of general comparison theorems. We conclude by mentioning another proof of the monotonicity of the energy in the recent preprint [@IP16], which was the result of a past collaborative work with Ivanov and Petkov. Remarkably, [@CW10] is also not acknowledged in [@IP16] despite the fact that the calculations in [@IP16] came after I introduced to Ivanov many of the interesting (sub-Riemannian) comparison problems and drew their attention to [@CW10]. While I can hardly wish to be associated with [@IP16], a quick look shows the line for line substantial overlap of [@IP16 Section 3] with Chang and Wu’ proof [@CW10 Lemma 3.3], the publication of collaborative work without a discussion with all sides is notable. Therefore, I decided to give my independent approach to the problem. [**Acknowledgments.**]{} Thanks are due to Qi Zhang for insightful comments on the S-Y Cheng eigenvalue estimate during the Beijing Workshop on Conformal Geometry and Geometric PDE in 2015, and to Jack Lee and Ben Chow for some useful discussions. The author also acknowledges the support of the Simons Foundation grant \#279381. Proofs of the Propositions ========================== Some preliminaries {#ss:prelim} ------------------ Throughout this section $M$ will be a qc manifold of dimension $4n+3$, [@Biq2], with horizontal space $H$ locally given as the kernel of a 1-form $\eta=(\eta_1,\eta_2,\eta_3)$ with values in $\mathbb{R}^3$, and Biquard connection $\nabla$ with torsion $T$. Below we record some of the properties needed for this paper, see also [@Biq1] and [@IV3] for a more expanded exposition. The $Sp(n)Sp(1)$ structure on $H$ is fixed by a positive definite symmetric tensor $g$ and a rank-three bundle $\mathbb{Q}$ of endomorphisms of $H$ locally generated by three almost complex structures $I_1,I_2,I_3$ on $H$ satisfying the identities of the imaginary unit quaternions and also the conditions $$g(I_s.,I_s.)=g(.,.)\qquad \text{and} \qquad 2g(I_sX,Y)\ =\ d\eta_s(X,Y).$$ Associated with the Biquard connection is the vertical space $V$, which is complementary to $H$ in $TM$. In the case $n=1$ we shall make the usual assumption of existence of Reeb vector fields $\xi_1, \xi_2, \xi_3$, so that the connection is defined following D. Duchemin [@D]. The fundamental 2-forms $\omega_s$ of the fixed qc structure will be denoted by $\omega_s$, $$2\omega_{s|H}\ =\ \, d\eta_{s|H},\qquad \xi\lrcorner\omega_s=0,\quad \xi\in V.$$ In order to give some idea of the involved quantities we list a few more essential for us details. Recall that $\nabla$ preserves the decomposition $H\oplus V$ and the $ Sp(n)Sp(1)$ structure on $H$, $$\nabla g=0, \quad \nabla\Gamma(\mathbb{Q}) \subset \Gamma( \mathbb{Q})$$ and its torsion on $H$ is given by $T(X,Y)=-[X,Y]_{|V}$. Furthermore, for a vertical field $\xi\in V$, the endomorphism $T_{\xi}\equiv T(\xi,.)_{|H}$ of $H$ belongs to the space $ (sp(n)\oplus sp(1))^{\bot}\subset gl(4n)$ hence $ T({\xi}, X,Y)=g(T_{\xi}X,Y)$ is a well defined tensor field. The two $Sp(n)Sp(1)$-invariant trace-free symmetric 2-tensors $T^0(X,Y)= g((T_{\xi_1}^{0}I_1+T_{\xi_2}^{0}I_2+T_{ \xi_3}^{0}I_3)X,Y)$, $U(X,Y) =g(uX,Y)$ on $H$, introduced in [@IMV], satisfy $$\label{propt} \begin{aligned} T^0(X,Y)+T^0(I_1X,I_1Y)+T^0(I_2X,I_2Y)+T^0(I_3X,I_3Y)=0, \\ U(X,Y)=U(I_1X,I_1Y)=U(I_2X,I_2Y)=U(I_3X,I_3Y). \end{aligned}$$ Note that when $n=1$, the tensor $U$ vanishes. The tensors $T^0$ and $U$ determine completely the torsion endomorphism due to the identity [@IV Proposition 2.3] $$4T^0(\xi_s,I_sX,Y)=T^0(X,Y)-T^0(I_sX,I_sY),$$ which in view of implies $$\label{need1} \sum_{s=1}^3T(\xi_s,I_sX,Y)= T^0(X,Y)-3U(X,Y).$$ The curvature of the Biquard connection is $R=[\nabla,\nabla]-\nabla_{[\ ,\ ]}$ with [qc-Ricci tensor]{} and *normalized* qc-scalar curvature, defined by respectively by $$\label{e:horizontal ricci} Ric(X,Y)=\sum_{a=1}^{4n}{g(R(e_a,X)Y,e_a)}, \qquad 8n(n+2)S=\sum_{a=1}^{4n}Ric(e_a,e_a).$$ According to [@Biq1] the Ricci tensor restricted to $H$ is a symmetric tensor. Remarkably, the torsion tensor determines the qc-Ricci tensor of the Biquard connection on $M$ in view of the formula, [@IMV], $$\label{e:Ric by torsion} Ric(X,Y) \ =\ (2n+2)T^0(X,Y) +(4n+10)U(X,Y)+\frac{S}{4n}g(X,Y).$$ Finally, ${\, Vol_{\eta}}$ will denote the volume form $$\label{e:volume form} {\, Vol_{\eta}}=\eta_1\wedge\eta_2\wedge\eta_3\wedge\Omega^n,$$ where $\Omega=\omega_1\wedge\omega_1+\omega_2\wedge\omega_2+\omega_3\wedge\omega_3$ is the [fundamental 4-form]{}. We note the integration by parts formula $$\label{div} \int_M (\nabla^*\sigma)\,\, Vol_{\eta}\ =\ 0,$$ where the (horizontal) divergence of a horizontal vector field $\sigma\in\Lambda^1\, (H)$ is given by $\nabla^* \sigma =-tr|_{H}\nabla\sigma= -\nabla \sigma(e_a,e_a) $ for an orthonormal frame $\{e_a\}_{a=1}^{4n}$ of the horizontal space. Proof of Proposition \[p:Paneitz estimate\] ------------------------------------------- We start with a formula for the change of the dependent function in the $P$-function of $f$. To this effect, with $f=f(F)$, a short calculation shows the next identity $$\begin{gathered} \nabla^3 f(Z,X,Y)=f^{\prime }\nabla^3F(Z,X,Y)+f^{\prime \prime \prime }dF(Z)dF(X)dF(Y) \\ +f^{\prime \prime }\nabla^2F(Z,X)dF(Y) +f^{\prime \prime }\nabla^2F(Z,Y)dF(X)+f^{\prime \prime }\nabla^2F(X,Y)dF(Z).\end{gathered}$$ Recalling definition we obtain $$\begin{gathered} \label{e:P from change variable} P_f(Z)=f^{\prime }P_F(Z) + f^{\prime \prime \prime }|{\nabla}F|^2dF(Z) + 2 f^{\prime \prime 2}F (Z, \nabla F)+ f^{\prime \prime }(\Delta F)dF(Z) \\ +f^{\prime \prime }\sum_{s=1}^3 g(\nabla^2 F, \omega_s)dF(I_s Z),\end{gathered}$$ which implies the identity $$\label{e:Paneitz change variable} P_f(\nabla f)=f^{\prime 2}P_F(\nabla F) + f^{\prime }f^{\prime \prime \prime }|{\nabla}F|^4 + 2f^{\prime }f^{\prime \prime }\nabla ^2F (\nabla F, \nabla F)+f^{\prime }f^{\prime \prime }|{\nabla}F|^2\Delta F.$$ In our case, since we are interested in expressing the integral of $uP_f(\nabla f)=e^{-f}P_f(\nabla f)$ in terms of the integral of a $P$-function of some function, equation leads to the ordinary differential equation $u\left (-\frac {u^{\prime }}{u}\right)^2=const$. Therefore, we let $u=F^2$ and find $$\label{e:Paneitz change variable F} uP_f(\nabla f)=4P_F(\nabla F) + 8F^{-2}|\nabla F|^4 -8F^{-1} \nabla^2F (\nabla F, \nabla F)-4F^{-1}|\nabla F|^2\Delta F.$$ Now, the last three terms will be expressed back in the variable $f$ which gives $$\label{e:Paneitz change variable f} uP_f(\nabla f)=4P_F(\nabla F) + \left [-\frac 14|\nabla f|^4+\frac 12 |\nabla f|^2\Delta f+ \nabla^2f (\nabla f, \nabla f)\right]u$$ At this point, we integrate the above identity and then apply the (integration by parts) divergence formula in order to show $$\int_M \nabla^2f (\nabla f, \nabla f)u \, Vol_{\eta}=\frac 12\int_M \left[ |\nabla f|^4-|\nabla f|^2\Delta f\right]u \, Vol_{\eta},$$ which leads to . The proof of Proposition \[p:Paneitz estimate\] is complete. Proof of Proposition \[p:energy ineq\] -------------------------------------- The initial step is identical to the Riemannian case [@Ni04], so we skip the computations. Let $w=2\Delta f - |\nabla f|^2$. Using the heat equation and integration by parts, exactly as in the Riemannian case, we have $$\label{e:E'} {\frac{d}{dt}}\mathcal{E}(t)=\int_M(\partial_t-\Delta)(uw)\, Vol_{\eta}$$ and also $$\label{e:key 1} (\partial_t-\Delta)(uw)=\left [2g\left ( \nabla\left (\Delta f \right), \nabla f\right)-\Delta|\nabla f|^2 \right]u.$$ Next, we apply the qc Bochner formula [@IPV1; @IPV2] $$\begin{gathered} \frac12\triangle |\nabla f|^2=|\nabla^2f|^2+g\left (\nabla (\triangle f), \nabla f \right )+2(n+2)S|\nabla f|^2 \\ +2(n+2)T^0(\nabla f,\nabla f) +4(n+1)U(\nabla f,\nabla f) + 4R_f(\nabla f),\end{gathered}$$ where $$R_f(Z)=\sum_{s=1}^3\nabla^2f(\xi_s,I_sZ).$$ Therefore, $$\begin{gathered} \label{e:dt-lap} \frac 12(\partial_t-\Delta)(uw) =\big[-|\nabla^2f|^2-2(n+2)S|\nabla f|^2-2(n+2)T^0(\nabla f,\nabla f) \\ -4(n+1)U(\nabla f,\nabla f) - 4R_f (\nabla f)\big]u\end{gathered}$$ The next step is the computation of $\int_M R_f (\nabla f)u\, Vol_{\eta}$ in two ways as was done in [@IPV1; @IPV2] for the Lichnerowicz type first eigenvalue lower bound but integrating with respect to ${\, Vol_{\eta}}$ rather than $u{\, Vol_{\eta}}$ as we need to do here. For ease of reading we will follow closely [@IV14 Section 8.1.1] but notice the opposite convention of the sub-Laplacian in [@IV14 Section 8.1.1]. First with the help of the $P$-function, working similarly to \[Lemma 3.2\][IPV2]{} where the integration was with respect to ${\, Vol_{\eta}}$, we have $$\begin{gathered} \label{e:Rf u1} \int_M R_f (\nabla f)u\, Vol_{\eta}=\int_M [- \frac{1}{4n}P_n(\nabla f)-\frac{1}{4n}(\triangle f)^2-S|\nabla f|^2 \\ + \frac{n+1}{n-1}U(\nabla f,\nabla f)]u\, Vol_{\eta} +\frac {1}{4n}\int_M |\nabla f|^2(\Delta f)u\, Vol_{\eta},\end{gathered}$$ with the convention that in the case $n=1$ the formula is understood by formally dropping the term involving (the vanishing) tensor $U$. Notice the appearance of a “new” term in the last integral in comparison to the analogous formula in [@IV14 Section 8.1.1, p. 310]. Indeed, taking into account the $Sp(n)Sp(1)$ invariance of $R_f(\nabla f)$ and Ricci’s identities we have, cf. [@IPV2 Lemma 3.2], $$R_f(X)=-\frac {1}{4n}\sum_{s=1}^3\sum_{a=1}^{4n} \nabla^3 f(I_sX, e_a, I_s e_a)+\left [ T^0(X,\nabla f)-3U(X, \nabla f)\right]$$ hence gives $$uR_f (\nabla f)= \big[- \frac{1}{4n}P_n(\nabla f)-S|\nabla f|^2 + \frac{n+1}{n-1}U(\nabla f,\nabla f)\big]u\newline +\frac {1}{4n}\sum_{a=1}^{4n}\nabla^3f(\nabla f, e_a, e_a)u.$$ An integration by parts shows the validity of . On the other hand, we have $$\begin{gathered} \label{e:Rf u2} \int_MR_f(\nabla f) u\, Vol_{\eta} =-\int_M \left[\frac {1}{4n}\sum_{s=1}^3 g(\nabla^2 f, \omega_s)^2 +T^0(\nabla f,\nabla f)-3U(\nabla f,\nabla f)\right]u{\, Vol_{\eta}},\end{gathered}$$ which other than using different volume forms is identical to the second formula in [IV14]{}. Indeed, following [@IPV1 Lemma 3.4], using Ricci’s identity $$\nabla^2f (X,\xi_s)-\nabla^2f(\xi_s,X)=T(\xi_s,X,\nabla f)$$ and , we have $$R_f({\nabla}f)=\left(\sum_{s=1}^3\nabla^2f(I_s{\nabla}f, \xi_s) \right)-\left[ T^0({\nabla}f, {\nabla}f)-3U({\nabla}f, {\nabla}f)\right]$$ An integration by parts gives , noting the term $\sum_{s=1}^3 df(\xi_s)df(I_s\nabla f)=0$ and taking into account that by Ricci’s identity $$\nabla^2f (X,Y)-\nabla^2f(Y,X)=-2\sum_{s=1}^3\omega_s(X,Y)df(\xi_s)$$ we have $g(\nabla^2f , \omega_s) {=}\sum_{a=1}^{4n}\nabla^2f(e_a,I_se_a)=-4ndf(\xi_s). $ Now, working as in [@IV14 Section 8.1.1, p. 310], we subtract and three times formula from which brings us to the following identity $$\begin{gathered} \label{e:E formula 1} \frac 12 \frac{d}{dt}\mathcal{E}(t)=\int_M\big[ -|(\nabla^2f)_{0}|^2-\frac {2n+1}{2}\mathcal{L}(\nabla f,\nabla f)\big]u\, Vol_{\eta} \\ +\frac {1}{4n}\int_M \big[3P_f(\nabla f)+2(\Delta f)^2-3|\nabla f |^2\Delta f \big]u {\, Vol_{\eta}},\end{gathered}$$ where $|(\nabla^2f)_{0}|^2$ is the square of the norm of the traceless part of the horizontal Hessian $$|(\nabla^2 f)_0|^2=|\nabla^2f|^2-\frac{1}{4n}\Big[(\triangle f)^2+\sum_{s=1}^{3}[g(\nabla^2f,\omega_s)]^2\Big].$$ Next, we consider $\int_M \big[2(\Delta f)^2-3|\nabla f |^2\Delta f \big]u\, Vol_{\eta}$. Using the heat equation we have the identical to the Riemannian case relation $$\frac{d}{dt} \mathcal{E}(t)=\frac{d}{dt} \int_M w \Delta u\, Vol_{\eta}=\int_M \big(-2(\Delta f)^2 + 3|\nabla f|^2\Delta f - |\nabla f|^4\big )u\, Vol_{\eta},$$ hence $$\label{e:E formula 2} \int_M \big( 2(\Delta f)^2-3|\nabla f|^2\Delta f \big) u\, Vol_{\eta}= -\frac{d}{dt} \mathcal{E}(t)-\int_M |\nabla f|^4u\, Vol_{\eta}.$$ A substitution of the above formula in gives $$\begin{gathered} \frac {2n+1}{4n}\frac{d}{dt} \mathcal{E}(t)=\int_M\big[ -|(\nabla^2f)_0 |^2-\frac {2n+1}{2}\mathcal{L}(\nabla f,\nabla f)\big]u\, Vol_{\eta} +\frac {1}{4n}\int_M \big[3P_f(\nabla f)- |\nabla f|^4\big]u\, Vol_{\eta}.\end{gathered}$$ Finally, we invoke Proposition \[p:Paneitz estimate\] in order to complete the proof. The CR case {#s:CR} =========== In this section we prove the monotonicity formula in the CR case stated in Proposition \[p:CR energy ineq\] following the method we employed in the qc case. This implies the monotonicity of the entropy like energy which was proved earlier in [@CW10]. Throughout the section $M$ will be a $(2n+1)$-dimensional strictly pseudoconvex (integrable) CR manifold with a fixed pseudohermitian structure defined by a contact form $\eta$ and complex structure $J$ on the horizontal space $H=Ker \, \eta$. The fundamental 2-form is defined by $\omega=\frac 12 \eta$ and the Webster metric is $g(X,Y)=-\omega (JX,Y)$ which is extended to a Riemannian metric on $M$ by declaring that the Reeb vector field associated to $\eta$ is of length one and orthonormal to the horizontal space. We shall denote by ${\nabla}$ the associated Tanaka-Webster connection [@T] and [@W; @W1], while $\triangle u=tr^g(\nabla^2 u)$ will be the negative sub-Laplacian with the trace taken with respect to an orthonormal basis of the horizontal $2n$-dimensional space. Finally, we define the Ricci type tensor $$\label{e:Lichnerowicz tensor CR} \mathcal{L}(X,Y)=\rho(JX,Y)+2nA(JX,Y)$$ recalling that on a CR manifold we have $$\label{rid} Ric(X,Y)=\rho(JX,Y)+2(n-1)A(JX,Y),$$ where $\rho$ is the $(1,1)$-part of the pseudohermitian Ricci tensor (the Webster Ricci tensor) while the $(2,0)+(0,2)$-part is the Webster torsion $A$, see [@IV3 Chapter 7] for the expressions in real coordinates of these known formulas [@W; @W1], see also [@DT]. With the above convention in place, as in [@CW10], for a positive solution of we consider the entropy and energy , where ${\, Vol_{\eta}}=\eta\wedge(d\eta)^{2n}$. We turn to the proof of Proposition \[p:CR energy ineq\]. For a function $f$ we define the one form, $$\label{e:Pdef} P_{f}(X)=\nabla ^{3}f(X,e_{b},e_{b})+\nabla ^{3}f(JX,e_{b},Je_{b})+4nA(X,J\nabla f)$$so that the fourth order CR-Paneitz operator is given by $$\begin{gathered} \label{e:Cdef} C(f)=-\nabla ^{\ast }P=(\nabla_{e_a} P)({e_a})=\nabla ^{4}f(e_a,e_a,e_{b},e_{b})+\nabla ^{4}f(e_a,Je_a,e_{b},Je_{b})\\ -4n\nabla^* A(J\nabla f)-4n\,g(\nabla^2 f,JA).\end{gathered}$$ By [@GL88], when $n>1$ a function $f\in \mathcal{C}^3(M)$ satisfies the equation $Cf=0$ iff $f$ is CR-pluriharmonic. Furthermore, the CR-Paneitz operator is non-negative, $$\int_M f\cdot Cf {\, Vol_{\eta}}=-\int_MP_f(\gr) {\, Vol_{\eta}}\geq 0.$$ On the other hand, in the three dimensional case the positivity condition is a CR invariant since it is independent of the choice of the contact form by the conformal invariance of $C$ proven in [@Hi93]. We turn to the proof of Proposition \[p:CR energy ineq\]. Taking into account and the CR Bochner formula [@Gr], $$\begin{gathered} \label{e:bohh} \frac12\triangle |\nabla f|^2=|\nabla^2 f|^2+g(\nabla(\triangle f),\nabla f)+Ric(\nabla f,\nabla f)+2A(J\nabla f,\nabla f) + 4R_f({\nabla}f),\end{gathered}$$ where $R_f(Z)=\nabla df(\xi,JZ)$, see [@IV14 Section 7.1] and references therein but note the opposite sign of the sub-Laplacian, we obtain the next identity $$\label{e:dt-CR lap} \frac 12(\partial_t-\Delta)(uw) =\big[-|\nabla^2f|^2-Ric(\nabla f,\nabla f) -2A(\nabla f,{\nabla}\nabla f) - 4R_f (\nabla f)\big]u.$$ Since still holds, working as in the qc case we compute $\int_M R_F({\nabla}f)u{\, Vol_{\eta}}$ in two ways [@Gr Lemma 4] and [@IVO Lemma 8.7] following the exposition [@IV14]. From Ricci’s identity $$\nabla^2f (X,Y)-\nabla^2f(Y,X)=-2\omega(X,Y)df(\xi)$$ it follows $df(\xi)=-\frac {1}{2n}g(\nabla^2 f, \omega)$. Hence $$\nabla^2f(JZ, \xi)=-\frac {1}{2n}\sum_{b=1}^{2n}\nabla^3 f(JZ, e_b,Je_b),$$ where $\{e_b\}_{b=1}^{2n}$ is an orthonormal basis of the horizontal space. Applying Ricci’s identity $$\nabla^2f (X,\xi)-\nabla^2f(\xi,X)=A(X,\nabla f)$$ it follow $$\label{e:CR rem} R_f(Z)=\nabla ^{2}f(\xi ,JZ)=-\frac{1}{2n}\sum_{b=1}^{2n}\nabla^3 f(JZ, e_b,Je_b) -A(JZ,\nabla f).$$ Taking into account the last formula gives $$R_f(Z)=-\frac{1}{2n}P_f(Z) +A(JZ,\nabla f)+\frac{1}{2n}\sum_{b=1}^{2n}\nabla^3 f(Z, e_b,e_b).$$ Now, an integration by parts shows the next identity $$\begin{gathered} \label{e:Rf u1 CR} \int_M R_f({\nabla}f) u{\, Vol_{\eta}}=\int_M\big[-\frac{1}{2n}P_f({\nabla}f) +A(J\nabla f,\nabla f)-\frac {1}{2n}(\Delta f)^2 +\frac {1}{2n}|{\nabla}f|^2 (\Delta f) \big]u{\, Vol_{\eta}}.\end{gathered}$$ On the other hand, using again but now we integrate and then use integration by parts we have $$\label{e:Rf u2 CR} \int_M R_f({\nabla}f) u{\, Vol_{\eta}}=\int_M\big[ -\frac {1}{2n}g(\nabla^2 f,\omega)^2-A(J\nabla f, \nabla f)\big ] u{\, Vol_{\eta}}.$$ At this point, exactly as in the qc case, we subtract and three times formula from , which gives $$\mathcal{E}'(t)=-\int_M\big[ |(\nabla^2f)_{0}|^2 +\mathcal{L}(\nabla f,\nabla f)\big]u{\, Vol_{\eta}}\\ +\frac {1}{2n}\int_M \big[3P_f(\nabla f)+2(\Delta f)^2-3|\nabla f |^2\Delta f \big]u{\, Vol_{\eta}},$$ where $|(\nabla^2f)_{0}|^2$ is the square of the norm of the traceless part of the horizontal Hessian $$|(\nabla^2 f)_0|^2=|\nabla^2f|^2-\frac{1}{2n}\Big[(\triangle f)^2+g(\nabla^2f,\omega)^2\Big].$$ Taking into account that the formulas in Proposition \[p:Paneitz estimate\] and hold unchanged we complete the proof. 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--- abstract: 'Hypothesis testing is a central statistical method in psychology and the cognitive sciences. However, the problems of null hypothesis significance testing (NHST) and $p$-values have been debated widely, but few attractive alternatives exist. This article introduces the `fbst` R package, which implements the *Full Bayesian Significance Test (FBST)* to test a sharp null hypothesis against its alternative via the $e$-value. The statistical theory of the FBST has been introduced by [@Pereira1999] more than two decades ago and since then, the FBST has shown to be a Bayesian alternative to NHST and $p$-values with both theoretical and practical highly appealing properties. The algorithm provided in the `fbst` package is applicable to any Bayesian model as long as the posterior distribution can be obtained at least numerically. The core function of the package provides the Bayesian evidence against the null hypothesis, the $e$-value. Additionally, $p$-values based on asymptotic arguments can be computed and rich visualisations for communication and interpretation of the results can be produced. Three examples of frequently used statistical procedures in the cognitive sciences are given in this paper which demonstrate how to apply the FBST in practice using the `fbst` package. Based on the success of the FBST in statistical science, the `fbst` package should be of interest to a broad range of researchers in psychology and the cognitive sciences and hopefully will encourage researchers to consider the FBST as a possible alternative when conducting hypothesis tests of a sharp null hypothesis.' author: - Riko Kelter bibliography: - 'library.bib' title: 'fbst: An R package for the Full Bayesian Significance Test for testing a sharp null hypothesis against its alternative via the e-value' --- Introduction {#introduction .unnumbered} ============ Hypothesis testing is a widely used method in the cognitive sciences and in experimental psychology. However, the recently experienced replication crisis troubles experimental sciences and the underlying problems are still widely debated [@Pashler2012; @Pashler2012a; @Wasserstein2019; @Haaf2019]. Among the identified problems is the inappropriate use and interpretation of $p$-values, which are used in combination with null hypothesis significance tests (NHST) [@Benjamin2019; @benjaminRedefineStatisticalSignificance; @Colquhoun2014; @Colquhoun2017]. As a consequence, in 2016 the American Statistical Association issued a statement about the identified problems and recommended to consider alternatives to $p$-values or supplement data analysis with further measures of evidence: > “All these measures and approaches rely on further assumptions, but they may more directly address the size of an effect (and its associated uncertainty) or whether the hypothesis is correct.” [@wasserstein2016 p. 132] Due to the problems with NHST and $p$-values, the editors of *Basic and Applied Social Psychology* even decided to ban p-values and NHST completely from their journal. In the recent literature various proposals have been made how to improve the reproducibility of research and the quality of statistical data analysis, in particular the reliability of statistical hypothesis tests. These proposals range from stricter thresholds for stating statistical significance [@benjaminRedefineStatisticalSignificance] to more profound methodological changes [@Kruschke2018b; @Wagenmakers2016; @Morey2016]. In the last category, an often stated solution is a shift towards Bayesian data analysis [@Wagenmakers2016; @Kruschke2018b; @Kruschke2012; @Ly2016a; @Ly2016]. The advantages of such a shift include the adherence of Bayesian methods to the likelihood principle [@Birnbaum1962], which has important implications. Some of them are the simplified interpretation and appealing properties of Bayesian interval estimates for quantifying the uncertainty in parameter estimates [@Morey2016c]. Others are given by the independence of results of the researcher’s intentions [@Kruschke2018; @Berger1988a; @Edwards1963] as well as the ability to make use of optional stopping [@Rouder2014]. The last property is, in particular, appealing in practical research, as it allows to stop recruiting participants and report the results based on the collected data in case they already show overwhelming evidence. Notice that this is not permitted when making use of NHST and $p$-values, which can lead to financial and ethical problems, in particular in the biomedical and psychological sciences. Considering Bayesian alternatives to NHST and $p$-values, the most prominent approach to Bayesian hypothesis testing is the Bayes factor which was invented by [@Jeffreys1931], see also [@Etz2015]. The Bayes factor is often advocated as a Bayesian alternative to the frequentist $p$-value when it comes to hypothesis testing, in particular in the cognitive sciences and psychology [@VanDeSchoot2017; @Wagenmakers2016; @Wagenmakers2010; @Ly2016; @VanDoorn2019; @VanDongen2019]. However, there are also other approaches like Bayesian equivalence testing based on the region of practical equivalence (ROPE) [@Kruschke2013; @Kruschke2015; @Kruschke2018; @Kruschke2018a; @Westlake1976; @Kirkwood1981; @Kelter2020BayesianPosteriorIndices; @Liao2020] which are based on an analogy to frequentist equivalence tests [@Lakens2017; @Lakens2018]. Also, there exist various other measures and alternatives to test hypotheses in the Bayesian approach, including the MAP-based $p$-value [@Mills2017], the probability of direction (PD) [@Makowski2019; @Makowski2019a] and the Full Bayesian Significance Test (FBST) [@Pereira1999; @Stern2003; @Madruga2001; @Madruga2003; @Pereira2008; @Pereira2020; @Esteves2019]. In contemporary literature, there is still a debate about which Bayesian measure to use in which setting for scientific hypothesis testing, and while some authors argue in favour of the Bayes factor [@Wagenmakers2016; @Etz2016; @Kelter2020BMCJasp], there is also criticism about the focus on the Bayes factor in the cognitive sciences [@Tendeiro2019; @Greenland2019]. By now, comparisons of different Bayesian posterior indices are rare, but the existing results show that it is useful to consider various different Bayesian approaches to hypothesis testing depending on the research goal and study design, see [@Kelter2020BayesianPosteriorIndices; @Makowski2019; @Liao2020]. In this paper, attention is directed to one specific Bayesian alternative to NHST and $p$-values, the Full Bayesian Significance Test (FBST) and the $e$-value, and the R package `fbst` is introduced. The FBST has been developed over two decades ago in the statistical literature [@Pereira1999], and since has been employed successfully in a broad range of scientific areas and applications. It is not possible to cover all theoretical and practical work which has been pursued concerning the FBST in the last two decades, and for a concise review, we refer the reader to [@Pereira2020]. The R package `fbst` introduced in this paper offers an intuitive and widely applicable software implementation of the FBST and the $e$-value. The package has been designed to work in combination with widely used R packages for fitting Bayesian models in the cognitive sciences and psychology and offers appealing visualisations to communicate and share the results of an analysis with colleagues. The structure of this paper is as follows: First, we describe the underlying theory of the FBST and the $e$-value. Second, we give information about the available functionality and software implementation of the package. Subsequently, we demonstrate with two examples of widely used statistical models in psychological research how the FBST can be applied in practice via the `fbst` package. Finally, we conclude by drawing attention to the benefits and limitations of the package and give some ideas about future extensions. In summary, the FBST and $e$-value could be an appealing Bayesian alternative to NHST and $p$-values which has been widely under-utilised by now in the cognitive sciences and psychology. This clearly can be attributed to the dearth of accessible software implementations, one of which is presented in form of the R package introduced in this paper. The `fbst` package hopefully will foster critical discussion and reflection about different approaches to Bayesian hypothesis testing and allow to pursue further research to investigate the relationship between different posterior indices for significance and effect size [@Kelter2020BayesianPosteriorIndices; @Makowski2019; @Liao2020]. The FBST and the $e$-value {#the-fbst-and-the-e-value .unnumbered} ========================== This section describes the statistical theory behind the FBST and the $e$-value in more detail. The philosophical basis (or conceptual approach) is first described briefly, and subsequently, the necessary notation is introduced. Conceptual approach of the FBST {#conceptual-approach-of-the-fbst .unnumbered} ------------------------------- The Full Bayesian Significance Test was first introduced by [@Pereira1999] more than two decades ago as a Bayesian alternative to traditional frequentist null hypothesis significance tests. It was invented to test a *sharp* (or precise) point null hypothesis $H_0$ against its alternative $H_1$. Traditional frequentist approaches measure the inconsistency of the observed data with a null hypothesis $H_0$ [@Kempthorne1976; @Cox1977]. Frequentist hypothesis tests employ $p$-values to order the *sample space* according to increasing inconsistency with the hypothesis. Notice, that a $p$-value is defined as the probability of obtaining a result (which, of course, is located in the sample space) equal to or more extreme than the one observed under the assumption of the null hypothesis $H_0$ [@Held2014]. In contrast, the $e$-value produced in the FBST aims at ordering the *parameter space* according to increasing inconsistency with the observed data [@Pereira2008]. In formulas, traditional frequentist significance tests use the $p$-value to reject the null hypothesis $H_0$: $$\begin{aligned} p=Pr(x\in C|\theta_0)\end{aligned}$$ Here, $C$ often is the set of sample space values $x\in \mathcal{X}$ (where $\mathcal{X}$ is the sample space) for which a test statistic $T_{\theta_0}$ (derived under the assumption of the null hypothesis value $\theta_0$) is at least as large as the test statistic value $t$ calculated from the observed data. The set $C$ can be interpreted as the sample space values $x\in \mathcal{X}$ which are at least as *inconsistent* with the null hypothesis $H_0$ as the observed data. The $p$-value now quantifies the evidence against $H_0$ by calculating the probability of sample space values $x$ being located precisely in this set [@Casella2002a]. The idea put forward in [@Pereira1999] and [@Pereira2008] is simple: Instead of considering the sample space, a Bayesian should inspect the *tangential set* $T$ of parameter values (which are, of course, located in the parameter space). This set consists of all parameter values which are *more consistent* with the observed data $x$ than $\theta_0$, which is the Bayesian evidence $ev$. Here, $\overline{ev}$ is defined as $$\begin{aligned} \overline{ev} = Pr(\theta \in T|x)\end{aligned}$$ and $ev=1-\overline{ev}$. $ev$ can be interpreted as the evidence *in favour* of the null hypothesis $H_0$, while $\overline{ev}$ is interpreted as the evidence *against* $H_0$. This latter value is the probability of all parameter values $\theta$ which are *more consistent* with the data $x$ than the null value $\theta_0$. The conceptual approach of the FBST consists, as a consequence, of constructing a duality Bayesian theory and frequentist sampling theory. This duality is constructed between frequentist significance measures which are based on ordering the *sample space* according to increasing inconsistency with the data, and the Bayesian e-value, which is based on ordering the *parameter space* according to increasing inconsistency with the observed data. This conceptual basis ensures that the FBST allows a seamless transition to Bayesian data analysis for researchers who are acquainted with NHST and $p$-values. The FBST produces the $e$-value which can be interpreted similarly to the frequentist $p$-value and little methodological changes are required. However, the consequences of the conceptual basis of the FBST are substantial: As the quantity $\overline{ev}$ is a fully Bayesian quantity, it allows statements in terms of probability to quantify the evidence. Traditional frequentist measures like $p$-values do not make probabilistic statements about the parameter (because they are computed over the sample space instead of the parameter space), which is questionable as the goal of the study or experiment is to quantify the uncertainty about a given research hypothesis, which naturally should be done via probability measures [@Howie2002; @Berger1988a]. As a consequence, the FBST and the $e$-value follow the likelihood principle [@Birnbaum1962; @Basu1975; @Berger1988a], which brings several advantages with it: - [Researchers can use optional stopping. This means that they are allowed to stop recruiting participants or even abort an experiment and readily report the results when only a fraction of the data already shows overwhelming evidence for or against the hypothesis under consideration [@Edwards1963; @Rouder2014].]{} - [Censored data (which are often observed in longitudinal studies or clinical trials in the cognitive sciences and psychology) can be interpreted easily [@Berger1988a]. The likelihood contribution of a single observation in a study where no censoring was possible is equal to the likelihood contribution of a single observation in a study where censoring is possible but did not occur (for the single observation considered). This simplifies the analysis and interpretation of statistical models which include censoring mechanisms, see [@Berger1988a Chapter 4].]{} - [As highlighted by [@Edwards1963], [@Wagenmakers2016], and [@Kruschke2018a], the result of a hypothesis test (in this case, the FBST), is not influenced by the researchers’ intentions. This last property is substantial for improving the reliability of research in the cognitive sciences and psychology, see [@McElreath2015].]{} Statistical theory of the FBST {#statistical-theory-of-the-fbst .unnumbered} ------------------------------ In this section we introduce the necessary mathematical notation for a rigid understanding of the FBST. The FBST can be used with any standard parametric statistical model, where $\theta \in \Theta \subseteq \mathbb{R}^p$ is a (vector-valued) parameter of interest, $p(x|\theta)$ is the model likelihood and $p(\theta)$ is the prior distribution for the parameter $\theta$ of interest. A sharp (or expressed equivalently, precise) hypothesis $H_0$ makes a statement about the parameter $\theta$: Specifically, the null hypothesis $H_0$ states that $\theta$ lies in the so-called *null set* $\Theta_{H_0}$. For simple point null hypotheses like $H_0:\theta=\theta_0$, which are often used in practice, this null set consists of the single parameter value $\theta_0$ so that the null set can be written as $\Theta_{H_0} = \{\theta_0 \}$. As detailed in the previous section, the conceptual approach of the FBST is to state the Bayesian evidence against $H_0$, the $e$-value. This value is the proposed Bayesian replacement of the traditional $p$-value. To construct the $e$-value, [@Pereira2008] introduced the posterior *surprise function* $s(\theta)$ which is defined as follows: $$\begin{aligned} s(\theta):=\frac{p(\theta|x)}{r(\theta)} \end{aligned}$$ The surprise function $s(\theta)$ is the ratio of the posterior distribution $p(\theta|x)$ and a suitable *reference function* $r(\theta)$. The first thing to note is that two important special cases are given by a flat reference function $r(\theta)=1$ or any prior distribution $p(\theta)$ for the parameter $\theta$. First, when a flat reference function is selected the surprise function recovers the posterior distribution $p(\theta|x)$. Second, when any prior distribution is used as the reference function, one can interpret parameter values $\theta$ with a surprise function value $s(\theta)\geq 1$ as being corroborated by the observed data $x$. In contrast, parameter values $\theta$ with a surprise function $s(\theta)<1$ indicate that they have not been corroborated by the data. The next step is to calculate the supremum $s^{*}$ of the surprise function $s(\theta)$ over the null set $\Theta_{H_0}$. $$\begin{aligned} s^{*}:=s(\theta^{*})=\sup\limits_{\theta \in \Theta_{H_0}}s(\theta)\end{aligned}$$ This supremum is subsequently used in combination with the tangential set, which has been introduced in the last section. [@Pereira2008] defined $$\begin{aligned} \label{eq:tangentialSet} T(\nu):=\{\theta \in \Theta|s(\theta)\leq \nu \}\end{aligned}$$ and the tangential set $\overline{T}(\nu)$ to the sharp null hypothesis $H_0$ is then given as follows: $$\begin{aligned} \overline{T}(\nu):=\Theta \setminus T(\nu)\end{aligned}$$ When setting $\nu=s^{*}$, the tangential set $\overline{T}(\nu)$ has its unique interpretation which has been discussed in the previous section: While $T(s^{*})$ includes all parameter values $\theta$ which are smaller or equal to the supremum value $s^{*}$ of the surprise function $s(\theta)$, the tangential set $\overline{T}(s^{*})$ includes all parameter values $\theta$ which attain a *larger* surprise function value than the supremum $s^{*}$ of the null set. The final step to obtain the $e$-value, the Bayesian evidence against $H_0$, is to make use of the *cumulative surprise function* $W(\nu)$ $$\begin{aligned} W(\nu):=\int_{T(\nu)}p(\theta|x)d\theta\end{aligned}$$ The cumulative surprise function $W(\nu)$ is simply an integral of the posterior distribution $p(\theta|x)$ over all parameter values with surprise function values $s(\theta)\leq \nu$. Setting $\nu=s^{*}$, the cumulative surprise function $W(s^{*})$ becomes the integral of the posterior $p(\theta|x)$ over $T(s^{*})$. This is the integral of the posterior $p(\theta|x)$ over all parameter values which have a surprise function value $s(\theta)\leq s^{*}$. The $e$-value is then given as $$\begin{aligned} \label{eq:evalue} \overline{\text{ev}}(H_0):=\overline{W}(s^*)\end{aligned}$$ ![image](fbst.pdf){width="100.00000%"} Here $\overline{W}(\nu):=1-W(\nu)$. Figure \[fig:fbst\]a visualises the FBST and the $e$-value $\overline{\text{ev}}(H_0)$. The solid line shows the posterior distribution $p(\delta|x)$ of the effect size $\delta$ after observing the data $x$, and is produced by a Bayesian two-sample t-test [@Kelter2020JORSBayest]. A flat reference function $r(\delta)=1$ was selected in figure \[fig:fbst\]a. The supremum over the null set $\Theta_{H_0}=\{0\}$ is $s^{*}=s(0)$, shown as the blue point. The horizontal blue dashed line visualises the boundary between $T(0)$ and $\overline{T}(0)$, and values with posterior density $p(\delta)>p(0)$ are located in $\overline{T}(0)$, while values with $p(\delta)\leq p(0)$ are located in $T(0)$. The blue shaded area is the cumulative surprise function $\overline{W}(0)$, which is the integral over the tangential set $\overline{T}(0)$ against $H_0:\delta =0$. This is the $e$-value $\overline{\text{ev}}(H_0)$ against $H_0$, the Bayesian evidence against the sharp null hypothesis. The red shaded area is the integral $W(0)$ over $T(0)$, which equals the $e$-value ev$(H_0)$ in favour of $H_0:\delta=0$. Figure \[fig:fbst\]b shows the same situation, but now the reference function is selected as a wide Cauchy prior $C(0,1)$, so that the surprise function becomes $$\begin{aligned} s(\delta)=p(\delta|x)/c(\delta)\end{aligned}$$ where $c(\delta)$ is the p.d.f. of the $C(0,1)$ Cauchy distribution. Although the situation seems similar to figure \[fig:fbst\]a, the scaling on the $y$-axis now is different. Also, the evidence has changed based on the new surprise function and the interpretation of the surprise function has changed, too. While in figure \[fig:fbst\]a, the surprise function could be interpreted as the posterior distribution, now it is interpreted as follows: If one assumes a Cauchy prior $C(0,1)$ on the effect size $\delta$, then parameters with a surprise function value $s(\delta)\geq 1$ can be interpreted as being corroborated by the data. Parameter values with a surprise function $s(\delta)<1$ are interpreted as not being corroborated by the data. [@Pereira1999] formally defined the $e$-value ev$(H_0)$ in *support* of $H_0$ as $$\begin{aligned} \text{ev}(H_0):=1-\overline{\text{ev}}(H_0)\end{aligned}$$ but notice that one can not interpret this value as evidence *against* $H_1$. This can be attributed to the fact that $H_1$ is not even a sharp hypothesis, see Definition 2.2 in [@Pereira2008]. It is crucial to note that it is not possible to utilise the $e$-value ev$(H_0)$ to *confirm* the null hypothesis $H_0$ [@Kelter2020BayesianPosteriorIndices]. However, the FBST can be generalized in to an extended framework which then allows for hypothesis confirmation and itself is an active topic of ongoing research [@Esteves2019]. Additionally, the $e$-value ev$(H_0)$ can be used to reject $H_0$ if ev$(H_0)$ is sufficiently small based on asymptotic arguments [@Pereira2008 Section 5]. [@Pereira2008] showed that the distribution of the $e$-value is a Chi-square distribution $$\begin{aligned} \label{eq:evPVal} \text{ev}(H_0) \sim \chi_k^2(||m-M||^2) \end{aligned}$$ where $M$ is the posterior mode calculated over the entire parameter space $\Theta$ and $m$ is the posterior maximum over $\Theta_{H_0}$. The $p$-value associated with the Bayesian evidence in support of $H_0$ is then calculated as the superior tail of the $\chi^2$ density with $k-h$ degrees of freedom, starting from $-2\lambda(m_0)$. Here, $k$ is the dimension of the parameter space $\Theta$ and $h$ is the dimension of the null set $\Theta_{H_0}$. The quantity $m_0$ is the observed value and $\lambda(t)=\ln l(t)$ is the logarithm of the relative likelihood function, where $l(t)=L(t)/L(M)$ is the relative likelihood. Denoting $F_{k-h}$ as the Chi-square distribution’s cumulative distribution function with $k-h$ degrees of freedom ($F_k$ analogue), the $p$-value associated with the Bayesian $e$-value $\text{ev}(H_0)$ is then computed as $$\begin{aligned} pv_{0}=1-F_{k-h}(-2\lambda(m_0)) \end{aligned}$$ This latter $p$-value has a frequentist interpretation. The $p$-value based on equation (\[eq:evPVal\]) can be expressed as $$\begin{aligned} \label{eq:evPVal2} ev_0 = F_k(||m_0-M_0||^2) \end{aligned}$$ and can be interpreted as a Bayesian significance value which quantifies the probability of obtaining $\text{ev}(H_0)$ or even *less* evidence in support of the null hypothesis $H_0$. Consequently, after observing $m_0$ and $M_0$ one only needs to calculate the euclidian distance $d_0=||m_0-M_0||^2$ and the value of the $\chi_{k}^2$ distribution’s cumulative distribution function of this distance is the corresponding $p$-value. Based on a threshold (like $0.05$) one can decide to reject the null hypothesis $H_0:\theta=\theta_0$ or not. However, if a $p$-value is required which is closest to the frequentist $p$-value in interpretation, one should use the standardized $e$-value $\text{sev}(H_0)$, as defined in [@Borges2007 Section 2.2] and in [@Pereira2020 Section 3.3]. The standardized $e$-value is defined as: $$\begin{aligned} \overline{\text{sev}}(H_0)=F_{k-h}(F^{-1}_{k}(\overline{\text{ev}}))\end{aligned}$$ Here, $F^{-1}_{k}$ is the quantile function of the cumulative distribution function of the $\chi_{k}^2$ distribution with $k$ degrees of freedom. $\overline{\text{sev}}(H_0)$ can, as a consequence, be interpreted as the probability of obtaining less evidence than $\overline{\text{ev}}(H_0)$ against the null hypothesis $H_0$. Defining $$\begin{aligned} \text{sev}(H_0)=1-\overline{\text{sev}}(H_0)\end{aligned}$$ $\text{sev}(H_0)$ can then be interpreted as the probability of obtaining $\overline{\text{ev}}(H_0)$ or more evidence against $H_0$, which is closely related to the interpretation of a frequentist $p$-value. However, the $p$-value operates in the sample space while the standardized $e$-value operates in the parameter space. The standardized $e$-value can be used as a Bayesian replacement of the frequentist $p$-value, while being very similar in interpretation. For theoretical properties of $\text{sev}(H_0)$ see [@Borges2007] and [@Pereira2020]. In the examples below, the Bayesian evidence against $H_0$, the $e$-value $\overline{\text{ev}}(H_0)$ is reported and also the standardized $e$-values $\text{sev}(H_0)$ are given. Overview and functionality of the fbst package {#overview-and-functionality-of-the-fbst-package .unnumbered} ============================================== The centerpiece of the `fbst` package is the `fbst()` function, which is used to perform the FBST. In addition to the `fbst()` function, the package provides customised `summary()` and `plot()` functions which allow users to print the results of a FBST or obtain a visualisation of their results to communicate and share the results. The `fbst()` function has the following structure: fbst(posteriorDensityDraws, nullHypothesisValue, FUN, par, dimensionTheta, dimensionNullset) Here, `posteriorDensityDraws` needs to be a numeric vector holding the posterior parameter draws obtained via MCMC or any other numerical method of choice.[^1] The argument `nullHypothesisValue` is the value specified in the null hypothesis $H_0:\theta=\theta_0$, and `dimensionTheta` is the dimension of the parameter space $\Theta$. `dimensionNullset` is the dimension of the null set $\Theta_{H_0}$, and `FUN` and `par` are additional arguments which only need to be specified when a user-defined reference function $r(\theta)$ is desired. In general, `FUN` should be the name of the reference function which should be used and `par` should be a list of parameters which this reference function utilises (e.g. the location and scale parameters when the reference function is a Cauchy prior). Details will be given in the examples below. The `fbst()` function returns an object of the class `fbst`, which stores several useful details and the results of the conducted FBST. To obtain a concise summary of the FBST, the `summary()` function of the class `fbst` can be used. To visualise the FBST, the `plot()` function of the `fbst` class can be used. Details are provided in the examples below. From an algorithmic perspective, the `fbst` package proceeds via the following steps when computing the e-value via the `fbst()` function: 1. [Based on the posterior parameter samples\ `posteriorDensityDraws`, the posterior density $p(\theta|x)$ is estimated via a Gaussian kernel density estimator, resulting in a posterior density estimate $\hat{p}(\theta|x)$. The Gaussian kernel is used due to well-known Bayesian asymptotics of posterior distributions, the Bernstein-von-Mises theorem [@Held2014].]{} 2. [Based on this posterior density estimate $\hat{p}(\theta|x)$, the surprise function $s(\theta)$ is estimated (i) as the posterior density estimate $\hat{p}(\theta|x)$ if no arguments `FUN` and `par` are supplied so that a flat reference function $r(\theta)=1$ is used as default, or (ii) as the ratio $\hat{p}(\theta|x)/r(\theta)$ if arguments `FUN` and `par` are supplied. The result is a surprise function estimate $\hat{s}(\theta)$.]{} 3. [The surprise function estimate $\hat{s}(\theta)$ is evaluated at the null hypothesis value supplied via the argument `nullHypothesisValue`, resulting in the value $\hat{s}_0$.]{} 4. [The $e$-value $\overline{\text{ev}}(H_0)$ is computed via numerical integration of the posterior density estimate $\hat{p}(\theta|x)$ over the tangential set $\overline{T}(H_0)$, which is determined via a linear search algorithm on the vector `posteriorDensityDraws` by including all values $\theta$ which fulfill the condition $\hat{s}(\theta)>\hat{s}_0$.]{} 5. [The $p$-value associated with the $e$-value $\text{ev}(H_0)$ in favour of the null hypothesis $H_0$ and the standardized $e$-values sev$(H_0)$ are computed.]{} In summary, the FBST is based only on simple numerical optimization and integration which makes it a computationally cheap option. This is a benefit, in particular, when the parameter space $\Theta$ is high-dimensional [@Pereira2020; @Stern2003; @Kelter2020BayesianPosteriorIndices]. Also, the presence of nuisance parameters does not trouble the computation unlike it is the case for example for the Bayes factor, as computing the marginal likelihoods can quickly become difficult then [@Stern2003]. Example 1: Two-sample Bayesian t-test {#example-1-two-sample-bayesian-t-test .unnumbered} ===================================== As a preliminary note, all analyses can be reproduced by following the provided code.[^2]. To demonstrate how to use the `fbst` package, we start with the two-sample t-test, a widely used statistical model in the cognitive sciences [@Nuijten2016]. We use the two-sample Bayesian t-test of [@Rouder2009] together with simulated data. The recommended medium Cauchy prior $C(0,\sqrt{2}/2)$ was assigned to the effect size $\delta$. Observations in the first group were simulated as $\mathcal{N}(0,1.7)$, and observations belonging to the second group were generated according to the $\mathcal{N}(0.8,3)$ distribution. As a consequence, the resulting true effect size $\delta$ according to [@cohen_statistical_1988] is given as $$\begin{aligned} \delta = \frac{0-0.8}{\sqrt{(1.7^2+3^2)/2}} \approx -0.33\end{aligned}$$ which equals a small effect size. The code to simulate the data is given in listing \[r:listing1\]. set.seed(57) grp1=rnorm(18,0,1.7) grp2=rnorm(18,0.8,3) The corresponding Bayes factor $BF_{10}$ for the alternative hypothesis $H_1:\delta \neq 0$ against the null hypothesis $H_0:\delta = 0$ is given as $BF_{10}=0.91$, which does not indicate evidence worth mentioning according to [@jeffreys1961] or [@VanDoorn2019]. The slight favour towards $H_0$ can be attributed to the medium Cauchy prior used, which centres the prior probability mass closely around small effect sizes (and no effect, too). Figure \[fig:example1priorposterior\] shows a prior-posterior plot for the example. The code to compute the Bayes factor is given in listing \[r:listing2\]. install.packages("BayesFactor") library(BayesFactor) p = BayesFactor::ttestBF(x=grp1,y=grp2, posterior = TRUE, iterations = 3000000, rscale = "medium") p = as.vector(p[,4]) BF_10 = BayesFactor::ttestBF(x=grp1,y=grp2, posterior = FALSE, rscale = "medium", paired = FALSE) BF_10 ![Prior-posterior plot for Example 1[]{data-label="fig:example1priorposterior"}](example1priorposterior.pdf){width="50.00000%"} To perform the FBST and compute the $e$-value, we first install and load the R package from CRAN by executing the code in listing \[r:listing3\]. install.packages("fbst") library(fbst) resFlatSim = fbst(posteriorDensityDraws = p, nullHypothesisValue = 0, dimensionTheta = 3, dimensionNullset = 2) Note that in the example, the parameter space $\Theta$ consists of three parameters: The mean $\mu_1$ in the first group, the mean $\mu_2$ in the second group, and the standard deviation $\sigma^2$. As a consequence, the argument `dimensionTheta` is therefore set to `dimensionTheta=3`. The null set $\Theta_{H_0}$ consists of the set $\{\mu_1=\mu_2,\sigma^2\}$, which is two-dimensional so that `dimensionNullset = 2`. The object stored in the variable `resFlatSim` is an object of the class `fbst`, which stores several values used in the `summary()` and `plot()` functions of the package. These are available to communicate and visualise the results of the FBST. For example, we can access the $e$-value $\overline{\text{ev}}(H_0)$ as follows (see listing \[r:listing3b\]): resFlatSim@eValue [1] 0.8305998 Instead of accessing each attribute manually, to obtain a summary of the FBST and print the relevant quantities the `summary()` function of the `fbst` package provides a more convenient option: summary(resFlatSim) Full Bayesian Significance Test for testing a sharp hypothesis against its alternative: Reference function: Flat Testing Hypothesis H_0:Parameter= 0 against its alternative H_1 Bayesian e-value against H_0: 0.8305998 p-value associated with the Bayesian e-value in favour of the null hypothesis: 0.1461029 Standardized e-value: 0.0248695 Based on the results, we can see that there is some evidence against the null hypothesis according to the Bayesian $e$-value $\overline{\text{ev}}(H_0)$ against $H_0$ (compare equation (\[eq:evalue\])). The corresponding $p$-value $ev_0 \approx 0.146$ is not significant if a standard threshold of $0.05$ is used, but the standardized $e$-value $\text{sev}(H_0) \approx 0.025 <0.05$ is. Note that when a $p$-value is used for hypothesis testing, it is recommended to use the standardized $e$-value [@Borges2007; @Pereira2020], so one would reject the null hypothesis $H_0:\delta = 0$ in this case. However, it is also possible to use only the Bayesian evidence $\overline{\text{ev}}(H_0)$ against $H_0$ without any $p$-value to quantify the evidence continuously. To visualise the results, we use the `plot()` function of the `fbst` package: plot(resFlatSim) The result is shown in figure \[fig:example1fbst\]a: The blue shaded area under the surprise function (which is by default the posterior distribution, that is, a flat reference function $r(\delta)=1$ is used by default by the `fbst()` function) is the Bayesian evidence against $H_0$, the $e$-value $\overline{\text{ev}}(H_0)\approx 0.83$ (compare listing \[r:listing4\]). The red shaded area is the $e$-value $\text{ev}(H_0)$ in favour of $H_0$, which is $ev(H_0) \approx 1-0.83=0.17$. ![image](example1fbst.pdf){width="100.00000%"} Instead of a flat reference function $r(\delta)=1$, one could also use a more reasonable prior distribution. For example, as small to medium effect sizes are to be expected in the cognitive sciences and psychology, [@Rouder2009] recommended to use a medium Cauchy prior $C(0,\sqrt{2}/2)$ as a default prior on the effect size. To see if parameter values $\delta$ have been corroborated (compared to this prior assumption) by observing the data, we can use this prior as the reference function $r(\delta)=C(0,\sqrt{2}/2)$, and the resulting surprise function is shown in figure \[fig:example1fbst\]b. The code to produce the FBST based on a Cauchy reference density is given in listing \[r:listing6\]: resMediumSim = fbst(posteriorDensityDraws = p, nullHypothesisValue = 0, dimensionTheta = 3, dimensionNullset = 2, FUN=dcauchy, par = list(location = 0, scale = sqrt(2)/2)) summary(resMediumSim) Full Bayesian Significance Test for testing a sharp hypothesis against its alternative: Reference function: User-defined Testing Hypothesis H_0:Parameter= 0 against its alternative H_1 Bayesian e-value against H_0: 0.9032063 p-value associated with the Bayesian e-value in favour of the null hypothesis: 0.1461029 Standardized e-value: 0.01189972 There, the `FUN` argument is supplied with the name of the density to be used and the `par` argument is supplied with a list of arguments for this density. As the Cauchy distribution has a `location` and `scale` parameter, we supply these here. Notice that the blue point which indicates the surprise function value $s(0)$ of the null hypothesis parameter $\delta=0$ is larger than one. This means that the null hypothesis value has been corroborated by the data. However, all parameter values in the tangential set have been corroborated *even more* by the data than the null value $\delta=0$. Based on the continuous quantification, there is again strong evidence against the null hypothesis when changing the reference function to a medium Cauchy prior: More than 90% of the posterior distribution’s parameter values attain a larger surprise function value than the null hypothesis value. The resulting standardized $e$-value $\text{sev}(H_0)$ is also significant. Example 2: Directional two-sample Bayesian t-test {#example-2-directional-two-sample-bayesian-t-test .unnumbered} ================================================= Example 1 showed how to apply the FBST in the setting of the Bayesian two-sample t-test. Example 2 is a slight modification of Example 1. Instead of testing a two-sided hypothesis, we now turn to directional hypotheses and show how these can easily be tested via the `fbst` package, too. We use data of [@Moore2012], which provides the reading performance of two groups of pupils: One control group and a treatment group which was given directed reading activities. The data are freely available in the built-in data library of the open-source software JASP[^3]. We test the hypothesis $H_0:\delta < 0$, which is equivalent to the hypothesis $H_0:\mu_1 < \mu_2$, where the measured quantity is the performance of pupils in the degree of reading power test (DRP) [@Moore2012]. First, we save the data in a .csv-file (which is called `DirectedReadingActivities.csv` in listing \[r:listing7\]), set the working directory and load the data[^4]: setwd(' ... ') # Change to where the data are stored on your machine library(dplyr) dra=read.csv("DirectedReadingActivities.csv",sep=",") head(dra) id group g drp 1 1 Treat 0 24 2 2 Treat 0 56 3 3 Treat 0 43 4 4 Treat 0 59 5 5 Treat 0 58 6 6 Treat 0 52 treat = (dra %>% filter(group=="Treat") %>% select(drp))$drp control = (dra %>% filter(group=="Control") %>% select(drp))$drp The code to perform a standard hypothesis test based on the Bayes factor is given in listing \[r:listing8\], which results in $BF_{10}=4.32$, indicating moderate evidence for the alternative $H_1:\delta <0$ according to [@VanDoorn2019]. library(BayesFactor) # BF closed-form of Rouder et al. (2009) p = BayesFactor::ttestBF(x=control,y=treat, posterior = TRUE, rscale = "medium", paired = FALSE, nullInterval = c(-Inf,0), iterations = 3000000) p = as.vector(p[,4]) BF_10 = BayesFactor::ttestBF(x=control,y=treat, posterior = FALSE, rscale = "medium", paired = FALSE, nullInterval = c(-Inf,0)) BF_10[1] Bayes factor analysis -------------- [1] Alt., r=0.707 -Inf<d<0 : 4.327919 ±0% Against denominator: Null, mu1-mu2 = 0 --- Bayes factor type: BFindepSample, JZS The code to perform the FBST with a flat reference function $r(\delta)=1$ is given in listing \[r:listing9\]: library(fbst) resFlatDRA = fbst(posteriorDensityDraws = p, nullHypothesisValue = 0, dimensionTheta = 3, dimensionNullset = 2) summary(resFlatDRA) Full Bayesian Significance Test for testing a sharp hypothesis against its alternative: Reference function: Flat Testing Hypothesis H_0:Parameter= 0 against its alternative H_1 Bayesian e-value against H_0: 0.9859827 p-value associated with the Bayesian e-value in favour of the null hypothesis: 0.06689926 Standardized e-value: 0.001123303 The dimensions of $\Theta$ and $\Theta_{H_0}$ are identical to Example 1, and the Bayesian $e$-value $\overline{\text{ev}}(H_0) \approx 0.986$ expresses strong evidence against the null hypothesis $H_0:\delta=0$. Also, the standardized $e$-value $\text{sev}(H_0) \approx 0.001 < 0.05$ is significant and leads to the same conclusion if a threshold of $0.05$ is applied. The results are visualised in figure \[fig:example2fbst\]. Figure \[fig:example2fbst\]a shows the FBST when a wide half-Cauchy prior $C_{+}(0,1)$ is used as the reference function $r(\delta)$ [@Rouder2009][^5]. Figure \[fig:example2fbst\]a is produced by the code in listing \[r:listing10\], where the additional parameter `rightBoundary = 0` needs to be added to inform the `plot()` function that a one-sided hypothesis was used. Should the alternative be $H_1:\delta >0$, one would supply the argument `leftBoundary = 0` to the `plot()` function. plot(resFlatDRA, rightBoundary = 0) ![image](example2fbst.pdf){width="100.00000%"} Based on the continuous quantification of evidence against $H_0$ in form of $\overline{\text{ev}}(H_0)$ and the standardized $e$-value $\text{sev}(H_0)$ one would reject the null hypothesis $H_0:\delta=0$ in favour of the alternative $H_1:\delta <0$. That is, the performance in the treatment group is better than in the control group which was not given directed reading activities. Example 3: Bayesian logistic regression {#example-3-bayesian-logistic-regression .unnumbered} ======================================= As a third example, we demonstrate how to use the FBST via the `fbst` package in the context of the Bayesian logistic regression model [@McElreath2020]. Notice that while we focus on the standard logistic model here, the procedure is applicable to any regression model of interest like probit or linear regression models. We use data from the Western Collaborative Group Study (WCGS) of [@Rosenman1975], in which $3154$ healthy young men aged $39-59$ from the San Francisco area were assessed for their personality type. All were free from coronary heart disease at the start of the research. Eight and a half years later change in this situation was recorded. We use a subset of $n=3140$ participants, where 14 participants have been excluded because of incomplete data. The data set is freely available in the `faraway` R package, so we first load and prepare the data as shown in listing \[r:listing11\]. library(faraway) data(wcgs) wcgs = wcgs[complete.cases(wcgs), ] For illustration purposes, we use a Bayesian logistic regression model which studies the influence of the covariates `age`, `height`, `weight`, systolic blood pressure (`sdp`), diastolic blood pressure (`dbp`), fasting serum cholesterol (`chol`) and the number of cigarettes smoked per day (`cigs`) on the outcome chronic heart disease (yes / no, variable `chd`) stored in the response variable `chd`. The model is fit via the Hamiltonian Monte Carlo sampler Stan [@Carpenter2017; @Kelter2020] which uses the No-U-Turn sampler of [@Hoffman2014] to sample from the posterior distribution. We obtain the posterior distribution of the intercept and the seven regression coefficients $\beta_1,...,\beta_7$, belonging to the six covariates included in the model. The default weakly informative $\sigma \sim \exp(1)$ prior is assigned to the standard deviation $\sigma$, see [@Gabry2020RstanarmPriorsVignette]. We use the `rstanarm` package [@Goodrich2020] for fitting the Bayesian logistic regression model, and the code to prepare the data for Stan is given in listing \[r:listing12\]. f1 = as.formula(paste('chd ~ age + height + weight + sdp + dbp + chol + cigs')) X1 <- model.matrix(f1, wcgs) # build model matrix standata_m1 <- list(y = as.numeric(wcgs$chd)-1, X = X1, N = nrow(X1), P = ncol(X1)) # format data as list for Stan stan_df1 <- as.data.frame(standata_m1) The standard weakly informative prior distribution $\beta_j \sim \mathcal{N}(0,2.5)$ is assigned to the regression coefficients $\beta_j, j=1,...,7$, and the intercept $\beta_0$ is assigned the weakly informative default prior $\beta_0 \sim \mathcal{N}(0,10)$ recommended by [@Gabry2020RstanarmPriorsVignette]. Listing \[r:listing13\] shows the code to fit the model via the `rstanarm` package, summarise and plot the results. library(rstanarm) post_m1 <- stan_glm(f1, data = wcgs, family = binomial(link = "logit"), prior = normal(0,2.5), prior_intercept = normal(0,10), QR=TRUE, iter = 4000, seed = 4711) summary(post_m1) Model Info: function: stan_glm family: binomial [logit] formula: chd ~ age + height + weight + sdp + dbp + chol + cigs algorithm: sampling sample: 8000 (posterior sample size) priors: see help('prior_summary') observations: 3140 predictors: 8 plot(post_m1, "areas", prob = 0.95, prob_outer = 1, pars=c("age", "height", "weight", "sdp", "dbp", "chol", "cigs")) Figure \[fig:example3\] shows the marginal posterior distributions of the regression coefficients $\beta_j$ for the Bayesian logistic regression model in Example 3. ![image](example3.pdf){width="100.00000%"} To compute the FBST on the regression coefficients, we need to extract the posterior MCMC sample first, as shown in listing \[r:listing14\]. For illustration purposes, we conduct the FBST on the regression coefficient belonging to the covariate weight. The FBST is computed using a normal prior $\mathcal{N}(0,2.5)$ as reference function, which was also used to fit the model. This way, the surprise function quantifies which parameter values $\beta_j$ have been corroborated more by observing the data than the null value $\beta_j=0$. posteriorDrawsMatrix = as.matrix(post_m1) weightDraws = posteriorDrawsMatrix[,4] resWeight = fbst(posteriorDensityDraws = weightDraws, nullHypothesisValue = 0, dimensionTheta = 8, dimensionNullset = 7, FUN=dnorm, par = list(mean = 0, sd = 2.5)) # Bayesian evidence against null hypothesis resWeight@eValue [1] 0.9758885 # Standardized e-value resWeight@sev_H_0 [1] 0.00002672151 plot(resWeight) The results are also shown in figure \[fig:example3fbst\], which is produced via the `plot()` function call in listing \[r:listing14\]. ![image](example3fbst.pdf){width="100.00000%"} Based on the standardized $e$-value $\text{sev}(H_0)\approx 0.0000267$ and the Bayesian evidence against $H_0$, the $e$-value $\overline{\text{ev}}(H_0)\approx 0.9759$ one would reject the null hypothesis $H_0:\beta_j=0$. Discussion {#discussion .unnumbered} ========== This paper introduced the R package `fbst` for computing the Full Bayesian Significance Test and the $e$-value for testing a sharp hypothesis against the alternative. The conceptual approach and the statistical theory of the FBST were detailed, and three examples of statistical models frequently used in psychology and the cognitive sciences highlighted how the FBST can be computed in practice via the `fbst` R package. It was shown that both one-sided and two-sided hypotheses can be tested with the `fbst` package. The package’s core function `fbst()` requires only a posterior MCMC sample so it should be applicable to a wide range of statistical models used in the cognitive sciences and psychology. The examples demonstrated that it is simple to combine the FBST via the `fbst` package with widely used libraries like `rstanarm` [@Goodrich2020] or the `BayesFactor` package [@BayesFactorPackage]. The provided summary and plot functions in the package allow intuitive use and produce appealing visualisation of the FBST results which simplifies sharing and communication of the results with colleagues. We omitted simulation studies in this paper, because these were recently conducted by [@Kelter2020BayesianPosteriorIndices] to which the interested reader is referred. For more details on the theoretical properties of the FBST, we also refer the reader to [@Pereira2020]. To conclude, we direct attention to some limitations and possible extensions of the FBST and the `fbst` package presented in this paper. First, the `fbst` package is widely applicable but this strength can also be interpreted as a limitation. The `fbst` package requires a posterior distribution which has been derived analytically or numerically to conduct the FBST and compute the $e$-value, so it is not a standalone solution. Second, the core functionality in the current form is restricted to computing, summarising and visualising the FBST. Future extensions could include more detailed analysis results like robustness checks depending on the reference function used, see [@VanDoorn2019]. Also, in its current form the package uses only posterior MCMC draws, and future versions could provide the option to provide the posterior as a closed-form function. Another option to extend the functionality would be to make various algorithms available to estimate the posterior density based on the posterior draws: By now, only Gaussian kernel density estimation is used. In small sample situations the asymptotics of Bayesian posterior distributions guaranteed by the Bernstein-von-Mises theorem can be questionable and other approaches like spline-based interpolation or non-Gaussian kernels may be more useful. Third, while the standardized $e$-values may be used as a replacement of frequentist $p$-values, they are also based on asymptotic arguments and future research is needed to study the behaviour of the standardized $e$-values $\text{sev}(H_0)$ for small samples. This is why we recommend a continuous interpretation of the Bayesian $e$-value $\overline{\text{ev}}(H_0)$ over a threshold-oriented interpretation via standardized $e$-values $\text{sev}(H_0)$. In closing, it must be emphasized that we do not argue against the appropriate use of $p$-values, Bayes factors or any other suitable method of hypothesis testing. However, the ongoing debate about the concept of statistical significance shows that it is useful to explore existing alternatives for statistical hypothesis testing and investigate the relationships between these approaches both from a theoretical and practical perspective [@Berger1987; @Makowski2019; @Liao2020]. The `fbst` R package introduced in this paper could contribute in particular to the former, as simulation studies can easily be carried out by employing the package, see for example [@Kelter2020BayesianPosteriorIndices]. There is much value in testing a sharp null hypothesis against its alternative in the cognitive sciences and psychology [@BergerBrownWolpert1994; @Berger1997; @Rouder2009]. While there are also other useful approaches such as equivalence testing – see [@Lakens2017; @Lakens2018; @Kruschke2018; @Kruschke2018a] – the FBST has shown to be an attractive alternative to NHST and $p$-values with desirable theoretical and practical properties [@Kelter2020BayesianPosteriorIndices; @Pereira2020; @Esteves2019]. It is hoped that this package will be useful to researchers from the cognitive sciences and psychologists who are interested in a fully Bayesian alternative to null hypothesis significance testing which requires little methodological changes, but offers all the benefits of a fully Bayesian data analysis. Conflict of interest {#conflict-of-interest .unnumbered} ==================== The authors declare that they have no conflict of interest. Open Practices {#open-practices .unnumbered} ============== The data and materials for all analyses are available at <https://osf.io/u6xnc/>. [^1]: If the posterior is available in closed form, one can directly sample from it and provide the argument with the samples. [^2]: However, a supplementary replication codebook is provided at the Open Science Foundation under <https://osf.io/u6xnc/>. [^3]: See [www.jasp-stats.org](www.jasp-stats.org) [^4]: The data set is also provided as a .csv-file at the OSF repository <https://osf.io/u6xnc/>. [^5]: A *left*-half Cauchy prior is used, as under $H_1:\delta <0$, so a priori only negative effect sizes are assumed under the alternative hypothesis.
--- abstract: 'We introduce a dissipative particle dynamics scheme for the dynamics of non-ideal fluids. Given a free-energy density that determines the thermodynamics of the system, we derive consistent conservative forces. The use of these effective, density dependent forces reduces the local structure as compared to previously proposed models. This is an important feature in mesoscopic modeling, since it ensures a realistic length and time scale separation in coarse-grained models. We consider in detail the behavior of a van der Waals fluid and a binary mixture with a miscibility gap. We discuss the physical implications of having a single length scale characterizing the interaction range, in particular for the interfacial properties.' author: - 'I. Pagonabarraga(\*) and D. Frenkel' title: Dissipative particle dynamics for interacting systems --- Introduction ============ There is a strong incentive to develop “mesoscopic” numerical techniques to model the dynamics of fluids with different characteristic length scales. Mesoscopic simulations make it possible to analyze processes that take place on length and time scales that are out of reach for purely atomistic simulations such as Molecular Dynamics (MD). In MD, one retains the full atomic details in the description of the system, but at the expense of restricting the studies to short times. In contrast, models that describe the system at mesoscopic scales, employ a certain degree of coarse graining, which allows one to analyze longer times. However, care should be taken that the loss of “atomic” information associated with the coarse-graining process does not lead to unrealistic features on larger length and time scales. In particular, the coarse-grained models should provide an adequate description of the equilibrium properties of the system. Some of the mesoscopic models that have been proposed previously in the literature were derived in a systematic way from underlying microscopic models, as is the case with the lattice-Boltzmann method[@LB], which can be viewed as a preaveraged lattice gas model[@LG]. Coming from the opposite side, smoothed particle dynamics was introduced as a Lagrangian discretization of the macroscopic equations of fluid motion[@SPH]. A different strategy to simulate structured fluids is to assume that the solvent is passive, and that the suspended objects have a diffusive dynamics with diffusion coefficients that are known [*a priori*]{}. This has led to the development of Brownian[@EMc] and Stokesian dynamics[@BB]. In the early nineties, Dissipative Particle Dynamics (DPD) was introduced as a novel way to simulate fluids at a mesoscopic scale[@HK]. In DPD, the fluid is represented by a large number ($N$) of point particles that have a pairwise additive interaction. The interparticle forces are the sum of three contributions. In addition to the usual conservative forces that can be derived from a Hamiltonian, DPD includes dissipative and random forces. These mimic the effect of viscous damping between fluid elements and the thermal noise of the fluid elements, respectively. Flekk[ø]{}y and Coveney [@Flekkoy] have shown that, in principle, a particular DPD-like model can be derived from an atomistic description. However, no such derivations exist for the commonly used DPD models. Nonetheless, even without such a link to the underlying microscopics, it has been shown that thermal equilibrium can be ensured by an appropriate choice of the ratio between dissipative and random forces[@EW]. The hydrodynamic behavior of the DPD model has been explored in some detail[@Colin; @IF; @Masters; @Serrano], although the link between the mesoscopic and the macroscopic description is not completely understood. In conventional DPD, all interparticle forces have the same finite interaction range $r_{c}$. Their amplitudes decay according to a weight function $w(r_{ij})$ that has been made to vanish at $r_{c}$ in order to avoid spurious jumps at the cut-off distance. In this paper we employ a more general description of the conservative interactions. In the existing literature, the conservative forces have usually been assumed to depend explicitly on the distance between a pair of particles. For the sake of computational convenience, the conservative forces between DPD particles are taken smooth and monotonic functions of the distance - in fact, the smoothness of the forces is one of the advantages of DPD. When the forces depend linearly on the interparticle separation, the equation of state (EOS) of the DPD fluid is approximately quadratic in the density and exhibits no fluid-fluid phase transition. Even though the forces between DPD particles are smooth, they still induce structure in the fluid (reminiscent of atomic behavior) on a length scale of order $r_{c}$. In this respect, the conventional DPD scheme is similar to other mesoscopic models for non-ideal fluids but differs from the - computationally more demanding - scheme of Flekk[ø]{}y and Coveney that was mentioned above[@Flekkoy]. Our aim in this paper is to arrive at a formulation of DPD that allows for a description of the behavior of non-ideal fluids and fluid-mixtures. To this end, we look for a model in which there is a direct link between the macroscopic equation of state and the effective interparticle forces. As we will show, as an additional advantage, our approach results in rather weak structural correlations in the fluid. In the next section we describe in detail the model and how conservative forces are derived. We will subsequently elaborate the general method on three characteristic examples: a non-ideal fluid without a gas-liquid phase transition that has been studied previously with a different choice of conservative forces, a van der Waals fluid, and a mixture with a miscibility gap. In section \[sect:inter\] we look at the interfacial properties of these examples to gain some insight in the physical meaning of the conservative forces that we introduce, and subsequently analyze their equilibrium behavior and compare with previous models. We conclude with a discussion of our main results. Model {#sect:model} ===== In DPD one has $N$ point particles of mass $\{m_{i}\}$ that interact through a sum of pairwise-additive conservative, dissipative and random forces. These particles can be interpreted as fluid elements, and the dissipative forces are introduced to mimic the viscous drag between them. The random force equilibrates the energy lost through friction between the particles, enabling the system to reach an equilibrium state. To be specific, if we call $\{{\bf r}_{k},{\bf p}_{k}\}$ the set of particle positions and momenta of the $N$ point particles, their dynamics are controlled by Newton equations of motion $$\begin{aligned} \frac{d{\bf r}_{k}}{dt} &=&{\bf v}_{k} \label{eqts-DPD1} \\ \frac{d{\bf p}_{k}}{dt} &=&\sum_{j\neq i}\left\{ {\bf F}^{C}({\bf r}_{ij})+ {\bf F}^{D}({\bf r}_{ij})+{\bf F}^{R}({\bf r}_{ij})\right\} \nonumber \\ &=&\sum_{j\neq i}\left\{ {\bf F}^{C}({\bf r}_{ij})-\gamma \omega ^{D}({\bf r} _{ij}){\bf v}_{ij}\cdot {\bf e}_{ij}{\bf e}_{ij}+\sigma \omega ^{R}({\bf r} _{ij}){\bf e}_{ij}\xi _{ij}\right\} \label{eqts-DPD}\end{aligned}$$ where we have used the notation ${\bf r}_{ij}\equiv {\bf r}_{i}-{\bf r}_{j}$ and ${\bf v}_{ij}\equiv {\bf v}_{i}-{\bf v}_{j}$. ${\bf e}_{ij}$ denotes a unit vector in the direction of ${\bf r}_{ij}$, and ${\bf v}_{i}={\bf p}% _{i}/m_{i}$ is the velocity of particle $i$. The dissipative force, ${\bf F}% ^{D}({\bf r}_{ij})$, depends both on the relative positions and velocities of the interacting pair of particles and its amplitude is characterized by the parameter $\gamma $. This parameter is related to the viscosity of the DPD fluid. The third term in eq.([\[eqts-DPD\]]{}), ${\bf F}^{R}( {\bf r}% _{ij})$, is a random force acting on each pair of DPD particles - $\xi $ stands for a random variable with Gaussian distribution and unit variance. The random force has an amplitude $\sigma $ and is also central. Central pair interactions ensure angular momentum conservation (although the dynamics can be generalized to account for non-central forces [@pepL]). The dissipative and random forces are completely specified once the weight functions, $\omega ^{D}(r_{ij})$ and $\omega ^{R}(r_{ij})$, are specified- these are smooth and of finite range. Although they can be chosen at will, Español and Warren showed[@EW] that $\omega ^{D}$ and $\omega ^{R}$ must be related to ensure that the probability to observe a particular configuration of DPD particles is given by the Boltzmann distribution in equilibrium. Specifically, if they are chosen such that $\omega ^{R}=% \sqrt{\omega ^{D}}$, then the correct equilibrium distribution is recovered, and the equilibrium temperature of the DPD fluid is fixed by the ratio of the amplitudes of the dissipative and random forces, $k_{B}T=\sigma ^{2}/(2\gamma )$. We stress that the DPD equations of motion, eq.(\[eqts-DPD1\]-\[eqts-DPD\]), cannot be derived from a Hamiltonian. Traditionally, and for simplicity, the conservative forces in DPD have been taken as pairwise-additive and central, with a weight function related to $% \omega ^{D}$, and with a variable amplitude that sets the temperature scale in the system. As long as the force is sufficiently weak that it does not induce appreciable inhomogeneities in the density around a DPD particle, it can only lead to an equation of state with a quadratic dependence in the density, irrespective of the precise choice for the weight function (see below). One consequence is that phase separation between disordered phases cannot occur in a pure system; at least a binary mixture of different kinds of particles is needed[@phasesep]. We will first consider the general form that the free energy of a DPD system can have, in order to elucidate the generic shape of consistent conservative forces. In agreement with the idea that the DPD particles refer to lumps of fluid, it seems natural to assume that the relevant energy associated to their configurations is a free energy, rather than a strictly “mechanical” potential energy. We can express quite generically the free energy ${\cal F}$ of an inhomogeneous system with density $\rho ({\bf r})$ as $${\cal F}=\int d{\bf r}\rho ({\bf r})f(n({\bf r})) \label{freevol}$$ where $f(\rho )$ is the expression for the local free energy per particle (in units of $k_{B}T$), and $n(\{{\bf r}\})$ is related to the density of the system at ${\bf r}$. This formulation is reminiscent of the strategy followed in density functional theory to study the equilibrium properties of the fluids [@Evans]. In fact, the particular case $n(\{{\bf r}\})=\rho (\{{\bf r}\})$ corresponds to the local density approximation in density functional theory, and if $n({\bf r})$ is chosen to be an average of the density over an interval around ${\bf r}$, it can be understood as a weighted density approximation for the true free energy. We can separate the total free energy, $f(\rho)=f^{id}(\rho)+f^{ex}(\rho)$, as the sum of the ideal $f^{id}(\rho )=\log (\Lambda ^{3}\rho)-1$ plus the excess contribution, where $\Lambda $ is the thermal de-Broglie wavelength. Our purpose is to obtain the equivalent expression for a DPD system, in which we have $N$ particles distributed in the space. Since the free energy is an extensive quantity, the total free energy of a DPD system can be obviously expressed in terms of the free energy per DPD particle, $\psi$, as $${\cal F}=\sum_{i=1}^N \psi(n_i)=\sum_{i=1}^N \int d{\bf r} \delta({\bf r}-% {\bf r}_i) \psi(n({\bf r}))=\int d{\bf r} \rho({\bf r} )\psi(n({\bf r}))) \label{free_part}$$ where we have introduced the symbol $n_i$ to refer to the generalized density defined above, although now expressed in terms of the positions of the discrete $N$ DPD particles (see below). Comparing eqs.(\[free\_part\]) and (\[freevol\]), we can easily identify $\psi(\rho)=f(\rho)$ which obviously implies that we can decompose $\psi$ into its ideal and excess contributions. If the free energy determines the relevant energy for a given configuration of DPD particles, we can then derive the force acting on each particle as the variation of such an energy when the corresponding particle is displaced. However, the motion of the particles themselves, due to the action of the dissipative and random forces, already accounts for the ideal contribution to the free energy of the system, which is not related to the interactions among the particles. Therefore, only the excess part of the free energy will be involved in the effective interactions between the DPD particles. Accordingly, we can write the conservative force acting on particle $i$, ${\bf F}^C_i$, as $${\bf F}_i=-\frac{\partial}{\partial{\bf r}_i}\sum_{j=1}^N \psi^{ex}(n_j) \label{force}$$ We have derived the generic form for the conservative force acting on a DPD particle as a function of the excess free energy that characterizes the system, which is in general not pair-wise additive. These forces are analogous to the ones derived from semi-empirical potentials[@FS] in MD, used to model effectively the many-body interactions in condensed systems. However, we have started from the macroscopic properties of the system, i.e. its free energy, rather than ensuring microscopic consistency. We can then fix the equilibrium thermodynamic properties of the system beforehand, and derive a set of conservative forces consistent with the desired equilibrium macroscopic behavior. This procedure is reminiscent of an approach used in other mesoscopic simulation techniques that deal with generic non-ideal fluids[@JY]. Given that the free energy has been defined as a functional of a certain local density, local variation in such a density are responsible for the effective forces among the DPD particles. The particular expression for the forces will then depend both on the specific form of the free energy and on the choice of the local density $n_i$. It seems natural to define the local density of a particle $i$ as its average on the corresponding interaction range. For simplicity, we weight this average with the same functions used to define the dissipative and random forces, as introduced in eq.(\[eqts-DPD\]). Therefore, we write $$n_{i}=\frac{1}{[w]}\sum_{j}w(r_{ij}) \label{dens}$$ where $[A]$ refers to the spatial integral of a given quantity $A$. The normalization factor $[w]$ ensures that $n_{i}$ is indeed a density, so that in a homogeneous region, $n=\rho$. This is in spirit similar to the weighted density approximation in density functional theory[@Evans]. The use of a continuous and smooth weight function that vanishes at the cut-off distance, $r_c$, ensures a smooth sampling of the environment of each particle, avoiding spurious jumps. There is no [*a priori*]{} reason to choose $w(r)$ equal to any of the other weight functions, although the particular case of a constant weight function constitutes a pathological limit - in this case the conservative force will only act when one particle enters or leaves the interaction range. The dependence of the energy of a particular configuration on the particles’ positions enters implicitly through the weighted densities. For densities of the form given by eq.(\[dens\]), the conservative force acting on particle $i$ can be rewritten as $${\bf F}_{i}=-\sum_{j=1}^{N}\frac{\partial \psi(n_j)}{ \partial {\bf r}_{i}}% =-\sum_{j}(\psi _{i}^{\prime }+\psi _{j}^{\prime }) \frac{w_{ij}^{\prime }}{% [w]}{\bf e}_{ij}\equiv \sum_{j}{\bf F}_{ij} \label{pairwise}$$ where we have introduced the notation, $\psi_i\equiv\psi^{ex}(n_i)$, and where the primes denote derivatives with respect to the corresponding variables. Although the free energy of each particle depends on the local density, and leads in general to many-body effective forces, for the particular local density introduced in eq.(\[dens\]), the forces between DPD particles can still be written down as additive pairwise forces- a computational advantage. The fact that the forces depend on the positions of many particles through their corresponding local weighted densities suggests that in general the local structure of the fluid phase will be smoother than in the case in which forces are derived from a pair-potential. This is an attractive feature of the present model; the local structure in a fluid should only be related to its microscopic structure, and should be smeared out at mesoscopic, coarse-grained, scales. In this respect, the density-dependent interactions of these DPD models enforce an appropriate length scale separation. In the next sections we will analyze these properties in detail. Before considering specific examples, as a consistency check, we will analyze the predictions for the pressure of a fluid following the free energy, $p^{th}$, and the virial, $p^{v}$, routes. If we start from the free energy per particle, eq.(\[free\_part\]), the pressure for a fluid will be $$p^{th}=-f+\rho\frac{\partial f}{\partial \rho}=k_BT\rho+ \rho^2 \frac{ \partial \psi^{ex}}{\partial \rho} \label{p:thermo}$$ On the other hand, since we have derived the force between particles from the free energy, we can also obtain the pressure of the fluid following the virial route. In this case the pressure is given $$p^{virial}=\rho k_{B}T+\frac{1}{2dV}\sum_{i}\sum_{j}{\bf r}_{ij}\cdot {\bf F} _{ij}=\rho k_{B}T+\frac{1}{2dV}\int \int d{\bf r}d{\bf r}^{\prime }\rho ( {\bf r},{\bf r}^{\prime })({\bf r}-{\bf r}^{\prime })\cdot {\bf F}( {\bf r}-{\bf r}^{\prime })$$ where we have approximated the discrete sum over the $N$ DPD particles by an integral. Introducing the pair correlation function, $g(r)$, we can rewrite the previous equation as $$p^{virial}=k_{B}T\rho +\frac{\rho ^{2}}{2d}\int d{\bf r}g(r)\frac{\partial \psi ^{ex}}{\partial \rho }{\bf r}\cdot \left\{ \frac{-2w^{\prime }(r) {\bf e% }}{[w]}\right\} =k_{B}T\rho -\frac{\rho ^{2}}{d}\frac{\partial \psi ^{ex}}{% \partial \rho }\frac{[rw^{\prime }]}{[w]} \label{p:virial}$$ In the last equality we have assumed that the density is nearly homogeneous, and that therefore $\partial \psi ^{ex}/\partial \rho $ is effectively a constant. Otherwise, it is not possible to express the force in terms of the relative coordinates only. If there is no local structure in the fluid, and $% w(r_c)=0$, then $[rw^{\prime }]=-d[w]$, and then eq.(\[p:virial\]) coincides with the prediction for the “thermodynamic” pressure, eq.(\[p:thermo\]) for any weight function[@note]. Otherwise, a discrepancy between the two pressures will appear because the averaged density $n_i$ is always centered on the corresponding DPD particle- a conditional density- and it is therefore related to the $g(r)$. We will see in some examples in subsequent sections how such local structure may develop. Theoretical studies have shown that in the fluid phase of DPD, in the hydrodynamic limit the usual Navier-Stokes equation is recovered[@Colin], and that the equilibrium pressure term is related to the pairwise forces through the usual virial expression, as we have derived previously. This corresponds to dynamics which conserves momentum locally (as in model-H[@H-H]), instead of being purely relaxational (as happens in certain dynamical models that start from density functional theories[@Marini]). By analogy with the usual non-ideal DPD models, in equilibrium we recover a probability distribution for a given configuration in agreement with Boltzmann fluctuation theorem: the probability of observing a fluctuation is proportional to the exponential of the deviation of the appropriate thermodynamic potential- the free energy (as introduced in eq.\[freevol\]) for DPD models at constant volume, temperature and number of particles. In the following subsections we will consider three particular examples, where we will compute explicitly the form of the conservative forces. Groot and Warren fluid {#subsec:gw} ---------------------- Let us first derive the expression for the conservative force that corresponds to the non-ideal fluid studied by Groot and Warren [@Groot]. They introduce a conservative force of the form $${\bf F}_{ij} = \left\{ \begin{array}{cc} a \left(1-\frac{r_{ij}}{r_c}\right){\bf e}_{ij} & ,r_{ij}<r_c \\ 0 & ,r_{ij}>r_c \end{array} \right. \label{GW_force}$$ For this conservative force, they have shown that the EOS is $% p=k_BT\rho+\alpha a\rho^2$, where by a numerical fit they found $% \alpha=0.101\pm0.001$. Using the expressions of the previous section, the corresponding pairwise force is $${\bf F}_{ij} = \left\{ \begin{array}{cc} 2 \alpha a \frac{w_{ij}^{\prime}}{[w]}{\bf e}_{ij} & ,r_{ij}<r_c \\ 0 & ,r_{ij}>r_c \end{array} \right. \label{EOSGW}$$ It corresponds to an excess free energy per particle $\psi^{ex} =\alpha a \rho$, which is linear in the density. As stated in the introduction, an interaction with a smooth, monotonic dependence in position does not induce a fluid-fluid phase separation. van der Waals fluid ------------------- The van der Waals fluid is the classic example of a fluid with a liquid-gas phase transition. It is characterized by the equation of state $p=\rho k_{B}T/(1-b\rho )-a\rho ^{2}$ (and excess free energy per particle, $% \psi^{ex} = -k_BT\log(1-b\rho)-a\rho$). We can recover this EOS in a DPD system with pairwise conservative forces of the form, $${\bf F}_{ij} = \left\{\left(\frac{k_BT b}{1-b n_i}-a\right)+\left(\frac{k_BT b}{1-b n_j}-a\right)\right\}\frac{w^{\prime}_{ij}}{[w]} {\bf e}_{ij} \label{EOSvdW}$$ For reasons that will be discussed below, it is helpful to generalize slightly the van der Waals fluid allowing for a contribution cubic in the density. The EOS then becomes $p=\rho k_BT/(1-b \rho)-a \rho^2-\alpha_3 a b \rho^3$. The critical point of this model corresponds to the parameters $$\begin{aligned} T_c&=& \frac{a}{b} b\rho_c(2+3\alpha_3 b\rho_c) (1-b\rho_c)^2 \nonumber \\ \rho_c &=&\frac{1}{b}\frac{\alpha_3-1+\sqrt{1+\frac{2}{3}\alpha_3+\alpha_3^2} }{4\alpha_3}\end{aligned}$$ $$\begin{aligned} \rho_c b&\equiv&x_c = \frac{-1+\alpha_3+\sqrt{1+\frac{2}{3}% \alpha_3+\alpha_3^2}}{ 4\alpha_3} \\ T_c b/a&\equiv& y_c = x_c(2+3\alpha_3 x_c) (1-x_c)^2\end{aligned}$$ The compressibility of the fluid, $\chi$, in turn, can be written down as $$\chi^{-1} = \frac{k_BT}{\rho}+\frac{k_BT b (2-b \rho)}{(1-b\rho)^2} -2a-3\alpha_3 a b \rho = \frac{y y_c}{x x_c (1-x x_c)^2}-2-3 \alpha_3 x x_c$$ In fig.\[fig:xi\] we show the behavior of the compressibility for two different values of the parameter $\alpha_3 $, for temperatures close to the critical temperature $T_{c}$. The increase in $\alpha_3$ reduces $\chi$ both above and below the critical temperature. As expected, $\chi$ becomes negative in a region below $T_{c}$ that is bounded by a spinodal. Controlling the compressibility of the fluid is a desirable feature; a low compressibility helps reducing fluctuations of the fluid interface, which may be useful in simulations. It also provides a way of modifying properties of the fluid, such as the speed of sound. Moreover, it gives an additional parameter to select the surface tension which, as we will explain, may even change sign in this DPD-van der Waals fluid. Finally, it proves useful to reduce the amplitude of the density fluctuations to compare with mean field theoretical predictions, as the ones developed in the next section. Binary mixture -------------- A binary mixture composed of particles of two species[@phasesep], $A$ and $B$, has also been considered by Groot and Warren[@Groot]. In this system, it is possible to induce demixing with usual pairwise forces by modifying the relative repulsions between the $A-A$, $B-B$ and $B-A$ pairs. Nevertheless, even in this case, a model in which the forces depend on local densities can be useful since if they induce less local structure, a relevant feature at a fluid-fluid interface. If the system consists of $N_{A}$ particles of type $A$ and $N_{B}$ particles of type $B$, then there are two relevant local density fields, $% n_{A}$ and $n_{B}$, that are the straightforward generalizations of eq.(\[dens\]), $$\begin{aligned} n_{A_i}&=&\sum_{j\in A}\frac{w(r_{ij})}{[w]} \\ n_{B_i}&=&\sum_{j\in B}\frac{w(r_{ij})}{[w]}\end{aligned}$$ $n_{A_i}$ and $n_{B_i}$ represent the concentration of $A$ and $B$ particles around particle $i$, respectively. Whenever it is appropriate, we will denote by $\rho_A$ and $\rho_B$ the continuum limit of the discrete densities $n_A$ and $n_B$, respectively. The simplest free energy that leads to a miscibility gap has an excess free energy of the form $${\cal F}^{ex}= \int d{\bf r} \left\{2 \lambda \rho_a({\bf r}) \rho_b(% {\bf r})+ \lambda_A \rho_a({\bf r})^2 + \lambda_B \rho_B({\bf r})^2\right\} = \left[\sum_{i\in A}\left(\lambda n_{B_i}+\lambda_A n_{A_i}\right)+ \sum_{i\in B}\left(\lambda n_{A_i}+\lambda_B n_{B_i}\right) \right] \label{freeen}$$ where the two sums run over particles of type $A$ and $B$, respectively. The corresponding conservative force acting on particle $i$ can be written down as $${\bf F}_j= \left\{\sum_{i\in A}\left\{\lambda\sum_{k\in B}+\lambda_A\sum_{k\in A}\right\}+ \sum_{i\in B}\left\{\lambda\sum_{k\in A}+\lambda_B\sum_{k\in B}\right\}\right\}\left[w^{\prime}_{ik}{\bf e}_{ik} (\delta_{ij}-\delta_{kj})\right]$$ Although in this case with two averaged local densities the conservative forces do not have the form of eq.(\[pairwise\]), they can still be expressed as pairwise additive forces, $${\bf F}_{ij}=\left\{ \begin{array}{ll} -2 \lambda_{A,B} w^{\prime}_{ij}{\bf e}_{ij} &\;\;\; , i\;j \mbox{ same type} \\ -2 \lambda w^{\prime}_{ij}{\bf e}_{ij} & \; \; \;, i\;j \mbox{ different type} \end{array} \right. \label{EOSmix}$$ This fluid will be miscible at high temperatures, and below a critical temperature $T_c$ a miscibility gap will develop. In terms of the parameters of the free energy, eq.(\[freeen\]), for a symmetric mixture $T_c$ is $$k_BT_c=\rho (\lambda-\lambda_A)\;\;\;\;,\;\;\;\;\;\;\frac{\rho_A}{% \rho_A+\rho_B}|_c\equiv c_c=\frac{1}{2} \label{eq:critbin}$$ Interfacial behavior {#sect:inter} ==================== In this section we develop a mean field theory for the interfacial properties for a non-ideal DPD fluid that gives some insight in the meaning of the conservative forces for these DPD models. For definiteness, we concentrate on the derivation of the surface tension, $% \tilde{\gamma}$. Since we are interested in the interfacial properties, we focus on the excess free energy, and will not write down the ideal gas contribution, which is local in the density and does not contribute to the interfacial properties. We start from the continuum limit of the appropriate free energy, and make an expansion in gradients. Therefore, we disregard correlations in the positions between the particles, hence the mean field character of the predictions of the present section. van der Waals fluid {#sect:FvdW} ------------------- For a van der Waals fluid we can express the continuum free energy of the fluid, that corresponds to the conservative forces introduced in eq.(\[EOSvdW\]), as $${\cal F}^{ex} =\int d{\bf r} \rho({\bf r}) \left(-k_BT\log(1-bn({\bf r}% ))-an( {\bf r})-\frac{\alpha_3}{2}a b n({\bf r})^2\right) \label{free_dft}$$ where $n({\bf r})$ is the continuum limit of eq.(\[dens\]), namely, $$n({\bf r}) = \frac{1}{[w]}\int d{\bf r}^{\prime}w(|{\bf r}-{\bf r} ^{\prime}|) \rho({\bf r}^{\prime}) \label{dens_cont}$$ In eq.(\[free\_dft\]), the density $\rho({\bf r})$ means the mean density at point ${\bf r}$. This is different from the density appearing in section \[sect:model\], where it referred to the instantaneous value of the density for a particular configuration. Due to this density preaveraging, the results of the present section constitute a mean field approximation. For a smooth planar interface, we can expand the density in eq.(\[dens\_cont\]) to second order in the gradients[@Evans], $$\rho({\bf r}-{\bf z})=\rho({\bf r})-{\bf z}\cdot{\bf \nabla}\rho({\bf r})+ \frac{1}{2}{\bf z}{\bf z}:{\bf \nabla}{\bf \nabla}\rho({\bf r}) \label{dens_exp0}$$ Inserting this expression in eq.(\[dens\_cont\]), and using the fact that the weight function is radially symmetric we get $$n({\bf r}) = \rho({\bf r})+\frac{[z^2w]}{2 d[w]} \nabla^2\rho({\bf r}) \label{dens_exp1}$$ With this expression, eq.(\[free\_dft\]) can be written down as $$\begin{aligned} {\cal F}^{ex} &=& \int d{\bf r} \rho({\bf r})\left\{-k_BT\ln\left(1-b\rho(% {\bf r})- \frac{b [z^2w]}{2 d[w]}\nabla^2\rho({\bf r})\right)-a\rho({\bf r}) \right. \nonumber \\ &-&\left. \frac{a [z^2 w]}{2 d[w]}\nabla^2\rho({\bf r})-\frac{\alpha_3 a b }{% 2} \left(\rho({\bf r})^2+ \frac{[z^2 w]}{d[w]}\rho({\bf r})\nabla^2\rho({\bf % r} )\right)\right\} \label{free_exp1}\end{aligned}$$ where terms containing derivatives higher than second order have been neglected. Collecting terms in powers of the density gradients, making use of the integration by parts we can rewrite eq.(\[free\_exp1\]) in the usual form $${\cal F}^{ex} = \int d{\bf r} \rho({\bf r}) \left(-k_BT\ln(1-b\rho({\bf r} ))- a\rho({\bf r})-\frac{\alpha_3}{2}a b \rho({\bf r})^2\right)+ \frac{[z^2 w]}{2 d[w]}\left(-\frac{k_BT b}{(1-b\rho({\bf r}))^2}+a+ 2\alpha_3 a b \rho( {\bf r})\right)|{\bf \nabla}\rho({\bf r})|^2 \label{free_exp2}$$ The first term in brackets gives the local contribution to the excess free energy. When the ideal contribution is added, it gives us the free energy for a homogeneous van der Waals fluid. The second term in brackets is the energy penalty to generate gradients in the system. It is this term that contains, to lowest order, the interfacial energy of the fluid. In particular, we can obtain from it an expression for the surface tension. If we assume that the profile is a hyperbolic tangent, and we estimate its width from the asymptotic bulk coexisting densities[@Godreche], we arrive at $$\tilde{\gamma}=\frac{\rho_l-\rho_g}{2}\sqrt{\frac{[z^2 w]}{d[w]}\left(-\frac{ k_BT b}{(1-b\rho_m)^2} +a+ 2\alpha_3 a b \rho_m\right) \frac{d^2 f}{d \rho^2} }$$ where $\frac{d^2 f}{d \rho^2}=1/\rho-2 a+k_BT (2-b \rho)/(1-b \rho)^2 -3\alpha_3 a b \rho$ is the second derivative of the homogeneous free energy with respect to the density evaluated at one of the coexisting phases. We have assumed for simplicity that the density difference between the two phases is small, so that we can approximate the density across the interface by its mean value, $\rho_{m}$ If we look at the structure of both the expansion of the free energy and the surface tension, we can recognize a qualitative difference with respect to the corresponding expressions for the standard van der Waals fluid. In the latter, the interfacial tension is a function only of the parameter $a$ characterizing the long range attraction between the particles, whereas now it depends on all the parameters, $a$, $b$ and $\alpha_3$. This qualitative difference can already be traced back to the coefficient of the gradient square term in free energy expansion, eq.(\[free\_exp2\]) - for the standard van der Waals fluid the gradient energy cost is only related to $a$. As a result, in this DPD van der Waals fluid there are different contributions to the gradient energy term with different signs. Therefore, depending on their relative strength, it is possible either to favor or penalize the appearance of density gradients in the fluid; hence, the sign of the interfacial tension may change. In an atomic fluid, the repulsion parameter, $b$, in the van der Waals EOS arises from the hard core repulsion, while the attraction parameter, $a$, comes from a long range weak attraction. Therefore, they appear in different length scales, and accordingly, only the parameter $a$- related to the long-range structure- is responsible for the behavior the interfacial tension. On the contrary, for a DPD fluid there is no excluded volume interaction, and all interactions between the particles take place at the same length scale, $r_c$. Then, the relative strength of the different contributions will determine their overall net effect. It is known that a microscopic model in which both attractions and repulsions are long ranged leads to a van der Waals equation of state in which the interfacial behavior can either favor or penalize the presence of interfaces[@Sear]. The van der Waals fluid introduced in this paper shares these same properties. Even if we can ensure a van der Waals EOS for a fluid, a careful tuning of the parameters $a$, $b$ and $\alpha_3$ may lead to a van der Waals model for lamellar fluid, whenever interfaces are favored. Although unrealistic for atomic fluids, this behavior is relevant, e.g for nanoparticles, for which repulsive and attractive interactions act on similar length scales[@Sear1]. Therefore, depending on the kind of fluid that needs to be modeled at mesoscopic scales, the parameters in the free energy should be chosen appropriately. For example, in order to get a positive surface tension, the densities of the fluid phases is restricted because one must ensure that both the pressure and the surface tension are positive. In fig.\[fig:stability\] we display the curves where the pressure and the surface tension vanish for two different values of $\alpha_3$. The area defined in between the corresponding set of curves defines the region of phase space where the fluid is mechanically stable with a positive surface tension. Remember that the values of $a$ and $b$ set the critical values $\rho_c$ and $T_c$. The allowed regions do not change very much as the parameter $\alpha_3$ is modified. If we make $b=0$, this model reduces to that of Groot and Warren. In this case, $\tilde{\gamma}$ becomes negative (remember that $a$ is negative now). As we have mentioned in subsection \[subsec:gw\], there is no fluid-fluid phase separation in this model; therefore this negative value of the surface tension does not lead to a proliferation of interfaces. However, the negative value of $\tilde{\gamma}$ implies that the structure factor will have a minimum at a finite wave vector. We can define a characteristic length, $\tilde{l}_0$, on which local structure in the fluid will develop. If we expand the free energy eq.(\[free\_dft\]) to next order in gradients we can estimate this length to be $$\tilde{l}_0 \sim 2\pi r_c \sqrt{\frac{[w r^4]}{12 [w r^2]}}$$ which does not depend on the amplitude $a$; only on the shape of the weight function $w$. Except for rapidly decaying weight functions, this length is of order of the interaction range $r_c$. This fact is consistent with the local structure observed in the radial distribution functions for this model (see section \[sec:gw\]). We have also verified numerically the presence of a minimum in the structure factor $S(k)$. Binary mixture {#sec:interbin} -------------- We can also compute the interfacial tension for a binary mixture following the procedure of the previous subsection. The excess free energy in the continuum limit is now $${\cal F}^{ex}=\lambda \int d{\bf r} (\rho_A({\bf r}) n_B({\bf r})+ \rho_B( {\bf r}) n_A({\bf r}))+ \lambda_A \int d{\bf r} \rho_A({\bf r}) n_A({\bf r}) + \lambda_B \int d{\bf r} \rho_B({\bf r}) n_B({\bf r})$$ It is useful to introduce the total density $\rho$ and the mole fraction $c$ of component A as the relevant variables. They are defined as usual, $$\begin{aligned} \rho_A &=& \rho c \\ \rho_B &=& \rho (1-c)\end{aligned}$$ If we expand the local densities $n({\bf r})$ in the same way as in eq.(\[dens\_exp1\]), we arrive at the square-gradient approximation for the free energy, $$\begin{aligned} {\cal F}^{ex}&=&\int d{\bf r} 2 \lambda \rho_A({\bf r}) \rho_B({\bf r}) +\lambda_A\rho_A({\bf r})^2+ \lambda_B\rho_B({\bf r})^2+\frac{[z^2 w]}{2 d [w]}\left\{\lambda \rho_A\nabla^2\rho_B+ \lambda \rho_B\nabla^2\rho_A +\lambda_A \rho_A\nabla^2\rho_A+ \lambda_B\rho_B\nabla^2\rho_B\right\} \nonumber \\ &=& 2\lambda \rho^2 c (1-c)+\lambda_A\rho^2 c^2+\lambda_B \rho^2 (1-c)^2+ \frac{[z^2 w]}{2 d[w]}\rho^2 \left\{-\lambda c\nabla^2 c-\lambda (1-c)\nabla^2 c+\lambda_A c\nabla^2c- \lambda_B(1-c)\nabla^2z\right\} \nonumber \\ &=& 2\lambda\rho^2 c(1-c)+\lambda_A\rho^2c^2+\lambda_B\rho^2(1-c)^2+\frac{ [z^2 w]}{2 d[w]}\rho^2 \left\{2\lambda-\lambda_A-\lambda_B)|\nabla c|^2\right\}\end{aligned}$$ Assuming that $\rho$ is constant, and for a symmetric mixture ($% \lambda_A=\lambda_B$), we get $${\cal F}^{ex}=\rho^2\int d{\bf r} \left\{ 2 (\lambda-\lambda_A) c (1-c)+2 \frac{[z^2 w]}{2 d[w]} (\lambda-\lambda_A) |\nabla c|^2\right\}$$ Again, the interfacial tension can have either a positive or negative sign, depending on the relative magnitudes of the $\lambda$ parameters. If $% \lambda_A=\lambda_B=0$, and only the repulsion between the particles belonging to different species is kept, then the surface tension has the same sign as $\lambda$, as expected. The interfacial width $\xi$ can be obtained taking into account that the concentration profiles converges exponentially to its bulk value. This gives us $\xi^2 = 4\kappa/F''$, where $\kappa/2$ is the amplitude of the $|\nabla c|^2$ in the gradient expansion of the free energy, and $F''$ is the second order derivative of the free energy with respect to the concentration evaluated at its bulk coexisting value. In the symmetric case, we get $$\xi^2 = \frac{ [z^2 \omega]}{[\omega]}\left(-1+\frac{T}{4 T_c c_{\infty} (1-c_{\infty})}\right)^{-1} \label{eq:xibin}$$ where $c_{\infty}$ is the value of the concentration in the bulk phase. The surface tension, $\gamma$, can be obtained integrating the diference between the free energy profile and its bulk value. In the small gradient limit, it reduces to $$\gamma = \int_{-\infty}^{\infty} \rho\frac{[z^2\omega]}{2 d[\omega]}T_c |\nabla c|^2 \label{eq:gammaexp}$$ If we assume that the concentration profile is a $\tanh$, we get the estimate $$\gamma = \frac{2 [z^2 \omega]}{3 [\omega]}\frac{\rho T_c}{\xi}( c_{\infty}-1/2)^2 \label{eq:sigmabin}$$ Close to the critical point, we recover the expected limiting behavior for the interfacial properties[@Godreche], $$\begin{aligned} \gamma &=&2 \rho k_{B}T_{c}\frac{T_{c}}{T}\sqrt{\frac{2 [z^2\omega]}{3 d [\omega]}\left( 1-\frac{T}{T_{c}}\right) ^{3}} \\ \xi &=&\sqrt{\frac{[z^{2}w]}{4d[w](1-T/T_{c})}}\end{aligned}$$ Equilibrium properties {#sect:comp} ====================== We will now analyze the equilibrium properties of the examples of non-ideal DPD systems introduced in section \[sect:model\] and will compare with the predictions of previous models performing numerical simulations. We take the interaction range $r_c$ as the unit of length, the mass of the DPD particles $m$ as the unit of mass, and the critical thermal energy, $k_BT_c$, of the corresponding free energy as the unity of energy. If no phase transition is present, then $k_BT$ is taken as the unit of energy. The equations of motion are integrated self-consistently to avoid spurious drifts in the thermodynamic properties[@IF]. Groot and Warren fluid {#sec:gw} ---------------------- Before studying a DPD model with fluid-fluid coexistence, we compare the results of our model for a Groot-Warren fluid with the original one, based on forces given by eq.(\[GW\_force\]). In this case, both models should coincide and we analyze it to see the effects of the weight function shape on the properties of non-ideal fluids.. We have performed simulations for a DPD fluid in 3 dimensions, taking as parameters $a=25$ and $\alpha =0.101$- which corresponds to those used in [@Groot]. In fig.\[fig:GW\] we compare the predictions for the EOS given by our model and by running a DPD simulation with the Groot-Warren model. Groot and Warren used the same weight function for all pairwise forces. The proposed model for this non-ideal fluid shows neatly that for the present class of models a linear weight function is not suitable to sample the local density of each DPD particle, because it leads to a pairwise conservative force that exhibits a discontinuity at the edge of the interaction region, $% r_c$. We have analyzed the effect of such a jump on the thermodynamic and structural properties of this system. To this end, we have considered both decreasing linear and quadratic $w$’s. In fig.\[fig:GW\] we compare the EOS obtained from simulations; for a quadratic $w$, our model coincides with that of Groot-Warren. However, for a linear $w$, the agreement survives only at low densities. This DPD model has a transition to a solid state at high densities, and the results obtained indicate that the location of such a transition is sensitive to the shape of the weight function - the characteristic force felt by each particle depends on the shape of $w$ for a given density. In figs.\[fig:gr\_GW\_n3\]-\[fig:gr\_GW\_n8\] we compare the radial distribution functions for our model and that of Groot-Warren, and for for different $w$’s, at increasing values of the density. It is clear that the shape of $w$ plays an important role in the local structure of the fluid, and will influence the location of the fluid-solid transition. In section \[sect:FvdW\] we have noted that for the present model there exists a characteristic length, $\tilde{l}_0$, associated with density fluctuations and which is of order $r_c$. Only for fairly narrow weight functions will this length become much smaller than $r_c $. At low densities, a linear $w$ generates less local structure, a pleasant feature for a mesoscopic model. However, as the density is increased, the local structure develops faster for a linear weight function, leading sooner to a transition to the ordered phase. The use of a quadratic weight function leads to results identical to those of the GW model, while a linear force tends to smooth the structure at short distances. The mean repulsion between particles is larger with a linear $w$ rather than with a quadratic one. Moreover, it seems plausible to assume that the discontinuity in the force induces a higher sensitivity to local density fluctuations. These results show how the modifications of the shape of the weight function can be used to tune fine details of the behavior of a fluid, once the EOS has been fixed. van der Waals fluid {#sec:lg} ------------------- Next, we focus on the liquid-gas equilibrium properties of a two-dimensional van der Waals fluid. Taking a homogeneous system, we can analyze the effect of the density fluctuations on the EOS, and compare it with the predictions coming from the macroscopically assumed EOS. In fig.\[fig:a3\], we show the pressure values obtained in simulations run at fixed homogeneous density, volume and temperature. In this case we can recover the characteristic van der Waals loop. The actual coexistence curve should be derived from it using the equal area Maxwell’s construction. The agreement with the expected EOS from the macroscopic free energy is very good, and only small deviations are observed, due to particle correlations. We have also analyzed the density and pressure profiles when we bring into contact a liquid and a gas in the coexistence region. As mentioned in section \[sect:model\], the compressibility of the fluid, especially in the coexistence region, is very sensitive to the parameter $\alpha_3$ that characterizes the amplitude of the term cubic in the pressure. For $% \alpha_3=0$ the density profiles tend to fluctuate substantially. Note that our estimates for the parameters and ranges of stability are all based on a mean field description, which may be no longer quantitatively correct under such conditions. Due to this, a series of simulations will be needed for each set of selected parameters whenever a detailed, quantitative comparison, may be required. When the parameter $\alpha_3$ is increased (we have taken the value $% \alpha_3=5$), imposing an initial slab of liquid in coexistence with a slab of gas the interface remains stable, and the density fluctuations in the liquid phase are not too large. In Fig.\[fig:templg\] we show the temperature, pressure and mean square displacement of the system during the extension of the simulation. One can see that the temperature does not shift, and corresponds to its nominally assigned value. The pressure exhibits important fluctuations, but if we subtract the normal and tangential components (in the figure we only display the averaged pressure), their difference, which is twice the surface tension, gives a value with a well-defined positive mean. Also the mean square displacement shows that particles have had the time to diffuse the interfacial width, which is roughly proportional to the interaction range, $% r_{c}$, indicating that the droplet is stabilized. Fig.\[fig:denslg\]a shows the density profiles obtained by starting with a step density profile in the liquid-gas coexistence region, where the numerical errors are smaller than the fluctuations, as in the rest of the plots. The shape of the drop is stable and the interfaces fluctuate around their initial location, as could be expected. The density ratio between the two fluid phases, $% \rho_{liq}/\rho_{gas}=4$ makes it reasonable to call the two phases liquid and gas. The density in the gas phase is $10 r_c^{-2}$, which ensures that in both phases the number of interacting particles is sufficiently high. By looking at the density profiles one can also observe that the density fluctuations in the dense phase are small, as expected on the basis of the small compressibility of the fluid. Finally, we have also computed the components of the pressure tensor across the profiles. For an inhomogeneous fluid there is no unambiguous way of computing the local components of the pressure tensor; we follow here the procedure described in ref.[@Irving] and display them in fig.\[fig:denslg\]b. They follow basically the increase in density, exhibiting larger fluctuations in the liquid phase. In the bulk phases, the two components of the pressure tensor have to be equal. This is clearly shown in fig.\[fig:diffpress\], where the differences in the two components are confined to the interfaces, if we compare the location of the differences with the density profiles of fig.\[fig:denslg\]a. Moreover, the increase in fluctuations in the dense phase is clearly displayed. The equilibration of the drop can also be monitored by analyzing the time scale at which the pressure profile becomes symmetric at both interfaces. Together with the pressure differences, we have also plotted in thin line the integral of the pressure difference across the profile. This quantity is the surface tension, and indeed, the values we get when the profile is equilibrated agree with the predictions extracted from the mean pressures, displayed in fig.\[fig:templg\]. We have also computed the excess free energy profile. Its integral gives us an alternative (thermodynamic) route to compute the surface tension. We have verified that the values of the surface tension obtained integrating the excess free energy profile coincides with the value presented above, computed along the virial route. Another appealing feature of these conservative interactions is that their density dependence induces smooth local structure. Indeed, if we analyze the radial distribution functions for a homogeneous phase, we can see that the structure in this case is almost non-existent. When an interface is present it is hard to assess the spurious structure that the model may induce through the density profile . All we can say is that the decay of the density is monotonic from one phase to the other, and avoids therefore spurious structure close to the interface. Such a structure would be spurious on the mesoscopic scale modeled by the DPD fluid. In contrast, the onset of structuring of the liquid-vapor interface on an atomic scale (beyond the Fisher-Widom line) is a real effect[@Evans2]. Binary mixture {#sec:bin} -------------- Finally, we have run simulations for a binary mixture corresponding to the model described in section \[sect:model\].C. As in the previous subsection, we concentrate on the equilibrium properties of the fluid in the coexistence region. We have simulated a 2D fluid, starting with an initial step profile in concentration. In fig.\[fig:tempmix\] we show the evolution of the temperature and pressure, which remain essentially constant through the simulation. We also display the root mean square displacement of the two species. One can clearly see that after a short initial period, when the species start to feel the presence of the interfaces their effective diffusion slows down. The fact that the mean square displacement is much larger than the interfacial width, which remains of the order of the interaction range, $r_c$, ensures that the initial configuration has relaxed to its proper equilibrium shape. We have computed the concentration profiles as a function of time. In fig.\[fig:densmix\] we show the concentration profiles of one of the species at an initial and late stage of the relaxation towards the equilibrium coexistence. As was the case in the van der Waals fluid, the fluctuations are greater in the concentrated phase. Although the concentration of each species goes basically to zero in one of the two coexisting phases, the interface does not broaden and keeps its width within $r_{c}$. Despite this large concentration gradient, the mean density barely changes across the interface. These normalized mean densities are displayed in also in fig.\[fig:densmix\] as thin curves. Although a small dip in the normalized mean density appears at the interfaces, its value is not large compared with the typical bulk density fluctuations (which are due to the compressibility of the fluid). Again, this indicates that the use of concentration dependent conservative forces suppresses the appearance of spurious structure at interfaces, while still being able to drive the phase separation. We can also test the predictions of section \[sec:interbin\] for the interfacial properties on the basis of a binary mixture. To this end, we have integrated numerically eq.(\[eq:gammaexp\]) using the concentration profiles obtained from the simulations, and we have compared the results with the theoretical prediction, eq.(\[eq:sigmabin\]). We display the results in fig.\[fig:tension\], where we have multiplied the theoretical curve by an overall numerical factor, since the numerical prefactors in eq.(\[eq:sigmabin\]) are approximate. One can observe that the overall good agreement is lost at small temperatures, where the interface is very sharp, and close to the critical point, where fluctuations are expected to play a relevant role. Conclusions {#sec:disc} =========== We have presented a new way of implementing conservative forces between DPD particles. Rather than assuming a force that depends on the interparticle separation, we have introduced a conservative interaction that depends on the local excess free energy. In this way, it is possible to fix beforehand, at the mean field level, the desired thermodynamic properties of the system. However, this procedure neglects the effect of particle correlations. Whenever an accurate quantitative comparison is needed, a set of numerical simulations will be required to determine accurately the appropriate phase diagram. We could equally use the free energy to carry out Monte Carlo simulations to analyze the static properties of fluids; this procedure will suffer from similar drawbacks as a result of the ignored particle correlations. When the free energy per particle depends on the averaged local density, it is possible to recover central pairwise additive forces- an important computational feature. The only assumption we have made is that the system is isothermal, although it should be straightforward to generalize it to include energy transport, along the lines developed previously[@Avalos]. These models can be viewed as a dynamical density functional theory (DFT) for smooth conservative forces with local momentum conservation. However, since the DPD particles do not have a local structure, these models can only describe the dynamics at a mesoscopic level, while the usual dynamical DFT can account for the dynamics down to the microscopic scale. In addition to the freedom in the choice of the free energy, this new type of proposed forces leads to weaker structure at short distances. Hence, we can enforce a proper length and time scale separation, avoiding the appearance of microscopic features of the system at distances of order $r_c$. At the mean-field level, and using standard techniques, it is easy to derive expressions for the interfacial properties. We have shown that the absence of internal structure of the DPD particles (implying that all forces act on the same length scale) leads to qualitatively new behavior not present in atomic fluids. From the physical point of view it shows that, for example, the same thermodynamic system can be tuned to favor macroscopic or microscopic phase separation. Although it may seem unrealistic, the competition of attractive and repulsive effective potentials on the same length scales correspond to certain physical situations, and they are probably more common on the mesoscopic than in the microscopic domain. In this respect, the models we have introduced are quite flexible because, for a given bulk thermodynamic behavior (e.g. a given EOS), it is still possible to modify the parameters to control other physical properties. For example, the mean interaction strength can be changed by modifying the way in which the local density is sampled, or for the van der Waals fluid, it is possible to modify the compressibility (and hence the speed of sound). As in any diffuse interface model, the typical interfacial width sets a minimum length scale in the system. For DPD, the natural scale is $r_c$, unless the parameters are chosen carefully. The work of the FOM Institute is part of the research program of “Stichting Fundamenteel Onderzoek der Materie” (FOM) and is supported by the Netherlands Organization for Scientific Research (NWO). We acknowledge Pep Español for sending us at the early stages of this work a preprint on a similar model for treating conservative forces in DPD, P.B. 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Marini Bettolo Marconi, and P. Tarazona, J. Chem. Phys. [**110**]{},8032 (1999). R. D. Groot and P. B. Warren, J. Chem. Phys. [**107**]{}, 4423 (1998). J. Langer, in [*Solids far from Equilibrium*]{}, C. Godrèche ed., (Cambridge Univ. Press, Cambridge, 1991). R. P. Sear, and W. M. Gelbart, J. Chem. Phys. [**110**]{}, 458 (1999). R. P. Sear, S. -W. Chung, G. Markovich, W. M. Gelbart, and J. R. Heath, Phys. Rev. E [**59**]{}, R6255 (1999). J. H. Irving and J. G. Kirkwood, J. Chem. Phys. [**18**]{}, 817 (1950). R. Evans, Mol. Phys. [**88**]{}, 579 (1996). J. Bonet Avalos, and A. D. Mackie, Europh. Lett. [**40**]{}, 141 (1997); P. Español, Europh. Lett. [**40**]{}, 631 (1997). ![Compressibilities of the van der Waals fluid around the critical point, for two different values of the parameter $\protect\alpha_3 $. a) Curves at $\protect T/T_{c}=1.1$; b) Curves at $\protect T/T_{c}=0.8$. []{data-label="fig:xi"}](./Figures/chi_t1.1.eps "fig:"){width="8cm"} ![Compressibilities of the van der Waals fluid around the critical point, for two different values of the parameter $\protect\alpha_3 $. a) Curves at $\protect T/T_{c}=1.1$; b) Curves at $\protect T/T_{c}=0.8$. []{data-label="fig:xi"}](./Figures/xi_t0.8.eps "fig:"){width="8cm"} ![Curves where the pressure and the surface tension vanish for two different values of $\protect\alpha_3$ for a van der Waals fluid. Above the solid curve the pressure is positive, and below the long dashed curves the surface tension is positive. The region contained in between the corresponding pair of curves corresponds to the portion of phase space where the fluid will be mechanically stable, with a positive surface tension. Above the long dashed curves the surface tension is negative. Two different values of $\protect\alpha_3$ are considered: $\protect\alpha_3=0$ and $\protect\alpha_3=5$.[]{data-label="fig:stability"}](./Figures/eos3d_scale.eps){width="12cm"} ![Pressure as a function of the density for a Groot-Warren fluid, using both the previously proposed pairwise force, eq.(\[GW\_force\]), and for the force of the present form, eq.(\[EOSGW\]). In the second case we compare the behavior for a linear and a quadratic weight function. $a=25$, $% \protect\alpha=0.101$. $L=6 r_c$, $k_BT=1 $, $\protect\gamma=1$ (See head of sec.\[sect:comp\] for units).[]{data-label="fig:GW"}](./Figures/eos_3d.eps){width="12cm"} ![Radial distribution for a Groot-Warren fluid, using both the previously proposed pairwise force, eq.(\[GW\_force\]), and for the force of the present form, eq.(\[EOSGW\]). In the second case, we compare the behavior for a linear and a quadratic weight function. Same parameters as in fig.\[fig:GW\]. The mean density is $\protect\rho_m=3$. []{data-label="fig:gr_GW_n3"}](./Figures/gr3d_GW_n3.eps){width="12cm"} ![Radial distribution function for a Groot-Warren fluid, using both the previously proposed pairwise force, eq.(\[GW\_force\]), and for the force of the present form, eq.(\[EOSGW\]). In the second case we compare the behavior for a linear and a quadratic weight function. Same parameters than in fig.\[fig:GW\]. The mean densities are: a) $\protect\rho_m=8$ and b) $\protect\rho_m=14$.[]{data-label="fig:gr_GW_n8"}](./Figures/gr3d_GW_n8.eps "fig:"){width="8cm"} ![Radial distribution function for a Groot-Warren fluid, using both the previously proposed pairwise force, eq.(\[GW\_force\]), and for the force of the present form, eq.(\[EOSGW\]). In the second case we compare the behavior for a linear and a quadratic weight function. Same parameters than in fig.\[fig:GW\]. The mean densities are: a) $\protect\rho_m=8$ and b) $\protect\rho_m=14$.[]{data-label="fig:gr_GW_n8"}](./Figures/gr3d_GW_n14.eps "fig:"){width="8cm"} ![ Equation of state for a 2-D van der Waals fluid. The different sets of data points correspond to different temperatures. $b = 0.016$, $% a=1.9 b$, $\protect\alpha_3 = 5 $, $L=7 r_c$, $\protect\gamma = 1$ (See head of sec.\[sect:comp\] for units).[]{data-label="fig:a3"}](./Figures/eosvdw2d_line.eps){width="12cm"} ![ Thermodynamic values of a DPD fluid with a van der Waals EOS, when a liquid is coexisting with the gas phase, in two dimensions. The initial condition corresponds to a slab of fluid in the $\protect y$ direction in coexistence with a slab of gas. $\tilde{\protect\gamma}$ is the interfacial tension, extracted from the mean pressures, $\tilde{\protect\gamma}= (L_y/2) (P_{yy}-P_{xx})$. Also displayed the mean square displacement in units of the interaction range $r_c$. $L_y=20$, $L_x=3$, $k_BT=0.75$, $a=1.9*b$, $% b=0.0156$, $\protect\alpha_3=5$. The unit of time is the time needed for a DPD particle to diffuse $\protect r_c$ initially. (See head of sec.\[sect:comp\] for units). []{data-label="fig:templg"}](./Figures/temp_2d_long.eps){width="12cm"} ![ Equilibrium a) density and b) pressure profiles for a 2-D van der Waals fluid. The initial profile is a step profile. Same parameters as in fig.\[fig:templg\] (See head of sec.\[sect:comp\] for units).[]{data-label="fig:denslg"}](./Figures/dens2d_prof.eps "fig:"){width="8cm"} ![ Equilibrium a) density and b) pressure profiles for a 2-D van der Waals fluid. The initial profile is a step profile. Same parameters as in fig.\[fig:templg\] (See head of sec.\[sect:comp\] for units).[]{data-label="fig:denslg"}](./Figures/pressure_prof_2d.eps "fig:"){width="8cm"} ![ Profiles of the difference between the normal and tangential components of the pressure tensor along the system, for the pressure profiles of fig.\[fig:denslg\]b. Same parameters as in fig.\[fig:templg\] (See head of sec.\[sect:comp\] for units).[]{data-label="fig:diffpress"}](./Figures/pressure_diff_2d.eps){width="12cm"} ![ Temperature, pressure and mean square displacements of the two species as a function of time, for a binary mixture below its critical temperature, $T/T_c=0.5$, and with $\protect\lambda =1$, $\protect\lambda% _A=0.2$ (See head of sec.\[sect:comp\] for units). []{data-label="fig:tempmix"}](./Figures/temp2d_mix.eps){width="12cm"} ![ Profiles of the relative amount of one of the species across the system, at two different times. These curves have been multiplied by $2$ to avoid confusion with the thin lines. The latter correspond to the normalized mean density at the same time (See head of sec.\[sect:comp\] for units).[]{data-label="fig:densmix"}](./Figures/dens_profiles1.eps){width="12cm"} ![Surface tension for a binary mixture at density $\rho=0.5$ with a critical temperature $T_c=8$ and quicomposed, as a function of the temperature. The squares correspond to the expression derived from the mean field free energy in the small gradient limit. []{data-label="fig:tension"}](./Figures/tension.eps){width="12cm"}
--- author: - | Minghao Hu, Yuxing Peng, Zhen Huang, Dongsheng Li\ National University of Defense Technology, Changsha, China\ [{huminghao09,pengyuxing,huangzhen,dsli}@nudt.edu.cn]{} bibliography: - 'sections/reference.bib' title: | Retrieve, Read, Rerank: Towards End-to-End\ Multi-Document Reading Comprehension ---
--- abstract: 'A primary goal for cosmology and particle physics over the coming decade will be to unravel the nature of the dark energy that drives the accelerated expansion of the Universe. In particular, determination of the equation-of-state of dark energy, $w\equiv p/\rho$, and its time variation, $dw/dz$, will be critical for developing theoretical understanding of the new physics behind this phenomenon. Type Ia supernovae (SNe) and cosmic microwave background (CMB) anisotropy are each sensitive to the dark energy equation-of-state. SNe alone can determine $w(z)$ with some precision, while CMB anisotropy alone cannot because of a strong degeneracy between the matter density $\Omega_M$ and $w$. However, we show that the Planck CMB mission can significantly improve the power of a deep SNe survey to probe $w$ and especially $dw/dz$. Because CMB constraints are nearly orthogonal to SNe constraints in the $\Omega_M$–$w$ plane, for constraining $w(z)$ Planck is more useful than precise determination of $\Omega_M$. We discuss how the CMB/SNe complementarity impacts strategies for the redshift distribution of a supernova survey to determine $w(z)$ and conclude that a well-designed sample should include a substantial number of supernovae out to redshifts $z \sim 2$.' author: - 'Joshua A. Frieman' - Dragan Huterer - 'Eric V. Linder' - 'Michael S. Turner' title: 'Probing Dark Energy with Supernovae: Exploiting Complementarity with the Cosmic Microwave Background' --- Introduction {#sec:intro} ============ Recent observations of Type Ia supernovae (SNe) have provided direct evidence that the Universe is accelerating [@perl99; @riess98], indicating the existence of a nearly uniform dark-energy component with negative effective pressure, $w\equiv p/\rho < -1/3$. Further evidence for dark energy comes from recent cosmic microwave background (CMB) anisotropy measurements pointing to a spatially flat, critical density Universe, with $\Omega_{0} = 1$ [@CMB], combined with a number of indications that the matter density $\Omega_M \simeq 0.3$ [@omega_m]; the ‘missing energy’ must also have sufficiently negative pressure in order to allow time for large-scale structure to form [@turner_white]. Together, these two lines of evidence indicate that dark energy composes 70% of the energy density of the Universe and has equation-of-state parameter $w < - (0.5 - 0.6)$ [@w_constraints]. Determining the nature of dark energy, in particular its equation-of-state, is a critical challenge for physics and cosmology. At present, particle physics theory provides little to no guidance about the nature of dark energy. A cosmological constant—the energy associated with the vacuum—is the simplest but not the only possibility; in this case, $w=-1$ and is time independent, and the dark energy density is spatially constant. Unfortunately, theory has yet to provide a consistent description of the vacuum: the energy density of the vacuum, at most $10^{-10}\,{\rm eV}^4$, is at least 57 orders of magnitude smaller than what one expects from particle physics—the cosmological constant problem [@weinberg]. In recent years, a number of other dark energy models have been explored, from slowly rolling, ultra-light scalar fields to frustrated topological defects [@demodels]. These models predict that $w\not= -1$, that $w$ may evolve in time, and that there may be small spatial variations in the dark energy density (of less than a part in $10^5$ on scales $\sim H_0^{-1}$ [@perturbations]). In all models proposed thus far dark energy can be characterized by its equation-of-state $w$. Measuring the present value of $w$ and its time variation will provide crucial clues to the underlying physics of dark energy. As far as we know, dark energy can only be probed directly by cosmological measurements, although it is possible that laboratory experiments could detect other physical effects associated with dark energy, e.g., a new long-range force arising from an ultra-light scalar field [@longrange]. Dark energy affects the expansion rate of the Universe and thereby influences cosmological observables such as the distance vs. redshift, the linear growth of density perturbations, and the cosmological volume element (see, e.g., [@HT]). Standard candles such as Type Ia supernovae offer a direct means of mapping out distance vs. redshift (see, e.g., [@WA2002]), while the CMB anisotropy can be used to accurately determine the distance to one redshift, the last scattering epoch ($z_{LS}\simeq 1100$). Because they measure distances at such different redshifts, the SNe and CMB measurements have complementary degeneracies in the $\Omega_M$–$\Omega_{\Lambda}$ and $\Omega_M$–$w$ planes [@HT; @WA2002; @CMB+SN]. More recently, Spergel & Starkman [@SS] have suggested that this complementarity argues for using supernovae at relatively low redshift, $z\sim 0.4$, to most efficiently probe dark energy. In so doing, they used a highly simplified model which did not consider a spread of SNe in redshift, systematic error, possible evolution of $w$, or the finite precision with which planned CMB missions can actually constrain $\Omega_M$ and $w$. By including these “real-world” effects, this paper clarifies the complementarity of the CMB and SNe and explores strategies for best utilizing it in SNe surveys to probe the properties of dark energy. We show that dark energy-motivated supernova surveys should target SNe over a broad range of redshifts out to $z \sim 2$, and that CMB/SNe complementarity in fact strengthens the case for deep SNe surveys. How Supernovae and the CMB Probe Dark Energy {#sec:2} ============================================ Supernovae and the CMB anisotropy probe dark energy in different ways and at different epochs. However, both do so through the effect of dark energy on the comoving distance vs. redshift relation, $r(z)$. For a spatially flat Universe and constant $w$: $$\begin{aligned} H_0r(z) & = & \int_0^z {dz\over H(z)/H_0} \nonumber\\[-0.4cm] &&\\[0.1cm] (H/H_0)^2 & = & \Omega_M(1+z)^3 + (1-\Omega_M)(1+z)^{3(1+w)}\nonumber\end{aligned}$$ where $\Omega_M$ is the present fraction of the energy density contributed by non-relativistic matter. This relation is easily generalized to non-constant $w$ and a curved Universe [@HT]; for notational simplicity we write this and succeeding formulae in terms of constant $w$, though we generalize them to the evolving case in our analysis. It is because $H_0r(z)$ depends upon only two quantities, $\Omega_M$ and $w$, that prior information about $\Omega_M$ (or two independent combinations of $\Omega_M$ and $w$) has such potential to improve the efficacy of a cosmological probe of dark energy based upon $H_0 r(z)$. CMB experiments can determine the positions and heights of the acoustic peaks in the temperature anisotropy angular power spectrum to high accuracy. The positions of the acoustic peaks in angular multipole space depend upon the physical baryon and matter densities $\Omega_Bh^2$ and $\Omega_Mh^2$, on $\Omega_M$, $w$, and to a lesser extent other cosmological parameters (e.g., [@HT; @husug]). Anisotropy measurements from the Planck [@Planck] mission, planned for launch later in the decade, should determine the positions of the peaks to better than 0.1%; the heights of the peaks will determine $\Omega_Mh^2$ and $\Omega_Bh^2$ (and other cosmological parameters) to roughly percent precision [@EHT]. Together, these measurements should constrain a combination of $\Omega_M$ and $w$ alone (e.g., [@HT; @SS]) to about 10% precision. In particular, in the vicinity of the fiducial values $w_0 = -1$ and $\Omega_{M0}=0.3$, the combination $$\begin{aligned} {\cal D} &\equiv& \Omega_M -0.94\,\Omega_{M0}(w-w_0)\nonumber\\[-0.25cm] &&\\[-0.15cm] &\approx& \Omega_M - 0.28(1+w) = 0.3\nonumber\end{aligned}$$ will be determined to about $\sigma_{\cal D} \simeq \pm 0.03 (\Omega_{M0}/0.3)$ (this result follows directly from Eq. 18 of Ref. [@HT] by setting $\Delta l/l= \Delta \Omega_0/\Omega_0 =0$). The resulting 68% CL error ellipse in the $\Omega_M$–$w$ plane predicted for Planck is shown in Fig. \[fig:Fig1\]. Polarization information could in principle improve the precision with which ${\cal D}$ is determined by about 50% [@HETW], absent problems with foregrounds or the polarization measurements themselves. The MAP CMB mission [@MAP] currently underway should determine $\cal D$ to a precision that is about 10 times worse than Planck, assuming temperature anisotropy information alone. This constraint is too weak to usefully complement the SNe measurements. However, if MAP polarization measurements are successful, this constraint could be improved by about a factor of two [@HETW]; we discuss the potential impact of MAP further in Sec. \[sec:4-1-3\]. As an aside, we note that the physical baryon and matter densities do not directly impact the determination of the properties of dark energy. Rather, together with other cosmological measurements, they can be used to determine $\Omega_M$. In the following Sections we illustrate how independent knowledge of $\Omega_M$ can improve the determination of $w$. Measurements of the energy fluxes and redshifts of Type Ia supernovae provide an estimate of the luminosity distance as a function of redshift, $d_L \equiv (1+z)r(z)$. As an example of a supernova survey, the Supernova/Acceleration Probe (SNAP) [@SNAP] is a proposed space-based telescope to observe $\sim 3000$ SNe Ia out to redshift $z \sim 1.7$, specifically designed to probe dark energy. To illustrate the essential principles for such a survey, though not all the details, we make the simplifying assumption that SNe Ia are nearly standard candles (after correction for the observed correlation between light-curve decline rate and peak luminosity [@phillips]). With this assumption, the mean peak energy flux from a supernova at redshift $z$ is: $$\begin{aligned} F(z) & = & {{\cal C} 10^{-0.4M} \over 4\pi d_L^2} = {(10^{10}{\cal C}/4\pi)10^{-0.4{\cal M}} \over H_0^2d_L^2} = \nonumber\\[-0.2cm] &&\\ && \hspace{-1.2cm} {(10^{10}{\cal C}/4\pi)10^{-0.4{\cal M}} \over (1+z)^2 \left[ {\displaystyle}{\int_0^z {dz\over \sqrt{\Omega_M(1+z)^3 +(1-\Omega_M)(1+z)^{3(1+w)}}}}\right]^2}\nonumber\end{aligned}$$ where ${\cal C} = 3.02\times 10^{35}\,{\rm erg\,sec^{-1}}$ is an unimportant constant, $M$ is the mean absolute peak magnitude of a Type Ia supernova, and ${\cal M} = M - 5 \log (H_0) + 25$, with distances measured in Mpc. It is important to note several things from Eq. (3). First the energy flux at fixed $H_0 d_L$ depends only upon the combination $\cal M$ and not upon $M$ and $H_0$ separately. Thus, the cosmological parameters $\Omega_M$ and $w$ can be determined by measuring ratios of fluxes at different redshifts, which are independent of $\cal M$, and so $\cal M$ is sometimes referred to as a nuisance parameter and can be easily marginalized over. Second, since $H_0d_L \rightarrow z$ for $z\rightarrow 0$, low-redshift supernovae can be used to determine $\cal M$, $$z^2F(z) \rightarrow {(10^{10}{\cal C}/4\pi)}10^{-0.4{\cal M}}\ \ {\rm as\ }z\rightarrow 0 .$$ For example, a sample of 300 low-redshift supernovae (e.g., as will be targeted by the Nearby SN Factory [@SN_Factory]) could be used to pin down $\cal M$ to a precision of $\pm (0.01-0.02)$. Finally, an absolute calibration of nearby SNe Ia luminosities by another reliable distance indicator (e.g., using Cepheid variables to determine distances to galaxies that host SNe Ia [@freedman]) can determine $M$; together, $M$ and $\cal M$ then fix the Hubble constant, but we emphasize that this is not needed to probe dark energy. For a survey of SNe Ia, the likelihood function for the three parameters the supernova energy flux depends upon is given by $${\cal L}_{\rm SNe} (\Omega_M ,w,{\cal M}) \propto \Pi_i \exp \left( -{\displaystyle}{[F_i-F(z_i)]^2\over 2\sigma_i^2} \right)$$ where $z_i$ are the redshifts of the supernovae, $F_i$ are their measured fluxes, and $\sigma_i$ are their measurement uncertainties (which also includes any random intrinsic spread in peak SNe Ia luminosities). Unlike the CMB, which probes the angular diameter distance at a single, fixed redshift $z_{LS}$, the efficacy of SNe for determining $w$ depends upon the redshift distribution of the supernovae. As a first example, Fig. \[fig:Fig1\] shows how well 3000 supernovae at a single redshift could constrain $\Omega_M$ and $w$, assuming a random flux error of 0.15 mag per supernova. Because the sensitivity of the comoving distance $r(z)$ to the dark energy equation-of-state (e.g., as measured by $dr/dw$) increases with redshift, the ellipse shrinks for SNe at higher redshift [@HT]. While Fig. \[fig:Fig1\] displays important trends, we note that a realistic survey would not target SNe all at one redshift. Such a delta-function redshift distribution is very much less than optimal for constraining $w$ (as we show in Sec. \[sec:4-1\]) and would be very inefficient, since large numbers of discovered SNe would have to be discarded. More importantly, a broad distribution of SNe redshifts is crucial for addressing systematic/evolutionary trends in the SNe population, which must be under control if SNe (or anything else) are to be valid probes of dark energy. In addition, there is much more to studying dark energy than determining the average value of $w$ in the most efficient manner. Constraining the time variation of the equation-of-state is critical for understanding the nature of dark energy. The CMB has no sensitivity to evolution of $w$; SNe can probe time variation of $w$, and a broad distribution of SNe redshifts (out to $z \sim 2$) is required to achieve it, as we show below. In Sec. \[sec:4\] we discuss strategies for the distribution of SNe redshifts and results for some plausible examples. Finally, determining cosmological parameters (here $\Omega_M$ and $w$) by two very different techniques has the virtue of providing consistency checks on the framework of dark energy as well as the Friedmann-Robertson-Walker cosmology [@Li02; @Max]. CMB/SNe Complementarity {#sec:3} ======================= Some trends in the CMB/SNe complementarity are illustrated in Fig. \[fig:Fig1\]. For the fiducial model ($w_0 = -1$, $\Omega_{M0} = 0.3$), the Planck error ellipse in the $\Omega_M$–$w$ plane is approximately oriented along the line $\Omega_M \simeq 0.3 + 0.28(1+w)$, as indicated by Eq. (2). By contrast, the error ellipse for 3000 SNe at fixed redshift has negative slope in this plane; with increasing redshift it rotates toward $\Omega_M = {\rm const}$, and its width narrows. The reason for the rotation is simple: at high redshift, matter becomes more dynamically important than dark energy, and the SNe are therefore probing the matter density. While the width of the SNe ellipse shrinks with increasing redshift, it becomes less complementary with the CMB ellipse. Fig. \[fig:Fig1\] also makes it clear why CMB anisotropy is more complementary than the matter density information: the matter density prior, which corresponds to a vertical stripe, is less orthogonal to the SNe ellipse. To be quantitative, it is useful to write down the joint likelihood function: $${\cal L}_{\rm joint} = {\cal L}_{\rm SNe} \times {\cal L}_{\rm CMB} \times {\cal L}_{\rm other}.$$ The CMB likelihood function can be approximated as $$\begin{aligned} {\cal L}_{\rm CMB} &=& {\cal L}_{\rm CMB,0} (\Omega_M,w) \times \exp \left[ -{\displaystyle}{(\rho_B - \rho_{B0})^2\over 2\sigma_{\rho_B}^2} \right]\nonumber\\[-0.3cm] &&\\[-0.2cm] &\times& \exp \left[ -{\displaystyle}{(\rho_M - \rho_{M0})^2\over 2\sigma_{\rho_M}^2} \right]\nonumber\end{aligned}$$ where $${\cal L}_{\rm CMB,0} \ \propto\ \exp \left [{\displaystyle}{ ({\cal D}-{\cal D}_0)^2\over 2\sigma^2_{\cal D} }\right ] ,$$ ${\cal D} = \Omega_M - 0.28(1+w)$, ${\cal D}_0 \simeq 0.3$ is the fiducial value of $\cal D$, $\sigma_{\cal D} \simeq 0.1 {\cal D}_0$ is the projected accuracy for Planck[^1], $\rho_{B} = 1.88\,\Omega_Bh^2 \times 10^{-29}\,{\rm g\,cm^{-3}}$, $\rho_M =1.88 \,\Omega_Mh^2 \times 10^{-29}\,{\rm g\,cm^{-3}}$, $\rho_{B0}$ is the fiducial value of the baryon density, and $\rho_{M0}$ is the fiducial value of the matter density. The accuracy of the CMB constraint in the $\Omega_M$-$w$ plane depends on three factors: the experiment (and whether polarization information is included), the parameter set considered, and the presence and nature of foregrounds. For comparison with SNe without systematics, we consider the CMB without foregrounds and assuming a moderate set of eight cosmological parameters and adopt the CMB constraint from ref. [@Hu_synergy]. Under these assumptions, $\Omega_M h^2$ and $\Omega_B h^2$ are determined to 1.6% and 0.8% respectively for Planck with temperature information only. Comparison with SNe with systematics requires the inclusion of foregrounds, and is obviously model-dependent. Tegmark et al. [@CMB_sys] have shown that the accuracy in all cosmological parameters of our interest degrades only slightly even in the presence of a fairly generous foreground model (their MID model). Nevertheless, the presence of foregrounds is expected to degrade the optimistic uncertainties computed without the systematics (e.g. [@Knox]). To account for that, we follow ref. [@HETW], where a more generous set of ten parameters is considered, but without such “luxury parameters” such as running of the spectral tilt and neutrino mass which were assumed in ref. [@CMB_sys]. As a result, our model of the CMB with systematics constrains the quantity $\cal{D}$ 30-40% worse than the CMB without systematics. As noted in Sec. \[sec:2\], the CMB determination of the baryon and matter densities is not directly useful for constraining dark energy: when the joint likelihood function is marginalized over the matter and baryon densities to obtain the one-dimensional probability distribution for $w$, the integrations over $\Omega_Bh^2$ and $\Omega_Mh^2$ are trivial. On the other hand, if we can obtain information about $M$ (from non-SNe distance measurements) and $\cal M$ (from low-redshift SNe) and thereby (or otherwise) constrain $H_0$, then the CMB determination of $\Omega_Mh^2$ constrains $\Omega_M$ as well, which would directly impact the joint determination of $w$. Of course, any other external determination of $\Omega_M$ would have the same effect; later, we will discuss how various $\Omega_M$ priors affect the determination of $w$. Assuming no information about $M$ (or equivalently $H_0$), the joint likelihood function becomes $${\cal L}_{\rm joint}(\Omega_M,w) = {\cal L}_{\rm CMB,0} \times {\cal L}_{\rm SNe}$$ From this function, we obtain one-dimensional probability distributions for $w$ by marginalizing over $\Omega_M$. As a first case, we again assume a baseline sample of 3000 SNe all at one redshift, with a random flux error of 0.15 mag per supernova. In Fig. \[fig:Fig2\], we show the effect of including CMB or $\Omega_M$ information in the determination of the dark energy equation-of-state, assuming $w = {\rm const}$. If the CMB measurement of $\cal D$ is assumed to be “perfect” ($\sigma_{\cal D} = 0$) as was done in Ref. [@SS], the predicted $\sigma_w$ drops significantly with increasing redshift and continues to do so out to $z\approx1.5$. The effect of a “perfect” matter density prior ($\sigma_{\Omega_M}=0$) is similar. This qualitative behavior can be understood by referring to Fig. \[fig:Fig1\] and considering the intersection of the CMB line (now an infinitely thin ellipse) with the SNe ellipses or of a vertical line (fixed $\Omega_M$) with the SNe ellipses. The decreasing width of the SNe ellipses wins out over the decreasing complementarity at higher redshift. The qualitative behavior changes, however, when finite precision for the CMB and matter density measurements is taken into account; as examples, for the CMB we use the projected Planck accuracy discussed above, and for the matter density we assume $\sigma_{\Omega_M} = 0.03$. Not only is the uncertainty $\sigma_w$ larger in these cases, but it now reaches a minimum at $z \sim 0.2$ and rises slightly at higher redshift. For finite widths of the matter density or CMB priors, the decreasing complementarity now wins out over the decreasing width of the SNe ellipse with increasing redshift. Thus far, we have not allowed for systematic error in measuring the supernova flux at a given redshift. This means that by measuring a large number of supernovae at a given redshift, the flux and thereby $r(z)$ can be determined to arbitrarily high accuracy. In reality, the presence of residual systematic uncertainty is likely to impose a floor to improvement. As a simple model for irreducible systematic error in the SNe measurements, we assume the flux error in a specified redshift interval is given by $\sqrt{ (0.02)^2 + (0.15)^2/N_i}$ mag, where 0.15 mag is the assumed statistical error per SN, 0.02 mag is the irreducible error,[^2] and $N_i$ is the number of supernovae observed in that redshift interval. This model penalizes observing large numbers of SNe at the same redshift since the irreducible error adds to the Poisson error: one reaches diminishing returns for $N_i \sim 100$, at which point the error is only $\sim 20$% larger than its asymptotic value. While this model is certainly simplistic, it captures in a straightforward way the essential point: increasing the number of SNe cannot decrease the measured error in $H_0r(z_i)$ to arbitrarily small values [@basa]. Figure \[fig:Fig2\] illustrates the effect of systematic error. At redshifts less than about $z\sim 0.5$, systematic error increases $\sigma_w$ significantly: without the irreducible flux error, the estimate for $\sigma_w$ was optimistically small because the flux error was allowed to decrease to a tiny value ($\sim 0.003$ mag). With systematic flux error included, the predicted error in $w$ from a combined Planck CMB measurement and a hypothetical sample of 3000 SNe (all at redshift $z$) flattens at $z\sim 1$, with an asymptotic amplitude $\sigma_w \simeq 0.05$. As noted in Sec. \[sec:2\], realistic survey would not target supernovae all at a single redshift, as assumed up to now. Moreover, since the orientation of the SN error ellipse in the $\Omega_M$–$w$ plane rotates with $z$ (see Fig. \[fig:Fig1\]), a spread of SNe redshifts helps break the degeneracy between $\Omega_M$ and $w$. In the next Section, we consider more realistic strategies for the supernova redshift distribution to optimally probe dark energy. Strategies for CMB/SNe Complementarity {#sec:4} ====================================== Optimal {#sec:4-1} ------- The issue of optimal strategies for determining dark energy properties using SNe in a realistic experiment has been addressed in Refs. [@HT; @HL]. Here, we extend these results to incorporate CMB anisotropy and other measurements. ### No systematic error {#sec:4-1-1} The optimization problem can be stated as follows: we have three cosmological parameters ($\cal M$, $\Omega_M$, and $w$; later we will add a fourth, $dw/dz$); we have “prior information” (from the CMB anisotropy and/or an independent determination of $\Omega_M$); and we wish to determine the redshift distribution of the SNe which minimizes the error on $w$, with the constraint that they are confined to the interval $[0,z_{\rm max}]$. For now, we assume that the total number of observed SNe is held fixed, and we do not include systematic error in the SNe measurements. Later we will relax both of these assumptions. Huterer & Turner [@HT] showed that for the $N$-parameter problem with no priors, the optimal redshift distribution comprises $N$ delta functions, with one at $z=0$, one at $z_{\rm max}$, and the others in between. The amplitudes of the delta functions and their positions relative to $z_{\rm max}$ vary little with the value of $z_{\rm max}$.[^3] Adding a “strong” prior on one, or a combination, of the three parameters reduces the number of delta functions by one, adding two “strong” priors reduces the number of delta functions by two, and so on. A “strong” prior is one that constrains one, or a combination, of the three parameters better than the SNe measurements alone would. In actuality, this is a continuous process, with the amplitude of one of the delta functions going to zero as the quality of the prior improves. Further, for smaller $z_{\rm max}$ it is easier to have a “strong” prior, since the SNe constrain the parameters less well. For illustration we consider a survey of about 3000 SNe with survey depth $z_{\rm max} = 1.7$. These choices are motivated by the proposed SNAP survey [@SNAP] and thus provide a useful benchmark (SNAP should obtain 3000 Type Ia SNe in about two years of observations). Figure \[fig:Fig3\] shows the optimal SNe redshift distribution with no CMB prior, a perfect CMB prior ($\sigma_{\cal D} = 0$), and the Planck prior (see Sec. \[sec:3\]). For comparison, we also show one of the redshift distributions currently proposed for SNAP (2812 SNe in the redshift interval $0.1-1.7$) combined with that for the Nearby SN Factory (300 SNe at $z< 0.1$). We see that a “perfect” CMB prior is a “strong” prior: the optimal SNe distribution in this case becomes two delta functions, one at $z=0$ and one at $z=z_{\rm max}$. The Planck prior is not strong: in this case, three delta functions remain, at $z=0, 0.5$, and 1.7. Figure \[fig:Fig4\] shows the optimal SNe redshift distribution using $\Omega_M$ instead of CMB priors, with $\sigma_{\Omega_M} = 0.005, 0.01,$ and 0.03. The $\Omega_M$ prior is only “strong” for $\sigma_{\Omega_M} \leq 0.005$. In Figs. \[fig:Fig3\] and \[fig:Fig4\], the $z \sim 0$ peaks in the optimal distributions serve mainly to determine $\cal M$. Indeed, the Nearby SN Factory redshift distribution is strongly peaked at $z\approx 0.05$, in part for this reason.[^4] We could have simply imposed a prior on $\cal M$ instead of including this portion of the redshift distribution. Finally, it is important to consider how much improvement the optimal redshift distribution actually provides compared to a uniform distribution or the SNAP+SN Factory distribution: for the cases shown in Figs. \[fig:Fig3\] and \[fig:Fig4\], $\sigma_w$ is typically 20% to 30% smaller for the optimal distribution. ### Inclusion of systematic error and evolution of $w$ {#sec:4-1-2} Now we consider the effect of systematic flux error on the optimal SNe redshift distribution. As before, we use the simple model of an irreducible flux error of 0.02 mag in each redshift interval of width $\Delta z = 0.1$. We should expect that this will broaden the optimal distribution, since it is more expedient to spread the remaining SNe to other redshift bins once the error in a given bin becomes comparable to the irreducible error. Figs. \[fig:Fig5\] and \[fig:Fig6\] show the optimal SNe redshift distributions, with and without CMB and $\Omega_M$ priors, in the presence of systematic errors. Figs. 5a and 6a show results for the $w={\rm const}$ case as before, while Figs. \[fig:Fig5\]b and \[fig:Fig6\]b allow for evolution of the equation-of-state, $w(z) = w_0 + w_1z$, with $w_1 = dw/dz|_{z=0}$. Comparison of Figs. \[fig:Fig5\]a and \[fig:Fig6\]a with Figs. \[fig:Fig3\] and \[fig:Fig4\] shows that inclusion of systematic error indeed changes the optimal distribution significantly, broadening it to become more uniform. For the case of constant $w$ (Figs. \[fig:Fig5\]a and \[fig:Fig6\]a), the gain in performance for the optimal SNe distribution vs. a uniform or SNAP+SN Factory distribution is reduced to only $3-5$% when systematic errors are included. We find that a number of qualitatively different redshift distributions yield essentially the same value of $\sigma_w$. In particular, in this case $\sigma_w$ is relatively insensitive to $z_{\rm max}$: there exist distributions with no SNe at $z>1$ which yield $\sigma_w$ only 3% larger than the optimal value (see also Fig. \[fig:Fig7\] below). The situation is markedly different if we allow for time variation in the equation-of-state. In Figs. \[fig:Fig5\]b and \[fig:Fig6\]b, we show the distributions that minimize $\sigma_{w_1}$ (the results are almost identical if $\sigma_{w_0}$ is minimized instead). In the presence of CMB or matter density priors, the optimal distributions now include larger numbers of SNe at high redshift. Furthermore, SNe in the high-redshift range $1 < z < 1.7$ are crucial for precision constraints to $w_1$, even in the presence of a strong prior. For example, as $z_{\rm max}$ increases from 1 to 1.7, $\sigma_{w_1}$ decreases by more than a factor of two, cf. Fig. \[fig:Fig9\]. ### Gains from complementarity {#sec:4-1-3} The preceding analysis shows that, for fixed $z_{\rm max}$, the error on $w$ is only weakly dependent on the SNe redshift distribution: in the presence of systematic error, distributions which are broadly spread over the range $0<z<z_{\rm max}$ differ only slightly in their performance. Therefore the chief determinant of the error is $z_{\rm max}$ itself, and we now address how the efficacy of SNe with complementary information depends on this maximum redshift. In Fig. \[fig:Fig7\], we show the effect of various CMB and matter density priors on the predicted value of $\sigma_w$ vs. $z_{\rm max}$, assuming $w={\rm const}$, with systematic error modeled as before and assuming a scaled version of the SNAP + SN Factory distribution of redshifts.[^5] (As noted above, the optimal redshift distribution with the same value of $z_{\rm max}$ would yield only slightly smaller $\sigma_w$.) Figure \[fig:Fig7\] also includes the error on $w$ for the case of no CMB prior or knowledge of the matter density (black curve). The primary effect of incorporating additional information, from either the CMB or the matter density, is to dramatically decrease $\sigma_w$ at redshifts less than one and thereby lessen the dependence of $\sigma_w$ on $z_{\rm max}$. With SNe only, $\sigma_w$ decreases from 0.8 to 0.15 as $z_{\rm max}$ is increased from 0.5 to 1.5. With the Planck or matter density prior, $\sigma_w$ decreases less rapidly and levels off at $z\sim 1$. Note that the Planck prior is more effective than either matter density prior shown. Even combining a $\sigma_{\Omega_M} = 0.01$ prior with Planck provides little improvement over the Planck prior alone. Although an independent determination of $\Omega_M$ to $\pm 0.03$ can substantially improve the precision with which $w$ can be determined if $z_{\rm max} \leq 1.5$ [@HT], the Planck CMB prior by itself does better by a factor of two. As mentioned at the end of Sec. \[sec:2\], time variation in the equation-of-state is generically expected and is a potentially important discriminator between dark energy models. Allowing for evolution, with $w(z) = w_0 + w_1 z$,[^6] there are now four parameters to determine: ${\cal M}, \Omega_M, w_0,$, and $w_1$. As Figs. \[fig:Fig8\] and \[fig:Fig9\] illustrate, without an additional prior, SNe have little leverage on $w_0$ and $w_1$ [@HT; @Maoretal]. An independent determination of the matter density to $\pm 0.03$ – not much more stringent than already achieved: 0.04 [@omega_m] – would allow $w_0$ and $w_1$ to be determined to precision of about $\pm 0.1$ and $\pm 0.35$ for $z_{\rm max} \sim 1.7$ [@HT]. The Planck prior is just as good as a $\sigma_{\Omega_M} = 0.03$ matter density prior for $w_0$ (if $z_{\rm max} \geq 1$) and better for $w_1$. Note that the improvement with survey depth in $\sigma_{w_1}$ (and to a lesser extent $\sigma_{w_0}$) continues out to $z_{\rm max}=2$ in all cases. That is, even in the presence of complementary information from the CMB or the matter density, a SNe survey aimed at detecting and constraining the evolution of the dark energy equation-of-state should extend out to high redshift, $z_{\rm max} \sim 1.5 - 2$. Thus far, our discussion of CMB anisotropy has been confined to the Planck mission. It is also worth considering what can be learned from the ongoing MAP experiment. As noted in Sec. \[sec:2\], with temperature anisotropy measurements alone, MAP can determine $\cal D$ about 10 times less accurately than Planck, $\sigma_{\cal D} \simeq 0.3$. In this case, MAP provides a far less useful prior than the matter density prior $\sigma_{\Omega_M} =0.03$ (about a factor of two worse for $\sigma_w$), cf. Fig. \[fig:Fig7\]. Even if MAP can achieve its full polarization capability (a factor of two improvement in $\sigma_{\cal D}$ [@HETW]), a MAP prior is still not as good as the matter density prior $\sigma_{\Omega_M} =0.03$. Moreover, mapping the polarization anisotropy on large angular scales — where it helps determine $w$ indirectly, by imposing an upper limit to the ionization optical depth $\tau$ — will be difficult in the presence of polarized synchrotron radiation from the Galaxy. Finally, we mention that while polarization measurements also have the potential to improve the Planck determination of $\cal D$ (by about 50%), this only improves the joint SNe/CMB determination of $w$ by about 15%. The reason is simple: it is the width of the SNe error ellipse that controls $\sigma_w$. Resource limited {#sec:4-2} ---------------- In the analysis so far, we have assumed a fixed total number of observed supernovae, $N_{SN} = 3112$. However, the resources required to discover and follow up a supernova depend in general upon its redshift. Thus, an important but more complicated problem involves the optimization of the determination of dark energy parameters with fixed total resources. Actually determining what these fixed resources are (e.g., discovery time, follow-up time, spectroscopy time) and how much each supernova ‘costs’ is beyond the scope of this paper (relevant ongoing studies can be found at [@SNAP]). We note that these costs will depend in detail upon a variety of technical factors: telescope aperture, pixel size and number, CCD quantum efficiency, sky brightness, atmospheric seeing (for ground-based observations), required signal to noise, etc. As a highly simplified model, let the normalized cost of each supernova observed at redshift $z$ be $(1+z)^m$, so that the total cost of a survey that follows up $N$ supernovae is $\sum_{i=1}^N (1+z_i)^m$. The problem is to find the optimal SNe redshift distribution for fixed total resources (total cost) $R$. For SNAP, the observing time cost for spectroscopy or photometry per supernova is estimated to scale as $(1+z)^6$ for fixed signal to noise [@SNAP]. In the case of wide field, multiplexing photometry that SNAP is designed for, simultaneously discovering and following up supernovae by repeatedly sweeping the same field could reduce this by a large factor. To span the plausible range of cost functions, we show results for $m=0, 3$, and 6. To fix the total resources $R$, we assume that there are sufficient resources to carry out a survey of 3112 SNe with the fiducial SNAP + SN Factory redshift distribution shown, e.g., in Fig. \[fig:Fig3\]. That is, for a given value of $m$, we fix $R$ by computing the total cost of the fiducial SNAP + SN Factory redshift distribution. Then we find the SN redshift distribution that minimizes $\sigma_w$ within the resource constraint, i.e., for the same value of $R$. If we place no upper bound on the number of SNe per redshift bin, the number of SNe at low redshifts would be driven to huge values as $m$ is increased. Clearly a distribution with many thousands of SNe in any redshift bin is not experimentally realistic, and the systematic error makes this an unwise choice: the gains in terms of reduced $\sigma_w$ are negligible once the number of SNe per bin goes much above 100. We therefore impose the further constraint that the number of SNe per redshift bin of width 0.25 not exceed (a very generous) 1000. The results for $m=0, 3$, and 6 are shown in Figs. \[fig:Fig10\] and \[fig:Fig11\], again for $z_{\rm max} = 1.7$, the same model for irreducible systematic error as above, and either no prior from the CMB (Figs. \[fig:Fig10\]a, \[fig:Fig11\]a) or the Planck prior (Figs. \[fig:Fig10\]b, \[fig:Fig11\]b). In Fig. \[fig:Fig10\], we assume constant $w$, while in Fig. \[fig:Fig11\] $w$ can evolve. We note that the performance of the optimal distribution in minimizing $\sigma_w$ (or $\sigma_{w_1}$) is only 2 to 10% better than the SNAP + SN Factory distribution in all cases. Consider first the constant $w$ case. Figure \[fig:Fig10\] shows that, as $m$ increases, SNe start filling up the lower redshift bins to the maximum allowed number; this continues until the resource limit is reached. While this is strictly true for the Planck prior, with no prior a significant fraction of SNe remain in the highest redshift bin. This behavior can be understood simply: without any priors, the high redshift SNe are crucial for breaking the degeneracy between $\Omega_M$ and $w$ (see Fig. \[fig:Fig7\]); the addition of the Planck prior partially breaks this degeneracy, and the number of SNe in the highest redshift bin therefore decreases. The case of evolving $w$ is qualitatively similar, with one important difference: the high-$z$ subsample of SNe is always present in the optimal distribution, regardless of the prior or the value of $m$. As Fig. \[fig:Fig11\] shows, the highest redshift bin always has a significant number of SNe ($\simeq 500$), even for $m=6$, when their cost is large. Although the exact optimal distribution for a given value of $m$, and the corresponding values of $\sigma_w$ and $\sigma_{w_1}$, will depend in practice on details of the optimization — the number of redshift bins and the maximum number of SNe allowed per bin — some clear trends emerge from this analysis. While the lower redshift bins become relatively more populated in the optimal distributions (reflecting the lower cost of low-redshift SNe), the importance of high redshift supernovae remains: in [*all*]{} cases, at least 800 SNe are at redshifts $z > 1$. For the constant $w$ case with no Planck prior, or for evolving $w$ regardless of prior, these high-redshift SNe are crucial to making the error on $w$ small enough to be useful. Clearly we have just scratched the surface with regard to resource-limited optimization; to proceed further, one would need a much more quantitative description of the resources available and the systematics. Summary and Conclusions {#sec:concl} ======================= Unraveling the nature of dark energy is one of the outstanding challenges in physics and astronomy. Determining its properties is critical to understanding the Universe and its destiny and may shed light on the fundamental nature of the quantum vacuum and perhaps even of space-time. Type Ia supernovae and CMB anisotropy can both probe the dark energy equation-of-state $w$, and we have explored in detail the synergy between the two. With the MAP mission in progress, the Planck mission slated for launch in 2007, and the design of dedicated SN surveys now underway, such a study is very timely. CMB anisotropy alone cannot tightly constrain the properties of dark energy because of a strong degeneracy between the average equation-of-state and the matter density. SNe can probe $w$ with a precision that improves significantly with knowledge of the matter density, because $H_0r(z)$ depends only upon $w$ and $\Omega_M$. A key result of this paper is that CMB anisotropy measurements by the upcoming Planck mission have even more potential for improving the ability of SNe to probe dark energy. The reason is simple: in the $\Omega_M$–$w$ plane (Fig. \[fig:Fig1\]), the CMB constraint is more complementary to the SNe constraint than is determination of $\Omega_M$. Compared to the matter density prior $\sigma_{\Omega_M} = 0.03$, Planck CMB data reduce the predicted error $\sigma_w$ (under the assumption of constant $w$) by about a factor of two (Fig. \[fig:Fig7\]). In probing possible variation of $w$ with redshift, the Planck prior is also significantly better than the same matter density prior (Fig. \[fig:Fig9\]). Given the concern expressed by some (e.g., [@Maoretal]) that a precise measurement of the matter density independent of dark energy properties may be difficult, this is good news. On the other hand, we find that even if MAP can successfully measure polarization on large scales, its potential for complementarity with SNe falls short of that for Planck and is not as good as the $\sigma_{\Omega_M}=0.03$ matter density prior. We have also explored how the SNe determination of the dark energy equation-of-state, with or without prior information from the CMB or the matter density, depends upon the redshift distribution of the survey, including the effects of systematic error and a realistic spread of SNe redshifts. For either constant or evolving $w$, the optimal strategy calls for significant numbers of SNe above redshift $z\sim 1$. For the constant $w$ case with no Planck prior, or for evolving $w$ regardless of prior, these high-redshift SNe are necessary for achieving $\sigma_w < 0.1$. Observing substantial numbers of SNe at these high redshifts also provides the only hope of probing time evolution of the equation-of-state with reasonable precision. Moreover, the improvement in $\sigma_{dw/dz}$ continues to high redshift: $\sigma_{dw/dz}$ falls by more than a factor of two when $z_{\rm max}$ increases from $1$ to $2$ (Fig. \[fig:Fig9\]). Since we currently have no prior information about (or consensus physical models which significantly constrain) the time variation of $w$, the design of a SNe survey aimed at probing dark energy should take into account the possibility that $w$ evolves. These conclusions about the need for high-redshift supernovae do not change significantly if we consider a hypothetical survey for which resources are constrained and a redshift-dependent cost is assigned to each supernova. Ref. [@SS] raised the question whether a shallow SNe survey is better than a deep one in determining the dark energy equation-of-state, given prior knowledge from the CMB. Our results indicate that it is not, once the SNe and CMB experiments are realistically modelled. 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Brustein, and P. J. Steinhardt, Phys. Rev. Lett. [**86**]{}, 6 (2001) [^1]: Note that this is merely illustrative. In fact we treat ${\cal D}$ by the exact expression for the distance to the last scattering surface, i.e., Eq. (1) generalized to evolving $w(z)$. [^2]: In practice, the level of the residual systematic error depends on survey design, e.g., telescope aperture and stability, wavelength coverage, observing cadence, point spread function, seeing, sky background, etc. The systematic error quoted here is based on the fact that SNAP is specifically designed to achieve 0.02 mag systematic error in redshift bins of width $\Delta z = 0.1$. [^3]: The optimization can be done with respect to the errors of the individual parameters or the determinant of the Fisher matrix (“area of error ellipse” for the two-parameter problem). The results in the two cases are similar. We will minimize $\sigma_w$ unless otherwise noted. [^4]: The SN Factory has another important purpose: the systematic study of Type Ia SNe to better establish their efficacy as standardizable candles. [^5]: When varying $z_{\rm max}$ from its fiducial value of 1.7, we truncate the fiducial SNAP distribution at the new $z_{\rm max}$ and scale it to preserve the total of 2812 SNe. The SN Factory distribution is then added unchanged – 300 SNe in the lowest redshift bin. [^6]: As discussed in Ref. [@HT], the exact form chosen for the parameterization is not essential.
--- abstract: 'An air pollution model is generally described by a system of PDEs on unbounded domain. Transformation of the independent variable is used to convert the problem for nonlinear air pollution on finite computational domain. We investigate the new, degenerated parabolic problem in Sobolev spaces with weights for well-posedness and positivity of the solution. Then we construct a fitted finite volume difference scheme. Some results from computations are presented.' author: - 'Tatiana P. Chernogorova, Lubin G. Vulkov' title: NUMERICAL SOLUTION OF A PARABOLIC SYSTEM IN AIR POLLUTION --- [**[Keywords:]{}**]{} Nonlinear air pollution, Infinite domain, Log-transformation, Degeneracy, Maximum principle, Finite volume method [**[2010 Mathematics Subject Classification:]{}**]{} 65N06 Introduction ============ Environmental problems are becoming more and more important for our world and their importance will even increase in the future. High pollution of air, water and soil may cause damage of plants, animals and humans. An air pollution model is generally described by a system of PDE-s for calculating the concentrations of a number of chemical species (pollutants and components of the air that interact with the pollutant) in a large 3-D domain (part of the atmosphere above the studied geographical region). The Danish Euler Model (DEM) is one of the most frequently used air pollution model and is mathematically represented by the following system PDE-s [[@13]]{}, [[@10]]{}, [[@8]]{}, [[@11]]{}: $$\begin{gathered} \frac{\partial c_s }{\partial t} = - \frac{\partial (uc_s )}{\partial x} - \frac{\partial (vc_s )}{\partial y} - \frac{\partial (wc_s )}{\partial z} + \frac{\partial }{\partial x}\left( {K^{x}_{s} \frac{\partial c_s }{\partial x}} \right) + \frac{\partial }{\partial y}\left( {K^{y}_{s} \frac{\partial c_s}{\partial y}} \right) \nonumber \\ + \frac{\partial }{\partial z}\left( {K_{s}^{z}\frac{\partial c_s }{\partial z}} \right) + F_s + R_s (c_1,c_2 ,\ldots ,c_S ) - (k_{1s} + k_{2s} )c_s ,\quad s =1,2,\ldots,S,\end{gathered}$$ where $c_s$ are the *concentrations* of the chemical species; $u$, $v$ and $w$ are wind velocities and $K_{s}^{x}$, $K_{s}^{y}$, $K_{s}^{z}$ are the diffusion components; $F_s$ are the emissions; $k_{1s}$, $k_{2s}$ are dry/wet deposition coefficients and $R_s(c_1, c_2,\ldots, c_S )$ are non-linear functions describing the chemical reactions between the species under consideration [[@11]]{}. Typical is the case $R_{s}(c_{1}, c_2,\dots, c_{s})=\sum_{i=1}^{S}\gamma_{s,i} c_{i} + \sum_{i=1}^{S}\sum_{j=1}^{S}\beta _{s,i,j} c_{i}c_{j}$, $s =1, 2, \dots, S,$ where $\gamma_{s,i}$ and $\beta_{s,i,j}$ are constants. For such complex models operator splitting is very often applied in order to achieve sufficient accuracy as well as efficiency of the numerical solution. Although the splitting is a crucial step in efficient numerical treatment of the model, after discretization of the large computational domain each sub-problem becomes itself a huge computational task. Here we will concentrate on a non-stationary sub-model of a horizontal advection-diffusion with chemistry, emissions and deposition, see [[@8]]{}, [[@11]]{}: $$\begin{gathered} \frac{\partial c_s }{\partial t} -\frac{\partial }{\partial z}\left( {K_s (z)\frac{\partial c_s }{\partial z}} \right)+ w\frac{\partial c_s }{\partial z} - R_s (c_1, c_2,\ldots, c_S ) = Q_s (t) \delta (z - z_s^\ast ),\label{1}\\ z \in (0,\infty),\quad t \in (0,T],\nonumber\\ \frac{\partial c_s }{\partial z}(t,0) = \delta _s c_s (t,0), \quad \delta _s = const \ge 0, \quad t \in [0,T],\label{2}\\ \mathop {\lim }\limits_{z \to \infty } c_s (t,z) = 0, \quad t \in [0,T] \label{3},\\ c_s(0,z)=c_{s,0},\quad z \in [0,\infty), \quad s = 1,2,\ldots,S. \label{4}\end{gathered}$$ The rest of the paper is organized as follows. In Section 2 we investigate the transformed differential problem and discuss its well-posedness and the properties of its solution. In Section 3 we derive the fitted finite volume discretization. In Section 4 we present some results from computational experiments. At the end we formulate some conclusions. Let us mention that construction and analysis of positive numerical methods is very important for models in medicine and finance, see e.g.[@K13; @K15]. The differential problem on bounded domain ========================================== In the numerical scheme it is not convenient to corporate the boundary condition at infinity. For the simplest case of (\[1\]) (one linear advection-diffusion equation) (discrete) transparent boundary conditions are constructed and analyzed in [[@3]]{} while in [[@4]]{} the transformation $$z = \frac{1}{2a}\log \left( {\frac{1 + \xi }{1 - \xi }} \right),\quad \xi \in \Omega = (0,1) \Leftrightarrow \xi = \frac{e^{2az} - 1}{e^{2az} +1},\quad z \in (0,\infty ),$$ is used. Here $a$ is a stretching factor. Using this transformation, the system (\[1\]) and the respective boundary and initial conditions (\[2\])–(\[4\]) in the computational domain become $$\begin{gathered} \frac{\partial C_s }{\partial t} - a^2\left( {1 - \xi ^2} \right)^2k_s (\xi) \frac{\partial ^2C_s }{\partial \xi ^2} \nonumber + a\left( {1 - \xi ^2}\right)\left( { 2a\xi k_s (\xi ) + w - a\left( {1 - \xi ^2} \right)\frac{\partial k_s (\xi )}{\partial \xi }} \right)\\ \times \frac{\partial C_s}{\partial \xi } - r_s (C_1,C_2 ,\ldots ,C_S ) = Q_s (t)\delta \left( {\xi - \xi_s ^\ast } \right),\quad \xi \in \Omega, \; t \in (0,T], \label{5} \\ a\frac{\partial C_{s}}{\partial \xi}(t,0)= \delta_{s}C_{s}(t,0), \quad t \in [0,T],\label{6}\\ C_{s}(t,1) = 0, \quad t \in [0,T], \label{7}\\ C_s(0,\xi)=C_{s,0},\quad \xi \in [0,1], \quad s = 1,2,\ldots,S.\label{8}\end{gathered}$$ We have denoted $C_{s}(t,\xi) \equiv c_{s}(t,z(\xi))$, $k_s(\xi)=K_{s}(z(\xi))$, $r_{s}(C_{1},C_{2},\dots,C_{s})= R_{s}(c_{1}(t,z(\xi)), c_{2}(t,z(\xi)),\dots,c_{S}(t,z(\xi))$. It is easy to seen that at $\xi=1$ the system (\[5\]) degenerates to the ODE system $$\label{9} \frac{\partial C_{s} (t,1)}{\partial z}-r_{s} (C_{1} (t,1), \dots, C_{S}(t,1)=0, \quad s = 1, 2, \dots, S, \quad t \in (0,T].$$ In the case $C_{s,0}=0$, the unique solution of (\[9\]) is the zero one, i. e. (\[7\]). By the Fichera and Oleinik-Radkevich theory [[@5]]{} for degenerate parabolic equations, at the degenerate boundary $\xi =1$, the boundary condition should not be given. But from physical motivation we have imposed the boundary condition (\[7\]). It is easy to check that if the functions $C_{s}(t,\xi)$ satisfy (\[7\]) then they also satisfy (\[9\]). Therefore (\[7\]) is a particular case of (\[9\]). The general theory [[@5]]{} does not provide existence and uniqueness of the solution to the problem (\[5\])–(\[8\]). Also, following the physical motivation we will discuss the non-negativity of the solution. For this we rewrite the system (\[5\]) in divergent form: $$\begin{gathered} \frac{\partial C_s }{\partial t} = \frac{\partial }{\partial \xi }\left({p_s (\xi )\frac{\partial C_s } {\partial \xi } + q_s (\xi )C_s } \right) +B_s (\xi, C_1,\ldots ,C_S ) + f_s(t,\xi),\label{10} \\ \quad (t,\xi) \in Q_{T} = [0,T] \times \Omega, \quad p_s (\xi ) = a^{2} ( 1 - \xi^{2} ) k_{s} ( \xi ),\nonumber \\ q_s(\xi )= a(1-\xi^{2})(2a \xi k_{s}(\xi)- w),\; B_s (\xi,C_1,\ldots,C_S ) =r_{s}(C_{1}, \dots, C_{S})-d_s(\xi)C_s,\nonumber\\ d_s(\xi) = 2 a^{2}(1- 3\xi^{2} )k_{s}(\xi)+2a^{2}\xi(1-\xi^{2}) \frac{\partial k_{s}(\xi)}{\partial \xi}+2aw \xi,\; s = 1,2,\ldots, S,\nonumber\end{gathered}$$ where $f_s(t,\xi)$ is a regularization of the Dirac delta-function, $s = 1,2,\ldots, S$. Further, to handle the degeneracy in the equation (\[10\]), we introduce the weighted inner product and corresponding norm on $L_{2,w} (\Omega )$ by $$(u,v)_{w} :=\int_{0}^{1}(1- \xi)^{2} u v d \xi, \quad \Vert v \Vert_{0,w}={\sqrt {(v,v)_{w}}}=\left(\int_{0}^{1}(1- \xi)^{2} v^{2} d \xi\right)^{ 1/2}.$$ The space of all weighted square-integrable functions is defined as $L_{2,w} ( \Omega ) : =\{ v: \Vert v \Vert_{0,w} < \infty \} $. By using a standard argument it is easy to show that the pair $( L_{2,w}(\Omega),(\cdot, \cdot)_{w})$ is a Hilbert space (cf., for example, [[@6]]{}). Using $L_{2}(\Omega)$ and $L_{2,w}(\Omega)$, we define the following weighted Sobolev space $$H_{w}^{1}(\Omega):=\{v \in L_{2}(\Omega), \; v' \in L_{2,w}(\Omega) \},\quad v'=\partial v / \partial \xi$$ with corresponding inner product on $H_{w}^{1}(\Omega):=(\cdot, \cdot)_{H_{w}^{1}}:=(\cdot, \cdot)_{w}+(\cdot, \cdot)$. Also, it is easy to prove that the pair $ ( H_{w}^{1} ( \Omega ), ( \cdot , \cdot )_{H_{w}^{1} } )$ is a Hilbert space with the norm $$\Vert v \Vert_{1,w}:=\{\Vert v' \Vert_{0,w}^{2} +\Vert v \Vert_{2}^{2} \}^{1/2}= \{ \left((1- \xi)^{2} v', v'\right) +(v,v) \}^{ 1/2}.$$ For $C_{s}, \eta_{s} \in H^{1}_w (\Omega )$ we define the bilinear forms $$A_{s} (C_s, \eta_s ;t) = \int_{0}^{1}\left[p_{s} ( \xi )\frac{ \partial C_{s} } {\partial \xi } \frac{ \partial \eta_{s} } {\partial \xi } +q_{s} ( \xi ) C_{s}\frac{ \partial \eta_{s} } {\partial \xi } -B_{s} ( \xi ,C_1, \ldots, C_S)C_s \eta_{s} \right]d \xi.$$ Now we are in position to define the following variational problem corresponding to (\[10\]) and (\[6\]), (\[8\]), (\[9\]): find $C=(C_1, \ldots, C_S)$, $C_{s} \in C ( [0,T]; H_{w}^{1} ( \Omega ))$, $s=1,2,\ldots,S$, satisfying the initial condition (\[6\]), such that for all $\eta_{s} \in H_{w}^{1} ( \Omega )$ $$\int_{0}^{1}\frac{ \partial C_{s} } {\partial t } \eta_{s} d \xi + A_{s} (C_s, \eta_s;t) = \int_{0}^{1} f_{s} \eta_{s} d \xi \; \; \mbox{a.~e in } (0,T).$$ [**[Theorem 1.]{}**]{} Let $B_{s} (\xi, C) $ be Lipshitz continuous with respect to $C$. There exists a unique solution $C$, $C_{s} \in H_{w}^{1} (\Omega )$, to problem (\[10\]) and (\[6\]), (\[8\]), (\[9\]) in the above sense. Considering the process of pollutant transport and diffusion in the atmosphere (and in the water) the concentrations $C_{1}, C_{2},\ldots,C_{S}$ of pollutants can not be negative if they are non-negative in the initial state $t=0$ for all $\xi \in (0,1)$. This property is called [*[non-negativity preservation]{}*]{} and it is well studied for single heat-diffusion equation. Unfortunately, one needs additional assumptions that the system has a (quasi-) monotonicity property, also called [*[cooperativeness]{}*]{}. This condition is rather restrictive and in this paper we use an idea from [[@13]]{} to establish maximum principle for the system (\[10\]). [**[Theorem 2.]{}**]{} Let $C$ be a solution of the system (\[10\]). Assume that for all $s=1,2, \dots,S$: a) $ f_{s} ( t, \xi ) \geq 0, \; (t,\xi) \in Q_T$; b) $B_{s} (\xi ,C)$ is Lipshitz continuous with respect to the concentrations $C$ and it satisfies the inequality $B_{s} ( \xi, C_1,\dots,C_{s-1},0,C_{s+1},\dots,C_S) \geq 0$, $\forall \; C \in R_{+}^{S} \equiv \{ C_{s} \geq 0 \}$. Then $C_{s} \geq 0$ for all $t \in [0,T]$ and $\xi \in [0,1]$. If we further assume that c) there exists $M_{s} \geq 0$ such that $B_{s} ( \xi, C_1,\dots,C_{s-1},M_s,C_{s+1},\dots,C_S) \leq 0,\; \forall \; C \in R_{+}^{S}$, then $0 \leq C_{s} ( t , \xi ) \leq M_{s} < \infty$, $s =1, \dots, S$. Fitted finite volume difference scheme for the transformed problem ================================================================== Let the interval $[0,1]$ be subdivided into $N$ intervals $I_i = [\xi _i ,\xi _{i + 1}]$, $i = 1,2,\ldots ,N$ with $0 = \xi _1 < \xi _2 < \ldots < \xi _N < \xi _{N + 1} = 1$ and $h_i = \xi _{i + 1} - \xi _i $. We set $\xi _{i - \frac{1}{2}} = 0.5\left( {\xi _{i - 1} + \xi _i } \right)$, $\xi_{i + \frac{1}{2}} = 0.5\left( {\xi _i + \xi _{i + 1} } \right)$, $\hbar _i = \xi _{i + \frac{1}{2}} - \xi _{i - \frac{1}{2}} $ for $i =2,3,\ldots,N$. [*[A. Internal nodes]{}.*]{} We integrate equation (\[10\]) on the cell $[\xi _{i - \frac{1}{2}} ,\xi _{i + \frac{1}{2}}]$: $$\begin{gathered} \label{11} \int\limits_{\xi _{i -\frac{1}{2}} }^{\xi _{i + \frac{1}{2}} } {\frac{\partial C_s}{\partial t}d\xi = \int\limits_{\xi _{i - \frac{1}{2}} }^{\xi _{i + \frac{1}{2}} } {\frac{\partial }{\partial \xi }\left( {p_s(\xi )\frac{\partial C_s }{\partial \xi } + q_s(\xi )C_s } \right)d\xi } } \\ + \int\limits_{\xi _{i - \frac{1}{2} } }^{\xi _{i + \frac{1}{2}} } {\left[ {B_s(\xi,C_1,\ldots,C_S) + f_s(t,\xi)} \right]d\xi}.\end{gathered}$$ Applying the mid-point quadrature rule to all the integrals in (\[11\]) with exception to the second one we obtain $$\begin{gathered} \label{12} \left. {\frac{\partial C_s }{\partial t}} \right|_{(t, \xi _i)} \hbar _i = \left. {\left( {p_s(\xi )\frac{\partial C_s }{\partial \xi } + q_s(\xi )C_s } \right)} \right|_{\left( {t, \xi _{i + \frac{1}{2}}} \right)}\\ - \left. {\left({p_s(\xi )\frac{\partial C_s}{\partial \xi } + q_s(\xi )C_s } \right)} \right|_{\left(t, {\xi _{i - \frac{1}{2}}} \right)} +\hbar _i \left. {\left[ {B_s(\xi,C_1,\ldots,C_S) + f_s(t,\xi)} \right] } \right|_{(t,\xi _i)}.\end{gathered}$$ Further, at the derivation of the discrete equations we follow the methodology in [[@7]]{}. Let us rewrite the equation (\[12\]) in the form $$\label{13} \frac{\partial C_{s,i }}{\partial t}\hbar _i = (1 - \xi _{i + \frac{1}{2} }^2 )\rho_{s,i + \frac{1}{2}} - (1 - \xi _{i - \frac{1}{2}}^2 )\rho_{s,i - \frac{1}{2}} + \hbar_i \left( {B_{s,i} + f_{s,i}} \right),$$ where $$\label{14} \rho_s \equiv \rho_s \left( C_s \right) = a^2(1 - \xi ^2)k_s(\xi )\frac{\partial C_s} {\partial \xi } + (2a^2\xi k_s(\xi ) - aw )C_s.$$ We need to derive an approximation of the continuous flux $\rho_s$ in the point $\xi _{i + 1 / 2} $, $i =2,3,\ldots ,N - 1$. To do this, we consider the [*[two-point BVP]{}*]{} $$\begin{gathered} \left( {l_{s,i + \frac{1}{2}} (1 - \xi ^2){V_s}' + m_{s,i + \frac{1}{2}} V_s} \right)^\prime = 0,\quad \xi \in I_i,\label{15}\\ V_s(\xi _i ) = C_{s,i} ,\quad \quad V_s(\xi _{i + 1} ) =C_{s,i+1},\label{16}\end{gathered}$$ where $l_s = a^2k_s(\xi )$, $m_s = 2a^2 \xi k_s(\xi ) - a$, $l_{s,i + \frac{1}{2}} = l_s(\xi _{i + \frac{1}{2}} )$, $m_{s,i + \frac{1}{2}} = m_s(\xi _{i + \frac{1}{2}} )$. Integrating (\[15\]) yields the first-order linear equation. Solving this equation and using the boundary conditions (\[16\]) gives $$\label{17} \rho_{s,i + \frac{1}{2}}=m_{s,i + \frac{1}{2}} \frac{\Delta_{s,i} \left( {\xi _{i + 1} } \right)C_{s,i + 1} - \Delta_{s,i} \left( {\xi _i } \right)C_{s,i} }{\Delta_{s,i} \left( {\xi _{i + 1} } \right) - \Delta_{s,i} \left( {\xi _i } \right)},$$ where $\alpha _{s,i} = \frac{m_{s,i + \frac{1}{2}} }{l_{s,i + \frac{1}{2}} }$, $\Delta_{s,i} \left( {\xi _i } \right) = \left( {\frac{1 + \xi _i }{1 - \xi _i }} \right)^{\frac{\alpha _{s,i}}{2}}$. In a similar way we approximate $\rho_{s,i - \frac{1}{2}}$ in (\[13\]) for $2 \le i \le N$. For approximation of $\rho_{s, N+\frac{1}{2}}$ we solve the BVP $$\begin{gathered} \left( {\bar {l}_{s,N + \frac{1}{2}} (1 - \xi ){V_s}' + m_{s,N + \frac{1}{2}} V_s} \right)^\prime = M_2, \quad V_s(\xi_N ) = C_{s,N},\quad V_s(\xi _{N + 1} ) =0, %C_{s,N + 1}=0,\end{gathered}$$ where $\bar {l}=a^2(1+\xi)k_s(\xi)$. After some calculation for the flow $\rho_{s,N + \frac{1}{2}}$ we get $$\rho _{s,N +\frac{1}{2}} = 0.5\left[{C_{s,N + 1} \left( {\bar {l}_{s,N + \frac{1}{2}} + m_{s,N + \frac{1}{2}} } \right) - C_{s,N} \left( {\bar {l}_{s,N + \frac{1}{2}} - m_{s,N + \frac{1}{2}} } \right)} \right].$$ [*[B. Boundary nodes.]{}*]{} To approximate the boundary condition on the left vertical boundary $\xi = 0$ we proceed as for the internal grid nodes, but integrating equation (\[10\]) on the interval $[\xi_1 ,\xi_{\frac{3}{2}} ]$ (i. e. in the semi-interval by $\xi$) to get $$\frac{\partial C_{s,1} }{\partial t}\frac{h_1 }{2} = (1 - \xi _{\frac{3}{2}}^2 )\rho_{s,\frac{3}{2}} - (1 - \xi _1^2 ) \rho_{s,1} + \frac{h_1 }{2}\left( {B_{s,1}+ f_{s,1}} \right).$$ &gt;From (\[17\]) for $i=1$ we get the approximation for $\rho_{s,\frac{3}{2}}$. For $(1 - \xi _1^2 )\rho_{s,1}$, where $\xi_1 = 0$, using the expression for $\rho_s$ (\[14\]) and the boundary condition (\[6\]) we find $$(1 - \xi_1^2 )\rho_{s,1} = a(\delta_s k_s(\xi _1 ) - w )C_{s,1}.$$ On the right vertical boundary $\xi = 1$ we have $C_{s,N + 1} = 0.$ Finally, for the space approximation we obtain the ODE non-linear system of equations for $C_{s,i}(t)$, $s=1,2,\ldots,S$, $i=1,2,\ldots,N+1$: $$\begin{gathered} \frac{\partial C_{s,1} }{\partial t}\frac{h_1 }{2} = - e_{s,1,1} C_{s,1} + e_{s,1,2} C_{s,2}+\frac{h_1 }{2}\left[B_s(\xi_1,C_{1,1},C_{2,1},\ldots,C_{S,1})+f_{s,1}(t)\right],\\ \frac{\partial C_{s,i} }{\partial t}\hbar_i = e_{s,i,i - 1} C_{s,i - 1} - e_{s,i,i}C_{s,i} + e_{s,i,i + 1} C_{s,i + 1}\\ +\hbar_i \left[ B_s(\xi_i,C_{1,i},C_{2,i},\ldots,C_{S,i})+f_{s,i}(t)\right],\quad i = 2,3,\ldots ,N , \\ C_{s,N + 1} =0,\end{gathered}$$ where $$\begin{gathered} e_{s,1,1} = \frac{(1 - \xi _{\frac{3}{2}}^2 )m_{s,\frac{3}{2}} \Delta _{s,1} (\xi _1 )}{\Delta _{s,1} (\xi_2 ) - \Delta_{s,1} (\xi_1 )} + a\left( {\delta _s k_s (\xi _1 ) - w} \right);\\ e_{s,i,i - 1} = \frac{(1 - \xi _{i - \frac{1}{2}}^2 )m_{s,i - \frac{1}{2}} \Delta _{s,i - 1} (\xi _{i - 1} )}{\Delta _{s,i - 1} (\xi _i ) - \Delta _{s,i - 1} (\xi _{i - 1} )},\; i = 2,3,\ldots ,N;\\ e_{s,i,i + 1} = \frac{(1 - \xi _{i + \frac{1}{2}}^2 )m_{s,i + \frac{1}{2}} \Delta _{s,i} (\xi _{i + 1} )}{\Delta _{s,i} (\xi _{i + 1} ) - \Delta _{s,i} (\xi _i )},\; i = 1,2,\ldots ,N - 1;\\ e_{s,i,i} = \frac{(1 - \xi _{i + \frac{1}{2}}^2 )m_{s,i + \frac{1}{2}} \Delta _{s,i} (\xi _i )}{\Delta _{s,i} (\xi _{i + 1} ) - \Delta _{s,i} (\xi _i )} + \frac{(1 - \xi _{i - \frac{1}{2}}^2 )m_{s,i - \frac{1}{2}} \Delta _{s,i - 1} (\xi _i )}{\Delta _{s,i - 1} (\xi _i ) - \Delta _{s,i - 1} (\xi _{i - 1} )},\\ i= 2,3,\ldots ,N - 1; \; e_{s,N,N + 1} = 0.5(1 - \xi _{N + \frac{1}{2}}^2 ) \left( {\bar {l}_{s,N +\frac{1}{2}} + m_{s,N + \frac{1}{2}} } \right),\\ e_{s,N,N} = 0.5(1 - \xi _{N + \frac{1}{2}}^2 )\left( {\bar {l}_{s,N +\frac{1}{2}} - m_{s,N + \frac{1}{2}} } \right) + \frac{(1 - \xi _{N - \frac{1}{2}}^2 )m_{s,N -\frac{1}{2}} \Delta _{s,N - 1} (\xi _N )}{\Delta _{s,N - 1} (\xi _N ) - \Delta _{s,N -1} (\xi _{N - 1} )}.\end{gathered}$$ In order to discretize the problem with respect to $t$ we introduce the mesh $$0 = t_1 < t_2 < \ldots < t_j < t_{j + 1} < \ldots < t_{M + 1} = T,\quad \Delta t_j = t_{j + 1} - t_j .$$ Then, the fully implicit scheme can be written in the form $$\begin{gathered} \frac{C_{s,1}^{j + 1} - C_{s,1}^j }{\Delta t_j }\frac{h_1 }{2} = - e_{s,1,1} C_{s,1}^{j + 1} + e_{s,1,2} C_{s,2}^{j + 1} \nonumber\\ + \frac{h_1}{2} \left[ {B_s (\xi _1 ,C_{1,1}^{j + 1},\ldots ,C_{S,1}^{j + 1} ) + f_{s,1}^{j + 1} } \right],\label{18} \\ \frac{C_{s,i}^{j + 1} - C_{s,i}^j }{\Delta t_j }\hbar _i = e_{s,i,i - 1} C_{s,i - 1}^{j + 1} - e_{s,i,i} C_{s,i}^{j + 1} + e_{s,i,i +1} C_{s,i + 1}^{j + 1} \nonumber \\ +\hbar _i \left[ {B_s (\xi _i ,C_{1,i}^{j + 1},\ldots ,C_{S,i}^{j + 1} ) + f_{s,i}^{j + 1} } \right], \quad i = 2,3,\ldots ,N, \label{19}\\ C_{s,N + 1}^{j + 1} = 0,\quad s=1,2,\ldots,S, \label{20} \end{gathered}$$ where $C_{s,i}^{j}=C_s(t_j,\xi_i)$. To solve the non-linear system (\[18\])–(\[20\]) we have used Newton’s method, which leads to a linear system of equations. Numerical Experiments ===================== To show the efficiency and usefulness of the discretization method, various test problems with different choices of parameters were solved. In the numerical experiment we approximate the Dirac-delta function by the function $$\begin{gathered} d_h (\xi ) = \left\{ {{\begin{array}{*{20}c} {\frac{2h - \left| {\xi^\ast - \xi } \right|}{4h^2},\quad \xi \in \left( {\xi^\ast - 2h,\xi ^\ast + 2h} \right),} \hfill \\ {0,\quad \quad \quad \quad \; \; \xi \notin \left( {\xi ^\ast - 2h,\xi ^\ast + 2h} \right).} \hfill \end{array} }} \right.\end{gathered}$$ ![Numerical solution $c_2(t,z)$.[]{data-label="f2"}](c11.eps){width="\textwidth"} ![Numerical solution $c_2(t,z)$.[]{data-label="f2"}](c21.eps){width="\textwidth"} ![Numerical solution $c_3(t,z)$.[]{data-label="f3"}](c31.eps){width="50.00000%"} For the numerical results presented here, we have used the following functions and values of the coefficients in the problem under consideration: $S=3$, $K_1(z)=1$, $K_2(z)=K_3(z)=5$, $w=1$, $Q_1(t)=t$, $Q_2(t)=1-t$, $Q_3(t)=0$, $z_1^\ast=20$, $z_2^\ast=85$, $\delta_1=\delta_2=\delta_3=0$, $T=1$, $c_{1,0}=c_{2,0}=0$, $c_{3,0}=2$, $a=0.005$, $R_s=\gamma_{s,2}c_2+\beta_{s,1,3}c_1c_3$, $s=1,2,3$, where $\gamma_{1,2}=-\gamma_{2,2}=\gamma_{3,2}=2000$, $\beta_{1,1,3}=-\beta_{2,1,3}=\beta_{3,1,3}=-1000$. A part of these data are taken from [[@8]]{}. It is easy to check that the conditions a), b) and c) of Theorem 2 are fulfilled. Fig. \[f1\], \[f2\] and \[f3\] show the numerical computed concentrations $c_1(t,z)$, $c_2(t,z)$ and $c_3(t,z)$. We have used the Runge method for practical estimation of the rate of convergence of the scheme with [*[respect to the space variable]{}*]{} at fixed value of $t=T=1$. We have used three inserted grids with 100, 200 and 400 subintervals respectively by $\xi$ and $\Delta t=\Delta t_j =0.001$. A part of the results from the calculations for the rate of convergence are presented in Table 1. [|p[10pt]{}|p[21pt]{}|p[21pt]{}|p[21pt]{}|p[22pt]{}|p[21pt]{}|p[17pt]{}|p[17pt]{}|p[17pt]{}|p[17pt]{}|p[17pt]{}|p[17pt]{}|p[17pt]{}|p[20pt]{}|p[17pt]{}|p[17pt]{}|]{} $\xi$& 0.00& 0.01& 0.02& 0.03& 0.04& 0.05& 0.06& 0.07& 0.08& 0.09& 0.10& 0.11& 0.12& 0.13& 0.14\ $n_1$& 11.04& 9.27& 7.63& 6.16& 4.89& 3.84& 3.04& 2.49& 2.26& 3.08& 0.08& 1.40& 2.12& 2.27& 2.71\ $n_2$& 11.04& 9.28& 7.63& 6.16& 4.89& 3.84& 3.04& 2.46& 2.07& 1.40& 1.25& 0.54& 1.57& 2.13& 2.69\ $n_3$& 1.74& 1.07& 2.17& -1.67& 5.15& 3.84& 3.04& 2.46& 2.07& 1.40& 1.25& 0.54& 1.57& 2.13& 2.69\ $\xi$& 0.26& 0.27& 0.28& 0.29& 0.30& 0.35& 0.36& 0.37& 0.38& 0.39& 0.40& 0.41& 0.42& 0.43& 0.44\ $n_1$& 23.97& 19.62& 17.23& 14.96& 12.79& 4.00& 2.90& 2.15& 1.89& 1.99& 1.67& 1.19& 1.59& 1.69& 2.36\ $n_2$& 23.98& 19.63& 17.24& 14.97& 12.80& 4.01& 2.93& 2.24& 1.97& 2.03& 1.79& 1.40& 1.68& 1.79& 2.44\ $n_3$& 2.01& 2.00& 1.94& 0.44& 5.25& 2.57& 2.06& 1.79& 1.82& 1.48& 2.79& 2.17& -1.96& 1.64& 1.71\ Conclusions ============= In this work we have considered a one-dimensional nonlinear problem of air pollution. We have used a log-transformation that makes the original problem, defined on a semi-infinite interval, to another one on the interval $(0,1)$. We have discussed the well-posedness of the transformed problem and the non-negativity of its solution. We have derived a fitted volume difference scheme that preserves the non-negativity property of the differential problem solution as numerical experiments show. Iterative algorithms and two-grid techniques for first solution of the non-linear system - , similar to those in [@K12; @K14; @K15] will be used in the future research. Detail experimental and theoretical analysis will be very interesting. [**[Acknowledgements:]{}**]{} The first author is supported by the Sofia University Foundation under Grant No 111/2014. The second author is supported by Bulgarian National Fund of Science under Project DID 02/37-2009. [99]{} W. Chen, K. Li, E. Wright. Commun. Pure Appl. Anal., 4 (4), 2005, 889-899. J. Karátson, T. Kurics. Math. Modelling and Anal., 18 (5), 2013, 641-653. A. Mamonov, Y.-H. R. Tsai. Inverse Problems, 29(3):035009, 2013. Z. Zlatev, I. Dimov. Computational and Numerical Challenges in Environmental Modeling, Elsevier, Amsterdam, 2006, p. 373 M.N. Koleva. Efficient numerical method for solving Cauchy problem for the Gamma equation, AIP CP, 1410, 2011, 120-27. M.N. Koleva. Efficient numerical method for solving 1D degenerate Keller-Segel systems, AIP CP, 1497, 2012, 168-175. Q. A. Dang, M. Ehrhardt. Math. and Comp. Modelling, 44, 2006, 834-856. T. Chernogorova, L. Vulkov. AIP Conf. Proc., 1497, 176, 2012, 176-183. Oleinik O. A., E. V. Radkevich. Second Order Equations with Nonnegative Characteristic Form, New York, Plenum Press, 1973, p. 267. A. Kufner. Weighted Sobolev spaces, J. Wiley and Sons, New York, 1985, p. 152. S. Wang. IMA J. of Numer. Anal., 24, 2004, 699-720. M.N. Koleva. Two-grid technique for semi-linear elliptic problems on complicated domains, Proceedings of the Fifth International Conference on Finite Difference Methods: Theory and Applications, 2011, 119-128. M.N. Koleva. Iterative methods for solving nonlinear parabolic problem in pension saving management, AIP CP, 1404, 2011, 457-463.
We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over ${{\mathbb C}}^n $. This class is unique (up to multiplication by a non zero holomorphic function), and is closely related to a Gaussian field over a Hilbert space of holomorphic functions on the reduced Heisenberg group. For a fixed random function of this class, we show that the probability that there are no zeros in a ball of large radius, is less than $e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to this result we also compute probability estimates for the event that a random function’s unintegrated counting function deviates significantly from its mean. Introduction ------------ Random polynomials and random holomorphic functions are studied as a way to gain insight into difficult problems such as string theory and analytic number theory. A particularly interesting case of random holomorphic functions is when the random functions can be defined so that they are invariant with respect to the natural isometries of the space in question. The class of functions that we will study are the unique Gaussian random holomorphic functions, up to multiplication by a nonzero holomorphic function, whose expected zero set is uniformly distributed on ${{\mathbb C}}^n$. For this class of random holomorphic functions we will determine the expected value of the unintegrated counting function for a ball of large radius and the chance that there are no zeros present. This pathological event is what is called the hole probability of a random function. In doing this we generalize a result of Sodin and Tsirelison, to n dimensions, in order to give the first nontrivial example, where the hole probability is computed in more than 1 complex variable. The topic of random holomorphic functions is an old one which has many results from the first half of the twentieth century, and is recently experiencing a second renaissance. In particular Kac determined a formula for the expected distribution of zeros of real polynomials in a certain case, [@Kac43]. This work was generalized throughout the years, and a terse geometric proof, and some consequences are presented by Edelman and Kostlan, [@EdelmanKostlan]. An excellent reference for other results regarding the general properties of random functions is Kahane’s text, [@Kahane85]. One series of papers, by Offord, is particularly relevant to questions involving the hole probability of random holomorphic functions and the distribution of values of random holomorphic functions, [@Offord67], [@Offord72], although neither is specifically used in this paper. Recently, there has been a flurry of interest in the zero sets of random polynomials and holomorphic functions which are much more natural objects than they may initially appear. For example Bleher, Shiffman and Zelditch show that for any positive line bundle over a compact complex manifold, the random holomorphic sections to $L^N$ (defined intrinsically) have universal high N correlation functions, [@BleherShiffmanZelditchUniv]. In addition to a plethora of results describing the typical behavior, there have also been several results in 1 (real or complex) dimension for Gaussian random holomorphic functions where the hole probability has been determined. For a specific class of real Gaussian polynomials of even degree 2n, Dembo, Poonen, Shao and Zeitouni have shown that for the event where there are no real zeros, $E_n$, the $\lim_{n\rightarrow \infty}\frac{Prob (E_n)}{\log (n)} n^{-b}= -b, \ b\in [0.4, 2]$, [@DemboPoonenShaoZeitouni02]. Hole probability for the complex zeros of a Gaussian random holomorphic function is a quite different problem. Let $Hole_r=\{f$, in a class of holomorphic functions, such that $\forall z\in B(0,r), \ f(z)\neq 0\}$. For the complex zeros in one complex dimension, there is a general upper bound for the hole probability: $Prob (Hole_r)\leq e^{-c \mu (B(0,r))}, \ \mu(z)= E[Z_{\psi_\omega}]$ as in theorem \[Onemoment\], [@Sodin00]. In one case this estimate was shown by Peres and Virag to be sharp: $Prob(Hole_r) = e^{-\frac{ \mu(B(0,r))}{24}+ o(\mu(B(0,r)))}$, [@PeresVirag04]. These last two results on hole probability might suggest that when the random holomorphic functions are invariant with respect to the local isometries, thus ensuring that $E[Z_\omega]$ is uniformly distributed on the manifold, the rate of decay of the hole probability would be the same as that which would be arrived at if the zeros where distributed according to a Poisson process. However, as the zeros repel in 1 dimension, [@Hannay96], one might expect there to be a quicker decay for hole probability of a random holomorphic function. This is the case for random holomorphic functions whose expected zero set is uniformly distributed on ${{\mathbb C}}^1$, [@SodinTsirelison03] : $$Prob(Hole_r)\leq e^{-c_1 r^4}= e^{-c \mu(B(0,r))^2}, \ and \ Prob(Hole_r)\geq e^{-c_2 r^4}= e^{-c \mu(B(0,r))^2}$$ The random holomorphic functions that we will study, can be written as $$\psi_\omega(z_1, \ z_2, \ldots, \ z_n)= {\displaystyle\sum}_{j\in {{\mathbb N}}^n}\omega_j \frac{z_1^{j_1} \cdot z_2^{j_2} \cdot \ldots \cdot z_n^{j_n}}{\sqrt{j_1 \cdot j_2 \cdot \ldots \cdot j_n}}= {\displaystyle\sum}_{j\in {{\mathbb N}}^n} \omega_j \frac{z^j}{\sqrt{j!}}$$ where $\omega_j$ are independent identically distributed standard complex gaussian random variables, and a.s. are holomorphic on ${{\mathbb C}}^n$. The second form is just the standard multi-index notation, and will frequently be used from here on out. Random holomorphic functions of this form are a natural link between Hilbert spaces of holomorphic functions on the reduced Heisenberg group and a similar Gaussian Hilbert Space. Further, these random functions will be the unique class (up to multiplication by a nonzero entire function) whose expected distribution of the zero set is: $$E[Z_\omega]= \frac{i}{2\pi}(dz_1\wedge d\overline{z}_1 + dz_2\wedge d\overline{z}_2+\ldots +dz_n\wedge d\overline{z}_n) .$$ The two main results of this paper are:\ \[Main\] If $$\psi_\omega (z_1, z_2 \ldots, z_n)= {\displaystyle\sum}_j \omega_j \frac{z_1^{j_1} z_2^{j_2} \ldots z_n^{j_n}}{\sqrt{j_1! \cdot j_n!}},$$ where $\omega_j$ are independent identically distributed complex Gaussian random variables, then for all $\delta > 0,$ there exists $c_{3,\delta}>0 \ and \ R_{n,\delta}$ such that for all $r> R_{n,\delta}$ $$Prob\left(\left\{\omega: \left|n_{\psi_\omega}(r) -\frac{1}{2}r^2 \right| \geq \delta r^2 \right\} \right) \leq e^{-c_{3,\delta} r^{2n+2}}$$ where $n_{\psi_\omega}(r)$ is the unintegrated counting function for $\psi_\omega$. \[Hole probability\] If $$Hole_r=\{ \omega:\forall z \in B(0,r), \ \psi_\omega (z)\neq 0 \},$$ then there exists $R_n, \ c_1, \ and \ c_2 >0$ such that for all $r>R_n$ $$e^{-c_2 r^{2n+2}} \leq Prob (Hole_r)\leq e^{-c_1 r^{2n+2}}$$ The proof of Theorems \[Main\] and \[Hole probability\], will use techniques from probability theory, several complex variables and an invariance rule for Gaussian random holomorphic functions which is derived from isometries of the reduced Heisenberg group. These results, using the mainly same techniques, were already proven in the case where n=1 by Sodin and Tsirelison, [@SodinTsirelison03]. [**Acknowledgement:**]{} I would like to thank Bernie Shiffman for many useful discussions. The link between random holomorphic functions, Gaussian Hilbert spaces and the reduced Heisenberg group ------------------------------------------------------------------------------------------------------- \[section:Background\] To develop the notion of a random holomorphic function on ${{\mathbb C}}^n$ we will need a way to place a probability measure on a space of holomorphic functions on ${{\mathbb C}}^n$. The definition we will use is that a random holomorphic function is a representative of a Gaussian field between two Hilbert spaces on the reduced Heisenberg group. Through this definition we will prove the crucial Lemma \[Invariance\] which gives a nice law to determine how random holomorphic functions behave under translation. Additionally, this definition is equivalent to defining a random holomorphic function as $\psi_\omega(z)= {\displaystyle\sum}_{j\in {{\mathbb N}}^n} \omega_j \left(\frac{z_1^{j_1} z_2^{j_2} \ldots z_n^{j_n}}{\sqrt{j_1! j_2! \ldots j_n! }}\right)$, where $\omega_j$ are independent identically distributed standard complex Gaussian random variables. We will start with the concept of a Gaussian Hilbert Space, as presented by Janson, [@Janson97], A Gaussian Linear space, $G$, is a linear space of random variables, defined on a probability space $(\Omega, d \nu )$, such that each variable in the space is Gaussian random variable. A Gaussian Hilbert Space, G is a Gaussian linear space that is complete with respect to the $L^2(\Omega, d\nu)$ norm. We will shortly apply these definitions to a Hilbert space of CR-holomorphic functions on the reduced Heisenberg group. The Heisenberg Group, as a manifold is nothing other than ${{\mathbb C}}^{n}\times {{\mathbb R}}$, and the reduced Heisenberg group is the circle bundle: $X=H^n_{red}= \left\{(z, \alpha), z\in {{\mathbb C}}^n, \alpha \in {{\mathbb C}}, \ |\alpha|= e^{\frac{-|z|^2}{2}} \right\}$. Consider holomorphic functions of the Heisenberg group, which are linear with respect to the $n+1^{st}$ variable. The restriction of these functions define functions on X. For functions on X there is the following inner product: ----------------------------------------------- ----------------------------------------------------------------------------------------------------- $\displaystyle{(F,G)= \int_X F \overline{G}}$ $\displaystyle{= \frac{1}{\pi^n} \int_X f(z) \overline{g(z)} |\alpha|^{2} d\theta (\alpha) dm(z) }$ $\displaystyle{= \frac{1}{\pi^n} \int_{{{\mathbb C}}^n} f(z) \overline{g(z)} e^{-|z|^2} dm(z) }$ ----------------------------------------------- ----------------------------------------------------------------------------------------------------- \ Here $dm$ is Lebesque measure. With respect to this inner product, $H_X=\{ F \in \mathcal{O}({{\mathbb C}}^{n+1}), \ F(z,\alpha) = f(z) \alpha, \ f \in {\mathcal{O}}({{\mathbb C}}^n) \}$ is a Hilbert Space, and $H_X \cong H^2({{\mathbb C}}^n, \ e^{-|z|^2} dm) $, as Hilbert Spaces. For $H_{X}, \ \left\{ \frac{z_1^{j_1}\cdot z_2^{j_2}\cdot \ldots z_n^{j_n} }{\sqrt{j_1! \cdot \ldots \cdot j_n! }} \alpha \right\}_{j\in {{\mathbb N}}^n}= \{ \psi_j(z) \alpha\}_{{{\mathbb N}}^n} $ is an orthonormal basis. The proof of this proposition is a straight forward computation. The isometries of the reduced Heisenberg group will play a crucial role in my computation of the hole probability. These isometries are of the form: $$\tau_{(\nu, \alpha)}: H^m_{red} \rightarrow H^m_{red}$$ $$\tau_{(\nu, \alpha)}: (\zeta, \beta) \rightarrow (\zeta + \nu, e^{-\zeta \overline{\nu} } \alpha \beta)$$ The inner product on $X$ is invariant with respect to the Heisenberg group law:\ ---------------- ----------------------------------------------------------------- $\tau^*\langle $= \frac{1}{\pi^n} \int \int f(\zeta+ \nu) \overline F,G\rangle $ {g(\zeta+ \nu)} |\beta \alpha e^{-\zeta \overline \nu}|^{2} d\theta \ dm(\zeta) $ $= \frac{1}{\pi^n} \int f(\zeta+ \nu) \overline{g(\zeta+ \nu)} e^{-|(\zeta + \nu)|^2} dm(\zeta + \nu) $ $= \frac{1}{\pi^n} \int f(\zeta) \overline{g(\zeta)} e^{-|\zeta|^2 } dm(\zeta) $ ---------------- ----------------------------------------------------------------- \ As such for $\alpha= e^{-\frac{|z|^2}{2}}$, $$\tau^* (\alpha \psi_j(z))= (\alpha e^{-{{\frac{1}{2}}}|\zeta|^2- z \overline{\zeta}} \psi_j(z+\zeta))= e^{-{{\frac{1}{2}}}|z+\zeta|^2-i\cdot Im (z \overline{\zeta})} \psi_j(z+\zeta)$$ and the collection of these, $\{e^{-{{\frac{1}{2}}}|z+\zeta|^2- i\cdot Im (z \overline{\zeta})} \psi_j(z+\zeta)\} $, is another orthonormal basis for $H_X$, as the inner product is invariant with respect to the group law. (Gaussian Hilbert Spaces) Let $G_{H_{X}}'= Closure(Span\left( \{\omega_j \psi_j(z) \alpha\}_{{{\mathbb N}}^n}\right))$, where the closure is taken with respect to the norm $ E[ (\| \cdot \|_{H_X})^2]^{{\frac{1}{2}}}$ and where $ \omega_j$ are independent identically distributed standard complex Gaussian random variables. $G_{H_{X}}'$ is not a Gaussian Hilbert space but is isometric to $G_{H_{X}}= Closure(Span\left( \{\omega_j \}_{{{\mathbb N}}^n}\right))$, which is. ------------ --------------------------------------------------------------------------------------- Of course, $\ \ \ \ \ \ \ \ H_X \ \ \ \ \ \rightarrow \ \ \ \ \ G_{H_X}'\ \ \ \ \ \ \ \ \ \rightarrow \ \ \ G_{H_X} $ $\ {\displaystyle\sum}a_j \psi_j(z)\alpha \longmapsto {\displaystyle\sum}a_j \omega_j \psi_j(z)\alpha \longmapsto {\displaystyle\sum}a_j \omega_j$ ------------ --------------------------------------------------------------------------------------- \ are isometries. $G_{H_X}'$ is in many ways more natural then $G_{H_X}$, and is closely related to random holomorphic functions. A Gaussian field is a linear isometry $L: H_X \rightarrow G_{H_X} $. As such, for all $\displaystyle{f\in H_X, L[f]= X_f, a \ standard \ complex \ Gaussian \ random \ variable}$\ $\displaystyle{ \ with \ Var = \| f \|_{H_X}^2}$. A Gaussian random function, is a representative for a Gaussian field $L$. In other words, $$if \ f\in H_X, L[f]=\langle \varphi_\omega , f \rangle_{H_X}$$ (Random holomorphic functions on ${{\mathbb C}}^n$ or (equivalently) Gaussian fields and functions between $H_{X}$ and $G_{H_{X}}$) Let $\psi_{\omega}(z) = {\displaystyle\sum}_{j\in {{\mathbb N}}} \omega_j \psi_j(z)= {\displaystyle\sum}_{j\in {{\mathbb N}}} \omega_j \frac{z^j}{\sqrt{j!}}$ , $\omega_j$ independent identically distributed standard complex Gaussian random variables. This will shortly be shown to a.s. be a holomorphic function on ${{\mathbb C}}^n$ by Theorem \[Convergence\]. Note that for $\alpha f(z)\in H_{X}, \ f(z)= {\displaystyle\sum}a_j \psi_j(z), \ \{a_j\}\in \ell^2$,$\langle \alpha \psi_\omega(z), \alpha f(z) \rangle= {\displaystyle\sum}\overline{a}_j \omega_j $, which is a complex Gaussian random variable with variance ${\displaystyle\sum}|a_j|^2= \|f\|_{H_{{{\mathbb C}}^n}} $, hence $\alpha \psi_\omega(z)$ is a random CR-holomorphic function on $X$. The variable $\alpha$ will be useful when we change bases. This occurs when we look at how random functions behave with respect to translation (Lemma \[Invariance\]). Abusing notation we will frequently drop it and we will call $\psi_\omega(z)$ a random holomorphic function on ${{\mathbb C}}^n$. There is a simple condition for when a function of the form ${\displaystyle\sum}\omega_j \psi_j(z)$ is a holomorphic condition, where $ \omega_j$ are independent identically distributed complex Gaussian Random variables. \[Convergence\] Let $\{\omega_j\}_{j\in {{\mathbb N}}}$ be a sequence of independent identically distributed, standard complex Gaussian random variables. If for $j\in {{\mathbb N}}, \newline \psi_j(z)\in \mathcal{O} ( \Omega)$, and for all $\ compact \ K\subset \Omega,$ $\displaystyle{{\displaystyle\sum}_{j\in {{\mathbb N}}} \max_{z \in K} |\psi_j(z)|^2< \infty}$ then for a.a.-$\omega, \ {\displaystyle\sum}_{j\in {{\mathbb N}}}\omega_j \psi_j(z)$ defines a holomorphic function on $\Omega$. This theorem can be proved easily by adapting a similar proof of convergence of “random sums” from [@Kahane85]. \[GRHF defnthm\] If $L$ is a Gaussian field, $L: H_X \rightarrow G_{H_X}$, and $\{ \phi_j\}$ is an orthonormal basis for $H_X$ then $L$ can be written as: $L[\cdot]= \langle \phi_X, \cdot \rangle $, where $\phi_X(z)= \sum X_i \phi_i$, and $\{X_i\} $ is a set of independent identically distributed standard Gaussian random variables. Let $L$ be a Gaussian random functional.\ Let $X_1 = L[\phi_1], \ X_2 = L[\phi_2], \ \ldots , \ X_j = L[\phi_j], \ldots $ We must only show that $X_j$ are independent identically distributed Gaussian Random variables, hence it suffices to prove independence as by the definition of Gaussian random field, $X_i, \ X_j$ are jointly normal, as $L\left[{\displaystyle\sum}a_j \psi_j\right]= {\displaystyle\sum}a_j X_j $ is normal.\ For $i\neq j$: ----- ------------------------------------------------------- $2$ $=E[|X_i +X_j|^2]= E[|X_i|^2]+ E[|X_j|^2] + E[X_i \overline{X}_j]+ E[X_j \overline{X}_i] $ $= 2 + E[X_i \overline{X}_j]+ E[X_j \overline{X}_i]$ ----- ------------------------------------------------------- \ Hence $Re(E[X_i \overline{X}_j])=0=Im(E[X_i \overline{X}_j])$, The result then follows.\ Common Results -------------- Let us briefly review properties of the zeros of random holomorphic functions. An elementary way to view the zeros of a holomorphic function is as a set: $Z_f= f^{-1}(\{0\})$, but this will be insufficient for my purpose, and we will instead view it as a (1,1) current. For $M^n$ an n dimensional manifold, and $f\in {\mathcal{O}}(M), \ f: M^n \rightarrow {{\mathbb C}}, \ f^{-1}(\{0\})$ is a divisor. Hence the regular points of $Z_f$ are a manifold, and by taking restriction we identify forms in $D^{(n-1,n-1)}_M$ with ones in $D^{(n-1,n-1)}_{Z_{f,reg}}$. As $Z_{f, reg}$ is an n-1 complex manifold, $\displaystyle{\int_{Z_{f, reg}} }$ is a (1,1) current on M, which we will denote $Z_f$ (abusing notation). As the singularities occur in real codimension 2. $Z_f= Z_{f,reg}$, and in general: \[Poincare-Lelong formula\]if $f\in{\mathcal{O}}(M^n)$, M an n complex manifold, then $Z_f= \frac{i}{2 \pi}\partial \overline{\partial} \log |f|^2$, as $(1,1)$ currents on M. Before we classify the atypical hole probability, we shall first describe the expected behavior. Many various forms of the following theorem have been proven, [@EdelmanKostlan], [@Kac43] and [@Sodin00]. For my purposes it is important that the proof is valid in n-dimensions, and for infinite sums. Many of the proofs resemble this one. After a conversation with Steve Zelditch, I was able to simplify a previously complicated argument into the current form. This simplification is already known to other researchers including Mikhail Sodin. For the following theorem let $\psi_j: \Omega \rightarrow {{\mathbb C}}$, $j \in \Lambda, \ \Lambda = \{0, 1, 2, \ldots, n \}$ or $\Lambda={{\mathbb N}}$, be a sequence of holomorphic functions on a domain of an n manifold to ${{\mathbb C}}$. \[Onemoment\] If $ E[|\psi_\omega|^2]={\displaystyle\sum}|\psi_j(z)|^2 \ converges \ locally \ uniformly \ in \ \Omega$\ then $E[Z_\omega] = {\frac}{i}{2\pi}\partial\overline{\partial} \log ||\psi(z)||_{\ell^2}^2$ Let $\beta \ \in \ D^{n-1,n-1}(\Omega)$ To simplify the notation, let $\beta=\phi \ dz_2 \wedge d\overline{z}_2 \wedge \ldots \wedge dz_n\wedge d\overline{z}_n$. -------------------------------------------- ------- ---------------------------------------------------------------- $ \langle Z_{\psi_\omega},\beta \rangle $ $ = $ $ \langle \frac{i}{2\pi} \partial\overline{\partial} \log (|\psi_\omega(z)|^2) , \beta \rangle$ $=$ $ \langle \frac{i}{2\pi} \log \left(|\psi_\omega(z)|^2\right) , \partial \overline{\partial} \beta \rangle$ $ =$ $ \langle \frac{i}{2\pi} \left(\log (||\psi(z)||_{\ell^2}^2) + \log \left(\frac{|\psi_\omega(z)|^2}{||\psi(z)||_{\ell^2}} \right)\right) , \partial \overline{\partial} \beta \rangle$ -------------------------------------------- ------- ---------------------------------------------------------------- Taking the expectation of both sides we compute: -------------------------------------------------------------- -------------------------------------------------------------------- $E[\langle Z_{\psi_\omega},\beta \rangle] = \frac{1}{2\pi}$ $ \int_{\omega} \int_{z \in \Omega} \log (||\psi(z)||_{\ell^2}^2) \frac{\partial^2 \phi} {\partial z_1 \partial \overline {z}_1} \ dm(z) \ d\nu(\omega)$ $+ \frac{1}{2\pi} \int_{\omega} \int_{z \in \Omega} \log \left(\frac{|\psi_\omega(z)|^2}{||\psi(z)||_{\ell^2}^2}\right) \frac{\partial^2 \phi} {\partial z_1 \partial \overline {z}_1} \ dm(z) \ d\nu(\omega)$ -------------------------------------------------------------- -------------------------------------------------------------------- The first term is the desired result (which by assumption is integrable and finite), while the second term will turn out to be zero. We first must establish that it is in fact integrable: ------------------------------------------------------------------ ------------------------------------------------------------------ $ \int_{z \in \Omega} \int_{\omega} |\log $ \left(\frac{|\psi_\omega(z)|^2}{||\psi(z)||_{\ell^2}^2} \right)$ \frac{\partial^2 \phi}{\partial z_1 \partial \overline {z}_1} d\nu(\omega) | \ dm(z)$ $\leq c \int_{z \in K} \int_{\omega} \left| \log \left(\frac{|\psi_\omega(z)|^2}{||\psi(z)||_{\ell^2}^2}\right) \ d\nu(\omega)\right| \ dm(z)$ $= c \int_{z \in K} \int_{\omega} \left| \log (|\omega'|^2) \right| \ d\nu(\omega') \ dm(z)$ ------------------------------------------------------------------ ------------------------------------------------------------------ where $\omega '$ is a standard centered Gaussian ($\forall z$), thusly proving integrability as: $$\int_{z \in \Omega} \int_{\omega} \left| \log (\frac{|\psi_\omega(z)|^2}{||\psi(z)||_{\ell^2}^2}) \right| \leq C \int_{|x|<1} | \log(x) | dm(x)+ c \int_{|x|>1} |xe^{-x^2}| dm(x)\leq c$$\ Finally, $$\int_\Omega \beta \wedge E[Z_{\psi_\omega}]= \frac{i}{2 \pi} \int_\Omega \partial \overline{\partial}\beta \log(\|\psi(z)\|_{\ell^2}^2) \ + \ \int_\Omega C \frac {\partial^2 \phi}{\partial z_i \partial \overline{z}_j} dm(z)$$ $$=\frac{i}{2 \pi} \int_{\Omega}\partial \overline{\partial} \beta \wedge \log(||\psi(z)||_{\ell^2}^2)$$ For $\psi_\omega $ a random holomorphic function on ${{\mathbb C}}^n$, $$E[Z_{\psi_\omega}] = \frac{i}{2\pi}(dz_1\wedge d\overline{z}_1 + dz_2\wedge d\overline{z}_2+\ldots +dz_n\wedge d\overline{z}_n)$$ In Theorem \[Onemoment\], we proved that the expected zero set is determined by the variance of a of the random function when evaluated at a point. More can be said: \[uniquenessOfIsoInvarGrhf\] For gaussian analytic functions the expected zero set determines the process uniquely (up to multiplication by nonzero holomorphic functions) on a simply connected domain. This theorem is proven in one dimension by Sodin [@Sodin00], and the same proof works in n-dimensions. Invariance of Gaussian random functions with respect to the isometries of the reduced Heisenberg group ------------------------------------------------------------------------------------------------------ \[InvarianceSection\] The invariance property of the random function in question with respect to the reduced Heisenberg group’s isometries plays a central role in proving that: $$\displaystyle{{\max_{z\in \partial B(0,r)}\left(\log(|\psi_\omega(z)| ) - {{\frac{1}{2}}}|z|^2\right)} = {\max_{z\in \partial B( \zeta ,r)} \left(\log(|\psi_\omega(z) |) - {{\frac{1}{2}}}|z|^2\right)} }$$ This invariance property, which was known and used in the 1 dimensional case, and makes sense (from the view that $\forall (z,\alpha)\in X, \ \alpha \psi_\omega(z)$) defines a standard complex Gaussian random variable for any fixed $z$. Apparently, however, there was no proof in the literature until recently, [@Sodin05]. I also independently came up with this same result by using the properties of the Heisenberg group, and this is presented here. This next result is that a random holomorphic function is well defined independent of basis chosen, and also will be shortly restated in order to give an important translation law for random holomorphic functions of ${{\mathbb C}}^n$. \[IndepOfBasis\] If $\{\phi_j \}_{j \in \Lambda} $ is an orthonormal basis for $H_X$ then there exists $\{\omega_j'\}_{j\in \Lambda}$ independent identically distributed standard complex Gaussian random variables such that $for \ all \ (z,\alpha) \in H^n_{red},$ $$\alpha \psi_\omega(z) = \phi_{\omega'}(z,\alpha), \ a.s.$$ where $\phi_{\omega'}(z,\alpha)= \sum \omega_j' \phi_j(z,\alpha)$. For all $j\in {{\mathbb N}}$, let $\omega_j' = \langle \alpha \psi_\omega(z), \phi_j (z,\alpha) \rangle_{H_X}$, which is a standard complex Gaussian random variable by Theorem \[GRHF defnthm\]. Further, for $j\neq k, \ \omega'_j$ and $\omega'_k$ are independent. Let $f\in H_X \Rightarrow f = \sum a_j \phi_j, \ \{a_j\}_{j\in {{\mathbb N}}} \in \ell^2$\ We now demonstrate that $\phi_{\omega'} = \psi_\omega$ as a Gaussian field: ------------------------------------- ---------------------------------------------------------------- $\langle \phi_{\omega'}, f\rangle $ $=\sum \overline{a}_j \omega_j'$ $\langle \psi_{\omega}, f\rangle $ $= \langle \psi_{\omega}, \sum a_j \phi_j \rangle$ $= \sum \overline{a}_j \langle \psi_{\omega}, \phi_j \rangle $ $= \sum \overline{a}_j \omega_j' $ ------------------------------------- ---------------------------------------------------------------- As $H_X= H_X^*$, for evaluation maps $eva_{(z_0,\alpha_0)}= \langle \sum b_n \phi_j, \cdot \rangle,$$ \sum b_n \phi_j\in H_X$ and therefore by the above work: $\langle \phi_{\omega'}, \sum \overline{b}_j \phi_j \rangle = \sum \omega_j' b_j = \sum \omega'_j \phi_j(z_0,\alpha_0)= \sum \omega_j \alpha_0 \psi_j(z_0)$ A Gaussian random function is invariant with respect to $\tau$ if both $\langle\cdot, \psi_\omega \rangle$ and $\langle\cdot, \tau^* \psi_\omega \rangle$ induce the same Gaussian field.\ A random CR-holomorphic function on X, will be invariant with respect to isometries of X. These will in turn be important for random holomorphic functions on $ {{\mathbb C}}^n$, which is illustrated in the following simple but important lemma. Let $\tau(z, \alpha)= (z+ \zeta, e^{-z \overline{\zeta}} \alpha \beta ), \ |\beta|=e^{-\frac{|\zeta|^2}{2}}$. Recall that $\tau$ is an isometry of $H_X$. \[Invariance\] For all $z \in {{\mathbb C}}^n,$ there exists $\ \omega_j'$ independent identically distributed standard complex Gaussian random variables, such that $$\ e^{-{{\frac{1}{2}}}|z|^2} \psi_\omega(z) = e^{-{{\frac{1}{2}}}|z+ \zeta|^2 - i \cdot Im(z\overline{\zeta})}\psi_{\omega'}(z+\zeta)$$\ Here, $\zeta$ is any fixed complex number and we set $\beta=e^{-|\zeta|^2}$. Both $\{\alpha \psi_j(z)\} $ and $\{\tau^*( \alpha \psi_j(z))\}$ are orthonormal bases of $H_X$, and they therefore induce the same Gaussian random function, as these are well defined independent of basis by Lemma \[IndepOfBasis\]. Hence ---------------------------------------- -------------------- ------------------------------------------------------------------------------ $\displaystyle{\alpha \psi_\omega(z)}$ $\displaystyle{=}$ $\alpha \beta e^{-z\overline{\zeta}}\psi_{\omega'}(z+\zeta)$ $\displaystyle{e^{-|z|^2} $\displaystyle{=}$ $\displaystyle{e^{-{{\frac{1}{2}}}(|z|^2+ |\zeta|^2 + 2 z \overline{\zeta})} \psi_\omega(z)}$ \psi_{\omega'}(z+\zeta)}$ $\displaystyle{=}$ $\displaystyle{e^{-{{\frac{1}{2}}}|z + \zeta|^2 - i \cdot Im( z \overline{\zeta})} \psi_{\omega'}(z+\zeta)}$ ---------------------------------------- -------------------- ------------------------------------------------------------------------------ The random variable: $\displaystyle{{\max_{z\in \partial B(0,r)} \left(\log(|\psi_\omega(z) |) - {{\frac{1}{2}}}|z|^2 \right) } }$ is invariant with respect to $\tau^*$. In other words $$\displaystyle{{\max_{z\in \partial B(0,r)}\left(\log( |\psi_\omega(z)| ) - {{\frac{1}{2}}}|z|^2\right)} = {\max_{z\in \partial B( \zeta ,r)} \left(\log( |\psi_\omega(z) |) - {{\frac{1}{2}}}|z|^2\right)} }$$ This corollary just specializes the previous lemma as,\ ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------ $\displaystyle{\max_{z\in \partial B(0,r)}} \left(\log | $= \psi_\omega(z) | - {{\frac{1}{2}}}|z|^2\right)$ \displaystyle{\max_{z\in \partial B(0,r)}} \left(\log| \psi_\omega(z) | - \log(e^{{{\frac{1}{2}}}|z|^2})\right)$ $=\displaystyle{\max_{z\in \partial B(0,r)}} \log(|\alpha \psi_\omega(z) |), \ |\alpha|= e^{-{{\frac{1}{2}}}|z|^2}$ $=\displaystyle{\max_{z\in \partial B(0,r)}} \log( \tau^*(|\alpha \psi_\omega(z) | ))$ $= \displaystyle{\max_{z\in \partial B(0,r)}} \log( |\beta \psi_\omega(z+\zeta) | ), \ |\beta|= e^{-{{\frac{1}{2}}}|z+\zeta|^2}$ $=\displaystyle{\max_{z\in \partial B(\zeta,r)}} \log( (|\beta' \psi_\omega(z) | )), \ |\beta'|= e^{-{{\frac{1}{2}}}|z|^2}$ $=\displaystyle{\max_{z\in \partial B(\zeta,r)}} \left(\log |\psi_\omega(z)|- {{\frac{1}{2}}}|z|^2 | \right)$ ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------ \ An estimate for the growth rate of random holomorphic functions on the reduced Heisenberg group ----------------------------------------------------------------------------------------------- In this section we begin working towards my main results. Lemma \[Anchises\] is interesting in and of itself as it proves that random functions for $H_X$ are of finite order 2, a.s. From hence forth we will work with ${{\mathbb C}}^n$, for any one fixed n. A family of events $\{E_r\}_{r\in {{\mathbb R}}^+}$, dependent on r, will be called a small family of events if exists $R, \ and \ c>0, \ such \ that \ $ for all $r>R, \ Prob(E_r)\leq e^{-cr^{2n+2}}$. We will be using properties of Gaussian random holomorphic functions to deduce typical properties of functions, and the size of the family of events where these typical properties will not work will always be small. Let $\displaystyle{M_{r, \omega}=\max_{\partial B(0,r)} \log |\psi_\omega(z)|}$ We will be able to compute this, adapting a strategy that Sodin and Tsirelison, [@SodinTsirelison03], used to solve the analogous 1 dimension problem, by using the Cauchy Integral Formula in conjunction with some elementary probability theory and computations: \[Gauss\] Let $\omega$ be a standard complex Gaussian RV (mean 0, Variance 1), with a probability distribution function $d\nu(\omega)$\ ------- ---------------------------------------------------------------- then: a-i) $\nu(\{ \omega : | \omega| \geq \lambda \}) = e^{-\lambda^2}$ a-ii) $\nu(\{ \omega : | \omega| \leq \lambda \}) =1- e^{-\lambda^2}\in [\frac{\lambda^2}{2}, \lambda^2], if \lambda \leq 1$ ------- ---------------------------------------------------------------- b\) If $\{\omega_j \}_{j\in {{\mathbb N}}^n}$ is a set of of independent identically distributed standard Gaussian random variables, then $\nu(\{ \omega: |\omega_j| < (1+ \varepsilon)^{|j|}\})=c>0$. Here, and throughout this paper for $j\in {{\mathbb N}}^n, \ |j|:=\sum j_i $ Lemma \[Gauss\]-a) is a straight forward computation using that the probability distribution for a standard complex Gaussian. of b) $\nu (\{\omega_j: |\omega_j| < (1+\varepsilon)^{|j|} \})= 1-e^{-(1+\varepsilon)^{2|j|}}$\ ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------ $\left(\nu(\{ $\Leftrightarrow {\displaystyle\prod}_{j\in {{\mathbb N}}^n, |j|=0, }^{|j|=\infty} 1-e^{-(1+\varepsilon)^{2|j|}} = c $ \omega: |\omega_j| < (1+ \varepsilon)^{|j|}\})>0\right)$ $\Leftrightarrow c {\displaystyle\sum}|j|^n |\log\left(1-e^{-(1+\varepsilon)^{2|j|}} \right)|<\infty $ ---------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------ \ as there are about $c |j|^n, \ j\in {{\mathbb N}}^n$ with a fixed value of $|j|$. $\forall |x|<1, \ \log(1-x)= - \int {\displaystyle\sum}(x)^m ={\displaystyle\sum}_{m\geq 0} \frac{x^{m+1}}{m+1}$ Therefore, ${\displaystyle\sum}|j| \left|\log\left(1-e^{-(1+\varepsilon)^{2|j|}} \right)\right| \leq c {\displaystyle\sum}_m m^n e^{-(1+\varepsilon)^{2m}}< \infty $ The following lemma is needed twice in this paper, including in the proof of Lemma \[Anchises\]. \[CalcII\] If $j \in {{\mathbb N}}^{+,n}$ then $\frac{|j|^{|j|}}{j^{j}} \leq n^{ |j|}$ Let $u_k = \frac{j_k}{|j|}\geq 0$, hence $\sum_{k=1}^n u_k =1$.\ As $\sum u_k \delta_{u_k}$ is a probability measure: --------------------------------------- ------------------------------- $\displaystyle{ \sum_{j=1}^n u_k \log $\displaystyle{\leq \log \sum \left(\frac{1}{u_k}\right)}$ \frac{1}{u_k} u_k }$, by Jensen’s inequality. $\displaystyle{= \log (n)}$ --------------------------------------- ------------------------------- \ ---------- -------------------------------------------------- Hence, $ $\displaystyle{ \geq \prod_k ( u_k)^{-|j| u_k}}$ n^{|j|}$ $=\frac{ |j|^{|j|}}{ j^j } $ ---------- -------------------------------------------------- \[Anchises\] (Probabilistic Estimate on the Rate of growth of the maximum of a random function on ${{\mathbb C}}^n $)\ For all $\delta >0$, $$E_{r,\delta}:= \left\{\omega : \left|\frac{log (M_{r,\omega})}{r^2}- \frac{1}{2} \right| \geq \delta \right\} \ is \ a \ small\ family \ of \ events$$ We will first prove that: $\nu(\{\omega : \frac{log (M_{r,\omega})}{r^2} \geq \frac{1}{2}+ \delta \} \leq e ^ {- c_{\delta, 1} r^{2n+2} }$ and we will prove this by specifying a set of measure almost 1 where the max grows at the appropriate rate.\ ------------------------------------ -------------------------------------------- ---------------------------------- Let $\Omega_r$ be the event where: $i) \ |\omega_j|\leq $ |j| \leq e^{\frac{\delta r^2}{4}},$ 2 e \cdot n \cdot r^2 $ $ii) |\omega_{j}|\leq 2^{\frac{|j|}{2}}, $ $ \ |j| > 2 e \cdot n \cdot r^2$ ------------------------------------ -------------------------------------------- ---------------------------------- \ ------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\nu(\Omega_r^c)$ $\displaystyle{\leq \sum_{|j|\leq 2 e \cdot n r^2} \nu ( \{|\omega_j|> e^{\frac{\delta r^2}{4}} \})+ \sum_{|j| > 2 e \cdot n r^2} \nu ( \{|\omega_j|> 2^{\frac{|j|}{2}} \}) } $ $\displaystyle{ \leq c_n r^{2n} e^{\left(-e^{\frac{\delta r^2}{2} }\right)}+{\displaystyle\sum}_{|j|_> 2 e \cdot n r^2} e^{-2^{|j|}}}$ $\leq e^{-e^{cr^2}} + c e^{-2^{c r^2}}, \ \forall r>R_0 $ $ \leq e^{-e^{c r}}$ ------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- We now have that $\Omega_r^c$ is contained in a small family of events (and in fact could make a stronger statement on the rate of decay in terms of r). It now remains for me to show that $\forall \omega\in \Omega_r, \ \frac{\log|M_{r,\omega}|}{r^2}\leq {{\frac{1}{2}}}+ {{\frac{1}{2}}}\delta.$ $\forall \omega \in \Omega_r,$ we have that:\ $M_{r,\omega} \leq {\displaystyle\sum}_{|j| = 0}^{|j|\leq 4 e \cdot n (\frac{1}{2} r^2)} |\omega_j| \frac{ |z|^{j}}{\sqrt{j!}}+ {\displaystyle\sum}_{|j| > 4 e \cdot n (\frac{1}{2}r^2)} |\omega_j| \frac{ |z|^{j} }{\sqrt{j!}} = \sum^1 + \sum^2$\ \ Using the Cauchy-Schwartz inequality: -------------------------- ---------------------------------------------------------------------------------------------------------------------------- ${\displaystyle\sum}^1 $ $ \leq (e^{\frac{1}{4} \delta r^2} ) \sqrt{c (r^2)^n} \left({\displaystyle\sum}_j \frac{|z^{2 j}|}{j!}\right)^\frac{1}{2}$ $\leq c_n e^{\frac{\delta r^2}{4}} r^n e^{\frac{1}{2}r^2} $ $ \leq e^{(r^2)(\frac{1}{2}+\frac{1}{3}\delta)}, \ \forall r> R_{n,\delta}$. -------------------------- ---------------------------------------------------------------------------------------------------------------------------- -------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $ {\displaystyle\sum}^2$ $ \leq {\displaystyle\sum}_{|j| > 4 e \cdot n r^2} (2)^{\frac{|j|}{2}} \frac{ |z^{j}| }{\sqrt{j!}}$ $ \leq {\displaystyle\sum}_{|j| > 4 e \cdot n r^2} (2)^{\frac{|j|}{2}} \left(\frac{|j|}{4 e n}\right)^{\frac{|j|}{2}} \prod_k \left(\frac{e}{j_k}\right)^{\frac{j_k}{2}} $, by Sterling’s Formula $\leq C $, by Lemma \[CalcII\]. -------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Hence, $\forall \omega \in \Omega_r, \ \log(M_{r,\omega})\leq (\frac{1}{2}+\frac{1}{2}\delta)r^2 $ It now remains for me to show that: $$\forall \delta< \Delta, \ \nu\left( \left\{\omega : \frac{log (M_{r,\omega})}{r^2} \leq \frac{1}{2}- \delta \right\}\right) \leq e ^ {- c_{\delta, 2} r^{2n+2} }$$ which we will do by using Cauchy’s integral formula to transfer information on $M_{r, \omega}$ to individual coefficients $\omega_j$. It suffices to prove this result only for small $\delta$ as $\delta< \delta' \Rightarrow E_{\delta', r}\subset E_{\delta, r}$. The constant $\Delta$ can be explicitly determined. It will be most convenient to prove this result for the polydisk, where the Cauchy Integral Formula applies. The notation for the polydisk is the standard one: $P(0,r):=\{z\in {{\mathbb C}}^n: \ \forall i, \ |z_i|<r \} $ Let $\displaystyle{M_{r,\omega}'= \max_{z\in P(0,r)} |\psi_\omega(z)}|$ The corresponding claim for a poly disk is that: $$\displaystyle{M_{r,\omega}'}\geq \frac{n}{2} r^2- \delta r^2$$except for a small family of events. We will now look at the probability of the event consisting of $\omega $ such that: $$\log(M_{r,\omega}') \leq \left(\frac{n}{2}-\delta \right)r^2$$ By Cauchy’s Integral Formula: $\left|\frac{\partial^{ j } \psi_\omega}{\partial z^j}\right|(0) \leq j! M_{r,\omega}' r^{-|j|}$\ By direct computation using the definition of $ \psi_\omega(z) $ in terms of a power series:$$\left|\frac{\partial^{ j } \psi_\omega}{\partial z^j}\right|(0)= |\omega_j| \sqrt{j!}$$ Therefore: $|\omega_j| \leq c M_{r,\omega}' \sqrt{j!} r^{-|j|},$\ and using Sterling’s formula ($j! \approx \sqrt{2 \pi} \sqrt{j} j^j e^{-j} $), we get that: $|\omega_j|\leq (2 \pi)^{\frac{n}{2}} (\prod_k j_k^{\frac{1}{4}}) e ^{{(\frac{n}{2}-\delta)r^2}+\sum \frac{{j_k}}{2} \log({j_k}) -(|j|) \log r- \frac{|j|}{2}} $, $\forall k , \ j_k\neq 0.$\ \ The $(2\pi)^{\frac{n}{2}}j^{\frac{1}{4}}$ term will not matter in the end so we will focus instead on the exponent.\ -------------------- -------------------------------------------------------------------------------------------------------------------------------------- $\displaystyle{A}$ $=\displaystyle{ \left(\frac{n}{2}-\delta\right)r^2-\frac{|j|}{2}+ {\displaystyle\sum}_k \left(\frac{j_k}{2}\log(j_k)\right) - (|j|) \log(r)}$ $\displaystyle{= {\displaystyle\sum}_{k=1}^{k=n} \left(\frac{j_k}{2}\right) \left(\left(1-\frac{2\delta}{n}\right)\frac{r^2}{j_k}-1+ \log(j_k) - 2 \log(r)\right)}$ -------------------- -------------------------------------------------------------------------------------------------------------------------------------- \ Let $j_k = \gamma_k r^2$ --------------------- ------------------------------------------------------------------------ $\displaystyle{ A}$ $\displaystyle{ = {\displaystyle\sum}_{k=1}^{k=n} \left(\frac{\gamma_k r^2}{2}\right) \left(\left(1-\frac{2\delta}{n}\right)\frac{1}{\gamma_k }-1+ \log(\gamma_k) \right)}$ $\displaystyle{= - \delta r^2+ n f(\gamma_k) \frac{r^2}{2}}$, where $f(\gamma_k) = 1- \gamma_k + \gamma_k \log (\gamma_k)$ --------------------- ------------------------------------------------------------------------ \ $f(\gamma_k) = (1-\gamma_k)^2 - (1-\gamma_k)^3 + o((1-\gamma)^4)$ near 1. Hence $\exists \Delta$ such that $\forall \delta\leq \Delta$ if $\gamma_k \in \left[1-\sqrt \frac{\delta}{n}, 1+\sqrt \frac{\delta}{n} \right]$ then $A\leq \frac{-\delta r^2}{2} $ Therefore for $j$ as above $|\omega_j|\leq (2 \pi)^{\frac{n}{2}} (\prod_k j_k^{\frac{1}{4}}) e^{-\frac{\delta r^2}{2}}\leq c r^{\frac{n}{2}} e^{-\frac{\delta r^2}{2}}.$ This holds true for all $\omega_j, \ j$ in terms of $r$. Specializing our work for large $r$, we have that $\forall \varepsilon>0, \ \exists R,$ such that $\forall r>R, |\omega_j| \leq e^{-\frac{1}{2}(\delta-\varepsilon) r^2}$. Note that the factor of $\varepsilon$ is used to compensate for the $\sqrt{2 \pi} j_k^{\frac{1}{4}}$ terms. The probability of which may be estimated using Lemma \[Gauss\] as: $$\nu(\{\omega: |\omega_{j}|\leq e^{-\frac{1}{2}(\delta-\varepsilon) r^2}\}) \leq e^{-(\delta-\varepsilon) r^2}.$$\ Hence $E_{\delta,r} $ is a small family of events as:\ $\nu(\{ \omega : \log M_{r,\omega}' \leq (\frac{1}{2}-\delta)r^2 \})$ $\leq \nu( \{ \omega : |\omega_{j}| \leq e^{-{{\frac{1}{2}}}(\delta-\varepsilon) r^2}, \ and \ j_k \ \in [(1- \sqrt \frac{\delta}{n}) r^2, \ (1+ \sqrt \frac{\delta}{n})r^2 ] \})$ $ \leq (e^{-(\delta-\varepsilon) r^2})^{(2\sqrt\frac{\delta}{n} r^{2})^n}= e^{-2 ^n(1+o(\delta))\delta^{\frac{n+2}{2}} r^{2n+2} }= e^{-c_{1,\delta} r^{2n+2}}$, using the independence of $\omega_{j}$.\ $\displaystyle{ M_{r,\omega}\geq M_{\frac{1}{\sqrt{n}}r,\omega}'\geq {{\frac{1}{2}}}r^2 -\delta r^2}$, except for small events thus proving the lemma. Results of this type can deceive one into thinking of random holomorphic functions as $ e^{{{\frac{1}{2}}}z^2}$. This absolutely is not the case, as they are weakly invariant with respect to the isometries of the reduced Heisenberg group. In particular, an analog of the previous theorem holds at any point (whereas this will be false for $ e^{{{\frac{1}{2}}}z^2}$). \[ValueEstimate\] For all $\delta > 0$ and $z_0 \in \overline{B(0,r)}\backslash B(0,\frac{1}{2}r)$, there exists $\zeta\in B(z_0,\delta r)$ s.t. $$\log |\psi_\omega(\zeta)| > \left(\frac{1}{2} - 3 \delta \right) |z_0 |^2$$ except for on a small family of events. By Lemma \[Anchises\]: $$\nu(\{\omega : \max_{z \in \partial B(0,r)}\log|\psi_\omega(z)|-\frac{1}{2}|z|^2\leq - \delta r^2 \})\leq e^{-c r^{2n+2} }$$ By Lemma \[Invariance\], we have that for $z_0 \in B(0,r)\backslash B(0,{{\frac{1}{2}}}r), \ z\in B(z_0,\delta r)$: $$\nu(\{\omega : \max_{z \in \partial B(0,\delta r)}\log|\psi_\omega(z-z_0)|-\frac{1}{2}|z-z_0|^2\leq - \delta (\delta r)^2 \})\leq e^{-cr^{2n+2} }$$\ Hence, $\exists \ z \in B(z_0,\delta r)$ s.t. $\log|\psi_\omega(z-z_0)|-\frac{1}{2} |z-z_0|^2\geq - \delta (\delta r)^2$, except for a small family of events. By hypothesis, $|z_0|\in [\frac{1}{2}r, r)$, hence $|z-z_0| \leq \delta r \leq \frac{1}{4}r= \frac{r}{2}\frac{1}{2}\leq \frac{1}{2}|z_0|$ Hence, $|z_0 - z|^2 \geq |z_0|^2 -\delta r^2 \geq |z_0|^2 (1-2\delta)$ Without loss of generality assume that $\delta< \frac{1}{4}$. [cl]{} $\log |\psi_\omega(z-z_0)|$ & $\geq \frac{1}{2} |z-z_0|^2-\delta^3 r^2\geq |z_0|^2 \frac{1}{2} (1-2\delta)^2 -4 \delta^3 |z_0|^2$\ & $\geq \frac{1}{2} |z_0|^2 -2\delta |z_0|^2- \frac{1}{4}\delta |z_0|^2$\ & $ \geq \frac{1}{2} |z_0|^2- 3 \delta |z_0|^2 $ \ And, setting $\zeta = z-z_0$ this is what we set out to prove. Using that $\log \displaystyle{ \max_{B(0,r)}} |\psi_\omega|$ is an increasing function in terms of r, we have the following corollary: \[GrowthOfLogMax\] For all $\delta > 0$ ---- ---------------------------------------------------------------------------------------------------------------------------------- a) $\displaystyle{Prob\left( \left\{ \omega : \ \lim_{r\rightarrow \infty} \frac{(\log \max_{z\in B(0,r)} |\psi_\omega (z)|)-{{\frac{1}{2}}}r^2}{r^2} \notin [-\delta, \delta] \right\}\right) =0}$ b) $\displaystyle{Prob\left( \left\{ \omega : \ \lim_{r\rightarrow \infty} \frac{(\log \max_{z\in B(0,r)} |\psi_\omega (z)|)-{{\frac{1}{2}}}r^2}{r^2} \neq 0 \right\}\right) =0}$ ---- ---------------------------------------------------------------------------------------------------------------------------------- \ This corollary as well as corollary \[FiniteOrder2\] have already been proven by more direct methods, [@ShiffmanZelditchVarOfZeros]. Part b follows immediately from part a, which we now prove: Let $E_{\delta, R} =\{\omega: \frac{\log \max_{B(0,R)} \psi_\omega(z) - {{\frac{1}{2}}}R^2}{R^2} \notin [-\delta, \delta ]\}$ Let $R_m= r + \delta (m+1) r, \ r>0$ Let $s_m \in [R_{m-1}, R_m ].$\ Claim: $\forall m> M_\delta, \ \forall s_m, \ E_{\delta, s_m} \subset E_{\frac{1}{3}\delta, R_m} \bigcup E_{\frac{1}{3}\delta, R_{m-1}}$ Let $M_\delta = \max \{ M_{1,\delta}, \ M_{2,\delta}\}$, which may be specifically determined.\ Case i: for $\omega\in E_{\delta, s_m}, \ \log \displaystyle{\max_{B(0,s_m)}} \psi_\omega \geq {{\frac{1}{2}}}s_m^2 +\delta s_m^2$\ ------------------------------------------------------- ------------------------------------------------------------------------------- $\log \displaystyle{ \max_{B(0,R_m)}} |\psi_\omega| $ $\geq {{\frac{1}{2}}}s_m^2 +\delta s_m^2$, $\geq {{\frac{1}{2}}}(1+ m\delta )^2 r ^2 +\delta (1+ m\delta )^2 r^2$ $>{{\frac{1}{2}}}R_m^2 +\frac{1}{3} \delta R_m^2, \ \forall m> M_{1,\delta} $ ------------------------------------------------------- ------------------------------------------------------------------------------- Therefore, $\omega\in E_{\frac{\delta}{3}, R_m} $\ Case ii: for $\omega\in E_{\delta, s_m}, \ \log \displaystyle{\max_{B(0,s_m)}} \psi_\omega \leq {{\frac{1}{2}}}s_m^2 -\delta s_m^2$\ ----------------------------------------------------------- ------------------------------------------------------------------------------------------- $\log \displaystyle{ \max_{B(0,R_{m-1})}} |\psi_\omega| $ $\leq {{\frac{1}{2}}}s_m^2 - \delta s_m^2$ $\leq {{\frac{1}{2}}}(1+ (m-1) \delta )^2 r ^2 - \delta (1+ m\delta )^2 r^2$ $\leq {{\frac{1}{2}}}R_{m-1}^2 -\frac{1}{3} \delta R_{m-1}^2, \ \forall m> M_{2,\delta} $ ----------------------------------------------------------- ------------------------------------------------------------------------------------------- Therefore, $\omega\in E_{\frac{\delta}{3}, R_{m-1}},$ Hence, $\forall m> M_\delta$ and $ \forall s \in [R_{m-1}, R_m], \ E_{\delta, s}\subset E_{\frac{1}{3}\delta, R_{m-1}} \cup E_{\frac{1}{3}\delta, R_m} $ Hence, $Prob (\bigcup_{s\in [R_{m-1}, R_m]} E_{\delta, s}) \leq 2 e^{-c_\delta r^{2n+2} m^{2n+2}}$, and ${\displaystyle\sum}_{m\in {{\mathbb N}}} Prob (\bigcup_{s\in[R_{m-1}, R_{m}]} E_{\delta, s}) = {\displaystyle\sum}_{m\in {{\mathbb N}}} e^{-c_\delta m^{2n+2}}<\infty$, and the result follows. The Second main lemma --------------------- Essentially to prove the main theorem that we are working towards we need only one more interesting lemma, Lemma \[Athena\], in which we will give an estimate for $\int \log |\psi_\omega|$. This will be proved first by obtaining a crude estimate for $\int |\log |\psi_\omega||$, except for a small family of events, and then by proving facts about the Poisson Kernel, which will allow me to approximate using Riemann integration the first integral with values of $\log|\psi_\omega(z)|$ at a number of fairly evenly spaced points. In order to establish notation I state the following standard result: \[Kundun\] For $\zeta \in B(0,r)$, h a harmonic function\ $$h(\zeta) = \int_{\partial B(0,r)} P_r(\zeta,z) h(z) d\sigma_r (z)$$ where $d\sigma_r$ is the Haar measure of the sphere $ S_r= \partial B(0,r)$ and $P_r$ is the Poisson kernel for $B(0,r)$. A proof of this can be found in many standard text books, [@Krantz]. It is convenient to normalize $\sigma_r$ so that $\sigma_r(S_r)=1$. For this normalization, the Poisson Kernel is: $$P_r(\zeta,z)= r^{2n-2}\frac{(r^2-|\zeta|^2)}{|\zeta-z|^{2n}}$$ \[oakley\] For all $r>R_n, \ \int_{\partial B(0,r)} |\log(|\psi_\omega|)| d \sigma_r(z) \leq (3^{2n}+1) r^2$ except for a small family of events. By Lemma \[Anchises\], with the exception of a small family of events, there exists $\zeta_0 \in \partial B(0,\frac{1}{2}r)$ such that $\log(|\psi_\omega (\zeta_0)|) > 0$. Combining this with Proposition \[Kundun\], $$\int_{\partial B(0,r)} P_r(\zeta_0, z) \log(|\psi_\omega(z)|) d \sigma_r (z)\geq \log(|\psi(\zeta_0)|)\geq 0.$$ Hence, $$\int_{\partial (B(0,r))} P_r(\zeta_0, z) \log^-(|\psi_\omega(z)|) \leq \int_{\partial (B(0,r))} P_r(\zeta_0, z) \log^+(|\psi_\omega(z)|)$$ Since $\zeta \in \partial B(0,{{\frac{1}{2}}}r)$ and $z \in \partial B(0,r)$, we have: ${{\frac{1}{2}}}r \leq |z- \zeta | \leq \frac{3}{2} r$.\ Hence by using the formula for the Poisson Kernel, $$\frac{1}{3}\left(\frac{2}{3}\right)^{2n-2} \leq P_r(\zeta, z)\leq (2)^{2n-2} 3$$ Therefore, $ \int_{\partial B(0,r)}\log^+(|\psi_\omega(z)|) d \sigma_r (z)\leq \log M_r \leq (\frac{1}{2}+ \delta) r^2\leq r^2$, except for a small family of events, by Lemma \[Anchises\].\ ------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------- $\int_{\partial (B(0,r))} P(\zeta_0, z)\log^+(|\psi_\omega(z)|)$ $\leq \sigma_r(S_r) \log (M_r) 3 (2)^{2n-2} $ $\leq 3 (2)^{2n-2} r^2 $ $\int_{\partial (B(0,r))} \log^-(|\psi_\omega(z)|)d \sigma_r(z)$ $\leq \frac{1}{\min_{z} P(\zeta_0, z)} \int_{\partial (B(0,r))} P(\zeta_0, z)\log^+(|\psi_\omega(z)|) $ $\leq 3\left(\frac{3}{2}\right)^{2n-2} \int_{\partial (B(0,r))} P(\zeta_0, z)\log^+(|\psi_\omega(z)|) $ $\leq 9 \left(\frac{3}{2}\right)^{2n-2}(2)^{2n-2} r^2 $ $\leq 3^{2n} r^2 $ ------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------- \ And, the result follows immediately. As we are already able to approximate $\log |\psi_\omega(z)|$ at any finite number of points in order to use Reimann integration to prove Lemma \[Athena\] we will need to be able to choose “evenly” spaced points on the sphere, as chosen according to the next proposition: \[Sphere\] (A partition of a Sphere)\ If $(2n) m^{2n-1} = N $\ then $S_r^{2n}\subseteq {{\mathbb R}}^{2n}$ can be “divided” into measurable sets $\{I^r_1, I^r_2, \ldots, I^r_N\}$ such that: 1\) $\bigcup_j I^r_j = S_r$ 2\) $\forall j \neq k, \ I^r_j \bigcap I^r_k = \emptyset$, 3\) $diam(I^r_j)\leq \frac{\sqrt{2n-1}}{m}r= \frac{c_n}{N^{\frac{1}{2n-1}}}r$ Surround $S_r$ with 2n pieces of planes: $P_{+,1}, P_{+,2},\ldots, P_{+,n}, P_{-,1},\ldots P_{-,n}$, where $$P_{+,j}= \{ x\in {{\mathbb R}}^{2n+1} : \ ||x||_{L^\infty}=r, \ x_j=r \}$$ $$P_{-,j}= \{ x\in {{\mathbb R}}^{2n+1} : \ ||x||_{L^\infty}=r, \ x_j=-r \}$$ Subdivide each piece into $m^{2n-1}$ even $2n-1$ cubes, in the usual way, and denote these sets $R_1, \ldots, \ R_N$. Let $I^r_j= \{ x\in S_r: \ \lambda x \in R_n, \ \lambda > 0 \}$ By design, $\lambda\geq 1$ and $x, \ y \in I_j \Rightarrow \ d(x,y)< d(\lambda_1 x, \lambda_2 y)\leq \frac{2}{m} r= diam(R_j) $. These sets can be redesigned to get that $I^r_j \bigcap I^r_k = \emptyset$ , $j \neq k$ by carefully defining $R_i$ so that $R_j \bigcap R_k= \emptyset$. The following elementary result is less well known then others and will be very useful in proving Lemma \[Athena\]. Note this integration is with respect to w, which is not the same variable of integration that is used in Proposition \[Kundun\]. This is done because the goal of this section, Lemma \[Athena\], is to estimate a surface integral, which corresponds to integration with respect to the first variable. \[poison\]For $\kappa<1$ $$\int_{w \in S_{\kappa r}^n} P_r(w,z) d\sigma_{\kappa r}(w)= 1$$ $P_r(w,z)= r^{2n-2} \frac{r^2-|w|^2}{|z-w|^{2n}}$, $z \in \partial B(0,r)$ If $w\in S_{\kappa r}^{2n} \subseteq {{\mathbb R}}^{2n}$, then the poisson Kernel can be rewritten as a function of $|z-w|$, and as such $\forall \Upsilon \in U_n ({{\mathbb R}}^n), \ P_r(\Upsilon w, \Upsilon z )=P_r(w,z)$ Let $f(z)= \int_{w \in S_{\kappa r}^n} P_r(w,z) d\sigma_{\kappa r}(w)$ -------- ------------------------------------------------------------------------------------------ $f(z)$ $=\int_{w \in S_{\kappa r}^n} P_r(w,z) d\sigma_{\kappa r}(w)$ $= \int_{w \in S_{\kappa r}^n} P_r(\Upsilon w,\Upsilon z) d\sigma_{\kappa r}(w)$, by the above work. $= \int_{w \in S_{\kappa r}^n} P_r(\Upsilon w,\Upsilon z) d\sigma_{\kappa r}(\Upsilon w)$, as $d\sigma_{\kappa r}$ is invariant under rotations. $= \int_{w \in S_{\kappa r}^n} P_r (w, \Upsilon z) d \sigma_{\kappa r}(w)$, by a change of coordinates. $= f(\Upsilon z) $ -------- ------------------------------------------------------------------------------------------ \ Hence $f(z)= c, \ \forall z \in S_r^n $\ By switching the order of integration we compute that: $$1= \int_{w \in S_{\kappa r}^n} \int_{z \in S_r^n} P_r(w,z)d\sigma_r(z) d\sigma_{\kappa r}(w)= c$$ Now we are able to prove our final lemma. \[Athena\] For all $\Delta > 0,$ $ \left\{\omega : \frac{1}{r^2} \displaystyle\int_{z\in\partial B(0,r) } \log |\psi_\omega | d \sigma_r(z) \leq \frac{1}{2} - \Delta \right\}$ is a small family of events.\ It suffices to prove the result for small $\Delta$. Let $\Delta>0$. Let $a_n = \frac{1}{2(2n +2) (2n-1)}$. Set $\displaystyle{\delta=\left(\frac{1}{ \lambda} \Delta\right)^{(\frac{1}{a_n})}<\frac{1}{6}}, \ \lambda>0$ to be determined later. Choose $m \in {{\mathbb N}}$ such that writing $N = (2n) m^{2n-1}, \ \frac{1}{N}\leq \delta$. Let $\kappa=1-\delta^{a_n}$. Choose $I^{\kappa r}_j$ measurable subsets of $ S_{\kappa r}$ as in Proposition \[Sphere\].\ In particular: ---- ----------------------------------------------------------- 1) $S_{\kappa r} = \cup I^{\kappa r}_j$, a disjoint union. 2) ${\displaystyle\sum}\sigma_r(I^{\kappa r}_j)=1 $ 3) $diam(I^{\kappa r}_j)\leq \frac{c_n}{N^{\frac{1}{2n-1}}} \kappa r\leq c \delta^{\frac{1}{2n-1}}r$ ---- ----------------------------------------------------------- \ Let $\sigma_j= \sigma_{\kappa r}(I^{\kappa r}_j)$, which does not depend on r. For all $j$ fix a point $x_j \in I_j^{\kappa r}$ By Lemma \[ValueEstimate\], $\exists \zeta_j \in B(x_j,\delta r)$ such that $$\log (|\psi_\omega(\zeta_j)|) > \left(\frac{1}{2} - 3 \delta \right)|x_j|^2= \left(\frac{1}{2} - 3 \delta\right)\kappa^2 r^2$$ Except, of course, on N different small families of events (the union of which remains a small family of events). ----------------------------------------- -------------------------------------------------------------------------------- $\displaystyle{(\frac{1}{2}-3 \delta)}$ $\displaystyle{(1-\delta^{a_n})^2 r^2 \leq {\displaystyle\sum}_{j=1}^N \sigma_j \log(|\psi_\omega (\zeta_j)|)}$ $\displaystyle{\leq \int_{\partial B(0,r)} \left( {\displaystyle\sum}_j \sigma_j P_r(\zeta_j, z) \log(|\psi_\omega (z)|) d \sigma_r (z)\right) }$ $\displaystyle{= \int_{\partial (B(0,r))} \left({\displaystyle\sum}_j \sigma_j (P_r(\zeta_j, z) -1) \right) \log(|\psi_\omega(z)|) d \sigma_r (z)}$ $\displaystyle{ \ \ \ \ + \int_{\partial (B(0,r))} \log(|\psi_\omega (z)|) d \sigma_r (z)}$ ----------------------------------------- -------------------------------------------------------------------------------- \ Hence,\ --------------------------- ------------------------------------------------------------------------------------------------------- $\int_{\partial B(0,r)} $ $ \log(|\psi_\omega|) d \sigma_r$ $\geq (\frac{1}{2}-3 \delta) (1-\delta^{a_n})^2 r^2 -\int |\log|\psi_\omega|| d \sigma_r \cdot \max_z |\sum_j \sigma_j (P_r(\zeta_j, z) -1) | $ $\geq (\frac{1}{2}-3 \delta) (1-\delta^{a_n}) r^2 - (3^{2n}+1)r^2\cdot C_n \delta^{\frac{1}{2(2n-1)}} \geq {{\frac{1}{2}}}r^2 - \lambda \delta^{a_n} r^2$ --------------------------- ------------------------------------------------------------------------------------------------------- \ by Lemmas \[oakley\] and the following claim. After proving this claim, the result will follow.\ Claim: $\displaystyle{ \max_{z\in \partial (B(0,r))} \left| \sum_j \sigma_j (P_r(\zeta_j,z) -1 )\right| \leq C_n \delta^{\frac{1}{2 (2n-1)}}}$\ Proof of claim: $\forall z\in \partial B(0,r), \ \int_{\zeta \in \partial B(0,\kappa r)} P_r(\zeta,z) d \sigma_{\kappa r} (\zeta) = 1$, by Lemma \[poison\]. Hence, $ 1 = \sum_{j=1}^{j=N} \sigma_j P_r(\zeta_j,z) + \sum_{j=1}^{j=N} \int_{\zeta\in I^{\kappa r}_j} (P_r(\zeta,z)-P_r(\zeta_j,z)) d\sigma_{\kappa r}(\zeta)$ --------------------------- -------------------------------------------------------------------- And, $ |\sum_{j=1}^{j=N}$ $ \sigma_j (P_r(\zeta_j,z)-1)|=|\sum_{j=1}^{j=N} \int_{\zeta\in I^{\kappa r}_j} (P_r(\zeta,z)-P_r(\zeta_j,z)) d\sigma_{\kappa r}(\zeta)|$ $ \displaystyle{\leq \max_{j, \ \zeta \in I^{\kappa r}_j}|\zeta - \zeta_j| \cdot \max_{w \in B(0, (\kappa + \delta) r) \backslash B(0, (\kappa - \delta) r)} \left|\frac{\partial P_r(w,z)}{\partial w} \right| }$ --------------------------- -------------------------------------------------------------------- $\displaystyle{\frac{\partial P_r( w, z )}{\partial w} = -r^{2n-2} \frac{\overline{w} |z-w|^2 + (r^2-|w|^2) n (\overline{z}-\overline{w}) }{|z-w|^{2n+2}}}$ As $|z|=r$, and $|w|=(1-\varepsilon) r\in [(\kappa - \delta)r, (\kappa + \delta)r]$ $\left|\frac{\partial P_r( w, z )}{\partial w}\right| \leq \frac{2+ 4 \varepsilon n}{r \varepsilon^{2n+2}}\leq \frac{c_n}{r \varepsilon^{2n+2}}=\frac{c_n}{r }\delta^{ -\frac{1}{2(2n-1)}}$ And, $\max_{\zeta} |\zeta-\zeta_j| \leq diam(I_j)+ \delta r \leq c \delta^{\frac{1}{2n-1}} r+\delta r\leq c' r \delta ^{\frac{1}{2n-1}}$ Therefore: $| \sum_{j=1}^{j=N} \sigma_j ( P_r(\zeta_j, z)-1)| \leq C \delta^{\frac{1}{2n-1}} \cdot \delta^{-\frac{1}{2(2n-1)}} =C \delta^{\frac{1}{2 (2n-1)}}$\ Proving the claim and the lemma. This lemma gives an alternate proof for the growth rate of the characteristic function. Let $T(f,r)= \int_{S_r} \log^+ |f(z)| d\sigma_r(z) $, the Nevanlina characteristic function. As $( \int_{S_r} \log |\psi_\omega| d\sigma_r)$ is increasing the proof of Corollary \[GrowthOfLogMax\] can be used in conjunction with Lemma \[Athena\] to prove that $\psi_\omega(z)$ is a.s. finite order 2. \[FiniteOrder2\] For all $\delta \in (0,\frac{1}{3}]$ ---- --------------------------------------------------------------------------------------------------------------------------------- a) $\displaystyle{Prob\left( \left\{ \omega : \ \lim_{r\rightarrow \infty} \frac{( \int_{S_r} \log |\psi_\omega| d\sigma_r)-{{\frac{1}{2}}}r^2}{r^2} \notin [-\delta, \delta] \right\}\right) =0}$ b) $\displaystyle{Prob\left( \left\{ \omega : \ \lim_{r\rightarrow \infty} \frac{( \int_{S_r} \log |\psi_\omega| d\sigma_r)-{{\frac{1}{2}}}r^2}{r^2} \neq 0 \right\}\right) =0}$ c) $\displaystyle{Prob\left( \left\{ \omega : \ \lim_{r\rightarrow \infty} \frac{T(\psi_\omega, r)-{{\frac{1}{2}}}r^2}{r^2} \neq 0 \right\}\right) =0}$ ---- --------------------------------------------------------------------------------------------------------------------------------- \ Proof of Main results --------------------- We will now be able to put the pieces together to estimate the number of zeroes in a large ball for a random holomorphic function $\psi_\omega(z)$. Further, This will help us to compute the hole probability. For $f\in {\mathcal{O}}(B(0,r)), \ B(0,r) \subset {{\mathbb C}}^n$, the unintegrated counting function,\ $n_{f}(r):= \int_{B(0,t)\bigcap Z_{f}} (\frac{i}{2 \pi} \partial \overline{\partial} \log |z|^2)^{n-1}= \int_{B(0,t)} (\frac{i}{2 \pi} \partial \overline{\partial} \log |z|^2)^{n-1} \wedge \frac{i}{2 \pi} \partial \overline{\partial} \log |f| $ The equivalence of these two definitions follows by the Poincare-Lelong formula. The above form ($(\frac{i}{2 \pi} \partial \overline{\partial} \log |z|^2)^{n-1} $) gives a projective volume, with which it is more convenient to measure the zero set of a random function. The Euclidean volume may be recovered as $\int_{B(0,t)\bigcap Z_{f}} (\frac{i}{2 \pi} \partial \overline{\partial} \log |z|^2)^{n-1}= \int_{B(0,t)\bigcap Z_{f}} (\frac{i}{2 \pi t^{2}} \partial \overline{\partial} |z|^2)^{n-1}$. \[Shabat\]If $u \in L^1(\overline B _r), \ and \ \partial \overline{\partial}u$ is a measure, then $$\int_{t=r\neq 0}^{t=R} \frac{dt}{t} \int_{B_t} \frac{i}{2 \pi} \partial \overline{\partial} u \wedge (\frac{i}{2 \pi} \partial \overline{\partial} \log |z|^2)^{m-1}= \frac{1}{2} \int_{S_R}u d\sigma_R - \frac{1}{2} \int_{S_r}u d\sigma_r$$ A proof of this result is available in the literature, [@ShiffmanEquidistTheort]. When applying this to random functions, my previous estimates of the surface integral will turn out to be extremely valuable.\ [**Theorem \[Main\]**]{} For all $\delta>0,$ $$F_r:=\left\{\omega: \left|n_{\psi_\omega}(r) -\frac{1}{2}r^2 \right| \geq \delta r^2 \right\} \ is \ a \ small \ family \ of \ events.$$ It suffices to prove the result for small $\delta$. We will start by estimating that: $$\nu \left( \left\{ \omega: \frac{n_{\psi_\omega}(r)}{r^2} \geq \frac{1}{2} + \delta \right\}\right) \leq e^{-c_\delta r^{2n+2}}$$ $n_{\psi_\omega}(r) \log(\kappa) \leq \int_{t=r}^{t=\kappa r} n_{\psi_\omega}(t)\frac{dt}{t} \leq n_{\psi_\omega}(\kappa r) \log(\kappa)$, as n(r) is increasing. let $\kappa = 1+ \sqrt{\delta} $. Except for a small family of events, we have:\ -------------------------------------------------- ----------------------------------------------------------------------------- $\displaystyle{n_{\psi_\omega}(r) \log(\kappa)}$ $\displaystyle{\leq \int_{t=r}^{t=\kappa r} n_{\psi_\omega}(t)\frac{dt}{t} }$ $\displaystyle{ = \int_{t=r}^{t= \kappa r} \int_{B(0,t)} \frac{i}{2 \pi} \partial \overline{\partial} \log|\psi_\omega(z)| \wedge \left(\frac{i}{2 \pi} \partial \overline{\partial} \log|z|^2\right)^{n-1} \frac{dt}{t} }$ $\displaystyle{ = \frac{1}{2} \int_{S_{\kappa r}} \log |\psi_\omega (z)| d\sigma- \frac{1}{2} \int_{S_r} \log |\psi_\omega (z)| d\sigma }$, by Lemma \[Shabat\]. $\displaystyle{ \leq \frac{1}{2}\left(\left(\frac{1}{2}+\delta\right) \kappa^2 r^2 - \int_{S_r} \log |\psi_\omega (z)| d\sigma\right) }$, by Lemma \[Anchises\]. $\displaystyle{ \leq \frac{1}{2}\left(\left(\frac{1}{2}+\delta\right) r^2 \kappa^2 - \left(\frac{1}{2} - \delta\right)r^2\right)} $, by Lemma \[Athena\]. -------------------------------------------------- ----------------------------------------------------------------------------- --------------------------------------------------- ------------------------------------------------------------- $\displaystyle{2 \frac{n_{\psi_\omega}(r)}{r^2}}$ $\displaystyle{\leq \frac{1}{\log (\kappa)} \left(\kappa^2 \left(\frac{1}{2} +\delta\right)-\left(\frac{1}{2}-\delta \right) \right) }$ $\displaystyle{= \frac{\kappa^2 -1}{2 \log(\kappa)}+ \delta \frac{\kappa^2+1}{\log(\kappa)} \leq 1 + c \sqrt{\delta}}$. --------------------------------------------------- ------------------------------------------------------------- \ This proves the probability estimate when the unintegrated counting function is significantly larger then expected. In order to prove the other probability estimate: $$\nu\left( \left\{\omega: \frac{n_{\psi_\omega}(r)}{r^2} \leq \frac{1}{2} - \delta \right\} \right) \leq e^{-c_\delta r^{2n+2}}$$ We start by using that: $\int_{t=\kappa^{-1} r}^{t= r} n_{\psi_\omega}(t)\frac{dt}{t} \leq n_{\psi_\omega}( r) \log(\kappa)$. We then use that, except for a small family of events, we have that: -------------------------------------------------- ---------------------------------------------------------------------------------- $\displaystyle{n_{\psi_\omega}(r) \log(\kappa)}$ $\displaystyle{\geq \int_{t=\kappa^{-1} r}^{t= r} n_{\psi_\omega}(t)\frac{dt}{t}}$ $\displaystyle{ = \int_{t=\kappa^{-1} r}^{t=r} \int_{B(0,t)} \frac{i}{2 \pi} \partial \overline{\partial} \log|\psi_\omega(z)| \wedge \left(\frac{i}{2 \pi} \partial \overline{\partial} \log|z|^2 \right)^{n-1}\frac{dt}{t}}$ $\displaystyle{= \frac{1}{2} \int_{S_r} \log |\psi_\omega (z)| d\sigma- \frac{1}{2} \int_{S_{\kappa^{-1}r}} \log |\psi_\omega (z)| d\sigma } $, by Lemma \[Shabat\]. $\displaystyle{\geq \frac{1}{2}[(\frac{1}{2}-\delta) r^2 - \int_{S_{\kappa^{-1}r}} \log |\psi_\omega (z)| d\sigma]}$, by Lemma \[Athena\]. $\displaystyle{\geq \frac{1}{2}[(\frac{1}{2}-\delta) r^2 - (\frac{1}{2} + \delta)r^2\kappa^{-2}]}$, by Lemma \[Anchises\]. -------------------------------------------------- ---------------------------------------------------------------------------------- --------------------------------------------------- --------------------------------------------------------------- $\displaystyle{2 \frac{n_{\psi_\omega}(r)}{r^2}}$ $\displaystyle{\geq \frac{1}{\log (\kappa)} \left( \left(\frac{1}{2} -\delta\right)-(\frac{1}{2}+\delta) \kappa^{-2} \right)} $ $\displaystyle{= \frac{1-\kappa^{-2}}{2 \log(\kappa)}- \delta \frac{1+\kappa^{-2}}{\log(\kappa)} \geq 1- 2 \sqrt \delta }$ --------------------------------------------------- --------------------------------------------------------------- \ Using this estimate for the typical measure of the zero set of a random function we get an upper bound for the hole probability, and putting this together with some elementary estimates we get an accurate estimate for the order of the decay of the hole probability:\ [**Theorem \[Hole probability\]**]{} [*If*]{} $$\psi_\omega (z_1, z_2 \ldots, z_n)= {\displaystyle\sum}_j \omega_j \frac{z_1^{j_1} z_2^{j_2} \ldots z_n^{j_n}}{\sqrt{j_1! \cdot j_n!}},$$ [*where $\omega_j$ are independent identically distributed complex Gaussian random variables, and* ]{} $$Hole_r=\{ \omega:\forall z \in B(0,r), \ \psi_\omega (z)\neq 0 \},$$ [*then there exists*]{} $c_1, c_2 >0$ [*such that for all*]{} $r>R_n$ $$e^{-c_2 r^{2n+2}} \leq Prob (Hole_r)\leq e^{-c_1 r^{2n+2}}$$ The upper estimate follows by the previous theorem, as if there is a hole then $n_{\psi_\omega}(r)=0$, and this can only occur on a small family of events. Therefore it suffices to show that the hole probability is bigger than a small set. Let $\Omega_r$ be the event where:\ $i) \ |\omega_{0}| \geq E_n + 1$,\ $ii) \ |\omega_{j}|\leq e^{-(1+ \frac{n}{2})r^2}, \ \forall j: 1\leq |j|\leq \lceil 24 n r^2 \rceil= \lceil(n \cdot 2 \cdot 12)r^2 \rceil $\ $iii) \ |\omega_{j}|\leq 2^{\frac{|j|}{2}}, \ |j| > \lceil 24 n r^2 \rceil \geq 24 n r^2 $\ $\nu( \{\omega | \ |\omega_{j}|\leq e^{-(1+ \frac{n}{2})r^2} \}) \geq \frac{1}{2}(e^{-(1+ \frac{n}{2})r^2})^2=\frac{1}{2} e^{-(2+ n)r^2} $, by Lemma \[Gauss\] $ \# \{j \in \ {{\mathbb N}}^n | 1\leq |j|\leq \lceil 24 n r^2 \rceil \} = ({\lceil 24 n r^2 \rceil + n \choose n} ) \approx c r^{2n} $ Hence, $\nu(\Omega_r)\geq C (e^{-c_n r^{2n+2}})$, by independence and Lemma \[Gauss\]. Therefore $\Omega_r$ contains a small family of events, and it now suffices to show that for $\omega \in \Omega_r, \ \psi_\omega$ has a hole in $B(0,r)$.\ $f(z) \geq |\omega_{0}| - {\displaystyle\sum}_{|j|=1}^{|j| \leq \lceil 24 n r^2 \rceil} |\omega_{j}| \frac{r^{|j|}}{\sqrt{j! }} - {\displaystyle\sum}_{|j| > \lceil 24 n r^2 \rceil} |\omega_{j}| \frac{r^{|j|}}{\sqrt{j!}}= |\omega_{0}|- \sum^1 - \sum^2$\ ------------------------- -------------------------------------------------------------------------------------------------------------------------- ${\displaystyle\sum}^1$ $\leq e^{-(1+ \frac{n}{2})r^2} {\displaystyle\sum}_{|j| = 1}^{|j| \leq \lceil 24 n r^2 \rceil} \frac{r^{|j|}}{\sqrt{j!}} $ $\leq e^{-(1+ \frac{n}{2})r^2} \sqrt{(24 n r^2 +1)^{n}} \sqrt{(e^{r^n})}$, by Cauchy-Schwarz inequality. $\leq C_n r^{n} e^{-r^2}\leq c e^{-0.9 r^2}<\frac{1}{2} $ for $r> R_n$ ------------------------- -------------------------------------------------------------------------------------------------------------------------- \ ------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\displaystyle\sum}^2$ $\leq {\displaystyle\sum}_{|j| > 24 n r^2} 2^{\frac{|j|}{2}} \left(\frac{|j|}{24 n}\right)^{\frac{|j|}{2}} \frac{1}{\sqrt{j! }}$, as $r<\sqrt{\frac{|j|}{24 n}} $ $\leq c {\displaystyle\sum}_{|j| > 24 n r^2} 2^{\frac{|j|}{2}} \left(\frac{|j|}{24 n}\right)^{\frac{|j|}{2}} \prod_{k=1}^{k=n} \left(\frac{e}{j_k}\right)^{\frac{j_k}{2}}$, by Sterling’s formula $=c{\displaystyle\sum}_{|j| > 24 n r^2} \frac{(|j|)^{\frac{|j|}{2}}}{\left(\prod_{k=1}^{k=n} j_k^\frac{j_k}{2}\right) n^{\frac{|j|}{2}}} \left(\frac{e}{12}\right)^{\frac{|j|}{2}} $ $\leq c {\displaystyle\sum}_{|j|>1} \left(\frac{1}{4}\right)^{{\frac{|j|}{2}}}$, by Lemma \[CalcII\]. $ \leq c {\displaystyle\sum}_{l>1} \left(\frac{1}{2}\right)^{l} l^n\leq E_n $ ------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \ Hence, $|\psi_\omega(z)| \geq E_n + 1- {\displaystyle\sum}^1-{\displaystyle\sum}^2\geq \frac {1}{2} $ [77]{} P. 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Kahane, [*Some random series of functions*]{} [**Second edition**]{}, (Cambridge, Great Britain: Cambridge University Press, 1985). S. Krantz, [*Function Theory of Several Complex Variables*]{} [**Second edition**]{}, (Pacific Grove, California: Wadsworth $\&$ Brooks/Cole Advanced Books $\&$ Software, 1992). F. Nazarov, M. Sodin, and A. Volberg, [*Transportation to Random Zeroes by the Gradient Flow*]{}, Israel J. Math.[**147**]{} (2005), 371-379 A. Offord [*The distribuition of zeros of powerseries whose coefficients are independent random variables*]{}, Indian J. Math. [**9**]{} (1967), 175-196. A. Offord [*The range of a random function in the unit disk*]{}, Studia Math. [**44**]{} (1972), 263-273. Y. Peres, B. Virag, [*Zeros of i.i.d. Gaussian powerseries: a conformally invariant determinental process*]{} arXiv:math.CV/0310297. S. Rice, Mathematical analysis of random noise, [*Bell System Tech. J.*]{} 23 (1944) 282-332. M. 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--- author: - '*Alberto Carrasco Ferreira*' date: 'April, 2011' title: '**Trapped surfaces in spacetimes with symmetries and applications to uniqueness theorems**' --- \[section\] \[thr\][Definition]{} \[thr\][Lemma]{} \[thr\][Corollary]{}\[thr\][Proposition]{} \#1\#2\#3\#4\#5\#6[\#1, “\#2”, [*\#3*]{} [**\#4**]{}, \#5 (\#6).]{} \#1\#2\#3[\#1, “\#2”, \#3.]{} Ł §[S]{} Ł §[S]{} ¶[P]{} A mis padres.\ A mi tía Nena. “The transition is a keen one, I assure you,\ from a schoolmaster to a sailor, and requires\ a strong decoction of Seneca and the Stoics\ to enable you to grin and bear it.\ But even this wears off in time."\ Agradecimientos =============== Quiero empezar expresando mi más sincero agradecimiento al profesor Marc Mars Lloret, director de esta tesis doctoral, por la atención que me ha prestado durante todos estos años en los que he tenido la suerte de trabajar a su lado; por haber compartido conmigo sus ideas y haberme mostrado las líneas de investigación a seguir en este trabajo; por tratar siempre de animar mi curiosidad y mi carácter crítico por encima de todo; por su confianza, su paciencia y su apoyo, sin los cuales nunca hubiera podido terminar este trabajo; por su incansable dedicación, su total disponibilidad, su trato siempre amable y su sana amistad. Quiero agradecer a todos los miembros del Departamento de Física Fundamental de la Universidad de Salamanca, por su trato cordial y por ayudarme siempre que lo he necesitado. Doy las gracias al profesor Walter Simon por su amistad y por compartir su saber conmigo. También agradezco al profesor Miguel Sánchez Caja del Departamento de Geometría y Topología de la Universidad de Granada, que fue mi tutor en los cursos de doctorado, por su hospitalidad, su apoyo y su sano interés por mi trabajo durante todos estos años. Agradezco a mis compañeros de doctorado (Marsopas y otras especies): Cuchi, Jorge, Álvaro Dueñas, Álvaro Hernández, Diego, Alberto Soria, Edu y Cristina por su ayuda, por las risas y por aguantarme todos los días. A Toni, el Ave Fénix de las Marsopas, por haber compartido conmigo tantas inquietudes y tanta magia. Sin duda, esto hubiera sido mucho más aburrido sin ellos. Agradezco también a la Escuela Kodokai, que me ha llenado de inspiración y me ha ayudado a estar en forma durante estos años. Estoy profundamente agradecido a mis padres, por darme una vida feliz y una buena educación, y por la confianza que siempre han depositado en mí. A Marta, por hacer que la vida sea más divertida, y a mi tía Nena, que siempre estuvo a nuestro lado. Y, por supuesto, a Raquel y a mi pijama azul. Gracias por existir y por quererme tanto. Finalmente, agradezco al MICINN por el apoyo económico prestado. Introduction {#ch:Introduction} ============ General Relativity, formulated by Einstein in 1915 [@Einstein], is up to the present date the most accurate theory to describe gravitational physics. Roughly speaking, this theory establishes that space, time and gravitation are all of them aspects of a unique structure: the spacetime, a four dimensional manifold whose geometry is closely related to its matter contents via the Einstein field equations. One of the most striking consequences of General Relativity is the existence of [*black holes*]{}, that is, spacetime regions from which no signal can be seen by an observer located infinitely far from the matter sources. Black holes in the universe are expected to arise as the final state of gravitational collapse of sufficiently massive objects, such as massive stars, as the works by Chandrasekhar, Landau and Oppenheimer and Volkoff [@Chandra] already suggested in the decade of the 1930’s. Despite the fact that many astronomical observations give strong indication that black holes really exist in nature, a definitive experimental proof of their existence is still lacking. Although black holes arose first as theoretical predictions of General Relativity, its modern theory was developed in the mid-sixties largely in response to the astronomical discovery of highly energetic and compact objects. During these years the works of Hawking and Penrose [@singularitythrs] showed that singularities (i.e. “points” where the fundamental geometrical quantities are not well-defined) are commonplace in General Relativity, in particular in the interior of black holes. Singularities have the potential danger of breaking the predictability power of a theory because basically anything can happen once a singularity is visible. However, for the singularities inside black holes the situation is not nearly as bad, because, in this case, the singularity is not visible from infinity and hence the predictability capacity of the observers lying outside the black hole region remains unaffected. This fact led Penrose to conjecture that naked singularities (i.e. singularities which do not lie inside a black hole) cannot occur in any reasonable physical situation [@CosmicCensor]. This conjecture, known as the [*cosmic censorship hypothesis*]{}, protects the distant observers from the lack of predictability that occurs in the presence of singularities. Whether this conjecture is true or not is at present largely unknown (see [@Wald1997] for an account of the situation in the late 90’s). Rigorous results are known only in spherical symmetry, where the conjecture has been proven for several matter models [@Christodoulou1999; @Dafermos2005]. In any case, the validity of (some form) of cosmic censorship implies that black holes are the generic end state of gravitational collapse, and hence fundamental objects in the universe. Of particular importance is the understanding of equilibrium configurations of black holes. The [*uniqueness theorems for static and stationary black holes*]{}, which are considered one of the cornerstones of the theory of black holes, also appeared during the sixties mainly motivated by the early work of Israel [@Israel]. These theorems assert that, given a matter model (for example vacuum), a static or a stationary black hole spacetime belongs necessarily to a specific class of spacetimes (in the vacuum case, they are Schwarzschild in the static regime and Kerr for the stationary case) which are univocally characterized by a few parameters that describe the fundamental properties of the black hole (for vacuum these parameters are the mass and the angular momentum of the black hole). Since, from physical principles, it is expected that astronomical objects which collapse into a black hole will eventually settle down to a stationary state, the black hole uniqueness theorems imply that the final state of a generic gravitational collapse (assuming that cosmic censorship holds) can be described by a very simple spacetime geometry characterized by a few parameters like the total mass, the electric charge or the angular momentum of the collapsing astronomical object (or, more precisely, the amount of these physical quantities which is kept by the collapsing object and does not get radiated away during the process). The resulting spacetime is therefore independent of any other of the properties of the collapsing system (like shape, composition, etc.). This type of result was, somewhat pompously, named [*“no hair"*]{} theorems for black holes by Wheeler [@Wheeler]. In 1973 Penrose [@Penrose1973] invoked cosmic censorship and the no hair theorems to deduce an inequality which imposes a lower bound for the total mass of a spacetime in terms of the area of the [*event horizon*]{} (i.e. the boundary) of the black hole which forms during the gravitational collapse. This conjecture is known as the [*Penrose inequality*]{}. The Penrose inequality, like the cosmic censorship conjecture on which it is based, has been proven only in a few particular cases. Both conjectures therefore remain, up to now, wide open. One of the intrinsic difficulties for their proof is that black holes impose, by its very definition (see e.g. Chapter 12 of [@Wald]), very strong global conditions on a spacetime. From an evolutive point if view, these objects are of teleological nature because a complete knowledge of the future is needed to even know if a black hole forms. Determining the future of an initial configuration (i.e. the metric and its first time derivative on a spacelike hypersurface) requires solving the spacetime field equations (either analytical or numerically) with such initial data. The Einstein field equations are non-linear partial differential equations, so determining the long time behavior of its solutions is an extremely difficult problem. In general, the results that can be obtained from present day technology do not give information on the global structure of the solutions and, therefore, they do not allow to study black holes in an evolutive setting. As a consequence, the concept of black hole is not very useful in this situation because, what does it mean that an initial data set represents a black hole? Since the concept of black hole is central in gravitation, it has turned out to be necessary to replace this global notion by a more local one that, on the one hand, can be studied in an evolutionary setting and, on the other, hopefully has something to do with the global concept of black hole. The objects that serve this purpose are the so-called [*trapped surfaces*]{}, which are, roughly speaking, compact surfaces without boundary for which the emanating null rays do not diverge (all the precise definitions will be given in Chapter \[ch:Preliminaires\]). The reason for this bending of light “inwards” is the gravitational field and, therefore, these surfaces reveal the presence of an intense gravitational field. This is expected to indicate that a black hole will in fact form upon evolution. More precisely, under suitable energy conditions, the maximal Cauchy development of this initial data is known to be causal geodesically incomplete (this is the content of one of the versions of the singularity theorems, see [@S97] for a review). [*If cosmic censorship holds*]{}, then a black hole will form. Moreover, it is known that in any black hole spacetime the subclass of trapped surfaces called [*weakly trapped surfaces*]{} and [*weakly outer trapped surface*]{} lie inside the black hole (see e.g. chapter 9.2 of [@HE] and chapter 12.2 of [@Wald]), and so they give an indication of where the back hole event horizon should be in the initial data (if it forms at all). In fact, the substitution of the concept of black hole by the concept of trapped surface is so common that one terminology has replaced the other, and scientists talk about black hole collision, of black hole-neutron star mergers to refer to evolutions involving trapped surfaces. However, it should be kept in mind that both concepts are completely different a priori. In the context of the Penrose inequality, the fact that, under cosmic censorship, weakly outer trapped surfaces lie inside the black hole was used by Penrose to replace the area of the event horizon by the area of weakly outer trapped surfaces to produce inequalities which, although motivated by the expected global structure of the spacetime that forms, can be formulated directly on the given initial data in a manner completely independent of its evolution. A particular case of weakly outer trapped surfaces, the so-called [*marginally outer trapped surfaces (MOTS)*]{} (defined as compact surfaces without boundary with vanishing outer null expansion $\theta^+$), are widely considered as the best quasi-local replacements for the event horizon. From what it has been said, it is clear that proving that these surfaces can replace black holes is basically the same as proving the validity of cosmic censorship, which is beyond present day knowledge. The advantage of seeing the problem from this perspective is that it allows for simpler questions that can perhaps be solved. One such question is the Penrose inequality already mentioned. Another one has to do with static and stationary situations. One might think that, involving no evolution at all, it should be clear that black holes, event horizons and marginally outer trapped surfaces are essentially the same in an equilibrium configuration. However, although certainly plausible, very little is known about the validity of this expectation. The aim of this thesis is precisely to study the properties of trapped surfaces in spacetimes with symmetries and their possible relation with the theory of black holes. Even this more modest goal is vast. We will concentrate on one aspect of this possible equivalence, namely [*whether the static black hole uniqueness theorems extend to static spacetimes containing MOTS*]{}. The main result of this thesis states that this question has an affirmative answer, under suitable conditions on the spacetime. To solve this question we will have to analyze in depth the properties of MOTS and weakly outer trapped surfaces in spacetimes with symmetries, and this will produce a number of results which are, hopefully, of independent interest. This study will naturally lead us to consider a second question, namely to study the Penrose inequality in static initial data sets which are not time-symmetric. Our main result here is the discovery of a counterexample of a version of the Penrose inequality that was proposed by Bray and Khuri [@BK] not long ago. It is worth to mention that most of the results we will obtain in this thesis do not use the Einstein field equations and, consequently, they are also valid in any gravitational theory of gravitation in four dimensions. In the investigations on stationary and static spacetimes there has been a tendency over the years of reducing the amount of global assumptions in time to a minimum. This is in agreement with the idea behind cosmic censorship of understanding the global properties as a consequence of the evolution. This trend has been particularly noticeable in black hole uniqueness theorems, where several conditions can be used to capture the notion of black hole (see e.g. Theorem \[thr:electrovacuniqueness0\] in Chapter \[ch:Preliminaires\]). In this thesis, we will follow this general tendency and work directly on slabs of spacetimes containing suitable spacelike hypersurfaces or, whenever possible, directly at the initial data level, without assuming the existence of a spacetime where it is embedded. It should be remarked that the second setting is more general than the former one. Indeed, in some circumstances the existence of such a spacetime can be proven, for example by using the notion of Killing development (see [@BC] and Chapter \[ch:Article1\]) or by using well-posedness of the Cauchy problem and suitable evolution equations for the Killing vector [@Coll]. The former, however, fails at fixed points of the static isometry and the second requires specific matter models, not just energy inequalities as we will assume. Nevertheless, although most of the results of this thesis will be obtained at the initial data level, we will need to invoke the existence of a spacetime to complete the proof of the uniqueness result (we emphasize however, that no global assumption in time is made in that case either). We will also try to make clear which is the difficulty that arises when one attempts to prove this result directly at the initial data level. The results obtained in this thesis constitute, in our opinion, a step forward in our understanding of how black holes evolve. Regarding the problem of establishing a rigorous relationship between black holes and trapped surfaces, the main result of this thesis (Theorem \[uniquenessthr\]) shows that, at least as far as uniqueness of static black holes is concerned, event horizons and MOTS do coincide. Our uniqueness result for static spacetimes containing MOTS is interesting also independently of its relationship with black holes. It proves that static configurations are indeed very rigid. This type of result has several implications. For instance, in any evolution of a collapsing system, it is expected that an equilibrium configuration is eventually reached. The uniqueness theorems of black holes are usually invoked to conclude that the spacetime is one of the stationary black holes compatible with the uniqueness theorem. However, this argument assumes implicitly that one has sufficient information on the spacetime to be able to apply the uniqueness theorems, which is far from obvious since the spacetime is being constructed during the evolution. In our setting, as long as the evolution has a MOTS on each time slice, if the spacetime reaches a static configuration, then it is unique. Related to this issue, it would be very interesting to know if these types of uniqueness results also hold in an approximate sense, i.e. if a spacetime is [*nearly*]{} static and contains a MOTS, then the spacetime is [*nearly*]{} unique. This problem is, of course, very difficult because it needs a suitable concept of “being close to”. In the particular case of the Kerr metric, there exists a notion of an initial data being close to Kerr [@Kroom] which is based on a suitable characterization of this spacetime [@Mars3]. This closeness notion is defined for initial data sets without boundary and has been extended to manifolds with boundary under certain circumstances [@Kroom2]. It would be of interest to extend it to the case with a non-empty boundary which is a MOTS. The static uniqueness result for MOTS is only a first step in this subject. Future work should try to extend this result to the stationary setting. The problem is, however, considerably more difficult because the techniques known at present to prove uniqueness of stationary black holes are much less developed than those for proving uniqueness of static black holes. Assuming however, that the spacetime is axially symmetric (besides being stationary) simplifies the black hole uniqueness proof considerably (the problem becomes essentially a uniqueness proof for a boundary value problem of a non-linear elliptic system on a domain in the Euclidean plane, see [@Heuslerlibro]). The next natural step would be to try and extend this uniqueness result to a setting where the black hole is replaced by a MOTS. The only result we prove in this thesis in the stationary (non-static) setting involves MOTS lying in the closure of the exterior region where the Killing is timelike. We show that in this case the MOTS cannot penetrate into the timelike exterior domain (see Theorem \[theorem1\]). In the remaining of this Introduction, we will try to give a general idea of the structure of the thesis and to discuss its main results.\ In rough terms, the typical structure of static black holes uniqueness theorems is the following:\ [*Let $(M,\gM)$ be a static solution of the Einstein equations for a given matter model (for example vacuum) which describes a black hole. Then $(M,\gM)$ belongs necessarily to a specific class of spacetimes which are univocally characterized by a number of parameters that can be measured at infinity (in the case of vacuum, the spacetime is necessarily Schwarzschild and the corresponding parameter is the total mass of the black hole).*]{}\ There exist static black hole uniqueness theorems for several matter models, such as vacuum ([@Israel], [@MzH], [@Robinson], [@BMuA], [@C]), electro-vacuum ([@Israel2], [@MzH2], [@Simon], [@Ruback], [@Simon2], [@MuA], [@C2], [@CT]) and Einstein-Maxwell dilaton ([@MuA2], [@MSimon]). As we will describe in more detail in Chapter \[ch:Preliminaires\] the most powerful method for proving these results is the so called [*doubling method*]{}, invented by Bunting and Masood-ul-Alam [@BMuA] to show uniqueness in the vacuum case. This method requires the existence of a complete spacelike hypersurface $\Sigma$ containing an exterior, asymptotically flat, region $\Sigma^{{ext}}$ such that the Killing is timelike on $\Sigma^{{ext}}$ and the topological boundary $\tbd \Sigma^{{ext}}$ is an embedded, compact and non-empty topological manifold. In static spacetimes, the condition that $(M,\gM)$ is a black hole can be translated into the existence of such a hypersurface $\Sigma$. In this setting, the topological boundary $\tbd \Sigma^{{ext}}$ corresponds to the intersection of the boundary of the domain of outer communications (i.e. the region outside both the black hole and the white hole) and $\Sigma$. This equivalence, however, is not strict due to the potential presence of non-embedded Killing prehorizons, which would give rise to boundaries $\tbd \Sigma^{{ext}}$ which are non-embedded. This issue is important and will be discussed in detail below. We can however, ignore this subtlety for the purpose of this Introduction. The type of uniqueness result we are interested in this thesis is of the form:\ [*Let $(M,\gM)$ be a static solution of the Einstein equations for a given matter model. Suppose that $M$ possesses a spacelike hypersurface $\Sigma$ which contains a MOTS. Then, $(M,\gM)$ belongs to the class of spacetimes established by the uniqueness theorem for static black holes for the corresponding matter model.*]{}\ The first result in this direction was given by Miao in 2005 [@Miao], who extended the uniqueness theorems for vacuum static black holes to the case of asymptotically flat and time-symmetric slices $\Sigma$ which contain a minimal compact boundary (it is important to note that for time-symmetric initial data, a surface is a MOTS if and only if it is a compact minimal surface). In this way, Miao was able to relax the condition of a time-symmetric slice $\Sigma$ having a compact topological boundary $\tbd \Sigma$ where the Killing vector vanishes to simply containing a compact minimal boundary. Miao’s uniqueness result is indeed a generalization of the static uniqueness theorem of Bunting and Masood-ul-Alam because the static vacuum field equations imply in the time-symmetric case that the boundary $\tbd \Sigma^{{ext}}$ is necessarily a totally geodesic surface, which is more restrictive than being a minimal surface. Miao’s result is fundamentally a uniqueness result. However, one of the key ingredients in its proof consists in showing that no minimal surface can penetrate into the exterior timelike region $\Sigma^{{ext}}$. As a consequence, Miao’s theorem can also be viewed as a confinement result for minimal surfaces. As a consequence, one can think of extending Miao’s result in three different directions: Firstly, to allow for other matter models. Secondly, to work with arbitrary slices and not just time-symmetric ones. This is important in order to be able to incorporate so-called degenerate Killing horizons into the problem. Obviously, in the general case minimal surfaces are no longer suitable and MOTS should be considered. And finally, try to make the confinement part of the statement as local as possible and relax the condition of asymptotic flatness to the existence of suitable exterior barrier. To that aim it is necessary a proper understanding of the properties of MOTS and weakly outer trapped surfaces in static spacetimes (or more general, if possible). For simplicity, let us restrict to the asymptotically flat case for the purpose of the Introduction. Consider a spacelike hypersurface $\Sigma$ containing an asymptotically flat end $\Sigma_{0}^{\infty}$. In what follows, let $\lambda$ be minus the squared norm of the static Killing $\vec{\xi}$. So, $\lambda>0$ means that $\vec{\xi}$ is timelike. Staticity and asymptotic flatness mean that this Killing vector is timelike at infinity. Thus, it makes sense to define $\ext$ as the connected component of $\{ \lambda > 0 \}$ which contains the asymptotically flat end $\Sigma_{0}^{\infty}$ (the set $\Sigma^{{ext}}$ in the Masood-ul-Alam doubling method is precisely $\ext$). Since we want to prove the expectation that MOTS and spacelike sections of the event horizon coincide in static spacetimes, we will firstly try to ensure that no MOTS can penetrate into $\ext$. This result will generalize Miao’s theorem as a confinement result and will extend the well-known confinement result of MOTS inside the black hole region (c.f. Proposition 12.2.4 in [@Wald])) to the initial data level. The main tool which will allow us to prove this result is a recent theorem by Andersson and Metzger [@AM] on the existence, uniqueness and regularity of the outermost MOTS on a given spacelike hypersurface. This theorem, which will be essential in many places in this thesis, requires working with trapped surfaces which are [*bounding*]{}, in the sense that they are boundaries of suitable regions (see Definition \[defi:boundingAM\]). Another important ingredient for our confinement result will be a thorough study of the causal character that the Killing vector is allowed to have on the outermost MOTS (or, more, generally on stable or strictly stable MOTS – all these concepts will be defined below –). For the case of [*weakly trapped surfaces*]{} (which are defined by a more restrictive condition than weakly outer trapped surfaces), it was proven in [@MS] that no weakly trapped surface can lie in the region where the Killing vector is timelike provided its mean curvature vector does not vanish identically. Furthermore, similar restrictions were also obtained for other types of symmetries, such as conformal Killing vectors (see also [@S3] for analogous results in spacetimes with vanishing curvature invariants). Our main idea to obtain restrictions on the Killing vector on an outermost MOTS $S$ consists on a geometrical construction [@CM1] whereby $S$ is moved first to the past along the integral lines of the Killing vector and then back to $\Sigma$ along the outer null geodesics orthogonal to this newly constructed surface, producing a new weakly outer trapped surface $S'$, provided the null energy condition (NEC) is satisfied in the spacetime. If the Killing field $\vec{\xi}$ is timelike anywhere on $S$ then we show that $S'$ lies partially outside $S$, which is a contradiction with the outermost property of $S$. This simple idea will be central in this thesis and will be extended in several directions. In particular, we will generalize the geometric construction to the case of general vector fields $\vec{\xi}$, not just Killing vectors. To ensure that $S'$ is weakly outer trapped in this setting we will need to obtain an explicit expression for the first variation of the outer null expansion $\theta^+$ along $\vec{\xi}$ in terms of the so called [*deformation tensor*]{} of the metric along $\vec{\xi}$ (Proposition \[propositionxitheta\]). This will allow us to obtain results for other types of symmetries, such as homotheties and conformal Killing vectors, which are relevant in many physical situations of interest (e.g. the Friedmann-Lemaître-Robertson-Walker cosmological models). Another relevant generalization involves analyzing the infinitesimal version of the geometric construction. As we will see, the infinitesimal construction is closely related to the stability properties of the the first variation of $\theta^{+}$ along $\Sigma$ on a MOTS $S$. This first variation defines a linear elliptic second order differential operator [@AMS] for which elliptic theory results can be applied. It turns out that exploiting such results (in particular, the maximum principle for elliptic operators) the conclusions of the geometric construction can be sharpened considerably and also extended to more general MOTS such as stable and strictly stable ones. (Theorem \[TrhAnyXi\] and Corollaries \[thrstable\] and \[shear\]). As an explicit application of these results, we will show that stable MOTS cannot exist in any slice of a large class of Friedmann-Lemaître-Robertson-Walker cosmological models. This class includes all classic models of matter and radiation dominated eras and also those models with accelerated expansion which satisfy the NEC (Theorem \[thrFRW\]). Remarkably, the geometric construction is more powerful than the elliptic methods in some specific cases. We will find an interesting situation where this is the case when dealing with homotheties (including Killing vectors) on outermost MOTS (Theorem \[thrkilling\]). This will allow us to prove a result (Theorem \[theorem1\]) which asserts that, as long as the spacetime satisfies the NEC, a Killing vector or homothety cannot be timelike anywhere on a bounding weakly outer trapped surface whose exterior lies in a region where the Killing vector is timelike. Another case when the elliptic theory cannot be applied and we resort to the geometric procedure deals with situations when one cannot ensure that the newly constructed surface $S'$ is weakly outer trapped. However, it can still occur that the portion of $S'$ which lies in the exterior of $S$ has $\theta^{+}\leq 0$. In this case, we can exploit a result by Kriele and Hayward [@KH97] in order to construct a weakly outer trapped surface $S''$ outside both $S$ and $S'$ by smoothing outwards the corner where they intersect. This will provide us with additional results of interest (Theorems \[thrnonelliptic\] and \[shear2\]). All these results have been published in [@CM2] and [@CMere2] and will be presented in Chapter \[ch:Article2\]. From then on, we will concentrate exclusively on [*static*]{} spacetimes. Chapter \[ch:Article1\] is devoted to extending Miao’s result as a confinement result. Since in this chapter we will work exclusively at the initial data level, we will begin by recalling the concept of a [*static Killing initial data (static KID)*]{}, (which corresponds to the data and equations one induces on any spacelike hypersurface embedded on a static spacetime, but viewed as an abstract object on its own, independently of the existence of any embedding into a spacetime). It will be useful to introduce two scalars $I_{1}, I_{2}$ which correspond to the invariants of the [*Killing form*]{} (or Papapetrou field) of the static Killing vector $\vec{\xi}$. It turns out that $I_{2}$ always vanishes due to staticity and that $I_{1}$ is constant on arc-connected components of $\tbd \{\lambda > 0 \}$ and negative on the arc-connected components which contains at least a fixed point (Lemma \[I1&lt;0\]). Fixed points are initial data translations of spacetime points where the Killing vector vanishes and, since $I_1$ turns out to be closely related to the surface gravity of the Killing horizons, this result extends a well-known result by Boyer [@Boyer] on the structure of Killing horizons to the initial data level. The general strategy to prove our confinement result for MOTS is to use a contradiction argument. We will assume that a MOTS can penetrate in the exterior timelike region. By passing to the outermost MOTS $S$ we will find that the topological boundary of $\tbd \ext$ must intersect both the interior and the exterior of $S$. It we knew that $\tbd \ext$ is a bounding MOTS, then we could get a contradiction essentially by smoothing outwards (via the Kriele and Hayward method) these two surfaces. However, it is not true that $\tbd \ext$ is a bounding MOTS in general. There are simple examples even in Kruskal where this property fails. The problem lies in the fact that $\tbd \ext$ can intersect both the black hole and the white hole event horizons (think of the Kruskal spacetime for definiteness) and then the boundary $\tbd \ext$ is, in general, not smooth on the bifurcation surface. To avoid this situations we need to assume a condition which essentially imposes that $\tbd \ext$ intersects only the black hole or only the white hole region. Furthermore, the possibility of $\tbd \ext$ intersecting the white hole region must be removed to ensure that this smooth surface is in fact a MOTS and not a [*past*]{} MOTS. The precise statement of this final condition is given in points (i) and (ii) of Proposition \[is\_a\_MOTS\], but the more intuitive idea above is sufficient for this Introduction. Since we will need to mention this condition below, we refer to it as ($\star$). In this way, in Proposition \[is\_a\_MOTS\], we prove that every arc-connected component of $\tbd \{\lambda>0\}$ is an injectively immersed submanifold with $\theta^{+}=0$. However, injectively immersed submanifolds may well not be embedded. Since, in order to find a contradiction we need to construct a bounding weakly outer trapped surface, and these are necessarily embedded, we need to care about proving that the injective immersion is an embedding (i.e. an homeomorphism with the induced topology in the image). In the case with $I_1 \neq 0$ this is easy. In the case of components with $I_1 =0$ (so-called [*degenerate*]{} components), the problem is difficult and open. This issue is very closely related to the possibility that there may exist non-embedded Killing prehorizons in a static spacetime which has already been mentioned before. This problem, which has remained largely overlooked in the black hole uniqueness theory until very recently [@Cc], is important and very interesting. However, it is beyond the scope of this thesis. For our purposes it is sufficient to assume an extra condition on degenerate components of $\tbd \ext$ which easily implies that they are embedded submanifolds. This condition is that every arc-connected component of $\tbd \ext$ with $I_1=0$ is topologically closed. This requirement will appear in all the main results in this thesis precisely in order to avoid dealing with the possibility of non-embedded Killing prehorizons. If one can eventually prove that such objects simply do not exist (as we expect), then this condition can simply be dropped in all the results below. Our main confinement result is given in Theorem \[theorem2\]. The results of Chapter \[ch:Article1\] have been published in [@CM1] and [@CMere1]. Theorem \[theorem2\] leads directly to a uniqueness result (Theorem \[uniquenessthr0\]) which already generalizes Miao’s result as a uniqueness statement. The idea of the uniqueness proof is to show that the presence of a MOTS boundary in an initial data set implies, under suitable conditions, that $\tbd\ext$ is a compact embedded surface [*without boundary*]{}. This is precisely the main hypothesis that is made in order to apply the doubling method of Bunting and Masood-ul-Alam. Thus, assuming that the matter model is such that static black hole uniqueness holds, then we can conclude uniqueness in the case with MOTS. The strategy is therefore to reduce the uniqueness theorem for MOTS to the uniqueness theorem for black holes. This idea is in full agreement with our main theme of showing that MOTS and black holes are the same in a static situation. Theorem \[uniquenessthr0\] is, however, not fully satisfactory because it still requires condition ($\star$) on $\tbd \ext$. Since $\tbd\ext$ is a fundamental object in the doubling method, it would be preferable if no conditions are a priori imposed on it. Chapter \[ch:Article4\] is devoted to obtaining a uniqueness result for static spacetimes containing weakly outer trapped surfaces with no a priori restrictions on $\tbd \ext$ (besides the condition on components with $I_1=0$ which we have already mentioned). In Chapter \[ch:Article1\] the fact that $\tbd \ext$ is closed (i.e. compact and without boundary) is proven as a consequence of its smoothness. However, when condition ($\star$) is dropped, we know that $\tbd \ext$ is not smooth in general, and in principle, it may have a non-empty manifold boundary. Therefore, we will need a better understanding of the structure of the set $\tbd \{\lambda>0\}$ when ($\star$) is not assumed. In this case, our methods of Chapter \[ch:Article1\] do not work and we will be forced to invoke the existence of a spacetime where the initial data set is embedded. By exploiting a construction by Rácz and Wald in [@RW] we show that, in an embedded static KID, the set $\tbd \{\lambda>0\}$ is a finite union of smooth, compact and embedded surfaces, possibly with boundary. Moreover, at least one of the two null expansions $\theta^+$ or $\theta^-$ vanishes identically on each one of these surfaces (Proposition \[proposition1\]). With this result at hand we then prove that the set $\tbd \ext$ coincides with the outermost bounding MOTS (Theorem \[mainthr\]) provided the spacetime satisfies the NEC and that the [*past weakly outer trapped region*]{} $T^{-}$ is included in the [*weakly outer trapped region*]{} $T^{+}$. It may seem that the condition $T^{-}\subset T^+$ is very similar to ($\star$): In some sense, both try to avoid that the slice intersects first the white hole horizon when moving from the outside. However, it is important to remark that $T^+$ and $T^-$ have a priori nothing to do with Killing horizons and that the condition $T^{-}\subset T^{+}$ is not a condition directly on $\tbd \ext$. Our main uniqueness theorem is hence Theorem \[uniquenessthr\], which states that, under reasonable hypotheses, MOTS and spacelike sections of Killing horizons do coincide in static spacetimes. If the static spacetime is a black hole (in the global sense) then the event horizon is a Killing horizon. This shows the equivalence between MOTS and (spacelike sections of) the event horizon in the static setting. The last part of this thesis is devoted to the study of the Penrose inequality in initial data sets which are not time-symmetric. The standard version of the Penrose inequality bounds the ADM mass of the spacetime in terms of the smallest area of all surfaces which enclose the outermost MOTS. The huge problem in proving this inequality has led several authors to propose more general and simpler looking versions of the Penrose inequality (see [@Mars2] for a review). In particular, in a recent proposal by Bray and Khuri [@BK], a Penrose inequality has been conjectured in terms of the area of so-called outermost [*generalized apparent horizon*]{} in a given asymptotically flat initial data set. Generalized apparent horizons are more general than weakly outer trapped surfaces and have interesting analytic and geometric properties. The Penrose inequality conjectured by Bray and Khuri reads $$\label{ineq} M_{\scriptscriptstyle ADM}\geq \sqrt{\frac{|S_{out}|}{16\pi}},$$ where $M_{\scriptscriptstyle ADM}$ is the total ADM mass of a given slice and $|S_{out}|$ is the area of the outermost generalized apparent horizon $S_{out}$. This new inequality has several appealing properties, like being invariant under time reversals, the fact that no minimal area enclosures are involved and that it implies the standard Penrose inequality. On the other hand, this version is not directly supported by any heuristic argument based on cosmic censorship, as the standard Penrose inequality. In fact, as a consequence of a theorem by Eichmair [@Eichmair] on the existence, uniqueness and regularity of the outermost generalized apparent horizon, there exist slices in the Kruskal spacetimes (for which $\tbd \ext$ intersects both the black hole and the white hole event horizons), with the property that its outermost generalized apparent horizon lies, at least partially, inside the domain of outer communications. In Chapter \[ch:Article3\] we present a counterexample of (\[ineq\]) precisely by studying this type of slices in the Kruskal spacetime. The equations that define a generalized apparent horizon are non-linear elliptic PDE. Thus, we intend to determine properties of the solutions of these equations for slices sufficiently close to the time-symmetric slice of the Kruskal spacetime. Since the outermost generalized apparent horizon in the time-symmetric slice is the well-known bifurcation surface, we can exploit the implicit function theorem to show that any solution of the linearized equation for the generalized apparent horizon corresponds to the linearization of a solution of the non-linear problem (Proposition \[proposition\]). With this existence result at hand, we find a generalized apparent horizon $\hat{S}$ which turns out to be located entirely inside the domain of outer communications and which has area larger than $16 \pi M^2_{\scriptscriptstyle ADM}$, this violating (\[ineq\]). This would give a counterexample to the Bray and Khuri conjecture provided $\hat{S}$ is either the outermost generalized apparent horizon $S_{out}$ or else, the latter has not smaller area than the former one. Finally, we will prove that the area of $S_{out}$ is, indeed, at least as large as the area of $\hat{S}$, which gives a counterexample to (\[ineq\]) (Theorem \[theorem\]). It is important to remark that the existence of this counterexample does not invalidate the approach given by Bray and Khuri in [@BK] to prove the standard Penrose inequality but it does indicate that the emphasis must not be on generalized apparent horizons. This result has been published in [@CM3] and [@CMere3]. Before going into our new results, we start with a preliminary chapter where the fundamental definitions and results required to understand this thesis are stated and briefly discussed. This chapter contains in particular, a detailed sketch of the Bunting and Masood-ul-Alam method to prove uniqueness of electro-vacuum static black holes. We have preferred to collect all the preliminary material in one chapter to facilitate the reading of the thesis. We have also found it convenient to include two mathematical appendices. One where some well-known definitions of manifolds with boundary and topology are included (Appendix \[ch:appendix1\]) and another one that collects a number of theorems in mathematical analysis (Appendix \[ch:appendix2\]) which are used as tools in the main text. Preliminaries {#ch:Preliminaires} ============= Basic elements in a geometric theory of gravity ----------------------------------------------- The fundamental concept in any geometric theory of gravity is that of spacetime. A [**spacetime**]{} is a connected $n$-dimensional smooth differentiable manifold $M$ without boundary endowed with a Lorentzian metric $\gN$. All manifolds considered in this thesis will be Hausdorff. (see Appendix \[ch:appendix1\] for the definition). A Lorentzian metric is a metric with signature $(-,+,+,...,+)$. The covariant derivative associated with the Levi-Civita connection of $\gN$ will be denoted by $\nabla^{(n)}$ and the corresponding Riemann, Ricci and scalar curvature tensors will be denoted by $\RN_{\mu\nu\alpha\beta}$, $\RN_{\mu\nu}$ and $\RN$, respectively (where $\mu,\nu,\alpha,\beta=0,...,n-1$). We follow the sign conventions of [@Wald]. We will denote by $T_{\p} M$ the tangent space to $M$ at a point $\p\in M$, by $TM$ the tangent bundle to $M$ (i.e. the collection of the tangent spaces at every point of $M$) and by $\mathfrak{X}(M)$ the set of smooth sections of $TM$ (i.e. vector fields on $M$). According to the sign of its squared norm, a vector $\vec{v}\in T_{\p}M$ is: - Spacelike, if $\left.\gN_{\mu\nu}v^{\mu}v^{\nu}\right|_{\p}>0$. - Timelike, if $\left.\gN_{\mu\nu}v^{\mu}v^{\nu}\right|_{\p}<0$. - Null, if $\left.\gN_{\mu\nu}v^{\mu}v^{\nu}\right|_{\p}=0$. - Causal, if $\left.\gN_{\mu\nu}v^{\mu}v^{\nu}\right|_{\p}\leq 0$. A spacetime $(M,\gN)$ is [**time orientable**]{} if and only if there exists a vector field $\vec{u}\in \mathfrak{X}(M)$ which is timelike everywhere on $M$.\ Consider a time orientable spacetime $(M,\gN)$. A [**time orientation**]{} is a selection of a timelike vector field $\vec{u}$ which is declared to be future directed.\ A [**time oriented**]{} spacetime is a time orientable spacetime after a time orientation has been selected. In a time oriented manifold, causal vectors can be classified in two types: future directed or past directed. Let $(M,\gN)$ be a spacetime with time orientation $\vec{u}$. Then, a causal vector $\vec{v}\in T_{\p}M$ is - future directed if $\left. \gN_{\mu\nu}u^{\mu}v^{\nu} \right|_{\p}\leq 0$. - past directed if $\left. \gN_{\mu\nu}u^{\mu}v^{\nu} \right|_{\p}\geq 0$. Throughout this thesis all spacetimes are oriented (see Definition \[defi:orientablemanifold\] in Appendix \[ch:appendix1\]) and time oriented. General Relativity is a geometric theory of gravity in four dimensions in which the spacetime metric $\gM$ satisfies the Einstein field equations, which in geometrized units, $G=c=1$ (where $G$ is the Newton gravitational constant and $c$ is the speed of light in vacuum), takes the form: $$\label{Einsteinequation} \GM_{\mu\nu}+\Lambda\gM_{\mu\nu}=8\pi T_{\mu\nu},$$ where $\GM_{\mu\nu}$ is the so-called Einstein tensor, $\GN_{\mu\nu}\equiv\RN_{\mu\nu}-\frac12 \RN\gN_{\mu\nu}$ (in $n$ dimensions), $\Lambda$ is the so-called cosmological constant and $T_{\mu\nu}$ is the stress-energy tensor which describes the matter contents of the spacetime. In such a framework, freely falling test bodies are assumed to travel along the causal (timelike for massive particles and null for massless particles) geodesics of the spacetime $(M,\gM)$. Due to general physical principles, it is expected that many dynamical processes tend to a stationary final state. Studying these stationary configurations is therefore an essential step for understanding any physical theory. This is the case, for example, in gravitational collapse processes in General Relativity which are expected to settle down to a stationary system. Since the fundamental object in gravity is the spacetime metric $\gM$, the existence of symmetries in the spacetime is expressed in terms of a group of isometries, that is, diffeomorphisms of the spacetime manifold $M$ which leave the metric unchanged. The infinitesimal generator of the isometry group defines a so-called [*Killing vector field*]{}. Conversely, a Killing vector field defines a local isometry, i.e. a local group of diffeomorphisms, each of which is an isometry of $(M,\gM)$. If the Killing vector field is complete then the local group is, in fact, a global group of isometries (or, simply, an isometry). Throughout this thesis, we will mainly work at the local level without assuming that the Killing vector fields are complete, unless otherwise stated. More precisely, consider a spacetime $(M,\gM)$ and a vector field $\vec{\xi}\in \mathfrak{X}(M)$. The Lie derivative $\mathcal{L}_{\vec{\xi}}\,\gM_{\mu\nu}$ describes how the metric is deformed along the local group of diffeomorphisms generated by $\vec{\xi}$. We thus define the [**metric deformation tensor**]{} associated to $\vec{\xi}$, or simply deformation tensor, as $$\label{mdt} a_{\mu\nu}(\vec{\xi}\,)\equiv \mathcal{L}_{\vec{\xi}}\,\gM_{\mu\nu}= \nablaM_{\mu}\xi_{\nu}+\nablaM_{\nu}\xi_{\mu},$$ where, throughout this thesis, $\nabla$ will denote the covariant derivative of $\gM$. If $a_{\mu\nu}(\vec{\xi}\,)=0$, then the vector field $\vec{\xi}$ is a [**Killing vector field**]{} or simply a Killing vector.\ If the Killing field is timelike on some non-empty set, then the spacetime is called [**stationary**]{}. If, furthermore, the Killing field is integrable, i.e. $$\label{integrablekilling} \xi_{[\mu}\nablaM_{\nu}\xi_{\alpha]}=0$$ where the square brackets denote anti-symmetrization, then the spacetime is called [**static**]{}.\ Other important types of isometries are the following. If the Killing field is spacelike and the isometry group generated is $U(1)$, then the spacetime has a [**cyclic symmetry**]{}. If, furthermore, there exists a regular axis of symmetry, then the spacetime is [**axisymmetric**]{}. If the isometry group is $SO(3)$ with orbits being spacelike 2-spheres (or points), then the spacetime is [**spherically symmetric**]{}.\ Other special forms of $a_{\mu\nu}(\vec{\xi}\,)$ define special types of vectors which are also interesting. In particular, $a_{\mu\nu}(\vec{\xi}\,)=2\phi \gM_{\mu\nu}$ (with $\phi$ being a scalar function) defines a [**conformal Killing vector**]{} and $a_{\mu\nu}(\vec{\xi}\,)=2 C \gM_{\mu\nu}$ (with $C$ being a constant) corresponds to a [**homothety**]{}.\ Regarding the matter contents of the spacetime, represented by $T_{\mu\nu}$, we will not assume a priori any specific matter model, such as vacuum, electro-vacuum, perfect fluid, etc. However, we will often restrict the class of models in such a way that various types of so-called energy conditions are satisfied (c.f. Chapter 9.2 in [@Wald]). These are inequalities involving $T_{\mu\nu}$ acting on certain causal vectors and are satisfied by most physically reasonable matter models. In fact, since in General Relativity without cosmological constant, the Einstein equations impose $\GM_{\mu\nu}=8\pi T_{\mu\nu}$, these conditions can be stated directly in terms of the Einstein tensor. We choose to define the energy conditions directly in terms of $\GM_{\mu\nu}$. This is preferable because then all our results hold in any geometric theory of gravity independently of whether the Einstein field equations hold or not. Obviously, these inequalities are truly energy conditions only in specific theories as, for instance, General Relativity with $\Lambda=0$. Throughout this thesis, we will often need to impose the so-called [*null energy condition (NEC)*]{}. A spacetime $(M,\gM)$ satisfies the [**null energy condition (NEC)**]{} if the Einstein tensor $\GM_{\mu\nu}$ satisfies $\GM_{\mu\nu} k^{\mu}k^{\nu} |_\p \geq 0$ for any null vector $\vec{k}\in T_{\p} M$ and all $\p\in M$. Other usual energy conditions are the [*weak energy condition*]{} and the [*dominant energy condition (DEC)*]{}. A spacetime $(M,\gM)$ satisfies the [**weak energy condition**]{} if the Einstein tensor $\GM_{\mu\nu}$ satisfies that $\GM_{\mu\nu} t^{\mu}t^{\nu}|_\p\geq 0$ for any timelike vector $\vec{t}\in T_{\p}M$ and all $\p\in M$. A spacetime $(M,\gM)$ satisfies the [**dominant energy condition (DEC)**]{} if the Einstein tensor $\GM_{\mu\nu}$ satisfies that $-{\GM}{}^{\nu}_{\mu} t^{\mu}|_\p$ is a future directed causal vector for any future directed timelike vector $\vec{t}\in T_{\p} M$ and all $\p\in M$. [**Remark.**]{} Obviously, the DEC implies the NEC. $\hfill \square$ Geometry of surfaces in Lorentzian spaces {#sc:GeometryOfSurfaces} ----------------------------------------- ### Definitions {#ssc:GeometryOfSurfacesDefinitions} In this subsection we will motivate and introduce several types of surfaces, such as trapped surfaces and marginally outer trapped surfaces, that will play an important role in this thesis. We will also discuss several relevant known results concerning them. For an extensive classification of surfaces in Lorentzian spaces, see [@S1]. Let us begin with some previous definitions and notation. In what follows, $M$ and $\Sigma$ are two smooth differentiable manifolds, $\Sigma$ possibly with boundary, with dimensions $n$ and $s$, respectively, satisfying $n\geq s$. Let $\Phi: \Sigma\rightarrow M$ be a smooth map between $\Sigma$ and $M$. Then $\Phi$ is an [**immersion**]{} if its differential has maximum rank (i.e. $rank(\Phi)=s$) at every point. The set $\Phi(\Sigma)$ is then said to be [*immersed*]{} in $M$. However $\Phi(\Sigma)$ can fail to be a manifold because it can intersect itself.\ To avoid self-intersections, one has to consider [*injective immersions*]{}. In fact, we will say that $\Phi(\Sigma)$ is a [**submanifold**]{} of $M$ if $\Sigma$ is injectively immersed in $M$. All immersions considered in this thesis will be submanifolds. For simplicity, and since no confusion usually arises, we will frequently denote by the same symbol ($\Sigma$ in this case) both the manifold $\Sigma$ (as an abstract manifold) and $\Phi(\Sigma)$ (as a submanifold). Similarly, and unless otherwise stated, we will use the same convention for contravariant tensors. More specifically, a contravariant tensor defined on $\Sigma$ and pushed-forward to $\Phi(\Sigma)$ will be usually denoted by the same symbol. Notice however that $\Phi(\Sigma)$ admits two topologies which are in general different: the induced topology as a subset of $M$ and the manifold topology defined by $\Phi$ from $\Sigma$. When referring to topological concepts in injectively immersed submanifolds we will always use the subset topology unless otherwise stated. Next, we will define the first and the second fundamental forms of a submanifold. Consider a smooth manifold $M$ endowed with a metric $\gN$ and let $\Sigma$ be a submanifold of $M$. Then, the [**first fundamental form**]{} of $\Sigma$ is the tensor field $g$ on $\Sigma$ defined as $$g=\Phi^{*}\left( \gN \right),$$ where $\Phi^{*}$ denotes the pull-back of the injective immersion $\Phi: \Sigma\rightarrow M$. According to the algebraic properties of its first fundamental form, a submanifold can be classified as follows. A submanifold $\Sigma$ of a spacetime $M$ is: - [**Spacelike**]{} if $g$ is non-degenerate and positive definite. - [**Timelike**]{} if $g$ is non-degenerate and non-positive definite. - [**Null**]{} if $g$ is degenerate. The following result is straightforward and well-known (see e.g. [@citesubmanifold]) Let $\Sigma$ be a submanifold of $M$. Then, the first fundamental form $g$ of $\Sigma$ is non-degenerate (and, therefore, a [*metric*]{}) at a point $\p\in \Sigma$ if and only if $$\label{nnsubmanifold} T_{\p} M=T_{\p}\Sigma \oplus (T_{\p} \Sigma)^{\perp},$$ where $(T_{\p} \Sigma)^{\perp}$ denotes the set of normal vectors to $\Sigma$ at $\p$. We will denote $(T_{\p}M)^{\perp}$ by $N_{\p}M$ and we will call this set the normal space to $\Sigma$ at $\p$. The collection of all normal spaces forms a vector bundle over $\Sigma$ which is called the normal bundle and is denoted by $N\Sigma$. From now on, unless otherwise stated, we will only consider submanifolds satisfying (\[nnsubmanifold\]) at every point. Let us denote by $\nablaSigma$ the covariant derivative associated with $g$. Next, consider two arbitrary vectors $\vec{X},\vec{Y}\in\mathfrak{X}(\Sigma)$. According to (\[nnsubmanifold\]), the derivative $\nabla^{(n)}_{\vec{X}}\vec{Y}$, as a vector on ${T M}$, can be split according to $$\nabla^{(n)}_{\vec{X}}\vec{Y}=\left( \nabla^{(n)}_{\vec{X}}\vec{Y} \right){}^{T} + \left( \nabla^{(n)}_{\vec{X}}\vec{Y} \right){}^{\perp},$$ where the superindices $T$ and $\perp$ denote the tangential and normal parts with respect to $\Sigma$. The following is an important result in the theory of submanifolds [@citesubmanifold]. With the notation above, we have $$\left( \nabla^{(n)}_{\vec{X}}\vec{Y} \right){}^{T}=\nablaSigma_{\vec{X}}\vec{Y}.$$ The extrinsic geometry of the submanifold is encoded in its second fundamental form. The [**second fundamental form vector**]{} $\vec{K}$ of $\Sigma$ in $M$ is a symmetric linear map $\vec{K}:\mathfrak{X}(\Sigma)\times \mathfrak{X}(\Sigma)\rightarrow N\Sigma$ defined by $$\vec{K}(\vec{X},\vec{Y})= - \left( \nabla^{(n)}_{\vec{X}}\vec{Y} \right){}^{\perp},$$ for all $\vec{X},\vec{Y}\in\mathfrak{X}(\Sigma)$. [**Remark.**]{} Our sign convention is such that the second fundamental form vector of a 2-sphere in the Euclidean 3-space points outwards. $\hfill \square$ The [**mean curvature vector**]{} of $\Sigma$ in $M$ is defined as $\vec{H}\equiv \tr_{\,\Sigma}\vec{K}$ (where $\tr_{\,\Sigma}$ denotes the trace with the induced metric $g$ on $T_{\p}\Sigma$ for any $\p\in \Sigma$). We will define an [**embedding**]{} $\Phi$ as an injective immersion such that $\Phi: \Sigma\rightarrow \Phi(\Sigma)$ is an homeomorphism with the topology on $\Phi(\Sigma)$ induced from $M$. The image $\Phi(\Sigma)$ will be called an embedded submanifold. A [**surface**]{} $S$ is a smooth, orientable, codimension two, embedded submanifold of $M$ with positive definite first fundamental form $\gamma$. From now on we will focus on 4-dimensional spacetimes $(M,\gM)$. For a surface $S\subset M$ we have the following result. \[thr:lk\] The normal bundle of $S$ admits two vector fields $\left\{ \vec{l}_{+},\vec{l}_{-} \right\}$ which are null and future directed everywhere, and which form a basis of $NS$ in $TM$ at every point $\p\in S$. [**Proof.**]{} Let $\p\in S$ and $(U_{\alpha},\varphi_{\alpha})$ be any chart at $\p$ belonging to the positively oriented atlas of $M$. Let us define $\{\vec{l}_{+}^{\,\, U_{\alpha}},\vec{l}_{-}^{\,\, U_{\alpha}}\}$ as the solution of the set of equations $$\begin{aligned} \label{lk} \left.\gM(\vec{l}_{\pm}^{\,\,U_{\alpha}},\vec{e}_{A})\right|_{\p}=0,\qquad \quad \left.\gM(\vec{l}_{\pm}^{\,\,U_{\alpha}},\vec{l}_{\pm}^{\,\,U_{\alpha}})\right|_{\p}=0,\nonumber\\ \left.\gM(\vec{l}_{+}^{\,\,U_{\alpha}},\vec{l}_{-}^{\,\,U_{\alpha}})\right|_{\p}=-2,\qquad\quad \left.\gM(\vec{l}_{+}^{\,\,U_{\alpha}},\vec{u})\right|_{\p}=-1,\\ \hspace{-3cm}\left. {\bf \eta}^{(4)}(\vec{l}_{-}^{\,\,U_{\alpha}},\vec{l}_{+}^{\,\,U_{\alpha}},\vec{e}_{1},\vec{e}_{2})\right|_{\p}>0. \nonumber\end{aligned}$$ where the vectors $\{\vec{e}_{A}\}$ ($A=1,2$) are the coordinate basis in $U_{\alpha}$, $\vec{u}$ is the timelike vector which defines the time-orientation for the spacetime and ${\bf \eta}^{(4)}$ is the volume form of $(M,\gM)$. It is immediate to check that $\{\vec{l}_{+}^{\,\,U_{\alpha}},\vec{l}_{-}^{\,\,U_{\alpha}}\}$ exists and is unique. The last equation is necessary in order to avoid the ambiguity $\vec{l}_{+}^{\,\,U_{\alpha}}\leftrightarrow\vec{l}_{-}^{\,\,U_{\alpha}}$ allowed by the previous four equations. The set $\{\vec{l}_{+}^{\,\,U_{\alpha}},\vec{l}_{-}^{\,\,U_{\alpha}}\}$ defines two vector fields if and only if this definition is independent of the chart. Select any other positively oriented chart $(U_{\beta},\varphi_{\beta})$ at $\p$. Let $\{\vec{e'}_{1},\vec{e'}_{2}\}$ be the corresponding coordinate basis, which is related with $\{\vec{e}_{1},\vec{e}_{2}\}$ by ${{e'_{A}}}^{\mu}=A_{\nu}^{\mu} e_{A}^{\nu}$ ($A,B=1,2$), where $A_{\nu}^{\mu}$ denotes the Jacobian. Since $U_{\alpha}$ and $U_{\beta}$ belong to the positively oriented atlas, we have that $\text{det} A >0$ everywhere. The first four equations in (\[lk\]) force that either $\vec{l}_{\pm}^{\,\,U_{\beta}}=\vec{l}_{\pm}^{\,\,U_{\alpha}}$ or $\vec{l}_{\pm}^{\,\,U_{\beta}}=\vec{l}_{\mp}^{\,\,U_{\alpha}}$. However, the second possibility would imply $$\left. {\bf \eta}^{(4)}(\vec{l}_{-}^{\,\,U_{\beta}},\vec{l}_{+}^{\,\,U_{\beta}},\vec{e'}_{1},\vec{e'}_{2})\right|_{\p}= \left. (\text{det} A)\, {\bf \eta}^{(4)}(\vec{l}_{+}^{\,\,U_{\alpha}},\vec{l}_{-}^{\,\,U_{\alpha}},\vec{e}_{1},\vec{e}_{2})\right|_{\p}<0,$$ which contradicts the fifth equation in (\[lk\]) for $U_{\beta}$. Consequently $\{\vec{l}_{+},\vec{l}_{-}\}$ does not depend on the chart, which proves the result. $\hfill \blacksquare$\ [**Remark.**]{} From now on we will take the vector fields $\vec{l}_{+}$, $\vec{l}_{-}$ to be partially normalized to satisfy ${l_{+}}_{\mu}l_{-}^{\mu}=-2$, as in the proof of the lemma. Note that these vector fields are then defined modulo a transformation $\vec{l}_{+}\rightarrow F\vec{l}_{+}$, $\vec{l}_{-}\rightarrow \frac 1F \vec{l}_{-}$, where $F$ is a positive function on $S$. $\hfill \square$\ For a surface $S$, $\nablaS$ will denote the covariant derivative associated with $\gamma$ and $\vec{\Pi}$ and $\vec{H}$ will denote the second fundamental form vector and the mean curvature of $S$ in $M$. The physical meaning of the causal character of $\vec{H}$ is closely related to the first variation of area, which we briefly discuss next. Let $\vec{ \nu}$ be a normal variation vector on $S$, i.e. a vector defined in a neighbourhood of $S$ in $M$ which, on $S$, is orthogonal to $S$. Choose $\vec{\nu}$ to be compactly supported on $S$ (which obviously places no restrictions when $S$ itself is compact). The vector $\vec{ \nu}$ generates a one-parameter local group $\{ \varphi_{\tau} \}_{\tau\in I}$ of transformations where $\tau$ is the canonical parameter and $I\subset \mathbb{R}$ is an interval containing $\tau=0$. We then define a one parameter family of surfaces $S_{\tau}\equiv \varphi_{\tau}(S)$, which obviously satisfies $S_{\tau=0}=S$. Let $|S_{\tau}|$ denote the area of the surface $S_{\tau}$. The formula of the first variation of area states (see e.g. [@Chavel]) $$\label{firstvariation} \delta_{\vec{\nu}}|S|\equiv\left.\frac{d|S_{\tau}|}{d\tau}\right|_{\tau=0}=\int _{S} H_{\mu}\nu^{\mu}\eta_{S}.$$ [**Remark.**]{} It is important to indicate that, when $S$ is boundaryless, expression (\[firstvariation\]) holds regardless of whether the variation $\vec{\nu}$ is normal or not. This formula is valid for any dimensions of $M$ and $S$, provided $\text{dim} M>\text{dim} S$. $\hfill \square$\ The first variation of area justifies the definition of a [*minimal surface*]{} as follows. A surface $S$ is [**minimal**]{} if and only if $\vec{H}=0$. According to (\[firstvariation\]), if $\vec{H}$ is timelike and future directed (resp. past directed) everywhere on $S$, then the area of $S$ will decrease along any non-zero causal future (resp. past) direction. If a surface is such that its area does not increase for any future variation, one may say that the surface is, in some sense, trapped. Thus, according to the previous discussion, we find that the [*trappedness*]{} of a surface is intimately related with the causal character and time orientation of its mean curvature vector $\vec{H}$. In what follows, we will introduce various notions of trapped surface. For that, it will be useful to consider a null basis $\{\vec{l}_{+},\vec{l}_{-}\}$ for the normal bundle of $S$ in $M$, as before. Then, the mean curvature vector decomposes as $$\label{MeanCurvature} \vec{H}=-\frac 12 \left( \theta^{-}\vec{l}_{+} + \theta^{+}\vec{l}_{-} \right),$$ where $\theta^{+}\equiv {l_{+}}^{\mu}H_{\mu}$ and $\theta^{-}\equiv {l_{-}}^{\mu}H_{\mu}$ are the null expansions of $S$ along $\vec{l}_{+}$ and $\vec{l}_{-}$, respectively. It is worth to remark that these null expansions $\theta^{\pm}$ are equal to the divergence on $S$ of light rays (i.e. null geodesics) emerging orthogonally from $S$ along $\vec{l}_{\pm}$. Thus, the negativity of both $\theta^{+}$ and $\theta^{-}$ indicates the presence of strong gravitational fields which bend the light rays sufficiently so that both are contracting. Thus, this leads to various concepts of trapped surfaces, as follows. \[defi:MTS\] A closed (i.e. compact and without boundary) surface is a: - [**Trapped surface**]{} if $\theta^{+}< 0$ and $\theta^{-}< 0$. Or equivalently, if $\vec{H}$ is timelike and future directed. - [**Weakly trapped surface**]{} if $\theta^{+}\leq 0$ and $\theta^{-}\leq 0$. Or equivalently, if $\vec{H}$ is causal and future directed. - [**Marginally trapped surface**]{} if either, $\theta^{+}=0$ and $\theta^{-}\leq 0$ everywhere, or, $\theta^{+}\leq 0$ and $\theta^{-}= 0$ everywhere. Equivalently, if $\vec{H}$ is future directed and either proportional to $\vec{l}_{+}$ or proportional to $\vec{l}_-$ everywhere. If the signs of the inequalities are reversed then we have trappedness along the past directed causal vectors orthogonal to $S$. Thus, A closed surface is a: - [**Past trapped surface**]{} if $\theta^{+}> 0$ and $\theta^{-}>0 $. Or equivalently if $\vec{H}$ is timelike and past directed. - [**Past weakly trapped surface**]{} if $\theta^{+}\geq 0$ and $\theta^{-}\geq 0$. Or equivalently if $\vec{H}$ is causal and past directed. - [**Past marginally trapped surface**]{} if either, $\theta^{+}=0$ and $\theta^{-}\geq 0$ everywhere, or $\theta^{+}\geq 0$ and $\theta^{-}=0$ everywhere. Equivalently, $\vec{H}$ is past directed and either proportional to $\vec{l}_{+}$ or proportional to $\vec{l}_-$ everywhere. We also define “untrapped" surface as a kind of strong complementary of the above. A closed surface is [**untrapped**]{} if $\theta^{+}\theta^{-}<0$, or equivalently if $\vec{H}$ is spacelike everywhere. Notice that, according to these definitions, a closed [*minimal*]{} surface is both weakly trapped and marginally trapped, as well as past weakly trapped and past marginally trapped. Because of their physical meaning as indicators of strong gravitational fields, trapped surfaces are widely considered as good natural quasi-local replacements for black holes. Let us briefly recall the definition of a black hole which, as already mentioned in the Introduction, involves global hypotheses in the spacetime. First, it requires a proper definition of asymptotic flatness in terms of the conformal compactification of the spacetime (see e.g. Chapter 11 of [@Wald]). Besides, it also requires that the spacetime is [*strongly asymptotically predictable*]{}, (see Chapter 12 of [@Wald] for a precise definition). A strongly asymptotically predictable spacetime $(M,\gM)$ is then said to contain a black hole if $M$ is not contained in the causal past of future null infinity $J^{-}(\mathscr{I}^{+})$. The [**black hole region**]{} ${\mathcal{B}}$ is defined as $\mathcal{B}=M\setminus J^{-}(\mathscr{I}^{+})$. The topological boundary $\mathcal{H}_{\mathcal{B}}$ of $\mathcal{B}$ in $M$ is called the [**event horizon**]{}. Similarly, we can define the [**white hole region**]{} $\mathcal{W}$ as the complementary of the causal future of past null infinity, i.e. $M\setminus J^{+}(\mathscr{I}^{-})$, and the [**white hole event horizon**]{} $\mathcal{H}_{\mathcal{W}}$ as its topological boundary. Finally, the [**domain of outer communications**]{} is defined as $M_{DOC}\equiv J^{-}(\mathscr{I}^{+})\cap J^{+}(\mathscr{I}^{-})$. Hawking and Ellis show (see Chapter 9.2 in [@HE]) that weakly trapped surfaces lie inside the black hole region in a spacetime provided this spacetime is future asymptotically predictable. However, as we already pointed out in the Introduction, the study of trapped surfaces is specially interesting when no global assumptions are imposed on the spacetime and the concept of black hole is not available. It is worth to remark that trapped surfaces are also fundamental ingredients in several versions of singularity theorems of General Relativity (see e.g. Chapter 9 in [@Wald]). Note that all the surfaces introduced above are defined by restricting both null expansions $\theta^+$ and $\theta^-$. When only one of the null expansions is restricted, other interesting types of surfaces are obtained: the [*outer*]{} trapped surfaces, which will be the fundamental objects of this thesis. Again, consider a surface $S$. Suppose that for some reason one of the future null directions can be geometrically selected so that it points into the “outer" direction of $S$ (shortly, we will find a specific setting where this selection is meaningful). In that situation we will always denote by $\vec{l}_{+}$ the vector pointing along this outer null direction. We will say that $\vec{l}_{+}$ is the future outer null direction, and similarly, $\vec{l}_{-}$ will be the future inner null direction. We define the following types of surfaces (c.f. Figure \[fig:MOTS\]). \[defi:MOTS\] A closed surface is: - [**Outer trapped**]{} if $\theta^+ < 0$. - [**Weakly outer trapped**]{} if $\theta^{+}\leq 0$. - [**Marginally outer trapped (MOTS)**]{} if $\theta^{+}=0$. - [**Outer untrapped**]{} if $\theta^{+}>0$. ![ This figure represents the normal space to $S$ in $M$ at a point $\p\in S$. If $S$ is outer trapped, the mean curvature vector $\vec{H}$ points into the shaded region. If $S$ is a MOTS, $\vec{H}$ points into the direction of the bold line. []{data-label="fig:MOTS"}](MOTS.eps){width="6cm"} As before, these definitions depend on the time orientation of the spacetime. If the time orientation is reversed but the notion of [*outer*]{} is unambiguous, then $-\vec{l}_{-}$ becomes the new future outer null direction. Since the null expansion of $-\vec{l}_{-}$ is $-\theta^{-}$, the following definitions become natural (c.f. Figure \[fig:pastMOTS\]). \[defi:PastMOTS\] A closed surface is: - [**Past outer trapped**]{} if $\theta^- > 0$. - [**Past weakly outer trapped**]{} if $\theta^{-}\geq 0$. - [**Past marginally outer trapped (past MOTS)**]{} if $\theta^{-}=0$. - [**Past outer untrapped**]{} if $\theta^{-}<0$. ![ On the normal space $N_{\p}S$ for any point $\p\in S$, the mean curvature vector $\vec{H}$ points into the shaded region if $S$ is past outer trapped, and into the direction of the bold line if $S$ is a past MOTS. []{data-label="fig:pastMOTS"}](pMOTS.eps){width="6cm"} As for weakly trapped surfaces, weakly outer trapped surfaces are always inside the black hole region provided the spacetime is strongly asymptotically predictable. In fact, in one of the simplest dynamical situations, namely the Vaidya spacetime, Ben-Dov has proved [@BenDov] that the event horizon is the boundary of the spacetime region containing weakly outer trapped surfaces, proving in this particular case a previous conjecture by Eardley [@Eardley]. On the other hand, Bengtsson and Senovilla have shown [@SB] that the spacetime region containing weakly trapped surfaces does not extend to the event horizon. This result suggests that the concept of weakly outer trapped surface does capture the essence of a black hole better than that of weakly trapped surface. Two other interesting classes of surfaces that also depend on a choice of outer direction are the so-called [*generalized trapped surfaces*]{} and its marginal case, [*generalized apparent horizons*]{}. They were specifically introduced by Bray and Khuri while studying a new approach to prove the Penrose inequality [@BK]. \[defi:GAH\] A closed surface is a: - [**Generalized trapped surface**]{} if $\left.\theta^{+}\right|_{\p}\leq 0$ or $\left.\theta^{-}\right|_{\p}\geq 0$ at each point $\p\in S$. - [**Generalized apparent horizon**]{} if either $\left.\theta^{+}\right|_{\p}=0$ with $\left.\theta^{-}\right|_{\p}\leq 0$ or $\left.\theta^{-}\right|_{\p}=0$ with $\left.\theta^{+}\right|_{\p}\geq 0$ at each point $\p\in S$. It is clear from Figures \[fig:MOTS\], \[fig:pastMOTS\] and \[fig:GAH\] that the set of generalized trapped surfaces includes both the set of weakly outer trapped surfaces and the set of past weakly outer trapped surfaces as particular cases. ![This figure represents the normal space of a surface $S$ in $M$ at a point $\p\in S$. For generalized trapped surfaces, the mean curvature vector $\vec{H}$ points into the shaded region. For generalized apparent horizons, $\vec{H}$ points into the direction of the bold line.[]{data-label="fig:GAH"}](GAH.eps){width="6cm"} In this thesis we will often consider surfaces embedded in a spacelike hypersurface $\Sigma\subset M$. For this reason, it will be useful to give a (3+1) decomposition of the null expansions and to reformulate the previous definitions in terms of objects defined directly on $\Sigma$. A [**hypersurface**]{} $\Sigma$ of $M$ is an embedded, connected spacelike submanifold, possibly with boundary, of codimension 1. Let us consider a hypersurface $\Sigma$ of $M$ and denote by $g$ its induced metric, by $\vec{K}$ its second fundamental form vector and by $K$ the second fundamental form, defined as $K(\vec{X},\vec{Y})=-{\bf n}(\vec{K}(\vec{X},\vec{Y}))$, where $\bf n$ is the unit, future directed, normal 1-form to $\Sigma$ and $\vec{X}, \vec{Y}\in \mathfrak{X}(\Sigma)$. Consider a surface $S$ embedded in $\id$ As before, we denote by $\gamma$, $\vec{\Pi}$ and $\vec{H}$ the induced metric, the second fundamental form vector and the mean curvature vector of $S$ as a submanifold of $(M,\gM)$, respectively. As a submanifold of $\Sigma$, $S$ will also have a second fundamental form vector $\vec{\kappa}$ and a mean curvature vector $\vec{p}$. From their definitions, we immediately have $$\vec{\Pi}(\vec{X},\vec{Y})=\vec{K}(\vec{X},\vec{Y})+\vec{\kappa}(\vec{X},\vec{Y}),$$ where $\vec{X},\vec{Y}\in\mathfrak{X}(S)$. Taking trace on $S$ we find $$\label{Hpvec} \vec{H}=\vec{p}+\gamma^{AB}\vec{K}_{AB},$$ where $\vec{K}_{AB}$ is the pull-back of $\vec{K}_{ij}$ ($i,j=1,2,3$) onto $S$. Assume that an outer null direction $\vec{l}_{+}$ can be selected on $S$. Then, after a suitable rescaling of $\vec{l}_{+}$ and $\vec{l}_{-}$, we can define $\vec{m}$ univocally on $S$ as the unit vector tangent to $\Sigma$ which satisfies $$\begin{aligned} \vec{l}_{+}&=&\vec{n}+\vec{m},\label{l+mn} \\ \vec{l}_{-}&=& \vec{n}-\vec{m}.\label{l-mn}\end{aligned}$$ By construction, $\vec{m}$ is normal to $S$ in $\Sigma$ and will be denoted as the [*outer normal*]{}. Multiplying (\[Hpvec\]) by $\vec{l}_+$ and by $\vec{l}_-$ we find $$\label{Hp} \theta^{\pm}=\pm p + q,$$ where $p\equiv p_{i}m^{i}$ and $q\equiv \gamma^{AB}K_{AB}$. All objects in (\[Hp\]) are intrinsic to $\Sigma$. This allows us to reformulate the definitions above in terms of $p$ and $q$. The following table summarizes the types of surfaces mostly used in this thesis. [Outer trapped surface]{} $p < -q$ --------------------------------------------------- ------------- [Weakly outer trapped surface]{} $p \leq -q$ Marginally outer trapped surface (MOTS) $p=-q$ Outer untrapped surface $p>-q$ Past outer trapped surface $p<q$ Past weakly outer trapped surface $p\leq q$ Past marginally outer trapped surface (past MOTS) $p=q$ Past outer untrapped surface $p>q$ Generalized trapped surface $p\leq |q|$ Generalized apparent horizon $p=|q|$ : Definitions of various types of trapped surfaces in terms of the mean curvature $p$ of $S\subset \Sigma$ and the trace $q$ on $S$ of the second fundamental form of $\Sigma$ in $M$. Having defined the main types of surfaces used in this thesis, let us next consider the important concept of stability of a MOTS. ### Stability of marginally outer trapped surfaces (MOTS) {#ssc:GeometryOfSurfacesStability} Let us first recall the concept of stability for minimal surfaces. Let $S$ be a closed minimal surface embedded in a Riemannian $3$-dimensional manifold $(\Sigma,g)$. From (\[firstvariation\]), $S$ is an extremal of area for all variations (normal or not). In order to study whether this extremum is a minimum, a maximum or a saddle point, it is necessary to analyze the second variation of area. A minimal surface is called [*stable*]{} if the second variation of area is non-negative for all smooth variations. This definition becomes operative once an explicit form for the second variation is obtained. For closed minimal surfaces the crucial object is the so-called [*stability operator*]{}, defined as follows. Consider a variation vector $\psi \vec{m}$ normal to $S$ within $\Sigma$. Let us denote by a sub-index $\tau$ the magnitudes which correspond to the surfaces $S_{\tau}=\varphi_{\tau}(S)$ (where, as before, $\{\varphi_{\tau}\}_{\tau\in I\subset\mathbb{R}}$ denotes the one-parameter local group of transformations generated by any vector $\vec{\nu}$ satisfying $\left.\vec{\nu} \right|_{S}=\psi\vec{m}$). For any covariant tensor $\Gamma$ defined on $S$, let us define the variation of $\Gamma$ along $\psi{\vec{m}}$ as $\delta_{\psi\vec{m}} \Gamma \equiv \left.\frac{d}{d\tau} \left[ \varphi_{\tau}^{*}(\Gamma_{\tau}) \right]\right|_{\tau=0}$, where $\varphi_{\tau}^{*}$ denotes the pull-back of $\varphi_{\tau}$ (this definition does not depend on the extension of the vector $\psi\vec{m}$ outside $S$). The stability operator $L^{min}_{\vec{m}}$ is then defined as $$\label{stabilityoperatorforminimalsurfaces} L^{min}_{\vec m} \psi\equiv\delta_{\psi\vec{m}}p=-\Delta_{S}\psi - (\RSigma_{ij}{m}^{i}{m}^{j}+{\kappa}_{ij}\kappa^{ij})\psi,$$ where $\Delta_{S}=\nablaS_{A}{\nablaS}^{A}$ is the Laplacian on $S$ and $\RSigma_{ij}$ denotes the Ricci tensor of $(\Sigma,g)$. The second equality follows from a direct computation (see e.g. [@Chavel]). In terms of the stability operator, the formula for the second variation of area of a closed minimal surface is given by $$\label{secondvariationforminimalsurfaces0} {\delta^{2}_{\psi\vec{m}}} |S|=\int_{S} \psi L^{min}_{\vec{m}}\psi \eta_{S}.\nonumber$$ The operator $L^{min}_{\vec{m}}$ is linear, elliptic and formally self-adjoint (see Appendix \[ch:appendix2\] for the definitions). Being self-adjoint implies that the principal eigenvalue $\auto$ can be represented by the Rayleigh-Ritz formula (\[Rayleigh-Ritz\]), and therefore the second variation of area can be bounded according to $$\label{secondvariationforminimalsurfaces} {\delta^{2}_{\psi\vec{ m}}} |S|\geq \auto\int_{S}\psi^{2}\eta_S ,\nonumber$$ where equality holds when $\psi$ is a principal eigenfunction (i.e. an eigenfunction corresponding to $\auto$). This implies that ${\delta^{2}_{\psi\vec{m}}} |S|\geq 0$ for all smooth variations is equivalent to $\auto\geq 0$. Thus, [*a minimal surface is stable if and only if $\auto\geq 0$*]{}. A related construction can be performed for MOTS. Consider a MOTS $S$ embedded in a spacelike hypersurface $\Sigma$ of a spacetime $M$. As embedded submanifolds of $\Sigma$, MOTS are not minimal surfaces in general. Consequently, any connection between stability and the second variation of area is lost. However, the stability for minimal surfaces involves the sign of the variation $\delta_{\psi\vec{ m}}p$ (see (\[stabilityoperatorforminimalsurfaces\])), so it is appropriate to define stability of MOTS in terms of the sign of first variations of $\theta^{+}$. A formula for the first variation of $\theta^{+}$ was derived by Newman in [@Newman] for arbitrary immersed spacelike submanifolds. The derivation was simplified by Andersson, Mars and Simon in [@AMS]. \[lema:variationtheta+general0\] Consider a surface $S$ embedded in a spacetime $(M,\gM)$. Let $\{\vec{l}_{+},\vec{l}_{-}\}$ be a future directed null basis in the normal bundle of $S$ in $M$, partially normalized to satisfy ${l_{+}}_{\mu}l_{-}^{\mu}=-2$. Any variation vector $\vec{\nu}$ can be decomposed on $S$ as $\vec{ \nu}=\vec{ \nu}^{\,\parallel}+b\vec{l}_{+}-\frac u2 \vec{l}_{-}$, where $\vec{ \nu}^{\,\parallel}$ is tangent to $S$ and $b$ and $u$ are functions on $S$. Then, $$\begin{aligned} \label{variationtheta+general0} \delta_{\vec{ \nu}}\theta^{+}&=&-\frac{\theta^{+}}{2}{l_{-}}^{\mu}\delta_{\vec{ \nu}}{{l_{+}}}_{\mu} +\vec{ \nu}^{\,\parallel}(\theta^{+}) - b\left( {\Pi^{\mu}}_{AB}{\Pi^{\nu}}^{AB}{l_{+}}_{\mu}{l_{+}}_{\nu}+G_{\mu\nu}{l_{+}}^{\mu}{l_{+}}^{\nu} \right) - \Delta_{S}u \nonumber\\ && + 2s^{A}\nablaS_{A}u+ \frac{u}{2}\left( R_{S}-H^{2}-G_{\mu\nu}{l_{+}}^{\mu}{l_{-}}^{\nu}-2s_{A}s^{A}+2\nablaS_{A}s^{A} \right),\end{aligned}$$ where $R_{S}$ denotes the scalar curvature of $S$, $H^{2}=H_{\mu}H^{\mu}$ and $s_{A}=-\frac12 {l_{-}}_{\mu}\nablaM_{\vec{e}_{A}}{l_{+}}^{\mu}$, with $\{\vec{e}_{A}\}$ being a local basis for $TS$. Expression (\[variationtheta+general0\]) can be particularized when the variation is restricted to $\Sigma$, i.e. when $\vec{\nu}=\psi\vec{m}$ for an arbitrary function $\psi$. Writing $\vec{l}_{\pm}=\vec{n}\pm\vec{m}$ as before, we have $\vec{\nu}=\frac{\psi}{2}(\vec{l}_{+}-\vec{l}_{-})$ and hence $\vec{\nu}^{\,\parallel}=0$, $b=\frac{\psi}{2}$, $u=\psi$. As a consequence of Lemma \[lema:variationtheta+general0\] we have the following [@AMS]. \[defi:stabilityoperator\] The [**stability operator**]{} $L_{\vec{m}}$ for a MOTS $S$ is defined by $$\label{stabilityoperator} L_{\vec{m}}\psi\equiv \delta_{\psi\vec{m}}\theta^{+}=-\Delta_{S}\psi + 2s^{A}\nablaS_{A}\psi + \left(\frac 12 R_{S}- Y- s_{A}s^{A}+\nablaS_{A}s^{A} \right)\psi,$$ where $$\label{Y} Y\equiv \frac 12 \Pi _{AB}^{\mu}{\Pi^{\nu}}^{AB}{l_{+}}_{\mu}{l_{+}}_{\nu}+G_{\mu\nu}l_{+}^{\mu}n^{\nu}.$$ [**Remark.**]{} In terms of objects on $\Sigma$, a simple computation using $\vec{l}_{\pm}=\vec{n}\pm\vec{m}$ shows that $s_{A}=m^{i}e_{A}^{j}K_{ij}$. $\hfill \square$\ If we consider a variation along $\vec{l}_{+}$, then (\[variationtheta+general0\]) implies that, on a MOTS, $$\delta_{\psi\vec{l}_{+}}\theta^{+}=-\psi W,\label{raychaudhuri}$$ where $$\label{W} W= \Pi _{AB}^{\mu}{\Pi^{\nu}}^{AB}{l_{+}}_{\mu}{l_{+}}_{\nu}+G_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}.$$ This is the well-known Raychaudhuri equation for a MOTS (see e.g. [@Wald]). Note that $W$ is non-negative provided the NEC holds and $Y$ is non-negative if the DEC holds (recall that $\vec{n}$ is timelike). The operator $L_{\vec{m}}$ is linear and elliptic which implies that it has a discrete spectrum. However, due to the presence of a first order term, it is not formally self-adjoint (see Appendix \[ch:appendix2\]) in general. Nevertheless, it is still true (c.f. Lemma (\[PrincipleEigenvalue\] in Appendix \[ch:appendix2\])) that there exists an eigenvalue $\auto$ with smallest real part. This eigenvalue is called the [*principal eigenvalue*]{} and it has the following properties: 1. It is real. 2. Its eigenspace (the set of smooth real functions $\psi$ on $S$ satisfying $L_{\vec{m}}\psi=\auto\psi$) is one-dimensional. 3. An eigenfunction $\psi$ of $\auto$ vanishes at one point $\p\in S$ if and only if it vanishes everywhere on $S$ (i.e. the principal eigenfunctions do not change sign). The stability of minimal surfaces could be rewritten in terms of the sign of the principal eigenvalue of its stability operator. In [@AMS2], [@AMS] the following definition of stability of MOTS is put forward. \[defi:stableMOTS\] A MOTS $S\subset\Sigma$ is [**stable in $\Sigma$**]{} if the principal eigenvalue $\auto$ of the stability operator $L_{\vec{m}}$ is non-negative. $S$ is [**strictly stable in $\Sigma$**]{} if $\auto>0$. For simplicity, since no confusion will arise, we will refer to [*stability in $\Sigma$*]{} simply as [*stability*]{}. For stable MOTS, there is no scalar quantity which is non-decreasing for arbitrary variations, like the area for stable minimal surfaces. However, in the minimal surface case, the formula $$<\phi,\psi>_{L^2}\auto=<L_{\vec{m}}^{min}\phi,\psi>_{L^2}=<\phi,L_{\vec{m}}^{min}\psi>_{L^2},$$ where $\phi$ is a principal eigenfunction of $L_{\vec{m}}^{min}$, implies that if there exists a positive variation $\psi\vec{m}$ for which $\delta_{\psi\vec{m}}p\geq 0$, then $\auto\geq 0$ and the minimal surface is stable. A similar result can be proven for MOTS [@AMS]: \[prop:stablevariation\] Let $S\subset \Sigma$ be a MOTS. Then $S$ is stable if and only if there exists a function $\psi\geq 0$, $\psi\not\equiv 0$ on $S$ such that $\delta_{\psi\vec{m}}\theta^{+}\geq 0$. Furthermore, $S$ is strictly stable if and only if, in addition, $\delta_{\psi\vec{m}}\theta^{+}\not\equiv 0$. [**Remark.**]{} For the case of [*past*]{} MOTS simply change $\vec{n}\rightarrow -\vec{n}$, $\vec{l}_{+}\rightarrow -\vec{l}_{-}$, $\vec{l}_{-}\rightarrow -\vec{l}_{+}$, $s_{A}\rightarrow -s_{A}$ and $\theta^{+}\rightarrow -\theta^{-}$ in equations (\[stabilityoperator\]), (\[Y\]), (\[raychaudhuri\]), (\[W\]) and, also, in Proposition \[prop:stablevariation\]. $\hfill \square$\ Thus, Proposition \[prop:stablevariation\] tells us that a (resp. past) MOTS $S$ is strictly stable if and only if there exists an outer variation with strictly increasing (resp. decreasing) $\theta^{+}$ (resp. $\theta^{-}$). This suggests that the presence of surfaces with negative $\theta^{+}$ (resp. positive $\theta^{-}$) outside $S$ may be related with the stability property of $S$. This can be made precise by introducing the following notion. \[defi:locallyoutermostMOTS\] A (resp. past) MOTS $S\subset\Sigma$ is [**locally outermost**]{} if there exists a two-sided neighbourhood of $S$ on $\Sigma$ whose exterior part does not contain any (resp. past) weakly outer trapped surface. The following proposition gives the relation between these concepts [@AMS2]. 1. A [*strictly stable*]{} MOTS (or past MOTS) is necessarily [*locally outermost*]{}. 2. A [*locally outermost*]{} MOTS (or past MOTS) is necessarily [*stable*]{}. 3. None of the converses is true in general. ### The trapped region {#ssc:GeometryOfSurfacesTrappedRegion} In this section we will extend the notion of locally outermost to a [*global*]{} concept and state a theorem by Andersson and Metzger [@AM] on the existence, uniqueness and regularity of the outermost MOTS on a spacelike hypersurface $\Sigma$. We will also see that an analogous result holds for the outermost generalized apparent horizon (Eichmair, [@Eichmair]). Both results will play a fundamental role throughout this thesis. The result by Andersson and Metzger is local in the sense that it works for any [*compact*]{} spacelike hypersurface $\Sigma$ with boundary $\bd \Sigma$ as long as the boundary $\bd \Sigma$ splits in two disjoint non-empty components $\bd \Sigma= \bd^{-}\Sigma \cup \bd^{+}\Sigma$. Neither of these components is assumed to be connected a priori. Andersson and Metzger deal with surfaces which are [*bounding with respect to*]{} the boundary $\bd^{+}\Sigma$ which plays the role of outer untrapped [*barrier*]{}. Both concepts are defined as follows. \[defi:barrier\] Consider a spacelike hypersurface $\Sigma$ possibly with boundary. A closed surface $\Sb\subset\Sigma$ is a [**barrier with interior $\Omegab$**]{} if there exists a manifold with boundary $\Omegab$ which is topologically closed and such that $\bd \Omegab=\Sb\bigcup\underset{a}{\cup}(\bd \Sigma)_{a}$, where $\underset{a}{\cup}(\bd \Sigma)_{a}$ is a union (possibly empty) of connected components of $\bd \Sigma$. [**Remark.**]{} For simplicity, when no confusion arises, we will often refer to a barrier $\Sb$ with interior $\Omegab$ simply as a [*barrier*]{} $\Sb$. $\hfill \square$\ The concept of a barrier will give us a criterion to define the exterior and the interior of a special type of surfaces called [*bounding*]{}. More precisely, \[defi:boundingAM\] Consider a spacelike hypersurface $\Sigma$ possibly with boundary with a barrier $\Sb$ with interior $\Omegab$. A surface $S\subset \Omegab\setminus \Sb$ is [**bounding with respect to the barrier**]{} $\Sb$ if there exists a compact manifold $\Omega\subset{\Omegab}$ with boundary such that $\bd\Omega= S\cup\Sb$. The set $\Omega\setminus S$ will be called the [**exterior**]{} of $S$ in $\Omegab$ and $(\Omegab\setminus\Omega)\cup S$ the [**interior**]{} of $S$ in $\Omegab$. [**Remark.**]{} Note that a surface $S$ which is bounding with respect to a barrier $S_{b}$ is always disjoint to $S_b$ and that its exterior is always not empty. Again, for simplicity and when no confusion arises, we will often refer to a surface which is bounding with respect a barrier simply as a [*bounding surface*]{}. Notice that, in the topology of $\Omegab$, the exterior of a bounding surface $S$ in $\Omegab$ is topologically open (because for every point $\p\in \bd\Omegab$ there exists an open set $U\subset \Omegab$ such that $\p\in U$), while its interior is topologically closed. For graphic examples of surfaces which are bounding with respect to a barrier see figures \[fig:boundingAM0\] and \[fig:boundingAM\]. $\hfill \square$\ The concept of bounding surface allows for a meaningful definition of [*outer null direction*]{}. For that, define the vector $\vec{m}$ as the unit vector normal to $S$ in $\Sigma$ which points into the exterior of $S$ in $\Omegab$. For $\Sb$, $\vec{m}$ will be taken to point outside of $\Omegab$. Then, we will select the outer and the inner null vectors, $\vec{l}_{+}$ and $\vec{l}_{-}$ as those null vectors orthogonal to $S$ or $\Sb$ which satisfy equations (\[l+mn\]) and (\[l-mn\]), respectively. ![In this graphic example, the surface $S_{b}$ (in red) is a barrier with interior $\Omegab$ (in grey). The surface $S_{1}$ is bounding with respect to $\Sb$ with $\Omega_{1}$ (the stripped area) being its exterior in $\Omega_{b}$. The surface $S_{2}$ fails to be bounding with respect to $S_{b}$ because its “exterior" would contain $\bd \Sigma$. []{data-label="fig:boundingAM0"}](boundingAM0.eps){width="6cm"} ![A manifold $\Sigma$ with boundary $\bd \Sigma= \bd^{-} \Sigma\cup \bd^{+}\Sigma$. The boundary $\bd^{+}\Sigma$ is a barrier whose interior coincides with $\Sigma$. The surface $S_{1}$ is bounding with respect to $\bd^{+}\Sigma$, while $S_{2}$ and $S_{3}$ fail to be bounding. The figure also shows the outer normal $\vec{m}$ as defined in the text. []{data-label="fig:boundingAM"}](boundingAM.eps){width="10cm"} \[defi:encloses\] Given two surfaces $S_1$ and $S_{2}$ which are bounding with respect to a barrier $\Sb$, we will say that $S_1$ encloses $S_2$ if the exterior of $S_2$ contains the exterior of $S_1$. \[defi:outermostsigmatilde\] A (past) MOTS $S\subset \Sigma$ which is bounding with respect to a barrier $\Sb$ is [**outermost**]{} if there is no other (past) weakly outer trapped surface in $\Sigma$ which is bounding with respect to $\Sb$ and enclosing $S$. Since bounding surfaces split $\Omegab$ into an exterior and an interior region, it is natural to consider the points inside a bounding weakly outer trapped surface $S$ as “trapped points". The region containing trapped points is called [*weakly outer trapped region*]{} and will be essential for the formulation of the result by Andersson and Metzger. More precisely, \[defi:trappedregion\] Consider a spacelike hypersurface containing a barrier $\Sb$ with interior $\Omegab$. The [**weakly outer trapped region**]{} $T^{+}$ of $\Omegab$ is the union of the interiors of all bounding weakly outer trapped surfaces in $\Omegab$. Analogously, \[defi:pasttrappedregion\] The [**past weakly outer trapped region**]{} $T^{-}$ of $\Omegab$ is the union of the interiors of all bounding past weakly outer trapped surfaces in $\Omegab$. The fundamental result by Andersson and Metzger, which will be an important tool in this thesis, reads as follows. \[thr:AM\] Consider a compact spacelike hypersurface $\Sigmatilde$ with boundary $\bd \Sigmatilde$. Assume that the boundary can be split in two non-empty disjoint components $\bd \Sigmatilde= \bd^{-}\Sigmatilde \cup \bd^{+}\Sigmatilde$ (neither of which are necessarily connected) and take $\bd^{+}\Sigmatilde$ as a barrier with interior $\Sigmatilde$. Suppose that $\theta^{+}[\bd^{-}\Sigmatilde]\leq 0$ and $\theta^{+}[\bd^{+}\Sigmatilde]>0$ (with respect to the outer normals defined above). Then the topological boundary $\tbd T^{+}$ of the weakly outer trapped region of $\Sigmatilde$ is a smooth MOTS which is bounding with respect to $\bd^{+}\Sigmatilde$ and stable. [**Remark.**]{} Since no bounding MOTS can penetrate into the exterior of $\tbd T^{+}$, by definition, this theorem shows the existence, uniqueness and smoothness of the outermost bounding MOTS in a compact hypersurface. Note also that another consequence of this result is the fact that the set $T^{+}$ is topologically closed (because it is the interior of the bounding surface $\tbd T^{+}$). $\hfill \square$\ The proof of this theorem uses the Gauss-Bonnet Theorem in several places and, therefore, this result is valid only in (3+1) dimensions. If we reverse the time orientation of the spacetime, an analogous result for the topological boundary of the past weakly outer trapped region $T^{-}$ follows. Indeed, if the hypotheses on the sign of the outer null expansion of the components of $\bd \Sigmatilde$ are replaced by $\theta^{-}[\bd^{-}\Sigmatilde]\geq 0$ and $\theta^{-}[\bd^{+}\Sigmatilde]<0$ then the conclusion is that $\tbd T^{-}$ is a smooth past MOTS which is bounding with respect to $\bd^{+}\Sigmatilde$ and stable.\ As we mentioned before, a similar result for the existence of the outermost generalized apparent horizon also exists. It has been recently obtained by Eichmair [@Eichmair]. \[thr:Eichmair\] Let $(\Sigmatilde, g, K)$ be a compact n-dimensional spacelike hypersurface in an (n+1)-dimensional spacetime, with $3\leq n\leq 7$ and boundary $\bd \Sigmatilde$. Assume that the boundary can be split in two non-empty disjoint components $\bd \Sigmatilde= \bd^{-}\Sigmatilde \cup \bd^{+}\Sigmatilde$ (neither of which are necessarily connected) and take $\bd^{+}\Sigmatilde$ as a barrier with interior $\Sigmatilde$. Suppose that the inner boundary $\bd^{-}\Sigmatilde$ is a generalized trapped surface, and the outer boundary satisfies $p>|q|$ with respect to the outer normals defined above. Then there exists a unique $C^{2,\alpha}$ (i.e. belonging to the Hölder space $C^{2,\alpha}$, with $0<\alpha\leq 1$, see Appendix \[ch:appendix2\]) generalized apparent horizon $S$ which is bounding with respect to $\bd^{+}\Sigmatilde$ and outermost (i.e. there is no other bounding generalized trapped surface in $\Sigmatilde$ enclosing $S$). Moreover, $S$ has smaller area than any other surface enclosing it. The proof of this result does not use the Gauss-Bonnet theorem or any other specific property of $3$-dimensional spaces, so it not restricted to (3+1) dimensions. However, it is based on regularity of minimal surfaces, which implies that the dimension of $\Sigmatilde$ must be at most seven (in higher dimensions minimal hypersurfaces need not be regular everywhere, see e.g. [@Giusti]). The area minimizing property of the outermost bounding generalized apparent horizon makes this type of surfaces potentially interesting for the Penrose inequality, as we will discuss in the next section. The Penrose inequality {#sc:PenroseInequality} ---------------------- The Penrose inequality involves the concept of the total ADM mass of a spacetime, so we start with a brief discussion about mass in General Relativity. The notion of [*energy*]{} in General Relativity is not as clear as in other physical theories. The energy-momentum tensor $T_{\mu\nu}$ represents the matter contents of a spacetime and therefore should contribute to the total energy of a spacetime. However, the [*gravitational field*]{}, represented by the metric tensor $\gM$, must also contribute to the total energy of the spacetime. In agreement with the Newtonian limit, a suitable [*gravitational energy density*]{} should be an expression quadratic in the first derivatives of the metric $\gM$. However, since at any point we can make the metric to be Minkowskian and the Christoffel symbols to vanish, there is no non-trivial scalar object constructed from the metric and its first derivatives alone. Therefore, a natural notion of energy density in General Relativity does not exist. The same problem is also found in other geometric theories of gravity. Nevertheless, there does exist a useful notion of the [*total energy*]{} in the so-called asymptotically flat spacetimes. The term [*asymptotic flatness*]{} was introduced in General Relativity to express the idea of a spacetime corresponding to an isolated system. It involves restrictions on the spacetime “far away" form the sources. There are several notions of asymptotic flatness according to the type of infinity considered (see e.g. Chapter 11.1 of [@Wald]), namely limits along null directions (null infinity) or limits along spacelike directions (spacelike infinity). The idea is to define the mass as integrals in the asymptotic region where the gravitational field is sufficiently weak so that integrals become meaningful (i.e. independent of the coordinate system). According to the type of infinity considered there are two different concepts: the [*Bondi energy-momentum*]{} where the integral is taken at null infinity and the [*ADM energy-momentum*]{} where the integral is taken at spatial infinity. Both are vectors in a suitable four dimensional vector space and transform as a Lorentz vector under suitable transformations. Moreover, the Lorentz length of this vector is either a conserved quantity upon evolution (ADM) or monotonically decreasing in advanced time (Bondi). An interesting and more precise discussion about the definitions of both Bondi and ADM energy-momentum tensors can be found in Chapter 11.2 of [@Wald]. Because of its relation with the Penrose inequality we are specially interested in the ADM energy-momentum. To make these concepts precise we need to define first [*asymptotic flatness*]{} for spacelike hypersurfaces. \[defi:asymptoticallyflatend\] An [**asymptotically flat end**]{} of a spacelike hypersurface $\id$ is a subset $\Sigma_{0}^{\infty}\subset\Sigma$ which is diffeomorphic to $\mathbb{R}^3 \setminus \overline{B_{R}}$, where $B_{R}$ is an open ball of radius $R$. Moreover, in the Cartesian coordinates $\{ x^i \}$ induced by the diffeomorphism, the following decay holds $$\label{asymptoticallyflatend} g_{ij}-\delta_{ij}=O^{(2)}(1/r),\quad\quad K_{ij}=O^{(2)}(1/r^{2}), $$ where $r=|x|=\sqrt{x^{i}x^{j}\delta_{ij}}$. Here, a function $f(x^i)$ is said to be $O^{(k)}(r^n), k\in\mathbb{N}\cup \{0\}$ if $f(x^i)=O(r^n)$, $\partial_{j}f(x^{i})=O(r^{n-1})$ and so on for all derivatives up to and including the $k$-th ones. \[defi:asymptoticallyflat\] A spacelike hypersurface $\id$, possibly with boundary, is [**asymptotically flat**]{} if $\Sigma=\mathcal{K}\cup \Sigma^{\infty}$, where $\mathcal{K}$ is a compact set and $\Sigma^{\infty}=\underset{a}{\cup}\Sigma_{a}^{\infty}$ is a finite union of asymptotically flat ends $\Sigma_{a}^{\infty}$. Consider a spacelike hypersurface $\id$ with a selected asymptotically flat end $\Sigma_{0}^{\infty}$. Then, the ADM energy-momentum ${\bf P}_{\scriptscriptstyle ADM}$ associated with $\Sigma_{0}^{\infty}$ is defined as the spacetime vector with components$$\begin{aligned} {P_{\scriptscriptstyle ADM}}_0=E_{\scriptscriptstyle ADM}\equiv \underset{r\rightarrow \infty}{lim} \frac{1}{16\pi} \overset{3}{\underset{j=1}{\sum}} \int_{S_{r}}\left( \partial_{j}g_{ij}-\partial_{i}g_{jj} \right)dS^{i}, \label{ADMenergy}\\ {P_{\scriptscriptstyle ADM}}_{i}= {p_{\scriptscriptstyle ADM}}_{i}\equiv \underset{r\rightarrow \infty}{lim} \frac{1}{8\pi}\int_{S_{r}}\left( K_{ij}-g_{ij}\tr{K} \right)dS^{j},\label{ADMmomentum}\end{aligned}$$ where $\{x^i\}$ are the Cartesian coordinates induced by the diffeomorphism which defines the asymptotically flat end, $S_r$ is the surface at constant $r$ and $dS^{i}=m^{i}dS$ with $\vec{m}$ being the outward unit normal and $dS$ the area element. The quantity $E_{\scriptscriptstyle ADM}$ is called the ADM energy while ${\bf p}_{\scriptscriptstyle ADM}$ the ADM spatial momentum. \[defi:ADMmass\] The ADM mass is defined as $$M_{\scriptscriptstyle ADM}=\sqrt{E_{\scriptscriptstyle ADM}^{2}-\delta^{ij}{P_{\scriptscriptstyle ADM}}_{i}{P_{\scriptscriptstyle ADM}}_{j}}.$$ A priori, these definitions depend on the choice of the coordinates $\{x^{i}\}$. However, the decay in $g$ and $K$ at infinity implies that ${\bf P}_{\scriptscriptstyle ADM}$ is indeed a geometric quantity provided $\GM_{\mu\nu}n^{\mu}$ decays as $1/r^{4}$ at infinity [@citeADM]. The notion of ADM mass is in fact independent of the coordinates as long as the decay (\[asymptoticallyflatend\]) is replaced by $$g_{ij}-\delta_{ij}=O^{(2)}(1/r^{\alpha}),\qquad\quad K_{ij}=O^{(1)}(1/r^{1+\alpha}),$$ with $\alpha>\frac{1}{2}$ [@Bartnik1]. A fundamental property of the ADM energy-momentum is its causal character. The Positive Mass Theorem (PMT) of Schoen and Yau [@SY] (also proven by Witten [@Witten] using spinors) establishes that the ADM energy is non-negative and the ADM mass is real (c.f. Section 8.2 of [@Strauman] for further details). More precisely, \[thr:PMT0\] Consider an asymptotically flat spacelike hypersurface $\id$ without boundary satisfying the DEC. Then the total ADM energy-momentum $\vec{P}_{\scriptscriptstyle ADM}$ is a future directed causal vector. Furthermore, $\vec{P}_{\scriptscriptstyle ADM}=0$ if and only if $(\Sigma,g,K)$ is a slice of the Minkowski spacetime. The global conditions required for the PMT were relaxed in [@CBartnik] where $\Sigma$ was allowed to be complete and contain an asymptotically flat end instead of being necessarily asymptotically flat (see Theorem \[thr:PMT1\] below). The PMT has also been extended to other situations of interest. Firstly, it holds for spacelike hypersurfaces admitting corners on a surface, provided the mean curvatures of the surface from one side and the other satisfy the right inequality [@Miao2]. It has also been proved for spacelike hypersurfaces [*with boundary*]{} provided this boundary is composed by either future or past weakly outer trapped surfaces [@GHHP]. Since future weakly outer trapped surfaces are intimately related with the existence of black holes (as we have already pointed out above), this type of PMT is usually referred to as [*PMT for black holes*]{}. Having introduced these notions we can now describe the Penrose inequality. During the seventies, Penrose [@Penrose1973] conjectured that the total ADM mass of a spacetime containing a black hole that settles down to a stationary state must satisfy the inequality $$\label{penrose0} M_{\scriptscriptstyle ADM}\geq \sqrt{\frac{|{\mathscr{H}}|}{16\pi}},$$ where $|{\mathscr{H}}|$ is the area of the event horizon at one instant of time. Moreover, equality happens if and only if the spacetime is the Schwarzschild spacetime. The plausibility argument by Penrose goes as follows [@Penrose1973]. Assume a spacetime $(M,\gM)$ which is globally well-behaved in the sense of being strongly asymptotically predictable and admitting a complete future null infinity $\mathscr{I}^{+}$ (see [@Wald] for definitions). Suppose that $M$ contains a non-empty black hole region. The black hole event horizon ${\mathcal{H_{B}}}$ is a null hypersurface at least Lipschitz continuous. Next, consider a spacelike Cauchy hypersurface $\Sigma\subset M$ (see e.g. Chapter 8 of [@Wald] for the definition of a Cauchy hypersurface) with ADM mass $M_{\scriptscriptstyle ADM}$. Clearly ${\mathcal{H_{B}}}$ and $\Sigma$ intersect in a two-dimensional Lipschitz manifold. This represents the event horizon at one instant of time. Let us denote by $\mathscr{H}$ this intersection and by $|\mathscr{H}|$ its area (the manifold is almost everywhere $C^{1}$ so the area makes sense). Consider now any other cut $\mathscr{H}_1$ lying in the causal future of $\mathscr{H}$. The black hole area theorem [@H1], [@H2], [@CDGH] states that $|\mathscr{H}_1|\geq |\mathscr{H}|$ provided the NEC holds. Physically, it is reasonable to expect that the spacetime settles down to some vacuum equilibrium configuration (if an electromagnetic field is present, the conclusions would be essentially the same). Then, the uniqueness theorems for stationary black holes (which hold under suitable assumptions [@CL-C], [@L-C]) imply that the spacetime must approach the Kerr spacetime. In the Kerr spacetime the area of any cut of the event horizon $\mathscr{H}_{Kerr}$ takes the value $|\mathscr{H}_{Kerr}|=8\pi M_{Kerr}\left( M_{Kerr}+\sqrt{{M_{Kerr}}^{2}-L_{Kerr}^{2}/{M_{Kerr}}^{2}} \right)$ where $M_{Kerr}$ and $L_{Kerr}$ are respectively the total mass and the total angular momentum of the Kerr spacetime (the angular momentum can be defined also as a suitable integral at infinity). This means that $M_{Kerr}$ is the asymptotic value of the Bondi mass along the future null infinite $\mathscr{I}^{+}$. Assuming that the Bondi mass tends to the $M_{\scriptscriptstyle ADM}$ of the initial slice, inequality (\[penrose0\]) follows because the Bondi mass cannot increase along the evolution. Moreover, equality holds if and only if $\Sigma$ is a slice of the Kruskal extension of the Schwarzschild spacetime. It is important to remark than inequality (\[penrose0\]) is global in the sense that, in order to locate the cut $\mathscr{H}$, it is necessary to know the global structure of the spacetime. Penrose proposed to estimate the area $|\mathscr{H}|$ from below in terms of the area of certain surfaces which can be defined independently of the future evolution of the spacetime. The validity of these estimates relies on the validity of the cosmic censorship. These types of inequalities are collectively called [*Penrose inequalities*]{} and they are interesting for several reasons. First of all, they would provide a strengthening of the PMT. Moreover, they would also give indirect support to the validity of cosmic censorship, which is a basic ingredient in their derivation. There are several versions of the Penrose inequality. Typically one considers closed surfaces $S$ embedded in a spacelike hypersurface with a selected asymptotically flat end $\Sigma_{0}^{\infty}$ which are [*bounding*]{} with respect to a suitable large sphere in $\Sigma_{0}^{\infty}$. This leads to the following definition: \[defi:bounding\] Consider a spacelike hypersurface $\id$ possibly with boundary with a selected asymptotically flat end $\Sigma_{0}^{\infty}$. Take a sphere $\Sb\subset\Sigma_{0}^{\infty}$ with $r=r_{0}=const$ large enough so that the spheres with $r\geq r_{0}$ are outer untrapped with respect to the direction pointing into the asymptotic region in $\Sigma_{0}^{\infty}$. Let $\Omegab=\Sigma\setminus\{r>r_{0}\}$, which is obviously topologically closed and satisfies $\Sb\subset\bd\Omegab$. Then $\Sb$ is a barrier with interior $\Omegab$. A surface $S\subset \Sigma$ will be called [**bounding**]{} if it is bounding with respect to $\Sb$. [**Remark 1.**]{} It is well-known that on an asymptotically flat end $\Sigma_{0}^{\infty}$, the surfaces at constant $r$ are, for large enough $r$, outer untrapped. Essentially, this definition establishes a specific form of selecting the barrier in hypersurfaces containing a selected asymptotically flat end. $\hfill \square$\ [**Remark 2.**]{} Obviously, the definitions of exterior and interior of a bounding surface (Definition \[defi:boundingAM\]), enclosing (Definition \[defi:encloses\]), outermost (Definition \[defi:outermostsigmatilde\]) and $T^{\pm}$ (Definitions \[defi:trappedregion\] and \[defi:pasttrappedregion\]), given in the previous section, are applicable in the asymptotically flat setting. Moreover, since $r_{0}$ can be taken as large as desired, the specific choice of $S_b$ and $\Omegab$ is not relevant for the definition of bounding (once the asymptotically flat end has been selected). Because of that, when considering asymptotically flat ends, we will refer to the exterior of $S$ in $\Omegab$ as the [*exterior of $S$ in $\Sigma$*]{}. $\hfill \square$\ ![The hypersurface $\Sigma$ possesses an asymptotically flat end $\Sigma_{0}^{\infty}$ but also other types of ends and boundaries. The surface $S_b$, which represents a large sphere in $\Sigma_{0}^{\infty}$ and is outer untrapped, is a barrier with interior $\Omega_b$ (in grey). The surface $S_1$ is bounding with respect to $S_{b}$ (c.f. Definition \[defi:boundingAM\]) and therefore is bounding. The surface $S_2$ fails to be bounding (c.f. Figure \[fig:boundingAM0\]). []{data-label="fig:bounding1"}](bounding1.eps){width="7cm"} The standard version of the Penrose inequality reads $$\label{penrose1} M_{\scriptscriptstyle ADM}\geq \sqrt{\frac{A_{{min}}(\tbd T^{+})}{16\pi}},$$ where $A_{min}(\tbd T^{+})$ is the minimal area necessary to enclose $\tbd T^{+}$. This inequality (\[penrose1\]) is a consequence of the heuristic argument outlined before because (under cosmic censorship) $\mathscr{H}$ encloses $\tbd T^{+}$ The minimal area enclosure of $\tbd T^+$ needs to be taken because $\mathscr{H}$ could still have less area than $\tbd T^{+}$ [@Horowitz]. By reversing the time orientation, the same argument yields (\[penrose1\]) with $\tbd T^{+}$ replaced by $\tbd T^{-}$. In general, neither $\tbd T^{+}$ encloses $\tbd T^{-}$ nor vice versa. In the case that $K_{ij}=0$, these inequalities simplify because $T^{+} = T^{-}$ and $\tbd T^{+}$ is the outermost minimal surface (i.e. a minimal surface enclosing any other bounding minimal surface in $\Sigma$) and, hence, its own minimal area enclosure. The inequality in this case is called [*Riemannian Penrose inequality*]{} and it has been proven for connected $\tbd T^+$ in [@HI] and in the general case in [@Bray] using a different method. In the non-time-symmetric case, (\[penrose1\]) is not invariant under time reversals. Moreover, the minimal area enclosure of a given surface $S$ can be a rather complicated object typically consisting of portions of $S$ together with portions of minimal surfaces outside of $S$. This complicates the problem substantially. This has led several authors to propose simpler looking versions of the inequality, even if they are not directly supported by cosmic censorship. Two of such extensions are $$\begin{aligned} M_{\scriptscriptstyle ADM} \geq \sqrt{\frac{A_{{min}}( \tbd (T^{+} \cup T^{-} ))}{16\pi}}, \quad M_{\scriptscriptstyle ADM} \geq \sqrt{\frac{|\tbd (T^{+} \cup T^{-} )|}{16\pi}}, \label{penrose2}\end{aligned}$$ (see e.g. [@Karkowski-Malec2005]). These inequalities are immediately stronger than (\[penrose1\]) and have the advantage of being invariant under time reversals. The second inequality avoids even the use of minimal area enclosures. Neither version is supported by cosmic censorship and at present there is little evidence for their validity. However, both reduce to the standard version in the Riemannian case and both hold in spherical symmetry. No counterexamples are known either. It would be interesting to have either stronger support for them, or else to find a counterexample. Recently, Bray and Khuri proposed [@BK] a new method to approach the general (i.e. non time-symmetric) Penrose inequality. The basic idea was to modify the Jang equation [@Jang], [@SY] so that the product manifold $\Sigma \times \mathbb{R}$ used to construct the graphs which define the Jang equation is endowed with a warped type metric of the form $-\varphi^2 dt^2 + g$ instead of the product metric. Their aim was to reduce the general Penrose inequality to the Riemannian Penrose inequality on the graph manifold. A discussion on the type of divergences that could possibly occur for the generalized Jang equation led the authors to consider a new type of trapped surfaces which they called [**generalized trapped surfaces**]{} and [**generalized apparent horizons**]{} (defined in Section \[ssc:GeometryOfSurfacesDefinitions\]). This type of surfaces have very interesting properties. The most notable one is given by Theorem \[thr:Eichmair\] [@Eichmair] which guarantees the existence, uniqueness and $C^{2,\alpha}$-regularity of the outermost generalized apparent horizon $S_{out}$. The Penrose inequality proposed by these authors reads $$\begin{aligned} \label{penroseBK} M_{\scriptscriptstyle ADM}\geq\sqrt{\frac{|S_{out}|}{16\pi}},\end{aligned}$$ with equality only if the spacetime is Schwarzschild. This inequality has several remarkable properties that makes it very appealing [@BK]. First of all, the definition of generalized apparent horizon, and hence the corresponding Penrose inequality, is insensitive to time reversals. Moreover, there is no need of taking the minimal area enclosure of $S_{out}$, as this surface has less area than any of its enclosures (c.f. Theorem \[thr:Eichmair\]). Since MOTS are automatically generalized trapped surfaces, $S_{out}$ encloses the outermost MOTS $\tbd T^{+}$. Thus, (\[penroseBK\]) is stronger than (\[penrose1\]) and its proof would also establish the standard version of the Penrose inequality. Moreover, Khuri has proven [@Khuri3] that no generalized trapped surfaces exist in Minkowski, which is a necessary condition for the validity of (\[penroseBK\]). Another interesting property of this version, and one of its motivations discussed in [@BK], is that the equality case in (\[penroseBK\]) covers a larger number of slices of Kruskal than the equality case in (\[penrose1\]). Recall that the rigidity statement of any version of the Penrose inequality asserts that equality implies that $(\Sigma,g,K)$ is a hypersurface of Kruskal. However, [*which*]{} slices of Kruskal satisfy the equality case may depend on the version under consideration. The more slices having this property, the more accurate the version can be considered. For any slice $\Sigma$ of Kruskal we can define $\Sigma^{+}$ as the intersection of $\Sigma$ with the domain of outer communications. Bray and Khuri noticed that whenever $\tbd \Sigma^{+}$ intersects both the black hole and the white hole event horizons, then the standard version (\[penrose1\]) gives, in fact, a strict inequality. Although (\[penroseBK\]) does not give equality for all slices of Kruskal, it does so in all cases where the boundary of $\Sigma^{+}$ is a $C^{2,\alpha}$ surface (provided this boundary is the outermost generalized apparent horizon). It follows that version (\[penroseBK\]) contains more cases of equality than (\[penrose1\]) and is therefore more accurate. It should be stressed that the second inequality in (\[penrose2\]) gives equality for [*all*]{} slices of Kruskal, so in this sense it would be optimal. Despite its appealing properties, (\[penroseBK\]) is [*not*]{} directly supported by cosmic censorship. The reason is that the outermost generalized apparent horizon need not always lie inside the event horizon. A simple example [@Mars2] is given by a slice $\Sigma$ of Kruskal such that $\tbd T^{+}$ (which corresponds to the intersection of $\Sigma$ with the black hole event horizon) and $\tbd T^{-}$ (the intersection $\Sigma$ with the white hole horizon) meet transversally. Since both surfaces are generalized trapped surfaces, Theorem \[thr:Eichmair\] implies that there must exist a unique $C^{2,\alpha}$ outermost generalized apparent horizon enclosing both. This surface must therefore penetrate into the exterior region $\Sigma^+$ somewhere, as claimed. We will return to the issue of the Penrose inequality in Chapter \[ch:Article3\], where we will find a counterexample of (\[penroseBK\]) precisely by studying the outermost generalized apparent horizon in this type of slices in the Kruskal spacetime. For further information about the present status of the Penrose inequality, see [@Mars2]. Uniqueness of Black Holes {#sc:UniquenessOfBlackHoles} ------------------------- According to cosmic censorship, any gravitational collapse that settles down to a stationary state should approach a stationary black hole. The [*black hole uniqueness theorems*]{} aim to classify all the stationary black hole solutions of Einstein equations. In this section we will first summarize briefly the status of stationary black hole uniqueness theorems. We will also describe in some detail a powerful method (the so-called [*doubling method*]{} of Bunting and Masood-ul-Alam) to prove uniqueness for [*static*]{} black holes which will be essential in Chapter \[ch:Article4\]. In the late sixties and early seventies the properties of equilibrium states of black holes were extensively studied by many theoretical physicists interested in the gravitational collapse process. The first uniqueness theorem for black holes was found by W. Israel in 1967 [@Israel], who found the very surprising result that a [static]{}, topologically spherical vacuum black hole is described by the Schwarzschild solution. In the following years, several works ([@MzH], [@Robinson], [@BMuA]) established that the Schwarzschild solution indeed exhausts the class of static vacuum black holes with non-degenerate horizons. The method of the proofs in [@Israel], [@MzH], [@Robinson] consisted in constructing two integral identities which were used to investigate the geometric properties of the level surfaces of the norm of the static Killing. This method proved uniqueness under the assumption of connectedness and non-degeneracy of the event horizon. The hypothesis on the connectedness of the horizon was dropped by Bunting and Masood-ul-Alam [@BMuA] who devised a new method based on finding a suitable conformal rescalling which allowed using the rigidity part of the PMT to conclude uniqueness. This method, known as the [*doubling method*]{} is, still nowadays, the most powerful method to prove uniqueness of black holes in the static case. Finally, the hypothesis on the non-degeneracy of the event horizon was dropped by Chruściel [@C] in 1999 who applied the doubling method across the non-degenerate components and applied the PMT for complete manifolds with one asymptotically flat end (Theorem \[thr:PMT1\] below) to conclude uniqueness (the Bunting and Masood-ul-Alam conformal rescalling transforms the degenerate components into cylindrical ends). The developments in the uniqueness of static electro-vacuum black holes go in parallel to the developments in the vacuum case. Some remarkable works which played an important role in the general proof of the uniqueness of static electro-vacuum black holes are [@Israel2], [@MzH2], [@Simon], [@Ruback], [@Simon2], [@MuA], [@C2], [@CT]. Uniqueness of static black holes using the doubling method has also been proved for other matter models, as for instance the Einstein-Maxwell-dilaton model [@MuA2], [@MSimon]. During the late sixties, uniqueness of [*stationary*]{} black holes also started to take shape. In fact, the works of Israel, Hawking, Carter and Robinson, between 1967 and 1975, gave an almost complete proof that the Kerr black hole was the only possible stationary vacuum black hole. The first step was given by Hawking (see [@HE]) who proved that the intersection of the event horizon with a Cauchy hypersurface has $\mathbb{S}^2$-topology. The next step, also due to Hawking [@HE] was the demonstration of the so-called Hawking Rigidity Theorem, which states that a stationary black hole must be static or axisymmetric. Finally, the work of Carter [@Carter] and Robinson [@Robinson2] succeeded in proving that the Kerr solutions are the only possible stationary axisymmetric black holes. Nevertheless, due to the fact that the Hawking Rigidity Theorem requires analyticity of all objects involved, uniqueness was proven only for analytic spacetimes. The recent work [@CL-C] by Chruściel and Lopes Costa has contributed substantially to reduce the hypotheses and to fill several gaps present in the previous arguments. Similarly, uniqueness of stationary electro-vacuum black holes has been proven for analytic spacetimes. Some remarkable works for the stationary electro-vacuum case are [@Carter2], [@Mazur] and, more recently, [@L-C], where weaker hypotheses are assumed for the proof. Uniqueness of stationary and axisymmetric black holes has also been proven for non-linear $\sigma$-models in [@BMG]. The Hawking Rigidity Theorem has not been generalized to non-linear $\sigma$-models and, hence, axisymmetry is required in this case. It is also worth to remark that, in the case of matter models modeled with Yang-Mills fields, uniqueness of stationary black holes is not true in general and counterexamples exist [@BartnikMcKinnon]. In this thesis we will be interested in uniqueness theorems for static [*quasi-local*]{} black holes and, particularly, in the doubling method of Bunting and Masood-ul-Alam. In the remainder of this chapter, we will describe this method in some detail by giving a sketch of the proof of the uniqueness theorem for static electro-vacuum black holes. ### Example: Uniqueness for electro-vacuum static black holes {#ssc:electro-vacuum} Let us start with some definitions. An electro-vacuum solution of the Einstein field equations is a triad $(M,\gM,{\bf F})$, where ${\bf F}$ is the source-free electromagnetic tensor, i.e. a 2-form satisfying the Maxwell equations which no sources, i.e. $$\begin{aligned} {\nablaM}^{\mu}F_{\mu\nu}=0, \\ \nablaM_{[\alpha}F_{\mu\nu]}=0,\end{aligned}$$ and $(M,\gM)$ is the spacetime satisfying the Einstein equations with energy-momentum tensor $$T_{\mu\nu}=\frac{1}{4\pi} \left(F_{\mu\alpha}{F_{\nu}}^{\alpha}-\frac14 F_{\alpha\beta}F^{\alpha\beta}\gM_{\mu\nu}\right).$$ We call a stationary electro-vacuum spacetime an electro-vacuum spacetime admitting a stationary Killing vector field $\vec{\xi}$, satisfying $\mathcal{L}_{\vec{\xi}} F_{\mu\nu}=0$. Let us define the electric and magnetic fields with respect to $\vec{\xi}$ as $$\begin{aligned} E_{\mu}&=&-F_{\mu\nu}\xi^{\nu},\\ B_{\mu}&=&(*F)_{\mu\nu}\xi^{\nu},\end{aligned}$$ respectively. Here, $*{\bf F}$ denotes the Hodge dual of ${\bf F}$ defined as $$(*F)_{\mu\nu}=\frac 12 \eta^{(4)}_{\mu\nu\alpha\beta}F^{\alpha\beta}.$$ From the Maxwell equations and $\mathcal{L}_{\vec{\xi}}F_{\mu\nu}=0$ it follows easily that $d {\bf E}=0$ and $d {\bf B}=0$ which implies that, at least locally, there exist two functions $\phi$ and $\psi$, called the [**electric**]{} and [**magnetic potentials**]{}, so that ${\bf E}=-d \phi$ and ${\bf B}=-d \psi$, respectively. These potentials are defined up to an additive constant and they satisfy $\vec{\xi}(\phi)=\vec{\xi}(\psi)=0$. A stationary electro-vacuum spacetime $(M,\gM,{\bf F})$ with Killing field $\vec{\xi}$ is said to be [**purely electric**]{} with respect to $\vec{\xi}$ if and only if ${\bf B}=0$. For simplicity, we will restrict ourselves to the purely electric case. In fact, the general case can be reduced to the purely electric case by a transformation called [*duality rotation*]{} [@Heusler2]. In the static case there exists an important simplification which allows to reduce the formulation of the uniqueness theorem for black holes in terms of conditions on a spacelike hypersurface instead of conditions on the spacetime. The fact is that, under suitable circumstances, the presence of an event horizon in a static spacetime implies the existence of an asymptotically flat hypersurface with compact topological boundary such that the static Killing field is causal everywhere and null precisely on the boundary. Then, the uniqueness theorem for static electro-vacuum black holes can be stated simply as follows. \[thr:electrovacuniqueness0\] Let $(M,\gM,F)$ be a static solution of the Einstein-Maxwell equations. Suppose that $M$ contains a simply connected asymptotically flat hypersurface $\Sigma$ with non-empty topological boundary such that ${\Sigma}$ is the union of an asymptotically flat end and a compact set, such that: - The topological boundary $\tbd \Sigma$ is a compact, 2-dimensional embedded topological submanifold. - The static Killing vector field is causal on $\Sigma$ and null only on $\tbd \Sigma$. Then, after performing a duality rotation of the electromagnetic field if necessary: - If $\tbd \Sigma$ is connected, then $\Sigma$ is diffeomorphic to $\mathbb{R}^{3}$ minus a ball. Moreover, there exists a neighbourhood of $\Sigma$ in $M$ which is isometrically diffeomorphic to an open subset of the Reissner-Nordström spacetime. - If $\tbd \Sigma$ is not connected, then $\Sigma$ is diffeomorphic to $\mathbb{R}^{3}$ minus a finite union of disjoint balls and there exists a neighborhood of $\Sigma$ in $M$ which is isometrically diffeomorphic to an open subset of the standard Majumdar-Papapetrou spacetime. [**Remark.**]{} The standard Majumdar-Papapetrou spacetime is the manifold $(\mathbb{R}^{3}\setminus \overset{n}{\underset{i=1}{\cup}}\p_{i}) \times \mathbb{R}$ endowed with the metric $ds^2=\frac{-dt^2}{u^2}+u^2(dx^2+dy^2+dz^2)$, where $u=1+\overset{n}{\underset{i=1}{\sum}}\frac{q_i}{r_{i}}$ with $q_i$ being a constant and $r_{i}$ the Euclidean distance to $\p_{i}$. $\hfill \square$\ In what follows we will give a sketch of the proof of the Theorem \[thr:electrovacuniqueness0\]. Firstly, we need some results concerning the boundary of the set $\{ \p\in M: \left.\lambda\right|_{\p}>0 \}$, where $\lambda \equiv -\xi_{\mu}\xi^{\mu}$, i.e. minus the squared norm of the stationary Killing field $\vec{\xi}$. Let us start with some definitions. Let $(M,\gM)$ be a spacetime with a Killing vector $\vec{\xi}$. A [**Killing prehorizon**]{} $\mathcal{H}_{\vec{\xi}}$ of $\vec{\xi}$ is a null, 3-dimensional submanifold (not necessarily embedded), at least $C^{1}$, such that $\vec{\xi}$ is tangent to $\mathcal{H}_{\vec{\xi}}$, null and different from zero. A [**Killing horizon**]{} is an embedded Killing prehorizon. Next, let us introduce a quantity $\kappa$ defined on a Killing prehorizon in any stationary spacetime. Clearly, on a Killing prehorizon $\mathcal{H}_{\vec{\xi}}$ we have $\lambda=0$. It implies that $\nablaM_{\mu}\lambda$ is normal to $\mathcal{H}_{\vec{\xi}}$. Now, since $\vec{\xi}$ is null and tangent to $\mathcal{H}_{\vec{\xi}}$, it is also normal to $\mathcal{H}_{\vec{\xi}}$. Since, moreover $ \vec{\xi}\,\big|_{\mathcal{H}_{\vec{\xi}}}$ is nowhere zero, it follows that there exists a function $\kappa$ such that $$\label{kappa0} \nablaM_{\mu}\lambda=2\kappa \xi_{\mu}.$$ $\kappa$ is called the [**surface gravity**]{} on $\mathcal{H}_{\vec{\xi}}$. The following result states the constancy of $\kappa$ on a Killing prehorizon in a static spacetime. \[lema:RW\] Let $\mathcal{H}_{\vec{\xi}}$ be a Killing prehorizon for an integrable Killing vector $\vec{\xi}$. Then $\kappa$ is constant on each arc-connected component of $\mathcal{H}_{\vec{\xi}}$. [**Remark.**]{} This lemma also holds in stationary spacetimes provided the DEC holds. Its proof can be found in Chapter 12 of [@Wald]. $\hfill \square$\ This lemma allows to classify Killing prehorizons in static spacetimes in two types with very different behavior. An arc-connected Killing prehorizon $\mathcal{H}_{\vec{\xi}}$ is called [**degenerate**]{} when $\kappa=0$ and [**non-degenerate**]{} when $\kappa\neq 0$. Since $\nablaM_{\mu}\lambda\neq 0$ on a non-degenerate Killing prehorizon, the set $\{\lambda=0\}$ defines an embedded submanifold (c.f. [@Cc]). Non-degenerate Killing prehorizons are Killing horizons. The next lemma guarantees the existence of a Killing prehorizon in a static spacetime. This lemma will be used several times along this thesis. For completeness, we find it appropriate to include its proof (we essentially follow [@C]). \[lema:VC\] Let $(M,\gM)$ be a static spacetime with Killing vector $\vec{\xi}$. Then the set $\mathcal{N}_{\vec{\xi}}\equiv \tbd \{\lambda>0\}\cap \{\vec{\xi}\neq 0\}$, if non-empty, is a smooth Killing prehorizon. [**Proof.**]{} Consider a point $\p\in \mathcal{N}_{\vec{\xi}}$. Due to the Fröbenius’s theorem (see e.g. [@Frobenius]), staticity implies that there exists a neighbourhood $\mathcal{V}_{0}\subset M$ of $\p$, with $\vec{\xi}\,\big|_{\mathcal{V}_{0}}\neq 0$, which (for $\mathcal{V}_{0}$ small enough) is foliated by a family of smooth embedded submanifolds $\Sigma_{t}$ of codimension one and orthogonal to $\vec{\xi}$. In particular, $\p\in \Sigma_{0}$, where $\Sigma_{0}$ denotes a leaf of this foliation. Now consider the leaves $\Sigma_\alpha$ of the $\Sigma_{t}$ foliation such that $\Sigma_{\alpha}\cap \{\lambda\neq0\}\neq \emptyset$. The staticity condition (\[integrablekilling\]) implies $$\xi_{[\nu}\nabla_{\mu]}\lambda=\lambda \nabla_{[\mu}\xi_{\nu]},$$ which on $\mathcal{V}_{0}\cap \{\lambda\neq0\}$ reads $$\label{eqVC} \xi_{[\nu}\nabla_{\mu]}(\ln{|\lambda|})=\nabla_{[\mu}\xi_{\nu]}.$$ Let $\vec{W}$ and $\vec{Z}$ be smooth vector fields on $\mathcal{V}_{0}$ such that $\vec{W}$ satisfies $\xi_{\mu}W^{\mu}=1$ and $\vec{Z}$ is tangent to the leaves $\Sigma_{t}$. At points of $\Sigma_{\alpha}$ on which $\lambda\neq0$, the contraction of equation (\[eqVC\]) with $Z^{\mu}W^{\nu}$ gives $$Z^{\mu}\nabla_{\mu}(\ln{|\lambda|})=2Z^{\mu}W^{\nu}\nabla_{[\mu}\xi_{\nu]}.$$ The right-hand side of this equation is uniformly bounded on $\Sigma_{\alpha}$, which implies that $\ln{|\lambda|}$ is uniformly bounded on $\Sigma_{\alpha} \cap \{\lambda\neq0\}$. This is only possible if $\Sigma_{\alpha} \cap \{\lambda=0\}= \emptyset$. Consequently, $\lambda$ is either positive, or negative, or zero in each leaf of the foliation $\Sigma_{t}$. In particular, it implies that $\{\lambda=0\}\cap\mathcal{V}_{0}$ is a union of leaves of the $\Sigma_{t}$ foliation. It only remains to prove that each arc-connected component of $\tbd \{\lambda>0\}\cap \mathcal{V}_{0}$ coincides with one of these leaves. For that, take coordinates $\{z,x^{A}\}$ in $\mathcal{V}_{0}$ in such a way that the coordinate $z$ characterizes the leaves of the foliation $\Sigma_t$ and $\p=(z=0,x^A=0)$ (this is possible because each leaf of $\Sigma_{t}$ is an embedded submanifold of $\mathcal{V}_{0}$). Note that the leaf $\Sigma_0\ni \p$ is then defined by $\{z=0\}$. In this setting, we just need to prove that $\{z=0\}$ coincides with an arc-connected component of $\tbd\{\lambda>0\}\cap\mathcal{V}_{0}$. Due to the fact that $\p\in \tbd \{\lambda>0\}\cap \mathcal{V}_{0}$, there exists a sequence of points $\p_{i}\in\mathcal{V}_{0}$ with $\lambda>0$ which converge to $\p$ and have coordinates $(z(\p_{i}),x^{A}(\p_{i}))$. Since the coordinate $z$ characterizes the leaves and $\lambda$ is either positive, or negative, or zero in each leaf, it follows that the sequence of points $\p'_{i}$ with coordinates $(z(\p_{i}),0)$ also has $\lambda>0$ and tends to $\p$. By the same reason, given any point $\q\in \{z=0\}$ with coordinates $(0,x_{0}^{A})$, the sequence of points $\q_{i}=(z(\p_{i}),x_{0}^{A})$ tends to $\q$ and lies in $\{\lambda>0\}$. Therefore, $\{z=0\}$ is composed precisely by the points of the arc-connected component of $\tbd\{\lambda>0\}\cap \mathcal{V}_{0}$ which contains $\p$. This implies that every arc-connected component of $\tbd \{\lambda>0\}\cap \mathcal{V}_{0}$ coincides with a leaf $\Sigma_{t}$ where $\lambda\equiv 0$ (and $\vec{\xi}\neq 0$). Finally, this local argument can be extended to the whole set $\mathcal{N}_{\vec{\xi}}$ simply by taking a covering of $\mathcal{N}_{\vec{\xi}}$ by suitable open neighbourhoods $\mathcal{V}_{\beta}\subset M$. $\hfill \blacksquare$\ [**Remark.**]{} Although each arc-connected component of $\tbd \{\lambda>0\}\cap \mathcal{V}_{\beta}$ is an embedded submanifold of $\mathcal{V}_{\beta}\subset M$, the whole set $\mathcal{N}_{\vec{\xi}}$ may fail to be embedded in $M$ (see Figure \[fig:spiral\]). Thus, a priori, degenerate Killing prehorizons may fail to be embedded. As mentioned before, this possibility has been overlooked in the literature until recently [@Cc]. The occurrence of non-embedded Killing prehorizons poses serious difficulties for the uniqueness proofs. One way to deal with these objects is to make hypotheses that simply exclude them. In Proposition \[prop:Chrusciel\] below, the hypothesis that $\tbd \Sigma$ is a compact and embedded topological manifold is made precisely for this purpose. Another possibility is to prove that these prehorizons do not exist. At present, this is only known under strong global hypotheses on the spacetime (c.f. Definition \[Iplusregularity\] below). It is an interesting open problem to either find an example of a non-embedded Killing prehorizon or else to prove that they do not exist. $\hfill \square$\ ![The figure illustrates a situation where $\mathcal{N}_{\vec{\xi}}=\tbd \{\lambda>0\}\cap \{\vec{\xi}\neq 0\}$ fails to be embedded. In this figure, the Killing vector is nowhere zero, causal everywhere and null precisely on the plotted line. Here, $\mathcal{N}_{\vec{\xi}}$ has three arc-connected components: two spherical and one with spiral form. The fact that the spiral component accumulates around the spheres implies that the whole set $\mathcal{N}_{\vec{\xi}}$ is not embedded. Moreover, the spiral arc-connected component, which is itself embedded, is not compact. []{data-label="fig:spiral"}](spiral.eps){width="9cm"} The hypotheses of Theorem \[thr:electrovacuniqueness0\] require the existence of a hypersurface $\Sigma$ with topological boundary such that $\lambda\geq 0$ everywhere and $\lambda=0$ precisely on $\tbd \Sigma$. It is clear then that $\tbd \Sigma \subset \tbd U$, where $U \equiv \{ \p\in M: \left.\lambda\right|_{\p}>0 \}$, but, in general, $\tbd \Sigma$ will not lie in a Killing prehorizon because it can still happen that $\vec{\xi}=0$ on a subset of $\tbd U$. However, the set of points where $\vec{\xi}= 0$ cannot be very “large" as the next result guarantees. \[thr:Boyer\] Consider a static spacetime $(M,\gM)$ with Killing vector $\vec{\xi}$. Let $\p\in\tbd\{\lambda>0\}$ be a fixed point (i.e. $\vec{\xi}\,\big|_{\p}=0$). Then $\p$ belongs to a connected, spacelike, smooth, totally geodesic, 2-dimensional surface $S_{0}$ which is composed by fixed points. Furthermore, $S_{0}$ lies in the closure of a non-degenerate Killing horizon $\mathcal{H}_{\vec{\xi}}$ Therefore, using Lemma \[lema:VC\] and Theorem \[thr:Boyer\], we can assert that [*$\tbd \{\lambda>0\}$ belongs to the closure of a Killing prehorizon*]{}. The manifold $\mbox{int}(\Sigma)$ admits, besides the induced metric, a second metric $h$ called [*orbit space metric*]{} which is a key object in the uniqueness proof. Let us first define the projector orthogonal to $\vec{\xi}$. On the open set $U\equiv \{ \lambda>0 \}\subset M$, the [**projector orthogonal**]{} to $\vec{\xi}$, denoted by $h_{\mu\nu}$, is defined as $$\label{h} h_{\mu\nu}\equiv \gM_{\mu\nu}+\frac{\xi_{\mu}\xi_{\nu}}{\lambda}.$$ This tensor has the following properties: - It is symmetric, i.e. $h_{\mu\nu}=h_{\nu\mu}$. - It has rank 3. - It satisfies $h_{\mu\nu}\xi^{\mu}=0$ On $U$ we can also define the function $V=+\sqrt{\lambda}$. The hypersurface $\mbox{int}(\Sigma)$ is fully contained in $U$. Let $\Phi: \mbox{int}(\Sigma)\rightarrow U\subset M$ denote the embedding of $\mbox{int}(\Sigma)$ in $U$, then the pull-back of the projector $\Phi^{*}(h)$ is a Riemannian metric on $\Sigma$. We will denote by the same symbols $h$, $V$ and $\phi$ both the objects in $U\subset M$ and their corresponding pull-backs in $\mbox{int}(\Sigma)$. The Einstein-Maxwell field equations for a purely electric stationary electro-vacuum spacetime are equivalent to the following equations on $\mbox{int}(\Sigma)$ see e.g. [@Heuslerlibro]. $$\begin{aligned} V \Delta_{h}\phi&=&D_{i}V{D}^{i}\phi,\label{EinsteinEV1}\\ V \Delta_{h} V&=&D_{i}\phi {D}^{i}\phi, \label{EinsteinEV2}\\ V R_{ij}(h)&=&D_{i}D_{j}V+\frac1V\left( D_{k}\phi {D}^{k}\phi h_{ij} - 2D_{i}\phi {D}_{j}\phi \right), \label{EinsteinEV3}\end{aligned}$$ where $D$ and $R_{ij}(h)$ are the covariant derivative and the Ricci tensor of the Riemannian metric $h$, respectively. Indices are raised and lowered with $h_{ij}$ and its inverse $h^{ij}$. In the asymptotically flat end $\Sigma^{\infty}_{0}$ of $\mbox{int}(\Sigma)$, the Einstein equations on $\mbox{int}(\Sigma)$ and (\[asymptoticallyflatend\]) that $V$ and $\phi$ decay as $$\label{VQ} V=1-\frac{M_{\scriptscriptstyle ADM}}{r} + O^{(2)}(1/r^{2}), \qquad \qquad \phi=\frac{Q}{r}+O^{(2)}(1/r^2),$$ where $Q$ is a constant (called the [**electric charge**]{} associated with $\Sigma^{\infty}_{0}$), and $M_{\scriptscriptstyle ADM}$ is the corresponding ADM mass. A crucial step for the uniqueness proof is to understand the behavior of the Riemannian metric $h$ near the boundary $\tbd \Sigma$. This is the aim of the following proposition. \[prop:Chrusciel\] Let $\Sigma$ be a spacelike hypersurface in a static spacetime $(M,\gM)$ with Killing vector $\vec{\xi}$. Suppose that $\lambda\geq 0$ on $\Sigma$ with $\lambda =0$ precisely on its topological boundary $\tbd \Sigma$ which is assumed to be a compact, 2-dimensional and embedded topological manifold. Then 1. Every arc-connected component $(\tbd \Sigma)_{d}$ which intersects a $C^2$ degenerate Killing horizon corresponds to a complete cylindrical asymptotic end of $(\Sigma,h)$. 2. $(\overline{\Sigma},h)$ admits a differentiable structure such that every arc-connected component $(\tbd \Sigma)_{n}$ of $\tbd \Sigma$ which intersects a non-degenerate Killing horizon is a totally geodesic boundary of $(\Sigma,h)$ with $h$ being smooth up to and including the boundary. This proposition shows that the Riemannian manifold $(\overline{\Sigma} \setminus \underset{d}{\cup} (\tbd \Sigma)_d,h)$ is the union of asymptotically flat ends, complete cylindrical asymptotic ends and compact sets with totally geodesic boundaries. Let us define $\Sigmatilde \equiv \overline{\Sigma} \setminus \underset{d}{\cup}(\tbd \Sigma)_d$. Now we are ready to explain the doubling method itself. Recall that the final aim is to show that the spacetime is either Reissner-Nordström or Majumdar-Papapetrou. Both have the property that $(\Sigmatilde,h)$ is conformally flat (i.e. there exists a positive function $\Omega$, called the conformal factor, such that the metric $\Omega^{2}h$ is the flat metric). Moreover, conformal flatness together with sufficient information on the conformal factor would imply, via the Einstein field equations, that the spacetime is in fact Reissner-Nordström or Majumdar-Papapetrou. A powerful method to prove that a given metric is flat is by using the rigidity part of the PMT. Unfortunately Theorem \[thr:PMT0\] cannot be applied directly to $(\Sigmatilde,h)$ because, first, $\Sigmatilde$ is a manifold with boundary, and second, $(\Sigmatilde,h)$ has in general cylindrical asymptotic ends and therefore it is not asymptotically flat. The presence of boundaries was dealt with by Bunting and Masood-ul-Alam who invented a method which constructs a new manifold without boundary to which the PMT can be applied. To simplify the presentation, let us assume for a moment that $(\Sigmatilde,h)$ has no cylindrical ends, so this manifold is the union of asymptotically ends and a compact interior with totally geodesic boundaries (by Proposition \[prop:Chrusciel\]). Next, find two conformal factors $\Omega_{+}>0$ and $\Omega_{-}>0$ such that - $h_{+}\equiv \Omega_{+}^{2} h$ is asymptotically flat, has vanishing mass and $R(h_{+})\geq 0$, where $R(h_{+})$ is the scalar curvature of $h_{+}$. - $h_{-}\equiv \Omega_{-}^{2}h$ admits a one point (let us denote it by $\Upsilon$) compactification of the asymptotically flat infinity, and $R(h_{-})\geq 0$. Then the idea is to glue the manifolds $(\Sigmatilde,h_{+})$ and $(\Sigmatilde\cup \Upsilon, h_{-})$ across the boundaries to produce a complete, asymptotically flat manifold $(\hat{\Sigma},\hat{h})$ with no boundaries, vanishing mass and non-negative scalar curvature $\hat{R}\geq 0$. In order to glue the two manifolds with sufficient differentiability, the following two conditions are required: - $\left. \Omega_{+}\right|_{\bd \Sigmatilde}=\left. \Omega_{-}\right|_{\bd \Sigmatilde}$, - $\left.\vec{m}(\Omega_{+})\right|_{\bd \Sigmatilde}=-\left.\vec{m}(\Omega_{-})\right|_{\bd \Sigmatilde}$. where $\vec{m}$ is the unit normal pointing to the interior $\Sigmatilde$ in each of the copies. ![The doubled manifold $(\hat{\Sigma},\hat{h})$ resulting from gluing $(\Sigmatilde,h_{+})$ and $(\Sigmatilde\cup \Upsilon, h_{-})$.[]{data-label="fig:doubling"}](doubling.eps){width="7cm"} Theorem \[thr:PMT0\] can be applied to $(\hat{\Sigma},\hat{h})$ to conclude that this space is in fact Euclidean. When the spacetime also has degenerate horizons the doubling method across non-degenerate components can still be done. The resulting manifold however is no longer asymptotically flat since it contains asymptotically cylindrical ends, so Theorem \[thr:PMT0\] cannot be applied directly. Fortunately, there exists a suitable generalization of the PMT that covers this case. The precise statement is the following. \[thr:PMT1\] Let $(\hat{\Sigma},\hat{h})$ be a smooth complete Riemannian manifold with an asymptotically flat end $\hat{\Sigma}^{\infty}_{0}$ and with a smooth one-form $\hat{\bf E}$ satisfying $\hat{D}_{i}\hat{E}^{i}=0$ and $\hat{E}_{i}dx^{i}=\frac{\hat{Q}}{r^2} dr + o(\frac{1}{r^2})$ in $\hat{\Sigma}^{\infty}_{0}$, where $\hat{Q}$ is a constant called electric charge. Suppose that $\hat{h}$ satisfies $R(\hat{h})\geq 2\hat{E}_{i} \hat{E}^{i}$ and that $$\int_{\hat{\Sigma}^{\infty}_{0}} \left( R(\hat{h})- 2\hat{E}_{i} \hat{E}^{i} \right) \eta_{\hat{h}}< \infty.$$ Then the ADM mass $\hat{M}_{\scriptscriptstyle ADM}$ of $\hat{\Sigma}^{\infty}_{0}$ satisfies $\hat{M}_{\scriptscriptstyle ADM}\geq |\hat{Q}|$ and equality holds if and only if locally $\hat{h}=u^{2}(dx^2+dy^2+dz^2)$, $\hat{\bf E}=\frac{du}{u}$ and $\Delta_{\delta}u=0$. [**Remark.**]{} As a consequence of this result, it is no longer necessary to require that $(\Sigmatilde,h_-)$ admits a one-point compactification. It is only necessary to assume that $(\overline{\Sigma_{0}^{\infty}},h_-)$ is complete. $\hfill \square$\ It is clear from the discussions above that the key to prove Theorem \[thr:electrovacuniqueness0\] is to find suitable conformal factors which allow to conclude that $({\Sigmatilde},{h})$ is conformally flat. For the static electro-vacuum case, two conformal factors have been considered, one due to Ruback [@Ruback], $\Omega_{\pm}=\frac{1\pm V+\phi}{2}$, and another proposed by Masood-ul-Alam [@MuA], $\Omega_{\pm}=\frac{(1\pm V)^{2}-\phi^2}{4}$. Recently, Chruściel has showed [@C2] that the Ruback conformal factor is the only one which works when degenerate Killing horizons are allowed a priori. We will therefore consider only the Ruback conformal factors $\Omega_{\pm}=\frac{1\pm V+\phi}{2}$. The first thing to do is to check that $\Omega_{\pm}$ are strictly positive on $\Sigmatilde$. This was shown by Ruback [@Ruback] and extended by Chruściel [@C2] and Chruściel and Tod [@CT] when there are degenerate horizons. \[proposition:positivityoftheconformalfactor\] On $\Sigmatilde$ it holds $|\phi|\leq 1-V$. Moreover, equality at one point only occurs when the spacetime is the standard Majumdar-Papapetrou spacetime. This proposition implies $\Omega_{-}>0$ unless we have Majumdar-Papapetrou. Moreover, since $V\geq 0$ on $\Sigmatilde$, we have $\Omega_{+}\geq \Omega_{-}>0$ except for the standard Majumdar-Papapetrou. The remaining ingredients are as follows: - The matching conditions for the gluing procedure follow easily from the fact that $\left.V\right|_{\bd \Sigmatilde}=0$, which immediately implies $\left.\Omega_{+}\right|_{\bd\Sigmatilde}=\left.\Omega_{-}\right|_{\bd \Sigmatilde}$ and $\left.\vec{m}(\Omega_{+})\right|_{\bd \Sigmatilde}=-\left.\vec{m}(\Omega_{-})\right|_{\bd \Sigmatilde}$. - The asymptotically flat end $(\Sigma_{0}^{\infty})$ becomes a complete end with respect to the metric $h_{-}$. This follows from the asymptotic form $\Omega_{-}=\frac{1}{4r}(M_{\scriptscriptstyle ADM}-Q)+O(1/r^{2})$ and the fact that $M_{\scriptscriptstyle ADM}> |Q|$ which follows from the positivity of $\Omega_{-}$. - The field ${\bf {E}}_{\pm}\equiv\frac{-(1+\phi)d\phi + VdV}{V(1+\phi\pm V)}$ has the following asymptotic behavior $${\bf E}_{+}=\frac12\frac{M_{\scriptscriptstyle ADM}+Q}{r^{2}}dr+o(1/r^{2}),$$ and satisfies, from the Einstein field equations, that $D^{\pm}_{i}{E_{\pm}}^{i}=0$ and $R(h_{\pm})=2E^{i}_{\pm}{E_{\pm}}_{i}$, where $R(h_{\pm})$ is the scalar curvature of $h_{\pm}$. - A direct computation gives that the ADM mass and the electric charge of $(\hat{\Sigma},\hat{h})$ satisfy, $$\hat{M}_{\scriptscriptstyle ADM}=\hat{Q}.$$ Therefore, the rigidity part of Theorem \[thr:PMT1\] can be applied, to conclude $\hat{h}=u^{2}g_{E}$, where $u$ is a specific function of $(V,\phi)$ and $g_{E}$ is the Euclidean metric. Consequently, $h$ (which was conformally related with $\hat{h}$) is conformally flat. The original proof used at this point the explicit form of $u(\phi,V)$ together with the field equations to conclude that $(\Sigmatilde,h)$ corresponds to the metric of the $\{t=0\}$ slice of Reissner-Nordström spacetime with $M>|Q|$. This last step has been simplified recently by González and Vera in [@GV] who show that the Reissner-Nordström and the Majumdar-Papapetrou spacetimes are indeed the only static electro-vacuum spacetimes for which $(\Sigmatilde,h)$ is asymptotically flat and conformally flat. Summarizing, we have obtained that, in the case when Theorem \[thr:PMT1\] can be applied, the spacetime is Reissner-Nordström, and in the cases when it cannot be applied the spacetime is already the standard Majumdar-Papapetrou spacetime. We conclude then that a static and electro-vacuum spacetime corresponding to a black hole must be either the Reissner-Nordström spacetime (where $\tbd \Sigma$ is connected) or the standard Majumdar-Papapetrou spacetime (where $\tbd \Sigma$ is non-connected), which proves Theorem \[thr:electrovacuniqueness0\]. [**Remark.**]{} The compactness assumption for the embedded topological submanifold $\tbd \Sigma$ is used in order to ensure that $(\hat{\Sigma},\hat{h})$ is complete. It would be interesting to study whether this condition can be relaxed or not. $\hfill \square$\ We will finish this chapter by giving a brief discussion about the global approach of Theorem \[thr:electrovacuniqueness0\]. In several works ([@C], [@Cc] and [@CG]) Chrúsciel and Galloway have studied sufficient hypotheses which ensure that a black hole spacetime possesses a spacelike hypersurface $\Sigma$ like the one required in Theorem \[thr:electrovacuniqueness0\] and, also, which assumptions are needed to conclude uniqueness for the whole spacetime (or at least for the domain of outer communications) The first work on the subject, namely [@C], deals with the vacuum case and requires, among other things, the spacetime to be analytic (although this hypothesis was not explicitly mentioned in [@C] and it was included only in the correction [@Cc]). This hypothesis is needed to avoid the existence of non-embedded degenerate Killing prehorizons, which implies that $\tbd \Sigma$ may fail to be compact and embedded as required in Theorem \[thr:electrovacuniqueness0\]. In [@Cc], Chruściel was able to drop the analyticity assumption by assuming a second Killing vector on $M$ generating a $U(1)$ action and a global hypothesis (named $I^{+}$-regularity in the later paper [@CL-C]). Finally, in [@CG] the assumption on the existence of a second Killing field was removed and the result was explicitly extended to the electro-vacuum case. Before giving the statement of such a result, let us define the property of $I^{+}$-regularity of a spacetime. \[Iplusregularity\] Let $(M,\gM)$ be a stationary spacetime containing an asymptotically flat end and let $\vec{\xi}$ be the stationary Killing vector field on $M$. $(M,\gM)$ is [**$I^{+}$-regular**]{} if $\vec{\xi}$ is complete, if the domain of outer communications $M_{DOC}$ is globally hyperbolic, and if $M_{DOC}$ contains a spacelike, connected, acausal hypersurface $\Sigma$ containing an asymptotically flat end, the closure $\overline{\Sigma}$ of which is a $C^0$ manifold with boundary, consisting of the union of a compact set and a finite number of asymptotically flat ends, such that $\tbd \Sigma$ is an embedded surface satisfying $$\tbd \Sigma\subset {\mathcal{E}^{+}}\equiv \tbd M_{DOC}\cap I^{+}(M_{DOC}),$$ with $\tbd \Sigma$ intersecting every generator of $\mathcal{E}^{+}$ just once. Then the result by Chruściel and Galloway states the following. \[thr:electrovacuniqueness1\] Let $(M,\gM)$ be a static solution of the electro-vacuum Einstein equations. Assume that $(M,\gM)$ is $I^{+}$-regular. Then the conclusions of Theorem \[thr:electrovacuniqueness0\] hold. Moreover, $M_{DOC}$ is isometrically diffeomorphic to the domain of outer communications of either the Reissner-Nordström spacetime or the standard Majumdar-Papapetrou spacetime. Stability of marginally outer trapped surfaces and symmetries {#ch:Article2} ============================================================= Introduction {#sc:A2sectionintroduction} ------------ As we have already mentioned in Chapter \[ch:Introduction\], although the main aim of this thesis is to study properties of certain types of trapped surfaces, specially weakly outer trapped surfaces and MOTS, in stationary and static configurations, isometries are not the only type of symmetries which can be involved in physical situations of interest. For instance, many relevant spacetimes admit other types of symmetries, such as conformal symmetries, e.g. in Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological models. Another interesting example appears when studying the critical collapse, which is a universal feature of many matter models. Indeed, the critical solution, which separates those configurations that disperse from those that form black holes, are known to admit either a continuous or a discrete self-similarity. Therefore, it is interesting to understand the relationship between trapped surfaces and several special types of symmetries. This is precisely the aim of this chapter. A recent interesting example of this interplay has been given in [@BenDov], [@SB], [@SB2] where the location of the boundaries of the spacetime set containing weakly trapped surfaces and weakly outer trapped surfaces was analyzed, firstly, in the Vaidya spacetime [@BenDov], [@SB] (which is one of the simplest dynamical situations) and, later, in spherically symmetric spacetimes in general [@SB2]. In these analyses the presence of symmetries turned out to be fundamental. In the important case of isometries, general results on the relationship between weakly trapped surfaces and Killing vectors were discussed in [@MS], where the first variation of area was used to obtain several restrictions on the existence of weakly trapped surfaces in spacetime regions possessing a causal Killing vector. More specifically, weakly trapped surfaces can exist in the region where the Killing vector is timelike only if their mean curvature vanishes identically. By obtaining a general identity for the first variation of area in terms of the deformation tensor of an arbitrary vector (defined in equation (\[mdt\])), similar restrictions were obtained for spacetimes admitting other types of symmetries, such as conformal Killing vectors or Kerr-Schild vectors (see [@CHS] for its definition). The same idea was also applied in [@S3] to obtain analogous results in spacetimes with vanishing curvature invariants. The interplay between isometries and dynamical horizons (which are spacelike hypersurfaces foliated by marginally trapped surfaces) was considered in [@AG05] where it was proven that dynamical horizons cannot exist in spacetime regions containing a nowhere vanishing causal Killing vector, provided the spacetime satisfies the NEC. Regarding MOTS, the relation between stable MOTS and isometries was considered in [@AMS], where it was shown that, given a strictly stable MOTS $S$ in a hypersurface $\Sigma$ (not necessarily spacelike), any Killing vector on $S$ tangent to $\Sigma$ must in fact be tangent to $S$. In the present chapter, we will study the interplay between stable and outermost properties of MOTS in spacetimes possessing special types of vector fields $\vec{\xi}$, including isometries, homotheties and conformal Killing vectors. In fact, we will find results involving completely general vector fields $\vec{\xi}$ and then, we will particularize them to the different types of symmetries. More precisely, we will find restrictions on $\vec{\xi}$ on stable, strictly stable and locally outermost MOTS $S$ in a given spacelike hypersurface $\Sigma$, or alternatively, forbid the existence of a MOTS in certain regions where $\vec{\xi}$ fails to satisfy those restrictions. In what follows, we give a brief summary of the present chapter. The fundamental idea which will allow us to obtain the results of this chapter will be introduced in Section \[sc:A2sectionbasics\]. As we will see, it will consist in a geometrical construction which can potentially restrict a vector field $\vec{\xi}$ on the outermost MOTS $S$. The geometrical procedure will involve the analysis of the stability operator $L_{\vec{m}}$ of a MOTS acting on a certain function $Q$. It will turn out that the results obtained by the geometric construction can, in most cases, be sharpened considerably by using the maximum principle of elliptic operators. This will also allow us to extend the validity of the results from the outermost case to the case of stable and strictly stable MOTS. However, the defining expression (\[stabilityoperator\]) for the stability operator $L_{\vec{m}}Q$ has a priori nothing to do with the properties of the vector field $\vec{\xi}$, which makes the method of little use. Our first task will be therefore to obtain an alternative (and completely general) expression for $L_{\vec{m}}Q$ in terms of $\vec{\xi}$, or more specifically, in terms of its deformation tensor $a_{\mu\nu}(\vec{\xi}\,)$. We will devote Section \[sc:A2variation\] to doing this. The result, given in Proposition \[propositionxitheta\], is thoroughly used in this chapter and also has independent interest. With this expression at hand, we will be able to analyze under which conditions our geometrical procedure gives restrictions on $\vec{\xi}$. In Section \[sc:A2LmQ\] we will concentrate on the case where $L_{\vec{m}}Q$ has a sign everywhere on $S$. The main result of Section \[sc:A2LmQ\] will be given in Theorem \[TrhAnyXi\], which holds for any vector field $\vec{\xi}$. This result will be then particularized to conformal Killing vectors (including homotheties and Killing vectors) in Corollary \[thrstable\]. Under the additional restriction that the homothety or the Killing vector is everywhere causal and future (or past) directed, strong restrictions on the geometry of the MOTS will be derived (Corollary \[shear\]). As a consequence, we will prove that in a plane wave spacetime any stable MOTS must be orthogonal to the direction of propagation of the wave. Marginally trapped surfaces will be also discussed in this section. As an explicit application of the results on conformal Killing vectors, we will show, in Subsection \[ssc:A2sectionFLRW\], that stable MOTS cannot exist in any spacelike hypersurface in FLRW cosmological models provided the density $\mu$ and pressure $p$ satisfy the inequalities $\mu \geq 0$, $\mu \geq 3 p $ and $\mu + p \geq 0$. This includes, for instance, all classic models of matter and radiation dominated eras and also those models with accelerated expansion which satisfy the NEC. Subsection \[ssc:A2sectiongeometric\] will deal with one case where, in contrast with the standard situation, the geometric construction does in fact give sharper results than the elliptic theory. One of these results, together with Theorem \[thr:AM\] by Andersson and Metzger, will imply an interesting result (Theorem \[theorem1\]) for weakly outer trapped surfaces in stationary spacetimes. In the case when $L_{\vec{m}} Q$ is not assumed to have a definite sign, the maximum principle loses its power. However, as we will discuss in Section \[sc:A2sectionnonelliptic\], a result by Kriele and Hayward [@KH97] will allow us to exploit our geometric construction again to obtain additional results. This will produce a theorem (Theorem \[thrnonelliptic\]) which holds for general vector fields $\vec{\xi}$ on any locally outermost MOTS. As in the previous section, we will particularize the result to conformal Killing vectors, and then to causal Killing vectors and homotheties which, in this case, will be allowed to change their time orientation on $S$ The results presented in this chapter have been published mainly in the papers [@CM2], [@CMere2] and partly in [@CM1] and [@CMere1]. Geometric procedure {#sc:A2sectionbasics} ------------------- Consider a spacelike hypersurface $(\Sigma,g,K)$ which is embedded in a spacetime $(M,\gM)$ with a vector field $\vec{\xi}$ defined on a neighbourhood of $\Sigma$. Assume that $\Sigma$ possesses a barrier $\Sb$ with interior $\Omegab$ and let $S\subset\Sigma$ be a bounding MOTS with respect to $\Sb$ (and therefore an [*exterior region*]{} of $S$ in $\Omegab$ can be properly defined). The idea we want to exploit consists in constructing under certain circumstances a new weakly outer trapped surface $\S_{\tau}\subset\Omegab$ which lies, at least partially, outside $\S$. This fact will provide a contradiction in the case when $\S$ is the outermost bounding MOTS and will allow us to obtain restrictions on the vector $\vec{\xi}$ on $\S$. As we will see below, this simple idea will allow us to obtain results also for stable, strictly stable and locally outermost MOTS, irrespectively of whether they are bounding or not, by using the theory of elliptic second order operators. The geometric procedure to construct the new surface $\S_{\tau}$ consists in moving $\S$ first along the integral lines of $\vec{\xi}$ a parametric amount $\tau$. This gives a new surface $S^{\prime}_{\tau}$. Next, take the null normal ${{\vec{l}_{+}}^{\prime}}(\tau)$ on this surface which coincides with the continuous deformation of the outer null normal $\vec{l}_{+}$ on $S$ normalized to satisfy $l_{+}^{\mu}n_{\mu}=-1$ (where $\vec{n}$ denotes the unit vector normal to $\Sigma$ and future directed) and consider the null hypersurface generated by null geodesics with tangent vector ${\vec{l}_{+}^{\prime}}(\tau)$. This hypersurface is smooth close enough to $\S^{\prime}_{\tau}$. Being null, its intersection with the spacelike hypersurface $\Sigma$ is transversal and hence defines a smooth surface $S_{\tau}$ (for $\tau$ sufficiently small). By this construction, a point $\p$ on $S$ describes a curve in $\Sigma$ when $\tau$ is varied. The tangent vector of this curve on $\S$, denoted by $\vec{\nu}$, will define the variation vector generating the one-parameter family $\{ \S_{\tau} \}_{\tau\in I\subset\mathbb{R}}$ on a neighbourhood of $S$ in $\Sigma$. Figure \[fig:construction\] gives a graphic representation of this construction. ![The figure represents how the new surface $S_t$ is constructed from the original surface $S$. The intermediate surface $S^{\prime}_{\tau}$ is obtained from $S$ by dragging along $\vec{\xi}$ a parametric amount $\tau$. Although $\vec{\xi}$ has been depicted as timelike here, this vector can be in fact of any causal character.[]{data-label="fig:construction"}](fig3.eps){width="9cm"} Let us decompose the vector $\vec{\xi}$ into normal and tangential components with respect to $\Sigma$, as $\vec{\xi}=N\vec{n}+\vec{Y}$ (see Figure \[fig:XiNY\]). ![The vector $\vec{\xi}$ decomposed into normal $N\vec{n}$ and tangential $\vec{Y}$ components.[]{data-label="fig:XiNY"}](XiNY.eps){width="9cm"} On $\S$ we will further decompose $\vec{Y}$ in terms of a tangential component $\vec{Y}^{\parallel}$, and a normal component $(Y_{i}m^{i})\vec{m}$, where $\vec{m}$ is the unit vector normal to $S$ in $\Sigma$ which points to the exterior of $S$ in $\Sigma$. Therefore, $\vec{\xi} |_S =N_S \vec{n}+ (Y_{i} m^{i}) \vec{m} + \vec{Y}^{\parallel}$, where $N_S$ is the value of $N$ on the surface. In order to study the variation vector $\vec{\nu}$, let us expand the embedding functions $\left\{ x^\mu\left( y^A,\tau \right) \right\}$ of the surface $\S_{\tau}$ (where $\left\{ y^A \right\}$ are intrinsic coordinates of $\S$) as $$\label{embedding2} x^{\mu}\left( y^A,\tau \right)=x^{\mu}\left( y^A,0 \right) + \xi^\mu\left( y^A,0 \right)\tau + F(y^A){l'_{+}}(\tau)^{\mu}\left( y^A\right)\tau + O(\tau^2),$$ where $F(y^A)$ is a function to be adjusted. Since $\vec{\nu}$ defines the variation of $\S$ to first order, equation (\[embedding2\]) implies that we only need to evaluate the vector ${\vec{l}_{+}^{\prime}}(\tau)$ to zero order in $\tau$, which obviously coincides with $\vec{l}_{+}$. It follows then that $\vec{\nu}$ is a linear combination (with functions) of $\vec{\xi}$ and $\vec{l}_{+}$. The amount we need to move $S^{\prime}_{\tau}$ in order to go back to $\Sigma$ can be determined by imposing $\vec{\nu}$ to be tangent to $\Sigma$. Since $\vec{\nu}(y^{A})=\vec{\xi}(y^{A})+F(y^{A})\vec{l}_{+}(y^{A})$, multiplication with $\vec{n}$ gives $0=N_{S}+F$. Thus, $F=-N_{S}$ and $\vec{\nu}=\vec{\xi}-N_{S}\vec{l}_{+}$. Using the previous decomposition for $\vec{\xi}$ and $\vec{l}_{+}=\vec{n}+\vec{m}$ we can rewrite $\vec{\nu}=Q\vec{m}+\vec{Y}^{\parallel}$, where $$\label{Q} Q= (Y_{i}m^{i}) -N_S = {\xi}_{\mu}l_{+}^{\mu}\,$$ determines at first order the amount and sense to which a point $\p\in S$ moves along the normal direction. Let us consider for a moment the simplest case that $\vec{\xi}$ is a Killing vector. Suppose $\S$ is a MOTS which is bounding with respect to a barrier $\Sb$ with interior $\Omegab$. Since the null expansion does not change under an isometry, it follows that the surface $S^{\prime}_{\tau}$ is also a bounding MOTS for the spacelike hypersurface obtained by moving $\Sigma$ along the integral curves of $\vec{\xi}$ an amount $\tau$. Moving back to $\Sigma$ along the null hypersurface gives a contribution to $\theta^{+} [S_\tau]$ which is easily computed to be $\left.\frac{d}{d\tau}\left[\hat{\varphi}_{\tau}^{*}(\theta^{+}[S_{\tau}]) \right]\right|_{\tau=0}=\left. \frac{1}{2}N{\theta^{+}}^{2}[S]+ N W\right|_{{S}}$ which is the well-known Raychaudhuri equation (which has already appeared before in equation (\[raychaudhuri\]) for the particular case of MOTS), where $\hat{\varphi}_{\tau}:S\rightarrow S_{\tau}$ is the diffeomorphism defined by the geometrical construction above and $W$ was defined in equation (\[W\]) and is non-negative provided the NEC holds. It implies that if $N_{S}<0$ and $W\neq 0$ everywhere, then $\theta^{+}[S_{\tau}]< 0$ provided $\tau$ is positive and sufficiently small and the NEC holds. Therefore, $S_{\tau}$ is a bounding (provided $\tau$ is sufficiently small) weakly outer trapped surface which lies partially outside $S$ if $Q>0$ somewhere. This is impossible if $S$ is an outermost bounding MOTS by Theorem \[thr:AM\] of Andersson and Metzger. Thus, the function $Q$ must be non-positive everywhere on any outermost bounding MOTS $S$ for which $N_{S}<0$ and $W\neq 0$ everywhere. Independently of whether $\vec{\xi}$ is a Killing vector or not, the more favorable case to obtain restrictions on the generator $\vec{\xi}$ on a given outermost bounding MOTS is when the newly constructed surface $S_{\tau}$ is bounding and weakly outer trapped. This is guaranteed for small enough $\tau$ when $\delta_{\vec{\nu}}\theta^{+}$ is strictly negative everywhere, because then this first order terms becomes dominant for small enough $\tau$. Due to the fact that the tangential part of $\vec{\nu}$ does not affect the variation of $\theta^{+}$ along $\vec{\nu}$ for a MOTS (c.f. (\[variationtheta+general0\])), it follows that $\delta_{\vec {\nu}}\theta^{+}=L_{\vec{m}}Q$, where $L_{\vec{m}}$ is the stability operator for MOTS defined in (\[stabilityoperator\]). Since the vector $\vec{\nu}=Q\vec{m}+\vec{Y}^{\parallel}$ determines to first order the direction to which a point $\p\in S$ moves, it is clear that $L_{\vec{m}}Q<0$ everywhere and $Q>0$ somewhere is impossible for an outermost bounding MOTS. This is precisely the argument we have used above and is intuitively very clear. However, this geometric method does not provide the most powerful way of finding this type of restriction. Indeed, when the first order term $L_{\vec{m}}Q$ vanishes at some points, then higher order coefficients come necessarily into play, which makes the geometric argument of little use. It is remarkable that using the elliptic results described in Appendix \[ch:appendix2\], most of these situations can be treated in a satisfactory way. Furthermore, since the elliptic methods only use infinitesimal information, there is no need to restrict oneself to outermost bounding MOTS, and the more general case of stable or strictly stable MOTS (not necessarily bounding) can be considered. Unfortunately, the general expression of $L_{\vec{m}}Q$ given in equation (\[stabilityoperator\]) is not directly linked to the vector $\vec{\xi}$, which is clearly unsuitable for our aims. In the case of Killing vectors, the point of view of moving $\S$ along $\vec{\xi}$ and then back to $\Sigma$ gives a simple method of calculating $L_{\vec{m}} Q$. For more general vectors, however, the motion along $\vec{\xi}$ will give a non-zero contribution to $\theta^+$ which needs to be computed (for Killing vectors this term was known to be zero via a symmetry argument, not from a direct computation). In order to do this, it becomes necessary to have an alternative, and completely general, expression for $\delta_{\vec{\xi}}\, \theta^+$ directly in terms of the deformation tensor $a_{\mu\nu}(\vec{\xi}\,)$ associated with $\vec{\xi}$. This is the aim of the following section. Variation of the expansion and the metric deformation tensor {#sc:A2variation} ------------------------------------------------------------ Let us derive an identity for $\delta_{\vec{\xi}}\, \theta^+$ in terms of $a_{\mu\nu}(\vec{\xi}\,)$. This result will be important later on in this chapter, and may also be of independent interest. We derive this expression in full generality, without assuming $\S$ to be a MOTS and for the expansion $\theta_{\vec{\eta}}$ along any normal vector $\vec{\eta}$ of $\S$ (not necessarily a null normal) i.e. $$\theta_{\vec{\eta}}\equiv H_{\mu}\eta^{\mu},$$ where $\vec{H}$ denotes the mean curvature of $S$ in $M$. To do this calculation, we need to take derivatives of tensorial objects defined on each one of $\S^{\prime}_{\tau}$. For a given point $\p \in \S$, these tensors live on different spaces, namely the tangent spaces of $\varphi_{\tau} (\p)$, where $\varphi_{\tau}$ is the one-parameter local group of diffeomorphisms generated by $\vec{\xi}$. In order to define the variation, we need to pull-back all these tensors to the point $\p$ before doing the derivative. We will denote the resulting derivative by $\mathscr{L}_{{\vec{\xi}}}$. In general, this operation is not the standard Lie derivative $\mathcal{L}_{\vec{\xi}}$ on tensors because it is applied to tensorial objects on each $S^{\prime}_{\tau}$ which may not define tensor fields on $M$ (e.g. when these surfaces intersect each other). Nevertheless, both derivatives do coincide when acting on spacetime tensor fields (e.g. the metric $\gM$) which will simplify the calculation considerably. Notice in particular that the definition of $\theta_{\vec{\eta}}$ depends on the choice of $\vec{\eta}$ on each of the surfaces $S^{\prime}_{\tau}$. Thus $\delta_{\vec{\xi}}\, \theta_{\vec{\eta}} \equiv \left. \mathscr{L}_{\vec{\xi\,}} \theta_{\vec{\eta}}\right|_{S}$ will necessarily include a term of the form $\mathscr{L}_{\vec{\xi}}\, \eta_{\alpha}$ which is not uniquely defined (unless $\vec{\eta}$ can be uniquely defined on each $S^{\prime}_{\tau}$, which is usually not the case). Nevertheless, for the case of MOTS and when $\vec{\eta} = \vec{l}_{+}$ this a priori ambiguous term becomes determined, as we will see. The general expression for $\delta_{\vec{\xi}}\, \theta_{\vec{\eta}}$ is given in the following proposition. \[propositionxitheta\] Let $\S$ be a surface on a spacetime $(M,\gM)$, $\vec{\xi}$ a vector field defined on $M$ with deformation tensor $a_{\mu\nu}(\vec{\xi}\,)$ and $\vec{\eta}$ a vector field normal to $\S$ and extend $\vec{\eta}$ to a smooth map $\vec{\eta}:(-\epsilon,\epsilon)\times S \rightarrow TM$ satisfying $\vec{\eta}(0,\p)=\vec{\eta}(\p)$ and $\vec{\eta}(\tau,\p)\in (T_{\varphi_{\tau}(\p)}S'_{\tau})^{\perp}$ where $\varphi_{\tau}$ is the local group of diffeomorphisms generated by $\vec{\xi}$ and $S'_{\tau}=\varphi_{\tau}(S)$. Then, the variation along $\vec{\xi}$ of the expansion $\theta_{\vec{\eta}}$ on $\S$ reads $$\begin{aligned} \label{xithetau} \delta_{\vec{\xi}}\,\theta_{\vec{\eta}}&=& H^{\mu}\mathscr{L}_{\vec{\xi}}\,\eta_{\mu}-a_{AB}(\vec{\xi}\,) \Pi^{AB}_{\mu}\eta^{\mu}\nonumber \\ &&\qquad\left . + \gamma^{AB}e_{A}^{\alpha}e_{B}^{\rho}\eta^{\nu} \left[ \frac{1}{2}\nabla_{\nu}a_{\alpha\rho}(\vec{\xi}\,) - \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)\right] \right|_{\S},\end{aligned}$$ where $\vec{\Pi}_{AB}$ denotes the second fundamental form vector of $S$ in $M$, and $a_{AB}(\vec{\xi}\,)\equiv e_{A}^{\alpha}e_{B}^{\beta}a_{\alpha\beta}(\vec{\xi}\,)$, with $\{\vec{e}_{A}\}$ being a local basis for $TS$. [**Proof.**]{} Since $\theta_{\vec{\eta}}=H_{\mu}\eta^{\mu}=\gamma^{AB}\Pi_{AB}^{\mu}\eta_{\mu}$, the variation we need to calculate involves three terms $$\label{Lietheta} \mathscr{L}_{\vec{\xi}}\,\theta_{\vec{\eta}}=\left(\mathscr{L}_{\vec{\xi}}\,\gamma^{AB}\right) \Pi_{AB}^{\mu}\eta_{\mu} + \gamma^{AB}\left(\mathscr{L}_{\vec{\xi}}\,\Pi_{AB}^{\mu}\right)+ H^{\mu}\left(\mathscr{L}_{\vec{\xi}}\,\eta_{\mu}\right).$$ In order to do the calculation, we will choose ${\varphi_{\tau}}_{\star} (\vec{e}_A)$ as the basis of tangent vectors at $\varphi_{\tau}(\p) \in S^{\prime}_{\tau}$ (we refer to ${\varphi_{\tau}}_{\star} (\vec{e}_A)$ merely as $\vec{e}_A$ in the following to simplify the notation). This entails no loss of generality and implies $\mathscr{L}_{\vec{\xi}}\,\vec{e}_{A}=0$, which makes the calculation simpler. Our aim is to express each term of (\[Lietheta\]) in terms of $a_{\mu\nu}(\vec{\xi}\,)$. For the first term, we need to calculate $\mathscr{L}_{\vec{\xi}}\,\gamma^{AB}$. We start with $\mathscr{L}_{\vec{\xi}}\,\gamma_{AB}=\mathscr{L}_{\vec{\xi}} \left ( \gM (\vec{e}_A, \vec{e}_B ) \right ) = ( \mathscr{L}_{\vec{\xi}\,} g ) \left (\vec{e}_A,\vec{e}_B \right ) = ( \mathcal{L}_{\vec{\xi}\,} g ) \left (\vec{e}_A,\vec{e}_B \right ) = a_{\mu\nu}(\vec{\xi}\,) e_{A}^{\mu}e_{B}^{\nu}\equiv a_{AB}(\vec{\xi}\,)$, which immediately implies $\mathscr{L}_{\vec{\xi}\,}\gamma^{AB}=-a_{CD}(\vec{\xi}\,)\gamma^{AC}\gamma^{BD}$, so that the first term in (\[Lietheta\]) becomes $$\label{firstterm} \mathscr{L}_{\vec{\xi}}\,\gamma^{AB} \Pi_{AB}^{\mu}\eta_{\mu}= - a_{AB}(\vec{\xi}\,)\Pi^{AB}_ {\mu}\eta^{\mu}.$$ The second term $\gamma^{AB}(\mathscr{L}_{\vec{\xi}}\Pi_{AB}^{\mu}) \eta_{\mu}$ is more complicated. It is useful to introduce the projector to the normal space of $S$, $h_{\nu}^{\mu}\equiv \delta^{\mu}_{\nu} - \gM_{\nu\beta}e_{A}^{\mu}e_{B}^{\beta}\gamma^{AB}$. From the previous considerations, it follows that $\mathscr{L}_{\vec{\xi}}\, h_{\nu}^{\mu}=e_{A}^{\mu}e_{B}^{\beta} ( a^{AB}(\vec{\xi}\,) \gM_{\nu\beta} - \gamma^{AB}a_{\nu\beta}(\vec{\xi}\,))$, which implies $\left( \mathscr{L}_{\vec{\xi}\,} h_{\nu}^{\mu} \right)\eta_{\mu}=0$ and hence $$\begin{aligned} \mathscr{L}_{\vec{\xi}}\, (\Pi^{\mu}_{AB} ) \eta_{\mu} = - \mathscr{L}_{\vec{\xi}} \left ( h^{\mu}_{\nu} e_{A}^{\alpha}\nabla_{\alpha}e_{B}^{\nu} \right) \eta_{\mu} = - \eta_{\nu}\mathscr{L}_{\vec{\xi}}\left( e_{A}^{\alpha}\nabla_{\alpha}e_{B}^{\nu}\right). \label{LiePi}\end{aligned}$$ Therefore we only need to evaluate $\mathscr{L}_{\vec{\xi}} \left( e_{A}^{\alpha}\nabla_{\alpha}e_{B}^{\nu}\right)$. It is well-known that for an arbitrary vector field $\vec{v}$, $\mathcal{L}_{\vec{\xi}}\nabla_{\alpha}v^{\nu}-\nabla_{\alpha}\mathcal{L}_{\vec{\xi}}\,v^{\nu} = v^{\rho}\nabla_{\alpha}\nabla_{\rho}\xi^{\nu}+{\RM}^{\nu}_{\,\,\,\rho\sigma\alpha}v^{\rho}\xi^{\sigma}$. However, this expression is not directly applicable to the variational derivative we are calculating and we need the following closely related lemma. \[lema:Liee\] $$\begin{aligned} \mathscr{L}_{\vec{\xi}}\left( {e}_A^{\alpha}\nabla_{\alpha}{e}_{B}^{\nu}\right) ={e}_A^{\alpha}{e}_{B}^{\rho}\nabla_{\alpha}\nabla_{\rho}\xi^{\nu} +{\RM}^{\nu}_{\,\,\,\rho\sigma\alpha}{e}_{A}^{\alpha}{e}_{B}^{\rho}\xi^{\sigma}. \label{Liee}\end{aligned}$$ [**Proof of Lemma \[lema:Liee\].**]{} Choose coordinates $y^A$ on $S$ and extend them as constants along $\vec{\xi}$. This gives coordinates on each one of $S'_{\tau}$. Define $e_{A}^{\alpha}=\frac{\partial x^{\alpha}}{\partial y^{A}}$, where $x^{\mu}(y^{A},\tau)$ are the embedding functions of $S'_{\tau}$ in $M$ in spacetime coordinates $x^{\mu}$. The map $\varphi_{-\tau}: M\rightarrow M$ relates every point $\p\in S_{\tau}$ with coordinates $\{x^{\alpha}\}$ to a point $\varphi_{-\tau}(\p)\in S$ with coordinates $\{\varphi_{-\tau}^{\,\alpha}(x^{\beta})\}$. By definition, $\mathscr{L}_{\vec{\xi}}(e^{\mu}_{A}\nabla_{\mu}e_{B}^{\nu})\equiv \frac{d}{d\tau}\left( (\varphi_{-\tau})_{*} (e_{A}^{\mu}\nabla_{\mu}e_{B}^{\nu}) \right)$. Using that $\frac{\partial \varphi_{-\tau}^{\,\alpha}(x^{\beta})}{\partial \tau}=-\xi^{\alpha}$, it is immediate to obtain $$\begin{aligned} &&\left.\frac{d}{d\tau}\left( (\varphi_{-\tau})_{*} (e_{A}^{\mu}\nabla_{\mu}e_{B}^{\nu}) \right)\right|_{\tau=0}= \left.\frac{d}{d\tau}\left[ (e_{A}^{\mu}\nabla_{\mu}e_{B}^{\alpha}) \frac{\partial \varphi_{-\tau}^{\,\nu}}{\partial x^{\alpha}} \right]\right|_{\tau=0}\\ &&\qquad =\frac{\partial}{\partial \tau} (e_{A}^{\mu}\nabla_{\mu}e_{B}^{\nu}(y^{C},\tau)) -\partial _{\alpha}\xi^{\nu}e_{A}^{\mu}\nabla_{\mu}e_{B}^{\alpha}\\ &&\qquad =\frac{\partial}{\partial \tau}\left[ \frac{\partial^{2}x^{\nu}}{\partial y^{A}\partial y^{B}} +\Gamma_{\alpha\rho}^{\nu}\frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^{B}}\right] -\partial_{\mu}\xi^{\nu}\left[ \frac{\partial^{2}x^{\mu}}{\partial y^{A}\partial y^{B}} +\Gamma_{\alpha\rho}^{\mu}\frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^{B}}\right].\end{aligned}$$ On the other hand, $$\begin{aligned} &&{e}_A^{\alpha}{e}_{B}^{\rho}\nabla_{\alpha}\nabla_{\rho}\xi^{\nu} +{\RM}^{\nu}_{\,\,\,\rho\sigma\alpha}{e}_{A}^{\alpha}{e}_{B}^{\rho}\xi^{\sigma}\\ &&\qquad= \frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^{B}}\left[ \partial_{\alpha} \partial_{\rho}\xi^{\nu}+ \Gamma_{\mu\rho}^{\nu} \partial_{\alpha}\xi^{\mu}+ \Gamma_{\mu\alpha}^{\nu} \partial_{\rho}\xi^{\mu} -\Gamma_{\alpha\rho} ^{\mu}\partial_{\mu}\xi^{\nu}+ \xi^{\sigma}\partial_{\sigma}\Gamma_{\alpha\rho}^{\nu} \right]\\ &&\qquad= \frac{\partial^{3}x^{\nu}}{\partial \tau\partial y^{A} \partial y^{B}}- \frac{\partial^{2}x^{\rho}}{\partial y^{A}\partial y^{B}}\partial_{\rho}\xi^{\nu}+ \frac{\partial x^{\rho}}{\partial y^{B}}\Gamma_{\mu\rho}^{\nu}\partial_{\tau} \left( \frac{\partial x^{\mu}}{\partial y^{A}} \right)+ \frac{\partial x^{\alpha}}{\partial y^{A}}\Gamma_{\mu\alpha}^{\nu}\partial_{\tau} \left( \frac{\partial x^{\mu}}{\partial y^{B}} \right)\\ &&\qquad\qquad + \frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^B}\left[ \partial_{\tau}\Gamma_{\alpha\rho}^{\nu}-\Gamma_{\alpha\rho}^{\mu}\partial_{\mu}\xi^{\nu}\right]\\ &&\qquad = \frac{\partial}{\partial \tau}\left[ \frac{\partial^{2}x^{\nu}}{\partial y^{A}\partial y^{B}} +\Gamma_{\alpha\rho}^{\nu}\frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^{B}}\right] -\partial_{\mu}\xi^{\nu}\left[ \frac{\partial^{2}x^{\mu}}{\partial y^{A}\partial y^{B}} +\Gamma_{\alpha\rho}^{\mu}\frac{\partial x^{\alpha}}{\partial y^{A}}\frac{\partial x^{\rho}}{\partial y^{B}}\right],\\\end{aligned}$$ where we have used $${{\RM}^{\nu}}_{\rho\sigma\alpha}=\partial_{\sigma}\Gamma_{\rho\alpha}^{\nu}-\partial_{\alpha}\Gamma_{\rho\sigma}^{\nu} +\Gamma_{\gamma\sigma}^{\nu}\Gamma_{\rho\alpha}^{\gamma}-\Gamma_{\gamma\alpha}^{\nu}\Gamma_{\rho\sigma}^{\gamma},$$ in the first equality and $\xi^{\mu}=\frac{\partial x^{\mu}(y^{A},\tau)}{\partial \tau}$ in the second one. This proves the lemma. $\hfill \blacksquare$\ We can now continue with the proof of Proposition \[propositionxitheta\]. It only remains to express the quantity $\nabla_{\alpha}\nabla_{\rho}\xi^{\nu} +{\RM}^{\nu}_{\,\,\,\rho\sigma\alpha}\xi^{\sigma}$ in terms of $a_{\mu\nu}(\vec{\xi}\,)$. To that end, we take a derivative of $\nabla_{\nu}\xi_{\rho}+\nabla_{\rho}\xi_{\nu}=a_{\nu\rho}(\vec{\xi}\,)$ to get $$\nabla_{\alpha}\nabla_{\nu}\xi_{\rho}+\nabla_{\alpha}\nabla_{\rho}\xi_{\nu}=\nabla_{\alpha}a_{\nu\rho}(\vec{\xi}),$$ and use the Ricci identity $\nabla_{\alpha}\nabla_{\nu}\xi_{\rho}-\nabla_{\nu}\nabla_{\alpha}\xi_{\rho}=-{\RM}_{\sigma\rho\alpha\nu}\xi^{\sigma}$ to obtain $$\nabla_{\nu}\nabla_{\alpha}\xi_{\rho} + \nabla_{\alpha}\nabla_{\rho}\xi_{\nu}= {\RM}_{\sigma\rho\alpha\nu}\xi^{\sigma} + \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,).$$ Now, write the three equations obtained from this one by cyclic permutation of the three indices. Adding two of them and subtracting the third one we find $$\begin{aligned} \nabla_{\alpha}\nabla_{\rho}\xi_{\nu} &=& \frac12({\RM}_{\sigma\rho\alpha\nu}+{\RM}_{\sigma\nu\rho\alpha}-{\RM}_{\sigma\alpha\nu\rho})\xi^{\sigma}\\ &&\qquad + \frac12\left[ \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)+\nabla_{\rho}a_{\alpha\nu}(\vec{\xi}\,)-\nabla_{\nu} a_{\alpha\rho}(\vec{\xi}\,) \right].\end{aligned}$$ which, after using the first Bianchi identity $\RM_{\sigma\rho\alpha\nu}+\RM_{\sigma\nu\rho\alpha}+\RM_{\sigma\alpha\nu\rho}=0$, leads to $$\nabla_{\alpha}\nabla_{\rho}\xi_{\nu} = {\RM}_{\sigma\alpha\rho\nu}\xi^{\sigma} + \frac12\left[ \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)+\nabla_{\rho}a_{\alpha\nu}(\vec{\xi}\,)-\nabla_{\nu} a_{\alpha\rho}(\vec{\xi}\,) \right].$$ Substituting (\[Liee\]) and this expression into (\[LiePi\]) yields $$\label{secondterm} \gamma^{AB}\mathscr{L}_{\vec{\xi}}\,\Pi_{AB}^{\mu}\eta_{\mu}= \gamma^{AB}e_{A}^{\alpha}e_{B}^{\rho}\eta^{\nu} \left[ \frac{1}{2}\nabla_{\nu}a_{\alpha\rho}(\vec{\xi}\,) - \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)\right].$$ Inserting (\[firstterm\]) and (\[secondterm\]) into equation (\[Lietheta\]) proves the proposition. $\hfill \blacksquare$\ We can now particularize to the outer null expansion in a MOTS. \[corollaryxitheta\] If $S$ is a MOTS then $$\begin{aligned} \label{xitheta} \delta_{\vec{\xi}}\,\theta^{+} &=&-\frac14\theta^{-}a_{\mu\nu}(\vec{\xi}\,)l_{+}^{\mu}l_{+}^{\nu}-a_{AB}(\vec{\xi}\,)\Pi^{AB}_{\mu}l_{+}^{\mu}\nonumber \\ && \qquad \left . + \gamma^{AB}e_{A}^{\alpha}e_{B}^{\rho} l_{+}^{\nu} \left[ \frac{1}{2}\nabla_{\nu}a_{\alpha\rho}(\vec{\xi}\,) - \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)\right] \right|_{\S}.\end{aligned}$$ [**Proof.**]{} The normal vector $\vec{l}_+^{\prime}(\tau)$ defined on each of the surfaces $S^{\prime}_{\tau}$ is null. Therefore, using $\mathscr{L}_{\vec{\xi}} \, {\gM}^{\mu\nu}=\mathcal{L}_{\vec{\xi}} \, {\gM}^{\mu\nu} = - a^{\mu\nu}(\vec{\xi}\,)$, $$\begin{aligned} 0 = \mathscr{L}_{\vec{\xi}} \left ( {l_{+}^{\prime}}_{\mu}(\tau) {l_{+}^{\prime}}_{\nu}(\tau) {\gM}^{\mu\nu} \right ) = 2 l_{+}^{\mu} \mathscr{L}_{\vec{\xi}} \,{l_{+}^{\prime}}_{\mu}(\tau) - a_{\mu\nu}(\vec{\xi}\,) l_{+}^{\mu} l_{+}^{\nu}. \label{null}\end{aligned}$$ Since, on a MOTS $\vec{H}=-\frac12 \theta^{-}\vec{l}_{+}$, it follows $H^{\mu} \mathscr{L}_{\vec{\xi}}\, {l_{+}^{\prime}}_{\mu}(\tau) = -\frac{1}{2} \theta^{-} l_{+}^{\mu} \mathscr{L}_{\vec{\xi}} \, {l_{+}^{\prime}}_{\mu}(\tau) = - \frac{1}{4} \theta^{-} a_{\mu\nu}(\vec{\xi}\,) l_{+}^{\mu} l_{+}^{\nu}$, and the corollary follows from (\[xithetau\]). $\hfill \blacksquare$\ [**Remark.**]{} Formula (\[xitheta\]) holds in general for arbitrary surfaces $\S$ at any point where $\theta^{+}=0$. $\hfill \square$ Results provided $L_{\vec{m}} Q$ has a sign on $\S$ {#sc:A2LmQ} --------------------------------------------------- In this section we will give several results provided $L_{\vec{m}}Q$ has a definite sign on $S$. In this case, a direct application of Lemma \[lemmaelliptic\] for a MOTS $\S$ with stability operator $L_{\vec{m}}$ leads to the following result. \[lemmaelliptic2\] Let $\S$ be a stable MOTS on a spacelike hypersurface $\Sigma$. If $\left. L_{\vec{m}}Q\right|_{\S}\leq 0$ (resp. $\left.L_{\vec{m}}Q\right|_{\S}\geq 0$) and not identically zero, then $\left. Q\right|_{\S}<0$ (resp. $\left. Q\right|_{\S}>0$). Furthermore, if $\S$ is strictly stable and $\left. L_{\vec{m}}Q\right|_{\S}\leq 0$ (resp. $\left.L_{\vec{m}}Q\right|_{\S}\geq 0$) then $\left. Q\right|_{\S}\leq 0$ (resp. $\left. Q\right|_{\S}\geq 0$) and it vanishes at one point only if it vanishes everywhere on $\S$. The general idea then is to combine Lemma \[lemmaelliptic2\] with the general calculation for the variation of $\theta^+$ obtained in the previous section to get restrictions on special types of generators $\vec{\xi}$ on a stable or strictly stable MOTS. Our first result is fully general in the sense that it is valid for any generator $\vec{\xi}$. \[TrhAnyXi\] Let $S$ be a stable MOTS on a spacelike hypersurface $\Sigma$ and $\vec{\xi}$ a vector field on $S$ with deformation tensor $a_{\mu\nu}(\vec{\xi}\,)$. With the notation above, define $$\begin{aligned} Z &=& -\frac14\theta^{-}a_{\mu\nu}(\vec{\xi}\,)l_{+}^{\mu}l_{+}^{\nu}-a_{AB}(\vec{\xi}\,)\Pi^{AB}_{\mu}l_{+}^{\mu}\nonumber\\ && \qquad \left . + \gamma^{AB}e_{A}^{\alpha}e_{B}^{\rho} l_{+}^{\nu} \left[ \frac{1}{2}\nabla_{\nu}a_{\alpha\rho}(\vec{\xi}\,) - \nabla_{\alpha}a_{\nu\rho}(\vec{\xi}\,)\right] + N W \right|_{\S}, \label{Z}\end{aligned}$$ where $W= \Pi _{AB}^{\mu}{\Pi^{\nu}}^{AB}{l_{+}}_{\mu}{l_{+}}_{\nu}+G_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$, and assume $Z \leq 0$ everywhere on $S$. - If $Z \neq 0$ somewhere, then $\xi_{\mu}l_{+}^{\mu} < 0$ everywhere. - If $S$ is strictly stable, then $\xi_{\mu}l_{+}^{\mu} \leq 0$ everywhere and vanishes at one point only if it vanishes everywhere. [**Proof.**]{} Consider the first variation of $S$ defined by the vector $\vec{\nu} = \vec{\xi} - N_S \vec{l}_{+} = Q \vec{m} + \vec{Y}^{\parallel}$. From equation (\[variationtheta+general0\]) and Definition \[defi:stabilityoperator\] we have $\delta_{\vec{\nu}}\, \theta^{+} = L_{\vec{m}} Q$. On the other hand, linearity of this variation under addition gives $\delta_{\vec{\nu}}\, \theta^{+} = \delta_{\vec{\xi}}\, \theta^{+} - \delta_{N_S \vec{l}_{+}} \theta^{+}$. The Raychaudhuri equation for MOTS establishes that $\delta_{N_S \vec{l}_{+}} \theta^{+} = - N_S W$ (see (\[raychaudhuri\]) and (\[W\])) and the identity (\[xitheta\]) gives $L_{\vec{m}} Q = Z$. Since $Q = \xi_{\mu}l_{+}^{\mu}$, the result follows directly from Lemma \[lemmaelliptic2\]. $\hfill \blacksquare$\ [**Remark.**]{} The theorem also holds if all the inequalities are reversed. This follows directly by replacing $\vec{\xi} \rightarrow - \vec{\xi}$. $\hfill \square$\ This theorem gives information about the relative position between the generator $\vec{\xi}$ and the outer null normal $\vec{l}_{+}$ and has, in principle, many potential consequences. Specific applications require considering spacetimes having special vector fields for which sufficient information about its deformation tensor is available. Once such a vector is known to exist, the result above can be used either to restrict the form of $\vec{\xi}$ in stable or strictly stable MOTS or, alternatively, to restrict the regions of the spacetime where such MOTS are allowed to be present. Since conformal vector fields (and homotheties and isometries as particular cases) have very special deformation tensors, the theorem above gives interesting information for spacetimes admitting such symmetries. \[thrstable\] Let $\S$ be a stable MOTS in a hypersurface $\Sigma$ of a spacetime $(M,\gM)$ which admits a conformal Killing vector $\vec{\xi}$, $\mathcal{L}_{\vec{\xi}} \gM_{\mu\nu} = 2 \phi \gM_{\mu\nu}$ (including homotheties $\phi=C$, and isometries $\phi=0$). - If $2 \vec{l}_{+} (\phi) + N W |_{S} \leq 0$ and not identically zero, then $\xi_{\mu}l_{+}^{\mu} |_S <0$. - If $S$ is strictly stable and $2 \vec{l}_{+} (\phi) + N W |_{S} \leq 0$ then $\xi_{\mu}l_{+}^{\mu} |_S \leq 0$ and vanishes at one point only if it vanishes everywhere [**Remark 1.**]{} As before, the theorem is still true if all inequalities are reversed. $\hspace*{1cm} \hfill \square$\ [**Remark 2.**]{} In the case of homotheties and Killing vectors, the condition of the theorem demands that $N_S W \leq 0$. Under the NEC, this holds provided $N_S \leq 0$, i.e. when $\vec{\xi}$ points below $\Sigma$ everywhere on $S$ (where the term “below” includes also the tangential directions). For strictly stable $S$, the conclusion of the theorem is that the homothety or the Killing vector must lie above the null hyperplane defined by the tangent space of $\S$ and the outer null normal $\vec{l}_{+}$ at each point $\p \in S$. If the MOTS is only assumed to be stable, then the theorem requires the extra condition that $\vec{\xi}$ points strictly below $\Sigma$ at some point with $W \neq 0$. In this case, the conclusion is stronger and forces $\vec{\xi}$ to lie strictly above the null hyperplane everywhere. By changing the orientation of $\vec{\xi}$, it is clear that similar restrictions arise when $\vec{\xi}$ is assumed to point [*above*]{} $\Sigma$. Figure \[fig:fig1\] summarizes the allowed and forbidden regions for $\vec{\xi}$ in this case. $\hfill \square$\ [**Proof.**]{} We only need to show that $Z = 2 \vec{l}_{+} (\phi) + N W |_{S}$ for conformal Killing vectors. This follows at once from (\[Z\]) and $a_{\mu\nu}(\vec{\xi}\,) = 2 \phi \gM_{\mu\nu}$ after using orthogonality of $\vec{e}_A$ and $\vec{l}_{+}$. Notice in particular that $Z$ is the same for isometries and for homotheties. $\hfill \blacksquare$\ ![The planes $T_{\p} \Sigma$ and $P\equiv T_{\p} S\oplus \mbox{span}\{ \vec{l}_{+}\, |_{\p} \}$ divide the tangent space $T_{\p}M$ in four regions. By Corollary \[thrstable\], if $S$ is strictly stable and $\vec{\xi}$ is a Killing vector or a homothety in a spacetime satisfying the NEC which points above $\Sigma$ everywhere, then $\vec{\xi}$ cannot enter into the forbidden region at any point (and similarly, if $\vec{\xi}$ points below $\Sigma$ everywhere). The allowed region includes the plane $P$. However, if there is a point with $W \neq 0$ where $\vec{\xi}$ is not tangent to $\Sigma$, then the result is also valid for stable MOTS with $P$ belonging to the forbidden region.[]{data-label="fig:fig1"}](fig1.eps){width="9cm"} This corollary has an interesting consequence in spacetime regions where there exists a Killing vector or a homothety $\vec{\xi}$ which is causal everywhere. \[shear\] Let a spacetime $(M,\gM)$ satisfying the NEC admit a causal Killing vector or homothety $\vec{\xi}$ which is future (or past) directed everywhere on a stable MOTS $\S \subset \Sigma$. Then, - The second fundamental form $\Pi^{+}_{AB}$ along $\vec{l}_{+}$ (i.e. $\Pi^{+}_{AB}\equiv \Pi_{AB}^{\mu}{l_{+}}_{\mu}$) and $\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$ vanish identically on every point $\p \in S$ where $\vec{\xi}|_{\p} \neq 0$. - If $S$ is strictly stable, then $\vec{\xi} \propto \vec{l}_{+}$ everywhere. [**Remark.**]{} If we assume that there exists an open neighbourhood of $\S$ in $M$ where the Killing vector or homothety $\vec{\xi}$ is causal and future (or past) directed everywhere then the conclusion (i) can be generalized to say that $\Pi^{+}_{AB}$ and $G_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$ vanish identically on $S$. The reason is that such a $\vec{\xi}$ cannot vanish anywhere in this neighbourhood (and consequently neither on $S$). For Killing vectors this result is proven in Lemma 3.2 in [@BEM][^1]. A simple generalization shows that the same holds for homotheties, as follows. Suppose that $\vec{\xi}\,|_{\p\in S}=0$. Take a timelike affine-parametrized geodesic $\gamma$ passing through $\p$ with future directed unit tangent vector $\vec{v}$. A simple computation gives that, if $\vec{\xi}$ is a homothety with constant $C$, $v^{\mu}\nabla_{\mu}(\xi_{\nu}v^{\nu})=-C$. Supposing $C>0$, this implies that the causal vector $\vec{\xi}$ is future directed on the future of $\p$ and past directed on the past of $\p$ contradicting the fact that $\vec{\xi}$ is future (past) directed everywhere on a neighbourhood of $\S$ in $M$. A similar argument works if $C<0$. Point (ii) can be generalized to locally outermost MOTS using a finite construction. We will prove this in Theorem \[corollaryextended\] below. $\hfill \square$\ [**Proof.**]{} We can assume, after reversing the sign of $\vec{\xi}$ if necessary, that $\vec{\xi}$ is past directed, i.e. $N_S \leq 0$. Under the NEC, $W$ is the sum of two non-negative terms, so in order to prove (i) we only need to show that $W =0$ on points where $\vec{\xi} \neq 0$, i.e. at points where $N_S <0$. Assume, on the contrary, that $W \neq 0$ and $N_S <0$ happen simultaneously at a point $\p \in S$. It follows that $N_S W \leq 0$ everywhere and non-zero at $\p$. Thus, we can apply statement (i) of Corollary \[thrstable\] to conclude $Q <0$ everywhere. Hence $N_S Q \geq 0$ and not identically zero on $S$. Recalling the decomposition $\vec{\xi} = N_S \vec{l}_{+} + Q \vec{m} + \vec{Y}^{\parallel}$, the squared norm of this vector is $$\begin{aligned} \label{normsquare} \xi_{\mu}\xi^{\mu} = 2N_{\S}Q+Q^2+ {{Y}^{\parallel}}_{\mu}{{Y}^{\parallel}}^{\mu}.\end{aligned}$$ This is the sum of non-negative terms, the first one not identically zero. This contradicts the condition of $\vec{\xi}$ being causal. To prove the second statement, we notice that point (ii) in Corollary \[thrstable\] implies $Q \leq 0$, and hence $N_S Q \geq 0$. The only way (\[normsquare\]) can be negative or zero is if $Q=0$ and $\vec{Y}^{\parallel} =0$, i.e. $\vec{\xi} \propto \vec{l}_{+}$. $\hfill \blacksquare$\ This corollary extends Theorem 2 in [@MS] to the case of stable MOTS and implies, for instance, that any strictly stable MOTS in a plane wave spacetime (which by definition admits a null and nowhere zero Killing vector field $\vec{\xi}\,$) must be aligned with the direction of propagation of the wave (in the sense that $\vec{\xi}$ must be one of the null normals to the surface). It also implies that any spacetime admitting a nowhere zero and causal Killing vector (or homothety) whose energy-momentum tensor satisfies the DEC and does not admit a null eigenvector cannot contain any stable MOTS. This is because $\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}=0$ and the DEC implies $\GM_{\mu\nu}l_{+}^{\mu}\propto l_{\nu}$ and $\GM_{\mu\nu}$ would have a null eigenvector. For perfect fluids this result holds even without the DEC provided $\mu+ p\neq 0$ (this is because in this case $\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}=(\mu+p)(l_{+}^{\mu}u_{\mu})^{2}\neq 0$ – where $\mu$ is the density, $p$ the pressure and $\vec{u}$ is the 4-velocity of the fluid–). The results above hold for stable or strictly stable MOTS. Among such surfaces, marginally trapped surfaces are of special interest. Our next result restricts (and in some cases forbids) the existence of such surfaces in spacetimes admitting Killing vectors, homotheties or conformal Killings. \[thrfirstvariation\] Let $\S$ be a stable MOTS in a spacelike hypersurface $\Sigma$ of a spacetime $(M,\gM)$ which satisfies the NEC and admits a conformal Killing vector $\vec{\xi}$ with conformal factor $\phi \geq 0$ (including homotheties with $C\geq0$ and Killing vectors). Suppose furthermore that either (i) $( 2\vec{l}_{+}(\phi)+NW ) |_S \not \equiv 0$ or (ii) $S$ is strictly stable and $\xi_{\mu}l_{+}^{\mu}|_S \not \equiv 0$. Then the following holds. - If $ 2\vec{l}_{+}(\phi)+NW |_S \leq 0$ then $S$ cannot be a marginally trapped surface, unless $\vec{H} \equiv 0$. The latter case is excluded if $\phi|_S \not \equiv 0$. - If $ 2\vec{l}_{+}(\phi)+NW |_S \geq 0$ then $S$ cannot be a past marginally trapped surface, unless $\vec{H} \equiv 0$. The latter case is excluded if $\phi|_S \not \equiv 0$. [**Remark.**]{} The statement obtained from this one by reversing all the inequalities is also true. This is a direct consequence of the freedom in changing $\vec{\xi} \rightarrow - \vec{\xi}$. $\hfill \square$\ [**Proof.**]{} We will only prove case (a). The argument for case (b) is similar. The idea is taken from [@MS] and consists of performing a variation of $S$ along the conformal Killing vector and evaluating the change of area in order to get a contradiction if $S$ is marginally trapped. The difference is that here we do not make any a priori assumption on the causal character for $\vec{\xi}$. Corollary \[thrstable\] provides us with sufficient information for the argument to go through. The first variation of area (\[firstvariation\]) gives $$\label{variationofareaMOTS} \delta_{\vec{\xi}\,} |S|=-\frac12\int_{S}\theta^{-} \xi_{\mu}l_{+}^{\mu} \eta_{\S},$$ where we have used $\vec{H} = - \frac{1}{2} \theta^{-} \vec{l}_{+}$. Now, since $2\vec{l}_{+}(\phi)+NW |_S \leq 0$, and furthermore either hypothesis (i) or (ii) holds, Corollary \[thrstable\] implies that $\xi_{\mu}{l}_{+}^{\mu} |_S < 0$. On the other hand, $\vec{\xi}$ being a conformal Killing vector, the induced metric on $S^{\prime}_{\tau}$ is related to the metric on $S$ by conformal rescaling. A simple computation gives $\delta_{\vec{\xi}\,}{\eta_{S}}= \frac12 \gamma^{AB}(\mathcal{L}_{\vec{\xi}\,}g)(\vec{e}_{A},\vec{e}_{B})\eta_{S}$ (see e.g. [@MS]), which for the particular case of conformal Killing vectors gives the following. $$\label{variationofareaconformalKilling} \delta_{\vec{\xi}\,} |S|=2\int_S \phi\eta_S,$$ This quantity is non-negative due to $\phi\geq 0$ and not identically zero if $\phi \neq 0 $ somewhere. Combining (\[variationofareaMOTS\]) and (\[variationofareaconformalKilling\]) we conclude that if $\theta^{-} \leq 0 $ (i.e. $S$ is marginally trapped) then necessarily $\theta^{-}$ vanishes identically (and so does $\vec{H}$). Furthermore, if $\phi |_S$ is non-zero somewhere, then $\theta^{-}$ must necessarily be positive somewhere, and $S$ cannot be marginally trapped. $\hfill \blacksquare$\ ### An application: No stable MOTS in Friedmann-Lemaître-Robertson-Walker spacetimes {#ssc:A2sectionFLRW} In this subsection we apply Corollary \[thrstable\] to show that a large subclass of Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes do not admit stable MOTS on [*any*]{} spacelike hypersurface. Obtaining this type of results for metric spheres only requires a straightforward calculation, and is therefore simple. The power of the method is that it provides a general result involving no assumption on the geometry of the MOTS or on the spacelike hypersurface where it is embedded. The only requirement is that the scale factor and its time derivative satisfy certain inequalities. This includes, for instance all FLRW cosmological models satisfying the NEC with accelerated expansion, as we shall see in Corollary \[corollaryFLRW\] below. Recall that the FLRW metric is $$\begin{aligned} \gM_{FLRW}=-dt^2+a^{2}(t)\left[ dr^2+\chi^{2}(r;k)d\Omega^{2} \right],\end{aligned}$$ where $a(t)>0$ is the scale factor and $\chi(r;k)= \{\sin{r}, r, \sinh r \}$ for $k=\{1,0,-1\}$, respectively. The Einstein tensor of this metric is of perfect fluid type and reads $$\begin{aligned} &&\GM_{\mu\nu} = (\mu + p) u_{\mu} u_{\nu} + p \gM_{\mu\nu}, \quad \vec{u} = \partial_t, \quad \mu = \frac{3(\dot{a}^2(t)+k)}{a^2(t)}, \label{EM_FLRW0}\\ &&\qquad\qquad \mu + p = 2\left ( \frac{\dot{a}^2(t)+k}{a^2(t)}- \frac{\ddot{a}(t)}{a(t)} \right) \label{EM_FLRW}\end{aligned}$$ where dot stands for derivative with respect to $t$. \[thrFRW\] There exists no stable MOTS in any spacelike hypersurface of a FLRW spacetime $(M,\gM_{FLRW})$ satisfying $$\label{conditionFLRW} \frac{{\dot{a}}^{2}(t)+k}{a(t)} \geq 0 , \quad -\frac{\dot{a}^{2}(t)+k}{a(t)}\leq \ddot{a}(t)\leq \frac{{\dot{a}}^{2}(t)+k}{a(t)}.$$ [**Remark.**]{} In terms of the energy-momentum contents of the spacetime, these three conditions read, respectively, $\mu \geq 0$, $\mu \geq 3 p$ and $\mu + p \geq 0$. As an example, in the absence of a cosmological constant they are satisfied as soon as the weak energy condition is imposed and the pressure is not too large (e.g. for the matter and radiation dominated eras). The class of FLRW satisfying (\[conditionFLRW\]) is clearly very large (c.f. Corollary \[corollaryFLRW\] below). We also remark that Theorem \[thrFRW\] agrees with the fact that the causal character of the hypersurface which separates the trapped from the non-trapped [*spheres*]{} in FLRW spacetimes depends precisely on the quantity $\mu^2(\mu+p)(\mu-3p)$ (c.f. [@S97]). $\hfill \square$\ [**Proof.**]{} The FLRW spacetime admits a conformal Killing vector $\vec{\xi}=a(t) \vec{u}$ with conformal factor $\phi=\dot{a}(t)$. Since this vector is timelike and future directed, it follows that $\xi_{\mu}l_{+}^{\mu}|_S<0$ for any spacelike surface $S$ embedded in a spacelike hypersurface $\Sigma$. If we can show that $2\vec{l}_{+}(\phi)+NW|_S \geq 0$, and non-identically zero for any $S$, then the sign reversed of point (i) in Corollary \[thrstable\] implies that $S$ cannot be a stable MOTS, thus proving the result. The proof therefore relies on finding conditions on the scale factor which imply the validity of this inequality on any $S$. First of all, we notice that the second fundamental form $\Pi^+_{AB}$ can be made as small as desired on a suitably chosen $S$. Thus, recalling that $W={\Pi^{+}}_{AB}{\Pi^{+}}^{AB}+\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$, it is clear that the inequality that needs to be satisfied is $$\label{conditioncorollary2} \left. 2\vec{l}_{+}(\phi)+N \GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}\right|_S \geq 0,$$ and positive somewhere. In order to evaluate this expression recall that $\vec{u} = a^{-1} \vec{\xi} = a(t)^{-1} N \vec{n} + a(t)^{-1} \vec{Y}$. Let us write $\vec{Y} = Y \vec{e}$, where $\vec{e}$ is unit and let $\alpha$ be the hyperbolic angle of $\vec{u}$ in the basis $\{\vec{n}, \vec{e}\, \}$, i.e. $\vec{u} = \cosh \alpha \, \vec{n} + \sinh \alpha \,\vec{e}$. It follows immediately that $N = a(t) \cosh \alpha$ and $Y = a(t) \sinh \alpha$. Furthermore, multiplying $\vec{u}$ by the normal vector to the surface $S$ in $\Sigma$ we find $u_{\mu}m^{\mu}= \cos \varphi \sinh \alpha$, where $\varphi$ is the angle between $\vec{m}$ and $\vec{e}$. With this notation, let us calculate the null vector $\vec{l}_{+}$. Writing $\vec{l}_{+} = A \vec{u} + \vec{b}$, with $\vec{b}$ orthogonal to $\vec{u}$, it follows $b_{\mu}b^{\mu} = A^2$ from the condition of $\vec{l}_{+}$ being null. On the other hand we have the decomposition $A \vec{u} + \vec{b} = \vec{l}_{+} = \vec{n} + \vec{m}$. Multiplying by $\vec{u}$ we immediately get $A = \cosh \alpha - \cos \varphi \sinh \alpha$, and, since $\phi = \dot{a}(t)$ only depends on $t$, $$\label{l(phi)FRW} \vec{l}_{+}(\phi)=\left( \cosh{\alpha}-\cos{\varphi}\sinh{\alpha} \right)\ddot{a}(t).$$ The following expression for $\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$ follows directly from $\vec{l}_{+} = A \vec{u} + \vec{b}$ and (\[EM\_FLRW0\]), (\[EM\_FLRW\]), $$\begin{aligned} \GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}&=&A^{2}(\mu+ p )\nonumber \\ \label{gll} &=& 2\left( \cosh{\alpha}-\cos{\varphi}\sinh{\alpha} \right)^2 \left ( \frac{\dot{a}^2(t)+k}{a^2(t)}-\frac{\ddot{a}(t)}{a(t)} \right).\end{aligned}$$ Inserting (\[l(phi)FRW\]) and (\[gll\]) into (\[conditioncorollary2\]) and dividing by $2 A^2 \cosh \alpha$ (which is positive) we find the equivalent condition $$\label{conditioncorollary2c} \left ( \frac{1}{\cosh{\alpha}\left(\cosh{\alpha}-\cos{\varphi}\sinh{\alpha}\right)}-1 \right )\ddot{a}(t)+\frac{\dot{a}^{2}(t)+k}{a(t)} \geq 0,$$ and non-zero somewhere. The dependence on $S$ only arises through the function $f(\alpha,\varphi) = \cosh \alpha ( \cosh \alpha - \cos \varphi \sinh \alpha)$. Rewriting this as $f = 1/2 ( 1 + \cosh (2 \alpha) - \cos \varphi \sinh (2\alpha) )$ it is immediate to show that $f$ takes all values in $(1/2, +\infty)$. Hence, $\left[ \cosh{\alpha}\left(\cosh{\alpha}-\cos{\varphi}\sinh{\alpha}\right)\right]^{-1} -1 $ takes all values between $-1$ and $1$. In order to satisfy (\[conditioncorollary2c\]) on all this range, it is necessary and sufficient that the three inequalities in (\[conditionFLRW\]) are satisfied. $\hfill \blacksquare$\ The following corollary gives a particularly interesting case where all the conditions of Theorem \[thrFRW\] are satisfied. \[corollaryFLRW\] Consider a FLRW spacetime $(M,\gM_{FLRW})$ satisfying the NEC. If $\ddot{a}(t)>0$, then there exists no stable MOTS in any spacelike hypersurface of $(M,\gM_{FLRW})$ [**Proof.**]{} The null energy condition gives $0 \leq \mu+p =2\left ( \frac{\dot{a}^2(t)+k}{a^2(t)} -\frac{\ddot{a}(t)}{a(t)} \right)$. This implies the first and third inequalities in (\[conditionFLRW\]) if $\ddot{a}>0$. The remaining condition $- \frac{\dot{a}^2(t)+k}{a(t)} \leq \ddot{a}$ is also obviously satisfied provided $\ddot{a} >0$. $\hfill \blacksquare$\ ### A consequence of the geometric construction of $S_{\tau}$ {#ssc:A2sectiongeometric} We have emphasized at the beginning of this section that the restrictions obtained directly by the geometric procedure of moving $S$ along $\vec{\xi}$ and then back to $\Sigma$ are intuitively clear but typically weaker than those obtained by using elliptic theory results. There are some cases, however, where the reverse actually holds, and the geometric construction provides stronger results. We will present one of these cases in this subsection. Corollary \[thrstable\] gives restrictions on ${\xi}_{\mu}{l}_{+}^{\mu} |_S$ for Killing vectors and homotheties in spacetimes satisfying the NEC, provided $\vec{\xi}$ is future or past directed everywhere. However, when $W$ vanishes identically, the result only gives useful information in the strictly stable case. The reason is that $W \equiv 0$ implies $L_{\vec{m}} Q \equiv 0$ and, for marginally stable MOTS (i.e. when the principal eigenvalue of $L_{\vec{m}}$ vanishes), the maximum principle is not strong enough to conclude that $Q$ must have a sign. There is at least one case where marginally stable MOTS play an important role, namely after a jump in the outermost MOTS in a (3+1) foliation of the spacetime (see [@AMMS] for details). As we will see next, the geometric construction does give restrictions in this case even when $W$ vanishes identically. \[thrkilling\] Consider a spacetime $(M,\gM)$ possessing a Killing vector or a homothety $\vec{\xi}$ and satisfying the NEC. Suppose $M$ contains a compact spacelike hypersurface $\Sigmatilde$ with boundary consisting in the disjoint union of a weakly outer trapped surface $\bd^{-} \Sigmatilde$ and an outer untrapped surface $\bd^{+} \Sigmatilde$ (neither of which are necessarily connected) and take $\bd^{+}\Sigmatilde$ as a barrier with interior $\Sigmatilde$. Without loss of generality, assume that $\Sigmatilde$ is defined locally by a level function $T=0$ with $T>0$ to the future of $\Sigmatilde$ and let $S$ be the outermost MOTS which is bounding with respect to $\bd^{+}\Sigmatilde$. If $\vec{\xi}(T) \leq 0 $ on some spacetime neighbourhood of $S$, then $\xi^{\mu}{l}_{\mu}^{+} \leq 0$ everywhere on $\S$. [**Remark 1.**]{} As usual, the theorem still holds if all the inequalities involving $\vec{\xi}$ are reversed. $\hfill \square$\ [**Remark 2.**]{} The simplest way to ensure that $\vec{\xi}(T)\leq 0 $ on some neighbourhood of $S$ is by imposing a condition merely on $\S$, namely ${\xi}_{\mu}{n}^{\mu} |_S > 0 $, because then $\vec{\xi}$ lies strictly below $\Sigmatilde$ on $S$ and this property is obviously preserved sufficiently near $S$ (i.e. $\vec{\xi}$ points strictly below the level set of $T$ on a sufficiently small spacetime neighbourhood of $S$). We prefer imposing directly the condition $\vec{\xi}(T)\leq 0$ on a spacetime neighbourhood of $S$ because this allows $\vec{\xi}\,|_{S}$ to be tangent to $\Sigma$. $\hfill \square$\ [**Proof.**]{} First note that the hypersurface $\Sigmatilde$ satisfies the assumptions of Theorem \[thr:AM\] which ensures that an outermost MOTS $S$ which is bounding with respect to $\bd^{+}\Sigmatilde$ does exist and, therefore, no weakly outer trapped surface can penetrate in its exterior region. Then, the idea is precisely to use the geometric procedure described above to construct $S_{\tau}$ and use the fact that $S$ is the outermost bounding MOTS to conclude that $S_{\tau}$ (${\tau}>0$) cannot have points outside $S$. Here we move $S$ a small but finite amount $\tau$, in contrast to the elliptic results before, which only involved infinitesimal displacements. We want to have information on the sign of the outer expansion of $S_{\tau}$ in order to make sure that a weakly outer trapped surface forms. The first part of the displacement is along $\vec{\xi}$ and gives $S^{\prime}_{\tau}$. Let us first see that all these surfaces are MOTS. For Killing vectors, this follows at once from symmetry arguments. For homotheties we have the identity $$\label{noindependentterm} \delta_{\vec{\xi}}\,\theta^{+}= \left ( -\frac{1}{2} l_{-}^{\alpha}\mathscr{L}_{\vec{\xi}}\, {l_{+}^{\prime}}_{\alpha}(\tau) -2C \right )\theta^{+},$$ which follows directly from (\[xithetau\]) with $\vec{\eta} = \vec{l}_{+}$ after using $l_{+}^{\mu} \mathscr{L}_{\vec{\xi}} \, {l_{+}^{\prime}}_{\mu}(\tau) = \frac{1}{2} a_{\mu\nu}(\vec{\xi}\,) l_{+}^{\mu} l_{+}^{\nu} = 0$, see (\[null\]). Expression (\[noindependentterm\]) holds for each one of the surfaces $\{S'_{\tau}\}$, independently of them being MOTS or not. Since this variation vanishes on MOTS and the starting surface $S$ has this property, it follows that each surface $S^{\prime}_{\tau}$ ($\tau>0$) is also a MOTS. Moving back to $\Sigmatilde$ along the null hypersurface introduces, via the Raychaudhuri equation (\[raychaudhuri\]), a non-positive term $N_S W$ in the outer null expansion, provided the motion is to the future. Hence, $S_{\tau}$ for small but finite ${\tau}>0$ is a weakly outer trapped surface provided $\vec{\xi}$ moves to the past of $\Sigmatilde$. This is ensured if $\vec{\xi} (T) \leq 0$ near $S$, because $T$ cannot become positive for small enough $\tau$. On the other hand, since a point $\p \in S$ moves initially along the vector field $\nu = \vec{\xi} - N_S \vec{l}_{+} = Q \vec{m} + \vec{Y}^{\parallel}$, where $Q={\xi}_{\mu}l_{+}^{\mu}$ as usual, it follows that $Q>0$ somewhere implies (for small enough $\tau$) that the bounding weakly outer trapped surface $S_{\tau}$ has a portion lying strictly to the outside of $S$ which, due to Theorem \[thr:AM\] by Andersson and Metzger, is a contradiction to $S$ being the outermost bounding MOTS. Hence $Q\leq 0$ everywhere and the theorem is proven. $\hspace*{1cm} \hfill \blacksquare$\ It should be remarked that the assumption of $\vec{\xi}$ being a Killing vector or a homothety is important for this result. Trying to generalize it for instance to conformal Killings fails in general because then the right hand side of equation (\[noindependentterm\]) has an additional term $2\vec{l}_{+}(\phi)$, not proportional to $\theta^{+}$. This means that moving a MOTS along a conformal Killing does not lead to another MOTS in general. The method can however, still give useful information if $\vec{l}_{+} (\phi)$ has the appropriate sign, so that $S^{\prime}_{\tau}$ is in fact weakly outer trapped. We omit the details. An immediate consequence of the finite construction of $S_{\tau}$ is the extension of point (ii) of Corollary \[shear\] to locally outermost MOTS. \[corollaryextended\] Let $(M,\gM)$ be a spacetime satisfying NEC and admitting a causal Killing vector or homothety $\vec{\xi}$ which is future (past) directed on a locally outermost MOTS $\S \subset \Sigma$. Then $\vec{\xi}\propto \vec{l}_{+}$ everywhere on $\S$. [**Proof.**]{} As before, let $\Sigma$ be defined locally by a level function $T=0$ with $T>0$ to the future of $\Sigma$. Assume that $\vec{\xi}$ is past directed (the future directed case is similar). Then, the assumption $\vec{\xi}(T) \leq 0 $ on some spacetime neighbourhood of $S$ of Theorem \[thrkilling\] is automatically satisfied. Then we can use the finite construction therein to find a weakly outer trapped surface which, due to the fact that $\vec{\xi}$ is causal (and past directed), does not penetrate in the interior part of the two-sided neighbourhood of $\S$. In fact, this new trapped surface will have points strictly outside $\S$ if on some point of $\S$ $\vec{\xi}\not\propto \vec{l}_+$ which proves the result. $\hfill \blacksquare$\ Finally, Theorem \[corollaryextended\] together with Theorem \[thr:AM\] lead to the following result. \[theorem1\] Consider a spacelike hypersurface $\id$ possibly with boundary in a spacetime satisfying the NEC and possessing a Killing vector or a homothety $\vec{\xi}$ with squared norm $\xi_{\mu}\xi^{\mu}=-\lambda$. Assume that $\Sigma$ possesses a barrier $\Sb$ with interior $\Omegab$ which is outer untrapped with respect to the direction pointing outside of $\Omegab$. Consider any surface $S$ which is bounding with respect to $\Sb$. Let us denote by $\Omega$ the exterior of $S$ in $\Omegab$. If $S$ is weakly outer trapped and ${\Omega}\subset \{\lambda>0\}$, then $\lambda$ cannot be strictly positive on any point $\p\in S$. [**Remark.**]{} When [*weakly outer trapped surface*]{} is replaced by the stronger condition of being a [*weakly trapped surface with non-vanishing mean curvature*]{}, then this theorem can be proven by a simple argument based on the first variation of area [@MS]. In that case, the assumption of $S$ being bounding becomes unnecessary. It would be interesting to know if Theorem \[theorem1\] holds for arbitrary weakly outer trapped surfaces, not necessarily bounding. $\hfill \square$\ [**Proof.**]{} We argue by contradiction. Suppose a weakly outer trapped surface $S$ satisfying the assumptions of the theorem and with $\lambda>0$ at some point. Theorem \[thr:AM\] implies that an outermost MOTS $\tbd T^{+}$ which is bounding with respect to $\Sb$ exists in the closure of the exterior $\Omega$ of $S$ in $\Omegab$. In particular, $\tbd T^{+}$ is a locally outermost MOTS. The hypothesis $\Omega\subset\{\lambda>0\}$ implies that the vector $\vec{\xi}$ is causal everywhere on $\tbd T^{+}$, either future or past directed. Moreover, the fact that $\lambda>0$ on some point of $S$ implies that the Killing vector is timelike in some non-empty set of $\tbd T^{+}$, which contradicts Theorem \[corollaryextended\]. $\hfill \blacksquare$\ ![Theorem \[theorem1\] excludes the possibility pictured in this figure, where $S$ (in blue) is a weakly outer trapped surface which is bounding with respect to the outer trapped barrier $\Sb$. The grey (both light and dark) regions represent the region where $\lambda>0$. The dark grey region represents the interior of $S_{b}$, while the striped area corresponds to $\Omega$, which is the exterior of $S$ in $\Omega_{b}$.[]{data-label="fig:theorem1"}](theorem1.eps){width="10cm"} The following result is a particularization of Theorem \[theorem1\] to the case when the hypersurface $\Sigma$ possesses an asymptotically flat end. Let $\id$ be a spacelike hypersurface in a spacetime satisfying the NEC and possessing a Killing vector or homothety $\vec{\xi}$. Suppose that $\Sigma$ possesses an asymptotically flat end $\Sigma_{0}^{\infty}$. Consider any bounding surface $S$ (see Definition \[defi:bounding\]). Let us denote by $\Omega$ the exterior of $S$ in $\Sigma$. If $S$ is weakly outer trapped and ${\Omega}\subset{\{\lambda>0\}}$, then $\lambda$ cannot be strictly positive on any point $\p\in S$. [**Proof.**]{} The result is a direct consequence of Theorem \[theorem1\]. $\hfill \blacksquare$\ Two immediate corollaries follow. \[corollary\] Consider a spacelike hypersurface $\id$ in a spacetime satisfying the NEC and possessing a Killing vector or a homothety $\vec{\xi}$. Assume that $\Sigma$ has a selected asymptotically flat end $\Sigma_{0}^{\infty}$ and $\lambda>0$ everywhere on $\Sigma$. Then there exists no bounding weakly outer trapped surface in $\Sigma$. \[corollary2\] Let $\id$ be a spacelike hypersurface of the Minkowski spacetime. Then there exists no bounding weakly outer trapped surface in $\Sigma$. The second Corollary is obviously a particular case of the first one because the vector $\partial_t$ in Minkowskian coordinates is strictly stationary everywhere, in particular on $\Sigma$. The non-existence result of a bounding weakly outer trapped surface in a Cauchy surface of Minkowski spacetime is however, well-known as this spacetime is obviously regular predictable (see [@HE] for definition) and then the proof of Proposition $9.2.8$ in [@HE] gives the result. So far, all the results we have obtained require that the quantity $L_{m}Q$ does not change sign on the MOTS $\S$. In the next section we will relax this condition. Results regardless of the sign of $L_{\vec{m}} Q$ {#sc:A2sectionnonelliptic} ------------------------------------------------- When $L_{\vec{m}} Q$ changes sign on $\S$, the elliptic methods exploited in the previous section lose their power. Moreover, for sufficiently small $\tau$, the surface $\S_{\tau}$ defined by the geometric construction above necessarily fails to be weakly outer trapped. Thus, obtaining restrictions in this case becomes a much harder problem. However, for locally outermost MOTS $S$, an interesting situation arises when $\S_{\tau}$ lies partially outside $S$ and happens to be weakly outer trapped in that exterior region. More precisely, if a connected component of the subset of $S_{\tau}$ which lies outside $S$ turns out to have non-positive outer null expansion, then using a smoothing result by Kriele and Hayward [@KH97], we will be able to construct a new weakly outer trapped surface outside $S$, thus leading to a contradiction with the fact that $S$ is locally outermost (or else giving restrictions on the generator $\vec{\xi}\, $). The result by Kriele and Hayward states, in rough terms, that given two surfaces which intersect on a curve, a new smooth surface can be constructed lying outside the previous ones in such a way that the outer null expansion does not increase in the process. The precise statement is as follows. \[lemasmoothness\] Let $S_{1},S_{2}\subset \Sigma$ be smooth two-sided surfaces which intersect transversely on a smooth curve $\gamma$. Suppose that the exterior regions of $S_1$ and $S_2$ are properly defined in $\Sigma$ and let $U_{1}$ and $U_{2}$ be respectively tubular neighbourhoods of $S_{1}$ and $S_{2}$ and $U^{-}_{1}$ and $U^{-}_{2}$ their interior parts. Assume it is possible to choose one connected component of each set $S_{1}\setminus \gamma$ and $S_{2}\setminus\gamma$, say $S_{1}^{+}$ and $S_{2}^{+}$ respectively, such that $S_{1}^{+}\cap U^{-}_{2}=\emptyset$ and $S^{+}_{2}\cap U^{-}_{1} =\emptyset$. Then, for any neighbourhood $V$ of $\gamma$ in $\Sigma$ there exists a smooth surface $\tilde{S}$ and a continuous and piecewise smooth bijection $\Phi\colon S_{1}^{+}\cup S_{2}^{+}\cup \gamma\rightarrow \tilde{S}$ such that 1. $\Phi(\p)=\p$, $\forall \p\in\left( S_{1}^{+} \cup S_{2}^{+}\right)\setminus V$ 2. $\left.{\theta}^{+}[\tilde{S}]\right|_{\Phi(\p)}\leq \left.{\theta}^{+}[S_{A}^{+}]\right|_{\p}$ $\forall \p\in S_{A}^{+}$ ($A=1,2$). Moreover $\tilde{S}$ lies in the connected component of $V\setminus\left( S^{+}_1\cup S^{+}_2\cup\gamma \right)$ lying in the exterior regions of both $S_1$ and $S_2$. ![The figure represents the two surfaces $S_{1}$ and $S_{2}$ which intersects in a curve $\gamma$, (where one dimension has been suppressed). The two intersecting grey regions are the tubular neighbourhoods $U_1$ and $U_2$ and, inside them, the stripped regions represents their interior parts, $U_{1}^{-}$ and $U_{2}^{-}$. The sets $S_{1}^{+}$ and $S_{2}^{+}$, in blue color, are then taken to be the connected components of $S_{1}\setminus \gamma$ and $S_{2}\setminus \gamma$ which do not intersect $U_{2}^{-}$ and $U_{1}^{-}$, respectively. Finally, the red line represents the smooth surface $\tilde{S}$ which has smaller $\theta^{+}$ than $S_{1}$ and $S_{2}$.. []{data-label="fig:KH"}](KH.eps){width="10cm"} [**Remark.**]{} It is important to emphasize that the statement of this result is slightly different from the one appearing in the original paper [@KH97] by Kriele and Hayward. Indeed, the assumptions made in [@KH97] are rather ambiguous and restrictive in the sense that the outer normals of $S_{1}$ and $S_{2}$ are required to form an angle (defined only by a figure), not smaller than 90 degrees. This condition is not necessary for the lemma to work. This result also appears quoted in [@AM] where the assumptions are wrongly formulated (although the result is properly used throughout the paper). In our paper [@CM2], where Lemma \[lemasmoothness\] is also formulated, the hypotheses are incomplete as well. $\hfill \square$\ This result will allow us to adapt the arguments above without having to assume that $L_{\vec{m}} Q$ has a constant sign on $S$. The argument will be again by contradiction, i.e. we will assume a locally outermost MOTS $S$ and, under suitable circumstances, we will be able to find a new weakly outer trapped surface lying outside $S$. Since the conditions are much weaker than in the previous section, the conclusion is also weaker. It is, however, fully general in the sense that it holds for any vector field $\vec{\xi}$ on $S$. Recall that $Z$ is defined in equation (\[Z\]). \[thrnonelliptic\] Let $S$ be a locally outermost MOTS in a spacelike hypersurface $\Sigma$ of a spacetime $(M,\gM)$. Denote by $\U_0$ a connected component of the set $\{\p\in\S ; \xi_{\mu}{l}_{+}^{\mu} |_{\p}>0 \}$. Assume $U_0\neq \emptyset$ and that its boundary $\gamma\equiv\tbd U_0$ is either empty, or it satisfies that the function $\xi_{\mu}{l}_{+}^{\mu}$ has a non-zero gradient everywhere on $\gamma$, i.e. $d(\xi_{\mu}{l}_{+}^{\mu}) |_{\gamma} \neq 0$. Then, there exists a point $\p\in\overline{U_0}$ such that $\left.Z\right|_{\p}\geq 0$. [**Proof.**]{} As mentioned, we will use a contradiction argument. Let us therefore assume that $$\label{conditionsubset} Z|_{\p} <0, \quad \forall {\p} \in \overline{\U_0}.$$ The aim is to construct a weakly outer trapped surface near $S$ and outside of it. This will contradict the condition of $S$ being locally outermost. First of all we observe that $Z$ cannot be negative everywhere on $S$, because then Theorem \[TrhAnyXi\] (recall that outermost MOTS are always stable) would imply $Q\equiv(\xi_{\mu}{l}_{+}^{\mu}) <0$ everywhere and $U_0$ would be empty against hypothesis. Consequently, under (\[conditionsubset\]), $U_0$ cannot coincide with $S$ and $\gamma\equiv \tbd \U_0 \neq \emptyset$. Since $\left.Q\right|_{\gamma}=0$ and, by assumption, $\left. dQ \right|_{\gamma}\neq 0$ it follows that $\gamma$ is a smooth embedded curve. Taking $\mu$ to be a local coordinate on $\gamma$, it is clear that $\{\mu,Q\}$ are coordinates of a neighbourhood of $\gamma$ in $S$. We will coordinate a small enough neighbourhood of $\gamma$ in $\Sigma$ by Gaussian coordinates $\{ u,\mu,Q\}$ such that $u=0$ on $S$ and $u>0$ on its exterior. By moving $S$ along $\vec{\xi}$ a finite but small parametric amount ${\tau}$ and back to $\Sigma$ with the outer null geodesics, as described in Section \[sc:A2sectionbasics\], we construct a family of surfaces $\{ S_{\tau} \}_{\tau}$. The curve that each point $\p \in S$ describes via this construction has tangent vector $\vec{\nu} = Q\vec{m}+\vec{Y}^{\parallel} |_{\S}$ on $S$. In a small neighbourhood of $\gamma$, the normal component of this vector, i.e. $Q \vec{m}$, is smooth and only vanishes on $\gamma$. This implies that for small enough $\tau$, $S_{\tau}$ are graphs over $S$ near $\gamma$. We will always work on this neighbourhood, or suitable restrictions thereof. In the Gaussian coordinates above, this graph is of the form $\{ u=\hat{u}(\mu,Q,\tau),\mu,Q \}$. Since the normal unit vector to $S$ is simply $\vec{m} = \partial_u$ in these coordinates and the normal component of $\vec{\nu}$ is $Q \vec{m}$, the graph function $\hat{u}$ has the following Taylor expansion $$\label{graph} \hat{u}(\mu,Q,\tau)=Q\tau+O(\tau^2).$$ Our next aim is to use this expansion to conclude that the intersection of $S$ and $S_\tau$ near $\gamma$ is an embedded curve $\gamma_\tau$ for all small enough $\tau$. To do that we will apply the implicit function theorem for functions to the equation $\hat{u}=0$. It is useful to introduce a new function $v(\mu,Q,\tau)=\frac{\hat{u}(\mu,Q,\tau)}{\tau}$, which is still smooth (thanks to (\[graph\])) and vanishes at $\tau=0$ only on the curve $\gamma$. Moreover, its derivative with respect to $Q$ is nowhere zero on $\gamma$, in fact $\left.\frac{\partial v}{\partial Q}\right|_{(\mu,0,0)}=1$ for all $\mu$. The implicit function theorem implies that there exist a unique function $Q=\varphi(\mu,\tau)$ which solves the equation $v(\mu,Q,\tau)=0$, for small enough $\tau$. Obviously, this function is also the unique solution near $\gamma$ of $\hat{u}(\mu,Q,\tau)=0$ for $\tau>0$. Consequently, the intersection of $S$ and $S_{\tau}$ ($\tau>0$) lying in the neighbourhood of $\gamma$ where we are working on is an embedded curve $\gamma_{\tau}$. Since $\gamma$ separates $S$ into two or more connected components, the same is true for $\gamma_{\tau}$ for small enough $\tau$ (note that $\gamma$ need not be connected and the number of connected components of $S\setminus \gamma$ may be bigger than two). Recall that $\gamma$ is the boundary of a connected set $\U_0$. Hence, by construction, there is only one connected component of $S_{\tau}\setminus \gamma_{\tau}$ which has $v (\mu,Q,{\tau}) >0$ near $\gamma$ (i.e. that lies in the exterior of $S$ near $\gamma$). Let us denote it by $S_{\tau}^{+}$. $S^{+}_{\tau}$ in fact lies fully outside of $S$, not just in a neighborhood of $\gamma$, as we see next. First of all, note that $Q>0$ on $\U_{0}$. We have just seen that $\gamma_{\tau}$ is a continuous deformation of $\gamma$. Let us denote by $U_{\tau}$ the domain in $S$ obtained by deforming $U_0$ when the boundary moves from $\gamma$ to $\gamma_{\tau}$ (See Figure \[fig:St\]). It is obvious that $S_{\tau}^+$ is obtained by moving $U_{\tau}$ first along $\vec{\xi}$ an amount $\tau$ and then back to $\Sigma$ by null hypersurfaces. The closed subset of $U_{\tau}$ lying outside the tubular neighbourhood where we applied the implicit function theorem is, by construction a proper subset of $U _0$. Consequently, on this closed set $Q$ is uniformly bounded below by a positive constant. Given that $Q$ is the first order term of the normal variation, all these points move outside of $S$. This proves that $S^+_{\tau}$ is fully outside $S$ for sufficiently small $\tau$. Incidentally this also shows that $S^+_{\tau}$ is a graph over $U_{\tau}$. ![The figure represents both intersecting surfaces $S$ and $S^{+}_{\tau}$ together with the curves $\gamma$ and $\gamma_{\tau}$. The shaded region corresponds to $U_0$ and the stripped region to $U_{\tau}$.[]{data-label="fig:St"}](St.eps){width="10cm"} The next aim is to show that the outer null expansion of $\S_{\tau}$ is non-positive everywhere on $\S^{+}_{\tau}$. To that aim, we will prove that, for small enough $\tau$, $Z$ is strictly negative everywhere on $U_{\tau}$. Since $Z$ is the first order term in the variation of $\theta^{+}$, this implies that the outer null expansion of $S^+_{\tau}$ satisfies $\theta^+[S^{+}_{\tau}] <0$ for $\tau>0$ small enough. By assumption (\[conditionsubset\]), $Z$ is strictly negative on $U_0$. Therefore, this quantity is automatically negative in the portion of $U_{\tau}$ lying in $U_0$ (in particular, outside the tubular neighbourhood where we applied the implicit function theorem). The only difficulty comes from the fact that $\gamma_{\tau}$ may move outside $U_0$ at some points and we only have information on the sign of $Z$ on $\overline{U_0}$. To address this issue, we first notice that $Q$ defines a distance function to $\gamma$ (because $Q$ vanishes on $\gamma$ and its gradient is nowhere zero). Consequently, the fact that $Z$ is strictly negative on $\gamma$ (by assumption (\[conditionsubset\])) and that this curve is compact imply that there exists a $\delta>0$ such that, inside the tubular neighbourhood of $\gamma$, $|Q|<\delta$ implies $Z <0$. Moreover, the function $Q = \varphi(\mu,\tau)$, which defines $\gamma_{\tau}$, is such that it vanishes at $\tau=0$ and depends smoothly on $\tau$. Since $\mu$ takes values on a compact set, it follows that for each $\delta^{\prime}>0$, there exists an $\epsilon(\delta^{\prime})>0$, independent of $\mu$ such that $|\tau| < \epsilon(\delta^{\prime})$ implies $|Q| = |\varphi(\mu,\tau)| < \delta^{\prime}$. By taking $\delta^{\prime} = \delta$, it follows that, for $|\tau| < \epsilon (\delta)$, $U_{\tau}$ is contained in a $\delta$-neighbourhood of $U_0$ (with respect to the distance function $Q$) and consequently $Z<0$ on this set, as claimed. We restrict to $0 < \tau < \epsilon(\delta)$ from now on. Summarizing, so far we have shown that $\S_{\tau}^{+}$ lies fully outside $\S$ and has $\theta^{+}[S_{\tau}^{+}]<0$. The final task is to use Lemma \[lemasmoothness\] to construct a weakly outer trapped surface strictly outside $\S$. Denote by $S_{\tau}^{*}$ the complement of $U_{\tau}$ in $S$, which may have several connected components. For any connected component $\gamma_{\tau}^{i}$ of $\gamma_{\tau}$ there exists a neighbourhood $W^{*}_{\tau ,i}$ of $\gamma_{\tau}^{i}$ in $S_{\tau}^{*}\subset S$ which lies in the exterior of $S_{\tau}$ (because the intersection between $S$ and $S_{\tau}$ is transverse). Similarly, there is a connected neighbourhood $W^{+}_{\tau ,i}$ of $\gamma_{\tau}^{i}$ in $S_{\tau}^{+}\subset S_{\tau}$ which lies in the exterior of $S$. The smoothing argument of Lemma \[lemasmoothness\] can be therefore applied locally on each union $W^{*}_{\tau, i}\cup \gamma_{\tau}^{i}\cup W^{+}_{\tau,i}$ to produce a weakly outer trapped surface $\tilde{S}$ which lies outside $S$, leading a contradiction. This surface $\tilde{S}$ is constructed in such a way that $\tilde{S}= S^{*}_{\tau}$ in $S^{*}_{\tau}\setminus \left(\underset{i}{\cup}W^{*}_{\tau, i}\right)$ and $\tilde{S}=S^{+}_{\tau}$ in $S^{+}_{\tau}\setminus \left(\underset{i}{\cup}W^{+}_{\tau, i}\right)$. $\hfill \blacksquare$\ [**Remark.**]{} As usual, this theorem also holds if all the inequalities are reversed. Note that in this case $U_0$ is defined to be a connected component of the set $\{\p\in\S; (\xi_{\mu}{l}_{+}^{\mu}) |_{\p}<0 \}$. For the proof simply take $\tau<0$ instead of $\tau>0$ (or equivalently move along $-\vec{\xi}$ instead of $\vec{ \xi}$). $\hfill \square$\ Similarly as in the previous section, this theorem can be particularized to the case of conformal Killing vectors, as follows (recall that $Z=2\vec{l}_{+}(\phi)+NW$ in the conformal Killing case, see Corollary \[thrstable\]). \[corollarynonelliptic\] Under the assumptions of Theorem \[thrnonelliptic\], suppose that $\ \vec{\xi}$ is a conformal Killing vector with conformal factor $\phi$ (including homotheties $\phi=C$ and isometries $\phi=0$). Then, there exists $\p\in\overline{U_0}$ such that $2\vec{l}_{+}(\phi)+N_{S}({\Pi^{+}_{AB}}{\Pi^{+}}^{AB}+\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}) |_{\p}\geq 0$. If the conformal Killing is in fact a homothety or a Killing vector and it is causal everywhere, the result can be strengthened considerably. The next result extends Corollary \[shear\] in a suitable sense to the cases when the generator is not assumed to be either future or past everywhere. Since its proof requires an extra ingredient we write it down as a theorem. \[shear2\] In a spacetime $(M,\gM)$ satisfying the NEC and admitting a Killing vector or homothety $\vec{\xi}$, consider a locally outermost MOTS $\S$ in a spacelike hypersurface $\Sigma$. Assume that $\vec{\xi}$ is causal on $S$ and that $W={\Pi^{+}_{AB}}{\Pi^{+}}^{AB}+\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$ is non-zero everywhere on $\S$. Define $U\equiv\{\p\in\S ; (\xi_{\mu}{l}_{+}^{\mu})|_{\p}>0\}$ and assume that this set is neither empty nor covers all of $S$. Then, on each connected component $U_{\alpha}$ of $U$ there exist a point $\p\in\tbd U_{\alpha}$ with $d(\xi_{\mu}{l}_{+}^{\mu})|_{\p}=0$. [**Remark 1.**]{} The same conclusion holds on the boundary of each connected components of the set $\{ \p\in\S ; (\xi_{\mu}{l}_{+}^{\mu})|_{\p}<0\}$. This is obvious since $\vec{\xi}$ can be changed to $-\vec{\xi}$. $\hfill \square$\ [**Remark 2.**]{} The case $\tbd U=\emptyset$, excluded by assumption in this theorem, can only occur if $\vec{\xi}$ is future or past everywhere on $S$. Hence, this case is already included in Corollary \[shear\]. $\hfill \square$\ [**Proof.**]{} We first show that on any point in $U$ we have $N_S < 0$, which has as an immediate consequence that $N_S \leq 0$ on any point in $\overline{U}$. The former statement is a consequence of the decomposition $\vec{\xi}=N\vec{l}_{+}+Q\vec{m}+\vec{Y}^{\parallel}$, where $Q=(\xi_{\mu}{l}_{+}^{\mu})$. The condition that $\vec{\xi}$ is causal then implies $\xi_{\mu}\xi^{\mu}=2N_{\S}Q+Q^{2}+{Y^{\parallel}}^{2} \leq 0$. This can only happen at a point where $Q>0$ (i.e. on $U$) provided $N_S < 0$ there. Moreover, if at any point $\q$ on the boundary $\tbd U$ we have $N_S |_{\q}=0$, then necessarily the full vector $\vec{\xi}$ vanishes at this point. This implies, in particular, that the geometric construction of $S_{\tau}$ has the property that $\q$ remains invariant. Having noticed these facts, we will now argue by contradiction, i.e. we will assume that there exists a connected component $U_{0}$ of $U$ such that $d(\xi_{\mu}{l}_{+}^{\mu}) |_{\tbd U_0}\neq 0$ everywhere. In these circumstances, we can follow the same steps as in the proof of Theorem \[thrnonelliptic\] to show that, for small enough $\tau$ the surface $S_{\tau}$ has a portion $S^{+}_{\tau}$ lying in the exterior of $S$ and which, in the Gaussian coordinates above, is a graph over a subset $U_{\tau}$ which is a continuous deformation of $U_0$. Moreover, the boundary of $U_{\tau}$ is a smooth embedded curve $\gamma_{\tau}$. The only difficulty with this construction is that we cannot use $N_S W = Z <0$ everywhere on $\overline{U}_0$, in order to conclude that $\theta^{+}[S^{+}_{\tau}] < 0$, as we did before. The reason is that there may be points on $\tbd U_0$ where $N_S=0$. However, as already noted, these points have the property that [*do not move at all*]{} by the construction of $S_{\tau}$, i.e. the boundary $\gamma_{\tau}$ (which is the intersection of $S$ and $S^+_{\tau}$) can only move outside of $U_0$ at points where $N_S$ is strictly negative. Hence on the interior points of $U_{\tau}$ we have $N_S <0$ everywhere, for sufficiently small $\tau$. Consequently the first order terms in the variation of $\theta^{+}$, namely $Z = N_s W$, is strictly negative on all the interior points of $U_{\tau}$. This implies that $S^{+}_{\tau}$ has negative outer null expansion everywhere except possibly on its boundary $\gamma_{\tau}$. By continuity, we conclude $\theta^{+}[S_{\tau}^{+}] \leq 0$ everywhere. The smoothing argument of the proof of Theorem \[thrnonelliptic\] implies that a smooth weakly outer trapped surface can be constructed outside the locally outermost MOTS $S$. This gives a contradiction. Therefore, there exists $\p\in\tbd U_0$ such that $d(\xi_{\mu}{l}_{+}^{\mu})|_{\p}=0$, as claimed. $\hfill \blacksquare$\ [**Remark**]{} The assumption $\left. dQ \right|_{ \gamma}\neq 0$ is a technical requirement for using the smoothing argument of Lemma \[lemasmoothness\]. This is why we had to include an assumption on $d Q |_{\gamma}$ in Theorem \[thrnonelliptic\] and also that the conclusion of Theorem \[shear2\] is stated in terms of the existence of critical points for $Q$. If Lemma \[lemasmoothness\] could be strengthened so as to remove this requirement, then Theorem \[shear2\] could be rephrased as stating that any outermost MOTS in a region where there is a causal Killing vector (irrespective of its future or past character) must have at least one point where the shear and $\GM_{\mu\nu}l_{+}^{\mu}l_{+}^{\nu}$ vanish simultaneously. In any case, the existence of critical points for a function in the boundary of [*every*]{} connected component of $\{ Q >0 \}$ and [*every*]{} connected component of $\{ Q <0 \}$ is obviously a highly non-generic situation. So, locally outermost MOTS in regions where there is a causal Killing vector or homothety can at most occur under very exceptional circumstances. $\hfill \square$\ Weakly outer trapped surfaces in static spacetimes {#ch:Article1} ================================================== Introduction {#introduction} ------------ In the next two chapters we will concentrate on [*static*]{} spacetimes. As we have remarked in Chapter \[ch:Introduction\], one of the main aims of this thesis is to extend the uniqueness theorems for static black holes to static spacetimes containing MOTS. This chapter is devoted to obtaining a proper understanding of MOTS in static spacetimes, which will be essential to prove the uniqueness result in the next chapter. The first answer to the question of whether the uniqueness theorems for static black holes extend to static spacetimes containing MOTS was given by Miao in 2005 [@Miao], who proved uniqueness for the particular case of time-symmetric, asymptotically flat and vacuum spacelike hypersurfaces possessing a minimal compact boundary (note that in a time-symmetric slice compact minimal surfaces are MOTS and vice versa). This result generalized the classic uniqueness result of Bunting and Masood-ul-Alam [@BMuA] for vacuum static black holes which states the following. \[thr:BMuA\] Consider a vacuum spacetime $(M,\gM)$ with a static Killing vector $\vec{\xi}$. Assume that $(M,\gM)$ possesses a connected, asymptotically flat spacelike hypersurface $(\Sigma,g,K)$ which is time-symmetric (i.e. $K=0$, $\vec{\xi}\perp \Sigma$), has non-empty compact boundary $\bd \Sigma$ and is such that the static Killing vector $\vec{\xi}$ is causal on $\Sigma$ and null only on $\bd \Sigma$.\ Then $(\Sigma,g)$ is isometric to $\left(\mathbb{R}^{3}\setminus B_{{M_{Kr}}/2}(0),{(g_{Kr})}_{ij}=\left( 1+\frac{M_{Kr}}{2|x|} \right)^{4}\delta_{ij}\right)$ for some $M_{Kr}>0$, i.e. the $\{t=0\}$ slice of the Kruskal spacetime with mass $M_{Kr}$ outside and including the horizon. Moreover, there exists a neighbourhood of $\Sigma$ in $M$ which is isometrically diffeomorphic to the closure of the domain of outer communications of the Kruskal spacetime. In other words, this theorem asserts that a time-symmetric slice $\Sigma$ of a non-degenerate static vacuum black hole must be a time-symmetric slice of the Kruskal spacetime. Miao was able to reach the same conclusion under much weaker assumptions, namely by simply assuming that the boundary of $\Sigma$ is a closed minimal surface. As in Bunting and Masood-ul-Alam’s theorem, Miao’s result deals with time-symmetric and asymptotically flat spacelike hypersurfaces embedded in static vacuum spacetimes. More precisely, \[thr:Miao\] Consider a vacuum spacetime $(M,\gM)$ with a static Killing vector $\vec{\xi}$. Assume that $(M,\gM)$ possesses a connected, asymptotically flat spacelike hypersurface $(\Sigma,g,K)$ which is time-symmetric and such that $\bd\Sigma$ is a (non-empty) compact minimal surface.\ Then $(\Sigma,g)$ is isometric to $\left(\mathbb{R}^{3}\setminus B_{M_{Kr}/2}(0),{(g_{Kr})}_{ij}=\left( 1+\frac{M_{Kr}}{2|x|} \right)^{4} \delta_{ij}\right)$ for some $M_{Kr}>0$, i.e. the $\{t=0\}$ slice of the Kruskal spacetime with mass $M_{Kr}$ outside and including the horizon. Moreover, there exists a neighbourhood of $\Sigma$ in $M$ which is isometrically diffeomorphic to the closure of the domain of outer communications of the Kruskal spacetime. A key ingredient in Miao’s proof was to show that the existence of a closed minimal surface implies the existence of an asymptotically flat end $\Sigma^{\infty}$ with smooth topological boundary $\tbd \Sigma^{\infty}$ such that $\vec{\xi}$ is timelike on $\Sigma^{\infty}$ and vanishes on $\tbd \Sigma^{\infty}$. Miao then proved that $\tbd \Sigma^{\infty}$ coincides in fact with the minimal boundary $\bd\Sigma$ of the original manifold. Hence, the strategy was to reduce Theorem \[thr:Miao\] to the Bunting and Massod-ul-Alam uniqueness theorem of black holes. As a consequence of the static vacuum field equations the set of points where the Killing vector vanishes in a time-symmetric slice is known to be a totally geodesic surface. Totally geodesic surfaces are of course minimal and in this sense Theorem \[thr:Miao\] is a generalization of Theorem \[thr:BMuA\]. In fact, Theorem \[thr:BMuA\] allows us to rephrase Miao’s theorem as follows: [*No minimal surface can penetrate in the exterior region where the Killing vector is timelike in any time-symmetric and asymptotically flat slice of a static vacuum spacetime.*]{} In this sense, Miao’s result can be regarded as a confinement result for MOTS in time-symmetric slices of static vacuum spacetimes. Here, it is important to remark that a general confinement result of this type was already known when suitable global hypotheses in time are assumed in the spacetime. In this case, weakly outer trapped surfaces must lie inside the black hole region (see e.g. Proposition 12.2.4 in [@Wald]). Consequently, Theorem \[thr:Miao\] can also be viewed as an extension of this result to the initial data setting (which drops completely all global assumptions in time) for the particular case of time-symmetric, static vacuum slices. We aim to generalize Miao’s theorem in three different directions. Firstly, we want to allow for non-vanishing matter as long as the NEC is satisfied. Secondly, the slices will no longer be required to be time-symmetric. In this situation the natural replacement for minimal surfaces are MOTS. And finally, we intend to relax the condition of asymptotic flatness to just assuming the presence of an outer untrapped surface (of course, this will not be possible for the uniqueness theorem, but it is possible when viewing Miao’s result as a confinement result). The proof given by Miao relies strongly on the vacuum field equations, so we must resort to different methods. Obviously, a fundamental step for our purposes is a proper understanding of MOTS in static spacetimes. In this chapter we explore the properties of MOTS in static spacetimes. The main result of this chapter is Theorem \[theorem2\] which extends Theorem \[thr:Miao\] as a confinement result for MOTS by asserting that no MOTS which are bounding can [*penetrate*]{} into the exterior region where the static Killing is timelike provided some hypotheses hold. In fact, this result for MOTS also holds for weakly outer trapped surfaces. It is important to note that Theorem \[theorem1\] in the previous chapter already forbids the existence of weakly outer trapped surfaces whose exterior lies in the region where the Killing vector is timelike, and which penetrates into the timelike region (recall that the exterior of $S$ does not contain $S$, by definition). However, this result does not exclude the existence of a weakly outer trapped surface penetrating into the timelike region but not lying entirely in the causal region. This is the situation we exclude in Theorem \[theorem2\]. The essential ingredients to prove this result will be a combination of the ideas that allowed us to prove Theorem \[theorem1\] together with a detailed study of the properties of the boundary of the region where the static Killing is timelike. Besides a confinement result, Miao’s theorem is also (and fundamentally) a uniqueness theorem. The generalization of Miao’s result as a uniqueness result will be studied in the next chapter, where several of the results of the present chapter will be applied. As we remarked in the introductory chapter, a general tendency in investigations involving stationary and static spacetimes over the years has been to relax the global hypotheses in time and work at the initial data level as much as possible. Good examples of this fact are the statements of Theorems \[thr:BMuA\] and \[thr:Miao\] above, where the existence of a spacelike hypersurface with suitable properties is, in fact, sufficient for the proof. Following this trend, all the results of this chapter will be proved by working directly on spacelike hypersurfaces, with no need of invoking a spacetime containing them. These spacelike hypersurfaces, considered as abstract objects on their own, will be called [*initial data sets*]{}. Some of these results generalize known properties of static spacetimes to the initial data setting and, consequently, can be of independent interest. We finish this introduction with a brief summary of the chapter. In Section \[preliminaries\] we define initial data set as well as Killing initial data (KID). Then we introduce the so-called Killing form and give some of its properties. In Section \[staticityKID\] we discuss the implications of imposing staticity on a Killing initial data set and state a number of useful properties of the boundary of the set where the static Killing vector is timelike, which will be fundamental to prove Theorem \[theorem2\]. Some of the technical work required in this section is related to the fact that we are not a priori assuming the existence of a spacetime. Finally, Section \[mainresults\] is devoted to stating and proving Theorem \[theorem2\]. The results presented in this chapter have been published in [@CM1], [@CMere1]. Preliminaries {#preliminaries} ------------- ### Killing Initial Data (KID) {#sKID} We start with the standard definition of initial data set [@BC]. An **initial data set** $(\Sigma,g,K;\rho,{\bf J})$ is a 3-dimensional connected manifold $\Sigma$, possibly with boundary, endowed with a Riemannian metric $g$, a symmetric, rank-two tensor $K$, a scalar $\rho$ and a one-form $\bf{J}$ satisfying the so-called [*constraint equations*]{}, $$\begin{aligned} 2 \rho & = & \RSigma+ (\tr_{\,\Sigma} K)^{2}-K_{ij}K^{ij}, \\ - J_{i} & = & {\nablaSigma}_j({K_{i}}^{j}- \tr_{\,\Sigma} K \delta_{i}^{j}),\end{aligned}$$ where $\RSigma$ and $\nablaSigma$ are respectively the scalar curvature and the covariant derivative of $(\Sigma,g)$ and $\tr_{\,\Sigma} K= g^{ij}K_{ij}$. For simplicity, we will often write $\id$ instead of $\idfull$ when no confusion arises. In the framework of the Cauchy problem for the Einstein field equations, $\Sigma$ is a spacelike hypersurface of a spacetime $(M,\gM)$, $g$ is the induced metric and $K$ is the second fundamental form. The [**initial data energy density**]{} $\rho$ and [**energy flux**]{} ${\bf J}$ are defined by $\rho \equiv \GM_{\mu\nu}n^{\mu}n^{\nu}, J_{i} \equiv -\GM_{\mu\nu}n^{\mu}e_{i}^{\nu}$, where $\GM_{\mu\nu}$ is the Einstein tensor of $\gM$, $\vec{n}$ is the unit future directed vector normal to $\Sigma$ and $\{ \vec{e}_i \}$ is a local basis for $\mathfrak{X}(\Sigma)$. When $\rho=0$ and ${\bf J}=0$, the initial data set is said to be [**vacuum**]{}. As remarked in the previous section, we will regard initial data sets as abstract objects on their own, independently of the existence of a spacetime where they may be embedded, unless explicitly stated. Consider for a moment a spacetime $(M,\gM)$ possessing a Killing vector field $\vec{\xi}$ and let $\id$ be an initial data set in this spacetime. We can decompose $\vec{\xi}$ along $\Sigma$ into a normal and a tangential component as $$\label{killingdecomposition} \vec{\xi}=N\vec{n}+Y^{i}\vec{e}_{i}$$ (see Figure \[fig:XiNY\]), where $N = -\xi^{\mu}n_{\mu}$. Note that with this decomposition $$\lambda\equiv -\xi_{\mu}\xi^{\mu}=N^{2}-Y^{2}.$$ Inserting (\[killingdecomposition\]) into the Killing equations and performing a 3+1 splitting on $\id$ it follows (see [@Coll], [@BC]), $$\begin{aligned} 2NK_{ij} + 2\nablaSigma_{(i}Y_{j)}&=&0, \hspace{98mm} \label{kid1} \\ \mathcal{L}_{\vec{Y}}K_{ij} + \nablaSigma_{i}\nablaSigma_{j}N&=&N\left( \RSigma_{ij}+ \tr_{\,\Sigma} K K_{ij}-2K_{il}K_{j}^{l}\right. -\tau_{ij}\nonumber\\ &&\left.\qquad\qquad\qquad\qquad\qquad\qquad + \frac{1}{2}g_{ij}(\tr_{\,\Sigma} \tau-\rho) \right), \label{kid2}\end{aligned}$$ where the parentheses in (\[kid1\]) denotes symmetrization, $\tau_{ij} \equiv \GM_{\mu\nu}e_{i}^{\mu}e_{j}^{\nu}$ are the remaining components of the Einstein tensor and $\tr_{\,\Sigma} \tau=g^{ij}\tau_{ij}$. Thus, the following definition of Killing initial data becomes natural [@BC]. An initial data set $\idfull$ endowed with a scalar $N$, a vector $\vec{Y}$ and a symmetric tensor $\tau_{ij}$ satisfying equations (\[kid1\]) and (\[kid2\]) is called a **Killing initial data** *(KID)*. In particular, if a KID has $\rho=0$, ${\bf J}=0$ and $\tau=0$ then it is said to be a [**vacuum KID**]{}. A point $\p \in \Sigma$ where $N=0$ and $\vec{Y}=0$ is a [**fixed point**]{}. This name is motivated by the fact that when the KID is embedded into a spacetime with a local isometry, the corresponding Killing vector $\vec{\xi}$ vanishes at $\p$ and the isometry has a fixed point there. A natural question regarding KID is whether they can be embedded into a spacetime $(M,\gM)$ such that $N$ and $\vec{Y}$ correspond to a Killing vector $\vec{\xi}$. The simplest case where existence is guaranteed involves “transversal” KID, i.e. when $N\ne0$ everywhere. Then, the following spacetime, called **Killing development** of $(\Sigma,g,K)$, can be constructed $$\label{killingdevelopment} \left( \Sigma\times\mathbb{R},\quad g^{(4)}=-\hat{\lambda}dt^{2} +2\hat{Y}_{i}dtdx^{i}+ \hat{g}_{ij}dx^{i}dx^{j} \right)$$ where $$\label{killingdevelopment2} \hat{\lambda}(t,x^{i})\equiv (N^2 - Y^i Y_i )(x^{i}), \quad \hat{g}_{ij}(t,x^{k})\equiv g_{ij}(x^{k}), \quad \hat{Y}^{i}(t,x^{j})\equiv Y^{i}(x^{j}).$$ Notice that $\partial_{t}$ is a complete Killing field with orbits diffeomorphic to $\mathbb{R}$ which, when evaluated on $\Sigma \equiv \{t=0 \}$ decomposes as $\partial_{t}=N\vec{n}+Y^{i}\vec{e}_{i}$, in agreement with (\[killingdecomposition\]). The Killing development is the unique spacetime with these properties. Further details can be found in [@BC]. Notice also that the Killing development can be constructed for any connected subset of $\Sigma$ where $N \neq 0$ everywhere. We will finish this subsection by giving the definition of asymptotically flat KID, which is just the same as for asymptotically flat spacelike hypersurface [*but adding*]{} the suitable decays for the quantities $N$ and $\vec{Y}$. \[asymptoticallyflat1\] A KID $\kid$ is [**asymptotically flat**]{} if $\Sigma=\mathcal{K} \cup \Sigma^{\infty}$, where $\mathcal{K}$ is a compact set and $\Sigma^{\infty}=\underset{a}{\bigcup}\Sigma^{\infty}_{a}$ is a finite union with each $\Sigma^{\infty}_{a}$, called an asymptotic end, being diffeomorphic to $\mathbb{R}^3 \setminus \overline{B_{R_{a}}}$, where $B_{R_{a}}$ is an open ball of radius $R_{a}$. Moreover, in the Cartesian coordinates $\{ x^i \}$ induced by the diffeomorphism, the following decay holds $$\begin{aligned} N-A_{a} = O^{(2)}(1/r),\qquad g_{ij}-{\delta}_{ij}&=&O^{(2)}(1/r),\nonumber\\ Y^{i}-C^{i}_{a}=O^{(2)}(1/r), \qquad\qquad K_{ij}&=&O^{(2)}(1/r^{2}).\nonumber\end{aligned}$$ where $A_{a}$ and $\{C^{i}_{a} \}_{i=1,2,3}$ are constants such that $A^2_{a}-{\delta}_{ij} C^{i}_{a} C^{j}_{a}>0$ for each $a$, and $r=\left(x^{i}x^{j}\delta_{ij} \right)^{1/2}$. [**Remark.**]{} The condition on the constants $A_a, C^i_a$ is imposed to ensure that the KID is timelike near infinity on each asymptotic end. $\hfill \square$ ### Killing Form on a KID {#subsectionkillingform} A useful object in spacetimes with a Killing vector $\vec{\xi}$ is the two-form $\nabla_{\mu}\xi_{\nu}$, usually called [**Killing form**]{} or also Papapetrou field. This tensor will play a relevant role below. Since we intend to work directly on the initial data set, we need to define a suitable tensor on $\id$ which corresponds to the Killing form whenever a spacetime is present. Let $\kid$ be a KID in $(M,\gM)$. Clearly we need to restrict and decompose $\nabla_{\mu}\xi_{\nu}$ onto $\kid$ and try to get an expression in terms of $N$ and $\vec{Y}$ and its spatial derivatives. In order to use (\[killingdecomposition\]) we first extend $\vec{n}$ to a neighbourhood of $\Sigma$ as a timelike unit and hypersurface orthogonal, but otherwise arbitrary, vector field (the final expression we obtain will be independent of this extension), and define $N$ and $\vec{Y}$ so that $\vec{Y}$ is orthogonal to $\vec{n}$ and (\[killingdecomposition\]) holds. Taking covariant derivatives we find $$\label{KF} \nabla_{\mu}\xi_{\nu}=\nabla_{\mu}Nn_{\nu}+N\nabla_{\mu}n_{\nu}+\nabla_{\mu}Y_{\nu}.$$ Notice that, by construction, $\nabla_{\mu}n_{\nu} |_{\Sigma}=K_{\mu\nu}-n_{\mu}a_{\nu} |_{\Sigma}$ where $a_{\nu}=n^{\alpha}\nabla_{\alpha}n_{\nu}$ is the acceleration of $\vec{n}$. To elaborate $\nabla_{\mu} Y_{\nu}$ we recall that $\nablaSigma$-covariant derivatives correspond to spacetime covariant derivatives projected onto $\Sigma$. Thus, from $\nablaSigma_{\mu} Y_{\nu} \equiv h^{\alpha}_{\mu} h^{\beta}_{\nu} \nabla_{\alpha} Y_{\beta}$, where $h^{\mu}_{\nu}=\delta^{\mu}_{\nu}+n^{\mu}n_{\nu}$ is the projector orthogonal to $\vec{n}$, and expanding we find $$\begin{aligned} \nabla_{\mu}Y_{\nu} |_{\Sigma} &=& \nablaSigma_{\mu}Y_{\nu}-n_{\mu}\left( n^{\alpha}\nabla_{\alpha}Y_{\beta} \right) h^{\beta}_{\nu} -n_{\nu}\left( n^{\beta}\nabla_{\alpha}Y_{\beta} \right) h^{\alpha}_{\mu} +n_{\mu}n_{\nu}n^{\alpha}n^{\beta}\nabla_{\alpha}Y_{\beta} |_{\Sigma}\\ &=& \nablaSigma_{\mu}Y_{\nu}-n_{\mu}\left( n^{\alpha}\nabla_{\alpha}Y_{\beta} \right) h^{\beta}_{\nu}+ n_{\nu}\left( Y^{\beta}\nabla_{\alpha}n_{\beta} \right) h^{\alpha}_{\mu} +n_{\mu}n_{\nu}n^{\alpha}n^{\beta}\nabla_{\alpha}Y_{\beta} |_{\Sigma}\\ &=& \nablaSigma_{\mu}Y_{\nu}-n_{\mu}\left( n^{\alpha}\nabla_{\alpha}Y_{\beta} \right) h^{\beta}_{\nu}+ K_{\mu\alpha}Y^{\alpha}n_{\nu}+n_{\mu}n_{\nu}n^{\alpha}n^{\beta}\nabla_{\alpha}Y_{\beta} |_{\Sigma},\end{aligned}$$ Substitution into (\[KF\]), using $\nabla_{\mu}N=\nablaSigma_{\mu}N-n_{\mu}n^{\alpha}\nabla_{\alpha}N$, gives $$\begin{aligned} \label{killingform1} \nabla_{\mu}\xi_{\nu}\big|_{\Sigma}&=&n_{\nu}\left( \nablaSigma_{\mu}N+K_{\mu\alpha}Y^{\alpha} \right) - n_{\mu}\left( Na_{\nu}+n^{\alpha}h^{\beta}_{\nu} \nabla_{\alpha}Y_{\beta} \right) \nonumber\\ && + (\nablaSigma_{\mu}Y_{\nu}+NK_{\mu\nu}) + n_{\mu}n_{\nu}\left( n^{\alpha}n^{\beta}\nabla_{\alpha}Y_{\beta}- n^{\alpha}\nabla_{\alpha}N \right) |_{\Sigma}.\end{aligned}$$ The Killing equations then require $ n^{\alpha}n^{\beta}\nabla_{\alpha}Y_{\beta} |_{\Sigma}= n^{\alpha}\nabla_{\alpha}N |_{\Sigma} $ and $ \nablaSigma_{\mu}N+K_{\mu\alpha}Y^{\alpha}|_{\Sigma}= Na_{\mu}+n^{\alpha}h^{\beta}_{\mu} \nabla_{\alpha}Y_{\beta} |_{\Sigma}$, so that (\[killingform1\]) becomes, after using (\[kid1\]), $$\label{killingform2} \left. \nabla_{\mu}\xi_{\nu} \right |_{\Sigma} = \left . n_{\nu}\left( \nablaSigma_{\mu}N+K_{\mu\alpha}Y^{\alpha} \right) - n_{\mu}\left( \nablaSigma_{\nu}N+K_{\nu\alpha}Y^{\alpha} \right) + \frac12\left( \nablaSigma_{\mu}Y_{\nu}-\nablaSigma_{\nu}Y_{\mu} \right) \right |_{\Sigma}.$$ This expression involves solely objects defined on $\Sigma$. However, it still involves four-dimensional objects. In order to work directly on the KID, we introduce an auxiliary four-dimensional vector space on each point of $\Sigma$ as follows (we stress that we are [*not*]{} constructing a spacetime, only a Lorentzian vector space attached to each point on the KID). At every point $\p\in\Sigma$ define the vector space $V_{\p}=T_{\p}\Sigma\oplus\mathbb{R}$, and endow this space with the Lorentzian metric $\gl |_{\p} =g |_{\p}\oplus\left( -\delta \right)$, where ${\delta}$ is the canonical metric on $\mathbb{R}$. Let $\vec{n}$ be the unit vector tangent to the fiber $\mathbb{R}$. Having a metric we can lower and raise indices of tensors in $T_{\p}\Sigma\oplus\mathbb{R}$. In particular define ${\bf n}=\gl(\vec{n},\cdot)$. Covariant tensors $Q$ on $T_{\p} \Sigma$ can be canonically extended to tensors of the same type on $V_{\p} = T_{\p}\Sigma\oplus\mathbb{R}$ (still denoted with the same symbol) simply by noticing that any vector in $V_{\p}$ is of the form $\vec{X} + a \vec{n}$, where $\vec{X} \in T_{\p} \Sigma$ and $a \in \mathbb{R}$. The extension is defined (for a type $m$ covariant tensor) by $Q (\vec{X_1} + a_1 \vec{n}, \cdots, \vec{X}_m + a_m \vec{n}) \equiv Q(\vec{X_1}, \cdots, \vec{X}_m)$. In index notation, this extension will be expressed simply by changing Latin to Greek indices. It is clear that the collection of $\left( T_{\p}\Sigma\oplus\mathbb{R},\gl \right)$ at every $ \p\in\Sigma$ contains no more information than just $(\Sigma,g)$. In particular, this construction allows us to redefine the energy conditions appearing in Chapter \[sc:A2sectionbasics\] at the initial data level. Let us give the definition of NEC for an initial data set. An initial data set $\id$ satisfies the [**null energy condition**]{} (NEC) if for all $\p\in\Sigma$ the tensor $\GM_{\mu\nu}\equiv \rho n_{\mu}n_{\nu}+J_{\mu}n_{\nu}+n_{\mu}J_{\nu}+\tau_{\mu\nu}$ on $T_{\p}\Sigma\times \mathbb{R}$ satisfies that $\GM_{\mu\nu}k^{\mu}k^{\nu} |_{\p}\geq 0$ for any null vector $\vec{k}\in T_{\p}\Sigma\oplus \mathbb{R}$. Motivated by (\[killingform2\]), we can define the Killing form directly in terms of objects on the KID The [**Killing form on a KID**]{} is the 2-form $F_{\mu\nu}$ defined on $\left( T_{\p}\Sigma\oplus\mathbb{R}, \gl \right)$ given by $$\label{killingform} F_{\mu\nu}= n_{\nu}\left( \nablaSigma_{\mu}N+K_{\mu\alpha}Y^{\alpha} \right) - n_{\mu}\left( \nablaSigma_{\nu}N+K_{\nu\alpha}Y^{\alpha} \right) + f_{\mu\nu},$$ where $f_{\mu\nu}= \nablaSigma_{[\mu}Y_{\nu]}$. In a spacetime setting it is well-known that for a non-trivial Killing vector $\vec{\xi}$, the Killing form cannot vanish on a fixed point. Let us show that the same happens in the KID setting. \[FixedPointF=0\] Let $\kid$ be a KID and $\p \in \Sigma$ a fixed point, i.e. $N |_{\p} = 0$ and $\vec{Y} |_{\p} =0$. If $F_{\mu\nu} |_{\p} = 0$ then $N$ and $\vec{Y}$ vanish identically on $\Sigma$. [**Proof.**]{} The aim is to obtain a suitable system of equations and show that, under the circumstances of the lemma, the solution must be identically zero. Decomposing $ \nablaSigma_{i}Y_{j}$ in symmetric and antisymmetric parts, $$\label{DY} \nablaSigma_{i}Y_{j}=-NK_{ij}+f_{ij},$$ and inserting into (\[kid2\]) gives $$\label{DDN} \nablaSigma_{i} \nablaSigma_{ j}N=NQ_{i j}-Y^{l}\nablaSigma_{l}K_{i j}-K_{il}{f_{ j}}^{l}-K_{ jl} {f_{i}}^{l},$$ where $Q_{i j}=\RSigma_{i j}+\tr_{\,\Sigma}K K_{i j}-\tau_{i j} + \frac{1}{2}g_{i j}(\tr_{\,\Sigma} \tau-\rho)$. In order to find an equation for $ \nablaSigma_{l}f_{i j}$, we take a derivative of (\[kid1\]) and write the three equations obtained by cyclic permutation. Adding two of them and subtracting the third one, we find, $$\nablaSigma_l \nablaSigma_i Y_j = \RSigma_{klij} Y^k + \nablaSigma_{j} ( N K_{li}) - \nablaSigma_i (N K_{lj}) - \nablaSigma_l (N K_{ij}),\nonumber$$ after using the Ricci and first Bianchi identities. Taking the antisymmetric part in $i,j$, $$\label{DF} \nablaSigma_{l} f_{i j}=\RSigma_{k l i j}Y^{k}+\nablaSigma_{ j}NK_{li}-\nablaSigma_{i}NK_{l j} + N\nablaSigma_{ j}K_{li}-N\nablaSigma_{i}K_{l j}.$$ If $F_{\mu\nu} |_{\p}=0$, it follows that $f_{ij} |_{\p} =0$ and $\nablaSigma_i N |_{\p} =0$. The equations given by (\[DY\]), (\[DDN\]) and (\[DF\]) is a system of PDE for the unknowns $N$, $Y_i$ and $f_{ij}$ written in normal form. It follows (see e.g. [@Eisenhart]) that the vanishing of $N$, $\nablaSigma_iN$, $Y_i$ and $f_{ij}$ at one point implies its vanishing everywhere (recall that $\Sigma$ is connected). $\hspace*{1cm} \hfill \blacksquare$\ ### Canonical Form of Null two-forms {#canonical} Let $F_{\mu\nu}$ be an arbitrary two-form on a spacetime $(M,g^{(4)})$. It is well-known that the only two non-trivial scalars that can be constructed from $F_{\mu\nu}$ are $I_{1}=F_{\mu\nu}F^{\mu\nu}$ and $I_{2}= F^{\star}_{\mu\nu}F^{\mu\nu}$, where $F^{\star}$ is the Hodge dual of $F$, defined by $F^{\star}_{\mu\nu}=\frac12\eta^{(4)}_{\mu\nu\alpha\beta}F^{\alpha\beta}$, with $\eta^{(4)}_{\mu\nu\alpha\beta}$ being the volume form of $(M,g^{(4)})$. When both scalars vanish, the two-form is called *null*. Later on, we will encounter Killing forms which are null and we will exploit the following well-known algebraic decomposition which gives its [**canonical form**]{}, see e.g. [@Israellibro] for a proof. \[lemmacanonicalform\] A null two-form $F_{\mu\nu}$ at a point $\p$ can be decomposed as $$\label{canonicalform} F_{\mu\nu} |_{\p} =l_{\mu} w_{\nu}-l_{\nu}w_{\mu} |_{\p},$$ where $\vec{l}\, |_{\p}$ is a null vector and $\vec{w} |_{\p}$ is spacelike and orthogonal to $\vec{l}\, |_{\p}$. Staticity of a KID {#staticityKID} ------------------ ### Static KID To define a static KID we have to decompose the integrability equation $\xi_{[\mu}\nabla_{\nu}\xi_{\rho]}=0$ according to (\[killingdecomposition\]). By taking the normal-tangent-tangent part (to $\Sigma$) and the completely tangential part (the other components are identically zero by antisymmetry) we find $$\begin{aligned} &&N\nablaSigma_{[i}Y_{j]}+2Y_{[i}\nablaSigma_{j]}N+2Y_{[i}K_{j]l}Y^{l}=0,\label{static5}\\ &&Y_{[i}\nablaSigma_{j}Y_{k]}=0.\label{statictwo}\end{aligned}$$ Since these expressions involve only objects on the KID, the following definition becomes natural. A KID $\kid$ satisfying (\[static5\]) and (\[statictwo\]) is called an [**integrable KID**]{}. Multiplying equation (\[static5\]) by $N$ and equation (\[statictwo\]) by $Y^{k}$, adding them up and using equation (\[kid1\]), we get the following useful relation, valid everywhere on $\Sigma$, $$\label{static1} \lambda \nablaSigma_{[i}Y_{j]}+Y_{[i}\nablaSigma_{j]}\lambda=0.$$ If $\lambda>0$ in some non-empty set of the KID, the Killing vector is timelike in some non-empty set of the spacetime. Hence A [**static KID**]{} is an integrable KID with $\lambda>0$ in some non-empty set. ### Killing Form of a Static KID {#subsectionFonstaticKID} In Subsection \[canonical\] we introduced the invariant scalars $I_1$ and $I_2$ for any two-form in a spacetime. In this section we find their explicit expressions for the Killing form of an integrable KID in the region $\{\lambda> 0 \}$. Although not necessary, we will pass to the Killing development (which is available in this case) since this simplifies the proofs. We start with a lemma concerning the integrability of the Killing vector in the Killing development. \[integrability\] The Killing vector field associated with the Killing development of an integrable KID is also integrable. [**Proof.**]{} Let $\kid$ be an integrable KID. Suppose the Killing development (\[killingdevelopment\]) of a suitable open set of $\Sigma$. Using $\vec{\xi} = \partial_t$ it follows $$\label{xidxi} {{\mbox{\boldmath $\xi$}}}\wedge d{{\mbox{\boldmath $\xi$}}}=-\hat{\lambda}\partial_{i}\hat{Y}_{j}dt\wedge dx^{i}\wedge dx^{j} - \hat{Y}_{i}\partial_{j}\hat{\lambda}dt\wedge dx^{i}\wedge dx^{j} + \hat{Y}_{i}\partial_{j}\hat{Y}_{k}dx^{i}\wedge dx^{j}\wedge dx^{k},$$ where $\hat{\lambda}$, $\hat{{\bf Y}}$ and $\hat{g}$ are defined in (\[killingdevelopment2\]). Integrability of $\vec{\xi}$ follows directly from (\[statictwo\]) and (\[static1\]). $\hfill \blacksquare$\ The following lemma gives the explicit expressions for $I_{1}$ and $I_{2}$. \[lemmaI1\] The invariants of the Killing form in a static KID in the region $\{ \lambda >0 \}$ read $$\label{I1} I_{1}=-\frac{1}{2\lambda}\left( g^{ij}-\frac{Y^{i}Y^{j}}{N^{2}} \right)\nablaSigma_{i}\lambda \nablaSigma_{j}\lambda,$$ and $$\label{I2} I_{2}=0.$$ [**Remark.**]{} By continuity $I_{2}\big|_{\tbd\{\lambda>0\}}=0$. $\hfill \square$\ [**Proof.**]{} Consider a static KID $\kid$ and let $\{ \lambda>0 \}_{0}$ be a connected component of $\{\lambda>0\}$. In $\{\lambda>0\}_{0}$ we have necessarily $N\neq 0$, so we can construct the Killing development $(\{ \lambda>0 \}_{0},g^{(4)})$ and introduce the so-called Ernst one-form, as $\sigma_{\mu}=\nabla_{\mu} \lambda -i\omega_{\mu}$ where $\omega_{\mu}=\eta^{(4)}_{\mu\nu\alpha\beta}\xi^{\nu}\nabla^{\alpha}\xi^{\beta}$ is the twist of the Killing field ($\eta^{(4)}$ is the volume form of the Killing development). The Ernst one-form satisfies the identity (see e.g. [@Mars]) $\sigma^{\mu}\sigma_{\mu}=-\lambda\left( F_{\mu\nu}+iF^{\star}_{\mu\nu} \right) \left( F^{\mu\nu}+i{F^{\star}}^{\mu\nu} \right)$, which in the static case (i.e. $\omega_{\mu}=0$) becomes $\nabla_{\mu} \lambda \nabla^{\mu} \lambda=-2\lambda\left(F_{\mu\nu}F^{\mu\nu} +iF_{\mu\nu}{F^{\star}}^{\mu\nu } \right)$ where the identity $F_{\mu\nu}F^{\mu\nu}=-F^{\star}_{\mu\nu}{F^{\star}}^{\mu\nu}$ has been used. The imaginary part immediately gives (\[I2\]). The real part gives $I_{1}=-\frac{1}{2\lambda}|\nabla \lambda|_{\gM}^{2}$. Taking coordinates $\{t,x ^{i}\}$ adapted to the Killing field $\partial_{t}$, it follows from (\[killingdevelopment2\]) that $|\nabla\lambda|_{\gM}^{2}={\gM}^{ij}\partial_{i}\lambda\partial_{j}\lambda$. It is well-known (and easily checked) that the contravariant spatial components of $g^{(4)}$ are ${g^{(4)}}^{ij} = g^{ij}-\frac{Y^{i}Y^{j}}{N^2}$, where $g^{ij}$ is the inverse of $g_{ij}$ and (\[I1\]) follows. $\hfill \blacksquare$\ This lemma allows us to prove the following result on the value of $I_1$ on the set ${\{\lambda>0\}}$. $I_{1}|_{{\{\lambda>0\}}}\leq 0$ in a static KID. [**Proof.**]{} Let $ \q\in \{ \lambda>0 \}\subset\Sigma$ and define the vector $\vec{\xi} \equiv N\vec{n}+\vec{Y}$ on the vector space $(V_{\q}, \gl)$ introduced in Section \[subsectionkillingform\]. Since $\vec{\xi}$ is timelike at $\q$, we can introduce its orthogonal projector $h_{\mu\nu}=\gl_{\mu\nu}+\frac{\xi_{\mu}\xi_{\nu}}{\lambda}$ which is obviously positive semi-definite. If we pull it back onto $T_{\q} \Sigma$ we obtain a positive definite metric, called [*orbit space metric*]{}, $$\label{quotientmetric} h_{ij}=g_{ij}+\frac{Y_{i}Y_{j}}{\lambda}.$$ It is immediate to check that the inverse of $h_{ij}$ is precisely the term in brackets in (\[I1\]). Consequently, $I_{1} |_{\q} \leq 0$ follows. $\hfill \blacksquare$\ [**Remark.**]{} By continuity $I_{1}|_{\tbd \{\lambda>0\}}\leq 0$. $\hfill \square$ Furthermore, for the fixed points on the closure of $\{ \lambda > 0 \}$ we have the following result. Notice that $\tbd\{\lambda>0 \} \subset \overline{\{N \neq 0\}}$. Since the result involves points where $N$ vanishes, we cannot rely on the Killing development for its proof and an argument directly on the initial data set is needed. \[I1&lt;0\] Let $\p \in \overline{\{\lambda >0 \}}$ be a fixed point of a static KID, then $I_1 |_{\p} < 0$. [**Proof.**]{} From the previous lemma it follows that $I_{1}|_{p}\leq 0$. It only remains to show that $I_1 |_{\p}$ cannot be zero. We argue by contradiction. Assuming that $I_1 |_{\p} =0$ and using $I_2 |_{\p} =0$ by Lemma \[lemmaI1\], it follows that $F_{\mu\nu}$ is null at $\p$. Lemma \[lemmacanonicalform\] implies the existence of a null vector $\vec{l}$ and a spacelike vector $\vec{w}$ on $V_{\p}$ such that (\[canonicalform\]) holds. Since $\vec{w}$ is defined up to an arbitrary additive vector proportional to $\vec{l}$, we can choose $\vec{w}$ normal to $\vec{n}$ without loss of generality. Decompose $\vec{l}$ as $\vec{l}=a\left( \vec{x} + \vec{n} \right)$ with $x^{\mu}x_{\mu}=1$. We know from Lemma \[FixedPointF=0\] that $a \neq 0$ (otherwise $F_{\mu\nu} |_{\p} =0$ and $\{ \lambda > 0 \}$ would be empty). Expression (\[killingform\]) and the canonical form (\[canonicalform\]) yield $$F_{\mu\nu} |_{\p} = 2n_{ [ \nu}\nablaSigma_{\mu ] }N + \nablaSigma_{ [ \mu}Y_{\nu ] } |_{\p} = 2a \left( x_{ [ \mu} w_{\nu ]}+n_{ [ \mu}w_{\nu ] } \right).$$ The purely tangential and normal-tangential components of this equation give, respectively $$\begin{aligned} \nablaSigma_{i}Y_{j}\big|_{\p}=2a x_{[i}w_{j]}, \quad \nablaSigma_{i}N\big|_{\p}=-aw_{i}, \label{ab}\end{aligned}$$ where $w_{i}$ is the projection of $w_{\mu}$ to $T_{\p} \Sigma$. The Hessian of $\lambda$ at $\p$ is then $$\begin{aligned} \nablaSigma_{i}\nablaSigma_{j}\lambda\big|_{\p}&=&2(\nablaSigma_{i}N\nablaSigma_{j}N- \nablaSigma_{i}Y^{k}\nablaSigma_{j}Y_{k})\big|_{\p}\\ &=& -2a^2 w^{k}w_{k} x_{i}x_{j},\end{aligned}$$ where we have used $x^{i}x_{i}=1$ and $x^{i}w_{i}=0$ (which follows from $\vec{w}$ being orthogonal to $\vec{l}$ ). This Hessian has therefore signature $\{-,0,0\}$. The Gromoll-Meyer splitting Lemma (see Appendix \[ch:appendix2\]) implies the existence of an open neighbourhood $U_{\p}$ of $\p$ and coordinates $\{x,z^{A}\}$ in $U_{\p}$ such that $\p=(x=0,z^{A}=0)$ and $\lambda=-\hat{a}^{2}x^{2}+\zeta(z^{A})$ where $\hat{a}>0$ and $\zeta$ is a smooth function satisfying $\zeta\big|_{\p}=0$, $\nablaSigma_{i}\zeta\big|_{\p}=0$ and $\nablaSigma_{i}\nablaSigma_{j}\zeta\big|_{\p}=0$. Since $\p\in\tbd\{\lambda>0\}$, there exists a curve $\mu(s)=(x(s),z^{A}(s))$ in $U_{\p}\cap \{\lambda>0\}$, parametrized by $s\in (0,\epsilon)$ such that $\mu(s)\underset{s\rightarrow 0}{\longrightarrow} \p$. Since $\lambda>0$ on the curve we have $-\hat{a}^{2}x^{2}(s)+\zeta(z^{A}(s))>0$, which implies $\zeta(z^{A}(s))>0$. It follows that the curve $\gamma(s)\equiv\left(x(s)=\frac{1}{\hat{a}}\sqrt{\zeta(z^{A}(s))},z^{A}(s)\right)$ (also parametrized by $s$) belongs to $\tbd \{\lambda>0\}$ and is composed by non-fixed points (because $\nablaSigma_{i}\lambda\big|_{\gamma(s)}\neq 0$). We can construct the Killing development (\[killingdevelopment\]) near this curve, which is a static spacetime (see Lemma \[integrability\]). Applying Lemma \[lema:VC\] by Vishveshwara and Carter it follows that $\gamma(s)$ (which belongs to $\tbd\{\lambda>0\}$ and has $N\neq 0$) lies in an arc-connected component of a Killing prehorizon of the Killing development. Projecting equation (\[kappa0\]), valid on a Killing prehorizon, onto $\Sigma$, we get the relation $$\label{kappa} \nablaSigma_{i}\lambda\big|_{\gamma(s)}=2\kappa Y_{i}\big|_{\gamma(s)},$$ where $\kappa$ is the surface gravity of the prehorizon. Therefore, $\kappa\big|_{\gamma(s)}\neq 0$. Since $I_{1}=-2\kappa^{2}$ (see e.g. equation (12.5.14) in [@Wald]) and $\kappa$ remains constant on $\gamma(s)$ (see Lemma \[lema:RW\]), it follows, by continuity of $I_{1}$, that $I_{1}\big|_{\p}=-2\kappa^{2}<0$.$\hfill \blacksquare$\ ### Properties of $\tbd \{ \lambda>0 \}$ on a Static KID In this subsection we will show that, under suitable conditions, the boundary of the region $\{\lambda> 0\}$ is a smooth surface. Our first result on the smoothness of $\tbd\{ \lambda > 0 \}$ is the following. \[surfaceNoFixed\] Let $\kid$ be a static KID and assume that the set $\E=\tbd\{ \lambda>0 \} \cap \{ N \neq 0 \}$ is non-empty. Then $\E$ is a smooth submanifold of $\Sigma$. Recall that in this thesis, a submanifold is, by definition, injectively immersed, but not necessarily embedded. Besides, it is worth to remark they are also not necessarily arc-connected. [**Proof.**]{} Since $N |_{\E}\ne0$, we can construct the Killing development (\[killingdevelopment\]) of a suitable neighbourhood of $\E\subset\Sigma$ satisfying $N \neq 0$ everywhere. Moreover, by Lemma \[integrability\], $\vec{\xi} = \partial_t$ is integrable. Applying Lemma \[lema:VC\] by Vishveshwara and Carter, it follows that the spacetime subset $\mathcal{N}_{\vec{\xi}}\equiv \tbd \{\lambda>0\}\cap \{\vec{\xi}\neq 0\}$ is a smooth null submanifold (in fact, a Killing prehorizon) of the Killing development and therefore transverse to $\Sigma$, which is spacelike. Thus, $\E = \Sigma \cap \mathcal{N}_{\vec{\xi}}$ is a smooth submanifold of $\Sigma$. $\hfill \blacksquare$\ This lemma states that the boundary of $\{ \lambda > 0 \}$ is smooth on the set of non-fixed points. In fact, for the case of boundaries having at least one fixed point, an explicit defining function for this surface on the subset of non-fixed points can be given: \[maldito\] Let $\kid$ be a static KID. If an arc-connected component of $\tbd\{\lambda>0 \}$ contains at least one fixed point, then $\nablaSigma_{i}\lambda\neq 0$ on all non-fixed points in that arc-connected component. [**Proof.**]{} Let $V$ be the set of non-fixed points in one of the arc-connected components under consideration. This set is obviously open with at least one fixed point in its closure. Constructing the Killing development as before, we know that $V$ belongs to a Killing prehorizon $\mathcal{H}_{\vec{\xi}}$. Projecting equation (\[kappa0\]) onto $\Sigma$ we get $\nablaSigma_{i}\lambda\big|_{\mathcal{H}_{\vec{\xi}\,}\cap\Sigma}= 2\kappa Y_{i}\big|_{\mathcal{H}_{\vec{\xi}\,}\cap\Sigma}$. Since the surface gravity $\kappa$ is constant on each arc-connected component of $\mathcal{H}_{\vec{\xi}}$ and $I_{1}=-2\kappa^2$, Lemma \[I1&lt;0\] implies $\kappa\big|_{V} \neq 0$ and consequently $\nablaSigma_{i}\lambda\big|_{V}\neq 0$. $\hfill \blacksquare$\ Fixed points are more difficult to analyze. We first need a lemma on the structure of $\nablaSigma_i N$ and $f_{ij}$ on a fixed point. \[lemmaFixed\] Let $\kid$ be a static KID and $\p \in \tbd\{ \lambda > 0 \}$ be a fixed point. Then $$\nablaSigma_i N |_{\p}\ne 0 \nonumber$$ and $$\left. f_{ij} |_{\p} = \frac{b}{Q} \left ( \nablaSigma_i N X_j - \nablaSigma_j N X_i \right )\right |_{\p} \label{fijp}$$ where $b$ is a constant, $X_i$ is unit and orthogonal to $\nablaSigma_i N |_{\p}$ and $Q = +\sqrt{\nablaSigma_i N {\nablaSigma}^i N}$. [**Proof.**]{} From (\[killingform\]), $$\label{FF} I_{1}=F_{\mu\nu}F^{\mu\nu}=f_{ij}f^{ij}-2 \left( \nablaSigma_{i}N+K_{ij}Y^{j} \right)\left( {\nablaSigma}^{i}N+K^{ik}Y_{k} \right).$$ Hence, $\nablaSigma_i N |_{\p} \neq 0$ follows directly from $I_1 |_{\p} < 0$ (Lemma \[I1&lt;0\]). For the second statement, let $u_i$ be unit and satisfy $\nablaSigma_i N = Q u_i$ in a suitable neighbourhood of $\p$. Consider (\[static5\]) in the region $N \neq 0$, which gives $$\begin{aligned} f_{ij} = -2 N^{-1} Y_{[i}\left( \nablaSigma_{j]}N+K_{j]k}Y^{k} \right). \label{fYN}\end{aligned}$$ Since $|\vec{Y}|/N$ stays bounded in the region $\{ \lambda > 0 \}$, it follows that the second term tends to zero at the fixed point $\p$. Thus, let $\vec{X}_1$ and $\vec{X}_2$ be any pair of vector fields orthogonal to $\vec{u}$. It follows by continuity that $f_{ij} X_1^i X_2^j |_{\p} =0$. Hence for any orthonormal basis $\{ \vec{u}, \vec{X}, \vec{Z}\}$ at $\p$ it follows $f_{ij} X^i Z^j |_{\p} =0$ (because $\vec{X}$ and $\vec{Z}$ can be extended to a neighbourhood of $\p$ while remaining orthogonal to $\vec{u}$). Consequently, $f_{ij} |_{\p} = (b/Q) ( \nablaSigma_i N X_j - \nablaSigma_j N X_i ) + ( c/Q) ( \nablaSigma_i N Z_j - \nablaSigma_j N Z_i ) |_{\p}$ for some constants $b$ and $c$. A suitable rotation in the $\{\vec{X},\vec{Z}\}$ plane allows us to set $c=0$ and (\[fijp\]) follows. $\hfill \blacksquare$\ As we will see next, a consequence of this lemma is that an open subset of fixed points in $\tbd \{\lambda>0\}$ is a smooth surface. In fact, we will prove that this surface is totally geodesic in $(\Sigma,g)$ and that the pull-back of the second fundamental form $K_{ij}$ vanishes there. This means from a spacetime perspective, i.e. when the initial data set is embedded into a spacetime, that this open set of fixed points is totally geodesic as a spacetime submanifold. This is of course well-known in the spacetime setting from Boyer’s results [@Boyer], see also [@Heuslerlibro]. In our initial data context, however, the result must be proven from scratch as no Killing development is available at the fixed points. \[Totally geodesic\] Let $\kid$ be a static KID and assume that the set $\tbd\{ \lambda>0 \}$ is non-empty. If $\E\subset \tbd\{\lambda>0 \}$ is open and consists of fixed points, then $\E$ is a smooth surface. Moreover, the second fundamental form of $\E$ in $(\Sigma,g)$ vanishes and $K_{AB} \big |_{\E}=0$ [**Proof.**]{} Consider a point $\p\in \E$. We know from Lemma \[lemmaFixed\] that $\nablaSigma_{i}N\big|_{\p}\neq 0$. This means that there exists an open neighbourhood $U_{\p}$ such that $\{N=\text{const}\}\cap U_{\p}$ defines a foliation by smooth and connected surfaces, and moreover that $\nablaSigma_{i}N\neq 0$ everywhere on $U_{\p}$. Restricting $U_{\p}$ if necessary we can assume that $\tbd\{\lambda>0\}\cap U_{\p}=\E\cap U_{\p}$ (because $\E$ is an open subset of $\tbd \{\lambda>0\}$). It is clear that $\E\cap U_{\p}\subset\{N=0\}\cap U_{\p}$ (because $N$ vanishes on a fixed point). We only need to prove that these two sets are in fact equal. Choose a continuous curve $\gamma:(-\epsilon,0)\rightarrow \{\lambda>0\}\cap U_{\p}$ satisfying $\text{lim}_{s\rightarrow 0}\gamma(s)=\p$. Assume that there is a point $\q\in\{N=0\}\cap U_{\p}$ not lying in $\tbd \{ \lambda>0 \}$. This means that there is an open neighbourhood $U_{\q}$ of $\q$ (which can be taken fully contained in $U_{\p}$) which does not intersect $\{\lambda>0\}$. Take a point $\r$ in $U_{\q}$ sufficiently close to $\q$ so that $N\big|_{\r}$ takes the same value as $N\big|_{\gamma(s_{0})}$ for some $s_{0}\in (-\epsilon,0)$ (this point $\r$ exists because $\nablaSigma_{i}N\big|_{\q}\neq 0$ and $N\big|_{\q}=0$). Since the surface $\{N=N\big|_{\r}\}\cap U_{\p}$ is connected and contains both $\r$ and $\gamma(s_{0})$, it follows that there is a path in $U_{\p}$ with $N=N\big|_{\r}$ constant and connecting these two points. This path must necessarily intersect $\tbd \{\lambda>0\}$ (recall that $\lambda\big|_{\gamma(s)}>0$ for all $s$). But this contradicts the fact that $\tbd\{\lambda>0\}\cap U_{\p}\subset \{N=0\}\cap U_{\p}$. Therefore, $\E\cap U_{\p}=\{N=0\}\cap U_{\p}$, which proves that $\E$ is a smooth surface. To prove the other statements, let us introduce local coordinates $\{ u,x^{A} \}$ on $\Sigma$ adapted to $\E$ so that $\E\equiv\{ u=0 \}$ and let us prove that the linear term in a Taylor expansion for $Y^i$ vanishes identically. Equivalently, we want to show that $u^{j}\nablaSigma_{j}Y_{i} |_{\E}=0$ for $\vec{u} = \partial_u$ (recall that on $\E$ we have $Y_{i} |_{\E}=0$ and this covariant derivative coincides with the partial derivative). Note that $\nablaSigma_{i}Y_{j} |_{\E}=f_{ij}$ (see (\[DY\])), so that $u^{i} u^j \nablaSigma_{i}Y_{j} |_{\E} =0 $ being the contraction of a symmetric and an antisymmetric tensor. Moreover, for the tangential vectors $e^i_A = \partial_A$ we find $u^{j} e^i_A \nablaSigma_{i}Y_{j} |_{\E} = u^j \partial_A Y_j =0$ because $Y_j$ vanishes all along $\E$. Consequently $u^{i}\partial_{i}Y_{j}|_{\E}=0$. Hence, the Taylor expansion reads $$\begin{aligned} \label{expansions} N&=&G(x^{A})u+O(u^{2}),\nonumber \\ Y_{i}&=&O(u^{2}).\end{aligned}$$ Moreover, $G\ne0$ everywhere on $\E$ because substituting this Taylor expansion in (\[I1\]) and taking the limit $u \rightarrow 0$ gives $I_{1} |_{\E}=-2g^{uu}G^{2}(x^{A})$ and we know that $I_{1} |_{\E} \neq 0$ from Lemma \[I1&lt;0\]. We can now prove that $\E$ is totally geodesic and that $K_{AB}=0$. For the first, the Taylor expansion above gives $$\label{fij0} f_{ij}|_{\E}=0$$ and obviously $N$ and $\vec{Y}$ also vanish on $\E$. Hence, from (\[DDN\]), $$\label{DDN=0} \nablaSigma_{i}\nablaSigma_{j}N |_{\E}=0.$$ Since, by Lemma \[lemmaFixed\], $\nablaSigma_{i}N |_{\E}$ is proportional to the unit normal to $\E$ and non-zero, then $\nablaSigma_{i}\nablaSigma_{j}N |_{\E}=0$ is precisely the condition that $\E$ is totally geodesic. In order to prove $K_{AB}|_{\E}=0$, we only need to substitute the Taylor expansion (\[expansions\]) in the $AB$ components of (\[kid1\]). After dividing by $u$ and taking the limit $u \rightarrow 0$, $K_{AB} |_{\E}=0$ follows directly. $\hfill \blacksquare$\ At this point, let us introduce a lemma on the constancy of $I_{1}$ on each arc-connected component of $\tbd \{\lambda>0\}$. \[lema:I1constant\] $I_{1}$ is constant on each arc-connected component of $\tbd \{\lambda>0\}$ in a static KID. [**Proof.**]{} For non-fixed points this is a consequence of the Vishveshwara-Carter Lemma (Lemma \[lema:VC\]) and it has already been used several times before. For an arc-connected open set $\E$ of fixed points, taking the derivative of equation (\[FF\]) we get $$\nablaSigma_{l}I_{1}=2f^{ij}\nablaSigma_{l}f_{ij}-4(\nablaSigma_{l}\nablaSigma_{i}N + \nablaSigma_{l}K_{ij}Y^{j}+K_{ij}\nablaSigma_{l}Y^{j})({\nablaSigma}^{i}N+K^{ik}Y_{k}).$$ Then, using the facts that $f_{ij}\big|_{\E}=0$ (equation (\[fij0\])), $\nablaSigma_{i}\nablaSigma_{j}N\big|_{\E}=0$ (equation (\[DDN=0\])) and $\nablaSigma_{i}Y_{j}=-NK_{ij}+f_{ij}$ (equation (\[DY\])), it is immediate to obtain that $\nablaSigma_{l}I_{1}\big|_{\E}=0$. Finally, continuity of $I_{1}$ leads to the result. $\hfill\blacksquare$\ We have already proved that both the open sets of fixed points and the open sets of non-fixed points are smooth submanifolds. Unfortunately, when $\tbd\{ \lambda >0 \}$ contains fixed points not lying on open sets, this boundary is [*not*]{} a smooth submanifold in general. Consider as an example the Kruskal extension of the Schwarzschild black hole and choose one of the asymptotic regions where the static Killing field is timelike in the domain of outer communications. Its boundary consists of one half of the black hole event horizon, one half of the white hole event horizon and the bifurcation surface connecting both. Take an initial data set $\Sigma$ that intersects the bifurcation surface transversally and let us denote by $\ext$ the connected component of the subset $\{ \lambda >0 \}$ within $\Sigma$ contained in the chosen asymptotic region. The topological boundary $\tbd \ext$ is non-smooth because it has a corner on the bifurcation surface where the black hole event horizon and the white hole event horizon intersect (see example of Figure \[fig:figure1\]). ![An example of non-smooth boundary $\E=\tbd\{\lambda>0\}$ in an initial data set $\Sigma$ of Kruskal spacetime with one dimension suppressed. The region outside the cylinder and the cone corresponds to one asymptotic region of the Kruskal spacetime. The initial data set $\Sigma$ intersects the bifurcation surface $S_{0}$ (in red). The shaded region corresponds to the intersection of $\Sigma$ with the asymptotic region, and is in fact a connected component of the subset $\{\lambda > 0 \} \subset \Sigma$. Its boundary is non-smooth at the point $\p$ lying on the bifurcation surface.[]{data-label="fig:figure1"}](figure1.eps){width="9cm"} We must therefore add some condition on $\tbd\ext$ in order to guarantee that this boundary does not intersect both a black and a white hole event horizon. In terms of the Killing vector, this requires that $\vec{Y}$ points only to one side of $\tbd\ext$. Lemma \[maldito\] suggests that the condition we need to impose is $Y^{i} \nablaSigma_{i} \lambda \big|_{\tbd\ext}\ge0$ or $Y^{i} \nablaSigma_{i} \lambda \big|_{\tbd\ext}\le0$. This condition is in fact sufficient to show that $\tbd\ext$ is a smooth surface. Before giving the precise statement of this result (Proposition \[C1\] below) we need to prove a lemma on the structure of $\lambda$ near fixed points with $f_{ij}\neq 0$. For this, the following definition will be useful. A fixed point $\p\in \tbd\{\lambda>0\}$ is called [**transverse**]{} if and only if $f_{ij} |_{\p} \neq 0$ and [**non-transverse**]{} if and only if $f_{ij} |_{\p} =0$ \[structurebneq0\] Let $\p\in\tbd \{\lambda>0\}$ be a transverse fixed point. Then, there exists an open neighbourhood $U_{\p}$ of $\p$ and coordinates $\{x,y,z\}$ on $U_{\p}$ such that $\lambda = \mu^2 x^2 - b^2 y^2$ for suitable constants $\mu>0$ and $b\neq 0$. [**Proof.**]{} From Lemma \[lemmaFixed\] we have $b\neq 0$. Squaring $f_{ij}$ we get $f_{il} f_{j}^{\,\,l} |_{\p} = \left . b^2 \left ( \frac{\nablaSigma_i N \nablaSigma_j N}{Q_0^2} + X_i X_j \right ) \right |_{\p}$ and $f_{ij} f^{ij}|_{\p} = 2 b^2$, where $Q_0 = Q(\p)$. Being $\p$ a fixed point, both $\lambda$ and its gradient vanish at $\p$ and we have a critical point. The Hessian of $\lambda$ at $\p$ is immediately computed to be $$\begin{aligned} \nablaSigma_i \nablaSigma_j \lambda |_{\p} &=& \left . 2 \nablaSigma_i N \nablaSigma_j N - 2 f_{il} f_{j}^{\,\,\,l} \right |_{\p} \nonumber \\ &=& \left . \frac{2 \left (Q_0^2 - b^2 \right )}{Q_0^2} \nablaSigma_i N \nablaSigma_j N - 2 b^2 X_i X_j \right |_{\p}. \label{Hessianbneq0}\end{aligned}$$ At a fixed point we have $I_1 |_{\p} = f_{ij} f^{ij} - 2 \nablaSigma_i N {\nablaSigma}^i N |_{\p} = 2 (b^2 - Q_0^2) < 0$ (Lemma \[I1&lt;0\]). Let us define $\mu >0$ by $\mu^2 = Q_0^2 - b^2$. The rank of the Hessian is therefore two and the signature is $(+,-,0)$. The Gromoll-Meyer splitting Lemma (see Appendix \[ch:appendix2\]) implies the existence of coordinates $\{x,y,z \}$ in a suitable neighbourhood $U'_{\p}$ of $\p$ such that $\p = \{x=0,y=0,z=0\}$ and $\lambda = \mu^2 x^2 - b^2 y^2 + h(z)$ on $U'_{\p}$. The function $h(z)$ is smooth and satisfies $h(0) = h'(0)= h''(0)=0$, where prime stands for derivative with respect to $z$. Moreover, evaluating the Hessian of $\lambda$ at $\p$ and comparing with (\[Hessianbneq0\]) we have $dx |_{\p} = Q_0^{-1} dN |_{\p} $ and $dy |_{\p} = {\mbox{\boldmath $X$}}$. This implies $N = Q_0 x + O(2)$. Moreover, since $\nablaSigma_i Y_j |_{\p} = f_{ij} |_{\p} = b (dx \otimes dy - dy \otimes dx )_{ij} |_{\p}$ we conclude $Y_x = - b y + O(2)$, $Y_y = b x + O(2)$, $Y_z = O(2)$. On the surface $\{z=0\}$, the set of points where $\lambda$ vanishes is given by the two lines $x=x_+ (y) \equiv b \mu^{-1} y $ and $x = x_{-} (y) \equiv - b \mu^{-1} y$. Computing the gradient of $\lambda$ on these curves we find $$\begin{aligned} d \lambda |_{(x=x_{\pm}(y), z=0)} = \pm 2 \mu b y dx - 2 b^2 y dy. \label{dlambda1}\end{aligned}$$ On the other hand, the Taylor expansion above for ${{\mbox{\boldmath $Y$}}}$ gives $$\begin{aligned} {\mbox{\boldmath $Y$}} |_{(x = x_{\pm}(y),z=0)} = -b y dx \pm \frac{b^2}{\mu} y dy + O(2). \label{yd}\end{aligned}$$ Let $\E$ be the arc-connected component of $\tbd\{\lambda>0\}$ containing $\p$. On all non-fixed points in $\E$ we have $d \lambda = 2 \kappa {\mbox{\boldmath $Y$}}$, with $\kappa^2 = -I_1/2$. Comparing (\[dlambda1\]) with (\[yd\]) yields $\kappa= - \mu$ on the branch $x= x_{+} (y)$ and $\kappa = +\mu$ on the branch $x= x_{-}(y)$ (this is in agreement with $I_1 = -2 \kappa^2 = - 2 \mu^2$ at every point in $\E$). We already know that $\kappa$ must remain constant on each arc-connected component of $\E \setminus F$, where $F = \{ \p \in \E, \p \,\,\, \mbox{fixed point} \}$. Let us show that this implies $h(z)=0$ on $U'_{\p}$. First, we notice that the set of fixed points on $\E$ are precisely those where $\lambda=0$ and $d \lambda=0$ (this is because in Lemma \[maldito\] we have shown that $d \lambda \neq 0$ on every non-fixed point of any arc-connected component of $\tbd\{\lambda >0\}$ containing at least one fixed point). From the expression $\lambda = \mu^2 x^2 - b^2 y^2 + h(z)$, this implies that the fixed points in $U'_{\p}$ are those satisfying $\{ x=0, y=0, h(z)=0, h'(z)=0 \}$. Assume that there is no neighbourhood $(-\epsilon,\epsilon)$ where $h$ vanishes identically. Then, there exists a sequence $z_n \rightarrow 0$ satisfying $h(z_n) \neq 0$. There must exist a subsequence (still denoted by $\{z_n \}$) satisfying either $h(z_n)>0$, $\forall n \in \mathbb{N}$ or $h(z_n) <0$, $\forall n \in \mathbb{N}$. The two cases are similar, so we only consider $h(z_n) = -a_n^2<0$. The set of points with $\lambda=0$ in the surface $\{z=z_n \}$ are given by $x = \pm \mu^{-1} \sqrt{b^2 y^2 + a_n^2}$. It follows that the points $\{\lambda=0\} \cap \{ z=z_n\} $ in the quadrant $\{ x>0, y >0\}$ lie in the same arc-connected component as the points $\{\lambda=0\} \cap \{ z=z_n \}$ lying in the quadrant $\{x> 0, y<0\}$. Since $z_n$ converges to zero, it follows that the points $\{x=x_+(y),y>0,z=0\}$ lie in the same arc-connected component of $\E \setminus F$ than the points $\{x=x_-(y),y< 0,z=0\}$. However, this is impossible because $\kappa$ (which is constant on $\E\setminus F$) takes opposite values on the branch $x=x_{+}(y)$ and on the branch $x=x_{-}(y)$. This gives a contradiction, and so there must exist a neighbourhood $U_{\p}$ of $\p$ where $h(z)=0$. $\hfill \blacksquare$\ Now, we are ready to prove a smoothness result for $\tbd \{\lambda>0\}$. \[C1\] Let $\kid$ be a static KID and consider a connected component $\{\lambda>0\}_{0}$ of $\{ \lambda>0 \}$. If $Y^i \nablaSigma_i \lambda \geq 0$ or $Y^i \nablaSigma_i \lambda \leq 0$ on an arc-connected component $\E$ of $\tbd \{\lambda>0\}_{0}$, then $\E$ is a smooth submanifold (i.e. injectively immersed) of $\Sigma$. [**Proof.**]{} If there are no fixed points in $\E$, the result follows from Lemma \[surfaceNoFixed\]. Let us therefore assume that there is at least one fixed point $\p \in \E$. The idea of the proof proceeds in three stages. The first stage will consist in showing that $Y^i \nablaSigma_i \lambda \geq 0$ (or $Y^i \nablaSigma_i \lambda \leq 0$) forces all fixed points in $\E$ to be non-transverse. The second one consists in proving that, in a neighbourhood of a non-transverse fixed point, $\E$ is a $C^1$ submanifold. In the third and final stage we prove that $\E$ is, in fact, $C^{\infty}$. [*Stage 1.*]{} We argue by contradiction. Assume the fixed point $\p$ is transverse. Lemma \[structurebneq0\] implies that either $\{\lambda>0\}_{0} \cap U_{\p} = \{x > \frac{|b| |y|}{\mu} \}$ or $\{\lambda>0\}_{0} \cap U_{\p} = \{ x < - \frac{|b| |y|}{\mu} \}$. We treat the first case (the other is similar). The boundary of $\{\lambda>0\}_{0} \cap U_{\p}$ is connected and given by $x = x_+(y)$ for $y>0$ and $x=x_{-}(y)$ for $y<0$. Using $d\lambda = 2 \kappa {\mbox{\boldmath $Y$}}$ on this boundary, it follows $Y^i \nablaSigma_i \lambda = 2 \kappa Y_i Y^i$. But $\kappa$ has different signs on the branch $x=x_{+}(y)$ and on the branch $x=x_{-}(y)$, so $Y^i \nablaSigma_i \lambda$ also changes sign, against hypothesis. Hence $\p$ must be a non-transverse fixed point. [*Stage 2.*]{} Let us show that there exists a neighbourhood of $\p$ where $\E$ is $C^1$. Being $\p$ non-transverse, we have $f_{ij} |_{\p} =0$ and, consequently, the Hessian of $\lambda$ reads $$\label{Hessianlambda} \nablaSigma_i \nablaSigma_j \lambda |_{\p} = 2 \nablaSigma_i N \nablaSigma_j N |_{\p},$$ which has signature $\{+,0,0\}$. Similarly as in Lemma \[I1&lt;0\], the Gromoll-Meyer splitting Lemma (see Appendix \[ch:appendix2\]) implies the existence of an open neighbourhood $U_{\p}$ of $\p$ and coordinates $\{x,z^A \}$ in $U_{\p}$ such that $\p = \{x=0, z^A =0\}$ and $\lambda = Q_0^2 x^2 - \zeta (z)$, where $\zeta$ is a smooth function satisfying $\zeta|_{\p} =0$, $\nablaSigma_i \zeta |_{\p} =0$ and $\nablaSigma_{i}\nablaSigma_{j} \zeta |_{\p} =0$, and $Q_{0}$ is a positive constant. Moreover, evaluating the Hessian of $\lambda=Q^{2}_{0}x^{2}-\zeta(z)$ and comparing with (\[Hessianlambda\]) gives $dx |_{\p} = Q_0^{-1} dN |_{\p}$. Let us first show that there exists a neighbourhood $V_{\p}$ of $\p$ where $\zeta \geq 0$. The surfaces $\{ N=0 \}$ and $\{ x=0 \}$ are tangent at $\p$. This implies that there exists a neighbourhood $V_{\p}$ of $\p$ in $\Sigma$ such that the integral lines of $\partial_x$ are transverse to $\{N=0 \}$. Assume $\zeta(z) <0$ on any of these integral lines. If follows that $\lambda = Q_0^2 x^2 - \zeta$ is positive everywhere on this line. But at the intersection with $\{N=0\}$ we have $\lambda = N^2 - Y^i Y_i = - Y^i Y_i \leq 0$. This gives a contradiction and hence $\zeta(z)\geq 0$ in $V_{\p}$ as claimed. The set of points $\{ \lambda > 0 \} \cap V_{\p}$ is given by the union of two disjoint connected sets namely $W_{+} \equiv \{ x > + \frac{\sqrt{\zeta}}{Q_{0}} \}$ and $W_{-} \equiv \{ x < - \frac{\sqrt{\zeta}}{Q_{0}} \}$. On a connected component of $\{\lambda >0\}$ (in particular on $\{\lambda>0\}_{0}$) we have that $N = \sqrt{\lambda + Y^i Y_i}$ must be either everywhere positive or everywhere negative. On the other hand, for $\delta >0$ small enough $N |_{(x=\delta, z^A=0)}$ must have different sign than $N |_{(x=-\delta, z^A=0)}$ (this is because $\partial_x N |_{\p}= dN (\partial _x)|_{\p} = Q_0 dx (\partial_x) |_{\p} >0)$. It follows that either $\{\lambda>0\}_{0} \cap V_{\p} = W_{+}$ (if $N>0$ in $\{\lambda>0\}_{0}$) or $\{\lambda>0\}_{0} \cap V_{\p} = W_{-}$ (if $N<0$ in $\{\lambda>0\}_{0}$). Consequently, $\E$ is locally defined by $x = \frac{\epsilon \sqrt{\zeta}}{Q_0}$, where $\epsilon$ is the sign of $N$ in $\{\lambda>0\}_{0}$. Now, we need to prove that $+ \sqrt{\zeta}$ is $C^1$. This requires studying the behavior of $\zeta$ at points where it vanishes. The set of fixed points $\p' \in V_{\p}$ is given by $\{ x=0, \zeta(z)=0\}$ (this is a consequence of the fact that fixed points in $\E$ are characterized by the equations $\lambda=0$ and $d\lambda=0$, or equivalently $x=0$, $\zeta=0$, $d\zeta=0$. Since, for non-negative functions, $\zeta=0$ implies $d\zeta=0$ the statement above follows). The Hessian of $\lambda$ on any fixed point $\p' \subset V_{\p}$ reads $\nablaSigma_i \nablaSigma_j \lambda |_{\p'} = 2 Q_0^2 (dx \otimes dx)_{ij} - \nablaSigma_{i}\nablaSigma_{j} \zeta |_{\p'}$. Since $\p'$ must be a non-transverse fixed point, we have $\nablaSigma_i Y_j |_{\p'} =f_{ij}|_{\p'}=0$ and hence $\nablaSigma_{i}\nablaSigma_{j}\lambda |_{\p'}= 2\nablaSigma_{i}N\nablaSigma_{j}N |_{\p'}$ which has rank 1. Consequently, $\nablaSigma_{i}\nablaSigma_{j} \zeta |_{\p'} =0$. So, at all points where $\zeta$ vanishes we not only have $d\zeta =0$ but also $\nablaSigma_{i}\nablaSigma_{j} \zeta =0$. We can now apply a theorem by Glaeser (see Appendix \[ch:appendix2\]) to conclude that the positive square root $u \equiv \frac{+\sqrt{\zeta}}{Q_{0}}$ is $C^1$, as claimed. [*Stage 3.*]{} Finally, we will prove that $\E$ is, in fact, $C^{\infty}$ in a neighbourhood of $\p$ (we already know that $\E$ is smooth at non-fixed points) This is equivalent to proving that the function $x=\epsilon u(z)$ is $C^{\infty}$. Since $u=\frac{+\sqrt{\zeta}}{Q_{0}}$ and $\zeta\geq 0$, it follows that $u$ is smooth at any point where $u>0$. The proof will proceed in two steps. In the first step we will show that $u$ is $C^{2}$ at points where $u$ vanishes and then, we will improve this to $C^{\infty}$. Let us start with the $C^{2}$ statement. At points where $u \neq 0$, we have $Y_{i} |_{(x= \epsilon u(z),z^A)}= \frac{1}{2\kappa} \nablaSigma_i \lambda |_{(x=\epsilon u(z),z^A)}$. Hence $Y_i$ is non-zero and orthogonal to $\E$ on such points. Pulling back equation $\nablaSigma_i Y_j + \nablaSigma_j Y_i + 2 N K_{ij}=0$ onto $\E\cap\{x\neq 0\}$, we get $$\label{kappaandK} \kappa_{AB} + \epsilon \sigma K_{AB}=0,$$ where $\sigma$ is the sign of $\kappa$ , $K_{AB}$ is the pull-back of $K_{ij}$ on the surface $\{x=\epsilon u(z)\}$ and $\kappa_{AB}$ is the second fundamental form of this surface with respect to the unit normal pointing inside $\{\lambda >0 \}_{0}$. By assumption $Y^i \nablaSigma_i \lambda$ has constant sign on $\E$. This implies that $\sigma$ is either everywhere $+1$ or everywhere $-1$. So, the graph $x= \epsilon u(z)$ satisfies the set of equations $\kappa_{AB} + \epsilon \sigma K_{AB}=0$ on the open set $\{ z^A ; u(z) > 0 \}\subset \mathbb{R}^{2}$. In the local coordinates $\{z^A\}$ these equations takes the form $$\begin{aligned} - \partial_{A} \partial_B u(z) + \chi_{AB}(u(z),\partial_C u(z), z )=0 \label{fulleq}\end{aligned}$$ where $\chi$ is a smooth function of its arguments which satisfies $\chi_{AB}(u=0,\partial_{C} u =0, z) = \epsilon\hat{\kappa}_{AB} (z) + \sigma \hat{K}_{AB} (z)$, where $\hat{\kappa}_{AB}$ is the second fundamental form of the surface $\{ x=0 \}$ (with respect to the outer normal pointing towards $\{x>0\}$) at the point with coordinates $\{z^{A}\}$ and $\hat{K}_{AB}$ is the pull-back of $K_{ij}$ on this surface at the same point. Take a fixed point $\p' \in \E$ not lying within an open set of fixed points (if $\p'$ lies on an open set of fixed points we have $u\equiv 0$ on the open set and the statement that $u$ is $C^{\infty}$ is trivial). It follows that $\p' \in \{x=0 \}$ and that the coordinates $z_0^A$ of $\p'$ satisfy $z_0^A \in \tbd\{ z^A ; u(z)>0 \} \subset \mathbb{R}^2$. By stage $2$ of the proof, the function $u(z)$ is $C^1$ everywhere and its gradient vanishes wherever $u$ vanishes. It follows that $u\big|_{z^A_0} = \partial_B u \big|_{z^A_0} = 0$. Being $u$ continuously differentiable, it follows that the term $\chi_{AB}$ in (\[fulleq\]) is $C^0$ as a function of $z^C$ and therefore admits a limit at $z^C_0$. It follows that $\partial_{A} \partial_{B} u$ also has a well-defined limit at $z^C_0$, and in fact this limit satisfies $$\begin{aligned} \partial_A \partial_B u \big|_{z^C_0} = \hat{\kappa}_{AB} \big|_{z^C_0} + \epsilon \sigma \hat{K}_{AB} \big|_{z^C_0}.\end{aligned}$$ This shows that $u$ is in fact $C^2$ everywhere. But taking the trace of $\kappa_{AB}+\epsilon\sigma K_{AB}=0$, we get $p + \epsilon\sigma q =0$, where $p$ is the mean curvature of $\E$ and $q$ is the trace of the pull-back of $K_{ij}$ on $\E$. This is an elliptic equation in the coordinates $\{z^{A}\}$ (see e.g. [@AMS]), so $C^2$ solutions are smooth as a consequence of elliptic regularity [@GilbargTrudinger]. Thus, the function $u (z)$ is $C^{\infty}$. $\hfill \blacksquare$\ Knowing that this submanifold is differentiable, our next aim is to show that, under suitable circumstances it has vanishing outer null expansion. This is the content of our next proposition. \[is\_a\_MOTS\] Let $\kid$ be a static KID and consider a connected component $\{\lambda>0\}_{0}$ of $\{ \lambda>0 \}$ with non-empty topological boundary. Let $\E$ be an arc-connected component $\tbd \{\lambda>0\}_{0}$ and assume - $N Y^i \nablaSigma_i \lambda|_{\E} \geq 0$ if $\E$ contains at least one fixed point. - $N Y^i m_i |_{\E} \geq 0$ if $\E$ contains no fixed point, where $\vec{m}$ is the unit normal pointing towards $\{\lambda>0\}_{0}$. Then $\E$ is a smooth submanifold (i.e. injectively immersed) with $\theta^{+}=0$ provided the outer direction is defined as the one pointing towards $\{\lambda>0\}_{0}$. Moreover, if $I_{1}\neq 0$ in $\E$, then $\E$ is embedded. [**Remark.**]{} If the inequalities in (i) and (ii) are reversed, then $\E$ has $\theta^{-}=0$. $\hfill \square$\ [**Proof.**]{} Consider first the case when $\E$ has at least one fixed point. Since, on $\E$, $N$ cannot change sign and vanishes only if $\vec{Y}$ also vanishes, the hypothesis $N Y^i \nablaSigma_i \lambda |_{\E} \geq 0$ implies either $Y^i \nablaSigma_i \lambda |_{\E} \geq 0$ or $Y^i \nablaSigma_i \lambda |_{\E} \leq 0$ and, therefore, Proposition \[C1\] shows that $\E$ is a smooth submanifold. Let $\vec{m}$ be the unit normal pointing towards $\{\lambda>0\}_{0}$ and $p$ the corresponding mean curvature. We have to show that $\theta^{+}=p + \gamma^{AB} K_{AB}$ (see equation (\[Hp\])) vanishes. Open sets of fixed points are immediately covered by Proposition \[Totally geodesic\] because this set is then totally geodesic and $K_{AB}=0$, so that both null expansions vanish. On the subset $V\subset \E$ of non-fixed points we have $Y_{i}\big|_{V}=\frac{1}{2\kappa}\nablaSigma_{i}\lambda\big|_{V}$ (see equation \[kappa\]) and, therefore, $Y_{i}\big|_{V}=|N|\text{sign}(\kappa)m_{i}\big|_{V}$. The condition $NY^{i}\nablaSigma_{i}\lambda\geq 0$ imposes $\text{sign}(N)\text{sign}(\kappa)=1$ or, in the notation of the proof of Proposition \[C1\], $\epsilon\sigma=1$. Equation $p+q=0$ follows directly from (\[kappaandK\]) after taking the trace. For the case $(ii)$, we know that $\E$ is smooth from Lemma \[surfaceNoFixed\] and, hence, $\vec{m}$ exists (this shows in particular that hypothesis (ii) is well-defined). Since $\E$ lies in a Killing prehorizon in the Killing development of the KID, it follows that $\vec{\xi}$ is orthogonal to $\E$ and hence that $\vec{Y}$ is normal to $\E$ in $\Sigma$. Since $\vec{Y}^2 = N^2$ on $\E$ it follows $\vec{Y}|_{\E}=N\vec{m}|_{\E}$ and the same argument applies to conclude $\theta^{+}=0$. To show that $\E$ is embedded if $I_{1}|_{\E}\neq 0$, consider a point $\p\in \E$. if $\p$ is a non-fixed point, we know that $\nablaSigma_{i}\lambda\big|_{\p}\neq 0$ and hence $\lambda$ is a defining function for $\E$ in a neighbourhood of $\p$. This immediately implies that $\E$ is embedded in a neighbourhood of $\p$. When $\p$ is a fixed point, we have shown in the proof of Proposition \[C1\] that there exists an open neighbourhood $V_{\p}$ of $\p$ such that, in suitable coordinates, $\overline{\{\lambda>0\}}\cap V_{\p}=\{x\geq u(z)\}$ or $\overline{\{\lambda>0\}}\cap V_{\p}=\{x\leq -u(z)\}$ for a non-negative smooth function $u(z)$. It is clear that the arc-connected component $\E$ is defined locally by $x=u(z)$ or $x=-u(z)$ and hence it is embedded. $\hfill \blacksquare$\ The confinement result {#mainresults} ---------------------- Now, we are ready to state and prove our confinement result. For simplicity, it will be formulated as a confinement result for outer trapped surfaces instead of weakly outer trapped surfaces. However, except for a singular situation, it can be immediately extended to weakly outer trapped surfaces (see Remark 1 after the proof). \[theorem2\] Consider a static KID $\kid$ satisfying the NEC and possessing a barrier $\Sb$ with interior $\Omegab$ (see Definition \[defi:barrier\]) which is outer untrapped and such that such that $\lambda\big|_{\Sb}>0$. Let $\{\lambda>0\}^{\text{ext}}$ be the connected component of $\{\lambda>0\}$ containing $\Sb$. Assume that every arc-connected component of $\tbd \ext$ with $I_{1}=0$ is topologically closed and 1. $NY^{i}\nablaSigma_{i}\lambda\geq 0$ in each arc-connected component of $\tbd \ext$ containing at least one fixed point. 2. $NY^{i}m_{i}\geq 0$ in each arc-connected component of $\tbd \ext$ which contains no fixed points, where $\vec{m}$ is the unit normal pointing towards $\{\lambda>0\}^{\text{ext}}$. Consider any surface $S$ which is bounding with respect to $\Sb$. If $S$ is outer trapped then it does not intersect $\{\lambda>0\}^{\text{ext}}$. ![Theorem \[theorem2\] forbids the existence of an outer trapped surface $S$ like the one in the figure (in blue). The striped area corresponds to the exterior of $S$ in $\Omegab$ and the shaded area corresponds to the set $\{\lambda>0\}^{\text{ext}}$ whose boundary is $\E_{0}$ (in red). Note that $\E_{0}$ may intersect $\bd \Sigma$.](figure3.eps){width="9cm"} [**Proof.**]{} We argue by contradiction. Let $S$ be an outer trapped surface which is bounding with respect to $\Sb$, satisfies the hypotheses of the theorem and intersects $\{\lambda>0\}^{\text{ext}}$. By definition of bounding, there exists a compact manifold $\Sigmatilde$ whose boundary is the disjoint union of the outer untrapped surface $\Sb$ and the outer trapped surface $S$. We work on $\Sigmatilde$ from now on. The Andersson and Metzger Theorem \[thr:AM\] implies that the topological boundary of the weakly outer trapped region $\tbd T^{+}$ in $\Sigmatilde$ is a stable MOTS which is bounding with respect to $\Sb$. We first show that $\tbd T^+$ necessarily intersects $\{\lambda>0\}^{\text{ext}}$. Indeed, consider a point $\r \in S$ with $\lambda |_{\r} >0$ (this point exists by hypothesis) and consider a path from $\r$ to $\Sb$ fully contained in $\{\lambda>0\}^{\text{ext}}$ (this path exists because $\{\lambda>0\}^{\text{ext}}$ is connected). Since $\r \in T^+$ it follows that this path must intersect $\tbd T^+$ as claimed. Furthermore, due to the maximum principle for MOTS (see Proposition \[maximumprincipleforMOTS\]), $\tbd T^{+}$ lies entirely in the exterior of $S$ in $\Omegab$ (here is where we use the hypothesis of $S$ being outer trapped instead of merely being weakly outer trapped). Let us suppose for a moment that $\tbd T^{+}\subset\overline{\{\lambda>0\}^{\text{ext}}}$. Then the Killing vector $N\vec{n}+\vec{Y}$ is causal everywhere on $\tbd T^{+}$, either future or past directed, and timelike somewhere on $\tbd T^{+}$. Since $\tbd T^{+}$ intersects $\{\lambda>0\}^{\text{ext}}$, there must be non-fixed points on $\tbd T^{+}$. If all points in $\tbd T^{+}$ are non-fixed, then we can construct the Killing development and Theorem \[corollaryextended\] can be applied at once giving a contradiction (note that $\tbd T^{+}$ is necessarily a locally outermost MOTS). When $\tbd T^{+}$ has fixed points we cannot construct the Killing development everywhere. However, let $V\subset\tbd T^{+}$ be a connected component of the set of non-fixed points in $\tbd T^+$ satisfying $V \cap \{\lambda >0 \} \neq \emptyset$ (this $V$ exists because $\lambda >0$ somewhere on $\tbd T^+$). Then, the Killing development still exists in an open neighbourhood of $V$. In this portion we can repeat the geometrical construction which allowed us to prove Theorem \[corollaryextended\] and define a surface $S'$ by moving $V$ a small, but finite amount $\tau$ along $\vec{\xi}$ to the past and back to $\Sigma$ along the outer null geodesics. Since $N$ and $\vec{Y}$ are smooth and approach zero at $\tbd V$ it follows that $S'$ and the set of fixed points in $\tbd T^{+}$ join smoothly and therefore define a closed surface $S''$. Clearly, $S''$ is weakly outer trapped and lies, at least partially, in the exterior of $\tbd T^{+}$, which is impossible. Until now, we have essentially applied the ideas of Theorem \[corollaryextended\]. When $\tbd T^{+}\not\subset\overline{\{\lambda>0\}^{\text{ext}}}$ new methods are required. However, the general strategy is still to construct a weakly outer trapped surface outside $\tbd T^{+}$ in $\Sigmatilde$. First of all, every arc-connected component $\E_{i}$ of $\tbd \ext$ with $I_1 \neq 0$ is embedded, as proven in Proposition \[is\_a\_MOTS\]. For an arc-connected component $\E_d$ with $I_1=0$ we note that, since no point on this set is a fixed point, it follows that there exists an open neighbourhood $U$ of $\E_d$ containing no fixed points. Thus, the vector field $\vec{Y}$ is nowhere zero on $U$. Staticity of the KID implies that ${\mbox{\boldmath $Y$}}$ is integrable (see (\[statictwo\])). It follows by the Fröbenius theorem that $U$ can be foliated by maximal, injectively immersed submanifolds orthogonal to $\vec{Y}$. $\E_d$ is clearly one of the leaves of this foliation because $\vec{Y}$ is orthogonal to $\E_d$ everywhere. By assumption, $\E_d$ is topologically closed. Now, we can invoke a result on the theory of foliations that states that any topologically closed leaf in a foliation is necessarily embedded (see e.g. Theorem 5 in page 51 of [@Neto]). Thus, each $\E_i$ is an embedded submanifold of $\Sigmatilde$. Since we know that $\tbd T^+$ intersects $\ext$ and we are assuming that $\tbd T^{+}\not\subset\overline{\{\lambda>0\}^{\text{ext}}}$, it follows that at least one of the arc-connected components $\{\E_i\}$, say $\E_0$, must intersect both the interior and the exterior of $\tbd T^+$ . In Proposition \[is\_a\_MOTS\] we have also shown that $\E_{0}$ has $\theta^{+}=0$ with respect to the direction pointing towards $\{\lambda>0\}^{\text{ext}}$. Thus, we have two intersecting surfaces $\tbd T^{+}$ and $\E_{0}$ which satisfy $\theta^{+}=0$. Moreover, $\tbd T^{+}$ is a stable MOTS. The idea is to use Lemma \[lemasmoothness\] by Kriele and Hayward to construct a weakly outer trapped surface $\hat{S}$ outside both $\tbd T^{+}$ and $\E_{0}$ and which is bounding with respect to $\Sb$. However, Lemma \[lemasmoothness\] can be applied directly only when both surfaces $\tbd T^{+}$ and $\E_{0}$ intersect transversally in a curve and this need not happen for $\E_{0}$ and $\tbd T^{+}$. To address this issue we use a technique developed by Andersson and Metzger in their proof of Theorems 5.1 and 7.6 in [@AM]. The idea is to use Sard Lemma (see Appendix \[ch:appendix2\]) in order to find a weakly outer trapped surface $\tilde{S}$ as close to $\tbd T^{+}$ as desired which does intersect $\E_{0}$ transversally. Then, the Kriele and Hayward smoothing procedure applied to $\tilde{S}$ and $\E_{0}$ gives a weakly outer trapped surface penetrating $\Sigmatilde\setminus T^{+}$, which is simply impossible. So, it only remains to prove the existence of $\tilde{S}$. Recall that $\tbd T^{+}$ is a stable MOTS. We will distinguish two cases. If $\tbd T^{+}$ is strictly stable, there exists a foliation $\{\Gamma_{s}\}_{s\in \left(-\epsilon,0\right]}$ of a one sided tubular neighbourhood ${\cal W}$ of $\tbd T^+$ in $T^+$ such that $\Gamma_{0}=\tbd T^+$ and all the surfaces $\{\Gamma_{s} \}_{s<0}$ have $\theta^{+}_{s}<0$. To see this, simply choose a variation vector $\vec{\nu}$ such that $\vec{\nu}\big|_{\tbd T^+}=\psi\vec{m}$ where $\psi$ is a positive principal eigenfunction of the stability operator $L_{\vec{m}}$ and $\vec{m}$ is the outer direction normal to $\tbd T^{+}$. Using $\delta_{\vec{\nu}}\theta^+=L_{\vec{m}}\psi=\lambda\psi>0$ it follows that the surfaces $\Gamma_{s}\equiv \varphi_{s}(\tbd T^+)$ generated by $\vec{\nu}$ are outer trapped for $s\in(-\epsilon,0)$. Next, define the mapping $\Phi: \E_{0}\cap ({\cal W} \setminus \tbd T^+) \rightarrow \left(-\epsilon,0\right)\subset\mathbb{R}$ which assigns to each point $\p\in \E_{0}\cap({\cal W} \setminus \tbd T^+)$ the corresponding value of the parameter of the foliation $s\in\left(-\epsilon,0\right)$ on $\p$. Sard Lemma (Lemma \[lema:Sard\]) implies that the set of regular values of the mapping $\Phi$ is dense in $\left(-\epsilon,0\right)\subset\mathbb{R}$. Select a regular value $s_{0}$ as close to $0$ as desired. Then, the surface $\tilde{S}\equiv \Gamma_{s_{0}}$ intersects transversally $\E_{0}$, as required. If $\tbd T^+$ is stable but [*not strictly stable*]{}, a foliation $\Gamma_s$ consisting on weakly outer trapped surfaces may not exist. Nevertheless, following [@AM], a suitable modification of the interior of $\tbd T^+$ in $\Sigma$ solves this problem. It is important to remark that, in this case, the contradiction which proves the theorem is obtained by applying the Kriele and Hayward Lemma in the modified initial data set. The modification is performed as follows. Consider the same foliation $\Gamma_{s}$ as defined above and replace the second fundamental form $K$ on the hypersurface $\Sigma$ by the following. $$\label{Ktilde} \tilde{K}=K-\frac 12 \phi(s) \gamma_{s},$$ where $\phi : \mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1,1}$ function such that $\phi(s)=0$ for $s\geq 0$ (so that the data remains unchanged outside $\tbd T^{+}$) and $\gamma_s$ is the projector to $\Gamma_s$. Then, the outer null expansion of $\Gamma_s$ computed in the modified initial data set $(\Sigma,g,\tilde{K})$ $${\tilde{\theta}^{+}}[\Gamma_{s}]={{\theta}^{+}}[\Gamma_{s}]-\phi(s),$$ where ${{\theta}^{+}}[\Gamma_{s}]$ is the outer null expansion of $\Gamma_{s}$ in $(\Sigma,g,K)$. Since $\tbd T^{+}$ was a stable but not strictly stable MOTS in $(\Sigma,g,K)$, ${\theta^+}[\Gamma_{s}]$ vanishes at least to second order at $s=0$. On $s \leq 0$, define $\phi(s)=bs^2$ with $b$ a sufficient large constant. It follows that for some $\epsilon >0$ we have ${\tilde{\theta}^{+}}[\Gamma_{s}]<0$ on all $\Gamma_s$ for $s \in (-\epsilon , 0)$. Working with this foliation, Sard Lemma asserts that a weakly outer trapped surface $\Gamma_{s_0}$ lying as close to $\tbd T^{+}$ as desired and intersecting $\E_0$ transversally can be chosen in $(\Sigma,g,\tilde{K})$. Furthermore, the surface $\E_{0}$ also has non-positive outer null expansion in the modified initial data, at least for $s$ sufficiently close to zero. Indeed, this outer null expansion $\tilde{\theta}^{+} [\E_0]$ reads $\tilde{\theta}^{+} [\E_0]=p [\E_0]+\tr_{\E_{0}}\tilde{K}$. By (\[Ktilde\]), we have $\tr_{\E_{0}}\tilde{K}\big|_{\r}=\tr_{\E_{0}}K\big|_{\r}-\frac12\phi(s_{\r})\tr_{\E_{0}}\gamma_{s_{\r}}$, at any point $\r\in\E_{0}$, where $s_{\r}$ is the value of the leaf $\Gamma_{s}$ containing $\r$, i.e. $\r \in \Gamma_{s_{\r}}$. Since $\tr_{\E_{0}} \gamma_{s}\geq 0$ (because the pull-back of $\gamma_{s}$ is positive semi-definite) we have $\tr_{\E_{0}}\tilde{K} = \tr_{\E_{0}} K$ for $s \geq 0$ and $\tr_{\E_{0}}\tilde{K} \leq \tr_{\E_{0}} K$ for $s<0$ (small enough). In any case $\tilde{\theta}^{+} (\E_0) \leq \theta^{+} (\E_0) =0$ and we can apply the Kriele and Hayward Lemma to $\Gamma_{s_{0}}$ and $\E_{0}$ to construct a weakly outer trapped surface which is bounding with respect to $\Sb$, lies in the topological closure of the exterior of $\tbd T^+$ and penetrates this exterior somewhere. Since the geometry outside $\tbd T^{+}$ has not been modified, this gives a contradiction. $\hfill\blacksquare$\ [**Remark 1.**]{} This theorem has been formulated for outer trapped surfaces instead of weakly outer trapped surfaces. The reason is that in the proof we have used a foliation in the [*inside*]{} part of a tubular neighbourhood of $\tbd T^{+}$. If $S$ satisfies $\theta^+=0$, it is possible that $S=\bd \Sigma = \tbd T^+$ and then we would not have room to use this foliation. It follows that the hypothesis of the theorem can be relaxed to $\theta^{+}\leq 0$ if one of the following conditions hold: 1. $S$ is not the outermost MOTS. 2. $S\cap \bd\Sigma=\emptyset$. 3. The KID $\kid$ can be isometrically embedded into another KID $(\hat{\Sigma},\hat{g},\hat{K},\hat{N},\vec{\hat{Y}},\hat{\tau})$ with $\bd\Sigma\subset \text{int}(\hat{\Sigma})$ In this case, Theorem \[theorem2\] includes Miao’s theorem in the particular case of asymptotically flat time-symmetric vacuum static KID with minimal compact boundary. This is because in the time-symmetric case all points with $\lambda=0$ are fixed points and hence there are no arc-connected components of $\tbd \{ \lambda > 0 \}$ with $I_1=0$ and $Y^{i}\nabla^{\Sigma}_{i}\lambda$ is identically zero on $\tbd \ext$. $\hfill \square$\ [**Remark 2.**]{} In geometric terms, hypotheses $1$ and $2$ of the theorem exclude a priori the possibility that $\tbd \ext$ intersects the white hole Killing horizon at non-fixed points. A similar theorem exists for initial data sets which do not intersect the black hole Killing horizon (more precisely, such that both inequalities in $1$ and $2$ are satisfied with the reversed inequality signs). The conclusion of the theorem in this case is that no bounding [*past*]{} outer trapped surface can intersect $\{\lambda>0\}^{\text{ext}}$ provided $\Sb$ is a [*past*]{} outer untrapped barrier (the proof of this statement can be obtained by applying Theorem \[theorem2\] to the static KID $(\Sigma,g,-K;-N,\vec{Y}; \rho, -\vec{J}, \tau)$). No version of this theorem, however, covers the case when $\tbd \ext$ intersects both the black hole and the white hole Killing horizon. The reason is that, in this setting, $\tbd \ext$ is, in general, not smooth and we cannot apply the Andersson-Metzger theorem to $\Sigmatilde$. In the next chapter we will address this case in more detail. $\hfill \square$\ For the particular case of KID possessing an asymptotically flat end we have the following corollary, which is an immediate consequence of Theorem \[theorem2\]. \[corollarytheorem2\] Consider a static KID $\kid$ with a selected asymptotically flat end $\Sigma_{0}^{\infty}$ and satisfying the NEC. Denote by $\{\lambda>0\}^{\text{ext}}$ the connected component of $\{\lambda>0\}$ which contains the asymptotically flat end $\Sigma_{0}^{\infty}$. Assume that every arc-connected component of $\tbd \ext$ with $I_{1}=0$ is closed and 1. $NY^{i}\nablaSigma_{i}\lambda\geq 0$ in each arc-connected component of $\tbd \{\lambda>0\}^{\text{ext}}$ containing at least one fixed point. 2. $NY^{i}m_{i}\geq 0$ in each arc-connected component of $\tbd \{\lambda>0\}^{\text{ext}}$ which contains no fixed points, where $\vec{m}$ is the unit normal pointing towards $\{\lambda>0\}^{\text{ext}}$. Then, any bounding (see Definition \[defi:bounding\]) outer trapped surface $S$ in $\Sigma$ cannot intersect $\{\lambda>0\}^{\text{ext}}$. Uniqueness of static spacetimes with weakly outer trapped surfaces ================================================================== \[ch:Article4\] Introduction {#sectionintroduction} ------------ In this chapter we will extend the classic static black hole uniqueness theorems to asymptotically flat static KID containing weakly outer trapped surfaces. As emphasized in the previous chapter, the first step for this extension was given by Miao for the particular case of asymptotically flat, time-symmetric, static and vacuum KID, with compact minimal boundary (Theorem \[thr:Miao\]). Indeed, our aim of extending the classic uniqueness theorems for static black holes to the quasi-local setting can be reformulated as generalizing Theorem \[thr:Miao\] to non-vanishing matter (as long as the NEC is satisfied) and arbitrary slices (not necessarily time-symmetric) containing weakly outer trapped surfaces. In the previous chapter we obtained a generalization of this result as a confinement result. In this chapter we address the extension of Miao’s theorem as a uniqueness result. As we already know, the most powerful method to prove uniqueness of static black holes is the [*doubling method*]{} of Bunting and Masood-ul-Alam. This method was described in some detail in Section \[sc:UniquenessOfBlackHoles\] where we gave a sketch of the proof of the uniqueness theorem for static electro-vacuum black holes. In the present chapter, our strategy will be precisely to recover the framework of the doubling method from an arbitrary static KID containing a weakly outer trapped surface. As it was discussed in Section \[sc:UniquenessOfBlackHoles\], this framework consists of an asymptotically flat spacelike hypersurface $\Sigma$ with topological boundary $\tbd \Sigma$ which is a closed (i.e. compact and without boundary) embedded topological manifold and such that the static Killing field is causal on $\Sigma$ and null only on $\tbd \Sigma$. As we pointed out in Section \[sc:UniquenessOfBlackHoles\], the existence of this topological manifold $\tbd \Sigma$ is ensured precisely by the presence of a black hole. Note that $\tbd \Sigma$ is not required to be smooth. Hence, our strategy to conclude uniqueness departing from a static KID $\kid$ with an asymptotically flat end $\Sigma_{0}^{\infty}$ which contains a bounding MOTS $S$ will be therefore to prove that the topological boundary $\tbd \ext$, where $\ext$ is the connected component of $\{\lambda>0\}$ in $\Sigma$ which contains $\Sigma_{0}^{\infty}$, is a closed embedded topological submanifold. Since a priori MOTS have nothing to do with black holes, $\tbd \ext$ may fail to be closed (see Figure \[fig:problem\]) as required in the doubling method. Consequently, throughout this chapter we will study under which conditions we can guarantee that $\tbd \ext$ is closed. In fact, it turns out that the confinement Theorem \[theorem2\] and its Corollary \[corollarytheorem2\] are already sufficient to conclude that $\tbd \ext$ is a closed surface. This leads to our first uniqueness result. ![The figure illustrates a situation where $\tbd \ext$ (in red) has non-empty manifold boundary (which lies in $\bd \Sigma$) and, therefore, is not closed. Here, $S$ (in blue) represents a bounding MOTS and the grey region corresponds to $\ext$. In a situation like this the doubling method cannot be applied.[]{data-label="fig:problem"}](problem.eps){width="9cm"} \[uniquenessthr0\] Consider a static KID $\kid$ with a selected asymptotically flat end $\Sigma^{\infty}_0$ and satisfying the NEC. Assume that $\Sigma$ possesses an outer trapped surface $S$ which is bounding. Denote by $\ext$ the connected component of $\{\lambda>0\}$ which contains the asymptotically flat end $\Sigma_{0}^{\infty}$. If 1. Every arc-connected component of $\tbd \ext$ with $I_1=0$ is topologically closed. 2. $NY^{i}\nablaSigma_{i}\lambda\geq 0$ in each arc-connected component of $\tbd \ext$ containing at least one fixed point. 3. $NY^{i}m_{i}\geq 0$ in each arc-connected component of $\tbd \ext$ which contains no fixed points, where $\vec{m}$ is the unit normal pointing towards $\{\lambda>0\}^{\text{ext}}$. 4. The matter model is such that Bunting and Masood-ul-Alam doubling method gives uniqueness of black holes. Then, $(\ext,g,K)$ is a slice of such a unique spacetime. [**Proof.**]{} Proposition \[is\_a\_MOTS\] implies that $\tbd \ext$ is a smooth submanifold with $\theta^+ =0$ with respect to the normal pointing towards $\ext$. We only need to show that $\tbd \ext$ is closed (i.e. embedded, compact and without boundary) in order to apply hypothesis 4 and conclude uniqueness. By definition of bounding in the asymptotically flat setting (see Definition \[defi:bounding\]) we have a compact manifold $\Sigmatilde$ with boundary $\partial \Sigmatilde = S \cup \Sb$, where $\Sb = \{ r =r_0 \}$ is a sufficiently large coordinate sphere in $\Sigma^{\infty}_0$. Take this sphere large enough so that $\{ r \geq r_0 \} \subset \ext$. We are in a setting where all the hypothesis of Theorem \[theorem2\] hold. In the proof of this theorem we have shown that $\tbd \ext$ is embedded and compact. Moreover, $\tbd T^+$ lies in the interior $\mbox{int} (\Sigmatilde)$ and does not intersect $\ext$. This, clearly prevents $\tbd \ext$ from reaching $S$, which in turn implies that $\tbd \ext$ has no boundary. $\hfill \blacksquare$\ [**Remark.**]{} This theorem applies in particular to static KID which are asymptotically flat, without boundary and have at least two asymptotic ends, as long as conditions 1 to 4 are fulfilled. To see this, recall that an asymptotically flat initial data is the union of a compact set and a finite number of asymptotically flat ends. Select one of these ends $\Sigma^{\infty}_0$ and define $S$ to be the union of coordinate spheres with sufficiently large radius on all the other asymptotic ends. This surface is an outer trapped surface which is bounding with respect to $\Sigma^{\infty}_0$ and we recover the hypotheses of Theorem \[uniquenessthr0\]. $\hfill \square$\ Theorem \[uniquenessthr0\] has been formulated for outer trapped surfaces instead of weakly outer trapped surfaces for the same reason as in Theorem \[theorem2\]. Consequently, the hypotheses of this theorem can also be relaxed to $\theta^{+}\leq 0$ if one of the following conditions hold: $S$ is not the outermost MOTS, $S\cap\bd \Sigma=\emptyset$, or the KID can be extended. Under these circumstances, this result already extends Miao’s theorem as a uniqueness result. Nevertheless, the theorem above requires several conditions on the boundary $\tbd \ext$. Since $\tbd\ext$ is a fundamental object in the doubling procedure, it is rather unsatisfactory to require conditions directly on this object. Out main aim in this chapter is to obtain a uniqueness result which does not involve any a priori restriction on $\tbd \ext$. As discussed in the previous chapter, $\tbd \ext$ is in general not a smooth submanifold (see e.g. Figure \[fig:figure1\]) and the techniques of the previous chapter cannot be applied to conclude that $\tbd \ext$ is a closed embedded topological submanifold. The key difficulty lies in proving that $\tbd \ext$ is a manifold without boundary. In the previous theorem, we used the non-penetration property of $\tbd T^+$ into $\ext$ in order to conclude that $\tbd \ext$ must lie in the exterior of the bounding outer trapped surface $S$ (which implies that $\tbd \ext$ is a manifold without boundary). In turn, this non-penetration property was strongly based on the smoothness of $\tbd \ext$, which we do not have in general. The main problem is therefore: How can we exclude the possibility that $\tbd \ext$ reaches $S$ in the general case? (see Figure \[fig:problem\]). To address this issue we need to understand better the structure of $\tbd \ext$ (and, more generally, of $\tbd \{ \lambda >0\}$) when conditions 2 and 3 are not satisfied. As we will discuss later, this will force us to view KID as hypersurfaces embedded in a spacetime, instead as abstract objects on their own, as we have done in the previous chapter. To finish this introduction, let us give a briefly summary of the chapter. In Section \[sc:preliminaries5\] we define the concept of an [*embedded static KID*]{} and present some known results on the structure of the spacetime in the neighbourhood of the fixed points of the isometry. In Section \[sectionproposition\] we will revisit the study of the properties of $\tbd\{\lambda>0\}$, this time for embedded static KID. Finally, Section \[sectionuniqueness\] is devoted to state and prove the uniqueness theorem for asymptotically flat static spacetimes containing a bounding weakly outer trapped surface. The results presented in this chapter have been summarized in [@CMere4] and will also be sent to publication [@CM4]. Embedded static KID {#sc:preliminaries5} ------------------- We begin this section with the definition of an embedded static KID. Recall that, according to our definitions, a spacetime has [**no boundary**]{}. An [**embedded static KID**]{} $\kid$ is a static KID, possibly with boundary, which is embedded in a spacetime $(M,\gM)$ with static Killing field $\vec{\xi}$ such that $\vec{\xi}\, |_{\Sigma}=N\vec{n}+\vec{Y}$, where $\vec{n}$ is the unit future directed normal of $\Sigma$ in $M$. [**Remark.**]{} If a static KID has no boundary and belongs to a matter model for which the Cauchy problem is well-posed (e.g. vacuum, electro-vacuum, scalar field, Yang-Mills field, $\sigma$-model, etc), it is clear that there exists a spacetime which contains the initial data set as a spacelike hypersurface. Whether this Cauchy development admits or not a Killing vector $\vec{\xi}$ compatible with the Killing data has only been answered in the affirmative for some special matter models, which include vacuum and electro-vacuum [@Coll]. Even in these circumstances, it is at present not known whether the spacetime thus constructed is in fact [*static*]{} (i.e. such that the Killing vector $\vec{\xi}$ is integrable). This property is obvious near points where $N \neq 0$ (i.e. points where $\vec{\xi}$ is transverse to $\Sigma$), but it is much less clear near fixed points, specially those with $I_1 <0$. Indeed, these points belong to a totally geodesic closed spacelike surface in the Cauchy development of the initial data set. The points lying in the chronological future of this surface cannot be reached by integral curves of the Killing vector starting on $\Sigma$. Proving that the Killing vector is integrable on those points is an interesting and, apparently, not so trivial task. In this thesis we do not explore this problem further and simply work with the definition of embedded static KID stated above. $\hfill \square$\ In what follows, we will review some useful results concerning the structure of the spacetime near fixed points of the static Killing $\vec{\xi}$. \[propositionRW\] Let $\kid$ be a static embedded KID and let $(M,\gM)$ be the static spacetime where the KID is embedded. Consider a fixed point $\p\in\tbd \{\lambda>0\} \subset \Sigma$ and let $S_0$ be the connected spacelike surface of fixed points in $M$ containing $\p$ (which exists by Theorem \[thr:Boyer\]). Then, there exists a neighbourhood ${\cal V}$ of $\p$ in $M$ and coordinates $\{u,v,x^{A} \}$ on ${\cal V}$ such that $\{x^{A}\}$ are coordinates for $S_0 \cap {\cal V}$ and the spacetime metric takes the Rácz-Wald-Walker form $$\label{RWmetric} \gRW=2Gdudv+\gamma_{AB}dx^A dx^B,$$ where $S_0 \cap {\cal V} = \{ u=v=0 \}$, $\partial_v$ is future directed and $G$ and $\gamma_{AB}$ are both positive definite and depend smoothly on $\{w \equiv uv,x^A\}$. [**Proof.**]{} Theorem \[thr:Boyer\] establishes that $\p$ belongs to a connected, spacelike, smooth surface $S_{0}$ which lies in the closure of a non-degenerate Killing horizon. Thus, we can use the Rácz-Wald-Walker construction, see [@RW], which shows that there exists a neighbourhood ${\cal V}$ of $\p$ and coordinates $\{ u,v,x^A\}$ adapted to $S_0 \cap {\cal V}$ such that the metric $\gM$ takes the form $$\label{RW} \gM=2Gdudv+2vH_{A}dx^{A}du+\gamma_{AB}dx^{A}dx^{B},$$ where $G$, $H_A$ and $\gamma_{AB}$ depend smoothly on $\{ w ,x^A \}$. In these coordinates, the Killing vector $\vec{\xi}$ reads $$\label{killingRW} \vec{\xi}=c^2\left( v\partial_{v}-u\partial_{u} \right),$$ where $c$ is a (non-zero) constant and $\partial_v$ is future directed. We only need to prove that staticity implies that $\{ u,v,x^A\}$ can be chosen in such a way that $H_{A}=0$. A straightforward computation shows that the integrability condition ${\mbox{\boldmath $\xi$}} \wedge d {\mbox{\boldmath $\xi$}} =0$ is equivalent to the following equations $$\begin{aligned} G \partial_{w} H_{A} - H_{A}\partial_{w}G&=&0, \label{RW1}\\ H_{[A}\partial_{B]}G + G\partial_{[A}H_{B]} &=& 0, \label{RW2} \\ H_{[A}\partial_{w}H_{B ]} & = & 0. \label{RW3}\end{aligned}$$ Equation (\[RW1\]) implies $H_{A}=f_{A}G$, where $f_{A}$ depend on $x^{C}$. Inserting this in (\[RW2\]), we get $\partial_{[A} f_{B]}=0$, which implies (after restricting ${\cal V}$ if necessary) the existence of a function $\zeta(x^{C})$ such that $f_{A}=\partial_{A}\zeta$. Equation (\[RW3\]) is then identically satisfied. Therefore, staticity is equivalent to $$\label{HG} H_{A}(w,x^{C})=G(w,x^{C}) \partial_{A}\zeta(x^{C}).$$ We look for a coordinate change $\{u,v,x^{C}\}\rightarrow \{u',v',x'^{C} \}$ which preserves the form of the metric (\[RW\]) and such that $H'_A=0$. It is immediate to check that an invertible change of the form $$\left\{ u=u(u'), v=v(v',{x'}^{C}),x^{A}={x'}^{A} \right\}$$ preserves the form of the metric and transforms $H_A$ as $$\begin{aligned} v'H'_{A}&=&\frac{d u}{d u'}\left(\frac{\partial v}{\partial x'^{A}}G + vH_{A}\right),\label{RW5}\end{aligned}$$ So, we need to impose $G \partial_A v + vH_{A} = 0$, which in view of (\[HG\]), reduces to $\partial_A v +v \partial_{A} \zeta=0$. Since $v=v' e^{-\zeta}$ (with $v'$ independent of $x^A$) solves this equation, we conclude that the coordinate change $$\left\{ u=u', v=v' e^{-\zeta(x'^{C})}, x^{A}=x'^{A}\right\}$$ brings the metric into the form (\[RW\]) (after dropping the primes). $\hfill \blacksquare$\ Now, let us consider an embedded static KID in a static spacetime with Rácz-Wald-Walker metric $({\cal V},\gRW)$. Since the vector $\partial_{v}$ is null on ${\cal V}$, it is transverse to $\Sigma \cap {\cal V}$ and, therefore, the embedding of $\Sigma \cap {\cal V}$ can be written locally as $$\label{embedding} \Sigma:(u,x^A)\rightarrow (u,v=\phi(u,x^A),x^A),$$ where $\phi$ is a smooth function. A simple computation using (\[killingRW\]) leads to $$\begin{aligned} \left.\lambda\right|_{\Sigma \cap {\cal V}} &=& 2c^{4}\hat{G} u\phi, \label{lambdaRW}\\ \left. N \right|_{\Sigma \cap {\cal V}} &=& \left( \phi+u\partial_{u}\phi \right)\sqrt{\frac{c^4 \hat{G}}{2\partial_{u}\phi-\hat{G} \partial_{A}\phi\partial^{A}\phi}}, \label{NRW}\\ \left. {\bf Y} \right|_{\Sigma \cap {\cal V}}&=& c^{2}\hat{G}\left( \phi du - u d\phi \right) \label{YRW}.\end{aligned}$$ where $\hat{G} \equiv G (w = u \phi, x^A)$ and indices $A,B,\dots$ are raised with the inverse of $\hat{\gamma}_{AB} \equiv \gamma_{AB} (w = u \phi, x^A ) $. Since $\Sigma$ is spacelike, the quantity ${2\partial_{u}\phi- \hat{G} \partial_{A}\phi\partial^{A}\phi}$ is positive. In particular, this implies that $$\label{partialuphi} \partial_{u}\phi>0,$$ which will be used later. For the sets $\{u=0\}$ and $\{\phi=0\}$ in $\Sigma \cap {\cal V}$ we have the following result. \[lemau0v0smooth\] Consider an embedded static KID $\kid$ and use Rácz-Wald-Walker coordinates $\{u ,v, x^A \}$ in a spacetime neighbourhood ${\cal V}$ of a fixed point $\p \in \tbd \{ \lambda > 0\} \subset \Sigma$ such that the embedding of $\Sigma$ reads (\[embedding\]). Then the sets $\{u=0\}$ and $\{\phi=0\}$ in $\Sigma \cap {\cal V}$ are both smooth surfaces (not necessarily closed). Moreover, a point $\p\in \tbd\{ \lambda>0\}$ in $\Sigma \cap {\cal V}$ is a non-fixed point if and only if $u\phi=0$ with either $u$ or $\phi$ non-zero. [**Proof:**]{} The lemma follows directly from the fact that both sets $\{ u=0 \}$ and $\{ \phi=0 \}$ in $\Sigma$ are the intersections between $\Sigma$ and the null smooth embedded hypersurfaces $\{u=0\}$ and $\{v=0\}$ in $({\cal V},\gRW)$, respectively. The second statement of the lemma is a direct consequence of equations (\[killingRW\]) and (\[lambdaRW\]). $ \hspace*{1cm} \hfill \blacksquare$\ Properties of $\tbd\left\{\lambda>0\right\}$ on an embedded static KID {#sectionproposition} ---------------------------------------------------------------------- In this section we will explore in more detail the properties of the set $\tbd\left\{\lambda>0\right\}$ in $\Sigma$. In particular, we will study the structure $\tbd\{\lambda>0\}$ in an embedded KID when no additional hypothesis are made. First, we will briefly recall some results of the previous chapter which will be used below. In Proposition \[Totally geodesic\] we showed that an open set of fixed points in $\tbd \{\lambda>0\}$ in a static KID $\kid$ is a smooth and totally geodesic surface. Moreover, Lemma \[surfaceNoFixed\] and Proposition \[is\_a\_MOTS\] imply that every arc-connected component of the open set of non-fixed points in $\tbd\{\lambda>0\}\subset\Sigma$ is a smooth submanifold (not necessarily embedded) of $\Sigma$ and has either $\theta^{+}=0$ or $\theta^{-}=0$. The structure of those arc-connected components of $\tbd \{\lambda>0\}$ having exclusively fixed points or exclusively non-fixed points is therefore clear with no need of additional assumptions. However, for the case of arc-connected components having both types of points an additional assumption on the sign of $NY^{i}\nablaSigma_{i}\lambda$ was required to conclude smoothness (see Propositions \[C1\] and \[is\_a\_MOTS\]). This hypothesis was imposed in order to avoid the existence of [*transverse*]{} fixed points in $\tbd \{\lambda>0\}$ (see stage 1 on the proof of Proposition \[C1\]). Actually, the existence of transverse points is, by itself, not very problematic. Indeed, as we showed in Lemma \[structurebneq0\], the structure of $\tbd\{\lambda>0\}$ on a neighbourhood of transverse fixed points is well understood and consists of two intersecting branches. The problematic situation happens when a sequence of transverse fixed points tends to a non-transverse point $\p$. In this case the intersecting branches can have a very complicated limiting behavior at $\p$. If we consider the non-transverse limit point $\p$, then we know from the previous chapter (see stage 2 on the proof of Proposition \[C1\]) that locally near $\p$ there exists coordinates such that $\lambda = Q_0^2x^2 - \zeta(z^A)$, with $\zeta$ a non-negative smooth function. In order to understand the behavior of $\tbd \{\lambda > 0 \}$ we need to take the square root of $\zeta$. Under the assumptions of Proposition \[C1\] we could show that the [*positive*]{} square root is $C^1$. For general non-transverse points, this positive square root is not $C^{1}$. In fact, is not clear at all whether there exists any $C^{1}$ square root (even allowing this square root to change sign). The following example shows a function $\zeta$ which admits no $C^{1}$ square root. It is plausible that the equations that are satisfied in a static KID forbid the existence of $\zeta$ functions with no $C^1$ square root. This is, however, a difficult issue and we have not been able to resolve it. This is the reason why we need to restrict ourselves to embedded static KID in this chapter. Assuming the existence of a static spacetime where the KID is embedded, it follows that, irrespectively of the structure of fixed points in $\Sigma$, a suitable square root of $\zeta$ always exists. [**Example.**]{} Non-negative functions do not have in general a $C^1$ square root. A simple example is given by the function $\rho = y^2 + z^2$ on $\mathbb{R}^2$. We know, however, that this type of example cannot occur for the function $\zeta$ because the Hessian of $\zeta$ must vanish at least on one point where $\zeta$ vanishes (and this is obviously not true for $\rho$). The following is an example of a non-negative function $\zeta$ for which the function and its Hessian vanish at one point and which admits no $C^1$ square root. Consider the function $\zeta(y,z)=z^2 y^{2}+z^{4}+f(y)$, where $f(y)$ is a smooth function such that $f(y)=0$ for $y\geq 0$ and $f(y)>0$ for $y<0$. Recall that the set of fixed points consists of the zeros of $\zeta$, and a fixed point is non-transverse if and only if the Hessian of $\zeta$ vanishes (see the proof of Proposition \[C1\]). It follows that the fixed points occur on the semi-line $\sigma\equiv \{y\geq 0,z=0\}$, with $(0,0)$ being non-transverse and $(y>0,z=0)$ transverse. Consider the points $\p=(1,-1)$ and $\q=(1,1)$. First of all take a curve $\gamma$ joining them in such a way that it does not intersect $\sigma$. It is clear that $\zeta$ remains positive along $\gamma$ and, therefore, its square root cannot change sign (if it is to be continuous). Now consider the curve $\gamma'= \{y=1,-1\leq z\leq 1 \}$ joining $\p$ and $\q$ (which does intersect $\sigma$). Since $\zeta\big|_{\gamma'}=z^{2}(1+z^{2})$, the only way to find a $C^{1}$ square root is by taking $u=z\sqrt{1+z^{2}}$, which changes sign from $\p$ to $\q$. This is a contradiction to the property above. So, we conclude that no $C^{1}$ square root of $\zeta$ exists. Let us see that, in the spacetime setting, this behavior cannot occur. Our first result of this section shows that the set $\tbd\{\lambda>0\}$ in an embedded KID is a union of compact, smooth surfaces which has one of the two null expansions equal to zero. \[proposition1\] Consider an embedded static KID $\kidtilde$, compact and possibly with boundary $\bd \Sigmatilde$. Assume that every arc-connected component of $\tbd \{ \lambda > 0 \}$ with $I_1 =0$ is topologically closed. Then $$\tbd \{\lambda>0\} = \underset{a}\cup S_a,$$ where each $S_a$ is a smooth, compact, connected and orientable surface such that its boundary, if non-empty, satisfies $\bd S_{a}\subset \bd\Sigmatilde$. Moreover, at least one of the two null expansions of $S_a$ vanishes everywhere. [**Proof.**]{} Let $\{ \SS_{\alpha} \}$ be the collection of arc-connected components of $\tbd \{ \lambda > 0\}$. We know that the quantity $I_1$ is constant on each $\SS_{\alpha}$ (see Lemma \[lema:I1constant\]). Consider an arc-connected component $\SS_d$ of $\tbd \{ \lambda > 0\}$ with $I_1 =0$. Since all points in this component are non-fixed, it follows that $\SS_d$ is a smooth submanifold. Using the hypothesis that arc-connected components with $I_1=0$ are topologically closed it follows that $\SS_d$ is, in fact, embedded. Choose $\vec{m}$ to be the unit normal satisfying $$\begin{aligned} \vec{Y} = N \vec{m}, \label{choice_m_degenerate}\end{aligned}$$ on $\SS_d$. This normal is smooth (because neither $\vec{Y}$ nor $N$ vanish anywhere on $\SS_d$), which implies that $\SS_d$ is orientable. Inserting $\vec{Y} = N \vec{m}$ into equation (\[kid1\]) and taking the trace it follows $$\begin{aligned} p + q =0. \label{theta_plus_degenerate}\end{aligned}$$ Consider now a $\SS_{\alpha}$ with $I_1 \neq 0$. At non-fixed points we know that $\SS_{\alpha}$ is a smooth embedded surface with $ \nablaSigma_i\lambda \neq 0$. On those points, define a unit normal $\vec{m}$ by the condition $$\begin{aligned} N \vec{m} (\lambda ) > 0 \label{defm}\end{aligned}$$ We also know that $\nablaSigma_i \lambda = 2 \kappa Y_i$ where $I_1 = - 2 \kappa^2$. Let us see that $\SS_{\alpha} = \SS_{1,\alpha} \cup \SS_{2,\alpha}$, where each $\SS_{1,\alpha}$ and $\SS_{2,\alpha}$ is a smooth, embedded, connected and orientable surface. To that aim, define $$\begin{aligned} \SS_{1,\alpha} & = & \{ \p \in \SS_{\alpha} \mbox{ such that } \kappa \big|_{\p} > 0 \} \cup \{ \mbox{ fixed points in } \SS_{\alpha} \}, \\ \SS_{2,\alpha} & = & \{ \p \in \SS_{\alpha} \mbox{ such that } \kappa \big|_{\p}< 0 \} \cup \{ \mbox{ fixed points in } \SS_{\alpha} \}.\end{aligned}$$ Notice that the fixed points are assigned to [*both*]{} sets. It is clear that at non-fixed points, both $\SS_{1,\alpha}$ and $\SS_{2,\alpha}$ are smooth embedded surfaces. Let $\q$ be a fixed point in $\SS_{\alpha}$ and consider the Rácz-Wald-Walker coordinate system discussed in Proposition \[propositionRW\]. The points in $\SS_{\alpha} \cap {\cal V}$ are characterized by $\{ u \phi =0 \}$ (due to (\[lambdaRW\])). Inserting (\[lambdaRW\]) and (\[YRW\]) into $\nablaSigma_i \lambda = 2 \kappa Y_i$ yields, at any non-fixed point $\q' \in \SS_{\alpha} \cap {\cal V}$, $$2c^2\left( \phi du+ud\phi \right) |_{\q'}= 2\kappa \left( \phi du -u d\phi \right) |_{\q'} \nonumber.$$ Since $du\neq 0$ (because $u$ is a coordinate) and $d\phi\neq 0$ (see equation (\[partialuphi\])) we have $$\begin{aligned} \label{sign(kappa)} \kappa>0 & \text{on}\quad \{u=0,\phi\neq 0\}, \nonumber\\ \kappa<0 & \text{on}\quad \{u\neq 0,\phi=0\}.\end{aligned}$$ Consequently, the non-fixed points in $\SS_{1,\alpha} \cap {\cal V}$ are defined by the condition $\{ u=0,\phi \neq 0 \}$ and the non-fixed points in $\SS_{2,\alpha} \cap {\cal V}$ are defined by the condition $\{ u \neq 0, \phi =0\}$. It is then clear that $\SS_{1,\alpha} \cap {\cal V} = \{ u=0 \}$ and $\SS_{2,\alpha} \cap {\cal V} = \{ \phi = 0 \}$, which are smooth embedded surfaces. It remains to see that the unit normal $\vec{m}$, which has been defined only at non-fixed points via (\[defm\]), extends to a well-defined normal to all of $\SS_{1,\alpha}$ and $\SS_{2,\alpha}$ (see Figure \[fig:cruz\]). ![In the Rácz-Wald-Walker coordinate system we define four open regions by $I=\{u>0\}\cap\{\phi> 0\}, II=\{u<0\}\cap\{\phi> 0\}, III=\{u<0\}\cap\{\phi< 0\}, IV=\{u>0\}\cap\{\phi< 0\}$. The normal on its boundaries which satisfies (\[defm\]) is depicted in red color. It is clear graphically that these normals extend smoothly to the fixed points on the hypersurfaces $\{u=0\}$ and $\{ \phi=0 \}$, such as $\q$ in the figure. This figure is, however, only schematic because one dimension has been suppressed and fixed points need not be isolated in general. A formal proof that $\vec{m}$ extends smoothly in all cases is given in the text.[]{data-label="fig:cruz"}](cruz.eps){width="6cm"} This requires to check that the condition (\[defm\]), when evaluated on ${\cal V}$ defines a normal which extends smoothly to the fixed points. Consider first the points $\{ u \neq 0, \phi =0 \}$. The unit normal to this surface is $\vec{m} = \epsilon | \nablaSigma \phi|^{-1}_{g} {\nablaSigma}\phi$ where $\epsilon = \pm 1$ and may, a priori, depend on the point. Since $$\begin{aligned} \left. N \right|_{\{ u \neq 0 ,\phi =0\}} &=& u\partial_{u}\phi \sqrt{\frac{c^4 \hat{G}}{2\partial_{u}\phi-\hat{G} \partial_{A}\phi\partial^{A}\phi}}, \\ \left . \nablaSigma_i \lambda \right|_{\{ u \neq 0 ,\phi =0\}} &=& 2 c^4 \hat{G} u \nablaSigma_i \phi,\end{aligned}$$ expression (\[defm\]) implies $$\begin{aligned} 0 < N \vec{m} (\lambda ) |_{ \{ u\neq0, \phi =0 \}} = 2 \epsilon c^4 \hat{G} u^2 \partial_u \phi |\nablaSigma \phi|_{g} \sqrt{\frac{c^4 \hat{G}}{2\partial_{u}\phi-\hat{G} \partial_{A}\phi\partial^{A}\phi}}.\end{aligned}$$ Hence $\epsilon = 1$ at all points on $\{ u \neq 0, \phi =0 \}$. Thus the normal vector reads $\vec{m} = | \nablaSigma \phi|^{-1}_{g} {\nablaSigma} \phi $ at non-fixed points, and this field clearly extends smoothly to all points on $\SS_{1,\alpha} \cap {\cal V}$. This implies, in particular, that $\SS_{1,\alpha}$ is orientable. The argument for $\SS_{2,\alpha}$ is similar. Consider now the points $\{ u = 0, \phi \neq 0 \}$. The unit vector normal to this surface is $\vec{m} = \epsilon^{\prime} | \nablaSigma u|^{-1}_{g} {\nablaSigma}u$ where $\epsilon^{\prime} = \pm 1$. Using (\[lambdaRW\]) and (\[NRW\]) in (\[defm\]) gives now $$\begin{aligned} 0 < N \vec{m} (\lambda ) |_{ \{ u = 0, \phi \neq 0 \}} = 2 \epsilon^{\prime} c^4 \hat{G} \phi^2 |\nablaSigma u|_{g} \sqrt{\frac{c^4 \hat{G}}{2\partial_{u}\phi-\hat{G} \partial_{A}\phi\partial^{A}\phi}},\end{aligned}$$ which implies $\epsilon^{\prime} = 1$ all points on $\{ u=0, \phi \neq 0\}$. The normal vector is $\vec{m} = |\nablaSigma u|^{-1}_{g} {\nablaSigma} u$ which again extends smoothly to all points on $\SS_{2,\alpha} \cap {\cal V}$. As before, $\SS_{2,\alpha}$ is orientable. Let us next check that $\SS_{1,\alpha}$ has $\theta^+ =0$ and $\SS_{2,\alpha}$ has $\theta^{-} =0$ (both with respect to the normal $\vec{m}$ defined above). On open sets of fixed points this is a trivial consequence of Proposition \[Totally geodesic\] which implies both $p= q =0$. To discuss the non-fixed points, we need an expression for $\vec{Y}$ in terms of $\vec{m}$. Let $\vec{Y} = \epsilon^{\prime\prime} N \vec{m}$, where $\epsilon^{\prime \prime} = \pm 1$. Using $\vec{Y} = \frac{1}{2\kappa} \nablaSigma \lambda$, we have $$\begin{aligned} \frac{\epsilon^{\prime\prime}}{2 \kappa} | \nablaSigma \lambda|^2_g = \epsilon^{\prime\prime} \vec{Y} \left ( \lambda \right ) = N \vec{m} \left ( \lambda \right ) > 0\end{aligned}$$ Hence $\epsilon^{\prime\prime} = \mbox{sign}(\kappa)$ and $$\begin{aligned} \vec{Y} = \mbox{sign} (\kappa) N \vec{m}. \label{Y_in_terms_of_m}\end{aligned}$$ Inserting this into (\[kid1\]) and taking the trace, it follows $$\begin{aligned} \mbox{sign} (\kappa ) p + q =0 \label{theta_plus_non_degenerate}\end{aligned}$$ This implies that $\theta^{+} = p + q =0$ at non-fixed points of $\SS_{1,\alpha}$ and $\theta^{-} = -p +q = 0$ at non-fixed points at $\SS_{2,\alpha}$. At fixed points not lying on open sets, equations $\theta^+ =0$ (resp. $\theta^{-}=0$) follow by continuity once we know that $\SS_{1,\alpha}$ (resp. $\SS_{2,\alpha}$) is smooth with a smooth unit normal. The final step is to prove that $\SS_{1,\alpha}$ and $\SS_{2,\alpha}$ are topologically closed. Let us first show that $\SS_{\alpha}$ is topologically closed. Consider a sequence of points $\{\p_i \}$ in $\SS_{\alpha}$ converging to $\p$. It is clear that $\p \in \tbd \{\lambda > 0\}$, so we only need to check that we have not moved to another arc-connected component. If $\p$ is a non-fixed point, then $\{ \lambda =0 \}$ is a defining function for $\tbd \{ \lambda > 0 \}$ near $\p$ and the statement is obvious. If $\p$ is a fixed point, we only need to use the Rácz-Wald-Walker coordinate system near $\p$ to conclude that no change of arc-connected component can occur in the limit. To show that each $\SS_{1,\alpha}$, $\SS_{2,\alpha}$ is topologically closed, assume now that $\p_i$ is a sequence on $\SS_{1,\alpha}$. If the limit $\p$ is a fixed point, it belongs to $\SS_{1,\alpha}$ by definition. If the limit $\p$ is a non-fixed point, we can take a subsequence $\{\p_i\}$ of non-fixed points. Since $\kappa$ remains constant on the sequence, it takes the same value in the limit, which shows that $\p \in \SS_{1,\alpha}$, i.e. $\SS_{1,\alpha}$ is topologically closed. The surfaces $S_a$ in the statement of the theorem are the collection of $\{ \SS_{d} \}$ having $I_1 =0$ and the collection of pairs $\{ \SS_{1,\alpha}$, $\SS_{2,\alpha}\}$ for the arc-connected components $\SS_{\alpha}$ with $I_1 \neq 0$. The statement that $\bd S_a \subset \bd \Sigmatilde$ is obvious. $\hfill \blacksquare$\ [**Remark 1.**]{} In this proof we have tried to avoid using the existence of a spacetime where $\kid$ is embedded as much as possible. The only essential information that we have used from the spacetime is that, near fixed points, $\lambda$ can be written as the product of two smooth functions with non-zero gradient, namely $u$ and $\phi$. This is the square root of $\zeta$ that we mentioned above. To see this, simply note that if a square root $h$ of $\zeta$ exists, then $\lambda = Q_0 x^2 - \zeta = Q_0^2 x - h^2 = \left (Q_0 x - h \right ) \left ( Q_0 x + h \right )$). The functions $Q_0 x \pm h$ have non-zero gradient and are, essentially, the functions $u$ and $\phi$ appearing the Rácz-Wald-Walker coordinate system. $\hfill \square$\ [**Remark 2.**]{} The assumption of every arc-connected component of $\tbd \{\lambda>0\}$ with $I_{1}=0$ being topologically closed is needed to ensure that these arc-connected components are embedded and compact. From a spacetime perspective, this hypothesis avoids the existence of non-embedded degenerate Killing prehorizons which would imply that, on an embedded KID, the arc-connected components of $\tbd \{\lambda>0\}$ which intersect these prehorizons could be non-embedded or non-compact (see Figure \[fig:spiral\] in Chapter \[ch:Preliminaires\]). Although it has not been proven, it may well be that non-embedded Killing prehorizons cannot exist. A proof of this fact would allow us to drop automatically this hypothesis in the theorem. $\hfill \square$\ We are now in a situation where we can prove that $\tbd \ext = \tbd T^+$ under suitable conditions on the trapped region and on the topology of $\Sigmatilde$. This result is the crucial ingredient for our uniqueness result later. The strategy of the proof is, once again, to assume that $\tbd \ext \neq \tbd T^+$ and to construct a bounding weakly outer trapped surface outside $\tbd T^+$. This time, the surface we use to perform the smoothing is more complicated than $\tbd \ext$, which we used in the previous chapter. The newly constructed surface will have vanishing outer null expansion and will be closed and oriented. However, we cannot guarantee a priori that it is bounding. To address this issue we impose a topological condition on $\mbox{int} ( \Sigmatilde)$ which forces that all closed and orientable surfaces separate the manifold into disconnected subsets. This topological condition involves the first homology group $H_1 ( \mbox{int} (\Sigmatilde), \mathbb{Z}_2)$ with coefficients in $\mathbb{Z}_2$ and imposes that this homology group is trivial. More precisely, the theorem that we will invoke is due to Feighn [@Feighn] and reads as follows \[Feighn\] Let ${\cal N}$ and ${\cal M}$ be manifolds without boundary of dimension $n$ and $n+1$ respectively. Let $f: {\cal N} \rightarrow {\cal M}$ be a proper immersion (an immersion is proper if inverse images of compact sets are compact). If $H_1 ({\cal M},\mathbb{Z}_2 ) = 0$ then ${\cal M} \setminus f ({\cal N})$ is not connected. Moreover, if two points $\p_1$ and $\p_2$ can be joined by an embedded curve intersecting $f ({\cal N})$ transversally at just one point, then $\p_1$ and $\p_2$ belong to different connected components of ${\cal M} \setminus f ({\cal N})$. The proof of this theorem requires that all embedded closed curves in ${\cal M}$ are the boundary of an embedded compact surface. This is a consequence of $H_1 ({\cal M}, \mathbb{Z}_2)=0$ and this is the only place where this topological condition enters into the proof. This allows us to understand better what topological restriction we are really imposing on ${\cal M}$, namely that every closed embedded curve is the boundary of a compact surface. Without entering into details of algebraic topology, we just notice that $H_1 ({\cal M},\mathbb{Z}_2 )$ vanishes if $H_1 ({\cal M}, \mathbb{Z} )=0$ (see e.g. Theorem 4.6 in [@Zomorodian]) and, in turn, this is automatically satisfied in simply connected manifolds (see e.g. Theorem 4.29 in [@Rotman]). \[mainthr\] Consider an embedded static KID $\kidtilde$ compact, with boundary $\bd \Sigmatilde$ and satisfying the NEC. Suppose that the boundary can be split into two non-empty disjoint components $\bd \Sigmatilde= \bd^{-}\Sigmatilde\cup\bd^+\Sigmatilde$ (neither of which are necessarily connected). Take $\bd^{+}\Sigmatilde$ as a barrier with interior $\Sigmatilde$ and assume $\theta^{+}[ \bd^{-}\Sigmatilde ] \leq 0$ and $\theta^{+}[\bd^{+}\Sigmatilde]>0$ Let $T^{+}, T^-$ be, respectively, the weakly outer trapped and the past weakly outer trapped regions of $\Sigmatilde$. Assume also the following hypotheses: 1. Every arc-connected component of $\tbd \ext$ with $I_1 =0$ is topologically closed. 2. $\left. \lambda \right|_{\bd^{+}\Sigmatilde}>0$. 3. $H_1\left( \mbox{int} (\Sigmatilde),\mathbb{Z}_2 \right)=0$. 4. $T^-$ is non-empty and $T^{-}\subset T^{+}$. Denote by $\ext$ the connected component of $\{ \lambda>0 \}$ which contains $\bd^{+}\Sigmatilde$. Then $$\tbd\ext = \tbd T^{+},$$ Therefore, $\tbd\ext$ is a non-empty stable MOTS which is bounding with respect to $\bd^{+}\Sigmatilde$ and, moreover, it is the outermost bounding MOTS. [**Proof.**]{} After replacing $\vec{\xi} \rightarrow - \vec{\xi}$ if necessary, we can assume without loss of generality that $N >0$ on $\ext$. From Theorem \[thr:AM\], we know that the boundary of the weakly outer trapped region $T^{+}$ in $\Sigmatilde$ (which is non-empty because $\theta^+[\bd^{-}\Sigmatilde] \leq 0$) is a stable MOTS which is bounding with respect to $\bd^{+}\Sigmatilde$. $\tbd T^-$ is also non-empty by assumption. Since we are dealing with embedded KID, and all spacetimes are boundaryless in this thesis, it follows that $\kid$ can be extended as a smooth hypersurface in $(M,\gM)$[^2]. Working on this extended KID allows us to assume without loss of generality that $\tbd T^+$ and $\tbd T^-$ lie in the [*interior*]{} of $\Sigmatilde$. This will be used when invoking the Kriele and Hayward smoothing procedure below. First of all, Theorem \[theorem1\] implies that $\tbd\ext$ cannot lie completely in ${T^{+}}$ and intersect the topological interior $\overset{\circ}{ T}{}^+$ (here is where we use the NEC). Therefore, either $\tbd\ext$ intersects the exterior of $\tbd T^{+}$ or they both coincide. We only need to exclude the first possibility. Suppose, that $\tbd\ext$ penetrates into the exterior of $\tbd T^{+}$. Let $\{\mathfrak{U}\}$ be the collection of arc-connected components of $\tbd \{\lambda >0\}$ which have a non-empty intersection with $\tbd \ext$. In Proposition \[proposition1\] we have shown that $\{\mathfrak{U}\}$ decomposes into a union of smooth surfaces $S_a$. Define its unit normal $\vec{m}'$ as the smooth normal which points into $\ext$ at points on $\tbd \ext$. This normal exists because all $S_a$ are orientable. By (\[defm\]) and the fact that $N>0$ on $\ext$, we have that on the surfaces $S_{a}$ with $I_1 \neq 0$, the normal $\vec{m}'$ coincides with the normal $\vec{m}$ defined in the proof of Proposition \[proposition1\]. On the surfaces $S_{a}$ with $I_1 =0$, this normal coincides with $\vec{m}$ provided $\vec{Y}$ points into $\ext$, see (\[choice\_m\_degenerate\]). Since, by assumption, $\tbd \ext$ penetrates into the exterior of $T^+$, it follows that there is at least one $S_{a}$ with penetrates into the exterior of $T^+$. Let $\{ S_{a'} \}$ be the subcollection of $\{ S_{a} \}$ consisting on the surfaces which penetrate into the exterior of $\tbd T^+$. A priori, none of the surfaces $S_{a'}$ need to satisfy $p +q=0$ with respect to the normal $\vec{m}'$. However, one of the following two possibilities must occur: 1. There exists at least one surface, say $S_{0}$, in $\{S_{a'}\}$ containing a point $\q \in \tbd \ext$ such that $\vec{Y} |_{\q}$ points inside $\ext$, or 2. All surfaces in $\{ S_{a'} \}$ have the property that, for any ${\q} \in S_{a'} \cap \tbd \ext$ we have $\vec{Y} |_{\q}$ is either zero, or it points outside $\ext$. In case 1, we have that $S_0$ satisfies $p+q =0$ with respect to the normal $\vec{m}'$. Indeed, we either have that $S_0$ satisfies $I_1=0$ or $I_1 \neq 0$. If $I_1=0$ then, since $\vec{Y}$ points into $\ext$, we have that $\vec{m}$ and $\vec{m'}$ coincide. Since $S_0$ satisfies $p +q=0$ with respect to $\vec{m}$ (see (\[theta\_plus\_degenerate\])) the statement follows. If $I_1 \neq 0$ then $\kappa>0$ on $S_0$ (from (\[Y\_in\_terms\_of\_m\]) and the fact that $\vec{m} = \vec{m}'$). Thus, $p+q =0$ follows from (\[theta\_plus\_non\_degenerate\]). In case 2, all surfaces $\{ S_{a'}\}$ satisfy $\theta^{-} = - p +q =0$ with respect to $\vec{m}'$ and we cannot find a MOTS outside $\tbd T^+$. However, under assumption 3, we have $T^{-} \subset T^+$ and hence each $S_{a'}$ penetrates into the exterior of $T^-$. We can therefore reduce case 2 to case 1 by changing the time orientation (or simply replacing $\theta^+$ and $T^+$ by $\theta^-$ and $T^-$ in the argument below). Let us therefore restrict ourselves to case 1. We know that $S_0$ either has no boundary, or the boundary is contained in $\bd^- \Sigmatilde$. If $S_0$ has no boundary, simply rename this surface to $S_1$. When $S_0$ has a non-empty boundary, it is clear that $S_0$ must intersect $\tbd \T^+$. We can then use the smoothing procedure by Kriele and Hayward (see Lemma \[lemasmoothness\]) to construct a closed surface $S_1$ penetrating into the exterior of $\tbd T^+$ and satisfying $\theta^+ \leq 0$ with respect to a normal $\vec{m}''$ which coincides with $\vec{m}'$ outside the region where the smoothing is performed (see Figure \[fig:mainthr\]). As discussed in the previous chapter, when $S_0$ and $\tbd T^{+}$ do not intersect transversally we need to apply the Sard Lemma to surfaces inside $\tbd T^+$. If $\tbd T^+$ is only marginally stable, a suitable modification of the initial data set inside $\tbd T^+$ is needed. The argument was discussed in depth at the end of the proof of Theorem \[theorem2\] and applies here without modification. ![The figure illustrates the situation when $S_{0}$ has boundary. The grey region represents the region with $\lambda>0$ in $\Sigmatilde$. In this case we use the smoothing procedure of Kriele and Hayward to construct a smooth surface $S_{1}$ from $S_{0}$ and $\tbd T^{+}$ (in blue). The red lines represent precisely the part of $S_{1}$ which comes from smoothing $S_{0}$ and $\tbd T^{+}$.[]{data-label="fig:mainthr"}](mainthr.eps){width="9cm"} So, in either case (i.e. irrespectively of whether $S_0$ has boundary of not), we have a closed surface $S_1$ penetrating into the exterior of $\tbd T^+$ and satisfying $\theta^{+}\leq 0$ with respect to $\vec{m}''$. Here we apply the topological hypothesis $3$ ($H_{1}(\mbox{int}(\Sigmatilde), \mathbb{Z}_{2})=0$). Indeed $S_1$ is a closed manifold embedded into $\mbox{int} (\Sigmatilde )$. Since $S_{1}$ is compact, its embedding is obviously proper. Thus, the theorem by Feighn [@Feighn] (Theorem \[Feighn\]) implies that $\mbox{int} (\Sigmatilde) \setminus S_{1}$ has at least two connected components. It is clear that one of the connected components $\Omega$ of $\mbox{int} (\Sigmatilde) \setminus S_1$ contains $\bd^+ \Sigmatilde$. Moreover, by Feighn’s theorem there is a tubular neighbourhood of $S_1$ which intersects this connected component only to one side of $S_1$. Consequently, $\overline{\Omega}$ is a compact manifold with boundary $\bd \overline{\Omega} = S_1 \cap \partial^+ \Sigma$. If follows that $S_1$ is bounding with respect to $\bd^+ \Sigmatilde$. The choice of $\vec{m}''$ is such that $\vec{m}''$ points towards $\bd^+ \Sigmatilde$. Consequently $S_1$ is a weakly outer trapped surface which is bounding with respect to $\bd^+ \Sigmatilde$ penetrating into the exterior of $\tbd T^+$, which is impossible. $\hfill\blacksquare$\ [**Remark 1.**]{} If the hypothesis $T^{-}\subset T^{+}$ is not assumed, then the possibility $2$ in the proof of the Theorem would not lead to a contradiction (at least with our method of proof). To understand this better, without the assumption $T^{-}\subset T^{+}$ it may happen a priori that all the surfaces $S_{a'}$ (which have $\theta^{-}=0$ and penetrate in the exterior of $\tbd T^{+}$) are fully contained in $T^{-}$. A situation like this is illustrated in Figure \[figT+subsetT-\], where $\tbd T^{-}$ intersects $\tbd T^{+}$. It would be interesting to either prove this theorem without the assumption $T^- \subset T^+$ or else find a counterexample of the statement $\tbd \ext = \tbd T^+$ when assumption 4 is dropped. The problem, however, appears to be difficult. $\hfill \square$\ ![The figure illustrates a hypothetical situation where $T^{+}\subset T^{-}$ does not hold and the conclusions of the Theorem \[mainthr\] would not be true. The red continuous line represents the set $\tbd \ext$ which is composed by a smooth surface with $\theta^{+}=0$, lying inside of $\tbd T^{+}$ (in blue) and partly outside of $\tbd T^{-}$ (in green), and a smooth surface with $\theta^{-}=0$, which lies partly outside of $\tbd T^{+}$ and inside of $\tbd T^{-}$.[]{data-label="figT+subsetT-"}](T+subsetT-.eps){width="12cm"} The uniqueness result {#sectionuniqueness} --------------------- Finally, we are ready to state and prove the uniqueness result for static spacetimes containing trapped surfaces. \[uniquenessthr\] Let $\kid$ be an embedded static KID with a selected asymptotically flat end $\Sigma_{0}^{\infty}$ and satisfying the NEC. Assume that $\Sigma$ possesses a weakly outer trapped surface $S$ which is bounding. Assume the following: 1. Every arc-connected component of $\tbd \ext$ with $I_1 =0$ is topologically closed. 2. $T^-$ is non-empty and $T^{-}\subset T^{+}$. 3. $H_{1}\left(\Sigma,\mathbb{Z}_{2} \right)=0$. 4. The matter model is such that Bunting and Masood-ul-Alam doubling method for time-symmetric initial data sets gives uniqueness of black holes. Then $\left( \Sigma \setminus {T}^{+},g,K \right)$ is a slice of such a unique spacetime. [**Proof.**]{} Take a coordinate sphere $S_b \equiv \{r = r_0 \}$ in the asymptotically flat end $\Sigma^{\infty}_0$ with $r_0$ large enough so that $\lambda >0$ on $\{ r \geq r_0 \} \subset \Sigma^{\infty}_0$ and all the surfaces $\{ r = r_1 \}$ with $r_1 \geq r_0$ are outer untrapped with respect to the unit normal pointing towards increasing $r$. $S_b$ is a barrier with interior $\Omegab=\Sigma\setminus\{r>r_{0}\}$. Take $\Sigmatilde$ to be the topological closure of the exterior of $S$ in $\Omega_b$. Then define $\bd^- \Sigmatilde = S$ and $\bd^{+}\Sigmatilde=S_{b}$. Let $\ext$ be the connected component of $\{\lambda >0\} \subset \Sigmatilde$ containing $\Sb$. All the hypothesis of Theorem \[mainthr\] are satisfied and we can conclude $\tbd \ext = \tbd T^+$. This implies that the manifold $\Sigma \setminus {T}^{+}$ is an asymptotically flat spacelike hypersurface with topological boundary $\tbd (\Sigma \setminus {T}^{+})$ which is compact and embedded (moreover, it is smooth) such that the static Killing vector is timelike on $\Sigma \setminus {T}^{+}$ and null on $\tbd(\Sigma \setminus{T}^{+})$. Under these assumptions, the doubling method of Bunting and Masood-ul-Alam [@BMuA] can be applied. Hence, hypothesis $4$ gives uniqueness. $\hfill \blacksquare$\ [**Remark 1.**]{} In contrast to Theorems \[theorem2\] and \[uniquenessthr0\], this result has been formulated for weakly outer trapped surfaces instead of outer trapped surfaces. As mentioned in the proof of Theorem \[mainthr\] this is because, $\kid$ being an embedded static KID, it can be extended smoothly as a hypersurface in the spacetime. It is clear however, that we are hiding the possible difficulties in the definition of [*embedded static KID*]{}. Consider, for instance, a [*static KID*]{} with boundary and assume that the KID is vacuum. The Cauchy problem is of course well-posed for vacuum initial data. However, since $\Sigma$ has boundary, the spacetime constructed by the Cauchy development also has boundary and we cannot a priori guarantee that the static KID is an embedded static KID (this would require extending the spacetime, which is as difficult – or more – than extending the initial data). Consequently, Theorem \[uniquenessthr\] includes Miao’s theorem in vacuum as a particular case only for vacuum static KID for which either (i) $S$ is not the outermost MOTS, (ii) $S \cap \bd \Sigma = \emptyset$ or (iii) the KID can be extended as a vacuum static KID. Despite this subtlety, we emphasize that all the other conditions of the theorem are fulfilled for asymptotically flat, time-symmetric vacuum KID with a compact minimal boundary. Indeed, condition 4 is obviously satisfied for vacuum. Moreover, the property of time-symmetry implies that all points with $\lambda =0$ are fixed points and hence no arc-connected component of $\tbd \{ \lambda > 0 \}$ with $I_1 =0$ exists. Thus, condition 1 is automatically satisfied. Time-symmetry also implies $T^- = T^+$ and condition 2 is trivial. Finally, the region outside the outermost minimal surface in a Riemannian manifold with non-negative Ricci scalar is $\mathbb{R}^3$ minus a finite number of closed balls (see e.g. [@HI]). This manifold is simply connected and hence satisfies condition 3. $\hfill \square$\ [**Remark 2.**]{} Condition 4 in the theorem could be replaced by a statement of the form - The matter model is such that static black hole initial data implies uniqueness, where a [*black hole static initial data*]{} is an asymptotically flat static KID possibly with boundary with an asymptotically flat end $\Sigma^{\infty}_0$ such that $\tbd \ext$ (defined as the connected component of $\{\lambda > 0 \}$ containing the asymptotic region in $\Sigma^{\infty}_0$) is a topological manifold without boundary and compact. The Bunting and Masood-ul-Alam method is, at present, the most powerful method to prove uniqueness under the circumstances of 4’. However, if a new method is invented, Theorem \[uniquenessthr\] would still give uniqueness. $\hfill \square$\ [**Remark 3.**]{} A comment on the condition $T^- \subset T^+$ is in order. First of all, in the static regime, $T^{+}$ and $T^{-}$ are expected to be the intersections of both the black and the white hole with $\Sigmatilde$. Therefore, the hypothesis $T^{-}\subset T^{+}$ could be understood as the requirement that the first intersection, as coming from $\bd^{+}\Sigmatilde$, of $\Sigmatilde$ with an event horizon occurs with the black hole event horizon. Therefore, this hypothesis is similar to the hypotheses on $\tbd \ext$ made in Theorem \[is\_a\_MOTS\]. However, there is a fundamental difference between them: The hypothesis $T^{-}\subset T^{+}$ is an hypothesis on the weakly outer trapped regions which, a priori, have nothing to do with the location and properties of $\tbd \ext$. In a physical sense, the existence of past weakly outer trapped surfaces in the spacetime reveals the presence of a white hole region. Moreover, given a (3+1) decomposition of a spacetime satisfying the NEC, the Raychaudhuri equation implies that $T^-$ shrinks to the future while $T^+$ grows to the future (see [@AMMS]) (“grow” and “shrink” is with respect to any timelike congruence in the spacetime). It is plausible that by letting the initial data evolve sufficiently long, only the black hole event horizon is intersected by $\Sigma$. The uniqueness theorem \[uniquenessthr\] could be applied to this evolved initial data. Although this requires much less global assumptions than for the theorem that ensures that no MOTS can penetrate into the domain of outer communications, it still requires some control on the evolution of the initial data. In any case, we believe that the condition $T^- \subset T^+$ is probably not necessary for the validity of the theorem. It is an interesting open problem to analyze this issue further. $\hfill \square$\ We conclude with a trivial corollary of Theorem \[uniquenessthr\], which is nevertheless interesting. Let $(\Sigma,g,K=0;N,\vec{Y}=0;\rho,\vec{J} =0,\tau_{ij};\vec{E})$ be a time-symmetric electrovacuum embedded static KID, i.e a static KID with an electric field $\vec{E}$ satisfying $$\begin{aligned} \nablaSigma_i E^i = 0, \quad \rho = |\vec{E}|^2_{g}, \quad \tau_{ij} = | \vec{E} |^2 g_{ij} - 2 E_i E_j.\end{aligned}$$ Let $\Sigma = \mathcal{K} \cup \Sigma^{\infty}_0$ where $\mathcal{K}$ is a compact and $\Sigma^{\infty}_0$ is an asymptotically flat end and assume that $\bd \Sigma \neq \emptyset$ with mean curvature with respect to the normal which points inside $\Sigma$ satisfying $p \leq 0$. Then $(\Sigma\setminus T^+,g, K=0; N, \vec{Y}=0, \rho, \vec{J} =0, \tau_{ij}, \vec{E})$ can be isometrically embedded in the Reissner-Nordström spacetime with $M > |Q|$, where $M$ is the ADM mass of $(\Sigma,g)$ and $Q$ is the total electric charge of $\vec{E}$, defined as $Q = \frac{1}{4 \pi} \int_{S_{r_0}} E^i m_i \eta_{S_{r_0}}$ where $S_{r_0} \subset \Sigma^{\infty}_{0}$ is the coordinate sphere $\{ r = r_0 \}$ and $\vec{m}$ its unit normal pointing towards infinity. [**Remark.**]{} The standard Majumdar-Papapetrou spacetime cannot occur because it possesses degenerate Killing horizons which are excluded in the hypotheses of the corollary (recall that, by Proposition \[prop:Chrusciel\], degenerate Killing horizons implies cylindrical ends in time-symmetric slices). $\hfill \square$ A counterexample of a recent proposal on the Penrose inequality =============================================================== \[ch:Article3\] Introduction {#introduction} ------------ In this chapter we will give a counter-example of the Penrose inequality proposed by Bray and Khuri in [@BK]. As discussed in Chapter \[ch:Preliminaires\], in a consistent attempt [@BK] to prove the standard Penrose inequality (equation (\[penrose1\])) in the general case (i.e. non-time-symmetric), Bray and Khuri were led to conjecture a new version of the Penrose inequality in terms of the outermost generalized apparent horizon (see Definition \[defi:GAH\]) as follows. $$\label{penroseBK2} M_{\scriptscriptstyle ADM}\geq \sqrt{\frac{|S_{{out}}|}{16\pi}},$$ where $M_{\scriptscriptstyle ADM}$ is the ADM mass of a spacelike hypersurface $\Sigma$, which contains an asymptotically flat end $\Sigma_{0}^{\infty}$, and $|S_{{out}}|$ denotes the area of the outermost bounding generalized apparent horizon $S_{{out}}$ in $\Sigma$. As we already remarked in Section \[sc:PenroseInequality\], this inequality has several convenient properties such as the invariancy under time reversals, no need of taking the minimal area enclosure of $S_{{out}}$, and the facts that it is stronger than (\[penrose1\]) and covers a larger number of slices of Kruskal with equality than (\[penrose1\]). Furthermore, it also has good analytical properties which potentially can lead to its proof in the general case. Indeed, Bray and Khuri proved that if a certain system of PDE admits solutions with the right boundary behavior, then (\[penroseBK2\]) follows. Nevertheless, as we also pointed out in Section \[sc:PenroseInequality\], inequality (\[penroseBK2\]) is not directly supported by cosmic censorship. In fact, it is not difficult to obtain particular situations where $S_{{out}}$ lies, at least partially, outside the event horizon, as for example for a slice $\Sigma$ in the Kruskal spacetime for which $\tbd T^{+}$ and $\tbd T^{-}$ intersect transversally. In this case, Eichmair’s theorem (Theorem \[thr:Eichmair\]) implies that there exists a $C^{2,\alpha}$ outermost generalized apparent horizon lying, at least partially, in the domain of outer communications of the Kruskal spacetime. Thus, it becomes natural to study the outermost generalized apparent horizon in slices of this type in order to check whether (\[penroseBK2\]) holds or not. Surprisingly, the result we will find is that there are examples for which inequality (\[penroseBK2\]) turns out to be violated. More precisely, \[theorem\] In the Kruskal spacetime with mass $M_{Kr}>0$, there exist asymptotically flat, spacelike hypersurfaces with an outermost generalized apparent horizon $S_{out}$ satisfying $|S_{out}|>16\pi M_{Kr}^{2}$. For the systems of PDE proposed in [@BK], this means that a general existence theory cannot be expected with boundary conditions compatible with generalized apparent horizons. However, simpler boundary conditions (e.g. compatible with future and past apparent horizons) are not ruled out. This may in fact simplify the analysis of these equations. The results on this chapter have been published in [@CM3], [@CMere3]. Construction of the counterexample ---------------------------------- Let us consider the Kruskal spacetime of mass $M_{Kr}>0$ with metric $$ds^2= \frac{32M_{Kr}^{3}}{r}e^{-r/2M_{Kr}}d\hat{u}d\hat{v}+ r^2 \left( d\theta^2 + \sin^2{\theta} d\phi^2 \right),$$ where $r(\hat{u}\hat{v})$ solves the implicit equation $$\label{uvr} \hat{u}\hat{v}= \frac{r-2M_{Kr}}{2M_{Kr}}e^{r/2M_{Kr}}.$$ In this metric $\partial_{\hat{v}}$ is future directed and $\partial_{\hat{u}}$ is past directed. The region $\{\hat{u}>0,\hat{v}>0\}$ defines the domain of outer communications and $\{\hat{u}=0\}$, $\{\hat{v}=0\}$ define, respectively, the black hole and white hole event horizons. Consider the one-parameter family of axially-symmetric embedded hypersurfaces $\Sigma_{\epsilon} = \mathbb{R} \times \mathbb{S}^2$, with intrinsic coordinates $\ycoor \in \mathbb{R}$, $x \in [-1,1]$, $\phi \in [0,2 \pi]$, defined by the embedding $$\begin{aligned} \Sigma_{\epsilon}\equiv \left\{ \hat{u}=\ycoor- \epsilon x, \hat{v}=\ycoor+ \epsilon x, \cos \theta = x , \phi = \phi\right\}.\end{aligned}$$ Inserting this embedding functions into equation (\[uvr\]) we get $$\label{uvrembedded} \ycoor^{2}-\epsilon^{2}x^{2}=\frac{r-2M_{Kr}}{2M_{Kr}}e^{r/2M_{Kr}},$$ from which it is immediate to show that, for $|\epsilon|<1$, $\Sigma_{\epsilon}$ does not touch the Kruskal singularity ($r=0$) for any value of $\{\ycoor,x\}$ in their coordinate range. It is also immediate to check that the hypersurfaces $\Sigma_{\epsilon}$ are smooth everywhere, included the north and south poles defined by $|x|=1$. It is straightforward to prove that the induced metric $g_{\epsilon}$ on $\Sigma_{\epsilon}$ is positive definite and satisfies (for $\epsilon$ is small enough) $g_{\epsilon}=dr^{2}+r^{2}\left( \frac{dx^2}{1-x^{2}} +(1-x^{2})d{\phi}^{2}\right)+O^{(2)}(\frac{1}{r})$, where $r$ is defined in (\[uvrembedded\]). Consequently, the hypersurfaces $\Sigma_{\epsilon}$ are spacelike and asymptotically flat. Let us select ${\Sigma_{\epsilon}}_{0}^{\infty}$ to be the asymptotically flat end of the region $\{\hat{u}>0,\hat{v}>0\}$. The discrete isometry of the Kruskal spacetime defined by $\left\{\hat{u},\hat{v}\right\}\rightarrow \left\{\hat{v},\hat{u}\right\}$ implies that under reflection with respect to the equatorial plane, i.e. $(\ycoor,x,\phi) \rightarrow (\ycoor,-x,\phi)$, the induced metric of $\Sigma_{\epsilon}$ remains invariant, while the second fundamental form of $\Sigma_{\epsilon}$ changes sign. The latter is due to the fact that $\Sigma_{\epsilon}$ is defined by $\hat{u}-\hat{v}+2\epsilon x=0$ and hence the future directed unit normal to $\Sigma_{\epsilon}$ is proportional (with metric coefficients which only depend on $uv$ and $x^{2}$) to $d\hat{u}-d\hat{v}+2\epsilon dx$ and, therefore, it changes sign under a reflection $(\ycoor,x,\phi) \rightarrow (\ycoor,-x,\phi)$ and a simultaneous spacetime isometry $\left\{\hat{u},\hat{v}\right\}\rightarrow \left\{\hat{v},\hat{u}\right\}$ (notice that this isometry reverses the time orientation). Let us denote by $\Sigma^{+}_{\epsilon}$ the intersection of $\Sigma_{\epsilon}$ with the domain of outer communications $\{\hat{u}>0,\hat{v}>0\}$, which is given by $\{\ycoor - | \epsilon x |>0\}$. For $\epsilon \neq 0$, $\tbd \Sigma^{+}_{\epsilon}$ is composed by a portion of the black hole event horizon and a portion of the white hole event horizon. Moreover, $\tbd \T^{+}$ is given by $\{\ycoor - \epsilon x=0\}$, while $\tbd \T^{-}$ is $\{\ycoor + \epsilon x= 0 \}$ so that these surfaces intersect transversally on the circumference $\{\ycoor=0, x= 0\}$ provided $\epsilon \neq 0$[^3]. By Eichmair’s theorem (Theorem \[thr:Eichmair\]), there exists a $C^{2,\alpha}$ outermost generalized apparent horizon $S_{{out}}$ which is bounding and contains both $\tbd \T^{+}$ and $\tbd \T^{-}$. Uniqueness implies that this surface must be axially symmetric and have equatorial symmetry. In what follows we will estimate the area of $S_{{out}}$ from below . To that aim we will proceed in two steps. Firstly, we will prove that an axial and equatorially symmetric generalized apparent horizon $\hat{S}_{\epsilon}$ of spherical topology and lying in a sufficiently small neighbourhood of $\{\ycoor =0\}$ exists (provided $\epsilon$ is small enough) and determine its embedding function. In the second step we will compute its area and prove that it is smaller or equal than the area of the outermost generalized apparent horizon $S_{{out}}$. ### Existence and embedding function This subsection is devoted to prove the existence of $\hat{S}_{\epsilon}$ and to calculate its embedding function up to first order in $\epsilon$. For that, we will consider surfaces $S_{\epsilon}$ of spherical topology defined by embedding functions $\{ \ycoor = \y (x, \epsilon), x=x, \phi = \phi \}$ in $\Sigma_{\epsilon}$ and satisfying $\y(-x,\epsilon)=\y(x,\epsilon)$. Since the outermost generalized apparent horizon is known to be $C^{2,\alpha}$ it is natural to consider the spaces of functions $$U^{m,\alpha} \equiv \left\{ \y\in C^{m,\alpha}(\mathbb{S}^2):\partial_{\phi}\y=0,\y(-x)= \y(x)\right\} ,$$ i.e. the spaces of $m$-times differentiable functions on the unit sphere, with Hölder continuous $m$-th derivatives with exponent $\alpha \in (0,1]$ and invariant under the axial Killing vector on $\mathbb{S}^2$ and under reflection about the equatorial plane. Each space $U^{m,\alpha}$ is a closed subset of the Banach space $C^{m,\alpha}(\mathbb{S}^2)$ and hence a Banach space itself. Let $I \subset \mathbb{R}$ be the closed interval where $\epsilon$ takes values. The expression that defines a generalized apparent horizon is $p-|q|=0$, where $p$ is the mean curvature of the corresponding surface $S_{\epsilon}$ in $\Sigma_{\epsilon}$ with respect to the direction pointing into ${\Sigma_{\epsilon}}_{0}^{\infty}$ and $q$ is the trace on $S_{\epsilon}$ of the pull-back of the second fundamental form $K$ of $\Sigma_{\epsilon}$. For each function $\y \in U^{2,\alpha}$ the expression $p - |q|$ defines a non-linear map $\f:U^{2,\alpha}\times I \rightarrow U^{0,\alpha}$. Thus, we are looking for solutions $\y\in U^{2,\alpha}$ of the equation $\f=0$. We know that when $\epsilon=0$, the hypersurface $\Sigma_{\epsilon}$ is totally geodesic, which implies $q=0$ for any surface on it. Consequently, all generalized apparent horizons on $\Sigma_{\epsilon=0}$ satisfy $p=0$ and are, in fact, minimal surfaces. The only closed minimal surface in $\Sigma_{\epsilon =0}$ is the bifurcation surface $S_{0}=\left\{ \hat{u}=0, \hat{v}=0 \right\}$. Thus, the equation $\f(\y,\epsilon)=0$ has $\y=0$ as the unique solution when $\epsilon=0$. It becomes natural to use the implicit function theorem for Banach spaces to show that there exists a unique solution $\y\in U^{2,\alpha}$ of $\f=0$ in a neighbourhood of $\y=0$ for $\epsilon$ small enough. To apply the implicit function theorem it will be necessary to know the explicit form of the linearization of the differential equation $\f(y,\epsilon)=0$. The following lemma gives precisely the explicit form of $\f$ up to first order in $\epsilon$. \[lema:explicitf\] Let $\Sigma_{\epsilon}$ be the one-parameter family of axially-symmetric hypersurfaces embedded in the Kruskal spacetime with mass $M_{Kr}>0$, with intrinsic coordinates $\ycoor \in \mathbb{R}$, $x \in [-1,1]$, $\phi \in [0,2 \pi]$, defined by $$\begin{aligned} \Sigma_{\epsilon}\equiv \big\{ \hat{u}=\ycoor- \epsilon x, \hat{v}=\ycoor+ \epsilon x, \cos \theta = x , \phi = \phi\big\}.\end{aligned}$$ Consider the surfaces $S_{\epsilon}\subset \Sigma_{\epsilon}$ defined by $\{\ycoor=y(x),x,\phi\}$ where the embedding function has the form $y=\epsilon Y$, with $Y\in U^{m,\alpha}(\mathbb{S}^2)$. Then, $p$ and $q$ satisfy $$\begin{aligned} p(y=\epsilon Y,\epsilon)&=&\frac{1}{M_{Kr}\sqrt{e}}\, L[Y(x)]\epsilon+O(\epsilon^{2}),\label{explicitp} \\ q(y=\epsilon Y,\epsilon)&=&-\frac{1}{M_{Kr}\sqrt{e}}\, 3x\epsilon+O(\epsilon^{2}), \label{explicitq}\end{aligned}$$ where $L[z(x)]\equiv -(1-x^{2})\ddot{z}(x)+2x\dot{z}(x)+z(x)$ and where the dot denotes derivative with respect to $x$. [**Proof.**]{} The proof is by direct computation. Let us define $H=\frac{32M_{Kr}^{3}}{r}e^{-r/2M_{Kr}}$, $Q=r^{2}$ and $x=\cos{\theta}$, so that the Kruskal metric takes the form $$\gM=Hd\hat{u}d\hat{v}+\frac{Q}{1-x^{2}}dx^{2}+(1-x^{2})Q d\phi^{2}.$$ The induced metric $g_{\epsilon}$ on $\Sigma_{\epsilon}$ is $$\label{explicitg} g_{\epsilon}=\hat{H}d\ycoor^{2}+\left( \frac{\hat{Q}}{1-x^{2}}-\epsilon^{2}\hat{H}\right)dx^{2}+(1-x^{2})\hat{Q}d\phi^{2},$$ where $\hat{H}$, $\hat{Q}$ are obtained from $H$, $Q$ by expressing $r$ in terms of $(\ycoor,x)$ according to (\[uvrembedded\]). The induced metric $\gamma_{\epsilon}$ on $S_{\epsilon}$ satisfies $$\label{explicitgamma} \gamma_{\epsilon}=\left[ \frac{\tilde{Q}}{1-x^{2}}+\epsilon^{2}\left(\dot{Y}^{2}(x)-1\right) \tilde{H} \right]dx^{2}+(1-x^{2})\tilde{Q}d\phi^{2},$$ where $\tilde{H}$, $\tilde{Q}$ are obtained from $\hat{H}$ and $\hat{Q}$ by inserting $\ycoor=\epsilon Y(x)$. Firstly, let us deal with the computation of $p=-m_{i}\gamma_{\epsilon}^{AB}\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{A}}e_{B}^{i}$, where $\bold{m}$ is the unit vector tangent to $\Sigma_{\epsilon}$ normal to $S_{\epsilon}$ which points to the asymptotically flat end in $\{\hat{u}>0,\hat{v}>0\}$ and $\{\vec{e}_{A}\}$ is a basis for $T S_{\epsilon}$. In our coordinates $$\begin{aligned} &&\vec{e}_{x}=\partial_{x}+\epsilon\dot{Y}(x)\partial_{\ycoor},\\ &&\vec{e}_{\phi}=\partial_{\phi}.\end{aligned}$$ The unit normal is therefore $$\label{explicitm} \bold{m}=\sqrt{\frac{\tilde{H}\left(\tilde{Q}-\epsilon^{2}(1-x^2)\tilde{H}\right)}{\tilde{Q}+\epsilon^{2}(1-x^2) (\dot{Y}^{2}-1)\tilde{H}}} \,\left(d\ycoor-\epsilon\dot{Y}(x)dx\right).$$ Since $\gamma_{\epsilon}$ is diagonal, we only need to calculate $\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{x}}e_{x}^{\ycoor}$, $\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{\phi}}e_{\phi}^{\ycoor}$, $\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{x}}e_{x}^{x}$ and $\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{\phi}}e_{\phi}^{x}$ up to first order. The results are the following. $$\begin{aligned} \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{x}}e_{x}^{\ycoor}&=& - \frac{\partial_{\ycoor}\hat{Q}}{2(1-x^2)\tilde{H}}+ \epsilon\left(\ddot{Y}+\dot{Y}\partial_{x}\ln{\hat{H}}\right)+ O(\epsilon^{2}), \label{explicitDxxy}\\ \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{x}}e_{x}^{x}&=& \frac{2x+(1-x^{2})\partial_{x}\ln{\hat{Q}}}{2(1-x^{2})}+ \epsilon\dot{Y}\partial_{\ycoor}\ln{\hat{Q}} +O(\epsilon^{2}) , \label{explicitDxxx}\\ \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{\phi}}e_{\phi}^{\ycoor}&=& -\frac{(1-x^{2})\partial_{\ycoor}\hat{Q}}{2\tilde{H}}, \label{explicitDphiphiy}\\ \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{\phi}}e_{\phi}^{x}&=& \frac{(1-x^{2})\left( 2x-(1-x^{2})\partial_{x}\ln{\hat{Q}}\right)}{2}+O(\epsilon^{2}), \label{explicitDphiphix}\end{aligned}$$ where $\partial_{\ycoor}\hat{Q}$ means taking derivative with respect to $\ycoor$ of $\hat{Q}$ and afterwards, substituting $\ycoor=\epsilon Y(x)$ (and similarly for the other derivatives). In order to compute the derivatives of $\hat{H}$ and $\hat{Q}$, we need to calculate the derivatives $\partial_{\ycoor}r(\ycoor,x)$ and $\partial_{x}r(\ycoor,x)$. This can be done by taking derivatives of (\[uvrembedded\]) with respect to $x$ and $\ycoor$, which gives, $$\begin{aligned} &&\partial_{\ycoor}r=\epsilon\frac{8M_{Kr}^{2}}{r}e^{-r/2M_{Kr}} Y, \label{Dyr}\\ &&\partial_{x}r=-\epsilon^{2}\frac{8M_{Kr}^{2}}{r}e^{-r/2M_{Kr}} x. \label{Dxr}\end{aligned}$$ At $\epsilon=0$ we have $y=0$ and equation (\[uvrembedded\]) gives $r\big|_{S_{\epsilon=0}}=2M_{Kr}$. Then $r\big|_{S_{\epsilon}}=2M_{Kr}+O(\epsilon)$ This allows us to compute the derivatives of $\hat{H}$ and $\hat{Q}$ up to first order in $\epsilon$. The result is $$\begin{aligned} \partial_{x}\hat{H}&=&O(\epsilon^{2}),\label{DxH}\\ \partial_{\ycoor}\hat{Q}&=&\epsilon\frac{16M_{Kr}^{2}}{e} Y +O(\epsilon^2),\label{DyQ}\\ \partial_{x}\hat{Q}&=&O(\epsilon^{2}).\label{DxQ}\end{aligned}$$ Inserting these equations into (\[explicitgamma\]), (\[explicitm\]), (\[explicitDxxy\]), (\[explicitDxxx\]), (\[explicitDphiphiy\]) and (\[explicitDphiphix\]), and putting all these results together, we finally obtain that $p=-m_{i}\gamma_{\epsilon}^{AB}\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{A}}e^{i}_{B}$ satisfies (\[explicitp\]). Next, we will study $q=\gamma_{\epsilon}^{AB}e^{i}_{A}e^{j}_{B}K_{ij}$, where $K$ is the second fundamental form of $\Sigma_{\epsilon}$ with respect to the future directed unit normal. Since, $\gamma_{\epsilon}$ is diagonal, we just have to compute $e^{i}_{x}e^{j}_{x}K_{ij}=\dot{y}^{2}K_{yy}+2\dot{y}K_{xy}+K_{xx}$ and $e^{i}_{\phi}e^{j}_{\phi}K_{ij}=K_{\phi\phi}$ up to first order. To that aim, it is convenient to take coordinates $\{T=\frac12(\hat{v}-\hat{u}),\ycoor=\frac12(\hat{v}+\hat{u}),x,\phi\}$ in the Kruskal spacetime for which the metric $\gM$ is diagonal. In these coordinates $\Sigma_{\epsilon}$ is defined by $\{T=\epsilon x,\ycoor,x,\phi\}$ and the future directed unit normal to $\Sigma_{\epsilon}$ reads $$\bold{n}=\sqrt{\frac{\hat{H}\hat{Q}}{\hat{Q}-\epsilon^{2}(1-x^{2})\hat{H}}}\left( -dT+\epsilon dx \right).$$ The computation of the second fundamental form is straightforward and gives $$\begin{aligned} \dot{y}^{2}K_{yy}&=&O(\epsilon^{2}),\label{Kyy}\\ 2\dot{y}K_{xy}&=&O(\epsilon^{2}),\label{Kxy}\\ K_{xx}&=&\sqrt{\tilde{H}} \left[ \frac{\partial_{\,T}Q'}{2(1-x^{2})\tilde{H}} - \epsilon\frac{x}{1-x^2}\right]+O(\epsilon^{2}) \label{Kxx}\\ K_{\phi\phi}&=&\sqrt{\tilde{H}} \left[ \frac{(1-x^{2})\partial_{\,T}Q' }{2\tilde{H}} -\epsilon (1-x^{2})x\right]+O(\epsilon^{2})\label{Kphiphi},\end{aligned}$$ where we have denoted by $Q'$ the function obtained from $Q$ by expressing $r$ in terms of $(T,\ycoor)$ according to $\hat{u}\hat{v}=\ycoor^{2}-T^{2}=\frac{r-2M_{Kr}}{2M_{Kr}}e^{r/2M_{Kr}}$. This expression also allows us to compute $\partial_{\,T}Q'$ which, on $\Sigma_{\epsilon}$ (where $T=\epsilon x$) and using $r=2M_{Kr}+O(\epsilon)$, takes the form $$\partial_{\,T}Q'=-\epsilon \frac{16M_{Kr}^{2}}{e} x+O(\epsilon^2).$$ Inserting this into (\[Kxx\]) and (\[Kphiphi\]), and using (\[explicitgamma\]), it is a matter of simple computation to show that $q=\gamma_{\epsilon}^{AB}e^{i}_{A}e^{j}_{B}K_{ij}$ satisfies (\[explicitq\]). $\hfill \blacksquare$\ From this lemma we conclude that $f(y=\epsilon Y, \epsilon)\equiv p(y=\epsilon Y, \epsilon)-|q(y=\epsilon Y, \epsilon)|$ reads $$\label{explicitf} \f(y= \epsilon Y,\epsilon)=\frac{1}{M_{Kr}\sqrt{e}}\,(L[Y(x)]-3|x|)\epsilon+O(\epsilon^{2}).$$ The implicit function theorem requires the operator $\f$ to have a continuous Fréchet derivative and the partial derivative $\left.D_{\y}\f\right|_{(\y=0,\epsilon=0)}$ to be an isomorphism (see Appendix \[ch:appendix2\]). The problem is not trivial in our case because the appearance of $|x|$ makes the Fréchet derivative of $\f$ potentially discontinuous[^4]. However, the problem can be solved considering a suitable modification of $\f$, as we discuss in detail next. \[proposition\] There exists a neighborhood $\tilde{I}\subset I$ of $\epsilon=0$ such that $\f(\y,\epsilon)=0$ admits a solution $\y(x,\epsilon) \in U^{2,\alpha}(\mathbb{S}^2)$ for all $\epsilon \in \tilde{I}$. Moreover, $\y(x,\epsilon)$ is $C^1$ in $\epsilon$ and satisfies $\y(x,\epsilon=0)=0$. [**Proof.**]{} Firstly, let us consider surfaces $S_{\epsilon}$ in $\Sigma_{\epsilon}$ defined by $\left\{ \ycoor=y(x,\epsilon),x,\phi \right\}$ such that the embedding function has the form $\y = \epsilon \Y$, where $\Y \in U^{2,\alpha}$. Since we are considering surfaces with axial symmetry, neither $p$ nor $q$ depend on $\phi$. Let $\eta^{\mu}$ denote the spacetime coordinates, $z^{i}$ the coordinates on $\Sigma_{\epsilon}$, $x^{A}$ the coordinates on $S_{\epsilon}$, $\eta^{\mu}(z^{i})$ the embedding functions of $\Sigma$ in $M$ (which depend smoothly on $z^{i}$), and $z^{i}(x^{A})$ the embedding functions of $S$ in $\Sigma$ (which depend smoothly on $x^A$). Thus, by definition, we have $$p(x,\epsilon)=-\gamma^{AB}m_{i} \left[ \frac{\partial^{2}z^{i}}{\partial x^{A}\partial x^{B}} +{\Gamma^{\Sigma_{\epsilon}}}_{jk}^{i}(z(x))\frac{\partial z^{j}}{\partial x^{A}}\frac{\partial z^{k}}{\partial x^{B}}\right],$$ where ${\Gamma^{\Sigma_{\epsilon}}}_{jk}^{i}$ are the Christoffel symbols of $\Sigma_{\epsilon}$. In this expression all terms depend smoothly on $(\dot y(x),y(x),x,\epsilon)$, except $\frac{\partial^{2}z^{i}}{\partial x^{A}\partial x^{B}}$ which also depends on $\ddot y(x)$. Therefore, $p$ can be viewed as a smooth function of $(\ddot{y}(x),\dot{y}(x),y(x),x,\epsilon)$. Similarly, by definition, $$q(x,\epsilon)=\left . -\gamma^{AB}n_{\mu}e^{i}_{A} e_{B}^{j} \left[ \frac{\partial^{2}\eta^{\mu}}{\partial z^{i}\partial z^{j}} +{\Gamma}_{\nu\beta}^{\mu}(\eta(z)) \frac{\partial \eta^{\nu}}{\partial z^{i}}\frac{\partial \eta^{\beta}}{\partial z^{j}} \right]\right |_{z^i = z^i (x^A)},$$ where all terms depend smoothly on $(\dot{y}(x),y(x),x,\epsilon)$ Therefore, setting $y=\epsilon Y$ and since both $p$ and $q$ are $O(\epsilon)$ (see equations (\[explicitp\]) and (\[explicitq\])), we can write $$p = \epsilon \Pcal (\Y(x),\dot{\Y}(x),\ddot{\Y}(x),x, \epsilon)$$ and $$q = \epsilon \Qcal (\Y(x),\dot{\Y}(x),x, \epsilon),$$ where $\Pcal : \mathbb{R}^3 \times[-1,1] \times I \rightarrow \mathbb{R}$ and $\Qcal : \mathbb{R}^2 \times[-1,1] \times I \rightarrow \mathbb{R}$ are smooth functions. Moreover, the function $\Qcal$ has the symmetry $\Qcal\left(x_1,x_2,x_3,x_4\right) = - \Qcal\left(x_1,-x_2,-x_3,x_4\right)$, which reflects the fact that the extrinsic curvature of $\Sigma_{\epsilon}$ changes sign under a transformation $x \rightarrow -x$ and the symmetry $Y(-x)=Y(x)$. Let us write $\P(Y,\epsilon)(x) \equiv \Pcal(\Y(x),\dot{\Y}(x),\ddot{\Y}(x),x,\epsilon)$ and similarly $\Q(Y,\epsilon)(x) \equiv \Qcal (\Y(x),\dot{\Y}(x),x,\epsilon)$. Now, instead of $\f$, let us consider the functional $\F : U^{2,\alpha} \times I \rightarrow U^{0,\alpha}$ defined by $\F (\Y,\epsilon)= \P (\Y,\epsilon) - |\Q (\Y,\epsilon)|$. This functional has the property that, for $\epsilon>0$, the solutions of $\F(\Y,\epsilon)=0$ correspond exactly to the solutions of $\f(\y,\epsilon)=0$ via the relation $\y = \epsilon \Y$. Moreover, the functional $\F$ is well-defined for all $\epsilon \in I$, in particular at $\epsilon=0$. Therefore, by proving that $\F=0$ admits solutions in a neighbourhood of $\epsilon=0$, we will conclude that $\f=0$ admits solutions for $\epsilon > 0$ and the solutions will in fact belong to a neighbourhood of $\y=0$ since $\y = \epsilon \Y$. In order to show that $\F$ admits solutions we will use the implicit function theorem. Equation (\[explicitf\]) yields $$\label{explicitF} \F (\Y,\epsilon=0)(x) = c \left ( L[\Y(x)] - 3|x| \right)$$ where $c$ is the constant $1/(M_{Kr}\sqrt{e})$ and $L[\Y] \equiv -(1-x^2)\ddot{\Y} +2x\dot{\Y}+\Y$. As it is well-known the eigenvalue problem $(1-x^{2})\ddot{z}(x)-2x\dot{z}(x)+\lambda z(x)=0$ has non-trivial smooth solutions on $[-1,1]$ (the Legendre polynomials) if and only if $\lambda=l(l+1)$, with $l\in\mathbb{N}\cup\{0\}$. Thus, the kernel of $L[Y]$ (for which $\lambda=1$) is $Y=0$. We conclude that $L$ is an isomorphism between $U^{2,\alpha}$ and $U^{0,\alpha}$. Let $\Ysol \in U^{2,\alpha}$ be the unique solution of the equation $L[\Y]=3|x|$. For later use, we note that $\Q(\Ysol,\epsilon=0) = -3 c x$ (see equation (\[explicitq\])). This vanishes [*only*]{} at $x=0$. This is the key property that allows us to prove that $\F$ is $C^1(U^{2,\alpha}\times I)$. The $C^1(U^{2,\alpha}\times I)$ property of the functional $\P (\Y,\epsilon)$ is immediate from Theorem \[thr:GT\] in the Appendix \[ch:appendix2\]. More subtle is to show that $|\Q|$ is $C^1(U^{2,\alpha}\times I)$ in a suitable neighbourhood of $(\Ysol,\epsilon=0)$. Let $r_0 >0$ and define $$\label{Nur0} {\cal V}_{r_0} = \{ (\Y,\epsilon)\in U^{2,\alpha}\times I: \| (\Y-\Ysol,\epsilon) \|_{U^{2,\alpha}\times I} \leq r_0 \}.$$ First of all we need to show that $|\Q|$ is Fréchet-differentiable on ${\cal V}_{r_0}$, i.e. that for all $(\Y,\epsilon) \in {\cal V}_{r_0}$ there exists a bounded linear map $D_{\Y,\epsilon}|\Q|:U^{2,\alpha}\times I \rightarrow U^{0,\alpha}$ such that, for all $(Z,\delta) \in U^{2,\alpha} \times I$, $|\Q(\Y+Z,\epsilon+\delta)|-|\Q(\Y,\epsilon)| =D_{\Y,\epsilon}|\Q| (Z,\delta)+R_{\Y,\epsilon}(Z,\delta)$ where $\| R_{\Y,\epsilon}(Z,\delta)\|_{U^{0,\alpha}}=o(\|(Z,\delta)\|_{U^{2,\alpha}\times I})$. The key observation is that, by choosing $r_0$ small enough in Definition \[Nur0\], we have $$\begin{aligned} |\Q(\Y,\epsilon)(x)|=-\sigma(x)\Q(\Y,\epsilon)(x) \quad \forall (Y,\epsilon) \in {\cal V}_{r_0}, \label{key}\end{aligned}$$ where $\sigma(x)$ is the [*sign*]{} function, (i.e. $\sigma(x)=+1$ for $x \geq 0$ and $\sigma(x)=-1$ for $x<0$). To show this we need to distinguish two cases: when $x$ lies in a sufficiently small neighbourhood $(-\varepsilon,\varepsilon)$ of $0$ and when $x$ lies outside this neighbourhood. Consider first the latter case. As already mentioned, we have $\Q(\Ysol,\epsilon=0)=-3cx$ which is negative for $x>0$ and positive for $x<0$. Taking $r_0$ small enough, and using that $\Qcal$ is a smooth function of its arguments it follows that the inequalities $\Q(\Ysol,\epsilon)<0$ for $x\geq \varepsilon$ and $\Q(\Ysol,\epsilon)>0$ for $x \leq - \varepsilon$ still hold for any $(\Y,\epsilon) \in {\cal V}_{r_0}$. For the points $x\in (-\varepsilon,\varepsilon)$, the function $\Q (\Y,\epsilon)(x)$ is odd in $x$, so it passes through zero at $x=0$. Hence, the relation (\[key\]) holds in $(-\varepsilon,\varepsilon)$ provided we can prove that $\Q(\Y,\epsilon)$ is strictly decreasing at $x=0$. But this follows immediately from the fact that $\frac{d \Q (\Ysol,\epsilon=0)}{dx} |_{x=0} = -3 c$ and $\Qcal$ is a smooth function of its arguments. From its definition, it follows that $\Q (\Y,\epsilon)(x)$ is $C^{1,\alpha}$ (note that only first derivatives of $\Y$ enter in $\mathcal{Q}$) and that the functional $\Q(\Y,\epsilon)$ has Fréchet derivative (see Theorem \[thr:GT\] in Appendix \[ch:appendix2\]) $$D_{\Y,\epsilon}\Q(Z,\delta)(x)=A_{\Y,\epsilon}(x)Z(x)+B_{\Y,\epsilon}(x)\dot{Z}(x)+ C_{\Y,\epsilon}(x)\delta, \nonumber$$ where $A_{\Y,\epsilon}(x) \equiv \partial_{1} \Qcal |_{(\Y(x),\dot{\Y}(x),x,\epsilon)}$, $B_{\Y,\epsilon}(x) \equiv \partial_{2} \Qcal |_{(\Y(x),\dot{\Y}(x),x,\epsilon)}$ and $C_{\Y,\epsilon}(x) \equiv \partial_{4} \Qcal |_{(\Y(x),\dot{\Y}(x),x,\epsilon)}$. We note that these three functions are $C^{1,\alpha}$ and that $A_{\Y,\epsilon}$, $C_{\Y,\epsilon}$ are odd, while $B_{\Y,\epsilon}$ is even (as a consequence of the symmetries of $\Qcal$). Defining the linear map $$D_{\Y,\epsilon}|\Q| (Z,\delta) \equiv -\sigma( A_{\Y,\epsilon}Z+B_{\Y,\epsilon}\dot{Z}+C_{\Y,\epsilon}\delta ),\nonumber$$ it follows from (\[key\]) that $$|\Q(\Y+Z,\epsilon+\delta)|-|\Q(\Y,\epsilon)|= D_{\Y,\epsilon} |\Q| (Z,\delta) + R_{\Y,\epsilon} (Z,\delta),$$ with $\| R(Z,\delta)\|_{U^{0,\alpha}}=o(\|(Z,\delta)\|_{U^{2,\alpha}\times I})$. In order to conclude that $D_{\Y,\epsilon} |\Q|$ is the derivative of $|\Q(\Y,\epsilon)|$, we only need to check that, it is (i) well-defined (i.e. that its image belongs to $U^{0,\alpha}$) and (ii) that it is bounded, i.e. that $\| D_{\Y,\epsilon} |\Q| (Z,\delta) \|_{U^{0,\alpha}} < C \|(Z,\delta) \|_{U^{2,\alpha}\times I}$ for some constant $C$. To show (i), let us concentrate on the most difficult term which is $-\sigma B_{\Y,\epsilon} \dot{Z}$ (because $B_{\Y,\epsilon}(x)$ is even and need not vanish at $x=0$). Since $\dot{Z}$ is an odd function, $-\sigma B_{\Y,\epsilon} \dot{Z}$ is continuous. To show it is also Hölder continuous, we only need to consider points $x_1 = -a$ and $x_2 = b$ with $0 < a < b$ (if $x_1\cdot x_2\geq 0$, the [*sign*]{} function remains constant, so $-\sigma B_{\Y,\epsilon} \dot{Z}$ is in fact $C^{1,\alpha}$). Calling $w(x) \equiv -\sigma(x) B_{Y,\epsilon} (x) \dot{Z}(x)$ and using that $w(x)$ is even, we find $$\begin{aligned} &&|w(x_2) - w(x_1)| = |w(b) - w(-a)| = |w(b) - w(a) | = \nonumber \\ &&\left |\left . \frac{d (B_{Y,\epsilon} \dot{Z})}{dx} \right |_{x=\zeta} \right | |b-a| = \left | \left . \frac{d (B_{Y,\epsilon} \dot{Z})}{dx} \right |_{x=\zeta} \right | |b-a|^{1-\alpha}|b-a|^{\alpha}\leq \nonumber\\ &&\left | \left . \frac{d (B_{Y,\epsilon} \dot{Z})}{dx} \right |_{x=\zeta} \right | |b-a|^{1-\alpha} |x_2 - x_1|^{\alpha}\leq \left | \left . \frac{d (B_{Y,\epsilon} \dot{Z})}{dx} \right |_{x=\zeta} \right | |x_2 - x_1|^{\alpha}\nonumber \\ &&\leq \sup_{x}\left | \left . \frac{d (B_{Y,\epsilon} \dot{Z})}{dx}\right. \right ||x_2 - x_1|^{\alpha} \label{estim}\end{aligned}$$ where the mean value theorem has been applied in the third equality and $\zeta \in (a,b)$. We also have used that $|b-a|^{\alpha} \leq |b+a|^{\alpha} = |x_2 - x_1|^{\alpha}$ and $|b-a| < 1$. This proves that $- \sigma \B \dot{Z}$ is Hölder continuous with exponent $\alpha$. The remaining terms $-\sigma(x)A_{\Y,\epsilon}(x)Z(x)$ and $-\sigma(x)C_{\Y,\epsilon}(x)\delta$ are obviously continuous because they vanish at $x=0$. To show Hölder continuity the same argument that for $-\sigma(x) B_{Y,\epsilon} (x) \dot{Z}$ works. To check (ii), we have to find and upper bound for the norm $\| w(x) \|_{U^{0,\alpha}}$. $$\begin{aligned} &&\| w(x) \|_{U^{0,\alpha}}=\sup_{x}|w(x)|+\sup_{x_1\neq x_2} \frac{|w(x_{2})-w(x_{1})|}{|x_{2}-x_{1}|^{\alpha}}\\ &&\leq \sup_{x}|B_{Y,\epsilon}(x)|\sup_{x}|\dot{Z}(x)|+\sup_{x}\left| \frac{d(B_{Y,\epsilon}\dot{Z})}{dx}\right| \\ &&\leq \sup_{x}|B_{Y,\epsilon}(x)|\sup_{x}|\dot{Z}(x)|+\sup_{x}| \dot{B}_{Y,\epsilon}(x)|\sup_{x}|\dot{Z}(x)|+\sup_{x} |B_{Y,\epsilon}(x)|\sup_{x}|\ddot{Z}(x)|\\ &&\leq ( 2\sup_{x}|B_{Y,\epsilon}(x)|+\sup_{x}|\dot{B}_{Y,\epsilon}(x)| ) \|(Z,\delta)\|_{U^{2,\alpha}\times I},\end{aligned}$$ where, in the first inequality, (\[estim\]) has been used. Since $B_{Y,\epsilon}(x)$ is $C^{1,\alpha}$, then $( 2\sup_{x}|B_{Y,\epsilon}(x)|+\sup_{x}|\dot{B}_{Y,\epsilon}(x)| )$ is bounded in the compact set $[-1,1]$ and, therefore, there exists a constant $C$ such that $\| -\sigma B_{Y,\epsilon} \dot{Z} \|_{U^{0,\alpha}} < C \|(Z,\delta) \|_{U^{2,\alpha}\times I}$. A similar argument applies to $- \sigma A_{\Y,\epsilon} Z$ and $- \sigma C_{\Y,\epsilon} \delta$ and we conclude that $D_{\Y,\epsilon} |\Q|$ is indeed a continuous operator. In order to apply the implicit function theorem, it is furthermore necessary that $|\Q|\in C^1(U^{2,\alpha}\times I)$ (i.e. that $D_{Y,\epsilon} |\Q|$ depends continuously on $(\Y,\epsilon)$). This means that given any convergent sequence $(\Y_n,\epsilon_{n})\in {\cal V}_{r_0}$, the corresponding operators $D_{\Y_n,\epsilon_n} |\Q|$ also converge. Denoting by $(\Y,\epsilon) \in {\cal V}_{r_0}$ the limit of the sequence, we need to prove that $$\| D_{\Y_n,\epsilon_n} |\Q| - D_{\Y,\epsilon} |\Q| \|_{\pounds (U^{2,\alpha} \times I, U^{0,\alpha})} \rightarrow 0,$$ where, for any linear operator $\mathscr{L}:U^{2,\alpha} \times I\rightarrow U^{0,\alpha}$, the operator norm is $$\| \mathscr{L} \|_{\pounds (U^{2,\alpha} \times I, U^{0,\alpha})}\equiv \sup_{\scriptscriptstyle{(Z,\delta)\neq (0,0)}} \frac{\| \mathscr{L}(Z,\delta) \|_{U^{0,\alpha}}} {\|(Z,\delta)\|_{U^{2,\alpha}\times I}}.$$ For that it suffices to find a constant $K$ (which may depend on $(\Y,\epsilon)$), such that $$\begin{aligned} &&\hspace{-18mm}\| ( D_{\Y_n,\epsilon_n} |\Q| - D_{\Y,\epsilon} |\Q| ) (Z,\delta) \|_{U^{0,\alpha}}\nonumber \\ &&\qquad\qquad \leq K \|(Z,\delta)\|_{U^{2,\alpha}\times I} \|( \Y_n-\Y,\epsilon_{n}-\epsilon)\|_{U^{2,\alpha} \times I} \label{convergence}\end{aligned}$$ for all $(Z,\delta)\in U^{2,\alpha} \times I$. Indeed, if (\[convergence\]) holds then the right-hand side tends to zero when $(Y_{n},\epsilon_{n})\rightarrow (Y,\epsilon)$ Again, the most difficult case involves $\sigma (B_{Y,\epsilon} - B_{Y_n,\epsilon_n}) \dot{Z}$, so let us concentrate on this term (the same argument works for the remaining terms in $D_{\Y_n,\epsilon_n} |\Q| - D_{\Y,\epsilon}|\Q|$). With the definition $z \equiv \sigma (B_{\Y,\epsilon} - B_{\Y_n,\epsilon_n}) \dot{Z}$, we have $$\sup_x | z(x)| \leq \sup_{x}|{B}_{\Y,\epsilon}(x) - {B}_{\Y_n,\epsilon_n}(x)|\sup_{x}|\dot{Z}(x)|.$$ To bound the $C^0$-norm of $z$ in terms of $\|(Z,\delta)\|_{U^{2,\alpha}\times I} \|( \Y_n-\Y,\epsilon_{n}-\epsilon)\|_{U^{2,\alpha} \times I}$, we have to use the mean value theorem on the function ${\cal B} \equiv \partial_2 \Qcal$ (recall that $B_{Y,\epsilon}(x) = {\cal B} |_{(Y(x),\dot{Y}(x),x, \epsilon)}$). By the definition of $\mathcal{V}_{r_{0}}$ (see (\[Nur0\])) any element $(Y,\epsilon)\in \mathcal{V}_{r_{0}}$ satisfies that $|Y-Y_{1}|(x)\leq r_{0}$ and $|\dot{Y}-\dot{Y}_{1}|(x)\leq r_{0}$ $\forall x\in[-1,1]$. This implies that there is a compact set $\mathbb{K}\subset \mathbb{R}^{4}$ depending only on $r_{0}$ and $Y_{1}$ such that $(\Y(x), \dot{\Y}(x),x,\epsilon) \in \mathbb{K}$, for all $x \in [-1,1]$ and $(Y,\epsilon)\in{\cal V}_{r_{0}}$. When applying the mean value theorem to the derivatives $\partial_{1}{\cal B}$, $\partial_{2}{\cal B}$ and $\partial_{4}{\cal B}$ all mean value points will therefore belong to $\mathbb{K}$. Taking the supremum of these derivatives in $\mathbb{K}$, we get the following bound. $$\begin{aligned} \hspace{-6mm} \sup_x | z(x)|\leq \sup_{\mathbb{K}}\left( |\partial_{1}{\cal B}|+|\partial_{2}{\cal B}|+|\partial_{4}{\cal B}| \right) \sup_x | \dot{Z} | \| (Y_n - Y,\epsilon_{n}-\epsilon) \|_{U^{2,\alpha}\times I}. \label{ineq1} $$ Since ${\cal B}$ is smooth, (\[ineq1\]) is already of the form (\[convergence\]). It only remains to bound the Hölder norm of $z$ in a similar way. As before, this is done by distinguishing two cases, namely when $x_1 \cdot x_2 \geq 0$ and when $x_1 \cdot x_2 <0$. If $x_1 \cdot x_2 \geq 0$ then $\sigma(x)$ is a constant function and therefore, to obtaining an inequality of the form $$\sup_{x_1 \neq x_2} \frac{| z(x_2) - z(x_1)|}{|x_2 - x_1|^{\alpha}} \leq K_1 \|(Z,\delta)\|_{U^{2,\alpha}\times I} \|(\Y_n-\Y,\epsilon_{n}-\epsilon)\|_{U^{2,\alpha} \times I}$$ is standard (and a consequence of Theorem \[thr:GT\]). When $x_1 \cdot x_2 <0$, we exploit the parity of the functions as in (\[estim\]) to get $$| z(x_2) - z(x_1) | \leq \left | \left . \frac{d ((B_{\Y_n,\epsilon_n} - B_{\Y,\epsilon}) \dot{Z})}{dx} \right |_{x=\zeta} \right | |x_2 - x_1|^{\alpha},$$ where $\zeta \in (a,b)$ and we are assuming $x_1 = -a, x_2 = b, 0 < a < b$ without loss of generality. Since the sign function $\sigma(x)$ has already disappeared, a bound for the right hand side in terms of $K_2 \|(Z,\delta)\|_{U^{2,\alpha}\times I} \|(\Y_n-\Y,\epsilon_{n}-\epsilon)\|_{U^{2,\alpha} \times I} |x_2 - x_1|^{\alpha}$ is guaranteed by Theorem \[thr:GT\]. This, combined with (\[ineq1\]) gives (\[convergence\]) and hence continuity of the derivative of $D_{\Y,\epsilon} |\Q|$ with respect to $(\Y,\epsilon) \in {\cal V}_{r_0}$. The final requirement to apply the implicit function theorem to $\F = \P - |\Q|$ is to check that $D_{\Y} \F |_{(\Ysol,\epsilon=0)}$ is an isomorphism between $U^{2,\alpha}$ and $U^{0,\alpha}$. This is immediate from equation (\[explicitF\]) that implies $$D_\Y \F |_{(\Ysol,\epsilon=0)} (Z) = \F(Y_{1}+Z,\epsilon=0)-\F(Y_{1},\epsilon=0)= c L (Z),$$ and we have already shown that $L$ is an isomorphism. Thus, the implicit function theorem can be used to conclude that there exists an open neighbourhood $\tilde {I} \subset I$ of $\epsilon=0$ and a $C^1$ map $\tilde{Y}: \tilde{I} \rightarrow U^{2,\alpha}$ such that $\tilde{Y} (\epsilon=0) = \Ysol$ and $\y = \epsilon \tilde{Y}(\epsilon)$ defines a $C^{2,\alpha}$ generalized apparent horizon embedded in $\Sigma_{\epsilon}$. $\hfill \blacksquare$\ We will denote by $\hat{S}_{\epsilon}$ the surface defined by this solution. The proposition above implies that we can expand $\y(x, \epsilon)= Y_{1}(x)\epsilon + o(\epsilon)$. From (\[explicitf\]) it follows that $Y_{1}$ satisfies the linear equation $L[Y_{1}(x)]= 3 |x|$. Decomposing $Y_{1}(x)$ into Legendre polynomials $P_{l}(x)$, as $Y_{1}(x)=\sum_{l=0}^{\infty} a_{l}P_{l}(x)$, where convergence is in $L^{2}[-1,1]$, this equation reads $$L[Y_{1}(x)]=\sum_{l=0}^{\infty} a_{l}L[P_{l}(x)]=3|x|.$$ The Legendre equation, $-(1-x^{2})\ddot{P}_{l}(x)+2x\dot{P}_{l}(x)-l(l+1)P_{l}(x)=0$, implies that $L[P_{l}(x)]=(l(l+1)+1)P_{l}(x)$. We can also decompose $|x|$ in terms of Legendre polynomials. This computation can be found in [@Bravo] and gives $$|x|=\frac12 + \sum_{l=1}^{\infty} b_{2l}P_{2l}(x),\nonumber$$ where $$b_{2l}=\frac{(4l+1)(-1)^{l+1}}{2^{2l}} \frac{(2l-2)!}{(l-1)!(l+1)!}, \qquad l\geq 1.\nonumber$$ It follows that the unique solution to the equation $L[Y_{1}(x)]=3|x|$ is $$\begin{aligned} \label{Y1} \hspace{-1cm} Y_1(x) = \frac32 + \sum_{l=1}^{\infty}a_{2l}P_{2l}(x),\end{aligned}$$ with $$\begin{aligned} \label{a2n} a_{2l} = \frac{3(4l+1) (-1)^{l+1}}{\left[ 2l(2l+1)+1 \right]2^{2l}}\frac{(2l-2)!}{(l-1)!(l+1)!}, \qquad l\geq 1\end{aligned}$$ (see Figure \[fig:GAHKr\]). ![Parametic plot of the solution $Y_{1}(\cos{\theta})$ (in blue) in coordinates $\mathcal{T}\equiv M_{Kr}\ln{\frac{\hat{v}}{\hat{v}}}$, $\mathcal{X}=r \cos{\theta}$ and $\mathcal{Y}=r \sin{\theta}$ where $\theta$ has been allowed to vary between $0$ and $2\pi$, $M_{Kr}=1$ and $\epsilon=0.5$. The figure also shows the set $\{r=2M\}$ (in gold) in these coordinates. Note that the solution lies entirely outside the region $\{r\leq 2M\}$ (i.e. the region inside the cylinder).[]{data-label="fig:GAHKr"}](GAHKr.eps){width="12cm"} ### Area of the outermost generalized trapped horizon In this subsection we will compute the area of $\hat{S}_{\epsilon}$, to second order in $\epsilon$, and we will obtain that it is greater than $16\pi M_{Kr}^{2}$. Then, we will prove that any generalized apparent horizon enclosing $\hat{S}_{\epsilon}$ has greater or equal area than $\hat{S}_{\epsilon}$ which will complete the proof of Theorem \[theorem\]. Integrating the volume element of $\hat{S}_{\epsilon}$, it is straightforward to get $$\begin{aligned} \label{area0} \hspace{-1cm}|\hat{S}_{\epsilon}|&=&\int_{-1}^{1}\int_{0}^{2\pi} r^{2}\sqrt{1+\epsilon^{2} \frac{32M_{Kr}^{3}}{r^{3}}e^{-r/2M_{Kr}} (1-x^2)(\dot{Y}_{1}^{2}-1)+O(\epsilon^{3})}d\phi dx\nonumber \\ \hspace{-1cm} &=&\int_{-1}^{1}\int_{0}^{2\pi}\left[ r^{2}+\epsilon^{2}\frac{16M_{Kr}^{3}}{r}e^{-r/2M_{Kr}}(1-x^2)(\dot{Y}_{1}^{2}-1)+ O(\epsilon^{3}) \right]d\phi dx,\nonumber\end{aligned}$$ where $r$ still depends on $\epsilon$. Let us expand $r=r_{0}+r_{1}\epsilon+r_{2}\epsilon^{2}+O(\epsilon^{3})$. Using equation (\[uvrembedded\]) and expanding the exponential therein, it follows $$\label{r(epsilon)} r=2M_{Kr}+\frac{2M_{Kr}}{e}(Y_{1}^{2}-x^{2})\epsilon^{2}+O(\epsilon^{3}).$$ Then, after inserting (\[Y1\]), (\[a2n\]) and (\[r(epsilon)\]) into the integral and using the orthogonality properties of the Legendre polynomials, we find $$|\hat{S}_{\epsilon}| = 16 \pi M_{Kr}^2 + \frac{8 \pi M_{Kr}^2 \epsilon^2}{e} \left ( 5 + 4 \sum_{l=1}^{\infty} \frac{2l(2l+1)+1}{4l+1}a_{2l}^2 \right) + O (\epsilon^3).$$ Since the second term is strictly positive, it follows that $|\hat{S}_{\epsilon}| > 16 \pi M_{Kr}^2$. This is not yet a counterexample of (\[penroseBK2\]) because $\hat{S}_{\epsilon}$ is not known to be the outermost generalized apparent horizon. Before turning into this point, however, let us give an alternative argument to show that the area increases. This will shed some light into the underlying reason why the area of $\hat{S}_{\epsilon}$ is larger than $16 \pi M_{Kr}^2$. To that aim, let us now use coordinates $\{\hat{u},x,\phi\}$ in $\Sigma_{\epsilon}$. Then, the embedding of $\Sigma_{\epsilon}$ becomes $\Sigma_{\epsilon}\equiv\left\{ \hat{u},\hat{v}=\hat{u}+2 \epsilon x,x,\phi \right\}$, and the corresponding embedding in $\Sigma_{\epsilon}$ for the surfaces $\hat{S}_{\epsilon}$ is $\hat{S}_{\epsilon}=\left\{ \hat{u} = u(x,\epsilon),x,\phi \right\}$. Again, $u$ admits an expansion $u = U_1(x) \epsilon + o(\epsilon)$. The relationship between $U_1$ and $Y_1$ is simply $Y_{1}=U_{1}+x$. It follows that $U_1$ satisfies $L[U_{1}(x)]= 3(|x|- x)$. Similarly, if we take $\left\{ \hat{v},x,\phi \right\}$ as coordinates for $\Sigma_{\epsilon}$, then the embedding of $\hat{S}_{\epsilon}$ reads $\hat{v} = V_1 (x) \epsilon + o (\epsilon)$, with $V_1$ satisfying $Y_{1}=V_{1}-x$ and therefore $L[V_1(x)]= 3 (|x|+ x)$. Thus, $L[U_{1}(x)]\geq 0$ and $L[V_1(x)] \geq 0 $ and neither of them is identically zero. Since $L$ is an elliptic operator with positive zero order term, we can use the maximum principle to conclude that $U_1(x)>0$ and $V_{1}(x)>0$ everywhere. Geometrically, this means that $\hat{S}_{\epsilon}$ lies fully in $\Sigma^{+}_{\epsilon}$ for $\epsilon$ small enough (c.f. Figure \[fig:GAHKr\]). In fact, the maximum principle applied to $L[Y_1] = 3 |x|$ also implies $Y_1 > 0$. This will be used below. We can now view $\hat{S}_{\epsilon}$ as a first order spacetime variation of the bifurcation surface $\hat{S}_{\epsilon=0}$. The variation vector $\partial_{\epsilon}$ is defined as the tangent vector to the curve generated when a point with fixed coordinates $\{x,\phi\}$ in $\hat{S}_{\epsilon}$ moves as $\epsilon$ varies. This vector satisfies $\partial_{\epsilon}=U_{1}\partial_{\hat{u}}+V_{1}\partial_{\hat{v}}+O(\epsilon)$ and is spacelike everywhere on the unperturbed surface $\hat{S}_{\epsilon=0}$. If we do a Taylor expansion of $|\hat{S}_{\epsilon}|$ around $\epsilon=0$, we see that the zero order term is $|\hat{S}_{\epsilon=0}| = 16 \pi M_{Kr}^2$, as this is the area of the bifurcation surface. The bifurcation surface is totally geodesic so that, in particular, its mean curvature vector vanishes. Consequently, the linear term in the expansion is identically zero as a consequence of the first variation of area (\[firstvariation\]). For any $\epsilon \geq 0$ we have $$\begin{aligned} \label{firstvariationarea} &&\hspace*{-1.6cm}\frac{d |\hat{S}_{\epsilon}|}{d \epsilon} = \int_{\hat{S}_{\epsilon}} ( \vec{H}_{\hat{S}_{\epsilon}}, \partial_{\epsilon} ) {\mbox{\boldmath $\eta_{\hat{S}_{\epsilon}}$}}\nonumber\\ &&\hspace*{-6mm} = \int_{\hat{S}_{\epsilon}} \left( -\frac 12 \left[(p+q)\vec{l}_{-}+(-p+q)\vec{l}_{+}\right], U_{1}\partial_{\hat{u}} +V_{1}\partial_{\hat{v}}+O(\epsilon)\right){\mbox{\boldmath $\eta_{\hat{S}_{\epsilon}}$}}\end{aligned}$$ where $\vec{H}_{\hat{S}_{\epsilon}}$ is the spacetime mean curvature vector of $\hat{S}_{\epsilon}$, $( \, , \, )$ denotes the scalar product with the spacetime metric, and $\vec{l}_{+}$ and $\vec{l}_{-}$ are the outer and the inner null vectors which are future directed and satisfy $(\vec{l}_{+},\vec{l}_{-})=-2$. Since on $\hat{S}_{\epsilon=0}$ the vectors $\partial_{\hat{v}}$ and $-\partial_{\hat{u}}$ are proportional to $\vec{l}_{+}$ and $\vec{l}_{-}$, we have $$\begin{aligned} \vec{l}_{+}\big|_{\hat{S}_{\epsilon}}&=&\sqrt{\frac{e}{8M_{Kr}^{2}}}\partial_{\hat{v}}+O(\epsilon),\\ \vec{l}_{-}\big|_{\hat{S}_{\epsilon}}&=&\sqrt{\frac{e}{8M_{Kr}^{2}}}(-\partial_{\hat{u}})+O(\epsilon),\end{aligned}$$ where the factor $\sqrt{\frac{e}{8M_{Kr}^{2}}}$ is due to the normalization $(l_{+},l_{-})=-2$. Besides, ${\mbox{\boldmath ${\eta}_{\hat{S}_{\epsilon}}$}}=4M_{Kr}^{2}dx\wedge d\phi+O(\epsilon)$. Then, inserting these expressions into the first variation integral (\[firstvariationarea\]) and taking the derivative with respect to $\epsilon$ at $\epsilon=0$, we obtain $$\begin{aligned} \left. \frac{d^2 |\hat{S}_{\epsilon}|}{d\epsilon^2} \right|_{\epsilon=0}= \frac{16\sqrt{2}\pi M_{Kr}^2}{e}\int _{-1}^1 \left[ \frac{}{} U_1(x)L[V_1(x)]+V_1(x)L[U_1(x)]\right] dx,\end{aligned}$$ where (\[explicitp\]), (\[explicitq\]) and the relations $Y_{1}=U_{1}+x$ and $Y_1=V_{1}-x$ has been used. Since $U_1$ and $V_1$ are strictly positive and $L[U_{1}(x)]$, $L[V_1(x)]$ are non-negative and not identically zero, it follows $\left. \frac{d^2 |\hat{S}_{\epsilon}|}{d\epsilon^2} \right|_{\epsilon=0}>0$ and hence that the area of $\hat{S}_{\epsilon}$ is larger than $16 \pi M_{Kr}^2$ for small $\epsilon$. We have obtained that the second order variation of area turns out to be strictly positive along the direction joining the bifurcation surface with $\hat{S}_{\epsilon}$, which is tied to the fact that $L[U_1]$ and $L[V_1]$ have a sign. The right hand sides of these operators are (except for a constant) the linearization of $|q| \pm q$ and these objects are obviously non-negative in all cases. We conclude, therefore, that the fact that the area of $\hat{S}_{\epsilon}$ is larger than $16\pi M_{Kr}^2$ is closely related to the defining equation $p = |q|$. It follows that the increase of area is a robust property which does not depend strongly on the choice of hypersurfaces $\Sigma_{\epsilon}$ that we have made. In fact, had we chosen hypersurfaces $\Sigma_{\epsilon}\equiv \left\{ u=y- \epsilon \beta(x) , v=y+ \epsilon \beta(x) , \cos \theta = x, \phi = \phi \right\}$, the corresponding equations would have been $L[U_{1}(x)]=|L[\beta(x)]| - L[\beta(x)] $ and $L[V_{1}(x)]=|L[\beta(x)]| + L[\beta(x)]$. The same conclusions would follow provided the right hand sides are not identically zero. Having shown that $|\hat{S}_{\epsilon}| > 16 \pi M_{Kr}^2$ for $\epsilon \neq 0$ small enough, the next step is to analyze whether $|\hat{S}_{\epsilon}|$ is a lower bound for the area of the outermost generalized apparent horizon. Indeed, in order to have a counterexample of (\[penroseBK2\]) we only need to make sure that no generalized apparent horizon with less area than $\hat{S}_{\epsilon}$ and enclosing $\hat{S}_{\epsilon}$ exists in $\Sigma_{\epsilon}$. We will argue by contradiction. Let ${S}_{\epsilon}'$ be a generalized apparent horizon enclosing $\hat{S}_{\epsilon}$ and with $|{S}_{\epsilon}' | < |\hat{S}_{\epsilon}|$. In these circumstances, $\hat{S}_{\epsilon}$ cannot be area outer minimizing. Thus, its minimal area enclosure $\hat{S}_{\epsilon}'$ does not coincide with it. Now, two possibilities arise: (i) either $\hat{S}_{\epsilon}'$ lies completely outside $\hat{S}_{\epsilon}$, or (ii) it coincides with $\hat{S}_{\epsilon}$ on a closed subset $\mathcal{K}$, while the complement $\hat{S}_{\epsilon}' \setminus \mathcal{K}$ (which is non-empty) has vanishing mean curvature $p$ everywhere. To exclude case (i), consider the foliation of $\Sigma_{\epsilon}$ defined by the surfaces $\{\ycoor=y_0,x,\phi\}$, where $y_0$ is a constant. We then compute the mean curvature $p_{y_{0}}$ of these surfaces. The induced metric is $$\gamma^{y_{0}}_{AB}=\left( \frac{r^{2}}{1-x^2}-\epsilon^{2}\frac{32M_{Kr}^{3}}{r}e^{-r/2M_{Kr}}\right) dx^{2} +(1-x^{2})r^{2}d\phi^{2}.$$ The tangent vectors and the unit normal one-form are $$\begin{aligned} \vec{e}_{x}=\partial_{x}, \quad \vec{e}_{\phi}=\partial_{\phi}, \quad \bold{m}=A d\ycoor,\end{aligned}$$ where $A=\sqrt{\frac{32M_{Kr}^{3}}{r}e^{-r/2M_{Kr}}}$ is the normalization factor. Since $\gamma^{\ycoor_{0}}$ is diagonal we just need the following derivatives $$\begin{aligned} \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{x}}e_{x}^{\ycoor}&=&- \frac{r^{3}+8\epsilon^{2}M_{Kr}^{2}(2M_{Kr}+r)(1-x^{2})e^{-r/2M_{Kr}}}{4M_{Kr}(1-x^{2})r^2}y_{0}\\ \nabla^{\Sigma_{\epsilon}}_{\vec{e}_{\phi}}e_{\phi}^{\ycoor}&=&-\frac{(1-x^{2})r}{4M_{Kr}}y_{0}.\end{aligned}$$ Inserting all these expressions in $p_{y_{0}}=-m_{i}\gamma^{AB}\nabla^{\Sigma_{\epsilon}}_{\vec{e}_{A}}e_{B}^{i}$ we obtain $$\begin{aligned} p_{y_{0}}=A\left( \frac{r^{3}+8\epsilon^{2} M_{Kr}^2(2M_{Kr}+r)(1-x^2)e^{-r/2M_{Kr}}}{4M_{Kr}r\left(r^{3}-32\epsilon^{2}M_{Kr}^{3}(1-x^{2})e^{-r/2M_{Kr}}\right)}+ \frac{1}{4M_{Kr}r} \right) y_{0}. $$ Thus, taking $-1<\epsilon<1$ small enough so that $$\label{epsilon} \epsilon^{2}<\frac{r^{3}_{\text{min}}e^{r_{\text{min}}/2M_{Kr}}}{32M_{Kr}^{3}},$$ where $r_{\text{min}}$ is the minimum value of $r$ in $\Sigma_{\epsilon}$ (recall that $r_{\text{min}}>0$ provided $|\epsilon|<1$), we can assert that $p_{y_{0}}>0$ for all $y_{0}>0$. We noted above that $Y_1(x) >0$ everywhere. Thus, for small enough positive $\epsilon$, the function $y(x, \epsilon)$ is also strictly positive. Since $\hat{S}_{\epsilon}'$ lies fully outside $\hat{S}_{\epsilon}$, the coordinate function $\ycoor$ restricted to $\hat{S}_{\epsilon}'$ achieves a positive maximum $y_{\epsilon}$ somewhere. At this point, the two surfaces $\hat{S}_{\epsilon}'$ and $\{\ycoor = y_{\epsilon}\}$ meet tangentially, with $\hat{S}_{\epsilon}'$ lying fully inside $\{ \ycoor = y_{\epsilon} \}$ (see Figure \[fig:BK1\]). This is a contradiction to the maximum principle for minimal surfaces (see Proposition \[maximumprincipleforMOTS\] with $K=0$ in Appendix \[ch:appendix2\]). ![If the minimal area enclosure $\hat{S}_{\epsilon}'$ (in red) lies completely outside $\hat{S}_{\epsilon}$ then $\hat{S}_{\epsilon}'$, which is a minimal surface, must touch tangentially from the inside a surface $\{\ycoor=y_{\epsilon}\}$ (in blue) which has $p_{y_{\epsilon}}>0$.[]{data-label="fig:BK1"}](BK1.eps){width="5cm"} It only remains to deal with case (ii). The same argument above shows that the coordinate function $\ycoor$ restricted to $\hat{S}_{\epsilon}' \setminus \mathcal{K}$ cannot reach a local maximum. It follows that the range of variation of $\ycoor$ restricted to $\hat{S}_{\epsilon}'$ is contained in the range of variation of $\ycoor$ restricted to $\hat{S}_{\epsilon}$ (see Figure \[fig:BK2\]). ![In the case (ii), the minimal area enclosure $\hat{S}_{\epsilon}'$ coincides with $\hat{S}_{\epsilon}$ in a compact set. The coordinate function $\ycoor$ restricted to $\hat{S}_{\epsilon}'$ cannot achieve a local maximum in the set where $\hat{S}_{\epsilon}'$ and $\hat{S}_{\epsilon}$ do not coincide (in red). Then, this set can be viewed as an outward variation of order $\epsilon$ of the corresponding points in $\hat{S}_{\epsilon}$.[]{data-label="fig:BK2"}](BK2.eps){width="6cm"} Since $\max_{\hat{S}_{\epsilon}} \ycoor - \min_{\hat{S}_{\epsilon}} \ycoor = O (\epsilon)$, it follows that we can regard $\hat{S}_{\epsilon}'$ as an outward variation of $\hat{S}_{\epsilon}$ of order $\epsilon$ when $\epsilon$ is taken small enough. The corresponding variation vector field $\vec{\nu}$ can be taken orthogonal to $\hat{S}_{\epsilon}$ without loss of generality, i.e. $\vec{\nu}=\nu \vec{m}$, where $\vec{m}$ is the outward unit normal to $\hat{S}_{\epsilon}$. The function $\nu$ vanishes on $\mathcal{K}$ and is positive in its complement $U \equiv \hat{S}_{\epsilon} \setminus \mathcal{K}$. Expanding to second order and using the first and second variation of area (see e.g. [@Chavel]) gives $$\begin{aligned} |\hat{S}_{\epsilon}' |&=& |\hat{S}_{\epsilon}| + \epsilon \int_{U} p_{\hat{S}_{\epsilon}}\nu {\mbox{\boldmath $\eta_{\hat{S}_{\epsilon}}$}} \\ && + \frac{\epsilon^2}{2} \int_{U} \left ( |\nabla_{\hat{S}_{\epsilon}}\nu|^2 + \frac{\nu^2}{2} \left( R^{\hat{S}_{\epsilon}}-R^{\Sigma_{\epsilon}}-|\kappa_{\hat{S}_{\epsilon}}|^{2} + p_{\hat{S}_{\epsilon}}^2 \right ) + p_{\hat{S}_{\epsilon}} \frac{d\nu}{d\epsilon} \right ) {\mbox{\boldmath $\eta_{\hat{S}_{\epsilon}}$}} + O (\epsilon^3),\end{aligned}$$ where $\nabla_{\hat{S}_{\epsilon}}$, $R^{\hat{S}_{\epsilon}}$ and $\kappa_{\hat{S}_{\epsilon}}$ are, respectively, the gradient, scalar curvature and second fundamental form of $\hat{S}_{\epsilon}$, and $R^{\Sigma_{\epsilon}}$ is the scalar curvature of $\Sigma_{\epsilon}$. Now, the mean curvature $p_{\hat{S}_{\epsilon}}$ of $\hat{S}_{\epsilon}$ reads $p_{\hat{S}_{\epsilon}} =\frac{3 \epsilon}{M_{Kr} \sqrt{e}} |x| + o (\epsilon)$ (see equation (\[explicitp\])) and both $R^{\Sigma_{\epsilon}} $ and $\kappa_{\hat{S}_{\epsilon}}$ are of order $\epsilon$ (because $\Sigma_{\epsilon=0}$ has vanishing scalar curvature and $\hat{S}_{\epsilon=0}$ is totally geodesic). Moreover, $R^{\hat{S}_{\epsilon}} = 1/(2M_{Kr}^2) + O(\epsilon)$. Thus, $$\begin{aligned} |\hat{S}_{\epsilon}' |= |\hat{S}_{\epsilon}| + \epsilon^2 \left\{ \int_{U} \left [ \frac{3|x|\nu}{M_{Kr}\sqrt{e}}+ \left(\frac{|\nabla_{\hat{S}_{\epsilon}}\nu|^2}{2}+\frac{\nu^2}{8M_{Kr}^2} \right) \right ] {\mbox{\boldmath $\eta_{\hat{S}_{\epsilon}}$}} \right\}+ O(\epsilon^3).\end{aligned}$$ It follows that, for small enough $\epsilon$, the area of $\hat{S}_{\epsilon}'$ is larger than $\hat{S}_{\epsilon}$ contrarily to our assumption. This proves Theorem \[theorem\] and, therefore, the existence of counterexamples to the version (\[penroseBK2\]) of the Penrose inequality. It is important to remark that the existence of this counterexample does not invalidate the approach suggested by Bray and Khuri to study the general Penrose inequality. It means, however, that the emphasis should not be put on generalized apparent horizons. It may be that the approach can serve to prove the standard version (\[penrose1\]) as recently discussed in [@BK2]. Conclusions =========== In this thesis we have studied some questions within the framework of the theory of General Relativity. In particular, we have concentrated on some of the properties of marginally outer trapped surfaces (MOTS) and weakly outer trapped surfaces in spacetimes with symmetries, specially static isometries, and its application to the uniqueness theorems of black holes and the Penrose inequality. We can summarize the main results of this thesis in the following list.\ 1. We have obtained a general expression for the first variation of the outer null expansion $\theta^+$ of a surface $S$ along an arbitrary vector field $\vec{\xi}$ in terms of the deformation tensor of the spacetime metric associated with the vector $\vec{\xi}$. This expression has been particularized when $S$ is a MOTS. 2. Starting from a geometrical idea that generates a family of surfaces by moving first along $\vec{\xi}$ and then along null geodesics, we have used the theory of linear elliptic second order operators to obtain restrictions on any vector field on stable and strictly stable MOTS. Using the expression mentioned in the previous point, these results have been particularized to generators of symmetries of physical interest, such as Killing vectors, homotheties and conformal Killing vectors. As an application we have shown that there exists no stable MOTS in any spacelike hypersurface of a large class of Friedmann-Lemaître-Robertson-Walker cosmological models, which includes all classic models of matter and radiation dominated eras and those models with accelerated expansion which satisfy the null energy condition (NEC). 3. For the situations when the elliptic theory is not useful, we have exploited the geometrical idea mentioned before to obtain similar restrictions for Killing vectors and homotheties on outermost and locally outermost MOTS. As a consequence of these results, we have shown that, on a spacelike hypersurface possessing an untrapped barrier $S_{b}$, a Killing vector or a homothety $\vec{\xi}$ cannot be timelike anywhere on a bounding weakly outer trapped surface whose exterior lies in the region where $\vec{\xi}$ is timelike, provided the NEC holds in the spacetime. For the more general cases when the elliptic theory simply cannot be applied, a suitable variation of the geometrical idea has allowed us to obtain weaker restrictions on any vector field $\vec{\xi}$ on locally outermost MOTS. This results have also been particularized to Killing vectors, homotheties and conformal Killing vectors. 4. Analyzing the Killing form in a static Killing initial data (KID) $\kid$ we have shown, at the initial data level, that the topological boundary of each connected component $\{\lambda>0\}_{0}$ of the region where the Killing vector is timelike is a smooth injectively immersed submanifold with $\theta^{+}=0$ with respect to the outer normal which points into $\{\lambda>0\}_{0}$, provided - $N Y^i \nablaSigma_i \lambda|_{\tbd\{\lambda>0\}_{0}} \geq 0$ if $\tbd\{\lambda>0\}_{0}$ contains at least one fixed point. - $N Y^i m_i |_{\tbd\{\lambda>0\}_{0}} \geq 0$ if $\tbd\{\lambda>0\}_{0}$ contains no fixed point, where $\vec{m}$ is the unit normal pointing towards $\{\lambda>0\}_{0}$. There are examples in the Kruskal spacetime where these conditions do not hold and $\tbd \{\lambda>0\}_{0}$ fails to be smooth and has $\theta^{+}\neq 0$. 5. Under the same hypotheses as before we have proven a confinement result for MOTS in arbitrary spacetimes satisfying the NEC and for arbitrary spacelike hypersurfaces, not necessarily time-symmetric. The hypersurfaces need not be asymptotically flat either and are only required to have an outer untrapped barrier $S_b$. This result, which also have been proved at the initial data level, asserts that no bounding weakly outer trapped surface can intersect $\ext$, where $\ext$ denotes the connected component of $\{\lambda>0\}$ which contains $\Sb$. A condition which ensures that all arc-connected components of $\tbd \{\lambda>0\}$ are topologically closed is required. This condition is automatically fulfilled in spacetimes containing no non-embedded Killing prehorizons. 6. We have proven that the set $\tbd \{\lambda>0\}$ in an embedded static KID is a union of smooth injectively immersed surfaces with at least one of the two null expansions equal to zero (provided the topological condition mentioned in the previous point is satisfied). 7. Using the previous result, we have shown that, in a static embedded KID which satisfies the NEC and possesses an outer untrapped barrier $S_{b}$ and a bounding weakly outer trapped surface, the set $\tbd \ext$ is the outermost bounding MOTS provided that every arc-connected component of $\tbd \ext$ is topologically closed, the past weakly outer trapped region $T^{-}$ is contained in the weakly outer trapped region $T^{+}$ and a topological condition which ensures that all closed orientable surfaces separate the manifold. 8. With the previous result at hand, we have obtained a uniqueness theorem for embedded static KID containing an asymptotically flat end which satisfy the NEC and possess a bounding weakly outer trapped surface. The matter model is arbitrary as long as it admits a static black hole uniqueness proof with the Bunting and Masood-ul-Alam doubling method. This result extends a previous theorem by Miao valid on vacuum and time-symmetric slices, and allows to conclude that, at least regarding uniqueness of black holes, event horizons and MOTS do coincide in static spacetimes. This result requires the same hypotheses as the result in the previous point. As we have mentioned before, the condition on the arc-connected components of $\tbd \ext$ is closely related with the non-existence of non-embedded Killing prehorizons and can be removed if a result on the non-existence of these type of prehorizons is found. The condition $T^{-}\subset T^{+}$ is needed for out argument to work. Trying to drop this hypotheses is a logical next step, but it would require a different method of proof. 9. Finally, we have proved that there exist slices in the Kruskal spacetime where the outermost generalized apparent horizon has area greater than $16\pi M_{Kr}^{2}$, where $M_{Kr}$ is the mass of the Kruskal spacetime. This gives a counterexample of a Penrose inequality recently proposed by Bray and Khuri (in terms of the area of the outermost apparent horizon) in order to address the general proof of the standard Penrose inequality. The existence of this counterexample does not invalidate the approach of these authors but indicate that the emphasis must not be on generalized apparent horizons. \[chapter\] \[thra\][Definition]{} \[thra\][Lemma]{} \[thra\][Corollary]{} \[thra\][Proposition]{} Differential manifolds {#ch:appendix1} ====================== In this Appendix, we will give a definition of a differentiable manifold which allows us to consider manifolds with and without boundary at the same time. We follow [@Hirsch].\ Consider the vector space $\mathbb{R}^{n}$ and let $\bf \omega_{\alpha}$ be a one-form defined on this vector space (the index $\alpha$ is simply a label at this point). Let us define the set $H_{\alpha}=\{ \vec{r}\in\mathbb{R}^{n}: {\bf \omega}_{\alpha}(\vec{r}\,)\geq 0 \}$, which is either a half plane if $\omega_{\alpha}\neq 0$ or the whole space if $\omega_{\alpha}=0$. The concept of differentiable manifold may be defined as follows. \[defi:manifold\] A [**differentiable manifold**]{} is a topological space $M$ together with a collection of open sets $U_{\alpha}\subset M$ such that: 1. The collection $\{U_{\alpha}\}$ is an open cover of $M$, i.e. $M=\underset{\alpha}{\bigcup}U_{\alpha}$. 2. For each $\alpha$ there is a bijective map $\varphi_{\alpha}: U_{\alpha}\rightarrow V_{\alpha}$, where $V_{\alpha}$ is an open subset of $H_{\alpha}$ with the induced topology of $\mathbb{R}^{n}$. Every set $(U_{\alpha},\varphi_{\alpha})$ is called a [*chart*]{} or a [*local coordinate system*]{}. The collection $\{ (U_{\alpha},\varphi_{\alpha}) \}$ is called an [*atlas*]{}. 3. Consider two sets $U_{\alpha}$ and $U_{\beta}$ which overlap, i.e. $U_{\alpha}\cap U_{\beta}\neq \emptyset$, and consider the map $\varphi_{\beta}\circ\varphi_{\alpha}^{-1}: \varphi_{\alpha}(U_{\alpha}\cap U_{\beta})\rightarrow \varphi_{\beta}(U_{\alpha}\cap U_{\beta})$. Then, there exists a map $\varphi_{\alpha\beta}: W_{\alpha}\rightarrow W_{\beta}$, where $W_{\alpha}$ and $W_{\beta}$ are open subsets of $\mathbb{R}^{n}$ which, respectively, contain $\varphi_{\alpha}(U_{\alpha}\cap U_{\beta})$ and $\varphi_{\beta}(U_{\alpha}\cap U_{\beta})$ such that $\varphi_{\alpha\beta}$ is a differentiable bijection, with differentiable inverse and satisfying $\left.\varphi_{\alpha\beta}\right|_{\varphi_{\alpha}(U_{\alpha}\cap U_{\beta})}=\varphi_{\beta}\circ \varphi_{\alpha}^{-1}$. [**Remark.**]{} Since no confusion arises, we will denote a differential manifold $(M,\{U_{\alpha}\})$ simply by $M$. Note that manifolds need not be connected according to this definition. $\hfill \square$ \[defi:smoothmanifold\] A differentiable manifold $M$ is [**of class $C^{k}$**]{} if the mappings $\varphi_{\alpha\beta}$ and their inverses are $C^{k}$.\ A differentiable manifold $M$ is [**smooth**]{} (or $C^{\infty}$) if it is $C^{k}$ for all $k\in\mathbb{N}$. \[defi:manifoldwithboundary\] $M$ is a [**differentiable manifold with boundary**]{} if for at least one chart $U_{\alpha}$, we have $\omega_{\alpha}\neq 0$. In this case, the [**boundary**]{} of $M$ is defined as $\bd M=\underset{\alpha ,\omega_{\alpha}\neq 0}{\bigcup} \{\p\in U_{\alpha} \text{ such that } \omega_{\alpha}\left( \varphi_{\alpha}(\p) \right)=0\}$ [**Remark.**]{} Along this thesis the sign $\bd$ will denote the boundary of a manifold while the sign $\tbd$ will refer to the [*topological*]{} boundary of any subset of a topological space (both concepts are in general completely different). $\hfill \square$ $M$ is a [**differentiable manifold without boundary**]{} if $\omega_{\alpha}=0$ for all $\alpha$. It can be proven that $\bd M$ is a differentiable manifold without boundary. The interior $\text{int}({M})$ of a manifold $M$ is defined as $\text{int}({M})=M\setminus \bd M$. We will denote by $\overline{U}$ the topological closure of a set $U$ and by $\overset{\circ}{U}$ its topological interior. \[defi:orientablemanifold\] A differentiable manifold, with or without boundary, is [**orientable**]{} if there exists an atlas such that for any two charts $(U_{\alpha},\varphi_{\alpha})$ and $(U_{\beta},\varphi_{\beta})$ which overlap, i.e. $U_{\alpha}\cap U_{\beta}\neq 0$, the Jacobian of $\left.\varphi_{\alpha\beta}\right|_{U_{\alpha}\cap U_{\beta}}$ on $U_{\alpha}\cap U_{\beta}$ is positive. Such an atlas will be called [**oriented atlas**]{}\ A differentiable manifold with an oriented atlas is said to be [**oriented**]{}. Consider an oriented manifold $M$ endowed with a metric $\gN$. The [**volume element**]{} ${\bf \eta}^{(n)}$ of $(M,\gN)$ is the $n$-form $\eta^{(n)}_{\alpha_{1}...\alpha_{n}}=\sqrt{|\text{det } \gN|} \epsilon_{\alpha_{1}...\alpha_{n}}$ in any coordinate chart of the oriented atlas. Here, $\epsilon_{\alpha_{1}...\alpha_{n}}$ is the totally antisymmetric symbol and $\text{det } \gN$ is the determinant of $\gN$ in this chart. All manifolds in thesis are assumed to be Hausdorff and paracompact. These concepts are defined as follows. A topological space $M$ is [**Hausdorff**]{} if for each pair of points $\p,\q$ with $\p\neq \q$, there exist two disjoint open sets $U_{\p}$ and $U_{\q}$ such that $\p\in U_{\p}$ and $\q\in U_{\q}$. Let $M$ be a topological space and let $\{U_{\alpha}\}$ be an open cover of $M$. An open cover $ \{ V_{\beta}\}$ is said to be a [*refinement*]{} of $\{U_{\alpha}\}$ if for each $V_{\beta}$ there exists an $U_{\alpha}$ such that $V_{\beta}\subset U_{\alpha}$. The cover $\{V_{\beta }\}$ is said to be [*locally finite*]{} if each $\p\in M$ has an open neighbourhood $W$ such that only finitely many $V_{\beta}$ satisfy $W\cap V_{\beta}\neq \emptyset$.\ The topological space $M$ is said to be [**paracompact**]{} if every open cover $\{ U_{\alpha} \}$ of $M$ has a locally finite refinement $\{ V_{\beta} \}$. Elements of mathematical analysis {#ch:appendix2} ================================= This Appendix is devoted to introducing some elements of mathematical analysis which are used throughout this thesis. Firstly, recall that a Banach space is a normed vector space which is complete. Let ${\cal X}$, ${\cal Y}$ be Banach spaces with respective norms $|| \cdot ||_{\cal X}$ and $|| \cdot ||_{\cal Y}$. Let $U_{\cal X} \subset {\cal X}$, $U_{\cal Y} \subset {\cal Y}$ be open sets. A function $f : U_{\cal X} \rightarrow U_{\cal Y}$ is said to be Fréchet-differentiable at $x \in U_{\cal X}$ if there exists a linear bounded map $D_x f: {\cal X} \rightarrow {\cal Y}$ such that $$\begin{aligned} \lim_{ h \rightarrow 0} \frac{ ||f (x+h) - f(x) - D_xf (h) ||_{\cal Y}}{||h||_{\cal X}}=0.\end{aligned}$$ $f$ is said to be $C^1$ if it is differentiable at every point $x \in U_{\cal X}$ and the map $Df: U_{\cal X} \rightarrow L ({\cal X},{\cal Y})$ defined by $Df (x) = D_x f$ is continuous. Here $L({\cal X},{\cal Y})$ is the Banach space of linear bounded maps between ${\cal X}$ and ${\cal Y}$ with the operator norm. A key tool in analysis is the [*implicit function theorem*]{}. \[thr:implicitfunction\] Let ${\cal X}$, ${\cal Y}$, ${\cal Z}$ be Banach spaces and $U_{\cal X}$, $U_{\cal Y}$, $U_{\cal Z}$ respective open sets with $0 \in U_{\cal Z}$. Let $f: U_{\cal X} \times U_{\cal Y} \rightarrow U_{\cal Z}$ be $C^1$ with Fréchet-derivative $D_{(x,y)} f$. Let $x_0 \in U_{\cal X}$, $y_0 \in {\cal Y}$ satisfy $f(x_0,y_0)=0$ and assume that the linear map $$\begin{aligned} D_y f |_{(x_0,y_0)} : {\cal Y} & \rightarrow & {\cal Z}, \\ \hat{y} & \rightarrow & D_{(x_0,y_0)} f (0,\hat{y})\end{aligned}$$ is invertible, bounded and with bounded inverse. Then there exist open neighbourhoods $x_0 \in {U}_{x_0} \subset U_{\cal X}$ and $y_0 \in {U}_{y_0} \subset U_{\cal Y}$ and a $C^1$ map $g : U_{x_0} \rightarrow U_{y_0}$ such that $f (x,g(x))=0$ and, moreover, $f(x,y)=0$ with $(x,y) \in U_{x_0} \times U_{y_0}$ implies $y = g(x)$. In the context of partial differential equations, one important class of Banach spaces are the Hölder spaces. Let $\Omega \subset \mathbb{R}^n$ be a domain and $f : \overline{\Omega} \rightarrow \mathbb{R}$. Let $\beta = (\beta_1, \cdots, \beta_n)$ be multi-index (i.e. $\beta_i \in \mathbb{N} \cup \{ 0 \}$ for all $i \in \{ 1, \cdots n\}$) and define $|\beta| = \sum_{i=1}^{n} \beta_i$ . Denote by $D^{\beta} f$ the partial derivative $D^{\beta} f = \partial_{x_1^{\beta_1}} \cdots \partial_{x_n^{\beta_n}} f$ when this exists. For $k \in \mathbb{N} \cup \{0 \}$ we denote by $C^k (\overline{\Omega})$ the set of functions $f$ with continuous derivatives $D^{\beta} f$ for all $\beta$ with $|\beta|\leq k$. Let $0 < \alpha \leq 1$. The function $f$ is [*Hölder continuous with exponent $\alpha$*]{} if $$\begin{aligned} [f]_{\alpha} \equiv \sup_{\underset{x \neq y}{x,y\in \overline{\Omega}}} \frac{|f(x) - f(y)|}{|x-y|^{\alpha}}\end{aligned}$$ is finite. When $\alpha =1$, the function is called Lipschitz continuous. For $0 < \alpha \leq 1$ and $k\in \mathbb{N}\cup\{0\} $ the Hölder space $C^{k,\alpha} (\overline{\Omega})$ is the Banach space of all functions $u \in C^k (\overline{\Omega})$ for which the norm $$\begin{aligned} [f]_{k,\alpha} = \sum_{|\beta|=0}^{k} \sup_{\overline{\Omega}} |D^{\beta} f | + \max_{|\beta| = k} [D^{\beta} f]_{\alpha}\end{aligned}$$ is finite. The definition extends to Riemannian manifolds if we replace $|x - y |$ by the distance function $d(x,y)$ between two points. The following result appearing in [@GilbargTrudinger] (pages 448-449 and problem 17.2) is useful when we apply the implicit function theorem in Chapter \[ch:Article3\]. \[thr:GT\] Let $\psi\in C^{2,\alpha}(\overline{\Omega})$ with $\Omega\subset \mathbb{R}$ a domain and consider the maps $$F:C^{2,\alpha}(\overline{\Omega})\longrightarrow C^{0,\alpha}(\overline{\Omega})$$ and $$\mathcal{F}:\Gamma=\overline{\Omega_{2}}\times \overline{\Omega}\longrightarrow \mathbb{R},$$ where $\Omega_2\subset \mathbb{R}^{3}$ is a domain, which are related by $$F(\psi)(x)=\mathcal{F}(\ddot{\psi}(x),\dot{\psi}(x),\psi(x),x).$$ Assume that $\mathcal{F}\in C^{2,\alpha}({\Gamma})$. Then $F$ has continuous Fréchet derivative given by $$\begin{aligned} D_{\psi}F(\varphi)&=&\partial_{1}\mathcal{F}\big|_{(\ddot{\psi}(x),\dot{\psi}(x),\psi(x),x)}\ddot{\varphi}(x) +\partial_{2}\mathcal{F}\big|_{(\ddot{\psi}(x),\dot{\psi}(x),\psi(x),x)}\dot{\varphi}(x)\\ &&\quad +\partial_{3}\mathcal{F}\big|_{(\ddot{\psi}(x),\dot{\psi}(x),\psi(x),x)}\varphi(x).\end{aligned}$$ Consider a manifold $S$ with metric $g$ and let $\nabla$ be the corresponding covariant derivative. Let $a^{ij}$ be a symmetric tensor field , $b^i$ a vector field and $c$ a scalar. Consider a linear second order differential operator $L$ on the form $$\label{ellipticoperator} L\psi=-a^{ij}(x) \nabla_{i} \nabla_{j}\psi+b^{i}(x) \nabla_{i}\psi+c(x)\psi,$$ L is [**elliptic**]{} at a point $x\in S$ if the matrix $[a^{ij}](x)$ is positive definite. Assume that $S$ is orientable and denote by $<,>_{L^{2}}$ the $L^2$ inner product of two functions $\psi,\phi : S \rightarrow \mathbb{R}$ defined by $<\psi,\phi>_{L^{2}}\equiv \int_{S}\psi\phi {\mbox{\boldmath $\eta$}}_S$, where ${\mbox{\boldmath $\eta$}}_S$ is the (metric) volume form on $S$. Given a second order linear differential operator, the formal adjoint $L^{\dagger}$ is the linear second order differential operator which satisfies $$<\psi,L^{\dagger}\phi>_{L^{2}}=<\phi,L\psi>_{L^{2}}.$$ for all pairs of smooth functions with compact support. A linear operator $L$ is [*formally self-adjoint*]{} with respect to the product $L^{2}$ if $L^{\dagger}=L$. When acting on the Hölder space $C^{2,\alpha}(S)$ for $0 < \alpha <1$, the linear second order operator $L$ becomes a bounded linear operator $L: C^{2,\alpha} (S) \rightarrow C^{0,\alpha}(S)$. The formal adjoint is also a map $L^{\dagger}: C^{2,\alpha} (S) \rightarrow C^{0,\alpha}(S)$. An [*eigenvalue*]{} of $L$ is a number $\mu \in \mathbb{C}$ for which there exist functions $u,v \in C^{2,\alpha} (S)$ such that $L [u]+ i L[v] = \mu \left ( u+ i v\ \right )$. The complex function $u + i v$ is called an [*eigenfunction*]{}. The following lemma concerns the existence and uniqueness of the [*principal eigenvalue*]{} (i.e. the eigenvalue with smallest real part) of $L$ and $L^{\dagger}$. This result is an adaptation of a standard result of elliptic theory to the case of compact connected manifolds without boundary (see Appendix B of [@AMS]). \[PrincipleEigenvalue\] Let $L$ be a linear second order elliptic operator on a compact manifold $S$. Then 1. There is a real eigenvalue $\auto$, called the principal eigenvalue, such that for any other eigenvalue $\mu$ the inequality $\text{Re}(\mu)\geq \auto$ holds. The corresponding eigenfunction $\phi$, $L\phi=\auto\phi$ is unique up to a multiplicative constant and can be chosen to be real and everywhere positive. 2. The formal adjoint $L^{\dagger}$ (with respect to the $L^{2}$ inner product) has the same principal eigenvalue $\auto$ as $L$. For formally self-adjoint operators, the principal eigenvalue $\auto$ satisfies $$\label{Rayleigh-Ritz} \auto= \underset{\underset{\scriptscriptstyle{\psi\neq 0}}{\scriptscriptstyle{\psi\in C^{2,\alpha}(S^{2})}}}{\mbox{inf}} \frac{<\psi,L\psi>_{L^{2}}}{<\psi,\psi>_{L^{2}}},$$ where the quotient $\frac{<\psi,L\psi>_{L^{2}}}{<\psi,\psi>_{L^{2}}}$ is called the Rayleigh-Ritz ratio of the function $\psi$. This formula, which reflects the connection between the eigenvalue problems and the variational problems, is also useful to obtain upper bounds for $\auto$. An important tool in the analysis of the properties of the elliptic operator $L$ is the maximum principle. The standard formulations of the maximum principle for elliptic operators requires that the coefficient $c$ in (\[ellipticoperator\]) is non-negative (see e.g. Section 3 of [@GilbargTrudinger]). The following formulation of the maximum principle, which is more suitable for our purposes, requires non-negativity of the principal eigenvalue. Its proof can be found in Section 4 of [@AMS]. \[lemmaelliptic\] Consider a linear second order elliptic operator $L$ on a compact manifold $\S$ with principal eigenvalue $\auto\geq 0$ and principal eigenfunction $\phi$ and let $\psi$ be a smooth function satisfying $L \psi \geq 0$ ($L \psi \leq 0$). 1. If $\auto=0$, then $L \psi \equiv 0$ and $\psi=C\phi$ for some constant $C$. 2. If $\auto>0$ and $L \psi \not\equiv 0$, then $\psi>0$ ($\psi<0$) all over $\S$. 3. If $\auto>0$ and $L \psi \equiv 0$, then $\psi\equiv 0$. For surfaces $S$ embedded in an initial data set $\id$, the outer null expansion $\theta^{+}$ (also the inner null expansion $\theta^{-}$) is a [*quasilinear*]{} second order elliptic operator[^5] acting on the embedding functions of $S$. In this case, there also exists a maximum principle which is useful (see e.g. [@AM]). \[maximumprincipleforMOTS\] Let $\id$ be an initial data set and let $S_1$ and $S_2$ be two connected $C^2$-surfaces touching at one point $\p$, such that the outer normals of $S_{1}$ and $S_{2}$ agree at $\p$. Assume furthermore that $S_2$ lies to the outside of $S_1$, that is in direction of its outer normal near $\p$, and that $$\sup_{S_1}\theta^{+}[S_1]\leq \inf_{S_2}\theta^{+}[S_2].$$ Then $S_1=S_2$. In particular, if two MOTS touch at one point and the outer normals agree there then the two surfaces must coincide. This maximum principle can be viewed as an extension of the maximum principle for minimal surfaces which asserts precisely that two minimal surfaces touching at one point are the same surface (see e.g. [@DHT]). We discuss next the Sard Lemma, which is needed at several places in the main text. First we define regular and critical value for a smooth map. Let $f : {\cal N} \rightarrow {\cal M}$ be a smooth map. A point $\p \in {\cal N}$ is a [**regular point**]{} if $D_{\p} f: T_{\p} {\cal N} \rightarrow T_{f(\p)} {\cal M}$ has maximum rank (i.e. $\text{rank}(D_{\p} f)=\mbox{min} (n,m)$, where $n$ is the dimension of ${\cal N}$ and $m$ is the dimension of ${\cal M}$). A [**critical point**]{} $\p \in {\cal M}$ is a point which is not regular. A point $\q \in {\cal M}$ is a [**regular value**]{} if $f^{-1} (\q)$ is either empty or all $\p \in f^{-1} (\q)$ are regular points. A point $\q \in {\cal M}$ is a [**critical value**]{} if it is not a regular value. We quote Theorem 1.2.2 in [@Artino] \[lema:Sard\] Let ${\cal N}$ and ${\cal M}$ be paracompact manifolds, then the set of critical values of a smooth map $f : {\cal N} \rightarrow {\cal M}$ has measure zero in ${\cal M}$. This theorem is equivalent to saying that the set of regular values of $f: {\cal N} \rightarrow {\cal M}$ is dense in ${\cal M}$. For maps $f : {\cal N} \rightarrow \mathbb{R}$ the definition above states that $\p\in\mathcal{N}$ is a critical point if and only if $df |_{\p} =0$. Let $\p \in {\cal N}$ be a critical point and $H_{\p}$ the Hessian at $\p$ (i.e. $H_{\p} (\vec{X}, \vec{Y}) = \vec{X} (\vec{Y} (f)) |_{\p}$). For any isolated critical point $\p \in {\cal N}$ with non-degenerate Hessian, the Morse Lemma (see e.g. Theorem 7.16 in [@Morse]) asserts that there exists neighbourhood $U_{\p}$ of $\p$ and coordinates $\{ x_1,\cdots, x_n\}$ on $U_{\p}$ such that $\p = (0, \cdots 0)$ and $f$ takes the form $f (x) = f(\p) - (x_1 )^2 - \cdots - (x_q)^2 + (x_{q+1})^2 + \cdots (x_{n})^2$ where the signature of $H_{\p}$ is $n - q$. For arbitrary critical points this Lemma has been generalized by Gromoll and Meyer [@GromollMeyer]. The generalization allows for Hilbert manifolds of infinite dimensions. In the finite dimensional case Lemma 1 in [@GromollMeyer] can be rewritten in the following form. Let ${\cal N}$ be a manifold of dimension $n$ and $f: {\cal N} \rightarrow \mathbb{R}$ a smooth map. Let $\p$ be a critical point (not necessarily isolated) and $H_{\p}$ the Hessian of $f$ at $\p$. Assume that the signature of $H_{\p}$ is $\{ \underbrace{+, \cdots, +}_{q}, \underbrace{-, \cdots, -}_{r}, \underbrace{0, \cdots, 0}_{n-q-r} \}$ Then, there exists an open neighbourhood $U_{\p}$ of $\p$ and coordinates $\{ x_1 , \cdots, x_{n} \}$ such that $ \p = \{ 0, \cdots 0 \}$ and $f$ takes the form $$\begin{aligned} f (x) = f(\p) + (x_1 )^2 + \cdots + (x_q)^2 - (x_{q+1})^2 - \cdots (x_{q+r})^2 + h (x_{q+r+1},\cdots,x_n )\end{aligned}$$ where $h$ is smooth and this function, its gradient and its Hessian vanishes at $(x_{q+r+1}=0,\cdots,x_n=0 )$. 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[^1]: We thank Miguel Sánchez Caja for pointing this out. [^2]: Simply consider $\bd \Sigmatilde$ as a surface in $(M,\gM)$ and let $\vec{m}$ the be the spacetime normal to $\bd \Sigmatilde$ which is tangent to $\Sigmatilde$. Take a smooth hypersurface containing $\bd \Sigmatilde$ and tangent to $\vec{m}$. This hypersurface extends $\kid$. It is clear that the extension can be selected as smooth as desired. [^3]: A graphic example of this type of hypersurface was already given in Figure \[fig:figure1\] (where one spatial dimension was suppressed), where the portions of $S$ intersecting the black hole event horizon and the white hole event horizon represent part of the sets $\tbd T^+$ and $\tbd T^-$, respectively. The tip of $S$ at the intersection with the bifurcation surface $S_{0}$ corresponds to the circumference $\{\ycoor=0, x=0\}$. [^4]: We thank M. Khuri for pointing out this issue. [^5]: A quasilinear second order elliptic operator $Q$ has the form $Q\psi=-a^{ij}(x,\psi,\nabla\psi)\nabla_{i} \nabla_{j}\psi+b(x,\psi,\nabla\psi)$, with the matrix $[a^{ij}]$ being positive definite.
--- abstract: 'We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood.' author: - Stefan Sommer - Alexis Arnaudon - Line Kuhnel - Sarang Joshi bibliography: - 'ss.bib' title: Bridge Simulation and Metric Estimation on Landmark Manifolds --- Introduction ============ Finite dimensional landmark configurations are essential in shape analysis and computational anatomy, both for marking and following anatomically important areas in e.g. changing brain anatomies and discretely represented curve outlines, and in being among the simplest non-linear shape spaces. This simplicity, in particular the finite dimensionality, makes landmarks useful for theoretical investigations and for deriving algorithms that can subsequently be generalized to infinite dimensional shape spaces. While probability distributions in Euclidean space can often be specified conveniently from their density function, e.g. the normal distribution with the density $p_{\mu,\Sigma}(x)\propto e^{-\frac12(x-\mu)^T\Sigma^{-1}(x-\mu)}$, the non-linear nature of shape spaces often rules out closed form functions. Indeed, a density defined in coordinates will be dependent on the chosen coordinate chart and thus not geometrically intrinsic, and normalization factors can be inherently hard to compute. A different approach defines probability distributions as transition distributions of stochastic processes. Because stochastic differential equations (SDEs) can be specified locally from their infinitesimal variations, it is natural to define them in geometric spaces. Belonging to this category, the present paper aligns with a range of recent research activities on nonlinear SDEs in shape analysis and geometric mechanics [@vialard_extension_2013; @trouve_shape_2012; @marsland_langevin_2016; @arnaudon_stochastic_2017; @arnaudon_geometric_2017]. We consider here observations distributed according to the transition distribution of a Brownian motion, which is arguably one of the most direct generalizations of the Gaussian distribution to nonlinear geometries. For the Brownian motion, each infinitesimal step defining the SDE can be considered normally distributed with isotropic covariance with respect to the Riemannian metric of the space. Then, from observations, we aim to infer parameters of this metric. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) setting, this can be framed as inferring parameters of the kernel mapping $K$ between the dual Lie algebra $\mathfrak g^*$ and the Lie algebra $\mathfrak g=\mathfrak X(\Omega)$ of the diffeomorphism group $\operatorname{\mathrm{Diff}}(\Omega)$ that acts on the domain $\Omega$ containing the landmarks. We achieve this by deriving a scheme for Monte Carlo simulation of Brownian landmark bridges conditioned on hitting the observed landmark configurations. Based on the Euclidean diffusion bridge simulation scheme of Delyon and Hu [@delyon_simulation_2006], we can compute expectation over bridges using the correction factor of a guided diffusion process to obtain the transition density of the Brownian motion. From this, we can take derivatives to obtain an iterative optimization algorithm for the most likely parameters. The scheme applies to the situation when the landmark configurations are considered observed at a fixed positive time $t=T$. The time interval $[0,T]$ will generally be sufficiently large that many time discretization points are needed to accurately represent the stochastic process. We begin in Section \[sec:background\] with a short survey of LDDMM landmark geometry, metric estimation, large deformation stochastics, and uses of Brownian motion in shape analysis. In Section \[sec:simulation\], we derive a scheme for simulating Brownian landmark bridges. We apply this scheme in Section \[sec:inference\] to derive an inference algorithm for estimating parameters of the metric. Numerical examples are presented in Section \[sec:experiments\] before the paper ends with concluding remarks. Landmarks Manifolds and Stochastic Landmark Dynamics {#sec:background} ==================================================== We start with a short survey of landmark geometry with the LDDMM framework as derived in papers including [@trouve_infinite_1995; @dupuis_variational_1998; @joshi_landmark_2000; @beg_computing_2005]. The framework applies to general shape spaces though we focus on configurations $\mathbf q=(q_1,\ldots,q_N)$ of $N$ landmarks $q_i\in\Omega\subseteq{\mathbb{R}}^d$. We denote the resulting manifold $Q$. Two sets of shapes $\mathbf q^0,\mathbf q^1$ are in LDDMM matched by minimizing the energy functional $$E(u_t) = \int_0^1l(u_t)dt + \frac{1}{2\lambda^2}\|g_1.\mathbf q^0-\mathbf q^1\|^2 \,. \label{eq:E}$$ The parameter of $E$ is a time-dependent vector field $u_t\in\mathfrak X(\Omega)$ that via a reconstruction equation $$\partial_tg_t = u_t\circ g_t \label{eq:rec}$$ generates a corresponding time-dependent flow of diffeomorphisms $g_t\in\operatorname{\mathrm{Diff}}(\Omega)$. The endpoint diffeomorphism $g_1$ move the landmarks through the action $g.\mathbf q=(g(q_1),\ldots,g(q_N))$ of $\operatorname{\mathrm{Diff}}(\Omega)$ on $Q$. The right-most term of measures the dissimilarity between $g_1.\mathbf q^0$ and $\mathbf q^1$ weighted by a factor $\lambda>0$. In the landmark case, the squared Euclidean distance when considering the landmarks elements of ${\mathbb{R}}^{Nd}$ is often used here. The Lagrangian $l$ on $u$ is often in the form $l(u)={\left\langle u,Lu \right\rangle}$ with the $L^2$-pairing and $L$ being a differential operator. Because $\mathfrak X(\Omega)$ can formally be considered the Lie algebra $\mathfrak g$ of $\operatorname{\mathrm{Diff}}(\Omega)$, $l$ puts the dual Lie algebra $\mathfrak g^*$ into correspondence with $\mathfrak g$ by the mapping $\frac{1}{2}\frac{\delta l}{\delta u}:\mathfrak g\rightarrow\mathfrak g^*$, $u\mapsto Lu$. The inverse of the mapping arise from the Green’s function of $L$, written as the kernel $K$. Such $l$ defines a right-invariant inner product on the tangent bundle $T\operatorname{\mathrm{Diff}}(\Omega)$ that descends to a Riemannian metric on $Q$. Because $Q$ can be considered a subset of ${\mathbb{R}}^{Nd}$ using the representation above, the metric structure can be written directly as a cometric $${\left\langle \xi,\eta \right\rangle}_\mathbf q = \xi^T K(\mathbf q,\mathbf q)\eta \label{eq:cometric}$$ using the kernel $K$ evaluated on $\mathbf q$ for two covectors $\xi,\eta\in T^*_\mathbf qQ$. The kernel is often specified directly in the form $K(\mathbf q_1,\mathbf q_2)={\mathrm{Id}}_dk(\|\mathbf q_1-\mathbf q_2\|^2)$ for appropriate kernels $k$. One choice of $k$ is the Gaussian kernel $k(x)=\alpha e^{-\frac{1}{2}x^T\Sigma^{-1} x}$ with matrix $\Sigma=\sigma\sigma^T$ specifying the spatial correlation structure, and $\alpha>0$ a scaling of the general kernel amplitude.  \ Estimating parameters of $K$, with $K$ as above $\alpha$ and the entries of $\Sigma$ or $\sigma$, has to our knowledge previously only been treated for landmarks in the small-deformation setting [@allassonniere_towards_2007]. While a linear vector space probability distribution is mapped to the manifold with small deformations, this paper concerns the situation when the probability distribution is constructed directly from the Riemannian metric on the nonlinear space $Q$. The approach has similarities with the estimation procedures derived in [@sommer_brownian_2017] where a metric on a finite dimensional Lie group is estimated to optimize likelihood of data on a homogeneous space arising as the quotient of the group by a closed subgroup. Though the landmark space can be represented as $\operatorname{\mathrm{Diff}}(\Omega)/H$ with $H$ the landmark isotropy subgroup [@sommer_reduction_2015], the approach of [@sommer_brownian_2017] can not directly be applied because of the infinite dimensionality of $\operatorname{\mathrm{Diff}}(\Omega)$. Brownian Motion --------------- A diffusion processes $\mathbf q_t$ on a Riemannian manifold $Q$ is said to be a Brownian motion if its generator is $\frac{1}{2}\Delta_g$ with $\Delta_g$ being the Laplace-Beltrami operator of the metric $g$. Such processes can be constructed in several ways, see e.g. [@hsu_stochastic_2002]. By isometrically embedding $Q$ in a Euclidean space ${\mathbb{R}}^p$, the process can be constructed as a process in ${\mathbb{R}}^p$ that will stay on $Q$ a.s. The process can equivalently be characterized in coordinates as being solution to the Itô integral $$dq_t^i = g^{kl}\Gamma(\mathbf q_t)\indices{_{kl}^i}dt + \sqrt{g^*(\mathbf q_t)}^idW_t \label{eq:brown-coords}$$ where $\sqrt{g^*}$ is a square root of the cometric tensor $[g^*]^{ij}=g^{ij}$, and the drift term arise from contraction of the Christoffel symbols $\Gamma\indices{_{kl}^i}$ with the cometric. The noise term is infinitesimal increments $dW$ of an ${\mathbb{R}}^{\mathrm{dim}(Q)}$-valued Brownian motion $W_t$. Equivalently, the Brownian motion can be constructed as a hypoelliptic diffusion processes in the orthonormal frame bundle $OQ$ where a set of globally defined horizontal vector fields $H_1,\ldots,H_{\mathrm{dim}(Q)}\in TOQ$ gives the Stratonovich integral equation $$du_t = H_i(u_t)\circ W_t^i \ . \label{eq:brown-om}$$ Note the sum over the $\mathrm{dim}(Q)$ horizontal fields $H_i$. The process $\mathbf q_t=\pi(u_t)$ where $\pi:OQ\rightarrow Q$ is the bundle map is then a Brownian motion. This is known as the Eells-Elworthy-Malliavin construction of Brownian motion. The fields $H_i$ evaluated at $u\in OQ$ model infinitesimal parallel transport of the vectors comprising the frame $u$ in the direction of the $i$th frame vector, see e.g. [@hsu_stochastic_2002]. While Brownian motion is per definition isotropic with equal variation in all directions, data with nontrivial covariance can be modeled by defining the SDE in the larger frame bundle $FQ$ [@sommer_anisotropic_2015; @sommer_modelling_2016] using nonorthonormal frames to model the square root of the local covariance structure. In this setup, the inference problem consists of finding the starting point of the diffusion and the square root covariance matrix. Estimators are defined via a Frechét mean like minimization in $FQ$ with square $FQ$ distances used as proxy for the negative log-transition density. In this paper, we remove this proxy by approximating the actual transition density, but only in the isotropic Brownian motion case. Large Deformation Stochastics ----------------------------- Several papers have recently derived models for Brownian motion [@markussen_large_2007] and stochastic dynamics in shape analysis and for landmark manifolds. [@trouve_shape_2012; @vialard_extension_2013] considered stochastic shape evolution by adding finite and infinite dimensional noise in the momentum equation of the dynamics. In [@marsland_langevin_2016], noise is added to the momentum equation to make the landmark dynamics correspond to a type of heat bath appearing in statistical physics. In [@arnaudon_stochastic_2017; @arnaudon_geometric_2017] a stochastic model for shape evolution is derived that descends to the landmark space in the same fashion as the right-invariant LDDMM metric descends to $Q$. The fundamental structure is here the momentum map that is preserved by the introduction of right-invariant noise. The approach is linked to parametric SDEs in fluid dynamics [@holm_variational_2015] and stochastic coadjoint motion [@arnaudon_noise_2016]. Brownian Bridge Simulation {#sec:simulation} ========================== Brownian motion can be numerically simulated on $Q$ using the coordinate Itô form . With a standard Euler discretization, the scheme becomes $$\mathbf q_{t_{k+1}} = \mathbf q_{t_k} + K(\mathbf q_{t_k},\mathbf q_{t_k})^{kl}\Gamma(\mathbf q_t)\indices{_{kl}}\Delta t + \sqrt{K(\mathbf q_{t_k},\mathbf q_{t_k})}_j\Delta W_{t_k}^j \label{eq:brown-coords-disc}$$ with time discretization $t_1,\ldots,t_k$, $t_k-t_{k-1}=\Delta t$ and discrete noise $W_{t_1},\ldots,W_{t_k}\in{\mathbb{R}}^{Nd}$, $\Delta W_{t_k}=\Delta W_{t_k}-\Delta W_{t_{k-1}}$. Alternatively, a Heun scheme for discrete integration of the Stratonovich equation results in $$\begin{split} & v_{t_{k+1}} = H_i(u_{t_k})\Delta W_{t_k}^i \\& u_{t_{k+1}} = u_{t_k} + \frac{v_{t_{k+1}}+H_i(u_{t_k}+v_{t_{k+1}})\Delta W_{t_k}^i}{2} \ . \end{split} \label{eq:brown-om-disc}$$ Because the horizontal fields represent infinitesimal parallel transport, they can be expressed using the Christoffel symbols of $g$. The Christoffel symbols for the landmark metric are derived in [@micheli_differential_2008] from which they can be directly implemented or implicitly retrieved from an automatic symbolic differentiation as done in the experiments in Section \[sec:experiments\]. Bridge Sampling --------------- The transition density $p_T(\mathbf v)$ of a Brownian motion $\mathbf q_t$ evaluated at $\mathbf v\in Q$ at time $T>0$ can be informally obtained by taking an expectation to get the “mass” of those of the sample paths hitting $\mathbf v$ at time $T$. We write $\mathbf q_t|\mathbf v$ for the process $\mathbf q_t$ conditioned on hitting $\mathbf v$ a.s. at $t=T$. Computing the expectation analytically is in nonlinear situations generally intractable. Instead, we wish to employ a Monte Carlo approach and thus derive a method for simulating from $\mathbf q_t|\mathbf v$. For this, we employ the bridge sampling scheme of Delyon and Hu [@delyon_simulation_2006]. We first describe the framework for a general diffusion process in Euclidean space before using it directly on the landmark manifold $Q$. Let $$dx_t= b(t,x_t)dt + \sigma(t,x_t)dW_t \label{eq:driftsde}$$ be an ${\mathbb{R}}^k$ valued Itô diffusion with invertible diffusion field $\sigma$. In order to sample from the conditioned process $x_t|v$, $v\in{\mathbb{R}}^k$, a modified processes is in [@delyon_simulation_2006] constructed by adding an extra drift term to the process giving the new process $$dy_t= b(t,y_t)dt - \frac{y_t-v}{T-t}dt + \sigma(t,y_t)dW_t \ . \label{eq:guideddriftsde}$$ As $t\rightarrow T$, the attraction term $-(y_t-v)/(T-t)dt$ becomes increasingly strong forcing the processes to hit $y$ at $t=T$ a.s. It can be shown that the process $y_t$ exists when $b$, $\sigma$ and $\sigma^{-1}$ are $C^{1,2}$ with bounded derivatives. The process is then absolutely continuous with respect to the conditioned process $x_t|v$. The Radon-Nikodym derivative between the laws $P_{x_t|v}$ and $P_y$ is $$\frac{dP_{x|v}}{dP_y}(y) = \frac{\varphi_T(y)}{E_{y}[\varphi_T]}$$ with $E_{y}[\cdot]$ denoting expectation with respect to $P_y$, and the correction factor $\varphi_T(y)$ defined as the $t\rightarrow T$ limit of $$\begin{split} \varphi_t(y) &= \exp \left( -\int_0^t \frac{\tilde{y}_s^TA(s,y_s)b(s,y_s)}{T-s}ds \right. \\ &\qquad\qquad \left. - \frac{1}{2} \int_0^t \frac{\tilde{y}_s^T(dA(s,x_s))\tilde{y}_s+\sum_{i,j}d\left<A^{ij}(s,y_s),\tilde{y}_s^i\tilde{y}_s^j\right>}{T-s} \right) \end{split} \label{eq:phit}$$ Here $\tilde{y}_t=y_t-v$, $A=(\sigma\sigma^T)^{-1}$, and quadratic variation is denoted by $\left<\cdot,\cdot\right>$. Then $E_{x}[f(x)|x_T=v]=E_{x|v}[f(x)]=E_{y}[f(y)\varphi_T(y)]/E_{y}[\varphi_T(y)]$ and $$E[f(y)\varphi_t] = \frac{T^{k/2}e^{\frac{\|\sigma^{-1}(0,x_0)(x_0-v)\|^2}{2T}}}{(T-t)^{k/2}} E\big[ f(x) e^{-\frac{\|\sigma^{-1}(t,x_t)(x_t-v)\|^2}{2(T-t)}} \big]$$ for $t<T$. The fact that the diffusion field $\sigma$ must be invertible for the scheme to work as outlined here can be seen explicitly from the use of the inverse of $\sigma$ and $A$ in these equations. We can use the guided process to take conditional expectation for general measurable functions on the Wiener space of paths $W({\mathbb{R}}^k,[0,T])$ by sampling from $y_t$. Taking the particular choice of the constant function, the expression $$p_T(v) = \sqrt{\frac{|A(T,v)|}{(2\pi T)^k}} e^{\frac{-\|\sigma(0,x_0)^{-1}(x_0-v)\|^2}{2T}} E_{y}[\varphi_T(y)] \label{eq:eucptapproxgeneral}$$ for the transition density of $x_t$ arise as shown in [@papaspiliopoulos_importance_2012]. Note that both the leading factors and the correction factor $\varphi_T$ are dependent on the diffusion field $\sigma$, the starting point $x_0$ of the diffusion, and the drift $b$. Again, we can approximate the expectation in by sampling from $y_t$. Landmark Bridge Simulation -------------------------- Because the landmark manifold has a global chart on ${\mathbb{R}}^{Nd}$ from the standard representation of each landmark position in ${\mathbb{R}}^d$, we can conveniently apply the bridge construction of [@delyon_simulation_2006]. Writing the Itô coordinate form of the Brownian motion $\mathbf q_t$ in the form , we have $b(t,\mathbf q)=K(\mathbf q,\mathbf q)^{kl}\Gamma(\mathbf q)\indices{_{kl}}$ and $\sigma(t,\mathbf q)=\sqrt{K(\mathbf q,\mathbf q)}$ giving the guided SDE $$d\mathbf y_t= K(\mathbf y_{t_k},\mathbf y_{t_k})^{kl}\Gamma(\mathbf y_t)\indices{_{kl}}dt - \frac{\mathbf y_t-\mathbf v}{T-t}dt + \sqrt{K(\mathbf y_{t_k},\mathbf y_{t_k})}dW_t \label{eq:guided-landmarks}$$ The attraction term $-(\mathbf y_t-\mathbf v)/(T-t)dt$ is the difference between the current landmark configuration $\mathbf y_t$ and the target configuration $\mathbf v$. The transition density becomes $$p_{T,\theta}(\mathbf v) = \frac{1}{\sqrt{|K(\mathbf v,\mathbf v)|(2\pi T)^{Nd}}} e^{-\frac{\|(\mathbf q_0-\mathbf v)^TK(\mathbf q_0,\mathbf q_0)^{-1}(\mathbf q_0-\mathbf v)\|^2)}{2T}} E_{\mathbf y_\theta}[\varphi_{\theta,T}(\mathbf y)] \label{eq:trans-dens-landmarks}$$ where we use the subscript $\theta$ to emphasize the dependence on the parameters $\mathbf q_0$ and the kernel $K$. As above, the expectation $E_{\mathbf y_\theta}[\varphi_{\theta,T}(\mathbf y)]$ can be approximated by drawing samples from $\mathbf y_\theta$ and evaluating $\varphi_{\theta,T}(\mathbf y)$. A similar scheme is used for the bridge simulation of the stochastic coadjoint processes of [@arnaudon_stochastic_2017; @arnaudon_geometric_2017]. In these cases, the flow is hypoelliptic in the phase space $(\mathbf q,\mathbf p)$ and observations are partial in that only the landmark positions $\mathbf q$ are observed. The momenta $\mathbf p$ are unobserved. In addition, the fact that the landmarks can carry a large initial momentum necessitates a more general form of the guidance term $-(\mathbf y_t-\mathbf v)/(T-t)dt$ that takes into account the expected value of $E_{\mathbf y}[\mathbf y_T|(\mathbf q_t,\mathbf p_t)]$ of the process at time $T$ given the current time $t$ position and momentum of the process. Inference Algorithm {#sec:inference} =================== Given a set of i.i.d. observations $\mathbf q^1,\ldots,\mathbf q^N$ of landmark configurations, we assume the configurations $\mathbf q^i$ are distributed according to the time $t=T$ transition distribution $\mathbf q_T$ of a Brownian motion on $Q$ started at $\mathbf q_0$. We now intend to infer parameters $\theta$ of the model. With the metric structure on $Q$ given by and kernel of the form $K(\mathbf q_1,\mathbf q_2)={\mathrm{Id}}_dk(\|\mathbf q_1-\mathbf q_2\|^2)$, $k(x)=\alpha e^{-\frac{1}{2}x^T\Sigma x}$, parameters are the starting position $\mathbf q_0$, $\alpha$, and $\Sigma=\sigma\sigma^T$, i.e. $\theta=(\mathbf q_0,\alpha,\sigma)$. The likelihood of the model given the data with respect to the Lebesgue measure on ${\mathbb{R}}^{Nd}$ is $$\mathcal{L}_\theta(\mathbf q^1,\ldots,\mathbf q^N) = \prod_{i=1}^N p_{T,\theta}(\mathbf q^i) \label{eq:likelihood} .$$ Using our ability to approximate by bridge sampling, we aim to find a maximum-likelihood estimate (MLE) $\hat{\theta}\in{{\operatorname{argmin}_{\theta}}}\mathcal{L}_\theta(\mathbf q^1,\ldots,\mathbf q^N)$. We do this by a gradient based optimization on $\theta$, see Algorithm \[alg:inference\]. Note that the likelihood and thus the MLE of $\theta$ are dependent on the chosen background measure, in this case coming from the canonical chart on ${\mathbb{R}}^{Nd}$. The inference Algorithm \[alg:inference\] optimizes the likelihood $\mathcal{L}_\theta$ directly by stochastic gradient descent. A different but related approach is an Expectation-Maximization approach where the landmark trajectories between $t=0$ and the observation time $t=T$ are considered missing data. The $E$-step of the EM algorithm would then involve the expectation $E_{\mathbf x|\mathbf q^i}[\log p(\mathbf x)]$ of the landmark bridges conditioned on the data with $p(\mathbf x)$ formally denoting a likelihood of an unconditioned sample path $\mathbf x$. This approach is used in e.g. [@arnaudon_geometric_2017]. While natural to formulate, the approach involves the likelihood $p(\mathbf x)$ of a stochastic path which is only defined for finite time discretizations. In addition, the expected correction factor $E_{\mathbf y}[\varphi_T(\mathbf y)]$ that arise when using the guided process $\mathbf y$ in the estimation appears as a normalization factor in the EM $Q$-function. This can potentially make the scheme sensitive to the stochasticity in the Monte Carlo sampling of the expected correction $E_{\mathbf y}[\varphi_T(\mathbf y)]$. While the differences between these approaches needs further investigation, we hypothesize that direct optimization of the likelihood is superior in the present context. Instead of taking expectations over $\mathbf q_T$, we can identify the most probable path of the conditioned process $\mathbf q_t|\mathbf v$. This results in the Onsager-Machlup functional [@fujita_onsager-machlup_1982]. In [@sommer_anisotropically_2016], a different definition is given that, in the isotropic Brownian motion situation, makes the set of Riemannian geodesics from $\mathbf q_0$ to $\mathbf v$ equal to the set of most probable paths of the conditioned process $\mathbf q_t|\mathbf v$. The sample Frechét mean $${{\operatorname{argmin}_{\mathbf q_0}}} \frac{1}{N}\sum_{i=1}^Nd_g(\mathbf q_0,\mathbf q_i)^2 \label{eq:frechet-mean}$$ is in that case formally also a minimizer of the negative log-probability of the most probable path to the data. Given that we are now able to approximate the density function of the Brownian motion, the MLE of the likelihood with respect to $\mathbf q_0$ is equivalent to $${{\operatorname{argmin}_{\mathbf q_0}}} -\frac{2}{N}\sum_{i=1}^N\log p_{T,\mathbf q_0}(\mathbf q_i) \ . \label{eq:density-frechet-mean}$$ Compared to , the negative log-probability of the data is here minimized instead of the squared geodesic distance. The estimator can therefore be considered a transition density equivalent of the sample Frechét mean. Numerical Experiments {#sec:experiments} ===================== We here present examples of the method on simulated landmark configurations, and an application of the method and algorithm to landmarks annotated on cardiac images of left ventricles. We aim for testing the ability of the algorithm to infer the parameters of the model given samples. We here take the first steps in this direction and leave a more extensive simulation study to future work. For the simulated data, we compare the results against the true values used in the simulation. In addition, we do simple model checking for both experiments by simulating with the estimated parameters and comparing the per-landmark sample mean and covariance. We use code based on the Theano library [@the_theano_development_team_theano:_2016] for the implementation, in particular the symbolic expression and automatic derivative facilities of Theano. The code used for the experiments is available in the software package *Theano Geometry* <http://bitbucket.org/stefansommer/theanogeometry>. The implementation and the use of Theano for differential geometry applications including landmark manifolds is described in [@kuhnel_deep_2017].  \ With 10 landmarks arranged in an ellipse configuration $\mathbf q_0$, we sample 64 samples from the transition distribution at time $T=1$ of a Brownian motion started at $\mathbf q_0$, see Figure \[fig:ellipse\_samples\]. Parameters for the kernel are $\sigma=\mathrm{diag}(\sigma_1,\sigma_2)$ with $\sigma_1,\sigma_2$ set to the average inter-point distance in $\mathbf q_0$, and the amplitude parameter $\alpha=0.01$. We run Algorithm \[alg:inference\] with initial conditions for $\mathbf q_0$ the pointwise mean of the samples. The parameter evolution trough the iterative optimization and the result of the inference can be seen in Figure \[fig:ellipse\_results\]. The algorithm is able estimate the initial configuration and the parameters of $\alpha$ and $\Sigma$ with a reasonable precision. The sample per-landmark covariance as measured on a new set of simulated data with the estimated parameters is comparable to the per-landmark covariance of the original dataset. Left Cardiac Ventricles ----------------------- To exemplify the approach on real data, we here use a set of landmarks obtained from annotations of the left ventricle in 14 cardiac images [@stegmann_extending_2001]. Each ventricle is annotated with sets of landmarks from which we select 17 from each configuration for use in this experiment. Figure \[fig:cardiac\] shows an annotated image along with the sets of annotations for all images. Figure \[fig:cardiac\_results\] shows the results of the inference algorithm with setup equivalent to Figure \[fig:ellipse\_results\]. While the parameters converges during the iterative optimization, we here have no ground-truth comparison. A subsequent sampling using the estimated parameters allows comparison of the per-landmark sample covariance. While the new sample covariance in magnitude and to some degree shape corresponds to the sample covariance from the original data, the fact that the Brownian motion is isotropic forces the covariance to be equivalent for all landmarks as measured by the Riemannian landmark metric. Including anisotropic covariance in the distribution or the right-invariant stochastics of [@arnaudon_stochastic_2017; @arnaudon_geometric_2017] would allow the per-landmark covariance to vary and result in a closer fit. Conclusion {#sec:conclusion} ========== In the paper, we have derived a method for maximum likelihood estimation of parameters for the starting point of landmark Brownian motions and for the Riemannian metric structure specified from the kernel $K$. Using the guided process scheme of [@delyon_simulation_2006] for sampling conditioned Brownian bridges, the transition density is approximated by Monte Carlo sampling. With this approximation of the data likelihood, we use a gradient based iterative scheme to optimize parameters. We show on synthetic and real data sets the ability of the method to infer the underlying parameters of the data distribution and hence the metric structure of the landmark manifold. A direct extension of the method presented here is to generalize to the anisotropic normal distributions [@sommer_anisotropic_2015] defined via Brownian motions in the frame bundle $FQ$. This would allow differences in the per-landmark covariance and thus improve results on datasets such as the presented left ventricle annotations. Due to the hypoellipticity of the anisotropic flows that must be conditioned on hitting fibers in $FQ$ above points $q\in Q$, further work is necessary to adapt the scheme presented here to the anisotropic case.  \ [**Acknowledgements**]{}We are grateful for the use of the cardiac ventricle dataset provided by Jens Chr. Nilsson and Bjørn A. Grønning, Danish Research Centre for Magnetic Resonance (DRCMR).
--- abstract: 'A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass model, at and above the critical temperature $T_{g}$. The data show a long time stretched exponential relaxation $q(t) \sim \exp[-(t/\tau(T))^{\beta(T)}]$ with an exponent $\beta(T)$ tending to $\approx 1/3$ at $T_{g}$. The results are compared to those which were observed by Ogielski in the $3d$ ISG model, and are discussed in terms of a phase space percolation transition scenario.' author: - Alain Billoire - 'I. A. Campbell' title: 'Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above $T_{g}$' --- Introduction ============ The Sherrington-Kirkpatrick (SK) Ising spin glass (ISG) model has been intensively studied for almost forty years. Introduced [@sherrington:75; @kirkpatrick:78] as a starting point for studying Edwards-Anderson-like [@edwards:75] spin glasses, it is a classical mean-field $N$-spin Ising model in which there are quenched random interactions between all pairs of spins. It has an ordered spin glass phase below the critical temperature $T_{g}=1$. The static properties such as the specific heat, the magnetic susceptibility and the spin glass order parameter are known to high precision in the ordered state and are well understood in terms of the Parisi Replica Symmetry Breaking (RSB) theory [@parisi:79] and its subsequent developments. The paramagnetic regime has been considered “trivial” in that there are simple exact expressions for the basic static physical properties in the large $N$ limit. In the ordered state the dynamics are very slow; the equilibrium relaxation time diverges exponentially when the number of sites $N$ goes to infinity [@billoire:01; @billoire:10; @billoire:11]. For the paramagnetic state, the dynamics were discussed in early work, Refs. [@kinzel:77; @kirkpatrick:78; @sompolinski:81; @sompolinski:82]. The ISG relaxation corresponds to a continuous spectrum of relaxation frequencies, for which an explicit linearized expression was given in Ref. [@kirkpatrick:78], leading to a relaxation function for Gaussian interactions and Glauber dynamics in the form of an integral, Ref. [@kirkpatrick:78], Eq. 5.27. It was concluded that in the large size limit at $T_{g}$ the relaxation at thermal equilibrium of the autocorrelation $q(t) =(1/N)\sum_{i} <S_{i}(0).S_{i}(t)>$ as a function of time $t$ (in Monte Carlo steps per spin) would be $q(t) \sim t^{-1/2}$, and above $T_{g}$ the very long time limit would take the form of a simple exponential $q(t) \sim \exp[-(t/\tau(T))]$ corresponding to a cutoff in the relaxation frequency spectrum. As far as we are aware, after exploratory numerical calculations on a very limited number of Gaussian interaction samples over a few Glauber time steps in Ref. [@kirkpatrick:78] no further numerical work on the dynamics in this temperature region has been reported. Here a detailed numerical study of the thermal equilibrium relaxation of the autocorrelation function $q(t,N)$ in the ordered state as a function of time $t$ in updates per spin and $N$ the number of spins [@billoire:11] is extended to temperatures at and above $T_{g}$. At $T_{g}$ the relaxation is always finite size limited; the expected scaling form $$q(t,N)N^{1/3} = F[-(t/N^{2/3})] \label{qtN33}$$ is observed. For the three temperatures above $T_{g}$ at which measurements were made the large size limit behavior \[$q(t,N)$ independent of $N$\] is reached with the largest samples studied. For $q(t)$ values down to at least $q(t) \approx 10^{-6}$ the autocorrelation function decay is strongly non-exponential. Satisfactory long $t$ fits are given both above $T_g$ and in the finite size scaling regime at $T_g$ by stretched exponentials $$q(t) \sim exp[-(t/\tau(T))^{\beta(T)}] \label{qt}$$ with a temperature dependent exponent $\beta(T)$ which is smaller than $1$ and which tends to $\beta(T) \sim 1/3$ at $T=T_g$ [@beta]. The Mean field ferromagnet ========================== Before discussing the SK data analysis it is instructive as an illustration to recall the (non-equilibrium) relaxation behavior for the mean field Ising ferromagnet, for which exact results by Suzuki and Kubo [@suzuki:68] were discussed in Ref. [@kirkpatrick:78]. At temperatures above $T_c=1$ the relaxation takes the form $$\begin{gathered} m(t,T)=\\(1-\beta)^{1/2}/[(1-\beta +\beta^{3}/3)\exp(2(1-\beta)t)-\beta^3/3]^{1/2} \label{ferro}\end{gathered}$$ where $\beta=1/T$, which becomes $$m(t,T_c) = (1+ (2/3)t)^{-1/2} \label{ferroTc}$$ at $T_c$. It is easy to see from Eq. \[ferro\] that above $T_{c}$ and beyond a time $t \simeq \tau(T) = 1/(1-\beta)$, $m(t,T)$ is essentially equal to $A(T)\exp(-t/\tau(T))$, a pure exponential with a time independent prefactor, $$A(T) = [(1-\beta)/(1-\beta +\beta^{3}/3)]^{1/2} \label{ferroA}$$ see Figure 1. Beyond $t \approx \tau$ only one single mode contributes to the relaxation in the mean field ferromagnet. At short times (time zero to time $\sim \tau(T)$) there is a more complex function which connects up between the starting point $m(0,T)=1$ and the asymptotic region. At no time for temperatures above $T_c$ is $m(t,T) \approx t^{-1/2}\exp(-t/\tau)$ a useful approximation. It can be helpful to show the data in the form of a derivative plot $-d\log(m(t))/d\log(t)$ against $t$. Then a pure exponential $m(t)=A\exp[-t/\tau]$ appears as a straight line of slope $1/\tau$ passing through the origin. ![(Color online) An example of an $m(t)$ plot for the mean field ferromagnet calculated using Eq. \[ferro\]. The points correspond to $m(t)$ at $T=1.05T_{c}$; the black line is the algebraic critical behavior, and the blue curve is the asymptotic pure exponential.](SK_fig1.eps){width="3.5in"} \[fig:1\] SK simulations ============== We now return to the SK simulations. Systems of size $N = 64, 128, 256, 512, 1024$ and $2048$ were equilibrated using the procedure described in Ref. [@billoire:11]. Simulations were made on $1024$ independent disorder samples at each size with two clones simulated in parallel. (As shown in Ref. [@billoire:10] two clones allow one to build up a noisy but unbiased estimate of the thermal fluctuations. Here they are only used to decrease the thermal fluctuations). We performed $10^7$ Monte Carlo updates per site after thermalization, with measurements of $q(t)$ made every 4 time intervals. Both Metropolis and heat bath relaxation runs were carried out. It can be noted that SK simulations are numerically demanding : there are about as many individual spin-spin interactions in an $N=2048$ SK sample as in an $L=90$ sample in dimension $3$. In the $N=256$ data set, and that set only, it turns out that one specific disorder sample $i$ gives $q_{i}(t,T)$ that are very far from the values for all the $1023$ other samples of the same size. This is true for sample $i$ with either Heat Bath or Metropolis updating schemes, and for all values of $T$. For example at $T=T_g=1$ we have $1024$ estimates for $q(256,t=1024)$ with a median equal to 0.0000040, first and third quartiles equal to -0.0001183 and 0.0001292 respectively, and an isolated maximum value equal to 0.2946040. This single outlier was omitted from the global analysis, since it would have led to a long $t$ distortion of our values of the mean $<q(256,t,T)>$ and the statistical error. At long $t$ when $<q(t,T)>$ is very small, if one single sample by statistical accident is very uncharacteristic it can have a big influence on the mean and the error. Note that in our simulation the thermal noise on $<q(t,T)>$ is much smaller than the disorder noise. The temperatures at which measurements were made are $T = T_{g}= 1$ and $T = 1.1, 1.2, 1.3$ SK analysis =========== The standard dynamic finite size scaling rule exactly at the critical point can be written $$q_{c}(L,t)= L^{-\beta/\nu} F[t/\tau(L)]$$ with $\tau(L) \sim L^z$. It is well known that quite generally finite size scaling is modified above the upper critical dimension $d_{u}$, with the combination $(T-T_g)L^{1/\nu}$ replaced by $(T-T_g)N^{1/(d_{u}\nu)}$ [@brezin], where $N$ is the number of sites, namely $L$ is replaced by $N^{1/d_{u}}$. The SK model is the infinite dimensional version of a model with upper critical dimension $d_{u}=6$ [@cirano], and its scaling behavior is accordingly described by the effective length scale $L_{eff}=N^{1/6}$. This result can be obtained directly by noting that in the critical region the Landau expansion of the free energy of the replicated model starts (symbolically) by $a(T-T_g)Q^{2}+bQ^{3}$ with $Q$ the $n$ by $n$ matrix $q_{a,b}$ where $n$ is the number of replica. The partition function takes the form $\int dQ \exp(-N(a(T-T_g)Q^{2}+b Q^{3}))$, and after the change of variable $Q=X/[(T-T_g)N]^{1/2}$ one obtains a free energy that is a function of $(T-T_g)N^{1/3}$, in agreement with the above statement since $\nu=1/2$ for this model. Therefore for dimension $6$ and above (including SK) the effective ISG “length” scale is $L_{eff} = N^{1/6}$. The mean field ISG critical exponents are $\beta=1$, $\nu=1/2$, and $z=4$. Hence in the SK case the dynamic scaling becomes $$q_{c}(N,t) \sim N^{-1/3}F[t/N^{2/3}] \label{qctN}$$ so that in a critical temperature scaling plot one should expect $q_{c}(N,t)N^{1/3}$ to be a function $F[t/N^{2/3}]$. In conditions where $q_{c}(N,t)$ approaches the $N \to \infty$ limit, the scaling rule becomes $$q_{c}(t)\sim N^{-1/3}(t/N^{2/3})^{-1/2} = t^{-1/2} \label{qct}$$ independent of $N$. This is the situation for short times and large $N$, Figure 2. ![(Color online) The short time scaled SK relaxation data $q(t)N^{1/3}$ against $t/N^{2/3}$ in logarithmic coordinates at the critical temperature $T=1$. The colors for $N=64, 128, 256, 512, 1024, 2048$ are yellow, pink, cyan, green, red and black. For clarity the $N=2048$ points are larger than the others. The blue line indicates $q_c(t)\sim t^{-1/2}$.](SK_fig2.eps){width="3.5in"} \[fig:2\] ![(Color online) The scaled SK relaxation data $q(t)N^{1/3}$ in logarithmic coordinates against $t/N^{2/3}$ at the critical temperature $T=1$. The color code is the same as in Figure 2. The blue curve is a stretched exponential $q(t)N^{1/3}=14\exp[-((t/N^{2/3})^{1/3})]$.](SK_fig3.eps){width="3.5in"} \[fig:3\] For reasons to be discussed below, in addition to testing this scaling form we want to test if the long time critical finite size scaling function $F[x]$ is consistent with the stretched exponential form $B\exp[-(x)^{\beta}]$ having the particular exponent value $\beta =1/3$. In figure 3 the scaled critical temperature relaxation data are plotted as $\log(q(t)N^{1/3})$ on the y-axis against $(t/N^{2/3})$ on the x-axis. The long time critical scaling rule is well obeyed over the whole range of sizes $N$ which have been studied; beyond times of a few Monte Carlo steps $q_{c}(N,t)N^{1/3}$ is indeed an $N$-independent function $F[t/N^{2/3}]$ to within the error bars for all $t$ and $N$, with no visible sign of corrections to finite size scaling. The initial very short time relaxation (Figure 2, invisible in Figure 3) follows approximately the $N$-independent form $q(t) \sim t^{-1/2}$ as predicted. As can be seen in Figure 3, within the numerical precision the finite size scaling function is from then on compatible with a stretched exponential having exponent $\beta = 1/3$. For the three temperatures above $T_{g}$, $T = 1.1, 1.2, 1.3$ the raw $q(N,t)$ data are shown in Figures 4, 5 and 6. It can be seen that at each of these temperatures the size dependence saturates with increasing $N$. The $q(N,t)$ curves for the largest sizes coincide within the numerical error bars, so these data can be taken to represent the “infinite” $N$ limit behavior. The $q(t)$ curves calculated with the expression given in Ref. [@kirkpatrick:78] Eq. 5.27, are almost identical to the limiting numerical $q(N,t)$ curves for $T=1.1$ and $T=1.2$ but the calculated curve for $T=1.3$ lies somewhat above the numerical data and has a rather different shape. The good quantitative agreement for the first two temperatures is perhaps fortuitous as the expression in Ref. [@kirkpatrick:78] corresponds to a model with Gaussian interactions, and a linearization approximation was made in the calculation. The authors expected the expression to become accurate only at temperatures well above $T_g$. Nevertheless the approach of Ref. [@kirkpatrick:78] is substantially validated by the numerical data. We can note that the simulations made with the Metropolis updating rules (not shown) provide $q(t,N)$ data which are very similar to the heatbath data but with a global shift in the time scale to times shorter by a factor of between $2$ and $3$ depending on the temperature. It can be shown that as a general rule $q(t)$ will fall faster with Metropolis updating than with heatbath/Glauber updating; the Metropolis acceptance rate is higher than that for heatbath/Glauber with a ratio of which depends on temperature. ![(Color online) The raw relaxation data $q(t)$ against $t$ at the temperature $T=1.1$. The color code for $N$ is the same as in Figure 2. The fit Eq. \[qt\] to the largest sizes is shown as the full black curve. $q(t)$ calculated from Ref. [@kirkpatrick:78] Eq. 5.27 is shown as the full blue curve.](SK_fig4.eps){width="3.5in"} \[fig:4\] ![(Color online) The raw relaxation data $q(t)$ against $t$ at the temperature $T=1.2$. The color code for $N$ is the same as in Figure 2. The fit Eq. \[qt\] to the largest sizes is shown as the full black curve. $q(t)$ calculated from Ref. [@kirkpatrick:78] Eq. 5.27 is shown as the full blue curve.](SK_fig5.eps){width="3.5in"} \[fig:5\] ![(Color online) The raw relaxation data $q(t)$ against $t$ at the temperature $T=1.3$. The color code for $N$ is the same as in Figure 2. The fit Eq. \[qt\] to the largest sizes is shown as the full black curve. $q(t)$ calculated from Ref. [@kirkpatrick:78] Eq. 5.27 is shown as the full blue curve.](SK_fig6.eps){width="3.5in"} \[fig:6\] ![(Color online) The derivative plot $-d\log(q(t))/d\log(t)$ against $t$ at $T=1.1$ for the two largest sizes, $N = 2048$ (black) and $1024$ (red). The fit (blue curve) is for the same parameters as in Figure 4.](SK_fig7.eps){width="3.5in"} \[fig:7\] The figures are of the same form as in the ferromagnetic example of Figure 1 with an initial short time regime followed by a long time asymptote, except that the long time ferromagnet pure exponential asymptote is replaced by a stretched exponential. Beyond the initial short time regime the stretched exponentials fit the large $N$ SK data to within the numerical precision. If the stretched exponential form Eq. \[qt\] is assumed, then $$-d\log(q(t))/d\log(t) = \beta(T)(t/\tau)^{\beta(T)} \label{dlogqt}$$ An example of a plot of $-d\log(q(t))/d\log(t)$ against $t$ is shown in Figure 7. For a pure exponential relaxation $\beta = 1$ the data points would lie on a straight line going through the origin. This is clearly not the case. By analogy with the ferromagnetic behavior, beyond an initial short time region the large $N$ data are fitted by a $B(T)t^{\beta(T)}$ curves with $\beta(T)$ less than $1$ and $B(T) = \beta(T)/\tau(T)^{\beta(T)}$. Satisfactory fits are obtained with : $\beta(1.1) \approx 0.52$, $\tau(1.1) \approx 6.8$, $\beta(1.2) \approx 0.61$, $\tau(1.2) \approx 4.2$,and $\beta(1.3) \approx 0.68$, $\tau(1.3) \approx 3.2$ for the heatbath updates. These fits are shown as the black curves in Figures 4 to 6. The relaxation time $\tau(T)$ is of course increasing as $T_{g}$ is approached but with only three temperature points it is difficult to estimate a functional form for $\tau(T)$. However the most significant result is that the relaxation above $T_{g}$ in the effectively infinite size limit can be represented by stretched exponentials having exponents $\beta(T)$ which are less than $1$ and which drop regularly as $T_{g}$ is approached. From this result for the large $N$ limit relaxation data at the temperatures above $T_g$ together with the scaled finite size limited relaxation at criticality it can thus be concluded that in the paramagnetic region the SK model thermodynamic limit equilibrium relaxation can be treated as essentially stretched exponential, with an exponent $\beta(T)$ which decreases continuously as the temperature is lowered. $\beta(T)$ tends to a value $\beta_{c} \approx 1/3$ at criticality. It should be underlined that this behavior has been followed down numerically though five orders of magnitude in $q(t)$ at each temperature. ISG in dimension three ====================== It is instructive to compare with the situation for a finite dimension model. Ogielski [@ogielski:85] fitted thermodynamic limit $q(t)$ data on the bimodal ISG in dimension $3$ using an empirical function of the form $$q(t) \sim t^{-x(T)}\exp[-(t/\tau(T))^{\beta(T)}] \label{qtOg}$$ where an algebraic prefactor multiplies a stretched exponential. Ogielski estimated $T_{g} \approx 1.175$ which is close to recent estimates $T_{g} \approx 1.12$ [@katzgraber:06; @hasenbusch:08]. The values estimated for the stretched exponential exponent $\beta(T)$ decrease regularly from $\beta(T) \approx 1$ at a temperature of about $4.0$ to $\beta(T) \approx 0.35$ at a temperature just above $T_{g}$. The relaxation time $\tau(T)$ diverges as $T \to T_{g}$ and the algebraic prefactor exponent $x(T)$ decreases from $\approx 0.5$ at high temperatures to $(d-2+\eta)/2z \sim 0.070$ at $T_g$. These equilibrium relaxation results were obtained from high quality simulations on very large samples (up to $L=64$) and have never been improved on since. Thus the general pattern of the paramagnetic state relaxation reported by Ogielski in the $3d$ ISG and the observed behavior of the SK model described above show striking similarities; above all both sets of data in the paramagnetic region are consistent with stretched exponential decay having an exponent $\beta(T)$ tending to a critical value $\beta_{c} \approx 1/3$ at $T_g$. These two models represent the lowest integer dimension at which finite temperature ISG ordering takes place and the infinite dimension mean field limit (for which the RSB theory is well established) respectively, so it would seem reasonable to expect that the same pattern of relaxation should hold for ISGs at all intermediate dimensions also, both above and below the upper critical dimension (see Ref. [@bernardi:94]). Shortly after Ogielski’s work was published it was noted that his data were consistent with $\beta(T_g)=1/3$, and it was conjectured [@campbell:85] that stretched exponential relaxation with an exponent tending to precisely $1/3$ at criticality could be universal in ISGs. The argument is briefly summarized in the next section. Percolation transition scenario =============================== A thermodynamic phase transition can be regarded as a qualitative change in the topology of the thermodynamically attainable phase space with decreasing temperature, i.e. with decreasing internal energy. Thus for a standard ferromagnetic transition a high temperature spherical phase space becomes more and more “elliptic” as the temperature is lowered and the number of attainable states drops; finally at $T_c$ it splits into two (up and down) mirror-image subspaces. In a naïve scenario based on RSB, for an ISG at $T_g$ phase space shatters into a large number of inequivalent clusters. In Euclidean space there is a well studied transition of this type, the percolation transition. For a concentration $p > p_c$ there is a giant cluster of sites while just below $p_c$ there are only small inequivalent clusters. The total phase space of an $N$-spin $S=1/2$ Ising system is an $N$ dimensional hypercube. Relaxation of any $N$-spin Ising system by successive single spin updates can be considered strictly as a random walk of the system point along near neighbor edges among the thermodynamically attainable vertices on this hypercube [@ogielski:85]. It was argued [@campbell:85] that as random walks on full \[hyper\]spherical surfaces result in pure exponential decay [@debye:29; @caillol:04] and random walks on threshold percolation clusters in Euclidean space lead to sub-linear diffusion $\langle R^2 \rangle \sim t^{\beta_{d}}$ [@alexander:82], random walks on threshold percolation clusters inscribed on \[hyper\]spheres would be characterized by “sub-exponential” relaxation of the form $q(t) = \langle \cos(\theta(t))\rangle = \exp[-(t/\tau)^{\beta_{d}}]$ with the same exponents $\beta_{d}$ as in the corresponding Euclidean space. This was demonstrated numerically for $d = 3$ to $8$ [@jund:01]. A hypercube being topologically equivalent to a hypersphere, on a diluted hypercube at threshold [@eordos:79; @borgs:06] random walks can be expected to lead to stretched exponential relaxation with exponent $\beta = 1/3$. Successive explicit numerical studies of random walks on the randomly occupied hypercube at $p_{c}$ [@campbell:87; @dealmeida:00; @lemke:11] have confirmed this. It is important to note that the limiting behavior for relaxation due to diffusion on sparse graphs has been shown analytically to take the form of a stretched exponential with exponent $\beta = 1/3$ [@bray:88; @samukhin:08]. It was further conjectured [@campbell:85] that in a complex Ising system such as an ISG the phase transition would be analogous to a percolation transition in configuration space. Detailed “rough landscape” models for the configuration space of complex systems have been widely invoked (see for instance Refs. [@angelani:98; @doye:02; @burda:07]); these models can be thought of in terms of linked basins with a gradual dilution of the links leading finally to a percolation threshold. As the critical properties of a percolation transition are very robust, if the configuration space percolation threshold scenario is valid the basic critical behavior should not be sensitive to model details; in particular stretched exponential relaxation with exponent $1/3$ should be observed generically in ISGs whatever the space dimension and possibly also in a wider class of complex systems. Conclusion ========== The present numerical SK results taken together with Ogielski’s $3d$ ISG analysis [@ogielski:85] can be taken as a strong empirical indication of a universal equilibrium relaxation pattern for ISGs in the paramagnetic regime : stretched exponential decay Eq. \[qt\] having an exponent $\beta(T)$ which tends to $1/3$ when the relaxation time diverges at the critical temperature $T_{g}$. 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[**[GEOMETRIC PROPERTIES OF QCD STRING FROM WILLMORE FUNCTIONAL]{}**]{} R.Parthasarathy[[^1]]{}\ The Institute of Mathematical Sciences\ C.I.T.Campus, Taramani Post\ Chennai 600 113, India.\ and\ K.S.Viswanathan[[^2]]{}\ Department of Physics\ Simon Fraser University\ Burnaby\ Canada, V5A 1S6.\ [[*[Abstract]{}*]{}]{} The extremum of the Willmore-like functional for $m$-dimensional Riemannian surface immersed in $d$-dimensional Riemannian manifold under normal variations is studied and various cases of interest are examined. This study is used to relate the parameters of QCD string action, including the Polyakov-Kleinert extrinsic curvature action, with the geometric properties of the world sheet. The world sheet has been shown to have [*[negative stiffness]{}*]{} on the basis of the geometric considerations. [[MSC indices: 51P05;53C42;53C80;81V05]{}]{} [[Keywords:Willmore functional, immersion in Riemann space, QCD string action, extrinsic curvature]{}]{} [**[I.Introduction]{}**]{} QCD strings is a string theory in 4-dimensions. It has been realized by Polyakov \[1\] and independently by Kleinert \[2\] that for QCD strings, added extrinsic curvature action to the usual Nambu-Goto (NG) area term is appropriate. In particular the theory with extrinsic curvature action alone has been shown to be asymptotically free \[1,2,3\] - a feature relevant to describe QCD. By considering the 1-loop multi-instanton effects in the theory of 2-dimensional world sheet in $R^3$ and $R^4$, the grand partition function has been found to be that of a 2-dimensional modified Coulomb gas system with long range order in the infra-red region and, in plasma phase in the ultra-violet region \[4\]. The above result uses the running coupling constant and the string world sheet is stable against small fluctuations along the normal (transverse) directions in the infra-red region and avoids crumpling. Thus the Polyakov-Kleinert string provides a relevant description of colour flux tubes between quarks in QCD. In order to remove the unphysical ghost poles and to realize a lowest energy state, Kleinert and Chervyakov \[5\] recently proposed a [*new*]{} string model with [*negative*]{} sign for the extrinsic curvature action i.e., they [*hypothesize*]{} negative [*stiffness*]{} for the gluonic flux tubes, inspired by properties of magnetic flux tubes in Type-II superconductor and of Nielsen-Olesen vortices in relativistic gauge models. They propose an effective string action as, $$\begin{aligned} S&=&\frac{(c-1)}{2}M^2\int d^2\xi \sqrt{g} g^{\alpha\beta} {\nabla}_{\alpha}X^{\mu}(\xi)\frac{1}{c-e^{ {\nabla}^2/{\mu}^2}} {\nabla}_{\beta}X^{\mu}(\xi), \nonumber \end{aligned}$$ where $X^{\mu}(\xi)$ ; $\mu\ =\ 1,2,3,4$ are the worldsheet coordinates, ${\xi}_1,{\xi}_2$ are the local isothermal coordinates on the surface, $g_{\alpha\beta}\ =\ {\partial}_{\alpha}X^{\mu}{\partial}_{\beta}X^{\mu}$ is the induced metric (first fundamental form) on the world sheet, ${\nabla}_{\alpha}$ is the covariant derivative on the surface, $M$ is dimensionfull (mass dimension) constant and $c$ is a dimensionless constant. The propagator from the quadratic part in $X^{\mu}$, in momentum space \[5\], is $$\begin{aligned} G(k^2)&=&\frac{1}{(c-1)}\ \frac{c-e^{-k^2/{\mu}^2}}{k^2}, \nonumber \end{aligned}$$ and for small momentum, this is $$\begin{aligned} G(k^2)&\simeq & \frac{1+k^2/{\Lambda}^2}{k^2}, \nonumber \end{aligned}$$ with ${\Lambda}^2\ =\ (c-1)/{\mu}^2$. This has a single pole at $k^2\ =\ 0$ with negative stiffness ${\alpha}_0\ =\ -{\Lambda}^2/M^2$, in contrast to the propagator in Polyakov-Kleinert model $1/(k^2(1+k^2/{\Lambda}^2)$, which has unphysical pole at $k^2\ =\ -{\Lambda}^2$ and which has positive stiffness of ${\Lambda}^2/M^2$. Approximating the full propagator by its low momentum expression, Kleinert and Chervyakov \[5\] have proposed an action at low momentum region as, $$\begin{aligned} S_{KC} &=& \frac{1}{2}M^2 \int d^2\xi\ \sqrt{g} \ g^{\alpha\beta} {\nabla}_{\alpha}X^{\mu}\ \frac{1}{1\ -\ \frac{{\nabla}^2}{{\Lambda}^2}}\ {\nabla}_{\beta}X^{\mu}. \end{aligned}$$ Such an action (1) has the the high temperature behaviour as that of large-N QCD \[6\]. The [*negative*]{} extrinsic curvature term can be seen from (1) by expanding the non-local term, using the Gauss equation $$\begin{aligned} {\nabla}_{\alpha}{\nabla}_{\beta}X^{\mu} &=& H^i_{\alpha\beta}N^{i\mu},\end{aligned}$$ where $i\ =\ 1,2$ and $H^{i}_{\alpha\beta}$ are the components of the extrinsic curvature (second fundamental form) along the two normals $N^{i\mu}$ to the world sheet, and the Weingarten equation \[4\] $$\begin{aligned} {\nabla}_{\alpha}N^{i\mu} &=& -H^{i\gamma}_{\alpha}{\partial}_{\gamma}X^{\mu},\end{aligned}$$ where the covariant derivative ${\nabla}_{\alpha}$ in (3) incorporates the connection in the normal frame as well. By expanding $(1\ -\ \frac{{\nabla}^2} {{\Lambda}^2})^{-1}$ in (1) and realizing $X^{\mu}$ is a scalar on the world sheet, (1) can be written as $$\begin{aligned} S_{KC} &=& \frac{1}{2} M^2 \int d^2\xi\ \sqrt{g}\ \{2\ +\ \frac{1}{{\Lambda}^2}g^{\alpha\beta}{\partial}_{\alpha}X^{\mu} \ g^{\gamma\delta}{\nabla}_{\gamma} ({\nabla}_{\delta}{\partial}_{\beta} X^{\mu}) - \cdots \}, \nonumber\end{aligned}$$ where we have retained up to the $\frac{1}{{\Lambda}^2}$ term for illustration. Upon using (2) and (3) and the fact ${\partial}_{\alpha}X^{\mu}\ N^{i\mu}\ =\ 0$, the above expression simplifies to $$\begin{aligned} S_{KC} &=& \frac{1}{2} M^2 \int d^2\xi\ \sqrt{g}\ \{2 \ -\ \frac{1}{{\Lambda}^2} H^{i\alpha\beta}H^i_{\alpha\beta}\}. \nonumber \end{aligned}$$ But then, $$\begin{aligned} H^{i\alpha\beta}H^i_{\alpha\beta} &=& 4{\mid H\mid}^2 + R, \nonumber\end{aligned}$$ where ${\mid H\mid}^2\ =\ H^iH^i$, with $H^i\ =\ \frac{1}{2}g^{\alpha\beta}H^i_{\alpha\beta}$ and $R$ is the scalar curvature of the world sheet. In view of this, the expression for $S_{KC}$ becomes $$\begin{aligned} S_{KC} &\simeq & M^2 \int \sqrt{g}\ d^2\xi - \frac{2M^2}{{\Lambda}^2}\int \sqrt{g}\ {\mid H\mid}^2\ d^2\xi - \frac{M^2}{{\Lambda}^2}\int \sqrt{g}\ R\ d^2\xi + \cdots \end{aligned}$$ In above $M^2$ plays the role of string tension. The extrinsic curvature action (the second term in (4)) has [*negative*]{} stiffness. The third term is just the Euler characteristic of the surface which is a topological invariant action. It is clear from (1) that unphysical poles can be avoided by appealing to surfaces with negative stiffness. It will be worthwhile to examine whether the negative stiffness is favoured from purely geometric considerations of the surface. In this context, the Willmore surfaces which extremize the Willmore functional \[7\] $$\begin{aligned} S_W &=& \int \sqrt{g}\ {\mid H\mid}^2\ d^2\xi, \nonumber \end{aligned}$$ become relevant. It is the purpose of this paper to first consider general Willmore functional for $m$-dimensional surface immersed in $d$-dimensional Riemannian space ($m\ <\ d$) and study various cases of interest. Then using the results, [*[we compare the classical equation of motion for (4) with immersion in flat space, with that of the Willmore functional for immersion in a Riemannian space]{}*]{}, thereby showing the effects of the Nambu-Goto term in (4) could be accounted for by considerations of the Willmore functional in a curved space. The extremum of Willmore functional for hypersurfaces in Euclidean space ($E^3$) has been dealt with in detail by Willmore \[7\] and, by Chen \[8\] for $m$-dimensional oriented closed hypersurface in Euclidean space $E^{m+1}$. Willmore and Jhaveri \[9\] extended to $m$-dimensional manifold immersed as a hypersurface of a general ($m+1$)-dimensional Riemannian manifold and Weiner \[10\] to that of 2-dimensional surface in a general Riemannian manifold. In this paper, we examine the Willmore functional for the general case of $m$-dimensional Riemann surface immersed in $d$-dimensional Riemann space and then consider various cases of interest. As an application of this study, we will compare the equation of motion of Willmore functional for $2$-dimensional surface immersed in $d$-dimensional space with the classical equation of motion of QCD string to relate the QCD string parameters, namely, the string tension and stiffness parameter to the geometrical properties of the surface. [**[II. EFFECTS DUE TO NORMAL VARIATIONS]{}**]{} For an $m$-dimensional surface $\Sigma $ immersed in a $d$-dimensional ($d\ >\ m$) Riemannian manifold [$\Sigma'$]{} with metric $h_{\mu\nu}\ ;\ (\mu,\nu\ =\ 1,2,\cdots d)$, we have the induced metric on $\Sigma$ as $$\begin{aligned} g_{\alpha\beta} &=& {\partial}_{\alpha}X^{\mu}\ {\partial}_{\beta}X^{\nu}\ h_{\mu\nu},\end{aligned}$$ where the indices $\alpha,\beta$ take values $1,2,\cdots m$ and $X^{\mu}\ =\ X^{\mu}({\xi}_1,{\xi}_2\cdots ,{\xi}_m)$, with ${\xi}_{\alpha}$’s as coordinates on $\Sigma$. There are $(d-m)$ unit normals at a point $P\in \Sigma$, denoted by $N^{i\mu};\ (i\ =\ 1,2,\cdots (d-m))$, chosen to satisfy $$\begin{aligned} N^{i\mu}\ N^{j\nu}\ h_{\mu\nu} &=& {\delta}^{ij} \nonumber \\ {\partial}_{\alpha}X^{\mu}\ N^{i\nu}\ h_{\mu\nu} &=& 0, \ \ \ \forall \ i\ =\ 1,2,\cdots (d-m);\ \ \forall \ \alpha\ =\ 1,2, \cdots ,m.\end{aligned}$$ Repeated indices will be appropriately summed over in this paper. The equation of Gauss \[11\] for $\Sigma$, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}_{\beta}X^{\mu} &\equiv & {\partial}_{\alpha}{\partial}_{\beta}X^{\mu}-{\Gamma}^{\gamma} _{\alpha\beta}{\partial}_{\gamma}X^{\mu}+{\tilde{\Gamma}}^ {\mu}_{\nu\rho}{\partial}_{\alpha}X^{\nu}{\partial}_{\beta} X^{\rho} \nonumber \\ & = & H^i_{\alpha\beta}N^{i\mu},\end{aligned}$$ defines the [*[second fundamental form]{}*]{} $H^i_{\alpha\beta}$. ${\Gamma}^{\gamma}_{\alpha\beta}$ and ${\tilde{\Gamma}}^{\mu}_{\nu\rho}$ are the connections on $\Sigma$ and $\Sigma'$ determined by $g_{\alpha\beta}$ and $h_{\mu\nu}$ respectively. The $(d-m)$ normals $N^{i\mu}$ satisfy the Weingarten equation $$\begin{aligned} {\nabla}_{\alpha}N^{i\mu} &\equiv& {\partial}_{\alpha}N^{i\mu}+{\tilde{\Gamma}}^{\mu}_{\nu\rho} {\partial}_{\alpha}X^{\nu}N^{i\rho}-A^{ij}_{\alpha}N^{j\mu} \nonumber \\ &=& -H^{i\beta}_{\alpha}{\partial}_{\beta}X^{\mu},\end{aligned}$$ where $A^{ij}_{\alpha}\ =\ N^{j\nu}({\partial}_{\alpha}N^{i\mu}+{\tilde{\Gamma}}^{\mu}_ {\nu\rho}{\partial}_{\alpha}X^{\nu}N^{i\rho})h_{\mu\nu}$ is the $m$-dimensional gauge field or connection in the normal bundle. We need the Gauss equation \[11\] $$\begin{aligned} {\tilde{R}}_{\mu\nu\rho\sigma}{\partial}_{\alpha}X^{\mu} {\partial}_{\beta}X^{\nu}{\partial}_{\gamma}X^{\rho} {\partial}_{\delta}X^{\sigma} &=& R_{\alpha\beta\gamma \delta}+H^{i}_{\beta\gamma}H^i_{\alpha\delta}-H^i_ {\beta\delta}H^i_{\alpha\gamma},\end{aligned}$$ where $R_{\alpha\beta\gamma\delta}$ and ${\tilde{R}}_{\mu\nu \rho\sigma}$ are the Riemann symbols of the first kind for $\Sigma$ and $\Sigma'$ respectively. We introduce the mean curvature $H^i$ (there are ($d-m$) such quantities) by $$\begin{aligned} H^i &=& \frac{1}{m} g^{\alpha\beta}H^i_{\alpha\beta}.\end{aligned}$$ The variations of the surface can be described by the variations of $X^{\mu}({\xi}_1,\cdots {\xi}_m)$ as $X^{\mu}({\xi}_1,\cdots {\xi}_m)\ +\ \delta X^{\mu} ({\xi}_1,\cdots {\xi}_m)$. In general $\delta X^{\mu}\ =\ {\phi}^i N^{i\mu}\ +\ {\partial}_{\alpha}X^{\mu} {\eta}^{\alpha}$, comprising of variations along $m$ tangent directions and $(d-m)$ normals. The tangential variations are related to the structure equations \[12\]. So, we consider only the normal variations and accordingly, $$\begin{aligned} \delta X^{\mu} &=& {\phi}^i N^{i\mu}.\end{aligned}$$ Using (5) it follows for normal variations $$\begin{aligned} \delta \sqrt{g} &=& -m\sqrt{g} {\phi}^i H^i, \nonumber \\ \delta g^{\alpha\beta} &=& 2{\phi}^i H^{i\alpha\beta}.\end{aligned}$$ From (10) it follows $$\begin{aligned} \delta H^i &=& \frac{2}{m} {\phi}^k H^{k\alpha\beta} H^i_{\alpha\beta} + \frac{1}{m} g^{\alpha\beta}\delta H^i_{ \alpha\beta},\end{aligned}$$ using the second equation in (12). For hypersurfaces, there will be only one normal and in such a case, the computation of $\delta H$ has been given in Ref.9. The evaluation of $\delta H^i_{\alpha\beta}$ for 2-dimensional surface in $d$-dimensional Riemannian manifold is described in Ref.12. The computation of $\delta H^i_{\alpha\beta}$ for $m$-dimensional surface in $d$-dimensional Riemannian manifold is involved and we give here the relevant steps for the sake of completeness. From (7) we have $$\begin{aligned} H^i_{\alpha\beta} &=& \{{\partial}_{\alpha}{\partial}_{\beta} X^{\mu} + {\tilde{\Gamma}}^{\mu}_{\rho\sigma}{\partial} _{\alpha}X^{\rho}{\partial}_{\beta}X^{\sigma}\}N^{i\nu}h_{ \mu\nu}.\end{aligned}$$ Using (6), we have $$\begin{aligned} \delta N^{j\mu}\ N^{i\nu}\ h_{\mu\nu} + N^{j\mu}\ \delta N^{i\nu}\ h_{\mu\nu} &=& -N^{j\mu}N^{i\nu}\ \delta h_{\mu\nu}, \nonumber \\ & & \nonumber \\ {\partial}_{\gamma}(\delta X^{\mu}) N^{i\nu}\ h_{\mu\nu} + {\partial}_{\gamma}X^{\mu}\ \delta N^{i\nu}\ h_{\mu\nu} &=& - {\partial}_{\gamma}X^{\mu} N^{i\nu}\ \delta h_{\mu \nu}, \nonumber\end{aligned}$$ and then, $$\begin{aligned} g^{\alpha\beta}H^i\ \delta H^i_{\alpha\beta} &=& g^{\alpha\beta}H^i{\tilde{R}}_{\rho\sigma\nu\lambda}{\partial} _{\alpha}X^{\nu}{\partial}_{\beta}X^{\rho}\ \delta X^{ \sigma} N^{i\lambda} \nonumber \\ &+& g^{\alpha\beta}H^i({\nabla}_{\alpha}{\nabla}_{\beta}\ \delta X^{\mu})\ N^i_{\mu} +\frac{m}{2}H^iH^j N^{j\mu} N^{i\nu}\ ({\partial}_ {\lambda}h_{\mu\nu})\ \delta X^{\lambda} \nonumber \\ &-& m H^iH^j{\tilde{ \Gamma}}^{\mu}_{\rho\lambda}N^{j\sigma}\ \delta X^{\lambda} \ N^i_{\mu}.\end{aligned}$$ The last two terms cancel each other after expanding ${\tilde{\Gamma}}^{\mu}_{\rho\lambda}$ and using $i\ \leftrightarrow j$ symmetry. Now using (11) and (8), we find $$\begin{aligned} g^{\alpha\beta}H^i\ \delta H^i_{\alpha\beta} &=& g^{\alpha\beta}H^i{\tilde{R}}_{\rho\sigma\nu\lambda}{\partial} _{\alpha}X^{\nu}{\partial}_{\beta}X^{\rho}{\phi}^k N^{k \sigma}N^{i\lambda} \nonumber \\ &+& H^k({\nabla}_{\alpha}{\nabla}^{\alpha}{\phi}^k) - H^i H^i_{\alpha\beta}H^{k\alpha\beta}{\phi}^k.\end{aligned}$$ We consider the extremum of the following functional $$\begin{aligned} W &=& \int \sqrt{g}\ \left( H^iH^i\right)^{\frac{m}{2}}\ d^m\xi,\end{aligned}$$ which reduces to Willmore functional for $m\ =\ 2$ and to that of Chen \[8\] for $m$-dimensional [*hypersurface*]{} as well with Willmore and Jhaveri \[9\] for $m$-dimensional [*hypersurface*]{} in $(m+1)$ dimensional Riemannian manifold. The normal variations of (17) give the equations of motion. Taking the normal variations of (17) and using (12), (13) and (16), we obtain $$\begin{aligned} \delta W &=& \int \sqrt{g} (H^jH^j)^{\frac{m}{2}-1}H^k ({\nabla}_{\alpha}{\nabla}^{\alpha}{\phi}^k)\ d^m\xi \nonumber \\ &-& m\int \sqrt{g}{\phi}^kH^k(H^jH^j)^{\frac{m}{2}}\ d^m\xi \nonumber \\ &+& \int \sqrt{g}(H^jH^j)^{\frac{m}{2}-1}{\phi}^k H^iH^i_ {\alpha\beta}H^{k\alpha\beta}\ d^m\xi \nonumber \\ &+&\int \sqrt{g} (H^jH^j)^{\frac{m}{2}-1}H^i g^{\alpha\beta} {\tilde{R}}_{\rho\sigma\nu\lambda}{\partial}_{\alpha}X^{\nu} {\partial}_{\beta}X^{\rho}{\phi}^k N^{k\sigma}N^{i\lambda}\ d^m\xi.\end{aligned}$$ Equating this to zero and using $$\begin{aligned} \int \ \sqrt{g}\ (H^jH^j)^{\frac{m}{2}-1}\ H^k\ ({\nabla}_{\alpha}{\nabla}^{\alpha}{\phi}^k)\ d^m\xi &=& \nonumber \\ \hspace{2.0cm} \int \ \sqrt{g}\ {\phi}^k\ {\nabla}_{\alpha}{\nabla}^ {\alpha}\left( (H^jH^j)^{\frac{m}{2}-1} H^k\right)\ d^m\xi,\end{aligned}$$ we obtain the equation of motion for (17) as $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}\left( (H^jH^j)^{\frac{m}{2}-1}H^k\right)-mH^k(H^jH^j)^{\frac{m}{2}} +(H^jH^j)^{\frac{m}{2}-1}H^iH^i_{\alpha\beta}H^{k\alpha\beta} \nonumber \\ +(H^jH^j)^{\frac{m}{2}-1}H^i g^{\alpha\beta}{\tilde{R}}_ {\rho\sigma\nu\lambda}{\partial}_{\alpha}X^{\nu}{\partial}_ {\beta}X^{\rho}\ N^{k\sigma}\ N^{i\lambda}\ =\ 0,\end{aligned}$$ since (18) must hold for all allowed ${\phi}^k$. We now consider various cases. [[*[Case.1$\ $ hypersurface in Euclidean space]{}*]{}]{} Let $\Sigma $ be an $m$-dimensional hepersurface in $d\ =\ m+1$ dimensional [*[Euclidean]{}*]{} space, i.e., ${\Sigma}'\ =\ E^{m+1}$. As there will be only one normal for a hypersurface, $H^jH^j\ =\ H^2$ and (20) reduces to $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}(H^{m-1}) - mH^{m+1} + H^{m-1} H_{\alpha\beta}H^{\alpha\beta} &=& 0,\end{aligned}$$ and in this case the Gauss equation (9) when contracted with $g^{\alpha\gamma}g^{\beta\delta}$ gives $$\begin{aligned} H^{\alpha\beta}H_{\alpha\beta} &=&-R + m^2H^2, \nonumber \end{aligned}$$ where $R$ is the curvature scalar of the hypersurface $\Sigma $. Then (21) becomes $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}(H^{m-1}) + m(m-1)H^{m+1} - H^{m-1}R &=& 0,\end{aligned}$$ which is the result of Chen \[8\] and agrees with Eqn.5.59 of Willmore \[7\]. [[*[Case.2$\ $ hypersurface in Riemannian space]{}*]{}]{} Let $\Sigma $ be an $m$-dimensional [*[hypersurface]{}*]{} immersed in $d\ =\ m+1$ dimensional [*[Riemannian]{}*]{} manifold $\Sigma '$. Then Eqn.20 becomes $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^{m-1} - mH^{m+1} + H^{m-1}H_{\alpha\beta}H^{\alpha\beta} \nonumber \\ + H^{m-1}g^{\alpha\beta}{\tilde{R}}_{\rho\sigma \nu\lambda}{\partial}_{\alpha}X^{\nu}{\partial}_{\beta}X^ {\rho}N^{\sigma}N^{\lambda}\ =\ 0. \end{aligned}$$ In this case, the equation of Gauss (9) is $$\begin{aligned} R_{\alpha\beta\gamma\delta}&=&H_{\beta\delta}H_{\alpha\gamma}- H_{\beta\gamma}H_{\alpha\delta}+{\tilde{R}}_{\mu\nu\rho \sigma}{\partial}_{\alpha}X^{\mu}{\partial}_{\beta}X^{\nu} {\partial}_{\gamma}X^{\rho}{\partial}_{\delta}X^{\sigma}, \nonumber \end{aligned}$$ which when contracted with $g^{\alpha\gamma}g^{\beta\delta}$ gives $$\begin{aligned} R&=&-H^{\alpha\beta}H_{\alpha\beta}+m^2H^2+{\tilde{R}}_{\mu \nu\rho\sigma}{\partial}_{\alpha}X^{\mu}{\partial}_{\beta} X^{\nu}{\partial}_{\gamma}X^{\rho}{\partial}_{\delta}X^ {\sigma}g^{\alpha\gamma}g^{\beta\delta}.\end{aligned}$$ The completeness relation found in Ref.12 will now be used and it is $$\begin{aligned} h^{\mu\nu} &=& g^{\alpha\beta}{\partial}_{\alpha}X^{\mu}{\partial}_{\beta}X^ {\nu} + \sum^{d-2}_{i}N^{i\mu}N^{i\nu}. \end{aligned}$$ For hypersurfaces, (25) is simply $$\begin{aligned} h^{\mu\nu} &=& g^{\alpha\beta}{\partial}_{\alpha}X^{\mu}{\partial}_{\beta} X^{\nu} + N^{\mu}N^{\nu},\end{aligned}$$ and using this in (24), we find $$\begin{aligned} H^{\alpha\beta}H_{\alpha\beta} &=& -R + m^2H^2 + \tilde{R}- 2{\tilde{R}}_{\mu\nu}N^{\mu}N^{\nu}.\end{aligned}$$ Using (26) in the last term of (23), we have $$\begin{aligned} g^{\alpha\beta}{\tilde{R}}_{\rho\sigma\nu\lambda}{\partial}_ {\alpha}X^{\nu}{\partial}_{\beta}X^{\rho}N^{\sigma}N^ {\lambda}&=& {\tilde{R}}_{\rho\sigma\nu\lambda}h^{\rho\nu} N^{\sigma}N^{\lambda}-{\tilde{R}}_{\rho\sigma\nu\lambda} N^{\nu}N^{\sigma}N^{\rho}N^{\lambda} \nonumber \\ &=& {\tilde{R}}_{\sigma\lambda}N^{\sigma}N^{\lambda}, \nonumber\end{aligned}$$ and so the Eqn.23 becomes, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^{m-1}+m(m-1)H^{m+1}+ H^{m-1}\{-R+\tilde{R}-{\tilde{R}}_{\mu\nu}N^{\mu}N^{\nu} \}&=& 0.\end{aligned}$$ We analyse (28), by choosing an orthogonal frame at $P\ \in \ \Sigma$ such that the matrix $H^{\alpha\beta}$ is diagonal. Then $H^{\alpha\beta}H_{\alpha\beta}\ =\ \sum^{m}_{i=1}h^{2}_i$, and $m^2H^2\ =\ (\sum^m_{i=1}h_i)^2$. (See \[9\]). Then, Eqn.28 can be written as, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^{m-1} &=& -H^{m-1}\{\sum^{m}_{i=1} h^2_i - \frac{1}{m}(\sum^m_{i=1}h)^2 +{\tilde{R}}_{\mu\nu}N^{\mu}N^{\nu}\},\end{aligned}$$ since (27) gives $$\begin{aligned} -R+{\tilde{R}}+m^2H^2-2{\tilde{R}}_{\mu\nu}N^{\mu}N^{\nu}&=& \sum^m_{i=1}h^2_i. \nonumber\end{aligned}$$ It is to be noted that $\sum^m_{i=1}h^2_i -\frac{1}{m}(\sum^m_{i=1}h_i)^2\ \geq \ 0$. For ${\tilde{R}} _{\mu\nu}$ positive-definite, it is seen from (29) that ${\nabla}_{\alpha}{\nabla}^{\alpha}H^{m-1}$ has the same sign as $-H^{m-1}$. [[*[Case.3$\ $ 2-dimensional surface in Riemannian space]{}*]{}]{} Let $\Sigma$ be a 2-dimensional surface immersed in ${\Sigma}'$. Then (20) becomes, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^k-2H^k(H^jH^j)+H^iH^i _{\alpha\beta}H^{k\alpha\beta}+ \nonumber \\ H^ig^{\alpha\beta}{\tilde{ R}}_{\rho\sigma\nu\lambda}{\partial}_{\alpha}X^{\nu} {\partial}_{\beta}X^{\rho}N^{k\sigma}N^{i\lambda}&=&0,\end{aligned}$$ agreeing with Eqn.35 of Ref.12. [[*[Case.4$\ $ Immersions in space of constant curvature]{}*]{}]{} Consider $\Sigma'$ space to be a space of constant curvature i.e., de-Sitter or anti-de-Sitter type. In this case \[13\] $$\begin{aligned} {\tilde{R}}_{\mu\nu\rho\sigma}&=& K(h_{\mu\rho}h_{\nu\sigma}-h_{\mu\sigma}h_{\nu\rho}).\end{aligned}$$ Then, $$\begin{aligned} {\tilde{R}}_{\nu\sigma}\ =\ h^{\mu\rho}{\tilde{R}}_{\mu\nu\rho\sigma}&=&K(d-1)h_{\nu \sigma}, \nonumber \\ & & \nonumber \\ \tilde{R} &=& h^{\mu\sigma}{\tilde{R}}_{\mu\sigma}\ =\ Kd(d-1).\end{aligned}$$ Then (20) becomes, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}\left( (H^jH^j)^{\frac{m}{2}-1}H^k\right)+mH^k(H^jH^j)^{\frac{m}{2}-1 }(K-H^{\ell}H^{\ell}) \nonumber \\ +(H^jH^j)^{\frac{m}{2}-1}H^iH^i_{\alpha\beta}H^{k\alpha\beta}\ \ =\ 0. \end{aligned}$$ Similarly Eqn.28 in Case.2, becomes, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^{m-1}+m(m-1)H^{m+1} -R H^{m-1}&=&0,\end{aligned}$$ for $m$-dimensional hypersurface immersed in $\Sigma'$ space of constant curvature. Eqn.30 of Case.3 i.e., 2-dimensional surface immersed in $\Sigma'$, a space of constant curvature, becomes, $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^k - 2H^k(H^jH^j) + H^i H^{i\alpha\beta}H^k_{\alpha\beta} + 2KH^k &=& 0.\end{aligned}$$ In spite of the simplifications, these equations are still difficult to solve explicitly without further choices for the geometry. [[**[III. QCD STRING AND WILLMORE FUNCTIONAL]{}**]{}]{} As explained in the Introduction, it appears that a candidate action for describing QCD string has to involve the extrinsic geometry of the world sheet, regarded as a 2-dimensional Riemannian surface immersed in $R^4$. With [*[negative]{}*]{} stiffness, Kleinert and Chervyakov \[5\] successfully obtained the correct hight temperature behaviour as in large-N QCD \[6\]. Further evidence for the role of the extrinsic geometry in QCD stems from the $U(N);\ \ N\ \rightarrow \ \infty$ lattice gauge theory calculations of Kazakov \[14\], Kostov \[15\] and O’Brien and Zuber \[16\]. These calculations confirm the equivalence of multicolour QCD and string theory in which the resulting surfaces intersect at self-intersections. It is known that the self-intersection number involves extrinsic geometry. We consider the action (4) without the Euler characteristic term, as $$\begin{aligned} S_{KC} &=& T \int \sqrt{g}\ d^2\xi + {\alpha}_0 \int \sqrt{g}\ {\mid H\mid}^2\ d^2\xi,\end{aligned}$$ where $T$ is the string tension, ${\alpha}_0$ a measure of stiffness of the QCD string immersed in $R^d$ (say) and $H^2\ =\ H^iH^i;\ i=1,2 \cdots (d-2)$. The extremum of (36) can be easily found using (12), (13), and (16) for normal variations. We find the equation of motion for (36) as $$\begin{aligned} {\nabla}_{\alpha}{\nabla}^{\alpha}H^k-\frac{2T}{{\alpha}_0}H^k -2H^kH^jH^j+H^iH^{k\alpha\beta}H^i_{\alpha\beta}&=&0,\end{aligned}$$ for the stiffness parameter ${\alpha}_0\ \neq\ 0$. This non-linear equation is complicated and it will be wothwhile hence to draw some information from this. Kholodenko and Nesterenko \[17\] proposed an approach in this direction by considering (36) for immersion in $R^3$ and relating to the extremum of the Willmore functional for immersion in $S^3$. We will generalize this approach here, by relating (37) to (20) for $m\ =\ 2$, which is (30). Eqn.30 has the same form as (37) provided we identify, $$\begin{aligned} -\frac{2T}{{\alpha}_0}H^k &=& H^i{\tilde{R}}_{\rho\sigma\nu\lambda}{\partial}_{\alpha}X^{\nu }{\partial}_{\beta}X^{\rho}N^{k\sigma}N^{i\lambda}g^{\alpha \beta}.\end{aligned}$$ Upon using (25), this becomes, $$\begin{aligned} -\frac{2T}{{\alpha}_0}H^k&=& H^i({\tilde{R}}_{\sigma\lambda}N^{k\sigma}N^{i\lambda}- {\tilde{R}}_{\rho\sigma\nu\lambda}N^{j\nu}N^{j\rho}N^{k\sigma} N^{i\lambda}),\end{aligned}$$ a [*[new relation]{}*]{} among the string tension, stiffness parameter, mean curvature scalar and the geometric properties of ${\Sigma}'$. In order to make (39) manageable, we take ${\Sigma}'$ a space of constant curvature as in (31) and (32). Then it can be seen, $$\begin{aligned} {\tilde{R}}_{\sigma\lambda}N^{k\sigma}N^{i\lambda}&=&K(d-1) {\delta}_{ik}, \nonumber \\ & & \nonumber \\ {\tilde{R}}_{\rho\sigma\nu\lambda}N^{j\nu}N^{j\rho}N^{k\sigma} N^{i\lambda}&=&K(d-3){\delta}^{ik},\end{aligned}$$ and so (39) becomes $$\begin{aligned} -\frac{2T}{{\alpha}_0}H^k &=& K(d-1)H^k-K(d-3)H^k\ =\ 2KH^k.\end{aligned}$$ Now as we have assumed that $H^k$’s are not zero, it follows $$\begin{aligned} \frac{T}{{\alpha}_0} &=& -K.\end{aligned}$$ It is noted here that the dimensionality of ${\Sigma}'$ does not directly appear in relating $T$, the string tension, and ${\alpha}_0$, the stiffness parameter, with $K$. From this, it follows that the stiffness parameter can be positive for $K<0$ (Anti-de-Sitter background) or negative for $K>0$ (de-Sitter background). [[**[VI. CONCLUSIONS]{}**]{}]{} The extremum of the Willmore functional for $m$-dimensional surface immersed in $d$-dimensional Riemannian space is studied under the normal variations of the immersed surface. Various cases of interest are examined. In particular the equation of motion for a 2-dimensional surface immersed in spaces of constant curvature, is compared with the equation of motion of the Polyakov-Kleinert action of the QCD string considered as a Riemann surface immersed in $R^4$, to obtain a new relation connecting the string tension and stiffness parameter of the QCD string on the one hand and the constant $K$ of the Riemann space (31). This relation $T/{\alpha}_0\ =\ -K$, is independent of the dimensionality of ${\Sigma}'$. For positive $K$, favoured by positive-definteness of ${\tilde{R}}_{\mu\nu}$ (see Case.2) from (32), it follows that negative stiffness is recommended by geometric considerations. This result agrees with the observation of Kleinert and Chervyakov \[5\] using (physical) QCD string. Thus the QCD string world sheet regarded as a 2-dimensional surface immersed in $R^4$ has been shown to favour negative stiffness by comparing its classical equation with that of a Willmore 2-dimensional surface immersed in a space of constant curvature. [[**[REFERNCES]{}**]{}]{} 1. A.M.Polyakov, Nucl.Phys. [**B268**]{} (1986) 406 ; [**B486**]{} (1997) 23. 2. H.Kleinert, Phys.Lett. [**B174**]{} (1986) 335;\ Phys.Rev.Lett. [**58**]{} (1987) 1915; Phys.Lett. [**B189**]{} (1987) 187. 3. K.S.Viswanathan, R.Parthasarathy and D.Kay,\ Ann.Phys. (N.Y), [**206**]{} (1991) 237. 4. K.S.Viswanathan and R.Parthasarathy, Phys.Rev. [**D51**]{} (1995) 5830. 5. H.Kleinert and A.M.Chervyakov, Phys.Lett. [**B381**]{} (1996) 286.\ See also, H.Kleinert, Phys.Lett. [**B211**]{} (1988) 151. 6. J.Polchinski, Phys.Rev.Lett. [**68**]{} (1992) 1267. 7. T.J.Willmore, [*Total Curvature in Riemannian Geometry*]{},\ Ellis Harwood, 1982. 8. B.Y.Chen, J.London.Math.Soc. (2) [**6**]{} (1973) 321. 9. T.J.Willmore and C.S.Jhaveri, Quart.J.Math.Oxford. (2) [**23**]{} (1972) 319. 10. J.L.Weiner, Indiana Univ. Math. Journal. [**27**]{} (1978) 19. 11. L.P.Eisenhart, [*[Riemannian Geometry]{}*]{},\ Princeton University Press. 1966. 12. K.S.Viswanathan and R.Parthasarathy, Phys.Rev. [**D55**]{} (1997) 3800. 13. S.Weinberg, [*[Gravitation and Cosmology]{}*]{},\ John-Wiley and Sons.Inc.N.Y. 1972. 14. V.A.Kazakov, Phys.Lett. [**B128**]{} (1983) 316. 15. I.K.Kostov, Phys.Lett. [**B138**]{} (1984) 191. 16. K.H.O’Brien and J.B.Zuber, Nucl.Phys. [**B253**]{} (1985) 621. 17. A.L.Kholodenko and V.V.Nesterenko, J.Geom.Phys. [**16**]{} (1995) 15. [^1]: e-mail address:[email protected] [^2]: e-mail address:[email protected]
--- abstract: | We show that a rank-1 quantum observable (POVM) ${\mathsf{M}}$ is jointly measurable with a quantum observable ${\mathsf{M}}'$ exactly when ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$. If ${\mathsf{M}}$ is extreme, rank-1 and discrete then ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent if and only if they are jointly measurable. PACS numbers: 03.65.Ta, 03.67.–a address: 'Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland' author: - 'Juha-Pekka Pellonpää' title: 'On coexistence and joint measurability of rank-1 quantum observables' --- Introduction ============ For observables (POVMs) ${\mathsf{M}}$ and ${\mathsf{M}}'$ of a quantum system, the question arises under which conditions it is possible to collect the experimental data of measurements of ${\mathsf{M}}$ and ${\mathsf{M}}'$ from a measurement of a single observable ${\mathsf{N}}$ of the same system. For example, if ${\mathsf{M}}$ and ${\mathsf{M}}'$ can be measured together then their measurement outcome statistics can be obtained from the marginals of the joint measurement distribution of their joint observable ${\mathsf{N}}$. In this case, ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent, that is, their ranges are contained in the range of ${\mathsf{N}}$. To clarify the above definitions, let us consider the case of a discrete POVM ${\mathsf{M}}$ in a finite-dimensional Hilbert space ${\mathcal{H}}$ (denote $d=\dim{\mathcal{H}}<\infty$). If ${\Omega}=\{x_1,\,x_2\,\ldots,\,x_N\}$ is the outcome space of ${\mathsf{M}}$ then (by fixing a basis) ${\mathcal{H}}\cong{\mathbb C}^d$ and ${\mathsf{M}}$ can be viewed as a collection $({\mathsf{M}}_1,\,{\mathsf{M}}_2,\ldots,\,{\mathsf{M}}_N)$ of positive $d\times d$–matrices ${\mathsf{M}}_i$ such that $\sum_{i=1}^N{\mathsf{M}}_i=I$ (the identity matrix). A state of the system is represented as a density matrix $\rho$, that is, a positive matrix of trace 1, and the number ${\mathrm{tr}\left[\rho{\mathsf{M}}_i\right]}\in[0,1]$ is interpreted as the probability of getting an outcome $x_i$ when a measurement of ${\mathsf{M}}$ is performed and the system is in the state $\rho$. Actually, ${\mathsf{M}}$ is a map which assigns to each subset $X$ of ${\Omega}$ a positive matrix ${\mathsf{M}}(X)=\sum_{x_i\in X}{\mathsf{M}}_i$ so that ${\mathrm{tr}\left[\rho{\mathsf{M}}(X)\right]}$ is the probability of getting an outcome belonging to the set $X$. Especially, ${\mathsf{M}}\big(\{x_i\}\big)={\mathsf{M}}_i$. Hence, the range of the POVM ${\mathsf{M}}$ (as a map) is the set $$\begin{aligned} {\rm ran}\,{\mathsf{M}}&=&\big\{{\mathsf{M}}(X)\,|\,X\subset{\Omega}\big\} \\ &=&\big\{{\mathsf{M}}_1,\,{\mathsf{M}}_2,{\mathsf{M}}_3,\ldots,\,{\mathsf{M}}_1+{\mathsf{M}}_2,\,{\mathsf{M}}_1+{\mathsf{M}}_3,\,{\mathsf{M}}_2+{\mathsf{M}}_3,\ldots,\,{\mathsf{M}}_1+{\mathsf{M}}_2+{\mathsf{M}}_2,\ldots\big\}.\end{aligned}$$ Two POVMs ${\mathsf{M}}=({\mathsf{M}}_i)$ and ${\mathsf{M}}'=({\mathsf{M}}'_j)$ (of the same Hilbert space) are jointly measurable if there exists a POVM ${\mathsf{N}}=({\mathsf{N}}_{ij})$ such that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are the marginals of ${\mathsf{N}}$, that is, $$\label{ghy} {\mathsf{M}}_i=\sum_j {\mathsf{N}}_{ij}, \qquad {\mathsf{M}}'_j=\sum_i {\mathsf{N}}_{ij}.$$ Immediately one sees that ${\rm ran}\,{\mathsf{M}}$ is a subset of the range of ${\mathsf{N}}$. Similarly, ${\rm ran}\,{\mathsf{M}}' \subseteq {\rm ran}\,{\mathsf{N}}$. More generally, if the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong to the range of some POVM ${\mathsf{N}}$, i.e. ${\rm ran}\,{\mathsf{M}}\cup{\rm ran}\,{\mathsf{M}}'\subseteq {\rm ran}\,{\mathsf{N}}$ (but equations do not necessarily hold) then ${\mathsf{M}}$ and ${\mathsf{M}}'$ are said to be coexistent. The notion of the coexistence was introduced by Ludwig [@Ludwig] and studied by many authors, see, e.g.[@Lahti; @BuKiLa; @LaPu] and references therein. Recently Reeb [*et al*]{}. [@lyhyt] were able to show that [*coexistence does not imply joint measurability.*]{} More specifically, they constructed two observables which are coexistent but cannot be measured together (in the sense of eq.  above). None of these observables is rank-1 (a discrete POVM ${\mathsf{M}}$ is rank-1 if every ${\mathsf{M}}_i$ is a rank-1 matrix, i.e.the maximum number of linearly independent row (or column) vectors of ${\mathsf{M}}_i$ is 1 or, equivalently, ${\mathsf{M}}_i$ is of the form ${|d_i \rangle\langle d_i|}$ where $d_i\in{\mathcal{H}}$). Recall that rank-1 observables have many important properties [@HeWo; @Part2; @Pell; @Pell']. For example, their measurements can be seen as state preparation procedures and the measurements break entanglement completely between the system and its environment. In this paper, we show that a rank-1 observable ${\mathsf{M}}$ is jointly measurable with an observable ${\mathsf{M}}'$ if and only if ${\mathsf{M}}'$ is a smearing (post-processing) of ${\mathsf{M}}$ (Theorem 1, Remark \[remmmm\]). For example, a discrete POVM ${\mathsf{M}}'$ is a smearing of a discrete POVM ${\mathsf{M}}$ if there exists a probability (or stochastic or Markov) matrix $(p_{kj})$ such that $0\le p_{kj}\le 1$, $\sum_j p_{kj}=1$, and ${\mathsf{M}}_j'=\sum_k p_{kj}{\mathsf{M}}_k$. Furthermore, we show that if ${\mathsf{M}}$ is extreme (i.e. an extreme point of the convex set of all observables), discrete and rank-1 then the coexistence of ${\mathsf{M}}$ and ${\mathsf{M}}'$ implies their joint measurability (Theorem 2). Finally, if the range of a discrete rank-1 observable ${\mathsf{M}}$ is contained in the range of an observable ${\overline}{\mathsf{M}}$, i.e. ${\rm ran}\,{\mathsf{M}}\subseteq {\rm ran}\,{\overline}{\mathsf{M}}$, then ${\mathsf{M}}$ and ${\overline}{\mathsf{M}}$ are jointly measurable (Proposition \[proposition1\]). Notations and basic results =========================== For any Hilbert space ${\mathcal{H}}$ we let ${\mathcal{L(H)}}$ denote the set of bounded operators on ${\mathcal{H}}$. The set of states (density operators i.e. positive operators of trace 1) is denoted by ${\mathcal{S(H)}}$ and $I_{\mathcal{H}}$ is the identity operator of ${\mathcal{H}}$. Throughout this article, we let ${\mathcal{H}}$ be a [separable]{}[^1] (complex) nontrivial Hilbert space and $(\Omega,\Sigma)$ be a measurable space (i.e. $\Sigma$ is a $\sigma$-algebra of subsets of a nonempty set $\Omega$).[^2] In the discrete case, ${\Omega}=\{x_1,\,x_2,\,\ldots\}$ and $\Sigma$ consists of [*all*]{} subsets of ${\Omega}$. We denote ${\mathbb N}:=\{0,1,\ldots\}$ and ${\mathbb N}_\infty:={\mathbb N}\cup\{\infty\}$. Let ${\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ be the set of [quantum observables]{}, that is, normalized positive operator (valued) measures (POVMs) ${\mathsf{M}}:\,\Sigma\to{\mathcal{L(H)}}$. Recall that a map ${\mathsf{M}}:\,\Sigma\to{\mathcal{L(H)}}$ is a POVM if and only if $X\mapsto{\mathrm{tr}\left[\rho{\mathsf{M}}(X)\right]}$ is a probability measure for all $\rho\in{\mathcal{S(H)}}$. Moreover, ${\mathrm{tr}\left[\rho{\mathsf{M}}(X)\right]}$ is interpreted as the probability of getting an outcome $x$ which belong to $X\in\Sigma$ when a measurement of ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ is performed and the system is in the state $\rho\in{\mathcal{S(H)}}$. The range of a POVM ${\mathsf{M}}:\,\Sigma\to{\mathcal{L(H)}}$ is the set $${\rm ran}\,{\mathsf{M}}= \big\{{\mathsf{M}}(X)\,|\,X\in\Sigma\big\}.$$ Any ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ is called a projection valued measure (PVM) or a sharp observable or a spectral measure if ${\mathsf{M}}(X)^2={\mathsf{M}}(X)$ for all $X\in\Sigma$. For any observables ${\mathsf{M}}^a,\,{\mathsf{M}}^b\in{\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ and a number $\lambda\in[0,1]$ one can form the convex combination $\lambda{\mathsf{M}}^a+(1-\lambda){\mathsf{M}}^b\in{\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ which corresponds to classical randomization between the two observables (or their measurements). Hence, the set ${\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ is convex. We say that a POVM ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$ is [*extreme*]{} if it is an extreme point of ${\mathrm{Obs}}(\Sigma,\,{\mathcal{H}})$, that is, if ${\mathsf{M}}=\lambda{\mathsf{M}}^a+(1-\lambda){\mathsf{M}}^b$ implies that ${\mathsf{M}}^a={\mathsf{M}}={\mathsf{M}}^b$. In other words, extreme POVMs do not allow (nontrivial) convex decompositions and are free from the classical noise arising from this type of randomization. It is easy to show that PVMs are extreme but there are other extreme POVMs too [@Pe11]. Next we consider minimal Naimark dilations of observables. We start with a simple example of a discrete POVM. Assume that $d=\dim{\mathcal{H}}<\infty$ and ${\mathsf{M}}=({\mathsf{M}}_1,\ldots,\,{\mathsf{M}}_N)$ is a discrete POVM. Since each ${\mathsf{M}}_i$ is a positive non-zero $d\times d$–matrix (bounded by the identity matrix), we may write $${\mathsf{M}}_i=\sum_{k=1}^{m_i}\lambda_{ik}{|{\varphi}_{ik} \rangle\langle {\varphi}_{ik}|}=\sum_{k=1}^{m_i}{|d_{ik} \rangle\langle d_{ik}|}$$ where the eigenvectors ${\varphi}_{ik}$, $k=1,\ldots,\,m_i$, form an orthonormal set, the eigenvalues $\lambda_{ik}$ are non-zero (and bounded by 1), and $d_{ik}:=\sqrt{\lambda_{ik}}{\varphi}_{ik}$. We say that $m_i\in{\mathbb N}$ is the multiplicity of the outcome $x_i$ or the rank of ${\mathsf{M}}_i$, and ${\mathsf{M}}$ is of rank 1 if $m_i=1$ for all $i=1,\ldots,N$. Note that always $m_i\le d$. Let then ${{\mathcal{H}_\oplus}}$ be a Hilbert space spanned by an orthonormal basis $e_{ik}$ where $i=1,\ldots,N$ and $k=1,\ldots,m_i$. Obviously, $\dim{{\mathcal{H}_\oplus}}=\sum_{i=1}^N m_i$. Define a discrete PVM ${\mathsf{P}}=({\mathsf{P}}_1,\ldots,{\mathsf{P}}_N)$ of ${{\mathcal{H}_\oplus}}$ via $${\mathsf{P}}_i=\sum_{k=1}^{m_i}{|e_{ik} \rangle\langle e_{ik}|}$$ so that ${\mathcal{H}}_{m_i}={\mathsf{P}}_i{{\mathcal{H}_\oplus}}$ is spanned by the vectors $e_{ik}$, $k=1,\ldots,m_i$, and we may write (the direct sum) $${{\mathcal{H}_\oplus}}=\bigoplus_{i=1}^N{\mathcal{H}}_{m_i}.$$ Define a linear operator $J:\,{\mathcal{H}}\to{{\mathcal{H}_\oplus}}$, $$J=\sum_{i=1}^N\sum_{k=1}^{m_i}{|e_{ik} \rangle\langle d_{ik}|}$$ for which $ J^*{\mathsf{P}}_iJ={\mathsf{M}}_i. $ Especially, $J^*J=J^*\sum_i{\mathsf{P}}_i J=\sum_i{\mathsf{M}}_i=I_{\mathcal{H}}$ showing that $J$ is an isometry. Hence, $\big({{\mathcal{H}_\oplus}},J,{\mathsf{P}}\big)$ is a Naimark dilation of ${\mathsf{M}}$. The dilation is minimal, that is, the span of vectors ${\mathsf{P}}_i J\phi$, $i=1,\ldots,N$, $\phi\in{\mathcal{H}}$, is the whole ${{\mathcal{H}_\oplus}}$. Indeed, this follows immediately from equation $\psi=\sum_{i=1}^N\sum_{k=1}^{m_i}\<e_{ik}|\psi\>e_{ik}= \sum_{i=1}^N\sum_{k=1}^{m_i}\<e_{ik}|\psi\>\lambda_{ik}^{-1}{\mathsf{P}}_i J d_{ik}$ where $\psi\in{{\mathcal{H}_\oplus}}$. It is well known that ${\mathsf{M}}$ is a PVM if and only if $J$ is unitary (i.e. $\{d_{ik}\}_{i,k}$ is an orthonormal basis of ${\mathcal{H}}$). In this case one can identify ${{\mathcal{H}_\oplus}}$ with ${\mathcal{H}}$ and ${\mathsf{P}}$ with ${\mathsf{M}}$ e.g. by setting $e_{ik}=d_{ik}$. Let then $\{h_n\}_{n=1}^d$ be an orthonormal basis of ${\mathcal{H}}$ and define (orthonormal) [*structure vectors*]{} $\psi_n:=Jh_n\in{{\mathcal{H}_\oplus}}$ so that $\<e_{ik}|\psi_n\>=\<d_{ik}|h_n\>$ and $$\label{hhiew47} {\mathsf{M}}(X)=\sum_{x_i\in X}{\mathsf{M}}_i=\sum_{n,m=1}^d\sum_{x_i\in X}\<h_n|{\mathsf{M}}_ih_m\>{|h_n \rangle\langle h_m|} =\sum_{n,m=1}^d\sum_{x_i\in X}\<\psi_n(x_i)|\psi_m(x_i)\>{|h_n \rangle\langle h_m|}$$ where $\psi_n(x_i)={\mathsf{P}}_i\psi_n={\mathsf{P}}_iJh_n$. Finally, an operator $E\in{\mathcal{L}}({{\mathcal{H}_\oplus}})$ is decomposable if it commutes with every ${\mathsf{P}}_i$, that is, $E=\sum_{i=1}^N E(x_i)$ where $E(x_i):={\mathsf{P}}_iE{\mathsf{P}}_i\in{\mathcal{L}}({\mathcal{H}}_{m_i})$. Now ${\mathsf{P}}_i(E\psi)=E(x_i){\mathsf{P}}_i\psi$. It is easy to generalize the results of this (discrete) example to the case of an arbitrary (e.g. continuous) POVM; just replace the sums $\sum_{x_i\in X}(\ldots)$ by integrals $\int_X(\ldots)\d\mu(x)$ (see, e.g. [@Pell']). This will be done next. In the rest of this article, we let ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,{\mathcal{H}})$ be an observable and $\big({{\mathcal{H}_\oplus}},J,{\mathsf{P}}\big)$ its minimal (diagonal) Naimark dilation [@Part1 Theorem 1]. Here $\mu:\,\Sigma\to[0,1]$ is a probability measure[^3] which can always be chosen to be $X\mapsto{\mathrm{tr}\left[\rho_0{\mathsf{M}}(X)\right]}$ where $\rho_0\in{\mathcal{S(H)}}$ has only non-zero eigenvalues, and $${{\mathcal{H}_\oplus}}=\int_\Omega^\oplus{\mathcal{H}}_{m(x)}\d\mu(x)$$ is a direct integral Hilbert space with $m(x)$-dimensional fibers (Hilbert spaces) ${\mathcal{H}}_{m(x)}$; recall that ${{\mathcal{H}_\oplus}}$ consists of square integrable ‘wave functions’ $\psi$ such that $\psi(x)\in{\mathcal{H}}_{m(x)}$. The operator $J:\,{\mathcal{H}}\to{{\mathcal{H}_\oplus}}$ is isometric, and $${\mathsf{M}}(X)=J^*{\mathsf{P}}(X) J,\qquad X\in\Sigma,$$ where ${\mathsf{P}}\in{\mathrm{Obs}}(\Sigma,{{\mathcal{H}_\oplus}})$ is the canonical spectral measure (or the ‘position observable’) of ${{\mathcal{H}_\oplus}}$, that is, for all $X\in\Sigma$ and $\psi\in{{\mathcal{H}_\oplus}}$, ${\mathsf{P}}(X)\psi={\ensuremath{ \chi\raisebox{-1ex}{$\scriptstyle X$} }}\psi$ where ${\ensuremath{ \chi\raisebox{-1ex}{$\scriptstyle X$} }}$ is the characteristic function of $X$. Moreover, the set of linear combinations of vectors ${\mathsf{P}}(X) J\psi$, $X\in\Sigma$, $\psi\in{\mathcal{H}}$, is dense in ${{\mathcal{H}_\oplus}}$. We say that $m(x)\in{\mathbb N}_\infty$, $m(x)\le\dim{\mathcal{H}}$, is the [*multiplicity*]{} of the measurement outcome $x\in\Omega$ since $x$ can be viewed as a collection of $m(x)$ outcomes $(x,1),\,(x,2),\ldots$ of some ‘finer’ observable (which can distinguish them) [@Pell]. If $m(x)=1$ for ($\mu$-almost)[^4] all $x\in\Omega$ then ${\mathsf{M}}$ is of [*rank 1,*]{} that is, the outcomes of ${\mathsf{M}}$ are ‘nondegenerate’ [@Part2 Section 4]. For any orthonormal (ON) basis ${{\bf h}}=\{h_n\}_{n=1}^{\dim{\mathcal{H}}}$ of ${\mathcal{H}}$, define [*structure vectors*]{} $\psi_n:=Jh_n$ of ${\mathsf{M}}$ so that one can write (weakly) $$\begin{aligned} \label{eq1} {\mathsf{M}}(X)= \sum_{n,m=1}^{\dim{\mathcal{H}}}\int_{X}\<\psi_n(x)|\psi_m(x)\>\d\mu(x){|h_n \rangle\langle h_m|}.\end{aligned}$$ (Compare this equation to above; indeed, in the discrete case, $\mu$ is just a counting measure[^5] or a sum of Dirac deltas so that all integrals reduce to sums [@Pell'].) If ${\mathsf{M}}$ is rank-1 then the fibers ${\mathcal{H}}_{m(x)}$ are just one-dimensional Hilbert spaces so that, without restricting generality, we may assume that ${\mathcal{H}}_{m(x)}\equiv{\mathbb C}$ and thus ${{\mathcal{H}_\oplus}}=L^2(\mu)$, the space of square integrable wave functions $\psi:\,\Omega\to{\mathbb C}$. Now, for example, the inner product $\<\psi_n(x)|\psi_m(x)\>$ of ${\mathcal{H}}_{m(x)}$ in eq.  is just $\overline{\psi_n(x)}\psi_m(x)$ (the inner product of the 1-dimensional Hilbert space ${\mathbb C}$). Consider a Hilbert space ${\mathcal{H}}=L^2({\mathbb R})$ spanned by the Hermite functions $h_n$, $n\in{\mathbb N}$. Denote briefly ${|n\rangle}=h_n$ and let $a=\sum_{n=0}^\infty\sqrt{n+1}{|n \rangle\langle n+1|}$ be the lowering operator. Let $|z\>=e^{-|z|^2/2}\sum_{n=0}^\infty z^n/\sqrt{n!}\,{|n\rangle}$, $z\in{\mathbb C}$, be a coherent state. The following rank-1 POVMs are physically relevant (see, eq. ): - The spectral measure[^6] ${\mathsf{Q}}(X)=\sum_{n,m=0}^{\infty}\int_{X}{\overline}{h_n(x)}h_m(x)\d x{|n \rangle\langle m|}$ of the position operator $Q=2^{-1/2}(a+a^*)$; now $\Omega={\mathbb R}$, $\d\mu(x)=\d x$ and $\psi_n(x)=h_n(x)$. - The spectral measure ${\mathsf{N}}(\{n\})={|n \rangle\langle n|}$ of the number operator $N=a^*a$; now $\Omega={\mathbb N}$, $\mu$ is the counting measure (discrete case) and $\psi_n(x)=\delta_{xn}$ (Kronecker delta) where $x\in{\mathbb N}$. - The phase space observable (associated with the $Q$-function) ${\mathsf{G}}(Z)=\int_Z|z\>\<z|\d^2 z/\pi=\sum_{n,m=0}^{\infty}\int_Z{{\overline}z}^n z^m/\sqrt{n!m!}\cdot\pi^{-1}e^{-|z|^2}\d^2z{|n \rangle\langle m|}$; now $\Omega={\mathbb C}$, $\d\mu(z)=\pi^{-1}e^{-|z|^2}\d^2z$ and $\psi_n(z)=z^n/\sqrt{n!}$. - The canonical phase observable $\Phi(X)=\sum_{n,m=0}^{\infty}\int_{X}e^{i(n-m)\theta}(2\pi)^{-1}\d\theta{|n \rangle\langle m|}$; now $\Omega=[0,2\pi)$, $\d\mu(\theta)=(2\pi)^{-1}\d\theta$ and $\psi_n(\theta)=e^{-in\theta}$. Recall that $E\in{\mathcal{L}}({{\mathcal{H}_\oplus}})$ is decomposable if there exists a (measurable) family of operators $E(x)\in{\mathcal{L}}({\mathcal{H}}_{m(x)})$, $x\in\Omega$, such that $\mu\text{-ess sup}_{x\in\Omega}\|E(x)\|<\infty$ and $(E\psi)(x)=E(x)\psi(x)$ for all $\psi\in{{\mathcal{H}_\oplus}}$ and $\mu$-almost all $x\in\Omega$. \[lemma\] Let ${\mathsf{E}}:\,\Sigma\to{\mathcal{H}}$ be a (possibly non-normalized) positive operator measure. Then ${\mathsf{E}}(X)\le{\mathsf{M}}(X)$ for all $X\in\Sigma$ if and only if there exists a (unique) $E\in{\mathcal{L}}({{\mathcal{H}_\oplus}})$, $0\le E\le I_{{\mathcal{H}_\oplus}}$, such that $[E,{\mathsf{P}}(X)]=0$ and ${\mathsf{E}}(X)=J^*{\mathsf{P}}(X) EJ$ for all $X\in\Sigma$ if and only if there exists a (unique) $E\in{\mathcal{L}}({{\mathcal{H}_\oplus}})$, $0\le E\le I_{{\mathcal{H}_\oplus}}$, which is decomposable, $E=\int_\Omega^\oplus E(x)\d\mu(x)$, and $${\mathsf{E}}(X)= \sum_{n,m=1}^{\dim{\mathcal{H}}}\int_{X}\<\psi_n(x)|E(x)\psi_m(x)\>\d\mu(x){|h_n \rangle\langle h_m|}$$ for all $X\in\Sigma$. The first part of the lemma follows immediately from [@HaHePe13 Proposition 1] (or from [@Part3 Lemma 1]) by noting that any POVM (of ${\mathcal{H}}$) can be seen as a completely positive map from an Abelian von Neumann algebra to ${\mathcal{L(H)}}$. For example, in the case of ${\mathsf{M}}$, the von Neumann algebra is $L^\infty(\mu)$ (the $\mu$-essentially bounded functions $\Omega\to{\mathbb C}$). Finally, it is well-known that any bounded operator on ${{\mathcal{H}_\oplus}}$ is decomposable if and only if it commutes with the canonical spectral measure ${\mathsf{P}}$ (see, e.g. [@Part1 Proposition 1]). \[rem\]Note that ${\mathsf{M}}$ is a PVM[^7] if and only if $J$ is unitary [@Part1 Theorem 1]. In this case, ${\mathsf{E}}(X)\le{\mathsf{M}}(X)$, $X\in\Sigma$, if and only if ${\mathsf{E}}(X)=J^*{\mathsf{P}}(X)JJ^*EJ={\mathsf{M}}(X)E'$ where $E':=J^*EJ={\mathsf{E}}(\Omega)$, see Lemma \[lemma\]. Moreover, $[E',{\mathsf{M}}(X)]=0$, $X\in\Sigma$. If ${\mathsf{E}}$ is normalized, i.e. $E'=I_{\mathcal{H}}$, then ${\mathsf{E}}={\mathsf{M}}$. Jointly measurable observables ============================== Let $(\Omega,\Sigma)$ and $(\Omega',\Sigma')$ be measurable spaces and ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,{\mathcal{H}})$ and ${\mathsf{M}}'\in{\mathrm{Obs}}(\Sigma',{\mathcal{H}})$. If $[{\mathsf{M}}(X),{\mathsf{M}}'(Y)]=0$ for all $X\in\Sigma$, $Y\in\Sigma'$, then ${\mathsf{M}}$ and ${\mathsf{M}}'$ are said to [*commute*]{} (with each other). Denote the product $\sigma$-algebra[^8] of $\Sigma$ and $\Sigma'$ by $\Sigma\otimes\Sigma'$. We say that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are [*jointly measurable*]{} if there exists an ${\mathsf{N}}\in{\mathrm{Obs}}(\Sigma\otimes\Sigma',{\mathcal{H}})$ such that ${\mathsf{N}}(X\times\Omega')={\mathsf{M}}(X)$ for all $X\in\Sigma$ and ${\mathsf{N}}(\Omega\times Y)={\mathsf{M}}'(Y)$ for all $Y\in\Sigma'$. In this case, ${\mathsf{N}}$ is called a [*joint observable*]{} of ${\mathsf{M}}$ and ${\mathsf{M}}'$. Physically, this means that the measurement outcome probabilities ${\mathrm{tr}\left[\rho{\mathsf{M}}(X)\right]}$ and ${\mathrm{tr}\left[\rho{\mathsf{M}}'(Y)\right]}$ can be obtained from a measurement of a single observable ${\mathsf{N}}$ in the state $\rho$ which gives all probabilities ${\mathrm{tr}\left[\rho{\mathsf{N}}(X\times Y)\right]}$. Recall that jointly measurable observables need not commute (see, Remark \[remmmm\]). Let $({{\mathcal{H}_\oplus}},J,{\mathsf{P}})$ be a minimal (diagonal) Naimark dilation of ${\mathsf{M}}$ where, e.g.${{\mathcal{H}_\oplus}}=\int_\Omega^\oplus{\mathcal{H}}_{m(x)}\d\mu(x)$. Following [@JePuVi Section 4.2] we say that $f:\,\Omega\times\Sigma'\to{\mathbb R}$ is a [*weak Markov kernel (with respect to $\mu$)*]{} if - $\Omega\ni x\mapsto f(x,Y)\in{\mathbb R}$ is $\mu$-measurable for all $Y\in\Sigma'$, - for every $Y\in\Sigma'$, $0\le f(x,Y)\le 1$ for $\mu$-almost all $x\in\Omega$, - $f(x,\Omega')=1$ and $f(x,\emptyset)=0$ for $\mu$-almost all $x\in\Omega$, - if $\{Y_i\}_{i=1}^\infty\subseteq\Sigma'$ is a disjoint sequence (i.e. $Y_i\cap Y_j=\emptyset$, $i\ne j$) then $ f(x,\cup_i Y_i)=\sum_i f(x,Y_i) $ for $\mu$-almost all $x\in\Omega$. If there exists a weak Markov kernel $f:\,\Omega\times\Sigma'\to{\mathbb R}$ such that ${\mathsf{M}}'(Y)=\int_\Omega f(x,Y)\d{\mathsf{M}}(x)$ for all $Y\in\Sigma$ then ${\mathsf{M}}'$ is a [*smearing*]{} or a [*post-processing*]{} of ${\mathsf{M}}$, or any measurement of ${\mathsf{M}}'$ is subordinate to a measurement of ${\mathsf{M}}$ [@Holevo]. Note that one can interpret $f(x,Y)$ as a (classical) conditional probability which is the probability of the event $Y$ assuming that $x$ is obtained. If ${\mathsf{M}}'$ is a post-prosessing of ${\mathsf{M}}$ then the measurement outcome probabilities ${\mathrm{tr}\left[\rho{\mathsf{M}}'(Y)\right]}=\int_\Omega f(x,Y){\mathrm{tr}\left[\rho{\mathsf{M}}(\d x)\right]}$, that is, they can be seen as a classical processing (integration) of the probability distribution ${\mathrm{tr}\left[\rho{\mathsf{M}}(\d x)\right]}$ related to a measurement of ${\mathsf{M}}$ in the state $\rho$. In the case of discrete POVMs, $\Omega=\{x_1,\ldots,x_N\}$ and $\Omega'=\{y_1,\ldots,y_{N'}\}$. Moreover, $\Sigma$ (resp. $\Sigma'$) consists of all subsets of $\Omega$ (resp. $\Omega'$) and $\mu$ is the counting measure of $\Omega$. Let $f:\,\Omega\times\Sigma'\to{\mathbb R}$ be a weak Markov kernel and denote $p_{kj}=f\big(x_k,\,\{y_j\}\big)\in[0,1]$. Now $$\sum_{j=1}^{N'} p_{kj}=\sum_{y_j\in\Omega'}f\big(x_k,\,\{y_j\}\big)=f\big(x_k,\,\cup_{y_j\in\Omega'}\{y_j\}\big) =f\big(x_k,\Omega'\big) =1$$ so that $(p_{kj})$ is a probability (or stochastic or Markov) matrix, see Introduction. \[remmmm\] Let $f:\,\Omega\times\Sigma'\to{\mathbb R}$ be a weak Markov kernel with respect to $\mu$ (associated with ${\mathsf{M}}$). Then ${\mathsf{M}}_f:\,\Sigma'\to{\mathcal{L(H)}},\;Y\mapsto \int_\Omega f(x,Y)\d{\mathsf{M}}(x)$ is an observable (i.e. a smearing of ${\mathsf{M}}$). Furthermore, ${\mathsf{M}}$ and ${\mathsf{M}}_f$ are jointly measurable, a joint observable ${\mathsf{N}}\in{\mathrm{Obs}}(\Sigma\otimes\Sigma',{\mathcal{H}})$ being defined by $${\mathsf{N}}(X\times Y)=\int_X f(x,Y)\d{\mathsf{M}}(x), \qquad X\in\Sigma,\;Y\in\Sigma',$$ if and only if, for each $\rho\in{\mathcal{S(H)}}$, the probability bimeasure $$\Sigma\times\Sigma'\ni (X,Y)\mapsto \int_X f(x,Y){\mathrm{tr}\left[\rho{\mathsf{M}}(\d x)\right]}\in[0,1]$$ extends to a probability measure on $\Sigma\otimes\Sigma'$. For this, one needs additional conditions for $f$, or for the measurable spaces [@LaYl Section 6]. If $f$ satisfies a slightly stronger condition - for each sequence $\{Y_i\}_{i=1}^\infty\subseteq\Sigma'$, there exists $\mu$-null set $N\subset X$ such that, for all $x\in\Omega\setminus N$ and for any disjoint subsequence $\{Y_{i_k}\}_{k=1}^\infty\subseteq \{Y_i\}_{i=1}^\infty$, $ f(x,\cup_k Y_{i_k})=\sum_k f(x,Y_{i_k}), $ then ${\mathsf{M}}$ and ${\mathsf{M}}_f$ are jointly measurable [@Tulcea Proposition 1]. Obviously, if $Y\mapsto f(x,Y)$ is a probability measure for ($\mu$-almost) all $x\in\Omega$ then (iv)’ holds (recall that, in this case, $f$ is called a [*Markov kernel*]{}). For example, by choosing $\Sigma'=\Sigma$ and $f(x,Y)={\ensuremath{ \chi\raisebox{-1ex}{$\scriptstyle Y$} }}(x)$ for all $x\in\Omega$ and $Y\in\Sigma$ then ${\mathsf{N}}(X\times Y)={\mathsf{M}}(X\cap Y)$ and ${\mathsf{M}}'(Y)={\mathsf{M}}(Y)$, $X,\,Y\in\Sigma$, that is, any observable is jointly measurable with itself even if it does not commute with itself. Clearly, in physical applications, we may always assume that (iv)’ holds. Hence, we have seen that (classical) post-processing ${\mathsf{M}}\mapsto {\mathsf{M}}_f$ can be viewed as a joint measurement of ${\mathsf{M}}$ and the smeared ${\mathsf{M}}_f$. Suppose then that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable, and let ${\mathsf{N}}$ be their joint observable. Then, for all $X\in\Sigma$ and $Y\in\Sigma'$, ${\mathsf{N}}(X\times Y)\le {\mathsf{N}}(X\times\Omega')={\mathsf{M}}(X)=J^*{\mathsf{P}}(X)J$ so that, by Lemma \[lemma\], $${\mathsf{N}}(X\times Y)=J^*{\mathsf{P}}(X){\mathsf{F}}(Y) J=\sum_{n,m=1}^{\dim{\mathcal{H}}}\int_{X}\<\psi_n(x)|{\mathsf{F}}(x,Y)\psi_m(x)\>\d\mu(x){|h_n \rangle\langle h_m|}$$ where ${\mathsf{F}}:\,\Sigma'\to{\mathcal{L}}({{\mathcal{H}_\oplus}})$ is a POVM which commutes with the canonical spectral measure ${\mathsf{P}}$, that is, for each $Y\in\Sigma'$, the operator ${\mathsf{F}}(Y)$ is decomposable, $${\mathsf{F}}(Y)=\int_\Omega^\oplus {\mathsf{F}}(x,Y)\d\mu(x).$$ Note that ${\mathsf{F}}(x,Y)\in{\mathcal{L}}({\mathcal{H}}_{m(x)})$ can be chosen to be positive for all $Y\in\Sigma'$ and all $x\in\Omega$. Moreover, $${\mathsf{M}}'(Y)=J^*{\mathsf{F}}(Y)J=\sum_{n,m=1}^{\dim{\mathcal{H}}}\int_{\Omega}\<\psi_n(x)|{\mathsf{F}}(x,Y)\psi_m(x)\>\d\mu(x){|h_n \rangle\langle h_m|},\qquad Y\in\Sigma'.$$ In addition, if ${\mathsf{M}}$ is rank-1 then $m(x)=1$, ${\mathcal{H}}_{m(x)}\cong{\mathbb C}$, and ${{\mathcal{H}_\oplus}}$ is isomorphic to $L^2(\mu)$, the Hilbert space of the $\mu$-square integrable complex functions on $\Omega$. In this case, $f(x,Y):={\mathsf{F}}(x,Y)\in[0,1]$ and ${\mathsf{F}}(Y)\in{\mathcal{L}}\big(L^2(\mu)\big)$ is a multiplicative operator, that is, $({\mathsf{F}}(Y)\psi)(x)=f(x,Y)\psi(x)$ for all $\psi\in{{\mathcal{H}_\oplus}}$ and for $\mu$-almost all $x\in\Omega$. Indeed, it is easy to check that $f:\,\Omega\times\Sigma'\to{\mathbb C},\;(x,Y)\mapsto f(x,Y)$, is a weak Markov kernel with respect to $\mu$ and, since $${\mathsf{N}}(X\times Y)=\int_X f(x,Y)\d{\mathsf{M}}(x),\qquad {\mathsf{M}}'(Y)=\int_\Omega f(x,Y)\d{\mathsf{M}}(x),\qquad X\in\Sigma,\;Y\in\Sigma',$$ ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$. Hence, we have proved the following theorem: Let ${\mathsf{M}}$ be a rank-1 observable and ${\mathsf{M}}'$ any observable. If ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable then ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$. Note that, in the general case, ${\mathsf{F}}(x,Y)$ is an operator valued ‘conditional probability’ which operates on the ‘eigenspace’ ${\mathcal{H}}_{m(x)}$ of $x$. One could say that ${\mathsf{F}}(x,Y)$ is a ‘quantum (weak) Markov kernel’ which reduces to a ‘classical’ kernel $f(x,Y)$ when ${\mathsf{M}}$ is rank-1, i.e. when ${\mathsf{M}}$ has the ‘nondegenerate’ outcomes $x$. Let ${\mathsf{M}}$ and ${\mathsf{M}}'$ be any jointly measurable POVMs, and let ${\mathsf{N}}$ be their joint observable as above. If ${\mathsf{M}}$ is a PVM then $J$ is unitary and $${\mathsf{N}}(X\times Y)=J^*{\mathsf{P}}(X)JJ^*{\mathsf{F}}(Y) J={\mathsf{M}}(X){\mathsf{M}}'(Y)={\mathsf{M}}'(Y){\mathsf{M}}(X),\qquad X\in\Sigma,\; Y\in\Sigma',$$ that is, ${\mathsf{M}}'$ commutes with ${\mathsf{M}}$. If ${\mathsf{M}}$ is also rank-1 then ${\mathsf{M}}'$ commutes with itself [@JePuVi Theorem 4.4]. If both ${\mathsf{M}}$ and ${\mathsf{M}}'$ are PVMs then also ${\mathsf{N}}$ is projection valued and, for all $Y\in\Sigma'$, ${\mathsf{F}}(Y)$ is a projection, that is, ${\mathsf{F}}(x,Y)$ is a projection of ${\mathcal{H}}_{m(x)}$ for $\mu$-almost all $x\in\Omega$. Hence, if ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable PVMs and ${\mathsf{M}}$ is rank-1 (i.e. ${\mathcal{H}}_{m(x)}={\mathbb C}$) then ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$ given by a weak Markov kernel $f$ such that, for all $Y\in\Sigma'$, $f(x,Y)\in\{0,1\}$ for $\mu$-almost all $x\in\Omega$. In this case, the kernel is ‘sharp’ in the sense that each conditional probability $f(x,Y)$ is either 1 or 0. Coexistent observables ====================== Let ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,{\mathcal{H}})$ and ${\mathsf{M}}'\in{\mathrm{Obs}}(\Sigma',{\mathcal{H}})$. We say that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are [*coexistent*]{} if there exists a $\sigma$-algebra ${\overline}\Sigma$ over a set ${\overline}{\Omega}$ and an observable ${\overline}{\mathsf{M}}:\,{\overline}\Sigma\to{\mathcal{L(H)}}$ such that the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong the range of ${\overline}{\mathsf{M}}$, ${\rm ran}\,{\mathsf{M}}\cup{\rm ran}\,{\mathsf{M}}'\subseteq {\rm ran}\,{\overline}{\mathsf{M}}$, that is, for any $X\in\Sigma$ and $Y\in\Sigma'$ there exists sets $Z,\,Z'\in{\overline}\Sigma$ such that ${\overline}{\mathsf{M}}(Z)={\mathsf{M}}(X)$ and ${\overline}{\mathsf{M}}(Z')={\mathsf{M}}'(Y)$. Physically, this means that (if the sets $Z$ and $Z'$ are known) one can obtain the probabilities ${\mathrm{tr}\left[\rho{\mathsf{M}}(X)\right]}$ and ${\mathrm{tr}\left[\rho{\mathsf{M}}'(Y)\right]}$ from a measurement of ${\overline}{\mathsf{M}}$ in the state $\rho$. However, there does not necessarily exist a simple ‘rule’ or ‘formula’ from which one could find sets $Z$ and $Z'$ corresponding to $X$ and $Y$. Obviously, if ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable[^9] then they are coexistent but the converse does not hold in general, see [@lyhyt Proposition 1]. Suppose then that ${\mathsf{M}}$ is discrete and rank-1. Then, without restricting generality, we may assume that the outcome space $(\Omega,\Sigma)$ of ${\mathsf{M}}$ is such that $\Omega$ is finite or countably infinite, i.e. $\Omega=\{x_1,x_2,\ldots\}$, $x_i\ne x_j$, $i\ne j$, and $\Sigma$ consists of all subsets of $\Omega$ (i.e. $\Sigma=2^\Omega$). Moreover, for all $i\in\{1,2\ldots\}$, $i<\#\Omega+1$, $${\mathsf{M}}\big(\{x_i\}\big)=|d_i\>\<d_i|\ne 0$$ where vectors $d_i\in{\mathcal{H}}$, $d_i\ne 0$, are such that (weakly) $$\sum_{i=1}^{\#\Omega}|d_i\>\<d_i|={\mathsf{M}}(\Omega)=I_{\mathcal{H}}.$$ Note that ${\mathsf{M}}$ is a PVM if and only if the vectors $d_i$ constitute an orthonormal basis of ${\mathcal{H}}$. We declare that $x_i\in\Omega$ is equivalent with $x_j\in\Omega$, and denote $x_i\sim x_j$, if there exists a $p>0$ such that $|d_i\>\<d_i|=p|d_j\>\<d_j|$. Clearly, $\sim$ is an equivalence relation. Let $$[x_i]:=\{x_j\in\Omega\,|\,x_j\sim x_i\}\subseteq\Omega$$ be the equivalence class of $x_i\in\Omega$ so that $\Omega$ is the disjoint union of the equivalence classes $[x_i]$. Let $\Omega/{\sim}$ be the quotient set of $\Omega$ by $\sim$. Now, for all $x_i\in\Omega$, $${\mathsf{M}}([x_i])=\sum_{j=1\atop x_j\sim x_i}^{\#\Omega}|d_j\>\<d_j|=p_i|d_i\>\<d_i|$$ where $p_i>0$. Let ${\mathsf{M}}'\in{\mathrm{Obs}}(\Sigma',{\mathcal{H}})$ be an arbitrary observable and assume that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent, i.e.  the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong the range of some observable ${\overline}{\mathsf{M}}\in{\mathrm{Obs}}({\overline}\Sigma,{\mathcal{H}})$. For any $[x_i]\in\Omega/{\sim}$, let $Z_{[x_i]}\in{\overline}\Sigma$ be such that ${\overline}{\mathsf{M}}(Z_{[x_i]})={\mathsf{M}}([x_i])$. If $[x_i]\ne[x_j]$ then ${\overline}{\mathsf{M}}(Z_{[x_i]}\cap Z_{[x_j]})\le {\overline}{\mathsf{M}}(Z_{[x_i]})=p_i|d_i\>\<d_i|$ and ${\overline}{\mathsf{M}}(Z_{[x_i]}\cap Z_{[x_j]})\le {\overline}{\mathsf{M}}(Z_{[x_j]})=p_j|d_j\>\<d_j|$ so that ${\overline}{\mathsf{M}}(Z_{[x_i]}\cap Z_{[x_j]})=\tilde p_i|d_i\>\<d_i|=\tilde p_j|d_j\>\<d_j|$ where $\tilde p_i,\,\tilde p_j\ge 0$. If, e.g. $\tilde p_i\ne 0$, then $|d_i\>\<d_i|=(\tilde p_j/\tilde p_i)|d_j\>\<d_j|$ yielding a contradiction. Hence, ${\overline}{\mathsf{M}}(Z_{[x_i]}\cap Z_{[x_j]})=0$ and $${\overline}{\mathsf{M}}\Big(\bigcup_{[x_i]\in{\Omega}/\sim}Z_{[x_i]}\Big)= \sum_{[x_i]\in{\Omega}/\sim}{\overline}{\mathsf{M}}\big(Z_{[x_i]}\big)= \sum_{[x_i]\in{\Omega}/\sim}{\mathsf{M}}\big([x_i]\big)=I_{\mathcal{H}}$$ implying that, for all $Z\in{\overline}\Sigma$, $${\overline}{\mathsf{M}}(Z)={\overline}{\mathsf{M}}\Big(Z\cap\bigcup_{[x_i]\in{\Omega}/\sim}Z_{[x_i]}\Big)= \sum_{[x_i]\in{\Omega}/\sim}{\overline}{\mathsf{M}}\big(Z\cap Z_{[x_i]}\big).$$ Since ${\overline}{\mathsf{M}}\big(Z\cap Z_{[x_i]}\big)\le{\overline}{\mathsf{M}}\big(Z_{[x_i]}\big)=p_i|d_i\>\<d_i|$, $${\overline}{\mathsf{M}}\big(Z\cap Z_{[x_i]}\big)=p([x_i],Z)p_i|d_i\>\<d_i|=p([x_i],Z){\mathsf{M}}\big([x_i]\big)$$ where $p([x_i],Z)\in[0,1]$. Now each mapping $Z\mapsto {\overline}{\mathsf{M}}\big(Z\cap Z_{[x_i]}\big)$ is $\sigma$-additive and, thus, $Z\mapsto p([x_i],Z)$ is a probability measure for any $[x_i]\in\Omega/{\sim}$. Define a mapping $f:\,\Omega\times{\overline}\Sigma\to[0,1]$, $(x_i,Z)\mapsto f(x_i,Z):=p([x_i],Z)$. It is easy to check that $f$ is a Markov kernel with respect to the counting measure[^10] $\#:\,2^\Omega\to[0,\infty]$ and, for all $Z\in{\overline}\Sigma$, $${\overline}{\mathsf{M}}(Z)=\sum_{[x_i]\in{\Omega}/\sim}p([x_i],Z){\mathsf{M}}\big([x_i]\big) =\sum_{[x_i]\in{\Omega}/\sim}p([x_i],Z)\sum_{j=1\atop x_j\sim x_i}^{\#\Omega}|d_j\>\<d_j| =\sum_{k=1}^{\#\Omega}f(x_k,Z)|d_k\>\<d_k|$$ so that ${\overline}{\mathsf{M}}$ is a smearing of ${\mathsf{M}}$. Hence, we have: \[proposition1\] If the range of a discrete rank-1 observable ${\mathsf{M}}$ is contained in the range of an observable ${\overline}{\mathsf{M}}$, then ${\mathsf{M}}$ and ${\overline}{\mathsf{M}}$ are jointly measurable. For each $Y\in\Sigma'$, let $Z_Y\in{\overline}\Sigma$ be such that ${\overline}{\mathsf{M}}(Z_Y)={\mathsf{M}}'(Y)$. Then $$\label{equ} {\mathsf{M}}'(Y)=\sum_{[x_i]\in{\Omega}/\sim}p([x_i],Z_Y){\mathsf{M}}\big([x_i]\big)=\sum_{k=1}^{\#\Omega}f'(x_k,Y)|d_k\>\<d_k|$$ where $f':\,\Omega\times\Sigma'\to[0,1]$ is defined by $f'(x_i,Y):=p([x_i],Z_Y)$. To show that $f'$ is a Markov kernel (with respect to the counting measure $\#:\,2^\Omega\to[0,\infty]$) one is left to check the $\sigma$-additivity of the mappings $Y\mapsto p([x_i],Z_Y)$. Assume that the operators ${\mathsf{M}}([x_i])=p_i|d_i\>\<d_i|$, $[x_i]\in\Omega/{\sim}$, are linearly independent in the sense that, for any (norm) bounded set of real numbers $r_{[x_i]}$, $[x_i]\in\Omega/{\sim}$, the condition $\sum_{[x_i]\in{\Omega}/\sim}r_{[x_i]}{\mathsf{M}}\big([x_i]\big)=0$ implies $r_{[x_i]}=0$ for all $[x_i]\in\Omega/{\sim}$. Then, for any disjoint sequence $\{Y_j\}_{j=1}^\infty\subseteq\Sigma'$, $$0={\mathsf{M}}'\big(\cup_{j=1}^\infty Y_j\big)-\sum_{j=1}^\infty{\mathsf{M}}'(Y_j)=\sum_{[x_i]\in{\Omega}/\sim}\Big\{p([x_i],Z_{\cup_j Y_j})-\sum_{j=1}^\infty p([x_i],Z_{Y_j})\Big\}{\mathsf{M}}\big([x_i]\big)$$ implying that $f'(x_i,\cup_j Y_j)=p([x_i],Z_{\cup_j Y_j})=\sum_{j=1}^\infty p([x_i],Z_{Y_j})=\sum_{j=1}^\infty f'(x_i,Y_j)$ and $Y\mapsto f'(x_i,Y)$ is a probability measure for all $x_i\in\Omega$. Since $f'$ is a Markov kernel, ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$ showing that ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable. Note that ${\mathsf{M}}^\sim:\,2^{\Omega/\sim}\to{\mathcal{L(H)}},\;X'\mapsto \sum_{[x_i]\in X'}{\mathsf{M}}([x_i])$ is a rank-1 observable, a [*relabeling*]{} of ${\mathsf{M}}$ [@HaHePe12]. If ${\mathsf{M}}^\sim$ is an extreme point of ${\mathrm{Obs}}(2^{\Omega/\sim},{\mathcal{H}})$ then the effects ${\mathsf{M}}([x_i])$ are linearly independent in the above sense [@HaHePe12] and ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable. Especially, if ${\mathsf{M}}$ is extreme in ${\mathrm{Obs}}(2^{\Omega},{\mathcal{H}})$ then the operators ${\mathsf{M}}(\{x_i\})$, $i<\#\Omega+1$, are linearly independent. Moreover, $[x_i]=\{x_i\}$ for all $i<\#\Omega+1$. Hence, we have: Let ${\mathsf{M}}$ be an extreme rank-1 discrete observable and ${\mathsf{M}}'$ any observable. Then ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent if and only if they are jointly measurable if and only if ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$. The next examples demonstrate that the extremality requirement (or the linear independence) is needed in the above proof. Consider a two-dimensional Hilbert space ${\mathbb C}^2$ and fix an orthonormal basis ${|0\rangle}:=(1,0)$ and ${|1\rangle}:=(0,1)$. Define unitary operators $U_k={|0 \rangle\langle 0|}+i^k{|1 \rangle\langle 1|}$, $k=1,2,3,4$. Let $|d\>:=\frac12({|0\rangle}+{|1\rangle})$ and ${\mathsf{M}}_k:=U_k|d\>\<d|U_k^*$. Now $\{{\mathsf{M}}_k\}_{k=1}^4$ is linearly dependent set (with exactly 3 linearly independent operators) and $ (2-\epsilon){\mathsf{M}}_1+\epsilon{\mathsf{M}}_2+(2-\epsilon){\mathsf{M}}_3+\epsilon{\mathsf{M}}_4=I_{{\mathbb C}^2} $ for all $\epsilon\in{\mathbb R}$. Clearly, $\big({\mathsf{M}}_1,{\mathsf{M}}_2,{\mathsf{M}}_3,{\mathsf{M}}_4\big)$ constitute a rank-1 POVM ${\mathsf{M}}$. Define a 2-outcome rank-2 POVM ${\mathsf{M}}'=\big({\mathsf{M}}'_1,{\mathsf{M}}'_2\big)=\big(\frac12 I_{{\mathbb C}^2},\frac12 I_{{\mathbb C}^2}\big)$ so that the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong to the range of ${\overline}{\mathsf{M}}={\mathsf{M}}$, that is, ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent. Now equation reduces to ${\mathsf{M}}'_1=\sum_{k=1}^4 p_{k1}{\mathsf{M}}_k$ and ${\mathsf{M}}'_2=\sum_{k=1}^4 p_{k2}{\mathsf{M}}_k$ where the coefficients $p_{kj}\in[0,1]$ are not unique. Indeed, one can write ${\mathsf{M}}'_1={\mathsf{M}}_1+{\mathsf{M}}_3$ and ${\mathsf{M}}'_2={\mathsf{M}}_2+{\mathsf{M}}_4$ showing that [*${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable,*]{} i.e. ${\mathsf{M}}'$ is a smearing of ${\mathsf{M}}$ by a Markov kernel $(p_{kj})$ whose non-zero elements $p_{11}$, $p_{31}$, $p_{22}$ and $p_{42}$ are equal to one. However, if one writes ${\mathsf{M}}'_1={\mathsf{M}}_1+{\mathsf{M}}_3$ and ${\mathsf{M}}'_2={\mathsf{M}}_1+{\mathsf{M}}_3$ then, e.g., $p_{11}=1$ and $p_{12}=1$. In this case, $p_{11}+p_{12}=2\ne1$ so that $(p_{kj})$ is not a Markov kernel. Let ${\mathcal{H}}={\mathbb C}^3$ and let $\{|1\>,\,|2\>,\,|3\>\}$ be its orthonormal basis. Define orthonormal unit vectors $\psi_1:=\big(|1\>+|2\>+|3\>\big)/\sqrt3$, $\psi_2:=\big(|1\>+\alpha|2\>+\alpha^2|3\>\big)/\sqrt3$ and $\psi_3:=\big(|1\>+\alpha^2|2\>+\alpha|3\>\big)/\sqrt3$ where $\alpha:=\exp(2\pi i/3)$ (so that $1+\alpha+\alpha^2=0$ and $\alpha^3=1$). Define a 6-outcome rank-1 POVM[^11] $${\mathsf{M}}=\big({\mathsf{M}}_1,{\mathsf{M}}_2,\ldots,{\mathsf{M}}_6\big)=\left(\frac12{|1 \rangle\langle 1|},\,\frac12{|2 \rangle\langle 2|},\,\frac12{|3 \rangle\langle 3|},\, \frac12{|\psi_1 \rangle\langle \psi_1|},\,\frac12{|\psi_2 \rangle\langle \psi_2|},\,\frac12{|\psi_3 \rangle\langle \psi_3|} \right)$$ and a 3-outcome rank-2 POVM $${\mathsf{M}}'=\big({\mathsf{M}}'_1,{\mathsf{M}}'_2,{\mathsf{M}}'_3\big)=\left( \frac12{|2 \rangle\langle 2|}+\frac12{|3 \rangle\langle 3|},\, \frac12{|1 \rangle\langle 1|}+\frac12{|3 \rangle\langle 3|},\, \frac12{|1 \rangle\langle 1|}+\frac12{|2 \rangle\langle 2|} \right).$$ Since the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong to the range of ${\mathsf{M}}$ ($={\overline}{\mathsf{M}}$) the observables ${\mathsf{M}}$ and ${\mathsf{M}}'$ are coexistent. If ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable then ${\mathsf{M}}'$ must be a smearing of ${\mathsf{M}}$ (Theorem 1) and, hence, of the form $$\label{djwhfvnds} {\mathsf{M}}'_j=\sum_{k=1}^6 p_{kj}{\mathsf{M}}_k,\qquad j=1,\,2,\,3,$$ where $p_{kj}\in[0,1]$ and $\sum_{j=1}^3 p_{kj}=1$. But $$0=\<j|{\mathsf{M}}'_j|j\>=\sum_{k=1}^6 \underbrace{p_{kj}\<j|{\mathsf{M}}_k|j\>}_{\ge\;0},\qquad j=1,\,2,\,3,$$ implying that $p_{kj}\<j|{\mathsf{M}}_k|j\>\equiv 0$. Since, for all $j=1,2,3$ and $k=4,5,6$, $\<j|{\mathsf{M}}_k|j\>=1/6$ (and thus $p_{kj}=0$) equation reduces to ${\mathsf{M}}'_j=\sum_{k=1}^3 p_{kj}{\mathsf{M}}_k$. Now ${\mathsf{M}}_1'={\mathsf{M}}_2+{\mathsf{M}}_3$, ${\mathsf{M}}_2'={\mathsf{M}}_1+{\mathsf{M}}_3$, ${\mathsf{M}}_3'={\mathsf{M}}_1+{\mathsf{M}}_2$, and the operators ${\mathsf{M}}_k$, $k=1,2,3$, are linearly independent so that one must have $p_{11}=0$, $p_{12}=1$ and $p_{13}=1$ yielding a contradiction $\sum_{j=1}^3p_{1j}=2\ne 1$. Hence, [*${\mathsf{M}}$ and ${\mathsf{M}}'$ are not jointly measurable.*]{} In literature, there exist two significant classes of observables for which coexistence and joint measurability are known to be equivalent (see, e.g. [@BuKiLa]): Let ${\mathsf{M}}\in{\mathrm{Obs}}(\Sigma,{\mathcal{H}})$ and ${\mathsf{M}}'\in{\mathrm{Obs}}(\Sigma',{\mathcal{H}})$ be coexistent and ${\overline}{\mathsf{M}}\in{\mathrm{Obs}}({\overline}\Sigma,{\mathcal{H}})$ be such that the ranges of ${\mathsf{M}}$ and ${\mathsf{M}}'$ belong to the range of ${\overline}{\mathsf{M}}$. Then ${\mathsf{M}}$ and ${\mathsf{M}}'$ are jointly measurable if[^12] 1. ${\mathsf{M}}$ (or ${\mathsf{M}}'$) is projection valued, or 2. ${\overline}{\mathsf{M}}$ is regular, that is, for any $Z\in{\overline}\Sigma$ such that $0\ne{\overline}{\mathsf{M}}(Z)\ne I_{\mathcal{H}}$ one has ${\overline}{\mathsf{M}}(Z)\not\le\frac12 I_{\mathcal{H}}$ and ${\overline}{\mathsf{M}}(Z)\not\ge\frac12 I_{\mathcal{H}}$. Note that (2) implies that also ${\mathsf{M}}$ and ${\mathsf{M}}'$ are regular but (2) does not imply (1). In both cases, one need not assume that ${\mathsf{M}}$ (or ${\mathsf{M}}'$) is discrete or rank-1. However, if ${\mathsf{M}}$ is projection valued then it is automatically extreme. Moreover, any rank-1 effect is of the form $p|d\>\<d|$ where $d\in{\mathcal{H}}$ is a unit vector and $p\in(0,1]$. It is regular[^13] (respectively, a projection) if and only if $p>\frac12$ (resp. $p=1$). Suppose then that $\{|0\>,\,|1\>\}$ is an orthonormal basis of ${\mathcal{H}}={\mathbb C}^2$ and ${\mathsf{M}}=\big({\mathsf{M}}_1,{\mathsf{M}}_2,{\mathsf{M}}_3\big)=\big({|0 \rangle\langle 0|},\,0.1{|1 \rangle\langle 1|},\,0.9{|1 \rangle\langle 1|}\big)$ which is not projection valued or regular. Since ${\mathsf{M}}$ has a projection valued (especially, extreme) relabeling ${\mathsf{M}}^\sim=\big({|0 \rangle\langle 0|},\,{|1 \rangle\langle 1|}\big)$ it follows that ${\mathsf{M}}$ and an [*arbitrary*]{} ${\mathsf{M}}'$ are jointly measurable if and only if they are coexistent. Discussion ========== It is shown in [@Part2] that any observable ${\mathsf{M}}$ can be maximally refined into a rank-1 observable ${\mathsf{M}}_1$ whose value space ‘contains’ also the multiplicities of the measurement outcomes of ${\mathsf{M}}$. We called a measurement of ${\mathsf{M}}_1$ as a [*complete*]{} measurement of ${\mathsf{M}}$ since (a) it gives information on the multiplicities of the outcomes, (b) it can be seen as a preparation of a new measurement, and (c) it breaks entanglement between the system and its environment [@Pell; @Pell']. Moreover, ${\mathsf{M}}_1$ can be measured by performing a sequential measurement of ${\mathsf{M}}$ and some discrete ‘multiplicity’ observable [@Pell]. 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J.-P. Pellonpää, “Complete quantum measurements break entanglement”, Phys. Lett. A [**376**]{}, 3495 (2012) J.-P. Pellonpää, “Complete measurements of quantum observables”, Found. Phys., in press, [DOI: 10.1007/s10701-013-9764-y]{}. D. Reeb, D. Reitzner, and M. M. Wolf, “Coexistence does not imply joint measurability”, J. Phys. A: Math. Theor. [**46**]{}, 462002 (2013). [^1]: That is, ${\mathcal{H}}$ has a finite $(d<\infty)$ or countably infinite basis $(d=\infty)$. If $d<\infty$ then ${\mathcal{H}}\cong{\mathbb C}^d$ and ${\mathcal{L(H)}}$ (resp. ${\mathcal{S(H)}}$) can be identified with the set of all $d\times d$–complex matrices (resp. density matrices). [^2]: Usually in physics, $\Omega$ is finite (or countably infinite) set or a manifold (e.g. ${\mathbb R}^n$) when $\Sigma$ contains e.g. open sets (the Borel $\sigma$-algebra of a manifold). [^3]: Or any $\sigma$-finite positive measure on $\Sigma$ such that $\mu$ and ${\mathsf{M}}$ are mutually absolutely continuous, that is, for all $X\in\Sigma$, $\mu(X)=0$ if and only if ${\mathsf{M}}(X)=0$. [^4]: Recall that if one says that some condition holds ‘for $\mu$-almost all $x\in\Omega$’ this means that the condition holds for all $x\in\Omega\setminus O$ where $O$ is a $\mu$-null set, i.e. a set of $\mu$-measure zero. [^5]: A counting measure counts the number of the elements of a (sub)set, i.e. $\mu(X)=\# X$ (the number of the elements of the set $X$). [^6]: That is, for any (suitable) wave function $\psi:\,{\mathbb R}\to{\mathbb C}$ one has $(Q\psi)(x)=x\psi(x)$ and $\big({\mathsf{Q}}(X)\psi\big)(x)={\ensuremath{ \chi\raisebox{-1ex}{$\scriptstyle X$} }}(x)\psi(x)$, i.e. $Q=\int_{\mathbb R}x\,\d{\mathsf{Q}}(x)$. [^7]: Hence, one can identify ${\mathsf{M}}$ with ${\mathsf{P}}$, i.e. diagonalize ${\mathsf{M}}$. If ${\mathsf{M}}$ is the spectral measure of a self-adjoint operator $S$ then ${\mathsf{P}}$ can be found by solving the eigenvalue equation of $S$. Now $m(x)$ is the usual multiplicity of the eigenvalue $x\in{\mathbb R}$. [^8]: That is, $\Sigma\otimes\Sigma'$ is the smallest $\sigma$-algebra over ${\Omega}\times{\Omega}'$ which contains the ‘rectangles’ $X\times Y$ where $X\in\Sigma$ and $Y\in\Sigma'$. [^9]: Now simply $Z=X\times{\Omega}'$ and $Z'={\Omega}\times Y$. [^10]: That is, $\#X$ is the number of the elements of $X\subseteq{\Omega}$. [^11]: Note that ${\mathsf{M}}$ is not extreme since ${\mathsf{M}}_1+{\mathsf{M}}_2+{\mathsf{M}}_3=\frac12I_{{\mathbb C}^3}={\mathsf{M}}_4+{\mathsf{M}}_5+{\mathsf{M}}_6$. Also, ${\mathsf{M}}'$ is not extreme. [^12]: Moreover, one must assume that the measurable spaces $({\Omega},\Sigma)$, $({\Omega}',\Sigma')$ and $({\overline}{\Omega},{\overline}\Sigma)$ are regular enough, e.g. locally compact metrizable and separable topological spaces equipped with their Borel $\sigma$-algebras. [^13]: If $\dim{\mathcal{H}}>1$.
--- abstract: 'We present the first results of a series of Monte-Carlo simulations investigating the imprint of a central black hole on the core structure of a globular cluster. We investigate the three-dimensional and the projected density profile of the inner regions of idealized as well as more realistic globular cluster models, taking into account a stellar mass spectrum, stellar evolution and allowing for a larger, more realistic, number of stars than was previously possible with direct N-body methods. We compare our results to other N-body simulations published previously in the literature.' --- Introduction ============ As recently as 10 years ago, it was generally believed that black holes (BHs) occur in two broad mass ranges: stellar ($M\sim 3-20 M_{\odot}$), which are produced by the core collapse of massive stars, and supermassive ($M\sim 10^6 - 10^{10} M_{\odot}$), which are believed to have formed in the center of galaxies at high redshift and grown in mass as the result of galaxy mergers (see e.g. Volonteri, Haardt & Madau 2003). However, the existence of BHs with masses intermediate between those in the center of galaxies and stellar BHs could not be established by observations up until recently, although intermediate mass BHs (IMBHs) were predicted by theory more than 30 years ago; see, e.g., Wyller (1970). Indirect evidence for IMBHs has accumulated over time from observations of so-called ultraluminous X-ray sources (ULXs), objects with fluxes that exceed the angle-averaged flux of a stellar mass BH accreting at the Eddington limit. An interesting result from observations of ULXs is that many, if not most, of them are associated with star clusters. It has long been speculated (e.g., Frank & Rees 1976) that the centers of globular clusters (GCs) may harbor BHs with masses $\sim 10^3 \rm{M_{\odot}}$. If so, these BHs affect the distribution function of the stars, producing velocity and density cusps. A recent study by Noyola & Gebhardt (2006) obtained central surface brightness profiles for 38 Galactic GCs from HST WFPC2 images. They showed that half of the GCs in their sample have slopes for the inner 0.5" surface density brightness profiles that are inconsistent with simple isothermal cores, which may be indicative of an IMBH. However, it is challenging to explain the full range of slopes with current models. While analytical models can only explain the steepest slopes in their sample, recent N-body models of GCs containing IMBHs (Baumgardt et al. 2005), might explain some of the intermediate surface brightness slopes. In our study we repeat some of the previous N-body simulations of GCs with central IMBHs but using the Monte-Carlo (MC) method. This gives us the advantage to model the evolution of GCs with a larger and thus more realistic number of stars. We then compare the obtained surface brightness profiles with previous results in the literature. Imprints of IMBHs {#ImprintsIMBH} ================= The dynamical effect of an IMBH on the surrounding stellar system was first described by Peebles (1972), who argued that the bound stars in the cusp around the BH must obey a shallow power-law density distribution to account for stellar consumption near the cluster center. Analyzing the Fokker-Planck equation in energy space for an isotropic stellar distribution, Bahcall & Wolf (1976) obtained a density profile with $n(r)\propto r^{-7/4}$, which is now commonly referred to as the Bahcall-Wolf cusp. The formation of this cusp has been confirmed subsequently by many different studies using different techniques and also, more recently, by direct $N$-body methods (Baumgardt et al. 2004). In Fig. \[Imprints\] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![The imprint of an IMBH on the stellar distribution of a GC. On the left-hand side, the radial number density profile of an evolved single-mass GC is shown (solid) together with a power-law fit to its inner region (dashed line). The right hand side shows its velocity dispersion profile (solid) and the Keplerian velocity profile of the IMBH. The dotted line marks its radius of influence.[]{data-label="Imprints"}](Umbreit_fig1a.eps "fig:"){width="45.00000%"} ![The imprint of an IMBH on the stellar distribution of a GC. On the left-hand side, the radial number density profile of an evolved single-mass GC is shown (solid) together with a power-law fit to its inner region (dashed line). The right hand side shows its velocity dispersion profile (solid) and the Keplerian velocity profile of the IMBH. The dotted line marks its radius of influence.[]{data-label="Imprints"}](Umbreit_fig1b.eps "fig:"){width="33.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- such a profile from one of our simulations is shown (for initial cluster parameters see Baumgard et al. (2004) (run16)). As can be clearly seen, the density profile of the inner region of the evolved cluster can be very well fitted by a power-law and the power-law slope $\alpha$ we obtain is, with $\alpha=1.72$, in good agreement with the value found by Bahcall & Wolf (1976). Also the extent of the cusp profile is given by the radius where the Keplerian velocity of a star around the central BH equals the velocity dispersion of the cluster core, the radius of influence of the IMBH. However, Baumgardt et al. (2005) found that such a cusp in density might not be easily detectable in a real star cluster, as it should be much shallower and difficult to distinguish from a standard King profile. They find that this is mainly an effect of mass segregation and stellar evolution, where the more massive dark stellar remnants are concentrated towards the center while the lower-mass main sequence stars that contribute most of the light are much less centrally concentrated. In their simulations they found power-law surface brightness slopes ranging from $\alpha=-0.1$ to $\alpha=-0.3$. Based on these results they identified 9 candidate clusters from the sample of galactic GCs of Noyola & Gebhardt (2006) that might contain IMBHs. However, the disadvantage of current N-body simulations is that for realistic cluster models, that take into account stellar evolution and a realistic mass spectrum, the number of stars is restricted to typically less than $2\times10^{5}$ as these simulations require a large amount of computing time. However, many GCs are known to be very massive, with masses reaching up to $1\times10^6 \rm{M_\odot}$ resulting in a much larger number of stars one has to deal with when modelling these objects. In previous N-body simulations, such large-N clusters have been scaled down to low-N systems. Scaling down can be achieved in two ways (e.g. Baumgardt et al. 2005): either the mass of the central IMBH $M_{BH}$ is kept constant and $N$ is decreased, effectively decreasing the total cluster mass $M_{Cl}$, or the ratio $M_{BH}/M_{Cl}$ is kept constant, while lowering both $M_{BH}$ and $M_{Cl}$. As both $M_{BH}/M_{Cl}$ and the ratio of $M_{BH}$ to stellar mass are important parameters that influence the structure of a cluster, but cannot be held constant simultaneously when lowering $N$, it is clear that only with the real $N$ a fully self-consistent simulation can be achieved. One such method that can evolve such large-$N$ systems for a sufficiently long time is the MC method. Monte-Carlo Method with IMBH ============================ The MC method shares some important properties with direct $N$-body methods, which is why it is also regarded as a randomized $N$-body scheme (see e.g. Freitag & Benz 2001). Just as direct $N$-body methods, it relies on a star-by-star description of the GC, which makes it particularly straightforward to add additional physical processes such as stellar evolution. Contrary to direct $N$-body methods, however, the stellar orbits are resolved on a relaxation time scale $T_{rel}$, which is much larger than the crossing time $t_{cr}$, the time scale on which direct $N$-body methods resolve those orbits. This change in orbital resolution is the reason why the MC method is able to evolve a GC much more efficiently than direct $N$-body methods. It achieves this efficiency, however, by making several simplifying assumptions: (i) the cluster potential has spherical symmetry (ii) the cluster is in dynamical equilibrium at all times (iii) the evolution is driven by diffusive 2-body relaxation. The specific implementation we use for our study is the MC code initially developed by Joshi et al. (2000) and further enhanced and improved by Fregau et al. (2003) and Fregau & Rasio (2007). The code is based on Hénon’s algorithm for solving the Fokker-Planck equation. It incorporates treatments of mass spectra, stellar evolution, primordial binaries, and the influence of a galactic tidal field. The effect of an IMBH on the stellar distribution is implemented in a manner similar to Freitag & Benz (2002). In this method the IMBH is treated as a fixed, central point mass while stars are tidally disrupted and accreted onto the IMBH whenever their periastron distances lie within the tidal radius, $R_{disr}$, of the IMBH. For a given star-IMBH distance, the velocity vectors that lead to such orbits form a so called loss-cone and stars are removed from the system and their masses are added to the BH as soon as their velocity vectors enter this region. However, as the star’s removal happens on an orbital time-scale one would need to use time-steps as short as the orbital period of the star in order to treat the loss-cone effects in the most accurate fashion. This would, however, slow down the whole calculation considerably. Instead, during one MC time-step a star’s orbital evolution is followed by simulating the random-walk of its velocity vector, which approximates the effect of relaxation on the much shorter orbital time-scale. After each random-walk step the star is checked for entry into the loss-cone. For further details see Freitag & Benz (2002). Comparison with $N$-body calculations have shown that in order to achieve acceptable agreement between the two methods, the MC time-step must be chosen rather small relative to the local relaxation time, with $dt \leq 0.01 T_{rel}(r)$ (Freitag et al. 2006). While choosing such a small time-step was still feasible in the code of Freitag & Benz (2002), to enforce such a criterion for all stars in our simulation would lead to a dramatic slow-down of our code and notable spurious relaxation. The reason is that our code uses a shared time step scheme, with the time-step chosen to be the smallest value of all $dt_i=f~T_{rel}(r_i)$, where $f$ is some constant fraction and the subscript $i$ refers to the individual star. In Freitag & Benz (2002) each star is evolved separately according to its local relaxation time, allowing for larger time steps for stars farther out in the cluster where the relaxation times are longer. In order to reduce the effect of spurious relaxation we are forced to choose a larger $f$, typically around $f=0.1$. This has the consequence that the time-step criterion is only strictly fulfilled for stars typically outside of $0.1 r_h$, where $r_h$ is the half-mass radius of the cluster. To arrive at the correct merger rate of stars with the IMBH, despite the larger time step for the stars in the inner region, we apply the following procedure: (i) for each star $i$ with $dt> 0.01 T_{rel}(r_i)$ we take $n= dt/(0.01T_{rel})$ sub-steps. (ii) during each of these sub-steps we carry out the random-walk procedure as in Freitag & Benz (2002) (iii) after each sub-step we calculate the star’s angular-momentum $J$ according to the new velocity vector (iv) we generate a new radial position according to the new J. By updating $J$ after each sub-step we approximately account for the star’s orbital diffusion in $J$ space during a full MC step, while neglecting any changes in orbital energy. This is, at least for stars with low $J$, legitimate (Shapiro & Marchant 1978), while for the other stars the error might not be significant as the orbital energy diffusion is still slower than the $J$ diffusion (Frank & Rees 1976). A further assumption is that the cluster potential in the inner cluster region does not change significantly during a full MC step, which constrains the MC step size. Comparison to N-body Simulations ================================ In Fig. \[mergerRates\] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Comparison of the stellar merger rate per crossing time with simulations of Baumgard et al. (2004). In both cases the evolution of single-mass clusters were calculated and the tidal radius of the IMBH was fixed to $1\times10^{-7}$ in N-body units. Full circles with error bars are results from our MC runs at selected times, while the solid line goes through all obtained points.[]{data-label="mergerRates"}](Umbreit_fig2a.eps "fig:"){width="38.00000%"} ![Comparison of the stellar merger rate per crossing time with simulations of Baumgard et al. (2004). In both cases the evolution of single-mass clusters were calculated and the tidal radius of the IMBH was fixed to $1\times10^{-7}$ in N-body units. Full circles with error bars are results from our MC runs at selected times, while the solid line goes through all obtained points.[]{data-label="mergerRates"}](Umbreit_fig2b.eps "fig:"){width="34.50000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- the rates of stellar mergers with the central BH per crossing time from two of our single-mass cluster simulations are compared to the corresponding results of Baumgardt et al. (2004) (run16 and run2). As can bee seen, the differences between our MC and the $N$-body results are within the respective error bars and thus in reasonable agreement with each other. However, the merger rates in the left panel of Fig. \[mergerRates\] seem to be consistently lower than in the $N$-body calculations. This might indicate that the agreement gets worse for other $M_{BH}/M_{Cl}$ than we considered here ($0.25\% -1\%$) and a different choice of time-step parameters for our MC code might be necessary in those cases. On the other hand, the differences might also be caused by differences in the initial relaxation phase before the cluster reaches an equilibrium state. This phase cannot be adequately modeled with a MC code because the code assumes dynamical equilibrium. Further comparisons to $N$-body simulations for different $M_{BH}/M_{Cl}$ and $N$ are necessary to test the validity of our method. Realistic Cluster Models ======================== In order to compare our simulations to observed GCs additional physical processes need to be included. Here we consider two clusters containing $1.3\times10^5$ and $2.6\times10^5$ stars with a Kroupa-mass-function (Kroupa 2001), and follow the evolution of the single stars with the code of Belczynski et al. (2002) (for all other parameter see Baumgardt et al. 2005). Fig. \[surfaceDensity\] ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ![Two-dimensional density profile of bright stars for two clusters with different numbers of stars and BH to cluster mass ratios. The dashed line in the right panel is a power-law fit to the inner region of the cluster, while the two dashed lines in the left panel are for orientation only.[]{data-label="surfaceDensity"}](Umbreit_fig3a.eps "fig:"){width="38.50000%"} ![Two-dimensional density profile of bright stars for two clusters with different numbers of stars and BH to cluster mass ratios. The dashed line in the right panel is a power-law fit to the inner region of the cluster, while the two dashed lines in the left panel are for orientation only.[]{data-label="surfaceDensity"}](Umbreit_fig3b.eps "fig:"){width="38.50000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- shows the two-dimensional density profiles of bright stars for the two clusters at an age of $12\rm{Gyr}$. The profile in the left panel can directly be compared to the corresponding result of Baumgardt et. al (2005) as $N$ is the same. As was expected from the discussion in §\[ImprintsIMBH\], the profile shows only a very shallow cusp with a power-law slope $\alpha$ between $-0.2$ and $-0.3$, consistent with the $N$-body results. The right panel shows the resulting profile for a cluster that is similar but twice as massive and, consequently, has twice as many stars as in the $N$-body simulation. We obtain a very similar profile with $\alpha=-0.23$ which is very close to the average $\alpha=-0.25$ found in Baumgardt et al. (2005). Therefore, based on these very preliminary results, there seems to be no significant difference in cusp slopes for larger-$N$ clusters compared to small-$N$ ones, which means that no new candidate cluster from the sample of Noyola & Gebhardt (2006) can be identified in addition to those found by Baumgardt et al. (2005). However, the parameter space must be explored much further in order to confirm this finding. This work was supported by NSF Grant AST-0607498 at Northwestern University. Bahcall, J. N., & Wolf, R. A. 1976, [*ApJ*]{}, 209, 214 Baumgardt, H., Makino, J., & Hut, P. 2005, [*ApJ*]{}, 620, 238 Baumgardt, H., Makino, J., & Ebisuzaki, T. 2004, [*ApJ*]{}, 613, 1133 Belczynski, K., Bulik, T., & Klu[ź]{}niak, W. [ł]{}. 2002, [*ApJ* (Letters)]{}, 567, L63 Frank, J., & Rees, M. J. 1976, [*MNRAS*]{}, 176, 633 Fregeau, J. M., & Rasio, F. A. 2007, [*ApJ*]{}, 658, 1047 Fregeau, J. M., G[ü]{}rkan, M. A., Joshi, K. J., & Rasio, F. A. 2003, [*ApJ*]{}, 593, 772 Freitag, M., & Benz, W. 2001, [*A&A*]{}, 375, 711 Freitag, M., & Benz, W. 2002, [*A&A*]{}, 394, 345 Freitag, M., Amaro-Seoane, P., & Kalogera, V. 2006, [*ApJ*]{}, 649, 91 Joshi, K. J., Rasio, F. A., & Portegies Zwart, S. 2000, [*ApJ*]{}, 540, 969 Kroupa, P. 2001, [*MNRAS*]{}, 322, 231 Noyola, E., & Gebhardt, K. 2006, [*AJ*]{}, 132, 447 Peebles, P. J. E. 1972, [*ApJ*]{}, 178, 371 Shapiro, S. L., & Marchant, A. B. 1978, [*ApJ*]{}, 225, 603 Volonteri, M., Haardt, F., & Madau, P. 2003, [*ApJ*]{}, 582, 559 Wyller, A. A. 1970, [*ApJ*]{}, 160, 443
--- author: - Zahra Eslami - Piotr Ryczkowski - Lauri Salmela - Goëry Genty bibliography: - 'low\_noise.bib' title: 'Low-noise octave-spanning mid-infrared supercontinuum generation in a multimode chalcogenide fiber' --- Supercontinuum (SC) generation in the mid-infrared has attracted significant interest over the past decade due to its wide range of potential applications from tissue imaging [@petersen2018mid; @seddon2013potential; @dupont2012ir] and spectroscopy [@islam2009mid; @eggleton2011chalcogenide] to defense and security [@kumar2012stand]. Supercontinuum sources are generally optimized in terms of spectral coverage or power for a particular application, but their noise characteristics are also of particular significance as large pulse-to-pulse fluctuations can severely degrade the performance and reduce the achievable resolution and contrast e.g. in sensing or imaging [@dupont2014ultra; @gonzalo2018ultra; @jensen2019noise]. Pulse-to-pulse instabilities in SC generation arise from nonlinear amplification dynamics of the input pulse fluctuations. These dynamics depend on various parameters including the pump pulse wavelength, energy, and duration [@newbury2003noise; @gaeta2002nonlinear; @moller2012power], such that by carefully selecting the pump pulse characteristics one can in principle reduce the SC fluctuations [@dudley2002coherence]. More specifically, when pumping in the anomalous dispersion regime, intensity (and phase) fluctuations can be minimised using short pump pulses with femtosecond duration to seed the initial spectral broadening dynamics from the pump spectral components. In the case of longer pump pulses (typically few hundred femtosecond and beyond), the broadening mechanism is triggered by the noise present on top of the input pulse, which leads to significant shot-to fluctuations and generally high intensity noise. The generation of a broadband supercontinuum in the mid-infrared requires using fibers made of soft glass with attenuation much lower than that of silica-based fibers, and in recent years there have been many studies of SC generation in fluoride [@li2019step], chalcogenide [@diouf2017super; @wang20171; @diouf2019numerical; @liu2016coherent], multi-compound glasses [@jiao2019mid; @nguyen2018highly; @zhang2019ultrabroadband] and other type of soft glass fibers [@klimczak2014coherent]. Most of these studies, however, use single-mode photonics crystal fibers or tapered fibers with small core size to enhance the nonlinearity and engineer the dispersion properties, thereby limiting the injected power due to the low damage threshold of soft glasses. For mid-infrared applications where high power is especially crucial due to the lack of sensitive detectors, the possibility to generate a broadband SC in multimode fibers with larger core enabling to inject more power has recently attracted attention [@eslami2019high; @swiderski2014high; @zhang2016broadband; @shi2016multi; @khalifa2016mid]. These studies typically use anomalous dispersion pumping as this particular regime has been shown to yield the broadest SC spectra [@dudley2006supercontinuum; @genty2007fiber]. However, this regime is also particularly sensitive to noise amplification dynamics, leading to large shot-to-shot intensity fluctuations [@dudley2006supercontinuum; @genty2007fiber; @klimczak2016direct] and the development of high-power broadband mid-infrared SC sources with low noise is thus still challenging. Here, we report the generation of a low noise, octave-spanning, mid-infrared supercontinuum in a multimode chalcogenide fiber using an all-normal dispersion pumping scheme. The all-normal pumping approach has the advantage to be less sensitive to noise amplification dynamics than the anomalous dispersion regime as the spectral broadening mechanism in this case arises mainly from self-phase modulation (SPM) allowing to preserve high stability even with few hundred femtosecond pump pulses [@dudley2006supercontinuum; @genty2007fiber; @hooper2011coherent]. We perform a systematic study of the noise performance by measuring the pulse-to-pulse intensity fluctuations in different wavelength bands across the SC spectrum and we compare the results with the noise characteristics measured for an octave-spanning supercontinuum generated in the anomalous dispersion regime of a multimode fluoride fiber with similar core size and length. Our study shows that pumping in the normal dispersion regime leads to the generation of a SC with significantly reduced noise level and opens novel perspectives for the generation of high-power broadband mid-infrared SC sources for applications such as remote sensing where the stability and power are more important than the beam profile. ![(a) Experimental setup for mid-infrared (MIR) SC generation and intensity noise measurement. OPA: optical parametric amplifier. L1-L4: plano-convex lenses. F1-F2: long-pass filters C: chopper. M: monochromator. S/I: signal/idler isolator. LA:  lock-in amplifier. PD: photodetector. MMF: multimode fiber. Intensity fluctuations measured over 5000 pulses corresponding to the OPA pulses at 1700 nm (b) and at 3500 nm (c).[]{data-label="fig:1"}](setup.png){width="\linewidth"} ![Average SC spectrum generated in a 1-m long multimode As$_2$S$_3$ fiber with 100 $\mu$m core diameter. The dashed line marks the spectral regions measured using an OSA and monochromator. The colored areas indicate the filtered wavelength bands where the intensity fluctuations are measured.[]{data-label="fig:2"}](chalcogenidespec.png){width="\linewidth"} The experimental setup is shown in Fig. \[fig:1\]. Using an anti-reflection (AR) coated Si plano–convex lens with 10 cm focal length, we inject 350 fs pulses (FWHM) from an optical parametric amplifier (OPA) at a repetition of 1 MHz into a 1-m long step-index As$_2$S$_3$ multimode fiber (IRflex, IRF-S-100) with numerical aperture and diameter of NA=0.30 and 100 $\mu$m, respectively. The OPA wavelength is tuned to 3500 nm which is located in the normal dispersion regime of the fiber. The SC spectrum is measured with an optical spectrum analyzer (OSA, ANDO AQ6317B) for wavelengths in the 1000–1750 nm range and the combination of a monochromator (DK 480) and lock-in detection for wavelengths beyond 1700 nm. A maximum throughput of 60% (calculated as a ratio of output to input power, and including coupling efficiency and attenuation along the fiber) was measured for our setup. Considering that the attenuation of the chalcogenide fiber is relatively low (about 0.05 dB/m at 2800 nm), the throughput is essentially limited by the Fresnel reflection losses due to the high refractive index contrast between the chalcogenide glass (n=2.4) and air. In order to characterize the SC pulse-to-pulse intensity fluctuations, the same monochromator was used to select wavelength-channels in narrow spectral bands of 6 nm bandwidth (experimentally measured during the calibration process of the monochromator). Light from the monochromator output was focused into a photodetector using MgF$_2$ plano–convex lens with 5 cm focal length and the electrical signal was recorded with a 9 MHz photodetector (InAsSb; PDA07P2) and 1 GHz real-time oscilloscope (LeCroy WaveRunner 6100A). Note that lock-in detection is removed for the intensity noise characterization to allow for shot-to-shot intensity measurements. Figure \[fig:2\] shows the normalized spectrum measured at the output of the 1-m long multimode As$_2$S$_3$ fiber for a pump pulse peak power of 380 kW. The SC spectrum corresponds to an average output power of 145 mW and spans from 1700 nm to 4800 nm (-30 dB bandwidth) with relatively flat spectrum (&lt;10 dB variation) in the 2200–4500 nm range. The dip in the spectrum at around 4200 nm is caused by the fiber glass attenuation at this particular wavelength. ![Numerically simulated (a) dispersion and (b) mode profile of fundamental along with selected higher-order modes. Indices represent the mode number and vertical black lines mark the ZDW of the modes.[]{data-label="fig:3"}](dispersion.png){width="\linewidth"} We performed numerical simulations of the dispersion profile associated with the fundamental and selected higher-order modes of the multimode As$_2$S$_3$ fiber. As can be seen in Fig. \[fig:3\], the zero-dispersion wavelength of the fundamental mode is located at 5000 nm (close to that of the bulk glass) while that of higher-order modes decreases towards shorter wavelength with the mode order. Although we only show here selected the higher-order modes dispersion profiles, even for a higher mode order the zero-dispersion wavelength is still located above 3500 nm such that the SC is essentially generated in the normal dispersion regime with the dominant spectral broadening mechanism being self-phase modulation. This regime is less susceptible to noise amplification compared to anomalous dispersion pumping as modulation instability and bright soliton dynamics do not occur in normal dispersion and this is indeed reflected in the spectral histogram of the intensity fluctuations shown in Fig. \[fig:4\] measured in different wavelength bands for 5000 consecutive pulses. ![Intensity fluctuations of the SC generated in the multimode As$_2$S$_3$ fiber measured for 5000 consecutive pulses at different wavelengths as indicated by the corresponding colors in Fig. \[fig:2\]. CV: coefficient of variation defined as the ratio of the standard deviation to the mean intensity.[]{data-label="fig:4"}](chalcnoise.png){width="\linewidth"} The fluctuations can be quantified by the coefficient of variation (CV) defined as the ratio of the intensity standard deviation to the mean intensity. One can see that the fluctuations in the SC spectrum are minimal in the central part from 3000 to 3500 nm with an increase by only a factor of 2 (CV=0.043 at 2800 nm) and 3 (CV=0.063 at 4000 nm) toward the short- and long-wavelength edges, respectively, when compared to the intensity fluctuations of the pump pulses (CV=0.022, see inset in Fig. \[fig:1\]). Interestingly, we also note that the noise level in the vicinity of the pump wavelength is lower than that of the input pump pulses. This is because in SPM-based SC generation, the spectral intensity in the vicinity of the pump wavelength varies proportionally to the pump pulse peak power and pulse duration. In the case of a mode-locked pump laser (such as the OPA used here), the peak power and duration are anti-correlated (i.e. an increase in one causes a decrease in another and vice-versa) such that during SC generation process the noise resulting from variations in the pump pulse duration and pump pulse peak power tend to cancel each other [@genier2019amplitude]. In order to assess further the noise characteristics of the multimode SC generated in the all-normal dispersion regime against other pumping scheme, we performed a set of additional experiments in a 1-m long InF$_3$ multimode fiber using the same OPA source. The fiber has a core diameter of 100 $\mu$m similar to the As$_2$S$_3$ fiber, and a numerical aperture of NA=0.26. The OPA was tuned to 1700 nm, which is located in the normal dispersion of the fundamental mode of the InF$_3$ fiber (ZDW at around 1850 nm) but, unlike for the As$_2$S$_3$ fiber, corresponds to the anomalous dispersion regime for a large number of higher-order modes [@eslami2019high]. Figure \[fig:5\] Shows the resulting SC spectrum extending from 1200 nm to 2400 nm (-40 dB bandwidth) for an input peak power of 1.4 MW. ![Average SC spectrum generated in 1-m long multimode InF$_3$ fiber with 100 $\mu$m core diameter. The dashed line marks the spectral regions measured using two different OSA. The colored areas indicate the filtered wavelength bands where the intensity fluctuations are measured.[]{data-label="fig:5"}](flouridespec.png){width="\linewidth"} ![Intensity fluctuations of the SC generated in the multimode InF$_3$ fiber measured for 5000 consecutive pulses at different wavelengths as indicated by the corresponding colors in Fig. \[fig:5\]. CV: coefficient of variation defined as the ratio of the standard deviation to the mean intensity.[]{data-label="fig:6"}](flournoise.png){width="\linewidth"} The intensity fluctuations of the SC were quantified in different output wavelengths using a 15 MHz photodetector (PbSe; PDA10D-EC) and the results are plotted in Fig. \[fig:6\]. One can see that the intensity noise in the SC spectrum is lowest in the vicinity of the pump wavelength and increases significantly toward both the short- and long-wavelength edges of the SC. The coefficient of variation is maximum at 2100 nm (CV=0.246) which is more than an order of magnitude compared to that of the pump pulses (CV=0.010, see inset in Fig. \[fig:1\]), which is in marked contrast with the all-normal dispersion regime SC. This can be explained from the fact that the spectral components near the pump wavelength are mostly generated by SPM but as energy is transferred to the anomalous dispersion regime of the higher-order modes the long and short-wavelength SC spectral components are generated through soliton dynamics and phase-matched dispersive waves, respectively [@eslami2019high], amplifying significantly the input pulses fluctuations and leading to the observed large intensity variations. In conclusion, we have demonstrated the generation of a low-noise, octave-spanning mid-infrared supercontinuum in a 1-m long multimode chalcogenide fiber with 100 $\mu$m core diameter by injecting 350 fs pulses from an OPA in the normal dispersion regime of the fiber. Systematic measurements of the pulse-to-pulse fluctuations in different wavelength bands show that the noise of the input pulses is at most amplified by factor of 3. Furthermore, comparison with an octave-spanning SC generated partially in the anomalous regime of a 1-m long multimode fluoride showed that the all-normal pumping scheme using few hundred femtosecond pulses is a promising approach to generate a high-power broadband SC source for noise-sensitive applications such as remote sensing or imaging. **[Funding.]{}** Horizon (2020) Framework Programme (722380).
--- abstract: 'This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of $1/h$, where $h$ is the sampling period. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves as $O(h^2)$, and in some cases as $O(h^4)$, which outperforms the state-of-the-art error bound of $O(h)$ for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.' bibliography: - 'PaperCollection2.bib' title: 'A Class of Prediction-Correction Methods for Time-Varying Convex Optimization' --- Time-varying optimization, non-stationary optimization, parametric programming, prediction-correction methods.
--- bibliography: - 'cdc\_ref.bib' - 'cdc\_epi.bib' --- [**[A Nonseparable Multivariate Space-Time Model for Analyzing County-Level Heart Disease Death Rates by Race and Gender]{}**]{} **Harrison Quick$^{*1}$, Lance A. Waller$^{2}$, and Michele Casper$^1$**\ $^{1}$ Division of Heart Disease and Stroke Prevention, Centers for Disease Control and Prevention, Atlanta, GA 30329\ $^2$ Department of Biostatistics, Emory University, 1518 Clifton Rd NE, Atlanta, GA 30322\ $^{*}$ *email:* [email protected] <span style="font-variant:small-caps;">Summary.</span> While death rates due to diseases of the heart have experienced a sharp decline over the past 50 years, these diseases continue to be the leading cause of death in the United States, and the rate of decline varies by geographic location, race, and gender. We look to harness the power of hierarchical Bayesian methods to obtain a clearer picture of the declines from county-level, temporally varying heart disease death rates for men and women of different races in the US. Specifically, we propose a nonseparable multivariate spatio-temporal Bayesian model which allows for group-specific temporal correlations and temporally-evolving covariance structures in the multivariate spatio-temporal component of the model. After verifying the effectiveness of our model via simulation, we apply our model to a dataset of over 200,000 county-level heart disease death rates. In addition to yielding a superior fit than other common approaches for handling such data, the richness of our model provides insight into racial, gender, and geographic disparities underlying heart disease death rates in the US which are not permitted by more restrictive models. <span style="font-variant:small-caps;">Key words:</span> Bayesian methods, Gender disparities in health, Heart disease, Nonseparable models, Racial disparities in health, Spatio-temporal data analysis Introduction ============ [Despite substantial reductions in death rates since the mid-1960s [e.g., @sempos; @ford:capewell; @young; @greenlund], heart disease remains the leading cause of death in the United States [US, @deaths].]{} Work by @casper:changes has identified that while the nation as a whole has experienced substantial declines in heart disease mortality rates, there has been a substantial geographic shift over time, as mortality rates in the northeast have declined at a much faster rate than those in the Deep South. Previous work has also shown disparities in heart disease death rates between the sexes [[e.g., @sempos; @kramer:apc]]{}, between races [[e.g., @kramer:apc]]{}, and geographically [[e.g., @gillum; @adam:geo; @adam:methods]]{}, yet accounting for these various sources of disparities simultaneously has yet to be considered. Here, we look to build upon the existing heart disease literature to obtain a broader picture of these declining death rates using a hierarchical Bayesian statistical approach which accounts for correlation spatially, temporally, and between race/gender groups. There is an extensive literature on the subject of space-time modeling, particularly in the Bayesian context. A common approach for modeling discrete — or areal — spatial data is the use of the conditionally autoregressive (CAR) model proposed by @besag and later popularized in the disease mapping context by @bym. Early uses of the CAR model in the space-time setting [include @waller:carlin and @knorr-held:besag]{} — both of which analyzed rates of lung cancer in Ohio counties — and @gelfand98, whose interest pertained to the sale prices of homes. While these methods have used *separable* model structures for space and time, @knorr-held:2000 discusses the use of nonseparable space-time models in a discrete space, discrete time setting with an application to lung cancer mortality rates in Ohio. In addition to space-time data models, @gelfand:mcar and @carlin:banerjee:2003 have developed methods for general multivariate spatial models. For a more complete coverage of the recent advances in spatial and space-time modeling, see @cressie:wikle and @BCG. The concept of multivariate space-time (MST) models for discrete spatial data has also been explored previously. For instance, @congdon modeled suicide mortality rates in the boroughs of London using spatially varying regression coefficients and a nonparametric specification of the random effects, and @daniels [developed a conditionally specified model]{} for the analysis of particulate matter and ozone data collected from monitoring sites in Los Angeles, CA. More recently, @jon proposed an alternative in which the authors use a shared component model [e.g., @knorr-held:best; @tzala] with a reduced-rank spatial domain, extending the approach of @hughes:haran to the MST setting. While a shared component model can offer substantial computational benefits by effectively reducing the complexity of a MST model to that of a reduced-rank space-time model, this assumption may not always be appropriate (e.g., when the available covariate information is insufficient to capture the differences in the geographic patterns) nor necessary (e.g., when number of groups in the multivariate structure is small). Due to the recent geographic and temporal evolutions in heart disease death rates, the methodological goal of this paper is to define a nonseparable multivariate space-time modeling framework to analyze the heart disease mortality rate data described in Section \[sec:data\]. We propose our model in Section \[sec:methods\] and demonstrate its ability to accurately estimate model parameters via simulation study in Section \[sec:sim\]. We then analyze our heart disease mortality data in Section \[sec:anal\], where we [observe temporally-evolving variance parameters inconsistent with the previously used separable model]{}. Finally, we summarize our findings and offer some concluding remarks in Section \[sec:disc\]. Data Description {#sec:data} ================ The study population for this analysis includes US residents, ages 35 and older, who were identified on a death certificate as either black or white — we restrict our analysis to these $N_g = 4$ groups because these are the only racial groups for whom data are available for the entire duration of our study period. Annual counts of heart disease deaths per county per race/gender group were obtained from the National Vital Statistics System (NVSS) of the National Center for Health Statistics (NCHS). Due to differences in the manner in which death records were processed by NCHS, we restrict the analysis to data from 1973–2010 to ensure valid comparisons across time. Deaths from heart disease were defined as those for which the underlying cause of death was “diseases of the heart” according to the 8th, 9th, and 10th revisions of the International Classification of Diseases (ICD)[^1]. Based on the works of @icd8:icd9 and @icd9:icd10, we assume that this definition is consistent over the 38 year study period. Annual projected population counts were obtained from the [@census]{}, and the numbers of heart disease deaths were age-standardized to the 2000 US Standard Population using 10 year age groups. The geographic unit used in this analysis was the county (or county equivalent). Given changes in county definitions during the study period (e.g., the creation of new counties), a single set of $N_s$ = 3,099 counties from the contiguous lower 48 states was used for the entire study period. In an attempt to stabilize the data, county-level age-standardized counts and populations were aggregated into $N_t = 19$ two year intervals (i.e., 1973–74, 1975–76, etc.). Methods {#sec:methods} ======= Review of methods for disease mapping {#sec:review} ------------------------------------- One of the seminal papers in the field of disease mapping was the work of @bym. Letting $Y_i$ and $n_i$ denote the incidence of disease and the population at risk in county $i$, the authors proposed a model of the form $$Y_{i} \sim {\mbox{$\text{Pois}$}}\left(n_i \exp\left[{ {\bf x} }_i{ \ensuremath{\boldsymbol{\beta}}}+ Z_i + \phi_i\right]\right), i=1,\ldots,N_s \label{eq:bym}$$ where ${ {\bf x} }_i$ denotes a $p$-vector of covariates with corresponding regression coefficients, ${ \ensuremath{\boldsymbol{\beta}}}$, $Z_{i}$ is a spatial random effect, and $\phi_i \sim N\left(0,\tau^2\right)$ is an exchangeable random effect. In their work, @bym modeled ${ {\bf Z} }=\left(Z_1,\ldots,Z_{N_s}\right)'$ as arising from an intrinsic conditional autoregressive (CAR) model, which has the conditional distribution $$Z_i{\,\vert\,}{ {\bf Z} }_{(i)},{ \ensuremath{\sigma}}^2 \sim N\left(\sum_{j=1}^{N_s} w_{ij} Z_j \slash \sum_{j=1}^{N_s} w_{ij}, { \ensuremath{\sigma}}^2\slash \sum_{j=1}^{N_s} w_{ij}\right) \label{eq:car} $$ where ${ {\bf Z} }_{(i)}$ denotes the vector ${ {\bf Z} }$ with the $i$th element removed and $W$ is an adjacency matrix with elements $w_{ij}=1$ if $i$ and $j$ are neighbors [(denoted $i\sim j$)]{} and 0 otherwise. Later work by @knorr-held:prec and @hodges:prec has shown that the joint distribution of ${ {\bf Z} }$ in  is of the form $$\pi\left({ {\bf Z} }{\,\vert\,}{ \ensuremath{\sigma}}^2\right) \propto \left({ \ensuremath{\sigma}}^2\right)^{-(N_s-1)\slash 2} \exp\left[-\frac{1}{2{ \ensuremath{\sigma}}^2}{ {\bf Z} }'(D-W){ {\bf Z} }\right],\label{eq:jcar}$$ where $D$ is an $N_s\times N_s$ diagonal matrix with elements $m_i = \sum_{j=1}^{N_s} w_{ij}$. Extending – to a setting consisting of multiple spatial surfaces is straightforward. For instance, suppose we wish to map $N_g$ diseases over an area consisting of $N_s$ counties. Letting $Y_{ik}$ denote the incidence of disease $k$ in county $i$, we may assume $$Y_{ik} \sim {\mbox{$\text{Pois}$}}\left(n_{ik} \exp\left[{ {\bf x} }_{ik}{ \ensuremath{\boldsymbol{\beta}}}_k + Z_{ik} + \phi_{ik}\right]\right), i=1,\ldots,N_s, \;k=1,\ldots,N_g. \label{eq:bym_k}$$ To model ${ {\bf Z} }= \left({ {\bf Z} }_{1\cdot}',\ldots,{ {\bf Z} }_{N_s\cdot}'\right)'$ where ${ {\bf Z} }_{i\cdot} = \left(Z_{i1},\ldots,Z_{iN_g}\right)'$, we may follow the example of @gelfand:mcar and let ${ {\bf Z} }\sim{\mbox{$\text{MCAR}$}}\left(1,{ \mbox{$\Sigma$} }_Z\right)$ which yields the following: $$\begin{aligned} \pi\left({ {\bf Z} }{\,\vert\,}{ \mbox{$\Sigma$} }_Z\right) &\propto \vert { \mbox{$\Sigma$} }_Z\vert^{(N_s-1)\slash 2} \exp\left[-\frac{1}{2}{ {\bf Z} }'\left\{(D-W)\otimes{ \mbox{$\Sigma$} }_Z^{-1}\right\}{ {\bf Z} }\right]\\ \text{and}\;\;{ {\bf Z} }_{i\cdot}{\,\vert\,}{ {\bf Z} }_{(i)\cdot},{ \mbox{$\Sigma$} }_Z &\sim N\left(\sum_{j\sim i} { {\bf Z} }_{j\cdot} \slash m_i, \frac{1}{m_i}{ \mbox{$\Sigma$} }_Z\right),\end{aligned}$$ where ${ \mbox{$\Sigma$} }_Z$ denotes the $N_g \times N_g$ covariance structure for our $N_g$ diseases and $\otimes$ denotes the Kronecker product. Extensions of  to the (multivariate) space-time setting follow similarly [e.g., @waller:meth], with the necessary specifications of the covariance matrix ${ \mbox{$\Sigma$} }_Z$. [While Poisson models like  are common, they can also pose computational challenges. For instance, the full conditional of ${ {\bf Z} }_i$, given by $$\begin{aligned} \pi\left({ {\bf Z} }_i{\,\vert\,}{ {\bf Y} },{ {\bf Z} }_{(i)\cdot},{ \ensuremath{\boldsymbol{\beta}}},{ \mbox{\boldmath $\phi$}},{ \mbox{$\Sigma$} }_Z\right) \propto \prod_{k=1}^{N_g}{\mbox{$\text{Pois}$}}\left(Y_{ik}{\,\vert\,}{ \ensuremath{\boldsymbol{\beta}}}_k, Z_{ik}, \phi_{ik}\right) \times \pi\left({ {\bf Z} }_i{\,\vert\,}{ {\bf Z} }_{(i)\cdot},{ \mbox{$\Sigma$} }_Z\right)\end{aligned}$$ is *not* a known distribution. That is, if we use a Markov chain Monte Carlo (MCMC) algorithm to estimate the posterior distribution of our model parameters, this model may require the use of Metropolis steps within our Gibbs sampler. When the number of groups is large — or in the space-time setting when $N_t$ is large — updating ${ {\bf Z} }$ can be cumbersome. @besag:poisson suggests a reparameterization of  which would involve integrating $\phi_{ik}$ out of the model, yielding a Gaussian full conditional for ${ {\bf Z} }_i$, though this model still consists of over twice as many parameters as data points. ]{} One alternative to modeling the counts using a Poisson likelihood is to model the *rates* as being log-normally distributed. For instance, suppose $Y_{ikt}$ and $n_{ikt}$ denote the number of heart disease-related deaths and the population at risk for the $k$th population in county $i$ at time $t$, respectively. We could then model $\theta_{ikt} = \log\left(Y_{ikt}\slash n_{ikt}\right)$ using a Gaussian distribution. This may be problematic, however, as our data consist of a large number of counties experiencing *zero* deaths related to heart disease for a given population in a given year. As such, this may require us to treat $Y_{ikt}=0$ as data below the limit of detection by substituting $Y_{ikt} = Y_{ikt}^* < 1$ or by multiply imputing values for $Y_{ikt}$ [e.g., see @fridley:dixon]. In order to avoid the computational burden associated with the Poisson model in  and the ill-handling of zeros in the log-normal model, we opt to model the rates themselves as Gaussian. That is, we let $Y_{ikt}$ denote the age-standardized death rate (per 100,000) in county $i\in\{1,\ldots,N_s\}$ during time interval $t=\{1,\ldots,N_t\}$ for race/gender group $k\in\{1,\ldots,N_g\}$ and we define ${ {\bf Y} }_{i\cdot t}$ to be the vector collecting the $N_g$ observations from time $t$ in the $i$th county, ${ {\bf Y} }_{i\cdot\cdot}=\left({ {\bf Y} }_{i\cdot1}',\ldots,{ {\bf Y} }_{i\cdot N_t}'\right)'$ to be the vector collecting the $\left(N_gN_t\right)$ observations from the $i$th county, and ${ {\bf Y} }= ({ {\bf Y} }_{1\cdot\cdot}',\ldots,{ {\bf Y} }_{N_s\cdot\cdot}')'$ to be the $N_sN_gN_t$-vector which stacks all of the age-standardized death rates. To model the death rates, we assume $$Y_{ikt} \sim N\left({ {\bf x} }_{ikt}'{ \ensuremath{\boldsymbol{\beta}}}_k + Z_{ikt},\tau_{ikt}^2\right),\;i=1,\ldots,N_s,\;k=1,\ldots,N_g,\;t=1,\ldots,N_t\label{eq:Y}$$ where ${ {\bf x} }_{ikt}$ is the $p\times1$ vector of covariates for the $i$th county at time $t$ with a corresponding $p\times1$ vector of regression coefficients, ${ \ensuremath{\boldsymbol{\beta}}}_k$, $Z_{ikt}$ is a random effect which accounts for the spatio-temporal dependence between and within the four race/gender groups, $\tau_{ikt}^2 = \tau_k^2\slash n_{ikt}$, and $n_{ikt}$ denotes the population of group $k$ in county $i$ at time $t$ divided by 100,000. A recent example of a model of this form is @qbc, where a Gaussian likelihood was used to model changes in county-level asthma hospitalization rates in California. [We provide a defense of the Gaussian assumption for these data in [Figure B.1]{} of the Web Appendix.]{} Choices for ${ \mbox{$\Sigma$} }_Z$ {#sec:special} ----------------------------------- Before we present our proposed ${\mbox{$\text{MSTCAR}$}}$ model for ${ \mbox{$\Sigma$} }_Z$ in Section \[sec:mstcar\], we begin by describing other natural choices: independence models and a separable model. Not only do these models have computational benefits, but they are also special cases of the ${\mbox{$\text{MSTCAR}$}}$. ### Independence models Based on the multivariate spatial models described in Section \[sec:review\], one could opt to fit a collection of $N_g$ independent space-time models (denoted ${\mbox{$\text{STCAR}$}}$) of the form $$\begin{aligned} { {\bf Z} }_{ik\cdot}{\,\vert\,}{ {\bf Z} }_{(i)k\cdot},{ \ensuremath{\sigma}}_k^2,\rho_k &\sim N\left(\frac{1}{m_i} \sum_{j\sim i} { {\bf Z} }_{jk\cdot}, \frac{{ \ensuremath{\sigma}}_k^2}{m_i} R\left(\rho_k\right)\right), k=1,\ldots,N_g\label{eq:ind}\end{aligned}$$ where $R\left(\rho_k\right)$ denotes an $N_t \times N_t$ temporal correlation matrix with parameter $\rho_k$ and ${ \ensuremath{\sigma}}_k^2$ is the variance parameter corresponding to race/gender group $k$. In addition to accounting for spatiotemporal correlation, a model of the form  with this structure for ${ {\bf Z} }$ has the added computational benefit of being able to be fit *in parallel* as $N_g$ separate models. This convenience, however, comes at the cost of failing to account for the correlation between groups. As we believe there to be a high degree of correlation between the heart disease mortality rates of our various race/gender groups, this drawback is particularly disappointing. We could also choose to fit $N_t$ independent multivariate spatial models of the form $$\begin{aligned} { {\bf Z} }_{i\cdot t}{\,\vert\,}{ {\bf Z} }_{(i)\cdot t},G_t &\sim N\left(\frac{1}{m_i} \sum_{j\sim i} { {\bf Z} }_{j\cdot t}, \frac{1}{m_i} G_t\right),\end{aligned}$$ where $G_t$ denotes a temporally-varying $N_g \times N_g$ multivariate covariance structure for our race/gender groups. While this model can also have substantial computational benefits, the assumption of *temporal* independence is especially damning. ### Separable model Driven by the desire to account for both temporal and between-group correlation in our spatial model, a *separable* model of the form $$\begin{aligned} { {\bf Z} }_{i\cdot\cdot}{\,\vert\,}{ {\bf Z} }_{(i)\cdot\cdot},G,\rho &\sim N\left(\frac{1}{m_i} \sum_{j\sim i} { {\bf Z} }_{j\cdot\cdot}, \frac{1}{m_i} R(\rho)\otimes G\right),\label{eq:sep}\end{aligned}$$ where we let $R\left(\rho\right)$ denote an $N_t \times N_t$ temporal correlation matrix and $G$ denote the $N_g \times N_g$ between-group covariance structure, may be attractive. The appeal of a separable model where ${ \mbox{$\Sigma$} }_Z = R(\rho)\otimes G$ is immediately clear: instead of accounting for multivariate temporal correlation using an unstructured $N_gN_t \times N_gN_t$ matrix, ${ \mbox{$\Sigma$} }_Z$, we can *separate* our problem into matrices of rank $N_g$ and $N_t$, reducing the computational complexity of inverting ${ \mbox{$\Sigma$} }_Z$ substantially. While the criticism of separable models in the spatiotemporal literature is primarily directed toward their use in the continuous space, continuous time setting where prediction at unobserved locations is of interest [e.g., see @stein2005], the lack of a temporally evolving $G_t$ or group-specific $\rho_k$ may be [undesirable]{}. ### The ${\mbox{$\text{MSTCAR}$}}$ model {#sec:mstcar} To construct our random effects, ${ {\bf Z} }$, we will begin by defining ${ {\bf v} }_{\iota\cdot t}{ \mbox{$\stackrel{\text{iid}}{\sim}$}}N({\textbf{0}},G_t)$ to be a collection of independent $N_g$-dimensional random variables with covariance $G_t$ for $\iota=1,\ldots,(N_s-1)$ and $t=1,\ldots, N_t$. Note the deliberate use of the subscript $\iota$ instead of the subscript $i$; this is to reinforce that ${ {\bf v} }_{\iota\cdot t}$ does not correspond to a particular county. From this, we define ${ {\bf v} }_{\iota k\cdot} = \left(v_{\iota k1},\ldots,v_{\iota kN_t}\right)'$ and construct $$\begin{aligned} { \mbox{\boldmath $ \eta $} }_{\iota k\cdot} = {\widetilde{R}}_k { {\bf v} }_{\iota k\cdot} \sim N\left({\textbf{0}},{\widetilde{R}}_k G_{k,k}^* {\widetilde{R}}_k'\right),\;\iota=1,\ldots,(N_s-1),\;k=1,\ldots,N_g,\label{eq:eta}\end{aligned}$$ where we define ${\widetilde{R}}_k$ to be the Cholesky decomposition of $R_k$ such that ${\widetilde{R}}_k {\widetilde{R}}_k' = R_k$, where $R_k \equiv R\left(\rho_k\right)$ is a temporal correlation matrix based on an autoregressive order 1 — or AR(1) — structure with correlation parameter $\rho_k$ and $G_{k,k}^*$ is the $N_t \times N_t$ diagonal matrix with elements $\left\{G_t\right\}_{k,k}$ for $t=1,\ldots,N_t$. Equivalently, we can define ${ \mbox{\boldmath $ \eta $} }_{\iota\cdot\cdot} \sim N\left({\textbf{0}},{ \mbox{$\Sigma$} }_{\eta}\right)$ where $$\begin{aligned} \label{eq:Sig_eta} { \mbox{$\Sigma$} }_{\eta} = \begin{bmatrix} {\widetilde{R}}_{1,1}^* & {\textbf{0}}& {\textbf{0}}\\ \vdots & \ddots & {\textbf{0}}\\ {\widetilde{R}}_{N_t,1}^* & \cdots & {\widetilde{R}}_{N_t,N_t}^* \end{bmatrix} \begin{bmatrix} G_1 & {\textbf{0}}& {\textbf{0}}\\ {\textbf{0}}& \ddots & {\textbf{0}}\\ {\textbf{0}}& {\textbf{0}}& G_{N_t} \end{bmatrix} \begin{bmatrix} {\widetilde{R}}_{1,1}^* & \cdots & {\widetilde{R}}_{N_t,1}^*\\ {\textbf{0}}& \ddots & \vdots \\ {\textbf{0}}& {\textbf{0}}& {\widetilde{R}}_{N_t,N_t}^* \end{bmatrix},\end{aligned}$$ where ${\widetilde{R}}_{t,t'}^*$ denotes the $N_g\times N_g$ diagonal matrix with elements $\left\{{\widetilde{R}}_k\right\}_{t,t'}$ for $k=1,\ldots,N_g$. Finally, we let ${ \mbox{$\Sigma$} }_Z \equiv { \mbox{$\Sigma$} }_{\eta}$ and define ${ {\bf Z} }$ in the form of an ${\mbox{$\text{MCAR}$}}\left(1,{ \mbox{$\Sigma$} }_{\eta}\right)$ of @gelfand:mcar with a conditional and (improper) joint distribution of $$\begin{aligned} { {\bf Z} }_{i\cdot\cdot}{\,\vert\,}{ {\bf Z} }_{(i)\cdot\cdot},G_{1},\ldots,G_{N_t},{ \mbox{\boldmath $\rho$} }\sim& \;N\left(\frac{1}{m_i} \sum_{j\sim i} { {\bf Z} }_j, \frac{1}{m_i} { \mbox{$\Sigma$} }_{\eta}\right),\;i=1,\ldots,N_s \label{eq:Zcar}\\ \pi\left({ {\bf Z} }{\,\vert\,}G_{1},\ldots,G_{N_t},{ \mbox{\boldmath $\rho$} }\right) \propto& \;\vert { \mbox{$\Sigma$} }_{\eta} \vert^{-(N_s-1)\slash2} \exp\left[-\frac{1}{2} { {\bf Z} }' \left\{(D-W) \otimes { \mbox{$\Sigma$} }_{\eta}^{-1}\right\} { {\bf Z} }\right], \label{eq:Zcar_joint}\end{aligned}$$ respectively. We denote the expression in (\[eq:Zcar\_joint\]) as ${ {\bf Z} }\sim {\mbox{$\text{MSTCAR}$}}\left(G_{1},\ldots,G_{N_t},{ \mbox{\boldmath $\rho$} }\right)$. Hierarchical model ------------------ We complete the hierarchial model by specifying prior distributions for the remaining parameters. As is common in Bayesian modeling, we place a flat, noninformative prior on ${ \ensuremath{\boldsymbol{\beta}}}_k$, and, following [@gelman2006]{}, assume an improper uniform prior over the positive real numbers for $\tau_k$. For each of the spatio-temporal covariance matrices, $G_t$, we assume an inverse Wishart distribution with positive definite scale matrix $G$ and $\nu>N_g-1$ degrees of freedom, and we use Beta priors for each of the $\rho_k$s. Finally, as many rural counties (particularly in the north-central states) have no data from the black populations, we decompose ${ {\bf Y} }$ as ${ {\bf Y} }_c = \left({ {\bf Y} }_o',{ {\bf Y} }_u'\right)'$, where ${ {\bf Y} }_o$ denotes the vector of counties with observed populations and ${ {\bf Y} }_u$ denotes the vector of counties with unobserved populations. Putting these pieces together, the full hierarchical model is as follows: $$\begin{aligned} \label{eq:hier} \pi\left({ \ensuremath{\boldsymbol{\beta}}},{ {\bf Z} },G_1,\ldots,G_t,{ \mbox{\boldmath $\rho$} },{ \mbox{\boldmath $\tau$}}^2,{ {\bf Y} }_u{\,\vert\,}{ {\bf Y} }_o\right) \propto& N\left({ {\bf Y} }{\,\vert\,}X{ \ensuremath{\boldsymbol{\beta}}}+{ {\bf Z} },{ \mbox{$\Sigma$} }_Y\right) \times {\mbox{$\text{MSTCAR}$}}\left({ {\bf Z} }{\,\vert\,}G_{1},\ldots,G_{N_t},{ \mbox{\boldmath $\rho$} }\right)\notag\\ &\times \prod_{k=1}^{N_g} \left[{\mbox{$\text{Beta}$}}\left(a_{\rho},b_{\rho}\right) \times \pi\left(\tau_k^2\right)\right] \notag\\ &\times \prod_{t=1}^{N_t} {\mbox{$\text{InvWish}$}}\left(G_t{\,\vert\,}G,\nu\right),\end{aligned}$$ where the notation $\pi(x)$ denotes the marginal distribution for a random variable $x$ and $\pi(x{\,\vert\,}y)$ denotes the conditional distribution of $x$ given $y$. Here, ${ \mbox{$\Sigma$} }_Y$ is a diagonal matrix with elements $\tau_{ikt}^2$, $X$ is the $(N_sN_gN_t\times p)$ matrix of covariates, and $\pi\left(\tau_k^2\right)$ is the density for $\tau_k^2$ which corresponds to a flat prior for $\tau_k$. In cases where it may be difficult to learn about each $G_t$ or each $\rho_k$, we may consider putting additional structure on the priors for these parameters. Note that in (\[eq:hier\]), ${ {\bf Y} }_u$ is treated as an unknown model parameter, and thus each $Y_{ikt}\in{ {\bf Y} }_u$ is sampled from (\[eq:Y\]) during each iteration of the MCMC algorithm. Furthermore, we assign a small value for each $n_{ikt}$ in the set $\{n_{ikt}: Y_{ikt}\in{ {\bf Y} }_u\}$. A detailed derivation of the MCMC sampler used for this analysis, as well as a description of the benefits of using an AR(1) model to account for temporal correlation, can be found in Web Appendix A. Simulation Study {#sec:sim} ================ To evaluate the ability of our model to accurately estimate all of our model parameters, we devised two simulation studies, each comprised of $L=100$ sets of data generated using our ${\mbox{$\text{MSTCAR}$}}$ model with $N_t=10$ timepoints, $N_g=3$ groups, and the $N_s=58$ counties of California as our spatial domain. This spatial domain offered a compromise between creating a computationally feasible simulation study (compared to using all 3,099 county equivalents) while representing a state with a moderate number of counties and variation in population density and geographic spread. The first simulation study assumes that $n_{ikt}\equiv n$ for all combinations of $(i,k,t)$, allowing us to focus on parameter estimation irrespective of the amount of information each county can provide. We will then relax this assumption by generating data using actual populations of California counties. In each simulation study, performance was primarily assessed via coverage (i.e., the percent of 95% credible intervals (95% CI) which cover the true parameter values) where values near 95% are desired. Furthermore, we will compare results from the ${\mbox{$\text{MSTCAR}$}}$ model proposed here to [those obtained using a separable model]{}. While the separable model will be incapable of providing accurate estimates for the many additional parameters which comprise ${ \mbox{$\Sigma$} }_{\eta}$, the focus here will be on model fit. Specifically, we will compare the coverage of ${ {\bf Z} }$ and the deviance information criterion (DIC) of @dic, where lower values indicate a better compromise of model fit and model complexity. Equal population sizes {#sec:sim1} ---------------------- The $\ell$th dataset is created by generating $Y_{ijk}^{(\ell)} \sim N(Z_{ijk}^{(\ell)},\tau_k^2)$ where $\tau_k^2=1$ for $k=1,\ldots,N_g$ and ${ {\bf Z} }^{(\ell)}$ is drawn from the ${\mbox{$\text{MSTCAR}$}}$ model in (\[eq:Zcar\_joint\]). To do this, we first let ${ \mbox{\boldmath $\rho$} }= (0.8, 0.85, 0.90)'$ and generated samples of $G_t$ from an inverse Wishart distribution with [$2*N_g+1$ degrees of freedom]{} and [scale matrix $20*N_g *I_{N_g}$]{}, where $I_{N_g}$ is the identity matrix of size $N_g$. Using these parameters to construct ${ \mbox{$\Sigma$} }_{\eta}$ (from which all $L$ datasets are based), we generated our latent variables ${ \mbox{\boldmath $ \eta $} }_{\iota\cdot\cdot}^{(\ell)} \sim N\left({\textbf{0}},{ \mbox{$\Sigma$} }_{\eta}\right)$. From these, we used the methods described in @rue:held to generate our ${ {\bf Z} }^{(\ell)}$; specifically, we found the eigenvalues and eigenvectors of the matrix $D-W$ (based on the adjacencies of counties in California) and used the linear dependence of the eigenvectors to generate our spatial structure. Each simulated dataset is then analyzed using the hierarchical model in (\[eq:hier\]) using MCMC. Using the priors described in the previous section, we initialized all of our parameters (including ${ {\bf Z} }$) at their true values, resulting in chains which were quick to converge and allowing us to assess the performance of our model using just 1,500 iterations of our MCMC algorithm, the last 500 of which were used as the basis for our results. In order to better visualize these results, we also display results from an arbitrarily selected dataset. Overall, our model performed quite well. Collectively, the $Z_{ikt}$ were well estimated, as demonstrated in Figure \[fig:sim\_Z\], with our model obtaining 94.4% coverage and offering an improvement in DIC in 82 of the 100 datasets. This accuracy is permitted due in part to the flexibility of our model to allow for temporally evolving $G_t$. As shown in Figure \[fig:sim\_G\], the randomly generated $G_t$ exhibited some irregular behavior. While the ${\mbox{$\text{MSTCAR}$}}$ model was able to estimate these $G_t$ quite well — with 95.4% coverage for the diagonal elements (i.e., the variances) and 95.3% coverage for the off-diagonal elements (i.e., the covariances) — the separable model fails to accommodate such a nuanced multivariate structure. We also achieved accurate estimates of the error variances, $\tau_k^2$, for which we obtained an average of 91.3% coverage. In contrast, coverage for $\rho_k$ was less than ideal (85%). ![Selected ${ {\bf Z} }_{ik\cdot}$ curves from one dataset of the first simulation study. Plots in the same row correspond to the same county, and plots in the same column correspond to the same group. Black lines denote posterior medians, red circles denote true values, and gray bands denote the 95% CI.[]{data-label="fig:sim_Z"}](sim_Z1.png){width=".95\textwidth"} ![Estimated $G_{t;k,k'}$ from one dataset of the first simulation study. Top row displays the diagonal elements, while the bottom row displays the off-diagonal elements. Black lines denote posterior medians, red circles denote true values, and gray bands denote the 95% CI. For comparison purposes, the green lines denote the analogous values from the separable model.[]{data-label="fig:sim_G"}](sim_G1.png){width=".95\textwidth"} Varying population sizes {#sec:sim2} ------------------------ In our second simulation study, we generated data using the same design as described in Section \[sec:sim1\], but here we assigned $n_{ikt}$ to be the population of the $i$th county at time $t$ for the following subpopulations: white men ($k=1$), white women ($k=2$), and black men and women ($k=3$). While white men and women have $n_{ikt}>200$ in all counties for all time periods, there are many counties with small black population sizes. As such, we combine black men and women to limit the number of counties with no data. In cases where a county has [no]{} population during time $t$, however, we assume $n_{ikt}=1$ and treat $Y_{ikt}$ as missing. Our model was again able to obtain accurate estimates for the $Z_{ikt}$ and the various elements of the $G_t$ while outperforming the separable model in all 100 datasets (based on DIC). Furthermore — aside from an expected increase in the width of the credible intervals — there does not appear to be any degradation in the estimation for these parameters as we shift from the well-populated groups to the third, less populated group. Unfortunately, our model again performs less well with respect to the temporal correlation parameters, $\rho_k$. It is understandable, though, how the problem from our first example would be exacerbated here, as the amount of information provided by each group depends on the county populations. General findings {#sec:sim_disc} ---------------- In both simulation studies, the ${\mbox{$\text{MSTCAR}$}}$ was able to obtain accurate estimates of both the $Z_{ikt}$ and the $G_t$. While the nonseparable model offered improved DIC when compared to the separable model, it is important to note that the differences were not substantial, with just over a 1% reduction on average. This suggests that the key benefit of the ${\mbox{$\text{MSTCAR}$}}$ model (with respect to model fit) is that it provides more precise results (i.e., narrower credible intervals) than the separable model while still achieving the desired coverage. Based on these results, the $\rho_k$ parameters appear to be difficult to identify. As such, if inference on the $\rho_k$ is desired, it may be necessary to run our MCMC algorithms for more iterations and consider *thinning* our samples to obtain samples which are less correlated over the course of the chain. Another option would be to consider respecifying our priors for the $\rho_k$. In these simulation studies, we had assumed a ${\mbox{$\text{Beta}$}}(9,1)$ prior for $\rho_k$, but a more informative prior may be appropriate, particularly in the case of varying population sizes. For instance, we could assume a multi-level model of the form $\rho_k \sim {\mbox{$\text{Beta}$}}\left(\upsilon_{\rho}\rho_0,\upsilon_{\rho}(1-\rho_0)\right)$ for $k=1,\ldots,N_g$, where $\rho_0 \sim {\mbox{$\text{Beta}$}}\left(a_0,b_0\right)$ and $\upsilon_{\rho}$ is a parameter which controls the informativeness of the prior. In extreme cases, we may even consider forcing $\rho_k \equiv \rho_0$, which can be induced by letting $\upsilon_{\rho}\to\infty$. In addition to improving the convergence of our MCMC algorithm, this may also lead to minor computational benefits while still yielding a model that is more flexible than the separable model in . Analysis of Heart Disease Death Rates {#sec:anal} ===================================== We fitted the nonseparable hierarchical model in (\[eq:hier\]) to the heart disease mortality data described in Section \[sec:data\] using covariates consisting of only an intercept term for each combination of 2-year time-interval and race/gender [as required, per @besag95], forcing the random effects to account for a substantial amount of the spatio-temporal variability in the data. We place a ${\mbox{$\text{Beta}$}}(9,1)$ prior on each of the $\rho_k$ to encourage higher temporal correlations in the model, and we use a vague inverse Wishart prior for each of the $G_t$. We ran the MCMC algorithm with a single chain for [6,000]{} iterations, diagnosing convergence via trace plots for many of the model parameters and discarding the first 1,000 iterations as burn-in. Following that, we thinned our posterior samples by removing 9 out of 10 samples — while this is not theoretically necessary, it [reduced the burden of storing excess samples for our over 200,000 random effects]{}. Estimates provided are based on posterior medians, and 95% credible intervals (95% CI) were obtained by taking the 2.5- and 97.5-percentiles from the thinned post-burn-in samples. To determine if the burden associated with fitting this nonseparable model was necessary, we compared our model to the separable model in  and the $N_g$ independent ${\mbox{$\text{STCAR}$}}$ models in . Table \[tab:dic\] displays the results of our model comparison. Here, it is clear that the independent ${\mbox{$\text{STCAR}$}}$ models — while computationally convenient — are inadequate for these data, as both the separable and ${\mbox{$\text{MSTCAR}$}}$ models offer improvements in DIC of over [94,000 units]{}. As seen in Section \[sec:sim\], the separable and ${\mbox{$\text{MSTCAR}$}}$ models appear to perform similarly, with the ${\mbox{$\text{MSTCAR}$}}$ model having a DIC only 5,828 units lower. Given the evidence in the literature that DIC tends to favor over-fitted models [e.g., @robert:dic:disc], it remains unclear if the flexibility of the ${\mbox{$\text{MSTCAR}$}}$ model is *required* here; nevertheless, we will henceforth focus our attention on results from the ${\mbox{$\text{MSTCAR}$}}$ model. Model DIC $p_D$ ---------------------------- ----------- -------- ${\mbox{$\text{STCAR}$}}$ 2,423,049 32,110 Separable 2,334,355 24,185 ${\mbox{$\text{MSTCAR}$}}$ 2,328,527 25,699 : [Model fit comparison between the independent ${\mbox{$\text{STCAR}$}}$ models, a separable model, and the nonseparable ${\mbox{$\text{MSTCAR}$}}$ model proposed here. Lower values of DIC indicate a better compromise of model fit and model complexity, where $p_D$ is a measure of model complexity.]{}[]{data-label="tab:dic"} Figure \[fig:trend\] displays the expected nationwide death rate trends for each group. These trend lines were computed by first computing the posterior distribution for the expected value for $Y_{ikt}$ as $\widehat{Y}_{ikt} = { {\bf x} }_{ikt}'{ \ensuremath{\boldsymbol{\beta}}}_k + Z_{ikt}$. We then estimated the nationwide death rate for group $k$ at time $t$ by constructing the posterior for $$\widehat{Y}_{\cdot kt} = \frac{\sum_i \widehat{Y}_{ikt} n_{ikt}}{\sum_i n_{ikt}}.$$ A number of important findings can be found from this figure. First and foremost, all four of our race/gender groups have experienced substantial declines, with death rates being more than cut in half. Secondly, men of both races experience significantly higher rate of heart disease-related death than women. That said, men and women of both races do not decline at the same rate; e.g., while white men began the study as the population with the highest risk, they were soon surpassed by black men, whose rates appear to be relatively stagnant for the period from 1975–76 to 1987–88. This trend is also visible for black women. To illustrate the changing geographic patterns, Figure \[fig:white\_men\] displays heart disease death rates for white men for four time-intervals. Here, we notice an interesting trend, as several major cities (e.g., Denver, CO; Washington, DC; Atlanta, GA; Minneapolis, MN) are consistently leading the charge toward lower rates of heart disease related death for white men in their respective regions. On the other hand, there are collections of counties in which rates are lagging behind, most prominently along the southern Mississippi River and much of the Deep South. Similar patterns can be found for the remaining race/gender groups. ![Heart disease death rates over time for each of the race/gender groups compared to the total population. Gray bands denote the 95% credible intervals for the estimates. []{data-label="fig:trend"}](time_trends.png){width=".6\textwidth"} \ ![image](col_bar2.png){width=".04\textwidth"} We now turn our attention to the numerous variance parameters permitted by the use of the nonseparable model. While in Section \[sec:sim\] we presented posterior distributions for the elements of $G_t$ (Figure \[fig:sim\_G\]), these parameters are not necessarily of direct interest as they are the variance parameters for ${ {\bf v} }_{\ell\cdot t}$, and thus they are *not* directly interpretable on the scale of the data. Instead, we need to use our posterior samples of $G_t$ and $\rho_k$ to construct ${ \mbox{$\Sigma$} }_{\eta}$ from . These values coincide to the conditional covariance matrix of ${ {\bf Z} }_{i\cdot\cdot}$ (when scaled by the number of neighbors, $m_i$), and thus *are* [interpretable on the scale of the data]{}. Figure \[fig:heart\_Sig\] displays the diagonal elements of ${ \mbox{$\Sigma$} }_{\eta}$ from the nonseparable model, as compared to the analogous estimates from the separable model. Here, we find — for all race/gender groups — that the variability of $Z_{ikt}$ has decreased substantially from the beginning of the study period to the end. More importantly, however, we note that the separable model severely underestimates the variance at the beginning of the study and severely overestimates the variance at the end. [As shown in Figure B.2 of the Web Appendix]{}, this can lead to oversmoothing when the rates are the highest (the 1970s) and undersmoothing when the rates are lowest (the 2000s), neither of which is desirable. [This may be due to the fact that the rates themselves decline over time.]{} Correlations between race/gender groups are all non-zero, with high correlations between genders of the same race and moderate correlations between races; these results can be found in [Figure B.3]{} of the Web Appendix. \ Discussion {#sec:disc} ========== In this paper, we have proposed a nonseparable framework for the purpose of modeling a dataset comprised of temporally-varying county-level heart disease death rates for multiple race/gender populations. We evaluated the validity of the proposed methodology — referred to as the ${\mbox{$\text{MSTCAR}$}}$ model — via simulation and demonstrated that the model was capable of providing a good fit to the data and obtaining accurate estimates for the many variance parameters. [Not only did the ${\mbox{$\text{MSTCAR}$}}$ model outperform two more conventional models, but we show our model can help control the degree of smoothing in data which undergo a substantial temporal evolution during the study period.]{} While the methods proposed here are much more sophisticated than more commonplace models like those discussed in Section \[sec:special\], there are a number of extensions which could be used to enhance the ${\mbox{$\text{MSTCAR}$}}$ model. For manageable values of $N_s$, for instance, one could envision models with region-specific parameters $G_{it}$ and $\rho_{ik}$. Implementing these models would likely require the use of a proper CAR model (e.g., the model proposed in Section \[sec:methods\] is constructed using only $N_s-1$ latent vectors), say by replacing $(D-W)$ in (\[eq:Zcar\_joint\]) with $(D-\alpha_{kt}W)$, where $\alpha_{kt}\in[0,1)$ ensures propriety and $\alpha_{kt}=1$ yields the improper CAR-based model used here. Furthermore, one may choose to use a multi-level modeling approach for specifying priors for many of these parameters, such as $$\begin{aligned} G_{it} \sim {\mbox{$\text{InvWish}$}}\left(\nu_i G_t,\nu_i\right),\;G_{t} \sim {\mbox{$\text{Wish}$}}\left(1\slash\nu G_0,\nu\right),\;\text{and}\;G_{0} &\sim {\mbox{$\text{InvWish}$}}\left(\nu_0 G,\nu_0\right)\end{aligned}$$ to facilitate additional borrowing-of-strength. [Computational burden and identifiability concerns notwithstanding]{}, such a model would be rather intuitive to specify and construct; i.e., one could let ${ \mbox{\boldmath $ \eta $} }_{i\cdot\cdot} \sim N\left({\textbf{0}},{ \mbox{$\Sigma$} }_{\eta_i}\right)$, where ${ \mbox{$\Sigma$} }_{\eta_i}$ is constructed as in (\[eq:Sig\_eta\]) with $i$ subscripts. Based on the results of @hcar — where the authors extended a separable space-time model to allow for region-specific variance parameters — there is evidence to believe that models of this sort may offer substantial improvements in fit. For cases where $N_s$ is large, one may also consider using dimension reduction techniques such as those proposed by @hughes:haran and extended by @jon. Unfortunately, it’s unclear whether or not this would actually result in computational gains in our setting without making additional assumptions, as the approach of @hughes:haran removes the conditional properties which make CAR models attractive. That is, when implementing the ${\mbox{$\text{MSTCAR}$}}$ model proposed here, one need only invert and manipulate matrices of rank $N_gN_t$ to sample the ${ {\bf Z} }_{i\cdot\cdot}$, albeit this requires looping through each of the $N_s$ areal regions. An analogous approach based on @hughes:haran, however, would replace this $N_s$ loop with a single $N_s^*N_gN_t$-dimensional update, where $N_s^*\ll N_s$ is the rank of the reduced spatial domain. Were we to reduce the dimension of our spatial domain from $N_s=3,099$ to $N_s^*=310$ (a 90% reduction), this would still require manipulating matrices of rank $N_s^*N_gN_t = 23,560$, which would not be feasible in our setting. While one could take advantage of the AR(1) structure to ease the burden, this would result in the manipulation of matrices of rank $N_s^*N_g$, which may *still* be too large to implement in practice without resorting to the shared component model of @jon. In the immediate future, we have two primary areas for next steps. Motivated by this and earlier work, we aim to investigate the observed geographic disparities in heart disease death rates by identifying potential factors which may be associated with the patterns observed here. In addition to further exploring the mechanics driving heart disease death rates, we plan to apply a similar modeling framework to data comprised of county-level *stroke*-related death rates. As stroke data are typically more erratic with much lower rates of incidence, these data will present additional challenges. In particular, the normal approximation used in this analysis will be less appropriate; as such, we aim to explore the possibility of implementing this methodology in a log-linear modeling framework using a Poisson likelihood. [^1]: ICD–8: 390-398, 402, 404, 410–429; ICD–9: 390–398, 402, 404–429; ICD-10: I00–I09, I11, I13, I20–I51
--- abstract: 'We report results on $f_B$ and semi-leptonic $B$ decay form factors using NRQCD. We investigate $1/M$ scaling behavior of decay amplitudes. For $f_B$ Effect of higher order relativistic correction terms are also studied.' address: - ' High Energy Accelerator Research Organization (KEK), Tsukuba 305, JAPAN' - ' Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739, JAPAN' author: - 'S. Hashimoto, K. Ishikawa, H. Matsufuru$^{\rm b}$, T. Onogi$^{\rm b}$ and N. Yamada$^{\rm b}$' title: | \ NRQCD Results on Form Factors[^1] --- Introduction ============ Weak matrix elements of $B$ meson such as $f_B$, $B_B$, and $B \rightarrow \pi(\rho)l\nu$ form factors are important quantities for the determination of Cabbibo-Kobayashi-Maskawa matrix elements. However simulating the $b$-quark with high precision is still a challenge in Lattice QCD, since the $b$-quark mass in the lattice unit is large, $a m_b \sim$ 2–3, even in recent lattice calculations. One approach to deal with the heavy quark is to extrapolate the matrix elements for heavy-light meson around the charm quark mass region to the $b$-quark mass assuming $1/m$ scaling. It is, however, rather difficult to control the systematic uncertainty in this approach, since the systematic error tends to become larger as increasing $am_Q$. An alternative approach is to use an effective nonrelativistic action. In Table \[tab:action\], we compare various features of NRQCD[@NRQCD], Fermilab[@FNAL], and ordinary Wilson/Clover actions as $b$-quark action. The first two actions are the effective nonrelativistic actions. Their advantage is that $b$-quark can be directly simulated. NRQCD Fermilab Wilson/Clover --------------------------------------------------------- --------------------------------------------------------- ----------------------------- Direct Direct Extrapolation Simulation Simulation from charm Error size Error size Error size $\frac{\alpha_s \Lambda}{m_Q} ,\frac{\Lambda^2}{m_Q^2}$ $\frac{\alpha_s \Lambda}{m_Q}, \frac{\Lambda^2}{m_Q^2}$ $(a m_Q)^2, \alpha_s a m_Q$ Error remains Error $\rightarrow$ 0 Error $\rightarrow$ 0 as $\beta \nearrow$ as $\beta \nearrow$ as $\beta \nearrow$ Easily Improvable Improvable : Fermion actions for heavy quark.[]{data-label="tab:action"} In this report, we present our study of the decay constant of $B$ meson and $B \rightarrow \pi l\overline{\nu}$ semi-leptonic decay form factors with NRQCD action. We study the mass dependence of these quantities by simulating heavy-light mesons over a wide range of the heavy quark mass. In section \[sec:decay\_const\], we study the 1/m dependence of the heavy-light decay constant using nonrelativistic action of $O(1/m_Q^2)$. The systematic errors due the truncation of higher order relativistic correction terms are estimated. In section \[sec:form\_factor\] we describe the first computation of the $B \rightarrow \pi l\overline{\nu}$ semi-leptonic decay form factors with NRQCD action of $O(1/m_Q)$. Section \[sec:discussion\] is devoted for discussion and future problems. $B$ meson decay constant $f_B$ {#sec:decay_const} ============================== In the NRQCD approach it is very important to investigate the size of the systematic error arising from the truncation of the action at a certain order of $1/m_{Q}$. Earlier studies on $f_B$ by Davies et al.[@fb1] and Hashimoto[@fb2] and the subsequent work by NRQCD group[@NRfB], where $b$-quark is simulated with the NRQCD action including up to $O(1/m_{Q})$ terms, showed that $1/m_P$ correction of $f_B$ from the static limit is significantly large. Thus the effect of the higher order correction of $O(1/m_Q^2)$ could be important. In this section, we compare the $B$ meson decay constant obtained from the action including $O(1/m)$ terms only and that including $O(1/m^2)$ terms entirely. NRQCD action and field rotation {#sec:action_fB} ------------------------------- We employ the following NRQCD action $$\begin{aligned} S & = & Q^{\dagger}(t,\mbox{\boldmath $x$}) \left[ Q(t,\mbox{\boldmath $x$})- \left( 1 - \frac{aH_0}{2n} \right)^n \right. \nonumber\\ & & \times \left( 1 - \frac{a\delta H}{2} \right) U_4^{\dag} \left( 1 - \frac{a\delta H}{2} \right) \nonumber\\ & & \times \left. \left( 1 - \frac{aH_0}{2n} \right)^n Q(t-1,\mbox{\boldmath $x$}) \right]\end{aligned}$$ where $$\begin{aligned} H_0 & = & - \frac{\Delta^{(2)}}{2m_Q}, \\ \delta H & = & \sum_i c_i \delta H^{(i)}, \\ \delta H^{(1)} & = & - \frac{g}{2m_Q} \mbox{\boldmath $\sigma$} \cdot \mbox{\boldmath $B$},\\ \delta H^{(2)} & = & \frac{ig}{8m_Q^2} ( \mbox{\boldmath $\Delta$} \cdot \mbox{\boldmath $E$} - \mbox{\boldmath $E$}\cdot \mbox{\boldmath $\Delta$} ),\\ \delta H^{(3)} & = & - \frac{g}{8m_Q^2} \sigma \cdot (\mbox{\boldmath $\Delta$}\times\mbox{\boldmath $E$} - \mbox{\boldmath $E$} \times \mbox{\boldmath $\Delta$}), \\ \delta H^{(4)} & = & - \frac{(\Delta^{(2)})^2}{8m_Q^3} ,\\ \delta H^{(5)} & = & \frac{a^2 \Delta^{(4)}}{24m_Q}, \\ \delta H^{(6)} & = & - \frac{a (\Delta^{(2)})^2}{16nm_Q^2},\end{aligned}$$ where $n$ denotes the stabilization parameter. The coefficients $c_i$ are unity at tree level and should be determined by perturbatively matching the action to that in relativistic QCD in order to include the 1-loop corrections. $\Delta$ and $\Delta^{(2)}$ denote the symmetric lattice differentiation in spatial directions and Laplacian respectively and $\Delta^{(4)} \equiv \sum_i (\Delta^{(2)}_i)^2$. $\mbox{\boldmath $B$}$ and $\mbox{\boldmath $E$}$ are generated from the standard clover-leaf field strength. The original 4-component heavy quark spinor $h$ is decomposed into two 2-component spinors $Q$ and $\chi$ after Foldy-Wouthuysen-Tani (FWT) transformation: $$\begin{aligned} h(x) & = & R \left( \begin{array}{c} Q(x) \\ \chi^{\dag}(x) \end{array} \right),\end{aligned}$$ where $R$ is an inverse FWT transformation matrix which has $4\times4$ spin and $3\times3$ color indices. After discretization, at the tree level $R$ is written as follows: $$\begin{aligned} R & = & \sum_{i}R^{(i)}, \\ \label{eq:R(1)} R^{(1)} & = & 1, \\ \label{eq:R(2)} R^{(2)} & = & - \frac{\mbox{\boldmath $\gamma$} \cdot \mbox{\boldmath $\Delta$} }{2m_Q}, \\ \label{eq:R(3)} R^{(3)} & = & \frac{\mbox{\boldmath $\Delta$}^{(2)}}{8m_Q^2}, \\ \label{eq:R(4)} R^{(4)} & = & \frac{g\mbox{\boldmath $\Sigma$} \cdot \mbox{\boldmath $B$}}{8m_Q^2} , \\ \label{eq:R(5)} R^{(5)} & = & - \frac{ig\gamma_4 \mbox{\boldmath $\gamma$} \cdot{\mbox{\boldmath $E$} }}{4m_Q^2}, \end{aligned}$$ where $$\begin{aligned} \Sigma^{j} = \left( \begin{array}{cc} \sigma^{j} & 0 \\ 0 & \sigma^{j} \\ \end{array} \right).\end{aligned}$$ We apply the tadpole improvement[@MFimp] to all link variables in the evolution equation and $R$ by rescaling the link variables as $U_{\mu} \rightarrow U_{\mu}/u_0$. We define two sets of action and current operator {$\delta H$,$R$} as follows, $$\begin{aligned} \mbox{setI} \equiv \{ \delta H_I,R_I \} \; & \mbox{and} & \; \mbox{setI$\!$I} \equiv \{ \delta H_{I\!I},R_{I\!I} \},\end{aligned}$$ where $$\begin{aligned} \delta H_I = \delta H^{(1)} \; & \mbox{and} & \; R_I = \sum_{i=1}^{2}R^{(i)}, \label{eq:R1} \\ \delta H_{I\!I} = \sum_{i=1}^{6}\delta H^{(i)} \; & \mbox{and} & \; R_{I\!I} = \sum_{i=1}^{5}R^{(i)}.\end{aligned}$$ $\delta H_1$ and $R_1$ include only $O(1/m_Q)$ terms while $\delta H_2$ and $R_2$ keep entire $O(1/m_Q^2)$ terms and the leading relativistic correction to the dispersion relation, which is an $O(1/m_Q^3)$ term. The terms improving the discretization errors appearing in $H_0$ and time evolution are also included. Using these two sets, we can realize two levels of accuracy of $O(1/m_Q)$ and $O(1/m_Q^2)$. Simulation methods ------------------ We have computed $f_B$ at $\beta=5.8$ on 120 $16^3\times32$ lattices with periodic boundary condition in the spatial direction and Dirichlet boundary condition in the temporal direction. The inverse lattice spacing $a^{-1}$ determined from $m_{\rho}$ is 1.714(63) GeV. For heavy quark, we use both mean-field improved $O(1/m_Q)$ and $O(1/m_Q^2)$ NRQCD action. We take six points for the heavy quark mass $a m_Q$ in a range 0.9–5.0 (1.5–8.5 GeV ). For light quark, we use Wilson action at $\kappa$= 0.1600, 0.1585, and 0.1570 ($k_{\rm crit}=0.16337$) which correspond to $m_s$–$2 m_s$. Results ------- We show our results of $f_P\sqrt{m_P}$ in Figure \[fig:fsqrtM\]. We find that the size of $O(1/m_P^2)$ correction is as small as about 3 % around the $B$ meson region and about 15 % around the $D$ meson region. =7.0cm In order to see how the correction terms in the rotation of operator (\[eq:R(2)\])–(\[eq:R(5)\]) changes the result, we show the contributions from each correction term to $f_P \sqrt{M_P}$ in Figure \[fig:correction\]. We find that $O(1/m_Q)$ corrections are rather large. On the other hand, each of the three $O(1/m_Q^2)$ correction is about 2 % around the $B$ meson region. There is a cancellation among the three corrections, and the total effect is of 3%. Although the effect of the higher order corrections is naively expected to be very small, there is no guarantee whether this cancellation takes place at higher order. We, therefore, estimate an upper bound for the $O(1/m_P^3)$ error to be of 6% at $B$ meson region. =7.0cm We study $1/m_P$ dependence of $f_P \sqrt{m_P}$ by fitting the data with the following form $$\begin{aligned} f_P \sqrt{m_P} & = & (f_P \sqrt{m_P})^{\infty} ( 1 + \frac{c_1}{m_P} + \frac{c_2}{m_P^2} + \cdots ), \nonumber\end{aligned}$$ for which we obtain $$\begin{aligned} (f_P \sqrt{m_P})^{\infty} & = & 0.308(20), \nonumber \\ c_1 & = & -0.87(11) \nonumber, \\ c_2 & = & 0.17(11) \nonumber. \end{aligned}$$ with the entire $O(1/m_Q^2)$ calculation. In physical units $c_1=-1.49(19)GeV$,$c_2=(0.71(23) {\rm GeV})^2$, To summarize, our analysis of $f_B$ shows that $O(1/m_Q^2)$ relativistic correction is of about 3 % and $1/m_P$ expansion from the static limit has a good behavior[@Hiroshima]. In order to obtain $f_B$ with higher precision, one has to control other systematic errors such as perturbative and discretization errors. We have not included one-loop correction for the renormalization constant, for which the calculation is underway. NRQCD group recently calculated the full one-loop renormalization factor for the decay constant with nonrelativistic heavy quark of $O(1/m_Q)$ and clover light quark[@Junko]. They find that the effect of operator $\partial P_5$ which appear at one-loop level significantly reduces the decay constant. We are also planning to carry out the simulations at higher $\beta$ values with $O(a)$-improved Wilson light quark to remove $O(a)$ error in near future. $B$ meson semi-leptonic decay form factors {#sec:form_factor} ========================================== In this section we report our study of $B \rightarrow \pi l \overline{\nu}$ semi-leptonic decay form factors. This is the first calculation with the NRQCD action. Earlier attempts to calculate the form factors were made by APE[@APE], UKQCD[@UKQCD], Wuppertal[@Wuppertal] by extrapolating the results in the D meson mass region obtained with the clover action assuming heavy quark scaling law. Direct simulation would be certainly necessary as an alternative approach just as in the calculation of $f_B$ in order to investigate the mass dependence and to obtain reliable results. The status of form factors with Wilson/Clover as well as Fermilab action is summarized in refs.[@Jim; @Flynn]. The semi-leptonic decay form factors $f^{+}$ and $f^{0}$ are defined as follows $$\begin{aligned} \lefteqn{ \langle\pi(k)|V_{\mu}|B(p)\rangle }\nonumber \\ & = & \left( p+k-q \frac{m_B^2-m_{\pi}^2}{q^2} \right)_{\mu} f^{+}(q^2) \nonumber \\ & & + q_{\mu} \frac{m_B^2-m_{\pi}^2}{q^2} f^{0}(q^2) \end{aligned}$$ where $q_{\mu}=p_{\mu}-k_{\mu}$ and $|B(p)\rangle$ has a normalization $$\label{eq:covariant_normalization} \langle B(p)|B(p^{\prime})\rangle = (2\pi)^3 2 E_B(p) \delta^3 (\mbox{\boldmath $p$}-\mbox{\boldmath $p$}^{\prime}).$$ The virtual $W$ boson mass $q^2$ takes a value in a region $0\le q^2 \le q_{max}^{2}$ where $q_{max}^2=(m_B-m_{\pi})^2$. In the rest frame of the initial $B$ meson, pion is also almost at rest in the large $q^2$ region ($q^2\approx q_{max}^2$), where the $W$ boson carries a large fraction of released energy from the $B$ meson. In the small $q^2$ region ($q^2\approx$ 0), on the other hand, the pion is strongly kicked and has a large spatial momentum ($\mbox{\boldmath $k$}_{\pi}\sim$ a few GeV/c). Lattice calculation is not reliably applicable for the small $q^2$ region, since the discretization error of $O(a\mbox{\boldmath $k$}_{\pi})$ becomes unacceptably large. This leads to a fundamental restriction in the kinematical region where lattice calculation may offer a reliable result. The differential decay rate is given as $$\begin{aligned} \lefteqn{ \frac{d\Gamma(\overline{B}^{0} \rightarrow \pi^{+}l^{-}\overline{\nu})}{dq^2} } \nonumber \\ & = & \frac{G_F^2 |V_{ub}|^2}{192\pi^3m_B^3} \lambda^{3/2}(q^2)|f^{+}(q^2)|^2\end{aligned}$$ where $$\lambda(q^2) = ( m_B^2 + m_{\pi}^2 -q^2 )^2 -4 m_B^2 m_{\pi}^2.$$ The decay rate vanishes at $q^2=q_{max}^2$ because the phase space gets smaller. It is, thus, essential to calculate $f^+(q^2)$ in a $q^2$ region where the experimental data will become available and the systematic error does not spoil the reliability, in order to determine $|V_{ub}|$ model independently. Another important feature of the $B$ meson semi-leptonic decay is the implication of the Heavy Quark Effective Theory (HQET)[@HQET]. In the heavy quark mass limit, it is more natural to normalize the heavy meson state as $$\langle \tilde{M}(p)|\tilde{M}(p^{\prime})\rangle = (2\pi)^3 2 \left( \frac{E_M(p)}{m_{M}} \right) \delta^3 (\mbox{\boldmath $p$}-\mbox{\boldmath $p$}^{\prime})$$ instead of the covariant normalization (\[eq:covariant\_normalization\]). With this normalization, the large mass scale $m_{M}$ is removed from the theory and one can use the heavy quark expansion. The amplitude may be expanded as $$\begin{aligned} \lefteqn{ \langle\pi(k)|V_{\mu}|\tilde{M}(p)\rangle } \nonumber \\ & = & \langle\pi(k)|V_{\mu}|M(p)\rangle / \sqrt{m_{M}} \nonumber \\ & = & X_{\mu}^{\infty}(v\cdot k) \nonumber \\ & & \times \left( 1 + \frac{c_1(v\cdot k)}{m_M} +\frac{c_2(v\cdot k)}{m_M^2} + \cdots \right) \label{eq:1/m-expansion}\end{aligned}$$ where $v$ is a velocity of the heavy meson and $c_1$, $c_2$, ... are functions of $v\cdot k$. It is worth to note that the heavy quark mass extrapolation and interpolation have to be done with $v\cdot k$ fixed. In the rest frame of the heavy meson, this condition implies fixed pion momentum, since $v\cdot k$ becomes $\sqrt{m_{\pi}^2 + \mbox{\boldmath $p$}_{\pi}^2}$. We propose to study the $1/M$ dependence of the following quantities $$\begin{aligned} \label{eq:v4} V_4(k,p) & \equiv & \frac{\langle\pi(k)|V^4(0)|B(p)\rangle} {\sqrt{2E_{\pi}}\sqrt{2E_{B}}}, \\ \label{eq:vi} V_k(k,p) & \equiv & \frac{ \langle\pi(k)| \frac{1}{k^2} \sum_i k^i \cdot V^i(0)|B(p)\rangle } {\sqrt{2E_{\pi}}\sqrt{2E_{B}}}, $$ which are natural generalization of $f_P \sqrt{M_P}$ for the the heavy-light decay constant. Indeed these quantities are almost raw number which one obtains in the lattice calculation as magnitudes of corresponding three-point functions, and then free from other ambiguities such as the choice of mass parameter of the heavy quark and the discretization of the spatial momenta. NRQCD action and simulation parameters -------------------------------------- The action we used for the semi-leptonic decay differs from that in the $f_B$ calculation. $$\begin{aligned} \lefteqn{ S = Q^{\dagger}(t,\mbox{\boldmath $x$}) [} \nonumber \\ & & \left( 1 - \frac{aH_0}{2n} \right)^{-n} U_4 \left( 1 - \frac{aH_0}{2n} \right)^{-n} Q(t+1,\mbox{\boldmath $x$}) \nonumber \\ & & - \left( 1 - a\delta H \right) Q(t,\mbox{\boldmath $x$}) ],\end{aligned}$$ where $$\begin{aligned} H_0 & = & - \frac{\Delta^{(2)}}{2m_Q}, \\ \delta H & = & - \frac{g}{2m_Q} \mbox{\boldmath $\sigma$} \cdot \mbox{\boldmath $B$}.\end{aligned}$$ Notation is the same as in section \[sec:action\_fB\]. The FWT transformation operator $R$ is identical to that in eq. (\[eq:R1\]) Numerical simulation has been done on the same 120 gauge configurations as was used for the decay constant. For the light quarks we use the Wilson fermion at $\kappa=$ 0.1570. The following six sets of parameters for the heavy quark mass and the stabilization parameter; $( m_Q, n )$ = (5.0,1), (2.6,1), (2.1,1), (1.5,2), (1.2,2), and (0.9,2). $m_Q=2.6$ and $0.9$ roughly correspond to $b$- and $c$-quark masses respectively. Extraction of three-point functions ----------------------------------- Matrix elements are extracted from the three-point correlation functions: $$\begin{aligned} \lefteqn{ C^{(3)}_{\mu}( \mbox{\boldmath $k$}, \mbox{\boldmath $p$}; t_B,t_V,t_{\pi} ) } \nonumber \\ & = & \sum_{x_f,x_s} e^{-i p\cdot x_B} e^{-i (k-p)\cdot x_V} \nonumber \\ & & \langle 0 | O_B(t_B,\mbox{\boldmath $x$}_B) V_{\mu}^{\dag}(t_V,\mbox{\boldmath $x$}_V) O^{\dag}_{\pi}(t_{\pi},0) | 0 \rangle \nonumber \\ & & \longrightarrow \frac{Z_{\pi}(k)}{ 2 E_{\pi}(k) } \frac{Z_B(p)}{ 2 E_B(p) } \nonumber \\ & & \times e^{ -E_{\pi}(k)(t_V-t_{\pi}) } e^{ -E_{B}(p)(t_B-t_V) } \nonumber \\ & & \times \langle B(p)| V_{\mu}^{\dag} | \pi(k) \rangle \nonumber \\ & & \;\;\; (\mbox{for}\;\;t_B \gg t_V \gg t_{\pi})\end{aligned}$$ where $t_B$, $t_V$ and $t_{\pi}$ indicate the location of the initial state $B$ meson, heavy-light current and the final state pion respectively. We fix $t_{\pi}=4$ and $t_V=14$, and $t_B$ is a variable. The amplitude $\langle\pi(k)|V_{\mu}|B(p)\rangle /\sqrt{2E_{\pi}}\sqrt{2E_B}$ is obtained by dividing the above expression by the corresponding two-point functions. Effective masses ---------------- In this section, we present our numerical results. In order to see whether the contamination from the excited state is sufficiently small, we examine the effective mass plots of the two-point functions of pion and $B$ meson, and three-point functions. =7.0cm Figure \[fig:efp\_pi\] shows the effective mass plot for the pion with finite spatial momenta up to $|a\mbox{\boldmath $k$}|=2(2\pi /16)$. Correlation functions seem to reach to the ground state beyond $t=14$, except for $a\mbox{\boldmath $k$}=(0,0,2)$ where statistical error becomes too large to extract reliable ground state energy. We, therefore, use momentum values up to $|a\mbox{\boldmath $k$}|=\sqrt{3}(2\pi /16)$ in the following analysis. The maximum momentum value corresponds to $\sim$ 1 GeV/c in the physical unit. =7.0cm The momentum dependence of the pion energy should obey the dispersion relation $$E_{\pi}(\mbox{\boldmath $k$})^2 = m_{\pi}^2 + \mbox{\boldmath $k$}^2 \label{eq:dr_cont}$$ in the continuum limit. Then the deviation of the lattice dispersion relation from the continuum one is a good indicator of the discretization error. In figure \[fig:light\_dr\] we plot the dispersion relation for $\pi$ and $\rho$ mesons measured in our simulations. Solid lines represent the above expression (\[eq:dr\_cont\]), which does not fit the measured point, showing the effect of the $O(a)$ discretization error. =7.0cm A similar plot for the $B$ meson is found in Figure \[fig:efp\_B\_L\], where the maximum momentum is $|a\mbox{\boldmath $k$}|=\sqrt{3}(2\pi /16)$. The fitting interval is chosen to be 16-24 where the contamination of excited state is negligible for each momentum values. =7.0cm =7.0cm Figures \[fig:effV4\] and \[fig:effVi\] show the effective mass plots for three point functions with temporal and spatial currents respectively. The location of the interpolating field of the $B$ meson is varied to define the effective mass, then the energy should coincide with that obtained from the two-point function if the ground state is sufficiently isolated. We observe clear plateau in a wide range for $t$ of 23-28 and the mass values consistent with that from the two-point functions, which are shown as solid lines in the figures. $1/m$ dependence ---------------- We are now confident that the ground state is reliably extracted for both two-point and three-point functions, and present the results for the matrix elements. Since $1/m$ dependence of the matrix elements is the main issue in this study, we plot $V_4$ and $V_k$, defined in (\[eq:v4\]) and (\[eq:vi\]) respectively, which obeys the simple heavy mass scaling law (\[eq:1/m-expansion\]), in Figures \[fig:V4\] and \[fig:Vki\]. We observe that $1/m$ dependence of matrix elements is rather small in contrast to the large $1/m$ correction for the heavy-light decay constant $f_{P}\sqrt{m_{P}}$. Although intuitive interpretation of this result is difficult, the smallness of the $1/m$ correction is a good news to obtain the form factor with high precision, because the error in setting the $b$-quark mass does not affect the prediction. And also this behavior is consistent with the previous works[@APE; @UKQCD; @Jim; @Wuppertal], in which the heavy quark mass is much smaller than ours. =7.0cm =7.0cm The $q^2$ dependence of the form factors $f^+$ and $f^0$ is shown in Figure \[fig:ff0201\]. The heavy quark mass is roughly corresponding to the $b$-quark. As we discussed previously the accessible $q^2$ region is rather restricted. It is, however, interesting that already at this stage one is able to see the momentum dependence which could really be tested by looking at the momentum spectrum data in the future $B$ factories. =7.0cm Discussion {#sec:discussion} ========== We have studied $1/m_Q$ dependence of the heavy-light decay constant and semi-leptonic decay form factors with NRQCD action. We find that the error of truncating higher order relativistic correction term is as small as 6% for the decay constant. We also find that the semi-leptonic decay form factor for $B \rightarrow \pi l\bar{\nu}$ has very small $1/m_Q$ dependence, which is consistent with the previous results in the Wilson/Clover approach. To obtain the physical result for extracting $V_{ub}$ matrix elements, chiral limit for the light quark must be taken and calculation of the renormalization constant $Z_V$ at one-loop is required, which is now underway. One of the largest problem is that so far, due to low statistics and the discretization errors of $O((a\mbox{\boldmath $k$})^2)$, lattice calculation works only for rather small recoil region, where the statistics of the experimental data is not high due to the phase space suppression. We are planning to carry out simulations with much higher statistics, with larger $\beta$ and with improved light quarks so that we can push up the accessible momentum region. It is also true that more data from CLEO, or future $B$ factories is required for relatively small recoil region $|\mbox{\boldmath $k$}_{\pi}| \sim$ 1 GeV, where the lattice calculation is most reliable. Acknowledgment ============== Numerical calculations have been done on Paragon XP/S at INSAM (Institute for Numerical Simulations and Applied Mathematics) in Hiroshima University. We are grateful to S. Hioki for allowing us to use his program to generate gauge configurations. We would like to thank J. Shigemitsu, C.T.H. Davies, J. Sloan and the members of JLQCD collaboration for useful discussions. H.M. would like to thank the Japan Society for the Promotion of Science for Young Scientists for a research fellowship. S.H. is supported by Ministry of Education, Science and Culture under grant number 09740226. and the members of JLQCD collaboration for useful discussions. H.M. would like to thank the Japan Society for the Promotion of Science for Young Scientists for a research fellowship. S.H. is supported by Ministry of Education, Science and Culture under grant number 09740226. [99]{} B. A. Thacker and G. P. Lepage, Phys. Rev. D[**43**]{}, 196 (1991); G.P. Lepage, L. Magnea, C. Nakhleh, U. Magnea and K. Hornbostel, Phys. Rev. [**D46**]{} 4052 (1992). A.X. El-Khadra, A.S. Kronfeld and P.B. Mackenzie, Phys. Rev. **D55** (1997) 3933. UKQCD Collaboration, presented by C. T. H. Davies, Nucl. Phys. B (Proc. Suppl.) [**30**]{}, 437 (1993). S. Hashimoto, Phys. Rev. [**D50**]{}, 4639 (1994). A. Ali Khan [*et al.*]{}, Nucl. Phys. B(Proc. Suppl.)[**47**]{}, 425 (1996); S. Collins [*et al.*]{}, Phys. Rev. [**D55**]{}, 1630 (1997); A. Ali Khan [*et al.*]{}, hep-lat/9704008. J. Shigemitsu, hep-lat/9705017. G.P. Lepage and P.B. Mackenzie, Phys. Rev. [**D48**]{}(1993)2250. K. Ishikawa, H.Matsufuru, T. Onogi, N. Yamada and S. Hashimoto hep-lat/9706008. A. Abada et al., Phys. Lett. [**B365**]{}, 275 (1996), C.R.Allton et al., Phys. Lett. [**B345**]{}, 513 (1995), A. Abada et al., Nucl. Phys. [**B416**]{}, 675 (1994). UKQCD collaboration, D.R. Burford et al., Nucl. Phys. [**B447**]{}, 425 (1995). G. Gusken, K. Schilling and G. Siegert, Nucl. Phys. B (Proc. Suppl.) [**47**]{}, 485 (1996). J.Simone, Nucl. Phys. B (Proc. Suppl.) [**47**]{}, 17 (1996). J.Flynn, Nucl. Phys. B (Proc. Suppl.) [**53**]{}, 168 (1997). For a review see, e.g. M. Neubert, Phys. Rep. [**245**]{}, 259 (1994). [^1]: Talk presented at the International workshop “Lattice QCD on Parallel Computers” ,March 1997, Tsukuba
--- abstract: 'Based on the hard-thermal-loop resummed improved ladder Dyson-Schwinger equation for the fermion mass function, we study how we can get the gauge invariant solution in the sense it satisfies the Ward identity. Properties of the “gauge-invariant” solutions are discussed.' address: | Institute for Natural Sciences, Nara University,\ Nara 631-8502, Japan\ E-mail: {nakk, yokotah, yoshidak}@daibutsu.nara-u.ac.jp author: - 'HISAO NAKKAGAWA, HIROSHI YOKOTA and KOJI YOSHIDA' title: | PHASE STRUCTURE OF THERMAL QED BASED ON THE HARD THERMAL LOOP IMPROVED LADDER DYSON-SCHWINGER EQUATION\ –A “GAUGE INVARIANT” SOLUTION– --- Introduction and summary {#aba:sec1} ======================== The Dyson-Schwinger equation (DSE) is a powerful tool to investigate with the analytic procedure the nonperturbative structure of field theories, such as the phase structure of gauge theories. However, the full DSEs are coupled integral equtions for several unknown functions, thus are hard to be solved without introducing appropriate approximations. We usually adopt the step-by-step approach to this problem, firstly approximate the integration kernel by the tree, or, ladder kernel, next use the improved ladder one, etc. Advantage of the DSE analysis lies in the possibility of such an systematic improvement through the analytic investigation. Analyses of the DSE have proven to be successful in studying the phase structure of vacuum gauge theories \[1-3\]. In the Landau gauge DSE with the ladder kernel for the fermion mass function in the vacuum QED, the fermion wave function renormalization constant is guaranteed to be unity \[1\], satisfying the Ward identity. Thus irrespective of the problem of the ladder approximation, the results obtained would be gauge invariant. Same analyses have been carried out in the finite temperature/density case with the ladder kernel \[4-6\], and with the hard-thermal-loop (HTL) resummed improved ladder kernel \[7\]. Results of Ref. \[7\] show that at finite temperature/density it is important to correctly analyse the physical mass function $\Sigma_R$, the mass function of the “unstable” quasiparticle in thermal field theories, and also to correctly take the dominant thermal effect into the interaction kernel. All the preceding analyses \[4-7\], however, suffer from the serious problem coming from the ladder approximation of the interaction kernel. Although in the vacuum case, despite the use of ladder kernel, in the analysis in the Landau gauge the Ward identity is guaranteed to be satisfied, at finite temperature /density there is no such guarantee. In fact, even in the Landau gauge the fermion wave function renormalization constant largely deviates from unity \[7,8\], being not even real. At finite temperature/density the results obtained from the ladder DSE explicitly violate the Ward identity, thus depend on the gauge, their physical meaning being obscure. In this paper, we worked out, in the analysis of the HTL resummed improved ladder DS equation for the fermion mass function in thermal QED, the procedure to get the gauge invariant solution in the sense it satisfies the Ward identity, and investigated the properties of the “gauge invariant” solution. Results of the present analysis are summarized as follows: \(1) We can determine the solution that satisfies the Ward identity, namely the fermion wave function renormalization constant being almost equal to unity. To get such a solution it is essential that the gauge parameter $\xi$ depends on the momentum of the gauge boson. \(2) The chiral phase transition in the massless thermal QED is confirmed to occur through the second order transition. \(3) Two critical exponents $\nu$ and $\eta$ are consistent with constant within the range of values of temperatures and couplingconstants under analysis: $\nu = 0.395, \eta = 0.518$. \(4) The effect of thermal fluctuation on the chiral symmetry breaking and/or restoration is smaller than that expected in the previous analysis in the Landau gauge \[7\]. The Dyson-Schwinger equation for the fermion self-energy function $\Sigma_R$ ============================================================================ The fermion self-energy function $\Sigma_R$ appearing in the fermion propagator $$S_R(P) = [ P\!\!\!\!/ + i \epsilon \gamma^0 - \Sigma_R(P)]^{-1}$$ can be decomposed at finite temperature and/or density as $$\Sigma_R(P) = (1 - A(P)) p_i \gamma^i - B(P) \gamma^0 + C(P)$$ with $A(P)$, $B(P)$ and $C(P)$ being the three scalar invariants to be determined. In the present analysis, we use the HTL resummed form $^*G^{\mu \nu}$ for the gauge boson propagator $G^{\mu \nu}$ $${}^*G^{\mu\nu} (K) = \frac{1}{{}^*\Pi_T -K^2 - i \epsilon k_0} A^{\mu \nu} + \frac{1}{{}^*\Pi_L -K^2 - i \epsilon k_0} B^{\mu \nu} - \frac{\xi}{K^2 + i \epsilon k_0} D^{\mu \nu}$$ where $^*\Pi_{L/T}$ is the HTL resummed longitudinal/transverse photon self-energy function \[9\], and $A^{\mu \nu}$, $B^{\mu \nu}$ and $D^{\mu \nu}$ are the projection tensors \[10\], $$A^{\mu \nu}=g^{\mu \nu} - B^{\mu \nu}- D^{\mu \nu}, B^{\mu \nu}=- \tilde{K}^{\mu} \tilde{K}^{\nu}/K^2, D^{\mu \nu}= K^{\mu} K^{\nu}/K^2,$$ where $\tilde{K}=(k, k_0{\bf \hat{k}})$, $k=\sqrt{{\bf k}^2}$ and ${\bf \hat{k}}={\bf k}/k$ denotes the unit three vector along ${\bf k}$. The parameter $\xi$ appearing in the term proportional to the projection tensor $D_{\mu \nu}$ represents the gauge-fixing parameter ($\xi=0$ in the Landau gauge). This gauge term plays an important role in the present analysis. The vertex function is approximated by the tree (point) vertex. With the instantaneous exchange approximation for the longitudinal photon mode, we get the DSEs for the three invariant functions $A(P)$, $B(P)$ and $C(P)$ $$\begin{aligned} & & -p^2[1-A(P)] = -e^2 \left. \int \frac{d^4K}{(2 \pi)^4} \right[ \{1+2n_B(p_0-k_0) \} \mbox{Im}[\ ^*G^{\rho \sigma}_R(P-K)] \times \nonumber \\ & & \Bigl[ \{ K_{\sigma}P_{\rho} + K_{\rho} P_{\sigma} - p_0 (K_{\sigma} g_{\rho 0} + K_{\rho} g_{\sigma 0} ) - k_0 (P_{\sigma} g_{\rho 0} + P_{\rho} g_{\sigma 0} ) + pkz g_{\sigma \rho} \nonumber \\ & & + 2p_0k_0g_{\sigma 0}g_{\rho 0} \}\frac{A(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } + \{ P_{\sigma} g_{\rho 0} \nonumber \\ & & + P_{\rho} g_{\sigma 0} - 2p_0 g_{\sigma 0} g_{\rho 0} \} \frac{k_0+B(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } \Bigr] \nonumber \\ & & + \{1-2n_F(k_0) \} \ ^*G^{\rho \sigma}_R(P-K) \mbox{Im} \Bigl[ \{ K_{\sigma}P_{\rho} + K_{\rho} P_{\sigma} - p_0 (K_{\sigma} g_{\rho 0} + K_{\rho} g_{\sigma 0} ) \nonumber \\ & & - k_0 (P_{\sigma}g_{\rho 0} + P_{\rho} g_{\sigma 0} ) + pkz g_{\sigma \rho} + 2p_0k_0g_{\sigma 0}g_{\rho 0}\} \times \nonumber \\ & & \frac{A(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2-C(K)^2 } + \{ P_{\sigma} g_{\rho 0} + P_{\rho} g_{\sigma 0} \nonumber \\ & & \left. - 2p_0 g_{\sigma 0} g_{\rho 0} \} \frac{k_0+B(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } \Bigr] \right] \ ,\end{aligned}$$ $$\begin{aligned} & & - B(P)= -e^2 \left. \int \frac{d^4K}{(2 \pi)^4} \right[ \{1+2_B(p_0-k_0)\} \mbox{Im}[\ ^*G^{\rho \sigma}_R(P-K)] \times \nonumber \\ & & \Bigl[ \{ K_{\sigma} g_{\rho 0} + K_{\rho} g_{\sigma 0} - 2k_0 g_{\sigma 0} g_{\rho 0} \} \frac{A(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } \nonumber \\ & & + \{ 2g_{\rho 0} 2g_{\sigma 0} - g_{\sigma \rho} \} \frac{k_0+B(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2-C(K)^2 } \Bigr] \nonumber \\ & & + \{1-2n_F(k_0) \} \ ^*G^{\rho \sigma}_R(P-K) \mbox{Im} \Bigl[ \frac{A(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } \nonumber \\ & & \times \{ K_{\sigma} g_{\rho 0} + K_{\rho} g_{\sigma 0} - 2k_0 g_{\sigma 0} g_{\rho 0} \} \nonumber \\ & & \left. + \frac{k_0+B(K)}{[k_0+B(K)+ i \epsilon]^2 - A(K)^2k^2-C(K)^2 } \{ 2g_{\rho 0} 2g_{\sigma 0} - g_{\sigma \rho} \} \Bigr] \right] \ , \\ & & C(P) = -e^2 \int \frac{d^4K}{(2 \pi)^4} g_{\sigma \rho} \{1+2_B(p_0-k_0) \} \mbox{Im}[\ ^*G^{\rho \sigma}_R(P-K)] \times \nonumber \\ & & \Bigl[ \frac{C(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } + \{1-2n_F(k_0) \} \times \nonumber \\ & & \left. \ ^*G^{\rho \sigma}_R(P-K) \mbox{Im} \Bigl[ \frac{C(K)}{[k_0+B(K)+i \epsilon]^2 - A(K)^2k^2 -C(K)^2 } \Bigr] \right] \ .\end{aligned}$$ The function $A(P)$ is nothing but the inverse of the fermion wave function renormalization constant $Z_2$, thus must be unity in order to satisfy the Ward identity in the ladder DSE analysis, where the vertex function receives no renormalization effect, $Z_1 =1$. We must solve the above DSEs and get the solution satisfying the Ward identity $Z_2 = Z_1 (=1)$, where $Z_2 = A(P)^{-1}$. The procedure to get the “gauge invariant” solution is as follows; \(1) Assume the nonlinear gauge such that the gauge parameter $\xi$ to be a function of the photon momentum $K = (k_0, {\bf k})$, and parametrize $\xi$ as $$\xi(k_0, k) = \sum \xi_{mn} H_m(k_0) L_n(k) ,$$ where $\xi_{mn}$ are unknown parameters to be determined, $H_m$ the Hermite functions and $L_n$ the Laguerre functions. \(2) In solving DSEs iteratively, impose the condition $A(P) =1$ by constraint for the input-functions at each step of the iteration. \(3) Determine $\xi_{mn}$ so as to minimize $|A(P) -1|^2$ for the out-put functions and find the solutions for $B(P)$ and $C(P)$. Results of the analysis „Ÿ“gauge invariant” solution„Ÿ ====================================================== Here we present the results obtained by allowing the gauge parameter $\xi$ to be a complex value. Number of parameters $\xi_{mn}$ to minimize $|A(P) -1|^2$ is $2 \times 5 \times 2=20$ (i.e., $m=1 \sim 4$ and $n=1, 2$). All the quantities with the mass dimension are evaluated in the unit of $\Lambda$, the cut-off parameter introduced as usual to regularize the DSEs. Now we present the solution consistent with the Ward identity, i.e., the “auge invariant” solution. Firstly in Fig.1 we show $\mbox{Re}{A(P)}$. For comparison, results in the constant $\xi$ analyses are also shown in the same figure. Next let us study the property of the phase transition. Fig.2(a) shows the real part of the fermion mass $\mbox{Re}{M (P)}$, $M(P) \equiv C(P)/A(P)$, obtained from the “gauge invariant” solution, as a function of the temperature $T$ . The mass is evaluated at $p_0=0$, $p=0.1$, to be consistent with the standard prescription to define the mass in the static limit, $p_0=0$, $p \to 0$.    The curves in the figure show the best-fit curves in determining the critical temperature $T_c$ and the critical exponent $\nu$, by fitting, at each coupling constant $\alpha$ and near the critical temperature $T_c$ , the temperature-dependent data of $\mbox{Re}{M (P)}$ to the functional form $$\mbox{Re}{M (P)} = C_T(T_c - T)^{\nu} .$$ Also shown in Fig.2(b) is the $\mbox{Re}{M (P)}$ obtained from the “gauge invariant" solution, as a function of the coupling constant $\alpha$. The mass is evaluated at $p_0=0$, $p=0.1$ as above. The curves in the figure show the best-fit curves in determining the critical coupling $\alpha_c$ and the critical exponent $\eta$, by fitting, at each temperature T and near the critical coupling $\alpha_c$, the coupling-dependent data of $\mbox{Re}{M (P)}$ to the functional form $$\mbox{Re}{M (P)} = C_{\alpha}(\alpha - \alpha_c)^{\eta} .$$ The determined critical exponents are given in Table 1. $\alpha$ $\nu$         $T$ $\eta$ ---------- --------- --------- ------- --------- 3.5 0.42800 0.115 0.54718 4.0 0.38126 0.120 0.57872 4.5 0.36420 0.125 0.51430 5.0 0.40579 0.130 0.46153 The averaged value of $\nu$ over the various coupling is $<\mu> = 0.395$, which fits to all the data $\mbox{Re}{M (P)}$ in Fig.2(a) irrespective of the value of the coupling constant. The averaged value of $\eta$ over various temperatures is $<\eta> = 0.518$, which fits to all the data $\mbox{Re}{M (P)}$ in Fig.2(b) irrespective of the value of the temperature. The phase boundary curve in the ($T,\alpha$)-plane thus determined shows that the region of the symmetry broken phase shrinks to the low-temperature-strong- coupling side compared with that of the Landau gauge. This fact means that the effect of thermal fluctuation on the chiral symmetry breaking/restoration is smaller than that expected in the previous analysis in the Landau gauge \[7\]. Discussion and comments ======================= Results presented in the present paper are preliminary, because of the rough analysis of the data processing. We are now refining the data analysis and soon get the results of the thorough reanalysis. Though the main conclusion will not be altered, several remarks should be added. \(1) Present analysis was performed by allowing the gauge parameter $\xi$ to be a complex value. Such a choice of gauge may correspond to studying the non-hermite dynamics, thus may cause some troubles. What happens if we restrict the gauge parameter to the real value? We are studying this case, finding a remarkable result: In both cases results completely agree, thus getting a solution totally independent of the choice of gauges. \(2) In the present analysis, the consistency of the solution with the Ward identity is respected only by imposing the condition $A(P) \approx 1$. Needless to say, in solving the (improved) ladder Dyson-Schwinger equation, there are no solutions totally consistent with the Ward identity. Despite that fact, following point should be closely examined: At least around or in the static limit, $p_0=0$, $p \to 0$, where we calculated (defined) the mass, each invariant function $A(P)$, $B(P)$ or $C(P)$ should not have big momentum dependence. This condition may be important in connection with the consistency of the obtained solution with the gauge invariance. Result of the present analysis shows that at least $B(P)$ and $C(P)$ satisfy this condition. [99]{} T. Maskawa and H. Nakajima, [*Prog. Theor. 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--- abstract: | To study operator algebras with symmetries in a wide sense we introduce a notion of [*relative convolution operators*]{} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolution etc.) and their different combinations. Basic properties of relative convolutions are given and a connection with usual convolutions is established. Presented examples show that relative convolutions provide us with a base for systematical applications of harmonic analysis to PDO theory, complex and hypercomplex analysis, coherent states, wavelet transform and quantum theory. address: 'School of Mathematics, University of Leeds, Leeds LS29JT, UK' author: - 'Vladimir V. Kisil' bibliography: - 'mrabbrev.bib' - 'akisil.bib' - 'analyse.bib' - 'aphysics.bib' date: 'September 2, 1994' title: | Relative Convolutions. I\ *Properties and Applications* --- [^1] [^2] Introduction ============ Convolution operators and different operators associated with them are very important in mathematics (for example, in complex analysis [@GreSte77]) and physics (for example, quantum mechanics [@Folland89] and [@Woodhouse80 Chap. 3]). The fundamental role of such operators may be easily explained. Indeed, a notion of symmetries and group transforms invariance are the basis of the contemporary science. It is well known, that operators, which are invariant under a transitive group operation, may be realized as convolution operators on the group. However, there are some limitations for an application of convolution operators: - It is not a rare case when a operators symmetry group and functions domain have different dimensions. - Group convolution operators are generated by transformations of function domains. However, mathematical objects are often connected not only with domain transforms but also with alterations of function range or even both of them. The paper introduces a notion of [*relative convolutions*]{}, which allows us to overcome these limitations. Naturally, such an interesting object cannot be totally unknown in mathematics and we will give a short description of connected ones in Remark \[re:origin\]. In this paper we work only with continuous groups of symmetry. Discrete groups or their mixtures with continuous ones may be also considered. A geometric group structure and related objects (especially the non-commutative Fourier transform) are obviously determining for properties of convolution algebra. If the group structure is used not only as a basis for particular calculations but also in a more general framework, then it is possible to find the proper level of generality for obtained results[^3]. Be found, that geometry of Lie groups and algebras jointly with properties of kernels determine representations of relative convolution algebras. Namely, all representations of relative convolution algebras will be induced by selected representations of corresponding Lie algebras (Lie groups) and the selection of representations will depend on kernels properties. The representation theory of Lie algebras is complicated and still unsolved completely. But, due to the author personal interest, the main examples are nilpotent Lie algebras fully described by the Kirillov theory (see [@Kirillov62] or [@MTaylor86 Chap. 6]). Considered examples will show that such a restriction still leaves enough space for very interesting applications. The layout is as follows. In Section \[se:relative\] we give necessary notations, introduce the main object of the paper — [*relative convolution*]{} — and show basic examples. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolutions etc.) and their different combinations. Basic properties of relative convolutions are given in Section \[se:properties\]. We prove a formula for the composition of relative convolutions and deduce from it that [*an algebra of relative convolutions induced by a Lie algebra is a representation of the algebra of group convolutions on the Lie group $\object{Exp}\algebra{g}$*]{}. We also state the universal role PDO and the Heisenberg group in the theory of relative convolution operators. Relative convolutions provide us with a tool for the systematic usage of harmonic analysis in different fields of pure and applied mathematics. This statement will be illustrated in Sections \[se:analysis\] and \[se:physics\] by many examples, which show applications of relative convolutions to the theory of PDO, complex and hypercomplex analysis, coherent states, wavelets and quantum mechanics. In Section \[se:coherent\] we will describe our view on a notion of coherent states. The main observation is: a direct employment of group structures of coherent states gives us the uniform theory for the Bargmann, Bergman and Szegö projectors at the Segal–Bargmann (Fock), Bergman and Hardy spaces respectively. Applications to wavelets (and other) theory are also possible. The author gratefully acknowledge his inspiration by papers [@Dynin75; @Dynin76; @Howe80a; @Howe80b; @MTaylor84]. Besides, there is some interference with the coherent recent paper of Folland [@Folland94]. The author is also grateful to Dr. V. V. Kravchenko and Prof. N. L. Vasilevski for helpful discussions. Relative Convolutions {#se:relative} ===================== Definitions and Notations {#ss:notation} ------------------------- Let $G$ be a connected simply connected Lie group, let $\algebra{g}$ be its finite-dimensional Lie algebra. By the usual way we can identify with for $N=\dim\algebra{g}$ as vector spaces. The exponential map $\exp: \algebra{g}\rightarrow G$ [@Kirillov76 § 6.4] identifies the group $G$ and its algebra . So $G$ as a real $C^\infty$-manyfold has the dimension $N$. Let fix a frame $\{X_j\}_{1\leq j\leq N}$ of . Via the exponential map we can write a decomposition $g=\sum_1^N x_j X_j$ in [*exponential coordinates*]{} for every $g\in G$. The group law on $G$ in the exponential coordinates may be expressed via the [*Campbell–Hausdorff formula*]{} [@Kirillov76 § 6.4]: $$\label{eq:Campbell-Hausdorff} g*h=\sum_{m=1}^\infty\frac{(-1)^{m-1}}{m} \sum_{\vbox{\hbox{\hfil\makebox[\width]{\ $\scriptstyle k_j+l_j\geq 1$}\hfil} \hbox{\makebox[\width]{$\scriptstyle k_j\geq 1,\, l_j\geq 1$}}}} \frac{[x^{k_1}y^{l_1}\ldots x^{k_m}y^{l_m}]}{k_1!l_1!\ldots k_m!l_m!},$$ where $[x_1x_2\ldots x_n]=\frac{1}{n}[\ldots[[x_1,x_2],x_3],\ldots,x_n]$ and $$g=\sum_1^N x_j X_j,\ h=\sum_1^N y_j X_j.$$ It seems reasonable to introduce a short notation for the right side of  , we select ${\object{CH}}[\sum_1^N x_j X_j,\sum_1^N y_j X_j]$ or ${\object{CH}}[ x,y]$ (if the frame is obvious). Let $S$ be a set and let be defined an operation $G: S\rightarrow S$ of $G$ on $S$. If we fix a point $s\in S$ then the set of elements $G_s=\{g\in G\such g(s)=s\}$ obviously forms the [*isotropy (sub)group of $s$ in $G$*]{} [@Lang69 § I.5]. There is an equivalence relation on $S$, say, $s_1\sim s_2 \Leftrightarrow \exists g\in G: gs_1=s_2$, with respect to which $S$ is a disjoint union of distinct orbits [@Lang69 § I.5]. Thus from now on, without lost of a generality, we assume that the operation of $G$ on $S$ is [*transitive*]{}, i. e. for every $s\in S$ we have $$Gs:=\relstack{\bigcup}{g\in G} g(s)=S.$$ If $G$ is a Lie group then the [*homogeneous space*]{} $G/G_s$ is also a Lie group for every $s\in S$. Therefore the one-to-one mapping $G/G_s \rightarrow S: g\mapsto g(s)$ induces a structure of $C^\infty$-manifold on $S$. Thus the class $\FSpace[\infty]{C}{0}(S)$ of smooth functions with compact supports on $S$ has the evident definition. A smooth measure $d\nu$ on $S$ is called [*invariant*]{} (the [*Haar measure*]{}) with respect to an operation of $G$ on $S$ if $$\int_S f(s) d\nu(s)\nonumber= \int_S f(g(s)) d\nu(s), \mbox{ for all } g\in G,\ f(s)\in\FSpace[\infty]{C}{0}(S).\label{eq:invar-m}$$ The Haar measure always exists and is uniquely defined up to a scalar multiplier [@MTaylor86 § 0.2]. An equivalent formulation of  is: [*$G$ operate on $\FSpace{L}{2}(S,d\nu)$ by unitary operators*]{}. We will transfer the Haar measure $d\mu$ from $G$ to via the exponential map $\exp: \algebra{g}\rightarrow G$ and will call it as the [*invariant measure on a Lie algebra* ]{}. Now we can define an operation of a Lie algebra in the space $\FSpace[\infty]{C}{0}(S)$ induced by an operation of $G$ on $S$. Let $X\in \algebra{g}$ and $f(s)\in \FSpace[\infty]{C}{0}(S)$ then $$\label{eq:alg-limit} \lim_{t\rightarrow 0}\frac{f(e^{tX}s)-f(s)}{it} \in \FSpace[\infty]{C}{0}(S),$$ where $\exp: X\mapsto e^{tX}$ is the exponential map $\exp: \algebra{g}\rightarrow G$. The value of limit  will be denoted by $[Xf](s)$. If $S$ is equipped by a measure $d\nu$ we can define an [*adjoint*]{} operation $X^*$ of on $\FSpace{L}{2}(S,d\nu)$ by natural formula $\scalar{f(s)}{[X^* g](s)}:=\scalar{[Xf](s)}{g(s)}$. The invariance  of $d\nu$ may be reformulated in the terms of the Lie algebra as follows: $$X_j^* =X_j,$$ for every $X_j,\ 1\leq j\leq N$ from a frame of . Summarizing, $X: f\rightarrow [Xf]$ is a selfadjoint (possibly unbounded) operator in $\FSpace{L}{2}(S,d\nu)$ with invariant measure $d\nu$. In general, we will speak on an [*operation*]{} of a Lie algebra on a manifold $S$ with a measure $d\nu$ if there is a linear representation of by selfadjoint operators on the linear space $\FSpace{L}{2}(S,d\nu)$. As usual, if $X$ is selfadjoint on $\FSpace{L}{2}$ then $e^{itX}$ is unitary on . Clearly, every operation of a Lie group on $S$ induces an operation of the corresponding Lie algebra, but inverse is not true generally speaking (see Example \[ex:multiplication\]). Therefore we will based on the notion of a [*Lie algebra*]{} operation. The succeeding object will be useful in our study of convolution algebras representations. We define the [*kernel*]{} of operation of on $S$ as follows: $$\label{eq:def-kernel} {\object{Ker}}(\algebra{g},S)=\{X\in \algebra{g} \such [Xf](x)=0 \mbox{ for all } f(x)\in\FSpace[\infty]{C}{0}(S)\}.$$ If an operation of on $S$ is induced by an operation of $G$ on $S$ then $X\in{\object{Ker}}(\algebra{g},S)$ if and only if for $e^X\in G$ and for any $s\in S$ we have $e^X(s)=s $. There is no doubt that ${\object{Ker}}(\algebra{g},S)$ is a two-sided ideal of (and therefore a linear subspace). Thus we can introduce the quotient Lie algebra ${\object{Ess}}(\algebra{g},S)=\algebra{g}/{\object{Ker}}(\algebra{g},S)$ (see [@Kirillov76 § 6.2]). An induced action of ${\object{Ess}}(\algebra{g},S)$ on $S$ is evidently specified. Now we can describe the main object of the paper. \[de:relative\] Let we have a (selfadjoint) operation of a Lie algebra on $S$ (possibly induced by an operation of a Lie group $G$ on a set $S$) and let $\{X_j\}_{1\leq j\leq N}$ be a fixed frame of . We will define the [*operator of relative convolution*]{} $K$ induced by on $E=\FSpace[\infty]{C}{0}(S)$ with a kernel $k(x)\in \FSpace[\infty]{C}{0}(\Space{R}{N})$ by the formula $$\label{eq:relative} K=(2\pi)^{-N/2} \int_{\Space{R}{N}} \widehat{k}(x_1,x_2,\ldots,x_N)\,e^{i\sum_1^N x_j X_j}\, dx,$$ where integration is made with respect to an invariant measure on $\algebra{g}\cong\Space{R}{N}$. Here $\widehat{k}(s)$ is the Fourier transform of the function $k(s)$ over : $$\widehat{k}(x)= (2\pi)^{-N/2}\int_{\Space{R}{N}}k(y)\,e^{-iyx}\,dy.$$ \[re:origin\] This definition has an origin at the [*Weyl functional calculus*]{} [@HWeyl] (or the Weyl quantization procedure), see Example \[ex:pdo\] for details. Feynman in [@Feynman51] proposed its extension — the [*functional calculus of ordered operators*]{} — in a very similar way. Anderson [@Anderson69] introduced a generalization of the Weyl calculus for arbitrary set of self-adjoint operators in a Banach space exactly by formula . A description of different operator calculuses may be found in [@Maslov73]. But it was shown by R. Howe [@Howe80a; @Howe80b], that success of the original Weyl calculus is intimately connected with the structure of the Heisenberg group and its different representations. Thus one can obtain a new fruitful branch in this direction making an assumption, that the operators $X_j$ in  are not arbitrary but are connected with some group structure. Such a treatment for the Heisenberg group and multipliers may be found at the Dynin paper [@Dynin75]. M. E. Taylor in [@MTaylor84] introduced “smooth families of convolution operators”, which technically coincide with relative convolutions in many important cases. However, full symmetry groups of such smooth families are not clear and thus cannot be exploited. Efforts to study simultaneously both the group action and operators of multiplication bring also a very similar object at the recent paper[^4] of Folland [@Folland94]. But general consideration of relative convolutions seems to be new (as well as systematical applications to new and already solved problems, see Sections \[se:analysis\] and \[se:physics\]). Due to paper [@Anderson69] our definition is correct for any representation of a Lie algebra in a Banach space. Thus we can use similar \[de:relative1\] Let we have a selfadjoint representation of a Lie algebra on a Banach space $B$ and let $\{X_j\}_{1\leq j\leq N}$ be operators represented a fixed frame of . We will define the [*operator of relative convolution*]{} $K$ induced by on $B$ by the formula $$\label{eq:relative1} K=(2\pi)^{-N/2} \int_{\Space{R}{N}} \widehat{k}(x_1,x_2,\ldots,x_N)\,e^{i\sum_1^N x_j X_j}\, dx.$$ Here integration is made with respect to an invariant measure on $\algebra{g}\cong\Space{R}{N}$. \[re:function\] As was shown at [@Anderson69], formula  may be treated as a definition of a function $k(X_1,X_2,\ldots,X_N)$ of operators $X_j$. See Remark \[re:riesz\] for an alternative Riesz-like definition of relative convolutions. We have defined relative convolutions only for a very restricted class of kernels $k(x)$ and a specific space $E$. Of course, both of them may be enlarged in the proper context. Another interesting modification may be required by a consideration of discrete groups and their actions. It may be achieved by a replacing integration in  by summation or integration by discrete measures. But in this paper we will not consider such cases. As we will see, relative convolutions are naturally defined not only for group operation on [*functions domains*]{} but also on [*ranges of function*]{} (see Examples \[ex:multiplication\], \[ex:fock\] and \[ex:clifford\]). Basic Examples -------------- The following Examples make clear the relationships between the relative convolutions and usual ones. \[ex:euclid-conv\] Let $G=\Space{R}{N}$ and $G$ operates on $S=\Space{R}{N}$ as a group of Euclidean shifts $g: y\rightarrow y+g$. The algebra consists of selfadjoint differential operators spanned on the frame $ X_j^e= \frac{1}{i} \frac{\partial}{\partial y_j},\ 1\leq j\leq N$. Then: $$\begin{aligned} [Kf](y)&=& (2\pi)^{-N/2}\int_{\Space{R}{N}} \widehat{k}(x)\,e^{i\sum_1^N -x_j (\frac{1}{i}\frac{\partial }{\partial y_j})}f(y)\,dx\\ &=& (2\pi)^{-N/2}\int_{\Space{R}{N}} \widehat{k}(x)\,e^{-\sum_1^N x_j\frac{\partial }{\partial y_j}}\,f(y)\,dx\\ &=&(2\pi)^{-N/2} \int_{\Space{R}{N}} \widehat{k}(x)\,f(y-x)\,dx .\end{aligned}$$ Otherwise, an operator of relative convolution with a kernel $k(x)$ evidently coincides with a usual (Euclidean) convolution on with the kernel $\widehat{k}(x)$. This Example can be obviously generalized for an arbitrary Lie group $G$ and the set $S=G$ with the natural left[^5] (or right) operation of $G$ on $S=G$ $$G: S \rightarrow GS\ (SG): g' \mapsto g^{-1}g'\ (g' \mapsto g'g),\ g\in G,\ g'\in S=G.$$ It is clear, that relative convolutions will coincide with the left (right) group convolutions on $G$ (with, may be, transformed kernels). \[ex:multiplication\] Let $G=\Space{R}{N}$ with the Lie algebra spanned on operators of multiplication $X_j=M_{y_j}$ by $y_j,\ 1\leq j \leq N$ on a space of functions on $S=\Space{R}{N}$. Then a relative convolution $K$ with a kernel $k(x)$ $$\begin{aligned} [Kf](y)&=&(2\pi)^{-N/2}\int_{\Space{R}{N}} \widehat{k}(x)\,e^{ixy}\,f(y)\,dx\\ &=& k(y)f(y)\end{aligned}$$ is simply an operator of multiplication by $k(y)$. In this case the operation of the Lie algebra on $S$ is not generated by an operation of a Lie group on $S$. Of course, it is possible to establish a connection with Example \[ex:euclid-conv\] through the Fourier transform, but it is not so simple in other cases. Particularly it will happen when one generalizes this Example by use more complicated transformations of range than a multiplication by scalars (see Example \[ex:clifford\]) or simultaneously applies transformations from this and the previous Examples (see Example \[ex:pdo\]). \[ex:two-sided\] Let be the Heisenberg group [@Dynin75; @MTaylor84; @MTaylor86]. The Heisenberg group is a step 2 nilpotent Lie group. As a $C^\infty -$manifold it coincides with $\Space{ R}{2n+1}$. If an element of it is given in the form $g=(u,v)\in \Space{H }n$, where $u\in \Space{ R}{}$ and $v=(v_1,\dots, v_n)\in \Space{ C}n$, then the group law on $ \Space{ H}{n}$ can be written as $$\label{eq:h-low} (u,v)*(u',v')=\left(u+u'-\frac{1}{2} \object{Im} \sum_1^n v'_k \bar{v}_k,\ v_1+v'_1,\ldots, v_n+v'_n\right) .$$ We single out on $\Space{ H}n$ the group of nonisotropic dilations $\{\delta_\tau\}$, $\tau\in \Space{ R}{}_+$: $$\delta_\tau(u,v)=(\tau^2u,\tau v).$$ Functions with the property $$\label{eq:homohen} (f\circ \delta_\tau)(g)=\tau^kf(g)$$ will be called [*$\delta_\tau$-homogeneous functions of degree $k$*]{}. The left and right Haar measure on the Heisenberg group coincides with the Lebesgue measure. Let us introduce the right $\pi_{r}$ and the left $\pi_{l}$ regular representations of on $\FSpace{L}{2}(\Heisen{n})$: $$\begin{aligned} {} [\pi_{l} (g) f] (h) &=& f(g^{-1}*h),\label{eq:lf-shift}\\ {} [\pi_{r} (g) f] (h) &=& f(h*g). \label{eq:rt-shift} \end{aligned}$$ Thereafter, $G=\Heisen{n}\times\Heisen{n}$, $S=\Heisen{n}$ and an operation of $G$ on $S$ is defined by the [*two-sided*]{} shift $$G: s \rightarrow g^{-1}_1*s*g_2,\ s\in S=\Heisen{n}, (g_1,g_2)\in G=\Heisen{n} \times\Heisen{n}.$$ The Lie algebra of $G$ is the direct sum $\algebra{g}=\algebra{h}_n\oplus\algebra{h}_n$ of two copies of the Lie algebra $\algebra{h}_n$ of the Heisenberg group . If in $S\cong\Space{R}{N}$ we define Cartesian coordinates $y_j,\ 0\leq j< N=2n+1$, then a frame of may be written as follows $$\begin{aligned} \displaystyle X^l_0=\frac{1}{i}\frac{\partial }{\partial y_0}, & \displaystyle X^l_j=\frac{1}{i}\left(\frac{\partial}{\partial y_{j+n}}-2y_j\frac{\partial }{\partial y_0 }\right), &\displaystyle X^l_{j+n}=\frac{1}{i}\left(\frac{\partial}{\partial y_{j}}+2y_{j+n}\frac{\partial }{\partial y_0 }\right),\nonumber \\ \label{eq:frame-left}\\ \displaystyle X^r_0=-\frac{1}{i}\frac{\partial }{\partial y_0},&\displaystyle X^r_j=\frac{1}{i}\left(\frac{\partial}{\partial y_{j+n}}+2y_j\frac{\partial }{\partial y_0 }\right), &\displaystyle X^r_{j+n}=\frac{1}{i}\left(\frac{\partial}{\partial y_{j}}- 2y_{j+n}\frac{\partial }{\partial y_0 }\right),\nonumber \\ \label{eq:frame-right}\end{aligned}$$ where $1\leq j\leq n$. Vector fields $X^l_j,\ 0\leq j < N$ generate the left shifts  on $S=\Heisen{n}$ and $X^r_j,\ 0\leq j < N$ generate the right ones . $X^l_j$ (correspondingly $X^r_j)$ satisfy to the famous Heisenberg commutation relation $$\label{eq:heisen-comm} [X^{l(r)}_j,X^{l(r)}_{j+n}]=-[X^{l(r)}_{j+n},X^{l(r)}_{j}]= X^{l(r)}_{0}$$ and all other commutators being zero. Particulary, all $X^l_j$ commute with all $X^r_j$. Vector fields $X^l_0$ and $-X^r_0$ are different as elements of , but have coinciding operations on functions. It is easy to see, that ${\object{Ker}}(\algebra{g},S)$ is a linear span of the vector $X^l_0+X^r_0$ and ${\object{Ess}}(\algebra{g},S)=\Heisen{2n}$. Now formula  with kernel $k(x),\ x\in\Space{R}{4n+2} $ defines the [*two-sided convolutions*]{} studied in [@VasTru94]: $$\begin{aligned} [Kf](y)&=&(2\pi)^{-N}\int_{\Space{R}{4n+2}} \widehat{k}(x)\,e^{i\sum_1^{2N}x_j X_j}\, f(y)\,dx\\ &\stackrel{(*)}{=}&(2\pi)^{-N}\int_{\Space{R}{4n+2}} \widehat{k}(x)\,e^{i\sum_0^{N-1}x'_j X^l_j}\, e^{i\sum_0^{N-1}x''_j X^r_j}\, f(y)\,dx\\ &=&(2\pi)^{-N}\int_{\Heisen{n}} \int_{\Heisen{n}} \widehat{k}(x',x'')\, f(x'^{-1}*y*x'')\,dx'dx''\\ &=&(2\pi)^{-N} \int_{\Heisen{n}} \int_{\Heisen{n}} \widehat{k}(x',x'')\pi_l(x')\pi_r(x'')\,dx'dx''\,f(y).\end{aligned}$$ Transformation $(*)$ is possible due the commutativity of $X_j^l$ and $X_j^r$. Reduction of two-sided convolutions to the usual group ones was done in [@Kisil94f] in a way very similar to the present consideration (see Corollary \[co:two-sided\]). The Heisenberg group here may be substituted by any non-commutative group $G$ (for a commutative group the left and the right shifts are the same) and we will obtain two-sided convolutions on $G$. These basic Examples form a frame for other ones: the number of examples may be increased both by simple compositions of convolutions from Examples \[ex:euclid-conv\]–\[ex:two-sided\] and by alterations of considered groups. Basic Properties {#se:properties} ================ The main purpose of the present introduction to relative convolutions is a sharp extension of harmonic analysis applications. Through relative convolutions we can transfer the knowledge on Lie groups and their representations to the different operator algebras. Properties of relative convolutions themselves are strictly depending on a structure of a concrete group. Thus, it seems unlikely that one can say too much about them in general. Nevertheless, results collected in this Section establish important properties of relative convolutions and will be forceful in the future. ${\object{Ker}}(\algebra{g},S)$ operates on $S$ trivially therefore we are interested to understand the effective part of a relative convolution operator. Let $m:=\dim{\object{Ker}}(\algebra{g},S)>0$. We have a decomposition $\algebra{g}={\object{Ker}}(\algebra{g},S)\oplus{\object{Ess}}(\algebra{g},S)$ as linear spaces. Let us introduce a frame $\{X_j\}_{1\leq j \leq N}$ of such that $X_j \in {\object{Ker}}(\algebra{g},S), 1\leq j \leq m$ and we assume that operator  is written through such a frame (it clearly can be obtained by a change of variables). Then the following Lemma is evident. \[le:decomposition\] Operator  of relative convolution is equal to an operator of relative convolution generated by the induced operation of ${\object{Ess}}(\algebra{g},S)$ on $S$ with the kernel $k'(x'')$ defined by the formula $$\label{eq:decomposition} \widehat{k'}(x'')=(2\pi)^{-m/2} \int_{\Space{R}{m}}\widehat{k}(x',x'')\,dx',$$ where $x'\in\Space{R}{m},\ x''\in\Space{R}{N-m}$. The effective decomposition allows us to consider always a basic case of ${\object{Ker}}(\algebra{g},S)=0,\ {\object{Ess}}(\algebra{g},S)=\algebra{g}$. After that the general case may be obtained by the obvious modification. Operators of relative convolution clearly form an algebra and we will denote by $(k_2*k_1)(x)$ the kernel of composed operator of two relative convolutions with kernels $k_1(x)$ and $k_2(x)$. Of course, $k_2*k_1\neq k_1*k_2$ generally speaking. \[pr:composition\] We have $$\label{eq:composition} (k_2*k_1)(x)=(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(y)\,k_1({\object{CH}}[-y,x])\,dy,$$ where ${\object{CH}}[-y,x]={\object{CH}}[y^{-1},x]$ is given by . For kernels $k_1, k_2$ satisfying the Fubini theorem we can change theintegration order and obtain formula : $$\begin{aligned} K_2 K_1&=& (2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(y)\,e^{i\sum_1^N y_j X_j}\,dy \times (2\pi)^{-N/2}\int_{\Space{R}{N}}k_1(z)\,e^{i\sum_1^N z_j X_j}\,dz\\ &=& (2\pi)^{-N}\int_{\Space{R}{N}}\int_{\Space{R}{N}}k_2(y)\,k_1(z)\, e^{i\sum_1^N y_j X_j}e^{i\sum_1^N z_j X_j}\,dy \,dz\\ &=& (2\pi)^{-N}\int_{\Space{R}{N}}\int_{\Space{R}{N}} k_2(y)\,k_1(z)\,e^{i{\object{CH}}[\sum_1^N y_j X_j,\sum_1^N z_j X_j]}\,dy \,dz\\ &=& (2\pi)^{-N/2}\int_{\Space{R}{N}}(2\pi)^{-N/2}\int_{\Space{R}{N}} k_2(y)\,k_1({\object{CH}}[-\sum_1^N y_j X_j,\sum_1^N x_j X_j])\,dy\\ &&{\qquad\qquad \times e^{i\sum_1^N x_j X_j}\,dx,}\end{aligned}$$ where $e^{i \sum_1^N x_j X_j}=e^{i \sum_1^N y_j X_j}e^{i \sum_1^N z_j X_j}$. Making the change of variables we used the invariance of the measure on . Note, that for an operation on $S=G$ induced by the operation of $G$ on $S$ formula  gives us the usual [*group convolution*]{} on $S=G$. Thus we can speak about a [*convolution-like*]{} calculus of relative convolutions. This remark gives us the following important result describing the nature of relative convolution algebras[^6]. \[th:nature\] Let be an algebra Lie operating on a set $S$ and let $G$ be the exponential Lie group of ${\object{Ess}}(\algebra{g},S)$. Let also be an algebra of relative convolutions induced by on $S$ and let $\widehat{\algebra{G}}$ be a group convolution algebra on $G$. Then is a linear representation of $\widehat{\algebra{G}}$. Due to Theorem \[th:nature\] we can give an alternative definition of a relative convolution algebra, namely, [*an operator algebra is a relative convolution algebra induced by a Lie algebra , if all representations of are subrepresentations of the convolution algebra on group $\object{Exp}\algebra{g}$*]{}. Theorem \[th:nature\] is the main tool for applications of harmonic analysis to every problem where relative convolutions occur. Particulary, it gives us an easy access to many conclusions obtained by direct calculations. Now we illustrate this by applications to two-sided convolutions from Example \[ex:two-sided\]. \[co:two-sided\] An algebra of two-sided convolutions on the Heisenberg group is a representation of the group convolution algebra on . Explicit descriptions of the established representation depend from properties of relative convolutions kernels. In the next paper will be shown that for kernels from $\FSpace{L}{1}(\Space{R}{N}\times\Space{R}{N})$ there is a one-to-one correspondence between representations of two-sided convolutions on and usual one-sided convolution on the $\object{Exp}{\object{Ess}}(\algebra{g}\times\algebra{g},\Space{R}{N})$ (for the Heisenberg group it was calculated in [@Kalyuzhny93]). If kernels have a symmetry group then some representations of $\object{Exp}{\object{Ess}}(\algebra{g}\times\algebra{g},\Space{R}{N})$ vanish, see, for example case of two-sided convolutions on with homogeneous kernels  in [@VasTru94]. There is another simple but important corollary of Theorem \[th:nature\] \[co:auto\] Let $\psi: \algebra{g} \rightarrow\algebra{g}$ be an automorphism of as a Lie algebra. Let $\Psi:\algebra{G}\rightarrow\algebra{G}$ be a mapping defined by the rule $\Psi: K \mapsto \Phi(K)$, where $K$ is a relative convolution with a kernel $k(x)$, $\Psi(K)$ is a relative convolution with the kernel $k(\psi x)\, J^{1/2}{\psi}(x)$ and $J{\psi}(x)$ is the Jacobian of $\psi$ at the point $x$. Then $\Psi$ is an automorphism of algebra . The key point of the proof is the invariance of  under $\Psi$ (the rest is almost evident). This invariance follows from a simple change of variables in the integral : $$\begin{aligned} \Psi[k_1*k_2](x)&=&(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(y)\,k_1({\object{CH}}[- y,\psi(x)])\, J^{1/2}{\psi(x)}\,dy\\ &=&(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(y)\,k_1({\object{CH}}[-\psi(\psi^{-1} (y)),\psi(x)])\, J^{1/2}{\psi(x)}\,dy\\ &=&(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(\psi(y'))\,k_1({\object{CH}}[- \psi(y'),\psi(x)])\, J^{1/2}{\psi(x)}\,dy\\ &\stackrel{(*)}{=}&(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(\psi(y'))\,k_1(\psi({\object{CH}}[- y',x])) \, J^{1/2}{\psi(x)}\,dy\\ &=&(2\pi)^{-N/2}\int_{\Space{R}{N}}k_2(\psi(y')) \, J^{1/2}{\psi(y')}\, k_1(\psi({\object{CH}}[-y',x]))\\ &&{\qquad\qquad \times} J^{1/2}{\psi(y'*x)} \, J^{-1}{\psi}(y')\,dy\\ &=&(2\pi)^{-N/2}\int_{\Space{R}{N}}\, \Psi k_2(y')\,\Psi k_1({\object{CH}}[- y',x]) \,dy'\\ &=& [\Psi k_2*\Psi k_1](x),\end{aligned}$$ where $y'=\psi^{-1}(y),\ y=\psi(y')$. Here transform $(*)$ is possible due to the automorphism property of $\psi$. We also employ the identity $$J^{1/2}{\psi(y')}\, J^{1/2}{\psi(y'*x)}\, J^{-1}{\psi}(y')= J^{1/2}{\psi(x)},$$ which follows from the chain rule. Next Lemma evidently follows from Definition \[de:relative1\] and Theorem \[th:nature\]. \[le:category\] Let an algebra $\widetilde{\algebra{G}}$ be a representation of an algebra of relative convolutions induced by a Lie algebra . Then $\widetilde{\algebra{G}}$ is also the algebra of relative convolutions induced by . Otherwise, algebras of relative convolutions induced by a Lie algebra form a closed sub-category $\mathcal G$ of the category of all operator algebras with morphisms defined by representations (up to unitary equivalence) of algebras. The universal repelling object of the category $\mathcal G$ is the algebra of group convolutions on $\object{Exp}\algebra{g}$. Besides a convolution-like calculus, there is another well developed type of calculus, namely, the calculus of pseudodifferential operators (PDO), which is highly useful in analysis. Let us remind, a PDO $\object{Op}a(x,\xi)$ [@Hormander85; @Shubin87; @MTaylor81; @HWeyl] with the [*Weyl symbol*]{} $a(x,\xi)$ is defined by the formula: $$\label{eq:pdo} [\object{Op}a](x,\xi)u(y)=\int_{\Space{R}{n}\times\Space{R}{n}} a(\frac{x+y}{2},\xi)\,e^{i\scalar{y-x}{\xi}}u(x)\,dx\,d\xi.$$ It was shown that relative convolutions are PDOs for many types studed before (for example, families of convolutions [@MTaylor84 Proposition 1.1] and meta-Heisenberg group [@Folland94]). But PDOs itself are relative convolutions induced by the Heisenberg group (see Example \[ex:pdo\]). Thus if we consider morphisms at categories of relative convolution algebras only up to smooth operators, then we have[^7] \[th:pdo\] The category $\mathcal G$ of relative convolution algebras induced by a Lie algebra , $\dim \algebra{g}= N$ is (up to smooth operators) a sub-category of the category ${\mathcal H}^N$ of relative convolution algebras induced by the Heisenberg group . It is well known [@MTaylor84 Proposition 1.1], that group convolutions on $\object{Exp}\algebra{g}$ are (up to smooth operators, at least) PDOs on and thus the algebra of convolutions belongs to ${\mathcal H}^N$. The rest of the assertion is given by Theorem \[th:nature\] and Lemma \[le:category\]. Of course, there is still a huge distance between Theorem \[th:pdo\] and the real PDO-like calculus of relative convolutions. Calculus of PDO is a representation [@Howe80b] of a group convolution calculus on the Heisenberg group (the simplest nilpotent Lie group). Thus Theorem \[th:pdo\] establishes connection between relative convolutions and convolutions on the Heisenberg group. Therefore it is not surprising, that for convolutions on nilpotent Lie groups this connection has a very simple form (see [@Kisil93e Theorem 1]). Applications to Complex and Hypercomplex Analysis {#se:analysis} ================================================= In this and the next Sections we would like to introduce a series of essential Examples, which show possible applications[^8] of relative convolutions. Due to the wide spectrum of Examples only brief descriptions will be given here. We are going to return to these subjects in the future papers in this series after an appropriate development of general theory of relative convolutions. As was pointed by Hörmander in [@Anderson69], formula  is closely connected with a partial differential equation with operator coefficients. Thus it is not surprizing, that most Examples within this Section are touching some spaces of holomorphic functions, which are solutions to corresponding equations. In Example \[ex:clifford\] this connection will be used directly. \[ex:pdo\] Let us consider a combination of Examples \[ex:euclid-conv\] and \[ex:multiplication\]. Namely, let has a frame consisting from vector fields $$\label{eq:pdo-frame} X_j=y_j,\ X_j^e=\frac{1}{i}\frac{\partial }{\partial y_j},\ 1\leq j\leq N,$$ which operate on $S=\Space{R}{N}$ by the obvious way. Note, that these vector fields have exactly the same commutators  as left (right) fields from  (or ) if we put $X_0^{l(r)}=iI$. Then an operator of relative convolution with the kernel $\widehat{k}(x,\xi),\ x,\xi\in\Space{R}{N}$ has form (see [@MTaylor86 § 1.3]) $$\begin{aligned} [Kf](y)&=&(2\pi)^{-N}\int_{\Space{R}{2N}} \widehat{k}(x,\xi)\, e^{i(\sum_1^Nx_jy_j-\sum_1^N \xi\frac{\partial }{i\partial y_j})}f(y) \,dx \,d\xi\\ &=&(2\pi)^{-N}\int_{\Space{R}{2N}} k(\frac{x+y}{2},\xi)\, e^{i(y-x)\xi} f(x)\,dx\,d\xi,\end{aligned}$$ i. e. it exactly defines the [*Weyl functional calculus*]{} (or the Weyl quantization) [@Hormander85; @Shubin87; @MTaylor81; @HWeyl]. This forms a very important tool for the theory of differential equations and quantum mechanics. Our definition of relative convolution operators obviously generalizes the Weyl functional calculus from the case of the Heisenberg group to an arbitrary exponential Lie group. A deep investigation of the role of the Heisenberg group (and its different representations) at PDO calculus and harmonic analysis on real line may be found at [@Howe80a; @Howe80b]. Unfortunately, usual definitions of PDO symbol classes $S^m$ destroy the natural symmetry between $x$ and $\xi$ and this restricts applications of harmonic analysis to the (standard) PDO theory. Particulary, $S^m(\Space{R}{2n} )$ is not invariant under all symplectic morphisms of , which are induced by automorphisms of the Heisenberg group (see Corollary \[co:auto\]). \[ex:toeplitz\] Now we follow the paper [@BoutetGuill85] footsteps but only in the original context of complex analysis (see also an elegant survey in [@Stein93 Chap. XII]). Let $\Space{U}{n}$ be an [*upper half-space*]{} $$\Space{U}{n}=\{z\in\Space{C}{n+1}\such \object{Im}z_{n+1} > \sum_{j+1}^n\modulus{z_j}^2\},$$ which is a domain of holomorphy of functions of $n+1$ complex variables. Its boundary $$b\Space{U}{n}=\{z\in\Space{C}{n+1}\such \object{Im}z_{n+1} = \sum_{j+1}^n\modulus{z_j}^2\}$$ may be naturally identified with the Heisenberg group . One can introduce the [*Szegö projector*]{} $R$ as the orthogonal projection of $\FSpace{L}{2}(\Heisen{n})$ onto its subspace $\FSpace{H}{2}(\Heisen{n})$ (the [*Hardy space*]{}) of boundary values of holomorphic functions on the upper half-space . Then a [*Toeplitz operator*]{} [@BoutetGuill85] on with the pre-symbol $Q$ is an operator of the form $T_Q=RQR $ where $Q: \FSpace{L}{2}(\Heisen{n}) \rightarrow \FSpace{L}{2}(\Heisen{n})$ is a pseudodifferential operator. Obviously $T_Q: \FSpace{H}{2}(\Heisen{n}) \rightarrow \FSpace{H}{2}(\Heisen{n})$. The invariance of the [*tangential Cauchy–Riemann equations*]{} under right shifts of implies that the Szegö projector can be realized as a (left) convolution operator on  [@Gindikin64] (see also Corollary \[co:szego\]). Thus the algebra of the Toeplitz operators on can be naturally imbedded into the algebra of (pseudodifferential) operators generated by left group convolutions on and PDO. First, let us consider a case of a pre-symbol $Q$ taken from usual Euclidean convolutions on $\Heisen{n}\cong\Space{R}{2n+1}$. Left convolutions on are generated by vector fields $X_j^l$ from  and Euclidean convolutions are induced by fields (see Example \[ex:euclid-conv\]) $$X_j^e=\frac{1}{i}\frac{\partial }{\partial y_j},\ 1\leq j\leq N.$$ But two frames of vector fields $\{X_j^l,X_j^e\}$ and $\{X_j^l,X_j^r\}$ define just the same action on . Therefore we have an embedding of the Toeplitz operators with Euclidean convolution pre-symbols to the algebra of two-sided convolutions on from Example \[ex:two-sided\]. If we now allow $Q$ to be a general PDO from Example \[ex:pdo\], then we should consider a joint operation of vector fields $X_j,\ X_j^e$ from  and $X_j^l$ from . Again one can pass to equivalent frame defined by $X_j,\ X_j^l,\ X_j^r$. The algebra of relative convolutions defined by the last frame is the algebra of operators generated by two-sided convolutions on and operators of multiplication by functions, which form a meta-Heisenberg group [@Folland94]. Such algebras for continuous multipliers were studied in [@Kisil93e; @Kisil94a]. There is our conclusion[^9]: The algebra of the Toeplitz operators with PDO pre-symbols is naturally imbedded into the algebra of relative convolutions generated by two-sided convolutions on and operators of multiplication by functions. For the first time the algebra of the Toeplitz operators with two-sided convolution pre-symbols was studied at [@Kisil91]. We are going to consider another problem from complex analysis. \[ex:fock\] Let $\FSpace{L}{2}(\Space{C}{n},d\mu_{n})$ be a space of all square-integrable functions on $\Space{C}{n}$ with respect to the Gaussian measure $$d\mu_{n}(z)=\pi^{-n}e^{-z\cdot\overline{z}}dv(z),$$ where $dv(z)=dx\,dy$ is the usual Euclidean volume measure on $\Space{C}{n}=\Space{R}{2n}$. Denote by $P_{n}$ the orthogonal Bargmann projector of $\FSpace{L}{2}(\Space{C}{n},d\mu_{n})$ onto the [*Segal–Bargmann*]{} or [*Fock space*]{} $\FSpace{F}{2}(\Space{C}{n})$, namely, the subspace of $\FSpace{L}{2}(\Space{C}{n},d\mu_{n})$ consisting of all entire functions. The Fock space $\FSpace{F}{2}(\Space{C}{n})$ was introduced by Fock [@Fock32] to give an alternative representation of the Heisenberg group in quantum mechanics. The rigorous theory of $\FSpace{F}{2}(\Space{C}{n})$ was developed by Bargmann [@Bargmann61] and Segal [@Segal60]. Such a theory is closely connected with representations of the Heisenberg group (see also [@Folland89; @Guillemin84; @Howe80b]), but studies of the Bargmann projector and the associated Toeplitz operators are usually based on the Hilbert spaces technique[^10]. There is a strong reason for this: the Bargmann projection $P_n$ is not a [*group*]{} convolution for any group. However, it is possible to consider $P_n$ as a [*relative*]{} convolution. Let us consider the group of Euclidean shifts $a: z\mapsto z+a$ of . To make unitary operators on $\FSpace{L}{2}(\Space{C}{n},d\mu)$ from the shifts we should multiply by the special weight function: $$\label{eq:fock-shift} a: f(z)\mapsto f(z+a)e^{-z\bar{a}-a\bar{a}/2}.$$ It is obvious, that  defines a unitary representation [@Howe80b] of the $(2n+1)$-dimensional Heisenberg group on $\FSpace{L}{2}(\Space{C}{n},d\mu)$, which preserves the Fock space $\FSpace{F}{2}(\Space{C}{n})$. Thereafter all operators  should commute with $P_n$. Unitary operators  have such infinitesimal generators $$i \sum_{k=1}^{n}\left( a_j'(\frac{\partial }{\partial z_j'}-z_j'- iz_j'')+ a_j''(\frac{\partial }{\partial z_j''}-z_j''+iz_j')\right),$$ where $a=(a_1,\ldots,a_n), z=(z_1,\ldots,z_n)\in\Space{C}{n}$ and $a_j=(a_j',a_j''), z=(z_j',z_j'')\in\Space{R}{2}$. This linear space of generators has the frame $$\label{eq:fshift-frame} A^{f\prime}_j=\frac{1}{i}\left(\frac{\partial }{\partial z_j'}-z_j'- iz_j''\right),\ A^{f\prime\prime}= \frac{1}{i}\left(\frac{\partial }{\partial z_j''}- z_j''+iz_j'\right).$$ Operators  still should commute with the Bargmann projector $P_n$ and we can expect that $P_n$ should be a relative convolution with respect to operators $$\label{eq:fconv-frame} X^{f\prime}_j=\frac{1}{i}\left(\frac{\partial }{\partial z_j'}- z_j'+iz_j''\right),\ X^{f\prime\prime}= \frac{1}{i}\left(\frac{\partial }{\partial z_j''}- z_j''-iz_j'\right),$$ which commute with all operators  and together with operator $2I$ form a self-adjoint representation of $\algebra{h}_n$. Indeed, we have \[pr:bargmann\] The Bargmann projector is a relative convolution induced by the Weyl-Heisenberg Lie algebra $\algebra{h}_n$, which have an operation on defined by . Its kernel $b(t,\zeta),\ t\in \Space{R}{},\ \zeta\in\Space{C}{n}$ is defined by the formula: $$\widehat{b}(t,\zeta)=2^{n+1/2} e^{-1} e^{- (t^2+\zeta\bar{\zeta}/2)}.$$ One can make an easy exercise with integral transforms: $$\begin{aligned} [P_nf](z)&=&(2\pi)^{-n-1/2}\int_{\Space{R}{}} \int_{\Space{C}{n}} 2^{n+1/2}\, e^{-(t^2+1+\zeta\bar{\zeta})/2} \nonumber \\ &&{\qquad\qquad \times} e^{-i(t\cdot 2I+\sum_{k=1}^n (\zeta_j'X_j^{f\prime}+\zeta_j''X_j^{f\prime\prime}))}\, f(z)\,dt\,d\zeta\nonumber \\ &=&\pi^{-n-1/2}\int_{\Space{R}{}}e^{-(t^2+1+2it)} \,dt\,\int_{\Space{C}{n}}\, e^{-(\zeta\bar{\zeta})/2}\, e^{-i\sum_{k=1}^n (\zeta_j'X_j^{f\prime}+\zeta_j''X_j^{f\prime\prime})}\, f(z)\,d\zeta\nonumber \\ &=&\pi^{-n}\int_{\Space{C}{n}} e^{-\zeta\bar{\zeta}/2}\, e^{- i\sum_{k=1}^n (\zeta_j'X_j^{f\prime}+\zeta_j''X_j^{f\prime\prime})}\, f(z)\,d\zeta\nonumber \\ &=&\pi^{-n}\int_{\Space{C}{n}} e^{-\zeta\bar{\zeta}/2}\, e^{- \sum_{k=1}^n (\zeta_j'\frac{\partial }{\partial z_j'} +\zeta_j''\frac{\partial }{\partial z_j''}- \zeta_j \bar{z}_j)}\, f(z)\,d\zeta\nonumber \\ &=&\pi^{-n}\int_{\Space{C}{n}} e^{-\zeta\bar{\zeta}/2}\, e^{- \zeta\bar{\zeta}/2}\, e^{\zeta \bar{z}}\, e^{-\sum_{k=1}^n(\zeta_j'\frac{\partial }{\partial z_j'} +\zeta_j''\frac{\partial }{\partial z_j''})}\, f(z)\,d\zeta\nonumber \\ &=&\pi^{-n}\int_{\Space{C}{n}} e^{(\bar{z}-\bar{\zeta})\zeta}\, f(z- \zeta)\, d\zeta\nonumber \\ &=&\pi^{-n}\int_{\Space{C}{n}}\, e^{\bar{w}(z-w)}\, f(w)\,dw\label{eq:bargmann},\end{aligned}$$ where $\zeta=(\zeta',\zeta'')\in\Space{R}{n} \times \Space{R}{n},\ w=z- \zeta$. Formula  is the well known expression for the Bargmann projector [@Bargmann61]. See also Corollary \[co:bargmann\] for an alternative proof. Following to papers [@BergCob86; @BergCob87; @Coburn90] we now consider the Toeplitz operator of the form $T_a=P_n a(z)I$, where $a(z)I$ is an operator of multiplication by a function $a(z)$. For the previous reasons we can handle them as relative convolutions generated by operators $X^f_j$ from  and operators $X_j=z_j I$. We can easily describe all non-zero commutators: $$\label{eq:fock-comm} [X^{f\prime}_j,X^{f\prime\prime}_j]=2iI,\ [X^{f\prime}_j,X'_j]=iI,\ [X^{f\prime\prime}_j,X''_j]=iI.$$ So they form a $(4n+1)$-dimensional nilpotent step 2 Lie algebra. Particulary sub-algebra spanned on the vectors $X^{f\prime}_j,\ X^{f\prime\prime}_j$ and $iI$ is isomorphic[^11] to our constant companion $\algebra{h}_n$. There is a natural embedding of the Toeplitz operator algebra on the Fock space into the algebra of relative convolutions induced by the Lie algebra with commutation relations . We will continue our discussion of Toeplitz operators on an “abstract nonsense”[@Howe80b] level at Section \[se:coherent\]. \[ex:clifford\] Let now $X_j$ be generators of the Clifford algebra (we use book [@DelSomSou92] as a standard reference within this Example). This means that the following [*anti-commutation*]{} relations hold (compare with ): $$\label{eq:anti-comm} \{X_i,X_j\}:=X_i X_j+X_j X_i=-2\delta_{ij}X_0,$$ where $X_0=I$. Function $f:\Space{R}{n}\rightarrow\Cliff{0}{n}$ is called [*monogenic*]{} if it satisfies the [*Dirac equation*]{} $$\label{eq:dirac} Df:=\frac{\partial f(y)}{\partial y_0} -\sum_{j=1}^n X_j\frac{\partial f(y)}{\partial y_j}=0\ \mbox{ or }\ \frac{\partial f(y)}{\partial y_0} =\sum_{j=1}^n X_j\frac{\partial f(y)}{\partial y_j}.$$ The success of Clifford analysis is mainly explained because the Dirac operator  factorizes the Laplace operator $\Delta=\sum_0^n\frac{\partial ^2}{\partial x_j^2}$. Hörmander’s remark from paper [@Anderson69] gives us by the fundamental solution to the Dirac equation in the form $$\begin{aligned} K(y)&=&\fourier{\eta\rightarrow y} e^{-iy_0\sum_{j=1}^n\eta_jX_j}\\ &=&\int_{\Space{R}{n}}e^{i\sum_{j=1}^n y_j\eta_j}\, e^{- iy_0\sum_{j=1}^n\eta_jX_j}\, d\eta.\end{aligned}$$ (“Simply take the Fourier transform with respect to the spatial variables, and solve the equation in $y_0$” [@Anderson69]). Otherwise, any solution $f(y)$ to  is given by a convolution of some function $\widetilde{f}(y)$ on and the fundamental solution $K(y)$. On the contrary, a convolution $K(y)$ with any function is a solution to . We have: $$\begin{aligned} [K*f](y)&=&\int_{\Space{R}{n}} K(y-t)\,f(t)\,dt\nonumber \\ &=&\int_{\Space{R}{n}}\int_{\Space{R}{n}} e^{-i\sum_{j=1}^n (y_j- t_j)\eta_j}\, e^{-iy_0\sum_{j=1}^n\eta_jX_j}\, d\eta f(t)\,dt\nonumber \\ &=&\int_{\Space{R}{n}}\int_{\Space{R}{n}}e^{i\sum_{j=1}^n t_j\eta_j}\, e^{-i\sum_{j=1}^n\eta_j(y_0 X_j-y_j X_0)}\, d\eta f(t)\,dt\nonumber \\ &=&\int_{\Space{R}{n}} e^{-i\sum_{j=1}^n\eta_j(y_0 X_j-y_j X_0)}\int_{\Space{R}{n}} e^{i\sum_{j=1}^n t_j\eta_j}\, f(t)\,dt\, d\eta \nonumber \\ &=&\int_{\Space{R}{n}} e^{-i\sum_{j=1}^n\eta_j(y_0 X_j-y_j X_0)}\, \widehat{f}(-\eta)\, d\eta. \label{eq:malonek}\end{aligned}$$ Equation  defines relative convolution  with the kernel $f(-y)$ and the Lie algebra of vector fields $$\label{eq:monom} \{y_0 X_j-y_j X_0\},\ 1\leq j \leq n.$$ Thus, at least formally, any solution to the Dirac equation  can be written as a function (see Remark \[re:function\]) of $n-1$ monomials : $$\label{eq:laville} \breve{f}(y_0,y_1,\ldots,y_n)= f(y_0 X_1-y_1 X_0, y_0 X_2-y_2 X_0,\ldots,y_0 X_n-y_n X_0).$$ Another significant remark: if we fix the value $y_0=0$ in  we easily obtain: $\breve{f}(0,y_1,\ldots,y_n)= f(-y_1 X_0, -y_2 X_0,\ldots,-y_n X_0)=f(-y_1 , -y_2 ,\ldots,-y_n )$. Thus we may consider the function $\breve{f}(y_0,y_1,\ldots y_n)$ of $n+1$ variables as [*analytical expansion*]{} for the function $f(y_1,\ldots,y_n)$ of $n$ variables (compare with [@Laville91]). Using the power series decomposition for the exponent one can see that formula  defines the permutational (symmetric) product of monomials . The significant role of such monomials and functions generated by them in Clifford analysis was recently discovered by Laville [@Laville87] and Malonek [@Malonek93]. But during our consideration we used only the commutation relation $[X_0,X_j]=0$ and never used the anti-commutation relations . Thus formula  as well. Any solution to equation , where $X_j$ are arbitrary self-adjoint operators, is given as arbitrary function of $n$ monomials  by the formula . It is possible also to introduce the notion of the [*differentiability*]{} [@Malonek93] for solutions to , namely, an increment of any solution to  may be approximated up to infinitesimals of the second order by a linear function of monomials . Due to physical application we will consider equation $$\label{eq:mass} \frac{\partial f}{\partial y_0}=(\sum_{j=1}^n X_j\frac{\partial }{\partial y_j}+M)f,$$ where $X_j$ are arbitrary self-adjoint operators and $M$ is a bounded operator commuting with all $X_j$. Then $X_j$ are generators  of the Clifford algebra and $M=M_\alpha $ is an operator of multiplication from the [*right*]{}-hand side by the Clifford number $\alpha $, differential operator  factorizes the Helmholtz operator $\Delta +M_{\alpha^2}$. Equation  is known in quantum mechanics as the [*Dirac equation for a particle with a non-zero rest mass*]{} [@BerLif82 §20], [@BogShir80 §6.3] and [@Kravchenko95a]. Simple modification of the previous calculations gives us the following result Any solution to equation , where $X_j$ are arbitrary self-adjoint operators and $M$ commutes with them, is given by the formula $$e^{y_0 M}\int_{\Space{R}{n}} e^{-i\sum_{j=1}^n\eta_j(y_0 X_j-y_j X_0)} \widehat{f}(-\eta)\, d\eta,$$ where $f$ is an arbitrary function on . \[re:riesz\] Connection between relative convolutions and Clifford analysis is two-sided. Not only relative convolutions are helpful in Clifford analysis but also Clifford analysis may be used for developing the relative convolution technique. Indeed, we have defined a relative convolution as a function of operators $X_j$ representing a Lie algebra . To do this we have used the Weyl function calculus from [@Anderson69]. Meanwhile, for a pair of self-adjoint operators $X_1,\ X_2$ the alternative Riesz calculus [@RieszNagy55 Chap. XI] is given by the formula $$\label{eq:riesz} f(X_1+iX_2)=\int_\Gamma f(\tau)(\tau I-(X_1+iX_2))^{-1}\,d\tau.$$ As shown in [@Anderson69], these two calculuses are essentially the same in the case of a pair of bounded operators. To extend the Riesz calculus for arbitrary $n$-tuples of bounded operators $\{X_j\}$ it seems natural to use Clifford analysis (see, for example, [@DelSomSou92]), which is an analogy to one-dimensional complex analysis. Then one can define a function of arbitrary $n$-tuple of bounded self-adjoint operators $\{X_j\}$ in such a way (compare with ): $$f(X_1,X_2,\ldots,X_n)=\int_\Gamma f(\tau_1,\tau_2,\ldots,\tau_n)\,K(\tau_1I-X_1,\tau_2I-X_2,\ldots,\tau_nI- X_n)\,d\tau,$$ where $K(\tau_1,\tau_2,\ldots,\tau_n)$ is the Cauchy kernel fron the Clifford analysis [@DelSomSou92] . Coherent States {#se:coherent} =============== This Section is a bridge between the previous and the next ones. Here we will give a new glance on some constructions of Examples \[ex:toeplitz\] and \[ex:fock\]. We also provide a foundation for the further investigation in Section \[se:physics\]. General Consideration {#ss:general} --------------------- Coherent states are a useful tool in quantum theory and have a lot of essentially different definitions [@Perelomov86]. Particulary, they were described by Berezin in [@Berezin72; @Berezin86; @Guillemin84] concerning so-called covariant and contravariant (or Wick and anti-Wick) symbols of operators (quantization). They say that the Hilbert space $H$ has a system of coherent states $\{f_\alpha \}, \alpha \in G$ if for any $f\in H$ $$\label{eq:norm} \scalar{f}{f}=\int_G \modulus{\scalar{f}{f_\alpha }}^2\, d\mu.$$ This definition does not take in account that within coherent states a group structure frequently occurs and is always useful [@Perelomov86]. For example, the original consideration of Berezin is connected with the Fock space, where coherent states are functions $e^{za-a\bar{a}/2}$. The representation of the Heisenberg group on the Fock space was exploited in Example \[ex:fock\]. Another type of coherent states with a group structure is given by the vacuum vector and operators of creation and annihilation, which represents group . Thus we would like to give an alternative definition. \[de:coherent1\] We will say that the Hilbert space $H$ has a system of coherent states $\{f_g \},\ g \in G$ if 1. \[it:group\] There is a representation $T: g\mapsto T_g$ of the group $G$ by unitary operators $T_g$ on $H$. 2. \[it:vector\] There is a vector $f_0\in H$ that for $f_g=T_g f_0$ and arbitrary $f\in H$ we have $$\label{eq:norm1} \scalar{f}{f}=\int_G \modulus{\scalar{f}{f_g }}^2\, d\mu,$$ where integration is taken over the Haar measure $d\mu$ on $G$. Because this construction independently arose in different contexts, vector $f_0$ has many various names: the [*vacuum vector*]{}, the [*ground state*]{}, the [*mother wavelet*]{} etc. Modifications of definition \[de:coherent1\] for other cases are discussed [@Perelomov86 § 2.1]. Equation  implies, that vector $f_0$ is a cyclic vector of the representation $G$ on $H$. Let $T$ is a unitary representation of a group $G$ in a Hilbert space $H$. Then there exists such $f_0\in H$ that equality  holds. Moreover, if the representation $T$ is irreducible, then one can take an arbitrary non-zero vector of $H$ (up to a scalar factor) as the vacuum vector. Let us fix some Haar measure $d\mu$ on $G$. (Different Haar measures are different on a scalar factor). If the representation $T$ is irreducible, then an arbitrary vector $f\in H$ is cyclic and we may put $f_0=c^{- 1/2}f$, where $$c=\frac{\int_G \modulus{\scalar{f}{T_g f}}^2\, d\mu(g)}{\scalar{f}{f}}.$$ It is easy to verify, that for the $f_0$ equality  holds [@Perelomov86 § 2.3]. Let now $T$ is an arbitrary representation, then we can decompose [@Kirillov76 § 8.4] it onto a direct integral [@Dixmier69 § 10] of irreducible representations $T=\int_Y T_\alpha\, d\alpha $ on the space $H=\int_Y H_\alpha\, d\alpha$. Again we can take an arbitrary vector $f=\int_Y f_\alpha\, d\alpha \in H=\int_Y H_\alpha\, d\alpha$, such that $f_\alpha $ are non-zero almost everywhere, and put $f_0=\int_Y f_\alpha c^{-1/2}_\alpha d\alpha$, where $$c_\alpha =\frac{\int_G \modulus{\scalar{f_\alpha }{T_g f_\alpha }_\alpha }^2 d\mu(g)}{\scalar{f_\alpha }{f_\alpha }_\alpha }.$$ In view of $\norm{f}=\int_Y \norm{f_\alpha }\,d\alpha $, the proof is complete. By the way, a polarization of  gives us the equality $$\label{eq:product} \scalar{f_1}{f_2}=\int_G \scalar{f_1}{f_g } \overline{\scalar{f_2}{f_g }}\, d\mu.$$ Thus we have an isometrical embedding $E: H\rightarrow \FSpace{L}{2}(G,d\mu)$ defined by the formula $$\label{eq:embedding} E: f \mapsto f(g)=\scalar{f}{f_g}=\scalar{f}{T_g f_0}= \scalar{T^*_g f}{f_0}= \scalar{T_{g^{-1}}f}{f_0}.$$ We will consider $\FSpace{L}{2}(G,d\mu)$ both as a linear space of functions and as an operator algebra with respect to the left and right group convolution operations: $$\begin{aligned} {}[f_1*f_2]_l(h)&=&\int_G f_1(g)f_2(g^{-1}h)\, d\mu(g), \label{eq:conv-l}\\ {}[f_1*f_2]_r(h)&=&\int_G f_1(g)f_2(hg)\, d\mu(g).\label{eq:conv-r}\end{aligned}$$ For a simplicity we will assume, that $G$ is unimodular (the left and the right Haar measures on $G$ coincide) and that $\FSpace{L}{2}(G,d\mu)$ is closed under the group convolution. Thus the construction under consideration may be regarded as a natural embedding of the linear space $H$ to the operator algebra ${{\mathcal B}}(H)$. Let $\FSpace{H}{2}(G,d\mu)\subset\FSpace{L}{2}(G,d\mu)$ will denote the image of $H$ under embedding $E$. It is clear, that $\FSpace{H}{2}(G,d\mu)$ is a liner subspace of $\FSpace{L}{2}(G,d\mu)$, which does not coincide with the whole $\FSpace{L}{2}(G,d\mu)$ in general. One can see, that \[le:h2-invar\] Space $\FSpace{H}{2}(G,d\mu)$ is invariant under left shifts on $G$. Indeed, for every $f(g)\in \FSpace{H}{2}(G,d\mu)$ the function $$f(h^{-1}g)=\scalar{f}{T_{h^{-1}g}f_0}= \scalar{f}{T_{h^{-1}}T_gf_0}=\scalar{T_{h}f}{T_gf_0}=[T_{h}f](g)$$ also belongs to $\FSpace{H}{2}(G,d\mu)$. If $P:\FSpace{L}{2}(G,d\mu)\rightarrow \FSpace{H}{2}(G,d\mu)$ the orthogonal projector on $\FSpace{H}{2}(G,d\mu)$, then due to Lemma \[le:h2-invar\] it should commute with all left shifts and thus we get immediately Projector $P:\FSpace{L}{2}(G,d\mu)\rightarrow \FSpace{H}{2}(G,d\mu)$ is a right convolution on $G$. The following Lemma characterizes [*linear subspaces*]{} of $\FSpace{L}{2}(G,d\mu)$ invariant under shifts in the term of [*convolution algebra*]{} $\FSpace{L}{2}(G,d\mu)$ and seems to be of the separate interest. \[le:ideal\] A closed linear subspace $H$ of $\FSpace{L}{2}(G,d\mu)$ is invariant under left (right) shifts if and only if $H$ is a left (right) ideal of the right group convolution algebra $\FSpace{L}{2}(G,d\mu)$. A closed linear subspace $H$ of $\FSpace{L}{2}(G,d\mu)$ is invariant under left (right) shifts if and only if $H$ is a right (left) ideal of the left group convolution algebra $\FSpace{L}{2}(G,d\mu)$. Of course we consider only the “right-invariance and right-convolution” case. Then the other three cases are analogous. Let $H$ be a closed linear subspace of $\FSpace{L}{2}(G,d\mu)$ invariant under right shifts and $k(g)\in H$. We will show the inclusion $$\label{eq:rt-convol} [f*k]_r(h)=\int_G f(g)k(hg)\,d\mu(g)\in H,$$ for any $f\in\FSpace{L}{2}(G,d\mu)$. Indeed, we can treat integral  as a limit of sums $$\label{eq:rt-sum} \sum_{j=1}^{N} f(g_j)k(hg_j)\Delta_j.$$ But the last sum is simply a linear combination of vectors $k(hg_j)\in H$ (by the invariance of $H$) with coefficients $f(g_j)$. Therefore sum  belongs to $H$ and this is true for integral  by the closeness of $H$. Otherwise, let $H$ be a right ideal in the group convolution algebra $\FSpace{L}{2}(G,d\mu)$ and let $\phi_j(g)\in\FSpace{L}{2}(G,d\mu)$ be an approximate unit of the algebra [@Dixmier69 § 13.2], i. e. for any $f\in\FSpace{L}{2}(G,d\mu)$ we have $$[\phi_j*f]_r(h)=\int_G \phi_j(g)f(hg)\, d\mu(g) \rightarrow f(h)\mbox{, when } j\rightarrow\infty.$$ Then for $k(g)\in H$ and for any $h'\in G$ the right convolution $$[\phi_j*k]_r(hh')=\int_G \phi_j(g)k(hh'g)\, d\mu(g)= \int_G \phi_j(h'^{-1}g')k(hg')\, d\mu(g'),\ g'=h'g,$$ from the first expression is tensing to $k(hh')$ and from the second one belongs to $H$ (as a right ideal). Again the closeness of $H$ implies $k(hh')\in H$ that proves the assertion. \[le:identity\] For any $f(g)\in\FSpace{H}{2}(G,d\mu)$ we have $$\begin{aligned} {}[f*f_0]_l(g)&=&f(g) \label{eq:l-reproduce}\\ {} [\bar{f}_0*f]_r(g)&=&f(g),\label{eq:r-reproduce}\end{aligned}$$ where $f_0(g)=\scalar{f_0}{T_gf_0}$ is the function corresponding to the vacuum vector $f_0\in H$. We again check only the left case and this is just a simple calculation: $$\begin{aligned} [f*f_0]_l(h)&=& \int_G f(g)\,f_0(g^{-1}h)\,d\mu(g)\\ &=& \int_G f(g)\scalar{f_0}{T_{g^{-1}h}f_0}\,d\mu(g)\\ &=& \int_G \scalar{f}{T_g f_0} \,\scalar{f_0}{T_{g^{-1}} T_h f_0}\, d\mu(g)\\ &=& \int_G \scalar{f}{T_g f_0}\, \scalar{T_g f_0}{ T_h f_0}\, d\mu(g)\\ &=& \int_G \scalar{f}{T_g f_0}\, \overline{\scalar{ T_h f_0}{T_g f_0}}\, d\mu(g)\\ &\stackrel{(*)}{=}& \scalar{f}{T_h f_0}\\ &=&f(h).\end{aligned}$$ Here transformation $(*)$ is based on  and we have used the unitary property of the representation $T$. The following general Theorem easily follows from the previous Lemmas \[th:projector\] The orthogonal projector $P: \FSpace{L}{2}(G,d\mu) \rightarrow \FSpace{H}{2}(G,d\mu)$ is a right convolution on $G$ with the kernel $\bar{f}_0(g)$ defined by the vacuum vector. let me a representation of group $G$ on $H$ with a cyclic vector $f_0$ and I will construct the projector $P: \FSpace{L}{2}(G,d\mu) \rightarrow \FSpace{H}{2}(G,d\mu)\cong H$. Let $P$ be the operator of right convolution  with the kernel $\bar{f}_0(g)$. By the Lemma \[le:h2-invar\] $\FSpace{H}{2}(G,d\mu)$ is an invariant linear subspace of $\FSpace{L}{2}(G,d\mu)$. Thus by Lemma \[le:ideal\] it is an ideal under convolution operators. Therefore the convolution operator $P$ with the kernel $\bar{f}_0(g)$ from $\FSpace{H}{2}(G,d\mu)$ has an image belonging to $\FSpace{H}{2}(G,d\mu)$. But by Lemma \[le:identity\] $P=I$ on $\FSpace{H}{2}(G,d\mu)$, so $P^2=P$ on $\FSpace{L}{2}(G,d\mu)$, i. e. $P$ is a projector on $\FSpace{H}{2}(G,d\mu)$. It is easy to see, that $f_0(g)$ has the property $f_0(g)=\bar{f}_0(-g)$, thus $P^*=P$, i. e. $P$ is orthogonal. It may be shown also in such a manner. Let $f(g)\in \FSpace{L}{2}(G,d\mu)$ be orthogonal to all functions from $\FSpace{H}{2}(G,d\mu)$. Particulary $f(g)$ should be orthogonal to $f_0(h^{-1}g)$ (due to the invariance of $\FSpace{H}{2}(G,d\mu)$) for any $h\in G$. Then $P(f)=[f*f_0]_l=0$ and we have shown the orthogonality again. This completes the proof. The stated left invariance of $\FSpace{H}{2}(G,d\mu)$ and the representation of $P$ as a right group convolution have a useful tie with differential equations. Really, let $X_j,\ j\leq m$ be left-invariant vector fields (i. e. left-invariant differential operators) on $G$. If $X_jf_0\equiv 0$ then $X_jf= 0$ for any $f\in H$. Thus space $\FSpace{H}{2}(G,d\mu)$ may be characterized as the space of solutions to the system of equations $X_jf= 0,\ 1\leq j\leq m$. Another connected formulation: we can think of $P$ as of an integral operator with the kernel $K(h,g)=f_0(g^{-1}h)=\overline{f_0(h^{- 1}g)}=\scalar{T_g f_0}{T_h f_0}$, then kernel $K(h,g)$ is an [*analytic*]{} function of $h$ and [*anti-analytic*]{} of $g$. \[ex:transform\] An important class of applications may be treated as follows. Let we have a space of functions defined on a domain $\Omega\in\Space{R}{n} $. Let we have also a transitive Lie group $G$ of automorphisms of $\Omega$. Then we can construct a unitary representation $T$ of $G$ on $\FSpace{L}{2}(\Omega)$ by the formula: $$\label{eq:unit-trans} T_g: f(x) \mapsto f(g(x))\, J^{1/2}_g(x),\ f(x)\in\FSpace{L}{2}(\Omega),\ g\in G,$$ where $J_g(x)$ is the Jacobian of the transformation defined by $g$ at the point $x$. If we fix some point $x_0\in \Omega$ then we can identify the homogeneous space $G/G_{x_0}$ with $\Omega$ (see notation at the beginning of Subsection \[ss:notation\]). Then left-invariant vector fields on $G$ may be considered as differential operators on $\Omega$ and convolution operators on $G$ as integral operators on $\Omega$. This is a way giving [*integral representations for analytic functions*]{}. Classical Results ----------------- We would like to show, how abstract Theorem \[th:projector\] and Example \[ex:transform\] are connected with classical results on the Bargmann, Bergman and Szegö projectors at the Segal-Bargmann (Fock), Bergman and Hardy spaces respectively. We will start from a trivial example. Let $\{\phi_j\},\ -\infty<j<\infty$ be an orthonormalized basis of a Hilbert space $H$. Then $$B=\sum_{j=-\infty}^\infty \ket{\phi_j}\bra{\phi_j}$$ is a reproducing operator, namely, $Bf=f$ for any $f\in H$. We will construct a unitary representation of group on $H$ by its action on the basis: $$T_k \phi_j = \phi_{j+k},\ k\in \Space{Z}{}.$$ If we equip with the invariant discreet measure $d\mu(k)=1$ and select the vacuum vector $f_0=\phi_0$, then all coherent states are exactly the basis vectors: $f_k=T_k\phi_0=\phi_k$. Equation  turns to be exactly the Plancherel formula $$\scalar{f_1}{f_2}=\sum_{j=-\infty}^\infty \scalar{f_1}{T_jf_0} \overline{\scalar{f_2}{T_jf_0}}=\sum_{j=-\infty}^\infty \scalar{f_1}{\phi_j} \overline{\scalar{f_2}{\phi_j}}$$ and we have obtained the usual isomorphism of Hilbert spaces $H\cong\FSpace{l}{2}(\Space{Z}{})$ by the formula $f(k)= \scalar{f}{\phi_{k}}$. Our construction gives $$\begin{aligned} f(k) &=& \sum_{j=-\infty}^\infty \overline{\scalar{f_0}{T_j f_0}}\scalar{f}{T_{j+k}f_0}\\ &=& \sum_{j=-\infty}^\infty \delta_{0j} \scalar{f}{\phi_{j+k}}\\ &=& \scalar{f}{\phi_{k}}.\end{aligned}$$ Thus operator $B$ is really identical on $H$. Note, that similar construction may be given for a case of not orthonormalized frame. In spite of simplicity of this construction, it was the (almost) unique tool to establish of various projectors (see [@Rudin80 3.1.4]). Following less trivial Corollaries bring us back to Section \[se:analysis\]. \[co:bargmann\] The Bargmann projector on the Segal–Bargmann space has the kernel $$\label{eq:kr-bargmann} K(z,w)=e^{\bar{w}(z-w)}.$$ Let us define a unitary representation of the Heisenberg group on $\Space{R}{n}$ by the formula [@MTaylor86 § 1.1]: $$g=(t,q,p): f(x) \rightarrow T_{(t,q,p)}f(x)=e^{i(2t-\sqrt{2}qx+qp)} f(x- \sqrt{2}p).$$ As “vacuum vector” we will select the original [*vacuum vector*]{} $f_0(x)=e^{-x^2/2}$. Then embedding $\FSpace{L}{2}(\Space{R}{n} ) \rightarrow \FSpace{L}{2}(\Heisen{n} ) $ is given by the formula $$\begin{aligned} \widetilde{f}(g)&=&\scalar{f}{T_gf_0}\nonumber \\ &=&\pi^{-n/4}\int_{\Space{R}{n}} f(x)\,e^{-i(2t-\sqrt{2}qx+qp)}\,e^{-(x- \sqrt{2}p)^2/2}\,dx\nonumber \\ &=&e^{-2it-(p^2+q^2)/2}\pi^{-n/4}\int_{\Space{R}{n}} f(x)\,e^{- ((p+iq)^2+x^2)/2+\sqrt{2}(p+iq)x}\,dx\nonumber \\ &=&e^{-2it-z\bar{z}/2}\pi^{-n/4}\int_{\Space{R}{n}} f(x)\,e^{- (z^2+x^2)/2+\sqrt{2}zx}\,dx, \label{eq:tr-bargmann}\end{aligned}$$ where $z=p+iq,\ g=(t,p,q)=(t,z)$. Then $\widetilde{f}(g)$ belong to $\FSpace{L}{2}(\Heisen{n}, dg)$. It is easy to see, that for every fixed $t_0$ function $\breve{f}(z)=e^{z\bar{z}/2}\widetilde{f}(t_0,z)$ belongs to the Segal-Bargmann space, say, is analytic by $z$ and square-integrable with respect the Gaussian measure $\pi^{-n}e^{- z\bar{z}}$. The integral in  is the well known Bargmann transform [@Bargmann61]. Then the projector $\FSpace{L}{2}(\Heisen{n}, dg)\rightarrow \FSpace{L}{2}(\Space{R}{n})$ is a convolution on the with the kernel $$\begin{aligned} P(t,q,p)&=&\scalar{f_0}{T_gf_0}\nonumber \\ &=&\pi^{-n/4}\int_{\Space{R}{n}} e^{-x^2/2}\,e^{-i(2t- \sqrt{2}qx+qp)}\,e^{-(x-\sqrt{2}p)^2/2}\,dx\nonumber \\ &=&\pi^{-n/4}\int_{\Space{R}{n}} e^{-x^2/2 -i2t+\sqrt{2}iqx-iqp- x^2/2+\sqrt{2}px -p^2}\,dx\nonumber \\ &=&e^{-i2t-(p^2+q^2)/2}\pi^{-n/4}\int_{\Space{R}{n}} e^{-(x- (p+iq)/\sqrt{2})^2}\,dx\nonumber \\ &=&\pi^{n/4}e^{-i2t-z\bar{z}/2}.\label{eq:conv-bargmann}\end{aligned}$$ It was shown during the proof of Proposition \[pr:bargmann\] that a convolution with kernel  defines the usual value of the Bargmann projector with kernel . \[co:bergman\] The orthogonal Bergman projector on the space of analytic functions on unit ball $\Space{B}{}\in \Space{C}{n}$ has the kernel $$K(\zeta,\upsilon)=(1-\scalar{\zeta}{\upsilon})^{-n-1},$$ where $\scalar{\zeta}{\upsilon}=\sum_1^n\zeta_j \bar{\upsilon}_j$ is the scalar product at . We will only rewrite material of Chapters 2 and 3 from [@Rudin80] using our vocabulary. The group of biholomorphic automorphisms $\object{Aut}(\Space{B}{} )$ of the unit ball acts on transitively. For any $\phi\in \object{Aut} (\Space{B}{})$ there is a unitary operator associated by  and defined by the formula [@Rudin80 2.2.6(i)]: $$\label{eq:unit-ball} [T_\phi f](\zeta)=f(\phi(\zeta))\left(\frac{\sqrt{1-\modulus{\alpha }^2}}{1-\scalar{\zeta}{\alpha }}\right)^{n+1},$$ where $\zeta\in\Space{B}{},\ \alpha =\phi^{-1}(0),\ f(\zeta)\in \FSpace{L}{2}(\Space{B}{})$. Operator $T_\phi$ from  obviously preserve the space $\FSpace{H}{2}(G)$ of square-integrable holomorphic functions on . The homogeneous space $\object{Aut}(\Space{B}{})/G_0$ may be identified with  [@Rudin80 2.2.5]. To distinguish points of these two sets we will denote points of $B=\object{Aut}(\Space{B}{})/G_0\cong\Space{B}{}$ by Roman letters (like $a,u,z$) and points of itself by Greek letters ($\alpha, \upsilon, \zeta$ correspondingly). We also always assume, that $a=\alpha, u=\upsilon, z=\zeta$ under the mentioned identification. We select the function $f_0(\zeta)\equiv 1$ as the vacuum vector. The mean value formula [@Rudin80 3.1.1(2)] gives us: $$\begin{aligned} \widetilde{f}(a)&=&\scalar{f(\zeta)}{T_\phi f_0} \nonumber \\ &=&\scalar{T_{\phi^{-1}}f(\zeta)}{ f_0} \nonumber \\ &=& \int_{\Space{B}{}} f(\phi(\zeta))\left(\frac{\sqrt{1- \modulus{\alpha }^2}}{1-\scalar{\zeta}{\alpha }}\right)^{n+1} \,d\nu(\zeta) \nonumber \\ &=& f(\phi(0))\left(\frac{\sqrt{1-\modulus{\alpha }^2}}{1- \scalar{0}{\alpha }}\right)^{n+1} \nonumber \\ &=& f(a)(\sqrt{1-\modulus{\alpha }^2})^{n+1} , \label{eq:f-trans}\end{aligned}$$ where $a=\alpha =\phi(0),\ \phi\in B$ and $\widetilde{f}(a)\in\FSpace{L}{2}(B)$. Particulary $$\begin{aligned} \widetilde{f}_0(a)&=&(\sqrt{1-\modulus{\alpha }^2})^{n+1} \nonumber, \\ \widetilde{f}_0(zu)&=&\left(\frac{\sqrt{1-\modulus{\zeta }^2}\sqrt{1- \modulus{\upsilon }^2}}{1-\scalar{\upsilon}{\zeta}}\right)^{n+1} \label{eq:f0-trans},\end{aligned}$$ An invariant measure on $B$ is given [@Rudin80 2.2.6(2)] by the expression: $$\label{eq:ball-measure} d\mu(z)=\frac{d\nu(\zeta)}{(1-\modulus{\zeta}^2)^{n+1}},$$ where $d\nu(\zeta)$ is the usual Lebesgue measure on $B\cong\Space{B}{}$. We will substitute expressions from , and  to the reproducing formula : $$\begin{aligned} \widetilde{f}(u)&=&f(\upsilon)(\sqrt{1-\modulus{\upsilon }^2})^{n+1} \label{eq:most}\\ &=&\int_B \widetilde{f}(z) \widetilde{f}_0(z^{-1}u)\, d\mu(z) \nonumber \\ &\stackrel{(*)}{=}&\int_B \widetilde{f}(z) \widetilde{f}_0(zu)\, d\mu(z) \nonumber \\ &=&\int_{\Space{B}{}} f(\zeta)(\sqrt{1-\modulus{\zeta }^2})^{n+1} \left(\frac{\sqrt{1-\modulus{\zeta }^2}\sqrt{1-\modulus{\upsilon }^2}}{1-\scalar{\upsilon}{\zeta}}\right)^{n+1} \frac{d\nu(\zeta)}{(1- \modulus{\zeta}^2)^{n+1}} \nonumber \\ &=&\int_{\Space{B}{}} f(\zeta) \left(\frac{\sqrt{1- \modulus{\upsilon }^2}}{1-\scalar{\upsilon}{\zeta}}\right)^{n+1}\, d\nu(\zeta) \nonumber \\ &=&(\sqrt{1-\modulus{\upsilon }^2})^{n+1}\int_{\Space{B}{}} \frac{f(\zeta)}{(1-\scalar{\upsilon}{\zeta})^{n+1}}\, d\nu(\zeta) \label{eq:almost}.\end{aligned}$$ Here transformation $(*)$ is possible because every element of $B$ is an involution [@Rudin80 2.2.2(v)]. It immediately follows from the comparison of  and  that: $$f(\upsilon)=\int_{\Space{B}{}} \frac{f(\zeta)}{(1- \scalar{\upsilon}{\zeta})^{n+1}}\, d\nu(\zeta).$$ The last formula is the integral representation with the Bergman kernel for holomorphic functions on unit ball in . \[co:szego\] The orthogonal projector Szegö on the boundary $b\Space{U}{n} $ of the upper half-space in has the kernel $$S(z,w)=(\frac{i}{2}(\bar{w}_{n+1} -z_{n+1})-\sum_{j=1}^nz_j\bar{w}_j)^{- n-1}.$$ It is well known [@Gindikin64; @GreSte77; @Stein93] and was described at Example \[ex:toeplitz\], that there is a unitary representation of the Heisenberg group as the simply transitive acting group of shift on $b\Space{U}{n} $ (see [@Stein93 Chap. XII, § 1.4]): $$\label{eq:u-shifts} (\zeta,t): (z',z_{n+1}) \mapsto (z'+\zeta, z_{n+1}+t+2i\scalar{z'}{\zeta} +i \modulus{\zeta}^2),$$ where $(\zeta,t)\in\Heisen{n}, z=(z',z_{n+1})\in\Space{C}{n+1}, \zeta,z'\in\Space{C}{n}, t\in \Space{R}{}$. We again apply the general scheme from Example \[ex:transform\]. This gives an identification of and $b\Space{U}{n}$ and act on $b\Space{U}{n}\cong\Heisen{n}$ by left group shifts. Left invariant vector fields are exactly the [*tangential Cauchy-Riemann equations*]{} for holomorphic functions on . Shifts  commute with the tangential Cauchy-Riemann equations and thus preserve the [Hardy space]{} $\FSpace{H}{2}(b\Space{U}{n} ) $ of boundary values of functions holomorphic on . As vacuum vector we select the function $f_0(z)=(iz_{n+1})^{-n- 1}\in\FSpace{H}{2}(b\Space{U}{n} )$. Then the Szegö projector $P:\FSpace{l}{2}(b\Space{U}{n} ) \rightarrow \FSpace{H}{2}(b\Space{U}{n} ) $ is the right convolution on $\Heisen{n} \cong b\Space{U}{n}$ with $f_0(z)$ and thus should have the kernel (see the group low formula  for ) $$S(z,w)=(\frac{i}{2}(\bar{w}_{n+1} -z_{n+1})-\sum_{j=1}^nz_j\bar{w}_j)^{- n-1}.$$ Reader may ask, [*why have we selected such a vacuum vector?*]{} The answer is: for a [*simplicity*]{} reason. Indeed, the [*Cayley transform*]{} ([@Stein93 Chap. XII, § 1.2] and [@Rudin80 § 2.3]) $$\label{eq:cayley} C(z)=i\frac{e_{n+1}+z}{1-z_{n+1}},\ e_{n+1}=(0,\ldots,0,1)\in\Space{C}{n+1}$$ establishes a biholomorphic map from unit ball $\Space{B}{}\in \Space{C}{n+1}$ to domain . We can construct an isometrical isomorphism of the Hilbert spaces $\FSpace{H}{2}(\Space{S}{2n+1})$ and $\FSpace{H}{2}(\Space{U}{n})$ based on  $$\label{eq:c-iso} f(z) \mapsto [Cf](z)= f(C(z))\frac{-2i^{n+1}z_{n+1}}{(1- z_{n+1})^{n+2}},\ f\in \FSpace{H}{2}(\Space{U}{n}),\ [Cf]\in\FSpace{H}{2}(\Space{S}{2n+1}).$$ Then the vacuum vector $f_0=(iz_{n+1})^{-n-1}$ is the image of function $\widetilde{f}_0(w)=(-2i/(w-i))^{n+2}\in\FSpace{H}{2}(\Space{S}{2n+1})$ under transformation . It seems to be one from the simplest functions from $\FSpace{H}{2}(\Space{S}{2n+1})$ with singularities on $\Space{S}{2n+1}$. Connections with Relative Convolutions {#ss:connect} -------------------------------------- Now we return to relative convolutions and will show their connections with coherent states. For any operator $A:\FSpace{L}{2}(G,d\mu)\rightarrow\FSpace{L}{2}(G,d\mu)$ we can construct the [*Toeplitz-like*]{} operator $P_A=PA:\FSpace{H}{2}(G,d\mu)\rightarrow\FSpace{H}{2}(G,d\mu)$. Of course, using the isomorphism $H\cong\FSpace{H}{2}(G,d\mu)$ we can think about $P_A$ as an operator $P_A:H\rightarrow H$. Particulary, operators of group convolution on $G$ will induce relative convolutions on $H$ (see Lemma \[le:category\]). Thus we again have a direct way for applications of harmonic analysis in every problem concerning coherent states. Although, coherent states are very useful in physics we will stop here[^12] and will only develop this theme in the next Example connected with wavelets. Applications to Physics and Signal Theory {#se:physics} ========================================= We are going to consider some Examples connected with physics, but our division between mathematics and physics is so fragile as in the real life. \[ex:wavelets\] Let us consider the “$ax+b$ group” [@MTaylor86 § 7.1] of affine transformations of the real line. We will denote this group by $A$ and its Lie algebra by . We will consider their operation on the real line $S=\Space{R}{} $. is spanned by two vector fields $X_s=\frac{1}{i}\frac{\partial }{\partial y}$ (which generate shifts) and $X_d=y\frac{1}{i}\frac{\partial }{\partial y}$ (which generate dilations). Their commutators are $[X_s,X_d]=X_s$. Then transformation  takes the form $$\begin{aligned} \widetilde{f}(x_1,x_2)&=&\frac{1}{2\pi}\int_{\Space{R}{}} \bar{f}(y)\, e^{i(x_1 X_s+ x_2 X_d)}f_0(y)\,dy\nonumber \\ &=&\frac{1}{2\pi}\int_{\Space{R}{}} \bar{f}(y)\, e^{-x_2/2}f_0(e^{-x^2}y-x_1)\,dy. \label{eq:wavelet}\end{aligned}$$ The last line is easily recognized as the [*wavelet transform*]{} [@HeilWaln89]. Similar expression in the spirit of Definition \[de:coherent1\] for the [*Gabor transform*]{} [@HeilWaln89] may be obtained if we replace $A$ by the Heisenberg group . Then, as was shown early, different signal alterations constructed in signal theory are relative convolution on $S=\Space{R}{}$ induced by or the meta-Heisenberg group from [@Folland94]. For example, signal filtration may be presented at the Gabor representation as a multiplication by characteristic functions of the wished time and frequency intervals. Using relative convolutions we can coherently introduce wavelets-like transform for every semi-direct product [@MTaylor86 § 5.3] of Lie group and Abelian one [@BernTayl94]. In view of applications to the signal theory it seems interesting to start from the Heisenberg group and its dilations. The one-parameter group $D=\{\delta_\tau \such \tau\in\Space{R}{} \}$ of dilations of the Heisenberg group is given by the formula $$\label{eq:dilation} \delta_\tau(t,z)=(e^{2\tau}t,e^\tau z),\ (t,z)\in\Heisen{n} \cong \Space{R}{}\times\Space{R}{2n}$$ and has the one-dimensional Lie algebra spanned on the vector field $$\label{eq:frame-dilat} X_h=\frac{1}{i}(2t\frac{\partial }{\partial t} + z\frac{\partial }{\partial z}).$$ If we introduce now the relative convolutions for Lie algebra generated by vector fields $X^l_j$ from  and $X_h$ from  then we obtain the [*Heisenberg Gabor-like transform*]{}, which should be useful to analyze of the radar ambiguity function [@Folland89 § 1.4]. \[ex:quantum\] We are going to describe [*group quantization*]{} from paper [@Kisil94d]. The usual “quantization” means some (more or less complete) set of rules for the construction of a quantum algebra from the classical description of a physical system. The group quantization is based on the Hamilton description and consists of the following steps. 1. Let $\Omega=\{x_j\}, 1\leq j\leq N$ be a set of physical quantities defining state of a classical system. Observables are real valued functions on the states. The most known and important case is the set $\{x_j=q_j,x_{j+n}=p_j\},\ 1\leq j\leq n, N=2n$ of coordinates and impulses of classical particle with $n$ degrees of freedom. Observables are real valued functions on . This example will be our main illustration during the present consideration. 2. We will complete the set $\Omega$ till $\bar{\Omega}$ by additional quantities ${x_j}, N<j\leq \bar{N}$, such that $\bar{\Omega}$ will form the smallest algebra containing $\Omega$ and closed under the Poisson bracket: $$\{x_i,x_j\}\in \bar{\Omega},\ \mbox{ for all } x_i,x_j\in \bar{\Omega}.$$ In the case of a particle we should add the function $x_{2n+1}=1$, which is equal to unit identically and one obtains the famous relations ($\bar{N}=2n+1$) $$\label{eq:poisson} \{x_j,x_{j+n}\}=-\{x_{j+n},x_j\}=x_{2n+1}$$ and all other Poisson brackets are equal to zero. 3. We form a $\bar{N}$-dimension Lie algebra $\algebra{p}$ with a frame $\{\widehat{x}_j\},\ 1\leq j \leq \bar{N}$ with the formal mapping $\hat{}: x_j\mapsto\widehat{x}_j$. Commutators of frame vectors of are formally defined throughout the formula $$\label{eq:hat-poisson} [\widehat{x}_i,\widehat{x}_j]=\widehat{\{x_i,x_j\}}$$ and we extend the commutator on whole algebra by the linearity. For a particle this step give us the Lie algebra $\algebra{h}_n$ of the Heisenberg group (compare  and ). 4. We introduce an algebra $\algebra{P}$ of relative convolutions  induced by . These operators are [*observables*]{} in the group quantization and by analogy to classic case they may be treated as functions of $\widehat{x}_j$ (see Remark \[re:function\]). A set $S$ which algebra acts on and type of kernels are depending on physically determining constraints. The family of all one-dimensional representations of is called [*classical*]{} mechanics and different noncommutative representations correspond to [*quantum*]{} descriptions with the different [*Planck constants*]{}. For particle we have the following opportunities: 1. $S=\Space{R}{n},\ \widehat{x}_j=X_j=M_{q_j},\ \widehat{x}_{j+n}=\hbar\frac{1}{i}\frac{\partial }{\partial q_j}$, relative convolutions are PDO from Example \[ex:pdo\] and we have obtained the [*Dirac–Heisenberg–Schrödinger–Weyl quantization*]{} by PDO. 2. $S=\Space{R}{2n},\ \widehat{x}_j=X_j=M_{q_j},\ \widehat{x}_{j+n}=M_{p_j}$, relative convolutions are operators of multiplication by functions (or just functions) from Example \[ex:multiplication\] and we have obtained the usual classical description, which we have started from. 3. $S=\Heisen{n},\ \widehat{x}_j=X^{l(r)}_j,\ 0\leq j\leq 2n+1$ and relative convolutions form the group convolution algebra on . This description (so-called [*plain mechanics*]{}) contain both the [*quantum*]{} and [*classical*]{} ones with the natural realization of the [*correspondence principle*]{} (see [@Kisil94d] for details). Group quantization is straightforward enough and obviously preserves the symmetry group of the classical system under investigation. Moreover, there is also other advantages of the proposed quantization, which distinguish it from the already known ones. - In contrary to the [*operator quantization*]{} of Berezin [@Berezin74] and the [*geometrical quantization*]{} of Kirillov–Souriau–Konstant [@Woodhouse80] we should not introduce [*a priori*]{} any Planck constants. Moreover, during the posterior analysis of relative convolution algebra representations a parameter corresponding to the Planck constant will appear naturally. By the way, a [*set of Planck constants*]{} should not necessary belong to $[0,+\infty[$ and may form more complicated topological spaces. - The problem of ordering of noncommutative quantities $\widehat{q}_j$ and $\widehat{p}_j$ does not occur under group quantization. Correspondence $$\mbox{function } k(x)\ \rightarrow \mbox{ convolution with the kernel }k(x)$$ is direct enough even for noncommutative groups. Meanwhile, in other quantization the “painful question of ordering” [@Berezin71] has generated many different answers: the $\widehat{q}\widehat{p}$-quantization, the $\widehat{p}\widehat{q}$-quantization, the Weyl-symmetrical, the Wick and the anti-Wick (Berezin) quantization. The presented group quantization has deep roots in the quantization procedure of Dirac [@Dirac67]. The main differences are - We recognize the Heisenberg commutation relations  only as a particular case among other possibilities. However, due to Theorem \[th:pdo\] they play the fundamental role. - We do not look only for irreducible representations of commutation relations. Conclusion ========== The paper tried to illustrate [*how the systematical usage of harmonic analysis in various applications may be useful for both: the analysis and applications*]{}. It seems, that relative convolutions form an appropriate tool for this purpose. Given Examples from different fields of mathematics and physics made reasonable studying of relative convolutions. Moreover, we have repeatedly met nilpotent Lie groups (and particularly the Heisenberg) group within important applications, so our primary interest in such groups should be excused. [^1]: This work was partially supported by CONACYT Project 1821-E9211, Mexico. [^2]: On leave from the Odessa State University. [^3]: See, for example, the deduction of a PDO–form for convolutions on step 2 nilpotent Lie groups in [@MTaylor84 p. 9–10] or Theorem \[th:projector\]. [^4]: In the mentioned paper consideration is restricted to nilpotent step 3 Lie algebras. [^5]: For a commutative group G the left and the right operations are the same. [^6]: Relative convolutions obviously include usual group convolutions and, on the other hand, due to Theorem \[th:nature\] they may be treated just as representations of group convolutions. Thus they are not [*generalized*]{} convolutions but simply [*relative*]{} convolutions. [^7]: “The Heisenberg group […]{} is basic to this paper (and much of the rest of the word)” [@Howe80b]. [^8]: “The most interesting aspect of the […]{} theory has to do with the application of this machinery to concrete examples” [@Guillemin84]. [^9]: This is an answer to the reasonable question of E. Stein:“Does the algebra of two-sided convolutions contain at least one interesting operator?” [^10]: Or, at least, do not use harmonic analysis directly. [^11]: This explains why “a critical ingredient in our analysis is an averaging operation over the Segal–Bargmann representation of the Heisenberg group” [@BergCob87]. [^12]: However, let us remind again that Example \[ex:fock\] forms an interesting application to physics.
--- abstract: 'Optical Feshbach resonances (OFRs) have generated significant experimental interest in recent years. These resonances are promising for many-body physics experiments, yet the practical application of OFRs has been limited. The theory of OFRs has been based on an approximate model that fails in important detuning regimes, and the incomplete theoretical understanding of this effect has hindered OFR experiments. We present the most complete theoretical treatment of OFRs to date, demonstrating important characteristics that must be considered in OFR experiments and comparing OFRs to the well-studied case of magnetic Feshbach resonances. We also present a comprehensive treatment of the approximate OFR model, including a study of the range of validity for this model. Finally, we derive experimentally useful expressions that can be applied to real experimental data to extract important information about the resonance structure of colliding atoms.' author: - 'T.L. Nicholson' - 'S. Blatt' - 'B.J. Bloom' - 'J.R. Williams' - 'J.W. Thomsen' - 'J. Ye' - 'Paul S. Julienne' bibliography: - 'references.bib' title: 'Optical Feshbach resonances: Field-dressed Theory and comparison with experiments' --- Introduction ============ Background {#sec:intro} ---------- Magnetic Feshbach resonances (MFRs) have become a staple of quantum gas experiments with alkali-metal atoms, allowing for unprecedented control of interatomic interactions [@Chin2010]. The MFR technique is so powerful that it has extended the reach of dilute quantum gas experiments to a variety of areas of physics. Examples of high impact experiments that utilize MFRs are the study of strongly correlated systems [@Bloch2008] such as unitary Bose [@Rem2013; @Wild2012] and Fermi [@Kinast2004; @Regal2004; @Zwierlein2004] gases, the discovery of exotic few-body bound states [@Kraemer2006; @Braaten2006; @Ferlaino2011; @Wang2013], the ability to make ultracold molecules [@Ni2008; @Quemener2012], and the engineering of novel quantum matter [@Chotia2012; @Yan2013]. Feshbach resonances based on laser fields—known as “optical Feshbach resonances” (OFRs) [@Fedichev1996; @Bohn1997]—have also been observed [@Fatemi2000; @Jones2006], but so far their utility has been limited. Since laser fields can be focused tighter and switched faster than magnetic fields, it is expected that OFRs could yield an MFR-like effect but with orders of magnitude better spatial and temporal control [@Yamazaki2010]. Furthermore, OFRs are better suited for alkaline-earth-metal atoms, which have magnetically insensitive electronic ground states. The study of alkaline earth atoms is now a rich field, attracting attention for metrology [@Bloom2014], quantum information [@Daley2008; @Gorshkov2009], and many-body physics [@Gorshkov2010]. Quantum degenerate gases of these atoms have also been realized [@Takasu2003; @Kraft2009; @Stellmer2013]. Many-body physics has been demonstrated in strontium lattice clocks [@Martin2013], and it has been shown that controlling many-body interactions in gases of alkaline earth metals could lead to better clock accuracy [@Swallows2011]. Without MFRs to facilitate the same many-body control enjoyed by alkali-metal experiments, OFRs have been suggested as an alternative [@Ciuryo2005]. OFRs have been the focus of several experiments. These resonances have been observed in alkali gases [@Fatemi2000; @Theis2004] and in alkaline-earth-like atoms [@Enomoto2008]. P-wave OFRs have been reported [@Yamazaki2012], and OFRs have been successfully applied to induce thermalization in Sr gases [@Blatt2011] and manipulate the condensate dynamics of a Sr Bose-Einstein condensate [@Yan2013a; @Yan2013b]. The theory used to describe these experiments was based on a quantum defect treatment by Bohn and Julienne, who used an isolated resonance approximation to derive the optically modified scattering length [@Bohn1999]. Although the isolated resonance theory has been successful in describing some observations of OFRs, it fails to explain OFR behavior in the large detuning regimes that are critical to a proposal for practically applying these resonances [@Ciuryo2005]. Attempts to experimentally realize this proposal did not succeed [@Blatt2011]; therefore, the limited theoretical understanding of OFRs has hindered experimental progress. To broaden the theoretical understanding of OFRs, we perform the most complete theoretical analysis of this effect to date. To this end, we treat OFRs with a numerical coupled channel method, which has been highly successful for treating MFRs [@Kohler2006; @Chin2010]. Like the MFR theory, our coupled channel approach is capable of treating multiple interacting resonances without being restricted to the more limited isolated resonance approximation. Consequently, this more general treatment allows us to study the range of validity of the isolated resonance approximation, and it also enables us to point out similarities and significant differences between OFRs and MFRs. Finally, we use the isolated resonance theory to derive experimentally useful formulas that can be used to understand real experimental OFR data. Basic Collision Theory {#subsec:Basic} ---------------------- In the context of cold-atom physics, a Feshbach resonance is a collisional resonance of two particles that is tunable by an external field. This is possible if a molecular bound state from an excited scattering channel (called the “closed channel") couples to the free atom continuum of the ground state scattering channel (called the “entrance channel" or the “background channel"). Furthermore, the bound state energy is tunable by an external magnetic or electromagnetic field. The presence of this bound molecular state modifies the $s$-wave scattering length of the atoms, thereby changing the atomic interactions as the external field is tuned. We emphasize that both MFRs and OFRs can be treated by the same scattering formalism, which accounts for the differences in their coupling and control mechanisms, as presented in the review by Chin [*et al.*]{} [@Chin2010]. A typical MFR is coupled to the entrance channel by internal short range spin-dependent couplings within the ground state manifold of Zeeman levels. MFRs are tuned by varying an external magnetic field $B$ to move a molecular bound state across a collision threshold, creating a pole in the scattering $S$-matrix as a function of $B$. An OFR involves coupling a bound molecular state to two colliding atoms in their ground states using a photon from a laser, hereafter called the “photoassociation laser" or “PA laser.” In this case, the coupling depends on both the PA laser detuning from a photoassociation resonance (the “molecular detuning”) and the PA laser intensity. In contrast to MFRs, which are often based on molecular states that have very long lifetimes, spontaneous decay of the excited molecular state in an OFR introduces an appreciable linewidth to the molecular transition. Any population transferred to the excited state undergoes spontaneous decay, which translates to inelastic loss collisions that must be minimized in order to utilize an OFR. However, resonance decay does not necessarily prevent the application of OFRs since MFRs with 2-body decay channels  [@Thompson2005b; @Kohler2005; @Naik2011] have proven experimentally useful  [@Cornish2000; @Donley2002; @Papp2008; @Trenkwalder2011; @Kohlstall2012]. In the OFR studies presented here, we consider bosonic [$^{88}$Sr]{} with the two ground state atoms providing the $^1S_0 + ^1\!\!S_0$ ground state entrance channel and the excited state $^1S_0 + ^3\!\!P_1$ providing the closed channels, schematically represented in Fig. \[fig:atoms\_in\_radiation\_field\]. Since bosonic isotopes of alkaline earths have no nuclear spin, the [$^{88}$Sr]{} resonance structure is considerably simpler than for atoms with hyperfine interactions, making it a good atom for an OFR experiment. The $^1S_0 \rightarrow ^3\!\!P_1$ atomic transition is an intercombination line with a natural linewidth of $\gamma_a = 2\pi \times$ 7.4 kHz. The narrowness of this transition means that all OFRs in [$^{88}$Sr]{} are well resolved from the atomic line, decreasing the severity of off-resonant atomic light scattering. To analyze the scattering of two colliding [$^{88}$Sr]{} atoms in a light field, we calculate the scattering *S*-matrix to determine the elastic and inelastic scattering cross sections. Since the [$^{88}$Sr]{} ground state is completely spinless, and since current OFR experiments are typically performed at temperatures of a few $\mu$K or below, the scattering is described by an $s$-wave collision with a single nondegenerate entrance channel. Therefore, we will develop our theory for this experimentally simple OFR system, for which we only need a single $s$-wave $S$-matrix element $S(k)=e^{2i\eta(k)}$, represented in general by a complex energy-dependent phase shift $\eta(k)$. Here $k$ is defined via the collision velocity $\hbar k / \mu$, $\mu = m/2$ is the reduced mass, and $m$ is the mass of an [$^{88}$Sr]{} atom. This phase shift in turn defines an energy-dependent scattering length $\alpha(k)$ as [@Hutson2007; @Idziaszek2010a; @Quemener2012], $$\label{eqn:scattering_length} \alpha(k) = a(k) - i b(k) = -\frac{\tan \eta(k)}{k} = \frac{1}{i k} \frac{1 - S(k)}{1 + S(k)} \,.$$ This expression is useful for small but nonvanishing collision energies, and the standard complex scattering length is the $k \to 0$ limit of this expression. The elastic and inelastic loss cross sections are $$\begin{aligned} \sigma_{el} & = \frac{\pi g}{k^{2}} |1 - S(k)|^{2} = 8 \pi |\alpha(k)|^{2} f^{2}(k), \label{eqn:el_cross_section} \\ \sigma_{in} & = \frac{\pi g}{k^{2}} \left( 1 - |S(k)|^{2} \right) = \frac{8 \pi}{k} b(k) f(k). \label{eqn:in_cross_section}\end{aligned}$$ Here $g$ is a collisional symmetry factor, which is equal to 2 for identical bosons (as assumed here). The function $$\label{eqn:f_factor} f(k) = \frac{1}{1 + k^{2} |\alpha(k)|^{2} + 2 k b(k)} \,$$ approaches unity when $k|\alpha| \ll 1$ for all detunings. For a trapped gas of atoms, this limit occurs when $k_B T/\hbar \gamma \ll 1$, where $k_B$ is Boltzmann’s constant and $T$ is the sample temperature. The elastic and inelastic collision rate coefficients are related to these cross sections as $$\begin{aligned} \label{eqn:rate_coefficients} K_{el} (k) & = \frac{\hbar k}{\mu} \sigma_{el} (k) \to 8\pi\frac{\hbar}{\mu} k|\alpha(k)|^2 \,\,\,\mathrm{as}\,\,\, k \to 0 \\ K_{in} (k) & = \frac{\hbar k}{\mu} \sigma_{in} (k) \to 8\pi\frac{\hbar}{\mu} b(k) \,\,\,\mathrm{as}\,\,\, k \to 0 \,. \label{eqn:loss_rate}\end{aligned}$$ These general expressions are valid in the $s$-wave limit for OFRs and for decaying or non-decaying MFRs. A sum over higher partial waves is needed when these begin to contribute at higher $k$, and a thermal average of $K_{el}(k)$ and $K_{in}(k)$ is needed when comparing to experiment. ![(Color online) a) The $^1S_0 + ^1\!S_0$ entrance channel of [$^{88}$Sr]{} couples to a bound state of the $^3P_1 + ^1\!S_0$ closed channel via the PA laser field. The atomic transition is a 7.5 kHz intercombination line. Here $E$ is the collision energy, $\omega_0$ is the atomic resonance frequency, and $\omega$ is the laser frequency. b) In the dressed state picture, two free atoms in the entrance channel are brought into resonance with a molecular bound state. Here $\delta$ is the “molecular detuning” (defined in Section \[sec:isolated\_res\]), and $n_p$ is the photon number. The Condon radius $R_c$ is defined as the value of $R$ where the two potentials cross.[]{data-label="fig:atoms_in_radiation_field"}](colliding_atoms.eps){width="\linewidth"} Coupled Channels Formulation of Optical Feshbach Resonances {#sec:CC} =========================================================== Background {#subsec:CC_background} ---------- The standard treatment for atomic collisions involving two or more internal states of the atoms is the coupled channels (CC) method. Numerical models based on CC methods have been very successful in treating collisions and MFRs of ground state alkali-metal atoms [@Chin2010; @Kohler2006]. The CC method involves setting up a basis set representing the “channels” of the electronic, spin, and rotation degrees of freedom of the colliding atoms and then solving the matrix Schr[ö]{}dinger equation for the amplitude of the radial motion in the interatomic separation coordinate $R$ for each of these channels. In MFR theory, the channels represent the states of the atoms in a magnetic field for $R \rightarrow \infty$, the Born-Oppenheimer potentials characterize the $R$-dependent interactions, and spin coupling matrix elements are approximated by their atomic values. An external $B$ field shifts the energies of the atomic and molecular energy levels. In the case of OFRs, the channels represent the field dressed atoms, where the ground and excited states are coupled by the light field in the dipole approximation, and the $R$-dependent molecular interactions are represented by the ground and excited state Born-Oppenheimer potentials together with any non-adiabatic coupling between them. The OFR case has the added complication of including the spontaneous emission of light by excited state atoms or molecules. To date, all cold atom OFRs have been treated by a resonant scattering formulation  [@Thorsheim1987; @Fedichev1996; @Bohn1999]. The next section will discuss the approximation of treating each OFR as an isolated resonance. Here we concentrate on giving a full CC treatment  [@Zimmerman1977; @DeVries1978a; @DeVries1978b; @Mies1981] that includes the effect of multiple overlapping resonances without restricting the theory to treating isolated single resonances. This enables us to establish the conditions under which the isolated resonance approximation is valid. We follow the field-dressed collision approach of Julienne [@Julienne1982a; @Julienne1982b], which was applied to explain experiments on the collisional redistribution of light  [@Julienne1984; @Julienne1986]. To do this it is necessary to properly represent the three-dimensional (3D) nature of the collision and the role of atomic degeneracy. References [@Julienne1982b; @Napolitano1997] treat the exchange of multiple photons during a collision, by which one partial wave is coupled to higher partial waves through the intrinsically anisotropic nature of the interaction with light. Reference [@Napolitano1997] adapts the CC dressed atom formalism to cold atom collisions in strong optical fields to explain the phenomena of optical shielding. Three effects need to be incorporated within a CC theory to describe OFRs in the collision of $^1$S$_0$ Sr atoms in a light field tuned near the $^1S_0 \rightarrow ^3\!\!P_1$ line: (1) the field dressing of the collision, (2) the inherently 3D nature of the collision, with a space axis defined for the separated atoms by the PA laser polarization but with a rotating interatomic axis needed for the excited molecular bound states, and (3) the spontaneous emission while in the excited state. If we make the approximation that the light field is weak, the total angular momentum $J$ is a good quantum number (the optical coupling matrix element remains small compared to the spacing of rotational levels in the excited state). In this case Refs. [@Julienne1984; @Julienne1986] showed that six CCs are needed to describe optically coupled $^1$S$_0+ ^1$S$_0 \to ^1$S$_0+ ^1$P$_1$ collisions. The same is true when we replace $^1$P$_1$ with $^3$P$_1$. One set of channels represents the ground state collision with partial wave $\ell=J$ and $n_p$ photons at an angular frequency $\omega$. Another set represents the excited $0_u$ and $1_u$ molecular states with $n_p-1$ photons at frequency $\omega$ and respective projection $\Omega=$ 0 and 1 of electronic angular momentum $j=1$ on the interatomic axis. These excited state channels have total angular momentum $J_e=J-1$ (two channels), $J$ (one channel), and $J+1$ (two channels). In the special case of $s$-wave collisions ($J=0$) of cold atoms, only three channels are needed, representing the ground state and the $0_u$ and $1_u$ states with $J=1$. Finally, spontaneous emission from the excited molecular state can be included with a complex potential  [@Napolitano1994], with a caveat that the imaginary decay part of the potential has to be turned off when the atoms are far apart in the free atom limit. We assume that the free atoms are weakly dressed—that is, the PA laser with photon energy $\hbar\omega$ is detuned from the atomic excitation energy $\hbar\omega_0$ by a large amount compared to the optical coupling strength $$V_\mathrm{opt}=\left (2\pi I/c\right )^{1/2} d \,, \label{eqn:Vopt}$$ where $I$ is PA laser intensity, $c$ the speed of light, and $d$ is a molecular transition dipole matrix element [@Julienne1986]. However, the short range molecular states can be strongly dressed, so that the peak of an on-resonant PA line at $\hbar\omega_n$, where $n$ is the molecular vibrational level, can be power broadened. The rotational quantum number $J_e$ will remain a good approximate quantum number as long as $V_\mathrm{opt}$ remains small compared to the rotational constant $B_n$ of level $n$. (The separations of the $J=0$ and 2 levels from the $J=1$ level are $2B_n$ and $4B_n$ respectively.) Formulation for $^{88}$Sr {#subsec:CC_formulation} ------------------------- We include in our treatment here the minimal number of three channels needed to get a basic description of near-threshold $s$-wave OFRs. This minimal treatment could be written in either of two basis sets representing the electronic, spin, rotational, and photon degrees of freedom. One basis set for the molecular degrees of freedom is the Hund’s case (c) basis represented as $|\Omega_sJM\rangle_c$, where the projection of electronic plus spin angular momentum $j$ on the rotating body-fixed axis is $\Omega$, $s$ represents the [*gerade*]{} or [*ungerade*]{} inversion symmetry of electronic coordinates, and $M$ is the projection of total angular momentum $J$ on a space-fixed axis. The other molecular basis is the Hund’s case (e) asymptotic basis of Refs. [@Julienne1982a; @Julienne1986] represented as $|j_s\ell JM\rangle_e$, where $j_s=0$ or 1 represents the separated atoms in the respective $^{1}S_{0} + ^{1} \! S_{0}$ and $^{1}S_{0} + ^{3} \! P_{1}$ channels with partial wave $\ell$, coupled to total angular momentum $J$ and projection $M$. The subscript $s$ on $j$ indicates that the electronic wavefunction is symmetrized with respect to the exchange of electronic coordinates. Table \[tab:basis\] shows the three basis functions for a dressed CC calculation in either representation. The transformation between the molecular and asymptotic representations is (see, for example, Eq. (36) of Ref. [@Julienne1982a]): $$\begin{aligned} |0_uJM\rangle_c &=& \left (\frac{J}{2J+1}\right)^{1/2} |1_u,J-1,JM\rangle_e \nonumber \\ & &- \left (\frac{J+1}{2J+1}\right)^{1/2} |1_u,J+1,JM\rangle_e \label{eqn:c-e1} \\ |1_uJM\rangle_c &=& \left (\frac{J+1}{2J+1}\right)^{1/2} |1_u,J-1,JM\rangle_e \nonumber \\ & & + \left (\frac{J}{2J+1}\right)^{1/2} |1_u,J+1,JM\rangle_e \,. \label{eqn:c-e2}\end{aligned}$$ [ccc]{} Channel & Case (c): $|\Omega_sJM\rangle|n_p\omega\sigma\rangle$ & Case (e): $|j\ell JM\rangle|n_p\omega\sigma\rangle$\ $|1\rangle_b$ & $|0_g00\rangle_c|n_p\omega\sigma\rangle$ & $|0_g000\rangle_e|n_p\omega\sigma\rangle$\ ------------------------------------------------------------------------ $|2\rangle_b$ & $|0_u1\sigma\rangle_c|n_p-1,\omega\sigma\rangle$ & $|1_u01\sigma\rangle_e|n_p-1,\omega\sigma\rangle$\ $|3\rangle_b$ & $|1_u1\sigma\rangle_c|n_p-1\omega\sigma\rangle$& $|1_u21\sigma\rangle_e|n_p-1\omega\sigma\rangle$\ Using the CC expansion of the full wavefunction at total energy $E$, $$\Psi(R,E) = \sum_{i=1}^3 |i\rangle_b F_{i,b}(R,E)/R \label{eqn:CC_Psi}$$ where $F_{i,b}$ represents the amplitude of the wavefunction projected on the basis function $| i \rangle_b$ for Hund’s case b = (c) or (e). The CC matrix Schr[ö]{}dinger equation for the $s$-wave collision of the two atoms in a (moderately) weak light field is $$\frac{\partial^2{\Psi}}{\partial R^2} + \frac{2\mu}{\hbar^2} \left (E\cdot \bf{I} - \bf{V}(R) \right )\Psi = 0 \label{eqn:CC}$$ where $\bf{I}$ is the identity matrix and the potential matrix $\bf{V}$ describes the diagonal and off-diagonal matrix elements of the collisional and optical interactions. Either the $b=$ (c) or (e) representations (Table \[tab:basis\]) of the excited state could be used to set up the expansion and $\bf{V}$ matrix in Eqs. (\[eqn:CC\_Psi\]) and (\[eqn:CC\]). Our numerical calculations use the Hund’s case (e) basis, for which the matrix elements are given in Table I of Ref. [@Julienne1986], and quoted in the supplemental online material for Ref. [@Blatt2011]: $$\label{eqn:CC_V} \bf{V} = \left( \begin{array}{ccc} V_g & V_\mathrm{opt} & 0 \\ V_\mathrm{opt} & \frac{1}{3}(V_{0u}+2V_{1u}) & \frac{\sqrt{2}}{3}(V_{1u}-V_{0u}) \\ 0 & \frac{\sqrt{2}}{3}(V_{1u}-V_{0u}) & \frac{1}{3}(2V_{0u}+V_{1u})+6 V_\mathrm{cen} \end{array} \right ) \, ,$$ where the $6 V_\mathrm{cen}$ term represents the $d$-wave centrifugal potential with $V_\mathrm{cen}=\hbar^2/(2\mu R^2)$. Here $V_g(R)$, $V_{0u}(R)$, and $V_{1u}(R)$ represent the ground state and $0_u$ and $1_u$ excited state BO potentials, each of which we model as a Lennard-Jones potential plus an additional long range term: $$\begin{aligned} V_g(R) &=& \left ( \left(\frac{R_{0,g}}{R}\right)^6 - 1 \right ) \frac{C_{6,g}}{R^6} -\frac{C_{8,g}}{R^8} +V_{g\infty} \label{eqn:Vg}\\ V_{0u}(R) &=& \left ( \left(\frac{R_{0,0u}}{R}\right)^6 - 1 \right ) \frac{C_{6,0u}}{R^6}-\frac{C_{3,0u}}{R^3} +V_{u\infty} \label{eqn:V0u} \\ V_{1u}(R) &=& \left ( \left(\frac{R_{0,1u}}{R}\right)^6 - 1 \right ) \frac{C_{6,1u}}{R^6}+\frac{C_{3,1u}}{R^3} +V_{u\infty} \label{eqn:V1u} \,,\end{aligned}$$ The $V_{s\infty}$ terms give the asymptotic values of the potentials as $R \to \infty$, as explained below. We use the excited state potential parameters from Zelevinsky [*et al.*]{} [@Zelevinsky2006; @footnote1]. The ground state $C_{6,g}$ and $C_{8,g}$ parameters come from Ref. [@Porsev2006], and $R_{0,g}$ was optimized to reproduce the measured bound state binding energies of Ref. [@Escobar2008] to within 0.4%  [@footnote2]. $V_g(R)$ has an $s$-wave scattering length of $-1.4$ a$_0$, consistent with that reported in Ref. [@Escobar2008]. The optical coupling matrix element in Eq. (\[eqn:CC\_V\]) is given by Eq. (\[eqn:Vopt\]) in the dipole approximation, where we neglect retardation (that is, the phase difference between the optical fields separated by distance $R \ll \lambda$, where $\lambda=2\pi c/\omega$ is the wavelength of the excitation light). Thus, since we use the symmetrized $g$ and $u$ electronic states, $d=\sqrt{2} d_a$, where the atomic transition dipole $d_a= 0.08682$ atomic units (1 a.u. $=$ $ea_0$, where $e$ is the electron charge and $a_0$ is the Bohr radius), corresponding to an atomic $^3$P$_1$ lifetime of 21.46 $\mu$s or linewidth of $\gamma_a$ $=$ $2 \pi \times$ 7.416 kHz. Thus, introducing units into Eq. (\[eqn:Vopt\]), $$V_\mathrm{opt}/h = 24.83 \, \mathrm{MHz} \, \times \, d_a \sqrt{I/\mathrm{(1 W/cm}^2)} \,,$$ where $d_a$ is in atomic units. The optical coupling in $\bf{V}$ conforms to the case (e) selection rule that $\Delta \ell =0, \Delta m_\ell =0$ (it is only the electronic $j$ quantum number that changes). This coupling is also independent of light polarization $\sigma$ for this transition. Using Eq. (\[eqn:CC\_V\]), there will be an asymptotic light shift $$V_\infty= \frac{\hbar(\omega_0-\omega)}{2} \left ( \sqrt{\left ( \frac{2 V_\mathrm{opt}}{\hbar(\omega_0-\omega)}\right )^2+1}-1 \right ) ,$$ which is negative for the ground state and positive for the excited state. Thus, taking $V_{g\infty}=V_\infty$ and $V_{u\infty}=\hbar(\omega_0-\omega)+V_\infty$ in Eqs. (\[eqn:Vg\])-(\[eqn:V1u\]) ensures that when $\bf{V}$ is diagonalized the lowest energy eigenvalue at large $R$ for the field-dressed ground state is zero. With this definition of the zero of energy, the total energy $E$ in the CC Schr[ö]{}dinger equation (\[eqn:CC\]) represents the relative collision kinetic energy $\hbar^{2} k^{2}/2 \mu$ of the dressed ground state atoms, and $E = \hbar^{2} k^{2}/2 \mu \to 0$ at the collision threshold. The matrix $\bf{V}$ in Eq. (\[eqn:CC\_V\]) could be transformed to the molecular case (c) representation by transforming the $2 \times 2$ excited state block using the (c) to (e) transformation matrix used in Eqs. (\[eqn:c-e1\]) and (\[eqn:c-e2\]). This would give the diagonal $J=1$ $0_u$ and $1_u$ potentials given in Eqs. (1) and (2) of Zelevinsky [*et al.*]{} [@Zelevinsky2006] and generate the body-frame Coriolis coupling term between these two states. The optical coupling in the case (c) molecular basis is different from that in the asymptotic case (e) basis. Using the transformations in Eqs. (\[eqn:c-e1\]) and (\[eqn:c-e2\]) shows that the optical couplings matrix elements between the ground $J=0$ $0_g$ state and the respective excited $J=1$ $0_u$ and $1_u$ states are determined from Eq. (\[eqn:Vopt\]) with the molecular dipole matrix elements $$\begin{aligned} d_{0u} &=& \sqrt{1/3} \sqrt{2}d_a \,, \label{eqn:d0u} \\ d_{1u} &=& \sqrt{2/3} \sqrt{2}d_a \,. \label{eqn:d1u}\end{aligned}$$ The $\sqrt{2}$ is the same homonuclear $ g \to u$ enhancement factor that affects the case (e) matrix element. The other factor corresponds to the usual H[ö]{}nl-London factor for R-branch ($J \to J+1$) molecular transitions. Treating an OFR requires that we include the decay from the excited state, which has a molecular decay rate $\gamma$. Our calculations assume $\gamma=\gamma_m$, where we define $\gamma_m=2\gamma_a$. This rate $\gamma_m$ is the rate of spontaneous emission from the excited molecular state in the long-range non-retarded dipole approximation. A nonzero value of $\gamma-\gamma_m$ would be due to other processes that induce decay of the excited state or change the emission rate from its long-range non-retarded dipolar value. While Bohn and Julienne [@Bohn1999] introduced artificial channels to simulate excited state decay, a simpler method is to introduce an imaginary term in the excited state potentials in Eqs. (\[eqn:V0u\]) and (\[eqn:V1u\]), replacing $V_{ju}$, $j=0,1$, with: $$V_{ju} -i\frac{\hbar \gamma}{2} \left ( 1+e^{\beta (R-R_\mathrm{cut})} \right )^{-1} , \label{eqn:ImV}$$ where $\gamma$ is an $R$-independent constant. The function in parenthesis ensures that molecular decay turns off at large distances when $R$ exceeds the arbitrary cutoff radius $R_\mathrm{cut}$ by an amount large compared to the length $1/\beta$. Furthermore, this function ensures that the full molecular decay rate turns on at small distances where $R_\mathrm{cut} - R$ is appreciably less than $1/\beta$. The constant $\beta$ parametrizes the distance over which molecular decay becomes appreciable. When the coupled equations are solved with this complex potential in Eqs. (\[eqn:V0u\]) and (\[eqn:V1u\]), the $S$-matrix is non-unitary, and $1 - |S(k)|^2$ in Eq. (\[eqn:in\_cross\_section\]) represents loss of ground state atoms due to molecular excitation followed by excited state decay. We assume that every spontaneous emission event represented by the imaginary term in $V_{ju}$ results in hot atom or molecular products that are lost from the trap. Our numerical studies show that this assumption is good for all the excited levels except the state nearest in energy to the atomic resonance (Section \[subsec:CC\_Results\]). We calculate that 60% of the emission from this state does not result in loss from a 10 $\mu$K trap [@Zelevinsky2006; @Reinaudi2012]. The cutoff ensures that there is no spurious excited state decay associated with the asymptotically dressed atoms. We find that in the core of a PA line, out to molecular detunings of several hundred line widths from molecular resonance, the loss associated with the imaginary part of the scattering length is not sensitive to the value chosen for $R_\mathrm{cut}$, as long as it is significantly outside the outer turning point of classical motion for the excited state vibrational level. Furthermore, the real part of the scattering length is completely insensitive to the choice of $R_\mathrm{cut}$. We typically choose $R_\mathrm{cut}=$ 500 a$_0$ and $\beta=$ 0.05 a$_0^{-1}$. We find that our numerical calculations were insensitive to the choice of $\beta$. Approximations and limitations {#subsec:CC_approx} ------------------------------ This three-channel model makes several approximations, but is capable of representing the essential qualitative and semi-quantitative effects associated with OFRs in the weak to moderate field regime. Our model only includes the minimal number of partial waves needed to represent the change in scattering length and molecular losses due to the OFR. It leaves out the coupling of the excited $J=1$ levels to the ground state $d$-waves as well as coupling to the doubly excited states associated with the $^3$P$_1$ $+$ $^3$P$_1$ separated atom limit. This means that the light shifts calculated from the three-channel model will not be accurate, since ground state $d$-waves are known to contribute to PA light shifts [@Simoni2002; @Ciurylo2006], and doubly excited states may contribute also. Furthermore, the effect of field-dressing in modifying the ground state threshold elastic scattering of partial waves with $\ell >0$ is not included. This modification is due to field dressing that brings in $1/R^3$ terms in the long range potential that will affect the $d$-wave collisions of like bosons or the $p$-wave collisions of like fermions or unlike species. Note that in our three-channel treatment, the field dressed $s$-wave interactions have the correct property that they have no long-range $1/R^3$ component, since such variation vanishes in the $V_{2,2}$ matrix element of Eq. (\[eqn:CC\_V\]) (if we had attempted only a two-channel field dressed treatment, the presence of the single $0_u$ excited state potential would have introduced a spurious $1/R^3$ term in the ground state dressed $s$-wave potential). It would be straightforward to introduce higher partial waves and strong field dressing into the calculation, using the formalism of Napolitano [*et al.*]{} [@Napolitano1997]. This formalism uses the “uncoupled" asymptotic basis $|j m_j \ell m_\ell\rangle$, which is better for treating strong field dressing than the “coupled” $|j \ell JM\rangle$ basis we use here. The subtle effects of retardation, switching off the dipole approximation, and including the weak coupling to the [*gerade*]{} states as $R$ increases [@Takasu2012] should be taken into account in a more complete theory. We do not perform a time-domain analysis, so we cannot reproduce the transient OFR dynamics [@Naidon2008] observed in Ref. [@Yan2013b]. Furthermore, a full treatment of excited state spontaneous emission during a collision is beyond the scope of CC methods, and would require treatment by stochastic Schr[ö]{}dinger equation methods (density matrix methods are computationally intractable) [@Suominen1994; @Suominen1998a]. Last, as we show in the next section, our analysis with a cutoff of the long range decay is sufficient for treating OFRs for molecular detunings that are of order 100 line widths (or less) from the center of a PA line. Coupled Channels Results {#subsec:CC_Results} ------------------------ The CC calculations to solve Eq. (\[eqn:CC\]) were carried out using the standard renormalized Numerov method [@Johnson1977] using complex variables so as to represent the effect of the complex potential in the excited state channels. A step-doubling algorithm was employed to optimize the number of steps needed as $R$ increases between the short and long range regions. The single $S$-matrix element $S(E,I,\omega)$ for the dressed ground state $s$-wave was extracted from the log derivative of the single open channel solution $F_{1,e}(R)$ of the three channel propagated wavefunction of Eq. (\[eqn:CC\_Psi\]) at a suitable large asymptotic value of $R$. Using Eq. (\[eqn:scattering\_length\]) then gives the complex energy-dependent scattering length $\alpha(k,I,\omega)$, which then gives the elastic and inelastic rate coefficients $K_{el}$ and $K_{in}$ (Eqns. \[eqn:rate\_coefficients\] and \[eqn:loss\_rate\]). ![(Color online) Real (solid line) and imaginary (dashed line) parts of the complex scattering length $\alpha=a-ib$. Here the detuning from atomic resonance $\nu-\nu_0$ (the “atomic detuning”) is measured with respect to the $^{88}$Sr intercombination line transition at $\nu_0$. Also, $E/k_B=4$ $\mu$K and $I=10$ W/cm$^2$. The dotted line shows the background $a_{bg}$ for $E/k_B=4$ $\mu$K and $I=0$. Inset: A close up of the off-resonant behavior of the $n = -2$ OFR (solid line). Also plotted is the scattering length predicted by treating the $n = -2$ OFR as an isolated resonance (dashed line).[]{data-label="fig:CC_abscan"}](a_and_b_10W_scan_4uK.eps){width="\linewidth"} Fig. \[fig:CC\_abscan\] shows the real and imaginary parts of $\alpha(k)$ as the PA laser frequency $\nu = \omega/2\pi$ is detuned from atomic resonance at $\nu_0=\omega_0/2\pi$ (where $\nu - \nu_0$ is the “atomic detuning”). This particular example was taken for a PA laser intensity of 10 W/cm$^2$ and a relative collision kinetic energy of $E/k_B=4$ $\mu$K. Here the background $a_\mathrm{bg}=$ 0.495 a$_0$ at $E/k_B=4$ $\mu$K differs from the background value -1.4 a$_0$ in the limit of $E=0$ due to the energy dependence of $a_\mathrm{bg}(k)$. The calculations were carried out for atomic detunings larger in magnitude than -20 MHz to avoid strong field-dressing effects at atomic detunings near resonance (the optical coupling matrix element $V_\mathrm{opt}/h=0.84$ MHz for this $I$). The decay rate was taken to be $\gamma =\gamma_m=2\gamma_a= 2 \pi \times$ 0.014833 MHz. The figure shows a series of four OFRs in this region. These four resonances correspond to the previously observed [@Zelevinsky2006] $n=$ -2, -3, and -4 members of the $0_u$ $J=1$ series at binding energies $E_n/h =$ 24 MHz, 222 MHz, and 1084 MHz and a single $n=$ -1 member of the $1_u$ $J=1$ series at 353 MHz. Here $n<0$ counts bound states down from the last level (of a given $0_u$ or $1_u$ symmetry) designated as $n=-1$. The scattering lengths show a series of overlapping resonances that cause a large change in scattering length near the poles of the resonances but return to $a_\mathrm{bg}$ between resonances. The fact that the stronger $n = -2$ resonance returns to its background value near the $n = -3$ line (Fig. \[fig:CC\_abscan\] inset) illustrates an important general feature of a vibrational sequence of OFRs: interfering resonances cause the scattering length to return to its background value in between resonances. Even the presence of a neighboring OFR that is comparatively weak will diminish the off-resonant magnitude of a stronger OFR (Fig. \[fig:CC\_abscan\] inset). This property imposes a constraint on OFR experiments, namely that molecular detunings cannot be so large as to be comparable to the frequency separation between the resonance of interest and the nearest resonance. In contrast, MFRs arising from neighboring spin-channel resonances interfere with one another in a manner that is qualitatively different than for a vibrational series (see Section \[subsec:OFR\_MFR\_multires\]). ![(Color online) Imaginary part of $\alpha = a - i b$ from Fig. \[fig:CC\_abscan\] shown on a log scale. The diamonds show a $1/(\nu-\nu_0)^2$ scaling. The inset shows an expanded view of the $0_u$ $n=-4$ resonance near $-1084$ MHz. The dashed, solid, and dotted lines show the results for $R_\mathrm{cut}=$ 200 a$_0$, 500 a$_0$, and 1000 a$_0$ respectively. Near the peak of the resonance, $b$ is independent of $R_\mathrm{cut}$.[]{data-label="fig:CC_bscan"}](b_10W_scan_4uK.eps){width="\linewidth"} The imaginary part $b$ of the scattering length that gives the loss rate coefficient, Eq. (\[eqn:loss\_rate\]), shows a series of sharp spikes near the poles of the resonances in Fig. \[fig:CC\_abscan\], and shows a value very near zero on the linear scale of the figure. Figure \[fig:CC\_bscan\] provides a better way to illustrate the basic features of $b$ by showing it on a log scale. Here the “background” on which the poles sit varies as $1/(\nu-\nu_0)^2$, with this functional form indicated by the diamonds on the figure. Furthermore, far detuned from a molecular resonance, the magnitude of this background is found to scale linearly with $R_\mathrm{cut}$ as $R_\mathrm{cut}$ increases. This is because away from resonance, most of the loss of flux in the collision due to the presence of a complex potential comes from the long range region, where decay should not be counted as loss, since it merely represents atomic light scattering that returns an atom to its ground state. Thus, loss is overcounted by use of a complex potential if $R_\mathrm{cut}$ is too large. In the core of the line spanning molecular detunings of $100 \gamma_m$, $b$ and $K_{in}$ are independent of $R_\mathrm{cut}$. For example at a molecular detuning of $100 \gamma_m$ (inset to Fig. \[fig:CC\_bscan\]) the values of $b$ calculated with $R_\mathrm{cut}=$ 200 a$_0$ or 500 a$_0$ differ by less than %. The difference grows to 10% if $R_\mathrm{cut}=$ 1000 a$_0$. The difference with $R_\mathrm{cut}=$ 200 a$_0$ or 500 a$_0$ only grows to 10% when the molecular detuning is over 250 line widths. Consequently, if $R_\mathrm{cut}$ is selected to have a small enough “physical” value where spontaneous decay for $R<R_\mathrm{cut}$ represents actual loss of atoms, the loss calculated for molecular detunings up to a few hundred line widths are meaningful and not sensitive to the choice of $R_\mathrm{cut}$. In any case, the scattering length $a$ given by the real part of $\alpha$ is completely insensitive at all detunings to the choice of $R_\mathrm{cut}$. Comparing our CC theory to experimental data taken at small molecular detunings, we are able to reproduce the resonance strengths measured in Refs. [@Blatt2011] and [@Yan2013a]. However, the atom loss rate of Ref. [@Yan2013a] is measured at a molecular detuning large enough for our theory to be sensitive to $R_\mathrm{cut}$; therefore, our theory is not designed to reproduce this rate. Within the inherent limitations of our approximations that we have outlined above, we expect our CC calculations to give the correct change in scattering length for all detunings and the atom loss rate coefficient for at least 100 line widths from the peak of a molecular resonance. Consequently, since the resonances are spaced by many thousands of line widths apart, we turn our attention in the next section to understanding the theory for single isolated OFRs for molecular detunings in the vicinity of a photoassociation resonance. Isolated Resonance Theory of Optical Feshbach Resonances {#sec:isolated_res} ======================================================== The OFR features in Figs. \[fig:CC\_abscan\] and \[fig:CC\_bscan\] tend to be well-isolated from one another and thus can be described quite successfully by theory designed to treat an isolated single resonance situated on a background. Isolated resonance theory has been used for cold atom OFRs since they were first proposed  [@Thorsheim1987; @Napolitano1994; @Fedichev1996; @Bohn1997; @Bohn1999]. This theory successfully explained alkali-metal atom PA spectra with hyperfine structure (in good agreement with experiment [@Napolitano1994; @Tiesinga2005]), and it also explained the saturation of PA lines at high intensity [@McKenzie2002; @Prodan2003; @Simoni2002]. In fact, both OFRs and MFRs can be treated by the same resonance scattering formalism when the possibility of decay of the closed channel resonance state is taken into account [@Chin2010]. The isolated resonance approximation assumes that each molecular bound state is far from the other molecular states in the closed channel and can be described by a strength parameter that is local to the resonance—that is, independent of energy and molecular detuning in the vicinity of the resonance. Bohn and Julienne give a general resonance scattering treatment for an OFR based on quantum defect theory [@Bohn1999]. They derive a general expression for the $S$-matrix element $S(k)$ for a single $s$-wave entrance channel coupled to an isolated resonance scattering bound state, including a decay rate $\gamma$ to exit channels that lead to atom loss. The elastic scattering $S$-matrix element $S(k)$ (equivalent to Eq. (3.13) of Ref. [@Bohn1999]) for an isolated decaying resonance is: $$\label{eqn:bohn_s_matrix} S(k) = \left( 1 - \frac{i \hbar\Gamma(k)}{ E - E_\mathrm{res}+ i\frac{1}{2}\hbar [\gamma + \Gamma(k)]} \right) e^{2 i \eta_{bg}(k)} \,.$$ where $$\label{eqn:Eres} E_\mathrm{res} =\hbar ( \omega_{n}+s_n I - \omega) = -\hbar\delta$$ is the energy of the field-dressed molecular resonance level $n$. Its “bare” location at $\hbar \omega_n=\hbar \omega_a-E_n$ is shifted by an intensity-dependent shift $\hbar s_n I$, where $E_n$ is the binding energy with respect to the excited separated atom limit when $I=0$. We define the molecular detuning $\delta$ so it is negative for red detuning, in which case a resonance peak occurs when $E=E_\mathrm{res}$. We assume low power, in which case the shift varies linearly with intensity. The coefficient $s_n$ can be either positive or negative  [@Bohn1999; @Ciurylo2006], where a positive value corresponds to a shift of the resonance peak closer to the atomic line. The closed channel resonance bound state is coupled to the entrance channel by the stimulated decay rate, $$\label{eqn:Gamma} \Gamma(k) = \frac{2\pi}{\hbar} |\langle n |V_\mathrm{opt}|E\rangle |^2 \,.$$ Here, $|n\rangle$ represents the excited bound state, which in general would be a mixture of the two $|2\rangle$ and $|3\rangle$ excited case (c) states in Table I. In practice $|n\rangle$ would be well-approximated by a single $0_u$ or $1_u$ $J=1$ vibrational state. The ground state scattering wavefunction $|E\rangle$ is assumed to be energy normalized [@Julienne2009], so that $$F_1(R,E) \to \left ( \frac{2\mu}{\pi \hbar^2 k} \right )^{1/2} \sin(kR +\eta_\mathrm{bg}) \,\,\mathrm{as} \,\, R \to \infty \,,$$ where the phase shift $\eta_\mathrm{bg}$ is related to the scattering length $a_\mathrm{bg}$ in the $k \to 0$ threshold limit as $\eta_\mathrm{bg} = -ka_\mathrm{bg}$. We emphasize that the form of the expression in Eq. (\[eqn:bohn\_s\_matrix\]) is completely general for any isolated threshold resonance and applies equally well for MFRs and OFRs, if the terms are identified properly. The Fermi golden rule width $\Gamma(k)$ expresses the strength of the resonance pole term with a tunable denominator. When $\Gamma(k)=0$, there is no Feshbach resonance, and we recover the standard expression $S(k)=e^{2i\eta_\mathrm{bg}}$ for the uncoupled entrance channel. The expression in Eq. (\[eqn:Gamma\]) ensures that $\Gamma(k)$ follows the standard threshold law, and thus for an $s$-wave entrance channel, $\Gamma(k) \propto k$. For non-decaying resonances, $\gamma=0$, and the imaginary term in the denominator vanishes as $k \to 0$. Comparison Between Optical Feshbach Resonances and Magnetic Feshbach Resonances =============================================================================== Isolated Resonance Theory {#subsec:OFR_MFR_isores} ------------------------- The resonance length formalism summarized in Section II.A.3 of Ref. [@Chin2010] shows how to relate MFR and OFR resonance strengths and compare OFRs to MFRs in a unified approach. All we need to note is that in the case of an isolated MFR, the threshold width $\Gamma(k)$ in Eq. (\[eqn:bohn\_s\_matrix\]) is given by an expression similar to Eq. (\[eqn:Gamma\]), except that $V_\mathrm{opt}$ needs to be replaced with an appropriate internal spin-dependent Hamiltonian [@Kohler2006; @Chin2010]. Furthermore, $E_\mathrm{res}$ would be replaced with a $B$-dependent tuning and shift [@Julienne2006; @Chin2010; @Jachymski2013], $E_\mathrm{res} = \delta\mu (B-B_c)+E_\mathrm{shift}$, where $B$ represents magnetic field, $B_c$ is the field where the bare resonance level crosses threshold, $\delta \mu$ represents the difference between the sum of the magnetic moments of the two atoms and the magnetic moment of the bare resonance level, and $E_\mathrm{shift}$ represents an energy-dependent shift term. The threshold law for $s$-wave collisions shows that as $k \to 0$ the quantity $\hbar \Gamma(k)/k$ (for either an OFR or the MFR analog) becomes a $k$-independent constant having the units of length times energy. Thus, for either an MFR or an OFR, we can decompose $\Gamma(k)/k$ into a product of a length factor $L_\mathrm{r}$ and an energy $E_\mathrm{r}$, $$\frac{\hbar \Gamma(k)}{2k} = L_\mathrm{r} E_\mathrm{r}\,, \label{eqn:res_constant}$$ Since only the $L_\mathrm{r} E_\mathrm{r}$ product is significant, we are free to choose either the length $L_\mathrm{r}$ or the energy $E_\mathrm{r}$ factor to yield a convenient expression for the scattering length. In the case of non-decaying MFRs, it is conventional to choose $L_\mathrm{r}= a_\mathrm{bg}$. The $E_\mathrm{r}$ factor is typically written as $ \delta\mu \, \Delta$, thereby defining the magnetic “width” $\Delta$ of the MFR: $$\label{eqn:Gamma_MFR} \frac{ \hbar \Gamma_\mathrm{MFR}(k)}{2k} = a_\mathrm{bg} ( \delta\mu \, \Delta ) \,.$$ Here the subscript “MFR” indicates the type of resonance. When this form is substituted in Eq. (\[eqn:bohn\_s\_matrix\]), $\gamma$ is set equal to zero, and the $k \to 0$ limit is taken with the shift term in Refs. [@Chin2010; @Julienne2006], Eq. (\[eqn:scattering\_length\]) reduces to the standard expression for an MFR, $$\label{eqn:a_MFR} a(B) = a_\mathrm{bg} - a_\mathrm{bg} \frac{\Delta}{B-B_0} \,,$$ where $B_0 = B_c-E_{\mathrm{shift}}/\delta\mu$ is the pole position. In the case $\gamma \neq 0$, this procedure would give the complex scattering length for a decaying MFR [@Chin2010; @Naik2011]. By analogy to MFRs, one can define an OFR resonance frequency width $w$ by $$\label{eqn:Gamma_OFR} \frac{ \hbar \Gamma_\mathrm{OFR}(k)}{2k} = a_\mathrm{bg} (-\hbar w ) \,.$$ Note that $-a_\mathrm{bg} w$ is positive definite. In the limit $|\delta| \gg \gamma$ where we can ignore the decay of the resonance, the scattering length due to an OFR is $$\label{eqn:a_OFR_abg} a(\omega) = a_\mathrm{bg} - a_\mathrm{bg} \frac{w}{\omega-(\omega_n+s_n I)} \,.$$ The standard way to express the $L_\mathrm{r} E_\mathrm{r}$ product in the case of an OFR is to define $E_\mathrm{r}$ to be the known quantity $\hbar \gamma_\mathrm{m}$ and call the length parameter the “optical length” $l_\mathrm{opt}$ [@Ciuryo2005; @Blatt2011], $$\label{eqn:Gamma_OFR_lopt} \frac{ \hbar \Gamma_\mathrm{OFR}(k)}{2k} = l_\mathrm{opt} (\hbar\gamma_{m}) \,.$$ For large detunings $|\delta| \gg \gamma$ the scattering length is $$\label{eqn:a_OFR_lopt} a(\omega) = a_\mathrm{bg} + l_\mathrm{opt} \frac{\gamma_{m}}{\omega-(\omega_n+s_n I)} \,.$$ A similar resonance length parameter has been defined for a decaying MFR by Hutson [@Hutson2007] by using the total decay width of the resonance for $E_\mathrm{r}$. In the case of an OFR that decays only to the ground state, choosing $E_\mathrm{r}=\hbar \gamma_m$ has the advantage of eliminating the dipole strength from the expression for $ l_\mathrm{opt}$. Using $\hbar \gamma_a= 32\pi^3 d_a^2/3 \lambda_a^3$, the definition $\gamma_m = 2\gamma_a$, taking $V_\mathrm{opt}$ in Eq. (\[eqn:Vopt\]), and assuming the $R$-independent molecular dipole moments of Eqs. (\[eqn:d0u\]) or (\[eqn:d1u\]), we find $$\label{eqn:l_opt} l_{\mathrm{opt}} =\frac{\Gamma(k)}{2 k \gamma_{m}} = \frac{\lambda_{a}^{3}}{16 \pi c} \frac{| \langle n | E \rangle |^{2}}{k} I f_{\mathrm{rot}} \,,$$ where $f_{\mathrm{rot}}$ is equal to 1 for $0_u$ states and 2 for $1_u$ states (due to the different rotational H[ö]{}nl-London factors for parallel and perpendicular transitions). Consequently, $L_r E_r$ is proportional to the product of a Franck-Condon factor and the square of the molecular electronic transition dipole moment. Equation (\[eqn:l\_opt\]) shows that $l_\mathrm{opt}$ varies linearly with PA laser power $I$. The only molecular physics parameter it depends on is the free-bound Franck-Condon factor $| \langle n | E \rangle |^{2}$, which varies linearly with $k$ at small $k$. As an example, direct calculation of $| \langle n | E \rangle |^{2}/k$ for the $J=0$ $n=-4$ $0_u$ level shows that this quantity decreases at a rate of 0.66% per $\mu$K as $E/k_B$ ranges from 0 to 10 $\mu$K. Thus, $l_\mathrm{opt}/I$ is only weakly dependent on collision energy in ultracold gases and may be approximated as a constant. Its weak variation with energy could be estimated from approximate theories based on the reflection approximation [@Bohn1999] or the stationary phase approximation [@Ciurylo2006]. A useful way to compare the strengths of MFRs and OFRs is to use a dimensionless resonance “pole strength” parameter that applies to either case: $s_\mathrm{res} = L_\mathrm{r} E_\mathrm{r}/\bar{a}\bar{E}$, where $\bar{a}$ is the mean scattering length of the van der Waals potential [@Gribakin1993] and $\bar{E}=\hbar^2/(2\mu \bar{a}^2)$ is the corresponding energy. These are $\bar{a}=$ 71.8 $a_0$ and $\bar{E}/h=$ 7.97 MHz for $^{88}$Sr collisions. Chin [*et al.*]{} [@Chin2010] used $s_\mathrm{res}$ to characterize and classify MFRs according to whether $s_\mathrm{res} >1$ (open channel dominated) or $s_\mathrm{res} <1$ (closed channel dominated), where the former tends to be “broad” and the latter “narrow.” Thus we have $$s_\mathrm{res}^\mathrm{MFR} = \frac{a_\mathrm{bg}}{\bar{a}} \frac{ \Delta \delta \mu}{\bar{E}} \,, \,\,\,\, s_\mathrm{res}^\mathrm{OFR} = \frac{l_\mathrm{opt}}{\bar{a}} \frac{ \hbar \gamma_m}{\bar{E}} \,.$$ One obvious difference between MFRs and OFRs is that the strength of an OFR can be controlled by increasing the PA laser power to increase $l_\mathrm{opt}$, whereas the strength of a MFR is fixed. However, $l_\mathrm{opt}$ cannot be increased too much since the light scattering loss rate due to either atomic or molecular processes also increases with $I$ [@Blatt2011]. Experimentally useful MFRs tend to have a pole strength parameter between unity and 100 [@Chin2010]. The width ratio $\hbar \gamma_m/\bar{E} =0.0019$ is much less than unity for the narrow OFRs near the intercombination line of $^{88}$Sr, so $s_\mathrm{res}^\mathrm{OFR} \ll 1$ unless it can be compensated by making $l_\mathrm{opt}/\bar{a}$ very large compared to unity. Thus, $^{88}$Sr OFRs tend to be weak, narrow, “closed channel dominated” resonances. An interesting comparison is with the experimentally useful broad open channel dominated MFR of two $^{85}$Rb atoms at 155.2 G, for which $s_\mathrm{res}=$ 28 [@Chin2010]. This is a decaying MFR in an excited spin channel [@Kohler2006], with a natural decay width of $\gamma/(2\pi)=$ 5.0 kHz due to spin relaxation of the “bare” closed channel state of the resonance. The lifetime of 32 $\mu$s [@Kohler2005] of this spin channel is comparable to that of the excited Sr $^3$P$_1$ state. The major difference between the $^{85}$Rb MFR and $^{88}$Sr OFRs is the much smaller resonance strength $s_\mathrm{res}$ of the latter at intensities where the atomic light scattering is not harmful. In contrast to $^{88}$Sr, OFRs for the species $^{172} \!\!$ Yb were found to have an $l_\mathrm{opt}$ on the order of $10^4 a_0$ at $I = 1 \mathrm{W/cm}^2$ for levels near 1 GHz atomic detuning [@Borkowski2009]. This implies that broad open-channel-dominated OFRs with $s_\mathrm{res} > 1$ may be realizable with $^{172} \!\!$ Yb. It is not yet known whether OFRs might exist with $s_\mathrm{res} > 1$ for frequencies near the alkaline earth intercombination line in mixtures of alkaline earth species and alkali metal species. This is a subject for future experimental and theoretical research. Multiresonance Theory {#subsec:OFR_MFR_multires} --------------------- It is useful to compare isolated OFRs and MFRs since isolated resonance theory is widely utilized in both cases; however, both OFRs and MFRs exist as sets of resonances that interfere with one another, so it is instructive to compare multiresonance treatments of the two effects. Although a thorough treatment of a multiresonance OFR-MFR comparison could be the subject of an entire publication, in this section we provide an overview of such a comparison using results from multichannel quantum defect theory (MQDT). When resonance interference is considered, significant qualitative differences between OFRs and MFRs emerge. There are two sources of such differences. First, the molecular physics that determines the resonance strength is due to short range spin-dependent interactions for MFRs and long-range photoassociation for OFRs. Second, OFRs typically span many vibrational levels of the same closed channel molecular state whereas experimentally utilized MFRs are typically different spin components rather than a vibrational progression. Sets of overlapping MFRs are well-studied for different alkali metal species [@Naik2011; @Berninger2013; @Takekoshi2012; @Tung2013; @Repp2013; @Jachymski2013], and overlapping MFRs have recently been shown to be important for few-body physics [@Wang2014]. The interference of overlapping MFRs can be quantitatively explained by MQDT [@Mies1984a; @Mies1984b; @Jachymski2013], with which one can derive an $S$-matrix that is a multiresonance generalization of the isolated resonance formula in Eq. (\[eqn:bohn\_s\_matrix\]). We introduce the MQDT theory here to highlight the differences between overlapping OFRs and MFRs. Considering one background channel and one closed channel, the background is characterized by the usual $E$-dependent phase shift $\eta_\mathrm{bg}(E)$ [@Jachymski2013; @Mies1984a; @Mies1984b; @MQDTnote1]. The closed channel c is characterized by a bound state phase function $\nu_\mathrm{c}(E-E_\mathrm{c})$. The energy $E_\mathrm{c}$ of the separated atoms in the closed channel is modified with the “field tuning,” which means varying the external magnetic field in the MFR case or the atomic detuning in the OFR case. Bound states of the closed channel exist where $\tan\nu_\mathrm{c}(E-E_\mathrm{c})=0$. Thus, the external field tuning moves the bound state spectrum relative to the background channel $E=0$ threshold, allowing bound states to be tuned across threshold. The coupling between the background and the closed channel is characterized by the dimensionless MQDT parameter $s_\mathrm{res}$, which may also depend on the external fields. If we follow Ref. [@Jachymski2013] and express energies as $\epsilon=E/\bar{E}$ and $\kappa = k \bar{a}$, then the equivalent MQDT expression to Eq. (\[eqn:bohn\_s\_matrix\]) for an MFR or an OFR can be written in a universal dimensionless form, $$S_{MQDT} = S_\mathrm{bg} \left(1-i\frac{2\kappa s_\mathrm{res}}{(\epsilon_\mathrm{m}/\pi)\tan\nu_\mathrm{c}+\epsilon_\mathrm{shift}+i\epsilon_\mathrm{width}}\right)\, , \label{eqn:MQDT_S}$$ where $S_\mathrm{bg}(\epsilon)=e^{2i\eta_\mathrm{bg}(\epsilon)}$ is the background term, $$\epsilon_\mathrm{m}(\epsilon)=\frac{\pi}{\partial \nu_\mathrm{c}(\epsilon)/\partial \epsilon}$$ represents the mean spacing between different eigenenergies, and $\epsilon_\mathrm{shift}(\epsilon)$ and $\epsilon_\mathrm{width}(\epsilon)=\frac12 \hbar \gamma/\bar{E}+\kappa s_\mathrm{res}$ represent the respective shift and decay parts of the complex energy of the interacting, decaying resonance. The shift term $\epsilon_\mathrm{shift}(\epsilon)$ will scale as $s_\mathrm{res}$, and both of these quantities are only very weakly dependent on energy near threshold. Here we need to view the MQDT parameter $s_\mathrm{res}$ as a continuous function of the external field that defines the Hamiltonian. Using Eq. (\[eqn:MQDT\_S\]), we can now describe some key differences of OFRs and MFRs. These come from the variation with field strength of both the numerator and the denominator of Eq. (\[eqn:MQDT\_S\]). In the MFR case, it is an excellent approximation to take $s_\mathrm{res}$ to be a constant, independent of $B$ and $E$, since the interactions that determine this parameter are short range, where $R \ll \bar{a}$ and the energy scale is large. Consequently, the matrix element that sets the magnitude of $s_\mathrm{res}$ is independent of small field tuning. On the other hand for an OFR, $s_\mathrm{res}$ scales linearly with laser power, and we must also think of $s_\mathrm{res}(\omega)$ as being highly sensitive to field tuning, since the optical coupling is determined by the Condon point at very long range (on the order of $\bar{a}$ or larger). The Condon point varies rapidly with PA laser frequency, so the crossing structure of the field-dependent Hamiltonian varies with laser frequency in a major way, changing the response of the system to the optical field. Another way of thinking about this variation is that for an isolated resonance $s_\mathrm{res}$ is proportional to a Franck-Condon factor, which will vary rapidly from level to level in the closed channel, so that the general MQDT coupling parameter can not be taken as a field-tuning-independent parameter [@MQDTnote2]. Secondly, note that the proper MQDT expression in the denominator of Eq. (\[eqn:MQDT\_S\]) that contains the effect of field tuning is the $\tan\nu_\mathrm{c}$ term, which vanishes at resonance poles. To get the normal isolated resonance approximation near a tunable eigenenergy $\epsilon_\mathrm{res}$, as in Eq. (\[eqn:bohn\_s\_matrix\]), it is necessary to expand this function in a Taylor series as [@Mies1984b; @Jachymski2013] $$\tan\nu_\mathrm{c}(\epsilon-\epsilon_\mathrm{c}) \approx \frac{\partial \nu_\mathrm{c}(\epsilon-\epsilon_\mathrm{c})}{\partial \epsilon}{|}_{\epsilon=\epsilon_\mathrm{res}} \left ( \epsilon - \epsilon_\mathrm{res} \right ) \,. \label{eqn:tan_nu}$$ In the ultracold case, $\epsilon$ tends to remain very close to 0 but $\epsilon_{res}$ is varied by tuning the field, so the expansion in Eq. (\[eqn:tan\_nu\]) should be made near $\epsilon=0$. This linearizing approximation is normally quite good as long as the range of expansion $\epsilon-\epsilon_\mathrm{res}$ remains small compared to the mean spacing $\epsilon_\mathrm{m}$ to adjacent levels. This is normally the case for MFRs, where the widths of even broad resonances tend to be small compared to the distance to the next vibrational level in the same spin channel [@MQDTnote3]. On the other hand, it is common to observe a series of vibrational levels of the same electronic state in the OFR case. In this case, the $\tan\nu_\mathrm{c}$ must be left unexpanded and $\nu_\mathrm{c}$ treated as a continuous function of field tuning if multiple resonances are present [@MQDTnote4]. Reference [@Jachymski2013] showed how to extend the MQDT formalism for Eq. (\[eqn:MQDT\_S\]) to multiple spin channels with overlapping resonances. Generally, for the reasons discussed above $s_\mathrm{res}$ for each separate channel can be well-approximated as a $E$- and $B$-independent constant, and the linearizing approximation in Eq. (\[eqn:tan\_nu\]) is used for detunings spanning multiple spin channels. Then the generalization of Eq. (\[eqn:tan\_nu\]) gives a sum of resonance terms similar to that in the pole term of Eq. (\[eqn:tan\_nu\]), where there is a global background scattering length $a_\mathrm{bg}$ for all channels and the shift terms in each denominator depend on all the poles simultaneously. The formula can be transformed to a form where each narrow resonance can be viewed as an isolated resonance having a “local” (in field tuning) background modified from the global one by nearby interfering resonances. An extension to OFRs from different electronic states may not be possible, because of the rapid variation of MQDT parameters with field tuning. Furthermore, the MFR theory should not be used for different members of the same vibrational series because of the inability to linearize the $\tan\nu_\mathrm{c}$ function across two or more vibrational levels. It may be possible to develop some approximations appropriate to the OFR case, but meanwhile the isolated resonance approximation or CC calculations remain the best tools for understanding OFRs. Elastic and Inelastic Collisions ================================ OFR isolated resonance formulas {#subsec:iso_res_formulas} ------------------------------- For detunings as large as hundreds of linewidths from the resonance center (but smaller than the separation between resonances), the complex scattering length $\alpha(k,\omega,I)$ and the elastic or inelastic collision rate coefficients derived from the $S$-matrix of Eq. (\[eqn:bohn\_s\_matrix\]) are in excellent agreement with the full CC calculations. In this regime, Eq. (\[eqn:bohn\_s\_matrix\]) fully describes an isolated OFR as a function of energy, detuning, and intensity. The expression in Eq. (\[eqn:bohn\_s\_matrix\]) gives the isolated resonance approximation to the entrance channel loss probability $P_\textrm{loss}$ in terms of only two parameters, $l_\mathrm{opt}$ and the resonance position $E_\mathrm{res}$: $$\label{eqn:loss} P_\textrm{loss} = 1 - |S(k)|^2 = \frac{2 k l_\textrm{eff}}{D^2 + \frac14(1+ 2 k l_\textrm{eff})^2} \,,$$ where $$\begin{aligned} l_\textrm{eff} &=& l_\mathrm{opt} (\gamma_m/\gamma), \label{eqn:x} \\ D &=& (E -E_\mathrm{res})/\hbar \gamma \label{eqn:D} \,.\end{aligned}$$ Here the effective optical length $l_\textrm{eff}$ determines the resonance coupling strength for the general case when $\gamma >\gamma_m$. $P_\textrm{loss}$ determines the inelastic cross section in Eq. (\[eqn:in\_cross\_section\]) and thus the loss rate coefficient $K_{in}$ in Eq. (\[eqn:loss\_rate\]). Note that for $2 k l_\textrm{eff} \gg 1$, Eq. \[eqn:loss\] describes power broadening. ![(Color online) (a) Coupled-channel calculated inelastic loss probability $1 - |S(k)|^2$ versus atomic detuning $\nu-\nu_0$ for different values of collision energy $E/k_B$. We have used $I=$ 5 W/cm$^2$ and the $0_u$ $J=1$ $n=-4$ feature, where $E_n/h = -1084.0763$ MHz. The black dots indicate the peak values calculated with $l_\mathrm{opt}=$161.5 a$_0$, $\gamma =\gamma_m$, and $\nu-\nu_0=(E_n-E)/h$ (where $D$ vanishes in Eq. (\[eqn:loss\])). Since $1 - |S(k)|^2$ calculated from the isolated resonance formula in Eq. (\[eqn:loss\]) is indistinguishable from the CC calculation on this graph, only the peak comparisons are shown by the black dots on the figure. Panel (b) Coupled channels calculated inelastic loss rate coefficient $K_{in}$ versus molecular detuning. Here we plot different values of PA laser intensity $I$. We use a collision energy $E/k_B=$ 1 nK and the $0_u$ $J=1$ $n=-4$ feature near $\nu - \nu_0 =$ $-1084$ MHz. The black dots show the predictions of the analytic formula in Eq. (\[eqn:loss\]). The line wings beyond around $|\delta/\gamma_m| = 6$ scale linearly with power. The black dotted line represents the unitarity limit where $K_{in}$ saturates.[]{data-label="fig:Iso_Ploss"}](broadening.eps){width="\linewidth"} Figures \[fig:Iso\_Ploss\](a) and  \[fig:Iso\_Ploss\](b) show the behavior of $P_\textrm{loss}$ as a function of detuning for different intensities and collision energies. In Fig. \[fig:Iso\_Ploss\](a), $P_\textrm{loss}$ is not saturated at low collision energy. As collision energy increases, $P_\textrm{loss}$ broadens, and its peak value (when $E=E_\mathrm{res}$) approaches its upper bound of unity. This only occurs for red molecular detunings. Fig. \[fig:Iso\_Ploss\](b) shows similar broadening in the inelastic rate coefficient $K_{in} \propto P_\mathrm{loss}$ as intensity is increased and collision energy is kept low. For large intensities, $K_{in}$ saturates at its value given by the unitarity limit. Note that according to Eq. \[eqn:Eres\], the intensity-dependent frequency shift of the resonance is accounted for in $\delta$. For temperatures in the $\mu$K range, thermal averaging of the line shape is essential to compare with experiment. As is well-known [@Jones2006], PA lines have a pronounced red-blue asymmetry when $k_B T$ is larger than the natural width of the PA line. The isolated resonance *S* matrix can be used to derive the complex $k$-dependent scattering length $\alpha(k)$. Combining Eqs. (\[eqn:bohn\_s\_matrix\]) and (\[eqn:scattering\_length\]), $$\alpha(k) = \alpha_\mathrm{bg}(k) + \frac{\frac{\hbar \Gamma(k)}{2k} \left ( 1 + k^2 \alpha_\mathrm{bg}(k)^2 \right )} {E-E_\mathrm{res} - k \alpha_\mathrm{bg}(k) \frac{\hbar \Gamma(k)}{2} + i \frac{\hbar \gamma}{2}} \,,$$ where $\alpha_\mathrm{bg}(k)$ is found by using the background $S_\mathrm{bg}(k)=e^{2i\eta_\mathrm{bg}}$ in Eq. (\[eqn:scattering\_length\]). Notice that this expression does not contain power broadening, which enters the elastic and inelastic cross sections through the $f(k)$ factor in Eqs. (\[eqn:el\_cross\_section\])-(\[eqn:f\_factor\]). In the limit that $k|\alpha_\mathrm{bg}| \ll 1$ (valid for a Bose-Einstein condensate), we obtain $$\alpha = a_\mathrm{bg} + l_\mathrm{eff} \frac{\delta \gamma}{\delta^{2} + \gamma^{2}/4} - \frac{i}{2} l_\mathrm{eff} \frac{\gamma^{2}}{\delta^{2} + \gamma^{2} / 4} \,, \label{eqn:Iso_alpha}$$ where we have taken $\alpha_\mathrm{bg}=a_\mathrm{bg}$ to be real. Fig. \[fig:Iso\_a+b\] illustrates the real and imaginary parts of $\alpha=a-ib$ calculated at $E/k_B=$ 1 nK. The results from Eq. (\[eqn:Iso\_alpha\]) are in excellent agreement with the CC calculations. The peak values of $b=2 l_\mathrm{opt}$ at $\delta=0$ and of $a=\pm l_\mathrm{opt}$ at $\delta=\pm \gamma/2$ are also plotted in Fig. \[fig:Iso\_a+b\]. ![(Color online) Coupled channel calculation of $a$ and $b$ versus molecular detuning in line width units $\delta/\gamma_\mathrm{m}$, with $\gamma=\gamma_\mathrm{m}$. We use the $0_u$ $J=1$ $n=-4$ feature and 10W/cm$^2$ PA laser intensity. The dots show the analytic predictions at the peak extrema using Eq. (\[eqn:Iso\_alpha\]) with $l_\mathrm{opt}=323.2$ a$_0$.[]{data-label="fig:Iso_a+b"}](a_and_b_vs_d_n-4_10W_1nK_lin2.eps){width="\linewidth"} Photoassociation {#subsec:PA} ---------------- The isolated resonance approximation also describes photoassociation [@Ciuryo2004], the process by which two ground state atoms and a photon combine to form an electronically excited molecule [@Jones2006]. This phenomenon can be used to measure $l_\mathrm{eff}$, which characterizes the strength of an OFR. This strength can be inferred from measurements of the trapped atom loss (into untrappable molecules) that results from driving a photoassociation resonance. Using Eqs. (\[eqn:in\_cross\_section\]) and (\[eqn:loss\_rate\]), the inelastic rate coefficient, which describes molecule formation, is $$\label{eqn:K_in} K_{in} (\delta,l_\textrm{eff},k) = \frac{4 \pi \hbar}{\mu} \frac{\gamma^{2} l_\textrm{eff}}{(\delta + E/\hbar)^{2} + \frac{\gamma^2}{4} (1 + 2 k l_\textrm{eff})^{2}}.$$ For a trapped ultracold thermal gas, one must introduce Boltzmann averages into the theory. To this end, we approximate that the PA laser interacts with an entire velocity class at each point in space within the trap. This approximation, which is good for large densities, means that one must momentum average $K_{in}$. Furthermore, we use the fact that photoassociation is a short-range phenomenon in the isolated resonance approximation [@Bohn1999]; therefore, we do not perform a Boltzmann spatial average in this treatment. The quantity $\overline{K}_{in}$, which is the momentum-averaged $K_{in}$, is given by $$\begin{aligned} \label{eqn:Kin_averaged} \overline{K}_{in} &=& \frac{1}{\pi^3 p_0^6} \int \!\! d^3 \vec{p}_1 \!\! \int \!\! d^3 \vec{p}_2 \, e^{-(p_1^2 + p_2^2)/p_0^2} K_{in}(\delta,l_\textrm{eff},k) \notag \\ &=& \frac{4}{\sqrt{\pi}k_{th}^3} \int_0^\infty \!\! dk \, k^2 \, e^{-k^2/k_{th}^2} K_{in}(\delta,l_\textrm{eff},k),\end{aligned}$$ where $p_0 = \sqrt{2 m k_B T}$, $k_{th} = \sqrt{2 \mu k_B T}/\hbar$, $\vec{p}_1$ and $\vec{p}_2$ are the momenta of the two collision partners, and $|\vec{p}_1 - \vec{p}_2| = \hbar k$. In order to use $\overline{K}_{in}$ to describe trap loss due to photoassociation, we must understand what fraction of molecules is ejected from the trap. As mentioned in Section \[subsec:CC\_formulation\], we approximate that every photoassociated molecule is lost to the trap. We have numerically studied this approximation, finding that it is good to 1% for all molecular states except for the least-bound $0_u$ level. In this case, the evolution of the in-trap atomic density is $$\label{eqn:two_body_rate_equation} \dot{\rho} = -2 \overline{K}_{in} \frac{\rho^{2}}{2} - \frac{\rho}{\tau},$$ where $\rho$ is the atomic density and $\tau$ is the one-body lifetime (due to loss mechanisms such as background gas molecules and atomic light scatter). Here the $\rho^{2}$ term arises from the number of pairs in an $N$-particle sample, $N(N-1)/2 \simeq N^{2}/2$. The signal in photoassociation experiments is typically the atom number $N$ after the application of the PA laser [@Blatt2011; @Escobar2008], given by integrating the solution to Eq. (\[eqn:two\_body\_rate\_equation\]) over space, $$\label{eqn:number_signal} N(\delta,l_\mathrm{eff}) = \int d^{3} \vec{r} \frac{\rho_{0} (\vec{r}) \,\, e^{-t_{PA} / \tau}}{1 + \overline{K}_{in}(\delta,l_\textrm{eff}) \,\, \rho_{0}(\vec{r}) \,\, \tau (1 - e^{-t_{PA}/\tau})}.$$ ![(Color online) Trap loss, given by Eq. \[eqn:number\_signal\], as a function of detuning for various cloud temperatures. Here $\gamma = \gamma_\textrm{m}$, $l_\textrm{eff} = 100 a_0$, $t_{PA} = 200 \mathrm{ms}$, and we take the limit of $\tau \gg t_{PA}$. We also assume a spherical trapped cloud with $N_0 = 6 \times 10^4$ atoms and a 20 $\mu$m r.m.s. radius. Note the broadening toward red detunings, which is a result of momentum averaging $K_{in}$.[]{data-label="fig:signal"}](signal.eps){width="\linewidth"} Here $t_{PA}$ is the pulse duration of the PA laser and $\rho_{0} (\vec{r})$ is the in-trap density before the PA laser is applied. Fig. (\[fig:signal\]) depicts $N(\delta,l_\mathrm{eff})$ for different temperatures and detunings. The density $\rho_{0}(\vec{r})$ can be determined by fitting experimental in-trap absorption images to a 3D Gaussian distribution (Ref. [@Blatt2011], supplementary online material). Unless a magic wavelength trap is employed, the ac Stark shift from optical traps causes a position-dependent frequency shift of the atomic resonance [@Blatt2011; @Escobar2008]. For photoassociation experiments, this effect results in a broadening of the lineshape feature in the signal $N$. To model this broadening, one must understand both the atomic response to the optical trap and the intensity profile $I_{trap}(\vec{r})$ of the trap laser (Ref. [@Blatt2011], supplementary online material). In the case of [$^{88}$Sr]{}, the $^1S_0$ and $^3P_1$ polarizabilities are known well enough to calculate the differential polarizability, $\alpha_{^3 \! P_1}(\omega_{trap}) - \alpha_{^1 \! S_0}(\omega_{trap})$, to better than 10% for typical trap laser wavelengths (such as 1064 nm). One can model $I_{trap}(\vec{r})$ with the Gaussian beam equation, using parametric heating measurements to obtain the beam waists [@Savard1997]. This broadening can be included in the photoassociation signal by adding a Stark shift, $$\omega_{Stark}(\vec{r}) = -\frac{1}{2 \hbar \epsilon_0 c} \left[ \alpha_{^3 \! P_1}(\omega_{trap}) - \alpha_{^1 \! S_0}(\omega_{trap}) \right] I_{trap}(\vec{r}),$$ to $\delta$ in Eq. \[eqn:K\_in\] and then carrying this term through to Eq. (\[eqn:number\_signal\]). With the trap ac Stark shift accounted for, we can fit experimental photoassociation data (Fig. \[fig:inelastic\_loss\_frequency\]). For these fits it is necessary to approximate the integrals in Eqs. (\[eqn:number\_signal\]) and (\[eqn:Kin\_averaged\]) as sums (Ref. [@Blatt2011], supplementary online material). ![(Color online) Atom number data from Ref. [@Blatt2011] as a function of the PA laser detuning from the atomic resonance. The false color pictures above the plot are the measured optical depths corresponding to the data points directly below the centers of the pictures. These measurements were performed in a non-magic wavelength trap, which caused broadening toward blue detunings since $\omega_{Stark}(0) = 2 \pi \times 327 \,\, \mathrm{kHz}$. The solid line is a fit using Eq. (\[eqn:number\_signal\]) with $\omega_{Stark}(\vec{r})$ included in $\overline{K}_{in}$. The quantities $T$ and $l_\mathrm{eff}$ were free parameters in the fit.[]{data-label="fig:inelastic_loss_frequency"}](inelastic_loss_frequency.eps){width="\linewidth"} A full treatment of the photoassociation lineshape would include Doppler broadening. However, according to theoretical studies of narrow-line photoassociation [@Ciuryo2004], since a $T$ of a few $\mu$K (typical for narrow-line laser cooled [$^{88}$Sr]{}) is greater than the PA laser photon recoil temperature, Doppler broadening is negligible compared to the momentum broadening shown in Fig. \[fig:signal\]. We numerically checked whether Doppler effects are significant for our analysis, and we find that Doppler broadening can only be neglected in the fits of Fig. \[fig:inelastic\_loss\_frequency\] due to the presence of both a large momentum broadening and an appreciable Stark shift broadening toward blue detunings. Treating collisions in this manner breaks down when elastic processes become important. The elastic-to-inelastic collision ratio is approximately $$\label{eqn:collision_ratio} \overline{K_{el} (k) / K_{in} (k)} \simeq 2 \overline{k} l_\textrm{eff} = 2 k_{th} l_\textrm{eff},$$ where the overline denotes thermal averaging. In deriving this formula, we made the approximation $e^{2 i \eta_{bg}} \simeq 1$, which is acceptable for the above estimate since $a_{bg}$ is only -1.4 $a_{0}$. Therefore, when $l_\textrm{eff} \sim 1 / 2 k_{th}$, elastic collisions must be treated. Elastic Collisions {#subsec:elastic_collisions} ------------------ If photoassociation can be minimized and elastic collision rate modifications can be made large, the OFR effect could be used to manipulate atomic interactions with relatively little particle loss. The usefulness of such manipulations is evident from experiments based on the MFR effect, which has proved to be a very fruitful technique that is central to numerous experiments [@Chin2010]. To access this regime in a quantum degenerate gas (for which $k \rightarrow 0$), the ratio of the optically modified elastic scattering length to the inelastic scattering length, $\left[ a(0)-a_{bg} \right]/b(0) = 2 \delta/\gamma$, must be much greater than unity. For a thermal gas, the rate coefficients determine the relevant limit, $\langle K_{el} (k) / K_{in} (k) \rangle \gg 1$, which implies via Eqn.\[eqn:collision\_ratio\] that $l_\textrm{eff} \gg 1/2 k_{th}$. We estimate from the latter condition that a thermal [$^{88}$Sr]{} gas at $T = 3~\mu\mathrm{K}$ (typical of narrow-line laser cooling) will require $l_\mathrm{eff}$ to be much greater than $400~a_0$. Using the isolated resonance approximation, large changes in elastic scattering were predicted to arise from an OFR based on the $n = -1$ vibrational state [@Ciuryo2005]. This prediction required high PA laser intensity and very large molecular detunings from the $n = -1$ state. However, as our CC theory has shown, these conditions will not yield an effect comparable to a lossless MFR because the requisite detunings are larger than the separation between resonances. Instead, the OFR physics is determined by the nearest resonance to a given detuning (Section \[subsec:CC\_Results\]). Experimentally, elastic collisions can be studied using cross-dimensional thermalization. For instance, in Ref. [@Monroe1993], a trapped atomic gas was prepared in a nonequilibrium state using parametric heating in 1D of the trap. Due to elastic collisions, the authors observed thermalization of the non-equilibrium gas. The thermalization time of the gas can be calculated from a simple treatment based on Enskog’s equation or a full molecular dynamics simulation [@Goldwin2005]. These treatments show that, on average, each particle participates in about three elastic collisions events during the $1/e$ thermalization time. Elastic collisions induced by OFRs were experimentally studied in Ref. [@Blatt2011] using cross-dimensional thermalization. In this work, an OFR was accessed in a trapped [$^{88}$Sr]{} gas prepared in a non-thermal state. In the absence of OFR-induced collisions, this gas would not thermalize over experimental time scales because of the negligible $a_\mathrm{bg} = - 1.4 a_0$ in [$^{88}$Sr]{}. With a PA laser applied, clear temperature changes were observed as a function of $\nu - \nu_0$. To understand whether these observations were caused by elastic collisions, we apply our theory to the experimental data. ![(Color online) Temperature data from Ref. [@Blatt2011] for horizontal (H) and vertical (V) trap eigenaxes. The solid lines are the results of a Monte Carlo simulation including elastic and inelastic collisions.[]{data-label="fig:simulation"}](simulation.eps){width="\linewidth"} Since the atomic samples in this measurement never reached thermal equilibrium over the time scale of the experiment, it is not possible to study the resulting data analytically. Instead, a time-dependent simulation of the phase-space density is necessary to understand the data quantitatively. Our analysis uses a numerically efficient method due to Bird [@Bird94] for simulating collisions between thousands of particles. The method discretizes the trap volume into small “collision volumes” containing much less than one particle on average. If there is more than one particle in a collision volume, the probabilities $P_{el}$ and $P_{in}$ of elastic and inelastic collisions between these atoms is calculated as $P_{el/in} = |\vec{v}_1 - \vec{v}_2| \sigma_{el/in} t_{step}/V$. Here $\vec{v}_1$ and $\vec{v}_2$ are the velocities of the two atoms, $t_{step}$ is the time step of the simulation, $V$ is the collision volume, and the cross sections are given by $$\begin{aligned} \label{eqn:collision_cross_sections} \sigma_\mathrm{el} &=& \frac{8 \pi}{1 + k^2 a_\mathrm{bg}^2} \frac{\left[l_\mathrm{eff} \gamma + a_\mathrm{bg} (\delta + \frac{E}{\hbar})\right]^2 + a_\mathrm{bg}^2 \gamma^2/4}{(\delta + E/\hbar)^2 + \frac{\gamma^2}{4} (1 + 2 k l_\mathrm{eff})^2} \\ \sigma_\mathrm{in} &=& \frac{4 \pi l_\mathrm{eff}}{k} \frac{\gamma^2}{(\delta + E/\hbar)^2 + \frac{\gamma^2}{4} (1 + 2 k l_\mathrm{eff})^2}.\end{aligned}$$ If an inelastic collision occurs, both particles are removed from the simulation. For an elastic collision event, the particles’ velocity vectors are rotated using a random rotation matrix. Between each of these collision steps, the particles evolve in the trap potential using an embedded Runge-Kutta method. We have checked this simulation against known results for cross-dimensional thermalization in harmonic traps and have also confirmed that, with elastic collisions removed, the simulation reproduces the results of photoassociation theory of the previous section [@Blatt2011thesis]. Fig. \[fig:simulation\] depicts temperature data from the experiments of Ref. [@Blatt2011] as well as our simulation results. Our simulation tells us that the temperature peaks for certain detunings because the PA laser is driving photoassociative loss of the coldest atoms, resulting in antievaporative heating. We also find that without including elastic collisions in our simulation, the simulation does not predict the dip in the horizontal temperature apparent in the data. Therefore, we conclude that this temperature dip indicates partial thermalization of the gas. The fact that antievaporation and thermalization have different detuning dependence arises because the elastic and inelastic collision rates average differently over the collision momentum $k$. The interplay between elastic and inelastic processes is sensitive to the value of $\gamma$ used in the simulation. The simulation only agrees with the experimental data if we set $\gamma = 2\pi\times 40~\mathrm{kHz} = 2.7 \gamma_m$. This leads us to conclude that the OFR effect in [$^{88}$Sr]{} is broadened beyond the natural spontaneous decay of the ${^{88}\mathrm{Sr}_2}$ molecules. Extra broadening has also been seen in other [$^{88}$Sr]{} [@Zelevinsky2006; @Yan2013a] and Rb [@Theis2004] OFR experiments. Summary and Conclusion ====================== We have developed CC and isolated resonance theories of OFRs. The CC theory predicts resonance interference for detunings between OFRs, causing the OFR effect to vanish between resonances. We conclude that OFR experiments have a “nearest resonance” constraint, meaning that the OFR effect is dictated by the nearest resonance to a given detuning. The isolated resonance theory agrees with the more complete CC theory for detunings near a molecular resonance. In this regime, it is possible to use the simpler isolated resonance theory to model photoassociation and OFR measurements and fit the data from these experiments. Such models have shown a broadening beyond the expected linewidth of the molecular state accessed by an OFR.
--- abstract: | We calculate merger rates of dark matter haloes using the Extended Press-Schechter approximation (EPS) for the Spherical Collapse (SC) and the Ellipsoidal Collapse (EC) models.\ Merger rates have been calculated for masses in the range $10^{10}M_{\odot}\mathrm{h}^{-1}$ to $10^{14}M_{\odot}\mathrm{h}^{-1}$ and for redshifts $z$ in the range $0$ to $3$ and they have been compared with merger rates that have been proposed by other authors as fits to the results of N-body simulations. The detailed comparison presented here shows that the agreement between the analytical models and N-body simulations depends crucially on the mass of the descendant halo. For some range of masses and redshifts either SC or EC models approximate satisfactory the results of N-body simulations but for other cases both models are less satisfactory or even bad approximations. We showed, by studying the parameters of the problem that a disagreement –if it appears– does not depend on the values of the parameters but on the kind of the particular solution used for the distribution of progenitors or on the nature of EPS methods.\ Further studies could help to improve our understanding about the physical processes during the formation of dark matter haloes. author: - 'N. Hiotelis' title: 'Merger rates of dark matter haloes: a comparison between EPS and N-body results' --- =1 Introduction ============ The development of analytical or semi-numerical methods for the problem of structure formation in the universe helps to improve our understanding of important physical processes. A class of such methods is based on the ideas of @prsc74 and on their extensions (Extended Press-Schechter Methods EPS, @boet91, @laco93): The linear overdensity $\delta(\textbf{x};R)\equiv [\rho(\textbf{x};R)-\rho_b(\textbf{x};R)]/\rho_b(\textbf{x};R)$ at a given point $\textbf{x}$ of an initial snapshot of the Universe fluctuates when the smoothing scale $R$ decreases. In the above relation, $\rho(\textbf{x};R)$ is the density at point $\textbf{x}$ of the initial Universe smoothed by a window function with smoothing scale $R$. The index $b$ denotes the density of the background model of the Universe. This fluctuation is a Markovian process when the smoothing is performed using a top-hat window in Fourier space. For any value of the smoothing scale $R$, the overdensity field is assumed to be Gaussian with zero mean value. The dispersion of these Gaussians is a decreasing function of the smoothing scale $R$ reflecting the large scale homogeneity of the Universe. The mass $M$ contained in a given scale $R$ depends on the window function used. For the top-hat window this relation is: $M=\frac{4}{3}\pi\rho_{b,i}R^3=\frac{\Omega_{m,i}H^2_i}{2G}R^3$, where $\rho_{b,i}$ and $\Omega_{m,i}$ are the values of the mean density and the density parameter of the Universe, $G$ is the gravitational constant and $H_i$ is the Hubble’s constant. The index $i$ indicates that all the above values are calculated at the initial snapshot. The dispersion in mass,$\sigma^2$, at scale $R$ is a function of mass $M$ and it is usually denoted by $S$, that is $S(M)\equiv\sigma^2[R(M)]$.\ In the plane $(S,\delta)$ random walks start from the point $(S=0,\delta=0)$ and diffuse as $S$ increases. Let the line $B=B_{SC}(z)$ that is a function of redshift $z$. In the case this line is parallel to $S$-axis in the $(S,\delta)$ plane, then it has a physical meaning as it can be connected to the spherical collapse model (SC): It is well known that in an Einstein-de Sitter Universe, a spherical overdensity collapses at $z$ if the linear extrapolation of its value up to the present exceeds $\delta_{sc}\approx 1.686$ (see for example @peeb80). All involved quantities (density, overdensities, dispersion) are linearly extrapolated to the present and thus the barrier in the spherical collapse model is written in the from $B(z)=1.686/D(z)$, where $D(z)$ is the growth factor derived by the linear theory, normalized to unity at the present epoch. It is clear that the line $B(z)$ is an increasing function of $z$. If a random walk crosses this barrier for first time at some value $S_0$ of $S$, then the mass element associated with the random walk is considered to belong to a halo of mass $M_0=S^{-1}(S_0)$ at the epoch with redshift $z$. However, the distribution of haloes mass $f_M$, at some epoch $z$, is connected to the first crossing distribution $f_S$, by the random walks, of the barrier that corresponds to epoch $z$ with the relation: $$f_M(M)\mathrm{d}M=f_S(S)|\frac{\mathrm{d}S(M)}{\mathrm{d}M}|\mathrm{d}M$$ A form of the barrier that results to a mass function that is in better agreement with the results of N-body simulations than the spherical model is the one given by the Eq. $$B_{EC}(S,z)=\sqrt{a}B_{SC}(z)[1+\beta[S/aB_{SC}^2(z)]^{\gamma}].$$ In the above Eq. $\alpha$, $\beta$ and $\gamma$ are constants. The above barrier represents an ellipsoidal collapse model (EC) [@shto99]. The barrier depends on the mass ($S=S(M)$) and it is called a moving barrier. The values of the parameters are $a=0.707$, $\beta=0.485$,  $\gamma=0.615$ and are adopted either from the dynamics of ellipsoidal collapse or from fits to the results of N-body simulations The spherical collapse model results for $a=1$ and $\beta=0.$\ In a hierarchical scenario of the formation of haloes, the following question is fundamental: Given that at some redshift $z_0$ a mass element belongs to a halo of mass $M_0$, what is the probability the same mass element at some larger redshift $z$ $(z >z_0)$ -that corresponds to an earlier time- was part of a halo with mass $M$ with $ M\leq M_0$? This question in terms of first crossing distributions and barriers can be written in the following equivalent form: Given a random walk passes for the first time from the point $(\delta_0,S_0)$ what is the probability this random walk crosses a barrier $B$ with $B > \delta_0$, for the first time between $S,~S+\mathrm{d}S$ with $S > S_0$?\ If we denote the above probability by $f(S/\delta_0,S_0)\mathrm{d}S$ it can be proved, [@zhhu06], that for an arbitrary barrier, $f$ satisfies the following integral equation: $$f(S/\delta_0,S_0)=g_1(S,\delta_0,S_0)+\int_0^Sg_2(S,S')f(S'/\delta_0,S_0)\mathrm{d}S'$$ where: $$g_1(S,\delta_0,S_0)=\left[\frac{B(S)-\delta_0}{S-S_0}-2\frac{\mathrm{d}B(S)}{\mathrm{d}S}\right]P_0[B(S),S/\delta_0,S_0]$$ $$g_2(S,S')=\left[2\frac{\mathrm{d}B(S)}{\mathrm{d}S}-\frac{B(S)-B(S')}{S-S'}\right]P_0[B(S),S/B(S'),S']$$ and $$P_0(x,y/x_0,y_0)=\frac{1}{\sqrt{2\pi(\Delta y)}}e^{- \frac{{\Delta x}^2}{2\Delta y}}$$ with $\Delta y=x-x_0$ and $\Delta y=y-y_0$.\ In the case of a linear barrier Eq.(3) admits an analytic solution. If $B(S)=\omega+qS$, where the coefficients $\omega$ and $q$ could be functions of the redshift $z$ in order to describe the dependence on the time, the solution is written: $$f(S /\delta_0,S_0)=\frac{B(S_0)-\delta_0}{\sqrt{2\pi(S-S_0)^3} }\exp\left[-\frac{[B(S)- \delta_0]^2}{2(S-S_0)}\right]$$ Thus, the spherical model which is of the form $B=B(z)=\omega(z)=1.686/D(z)$ leads to the solution: $$f_{SC}(S,z/S_0,z_0)=\frac{\Delta \omega}{\sqrt{2\pi(\Delta S)^3}}\exp\left[-\frac{(\Delta \omega)^2}{2\Delta S}\right]$$ where, $\Delta S\equiv S-S_0$, and $\Delta \omega=\omega(z)-\omega(z_0)$ .\ Unfortunately, no analytical solution exists for the ellipsoidal model. The exact numerical solution of Eq.(3) is well approximated by the expression proposed by @shto02 that is: $$f_{EC}(S,z/S_0,z_0)=\frac{1}{\sqrt{2\pi}}\frac{|T(S,z/S_0,z_0)|}{(\Delta S)^{3/2}} \exp\left[-\frac{(\Delta B)^2}{2\Delta S}\right]\mathrm{d}S$$ where, $\Delta B=B(S,z)-B(S_0,z_0)$, and the function $T$ is given by: $$T(S,z/S_0,z_0)=B(S,z)-B(S_0,z_0)+\sum_{n=1}^{5}\frac{[S_0-S]^n}{n!} \frac{\mathrm{\partial ^n}}{\partial S^n}B(S,z).$$ According to the hierarchical clustering any halo is formed by smaller haloes (progenitors). A number of progenitors merge at $z$ and form a larger halo of mass $M_0$ at $z_0$ ($z_0 <z$). Obviously, the sum of the masses of the progenitors equals to $M_0$. Given a halo of mass $M_0$ at $z_0$ the average number of its progenitors in the mass interval $[M,M+\mathrm{d}M]$ present at $z$ with $z > z_0$ is : $$\frac{\mathrm{d}N}{\mathrm{d}M}(M/M_0,\Delta \omega)\mathrm{d}M=\frac{M_0}{M}f(S,z/S_0,z_0)\mathrm{d}M$$ Recent comparisons show that the use of EC model improves the agreement between the results of EPS methods and those of N-body simulations. For example, @yaet04 showed that the multiplicity function resulting from N-body simulations is far from the predictions of spherical model while it shows an excellent agreement with the results of the EC model. On the other hand, @liet03 compared the distribution of formation times of haloes formed in N-body simulations with the formation times of haloes formed in terms of the spherical collapse model of the EPS theory. They found that N-body simulations give smaller formation times. @hipo06 showed that using the EC model, formation times are shifted to smaller values than those predicted by a spherical collapse model. Additionally, the EC model combined with the stable “clustering hypothesis" has been used by [@hi06] in order to study density profiles of dark matter haloes. Interesting enough,. the resulting density profiles at the central regions are closer to the results of observations than are the results of N-body simulations. Consequently, the EC model is a significant improvement of the spherical model and therefore we are well motivated to study merger-rates of dark matter haloes for both the SC and the EC model. This study depends upon the accurate construction of a set of progenitors for any halo for a very small “time step" $\Delta \omega$. The set of progenitors are created using the method proposed by @nede08 that we describe in Sect.3. In Sect. 2 we define merger rates and we recall fitting formulae resulting from N-body simulations. In Sect.4 our results are presented and discussed. Definition of merger rates and analytical formulae ================================================== We examine descendant haloes from a sample of $N_d$ haloes with masses in the range $M_d, M_d+\mathrm{d}M_d$ present at redshift $z_d$. For a single descendant halo the procedure is as follows: Let $M_{p,1},M_{p,2}...M_{p,k}$ be the masses of its $k$ progenitors at redshift $z_p > z_d$. For matter of simplicity we assume that the most massive progenitor is $M_{p,1}$. We define $\xi_i=M_{p,i}/M_{p,1}$ for $i\geq 2$ and we assume that the descendant halo is formed by the following procedure: During the interval $dz=z_p-z_d$ every one of the progenitors with $i\geq 2$ merge with the most massive progenitor $i=1$ and form the descendant halo we examine. We repeat the above procedure for all haloes in the range $M_d, M_d+\mathrm{d}M_d$ found in a volume $V$ of the Universe. Then, we find the number denoted by $N$ of all progenitors with $\xi_i,~i\geq 2$ in the range $(\xi, \xi+\mathrm{d}\xi)$ and we calculate the ratio $N/(V\mathrm{d}z\mathrm{d}M_d\mathrm{d}\xi)$. We define the merger rate $B_m$ as follows: $$B_m(M_d,\xi,z_p:z_d)=\frac{N}{V\mathrm{d}z\mathrm{d}M_d\mathrm{d}\xi}$$ Let the number density of haloes with masses in the range $M_d, M_d+\mathrm{d}M_d$ at $z_d$ be $n(M_d,z_d)=\frac{N_d(M_d,z_d)}{V\mathrm{d}M_d}$. The ratio $B_m/n=N/(N_d\mathrm{d}z\mathrm{d}\xi)$ measures the mean number of mergers per halo, per unit redshift, for descendant haloes in the range $M_d, M_d+\mathrm{d}M_d$ with progenitor mass ratio $\xi$.\ @fama08 analyzed the results of the Millennium simulation of @spet05. The fitting formula proposed by the above authors is separable in the three variables, mass $M_d$, progenitor ratio $\xi$ and redshift $z$:\ $$\frac{B(M_d,\xi,z_p:z_d)}{n(M_d,z)}=A\cdot F(M_d)G(\xi)H(z)$$ with\ $F(M_d)=\left(\frac{M_d}{\tilde{M}}\right)^{a_1}, ~G(\xi)={\xi}^{a_2}\exp\left[\left(\frac{\xi}{\tilde{\xi}}\right)^{a_3}\right], ~H(z)=\left(\frac{\mathrm{d}\delta_c}{\mathrm{d}z}\right)^{a_4}_{_{z=z_d}}$ and the values of the parameters are $\tilde{M}=1.2\times10^{12}M_{\odot}, A=0.0289, \tilde{\xi}=0.098, a_1=0.083, a_2=-2.01, a_3=0.409, a_4=0.371$.\ @laco93 showed that in the spherical model the transition rate is given by: $$\begin{aligned} r(M\longrightarrow M_d/z_d)\mathrm{d}M_d= \left(2/\pi\right)^{1/2}\left[\frac{\mathrm{d}\delta_c(z)}{{\mathrm{d}z}}\right]_{_{z=z_d}} \frac{1}{\sigma^2(M_d)}\left[\frac{\mathrm{d}\sigma(M)}{\mathrm{d}M}\right]_{M=M_d}\nonumber\\ \times\left[1-\frac{\sigma^2(M_d)}{\sigma^2(M)}\right]^{-3/2} \exp\left[-\frac{\delta^2_c(t)}{2} \left(\frac{1}{\sigma^2(M_d)}-\frac{1}{\sigma^2(M)}\right)\right]\mathrm{d}M_d~~~~~~~ \end{aligned}$$ This provides the fraction of the mass belonging to haloes of mass $M$ that merge instantaneously to form haloes of mass in the range $M_d, M_d+\mathrm{d}M_d$ at $z_d$. The product $r\cdot f_{sc}(M,z_d)\mathrm{d}M$, where $f_{sc}(M,z)$ is the unconditional first crossing distribution for the spherical model, gives the above fraction of mass as a fraction of the total mass of the Universe and successively multiplying by $(\rho_b/M)\cdot V$ the number of those haloes is found. Then, dividing by $(\rho_b/M_d)\cdot V\cdot f_{sc}(M_d,z_d)\mathrm{d}M_d$ (that equals to the number of the descendant haloes) we find: $$\begin{aligned} \frac{N}{N_d\mathrm{d}z}=\sqrt{\frac{2}{\pi}}\left[\frac{\mathrm{d}\delta_c(z)}{{\mathrm{d}z}}\right]_{_{z=z_d}} \frac{M_d}{M}\frac{1}{\sigma^2(M)}\left[\frac{\mathrm{d}\sigma(M)}{\mathrm{d}M}\right]_{M=M_d}\nonumber\\ \times \left[1-\frac{\sigma^2(M_d)}{\sigma^2(M)}\right]^{-3/2}\mathrm{d}{M} \end{aligned}$$ Assuming a strictly binary merger history *i.e.* every halo has two progenitors, and denoting by $\xi$ the mass ratio of the small progenitor to the large one ($\xi=(M_d-M)/M$), using $\mathrm{d}M=\frac{M^2}{M_d} \mathrm{d}\xi$ and substituting in (15) we have the final expression for the binary spherical case, that is: $$\begin{aligned} \frac{B_m}{n}= \frac{N}{N_d\mathrm{d}z\mathrm{d}\xi}=\sqrt{\frac{2}{\pi}}\left[\frac{\mathrm{d}\delta_c(z)}{{\mathrm{d}z}}\right]_{_{z=z_d}} \frac{M}{\sigma^2(M)}\left[\frac{\mathrm{d}\sigma(M)}{\mathrm{d}M}\right]_{M=M_d}\nonumber\\ \times\left[1-\frac{\sigma^2(M_d)}{\sigma^2(M)}\right]^{-3/2} \end{aligned}$$ Construction of the set of progenitors ====================================== The construction of progenitors of a halo can be based either or Eqs (8) and (9) or else on Eq. (11). For the first case a procedure is as follows: A halo of mass $M_0$ at redshift $z_0$ is considered. A new redshift $z$ is chosen. Then, a value $\Delta S$ is chosen from the desired distribution given by Eq.(8) or (9). The mass $M_p$ of a progenitor is found by solving for $M_p$ the equation $\Delta S=S(M_p)-S(M_0)$. If the mass left to be resolved $M_0-M_p$ is large enough (larger than a threshold), the above procedure is repeated so a distribution of the progenitors of the halo is created at $z$. If the mass left to be resolved -that equals to $M_0$ minus the sum of the masses of its progenitors- is less than the threshold, then we proceed to the next time step , and re-analyze using the same procedure.\ A complete description of the above numerical method is given in @hipo06. The algorithm - known as N-branch merger-tree- is based on the pioneer works of @laco93, @soko99 and @bo02.\ We have to note that the construction of a set of progenitors for an initial set of haloes after a “time step” $\Delta\omega$ is a problem that has not a unique solution. Consequently, it is interesting to compare different solutions with the results of N-body simulations in order to find those which show a better agreement. We note that any of the above proposed algorithms has a number of drawbacks. The algorithm to be used has to be suitable for the particular problem. If for example the algorithm assumes an initial set of descendant haloes of the same mass, it cannot be used for more than one time steps since the set of progenitors predicted at the first time step does not consist of haloes of the same mass. Since our purpose is the derivation of merger rates, we used the method proposed by @nede08 that is suitable for the calculation of a set of progenitors for descendant haloes of the same mass for a single time step. A description is given below:\ We assume a set of $N_d$ haloes of the same mass $M_0$ at $z_0=z_d$. We use the variables $M_1,M_2,M_3..$ to denote the masses of their progenitors at redshift $z_p$, after a time step $\Delta\omega=\omega(z_d)-\omega(z_p)$. We assume that $M_1 > M_2 >M_3,...$ and we denote by $P_i(M)$ the probability that the $i^{th}$ progenitor has mass $M$. We also assume that the value of $M_1$, that is the mass of the most massive progenitor of a halo, defines with a unique way the masses of all its rest progenitors. Additionally, $P_{i/1}(M_i/M_1)$ is the constrained probability that the $i^{th}$ progenitor of a halo equals $M$ given that its most massive progenitor is $M_1$. Obviously the following Eqs. hold: $$P_i(M)=\int P_{i/1}(M/M_1)P_1(M_1)\mathrm{d}M_1$$ $$P_{tot}(M)=\sum_{i}P_i(M)$$ $$P(M_1,M_2,...)=0~~ \mathrm{if}~~ \sum_{i}M_{i} > M_0$$ These are the key equations for the construction of the set of progenitors. We use the following three steps:\ **1st step**: The distribution of the most massive progenitors.\ We define $P_{tot}(M)$ using Eq.(11), that is: $$P_{tot}(M)\mathrm{d}M\equiv\frac{\mathrm{d}N}{\mathrm{d}M}(M/M_0,\Delta \omega)\mathrm{d}M$$ The value of the integral $\int_{M_{min}}^{M_0}P_{tot}(M)\mathrm{d}M$ depends on both $M_{Min}$ and $\Delta\omega$. For $M_{min}\rightarrow 0$ it declines due to the presence of the large number of very small progenitors. The value of the integral increases for increasing $\Delta\omega$. Thus, for reasonable choice of $M_{min}$ the values of the above integral is larger than unity. Then, the distribution of $M_1$ can be found by the following procedure: First, we solve the Eq. $$\int_{x_*M_0}^{M_0}P_{tot}(M)\mathrm{d}M=1$$ with respect to $x_*$. The resulting values of $M_{*}\equiv x_*M_0$ are larger than $M_{min}$. Then, we pick $M_1$ from the distribution: $$P_1(M_1)=\left\{ \begin{array}{l l} P_{tot}(M_1),~~ M_1\geq M_{*}\\ 0,~~ \mathrm{otherwise}\\ \end{array}\right.$$ This is done by the following procedure: A random number $r$ is chosen in the interval $[0,1]$ and the equation $\int_{M_1}^{M_0}P_{tot}(M)\mathrm{d}M=r$ is solved for $M_1$. The resulting values of $M_1$ have the above described distribution.\ If $M_{left}\equiv M_0-M_1 > M_{min}$ we proceed with the second progenitor. Otherwise, the halo has just one progenitor and we proceed with the next halo.\ **2st step**: The distribution of $M_2$.\ Let $f_i(M_1)$ be the mass of the $i^{th}$ progenitor given that the mass of the most massive progenitor equals to $M_1$. We assume that $$P_{i/1}(M_i/M_1)=\delta[M_i-f_i(M_1)]$$ where $\delta$ is a delta function and $f_i$ a monotonically decreasing function of $M_1$.\ We consider the differential equation: $$\frac{\mathrm{d}f_i(M_1)}{\mathrm{d}M_1}=-\frac{P_1(M_1)}{P_i[f_i(M_1)]}$$ Using (23) and (24) the right hand side of (17) is written: $$\begin{aligned} \int_{-\infty}^\infty \delta[M_1-f_i(M_1)]P_1(M_1)\mathrm{d}M_1=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \nonumber\\ -\int_\infty^{-\infty}\delta[M_i-f_i(M_1)P_i[f_i(M_1)]\mathrm{d}f_i(M_1)= \int_{-\infty}^{\infty}\delta[M_i-f_i(M_1)]P_i[f_i(M_1)]\mathrm{d}f_i(M_1)=\nonumber\\ P_i[f_i(M_1)]=P_i(M_i)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ \end{aligned}$$ and thus the solution of the differential Eq. (24) satisfies Eq. (17). ![The distribution of the first and the second progenitors $M_1$ and $M_2$, respectively for both the SC and EC models. Filled gradients show the distribution of $M_1$ and empty gradients show the distribution of $M_2$ for the spherical model for $M_0=100,~\Delta\omega=0.1$ and $z_d=0$. Squares show the same distributions for the ellipsoidal model. Solid and dashed lines are the predictions of Eq.(11) with $f$ given by (8) and (9), respectively.[]{data-label="fig1"}](hiotelis-fig1){width="16cm"} ![The function $f_2(M_1)$, the solution of the differential equation (26), equals the mass $M_2$ of the second progenitor and is plotted for $M_0=100$ and $\Delta \omega =0.1$. For the same mass $M_1$ of the most massive progenitor $M_2$ is smaller for EC model than for SC model. Additionally, the ellipsoidal model extends to lower values than the spherical model. The lowest values of $M_1$ shown, are about $0.36M_0$ for the ellipsoidal model and $0.41M_0$ for the spherical model, respectively.[]{data-label="fig2"}](hiotelis-fig2){width="16cm"} Thus, the mass of the second progenitor can be found by integrating numerically (24) for $i=2$: $$\frac{\mathrm{d}f_2(M_1)}{\mathrm{d}M_1}=-\frac{P_1(M_1)}{{P}_2[f_2(M_1)]}$$ The function $P_2$ involved is unknown. So a trial function $\tilde{P}_2(y)\equiv P_{tot}(y)-P_1(y)$ is used and the Eq. $$\frac{\mathrm{d}y}{\mathrm{d}x}=G(x,y)$$ where $G(x,y)\equiv -P_1(x)/\tilde{P}_2(y)$ is solved numerically for $y$ using a classical $4^{th}$ order Runge-Kutta with initial conditions $x_{in}=M_{*}, y_{in}=M_{2,0}=M_{*}$ (called solution I in @nede08). We used a step $\Delta x=[M_1-M_{*}]/N_s$, where $N_s$ defines the number of steps. We used various values of $N_s$ from 100 to 10000 and we found that the results are essentially the same.\ In the case the solution of the above differential equation is $M_2 < M_{min}$ then we enforce $M_2=M_0-M_1$. Finally, the resulting values of $M_2$ are used for the numerical construction of $P_2$.\ In our calculations, we used a flat model for the Universe with present day density parameters $\Omega_{m,0}=0.3$ and $ \Omega_{\Lambda,0}\equiv \Lambda/3H_0^2=0.7$. $\Lambda$ is the cosmological constant and $H_0$ is the present day value of Hubble’s constant. We used the value $H_0=100\mathrm{hKms^{-1}Mpc^{-1}}$ and a system of units with $m_{unit}=10^{12}M_{\odot}h^{-1}$, $r_{unit}=1h^{-1}\mathrm{Mpc}$ and a gravitational constant $ G=1$. At this system of units, $H_0/H_{unit}=1.5276.$\ As regards the power spectrum, we used the $\Lambda CDM$ form proposed by @smet98. The power spectrum is smoothed using the top-hat window function and is normalized for $\sigma_8\equiv\sigma(R=8h^{-1}\mathrm{Mpc})=0.9$.\ We used a number $N_{res}=10^5$ haloes of the same mass $M_0$ at $z_0=z_d$ and we found their progenitors at $z_p$ that is after a “time-step” $\Delta \omega=\omega(z_p)-\omega(z_d)$. We studied three values of $z_d$ that are $z_d=0,1$ and $3$ respectively. We examined values of $M_0$ in the range $0.01$ to $100$ in our system of units. These values correspond to masses in the range $M_{0}=10^{10}M_{\odot}h^{-1}$ to $M_{0}=10^{14}M_{\odot}h^{-1}$. We studied three values of $\Delta \omega$ namely $0.1, 0.05$ and $0.025$. We also used $M_{min}=10^{-3}M_0$ for $\Delta \omega=0.1$ and $M_{min}=5\cdot10^{-4}M_0$ for $\Delta \omega=0.05$ and $0.025$.\ Fig.1 compares the distributions of progenitors for $M_0=100$, $\Delta \omega=0.1$ and $z_d=0.0$ with the analytical ones given by Eq. (11) for both the spherical and the ellipsoidal models. Up to this step every halo has at most two progenitors. It is clear that the agreement is very satisfactory.\ Fig.2 shows the solution $f_2(M_1)$ of Eq. (26). It presents $M_2=f_2(M_1)$ as a function of $M_1$ both normalized to $M_0$. It corresponds to $z_d=3.0$ and $z_p=3.078$, that is to $\Delta \omega =0.1 $. It is shown that the distribution of most massive progenitors extends to smaller values in the ellipsoidal model. The reflection of this different behavior to merger rates will be studied in Sect.4\ ![The distribution of progenitors $\mathrm{d}N/\mathrm{d}M$ versus their mass $M$ normalized to the mass $M_0$ of the descendant halo. The first column corresponds to the spherical model for $z_d=0$, $z_d=1$ and $z_d=3$ (from top to bottom) and the second column to the ellipsoidal model. A value $\Delta\omega=0.1$ is used. Dashed lines are the predictions of the method studied in this paper while solid lines are the predictions of Eq. (11)[]{data-label="fig3"}](hiotelis-fig3){width="16cm"} The satisfactory agreement between the distributions of progenitors predicted by the method studied and by Eq.(11) holds also for various values of the descendant halo and various redshifts. This is shown in Fig.3 where the distribution of progenitors for both SC and EC models for $z_d=0,1$ and $3$ are presented. The value for the “time-step” is $\Delta\omega =0.1$. The corresponding values of $z_p$ are $z_p=0.1097,~1.083$ and $3.078$, respectively.\ ![The distribution of progenitors $\mathrm{d}N/\mathrm{d}M$ for small values of $M/M_0$ for $M_0=100,~z_d=0,~\Delta\omega=0.1$ and for the SC model. Thin solid line is the prediction of Eq. (11) and the thick solid line is the prediction of the two first steps of the method studied, that is without progenitors $M_i$ with $i > 2$. Dashed line is the final distribution after the third step, that is after the prediction of the full set of progenitors.[]{data-label="fig4"}](hiotelis-fig4){width="16cm"} However, if we focus on small values of $M/M_0$ we see that the distribution of progenitors there differs significantly from the theoretical one. Such an example is given in Fig.4 where the thin solid line is the theoretical distribution and the thick solid line is the distribution that results after the above two steps. (Dashed line is the final distribution after the completeness of $3^d$ step and it will be discussed later.) This disagreement shows clearly that the number of small progenitors is underestimated when every halo is analyzed to two progenitors and the need of more progenitors is clear. Although the disagreement appears only for small values $M/M_0$ is important for the calculation of merger rates as it will be shown below.\ The above two steps are completed for the whole sample of descendant haloes. Thus, after the completion of the second step, the distribution $P_2$ is found numerically and is expressed by a polynomial in order to be used in the $3^d$ step below.\ **$3^d$ step**: The distribution of $M_i, i >2$.\ Obviously, a halo has a progenitor $i$ if the mass left to be analyzed $M_{left,i}\equiv M_0-\sum_{n=1}^{n=i-1}M_n$ is $M_{left,i} \geq M_{min}$. We found the distribution of the rest progenitors using the following: First, we found the solution $R$ of the equation $P_2(y)=P_0(y)$. Obviously for $y < R$ we have $P_2(y)<P_0(y)$. Then, we define: $$P_i(x)=\left\{ \begin{array} {l l} P_0(x)-P_2(x),~~ \mathrm{when}~~ M_{min} \leq x \leq M_{high,i}\\ 0,~~ \mathrm{otherwise}\\ \end{array}\right.$$ where $$M_{high,i}=\min\{M_{left,i},M_{i-1}\}~~ \mathrm{for}~~~ i > 3~~\mathrm{and}~~ M_{high,3}=R\cdot M_0$$ Finally, we solve Eq. (24) for $f_i(M_1)$.\ Dashed line in Fig.4 is the distribution of progenitors after the completeness of the third step. It is clear that this distribution is much closer to the theoretical one given by the thin solid line than the distribution -that is described by the thick solid line- that results using only the first two progenitors $M_1$ and $M_2$.\ Results ======= We have already mentioned that distributing progenitors according to Eq. (11) is a problem that has not a unique solution. Additionally, the calculation of merger rates of dark matter haloes using analytical methods involves a large number of parameters. These are: the background cosmology, the model of collapse used (SC or EC), the mass of the descendant haloes $M_0$, the redshift $z_d$ and the “time step” $\Delta\omega$.\ The background cosmology used has been described in the previous section. The distribution of progenitors is done according to the method analyzed through this paper. So the parameters that were studied are: the model of collapse, the mass of the descendant haloes $M_0$, the redshift $z_d$ and the time step $\Delta\omega$.\ ![Merger rate for $M_0=100$ for $z_d=0$ and $z_p=0.02804$ (that corresponds to $\Delta\omega=0.025$). Solid line corresponds to the formula proposed to fit the results of N-body simulation that is given by Eq.(13). Thick long dashes show the predictions of the spherical binary model given by Eq.(16). Squares are the predictions of the method studied in this paper for the SC, using only two progenitors $M_1$ and $M_2$. Thick small dashes show the prediction of the above method for the whole set of progenitors.[]{data-label="fig5"}](hiotelis-fig5){width="16cm"} We give a first result in Fig.5. Solid line shows the predictions of the formula given by Eq.(13), proposed to fit the results of N-body simulation. Thick long dashes show the predictions of the spherical binary model given by Eq (16). It is shown that the spherical binary model overestimates merge rate for large values of $\xi$ while it underestimates the merger rate for values of $\xi$ smaller than $10^{-2}$. The predictions of the method studied are shown by squares and thick small dashes. Squares show the results after the first two steps described in Sect.3, that is after the distribution of the two first progenitors $M_1$ and $M_2$ only, while thick small dashes show the prediction for the whole set of progenitors. The third step in the procedure described in Sect.3 adds progenitors that have small masses. This increases the number of progenitors with small $\xi$ and rises the curve of the merger rate. This result agrees better with the predictions of N-body simulations. The results correspond to $z_d=0$. We used $\Delta\omega=0.025$ that results to $z_p=0.02804$.\ ![The role of the ’time-step’ in the estimation of merger rates: For the EC model, $M_0=100$ and $z_d=0$ solid and dashed lines correspond to $\Delta\omega=0.1$ and $\Delta\omega=0.025$, respectively. It is clear that differences are negligible.[]{data-label="fig6"}](hiotelis-fig6){width="16cm"} The accurate calculation of the merger rates requires that $\Delta\omega\rightarrow 0$. However, we examined different values of $\Delta\omega$ and we verified that the results do not depend crucially on this parameter. We used three values of $\Delta\omega$ namely $0.025, 0.05$ and $ 0.1$. Differences in merger rates due to the different values of $\Delta\omega$ are negligible. As an example we present Fig.6. It refers to the SC model for $z_d=0$ and for a descendant halo with mass $M_0=100$, for $\Delta\omega =0.1$ and $\Delta\omega=0.025$ (solid line and dashed line respectively). The corresponding values of $z_p$ are $0.1097$ and $0.02804$. Thus $\Delta z$ is about four times smaller in the second case. It is clear that only negligible differences are present.\ ![Merger rates for various models and various values of $M_0$ and $z_d$. The two snapshots of the first row show merger rates for $M_0=0.01$ at $z_d=0$ and $z_d=1$, respectively. Squares are the predictions of N-body given by Eq. (13). Thick dashes show the results of SC model and solid lines the results of EC model by the method used in this paper for $\Delta\omega=0.1$. Snapshots of the second row correspond to $M_0=1$ and those of the third row to $M_0=100$. []{data-label="fig7"}](hiotelis-fig7){width="16cm"} ![Merger rates for various models and various values of $M_0$ and $z_d$. The two snapshots of the first row show merger rates for $M_0=0.1$ at $z_d=0$ and $z_d=3$, respectively. Squares are the predictions of N-body given by Eq. (13). Thick dashes show the results of SC model and solid lines the results of EC model by the method used in this paper for $\Delta\omega=0.025$. Snapshots of the second row correspond to $M_0=1$ and those of the third row to $M_0=100$.[]{data-label="fig8"}](hiotelis-fig8){width="16cm"} ![Detailed comparisons between the EPS and N-body results. The relative difference $(R_{EPS} - R_{NB})/R_{NB}$, where $R_{EPS}$ and $R_{NB}$ are the merger rates predicted by the EPS and by N-body results respectively, is plotted as a function of $\xi$. SC model gives merger rates that are in good agreement with N-body results for small haloes (in the range $0.01-1$) while EC model approximates better the merger rates of heavy haloes ($M_0=100$).[]{data-label="fig9"}](hiotelis-fig9){width="16cm"} In Figs 7 and 8 we present results for different masses, redshifts and time-steps. In all snapshots dashed lines are the predictions of the SC model and solid lines show the results of the EC model. Squares are the predictions of the N-body fitting formula formula given by Eq.(13).\ From the results presented in the first row of Fig.7 is clear that the EC model results to merger rates that are not in agreement with the results of N-body simulations, for a descendant halo of small mass $M_0=0.01$. Instead the results of SC seem to be satisfactory. For larger masses the agreement between EC model and N-body results becomes better. For large haloes, $M_0=100$, EC model approximates N-body simulations better than SC model. All the results of Fig.7 have been calculated for $\Delta\omega=0.1$. Fist column shows results for $z_d=0$ while the second one for $z_d=1$.\ All curves in Fig.8 have been predicted for $\Delta\omega =0.025$. Three rows correspond to $M_0=0.1, M_0=1$ and $M_0=100$, respectively. As in Fig.7, different lines represent different models. Thick dashes show the results of SC model, solid line the results of EC model and squares the results of the fitting formula give by Eq. (13).\ A more detailed comparison between the results of EPS and N-body simulations is given in Fig.9. We calculated the relative difference $(R_{EPS} - R_{NB})/R_{NB}$ where $R_{EPS}$ and $R_{NB}$ are the merger rates predicted by the EPS and by N-body results, respectively. The results presented in this Fig. can be summarized as follows:\ For low redshifts ($z=0$ to $z=1$), merger rates of haloes with descendant mass in the range $10^{10}M_{\odot}\mathrm{h}^{-1}$ to $10^{12}M_{\odot}\mathrm{h}^{-1}$ derived by the SC model fit very satisfactory the results of N-body simulations. For example, for $z_d=0$ and $\Delta\omega=1$ the difference is less than $15$ percent, except for some very small values of $x$. Instead, for the same range of masses and redshifts, merger rates derived by the EC are significantly lower than those predicted by N-body simulations.\ For the above range of redshifts and for haloes of mass $10^{14}M_{\odot}\mathrm{h}^{-1}$ the fits by EC model are very satisfactory (in general the relative difference is smaller than $20$ percent) while the results of SC are significantly higher than those of N-body simulations.\ For a higher redshift ($z=3$) both SC and EC model overestimate the merger rate of large haloes. Merger rates of smaller haloes are overestimated by the SC model and underestimated by the EC model. The above conclusion seems not to depend, at least significantly, on the values of redshift $z_d$ and time-step $\Delta\omega$.\ We have to note here that both N-body simulations and analytical methods have problems in describing very accurately some physical properties of dark matter haloes. This is due to either technical difficulties or to the fact that some physical mechanisms are not taken into account. For example, in a recent paper @fama10 use the results of the Millenium - II simulation, [@boy09], to derive a formula of the same form of that given in Eq.(13). Millenium - II simulation has better resolution than Millenium, [@spet05], simulation. Due to the better resolution, the best fitting values of the parameters in Eq.(13) are changed. For example, the value of $a_1$, that is the exponent of the mass of the halo, from $0.083$ becomes now $0.133$. Obviously the dependence of mass remains weak but such a change in the value of $a_1$ results, for a halo with $M_d=100$, to a new merger rate that is $26\%$ larger. This percentage is too large since it can change the whole picture, at least for large haloes, resulting from our comparison. Additionally, it is interesting to notice Fig. A1 in the appendix of the above paper. It describes merger rates given by five different algorithms. These algorithms are used to analyze the results of the same simulation and to study fragmentation effects in FOF (friends of friends) merger trees. From this Fig. it is clear that differences due the use of different algorithms may be larger than the differences between analytical methods and N-body simulations derived by our study and shown in our Fig.9.\ From the above discussion it is clear that the results of N-body simulations are very sensitive not only to the resolution but also to the halo finding algorithm. This sensitivity can lead to completely different results. The following example is very characteristic: @bet07 studied, among other things, the value of the spin parameter as a function of the mass of dark matter haloes. They found that the FOF algorithm results to a spin parameter that is an increasing function of mass while a more advanced halo finding algorithm, that they been proposed, results to a spin parameter that is a decreasing function of mass! On the other hand, N-body simulations have the ability to deal with complex physical process. For example the destruction of dark matter haloes as well as the the role of the environment are factors that are not taken into account in most of the analytical methods. This is an additional reason for the presence of differences between the results.\ Summarizing our results we could say that: SC approximates better the merger rates of small haloes while EC the merger rates of heavy haloes. This is obviously an interesting information, but since it has been resulted from a specific solution for the problem of the distribution of progenitors, a further study of different solutions is required. The finding of a solution that approximates satisfactory merger rates from N-body simulations, independently on the redshift and mass should be an important achievement. Such a trial requires future comparisons and obviously improvements on both kind of methods.\ Acknowledgements ================  We acknowledge K. Konte and G. Kospentaris for assistance in manuscript preparation and the *Empirikion* Foundation for financial support. 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--- author: - | Zhijie Deng, Yucen Luo, Jun Zhu[^1]\ Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Lab, THBI Lab, Tsinghua University\ [{dzj17, luoyc15}@mails.tsinghua.edu.cn, [email protected]]{} title: | Supplementary Material:\ Cluster Alignment with a Teacher for Unsupervised Domain Adaptation --- Class-conditional cluster structure on more tasks ================================================= First, we visualize the learned feature spaces of CAT, RevGrad \[7\] and MSTN \[49\] on the imbalanced *SVHN* to *MNIST* task using t-SNE \[27\], as shown in Fig.\[afig:1\]. It is obvious that CAT can force the samples from the same class to concentrate together to form tighter clusters than those of RevGrad and MSTN, and the clusters present strip pattern in the 2-D space. CAT can also align the class-conditional distributions of the source and the target domains correctly. However, RevGrad and MSTN tend to align the ‘0’ images in *SVHN* with the ‘1’ images in *MNIST*, thus the learned feature spaces of them are confusing and not discriminative. This visualization verifies the results in Table. 1. Second, we plot the feature spaces learned by CAT+rRevGrad and RevGrad on *MNIST* to *USPS* and *USPS* to *MNIST* tasks in Fig. \[afig:2\] using t-SNE \[27\]. CAT+rRevGrad can deliver more discriminative feature spaces with separable and tight class-conditional clusters. Therefore, it is sufficient to use the first-order statistics based matching loss $\mathcal{L}_a$ to match the class-conditional distributions of the two domains. The aligned clusters of the source and the target domains also verify the effectiveness of the loss $\mathcal{L}_a$. Furthermore, we examine the feature space learned by CAT on more challenging tasks in *Office-31* dataset and *ImageCLEF-DA* dataset, and results are demonstrated in Fig. \[afig:5\]. These features are outputs of AlexNet trained with rRevGrad+CAT. The class-conditional distributions are shaped to be tight and separable clusters, and the corresponding cluters from the source domain and the target domain are aligned. Therefore, CAT can achieve the objectives of discriminative learning and class-conditional alignment, thus can perform well on the extensive experiments on *Office-31* and *ImageCLEF-DA* datasets. [0.2]{} ![(Best viewed in color.) Feature space learned on imbalanced *SVHN* to *MNIST* task. Green, red, blue and orange points represent ‘0’ images from *SVHN*, ‘1’ images from *SVHN*, ‘0’ images from *MNIST* and ‘1’ images from *MNIST*, respectively.](catim.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on imbalanced *SVHN* to *MNIST* task. Green, red, blue and orange points represent ‘0’ images from *SVHN*, ‘1’ images from *SVHN*, ‘0’ images from *MNIST* and ‘1’ images from *MNIST*, respectively.](rgim.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on imbalanced *SVHN* to *MNIST* task. Green, red, blue and orange points represent ‘0’ images from *SVHN*, ‘1’ images from *SVHN*, ‘0’ images from *MNIST* and ‘1’ images from *MNIST*, respectively.](mstnim.pdf "fig:"){width="\linewidth"} \[afig:1\] [0.2]{} ![(Best viewed in color.) Feature space learned on *MNIST* to *USPS* (Fig. \[afig:2-1\] and Fig. \[afig:2-2\]) and *USPS* to *MNIST* (Fig. \[afig:3-1\] and Fig. \[afig:3-2\]) tasks. Blue violet denotes the source domain and the other colors denote different classes of target domain.[]{data-label="afig:2"}](m2urg.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on *MNIST* to *USPS* (Fig. \[afig:2-1\] and Fig. \[afig:2-2\]) and *USPS* to *MNIST* (Fig. \[afig:3-1\] and Fig. \[afig:3-2\]) tasks. Blue violet denotes the source domain and the other colors denote different classes of target domain.[]{data-label="afig:2"}](m2ucat.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on *MNIST* to *USPS* (Fig. \[afig:2-1\] and Fig. \[afig:2-2\]) and *USPS* to *MNIST* (Fig. \[afig:3-1\] and Fig. \[afig:3-2\]) tasks. Blue violet denotes the source domain and the other colors denote different classes of target domain.[]{data-label="afig:2"}](u2mrg.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on *MNIST* to *USPS* (Fig. \[afig:2-1\] and Fig. \[afig:2-2\]) and *USPS* to *MNIST* (Fig. \[afig:3-1\] and Fig. \[afig:3-2\]) tasks. Blue violet denotes the source domain and the other colors denote different classes of target domain.[]{data-label="afig:2"}](u2mcat.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on four challenging tasks. Blue violet (in (a) and (b)) and deep sky blue (in (c) and (d)) denote the source domain and the other colors denote different classes of target domain.](a2wcat.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on four challenging tasks. Blue violet (in (a) and (b)) and deep sky blue (in (c) and (d)) denote the source domain and the other colors denote different classes of target domain.](a2dcat.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on four challenging tasks. Blue violet (in (a) and (b)) and deep sky blue (in (c) and (d)) denote the source domain and the other colors denote different classes of target domain.](p2icat.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space learned on four challenging tasks. Blue violet (in (a) and (b)) and deep sky blue (in (c) and (d)) denote the source domain and the other colors denote different classes of target domain.](i2pcat.pdf "fig:"){width="\linewidth"} \[afig:5\] [0.22]{} ![Jensen-Shannon divergence (JSD) curves during training.[]{data-label="afig:4"}](latex/d2ajsd.pdf "fig:"){width="\linewidth"} [0.22]{} ![Jensen-Shannon divergence (JSD) curves during training.[]{data-label="afig:4"}](latex/a2djsd.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space in the early stages of training. **Different** from the above feature spaces, blue violet (in (a) and (b)) and deep sky blue (in (c)) denote the **target** domain and the other colors denote different classes of **source** domain.[]{data-label="afig:7"}](s2mearly.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space in the early stages of training. **Different** from the above feature spaces, blue violet (in (a) and (b)) and deep sky blue (in (c)) denote the **target** domain and the other colors denote different classes of **source** domain.[]{data-label="afig:7"}](a2dearly.pdf "fig:"){width="\linewidth"} [0.2]{} ![(Best viewed in color.) Feature space in the early stages of training. **Different** from the above feature spaces, blue violet (in (a) and (b)) and deep sky blue (in (c)) denote the **target** domain and the other colors denote different classes of **source** domain.[]{data-label="afig:7"}](p2iearly.pdf "fig:"){width="\linewidth"} [0.2]{} ![The selection rate of the confidence-thresholding technique on different tasks.[]{data-label="afig:6"}](s2mcr.pdf "fig:"){width="\linewidth"} [0.2]{} ![The selection rate of the confidence-thresholding technique on different tasks.[]{data-label="afig:6"}](a2dcr.pdf "fig:"){width="\linewidth"} [0.2]{} ![The selection rate of the confidence-thresholding technique on different tasks.[]{data-label="afig:6"}](p2icr.pdf "fig:"){width="\linewidth"} Quantitative estimate of the divergence between domains ======================================================= When aligning the source domain and target domain via the combination of RevGrad and CAT, the loss $\mathcal{L}_d$ which is maximized w.r.t. the critic $c$ can be viewed as a lower bound of $2JSD(s, t)-2\log2$ (see \[9\] for the details) where $JSD$ denotes the Jensen-Shannon divergence between distributions. Therefore, we plot $\frac{1}{2}\mathcal{L}_d+\log2$ to quantitatively estimate the divergence between the two domains, following \[49\]. The results are shown in Fig. \[afig:4\] and we use the AlexNet as the classifier here. CAT can boost RevGrad significantly, leading to faster and better convergence. This group of experiments verifies that when combining CAT with the marginal distribution alignment approaches, it can provide a discriminative class-conditional alignment and bias the existing approaches to align the cluster-structure marginal distributions better. Verification of confidence-thresholding technique ================================================= Since the source classification loss and the source discriminative clustering loss can produce strong gradients and converge quickly, the discriminative cluster structure will form in the source domain in the early stages of training. However, the classifier has not been adapted for the target domain, so a notable part of the target features will lie in the gaps between the source clusters and have low classification confidence. Therefore, the marginal alignment approaches may easily map these features into incorrect clusters, as stated in Sec. 3.2.3. To address this problem, we propose the confidence-thresholding technique which includes the fine-level structure information into marginal alignment approaches. We claim that in the training procedure, the discriminative class-conditional alignment between the two domains forms gradually, so more and more samples are going to be selected into the marginal alignment training. Here we prove these through experiments on tasks in *SVHN-MNIST-USPS*, *Office-31* and *ImageCLEF-DA*. At first, we train the RevGrad+CAT models on the three tasks with limited iterations (, 2000 iterations on *SVHN* to *MNIST* task, 100 iterations on Amazon to DSLR task and 100 iterations on p to i task) and plot the feature spaces of them in Fig \[afig:7\]. Obviously, a notable part of target samples lie in the gaps between the source clusters, especially on the *SVHN* to *MNIST* task which has large source and target domains. Then, we train the rRevGrad+CAT models on these tasks following the same settings, and we plot the selection rate of the confidence-thresholding technique w.r.t. the number of iterations in Fig. \[afig:6\]. When using this technique, we note that the selection rate monotonically increases with the number of iterations and after several thousands of iterations, the selection rate will be almost $100\%$ on the Amazon to DSLR and p to i tasks. On *SVHN* to *MNIST* task, we use a ramp-up function $exp(-10*(1 - min(\frac{ite-5000}{10000}, 1.)))$ as $\alpha$ after 5000 iterations, suggested by related SSL works. Therefore, after around 15000 iterations, the discriminative clustering structure forms, and then the samples are pushed far away from the decision boundaries. So almost all the samples will have confidence more than $p$ and will be selected into the domain adversarial training. Convergence =========== To inspect how CAT converges, we plot the test accuracy with respect to the number of iterations in Fig. \[fig:4\]. On the two adaptation tasks using AlexNet, CAT shows similar convergence rate with RevGrad \[7\] but better performance. [0.22]{} ![Test accuracy curves.](d2a.pdf "fig:"){width="\linewidth"} [0.22]{} ![Test accuracy curves.](a2d.pdf "fig:"){width="\linewidth"} \[fig:4\] Experimental details ==================== On digits adaptation tasks, we use the simple LeNet with Batch Normalization after the convolutional layers and use the probability logits as features for adaptation, following \[49, 44\]. When combining with RevGrad \[7\] and rRevGrad, the critic model has a $10\rightarrow500\rightarrow500\rightarrow1$ architecture. On more challenging tasks, we conduct experiments based on the AlexNet \[15\] and ResNet-50 \[12\] equipped with 256-D bottleneck layers after the $fc7$ and $pool5$ layers respectively (following \[24, 49\]). We use the features outputted by the bottleneck layers as image representations for adaptation and use a three-layer critic with $256\rightarrow1024\rightarrow1024\rightarrow1$ architecture. We finetune all the layers before the bottleneck layers in AlexNet and ResNet-50 and train the bottleneck layers and the classification layers via back propagation. We use the stochastic gradient descent with 0.9 momentum with an annealed learning rate $\mu_p=\frac{0.01}{(1+10p)^{0.75}}$ where p changes from 0 to 1 in the training progress \[7, 49\] when using LeNet and AlexNet as the classifiers. The learning rate for finetuned layers is set to be the ten percent of that for layers trained from scratch. We use batches with 128 elements in experiments using LeNet, batches with 200 elements in experiments using AlexNet and batches with 36 elements in experiments using ResNet-50. We use the same architectures and optimization settings (, batch size, learning rate, optimizer and weight decay) as those of the original methods \[41, 37\] when combining CAT with them. The pseudo labels are not initialized randomly. Specifically, in the first 5000 iterations, we pre-train CAT by setting $\alpha = 0$. During this, the classifier is trained to fit source data but won’t overfit, thus its implicit ensemble can perform well on some target samples and provide a reliable initial set of pseudo labels. Then, we ramp-up α to activate the clustering and alignment losses to impose conditional alignment. [^1]: Corresponding author.
--- abstract: 'A novel way to realize a $\pi$ Josephson junction is proposed, based on a weak link in an unconventional $d$-wave superconductor with appropriately chosen boundary geometry. The critical current of such a junction is calculated from a fully selfconsistent solution of microscopic Eilenberger theory of superconductivity. The results clearly show, that a transition to a $\pi$ Josephson junction occurs for both low temperatures and small sizes of the geometry.' author: - 'A. Gumann' - 'C. Iniotakis' - 'N. Schopohl' date: 'August 29, 2007' title: 'Geometric $\pi$ Josephson junction in $d$-wave superconducting thin films' --- Josephson junctions with an intrinsic phase shift of $\pi$ ($\pi$ Josephson junctions) open up promising possibilities in superconducting electronics. Including them in a closed superconducting loop allows to create a degenerate current ground state. If they are combined with standard Josephson junctions, complementary superconducting quantum interference devices (SQUIDs) with characteristics $I_c(\Phi)$ and $V_{dc}(\Phi)$ shifted by $\Phi_0/2$ with respect to the standard SQUID can be realized [@Ter01] ($\Phi_0=h/2e$ is the flux quantum). These devices permit various improvements in rapid single flux quantum logic (RSFQ, [@Lik01]). Using $\pi$ Josephson junctions, complementary logic devices can be realized without the need for additional current bias lines, with improved device symmetry and enhanced operation margins [@Ort01]. At the same time, the size of the logic cells can be significantly reduced [@Ust01]. The first proposal for a $\pi$ Josephson junction was based on a tunnel junction with magnetic impurities [@Bul01]. Experimental realizations in the form of superconductor-ferromagnet-superconductor [@Rya01; @Bau01] or superconductor-isolator-ferromagnet-superconductor [@Kon01; @Blu01; @Wei02] mulitlayered systems have been presented. A second realization of $\pi$ Josephson junctions makes use of the $d_{x^2-y^2}$ ($d$-wave) pairing symmetry of the cuprates via grain boundaries intersecting domains with different orientation of the crystal lattice [@Tsu01; @Bar01; @Schu01; @Il02]. Another method exploits the pairing symmetry of the cuprates by combining high-T$_c$ and low-T$_c$ materials [@Wol01; @Bra01; @Smi01]. Furthermore, $\pi$ Josephson junctions have been realized in superconductor-normal conductor-superconductor structures with a nonequilibrium energy distribution of the current-carrying states in the normal region [@Bas01]. In this letter, we propose a novel realization of a $\pi$ Josephson junction, which is solely based on the boundary geometry of a $c$-axis oriented $d$-wave superconductor thin film. We consider an epitactic film of the superconducting material exhibiting a weak link of width $w$ as displayed in Fig. \[cap:geometry\]. The geometry of this weak link consists of a straight line on one side and a wedge-shaped incision of angle $2\beta$ on the other. We point out that the weak link defined by this geometry is a $\pi$ Josephson junction if (1) the crystal orientation is appropriately chosen and (2) the residual width $w$ is sufficiently small. The intrinsic phase shift of $\pi$ following from the proposed geometry is a direct consequence of the unconventional $d$-wave pairing symmetry. If the orientation angle between $d$-wave and geometry is chosen to be $\alpha=\pi/4$ as indicated in Fig. \[cap:geometry\], quasiparticles travelling through the constriction, which get reflected at the straight boundary line (opposite to the wedge), simultaneously suffer a sign change of the pairing potential. Thus, they generate zero energy Andreev bound states [@Hu01] in the junction associated with a phase shift of $\pi$. In contrast, quasiparticles passing through the constriction without such a reflection do not contribute to the zero energy Andreev bound states. If the residual width $w$ is sufficiently small, however, the contribution of the reflected quasiparticles dominates the total current across the junction resulting in a $\pi$ Josephson junction behaviour. ![\[cap:geometry\](Color online) Geometry of the $\pi$ Josephson junction based on a thin film of a $d$-wave superconductor.](fig1){width="0.99\columnwidth"} In the following, we calculate the critical current $I_c$ of the junction according to Fig. \[cap:geometry\], as a function of width $w$ and temperature $T$. For this purpose, we solve the Eilenberger equations of superconductivity [@Eilenberger; @Larkin] to take into account the effect of Andreev bound states quantitatively. We assume a cylindrical Fermi surface of the superconductor, which is aligned parallel to the $z$ axis and parametrized by the polar angle $\theta$. Accordingly, the Fermi velocity is given by $\mathbf{v}_F=v_F(\hat{\mathbf{x}}\cos{\theta}+\hat{\mathbf{y}}\sin{\theta})$. The pairing potential in the superconductor may be factorized as $\Delta(\mathbf{r},\hat{\mathbf{k}})= \psi(\mathbf{r}) \chi(\hat{\mathbf{k}})$ with $\chi(\hat{\mathbf{k}})=\cos(2\theta-2\alpha)$ representing the $d$-wave symmetry. Then, the selfconsistency equation for the pairing potential according to Eilenberger theory is given by $$\label{eq:gapeq} \psi(\mathbf{r})= 2\pi N(0) V k_B T\sum_{\varepsilon_n>0}^{\omega_c}\big<\chi(\hat{\mathbf{k}})f(\mathbf{r},\hat{\mathbf{k}},i\varepsilon_n)\big>_{\theta},$$ and the current density can be computed from $$\label{eq:curreq} \mathbf{j}(\mathbf{r})= 4\pi e N(0) k_B T\sum_{\varepsilon_n>0}^{\omega_c}\big<\mathbf{v}_F\cdot g(\mathbf{r},\hat{\mathbf{k}},i\varepsilon_n)\big>_{\theta}.$$ In these equations $N(0)$ is the normal density of states at the Fermi surface, $V$ is the coupling constant, $\varepsilon_n=(2n+1)\pi k_B T$ are Matsubara frequencies, and $\langle ... \rangle_\theta$ denotes Fermi surface averaging. The propagators $f$ and $g$ in the integrands of Eqs. (\[eq:gapeq\]) and (\[eq:curreq\]), respectively, can easily be calculated using the Riccati parametrization [@Schop02; @Schop01]. We numerically solve the selfconsistency Eq. (\[eq:gapeq\]) in the $xy$ plane for a two-dimensional section of the superconducting thin film enclosing the constriction shown in Fig. \[cap:geometry\]. This section has an area of more than $12.5\xi_0\times12.5\xi_0$, and a grid width of about $0.15\xi_0$ is used ($\xi_0=\hbar v_F/\pi\Delta(T\!=\!0)$ is the coherence length). All the boundaries of the geometry are assumed to be impenetrable, leading to specular reflection conditions which are incorporated appropriately [@She01]. The left and right sides are considered to be open ends, exhibiting a fixed phase difference $\Delta\phi$ between them as a constraint. Details of the iterative numerical procedure which we use in order to find the selfconsistent solution can be found in Ref. [@Gum01]. In order to find the critical current, these solutions have to be calculated for a large number of phase differences. In a second step, the current Eq. (\[eq:curreq\]) is used to access the whole current-phase relation of the junction. Finally, the critical current $I_c$ is extracted as the absolute maximum of the current-phase relation. ![\[cap:Ic0bT\](Color online) Critical current $I_c(w,T)$ for $\alpha=\pi/4$ and $\beta=0$. The two plots (a) and (b) correspond to two experimental situations, which can be thought of to verify the $0$-$\pi$-transition, i.e. by variation of $T$ or $w$, respectively. The lines are guides for the eye.](fig2){width="0.99\columnwidth"} In Fig. \[cap:Ic0bT\], we present our results for the critical current $I_c(w,T)$ of the weak link for $\alpha=\pi/4$ and $\beta=0$. Here, $T_c$ is the transition temperature and $d$ is the thin film thickness. In Fig. \[cap:Ic0bT\](a), the data is plotted for fixed width $w$ over temperature $T$, whereas, in Fig. \[cap:Ic0bT\](b), the same data is plotted for fixed temperature $T$ over width $w$. The results in Fig. \[cap:Ic0bT\](a) show that the $\pi$ state (indicated by a negative critical current) is predominantly entered at low temperatures. For very small values of $w$ however, the $\pi$ state survives up to the highest temperatures that we considered ($0.9\,T_c$), featuring small absolute values of the critical current. From the results in Fig. \[cap:Ic0bT\](b), we find that at $T=0.1\,T_c$, the $0$-$\pi$-transition occurs at about $b \simeq 4\xi_0$. With increased temperature, the critical width of the $0$-$\pi$-transition is shifted to smaller values of $w$. For the geometry $\alpha=\pi/4$ and $\beta=\pi/4$, we find similar results with a $0$-$\pi$-transition at slightly smaller values of $w$ (not shown here). This can easily be understood since only quasiparticles from a reduced angular interval contribute to the $\pi$ state. Nevertheless, the occurence of the $\pi$ state only weakly depends on the angle of the wedge $\beta$. Based on our results for the critical current $I_c(w,T)$, we expect the experimental realization of the proposed Josephson device to be challenging, but feasible. The size of the coherence length of the superconducting material directly corresponds to the necessary size of the structures. As stated above, the occurence of the $\pi$ state of the proposed Josephson junction hardly depends on the angle of the wedge $\beta$. Furthermore, previous studies indicate that the occurence of surface Andreev bound states in $d$-wave superconductors is not suppressed by microscopic surface roughness [@Ini01]. Accordingly, also the $\pi$ state of the Josephson junction proposed here should exhibit some robustness regarding surface roughness. ![\[cap:squidsqif\](Color online) In (a), a combination of two of the proposed Josephson devices to a SQUID is shown. Since the widths of the two junctions $w_1$ and $w_2$ can be chosen independently, a $0$-$0$-, $0$-$\pi$- or $\pi$-$\pi$-SQUID is viable. In (b), a larger number of the proposed Josephson devices is combined to a SQIF (6 junctions shown, more indicated by the black dots).](fig3){width="0.99\columnwidth"} The proposed realization of $\pi$ Josephson junctions based on the boundary geometry of $d$-wave superconducting thin films opens up interesting possibilities for application. It allows for the fabrication of $0$ and $\pi$ Josephson junctions with similar characteristics in the same process. The material can be epitactic thin films of any superconductor with $d$-wave pairing symmetry. A cuprate high temperature superconductor can be employed as well as a $d$-wave heavy-fermion superconductor like CeCoIn$_5$ [@Mat01]. Because of the simple planar geometry, the combination of two of the proposed Josephson devices to $0$-$0$-, $0$-$\pi$- or $\pi$-$\pi$-SQUIDs is straightforward. An example for a possible geometry of such a device is shown in Fig. \[cap:squidsqif\](a). Furthermore, the application of a large number of the proposed Josephson devices for superconducting quantum interference filters (SQIFs, [@Schu02]) containing both $0$-$0$- and $0$-$\pi$-SQUID loops offers new possibilities in the synthesis of the voltage response of such a device. A possible geometry for a serial SQIF is shown in Fig. \[cap:squidsqif\](b). Depending on temperature $T$ and the width of the junction $w$, the critical current of the proposed Josephson device can be comparatively small. This may pose a problem in some applications, but can be advantageous in others. In the case of a single SQUID, a small critical current implies a small value of the $\beta_L$ parameter. Thus, SQUIDs with larger dimensions can be designed, leading to an increased sensitivity. In the case of a serial SQIF, whose total voltage response is the sum of the voltage outputs of a large number of single SQUIDs, lower noise figures are expected. In conclusion, we proposed a new way to realize a $\pi$ Josephson junction, just consisting of a single layer of a $d$-wave superconductor thin film with appropriately chosen boundary geometry. A fabrication of this geometric $\pi$ Josephson device is in the reach of modern fabrication technology. This method may allow to create more evolved superconducting circuits containing both normal and $\pi$ Josephson junctions closely packed on a single substrate. [10]{} E. Terzioglu, M. R. Beasley, IEEE Trans. Appl. Supercond. **8**, 48 (1998). K. K. Likharev, V. K. Semenov, IEEE Trans. Appl. Supercond. **1**, 3 (1991). 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--- abstract: 'Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterise uncertainty in model inputs and how that propagates through to outputs or predictions. In this perspective piece we draw attention to an important and under-addressed source of uncertainty in our predictions — that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed *model discrepancy*, and we are often uncertain as to the size and consequences of this discrepancy. Here we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes (GPs) and autoregressive-moving-average (ARMA) models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided.' author: - | Chon Lok Lei$^{1}$, Sanmitra Ghosh$^{2}$, Dominic G. Whittaker$^{3}$, Yasser Aboelkassem$^{4}$,\ Kylie A. Beattie$^{5}$, Chris D. Cantwell$^{6}$, Tammo Delhaas$^{7}$, Charles Houston$^{6}$,\ Gustavo Montes Novaes$^{8}$, Alexander V. Panfilov$^{9,10}$, Pras Pathmanathan$^{11}$,\ Marina Riabiz$^{12}$, Rodrigo Weber dos Santos$^{8}$, Keith Worden$^{13}$,\ Gary R. Mirams$^{3}$ and Richard D. Wilkinson$^{14}$ date: title: Considering discrepancy when calibrating a mechanistic electrophysiology model --- [**Keywords:** Model discrepancy, Uncertainty quantification, Cardiac model, Bayesian inference]{} Introduction ============ This perspective paper discusses the issue of model discrepancy — the difference between a model’s equations and reality, even with the best possible set of values given for parameters within these equations. The concepts and issues we highlight are widely applicable to any modelling where governing equations are approximations or assumptions; thus our perspective paper is intended for computational, mathematical and statistical modellers within many other fields as well as within and outside biological modelling. The focus of the examples is in cellular cardiac electrophysiology, a well-developed area of systems biology [@noble_how_2012]. Cardiac modelling ----------------- Generally speaking, cardiac models are sets of mathematical functions governed by continuous sets of ordinary and/or partial (when spatial dimensions are considered) differential equations, integrated using computational discrete techniques which produce responses that depend on the model inputs. Inputs can include model parameters, initial conditions, boundary conditions and cellular, tissue or whole organ geometrical aspects. Such inputs are often calibrated using experimental data, and those which have physiological meaning can sometimes be obtained by direct measurement. Cardiac mathematical modelling and computational simulations have been remarkably successful tools that have provided insights into cardiac physiological mechanisms at cellular, tissue and whole organ scales. In the majority of these quantitative efforts, models are derived based on simplified representations of complex biophysical systems and use [*in vitro*]{} and [*in vivo*]{} experimental data for calibration and validation purposes. Quantitative cardiac models have been an essential tool not only for basic research, but have been proposed for transition into clinical and safety-critical applications [@Relan2011; @Sermesant2012; @Mirams2012; @Nied2018; @Li2018]. The translation of cardiac mathematical models for such applications will require high levels of credibility of predictive model outputs. One important phase of cardiac model credibility assessment is the study of uncertainty. Parameters in cardiac models are often uncertain, mainly due to measurement uncertainty and/or natural physiological variability [@Mirams2016]. Thus, uncertainty quantification (UQ) methods are required to study uncertainty propagation in these models and help to establish confidence in model predictions. Parametric UQ is the process of determining the uncertainty in model inputs or parameters, and then estimating the resultant uncertainty in model outputs. This tests the robustness of model predictions given our uncertainty in their inputs. There has been increasing recent interest and research into UQ of cardiac models to outline their predictive capability and credibility [@Nied2011; @Krish2014; @Pras2013; @Pras2014; @Pras2018; @Pras2019]. However, another major source of uncertainty in modelling is uncertainty in the model structure itself. There is always a difference between the imperfect model used to approximate reality, and reality itself; this difference is termed model discrepancy. What has received little attention in this field (and mathematical/systems biology more generally) is assessment of robustness of model predictions given our uncertainty in the model structure, and methods to characterise model discrepancy. We have found only one published explicit treatment of discrepancy in cardiac electrophysiology models, in papers by Plumlee *et al.* [@Plumlee2016; @Plumlee2017]. In these studies, the assumption of ion channel rate equations following an explicit form (such as that given, as we will see later, by Eq. (\[eq:rate\_equation\])) was relaxed, and rates were allowed to be Gaussian processes (GPs) in voltage. A two-dimensional GP (in time and voltage) was then also added to the current prediction to represent discrepancy in current for a single step to any fixed voltage. Notation and terminology ------------------------ Before discussing model discrepancy in detail, we introduce some notation and terminology. As the concepts introduced here are intended to be understood not just by a cardiac modelling audience, we provide a non-exhaustive list of terminology we have encountered in different fields to describe useful concepts relating to calibration and model discrepancy (and mathematical/computational modelling in general) in Table \[tab:terminology\]. We here delve into some of those concepts in more detail. Suppose a physiological system is modelled as $y = f(\boldsymbol{\theta}, u)$, where $f$ represents all governing equations used to model the system (also referred to as model form or model structure), $\boldsymbol{\theta}$ is a vector of parameters characterising the system, and $u$ are known externally applied conditions or control variables applied in the particular experimental procedure. In a cardiac modelling context, these might represent a stimulus protocol, a drug concentration, or the applied voltage protocol in a simulated voltage clamp experiment. In general, $\boldsymbol{\theta} = \{\boldsymbol{\theta}_D, \boldsymbol{\theta}_C\}$, where values of $\boldsymbol{\theta}_D$ are directly measured (or determined using a different model, as is often the case when inheriting already-parameterised equations for particular currents into cardiac action potential models [@niederer2009meta]) and are not determined using the model $f$, and where values of $\boldsymbol{\theta}_C$ are determined by calibration using the model $f$. Here, for simplicity of exposition, we assume $\boldsymbol{\theta}_D$ is fixed (and known) and $\boldsymbol{\theta} = \boldsymbol{\theta}_C$. We can distinguish between external conditions used for calibration, validation, and prediction (that is, the application of the model, or context of use (CoU)), $u_C$, $u_V$, $u_{CoU}$, say. Suppose we have experimental data $Y_C$ for calibration and $Y_V$ for validation. A typical workflow, without UQ, is: - **Calibration**: find $\boldsymbol{\theta}^* = \textrm{argmin}_{\boldsymbol{\theta} \in \boldsymbol{\Theta}} d_C \left( L(f(\boldsymbol{\theta}, u_C)), L(Y_C) \right)$, using some calibration metric $d_C(\cdot,\cdot)$ (e.g. it can be as simple as a vector norm: $d_C(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\|$), some postprocessing operator $L$, and some subset of parameter space $\boldsymbol{\Theta}$; - **Validation**: compare $y_V = f(\boldsymbol{\theta}^*, u_V)$ against $Y_V$, either qualitatively or using a suitable validation metric $d_V(f(\boldsymbol{\theta}^*, u_V), Y_V)$; - **Context of use**: compute $y_{CoU} = f(\boldsymbol{\theta}^*,u_{CoU})$, or some quantity derived from this, to learn about the system or to make a model-based decision. The calibration stage has many different names, see Table \[tab:terminology\] for different terms. Consideration of uncertainty in the parameters (parametric UQ) introduces additional complexity. There are various possibilities that should be distinguished. One potential source of uncertainty in the ‘correct’ value of $\boldsymbol{\theta}$ is measurement error in $Y_C$. Computing the uncertainty about $\boldsymbol{\theta}$ based on measurement error in $Y_C$ is referred to as ‘inverse UQ’, and requires a model of the experimental error to be specified. For example, a common choice is to assume independent zero-mean Gaussian measurement error which is the same on all data points, in which case (neglecting model discrepancy; see later) our model for the $i$^th^ data point ($i$^th^ realisation of the vector $Y_C$) is: $$(Y_{C})_i = f_i(\boldsymbol{\theta}, u_C) + \epsilon_i, \label{eq:iid-noise}$$ where $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$. There are many methods available to solve inverse UQ problems, which we cannot review in any detail here, but most are based on inferring or approximating a probability distribution of possible parameter sets that would be in agreement with the available data. Though a number of methods to solve inverse UQ problems have been applied in cardiac electrophysiology [@Mirams2016], the most common is a Bayesian approach, which combines prior information about the parameters, $\pi(\boldsymbol{\theta})$, with the likelihood of the data given each parameter $\pi(Y_C \mid \boldsymbol{\theta})$, to find the posterior distribution over the parameters: $$\pi(\boldsymbol{\theta} \mid Y_{C}) = \frac{\pi(Y_C \mid \boldsymbol{\theta})\pi(\boldsymbol{\theta})}{\pi(Y_C)}.\label{eq:Gaussian}$$ For an introduction to Bayesian methods, see [@Gelman2013; @lambert2018BayesBook]. For the above i.i.d. Gaussian error model (Eq. (\[eq:iid-noise\])), the likelihood is given by $$\pi(Y_C \mid \boldsymbol{\theta}) =(2\pi \sigma^2)^{-n/2}\exp\left(-\frac{||Y_C - f(\boldsymbol{\theta}, u_C)||_2^2}{2\sigma^2}\right),\label{eq:ll}$$ where $||x||_2^2=\sum_i x_i^2$, and $n$ is the number of data points. A second potential source of uncertainty about $\boldsymbol{\theta}$ can occur when the parameter varies across the (or a) population. Estimating population variability in $\boldsymbol{\theta}$ requires multiple $Y_C$ recordings, $\{Y^{(1)}_C, Y^{(2)}_C, \ldots\}$. Multilevel or hierarchical models can then be used: we assume the parameters for population $i$ are drawn from some distribution $\boldsymbol{\theta}^{(i)} \sim \pi(\boldsymbol{\theta} \mid \psi)$, and infer the population parameters $\psi$, see [@lei_rapid_2019-1]. Once uncertainty in $\boldsymbol{\theta}^*$ has been determined, the impact of this uncertainty on validation simulations $y_V$ or CoU simulations $y_{CoU}$ can be computed by propagating the uncertainty through the model $f$ in the validation/CoU simulations. This is referred to as ‘uncertainty propagation’ or ‘forward UQ’. Uncertainty in $y_V$ helps provide a more informed comparison to $Y_V$ in the validation stage (especially if experimental error in $Y_V$ is also accounted for). Uncertainty in $y_{CoU}$ enables a more informed model-based decision-making process. Model discrepancy ----------------- UQ as outlined above does not account for the fact that the model is always an imperfect representation of reality, due to limited understanding of the true data-generating mechanism and perhaps also any premeditated abstraction of the system. The model discrepancy is the difference between the model and the ‘true’ data-generating mechanism, and its existence has implications for model selection, calibration and validation, and CoU simulations. For calibration, the existence of model discrepancy can change the meaning of the estimated parameters. If we fail to account for the model discrepancy in our inference, our parameter estimates, instead of being physically meaningful quantities, will have their meaning intimately tied to the model used to estimate them. The estimated parameter values depend on the chosen model form, and the uncertainty estimates obtained during inverse parameter UQ tell us nothing about where the ‘true’ value is. In other words, there is no guarantee the obtained $\boldsymbol{\theta}$ will match ‘true’ physiological values of any parameters which have a clear physiological meaning. We can try to restore meaning to the estimated parameters by including a term to represent the model discrepancy in our models. Validation in particular, provides an opportunity for us to identify possible model discrepancy. In fact, validation, rather than being considered as an activity for confirming a ‘model is correct’, is better considered as a method for estimating the model discrepancy. To maximise the likelihood that the validation can discern model discrepancy, the validation data should ideally be ‘far’ from the calibration data, and as close to the CoU as possible. A motivating example of discrepancy {#sect:motivation} =================================== To illustrate the concept of model discrepancy and some of its potential consequences we have created a cardiac example inspired by previous work [@Brynjarsdottir2014]. We will assume that the Ten Tusscher *et al.* ventricular myocyte electrophysiology model [@ten2004model] (Model T) represents the ground truth. We use Model T to generate some data traces in different situations: (i) the action potential under pacing; (ii) under 2 Hz pacing; and (iii) under 1 Hz pacing with 75% [$I_\text{Kr}$]{} block ($g_{Kr}$ is multiplied by a scaling factor of $0.25$). We split the data traces into calibration (i), validation (ii), and ‘context of use’ (CoU) (iii). We might construct a candidate model based on comparison with only the calibration and validation dataset, and then use it to predict the CoU situation. So we assume we do not know the ground truth and instead fit an alternative model, the Fink *et al.* model [@fink2008contributions] (Model F), to the synthetic data generated from Model T. Since both models were built for human ventricular cardiomyocytes, comparing these two models is an example which highlights the potential problems associated with fitting any such model under model discrepancy. Note, that Model F is a modification of Model T which improves the descriptions of repolarising currents (see Supplementary Section \[supp-sec:model-t-f-equations\]), especially of hERG (or [$I_\text{Kr}$]{}), as this channel is a major focus for Safety Pharmacology. ![\[fig:tutorial-action-potential-models\] A comparison of Ten Tusscher (Model T [@ten2004model], blue) and Fink (Model F [@fink2008contributions], green) kinetics. These currents are voltage clamp simulations under the same action potential clamp, shown on the top panels. Here only those currents with different kinetics are shown; the kinetics of $I_\text{Na}$, $I_\text{NaCa}$, and $I_\text{NaK}$ are identical in both models. Two of the gates in $I_\text{CaL}$ are identical in the two models, one gate has a different formulation, and Model F has one extra gate compared to Model T. The two models use different formulations for $I_\text{Kr}$, different parameterisations of the kinetics for $I_\text{Ks}$ and $I_\text{to}$, and different equations for $I_\text{K1}$ steady state. Currents are normalised in this plot by minimising the squared-difference between the two models’ currents such that we emphasise the differences in kinetics rather than the conductances (which are rescaled during the calibration). Only $I_\text{CaL}$ shows what we would typically consider to be a ‘large’ difference in kinetics, with the rest of the currents apparently being close matches between Model T and Model F. ](model-differences.pdf){width="\textwidth"} We use Model T to generate synthetic current clamp experiments by simulating the different protocols then adding i.i.d. Gaussian noise $\sim \mathcal{N}(0,\sigma^2)$ to the resulting voltage traces, with $\sigma$ chosen to be . We then fit eight maximal conductance/current density parameters for $I_\text{Na}$, $I_\text{CaL}$, $I_\text{Kr}$, $I_\text{Ks}$, $I_\text{to}$, $I_\text{NaCa}$, $I_\text{K1}$, and $I_\text{NaK}$ to the synthetic data. Such an exercise assumes that the parameters and equations describing the kinetics/gating of the ion currents do not need to be re-calibrated to this particular dataset and have been fitted previously (a common assumption in electrophysiology modelling [@Kaur2014; @Groenendaal2015; @Johnstone2016; @lei_tailoring_2017; @Pouranbarani2019]). Here, we know that this assumption is incorrect, as we purposefully introduced discrepancy between the current kinetics. A comparison of the differences in current kinetics between Models T and F is shown in Figure \[fig:tutorial-action-potential-models\], also see equations for each current in Supplementary Section \[supp-sec:model-t-f-equations\] for reference. Only five currents have kinetics that vary between the two models, and importantly no currents or compartments are missing (unlike when attempting to fit a model in reality). We will investigate whether we can trust the calibrated Model F to make good predictions in new situations. The code to reproduce the results in this example is available at <https://github.com/CardiacModelling/fickleheart-method-tutorials>. Model calibration ----------------- We calibrate the model using a train of five action potentials stimulated under a pacing protocol as the calibration data. Before attempting to do this fitting exercise, the appropriately sceptical reader might ask whether we are attempting to do something sensible. Will we get back information on all the parameters we want, or will we just find one good fit to the data amongst many equally plausible ones, indicating non-identifiability of parameters? To address these questions we first look at inferring the parameters of the original Model T. We use Eq. (\[eq:iid-noise\]) with Gaussian noise giving the likelihood in Eq. (\[eq:Gaussian\]), together with a uniform prior distribution from $0.1\times$ to $10\times$ the original parameters of Model T (additionally inferring the noise model parameter, $\sigma$). We perform both non-UQ based and UQ-based calibration: for the first, we use a global optimisation algorithm [@Hansen2006] to find the optimal model parameters; for the second, we generate samples to approximate the full posterior distribution using Markov chain Monte Carlo (MCMC). All inference is done using an open source Python package, PINTS [@Clerx2019Pints], and simulations are performed in Myokit [@clerx_myokit_2016]. The results are shown in Supplementary Figure \[supp-fig:tutorial-action-potential-no-discrepancy\]. This exercise results in a narrow plausible distribution of parameters very close to the ones that generated the data. Therefore, the ‘true’ model’s parameters are identifiable with the given data. Additionally, Supplementary Figure \[supp-fig:tutorial-action-potential-no-discrepancy\] shows that when using samples of these distributions to make predictions, all of the forward simulations are very closely grouped around the synthetic data for the [$I_\text{Kr}$]{} block CoU. We now move on to attempt the fitting exercise using Model F. The fitting result, the maximum a-posteriori probability (MAP) estimate, is shown in Figure \[fig:tutorial-action-potential-fitting-and-prediction\](Top). The agreement between the calibrated model output and the synthetic data would be considered excellent if these were real experimental data. Therefore, it would be tempting for modellers to conclude that this calibrated model were a good model, due to small model discrepancy. However, can we truly trust the predictive power of the model based on the result we see in Figure \[fig:tutorial-action-potential-fitting-and-prediction\](Top)? (fig) \[matrix of nodes\][ ![\[fig:tutorial-action-potential-fitting-and-prediction\] **Model F fitting and validation results**. (**Top**) The model is fitted to the ground truth synthetic data (generated from Model T), using a five action potential recording under a pacing protocol. The calibrated Model F (blue dashed line) shows an excellent fit to the data (grey solid line). (**Bottom**) Model F validation and prediction for a different context of use (CoU). (*Left*) The calibrated Model F can match some validation data ( pacing) very well, giving us a (false) confidence in the model performance. (*Right*) Notably, despite the calibrated Model F giving an excellent fit and validation, it gives a catastrophic (posterior sample) prediction for the $I_\text{Kr}$ block (CoU) experiments (suggesting in this example (*Left*) is not a good validation experiment!). The posterior predictions are model predictions simulated using the parameter samples from the posterior distribution (Figure \[fig:tutorial-action-potential-posterior\]); here, 200 samples/predictions are shown. ](fink-2008-stim1hz-stim1hz.pdf "fig:"){width="90.00000%"}\ ]{}; (fig-1-1.north) – (fig-1-1.north) node\[midway,above\][Calibration]{}; (fig) \[matrix of nodes\][ ![\[fig:tutorial-action-potential-fitting-and-prediction\] **Model F fitting and validation results**. (**Top**) The model is fitted to the ground truth synthetic data (generated from Model T), using a five action potential recording under a pacing protocol. The calibrated Model F (blue dashed line) shows an excellent fit to the data (grey solid line). (**Bottom**) Model F validation and prediction for a different context of use (CoU). (*Left*) The calibrated Model F can match some validation data ( pacing) very well, giving us a (false) confidence in the model performance. (*Right*) Notably, despite the calibrated Model F giving an excellent fit and validation, it gives a catastrophic (posterior sample) prediction for the $I_\text{Kr}$ block (CoU) experiments (suggesting in this example (*Left*) is not a good validation experiment!). The posterior predictions are model predictions simulated using the parameter samples from the posterior distribution (Figure \[fig:tutorial-action-potential-posterior\]); here, 200 samples/predictions are shown. ](fink-2008-stim1hz-stim2hz.pdf "fig:"){width="49.00000%"} & ![\[fig:tutorial-action-potential-fitting-and-prediction\] **Model F fitting and validation results**. (**Top**) The model is fitted to the ground truth synthetic data (generated from Model T), using a five action potential recording under a pacing protocol. The calibrated Model F (blue dashed line) shows an excellent fit to the data (grey solid line). (**Bottom**) Model F validation and prediction for a different context of use (CoU). (*Left*) The calibrated Model F can match some validation data ( pacing) very well, giving us a (false) confidence in the model performance. (*Right*) Notably, despite the calibrated Model F giving an excellent fit and validation, it gives a catastrophic (posterior sample) prediction for the $I_\text{Kr}$ block (CoU) experiments (suggesting in this example (*Left*) is not a good validation experiment!). The posterior predictions are model predictions simulated using the parameter samples from the posterior distribution (Figure \[fig:tutorial-action-potential-posterior\]); here, 200 samples/predictions are shown. ](fink-2008-stim1hz-hergblock-pp.png "fig:"){width="49.00000%"}\ ]{}; (fig-1-1.north) – (fig-1-1.north) node\[midway,above\][Validation?]{}; (fig-1-2.north) – (fig-1-2.north) node\[midway,above\][Context of Use]{}; Discrepant model predictions ---------------------------- Interestingly, the calibrated Model F gives an excellent validation prediction for a pacing protocol, as shown in Figure \[fig:tutorial-action-potential-fitting-and-prediction\](Left). Such rate-adaptation predictions are used commonly as validation evidence for action potential models. At this stage we might be very tempted to say we have a good model of this system’s electrophysiology. But if one now uses the model to predict the effect of drug-induced [$I_\text{Kr}$]{} block, the catastrophic results are shown in the bottom panel of Figure \[fig:tutorial-action-potential-fitting-and-prediction\](Right). The calibrated Model F fails to repolarise, completely missing the ‘true’ $I_\text{Kr}$ block response of a modest APD prolongation. This example highlights the need for thorough validation and the CoU-dependence of model validation, but also the difficulty in choosing appropriate validation experiments. We quantify uncertainty in parameters and predictions whilst continuing to ignore the discrepancy in Model F’s kinetics. Again, we use Eq. (\[eq:Gaussian\]) together with a uniform prior to derive the posterior of the parameters. The posterior distributions estimated by MCMC and the point estimates obtained by optimisation, are shown in Figure \[fig:tutorial-action-potential-posterior\]. The posterior distribution is very narrow (note the scale), which suggests that we can be confident about the parameter values. The resulting posterior predictions, shown in Figure \[fig:tutorial-action-potential-fitting-and-prediction\](Right), give a very narrow bound. Therefore, by ignoring model discrepancy we could become very (and wrongly) certain that the catastrophically bad predictions are correct. ![ Marginal posterior distribution of Model F parameters, in terms of scaling factors for the conductances in Model T ($s_i = g^\text{Model~F}_i/g^\text{Model~T}_i$). Values of would represent the parameters of Model T that generated the data; notably, none of the inferred parameters for Model F is close to a value of 1. The red dashed lines indicate the result of the global optimisation routine. Note that two of these parameters, $S_{Ks}$ and $S_{NaK}$, have distributions hitting the lower bound that was imposed by the prior, indicating that the calibration process is attempting to make them smaller than 10% of the original Model F parameter values. []{data-label="fig:tutorial-action-potential-posterior"}](fink-2008-stim1hz-hergblock-hist.pdf){width="\textwidth"} It is worth noting that all the issues above arise from the fact that model discrepancy was ignored. In the scenario of no model discrepancy, i.e. when fitting Model T to the data, all of the issues above were solved, as shown in Supplementary Figure \[supp-fig:tutorial-action-potential-no-discrepancy\]. To conclude our motivation of this paper, we can see that neglecting discrepancy in the model’s equations is dangerous and can lead to false confidence in predictions for a new context of use, so we will discuss methods that have been suggested to remedy this. A statistical explanation of the issue -------------------------------------- To understand what is happening, consider the well-specified situation where the data generating process (DGP, the process that produces the experimental data in reality) has probability density function (pdf) $g(y)$, and for which we have data $y_i \sim g(\cdot)$ for $i=1,\ldots, n$. Then suppose we are considering the models $\mathcal{F}=\{f_{\boldsymbol{\theta}}(y): \boldsymbol{\theta}\in \boldsymbol{\Theta}\}$, i.e., a collection of pdfs parameterized by unknown parameter $\boldsymbol{\theta}$. If the DGP $g$ is in $\mathcal{F}$, i.e., we have a well-specified model so that for some $\boldsymbol{\theta}_0\in \boldsymbol{\Theta}$, we have $g =f_{\boldsymbol{\theta}_0}$, then asymptotically, as we collect more data (and under suitable conditions [@van2000]), the maximum likelihood estimator converges to the true value $\boldsymbol{\theta}_0$ almost surely: $$\hat{\boldsymbol{\theta}}_n =\operatorname*{arg max}_{\boldsymbol{\theta}} \sum_{i=1}^n \log f_{\boldsymbol{\theta}}(y_i) \longrightarrow \boldsymbol{\theta}_0, \mbox{ almost surely as } n \longrightarrow \infty,$$ or equivalently $f_{\hat{\boldsymbol{\theta}}_n}$ converges to $g$. Similarly, for a Bayesian analysis (again under suitable conditions [@bernardo2009]), the posterior will converge to a Gaussian distribution centered around the true value $\boldsymbol{\theta}_0$, with variance that shrinks to zero at the asymptotically optimal rate (given by the Cramér-Rao lower bound), i.e. $$\pi(\boldsymbol{\theta} \mid y_{1:n}) \xRightarrow{d} \mathcal{N}(\boldsymbol{\theta}_0, \frac{1}{n}\mathcal{I}(\boldsymbol{\theta}_0)^{-1}),$$ where $y_{1:n}= (y_1,\ldots,y_n)$, and $\mathcal{I}(\boldsymbol{\theta}_0)$ is the Fisher information matrix. However, when our model is misspecified, i.e., $g \not\in \mathcal{F}$ (there is no $\boldsymbol{\theta} \in \boldsymbol{\Theta}$ for which $g=f_{\boldsymbol{\theta}}$), if we do inference for $\boldsymbol{\theta}$ ignoring the discrepancy, then we usually still get asymptotic convergence of the maximum likelihood estimator and Bayesian posterior [@Kleijn2006; @DeBlasi2013]. However, instead of converging to a true value (which does not exist), we converge to the [*pseudo-true*]{} value $$\boldsymbol{\theta}^{*} = \operatorname*{arg min}_{\boldsymbol{\theta}\in\boldsymbol{\Theta}} \operatorname*{KL}( g|| f_{\boldsymbol{\theta}})$$ where $\operatorname*{KL}(g||f) = \int g(x) \log \frac{g(x)}{f(x)} \mathrm{d} x$ is the Kullback-Leibler divergence of $f$ from $g$ (a measure of the difference between two distributions). In other words, we converge upon the model, $f_{\boldsymbol{\theta}^*}$, which is closest to the DGP as measured by the Kullback-Leibler divergence (see Figure \[fig:schematic-model-discrepancy\]). Perhaps more importantly from a UQ perspective, as well as getting a point estimate that converges to the wrong value, we still usually get asymptotic concentration at rate $1/n$, i.e., the posterior variance shrinks to zero. That is, we have found model parameters that are wrong, and yet we are certain about this wrong value. The way to think about this is that the Bayesian approach is not quantifying our uncertainty about a meaningful physical parameter $\boldsymbol{\theta}_0$, but instead, it gives our uncertainty about the pseudo-true value $\boldsymbol{\theta}^*$. Consequently, we can not expect our calibrated predictions $$\pi(y' \mid y) = \int f_{\boldsymbol{\theta}}(y') \pi(\boldsymbol{\theta} \mid y_{1:n}) {\rm d} \boldsymbol{\theta}$$ to perform well, as we saw in the action potential example above. This leaves us with two options. We can either extend our model class $\mathcal{F}$ in the hope that we can find a class of models that incorporates the DGP (and which is still sufficiently simple that we can hope to learn the true model from the data), or we can change our inferential approach. ![\[fig:schematic-model-discrepancy\] **A cartoon to illustrate the effect of model discrepancy on parameter fits in different models**. Each cloud represents a range of possible outputs from each model, which they can reach with different parameter values. The true data generating process (DGP) lies outside either of our imperfect model classes 1 and 2, and neither can fit the data perfectly due to model discrepancy. When we attempt to infer parameters, we will converge upon models that generate outputs closest to the true DGP under the constraint of being in each model. Adding more data just increases the confidence in being constrained to model parameterisations on the boundary of the particular model. This, however, does not move the model parameters closer to the true parameters used in the data generating process, i.e. we become certain about $f_{\boldsymbol{\theta}^*}$, the model using the pseudo-true parameter value. Note that a different experiment will lead to a different model output, shifting these clouds to a different shape, and then different parameter sets may give outputs that are closer to the data generating process. ](fig4-fht.pdf){width="80.00000%"} Accounting for model discrepancy {#sect:tutorial} ================================ Once we have acknowledged that a model is misspecified, we are then faced with the challenge of how to handle the misspecification. The approach taken should depend upon the aim of the analysis. Using the model to predict independent events, for example, a current time-series for some experimental protocol, will require a different approach to if our aim is inference/calibration, i.e., if interest lies in the physical value of a particular parameter. In the first case (prediction), it can often suffice to fit the model to the data ignoring discrepancy, and then to correct the predictions in some way[^1], although this may not work well if the prediction involves extrapolating into a regime far away from the data. The latter case (calibration) is more challenging, as we need to jointly fit the model and the discrepancy model, which can lead to problems of non-identifiability. The most common approach for dealing with discrepancy is to try to correct the simulator by expanding the model class. The simplest approach is simply to add a flexible, non-parametric term to the simulator output, i.e. instead of assuming the data arose from Eq. (\[eq:iid-noise\]), to assume $$y = f(\boldsymbol{\theta}, u_C)+\delta(v_C)+\epsilon.\label{eqn:KOH}$$ Here, $\delta(v_C)$ is the model discrepancy term, and $\epsilon$ remains an unstructured white noise term. Note that $v_C$ is used as the input to $\delta$ as it is not necessary to have the same input as the mechanistic model. To train this model, one option is to first estimate $\theta^*$ assuming Eq. (\[eq:iid-noise\]), and then to train $\delta$ to mop up any remaining structure in the residual. However, a better approach is to jointly estimate $\delta$ and $\theta$ in a Bayesian approach [@Kennedy2001]. Unfortunately, as demonstrated below, this often fails as it creates a non-identifiability between $\boldsymbol{\theta}$ and $\delta$ when $\delta$ is sufficiently flexible: for any $\boldsymbol{\theta}$, there exists a functional form $\delta(\cdot)$ for which Eq. (\[eqn:KOH\]) accurately represents the data generating process. Brynjarsd[ó]{}ttir et al. [@Brynjarsdottir2014] suggested that the solution is to strongly constrain the functional form of $\delta(\cdot)$ using prior knowledge. They present a toy situation in which $\delta(0)=0$ and $\delta(x) $ is monotone increasing, and show that once armed with this knowledge, the posterior $\pi(\boldsymbol{\theta} \mid y)$ more accurately represents our uncertainty about $\boldsymbol{\theta}$. However, knowledge of this form is not available in many realistic problems. Ion channel model example ------------------------- We now illustrate the difficulty of accounting for model discrepancy, in a tutorial example. We demonstrate that it can be hard to determine the appropriate information to sufficiently constrain $\delta$, and that different functional forms can lead to different parameter estimates. The code to reproduce the results in the tutorials is available at <https://github.com/CardiacModelling/fickleheart-method-tutorials>. We consider three structurally different models: Models A, B, and C. We take Model C as the ground truth model in this particular example, and use it to perform synthetic voltage clamp experiments and generate synthetic data. The goal is to use Models A and B to explain the generated synthetic data, assuming we have no knowledge about the ground truth Model C. This tutorial aims to demonstrate the importance of consideration of model discrepancy, jointly with model selection, to represent given data with unknown true DGP. In this tutorial, we use the hERG channel current as an example. Models A, B, and C are various model structures proposed in the literature. Model A is a variant of the traditional Hodgkin-Huxley model, described in Beattie *et al.* [@beattie2018sinusoidal]; Model B is a Markov model structure used in Oehmen *et al.* [@oehmen2002mathematical]; and Model C is a Markov model structure adapted from Di Veroli *et al.* [@di2012high]. The model structures are shown in Figure \[fig:tutorial-ion-channel-models\]. ![\[fig:tutorial-ion-channel-models\] Markov model representation of Models A, B, and C used in the ion channel model tutorial where Model C is taken as ground truth and used to generate synthetic data whilst Models A and B are candidate models that we attempt to fit and use for predictions, demonstrating both the model discrepancy and model selection challenge. ](model-structure.pdf){width="\textwidth"} All three ion channel models can be expressed using a Markov model representation. For a model with a state vector, $\boldsymbol{y} = (y_1, y_2, \cdots)^T$, then $\boldsymbol{y}$ evolves according to $$\frac{\text{d}\boldsymbol{y}}{\text{d}t} = \mathbf{M} \boldsymbol{y},$$ where $\mathbf{M}$ is the Markov matrix describing the transition rates between states. Markov models are linear coupled ordinary differential equations (ODEs) with respect to time, $t$, and states, $\boldsymbol{y}$; whilst typically the components in the Markov matrix, $\mathbf{M}$, are nonlinear functions of voltage, $V$. The observable, the macroscopic ionic current, $I$, measured under $V(t)$, which in these voltage-clamp experiments is an externally prescribed function of time known as the ‘voltage clamp protocol’ (i.e. $u_C$ in Eq. (\[eq:iid-noise\])), is $$\label{eq:ion-current} I(t, V) = g \cdot \mathcal{O} \cdot (V - E),$$ where $g$ is the maximum conductance, $E$ is the reversal potential, and $\mathcal{O}$ is the sum of all ‘open states’ in the model (frequently, and in our examples, this is just one state, but more than one open state is possible). Take Model B as an example. Its state vector, $\boldsymbol{y}$, and Markov matrix, $\mathbf{M}$, can be written as $$\begin{aligned} \boldsymbol{y} &= \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \end{pmatrix} = \begin{pmatrix} \text{C}_2 \\ \text{C}_1 \\ \text{O} \\ \text{I} \end{pmatrix}; & \mathbf{M} &= \begin{pmatrix} -k_{1,2} & k_{2,1} & 0 & 0 \\ k_{1,2} & -k_{2,1}-k_{2,3} & k_{3,2} & 0 \\ 0 & k_{2,3} & -k_{3,2}-k_{3,4} & k_{4,3} \\ 0 & 0 & k_{3,4} & -k_{4,3} \\ \end{pmatrix},\end{aligned}$$ where $k_{i,j}$ represents the transition rate from state $y_i$ to state $y_j$. For all models each transition rate, $k_{i,j}$, is voltage dependent, and takes the form $$k_{i,j}(V) = A_{i,j} \exp(B_{i,j} V), \label{eq:rate_equation}$$ with two parameters ($A_{i,j}, B_{i,j}$) to be inferred. This yields a total of parameters for Model B which we denote as $\{p_1, \dots, p_{12}\}$, together with the maximum conductance, $g$, to be found. Similarly for Model A, it has parameters $\{p_1, \dots, p_{8}\}$ together with $g$, to be inferred. Synthetic experiments --------------------- We say that Model C represents our (hidden) ground truth and simulate data from it under multiple voltage clamp protocols, using parameters fitted to room temperature data from Beattie *et al.* [@beattie2018sinusoidal] (where $g=204$). We introduce i.i.d. Gaussian noise with a standard deviation $\sigma = 25$ to the simulated data. We generate data under three different voltage clamp protocols, $V(t)$. These are a sinusoidal protocol (see top plot in Figure \[fig:tutorial-ion-channel-prediction\]) and an action potential series protocol from Beattie *et al.* [@beattie2018sinusoidal] (see Figure \[supp-fig:tutorial-ion-channel-prediction2a\] in Supplementary Material), and the staircase protocol from Lei *et al.* [@lei_rapid_2019-1; @lei_rapid_2019-2] (see bottom plot in Figure \[fig:tutorial-ion-channel-prediction\]). Standard calibration ignoring model discrepancy ----------------------------------------------- To calibrate the model (without considering any model discrepancy), we assume a statistical model of the form of Eq. (\[eq:iid-noise\]), which has the same observation noise model as our synthetic data. The likelihood of observing the data, $\mathbf{y}=y_{1:n}$, given model parameters $\boldsymbol{\theta}$ is given by Eq. (\[eq:ll\]). We use the sinusoidal protocol as the calibration protocol; the action potential series protocol and the staircase protocol are used as validation. We employ a global optimisation algorithm [@Hansen2006] to fit the model parameters, and all inference is done using PINTS [@Clerx2019Pints]. The fitting results of Models A and B are shown in Figure \[fig:tutorial-ion-channel-prediction\]. Repeats of fitting with different starting points gave almost the same parameter sets. Although both models fit the calibration data reasonably well, neither match perfectly, due to model discrepancy. While the exact forms of the model discrepancy differ between the two models, both models notably fail to reproduce the correct form of the current decay following the step to shortly after . The validation predictions for the staircase protocol are also shown in Figure \[fig:tutorial-ion-channel-prediction\]. Unlike in the sinusoidal protocol, where Model A generally gives a better prediction than Model B, in the staircase protocol different traits of model discrepancy arising from different models are more evident. For example, whereas Model B appears to give slightly better predictions of the current during the first , after this point Model A begins to give better predictions. (fig) \[matrix of nodes\][ ![\[fig:tutorial-ion-channel-prediction\] **(Top)** Models A (blue) and B (orange) fitting results for the ion channel model example. The two models are fitted to the same synthetic data (grey) generated using a third model, Model C, with i.i.d. Gaussian noise added to it. The voltage clamp protocol for calibration is the sinusoidal protocol [@beattie2018sinusoidal]. **(Bottom)** Models A (blue) and B (orange) validation results for the ion channel model tutorial. We show predictions for the two fitted models for the unseen ground truth (grey), generated from Model C under the staircase protocol [@lei_rapid_2019-1]. Note that there are significant discrepancies around . ](compare-sinewave-sinewave.png "fig:"){width="4.5in"}\ ![\[fig:tutorial-ion-channel-prediction\] **(Top)** Models A (blue) and B (orange) fitting results for the ion channel model example. The two models are fitted to the same synthetic data (grey) generated using a third model, Model C, with i.i.d. Gaussian noise added to it. The voltage clamp protocol for calibration is the sinusoidal protocol [@beattie2018sinusoidal]. **(Bottom)** Models A (blue) and B (orange) validation results for the ion channel model tutorial. We show predictions for the two fitted models for the unseen ground truth (grey), generated from Model C under the staircase protocol [@lei_rapid_2019-1]. Note that there are significant discrepancies around . ](compare-sinewave-staircase.png "fig:"){width="4.5in"}\ ]{}; (fig-1-1.south west) – (fig-1-1.north west) node\[midway,above,sloped\][Calibration]{}; (fig-2-1.south west) – (fig-2-1.north west) node\[midway,above,sloped\][Validation]{}; Calibration with model discrepancy ---------------------------------- Next we consider methods that allow us to incorporate or acknowledge the model discrepancy when doing parameter inference and model predictions. First, we adapt the method proposed in [@Kennedy2001] and instead of assuming Eq. (\[eq:iid-noise\]), we consider an additive discrepancy model of the form given by Eq. (\[eqn:KOH\]). We consider three different choices for the discrepancy $\delta(v_C)$, and jointly infer $\theta$ and $\delta$. Note that we allow for a different choice of input $v_C$, compared to the input of model $f$, $u_C$. First, we model $\delta$ as a sparse-Gaussian process (GP), for which we adapted the implementation in PyMC3 [@salvatier2016probabilistic] using Theano [@2016arXiv160502688short]. We explored two possibilities, choosing $v_C$ to be either (i) $t$ (time); or (ii) $O$,$V$ (the open probability, $\mathcal{O}$ in Eq. (\[eq:ion-current\]), and the voltage, $V$). For details of the method, please refer to Supplementary Section \[supp-sec:gp-detail\]. Second, we model discrepancy $\delta$ and the white noise error $\epsilon$, as an autoregressive-moving-average (ARMA) model of order $p, q$ [@durbin2012time]. If $e_t = \delta_t(v_c) + \epsilon_t$ is the residual at time $t$, then an $\operatorname*{ARMA}(p, q)$ model for $e_t$ is $$\label{eq:arma-residual} e_t= \nu_t + \sum_{t'=1}^p \varphi_t e_{t - t'} + \sum_{t'=1}^q\zeta_{t'} \nu_{t - t'}$$ where $\nu \sim \mathcal{N}(0, \tau^2)$, and $\varphi_t, \zeta_t$ are, respectively, the coefficients of the autoregressive and moving-average part of the model. We used the StatsModels [@seabold2010statsmodels] implementation, and assumed $p=q=2$ throughout. Note that the ARMA model, unlike the GP, does not attempt to learn any structure for the discrepancy (the mean of the ARMA process remains zero even once conditioned upon data). The ARMA process is a simple approach for introducing correlation into the residuals. The motivation is that if the mechanistic model is correct, the residuals should be uncorrelated, but for misspecified models, they are typically correlated. For further details of the method, please refer to supplementary information, Section \[supp-sec:arma-detail\]. For all methods, i.i.d. noise, $\operatorname*{GP}(t)$, $\operatorname*{GP}(O, V)$, and $\operatorname*{ARMA}(2, 2)$, we infer the parameters using Eq. (\[eq:Gaussian\]), where the priors are specified in Supplementary Section \[supp-sec:priors\]. The posteriors are approximated by using an Adaptive Covariance MCMC method in PINTS [@Clerx2019Pints; @Johnstone2016]. The inferred (marginal) posterior distributions for Model A are shown in Figure \[fig:tutorial-ion-channel-posteriors\], and they are used to generate the posterior predictive shown below. Supplementary Figure \[supp-fig:tutorial-ion-channel-posteriors-b\] shows the same plot for Model B. Note that the choice of the discrepancy model can shift the posterior distribution significantly, both in terms of its location and spread. In particular, the $\operatorname*{ARMA}(2, 2)$ model gives a much wider posterior than the other discrepancy models. ![\[fig:tutorial-ion-channel-posteriors\] Model A inferred marginal posterior distributions for the conductance, $g$ in Eq. (\[eq:ion-current\]), and kinetic parameters $p_1,\dots,p_8$ (a list of parameters referring to $A_{i,j}$ and $B_{i,j}$ in Eq. (\[eq:rate\_equation\])) with different discrepancy models: i.i.d. noise (blue), $\operatorname*{GP}(t)$ (orange), $\operatorname*{GP}(O, V)$ (green), and $\operatorname*{ARMA}(2, 2)$ (red). ](compare-model_A-sinewave-posteriors.png){width="\textwidth"} Figure \[fig:tutorial-ion-channel-fitting2\] shows the posterior predictives of the calibration for Model A using different discrepancy models (Supplementary Figure \[supp-fig:tutorial-ion-channel-fitting2-b\] for Model B). The top panel shows the sinusoidal calibration voltage protocol, and the panels underneath are calibrated models with i.i.d. noise (blue), $\operatorname*{GP}(t)$ (orange), $\operatorname*{GP}(O, V)$ (green), and $\operatorname*{ARMA}(2, 2)$ (red). The calibration data are shown in grey in each panel. Visually, we can already see that the two GP models, $\operatorname*{GP}(t)$ (orange) and $\operatorname*{GP}(O, V)$ (green), are able to fit to the data with very high accuracy; we will see one of them is overfitting while the other is not. The $\operatorname*{ARMA}(2, 2)$ model (red) is able to increase the width of the posterior, although its posterior mean prediction does not follow the data as closely as the two GP models. ![\[fig:tutorial-ion-channel-fitting2\] Model A fitted to the sinusoidal calibration protocol using the different discrepancy models: i.i.d. noise, $\operatorname*{GP}(t)$, $\operatorname*{GP}(O, V)$, and $\operatorname*{ARMA}(2, 2)$. The plots show the mean (solid lines) and the 95% credible intervals (shaded) of the posterior predictive for each model. ](compare-model_A-sinewave-sinewave-pp.png){width="\textwidth"} Tables \[fig:tutorial-ion-channel-rmse-table\] and \[fig:tutorial-ion-channel-likelihood-table\] show two quantifications of the goodness of fits, which is coloured so that yellow shows the best performing model and red shows the worst. Table \[fig:tutorial-ion-channel-rmse-table\] shows the RMSE values of the posterior mean predictions for all of the models, and Table \[fig:tutorial-ion-channel-likelihood-table\] shows the marginal log-likelihoods (a proper scoring rule [@Gneiting2007], which assesses the entire predictive distribution, not just the mean). The first row of the two tables shows the results for calibration (sine wave), and it is clear that the $\operatorname*{GP}(t)$ and $\operatorname*{GP}(O, V)$ models give the best RMSE values, while the $\operatorname*{ARMA}(2, 2)$ and $\operatorname*{GP}(O, V)$ models give the best marginal log-likelihoods. ![image](compare-model_A-prediction-error-mean.pdf){width="\textwidth"}\ ![image](compare-model_B-prediction-error-mean.pdf){width="\textwidth"}\ ![image](compare-model_A-evidence.pdf){width="49.50000%"} ![image](compare-model_B-evidence.pdf){width="49.50000%"} Figure \[fig:tutorial-ion-channel-prediction2\] shows the prediction results for the staircase validation protocol for Model A (Supplementary Figure \[supp-fig:tutorial-ion-channel-prediction2-b\] for Model B) using different discrepancy models, with the same layout as Figure \[fig:tutorial-ion-channel-fitting2\]. Similar figures for AP protocol predictions are shown in Supplementary Figures \[supp-fig:tutorial-ion-channel-prediction2a\] (for Model A) and \[supp-fig:tutorial-ion-channel-prediction2a-b\] (for Model B). $\operatorname*{GP}(t)$ does not have any information that we have changed from calibration protocol to prediction protocol, and it ‘predicts’ as if it were still under the sinusoidal protocol. Thus, there is some residual from the calibration shown in the $\operatorname*{GP}(t)$ (orange) prediction, e.g. see ‘wobbly’ current at $\sim7000$ as pointed at by the blue arrow. For Model A, it is interesting to see that none of the discrepancy models are able to predict the data better than the i.i.d. noise model according to the RMSE value of the posterior mean prediction in Table \[fig:tutorial-ion-channel-rmse-table\], and the next best is the $\operatorname*{GP}(O, V)$ model. However, it is also interesting to notice that the $\operatorname*{GP}(O, V)$ model is able to capture and predict very nicely the tail current after the two activation steps, as indicated by the red arrows on Figure \[fig:tutorial-ion-channel-prediction\] — a visible area of model mismatch in our calibration without model discrepancy. For Model B, the $\operatorname*{GP}(O, V)$ discrepancy model gives the best overall predictions for both the staircase and the AP protocols, although when we examine the contributions of the mechanistic and discrepancy models, we see that an element of unidentifiability between them has arisen (Supplementary Section \[supp-sec:ion-channel-supp\]\[supp-sub:ModelBFullResults\]). In terms of the marginal log-likelihood, Table \[fig:tutorial-ion-channel-rmse-table\] (bottom) again highlights that the $\operatorname*{ARMA}(2, 2)$ and $\operatorname*{GP}(O, V)$ models tend to be better than the i.i.d. noise and $\operatorname*{GP}(t)$ models. ![\[fig:tutorial-ion-channel-prediction2\] Models A prediction with different discrepancy models: i.i.d. noise, $\operatorname*{GP}(t)$, $\operatorname*{GP}(O, V)$, and $\operatorname*{ARMA}(2, 2)$. The voltage clamp protocol for calibration is the staircase protocol [@lei_rapid_2019-1]. We plot the posterior predictive with the mean (solid lines) and the bounds showing the 95% credible interval (shaded). The red arrows point to the tail current after the two activation steps which mark a visible area of model mismatch when calibrated without model discrepancy (i.i.d. noise, blue), and how the $\operatorname*{GP}(O, V)$ and $\operatorname*{ARMA}(2,2)$ models handle the mismatch differently. The blue arrow points to an obvious artefact at $\sim7000$ induced by the $\operatorname*{GP}(t)$ prediction which was trained on the sinusoidal protocol, and doesn’t know anything about this staircase protocol. ](compare-model_A-sinewave-staircase-pp.png){width="\textwidth"} Supplementary Figures \[supp-fig:tutorial-ion-channel-fitting2-residual\], \[supp-fig:tutorial-ion-channel-prediction2a-residual\], and \[supp-fig:tutorial-ion-channel-prediction2b-residual\] show the model discrepancy for the sine wave protocol, AP protocol, and staircase protocol, respectively, for Model A; Supplementary Figures \[supp-fig:tutorial-ion-channel-fitting2-residual-b\], \[supp-fig:tutorial-ion-channel-prediction2a-residual-b\], and \[supp-fig:tutorial-ion-channel-prediction2b-residual-b\] show the same plots for Model B. Supplementary Figures \[supp-fig:tutorial-ion-channel-prediction2b-residual\] and \[supp-fig:tutorial-ion-channel-prediction2b-residual-b\] in particular highlight that the $\operatorname*{GP}(t)$ model has, by design, learnt nothing of relevance about model discrepancy for extrapolation under an independent validation protocol (in which $V(t)$, and indeed the range of $t$, differs from that of the training protocol). Furthermore, the discrepancy model is based only on information extending to (the duration of the training protocol), after which the credible interval resorts to the width of the GP prior kernel. In contrast, $\operatorname*{GP}(O, V)$ learns independently of $t$ about the discrepancy under combinations of $(O,V)$ present in the training data (such as the activation step to followed by a step to ), which is why it is able to better predict the tail current after the two activation steps. Finally, the $\operatorname*{ARMA}(2,2)$ model looks very similar to the i.i.d. noise model in terms of the 95% credible interval of the discrepancy term only, as it is defined to have zero mean. The ion channel (ODE) model-only predictions for the sine wave protocol, AP protocol, and staircase protocol are shown in Supplementary Figures \[supp-fig:tutorial-ion-channel-fitting2-model\], \[supp-fig:tutorial-ion-channel-prediction2a-model\], and \[supp-fig:tutorial-ion-channel-prediction2b-model\] for Model A and Supplementary Figures \[supp-fig:tutorial-ion-channel-fitting2-model-b\], \[supp-fig:tutorial-ion-channel-prediction2a-model-b\], and \[supp-fig:tutorial-ion-channel-prediction2b-model-b\] for Model B. Comparing the RMSE and likelihood in Tables \[fig:tutorial-ion-channel-rmse-table\] & \[fig:tutorial-ion-channel-likelihood-table\], it is interesting to see the differences in performance when applied to different models (Models A and B). Note that, for a given dataset, the RMSE and likelihood values in the tables are comparable across models. First we notice that with the i.i.d. noise model, Model A has a lower RMSE value than Model B for both the calibration and the two predictions. With Model A, none of the discrepancy models that we tried are able to outperform the simplest i.i.d. noise model when it comes to the mean predictions for the staircase and AP protocols; but $\operatorname*{GP}(O,V)$ has a very similar RMSE value as compare to the i.i.d. noise model while being able to capture some of the nonlinear dynamics that Model A misses as discussed above. However, with Model B, the $\operatorname*{GP}(O,V)$ model has the best RMSE value for the predictions, and the second best for the calibration where the best one is the $\operatorname*{GP}(t)$ model that overfits the data. The $\operatorname*{ARMA}(2,2)$ model consistently gives the best likelihood value for Models A and B, as it gives a wider posterior distribution compared to other methods (Figure \[fig:tutorial-ion-channel-posteriors\]). To conclude, in this example, we have used two different, incorrect model structures (Models A, B) to fit to a third model structure (Model C) generated synthetic data. We considered both ignoring discrepancy when calibration and incorporating discrepancy when calibration. Depending on the model, calibration with discrepancy could improve predictions notably as compared to calibration ignoring discrepancy (for the case of Model B), but not all (for Model A). Although our problem was a time-dependent (ODE) problem, constructing the discrepancy model as a pure time-series based function might not be useful in predicting unseen situations; the $\operatorname*{GP}(O, V)$ model performed the best compared with the other two time-series based models $\operatorname*{GP}(t)$ and $\operatorname*{ARMA}(2, 2)$. Discussion ========== In this review and perspective piece we have drawn attention to an important and under-appreciated source of uncertainty in mechanistic models — that of uncertainty in the model structure or the equations themselves (model discrepancy). Focusing on cardiac electrophysiology models, we provided two examples of the consequences of ignoring discrepancy when calibrating models at the ion channel and action potential scales, highlighting how this could lead to wrongly-confident parameter posterior distributions and subsequently spurious predictions. Statistically we can explicitly admit discrepancy exists, and include it in the model calibration process and predictions. We attempted to do this by modelling discrepancy using two proposals from the literature — Gaussian processes (GPs) trained on different inputs and an autoregressive-moving-average (ARMA) model. We saw how GPs could successfully describe discrepancy in the calibration experiment. A two-dimensional GP in voltage and time was used previously by Plumlee *et al.* [@Plumlee2016; @Plumlee2017] where it could extrapolate to new voltages for a given single step voltage-clamp experiment. To be most useful in making new predictions for unseen situations, the discrepancy model needs to be a function of something other than time, otherwise features specific to the calibration experiment are projected into new situations. One promising example of such a discrepancy model was our two-dimensional GP as a function of the mechanistic model’s open probability and voltage, although in the Model B case this led to ambiguity between the role of the ODE system and the role of the discrepancy (see Supplementary Section \[supp-sec:ion-channel-supp\]\[supp-sub:ModelBFullResults\]). The modelling community would hope to study any discrepancy model that is shown to improve predictions, and use insights from this process to iteratively improve the mechanistic model. How we handle model discrepancy may depend on whether we are more interested in learning about what is missing in the model, or in making more reliable predictions: both related topics are worthy of more investigation. Recommendations --------------- Very rarely do computational studies use more than one model to test the robustness of their predictions to the model form. We should bear in mind that all models are approximations and so when we are comparing to real data, all models have discrepancy. Here we have seen, using synthetic data that pretends we know a true data-generating mechanistic model, how fragile the calibration process can be for models with discrepancy and how this discrepancy manifests itself in predictions for new, unseen situations. Synthetic data studies, simulating data from different parameter sets and different model structures, allow the modeller to test how well the inverse problem can be solved and how robust predictions from the resulting models are. We strongly recommend performing such studies to learn more about your chosen model, and alternative models; as well as the effects of your model choice on parameter calibration and your subsequent predictions. To develop our field further, it will be important to document the model-fitting process, and to make datasets and infrastructure available to perform and reproduce these fits with different models [@Daly2018WebLab]. Open questions and future work ------------------------------ The apparent similarity of the action potential models we looked at (summarised in Figure \[fig:tutorial-action-potential-models\]) is a challenge for model calibration. A number of papers have emphasised that more information can be gained to improve parameter identifiability with careful choice of experimental measurements, in particular by using membrane resistance [@Kaur2014; @Pouranbarani2019], or other protocols promoting more information-rich dynamics [@Groenendaal2015; @Johnstone2016] and some of these measurements may be more robust to discrepancy than others. In synthetic data, fitting the model used to generate the data will recover the same parameter set from any different protocol (where there is sufficient information to identify the parameters). But in the presence of discrepancy, fitting the same model to data from different protocols/experiments will result in different parameter sets, as the models make the best possible compromise (as shown schematically in Figure \[fig:schematic-model-discrepancy\]). This phenomenon may be an interesting way to approach and quantify model discrepancy. If the difference between imperfect model predictions represented the difference between models and reality then this may also provide a way to estimate discrepancy. For instance, the largest difference between the ion channel Model A & B predictions in the staircase protocol was at the point in time that both of them showed largest discrepancy (Figure \[fig:tutorial-ion-channel-prediction\]). Some form of Bayesian Model Averaging [@Hoeting1999], using variance-between-models to represent discrepancy, may be instructive if the models are close enough to each other and reality, but can be misleading if the ensemble of models is not statistically exchangeable with the data generating process [@Chandler2013; @Rougier2013] or if there is some systematic error (bias) due to experimental artefacts [@lei2020b]. In time-structured problems, rather than adding a discrepancy to the final simulated trajectory, as we have done here, we can instead change the dynamics of the model directly. It may be easier to add a discrepancy term to the differential equations to address misspecification, than it is to correct their solution, but the downside is that this makes inference of the discrepancy computationally challenging. One such approach is to ‘noise-up’ the ODE by converting it to a stochastic differential equation [@Crucifix2009; @Carson2018], i.e., replace $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = f_\theta(\mathbf{x},t)$ by $\mathrm{d}\mathbf{x} = f_\theta(\mathbf{x},t) \mathrm{d}t + \Sigma^{\frac{1}{2}} \mathrm{d}W_t$ where $W_t$ is a Brownian motion with a variance matrix $\Sigma$. This turns the deterministic ODE into a stochastic model and can improve the UQ, but cannot capture any structure missing from the dynamics. We can go further and attempt to modify the underlying model equations, by changing the ODE system to $$\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = f_\theta(\mathbf{x},t) + \delta(\mathbf{x})$$ where again $\delta(\mathbf{x})$ is an empirical term to be learnt from the data. For example, this has been tried with a discretized version of the equations using a parametric model for $\delta$ [@wilkinson2011quantifying], with GPs [@Frigola2013], nonlinear autoregressive exogenous (NARX) models [@Worden2018], and deep neural networks [@Meeds2019]. Computation of posterior distributions for these models is generally challenging, but is being made easier by the development of automatic-differentiation software, which allows derivative information to be used in MCMC samplers, or in variational approaches to inference (e.g., [@Neal2011; @Ryder2018]). Ultimately, modelling our way out of trouble, by expanding the model class, may prove impossible given the quantity of data available in many cases. Instead, we may want to modify our inferential approach to allow the best judgements possible about the parameters given the limitation of the model and data. Approaches such as approximate Bayesian computation (ABC) [@Sisson2018] and history-matching [@Craig1997; @Craig2001] change the focus from learning a statistical model within a Bayesian setting, to instead only requiring that the simulation gets within a certain distance of the data. This change, from a fully specified statistical model for $\delta$ to instead only giving an upper bound for $\delta$, is a conservative inferential approach where the aim is not to find the best parameter values, but instead rule out only obviously implausible values [@Wilkinson2013; @Holden2018]. For example, in the action potential model from Section \[sect:motivation\], instead of taking a Bayesian approach with an i.i.d. Gaussian noise model, we can instead merely try to find parameter values that get us within some distance of the calibration data (see Supplement and Figure \[supp-fig:tradeoff1\] for details). In the Supplement, we describe a simple approach, based on the methods presented in [@Novaes2019], where we find candidate parameter sets that give a reasonable match to the calibration data. When we use these parameters to predict the validation data, and the 75% [$I_\text{Kr}$]{} block CoU data, we get a wide range of predictions that incorporate the truth (Figure \[supp-fig:tradeoff2\]) — for a small subset of 70 out of 1079, we get good predictions and not the catastrophic prediction shown in Figure \[fig:tutorial-action-potential-fitting-and-prediction\]. By acknowledging the existence of model discrepancy, the use of wider error bounds (or higher-temperature likelihood functions) during the fitting process may avoid fitting parameters overly-precisely. However, we have no way of knowing which subset of remaining parameter space is more plausible (if any) without doing these further experiments; testing the model as close as possible to the desired context of use helps us spot such spurious behaviour. Conclusions =========== In this paper we have seen how having an imperfect representation of a system in a mathematical model (discrepancy) can lead to spuriously certain parameter inference and overly-confident and wrong predictions. We have examined a range of methods that attempt to account for discrepancy in the fitting process using synthetic data studies. In some cases we can improve predictions using these methods, but different methods work better for different models in different situations, and in some cases the best predictions were still made by ignoring discrepancy. A large benefit of the calibration methods which include discrepancy is that they better represent uncertainty in predictions, although all the methods we trialled still failed to allow for a wide enough range of possible outputs in certain parts of the protocols. Methodological developments are needed to design reliable methods to deal with model discrepancy for use in safety-critical electrophysiology predictions. Data access {#data-access .unnumbered} ----------- Code to reproduce the results in the tutorials is available at <https://github.com/CardiacModelling/fickleheart-method-tutorials>. Author contributions {#author-contributions .unnumbered} -------------------- CLL and SG wrote the code to perform the examples in the main paper. CLL, SG, DGW, GRM and RDW drafted the manuscript. All authors conceived and designed the study. All authors read and approved the manuscript. Competing interests {#competing-interests .unnumbered} ------------------- The authors declare that they have no competing interests. Funding {#funding .unnumbered} ------- This work was supported by the Wellcome Trust \[grant numbers 101222/Z/13/Z and 212203/Z/18/Z\]; the Engineering & Physical Sciences Research Council \[grant numbers EP/R014604/1, EP/P010741/1, EP/L016044/1, EP/R006768/1, EP/S014985/1, and EP/R003645/1\]; the British Heart Foundation \[grant numbers PG/15/59/31621, RE/13/4/30184, and SP/18/6/33805\]; the Russian Foundation for Basic Research \[grant number 18-29-13008\]. CLL acknowledges support from the Clarendon Scholarship Fund; and the EPSRC, MRC and F. Hoffmann-La Roche Ltd. for studentship support. CDC and CH were supported by the BHF. MR acknowledges a BHF Turing Cardiovascular Data Science Award. AVP was partially supported by RF Government Act No.211 of March 16, 2013, and RFBR. RWS was supported by the Brazilian Government via CAPES, CNPq, FAPEMIG, and UFJF, and by an Endeavour Research Leadership Award from the Australian Government Department of Education. KW would like to acknowledge the support of the UK EPSRC. GMN was supported by CEFET-MG and CAPES. GRM & SG acknowledge support from the Wellcome Trust & Royal Society via a Sir Henry Dale Fellowship to GRM. GRM & DGW acknowledge support from the Wellcome Trust via a Wellcome Trust Senior Research Fellowship to GRM. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the ‘Fickle Heart’ programme. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank all the participants at the Isaac Newton Institute ‘Fickle Heart’ programme for helpful discussions which informed this manuscript. input[xx.tex]{} [^1]: Note that jointly fitting model and discrepancy can make the problem easier, for example, by making the discrepancy a better behaved function more amenable to being modelled.
--- abstract: | The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the *eigenvalue spectrum* over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined. A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (*drawdowns*), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (*drawups*), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies. address: 'Dublin City University, Glasnevin, Dublin 9, Ireland' author: - 'T. Conlon' - 'H.J. Ruskin' - 'M. Crane' title: 'Cross-Correlation Dynamics in Financial Time Series' --- , , , Correlation Matrix ,Eigenspectrum Analysis ,Econophysics , [89.65.-s]{} , [89.75.-k]{} Introduction ============ In recent years, the analysis of the equal-time cross-correlation matrix for a variety of multivariate data sets such as financial data, [@Laloux_1999; @Plerou_1999; @Laloux_2000; @Plerou_2000; @Gopikrishnan_2000; @Utsuki_2004; @Bouchaud_book; @Wilcox_2004; @Sharifi_2004; @Conlon_2007; @Conlon_2008; @Podobnik_2008], electroencephalographic (EEG) recordings, [@Schindler_2007; @Schindler_2007_b], magnetoencephalographic (MEG) recordings, [@Kwapien_2000], and others, has been studied extensively. Other authors have investigated the relationship between stock price changes and liquidity or trading volume, [@Ying_1966; @Karpoff_1987; @LeBaron_1999]. In particular, Random Matrix Theory, (RMT), has been applied to filter the relevant information from the statistical fluctuations, inherent in empirical cross-correlation matrices, for various financial time series, [@Laloux_1999; @Plerou_1999; @Laloux_2000; @Plerou_2000; @Gopikrishnan_2000; @Utsuki_2004; @Bouchaud_book; @Wilcox_2004; @Sharifi_2004; @Conlon_2007]. By comparing the eigenvalue spectrum of the correlation matrix to the analytical results, obtained for random matrix ensembles, significant deviations from RMT eigenvalue predictions provide genuine information about the correlation structure of the system. This information has been used to reduce the difference between predicted and realised risk of different portfolios. Several authors have suggested recently that some real correlation information may be hidden in the RMT defined random part of the eigenvalue spectrum. A technique, involving the use of power mapping to identify and estimate the noise in financial correlation matrices, [@Guhr_2003], allows the suppression of those eigenvalues, associated with the noise, in order to reveal different correlation structures buried underneath. The relationship, between the eigenvalue density $c$ of the true correlation matrix, and that of the empirical correlation matrix $C$, was derived to show that correlations can be measured in the random part of the spectrum, [@Burda_2004_a; @Burda_2004_b]. A Kolmogorov test was applied to demonstrate that the bulk of the spectrum is not in the Wishart RMT class, [@Malevergne], while the existence of factors, such as an overall market effect, firm size and industry type, is due to collective influence of the assets. More evidence that the RMT fit is not perfect was provided, [@Kwapien_2006], where it was shown that the dispersion of “noise” eigenvalues is inflated, indicating that the bulk of the eigenvalue spectrum contains correlations masked by measurement noise. The behaviour of the largest eigenvalue of a cross-correlation matrix for small windows of time, has been studied, [@Drozdz_2000], for the DAX and Dow Jones Industrial average Indices (DJIA). Evidence of a time-dependence between ‘drawdowns’ (‘draw-ups’) and an increase (decrease) in the largest eigenvalue was obtained, resulting in an increase of the *information entropy*[^1]of the system. Similar techniques were used, [@Drozdz_2001], to investigate the dynamics between the stocks of two different markets (DAX and DJIA). In this case, two distinct eigenvalues of the cross-correlation matrix emerged, corresponding to each of the markets. By adjusting for time-zone delays, the two eigenvalues were then shown to coincide, implying that one market leads the dynamics in the other. Equal-time cross-correlation matrices have been used, [@Muller_2005], to characterise dynamical changes in nonstationary multivariate time-series. It was specifically noted that, for increased synchronisation of $k$ series within an $M-$dimensional multivariate time series, a repulsion between eigenstates of the correlation matrix results, in which $k$ levels participate. Through the use of artifically-created time series with pre-defined correlation dynamics, it was demonstrated that there exist situations, where the relative change in eigenvalues from the lower edge of the spectrum is greater than that for the large eigenvalues, implying that information drawn from the smaller eigenvalues is highly relevant. The technique was subsequently applied to the eigenvalue spectrum of the equal time cross-correlation matrix of multivariate Epileptic Seizure time series and information on the correlation dynamics was found to be visible in *both* the lower and upper eigenstates. Further studies of the equal-time correlation matrix of EEG signals, [@Schindler_2007], investigated temporal dynamics of focal onset epileptic seizures[^2] and showed that zero-lag correlations between multichannel EEG signals tend to decrease during the first half of a seizure and increase gradually before the seizure ends. A further extension (to the case of *Status Epilepticus*, [@Schindler_2007_b]), used the equal-time correlation matrix to assess neuronal synchronisation prior to seizure termination. Examples have demonstrated, [@Muller_2006_b], that information about cross correlations can be found in the RMT bulk of eigenvalues and that the information extracted at the *lower* edge is statistically *more significant* than that extracted from the larger eigenvalues. The authors introduced a method of unfolding the eigenvalue level density, through the normalisation of each of the level distances by its ensemble average, and used this to calculate the corresponding individual nearest-neighbour distance. Those parts of the spectrum, dominated by noise, could be distinguished from those containing information about correlations. Application of this technique to multichannel EEG data showed the smallest eigenvalues to be more sensitive to detection of subtle changes in the brain dynamics than the largest. In this paper, we examine eigenvalue dynamics of the cross-correlation matrix for multivariate financial data with a view to characterising market behaviour. Methods are reviewed in Section \[methods\], the data described in Section \[Data\] and in Section \[Eresults\] we look at the results obtained both for the empirical correlation matrix and the model correlation matrices described. Methods ======= Empirical Dynamics {#Emp_Dyn} ------------------ The equal-time cross-correlation matrix, between time series of equity returns, is calculated using a sliding window where the number of assets, $N$, is smaller than the window size $T$. Given returns $G_{i} \left(t\right)$, $i = 1, \ldots,N$, of a collection of equities, we define a normalised return, within each window, in order to standardise the different equity volatilities. We normalise $G_{i}$ with respect to its standard deviation $\sigma_{i}$ as follows: $$g_{i}\left(t\right) = \frac{G_{i} \left(t\right) - \widehat{G_{i} \left(t\right)}}{\sigma_{i}}$$ Where $\sigma_{i}$ is the standard deviation of $G_{i}$ for assets $i = 1, \ldots,N$ and $\widehat{G_{i}}$ is the time average of $G_{i}$ over a time window of size $T$. Then, the equal-time cross-correlation matrix is expressed in terms of $g_{i}\left(t\right)$ $$C_{ij} \equiv \left\langle g_{i}\left(t\right) g_{j}\left(t\right) \right\rangle \label{cross_corr}$$ The elements of $C_{ij}$ are limited to the domain $-1 \leq C_{ij} \leq 1$, where $C_{ij} = 1$ defines perfect positive correlation, $C_{ij} = -1$ corresponds to perfect negative correlation and $C_{ij} = 0$ corresponds to no correlation. In matrix notation, the correlation matrix can be expressed as $$\mathbf{C} = \frac{1}{T} \mathbf{GG}^{\tau}$$ Where $\bf{G}$ is an $N\times T$ matrix with elements $g_{it}$. The eigenvalues $\mathbf{\lambda}_{i}$ and eigenvectors $\mathbf{\hat{v}}_{i}$ of the correlation matrix $\mathbf{C}$ are found from the following $$\mathbf{C} \mathbf{\hat{v}}_{i} = \mathbf{\lambda}_{i} \mathbf{\hat{v}}_{i}$$ The eigenvalues are then ordered according to size, such that $\mathbf{\lambda}_{1} \leq \mathbf{\lambda}_{2}\leq \ldots \leq \mathbf{\lambda}_{N}$. The sum of the diagonal elements of a matrix, (the Trace), must always remain constant under linear transformation. Thus, the sum of the eigenvalues must always equal the Trace of the original correlation matrix. Hence, if some eigenvalues increase then others must decrease, to compensate, and vice versa (*Eigenvalue Repulsion*). There are two limiting cases for the distribution of the eigenvalues [@Schindler_2007; @Muller_2005]. When all of the time series are perfectly correlated, $C_{i} \approx 1$, the largest eigenvalue is maximised with a value equal to $N$, while for time series consisting of random numbers with average correlation $C_{i} \approx 0$, the corresponding eigenvalues are distributed around $1$, (where any deviation is due to spurious random correlations). For cases between these two extremes, the eigenvalues at the lower end of the spectrum can be much smaller than $\lambda_{max}$. To study the dynamics of each of the eigenvalues using a sliding window, we normalise each eigenvalue in time using $${\tilde{\lambda}}_{i}(t) = \frac{\left(\mathbf{\lambda}_{i}(t) - {\widehat{\lambda_{i}(\tau)}}\right)}{\sigma^{\lambda_{i}\left(\tau\right)}} \label{normalise}$$ where $\mathbf{\widehat{\lambda}}$ and $\sigma^{\lambda}$ are the mean and standard deviation of the eigenvalue $i$ over a particular reference period, $\tau$. This normalisation allows us to visually compare eigenvalues at both ends of the spectrum, even if their magnitudes are significantly different. The reference period, used to calculate mean and standard deviation of the eigenvalue spectrum, can be chosen to be a low volatility sub-period, (which helps to enhance the visibility of high volatility periods), or the full time period studied. One-factor Model {#OneFModel} ---------------- In the *one-factor model* of stock returns, we assume a *global correlation* with the cross-correlation between all stocks the same, $\rho_{0}$. The spectrum of the associated correlation matrix consists of only two values, a large eigenvalue of order $(N-1)\rho_{0} + 1$, associated with the market, and an $(N-1)-$fold degenerate eigenvalue of size $1-\rho_{0}<1$. Any deviation from these values is due to the finite length of time series used to calculate the correlations. In the limit $N\rightarrow\infty$ (even for *small correlation*, i.e. $\rho\rightarrow0$) a large eigenvalue appears, which is associated with the eigenvector $v_{1} = \left(\frac{1}{\sqrt{N}}\right)\left(1,1,1\ldots1\right)$, and which dominates the correlation structure of the system. ‘Market plus sectors’ model {#MultiFModel} --------------------------- To expand the above to a ‘market plus sectors’ model, we perturb a number of pairs $N$ of the correlations $\rho_{0} + \rho_{n}$, where $-1-\rho_{0}\leq\rho_{n}\leq1-\rho_{0}$. Additionally, we impose a constraint $\displaystyle\sum_{N}\rho_n = 0$, ensuring that the average correlation of the system remains equal to $\rho_{0}$. These perturbations allow us to introduce *groups of stocks* with similar correlations, (corresponding to Market Sectors). Using the correlation matrix from the “one-factor model” and the “market plus sectors model”, we can construct correlated time series using the Cholesky decomposition $A$ of a correlation matrix $C = {AA}^{\tau}$. We can then generate finite correlated time series of length $T$, $$x_{it} = \sum_{j} A_{ij} y_{jt} \quad \quad t = 1,\dots,T \label{chol}$$ where $y_{jt}$ is a random Gaussian variable with mean zero and variance $1$ at time $t$. Using Eqn. \[cross\_corr\] we can then construct a correlation matrix using the simulated time series. The finite size of the time series introduces ‘noise’ into the system and hence empirical correlations will vary from sample to sample. This ‘noise’ could be reduced through the use of longer simulated time series or through averaging over a large number of time series. In order to compare the eigenvectors from each of the Model Correlation matrices to that constructed from the equity returns time series, we use the Inverse Participation Ratio (IPR) [@Plerou_2000; @Noh]. The IPR allows quantification of the number of components that participate significantly in each eigenvector and tells us more about the level and nature of deviation from RMT. The IPR of the eigenvector $u^{k}$ is given by $ I^{k} \equiv \sum^{N}_{l = 1} \left( u^{k}_{l}\right)^{4} $ and allows us to compute the inverse of the number of eigenvector components that contribute significantly to each eigenvector. Data {#Data} ==== In order to study the dynamics of the empirical correlation matrix over time, we analyse two different data sets. The first data set comprises the $384$ equities of the Standard & Poors (S&P) 500 where full price data is available from January $1996$ to August $2007$ resulting in 2938 daily returns. The S&P 500 is an index consisting of 500 large capitalisation equities, which are predominantly from the US. In order to demonstrate that our results are not market specific, however, we examine a second data set, made up of the 49 equities of the Dow Jones Euro Stoxx 50 where full price data is available from January $2001$ to August $2007$ resulting in $1619$ daily returns. The Dow Jones Euro Stoxx 50 is a stock index of Eurozone equities designed to provide a blue-chip representation of supersector leaders in the Eurozone. Results {#Eresults} ======= We analyse the eigenvalue dynamics of the correlation matrix of a subset of 100 S&P equities, chosen randomly, using a sliding window of 200 days. This subsector was chosen such that $Q = \frac{T}{N} = 2$, thus ensuring that the data would be close to non-stationary in each sliding window. Figure \[S&Peigs\](a) shows broadly similar sample dynamics from the $5^{th}$, $15^{th}$ and $25^{th}$ largest eigenvalues over each of these sliding windows. The sum of the 80 smallest eigenvalues are shown in Figure \[S&Peigs\](b), while the dynamics of the largest eigenvalue is displayed in Figure \[S&Peigs\](c). The repulsion between the largest eigenvalue and the small eigenvalues is evident here, (comparing \[S&Peigs\](b) and \[S&Peigs\](c)), with the dynamics of the small eigenvalues contrary to those of the largest eigenvalue. As noted earlier (Section \[Emp\_Dyn\]), this is a consequence of the fact that the trace of the correlation matrix must remain constant under transformations and any change in the largest eigenvalue must be reflected by a change in one or more of the other eigenvalues. Similar results were obtained for different subsets of the S&P and also for the members of the Dow Jones Euro Stoxx 50. ![Time Evolution of (a) Three small eigenvalues (b) Sum of the 80 smallest eigenvalues (c) The largest eigenvalue[]{data-label="S&Peigs"}](SandPEigenvalues.eps){height="100mm" width="130mm"} Normalised Eigenvalue Dynamics ------------------------------ Using normalised eigenvalues as described above, (Eqn. \[normalise\]), we performed a number of experiments to investigate the dynamics of a set of small eigenvalues versus the largest eigenvalue. The various experiments are described below: 1. As in Section \[Eresults\], the dynamics for the same subset of 100 equities are analysed using a sliding window of 200 days. The normalisation is carried out using the mean and standard deviation of each of the eigenvalues over the entire time-period. Figure \[S&PeigsNorm\](a) shows the value of the S&P index from $1997$ to mid$-2007$. The normalised largest eigenvalue is shown in Figure \[S&PeigsNorm\](b) together with the average of the 80 normalised small eigenvalues. The compensatory dynamics mentioned earlier are shown more clearly here, with the largest and average of the smallest 80 eigenvalues having opposite movements. The normalised eigenvalues for the entire eigenvalue spectrum are shown in Figure \[S&PeigsNorm\](c), where the colour indicates the number of standard deviations from the time average for each of the eigenvalues over time. As shown, there is very little to differentiate the dynamics of the $80-90$ or so smallest eigenvalues. In contrast, the behaviour of the largest eigenvalue is clearly opposite to that of the smaller eigenvalues. However, from the $90^{th}$ and subsequent eigenvalue there is a marked change in the behaviour, (Figure \[S&PeigsNorm\](d)), and the eigenvalue dynamics are distinctly different. This may correspond to the area outside the “Random Bulk” in RMT. Similar to [@Drozdz_2000; @Drozdz_2001], we also find evidence of an increase/decrease in the largest eigenvalue with respect to ‘drawdowns’/‘draw-ups’. Additionally, we find the highlighted *compensatory dynamics* of the small eigenvalues. These results were tested for various time windows and normalisation periods, with smaller windows and normalisation periods found to capture and emphasise additional features. ![(a) S&P Index (b) Normalised Largest Eigenvalue vs. Average of 80 smallest normalised eigenvalues (c) All Normalised Eigenvalues (d) Largest 12 Normalised Eigenvalues[]{data-label="S&PeigsNorm"}](SandPEigenvaluesNormalised.eps){height="120mm" width="130mm"} 2. To demonstrate the above result for a *different level of granularity*, we chose $50$ equities randomly with a time window of $500$ days, giving $Q = \frac{T}{N} = 10$. The results obtained, (Figure \[S&Peigs50\]), are in keeping with those for $Q = 2$ earlier, with a broad-band increase (decrease) of the $40$ smallest eigenvalues concurrent to a decrease (increase) of the largest eigenvalue. ![(a) Normalised Largest Eigenvalue vs. Average of 40 smallest normalised eigenvalues (b) All Normalised Eigenvalues[]{data-label="S&Peigs50"}](SandPEigenvaluesLargeQ.eps){height="70mm" width="130mm"} 3. The previous examples used random subsets of the S&P universe in order to keep $Q= \frac{T}{N}$ as large as possible. To demonstrate that the above results were not sampling artifacts, we also looked at the full sample of 384 equities, (that survived the entire $11$ year period), with a time window of $500$ days ($Q = 1.30$). The results, as shown in Figure \[S&Peigs384\], are similar to those above, with the majority of the small eigenvalues compensating for changes in the large eigenvalue. As indicated previously, however, there is a small band of large eigenvalues, for which behaviour is different to that of both the band of small eigenvalues and the largest eigenvalue. ![(a) Normalised Largest Eigenvalue vs. Average of 325 smallest normalised eigenvalues (b) All Normalised Eigenvalues[]{data-label="S&Peigs384"}](SandPEigenvaluesAllStocks.eps){height="70mm" width="130mm"} 4. All examples discussed so far have focused on the universe of equities from the S&P $500$ that have survived since $1997$. To ensure that the results obtained were not exclusive to the S&P $500$, we also applied the same technique to the $49$ equities of the EuroStoxx 50 index that survived from January $2001$ to August $2007$. The sliding window used was $200$ days, such that $Q = 4.082$. The results, (Figure \[SX5E\]), were again similar to before, with a wide band of small eigenvalues “responding to” movements in the largest eigenvalues. In this case, the band of deviating large eigenvalues (ie. those which correspond to the area outside the “Random Bulk” in RMT), (Figure \[SX5E\](d)), is not as marked as in the previous example. This effectively implies that equities in this index are dominated by “the Market”. ![(a) EuroStoxx 50 Index, Jan 2001 - Aug 2007 (b) Normalised Largest Eigenvalue vs. Average of 40 smallest normalised eigenvalues for EuroStoxx 50 (c) All Normalised Eigenvalues (d) The 9 Largest Normalised Eigenvalues[]{data-label="SX5E"}](SX5EEigenvalues.eps){height="120mm" width="130mm"} Model Correlation Matrix {#corr_model} ------------------------ The results, described, demonstrate that the time dependent dynamics of the small eigenvalues of the correlation matrix of stock returns move counter to those of the largest eigenvalue. Here, we show how a simple one-factor model, Section \[OneFModel\], of the correlation structure reproduces much of this behaviour. Furthermore, we show how additional features can be captured by including perturbations in this model, essentially a *“market plus sectors”* model, Section \[MultiFModel\], [@Malevergne; @Noh; @Papp]. In order to compare the empirical results, Section \[Eresults\], to those of the single factor model, we first constructed a correlation matrix where each non-diagonal element was equal to the average correlation of the empirical matrix in each sliding window. We then calculated the eigenvalues of this matrix over each sliding window and normalised these as before, (Section \[methods\]). The dynamics of the largest eigenvalue for the single-factor model are displayed in Figure \[modelEuroStoxx\](a) for the EuroStoxx 50 index using a sliding window of $200$ days. As can be seen, the main features of the dynamics are in agreement with those of Figure \[SX5E\] for the empirical data. The dynamics of the remaining eigenvalues, shown in figure \[modelEuroStoxx\](c), are found to be within a narrow range, with any change in time due to compensation for change in the largest eigenvalue. For the ‘market plus sectors’ model, we introducted perturbations with two groups of $5$ stocks having correlation $\rho_{0}-0.15$ and $\rho_{0}+0.15$, with the average correlation at each time window remaining the same. The largest eigenvalue dynamics for the ‘market plus sectors’ model is shown in \[modelEuroStoxx\](b), with the dynamics almost identical to those for the largest eigenvalue of the ‘one-factor’ model, (any differences are due to random fluctuations induced in the Cholesky decomposition, Section \[MultiFModel\]). However, looking at the remaining eigenvalues, \[modelEuroStoxx\](d), we see that a number are found to be deviate significantly from the bulk. These deviations are due to the additional sectorial information included in the market plus sectors model. This agrees with previous results, [@Plerou_2000; @Utsuki_2004], where small eigenvalues were found to deviate from the random bulk, in addition to large deviating eigenvalues containing sectorial information. ![(a) Largest Eigenavalue, one factor model (b) Largest Eigenavalue, Market plus sectors model (c) Eigenvalues 1-48, one factor model (d) Eigenvalues 1-48, Market plus sectors model[]{data-label="modelEuroStoxx"}](CompareSingleSectorModel.eps){height="95mm" width="135mm"} To examine properties of the eigenvector components themselves, we use the Inverse Participation Ratio (IPR). Figure \[IPR\](a), displays the IPR found using the empirical data from the Eurostoxx $50$. This has similar characteristics to those found for different indices, [@Plerou_2000], with the IPR for the largest eigenvalue much smaller than the mean, a large IPR corresponding to sectorial information in the $2^{nd}$ or $3^{rd}$ largest eigenvalues and the IPR for the small eigenvalues raised. For the single factor model, we created a *synthetic* correlation matrix using Eqn. (\[chol\]), with average correlation $(0.204)$ equal to that of the Euro Stoxx 50 over the time period studied. As shown in Figure \[IPR\](b), the IPR retains some of the features found for empirical data, [@Plerou_2000; @Noh], with that corresponding to the largest eigenvector having a much smaller value than the mean. This corresponds to an eigenvector to which many stocks contribute, (effectively the market eigenvector), [@Plerou_2000; @Noh]. In an attempt to include additional empirical features, such as the band of deviating large eigenvalues between the bulk and the largest eigenvalue, we also considered a perturbation, with two groups of stocks having correlation $\rho_{0}-0.15$ and $\rho_{0}+0.15$, and the same average correlation. In this case, Figure \[IPR\](b), additional features are found, with larger IPR for both smallest and second largest eigenvalue. This agrees with [@Plerou_2000] where, for empirical data, the group structure resulted in a number of small and large eigenvalues with larger IPR than that of the bulk of eigenvalues. These large eigenvalues were shown, [@Plerou_2000; @Conlon_2007], to be associated with correlation information related to the group structure. ![(a) IPR Empirical Data (b) IPR Simulated Data[]{data-label="IPR"}](IPRsim.eps){height="70mm" width="130mm"} Drawdown Analysis ----------------- As described, [@Drozdz_2000], and discussed above, drawdowns (periods of large negative returns) tend to reflect an increase of one eigenstate of the cross-correlation matrix, with the opposite occurring during drawups. In this section, we attempt to characterise the market according to the relative size of both the largest and small eigenvalues using stocks of the Dow Jones Euro Stoxx 50. Returns, correlation matrix and eigenvalue spectrum time series for overlapping windows of $200$ days were calculated and normalised using the mean and standard deviation over the entire series, (Eqn. \[normalise\]). Representing normalised eigenvalues in terms of standard deviation units (SDU) allows partitioning according to the magnitude of the eigenvalues. Table \[evalueReturns\] shows the average return of the index, (during periods when the largest eigenvalue is $\pm1$ SDU), for both the largest eigenvalue and the average of the normalised $40$ smallest eigenvalues. The results, Table \[evalueReturns\], demonstrate that when the largest eigenvalue is $>1$ SDU, the average index return over a $200$ day period is found to be $-16.80\%$. When it is small ($<-1$ SDU), the average index return is $18.46\%$. Hence, the largest eigenvalue can be used to characterise markets, with the eigenvalue peaking during periods of negative returns (Drawdowns) and bottoming out when the market is rising (Drawup). For the average of the $40$ smallest eigenvalues, the partition also reflected drawdowns and drawups, but with opposite signs. This indicates that information about the correlation dynamics of financial time series is visible in both the lower and upper eigenstates, in agreement with the results found by [@Schindler_2007; @Muller_2005] for both synthetic data and EEG seizure data. *Eigenvalues* *No. Std* *Index Return* --------- --------------- ----------- ---------------- $<$-1 18.46% \[0ex\] $>$1 -16.80% \[0ex\] $<$-1 -23.90% \[0ex\] $>$1 19.32% \[0ex\] Conclusions =========== The correlation structure of multivariate financial time series was studied by investigation of the eigenvalue spectrum of the equal-time cross-correlation matrix. By filtering the correlation matrix through the use of a sliding window we have examined behaviour of the largest eigenvalue over time. As shown, Figures (\[S&PeigsNorm\] - \[SX5E\]), the largest eigenvalue moves counter to that of a band of small eigenvalues, due to *eigenvalue repulsion*. A decrease in the largest eigenvalue, with a corresponding increase in the small eigenvalues, corresponds to a redistribution of the correlation structure across more dimensions of the vector space spanned by the correlation matrix. Hence, additional eigenvalues are needed to explain the correlation structure in the data. Conversely, when the correlation structure is dominated by a smaller number of factors (eg. the “single-factor model” of equity returns), the number of eigenvalues needed to describe the correlation structure in the data is reduced. In the context of previous work, [@Drozdz_2000; @Drozdz_2001], this means that fewer eigenvalues are needed to describe the correlation structure of ‘drawdowns’ than that of ‘draw-ups’. By introducing a simple ‘one-factor model’ of the correlation in the system (Section \[corr\_model\]), we were able to reproduce the main results of the empirical study. The compensatory dynamics, described, were clearly seen for a correlation matrix with all elements equal to the average of the empirical correlation matrix. The model was then adapted, by the addition of pertubations to the correlations, with the average correlation remaining unchanged. This ‘markets plus sectors’ type model was able to reproduce additional features of the empirical correlation matrix with eigenvalues deviating from below and above the bulk. The Inverse Participation Ratio of the “markets plus sectors” model was also shown to have group characteristics typically associated with known Industrial Sectors, with a larger than average value for the smallest eigenvalue and for the second largest eigenvalue. Through a partition of the time-normalised eigenvalues, it was then shown quantitatively that the largest eigenvalue is greatest/smallest during drawdowns/drawups, and vice versa for the small eigenvalues. Suggested future work includes a more detailed study of the relationship between the direction of the market and magnitude of the eigenvalues of the correlation matrix. Studying the multiscaled correlation dynamics over *different granularities* may shed some light on the different collective behaviour of traders with different strategies and time horizons. Additional analysis of high frequency data may also be useful in the characterisation of correlation dynamics, especially prior to market crashes. 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--- abstract: 'Many scenarios require a robot to be able to explore its 3D environment online without human supervision. This is especially relevant for inspection tasks and search and rescue missions. To solve this high-dimensional path planning problem, sampling-based exploration algorithms have proven successful. However, these do not necessarily scale well to larger environments or spaces with narrow openings. This paper presents a 3D exploration planner based on the principles of Next-Best Views (NBVs). In this approach, a Micro-Aerial Vehicle (MAV) equipped with a limited field-of-view depth sensor randomly samples its configuration space to find promising future viewpoints. In order to obtain high sampling efficiency, our planner maintains and uses a history of visited places, and locally optimizes the robot’s orientation with respect to unobserved space. We evaluate our method in several simulated scenarios, and compare it against a state-of-the-art exploration algorithm. The experiments show substantial improvements in exploration time ($2\times$ faster), computation time, and path length, and advantages in handling difficult situations such as escaping dead-ends (up to $20\times$ faster). Finally, we validate the on-line capability of our algorithm on a computational constrained real world MAV.' author: - 'Christian Witting$^{1}$, Marius Fehr$^{2}$, Rik B[ä]{}hnemann$^{2}$, Helen Oleynikova$^{2}$, and Roland Siegwart$^{2}$[^1]' bibliography: - 'IEEEabrv.bib' - 'bibliography/references.bib' title: | **History-aware Autonomous Exploration\ in Confined Environments using MAVs** --- Introduction {#sec:introduction} ============ Autonomous mobile robot exploration has a wide variety of applications such as visual inspection tasks, search-and-rescue missions, 3D reconstruction, or mining [@thrun2010a; @calisi2007a; @yoder2016a; @rosenblatt2002a]. More specifically, Micro-Aerial Vehicles (MAVs) allow unique viewpoints and access to confined and narrow environments [@Zang-2016-4518]. The common goal in all exploration applications is to efficiently explore an unknown environment completely. The research in exploration is classically divided into two fundamental approaches: the frontier-based approach and the sampling-based approach. The frontier approach seeks to maximize the map coverage by identifying and exploring the boundaries between the known and unknown parts of a map, while the sampling-based approach randomly searches the robot’s configuration space for sensor poses which maximize a given objective function. We base our algorithm on the sampling-based approach as it has been proven successful in 3D exploration and can be formulated independently of the underlying objective function, e.g., it allows multi-objective optimization of exploration, scene reconstruction and localization [@bircher2016a; @papachristos2017a; @gonza2002a]. Unfortunately, sampling-based planner performance deteriorates in large environments or confined scenarios featuring small openings or bottle-necks. In particular tree-based exploration has the tendency to get stuck in dead-end situations where the robot needs to reevaluate and revisit already traversed sections of the map in order to find new exploration gain. Thus a large amount of computation time is wasted on searching already visited places. In our Next-Best View Planner (NBVP) we introduce three new components to cope with the curse of dimensionality. First, our planner reduces the sampling space by locally optimizing the orientation of the sensor instead of sampling it randomly. Second, we use path simplification and smoothing methods to shorten traversal time. Third, we maintain and use a history of previously visited positions as seeds of the RRT to quickly find informative regions when the mapped area increases, and the next gain is far away. ![A map build from sensor data of a partly explored simulation of the willowgarage building. The MAV’s position is the axis marker in red, green and blue, with the Rapidly-exploring Random Tree (RRT) shown in dark blue. A sparse version of the history graph is shown in red.[]{data-label="fig:intro_fig"}](images/almost_full.png){width="\linewidth"} We evaluate the proposed components by comparing our planner against a state-of-the art information gain exploration algorithm, both quantitatively and qualitatively in small- and large-scale simulation scenarios. Furthermore, we validate the on-board planning capability in a real world MAV experiment. The main contributions resulting from this work are: - Boosting the RRT planning performance by using a history of exploration potential as seeds. - Increasing the sampling efficiency by maximizing the local sample gain w.r.t. orientation. - Employing dynamics-aware trajectory optimization techniques for trajectory refinement. - Evaluation and validation of our approach in both simulated and real world scenarios. We organize the paper as follows. Section \[otherwork\] presents related work. Section \[nbvp\] introduces NBV planning and elaborates on the particular issues in exploration. In section \[optimized\_nbvp\] we present our planning algorithm. In section \[results\] we benchmark the planner against an existing NBVP in simulation and show a real world application before we close the article with concluding remarks in section \[conclusion\]. Related works {#otherwork} ============= Exploration planning deals with the problem of finding a set of sensor poses along the border of an unknown volume such that eventually the whole volume is explored respecting some path cost, e.g., time, length or energy. As such, the problem is closely related to the art gallery problem, the traveling salesman problem, and the problem of finding a shortest path in 3D environments, which are NP-complete individually [@o1983some; @papadimitriou1977euclidean; @canny1987new]. Additional planning constraints arise from the MAV’s restricted computational and battery resources and its limitations in perception and actuation. The two most established heuristics in MAV exploration planning are frontier-based approaches and sampling-based information gathering approaches. Frontier-based -------------- The frontier-based approach is the classical approach to the exploration problem originally introduced in [@frontier1997a] who also extended it to multiple robots [@yamauchi1998a]. In a partly explored environment there exists a boundary between known and unknown free space which denotes the frontier. Frontier-based methods extract this boundary from a map and plan a path which visits the nearest boundary. In [@holz2010a] and [@julia2012a] the frontier method has been compared to several different exploration algorithms, and [@fraundorfer2012a] proposes a system which applies frontier exploration to MAVs. However, the MAV is kept at an constant height, and the comparisons are mainly made for the 2D case, whereas our method explores freely in 3D. Traditionally, the frontier-based methods pick the closest frontier [@frontier1997a]. [@titus2017a] takes the full 3D movement of the MAV into account when extracting the route to the next frontier. Their planner does not choose the closest frontier, but the one which requires the least change in velocity within the current field of view of the robot. While this approach also achieves faster exploration rates than the NBVP we are comparing against [@bircher2016a], our planner follows the sampling-based approach and thus potentially allows different exploration objectives. Sampling-based -------------- The opposing approach is sampling-based information gathering. Here the fundamental idea is to sample viewpoints in the explored map which could potentially contribute towards the exploration objective. This avoids explicit calculations of frontiers. Since the configuration space is sampled randomly, these planners also allow different optimization objectives without changing the underlying motion planning algorithm [@hollinger2014sampling]. [@bahnemann2017sampling] for example uses this approach to generate MAV system identification trajectories. The concept of NBVs was first introduced in [@connolly1985a], where the authors goal was to obtain a complete model of a scene by calculating a series of covering views. While not dealing explicitly with exploration, the idea of NBVs has been carried over into the exploration domain. [@bircher2016a] uses NBVs in a 3D exploration algorithm. In a receding horizon fashion, their approach iterates between sampling accessible viewpoints in an RRT and executing the most informative path. Our algorithm builds up on theirs but introduces a memory of already visited spaces, local gain optimization, and trajectory optimization which leads to a significantly better sampling efficiency and shorter and faster explorations. [@papachristos2017a] extends [@bircher2016a]’s NBVP to account for uncertainty in the localization and mapping between viewpoints. In [@gonza2002a] the uncertainty and quality of NBVs has been incorporated with respect to a simultaneous localization and mapping (SLAM) framework. In our work we assume reliable state estimation, and focus on rapid exploration. Other Approaches ---------------- [@charrow2015a] and [@visser2008a] combine frontier- and sampling-based approaches. Their algorithms calculate the frontiers explicitly, but sample the poses from which frontiers can be observed. [@charrow2015a] also uses trajectory optimization methods to convert a piece-wise linear planned trajectory into a smooth path that obeys robot dynamics. [@shen2012a] uses particles defined by a stochastic differential equation to explore the environment. [@grabowski2003a] calculates an explicit region-of-interest map based on the ideas of the NBVs. [@xu2017a] uses time-varying tensor fields to construct a topological skeleton of the map. Here the exploration gain is chosen based on the unknown area which can be scanned from the topological skeleton. This method resembles ours in maintaining the history of exploration potential. However, the tensor fields does not directly translate to the 3D case needed for MAV platforms which our work is based on. Limitations of RRT-based NBV planning {#nbvp} ===================================== In this section we introduce sampling-based NBV planning, discuss its limitations and motivate the proposed algorithmic changes. Algorithm --------- 3D NBV planning as introduced in [@bircher2016a] follows a receding horizon approach. Here the robot iterates between sampling new viewpoints uniformly in the current voxel map, and navigating towards the NBV. In the initial step the robot grows an RRT from the current position in $(x, y, z, \theta)$-space, where $x$, $y$ and $z$ describe the position and $\theta$ describes the yaw orientation [@kuffner2000rrt]. The RRT is the backbone of the planning part, and as such is responsible for finding a collision free straight-line path from the current position to a sample with gain. For each new sample that is connected to the nearest neighbor in the tree the algorithm calculates the expected information gain. In [@bircher2016a]’s NBVP, the immediate information gain is calculated by counting the number of visible unknown voxels in the sampled camera frustum exponentially discounted by its shortest distance to the current MAV position. The total gain of a node is the summation of all immediate gains along the RRT branch to the node. This approach of calculating samples and adding them to the tree is repeated for a predefined number of iterations or until a sample with sufficient gain is found. Once a potential gain is found, the MAV navigates edge by edge along the shortest path found in the RRT towards the node with the highest exploration gain. In the mean time a new RRT iteration begins, discarding all previous samples but the best branch. When a better branch is found the robot will update its goal pose. Drawbacks {#sec:drawbacks} --------- One of the limitations of the traditional NBVP for MAVs is the fact that the sampling takes place in the $(x, y, z, \theta)$ configuration space. While the sampling in $x$, $y$, $z$ is obviously desired in order for the RRT algorithm to propagate through the mapped volume, the sampling of the yaw component limits the sample efficiency of the exploration. For a given sampled position near unobserved voxels it is unlikely that the planner will also sample a yaw orientation that faces the camera in the optimal direction into the unobserved region and thus creates a high gain. This is evident in Figure \[fig:gain\_random\] which visualizes the expected gain when sampling the yaw randomly. Additionally, RRTs have a bias toward large Voronoi regions[^2]. Thus, they are sample efficient to span the Euclidean space $\mathbb{R}^3$ but take long to locally refine which is necessary to obtain different sensor viewpoints in the vicinity of the robot. A second limitation of [@bircher2016a] is that the extracted waypoint list from the RRT is used directly as trajectory. However, this has the effect that the resulting movement is jagged due to the randomness of the RRT (illustrated in Figure \[fig:branch\_bad\]). Which in turn results in both long routes, and high start-stop energy consumption. Last but not least, the RRT is a tree-based planning structure that always has its root in the current position of the MAV. This results in a behavior where the RRT needs to be discarded and recalculated after finishing a branch. As time passes and the mapped area increases, the distances to the next gain and thus the sampling time increases significantly. This issue is evident in Figure \[fig:deadend\_old\] where the next gain is far away from the current position and thus the RRT tree grows immensely large. The augmented NBVP {#optimized_nbvp} ================== Based on the limitations identified in Section \[sec:drawbacks\] we propose an augmented sampling-based NBVP. The main intuition behind our changes compared to [@bircher2016a] is to reduce the sampling space to interesting areas such that the RRT can quickly find a NBV. Our NBVP has a three-step sampling approach, a deterministic yaw policy, it generates simplified and smooth shortest path trajectories and maintains a graph of potential RRT seeds in free space as a hot-start in dead-end situations. The exploration algorithm is described in Algorithm \[alg:explore\]. In the initial step the algorithm samples a random tree with potential NBVs within the free space of a user-defined vicinity of the MAV. Each new node gets assigned an exploration gain based solely on the number of unobserved voxels in the expected sensor frustum. If no view with a certain gain is found in the vicinity, the robot may be stuck in a dead-end and will reseed the RRT to the closest node with a potential in the history graph. If the algorithm is still unable to find a gain within the vicinity of the new seed, the sampling will extend to the full free workspace until a sample with gain is found. Set sampling bounds to root vicinity Seed RRT with closest node in history Update sampling bounds Increase sampling bounds to full free space Sample within bounds Grow RRT Extract best branch from RRT Simplify and smooth trajectory Carry out trajectory Once a viewpoint with sufficient gain is found the MAV computes a smoothed trajectory to the node and repeats the sampling approach. Concurrent to the exploration, the robot updates the voxel map and maintains a potential seed graph. Figure \[fig:system\] shows the system in comparison to the receding horizon based planner [@bircher2016a]. ![A comparison of the system structure of the traditional NBVP [@bircher2016a] and our proposed method.[]{data-label="fig:system"}](images/system1.pdf){width="0.95\linewidth"} The mapping framework used for the exploration algorithm is based on Voxblox [@voxblox2017a]. Voxblox takes a planning-centric approach to dense mapping, by maintaining both a Truncated Signed Distance Field (TSDF) and Euclidean Signed Distance Field (ESDF). We use this representation to perform ray casting in the frustum optimization, collision checking in trajectory optimization and to refine the history graph. The three-step sampling approach increases the chances of finding a close NBV and escape dead-ends early. In the following we will elaborate on the extensions in orientation optimization, trajectory smoothing, and history maintainance. Local gain optimization {#sec:gain_theory} ----------------------- Instead of sampling in $(x, y, z, \theta)$-space, we propose to sample only in Euclidean $(x, y, z)$-space and set $\theta$ deterministically to the optimal direction for each sample. The optimal yaw direction is found by ray casting the sensor frustum in the set of $N$ discrete orientations $\theta \in \left\{-\pi, -\pi + \Delta \pi, \ldots, \pi - \Delta \pi \right\}$, where the discrete step size should ideally be set to $\Delta\theta = \frac{r}{R}$ to ensure that all voxels within the view frustum are considered. However, to balance the computational complexity with accuracy this value was increased to $\SI{5}{\degree}$ in the experiments here. Here $r$ is the voxel size of the map in meters, and $R$ is the desired projection range in meters. We approximate the viewing frustum as a circular sector of a cylinder, with radius $R$, height $H = 2 \, R \, \sin\left(\frac{v_{\mathrm{fov}}}{2}\right)$ and central angle $h_{\mathrm{fov}}$, where $v_{fov}$ and $h_{fov}$ are the vertical and horizontal field of view of the camera respectively. To compute the exploration gain for all $N$ frusta, we first precompute the gain of $N$ vertical slices over the whole cylinder originating from each sensor pose. For each of these slices we cast rays from the sensor origin to the voxels on the vertical boundary line of the cylinder. The gain of the slice is the summation of the number of unknown voxels along the rays. If a ray intersects with an obstacle in the map it is stopped short, in order to not count occluded unknown areas behind obstacles. The total gain for each direction is found by summing the slice gains over a window with angular width $h_{fov}$. The direction with the largest gain is then the optimal direction to face for that specific sample. Figure \[fig:comp\_gain\] shows a visual comparison of the expected gain between randomly sampled orientations and optimized orientations. As mentioned in Section \[sec:drawbacks\] it is obvious from Figure \[fig:gain\_random\] that the random sampling misses a lot of the gain in several cases leading to degraded performance. Furthermore, it can be seen from Figure \[fig:gain\_opti\] that the yaw optimization results in finding viewpoints that consistently point towards the frontiers of the map to achieve map coverage. Solution selection and trajectory generation -------------------------------------------- In the original NBVP, [@bircher2016a], the MAV chooses a straight-line path according to the distance and immediate gains along the branch of the tree (see Figure \[fig:branch\_bad\]). In our solution, we do not consider intermediate gain or distance along the path towards the NBV. Our planner directly navigates to the first viewpoint with sufficient exploration gain. Since we locally optimize the orientation, the first informative sampled pose is expected to automatically be the closest gain to the seed of the RRT. This simplification allows discarding intermediate waypoints and performing trajectory optimization. The first step takes the jagged branch and converts it into a minimum straight-line trajectory. The sparse waypoint list is then interpolated by continuous polyonomial trajectories as presented in [@richter2016polynomial] and implemented in [@burri2015real][^3]. This results in short, smooth, and dynamically feasible trajectories as illustrated in Figure \[fig:branch\_bad\_better\_best\]. History maintenance {#sec:history} ------------------- As mentioned in the drawbacks in Section \[sec:drawbacks\], the RRT exploration degrades in large and confined areas when the robot has to find its way towards distant information sources. In this section we specifically address this issue by introducing a graph in free space that stores knowledge about the spatial distribution of information in already explored parts of the map. The nodes of the graph are used to reseed the RRT when no gain is found in the vicinity of the current root. Algorithm \[alg:history\] shows the full history graph algorithm that runs concurrently to the exploration algorithm. The graph nodes consist of positions in free space that are sampled along the travelled path. For every node, we store a measure of the exploration potential while the MAV is navigating. This potential is a representation of exploration gain which is nearby, but not necessarily in view, and is specific to the chosen exploration objective. To update a node, we perform a Breadth First Search (BFS) over the free voxels in its vicinity to count the number of voxel with potential gain (in our case, we count frontier voxels). Here the BFS ensures that there exist a collision-free but not necessarily direct connection to the gain. By doing this for all nodes in the history graph, the potential for all the previous positions is kept up to date. To mitigate the effects of changes in the explored map, we attempt to maximize the clearance to obstacles of nodes in the history graph. This is done by ascending the obstacle distance gradient in the Voxblox ESDF map when refining the poses. This effectively warps the graph into the points of equal distance to the obstacles ensuring the connections remain collision free, approximating a Voronoi graph of the map. As the exploration progresses, some of the nodes in the history graph become redundant when they collapse onto the same position during the refinement. These are continuously combined while keeping the connectivity during the maintenance step. Figure \[fig:large\_maze\] shows a complete history graph in an explored scenario. ### Seeding As described in Algorithm \[alg:explore\] the RRT is reseeded whenever the robot cannot find informative poses in its vicinity. In this case the closest node with potential is extracted from the history graph, and used as a seed for the RRT. This puts the root of the RRT in the vicinity of unexplored gain, and thus the sampling time is significantly reduced, as the RRT no longer needs to sample all the way from the current position. The shortest path from the current position to the root of the RRT can then be extracted from the history graph. Figure \[fig:graph\_seed\] illustrates this process. ![Illustration of the history seeding. Gray denotes unknown area with the current frontier in red. The large golden circle is the current position, and the red graph is the history graph. The golden nodes are previous poses which do not have any exploration potential, while the green node is the closest that does. It has thus been chosen as seed for the RRT tree (shown in blue). The green branch is the best branch through the history graph and RRT to the large blue node which has a exploration gain.[]{data-label="fig:graph_seed"}](images/graph_seed1.png){width="0.9\linewidth"} A comparison of a specific scenario in a dead-end can be seen on Figures \[fig:deadend\_old\] and \[fig:deadend\_new\]. Save and connect current pose to graph Refine position of node with gradient from ESDF Discard refinement Recalculate potential Add to set without potential Combine collapsed nodes Results ======= We tested our algorithm both in simulation and on a real platform. In simulation we benchmark it in a small and a large maze scenario against [@bircher2016a]’s sampling-based NBVP, stressing the reduced computation time, exploration time and exploration path length of our approach. In the real-world experiment, we validate the online-capability of the planner. A video of the system in a range of these experiments can be found on <https://youtu.be/Rp2bIH_e9ig> Small scenario -------------- We use the small maze scenario shown in Figure \[fig:small\_maze\] to quantitatively compare the two approaches. As both the approaches are stochastic, they are run multiple times to account for the variance in the approach. In this scenario the algorithms were executed 20 times each from several different starting spots. We omitted the reseeding in our algorithm in this scenario as it is too small to take effect. Figure \[fig:boxplot\] presents the resulting exploration times. The proposed method is on average approximately $2\times$ faster, and has a significantly smaller variance in the exploration time, showing the merits of the proposed yaw optimization, trajectory smoothing, and vicinity sampling. Especially the yaw optimization contributes to the more deterministic behavior of the algorithm. ![The map of the small test scenario ($\SI{15.5}{\meter} \, \times \, \SI{6.5}{\meter}$). The red dots are the different starting positions used for the simulations.[]{data-label="fig:small_maze"}](images/small_scenario1.png){width="0.9\linewidth"} ![The boxplot for 20 exploration runs in the small scenario with our planner without history and [@bircher2016a] at the same maximum velocity $V_{max}=\SI{1.2}{\meter\per\second}$. Our improvements clearly reduces the exploration time and variance of the approach.[]{data-label="fig:boxplot"}](images/boxplot_small_flat.eps){width="0.95\linewidth"} Large scenario -------------- In the larger maze scenario shown in Figure \[fig:large\_maze\] we compare both approaches qualitatively. The resulting statistics from a single experiment can be seen in Figure \[fig:big\_maze\_plot\]. It shows that the proposed method finds a solution that is both faster and shorter. It also shows the advantage of trajectory simplification and smoothing which results in shorter trajectories with consistent velocity. ![A comparisons with the NBVP and the proposed method in the large scenario with $V_{max}=\SI{1.2}{\meter\per\second}$. The proposed method clearly outperforms the NBVP. Furthermore, trajectory optimization results in faster and shorter exploration.[]{data-label="fig:big_maze_plot"}](figures/new_old_nbvp.eps){width="0.95\linewidth"} A more thorough comparison of the proposed method and the history graph can be seen in Figure \[fig:boxplot\_large\]. Here we ran the large scale experiment at higher velocities which is feasible for the controller with the trajectory optimization but infeasible with [@bircher2016a]’s waypoint following. The plot shows that the history graph results in reduction of the time needed to explore the large scenario. ![The fully explored map of the large scenario ($\SI{30}{\meter} \, \times \, \SI{30}{\meter}$), together with the history graph.[]{data-label="fig:large_maze"}](images/maze_hist.png){width="0.9\linewidth"} \ Computation time ---------------- As mentioned in Section \[sec:drawbacks\] one of the issues with the NBVP was its performance in dead-ends. We have specifically evaluated this scenario here in the case of the large experiment. Here the MAV’s ability to escape the dead-end was evaluated. The result in Figure \[fig:deadend\], show how the seeding of the RRT in this case reduced the computation time significantly from around $\SI{35}{\second}$ to around $\SI{1.4}{\second}$. Furthermore, Figure \[fig:deadend\_old\] shows the excessive tree structure from the RRT without history is shown in blue. The equivalent tree in Figure \[fig:deadend\_new\] is much smaller. Figure \[fig:boxplot\_comp\] shows the computation time for the $10$ experiments on the large scale scenario which exploration results is in Figure \[fig:boxplot\_large\]. The plot shows a significant reduction of the expected maximum computation time per iteration from $\SI{38}{\second}$ to $\SI{2.2}{\second}$. Real world experiment --------------------- The algorithm was also put to test in the real world. Here it was tested in a small room to validate that it could be run onboard a MAV. Everything was run on-board with the robot using a limited field-of-view depth sensor and VoxBlox [@voxblox2017a] for mapping. The on-board state estimation was done with Rovio [@bloesch2017a] with the recently developed localization extension Rovioli [@maplab2017a], and thus no external sensing was used. The experiment ran in a semi-autonomous fashion. This was done by first letting the safety pilot map a small section of the room to give a starting point for the algorithm. As the room which the experiments was conducted in was open the history feature was disabled, but as there was always a potential in the MAV’s current position it would not have had any effect. On each iteration the chosen trajectory was manually approved by a human supervisor. This was done from a safety perspective such that the safety pilot can always operate from a safe position. A figure of the constructed Voxblox map in this scenario can be seen on Figure \[fig:real\_map\]. ![The map constructed in the real world scenario together with the MAV platform used for the experiment. The dimensions of the room was approximately $\SI{9}{\meter} \, \times \, \SI{6}{\meter}$. The pink arrows show edges in the RRT and the yellow arrows show next best sensor poses and gains.[]{data-label="fig:real_map"}](images/real_intro.png){width="0.9\linewidth"} Conclusion ========== In this work we presented an improved sampling-based NBVP for autonomous exploration. Statistical simulations show a reduction in exploration time by a factor of $2$ over an existing sampling-based planner. Our planner overcomes the curse of dimensionality of sampling-based exploration by guiding the sampling towards informative regions. This was accomplished by introducing a history of interesting places to hot-start the exploration algorithm, which reduces the worst-case computation time to find the NBV from $\SI{38}{\second}$ to $\SI{2.2}{\second}$ in our experiments. We formulated a geometrically optimized orientation policy which reduces the sampling space and thus the variance of the approach. The planned trajectories were optimized to be fast, short, and dynamically feasible which allows exploration velocities of up to $V_{max}=\SI{4.5}{\meter\per\second}$ and reduces the overall exploration time. Furthermore, we showed on-board capability in a real exploration experiment on a MAV. [^1]: $^{1}$C. Witting is a Master student at the Faculty of Electrical Engineering, Technical University of Denmark, Lyngby, Denmark. [[email protected]]{} $^{2}$M. Fehr, R. B[ä]{}̈hnemann, H. Oleynikova, R. Siegwart are with the Autonomous Systems Lab (ASL), ETH Z[ü]{}rich, Z[ü]{}rich, Switzerland. [mfehr, brik, oelena, rsiegwart]{}@ethz.ch This research was supported in part by the European Community’s Seventh Framework Programme (grant number n.608849). [^2]: See <http://msl.cs.uiuc.edu/rrt/gallery_2drrt.html>. [^3]: See [github.com/ethz-asl/mav\_trajectory\_generation](github.com/ethz-asl/mav_trajectory_generation).
--- abstract: 'We develop a new fully quantum method for determination of widths for nuclear decay by proton emission where multiple internal reflections of wave packet describing tunneling process inside proton–nucleus radial barrier are taken into account. Exact solutions for amplitudes of wave function, penetrability $T$ and reflection $R$ are found for $n$-step barrier (at arbitrary $n$) which approximates the realistic barrier. In contrast to semiclassical approach and two-potential approach, we establish by this method essential dependence of the penetrability on the starting point $R_{\rm form}$ in the internal well where proton starts to move outside (for example, for $^{157}_{73}{\rm Ta}$ the penetrability is changed up to 200 times; accuracy is $|T+R-1| < 1.5 \cdot 10^{-15}$). We impose a new condition: in the beginning of the proton decay the proton starts to move outside from minimum of the well. Such a condition provides minimal calculated half-life and gives stable basis for predictions. However, the half-lives calculated by such an approach turn out to be a little closer to experimental data in comparison with the semiclassical half-lives. Estimated influence of the external barrier region is up to 1.5 times for changed penetrability.' author: - 'Sergei P. Maydanyuk[^1] and Sergei V. Belchikov[^2]\' title: Calculations of widths in the problem of decay by proton emission --- **PACS numbers:** 03.65.Xp, 23.50.+z, 27.70.+q **Keywords:** tunneling, multiple internal reflections, wave packet, decay by proton emission, penetrability and reflection, half-life Introduction \[introduction\] ============================= Nuclei beyond the proton drip line are ground-state proton emitters, i. e. nuclei unstable for emission of proton from the ground state. Associated lifetimes, ranging from $10^{-6}$ sec to few seconds, are sufficiently long to obtain wealth of spectroscopic information. Experimentally, a number of proton emitters has been discovered in the mass region $A \approx 110$, 150, and 160 (see [@Hofmann.1989; @Hofmann.1995.RCA; @Hofmann.1996; @Davids.1996.PRL] and references in cited papers). A new regions of proton unstable nuclei is supposed to be explored in close future using radioactive nuclear beams. Initially, the parent nucleus is in quasistationary state, and the proton decay may be considered as a process where the proton tunnels through potential barrier. In theoretical study one can select three prevailing approaches [@Aberg.1997.PRC]: approach with distorted wave Born approximation (DWBA), two-potential approach (TPA), and approach for description of penetration through the barrier in terms of one-dimensional semiclassical method (WKBA). In systematical study these approaches are correlated between themselves, while calculation of penetrability of the barrier is keystone in successful estimation of gamma widths. While the third approach studies such a question directly, in the first and second approaches the penetrability of the barrier is not studied and the width is based on correlation between wave functions in the initial and final states. However, the most accurate information on amplitudes and phases of these wave functions and correspondence between them can be obtained from unite picture of penetration of proton through the barrier, which is used in the WKBA approach (up to the second order approximation). The main objective of this paper is to pass from semiclassical unite description of the process of penetration of proton through the barrier used in the WKBA approach to its fully quantum analogue, to put a fully quantum grounds for determination of the penetrability in this problem. In order to provide such a formalism, we have improved method of multiple internal reflections (MIR, see Refs. [@Maydanyuk.2000.UPJ; @Maydanyuk.2002.JPS; @Maydanyuk.2002.PAST; @Maydanyuk.2003.PhD-thesis; @Maydanyuk.2006.FPL]) generalizing it on the radial barriers of arbitrary shapes. In order to realize this difficult improvement, we have restricted ourselves by consideration of the spherical ground-state proton emitters, while nuclear deformations are supposed to be further included by standard way. This advance of the method never studied before allows to describe dynamically a process of penetration of the proton through the barrier of arbitrary shape in fully quantum consideration, to calculate penetrability and reflection without the semiclassical restrictions, to analyze abilities of the semiclassical and other models on such a basis. This paper is organized in the following way. In Sec. \[sec.2\], formalism of the method of multiple internal reflections in description of tunneling of proton through the barrier in proton decay is presented. Here, we give solutions for amplitudes, define penetrability, width and half-life. In Sec. \[sec.3\], results of calculations are confronted with experimental data and are compared with semiclassical ones. Here, using the fully quantum basis of the method, we study a role of the barrier shape in calculations of widths in details. In particular, we observe essential influence of the internal well before the barrier on the penetrability. We discuss shortly possible interconnections between the proposed approach and other fully quantum methods of calculation of widths. In Sec. \[conclusions\], we summarize results. Appendixes include proof of the method MIR and alternative standard approach of quantum mechanics used as test for the method MIR and for the results presented. Theoretical approach \[sec.2\] ============================== An approach for description of one-dimensional motion of a non-relativistic particle above a barrier on the basis of multiple internal reflections of stationary waves relatively boundaries has been studied in number of papers and is known (see [@Fermor.1966.AJPIA; @McVoy.1967.RMPHA; @Anderson.1989.AJPIA] and references therein). Tunneling of the particle under the barrier was described successfully on the basis of multiple internal reflections of the wave packets relatively boundaries (approach was called as *method of multiple internal reflections* or *method MIR*, see Refs. [@Maydanyuk.2000.UPJ; @Maydanyuk.2002.JPS; @Maydanyuk.2002.PAST; @Maydanyuk.2003.PhD-thesis]). In such approach it succeeded in connecting: 1) continuous transition of solutions for packets after each reflection, total packets between the above-barrier motion and the under-barrier tunneling; 2) coincidence of transmitted and reflected amplitudes of stationary wave function in each spatial region obtained by approach MIR with the corresponding amplitudes obtained by standard method of quantum mechanics; 3) all non-stationary fluxes in each step, are non-zero that confirms propagation of packets under the barrier (i. e. their “tunneling”). In frameworks of such a method, non-stationary tunneling obtained own interpretation, allowing to study this process at interesting time moment or space point. In calculation of phase times this method turns out to be enough simple and convenient [@Maydanyuk.2006.FPL]. It has been adapted for scattering of the particle on nucleus and $\alpha$-decay in the spherically symmetric approximation with the simplest radial barriers [@Maydanyuk.2000.UPJ; @Maydanyuk.2002.JPS; @Maydanyuk.2003.PhD-thesis] and for tunneling of photons [@Maydanyuk.2002.JPS; @Maydanyuk.2006.FPL]. However, further realization of the MIR approach meets with three questions. 1) *Question on effectiveness.* The multiple reflections have been proved for the motion above one rectangular barrier and for tunneling under it [@Anderson.1989.AJPIA; @Maydanyuk.2002.JPS; @Maydanyuk.2006.FPL]. However, after addition of the second step it becomes unclear how to separate the needed reflected waves from all their variety in calculation of all needed amplitudes. After obtaining exact solutions of the stationary amplitudes for two arbitrary rectangular barriers [@Maydanyuk.2003.PhD-thesis; @Maydanyuk.2000.UPJ], it becomes unclear how to generalize such approach for barriers with arbitrary complicate shape. In Ref. [@Esposito.2003.PRE] multiple internal reflections of the waves were studied for tunneling through a number of equal rectangular steps separated on equal distances. However, the amplitudes were presented for two such steps only, in approximation when they were separated on enough large distance, and these solutions in approach of multiple internal reflections were based of the amplitudes of total wave function obtained before by standard method (see Appendix A, eqs. (7), (18), (19) in this paper). So, *we come to a serious unresolved problem of realization of the approach of multiple reflections in real quantum systems with complicated barriers*, and clear algorithms of calculation of amplitudes should be constructed. 2\) *Question on correctness.* Whether is interference between packets formed relatively different boundaries appeared? Whether does this come to principally different results of the approach of multiple internal reflections and direct methods of quantum mechanics? Note that such interference cannot be appeared in tunneling through one rectangular barrier and, therefore, it could not visible in the previous papers. 3\) *Question on uncertainty in radial problem.* Calculations of half-lives of different types of decays based on the semiclassical approach are prevailing today. For example, in Ref. [@Buck.1993.ADNDT] agreement between experimental data of $\alpha$-decay half-lives and ones calculated by theory is demonstrated in a wide region of nuclei from $^{106}{\rm Te}$ up to nucleus with $A_{d}=266$ and $Z_{d}=109$ (see Ref. [@Denisov.2005.PHRVA] for some improved approaches). In review [@Sobiczewski.2007.PPNP] methodology of calculation of half-lives for spontaneous-fission is presented (see eqs. (21)–(24) in p. 321). Let us consider proton-decay of nucleus where proton penetrates from the internal region outside with its tunneling through the barrier. , reflected and incident waves turn out to be defined with uncertainty. How to determine them? The semiclassical approach gives such answer: *according to theory, in construction of well known formula for probability we neglect completely by the second (increasing) item of the wave function inside tunneling region* (see Ref. [@Landau.v3.1989], eq. (50.2), p. 221). In result, equality $T^{2} + R^{2} = 1$ has no any sense (where $T$ and $R$ are coefficients of penetrability and reflection). Condition of continuity for the wave function and for total flux is broken at turning point. So, we do not find reflection $R$. We do not suppose on possible interference between incident and reflected waves which can be non zero. The penetrability is determined by the barrier shape inside tunneling region, while internal and external parts do not take influence on it. The penetrability does not dependent on depth of the internal well (while the simplest rectangular well and barrier give another exact result). But, the semiclassical approach is so prevailing that one can suppose that it has enough well approximation of the penetrability estimated. It turns out that if in fully quantum approach to determine the penetrability through the barrier (constructed on the basis of realistic potential of interaction between proton and daughter nucleus) then one can obtain answer “no”. Fully quantum penetrability is a function of new additional independent parameters, it can achieve essential difference from semiclassical one (at the same boundary condition imposed on the wave function). This will be demonstrated below. Decay with radial barrier composed from arbitrary number of rectangular steps \[sec.2.2\] ----------------------------------------------------------------------------------------- Let us assume that starting from some time moment before decay the nucleus could be considered as system composite from daughter nucleus and fragment emitted. It‘s decay is described by a particle with reduced mass $m$ which moves in radial direction inside a radial potential with a barrier. We shall be interesting in the radial barrier of arbitrary shape, which has successfully been approximated by finite number $N$ of rectangular steps: $$V(r) = \left\{ \begin{array}{cll} V_{1}, & \mbox{at } R_{\rm min} < r \leq r_{1} & \mbox{(region 1)}, \\ V_{2}, & \mbox{at } r_{1} \leq r \leq r_{2} & \mbox{(region 2)}, \\ % V_{3}, & \mbox{at } r_{2} \leq r \leq r_{3} & \mbox{(region 3)}, \\ \ldots & \ldots & \ldots \\ % V_{N-1}, & \mbox{at } r_{N-2} \leq r \leq r_{N-1} & \mbox{(region N-1)}, \\ V_{N}, & \mbox{at } r_{N-1} \leq r \leq R_{\rm max} & \mbox{(region N)}, \end{array} \right. \label{eq.2.2.1}$$ where $V_{i}$ are constants ($i = 1 \ldots N$). We define the first region 1 starting from point $R_{\rm min}$, assuming that the fragment is formed here and then it moves outside. We shall be interesting in solutions for above barrier energies while the solution for tunneling could be obtained after by change $i\,\xi_{i} \to k_{i}$. A general solution of the wave function (up to its normalization) has the following form: $$\psi(r, \theta, \varphi) = \frac{\chi(r)}{r} Y_{lm}(\theta, \varphi), \label{eq.2.2.2}$$ $$\chi(r) = \left\{ \begin{array}{lll} e^{ik_{1}r} + A_{R}\,e^{-ik_{1}r}, & \mbox{at } R_{\rm min} < r \leq r_{1} & \mbox{(region 1)}, \\ \alpha_{2}\, e^{ik_{2}r} + \beta_{2}\, e^{-ik_{2}r}, & \mbox{at } r_{1} \leq r \leq r_{2} & \mbox{(region 2)}, \\ % \alpha_{3}\, e^{ik_{3}r} + \beta_{3}\, e^{-ik_{3}r}, & \mbox{at } r_{2} \leq r \leq r_{3} & \mbox{(region 3)}, \\ \ldots & \ldots & \ldots \\ \alpha_{n-1}\, e^{ik_{N-1}r} + \beta_{N-1}\, e^{-ik_{N-1}r}, & \mbox{at } r_{N-2} \leq r \leq r_{N-1} & \mbox{(region N-1)}, \\ A_{T}\,e^{ik_{N}r}, & \mbox{at } r_{N-1} \leq r \leq R_{\rm max} & \mbox{(region N)}, \end{array} \right. \label{eq.2.2.3}$$ where $\alpha_{j}$ and $\beta_{j}$ are unknown amplitudes, $A_{T}$ and $A_{R}$ are unknown amplitudes of transmission and reflection, $Y_{lm}(\theta, \varphi)$ is spherical function, $k_{i} = \frac{1}{\hbar}\sqrt{2m(E-V_{i})}$ are complex wave vectors. We shall be looking for solution for such problem in approach of multiple internal reflections (we restrict ourselves by a case of orbital moment $l=0$ while its non-zero generalization changes the barrier shape which was used as arbitrary before in development of formalism MIR and, so, is absolutely non principal). According to the method of multiple internal reflections, scattering of the particle on the barrier is considered on the basis of wave packet consequently by steps of its propagation relatively to each boundary of the barrier (idea of such approach can be understood the most clearly in the problem of tunneling through the simplest rectangular barrier, see [@Maydanyuk.2002.JPS; @Maydanyuk.2003.PhD-thesis; @Maydanyuk.2006.FPL] and Appendix A where one can find proof of the method and analysis of its properties). Each step in such consideration of propagation of the packet will be similar to on from the first $2N-1$ steps, independent between themselves. From analysis of these steps recurrent relations are found for calculation of unknown amplitudes $A^{(n)}$, $S^{(n)}$, $\alpha^{(n)}$ and $\beta^{(n)}$ for arbitrary step $n$, summation of these amplitudes are calculated. We shall be looking for the unknown amplitudes, requiring wave function and its derivative to be continuous at each boundary. We shall consider the coefficients $T_{1}^{\pm}$, $T_{2}^{\pm}$, $T_{3}^{\pm}$ and $R_{1}^{\pm}$, $R_{2}^{\pm}$, $R_{3}^{\pm}$ as additional factors to amplitudes $e^{\pm i\,k\,x}$. Here, bottom index denotes number of the region, upper (top) signs “$+$” and “$-$” denote directions of the wave to the right or to the left, correspondingly. At the first, we calculate $T_{1}^{\pm}$, $T_{2}^{\pm}$ …$T_{N-1}^{\pm}$ and $R_{1}^{\pm}$, $R_{2}^{\pm}$ …$R_{N-1}^{\pm}$: $$\begin{array}{ll} \vspace{2mm} T_{j}^{+} = \displaystyle\frac{2k_{j}}{k_{j}+k_{j+1}} \,e^{i(k_{j}-k_{j+1}) r_{j}}, & T_{j}^{-} = \displaystyle\frac{2k_{j+1}}{k_{j}+k_{j+1}} \,e^{i(k_{j}-k_{j+1}) r_{j}}, \\ R_{j}^{+} = \displaystyle\frac{k_{j}-k_{j+1}}{k_{j}+k_{j+1}} \,e^{2ik_{j}r_{j}}, & R_{j}^{-} = \displaystyle\frac{k_{j+1}-k_{j}}{k_{j}+k_{j+1}} \,e^{-2ik_{j+1}r_{j}}. \end{array} \label{eq.2.2.4}$$ Using recurrent relations: $$\begin{array}{l} \vspace{1mm} \tilde{R}_{j-1}^{+} = R_{j-1}^{+} + T_{j-1}^{+} \tilde{R}_{j}^{+} T_{j-1}^{-} \Bigl(1 + \sum\limits_{m=1}^{+\infty} (\tilde{R}_{j}^{+}R_{j-1}^{-})^{m} \Bigr) = R_{j-1}^{+} + \displaystyle\frac{T_{j-1}^{+} \tilde{R}_{j}^{+} T_{j-1}^{-}} {1 - \tilde{R}_{j}^{+} R_{j-1}^{-}}, \\ \vspace{1mm} \tilde{R}_{j+1}^{-} = R_{j+1}^{-} + T_{j+1}^{-} \tilde{R}_{j}^{-} T_{j+1}^{+} \Bigl(1 + \sum\limits_{m=1}^{+\infty} (R_{j+1}^{+} \tilde{R}_{j}^{-})^{m} \Bigr) = R_{j+1}^{-} + \displaystyle\frac{T_{j+1}^{-} \tilde{R}_{j}^{-} T_{j+1}^{+}} {1 - R_{j+1}^{+} \tilde{R}_{j}^{-}}, \\ \tilde{T}_{j+1}^{+} = \tilde{T}_{j}^{+} T_{j+1}^{+} \Bigl(1 + \sum\limits_{m=1}^{+\infty} (R_{j+1}^{+} \tilde{R}_{j}^{-})^{m} \Bigr) = \displaystyle\frac{\tilde{T}_{j}^{+} T_{j+1}^{+}} {1 - R_{j+1}^{+} \tilde{R}_{j}^{-}}, \end{array} \label{eq.2.2.5}$$ and selecting as starting the following values: $$\begin{array}{ccc} \tilde{R}_{N-1}^{+} = R_{N-1}^{+}, & \tilde{R}_{1}^{-} = R_{1}^{-}, & \tilde{T}_{1}^{+} = T_{1}^{+}, \end{array} \label{eq.2.2.6}$$ we calculate successively coefficients $\tilde{R}_{N-2}^{+}$ …$\tilde{R}_{1}^{+}$, $\tilde{R}_{2}^{-}$ …$\tilde{R}_{N-1}^{-}$ and $\tilde{T}_{2}^{+}$ …$\tilde{T}_{N-1}^{+}$. At finishing, we determine coefficients $\beta_{j}$: $$\begin{array}{l} \vspace{1mm} \beta_{j} = \tilde{T}_{j-1}^{+} \Bigl(1 + \sum\limits_{m=1}^{+\infty} (\tilde{R}_{j}^{+} \tilde{R}_{j-1}^{-})^{m} \Bigr) = \displaystyle\frac{\tilde{T}_{j-1}^{+}} {1 - \tilde{R}_{j}^{+} \tilde{R}_{j-1}^{-}}, \end{array} \label{eq.2.2.7}$$ the amplitudes of transmission and reflection: $$\begin{array}{cc} A_{T} = \tilde{T}_{N-1}^{+}, & A_{R} = \tilde{R}_{1}^{+} \end{array} \label{eq.2.2.8}$$ and corresponding coefficients of penetrability $T$ and reflection $R$: $$\begin{array}{cc} T_{MIR} = \displaystyle\frac{k_{n}}{k_{1}}\; \bigl|A_{T}\bigr|^{2}, & R_{MIR} = \bigl|A_{R}\bigr|^{2}. \end{array} \label{eq.2.2.9}$$ We check the property: $$\begin{array}{ccc} \displaystyle\frac{k_{n}}{k_{1}}\; |A_{T}|^{2} + |A_{R}|^{2} = 1 & \mbox{ or }& T_{MIR} + R_{MIR} = 1. \end{array} \label{eq.2.2.10}$$ which should be the test, whether the method MIR gives us proper solution for wave function. Now if energy of the particle is located below then height of one step with number $m$, then for description of transition of this particle through such barrier with its tunneling it shall need to use the following change: $$k_{m} \to i\,\xi_{m}. \label{eq.2.2.11}$$ For the potential from two rectangular steps (with different choice of their sizes) after comparison between the all amplitudes obtained by method of MIR and the corresponding amplitudes obtained by standard approach of quantum mechanics, we obtain coincidence up to first 15 digits. Increasing of number of steps up to some thousands keeps such accuracy and fulfillment of the property (\[eq.2.2.10\]) (see Appendix \[sec.app.2\] where we present shortly the standard technique of quantum mechanics applied for the potential (\[eq.2.2.1\]) and all obtained amplitudes). This is important test which confirms reliability of the method MIR. So, we have obtained full coincidence between all amplitudes, calculated by method MIR and by standard approach of quantum mechanics, and that is way we generalize the method MIR for description of tunneling of the particle through potential, consisting from arbitrary number of rectangular barriers and wells of arbitrary shape. Width $\Gamma$ and half-life \[sec.2.3\] ---------------------------------------- We define width $\Gamma$ of the decay of the studied quantum system by following the procedure of Gurvitz and Kälbermann [@Gurvitz.1987.PRL]: $$\Gamma = P_{p}\, F\: \displaystyle\frac{\hbar^{2}}{4m}\; T, \label{eq.2.3.1}$$ where $P_{p}$ is the preformation probability and $F$ is the normalization factor. $T$ is the penetrability coefficient in propagation of the particle from the internal region outside with its tunneling through the barrier, which we shall calculate by approach MIR or by approach WKB. In approach WKB we define it so: $$T_{WKB} = \exp\; \Biggl\{ -2 \displaystyle\int\limits_{R_{2}}^{R_{3}} \sqrt{\displaystyle\frac{2m}{\hbar^{2}}\: \Bigl(Q - V(r)\Bigr)} \; dr \Biggr\} \label{eq.2.3.2}$$ where $R_{2}$ and $R_{3}$ are the second and third turning points. According to [@Buck.1993.ADNDT], the *normalization factor* $F$ is given by simplified way by $F_{1}$ or by improved way by $F_{2}$ so: $$\begin{array}{ll} F_{1} = \Biggl\{\: \displaystyle\int\limits_{R_{1}}^{R_{2}} \displaystyle\frac{dr}{2k(r)} \Biggr\}^{-1}, & F_{2} = \Biggl\{\: \displaystyle\int\limits_{R_{1}}^{R_{2}} \displaystyle\frac{1}{k(r)}\; \cos^{2} \Biggl[\:\displaystyle\int\limits_{R_{1}}^{r} k(r')\; dr' - \displaystyle\frac{\pi}{4} \Biggr]\; dr \Biggr\}^{-1}. \end{array} \label{eq.2.3.3}$$ The half-life $\tau$ of the decay is related to the width $\Gamma$ by well known expression: $$\tau = \hbar\; \ln 2 / \Gamma. \label{eq.2.3.5}$$ For description of interaction between proton and the daughter nucleus we shall use the spherical symmetric proton–nucleus potential (at case $l=0$) in Ref. [@Becchetti.1969.PR] having the following form: $$V (r, l, Q) = v_{C} (r) + v_{N} (r, Q) + v_{l} (r), \label{eq.2.4.1}$$ where $v_{C} (r)$, $v_{N} (r, Q)$ and $v_{l} (r)$ are Coulomb, nuclear and centrifugal components $$\begin{array}{lll} v_{N} (r, Q) = \displaystyle\frac{V_{R}(A,Z,Q)} {1 + \exp{\displaystyle\frac{r-r_{m}} {d}}}, & v_{l} (r) = \displaystyle\frac{l\,(l+1)} {2mr^{2}}, & v_{C} (r) = \left\{ \begin{array}{ll} \displaystyle\frac{Z e^{2}} {r}, & \mbox{for } r \ge r_{m}, \\ \displaystyle\frac{Z e^{2}} {2 r_{m}}\; \biggl\{ 3 - \displaystyle\frac{r^{2}}{r_{m}^{2}} \biggr\}, & \mbox{for } r < r_{m}, \end{array} \right. \end{array} \label{eq.2.4.2}$$ Here, $A$ and $Z$ are the nucleon and proton numbers of the daughter nucleus, $Q$ is the $Q$-value for the proton-decay, $V_{R}$ is the strength of the nuclear component, $R$ is radius of the daughter nucleus, $r_{m}$ is the effective radius of the nuclear component, $d$ is diffuseness. All parameters are defined in Ref. [@Becchetti.1969.PR]. Note that in this paper we are concentrating on the principal resolution of question to provide fully quantum basis for calculation of the penetrability and half-life in the problem of the proton decay, while the proton–nucleus potential can be used in simple form that does not take influence on the reliability of the developed methodology of multiple internal reflections absolutely and could be naturally included for modern more accurate models. Results \[sec.3\] ================= Today, there are a lot of modern methods able to calculate half-lives, which have been studied experimentally well. So, we have a rich theoretical and experimental material for analysis. We shall use these nuclei: $^{157}_{73}{\rm Ta}$, $^{161}_{75}{\rm Re}$ and $^{167}_{77}{\rm Ir}$. Such a choice we explain by that they have small coefficient of quadruple deformation $\beta_{2}$ and at good approximation can be considered as spherical (we have $l=0$). We shall study proton-decay on the basis of leaving of the particle with reduced mass from the internal region outside with its tunneling through the barrier. This particle is supposed to start from $R_{\rm min} \le r \le R_{1}$ and move outside. Using technique of the $T^{\pm}_{j}$ and $R^{\pm}_{j}$ coefficients in eqs. (\[eq.2.2.4\])–(\[eq.2.2.6\]), we calculate total amplitudes of transmission $A_{T}$ and reflection $A_{R}$ by eqs. (\[eq.2.2.8\]), the penetrability coefficient $T_{MIR}$ by eqs. (\[eq.2.2.9\]). We check the found amplitudes, coefficients $T_{MIR}$ and $R_{MIR}$ comparing them with corresponding amplitudes and coefficients calculated by standard approach of quantum mechanics presented in Appendix \[sec.app.2\]. We restrict ourselves by eq. (\[eq.2.3.3\]) for $F_{1}$ and find width $\Gamma$ by eq. (\[eq.2.3.1\]) and half-live $\tau_{MIR}$ by eq. (\[eq.2.3.5\]). We define the penetrability $T_{WKB}$ by eq. (\[eq.2.3.2\]), calculate $\Gamma$-width and half-live $\tau_{WKB}$ by eqs. (\[eq.2.3.1\]) and (\[eq.2.3.5\]). Dependence of the penetrability on the starting point \[sec.3.2\] ----------------------------------------------------------------- The first interesting result which we have obtained is *essential dependence of penetrability on the position of the first region where we localize the wave incidenting on the barrier*. In particular, we have analyzed how much the internal boundary $R_{\rm min}$ takes influence on the penetrability. Taking into account that width of each interval is 0.01 fm, we consider point $R_{\rm min}$ as a *starting point* (with error up to 0.01 fm), from here proton begins to move outside and is incident on the internal part of the barrier in the first stage of the proton decay. In the Fig. \[fig.1\] \[left panel\] one can see that half-live of the proton decay of $^{157}_{73}{\rm Ta}$ is changed essentially at displacement of $R_{\rm min}$. So, we establish *essential dependence of the penetrability on the starting point $R_{\rm start}$, where the proton starts to move outside by approach MIR.* ![ Proton-decay for the $^{157}_{73}{\rm Ta}$ nucleus: the left panel is for dependence of the half-life $\tau_{MIR}$ on the starting point $R_{\rm min}$, the central panel is for dependence on penetrability $T_{MIR}$ on the external boundary $R_{\rm max}$, the right panel is for dependence of the half-live $\tau_{MIR}$ on $R_{\rm max}$ (here, we use $R_{\rm form}=7.2127$ fm where calculated $\tau_{MIR}$ at $R_{\rm max}=250$ fm coincides with experimental data $\tau_{\rm exp}$ for this nucleus). In all calculations factor $F$ is the same. \[fig.1\]](Figures/Proton-decay_Ta157-73_starting_point_fig1.eps "fig:"){width="60mm"} ![ Proton-decay for the $^{157}_{73}{\rm Ta}$ nucleus: the left panel is for dependence of the half-life $\tau_{MIR}$ on the starting point $R_{\rm min}$, the central panel is for dependence on penetrability $T_{MIR}$ on the external boundary $R_{\rm max}$, the right panel is for dependence of the half-live $\tau_{MIR}$ on $R_{\rm max}$ (here, we use $R_{\rm form}=7.2127$ fm where calculated $\tau_{MIR}$ at $R_{\rm max}=250$ fm coincides with experimental data $\tau_{\rm exp}$ for this nucleus). In all calculations factor $F$ is the same. \[fig.1\]](Figures/Proton-decay_Ta157-73_penetrability_Rmax.eps "fig:"){width="60mm"} ![ Proton-decay for the $^{157}_{73}{\rm Ta}$ nucleus: the left panel is for dependence of the half-life $\tau_{MIR}$ on the starting point $R_{\rm min}$, the central panel is for dependence on penetrability $T_{MIR}$ on the external boundary $R_{\rm max}$, the right panel is for dependence of the half-live $\tau_{MIR}$ on $R_{\rm max}$ (here, we use $R_{\rm form}=7.2127$ fm where calculated $\tau_{MIR}$ at $R_{\rm max}=250$ fm coincides with experimental data $\tau_{\rm exp}$ for this nucleus). In all calculations factor $F$ is the same. \[fig.1\]](Figures/Proton-decay_Ta157-73_half-life_Rmax.eps "fig:"){width="60mm"} At $R_{\rm form}=7.2127$ fm this dependence allows us to achieve very close coincidence between the half-live calculated by the approach MIR and experimental data. Dependence of the penetrability on the external region \[sec.3.3\] ------------------------------------------------------------------ The region of the barrier located between turning points $R_{2}$ and $R_{3}$ is main part of the potential used in calculation of the penetrability in the semiclassical approach (up to the second correction), while the internal and external parts of this potential do not take influence on it. Let us analyze whether convergence exists in calculations of the penetrability in the approach MIR if to increase the external boundary $R_{\rm max}$ ($R_{\rm max} > R_{3}$). Keeping width of each interval (step) to be the same, we shall increase $R_{\rm max}$ (through increasing number of intervals in the external region), starting from the external turning point $R_{3}$, and calculate the corresponding penetrability $T_{MIR}$. In Fig. \[fig.1\] \[central panel\] one can see how the penetrability is changed for $^{157}_{73}{\rm Ta}$ with increasing $R_{\rm max}$. Dependence of the half-life $\tau_{MIR}$ on $R_{\rm max}$ is shown in the next figure \[fig.1\] \[right panel\]. One can see that the method MIR gives convergent values for the penetrability and half-life at increasing of $R_{\rm max}$. From such figures we find that *inclusion of the external region into calculations changes the half-life up to 1.5 times* ($\tau_{\rm min}=0.20$ sec is the minimal half-life calculated at $R_{3} \le R_{\rm max}\le 250$ fm, $\tau_{\rm as}=0.30$ sec is the half-life calculated at $R_{\rm max}=250$ fm, ${\rm error} = \tau_{\rm as} / \tau_{\rm min} \approx 1.5$ or 50 percents). So, *error in determination of the penetrability in the semiclassical approach (if to take the external region into account) is expected to be the same as a minimum on such a basis*. Results for the proton emitters $^{157}_{73}{\rm Ta}$, $^{161}_{75}{\rm Re}$ and $^{167}_{77}{\rm Ir}$ \[sec.3.4\] ------------------------------------------------------------------------------------------------------------------ So, the fully quantum study of the penetrability of the barrier for the proton decay give us its large dependence on the starting point. In order to give power of predictions of half-lives calculated by the approach MIR, we need to find recipe able to resolve such uncertainty in calculations of the half-lives. We shall introduce the following hypothesis: *we shall assume that in the first stage of the proton decay proton starts to move outside the most probably at the coordinate of minimum of the internal well*. If such a point is located in the minimum of the well, the half-live obtains minimal value. So, as criterion we could use minimum of half-live for the given potential, which has stable basis. Let us analyze which results such approach gives. We shall compare the half-lives calculated by approach MIR and by the semiclassical approach with experimental data. We should take into account that the half-lives obtained before are for the proton occupied ground state while it needs to take into account probability that this state is empty in the daughter nucleus. In order to obtain proper values for the half-lives we should divide them on the spectroscopic factor $S$ (which we take from [@Aberg.1997.PRC]), and then to compare them with experimental data. Results of such calculations are presented in Table 1. One can see that the calculated half-lives by MIR approach turn out to be a little closer to experimental data in comparison with half-lives obtained by the semiclassical approach. ----------------------- ---------- ------------------ ----------------------- ----------------------- ---------------------- ---------------------- --------------------- ---------------- ---------------- Nucleus $Q$, MeV $S_{p}^{\rm th}$ $\tau_{WKB}$ $\tau_{MIR}$ $\tilde{\tau}_{WKB}$ $\tilde{\tau}_{MIR}$ $\tau_{\rm exp}$ $R_{\rm form}$ $R_{\rm 2,tp}$ $^{157}_{73}{\rm Ta}$ 0.947 0.66 $1.313\cdot 10^{-1}$ $1.369\cdot 10^{-1}$ $1.99\cdot 10^{-1}$ $2.074\cdot 10^{-1}$ $3.00\cdot 10^{-1}$ $3.1$ 7.43 $^{161}_{75}{\rm Re}$ 1.214 0.59 $1.5352\cdot 10^{-4}$ $1.5314\cdot 10^{-4}$ $2.602\cdot 10^{-4}$ $2.596\cdot 10^{-4}$ $3.70\cdot 10^{-4}$ 3.32 7.34 $^{167}_{77}{\rm Ir}$ 1.086 0.51 $2.981\cdot 10^{-2}$ $2.979\cdot 10^{-2}$ $5.85\cdot 10^{-2}$ $5.84\cdot 10^{-2}$ $1.10\cdot 10^{-2}$ 3.41 7.46 ----------------------- ---------- ------------------ ----------------------- ----------------------- ---------------------- ---------------------- --------------------- ---------------- ---------------- : Experimental and calculated half-lives of the ground state proton emitters $^{157}_{73}{\rm Ta}$, $^{161}_{75}{\rm Re}$ and $^{167}_{77}{\rm Ir}$. Here, $S_{p}^{\rm th}$ is theoretical spectroscopic factor, $\tau_{WKB}$ is half-life calculated by in the semiclassical approach, $\tau_{MIR}$ is half-life calculated by in the approach MIR, $\tilde{\tau}_{WKB} = \tau_{WKB}/\, S_{p}^{\rm th}$, $\tilde{\tau}_{MIR} = \tau_{MIR} / S_{p}^{\rm th}$, $\tau_{\rm exp}$ is experimental data, $R_{\rm form}$ is starting point in the internal well where the proton begins to move outside in the first stage of the proton decay, $R_{\rm tp}$ is turning point (values for $S_{p}^{\rm th}$, $\tau_{\rm exp}$ are used from Table IV in Ref. [@Aberg.1997.PRC], p. 1770; in calculations for each nucleus we use: $R_{\rm min}=0.11$ fm, $R_{\rm max}=250$ fm; number of intervals in region from $R_{\rm min}$ to 5 fm is 2000, in region from 5 fm to 8 fm is 500, in region from 8 fm to $R_{\rm max}$ is 5000) \[table.1\] Comparison with other approaches of calculations of widths of proton decay \[sec.3.5\] -------------------------------------------------------------------------------------- Half-life of the proton decay is defined on the basis of width $\Gamma$ which can be calculated by different approaches. For determination of width we shall use systematics of different approaches proposed in Ref. [@Aberg.1997.PRC]. The proton emitters are narrow resonances with extremely small widths. Perturbative approach based on standard reaction theory could be expected to be accurate. Let us analyze two following approaches in such a direction. ### The distorted wave Born approximation method \[sec.3.5.1\] The resonance width can be expressed through transition amplitude, which in the distorted wave Born approximation (DWBA) is given so [@Aberg.1997.PRC]: $$T_{A+1,Z+1;\, A,Z} = \langle \psi_{Ap}\, \Psi_{Ap}\, \bigl| V_{Ap} \bigr|\, \Psi_{A+1} \rangle. \label{eq.2.1.1}$$ The DWBA calculations of the decay width requires knowledge of the quasistationary initial state wave function, $\Psi_{A+1}$, the final state wave function, $\Psi_{Ap}\,\psi_{Ap}$, and interaction potential. The initial state wave function, $\Psi_{A+1}$, is written as a product of the daughter-nucleus wave function, $\Phi_{A}$, and the proton wave function, $\Phi_{nlj}$. The radial wave function of the proton $\psi_{l}(r) = \Psi_{l}(r)/r$ is found by numerically integrating the Schrödinger equation with one-body potential, and it should be irregular part of the Coulomb wave function, $G_{l}(r)$, in asymptotic region. So, such wave function is complex and it defines non-zero flux. As we use condition of continuity of total flux (i.e. absence of sources inside spatial region) we cannot obtain zero wave function in whole region of its definition, and at $r=0$, in particular. In the final state the wave function of the decaying nuclear system can be written as a product of the intrinsic wave function of the proton and the daughter nucleus (an inner core). Radial part of the proton wave function is $\psi_{l}(r) \sim F_{l}(r)/r$, where $F_{l}(r)$ is the regular Coulomb function. By other words, this wave function is real, and, therefore, it gives zero flux determined on the basis of the total wave function in the initial state. The total wave functions in the initial and final states correspond to different processes (with different total fluxes). This confirms that they, complete wave functions, do not take reflection from the barrier inside the internal region into account (but they are defined by different boundary conditions in the initial and final states only). Here, question about determination of the decay width is passed on successful determination of perturbation of the potential (that has another basis for the definition of the decay width as definition on the basis of the penetrability of the barrier). However, *the question about separation of the total wave function in the internal region before the barrier into the incident and reflected waves remains unresolved in the DWBA method.* Now, if we pass from real radial potential in optical model approach to complex one, then we shall introduce new additional independent parameter into our problem while the penetrability could be calculated for real radial barrier. Essential point in determination of the decay width in the DWBA method is accurate normalization of the wave functions in the initial and final states. It could introduce some (essential) uncertainty in calculation of width also while the penetrability is independent on such normalization absolutely. One can calculate the decay width through time-reversed capture process. However, in such calculations shape of the barrier is approximated by inverse oscillator (or other potentials with knowing exact solutions of the wave function) and the penetrability for such a barrier could be calculated. It is clear that both internal well and external region do not take influence on results absolutely (like calculations in semiclassical approach). But, this is possible to resolve this problem accurately and taking whole studied shape of the potential barrier into account that we have demonstrated above in the fully quantum approach MIR. ### The two-potential method \[sec.3.5.2\] In the modified two-potential approach (TPA) introduced by Gurvitz and Kalbermann in [@Gurvitz.1987.PRL] (details and examples can be found in [@Gurvitz.1988.PRA], see also [@Jackson.1977.AP; @Aberg.1997.PRC; @Gurvitz.2004.PRA]) the decay width is defined so (see (16) in [@Aberg.1997.PRC], and some details): $$\Gamma = \displaystyle\frac{4\mu}{\hbar^{2}k}\: \Biggl| \displaystyle\int\limits_{r_{B}}^{\infty} \psi_{nlj}(r)\, W(r)\, \chi_{l}(r)\; dr \Biggr|^{2}, \label{eq.2.2.1}$$ where $k=\sqrt{2\mu E_{0}} / \hbar$, $\mu$ is reduced mass, $r_{B}$ is radial coordinate of the barrier height, $\psi_{nlj}(r)$ is the radial wave function for the first radial potential including internal well up to point $r_{B}$, $\chi_{l}(r)$ is the regular radial wave function for the second radial potential including external region, starting from point $r_{B}$ and without the internal well and with asymptotic behavior $$\begin{array}{ccc} \chi_{l} (0) = 0, & \chi_{l} (r) \to \sin(kr - \pi l/2 + \delta_{l}) & {\rm at}\; r \to \infty. \end{array} \label{eq.2.2.2}$$ Both wave functions are real and defined at different energy levels. So, in the TPA approach we do not consider fluxes and do not calculate penetrability. We do not study possible reflection of proton from the barrier in the internal well. We escape from a problem of separation of the total wave function in the internal well into the incident and reflected waves which takes influence on the resulting penetrability essentially (for example, *for the simplest rectangular barrier with rectangular well such an uncorrect separation of the same exact wave function can give infinite penetrability* that is explained by increased role of interference between incident and reflected waves). Success in obtaining the resulting width $\Gamma$ is dependent on accuracy of correspondence between internal and external wave functions $\psi_{nlj}(r)$ and $\chi_{l}(r)$ which should be calculated from different Schrödinger equations with independent normalization. The correspondence between these wave functions is determined concerning only one boundary point $r_{B}$ (or it possible shift [@Gurvitz.2004.PRA]) separating two potentials and boundary conditions at $r=0$ or at $r \to \infty$. In contrary, the correspondence between the incident, transmitted and reflected waves in the MIR approach is determined concerning the barrier as the whole potential (with needed restrictions of the radial problem) that corresponds to fully quantum and unified consideration of penetration of the proton through the barrier shown in principle of *non-locality* of quantum mechanics. In particular, the transmitted wave in the MIR approach is dependent on the depth of the internal well and its shape, while the external wave function $\chi_{l}(r)$ in the TPA approach is absolutely independent on these depth and shape (such a dependence can be found in the wave function $\varphi_{nlj}(r)$, but starting from the simplest WKB approach factor F directly includes it also). By other words, we have strong correspondence between incident, reflected and transmitted waves in the MIR approach and a possible week correspondence between the internal and external wave functions in the TPA approach. This plays the essential role in calculations of the decay widths and explains so large difference between the essential dependence of penetrability on the starting point in the MIR approach and practically full absence of such a dependence in the TPA approach. The simplest example demonstrated why this dependence really exists and it could be not small, can be found in classical tasks of quantum mechanics. Let us consider definition of the penetrability in [@Landau.v3.1989] (see eq. (25.3), p. 103): $$D = \displaystyle\frac{k_{2}}{k_{1}}\: |A|^{2}, \label{eq.2.1.1}$$ where $D$ is the penetrability, $k_{1}$ and $k_{2}$ are wave numbers of transmitted and incident waves, i.e. concerning the left asymptotic part of the potential and its asymptotic right part (see Fig. 5 in [@Landau.v3.1989], p. 103), $A$ is the transmitted amplitude of the wave function. This formula demonstrates that decreasing of the left part of potential increases the wave number $k_{1}$ (as is connected with asymptotic presentations (25.1) and (25.2) of waves) and, so, changes the total penetrability $D$. Result on the essential dependence of the penetrability of the starting point $R_{\rm form}$ above has the similar sense, but has been obtained concerning the realistic barrier with the internal well and takes into account change of the internal amplitudes also. This contradicts with a possible little dependence of penetrability on the shape of the internal well in the TPA approach. So, these points seem to be reduction of the TPA approach, and confirm that *this approach does not determine the penetrability in the fully quantum consideration in the problem of proton decay*. At the same time, comparison of results obtained by such approach and results obtained by principally other fully quantum developments sometimes leads to some confusion as the TPA approach has been called as the fully quantum. So, approaches for determination of the decay widths on the basis of penetrability are physically motivated, could be more accurate and have perspective for research. Conclusions \[conclusions\] =========================== The new fully quantum method (called as the method of multiple internal reflections, or MIR) for calculation of widths for the decay of the nucleus by emission of proton in the spherically symmetric approximation and the realistic radial barrier of arbitrary shape is presented. Note the following: - Solutions for amplitudes of wave function (described motion of the proton from the internal region outside with its tunneling through the barrier), penetrability $T$ and reflection $R$ are found by the method MIR for $n$-step radial barrier at arbitrary $n$. These solutions are *exactly solvable* and have been obtained in *the fully quantum approach* for the first time. At limit $n \to \infty$ these solutions could be considered as exact ones for the realistic proton–nucleus potential with needed arbitrary barrier and internal hole. Estimated error of the achieved results is $|T+R-1| < 1.5 \cdot 10^{-15}$. - In contrast to the semiclassical approach and the TPA approach, the approach MIR gives essential dependence of the penetrability on the starting point $R_{\rm form}$ inside the internal well where proton starts to move outside in the beginning of the proton decay. For example, the penetrability of the barrier calculated by MIR approach for $^{157}_{73}{\rm Ta}$ is changed up to 200 times in dependence on position of $R_{\rm form}$ (see Fig. \[fig.1\], the left panel). The amplitudes calculated by MIR approach we compared with the corresponding amplitudes obtained (for the same potential) by independent standard stationary approach of quantum mechanics presented in Appendix \[sec.app.2\] and we obtained coincidence up to first 15 digits for all considered amplitudes. This important test confirms that *presence of the essential dependence of the penetrability of the starting point $R_{\rm form}$ is result independent on the fully quantum method applied*. Such a result could be connected with a possibility to introduce initial condition which could be imposed on proton decay in the fully quantum consideration. Comparison with the WKB and TPA approaches shows that such approaches have no such a perspective (having physical sense and opening a possibility to obtain a new information about the proton decay), which fully quantum study of the penetrability gives. - In order to resolve uncertainty in calculations of the half-lives caused by the dependence of the penetrability on $R_{\rm form}$, we have introduced the hypothesis: *in the first stage of the proton decay the proton starts to move outside at the coordinate of minimum of the internal well*. Such condition provides minimal value for the calculated half-life and gives stable basis for predictions in the MIR approach. However, the half-lives calculated by the MIR approach turn out to be a little closer to experimental data in comparison with the half-lives obtained by the semiclassical approach (see Tabl. 1). - Taking the external region of the potential after the barrier into account, half-live calculated by the MIR approach is changed up to 1.5 times (see Fig. \[fig.1\], the right panel). A main advance of the MIR method developed in this paper is not a new attempt to describe experimental data of half-lives more accurately than other approaches do this, but rather this method seems to be the first tools for estimation of the penetrability of any desirable barrier of the proton decay in the fully quantum consideration. Tunneling of packet through one-dimensional rectangular step \[sec.app.1\] ========================================================================== Main ideas and formalism of the multiple internal reflections can be the most clearly analyzed in the simplest problem of tunneling of the particle through one-dimensional rectangular barrier in whole axis [@Maydanyuk.2000.UPJ; @Maydanyuk.2002.JPS; @Maydanyuk.2002.PAST; @Maydanyuk.2003.PhD-thesis; @Maydanyuk.2006.FPL]. Let us consider a problem of tunneling of a particle in a positive $x$-direction through an one-dimensional rectangular potential barrier (see Fig. \[fig.2\]). Let us label a region I for $x < 0$, a region II for $0 < x < a$ and a region III for $x > a$, accordingly. We shall study an evolution of its tunneling through the barrier. ![Tunneling of the particle through one-dimensional rectangular barrier \[fig.2\]](Figures/Mir_fig_2_1.eps){width="50mm"} In standard approach, with energy less than the barrier height the tunneling evolution of the particle is described using a non-stationary propagation of WP $$\psi(x, t) = \int\limits_{0}^{+\infty} g(E - \bar{E}) \varphi(k, x) e^{-iEt/\hbar} dE, \label{eq.2.1.1}$$ where stationary WF is: $$\varphi(x) = \left\{ \begin{array}{ll} e^{ikx}+A_{R}e^{-ikx}, & \mbox{for } x<0, \\ \alpha e^{\xi x} + \beta e^{-\xi x}, & \mbox{for } 0<x<a, \\ A_{T} e^{ikx}, & \mbox{for } x>a \end{array} \right. \label{eq.2.1.2}$$ and $k = \frac{1}{\hbar}\sqrt{2mE}$, $\xi = \frac{1}{\hbar}\sqrt{2m(V_{1}-E)}$, $E$ and $m$ are the total energy and mass of the particle, accordingly. The weight amplitude $g(E - \bar{E})$ can be written in the standard gaussian form and satisfies to a requirement of the normalization $\int |g(E - \bar{E})|^{2} dE = 1$, value $\bar{E}$ is an average energy of the particle. One can calculate coefficients $A_{T}$, $A_{R}$, $\alpha$ and $\beta$ analytically, using a requirements of a continuity of WF $\varphi(x)$ and its derivative on each boundary of the barrier. Substituting in eq. (\[eq.2.1.1\]) instead of $\varphi(k, x)$ the incident $\varphi_{inc}(k, x)$, transmitted $\varphi_{tr}(k, x)$ or reflected part of WF $\varphi_{ref}(k, x)$, defined by eq. (\[eq.2.1.2\]), we receive the incident, transmitted or reflected WP, accordingly. We assume, that a time, for which the WP tunnels through the barrier, is enough small. So, the time necessary for a tunneling of an $\alpha$-particle through a barrier of decay in $\alpha$-decay of a nucleus, is about $10^{-21}$ seconds. We consider, that one can neglect a spreading of the WP for this time. And a breadth of the WP appears essentially more narrow on a comparison with a barrier breadth. Considering only sub-barrier processes, we exclude a component of waves for above-barrier energies, having included the additional transformation $$g(E - \bar{E}) \to g(E - \bar{E}) \theta(V_{1} - E), \label{eq.2.1.3}$$ where $\theta$-function satisfies to the requirement $$\theta(\eta) = \left\{ \begin{array}{ll} 0, & \mbox{for } \eta<0; \\ 1, & \mbox{for } \eta>0. \end{array} \right.$$ The method of multiple internal reflections considers the propagation process of the WP describing a motion of the particle, sequentially on steps of its penetration in relation to each boundary of the barrier [@Fermor.1966.AJPIA; @McVoy.1967.RMPHA; @Anderson.1989.AJPIA]. Using this method, we find expressions for the transmitted and reflected WP in relation to the barrier. At the first step we consider the WP in the region I, which is incident upon the first (initial) boundary of the barrier. Let’s assume, that this package transforms into the WP, transmitted through this boundary and tunneling further in the region II, and into the WP, reflected from the boundary and propagating back in the region I. Thus we consider, that the WP, tunneling in the region II, is not reached the second (final) boundary of the barrier because of a terminating velocity of its propagation, and consequently at this step we consider only two regions I and II. Because of physical reasons to construct an expression for this packet, we consider, that its amplitude should decrease in a positive $x$-direction. We use only one item $\beta\exp(-\xi x)$ in eq. (\[eq.2.1.2\]), throwing the second increasing item $\alpha\exp(\xi x)$ (in an opposite case we break a requirement of a finiteness of the WF for an indefinitely wide barrier). In result, in the region II we obtain: $$\psi^{1}_{tr}(x, t) = \int\limits_{0}^{+\infty} g(E - \bar{E}) \theta(V_{1} - E) \beta^{0} e^{-\xi x -iEt/\hbar} dE, \mbox{for } 0<x<a. \label{eq.2.1.4}$$ Thus the WF in the barrier region constructed by such way, is an analytic continuation of a relevant expression for the WF, corresponding to a similar problem with above-barrier energies, where as a stationary expression we select the wave $\exp(ik_{2}x)$, propagated to the right. Let’s consider the first step further. One can write expressions for the incident and the reflected WP in relation to the first boundary as follows $$\begin{array}{lcll} \psi_{inc}(x, t) & = & \int\limits_{0}^{+\infty} g(E - \bar{E}) \theta(V_{1} - E) e^{ikx -iEt/\hbar} dE, & \mbox{for } x<0, \\ \psi^{1}_{ref}(x, t) & = & \int\limits_{0}^{+\infty} g(E - \bar{E}) \theta(V_{1} - E) A_{R}^{0} e^{-ikx -iEt/\hbar} dE, & \mbox{for } x<0. \end{array} \label{eq.2.1.5}$$ A sum of these expressions represents the complete WF in the region I, which is dependent on a time. Let’s require, that this WF and its derivative continuously transform into the WF (\[eq.2.1.4\]) and its derivative at point $x=0$ (we assume, that the weight amplitude $g(E - \bar{E})$ differs weakly at transmitting and reflecting of the WP in relation to the barrier boundaries). In result, we obtain two equations, in which one can pass from the time-dependent WP to the corresponding stationary WF and obtain the unknown coefficients $\beta^{0}$ and $A_{R}^{0}$. At the second step we consider the WP, tunneling in the region II and incident upon the second boundary of the barrier at point $x = a$. It transforms into the WP, transmitted through this boundary and propagated in the region III, and into the WP, reflected from the boundary and tunneled back in the region II. For a determination of these packets one can use eq. (\[eq.2.1.1\]) with account eq. (\[eq.2.1.3\]), where as the stationary WF we use: $$\begin{array}{lcll} \varphi_{inc}^{2}(k, x) & = & \varphi_{tr}^{1}(k, x) = \beta^{0} e^{-\xi x}, & \mbox{for } 0<x<a, \\ \varphi_{tr}^{2}(k, x) & = & A_{T}^{0}e^{ikx}, & \mbox{for } x>a, \\ \varphi_{ref}^{2}(k, x) & = & \alpha^{0} e^{\xi x}, & \mbox{for } 0<x<a. \end{array} \label{eq.2.1.6}$$ Here, for forming an expression for the WP reflected from the boundary, we select an increasing part of the stationary solution $\alpha^{0} \exp(\xi x)$ only. Imposing a condition of continuity on the time-dependent WF and its derivative at point $x = a$, we obtain 2 new equations, from which we find the unknowns coefficients $A_{T}^{0}$ and $\alpha^{0}$. At the third step the WP, tunneling in the region II, is incident upon the first boundary of the barrier. Then it transforms into the WP, transmitted through this boundary and propagated further in the region I, and into the WP, reflected from boundary and tunneled back in the region II. For a determination of these packets one can use eq.  (\[eq.2.1.1\]) with account eq. (\[eq.2.1.3\]), where as the stationary WF we use: $$\begin{array}{lcll} \varphi_{inc}^{3}(k, x) & = & \varphi_{ref}^{2}(k, x), & \mbox{for } 0<x<a, \\ \varphi_{tr}^{3}(k, x) & = & A_{R}^{1}e^{-ikx}, & \mbox{for } x<0, \\ \varphi_{ref}^{3}(k, x) & = & \beta^{1} e^{-\xi x}, & \mbox{for } 0<x<a. \end{array} \label{eq.2.1.7}$$ Using a conditions of continuity for the time-dependent WF and its derivative at point $x = 0$, we obtain the unknowns coefficients $A_{R}^{1}$ and $\beta^{1}$. Analyzing further possible processes of the transmission (and the reflection) of the WP through the boundaries of the barrier, we come to a deduction, that any of following steps can be reduced to one of 2 considered above. For the unknown coefficients $\alpha^{n}$, $\beta^{n}$,$A_{T}^{n}$ and $A_{R}^{n}$, used in expressions for the WP, forming in result of some internal reflections from the boundaries, one can obtain the recurrence relations: $$\begin{array}{lll} \beta^{0} = \displaystyle\frac{2k}{k+i\xi}, & \alpha^{n} = \beta^{n} \displaystyle\frac{i\xi-k}{i\xi+k}e^{-2\xi a}, & \beta^{n+1} = \alpha^{n} \displaystyle\frac{i\xi-k}{i\xi+k}, \\ A_{R}^{0} = \displaystyle\frac{k-i\xi}{k+i\xi}, & A_{T}^{n} = \beta^{n} \displaystyle\frac{2i\xi}{i\xi+k}e^{-\xi a-ika}, & A_{R}^{n+1} = \alpha^{n} \displaystyle\frac{2i\xi}{i\xi+k}. \end{array} \label{eq.2.1.8}$$ Considering the propagation of the WP by such way, we obtain expressions for the WF on each region which can be written through series of multiple WP. Using eq. (\[eq.2.1.1\]) with account eq. (\[eq.2.1.3\]), we determine resultant expressions for the incident, transmitted and reflected WP in relation to the barrier, where one can need to use following expressions for the stationary WF: $$\begin{array}{lcll} \varphi_{inc}(k, x) & = & e^{ikx}, & \mbox{for } x<0, \\ \varphi_{tr}(k, x) & = & \sum\limits_{n=0}^{+\infty} A_{T}^{n} e^{ikx}, & \mbox{for } x>a, \\ \varphi_{ref}(k, x) & = & \sum\limits_{n=0}^{+\infty} A_{R}^{n} e^{-ikx}, & \mbox{for } x<0. \end{array} \label{eq.2.1.9}$$ Now we consider the WP formed in result of sequential $n$ reflections from the boundaries of the barrier and incident upon one of these boundaries at point $x = 0$ ($i = 1$) or at point $x = a$ ($i = 2$). In result, this WP transforms into the WP $\psi_{tr}^{i}(x, t)$, transmitted through boundary with number $i$, and into the WP $\psi_{ref}^{i}(x, t)$, reflected from this boundary. For an independent on $x$ parts of the stationary WF one can write: $$\begin{array}{ll} \displaystyle\frac{\varphi_{tr}^{1}}{\exp(-\xi x)} = T_{1}^{+} \displaystyle\frac{\varphi_{inc}^{1}}{\exp(ikx)}, & \displaystyle\frac{\varphi_{ref}^{1}}{\exp(-ikx)} = R_{1}^{+} \displaystyle\frac{\varphi_{inc}^{1}}{\exp(ikx)}, \\ \displaystyle\frac{\varphi_{tr}^{2}}{\exp(ikx)} = T_{2}^{+} \displaystyle\frac{\varphi_{inc}^{2}}{\exp(-\xi x)}, & \displaystyle\frac{\varphi_{ref}^{2}}{\exp(\xi x)} = R_{2}^{+} \displaystyle\frac{\varphi_{inc}^{2}}{\exp(-\xi x)}, \\ \displaystyle\frac{\varphi_{tr}^{1}}{\exp(-ikx)} = T_{1}^{-} \displaystyle\frac{\varphi_{inc}^{1}}{\exp(\xi x)}, & \displaystyle\frac{\varphi_{ref}^{1}}{\exp(-\xi x)} = R_{1}^{-} \displaystyle\frac{\varphi_{inc}^{1}}{\exp(\xi x)}, \end{array} \label{eq.2.1.10}$$ where the sign “+” (or “-”) corresponds to the WP, tunneling (or propagating) in a positive (or negative) $x$-direction and incident upon the boundary with number $i$. Using $T_{i}^{\pm}$ and $R_{i}^{\pm}$, one can precisely describe an arbitrary WP which has formed in result of $n$-multiple reflections, if to know a “path” of its propagation along the barrier. Using the recurrence relations eq. (\[eq.2.1.8\]), the coefficients $T_{i}^{\pm}$ and $R_{i}^{\pm}$ can be obtained. $$\begin{array}{lll} T_{1}^{+} = \beta^{0}, % = \displaystyle\frac{2k}{k+i\xi}, & T_{2}^{+} = \displaystyle\frac{A_{T}^{n}}{\beta^{n}}, % = \displaystyle\frac{2i\xi}{i\xi+k}e^{-\xi a-ika}, & T_{1}^{-} = \displaystyle\frac{A_{R}^{n+1}}{\alpha^{n}}, % = \displaystyle\frac{2i\xi}{i\xi+k}, \\ R_{1}^{+} = A_{R}^{0}, % = \displaystyle\frac{k-i\xi}{k+i\xi}, & R_{2}^{+} = \displaystyle\frac{\alpha^{n}}{\beta^{n}}, % = \displaystyle\frac{i\xi-k}{i\xi+k}e^{-2\xi a}, & R_{1}^{-} = \displaystyle\frac{\beta^{n+1}}{\alpha^{n}}. % \displaystyle\frac{i\xi-k}{i\xi+k}. \end{array} \label{eq.2.1.11}$$ Using the recurrence relations, one can find series of coefficients $\alpha^{n}$, $\beta^{n}$, $A_{T}^{n}$ and $A_{R}^{n}$. However, these series can be calculated easier, using coefficients $T_{i}^{\pm}$ and $R_{i}^{\pm}$. Analyzing all possible “paths” of the WP propagations along the barrier, we receive: $$\begin{array}{lcl} \sum\limits_{n=0}^{+\infty} A_{T}^{n} & = & T_{2}^{+}T_{1}^{-} \biggl(1 + \sum\limits_{n=1}^{+\infty} (R_{2}^{+}R_{1}^{-})^{n} \biggr) = \displaystyle\frac{i4k \xi e^{-\xi a-ika}}{F_{sub}}, \\ \sum\limits_{n=0}^{+\infty} A_{R}^{n} & = & R_{1}^{+} + T_{1}^{+}R_{2}^{+}T_{1}^{-} \biggl(1 + \sum\limits_{n=1}^{+\infty}(R_{2}^{+}R_{1}^{-})^{n} \biggr) = \displaystyle\frac{k_{0}^{2}D_{-}}{F_{sub}}, \\ \sum\limits_{n=0}^{+\infty} \alpha^{n} & = & \alpha^{0} \biggl(1 + \sum\limits_{n=1}^{+\infty} (R_{2}^{+}R_{1}^{-})^{n} \biggr) = \displaystyle\frac{2k(i\xi - k)e^{-2\xi a}}{F_{sub}}, \\ \sum\limits_{n=0}^{+\infty} \beta^{n} & = & \beta^{0} \biggl(1 + \sum\limits_{i=1}^{+\infty} (R_{2}^{+}R_{1}^{-})^{n} \biggr) = \displaystyle\frac{2k(i\xi + k)}{F_{sub}}, \\ \end{array} \label{eq.2.1.12}$$ where $$\begin{array}{lll} F_{sub} & = & (k^{2} - \xi^{2})D_{-} + 2ik\xi D_{+}, \\ D_{\pm} & = & 1 \pm e^{-2\xi a}, \\ k_{0}^{2} & = & k^{2} + \xi^{2} = \displaystyle\frac{2mV_{1}}{\hbar^{2}}. \end{array} \label{eq.2.1.13}$$ All series $\sum \alpha^{n}$, $\sum \beta^{n}$, $\sum A_{T}^{n}$ and $\sum A_{R}^{n}$, obtained using the method of multiple internal reflections, coincide with the corresponding coefficients $\alpha$, $\beta$, $A_{T}$ and $A_{R}$ of the eq. (\[eq.2.1.2\]), calculated by a stationary methods [@Landau.v3.1989]. Using the following substitution $$i\xi \to k_{2}, \label{eq.2.1.14}$$ where $k_{2}= \frac{1}{\hbar}\sqrt{2m (E-V_{1})}$ is a wave number for a case of above-barrier energies, expression for the coefficients $\alpha^{n}$, $\beta^{n}$, $A_{T}^{n}$ and $A_{R}^{n}$ for each step, expressions for the WF for each step, the total eqs. (\[eq.2.1.12\]) and (\[eq.2.1.13\]) transform into the corresponding expressions for a problem of the particle propagation above this barrier. At the transformation of the WP and the time-dependent WF one can need to change a sign of argument at $\theta$-function. Besides the following property is fulfilled: $$\biggl|\sum\limits_{n=0}^{+\infty} A_{T}^{n}\biggr|^{2} + \biggl|\sum\limits_{n=0}^{+\infty} A_{R}^{n}\biggr|^{2} = 1. \label{eq.2.1.15}$$ Direct method \[sec.app.2\] =========================== We shall add shortly solution for amplitudes of the wave function obtained by standard technique of quantum mechanics which could be obtained if to use only condition of continuity of the wave function and its derivative at each boundary, but on the whole region of the studied potential. At first, we find functions $f_{2}$ and $g_{2}$ (from the first boundary): $$\begin{array}{cc} f_{2} = \displaystyle\frac{k_{2}+k}{k_{2}-k} \,e^{2ik_{2}x_{1}}, & g_{2} = \displaystyle\frac{2k}{k-k_{2}} \,e^{i(k+k_{2})x_{1}}. \end{array} \label{eq.2.3.2.1}$$ Then, using the following recurrent relations: $$\begin{array}{ccl} f_{j+1} & = & \displaystyle\frac {(k_{j+1}-k_{j})\, e^{2ik_{j}x_{j}} + f_{j}\, (k_{j+1}+k_{j}) } {(k_{j+1}+k_{j})\, e^{2ik_{j}x_{j}} + f_{j}\, (k_{j+1}-k_{j}) } \cdot e^{2ik_{j+1}x_{j}}, \end{array} \label{eq.2.3.2.2}$$ we calculate next functions $f_{3}$, $f_{4}$, $f_{5}$ …$f_{n}$, and by such a formula: $$\begin{array}{ccl} g_{j+1} & = & g_{j} \cdot \displaystyle\frac{2k_{j}\, e^{i(k_{j+1}+k_{j})x_{j}}} {(k_{j+1}+k_{j})\, e^{2ik_{j}x_{j}} + f_{j}\, (k_{j+1}-k_{j})} \end{array} \label{eq.2.3.2.3}$$ the functions $g_{3}$, $g_{4}$, $g_{5}$ …$g_{n}$. From $f_{n}$ and $g_{n}$ we find amplitudes $\alpha_{n}$, $\beta_{n}$ and amplitude of transmission $A_{T}$: $$\begin{array}{cc} \beta_{n} = 0, & A_{T} = \alpha_{n} = -\displaystyle\frac{g_{n}} {f_{n}}. \end{array} \label{eq.2.3.2.4}$$ Now using the recurrent relations: $$\begin{array}{ccl} \alpha_{j-1} & = & \displaystyle\frac {\alpha_{j}\, e^{ik_{j}x_{j-1}} + \beta_{j}\, e^{-ik_{j}x_{j-1}} - g_{j-1}\, e^{-ik_{j-1}x_{j-1}}} {e^{ik_{j-1}x_{j-1}} + f_{j-1}\, e^{-ik_{j-1}x_{j-1}}} \end{array} \label{eq.2.3.2.5}$$ and such a formula: $$\beta_{j} = \alpha_{j} \cdot f_{j} + g_{j}, \label{eq.2.3.2.6}$$ we consistently calculate the amplitudes $\alpha_{n-1}$, $\beta_{n-1}$, $\alpha_{n-2}$, $\beta_{n-2}$ …$\alpha_{2}$, $\beta_{2}$. At finishing, we find amplitude of reflection $A_{R}$: $$A_{R} = \alpha_{2}\,e^{i(k+k_{2})x_{1}} + \beta_{2}\,e^{i(k-k_{2})x_{1}} - e^{2 ikx_{1}}. \label{eq.2.3.2.7}$$ As test we use condition: $$% \begin{array}{cc} \displaystyle\frac{k_{n}}{k_{1}}\; |A_{T}|^{2} + |A_{R}|^{2} = 1. % & \mbox{ or } & T_{MIR} + R_{MIR} = 1. % \end{array} \label{eq.2.3.2.8}$$ Studying the problem of proton decay, we used such a techniques for check the amplitudes obtained previously by the MIR approach and obtained coincidence up to first 15 digits for all considered amplitudes. In particular, we reconstruct completely the pictures of the probability presented in Fig. \[fig.1\] (a) and (b), but using standard technique above. So, *result on the large dependence of the penetrability of the position of the starting point $R_{\rm form}$ in such figures is independent on the used method*. 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