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+{"text":"\n\\section{Acknowledgement}\nWe thank Junjie Chen and Guancheng Wang for computing the number of distinct bugs and helping with the experiments related to HiCOND.\n\n\\section{Approach}\n\\label{sec:approach}\n\nFigure~\\ref{fig:workflow} depicts the overall workflow of our approach that constitutes the following three main steps: \n(1) extracting configurable test features from the initial test suite, \n(2) using the test features to cluster test program\\xspace{}s into similar groups, and\n(3) generating configurations based on the centroid of clusters.\nWe describe each step in the rest of this section.\n\n\\subsection{Extracting configurable test features}\n\n\\Part{Test features.}\nWe use the term \\emph{test features}, or \\emph{features} for short, similar to Groce et al.~\\cite{Groce:ISSRE:2013}: ``Test features are basic \\emph{compositional units} used to describe what a test does (and does not) involve.'' Test features are essentially building blocks of test cases. For example, in testing libraries, the API calls are the functions; in grammar-based testing, the production rules in the grammar can be features; and, in testing compilers or interpreters, programming structures can be test features. In Csmith\\xspace, test features are the same as C language features, therefore, in the rest of the paper, we use terms test feature, language feature, and feature interchangeably.\n\n\\Part{Configurable test generator.}\nA test generator essentially composes new test programs\\xspace with or based on the test features. \nSome test generators allow users to use configuration files or command-line options to customize test generation by modifying the test generation parameters. \nWe call such test generators \\emph{configurable}.\nCsmith\\xspace{}~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} is a configurable test generator for C compilers. \nCsmith\\xspace{} then uses these parameters to decide which features to include in the test programs.\n\n\\Part{Controllable test features in Csmith\\xspace.}\nCsmith\\xspace{} allows choosing the C programming constructs to be included in the test programs\\xspace through command-line options.\nThe order and number of the constructs are, however, chosen randomly and developers cannot control them---mainly because Csmith\\xspace{} is a random generator that uses grammar to generate test program\\xspace{}s. \nAs of Csmith\\xspace{} version $2.3.0$ \\footnote{\\label{csmith}\\url{https:\/\/embed.cs.utah.edu\/csmith\/}}, the Csmith\\xspace{} provides $32$ configurable features in the form of \\texttt{--ft} and \\texttt{--no-ft} command-line options, where $ft$ denotes a feature, and \\texttt{--ft} indicates inclusion and \\texttt{--no-ft} indicates exclusion of $ft$ in the generation of the test programs\\xspace. \nFigure ~\\ref{fig:feature_count_testinputs} shows the list of available controllable features in Csmith\\xspace. \nThese features describe a wide range of language features from programming constructs in C language, e.g., \\texttt{compound\\_assignment} for compound assignments, and \\texttt{float} for floating-point data types.\nDifferent features may impact the behavior of the compiler under test in different ways. \nFor example, Groce et al.~\\cite{Groce:ISSRE:2013} observed that pointer manipulation in a test program\\xspace can suppress certain class of bugs in the C compilers, as most compilers would conservatively avoid certain optimizations for pointers in the input program, consequently, masking potential bugs in those optimizations.\n\n\\Part{Extracting features.}\nIn the context of this paper, we only consider test programs\\xspace that have been reported in the GCC bug reports.\nFigure ~\\ref{fig:bug_report} shows an example of bug report (\\#11492) in GCC Bugzilla \\footnote{\\url{https:\/\/gcc.gnu.org\/bugzilla\/show_bug.cgi?id=11492}}.\nWe use the number of occurrences of features in each test program\\xspace to create a feature vector for the test program\\xspace.\nFor this, we use ClangAST\\xspace{} \\footnote{\\url{https:\/\/clang.llvm.org\/docs\/IntroductionToTheClangAST.html}} to extract the abstract syntax tree (AST) of the programs in the initial failing test suite.\nFeature vectors are normalized to the range of $[0,1]$ using min-max normalization~\\cite{Pedregosa:ScikitLearn:2011}, i.e.,\n$ z_{ij} = \\frac { x_{ij} \\, - \\, \\min(x_{.j}) } { \\max(x_{.j}) \\, - \\, \\min(x_{.j}) } $, where $x_{ij}$ is the value of the $j^{th}$ feature of the $i^{th}$ test program\\xspace $x_{i}=(x_{i1},...,x_{in})$, $x_{.j}$ is the list of $j^{th}$ feature from all test programs\\xspace, and $z_{ij}$ is the corresponding min-max normalized value of $x_{ij}$.\n\n\\subsection{Clustering test programs\\xspace}\nTo identify the clusters, we use X-Means\\xspace{} clustering algorithm \\cite{Pelleg:Cluster:xmeans:ICML:2000} that estimates the optimal number of clusters in the underlying distribution of data.\nThe X-Means\\xspace{} clustering is an unsupervised machine learning algorithm that performs clustering of unlabeled data without the need for presetting the number of clusters.\nEach cluster is represented by a \\emph{centroid} that has a minimum distance to the data points of the cluster.\nSince the feature vectors contain the value in the range of $[0,1]$, the values in the centroids would be a number between $0$ and $1$ as well.\n\n\\subsection{Generating test programs\\xspace}\nOur implementation of \\textsc{K-Config}\\xspace uses the centroid in X-Means\\xspace{} clustering \\cite{Pelleg:Cluster:xmeans:ICML:2000} to suggest configurations for the test generator.\nTo generate configurations, we use the corresponding value of a feature in the cluster as the probability of including the feature in a test program\\xspace{}.\nThe feature is more dominant in the test program\\xspace{}s if the corresponding value of a feature is closer to $1$, and conversely, if the corresponding value is closer to $0$, it denotes that the feature is less prevalent in the test program\\xspace{}s.\nTherefore, in this approach, for each cluster, we create a configuration wherein the probability of inclusion of a feature in the test program\\xspace is equal to its corresponding value in the centroid of that cluster.\nFor instance, suppose the centroid of a cluster is $(0.1, 0.7)$ for features \\texttt{f1} and \\texttt{f2}, respectively, \nthe configuration generation algorithm in \\textsc{K-Config}\\xspace, whenever called, it includes \\texttt{--f1} with probability $0.1$ and \\texttt{--no-f1} with probability $0.9$, similarly it would \\texttt{--f2} with $0.7$ probability and \\texttt{--no-f2} with $0.3$ probability.\nIt is in contrast with the swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} that uses the simplest form of fair coin-toss probability (i.e., $0.5$) for the inclusion of a feature.\n\nAlgorithm~\\ref{alg:generator} describes the process of generating new test program\\xspace{}s using \\textsc{K-Config}\\xspace in more detail.\nGiven a testing budget $totalBudget$ and a set of centroids $CS$, the algorithm calls $ConfigGen$ in round-robin fashion until the test budget expires.  \nProcedure $ConfigGen$ takes a centroid $C\\in CS$ and generates a new configuration.\nIn generating a new configuration, $ConfigGen$ chooses to include feature $f_i$ with a random probability $c_i$ where $f_i$ is represented by the element $c_i$ in the centroid $C$. \nFinally, $TS$ will have all the generated failure-inducing test programs.\n\n\\begin{algorithm} [ht]\n\\caption{\\textsc{K-Config}\\xspace}\n\\label{alg:generator}\n $totalBudget$  $\\leftarrow$ Testing budget (i.e. 10,000 test count)\\;\n $CS$ $\\leftarrow$ Set of centroids\\;\n $TS \\leftarrow \\{\\}$\\; \n \\DontPrintSemicolon\\;\n \n \\While{$spentBudget \\leq totalBudget$}{\n \\ForAll{centroid $C$ $\\in$ $CS$}{\n     $config \\leftarrow ConfigGen(C)$\\;\n     $test \\leftarrow Csmith(config)$\\;\n     \\If{doesFail(test,GCC)}{\n      $TS \\leftarrow TS \\cup test$\\;\n     }\n    }\n}\n     \n\\DontPrintSemicolon\\;\n\\SetKwProg{Fn}{Function}{:}{\\KwRet}\n\\Fn{ConfigGen($C$)}{\n    $features \\leftarrow \\emptyset$\\;\n    \\ForAll{value $c$ $\\in$ $C$}{\n      $randProb \\leftarrow [0:1]$\\;\n    \\eIf{randProb $\\leq$ c}{\n        $features.put(1)$\\;\n    }{\n        $features.put(0)$\\;\n    }\n }\n}\n\\end{algorithm}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThis paper proposed \\textsc{K-Config}\\xspace, a test configuration generation approach, that uses the code snippets in the bug reports to guide the test generation.\nGiven a \\textsc{Seed}\\xspace test suite and a configurable test generator, \\textsc{K-Config}\\xspace computes configurations for the test generator.\n\nWe extensively evaluated \\textsc{K-Config}\\xspace with GCC regression test suite and Csmith\\xspace{} on eight GCC versions. \nThe results suggest that the configurations computed by \\textsc{K-Config}\\xspace approach can help Csmith\\xspace{} to trigger more miscompilation failures than the state-of-the-art approaches. \nThe results signify the benefits of analyzing bug reports in generation of new test programs\\xspace.\nOur source code for this work is publicly available at \\url{https:\/\/github.com\/mdrafiqulrabin\/kconfig\/}.\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\\KC-Round-Robin\\xspace and \\textsc{Swarm}{} are complementary approaches in proposing configurations: \\textsc{Swarm}{} chooses the configurations randomly, while \\KC-Round-Robin\\xspace takes into account the bug reports in determining the configurations.\n\\KC-Round-Robin\\xspace generally finds more failures and provides higher code coverage than \\textsc{Swarm}. Moreover, the number of distinct bugs detected by the techniques are: $4$ by both, $3$ only by \\KC-Round-Robin\\xspace, and $3$ only by \\textsc{Swarm}.\n\nThe particle swarm optimization technique used in HiCOND\\xspace requires a considerable amount of failing and passing test programs\\xspace generated by Csmith\\xspace to find an optimal configuration. Unfortunately, as GCC matures, it becomes harder for Csmith\\xspace to trigger a failure in it---e.g., in Table \\ref{table:result_10k}, compare the number of failures triggered by Default Csmith\\xspace in GCC 4.3.0 and 4.8.2. Therefore, HiCOND\\xspace loses its applicability to newer, more stable versions of GCC. Notice that in \\cite{Junjie:PSO:ASE:2019}, HiCOND\\xspace is only trained on an older version (i.e. GCC-4.3.0), and in the cross-version experiment is tried in other versions (Section IV-F in \\cite{Junjie:PSO:ASE:2019}).\n\n\n\\Part{Searching around the lamp posts}.\n\\textsc{K-Config}\\xspace analyzes features of failing test program\\xspace{}s to create new \\emph{configurations} for a test generator.\nThe result of our experiment shows that \\textsc{K-Config}\\xspace could find up to \\KMC{} miscompilations in GCC, and outperforms \\textsc{Swarm}{} and HiCOND\\xspace in triggering miscompilation failures.\n\nIt suggests that analysis of the regression test suite can enable designing techniques to guide random test generators such as Csmith\\xspace. \nThe result indicates that processing failing test programs\\xspace can provide insights into the regions of code that are susceptible to bugs.\nIt might suggest that many bug fixes are incomplete~\\cite{Yin:IncompleteBugFix:ESEC:FSE:2011}, but we are unable to verify the similarity between root causes of the bugs that \\textsc{K-Config}\\xspace triggers and the test programs\\xspace of regression test suite.\n\n\n\\Part{\\textsc{Seed}\\xspace vs. generated test suite}.\nThe coverage of GCC for the test programs\\xspace in the regression test suite shows the power of small, directed test programs\\xspace.\nAlthough the size of the regression test suite was smaller than the test suites generated by Csmith\\xspace, they significantly outperformed those test suites in the coverage of GCC in terms of statements, branches, and functions.\nThe coverage of GCC for the regression test suite can serve as a benchmark to measure the shortcomings of the generated test suites.\nFrom this experiment, the gap still is very wide. \nNote that we only could compile the test programs\\xspace in the regression test suite without linking them because they did not accompany a \\texttt{main} function.\n\n\n\\Part{Difference with deep learning-based approaches}.\nThis approach is different from (deep) learning-based approaches such as DeepSmith~\\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018} that build a generative model for the test programs\\xspace of the programs. \nMoreover, learning-based techniques face two challenges. \nFirst, learning-based approaches require many test programs\\xspace with millions of tokens to train a model. \nSecond, learning-based approaches tend to converge to a restrictive language model of test program\\xspace{} that overly restricts the type of test program\\xspace{}s that can be produced ~\\cite{Godefroid:FeatureBased:LearnFuzz:ASE:2017}. \n\\textsc{K-Config}\\xspace instead uses the configuration of test generators to guide testing which is less constrained than the generation of test program\\xspace{}s in learning-based approaches. \nIn particular, \\textsc{K-Config}\\xspace only specifies the programming constructs that should be present in the generated test program\\xspace{}s, and the order or number of those constructs are determined by the test generator. \n\n\n\\Part{Limiting requirements}.\nThere are two main limitations to the application of \\textsc{K-Config}\\xspace. First, it assumes that a stable test generator exists. Second, it requires a set of failing test programs\\xspace. GCC compiler has been under development for decades and the bug reports are available.\nMoreover, GCC has Csmith\\xspace \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011}, a mature and well-engineered test generator, that allowed us to evaluate the effectiveness of \\textsc{K-Config}\\xspace in testing GCC compilers.\n\n\\section{Evaluation Setup}\n\\label{sec:evaluation}\n\n\\input{data\/possible_failures.tex}\n\\input{img\/difftest}\n\nThis section discusses the evaluation setup we used for the \\textsc{K-Config}\\xspace approach.\n\n\\Part{Test oracles}.\nThe common practice for testing compilers is differential testing \\cite{mckeeman1998differential, Regehr:TestGenGrammar:Csmith:PLDI:2011}.\nThat is, a test program\\xspace{} is compiled and executed by two or more versions of compilers, or two or more optimization levels, and the results are compared.\nThe differential test oracle specifies the result of the output of the compiled programs by all compilers and optimization levels must be the same.\nUndefined behaviors \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} in the C language can complicate this process, however, Csmith\\xspace{} does the best effort to avoid the undefined behaviors in C.\n\nWe use differential testing to evaluate the behavior of the compilers on the test programs\\xspace.\nFigure ~\\ref{fig:diff_test} shows the differential testing in optimization level.\nMore specifically, we compile a program with different levels of optimization and compare the output of the executed programs. \nDue to the large scale of the experiment, and to optimize the experimentation time, we only use the lowest \\texttt{-O0} and highest \\texttt{-O3} optimization levels. We do not include the intermediate optimization levels \\texttt{-O1} and \\texttt{-O2}, as most optimization passes that a compilation of a program can invoke can be captured in \\texttt{-O3}.\nNote that although almost all optimization passes are enabled in \\texttt{-O3}, there may be cases that the interactions of viable, competing optimizations may put compiler in a position to choose ones over another one, hence not exercising some of the passes that would have been otherwise exercised in \\texttt{-O1} or \\texttt{-O2} levels. Our experiment does not capture such cases, and we speculate that such cases would be rare and hence negligible.\n\n\n\\Part{Failure types}.\nWe expect three types of compiler failures in the compilation of a program: (1) Crash when compiler terminates the compilation abruptly with a crash report on screen, (2) Timeout when compiler gets into non-termination on the compilation of a program, and (3) Miscompilation when the executable generated by the compiler produced different results on different optimizations.\nWe check the behavior of the compiler (i.e., exit code) to identify the crashes.\nWe use $10$ seconds timeout in compiling test programs\\xspace.\nWe identify miscompilation by comparing the output of executables generated by the compiler on different optimization levels.\n\n\n\\Part{Experiment parameters}.\nOverall we evaluate five configuration generation for Csmith\\xspace{}: \\textsc{Default}\\xspace, \\textsc{Swarm}, HiCOND\\xspace, \\KC-Round-Robin\\xspace and \\KC-Weighted\\xspace.\n\\textsc{Default}\\xspace{} denotes the default configuration that includes all the features by default.\n\\textsc{Swarm}{} represents the swarm testing configuration where the probability of inclusion of a feature in a test program\\xspace is $0.5$, same as the fair coin-toss approach \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}.\nHiCOND\\xspace denotes the test configurations computed from the history-guided approach proposed by Chen et al.~\\cite{Junjie:PSO:ASE:2019}.\nIn \\textsc{K-Config}\\xspace, this probability is equal to the corresponding value in the centroid of the cluster.\nFor example, if the corresponding value for \\texttt{volatile} in the centroid of a cluster is $0.2$, there is $20\\%$ chance that \\textsc{K-Config}\\xspace to use \\texttt{--volatile} and $80\\%$ chance that it includes \\texttt{--no-volatile} in the configuration.\n\\KC-Round-Robin\\xspace{} adopts the round-robin strategy with uniform weights for all clusters, while \\KC-Weighted\\xspace{} uses a weighted strategy based on the size of clusters.\nNote that we do not evaluate the regression test suite, as only a few of its test programs\\xspace have \\texttt{main} functions that can generate an executable.\n\n\n\\Part{Clustering parameters}.\nWe use the implementation of X-Means\\xspace{} algorithm in \\texttt{pyclustring} data mining library ~\\cite{Novikov:PyClustering:2019} with default hyperparameters.\nWe use the default Bayesian information criterion (BIC)~\\cite{BIC} for the splitting type to approximate the number of clusters.\nThe stop condition for each iteration is $0.025$;\nthe algorithm stops processing whenever the maximum value of change in centers of clusters is less than this value.\nMoreover, the default distance metric used is the sum of squared errors (SSE) \\cite{kwedlo2011clustering} which is the distance between data points and its centroid.\n\n\n\\Part{Test budget}.\nWe create $10,000$ test programs\\xspace for each approach and evaluate the effectiveness of the test (i.e., bug finding, coverage, and the number of distinct bugs in each approach).\nTo account for the random effects in the approaches, we run each experiment three times, average them, and round the values to the closest integer.\nWe use $10$ seconds as the timeout for GCC to compile a test program\\xspace{}.\nA small experiment with 6 hours test budget and 30 seconds timeout yielded similar observations.\nOverall, we evaluated the techniques for over $1,500$ hours (approximately $62$ days).\n\n\\section{Experimental Setting}\n\\label{sec:experimental}\nThis section discusses the experimental setting we used for the \\textsc{K-Config}\\xspace approach.\n\n\n\\Part{Reference test suite}.\nA reference test suite is a collection of test programs\\xspace that exhibit interesting behaviors, e.g., high code coverage or triggering failures.\nWe use the GCC regression test suite that has \\N{} parsable C code snippets.\nThese code snippets have been collected from the confirmed bug reports in GCC Bugzilla $^{\\ref{bugzilla}}$ and Testsuite $^{\\ref{testsuite}}$.\nThe code snippets in the bug reports are small and usually do not include \\texttt{main} functions.\nWe call this test suite \\textsc{Seed}\\xspace henceforth.\nWe, hereby, use terms regression test suite and \\textsc{Seed}\\xspace test suite interchangeably.\n\nFigure ~\\ref{fig:feature_count_testinputs} shows the number of test program\\xspace{}s in \\textsc{Seed}\\xspace test suite for each feature.\nIt shows that the distribution of test features in failing test program\\xspace{}s is not uniform. \nFeatures such as comma operators, global variables, and pointers occurred more frequently than features such as unary plus operator (\\texttt{+}), safe math, or int8. \nWe observe that all Csmith\\xspace{} configurable test features appear in one or more failing test programs\\xspace in the regression test suite. Each feature has been present in $6$ to $3871$ test programs\\xspace.\nNote that it is difficult to identify the role of individual features in the failures in the large, complex systems such as GCC, without substantial simplification of the test~\\cite{DD} and close inspection of the program execution. \n\n\n\\Part{Test generation tool}.\nWe use Csmith\\xspace{} \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} that is an open-source automatic test generation tool for C compilers. \nGiven a set of C language features as options, Csmith\\xspace{} can generate a random C program that contains those features. \nThe programs generated by Csmith\\xspace{} are \\emph{closed}; that is, all variables are initialized in the source code by Csmith\\xspace and they do not require any external inputs for the execution. \nWe use Csmith\\xspace{} $2.3.0$ $^{\\ref{csmith}}$ in our experiments.\n\n\n\\Part{Test generation techniques}.\nWe compare the effectiveness of configurations proposed by \\textsc{K-Config}\\xspace with the effectiveness of configurations in three comparable techniques: (1) Csmith\\xspace default configuration, (2) swarm configurations, and (3) HiCOND\\xspace configurations. \nIn the \\emph{default configuration}, we use the default configuration of Csmith\\xspace wherein all C language features are enabled by default.\n\\emph{Swarm configurations} are configurations that the Csmith\\xspace features are enabled by a fair coin-toss as in ~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}.\n\\emph{HiCOND\\xspace configurations} are created by particle swarm optimization as in ~\\cite{Junjie:PSO:ASE:2019} that attempts to optimize the effectiveness of Csmith\\xspace by systematically exploring the bug-finding capability of configurations in Csmith\\xspace.\n\n\n\\Part{Configuration generation strategies}.\nWe evaluate \\textsc{K-Config}\\xspace with two configuration generation strategies: round-robin, and weighted.\nIn the round-robin strategy, centroids of the clusters are used in a round-robin fashion to generate test programs\\xspace.\nRound-robin strategy ignores the size of clusters.\nThe weighted \\textsc{K-Config}\\xspace strategy is similar to the round-robin, except the number of test programs\\xspace generated by each centroid is proportional to the size of the cluster that the centroid represents. \nIn weighted \\textsc{K-Config}\\xspace, larger clusters will have more test programs\\xspace generated. \n\n\n\\Part{GCC versions under test}.\nWe use eight versions of GCC to evaluate the effectiveness of the \\textsc{K-Config}\\xspace approach: GCC $4.3.0$, $4.8.2$, $5.4.0$, $6.1.0$, $7.1.0$, $8.1.0$, $9.1.0$, and GCC trunk (as of $09\/03\/2019$).\nThe official releases are mature and have been widely in use for building various programs and operating systems,\nwhile the trunk version contains the newest experimental features and, as a result, is not as stable as the official releases. \n\n\n\n\n\\section{Introduction}\n\nCompilers are key parts of software development infrastructure that translate high-level programs understandable by developers to low-level programs that machines can execute.\nDevelopers \\emph{rely} on compilers to build, profile, and debug the programs; therefore any bugs in compilers threaten the integrity of software development process.\nFor this reason, researchers have been focused on testing compilers to uncover bugs, \\cite{Alipour:FocusedTesting:ISSTA:2016,Cummins:TestGenLearn:DeepSmith:ISSTA:2018,Regehr:TestGenGrammar:Csmith:PLDI:2011}. \n\nTesting compilers is particularly difficult due to their sheer size and complexity. \nTesting such massive, sophisticated systems is a non-trivial task, and researchers and developers still can find bugs in them\n\\cite{CompilerSurvey:2020}.\nRandom testing, also know as fuzzing, is a common, lightweight approach for generating test programs\\xspace for compilers \\cite{miller1990empirical}.\nMany programming languages have specialized random test generators that use the language grammar and language-specific heuristics to produce test programs in those languages; Csmith\\xspace~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} for C, jsFunFuzz~\\cite{Link:jsFunFuzz} for JavaScript, and Go-Fuzz~\\cite{Link:goFuzz} for Go are good examples of such test generators that have been successful in helping developers to find hundreds of bugs in various compilers and interpreters. \n\nMature test generators, like Csmith\\xspace, are configurable and allow developers to guide the test generation through configurations. \nPrior studies have proposed different techniques to exploit configurations to generate more effective test programs.\nFor example, swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} configures Csmith\\xspace such that test programs contain a random subset of the language features, focused random testing ~\\cite{Alipour:FocusedTesting:ISSTA:2016} uses statistical analysis of the programs generated by Csmith\\xspace and their coverage to suggest Csmith\\xspace configurations that target specific blocks in GCC, or\nHiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019} analyzes on the historical test programs to create Csmith\\xspace configurations that are more likely to find bugs.\n\nIn this paper, we propose \\textsc{K-Config}\\xspace, an approach that uses the bugs reported by users to propose configurations for Csmith\\xspace.\nMore specifically, \\textsc{K-Config}\\xspace clusters the programs in the bug reports by X-Means\\xspace{} algorithm \\cite{Pelleg:Cluster:xmeans:ICML:2000} and uses the centroids of clusters as a basis for proposing configurations for Csmith\\xspace.\nWe implemented \\textsc{K-Config}\\xspace for GCC C compiler and Csmith\\xspace test generator. \nWe collected \\N{} failing test program\\xspace{}s from the GCC Bugzilla~\\footnote{\\label{bugzilla}\\url{https:\/\/gcc.gnu.org\/bugzilla\/}} and Testsuite~\\footnote{\\label{testsuite}\\url{https:\/\/github.com\/gcc-mirror\/gcc\/blob\/master\/gcc\/testsuite\/}}.\nWe performed an extensive experiment to evaluate the effectiveness of \\textsc{K-Config}\\xspace on eight versions of GCC. We compared it with swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, HiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019} and the default configuration of Csmith\\xspace~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011}.\n\nTo the best of our knowledge, the other comparable techniques do not use the information available in the bug reports in the generation of new test programs\\xspace. \\textsc{K-Config}\\xspace is the first attempt in that direction \\cite{rabin2019kconfig}. The reasoning it uses is that the code snippets in the bug reports that triggered bugs earlier are more likely to be of interest to developers. We believe bug reports contain insights that can be extracted to improve the test generation for compilers.\n\nThe results of our experiments suggest that \\textsc{K-Config}\\xspace could find up to \\KMC{} miscompilations, while swarm finds up to \\SMC{} miscompilations, and the default configurations couldn't find any miscompilations. Moreover, the coverage of \\textsc{K-Config}\\xspace is higher than other techniques in all versions of GCC compilers in our experiment.\nOur results also show that the coverage of regression test programs\\xspace is still higher than any automatically generated test suite. It suggests that despite the advances in generating test programs for compilers, there is still a wide gap between the effectiveness of the generated test programs, and the small, regression test suites.\n\nThe intuition behind \\textsc{K-Config}\\xspace is that the language features that have participated in previous bugs are likely to participate in new bugs as well. \nSimilar intuition has been used in LangFuzz~\\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} and HiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019}.\nLangFuzz transplants fragments of failing test program\\xspace{}s to generate new test programs\\xspace for JavaScript.\nAlthough similar in the intuition, instead of manipulating existing test programs, \\textsc{K-Config}\\xspace uses configurations to guide the test generator to generate new test programs\\xspace.\nHiCOND\\xspace uses the configuration of Csmith\\xspace \\emph{generated} test programs to optimize and propose diverse test configurations.\nWhile \\textsc{K-Config}\\xspace and HiCOND\\xspace both are similar in the sense that they use an initial test suite to infer effective configurations for Csmith\\xspace, they differ in two main ways. \nFirst, HiCOND\\xspace requires the configuration of the Csmith\\xspace in the generation of each test program. In other words, all initial test programs must be generated by Csmith\\xspace. Therefore, the test programs that are not created by Csmith\\xspace, like the ones available in the bug reports, cannot be used by HiCOND\\xspace.\nSecond, the particle swarm optimization technique used in HiCOND\\xspace requires a large initial test suite of failing and passing test programs\\xspace generated by Csmith\\xspace. Unfortunately, as GCC matures, it becomes harder for Csmith\\xspace to trigger failures in GCC hence, which negatively impacts HiCOND\\xspace applicability on newer, more stable versions of GCC. \n\n\\Part{Contributions}.\nThis paper makes the following contributions.\n\\begin{itemize}\n    \\item We propose an approach for computing configuration of the Csmith\\xspace test generator by processing the code snippets in the bug reports.\n    \\item We perform a large-scale case study on the effectiveness of our approach on eight versions of GCC---a large, complex C compiler.\n\\end{itemize}\n\n\\Part{Paper Organization}.\nThe paper is organized as follows.\nSection~\\ref{sec:approach} describes the proposed \\textsc{K-Config}\\xspace approach. \nSection~\\ref{sec:experimental} and Section~\\ref{sec:evaluation} describe the experimental setting and evaluation setup.\nSection~\\ref{sec:results} presents the results of the experiments. \nSection~\\ref{sec:discussion} discusses the results. \nSection~\\ref{sec:related} presents the related works.\nSection~\\ref{sec:threat} describes the main threats to validity. \nFinally, section~\\ref{sec:conclusion} concludes the paper.\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nSeveral approaches for testing compilers have been proposed; for example, \ncode mutation \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012}, \nrandom test generation \\cite{Regehr:TestSelection:Rank:PLDI:2013}, \nmetamorphic testing \\cite{Zhendong:TestOracle:classfuzz:PLDI:2016, Zhendong:TestGenMutation:Orion:PLDI:2014, Zhendong:TestGenMutation:Athena:OOPSLA:2015,Zhendong:TestGenMutation:Hermes:OOPSLA:2016},\nlearning-based testing \\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018}, \nand swarm testing \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, to name few. \nThese approaches either generate test programs\\xspace from scratch by grammar \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} and learning \\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018}, \nor they create new test programs\\xspace by manipulating \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} or transforming the existing test programs\\xspace, e.g., \\cite{Zhendong:TestGenMutation:Orion:PLDI:2014}.\n\nSwarm testing \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} randomly chooses a subset of features available to generate new test cases. \nThe generated test cases are very diverse and the evaluation result shows that this approach outperforms Csmith\\xspace{}'s default configuration in both code coverage and crash bug finding. \nSPE \\cite{Zhendong:TestSelection:SPE:PLDI:2017} where authors enumerate a set of programs with different variable usage patterns. \nThe generated diverse test cases exploit different optimization and the evaluation result shows that the skeletal program enumeration has confirmed bugs in all tested compilers. \nTwo more related studies in this area are LangFuzz \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} and Learn\\&Fuzz \\cite{Godefroid:FeatureBased:LearnFuzz:ASE:2017}. \nThe LangFuzz approach extracts code fragments from a given code sample that triggered past bugs and then apply random mutation within a pool of fragments to generate test inputs. \nOn the other hand, the Learn\\&Fuzz approach uses the learnt seq2seq model to automate the generation of an input grammar suitable for PDF objects using different sampling strategies. \nBoth approaches have revealed several previously unknown bugs in popular compilers and interpreters.\n\nSeveral approaches to accelerate the speed of test selection and triage have been proposed; for example, \\cite{Regehr:TestSelection:Rank:PLDI:2013}, \\cite{Chen:TestSelection:LET:ICSE:2017}, \\cite{Chen:TestSelection:4steps:ICSE:2018}, \\cite{Chen:TestSelection:COP:IEEE-TSE:2018}.\nChen et al. \\cite{Regehr:TestSelection:Rank:PLDI:2013} evaluate the impact of several distance metrics on test case selection and prioritization.\nChen et al. \\cite{Chen:TestSelection:LET:ICSE:2017} proposed LET where authors use machine learning to schedule the test inputs. This learning-to-test approach has two steps: learning and scheduling. In learning steps, LET extracts a set of features from the past bug triggering test cases and then trains a capability model to predict the bug triggering probability of the test programs, and trains another time model to predict the execution time of the test programs. In scheduling steps, LET ranks the target test programs based on the probability of bug triggering in unit time. The evaluation result shows that the scheduled test inputs significantly accelerate compiler testing.\nAnother example in this area is COP \\cite{Chen:TestSelection:COP:IEEE-TSE:2018} where authors predict the coverage information of compilers for test inputs and prioritize test inputs by clustering them according to the predicted coverage information. The result shows that COP significantly outperforms state-of-the-art acceleration approaches in test acceleration.\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\input{data\/result_10k.tex}\n\\subsection{Comparison with \\textsc{Default}\\xspace and \\textsc{Swarm}}\n\nIn Table \\ref{table:result_10k}, $10,000$ test program\\xspace{}s of \\KC-Round-Robin\\xspace{}, on average, triggered up to $96$ miscompilations, while \\textsc{Swarm}{} triggered up to $70$ miscompilations, on average. The default configuration of Csmith\\xspace{} (\\textsc{Default}\\xspace) triggered $16$ miscompilations in GCC-4.3.0 but did not find any miscompilations in other versions of GCC.\n\nThe \\textsc{Default}\\xspace found $1708$ crashes in GCC-4.3.0 compared to $755$ by \\KC-Round-Robin\\xspace{} and $666$ by \\textsc{Swarm}{}. However, the \\textsc{Default}\\xspace did not find any crashes in GCC-4.8.2, where the \\KC-Round-Robin\\xspace{} found $111$ crashes and the \\textsc{Swarm}{} found $125$ crashes.\n\nIn all cases, \\KC-Round-Robin\\xspace{} triggered the highest number of miscompilations compared to \\textsc{Default}\\xspace and \\textsc{Swarm}.\n\n\n\\subsection{Comparison with HiCOND\\xspace}\n\nHiCOND\\xspace uses particle swarm optimization to search for configurations that can find more bugs. \nWe received the configurations of HiCOND\\xspace for GCC-4.3.0 from the authors.\nIn GCC-4.3.0, HiCOND\\xspace finds the highest number of crashes on \\texttt{-O3}, while swarm followed by \\KC-Round-Robin\\xspace{} discover the highest number of crashes under \\texttt{-O1}. \\KC-Round-Robin\\xspace{} finds the highest number of miscompilations.  \nConfigurations proposed by HiCOND\\xspace do not find any crashes in GCC-4.8.2, as it is highly optimized for bugs in GCC-4.3.0. \nMoreover, HiCOND\\xspace relies on a set of bugs generated by \\textsc{Default}\\xspace to search for the optimal configurations. \n\\textsc{Default}\\xspace does not trigger any failures on GCC-4.8.2, we, therefore, could not compute new configurations using the approach in \\cite{Junjie:PSO:ASE:2019} for GCC-4.8.2 and beyond.\n\n\n\\subsection{Number of distinct bugs}\n\n\nTo identify the number of distinct bugs we used the correcting commits heuristic that has been used in previous studies~\\cite{Regehr:TestSelection:Rank:PLDI:2013, Chen:TestSelection:LET:ICSE:2017}.\nThis heuristic uses the commit that a test program switched from failing to passing as the proxy for the minimum number of distinct bugs.\nNote that in our results, there are test programs that still exhibit miscompilation characteristics on the latest versions of the GCC, therefore, this heuristic can not be used for them.\nDue to the submission deadline, we only compute the distinct bugs for \\textsc{Swarm}{} and \\KC-Round-Robin\\xspace in GCC-4.8.2.\nFor the crash and miscompilation of failing test programs\\xspace, the numbers of distinct bugs detected by the techniques are: 3\\xspace only by \\KC-Round-Robin\\xspace, 3\\xspace only by \\textsc{Swarm}, and 4\\xspace by both.\n\n\n\\subsection{Effectiveness of cluster weighting strategies}\n\nWe evaluated the effectiveness of \\textsc{K-Config}\\xspace for two cluster weighting strategies: \\KC-Round-Robin\\xspace{} that ignores the size of clusters, and \\KC-Weighted\\xspace{} which generates test programs proportional to the size of clusters.\nThe results in Table \\ref{table:result_10k} show that \\KC-Round-Robin\\xspace{} was more effective than \\KC-Weighted\\xspace{} in finding bugs. \n\\KC-Round-Robin\\xspace{} could trigger twice as more miscompilations, and thrice as more crashes than \\KC-Weighted\\xspace{}. \nThe potential reason can be that larger clusters represent prominent combinations of features in the bug reports, it is likely that developers already noticed them and addressed them. Therefore, test programs\\xspace generated based on \\KC-Weighted\\xspace{} would not lead to new failures.\nDue to the poor performance of \\KC-Weighted\\xspace, we excluded the \\KC-Weighted\\xspace{} from subsequent experiments to save computing time.\n\n\n\\subsection{Effectiveness of individual clusters}\n\nTo measure the effectiveness of individual clusters in \\KC-Round-Robin\\xspace, we count the number of failures triggered by the test programs\\xspace generated based on their centroid configuration.\nFigure ~\\ref{fig:nbug} shows the number of crashes and miscompilations triggered by each cluster. We excluded the timeouts from the figure due to their sheer numbers.\nThe $x$-axis in this figure denotes the clusters, and the $y$-axis denotes the number of miscompilation and crash failures triggered by each cluster.\n\nAmong $134$ clusters, $50$ did not contribute to finding any failures, $29$ triggered only one crash, $18$ triggered only one miscompilation, while there are configurations with $20$ to $26$ (crash or miscompilation) failures.\nTable~\\ref{table:top5_nbug} shows the top-$5$ most effective configurations in triggering failures (crash and miscompilation combined).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{img\/nbug.png}\n    \\caption{Number of failures in each configuration.}\n    \\label{fig:nbug}\n\\end{figure}\n\n\\input{data\/top5_centroids.tex}\n\n\n\\subsection{Coverage across test suites}\n\nTable~\\ref{table:coverage_10k} shows the number of covered statements, branches, and functions on GCC versions 4.3.0, 4.8.2, 5.4.0, 6.1.0, and 7.1.0 for the \\textsc{Seed}\\xspace{}, \\textsc{Default}\\xspace{}, \\textsc{Swarm}{}, and \\KC-Round-Robin\\xspace{} test suites.\nWe computed the coverage with -O3 optimization. \nAt the time of writing, we did not extract the coverage information for most recent versions of GCC, i.e., 8.1.0, 9.1.0, and trunk, due to difficulties stemming from recent changes in the file structure in GCC project.\nThe coverage shows that, in the measured GCC versions, the regression test suite has higher coverage than the generated test suites. \nIn the generated test suite, the \\KC-Round-Robin\\xspace{} has higher coverage than others (\\textsc{Default}\\xspace{}, \\textsc{Swarm}{}, and HiCOND\\xspace{}).\n\n\\input{data\/coverage_10k.tex}\n\n\n\\subsection{The probability of features}\n\nIn the swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, the probability of inclusion of a test feature in the test program depends on the fair coin-toss for all test features. Therefore, the probability is the same for all test features, it is 0.5. \nFigure~\\ref{fig:centroid:probabilities}, on the other hand, shows the distributions of probabilities per test feature in the \\KC-Round-Robin\\xspace approach.\nThe mean and median of the probabilities of all features are less than 0.5. However, 28 out of 32 features have been in one or more configurations with probability 1.\nIn more than $25\\%$ of configurations, $75\\%$ or more of features are excluded.\nIt highlights the importance of a few test features that dominate the test programs\\xspace generated by those clusters.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{img\/centroids_boxplots.png}\n    \\caption{Distribution of probability of inclusion for individual GCC features in \\KC-Round-Robin\\xspace.}\n    \\label{fig:centroid:probabilities}\n\\end{figure}\n\n\n\n\\section{Threats to Validity}\n\\label{sec:threat}\n\nIn this section, we describe several threats to validity for our study.\n\n\\Part{Internal Validity.} \nDespite our best effort, some bugs may exist in the tools and scripts used in this paper.\nHowever, we used the well-tested programs in the implementation and evaluation of \\textsc{K-Config}\\xspace to reduce the chance of mistakes.\nWe have taken care to ensure that our results are unbiased, and have tried to eliminate the effects of variability by repeating the experiments multiple times.\n\n\\Part{External Validity.}\nWe evaluated the \\textsc{K-Config}\\xspace approach on Csmith\\xspace and several GCC versions, therefore, the extent to which it generalizes to other compilers and test generators is quite unknown.\nIn this approach, the feature set is restricted to the ones that can be translated to Csmith\\xspace configuration options, therefore the feature space is also limited.\nAdditionally, \\textsc{K-Config}\\xspace relies on the features that have triggered compiler bugs in the past, therefore the effectiveness may decrease in future GCC versions for the bug fixes.\nAnother potential issue can be that the bug reports to build the \\textsc{Seed}\\xspace test suite may not be representative of all GCC bugs and can impact the clusters.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\paragraph{A factor of two.}\n\nThe early universe from at least big bang nucleosynthesis onwards\nis well described by a model where the geometry is locally\nspatially homogeneous and isotropic up to linear perturbations,\nthe matter consists of a gas of particles with\npositive pressure, and the relation between geometry\nand matter is given by the Einstein equation\nbased on the four-dimensional Einstein-Hilbert action.\nHowever, at late times such a model underpredicts the\ndistance to far-away sources and the expansion rate.\nCompared to the simplest possibility, the spatially flat\nmatter-dominated model, the discrepancy is a factor\nof about two in both the distance\n(for a fixed Hubble constant) and the expansion rate\n(for a fixed energy density or age of the universe).\nTherefore at least one of the three assumptions\n--homogeneity and isotropy, standard matter content\nand standard gravity-- is wrong, assuming that light\npropagation is correctly modeled by null geodesics.\nNo deviations from standard gravity have been observed in\nlocal physics, not in the solar system\n(apart from the Pioneer anomaly and the flyby anomaly, where\nthe possibility of systematics is not ruled out) nor\nin pulsars \\cite{solar, Will:2005}.\nNeither is there any detection of an effect of exotic\nmatter with negative pressure on local physics.\nThe factor of two discrepancy only appears in\nobservations of distance and expansion rate\nwhich involve quantities integrated over\nlarge scales\\footnote{It has been argued that locally\nrepulsive gravity has been observed in the motions\nof galaxies near the Local Group \\cite{localde}.\nThis is an interesting possibility, but the present data\nis not precise enough for such a detection.}.\nThis situation is quite different from that of dark matter,\nfor which there is evidence from various different systems\non several scales.\n\nWhile there is no evidence against standard general relativity\nor standard matter apart from the increased distance and\nexpansion rate,\nthe universe is known to be locally far from homogeneity and\nisotropy due to the formation of non-linear structures at late\ntimes. It is possible that the breakdown of the homogeneous\nand isotropic approximation could explain the failure of\nthe prediction of homogeneous and isotropic models with\nordinary matter and gravity\n\\cite{Buchert:2000, Tatekawa:2001, Wetterich:2001, Schwarz, Rasanen, Kolb:2004}.\nThe effect of inhomogeneity and\/or anisotropy on the\nevolution of the universe was first discussed in detail\nin \\cite{fitting} under the name ``the fitting problem'',\nand the effect on the average expansion rate is known as\nbackreaction\n\\cite{Buchert:1999, Buchert:2001, Ellis:2005, Rasanen:2006b, Buchert:2007}.\nIt has been shown with toy models that inhomogeneities\ncan lead to accelerating expansion\n\\cite{Rasanen:2006b, Chuang:2005, Paranjape:2006a, Kai:2006, Rasanen:2006a},\nbut whether this happens for the distribution of structures\npresent in the real universe is not yet clear.\nThe order of magnitude of the observed change in\nthe expansion rate and the correct timescale\nof around 10 billion years do emerge from the physics of\nstructure formation in a semi-realistic model\n\\cite{Rasanen:2008a, peakrevs}, but there is no fully\nrealistic calculation yet.\n\n\\paragraph{Light propagation and statistical homogeneity and isotropy.}\n\nMost cosmological observations probe quantities\nrelated to light propagation, such as the redshift,\nthe angular diameter distance (or equivalently the\nluminosity distance) and image distortion.\nIn linearly perturbed homogeneous and isotropic\nFriedmann-Robertson-Walker (FRW) models, the redshift\nand the distance are to leading order determined by the\nexpansion rate and the spatial curvature.\nThe corrections due to the perturbations are small for\ntypical light rays, and are important only\nfor the cosmic microwave background (CMB), whose redshift\nanisotropies are very accurately measured, and image\ndistortion, which is zero for the background and remains\nsmall when perturbations are included. (At least this\nis the case when the perturbations are statistically\nhomogeneous and isotropic and the homogeneity scale is small;\nsee \\cite{Enqvist:2009} for a counterexample with a large\nspherically symmetric structure.)\n\nThe fact that the optical properties of a FRW universe can be\nexpressed in terms of the expansion rate and the spatial curvature\nis rather obvious, because light propagation is, for small\nwavelengths, purely geometrical, and these are the only\ndegrees of freedom in the FRW geometry.\nIn a general spacetime, the situation is more involved.\nNevertheless, if the distribution of the geometry is statistically\nhomogeneous and isotropic, it could be expected that light\npropagation over distances longer than the homogeneity scale\ncan to a first approximation be similarly described with a few\nquantities related to the overall geometry, regardless of complicated\nlocal details \\cite{Rasanen:2008a}.\nLight propagation in statistically homogeneous and isotropic\nuniverses with irrotational dust as the matter content was studied\nin \\cite{Rasanen:2008b}, where it was argued that if the distribution\nevolves slowly compared to the time it takes for light\nto cross the homogeneity scale, then the redshift and the\nangular diameter distance are determined by the\nexpansion rate as a function of redshift and the\nmatter density today.\nThe study \\cite{Rasanen:2008b} had three shortcomings.\n\nFirst, it was assumed that the variation in the spatial\ndirection of the null geodesics (i.e. light deflection) is small.\nThe magnitude of the null shear was also left undetermined.\nObservationally, both light deflection and image distortion\nare known to be small for typical light rays \\cite{Munshi:2006},\nand it should be established that this follows from the\nsymmetry properties of the spacetime.\nSecond, the treatment of matter as irrotational dust\nis not locally valid \\cite{Buchert:2005, Pueblas:2008},\nbecause effects such as rotation and velocity dispersion are\nimportant for stabilising structures on small scales.\nThe vorticity and non-dust nature of the matter content\nmay be expected to be unimportant for the overall cosmological\nevolution in the real universe at late times.\nNevertheless, such effects should be included to establish under\nwhich conditions they can be neglected, and put the dust\napproximation on better footing. Including matter\nother than dust is also necessary for treating backreaction\nin the early universe such as during inflation\n\\cite{Buchert:2001, Woodard, Unruh, Geshnizjani, Brandenberger:2002}\nor preheating.\nThird, the arguments were qualitative and the corrections\nto the mean behaviour were not determined.\n\nWe now remedy the first and second problems.\nWe derive results for light propagation, including\ndeflection and shear, using only assumptions\nabout the symmetry of the spacetime geometry and matter content.\nWe consider general matter content and include rotation.\nConcentrating on observable quantities related to light propagation,\nwe show how the relevant averaging hypersurface is\ngiven by the statistical symmetry of the spacetime.\nHowever, our analysis is not more quantitative\nthan \\cite{Rasanen:2008b}, and the arguments should\nbe followed up with a more rigorous study.\n\nIn section 2 we go through our assumptions, set up\nthe covariant formalism and derive\nresults for the redshift, the deflection,\nthe null shear and the angular diameter distance.\nIn section 3 we derive the evolution equations\nfor the scale factor, which generalise the\nBuchert equations of the irrotational\ndust case \\cite{Buchert:1999}, and consider\nthe validity of the dust approximation.\nIn section 4 we discuss the possible effect\nof the discreteness of the matter content,\nthe relevance of average quantities and the FRW\ndescription, and summarise the situation.\n\n\\section{Light propagation} \\label{sec:light}\n\n\\subsection{Spacetime geometry} \\label{sec:geom}\n\n\\paragraph{Statistical symmetry.}\n\nWe assume that there exists a foliation of the\nspacetime into spatial hypersurfaces of statistical\nhomogeneity and isotropy, which we denote by $\\mathcal{N}$.\nThe time which is constant on such a hypersurface\nis denoted $t$, and when referring to a particular\nhypersurface, we use the notation $\\mathcal{N}(t)$.\nBy this we mean that when we consider\nany region larger than the homogeneity scale, the average\nquantities within the region do not depend on its\nlocation, orientation or size.\nIn other words, over large scales, there are no\npreferred locations or directions, and no correlations.\nLocally the dynamics can be complex, as the assumption\nof statistical homogeneity and isotropy only concerns\naverage quantities evaluated over large scales.\nThe frame of statistical homogeneity and isotropy\nmay not locally coincide with either\nthe Eckart frame (where there particle number flux is zero) or\nthe Landau-Lifshitz frame (where the energy flux is zero)\n\\cite{Maartens:1998}.\nHowever, statistical homogeneity and isotropy does imply\nthat the integrated flux of any quantity through the\nboundary of a volume larger than the homogeneity scale\nvanishes.\n\nIn this view, the universe consists of identical (up\nto statistical fluctuations) boxes stacked next\nto each other.\nIn the real universe, there are correlations even\nover scales longer than the Hubble scale,\ndue to inflation (or some other process in the early universe)\nwhich produces a large region that is exactly homogeneous\nand isotropic except for linear perturbations.\nThe distribution of the perturbations\nis statistically homogeneous and isotropic.\nWhen the perturbations become non-linear at late\ntimes (in typical supersymmetric dark matter models,\nthe first structures form around a redshift of 40--60 \\cite{SUSYCDM}),\nlocal homogeneity and isotropy are lost, but the distribution\nof non-linear structures remains statistically homogeneous\nand isotropic, and the amplitude of\ncorrelations is small beyond the homogeneity scale.\nWhat one finds as the homogeneity scale depends on\nthe limit that one sets for this amplitude.\nBased on the fractal dimension of the point set of galaxies,\nit has been argued that the distribution becomes homogeneous on a\nscale of around 100 Mpc to an accuracy of about 10\\% \\cite{hom}.\nHowever, there are still large fluctuations on 100 Mpc scales,\nand it has been argued that the sample size is not large\nenough to establish that the distribution is self-averaging,\nwhich is a necessary condition for statistical homogeneity\n\\cite{inhom, SylosLabini:2009}.\nStudies of morphology also suggest that the homogeneity\nscale could be 300 Mpc or more \\cite{morphology}.\n\nStatistical homogeneity and isotropy is\nformulated in terms of spatial hypersurfaces, but\nlight travels along null geodesics, not in a spacelike\ndirection. Therefore we also need information about the\nevolution which relates one hypersurface to the next.\nWe assume that the evolution is slow in the sense\nthat the timescale of change in the spatial distribution\nis much larger than the homogeneity scale.\nPhrased differently, the variation of the geometry\nalong a null geodesic is rapid compared to the\nscale over which the mean varies significantly.\nIn the real universe, the timescale for change\nin the distribution of matter and geometry\nis the Hubble time $H^{-1}$.\nToday $H_0^{-1}=3000$\\mbox{$h^{-1}$Mpc}{} (with $h$ somewhat below unity\n\\cite{Hubble}), much larger than 100--300 Mpc.\nIn the past, the homogeneity scale was even smaller\nrelative to the Hubble scale, as structure formation\nwas less advanced.\n\nThe combination of statistical homogeneity and\nisotropy on spatial hypersurfaces and slow evolution\nfrom one hypersurface to the next can be heuristically\nthought of as a distribution that is statistically\napproximately homogeneous and isotropic in four dimensions\nwhen considering scales larger than the homogeneity scale,\nbut smaller than the timescale of change in the distribution.\nThe notion of statistical homogeneity and isotropy\nin general spacetimes should be made more rigorous,\nand the role of slow evolution in the arguments we\nmake below on light propagation should be quantified.\n\n\\paragraph{The two frames.}\n\nWe denote the vector normal to $\\mathcal{N}$ by $n^\\a$ and\nthe velocity of the observers by $u^\\a$.\nBoth are normalised to unity, $n_\\a n^\\a=u_\\a u^\\a=-1$.\nThe observer velocity is completely general, it\nis not assumed to be geodesic or irrotational.\nFor reviews of the covariant approach we use, see\n\\cite{Ehlers:1961, Ellis:1971, Ellis:1998c, Clarkson:2000, Tsagas:2007};\nfor the relation to the ADM formalism \\cite{Arnowitt:1962},\nsee \\cite{Jantzen:2001}.\nThe tensors which project on the hypersurface\northogonal to $n^\\a$ and the rest space\northogonal to $u^\\a$ are, respectively,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{h}\n  h_{\\a\\b} &\\equiv& g_{\\a\\b} + n_\\a n_\\b \\nonumber \\\\\n  h_{\\a\\b}^{(u)} &\\equiv& g_{\\a\\b} + u_\\a u_\\b \\ ,\n\\eea\n\n\\noindent where $g_{\\a\\b}$ is the spacetime metric.\nThe restriction of the projection tensor\n$h_{\\a\\b}$ to $\\mathcal{N}$ is the metric on $\\mathcal{N}$.\nThe spatial derivative of a scalar is defined as\n$\\hat{\\nabla}_\\a f\\equiv h_{\\a}^{\\ \\b} \\nabla_\\b f$, for vectors we have\n$\\hat{\\nabla}_\\b f_\\a \\equiv h_{\\b}^{\\ \\d} h_{\\a}^{\\ \\c} \\nabla_\\d f_\\c$,\nand similarly for higher order tensors.\nThe spatially projected traceless part of a tensor is\n$f_{\\langle\\a\\b\\rangle}\\equiv h_{(\\a}^{\\ \\ \\c} h_{\\b)}^{\\ \\ \\d} f_{\\c\\d} - \\frac{1}{3} h_{\\a\\b} h^{\\c\\d} f_{\\c\\d}$.\nThe volume element on $\\mathcal{N}$ is $\\epsilon_{\\a\\b\\c}\\equiv\\eta_{\\a\\b\\c\\d} n^\\d$, \nwhere $\\eta_{\\a\\b\\c\\d}$ is the spacetime volume element.\nThe derivative with respect to the proper time $s$\nof the frame of statistical homogeneity and isotropy\nis $n^\\a\\nabla_\\a$, and it is denoted by an overdot.\nUnless $\\dot{n}^\\a=0$, the proper time $s$ does not coincide\nwith the time $t$ which is constant on $\\mathcal{N}$.\nWe can write $n_\\a=-\\dot{t}^{-1}\\partial_\\a t$.\nThe derivative with respect to $t$ is\n$m^\\a \\nabla_\\a$, with $m^\\a=\\dot{t}^{-1} n^\\a$.\nWe define $\\Gamma\\equiv-n_\\a m^\\a$, so\n$\\Gamma=\\dot{t}^{-1}=\\partial_t s$ and $m^\\a=\\Gamma n^\\a$.\nPhysically, $\\Gamma$ describes the time dilation due to\nthe non-geodesic motion of the $n^\\a$ frame; we have\n$\\dot{n}_\\a=\\hat{\\nabla}_\\a\\ln\\Gamma$.\n(Note that a $\\Gamma$ which depends only on $t$\ncorresponds to a different time coordinate, not\ndifferent physics.)\nOnly if $\\dot{n}^\\a=0$ can we choose $\\Gamma=1$ and $s=t$,\nwhich is equivalent to the statement that $\\mathcal{N}$ is a\nhypersurface of constant proper time.\nIn addition to $s$ and $t$, we also have the\nproper time of the observers, defined by $u^\\a$.\n\nBecause the timescale for the evolution of structures\nis determined by their proper time, the hypersurface of statistical\nhomogeneity and isotropy could be expected to coincide with the\nhypersurface of constant proper time of observers\ncomoving with the structures, as argued in\n\\cite{Rasanen:2006b, Rasanen:2008a, Rasanen:2008b}.\nHowever, if the matter consists of several components\nwhich form structures differently, the situation is not\nso simple. For example, in the real universe,\ndark matter and baryons cluster differently (though the differences\nare not expected to be important on scales larger\nthan the homogeneity scale).\nAnd on small scales, dark matter is multistreaming,\nso there is more than one proper time\nassociated with the matter flow at a single point.\nWe keep the hypersurface of statistical homogeneity\nand isotropy arbitrary.\n\nWithout loss of generality, we write the observer\nvelocity $u^\\a$ in terms of $n^\\a$ and a component\northogonal to $n^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{n}\n  u^\\a = \\gamma ( n^\\a + v^\\a ) \\ ,\n\\eea\n\n\\noindent where $\\gamma\\equiv -n_\\a u^\\a=(1-v^2)^{-1\/2}$, with\n$v^2\\equiv v_\\a v^\\a$ and $v_\\a n^\\a=0$.\nNote that $v^\\a$ is not the peculiar velocity, either\nin the perturbation theory sense of a velocity with\nrespect to a fictitious background, or in the\nphysical sense of deviation from\na shearfree velocity field \\cite{peculiar}.\nThe quantity $v^\\a$ measures the deviation of the local\nobserver velocity from the time direction set by the\nframe of statistical homogeneity and isotropy.\nEven if $v^\\a$ is zero, there can be arbitrarily large\nspatial variations in the expansion rate.\nWe will see that a large $v$ implies significant anisotropy\nin the CMB. We therefore often take $v$ to be small,\nand expand to first order in $v$. We use $\\simeq$ to indicate\nequality up to and including terms first order in $v^\\a$.\n(We do not assume that derivatives of $v^\\a$ are small.)\nPhysically, this means that the motion of the observers\nwith respect to the frame of homogeneity and isotropy is\nnon-relativistic.\n\n\\paragraph{Fluid kinematics.}\n\nThe covariant derivative of $n^\\a$ can be decomposed as\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{gradn}\n  \\nabla_\\b n_\\a\n  &=& \\frac{1}{3} h_{\\a\\b} \\theta + \\sigma_{\\a\\b} - \\dot{n}_\\a n_\\b \\ ,\n\\eea\n\n\\noindent where\n$\\theta\\equiv\\nabla_\\a n^\\a=\\hat{\\nabla}_\\a n^\\a$\nis the volume expansion rate and\n$\\sigma_{\\a\\b}\\equiv\\nabla_{\\langle\\b} n_{\\a\\rangle}=\\hat{\\nabla}_{\\langle\\b} n_{\\a\\rangle}$\nis the shear tensor.\nThe tensor $\\sigma_{\\a\\b}$ and the acceleration vector\n$\\dot{n}^\\a$ are spatial in the sense that they are orthogonal\nto $n^\\a$, $\\sigma_{\\a\\b} n^\\b=0$, $\\dot{n}_\\a n^\\a=0$.\nThe shear scalar is defined as\n$\\sigma^2\\equiv\\frac{1}{2}\\sigma_{\\a\\b}\\sigma^{\\a\\b}$.\nBecause $n^\\a$ is hypersurface-orthogonal, it follows\nfrom Frobenius' theorem that the vorticity \n$\\omega_{\\a\\b} \\equiv \\nabla_{[\\b} n_{\\a]} + \\dot{n}_{[\\a} n_{\\b]}=\\hat{\\nabla}_{[\\b} n_{\\a]}$\nis zero\n\\cite{Ehlers:1961, Ellis:1971, Ellis:1998c}, \\cite{Wald:1984} (page 434).\n\nThe covariant derivative of the observer velocity $u^\\a$ can be\nanalogously decomposed with respect to itself,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{gradu}\n  \\nabla_\\b u_\\a\n  &=& \\frac{1}{3} h^{(u)}_{\\a\\b} \\theta^{(u)} + \\sigma^{(u)}_{\\a\\b} + \\omega^{(u)}_{\\a\\b} - A_\\a u_\\b \\ ,\n\\eea\n\n\\noindent where $\\theta^{(u)}\\equiv\\nabla_\\a u^\\a$,\n$\\sigma_{\\a\\b}^{(u)}\\equiv h^{(u)}_{\\a\\c} h^{(u)}_{\\b\\d} \\nabla^\\d u^\\c-\\frac{1}{3}\\theta^{(u)} h^{(u)}_{\\a\\b}$,\n$\\omega^{(u)}_{\\a\\b} \\equiv \\nabla_{[\\b} u_{\\a]} + A_{[\\a}u_{\\b]}$\nand $A^\\a\\equiv u^\\b\\nabla_\\b u^\\a$.\n\nGiven \\re{n}, the expansion rates in the two frames \nare related as (see \\cite{Tsagas:2007} for the expressions\nfor the acceleration, shear and vorticity)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{reltheta} \\theta^{(u)} &=& \\gamma \\theta + \\gamma ( \\hat{\\nabla}_\\a v^\\a + \\dot{n}_\\a v^\\a ) + \\gamma^3 ( \\dot{v}_\\a v^\\a + v^\\a v^\\b \\hat{\\nabla}_\\a v_\\b ) \\nonumber \\\\\n  &\\simeq& \\theta + \\hat{\\nabla}_\\a v^\\a + \\dot{n}_\\a v^\\a + \\dot{v}_\\a v^\\a \\ .\n\\eea\n\n\\paragraph{The energy-momentum tensor.}\n\nIn the geometrical optics approximation, light propagation\nis kinematical, and independent of the laws which determine\nthe evolution of the geometry.\nHowever, we prefer to replace the Einstein tensor\nwith the energy-momentum tensor which describes the\nmatter content, and to do that we assume that the geometry\nis related to the matter by the Einstein equation,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Einstein}\n  G_{\\a\\b} &=& 8 \\piG_{\\mathrm{N}} T_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $G_{\\a\\b}$ is the Einstein tensor, $G_{\\mathrm{N}}$\nis Newton's constant, and $T_{\\a\\b}$ is the energy-momentum tensor.\n\nWithout loss of generality, the energy-momentum tensor can\nbe decomposed with respect to $n^\\a$ as\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{emdecn}\n  T_{\\a\\b} = \\rho^{(n)} n_\\a n_\\b + p^{(n)} h_{\\a\\b} + 2 q^{(n)}_{(\\a} n_{\\b)} + \\pi^{(n)}_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $\\rho^{(n)}\\equiv n^\\a n^\\b T_{\\a\\b}$ is the energy density,\n$p^{(n)}\\equiv\\frac{1}{3} h^{\\a\\b} T_{\\a\\b}$ is the pressure,\n$q^{(n)}_\\a\\equiv -h_\\a^{\\ \\b} n^\\c T_{\\b\\c}$ is the energy\nflux and\n$\\pi^{(n)}_{\\a\\b}\\equiv h_{\\a}^{\\ \\c} h_{\\b}^{\\ \\d} T_{\\c\\d} - \\frac{1}{3} h_{\\a\\b} h^{\\c\\d} T_{\\c\\d}=T_{\\langle\\a\\b\\rangle}$\nis the anisotropic stress.\nBoth $q^{(n)}_\\a$ and $\\pi^{(n)}_{\\a\\b}$ are spatial in\nthe sense that $q^{(n)}_\\a n^\\a=0, \\pi^{(n)}_{\\a\\b} n^\\b=0$.\nThe quantities measured by the observers\nare given by the decomposition with respect to $u^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{emdecu}\n  T_{\\a\\b} = \\rho^{(u)} u_\\a u_\\b + p^{(u)} h^{(u)}_{\\a\\b} + 2 q^{(u)}_{(\\a} u_{\\b)} + \\pi^{(u)}_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $\\rho^{(u)}, p^{(u)}, q^{(u)}_\\a$ and $\\pi^{(u)}_{\\a\\b}$\nare defined analogously to the $n^\\a$ frame quantities.\nLocally, dust is defined as matter for which\n$p^{(u)}$, $q^{(u)}_\\a$ and $\\pi^{(u)}_{\\a\\b}$ are zero;\nit then follows from the equations of motion that $A^\\a$\nis also zero.\nIn the $u^\\a$ frame, the non-dust terms have a clear physical\ninterpretation in terms of what the observers measure.\nSuch terms can arise from the properties of matter\n(it may be that the matter cannot be treated as dust in any frame)\nand from the fact that an ideal fluid looks non-ideal to a\nnon-comoving observer. We discuss treating the matter\napproximately as dust in \\sec{sec:evo}.\n\nWe could equally take \\re{emdecn} and \\re{emdecu}\nas decompositions of the Einstein tensor rather than the\nenergy-momentum tensor.\nWe use assumed symmetry properties of \\re{emdecn}\nsuch as the absence of preferred directions over large\ndistances, and these could be equally phrased in terms of the\ngeometry expressed in $G_{\\a\\b}$.\nHowever, $T_{\\a\\b}$ is more transparent because it can be\nunderstood in terms of a matter model.\n\n\\subsection{Photon energy and redshift}\n\n\\paragraph{The photon momentum.}\n\nWe want to relate quantities integrated along null\ngeodesics to average quantities which characterise\nthe spatial geometry.\nWe use assumptions about the symmetry properties\nof the spacetime, so averages are most meaningfully\ndiscussed in terms of quantities on $\\mathcal{N}$\nand the vector $n^\\a$.\nIn contrast, the observable redshift and light\ndeflection are defined by the observer velocity $u^\\a$.\n(The angular diameter distance and the null shear\nscalar are independent of the velocity field \\cite{Sachs:1961}.)\n\nIn the geometrical optics approximation light travels on null geodesics\n\\cite{Misner:1973} (page 570), \\cite{Schneider:1992} (page 93).\nWe do not consider caustics, which are not expected to be\nimportant for typical light rays in cosmology (though see \\cite{caustic}).\nFor treatment of the CMB in the covariant formalism, see\n\\cite{Maartens:1998, Dunsby:1997, Zibin:2008a}.\nThe null geodesic tangent vector\nis given by the gradient of the phase of the wave,\nidentified with the photon momentum, and denoted by $k^\\a$.\nIt satisfies $k_\\a k^\\a=0$ and $k^\\a \\nabla_\\a k^\\b=0$.\nThe redshift plus one is proportional to the energy\nmeasured by the observer, $1+z\\propto E^{(u)}$, which in turn is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eu}\n  E^{(u)} = - u_\\a k^\\a \\ .\n\\eea\n\nThe photon momentum can be decomposed into an\namplitude and the direction, and the direction\ncan be split into components parallel and orthogonal to $u^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kdecu}\n  k^\\a = E^{(u)} ( u^\\a + r^\\a ) \\ ,\n\\eea\n\n\\noindent with $u_\\a r^\\a=0$, $r_\\a r^\\a=1$.\n\nBecause the vector $n^\\a$ is adapted to the\nsymmetry of the spacetime, it is more convenient\nto calculate quantities in the $n^\\a$ frame and then transform\nto the $u^\\a$ frame.\nThe decomposition of $k^\\a$ with respect to $n^\\a$ reads\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kdecn}\n  k^\\a = E^{(n)} ( n^\\a + e^\\a ) \\ ,\n\\eea\n\n\\noindent with $E^{(n)} \\equiv - n_\\a k^\\a, n_\\a e^\\a=0$, $e_\\a e^\\a=1$.\nThe quantities $E^{(n)}$ and $e^\\a$ do not have a\nstraightforward observational interpretation, unlike $E^{(u)}$ and $r^\\a$.\n\nThe observed energy $E^{(u)}$ is given in terms of $E^{(n)}$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{EuEn}\n  E^{(u)} &=& \\gamma ( 1 - v_\\a e^\\a ) E^{(n)} \\ ,\n\\eea\n\n\\noindent and the observed direction $r^\\a$ is related to $e^\\a$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{re}\n  r^\\a &=& \\frac{1}{\\gamma ( 1 - v_\\b e^\\b )} ( n^\\a + e^\\a ) - \\gamma ( n^\\a + v^\\a ) \\nonumber \\\\\n  &\\simeq& ( 1 + v_\\b e^\\b ) e^\\a + v_\\b e^\\b n^\\a - v^\\a \\ .\n\\eea\n\n\\noindent The inverse relation is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{er}\n  e^\\a &=& \\frac{1}{ \\gamma + v_\\b r^\\b } ( u^\\a + r^\\a ) - \\gamma^{-1} u^\\a + v^\\a \\nonumber \\\\\n  &\\simeq& ( 1 - v_\\b r^\\b ) r^\\a - v_\\b r^\\b u^\\a + v^\\a \\ .\n\\eea\n\n\\paragraph{Statistical homogeneity and isotropy.}\n\nWe obtain the evolution of $E^{(n)}$ by operating with the derivative\nalong the null geodesic, $\\patl{}\\equiv k^\\a\\nabla_\\a$.\nDenoting $\\pate{} \\equiv (n^\\a + e^\\a) \\partial_\\a$ and using\n\\re{gradn} and \\re{kdecn}, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eder}\n  E^{(n)} \\pate{E^{(n)}} &=& k^\\b \\nabla_\\b E^{(n)} \\nonumber \\\\\n  &=& - k^\\a k^\\b \\nabla_\\b n_\\a \\nonumber \\\\\n  &=& - {E^{(n)}}^2 \\left( \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right) \\ ,\n\\eea\n\n\\noindent which integrates into\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eint}\n  E^{(n)}(\\eta) &=& E^{(n)}(\\eta_0) \\exp\\left( \\int_{\\eta}^{\\eta_0} \\mathrm{d} \\eta \\left[ \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right] \\right) \\nonumber \\\\\n  &=& E^{(n)}(t_0,\\bi{x}_0) \\exp\\left( \\int_{t}^{t_0} \\mathrm{d} t \\Gamma \\left[ \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right] \\right) \\nonumber \\\\\n  &\\approx& E^{(n)}(t_0,\\bi{x}_0) \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ ,\n\\eea\n\n\\noindent where the integral is along the null geodesic and\nthe subscript $0$ refers to the observer's position and time.\nOn the second line we have taken the time $t$ as the integration\nvariable; the spatial coordinates $\\bi{x}$ on $\\mathcal{N}$ are understood\nas functions of $t$ on the null geodesic.\nWe have then taken into account\nthat if there are no preferred directions in the\ngeometry of $\\mathcal{N}$ over long distances, and the\ndirection $e^\\a$ changes only little or\nevolves much more slowly than the distribution\nof the geometry (we discuss this in \\sec{sec:def}),\nthe dominant contribution is given by the average\nexpansion rate. (We use the symbol $\\approx$\nto indicate dropping terms which are suppressed due\nto statistical homogeneity and isotropy, in contrast\nto $\\simeq$, which indicates dropping terms which\nare small because $v\\ll1$.)\nThe argument for this is the following \\cite{Rasanen:2008b}.\nIf $\\Gamma\\dot{n}^\\a$ has no preferred orientation,\nit points equally in the directions along\nand opposite to $e^\\a$, so its contribution vanishes.\nSimilarly, $\\Gamma\\sigma_{\\a\\b}$ contributes only via\nits trace, which is zero.\nThe term $\\frac{1}{3}\\Gamma\\theta$ then gives the dominant contribution.\nUnder the assumption that the timescale for the evolution of\nthe distribution of the geometry is much larger than the time\nit takes for light to cross the homogeneity scale, the\nintegral is dominated by the average value of $\\Gamma\\theta$,\nas the contributions of the variation around the average cancel.\n(See \\sec{sec:av} for details of the averaging.)\nIn reality, there is some evolution of the quantities\nalong the null geodesic, and the cancellations are not\nperfect, so the contributions of $\\Gamma\\dot{n}_\\a e^\\a$,\n$\\Gamma\\sigma_{\\a\\b} e^\\a e^\\b$ and of the variation of\n$\\Gamma\\theta$ are only suppressed instead of zero.\n\nThe observed energy $E^{(u)}$ is, using \\re{EuEn},\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  E^{(u)} &\\approx& E^{(n)}(t_0,\\bi{x}_0) \\gamma ( 1 - v_\\a e^\\a ) \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ ,\n\\eea\n\n\\noindent and the redshift $1+z=E^{(u)}(\\eta)\/E^{(u)}(\\eta_0)$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{z}\n  1+z &\\approx& \\frac{ \\gamma ( 1 - v_\\a e^\\a ) }{ \\gamma_0 ( 1 - v_\\a e^\\a )|_0 } \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent Expressing $e^\\a$ in terms of the observed direction\n$r^\\a$ with \\re{er}, it is transparent that there are\nlarge observed anisotropies in the redshift of isotropically\ndistributed sources unless $v$ is small or constant or there\nis a conspiracy of cancellations.\nConversely, if $v$ is small, the anisotropy is small,\neven though the variations in the geometry can be large.\nIn particular, the near-isotropy of the CMB does\nnot imply that the universe would be nearly FRW\n\\cite{Rasanen:2008b, Rasanen:2009}.\nAssuming $v\\ll1$, the correction due to $v$ reduces to \n$v_\\a e^\\a|_0-v_\\a e^\\a\\simeq v_\\a r^\\a|_0-v_\\a r^\\a$.\nThe first term is the dipole due to the motion of the observer\nwith respect to the frame of statistical homogeneity and isotropy,\nand the second, which can have arbitrary angular dependence,\nis the corresponding term at the source. These are in addition\nto the usual dipole due to the difference between the velocity\nof the observer and the source.\nAs long as the difference between $u^\\a$ and $n^\\a$ is small,\nthe difference in the redshift between the two frames is small,\neven though the expansion rates $\\theta$ and $\\theta^{(u)}$\ncan be very different, as the gradient of $v^\\a$ can be\nlarge even when $v$ is small.\n\n\\paragraph{The local environment.}\n\nWhen we argue for the cancellation of terms other than\n$\\av{\\Gamma\\theta}$ in the integral \\re{Eint} due to symmetry,\nthis only applies to propagation over distances longer than the\nhomogeneity scale, and deviations due to the local environment\nare not accounted for.\nFor example, in linearly perturbed FRW spacetimes, the shear term\n$\\sigma_{\\a\\b} e^\\a e^\\b$ contains the usual local dipole,\nwhich we have neglected.\nTo be consistent in our approximation of concentrating on\npropagation over long distances and neglecting the\neffect of the local environments near the source and the observer,\nwe should approximate $1 - e^\\a v_\\a + v_\\a e^\\a|_0 \\simeq 1$.\nIn the real universe, this approximation seems to hold well.\nThe velocity difference between the CMB frame and our\nrest frame is of the order $10^{-3}$, and the rest frame\nof local large-scale structures is also near the CMB frame \\cite{dipole}.\nThe effect of the local environment is likely to be small\nas long as structures are small compared to the\ndistance the light travels and the observer is not in a\nspecial location \\cite{Rasanen:2008a}.\nThis is true for the structures which are known to exist\nand which are expected in usual models of structure formation,\nbut may not be valid for speculative large spherical structures,\noften described with the Lema\\^{\\i}tre-Tolman-Bondi (LTB)\nmodel \\cite{LTB, February:2009}.\n\nFor the CMB anisotropies, the corrections\ndue to the local environment and the deviations around the mean\ncannot be neglected. They are important\nfor the low multipoles, as in the Integrated Sachs-Wolfe\neffect and the Rees-Sciama effect, and could be\nrelated \\cite{asymmodels} to observed violations of\nstatistical isotropy of the CMB \\cite{asymobs}.\nFormalism for the CMB in the case when the geometry is\nnot perturbatively near FRW has been developed in \\cite{Maartens:1998}.\n\n\\paragraph{The mean redshift and the scale factor.}\n\nThe redshift characterises a single geodesic (or more accurately,\ntwo points and two frames along a single geodesic), so its spatial\naverage is not well defined. However, it is useful to introduce the \n``mean redshift'' $\\bar{z}$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  1 + \\bar{z} \\equiv \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent The physical interpretation of $1+\\bar{z}$ is that\nif we take any two points on $\\mathcal{N}(t)$ and $\\mathcal{N}(t_0)$ which are\nconnected by a null geodesic (or several), the redshift\nalong the null geodesic(s) is $\\bar{z}$ plus small corrections\n(assuming that the rest frames of the source and the observer\nare close to the frame of statistical homogeneity and isotropy).\nFrom the arguments above and the observational fact that the CMB\ndeviations from isotropy and from the blackbody shape of the\nspectrum are small \\cite{Fixsen:1996} we know that temperature\ndifferences between different spatial locations are small,\nand the mean value of the redshift gives the dominant contribution.\n\nWe define the scale factor $a$ as (setting $a(t_0)=1$)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{a}\n  a(t) &\\equiv& (1 + \\bar{z})^{-1} = \\exp\\left( - \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent The quantity $\\theta$ gives the change of rate of the\nlocal volume element with respect to the proper time $s$, so\n$\\Gamma\\theta$ gives the rate of change with respect to $t$.\nTherefore $a(t)^3$ is proportional to the\nvolume of $\\mathcal{N}$: the mean redshift is determined by the\nchange of the overall volume of space.\n\nThe change of the redshift of a given source with time,\ncalled redshift drift \\cite{Sandage:1962}, has been suggested\nas a test of the FRW metric and LTB\nmodels \\cite{Uzan:2008, Quartin:2009}. Essentially, the change of\nredshift with time tests the relationship $1+z=a(t)^{-1}$ between\nthe redshift and the scale factor, with the scale factor\nassociated with an average expansion rate. In the present case,\nunlike in LTB models, the relationship between\nthe mean redshift and the average expansion rate is the\nsame as in FRW models. (See \\sec{sec:avexp} for discussion\nof the average expansion rate.)\nHowever, because redshift drift is a small effect,\nthe variations around the mean would have to be considered\ncarefully to make a prediction.\n\n\\subsection{Deflection} \\label{sec:def}\n\n\\paragraph{Picard's proof.}\n\nIn deriving \\re{Eint} it was assumed that the\nspatial direction of the null geodesic does not change\nrapidly along the geodesic.\nThe direction $e^\\a$ is the direction of the null geodesic\nprojected on $\\mathcal{N}$, so it enters into the arguments\nabout cancellation due to symmetry.\nThe direction $r^\\a$ in turn describes the apparent position\nof the source as seen by the observer, so it is an observable,\nand the change in $r^\\a$ is called the deflection.\nAs we do not have information about the 'original'\nposition of the source, i.e. the position in the hypothetical\nsituation that the spacetime would be flat along the photon path,\nthe deflection can only be measured statistically, unless the\napparent position changes on the timescale of the observation,\nas in microlensing.\n\nLet us look at the change of $e^\\a$ and $r^\\a$ along the\nnull geodesic. As with the redshift, it is simpler to\nconsider $e^\\a$ and then relate it to $r^\\a$.\nIt is convenient to choose a non-coordinate basis and introduce\ntetrads adapted to the 1+3 decomposition\n\\cite{Ellis:1967, Ellis:1998c, vanElst:1996}.\nWe denote the tetrad basis by ${t^\\a_{\\ A}}$, with\n$\\eta^{AB} t_{\\a A} t_{\\b B}=g_{\\a\\b}, t^\\a_{\\ A} t_{\\a B}=\\eta_{AB}$\nas usual, where $\\eta_{AB}=\\mathrm{diag}(-1,1,1,1)$.\nWe use capital Latin letters to denote components in the tetrad\nbasis, e.g. $e^A\\equiv e^\\a t_{\\a}^{\\ A}$.\n\nThe change of $e^A$ along the null geodesic is given by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{eeq1}\n  {E^{(n)}}^{-1} \\patl{e^A} &=& ( n^B + e^B ) \\nabla_B e^A \\nonumber \\\\\n  &=& n^A \\left( \\frac{1}{3} \\theta + \\dot{n}_B e^B + \\sigma_{BC} e^B e^C \\right) - \\dot{n}^{A} \\nonumber \\\\\n  && + e^A ( \\dot{n}_B e^B + \\sigma_{BC} e^B e^C ) - \\sigma^A_{\\ \\, B} e^B \\ .\n\\eea\n\n\\noindent where we have on the second line inserted\n$e^A= {E^{(n)}}^{-1} k^A - n^A$ inside the covariant derivative\nand used \\re{gradn} and \\re{Eder}.\nWe specialise the choice of basis by taking $t_\\a^{\\ 0}=n_\\a$,\nso that $n^A=\\delta^{A0}, n_A=-\\delta_{A0}$.\n(In a coordinate basis, this choice is not\npossible in general \\cite{Ehlers:1961, Ellis:1971}.)\nThen $e^A$ is zero for $A=0$, while for the\nspatial components (which we denote by small Latin letters from\nthe middle of the alphabet, $e^i\\equiv h^i_{\\ A} e^A$)\nwe obtain, using the definition of the covariant derivative,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{grade}\n  {E^{(n)}}^{-1} \\patl{e^i} &=& ( n^B + e^B ) \\nabla_B e^i \\nonumber \\\\\n  &=& \\pate{e^i} + ( n^B + e^B ) \\Gamma^i_{\\ CB} e^C \\nonumber \\\\\n  &=& \\pate{e^i} + a^i + \\Omega^i_{\\ j} e^j - a_j e^j e^i - \\epsilon^{i}_{\\ j k} N^j_{\\ l} e^k e^l \\ ,\n\\eea\n\n\\noindent where the connection components $\\Gamma^A_{\\ BC}$\nhave been expressed in terms of the decomposition \\re{gradn}\nof $\\nabla_B n_A$ as well as an object $a_i$, a symmetric object $N_{ij}$\nand an antisymmetric object $\\Omega_{ij}$; see\n\\cite{vanElst:1996, Ellis:1998c} for details\\footnote{In the\nnotation of \\cite{vanElst:1996, Ellis:1998c},\n$\\Omega_{ij}\\equiv\\epsilon_{ijk}\\Omega^k$.}.\nTogether with the 9 degrees of freedom\nin $\\{ \\theta, \\dot{n}^i, \\sigma^i_{\\ j} \\}$, the 12 degrees\nof freedom in $\\{ a^i, N^i_{\\ j}, \\Omega^i_{\\ j} \\}$ completely\ncharacterise the spacetime geometry.\n\nPutting \\re{eeq1} and \\re{grade} together, we have a system\nof ordinary differential equations for the components $e^i$\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{eeq}\n  \\partial_\\eta e^i &=& - \\dot{n}^i - a^i - ( \\sigma^i_{\\ j} + \\Omega^i_{\\ j} ) e^j + ( \\dot{n}_j e^j + a_j e^j + \\sigma_{jk} e^j e^k ) e^i + \\epsilon^{i}_{\\ j k} N^j_{\\ l} e^k e^l \\nonumber \\\\\n  &\\equiv& f^i(\\eta, e^j) \\ .\n\\eea\n\n\\noindent If the geometry is statistically homogeneous and\nisotropic and its distribution evolves slowly, the change in\n$e^i$ remains small due to the lack of preferred directions in\n$\\{\\dot{n}^i, a^i, \\sigma^i_{\\ j}, N^i_{\\ j}, \\Omega^i_{\\ j} \\}$.\nThis can be expressed more formally as follows.\nAccording to Picard's theorem, the system of equations \\re{eeq}\nhas a unique solution given by an iteration\n(see e.g. \\cite{Duff:1966}, page 19). Let us define\n$e^i_{(N+1)}(\\eta)\\equiv e^i_{(0)} + \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' f^i [\\eta',e^i_{(N)}(\\eta')]$,\nwith $e^i_{(0)}\\equiv e^i(\\eta_0)$. The solution\nto \\re{eeq} is given by the $N\\rightarrow\\infty$ limit.\nAt first step, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{esol1}\n  e^i_{(1)} &=& e^i_{(0)} - \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' ( \\dot{n}^i + a^i ) -  e_{(0)}^j \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta'( \\sigma^i_{\\ j} + \\Omega^i_{\\ j} ) + e_{(0)}^i e_{(0)}^j \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' ( \\dot{n}_j + a_j ) \\nonumber \\\\\n  && + e^i_{(0)} e_{(0)}^j e_{(0)}^k \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' \\sigma_{jk}  + e_{(0)}^k e_{(0)}^l \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' \\epsilon^{i}_{\\  j k} N^j_{\\ l} \\ .\n\\eea\n\n\\noindent As with \\re{Eint}, we could write $\\mathrm{d}\\eta=\\Gamma\\mathrm{d} t$,\nwith the understanding that the spatial coordinates are functions\nof $t$ along the null geodesic.\nIf $\\dot{n}^i$ and $a^i$ have no preferred direction\nand evolve slowly, their integral vanishes,\nprovided the distance over which the integral is taken\nis longer than the homogeneity scale.\nIn practice, the cancellation is not perfect, because\nthere is evolution and statistical fluctuations,\nso the integral is simply suppressed.\nSimilarly, all diagonal components of any tensor\nshould contribute almost equally to the integrals\nand the contribution from non-diagonal components should\nbe suppressed.\nBecause $\\sigma^i_{\\ j}, \\Omega^i_{\\ j}$\nand $\\epsilon^{i}_{\\ j k} N^j_{\\ l}$ are traceless, their\ncontributions are suppressed.\nOne can repeat the argument at every iteration to conclude\nthat the solution $e^i$ is $e^i_{(0)}$ plus a small deviation.\n(For the linear term in \\re{eeq} this is obvious,\nas it simply gives the $\\eta$-ordered exponential of the\nintegral of $\\sigma^i_{\\ j} + \\Omega^i_{\\ j}$.)\nThe change in the observed position of the source\n$r^A$ is then also small as long as $v$ is small, as\nwe see from \\re{re}.\nThis qualitative understanding should be made more\nrigorous, and the amplitude and the distribution of\nthe deflection should be evaluated.\n\n\\subsection{Null shear and angular diameter distance} \\label{sec:da}\n\n\\paragraph{The null shear.}\n\nThe distortion of the size and shape of the source\nimage are described by the null expansion rate $\\tilde{\\theta}$\n(or equivalently the angular diameter distance $D_A$)\nand the null shear tensor $\\tilde{\\sigma}_{\\a\\b}$.\nTo find these quantities, we need\nto introduce a tensor $\\tilde{h}_{\\a\\b}$ which projects onto\na two-space orthogonal to the null geodesic, $\\tilde{h}_{\\a\\b} k^\\b=0$.\nAnalogously to \\re{gradn} and \\re{gradu}, the covariant\nderivative of $k^\\a$ can be decomposed as follows:\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kgrad}\n  \\nabla_\\b k_\\a &=& \\tilde{\\theta}_{\\a\\b} \\nonumber \\\\\n  &=& \\frac{1}{2} \\tilde{h}_{\\a\\b} \\tilde{\\theta} + \\tilde{\\sigma}_{\\a\\b} + k_{(\\a} P_{\\b)} \\ ,\n\\eea\n\n\\noindent where the trace\n$\\tilde{\\theta}=\\tilde{h}^{\\a}_{\\ \\b} \\nabla_\\a k^\\b=\\nabla_\\a k^\\a$\nis the expansion rate of the area of the null geodesic bundle,\n$\\tilde{\\sigma}_{\\a\\b}= \\tilde{h}_{\\a}^{\\ \\d} \\tilde{h}_{\\b}^{\\ \\c} \\nabla_\\c k_\\d - \\frac{1}{2} \\tilde{h}_{\\a\\b} \\tilde{\\theta}$\nis the null shear and $P_\\a$ is a vector which depends on\nthe choice of $\\tilde{h}_{\\a\\b}$ and plays no role in what follows. \nWe have $\\tilde{\\sigma}_{\\a\\b} k^\\b=0$, $P_\\a k^\\a=0$.\nThe null geodesic vorticity is zero,\nbecause $k^\\a$ is a gradient.\nThe null shear scalar is defined as\n$\\tilde{\\sigma}^2\\equiv\\frac{1}{2}\\tilde{\\sigma}_{AB}\\tilde{\\sigma}^{AB}$.\nThe area expansion rate $\\tilde{\\theta}$ is related to the\nangular diameter distance by (see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DA}\n  D_A \\propto \\exp \\left( {\\frac{1}{2} \\int \\mathrm{d}\\l \\tilde{\\theta}} \\right) \\ .\n\\eea\n\nThe angular diameter distance $D_A$ and the shear amplitude\n$\\tilde{\\sigma}_{\\a\\b}\\tilde{\\sigma}^{\\a\\b}$ do not depend on $\\tilde{h}_{\\a\\b}$.\nThey also do not depend on the observer velocity, so it is\nenough to look at the $n^\\a$ frame without having to transform\nto the $u^\\a$ frame at the end.\nIt is again convenient to use tetrads instead of sticking to\na coordinate basis. We choose $\\tilde{h}_{AB}$ to be orthogonal\nto both $n^A$ and $e^A$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{htt}\n  \\tilde{h}_{AB} &=& g_{AB} + n_{A} n_{B} - e_{A} e_{B} \\nonumber \\\\\n  &=& g_{AB} - {E^{(n)}}^{-2} k_{A} k_{B} + 2 {E^{(n)}}^{-1} k_{(A} n_{B)} \\ ,\n\\eea\n\n\\noindent We proceed with $\\tilde{\\sigma}_{AB}$ the same way\nas we did with $e^A$ in \\sec{sec:def}. Taking the derivative\nalong the null geodesic and projecting, we obtain\n(see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{shear1}\n  \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\patl{\\tilde{\\sigma}_{CD}} = - \\tilde{\\theta} \\tilde{\\sigma}_{AB} + C_{AB} \\ ,\n\\eea\n\n\\noindent where \n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  C_{AB} &\\equiv& k^M k^N \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} C_{MCND} \\nonumber \\\\\n  &=& 2 {E^{(n)}}^2 \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\left( E_{CD} + \\frac{1}{2} \\tilde{h}_{CD} e^M e^N E_{MN} - \\tilde\\epsilon_{CM} H^{M} _{\\ \\ D} \\right) \\,\n\\eea\n\n\\noindent where $C_{AB}$ has been expressed in terms\nof the electric and magnetic components of the Weyl tensor,\n$E_{AB} \\equiv C_{ACBD} n^C n^D=C_{A0B0}$,\n$H_{AB} \\equiv \\frac{1}{2} \\epsilon_A^{\\ \\ CD} C_{CDBE} n^E=\\frac{1}{2} \\epsilon_A^{\\ \\ CD} C_{CDB0}$, and $\\tilde\\epsilon_{AB}\\equiv\\epsilon_{ABC} e^C$.\n\nOn the other hand, from the definition of the covariant derivative\nwe obtain (see \\cite{Ellis:1998c, vanElst:1996} for the\ndecomposition of $\\Gamma^A_{\\ BC}$)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{shear2}\n  {E^{(n)}}^{-1} \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\patl{\\tilde{\\sigma}_{CD}} &=& \\pate \\tilde{\\sigma}_{AB} - 2 \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} ( n^E + e^E ) \\Gamma^F_{\\ C E} \\tilde{\\sigma}_{FD} \\nonumber \\\\\n  &=& \\pate \\tilde{\\sigma}_{AB} + \\left( 2 \\Omega_{ij} + 2 \\tilde\\epsilon_{k[i} N_{j]}^{\\ k} + \\epsilon_{ijk} N^{k}_{\\ l} e^l \\right) \\tilde{h}_{(A}^{\\ \\ \\, i} \\tilde{\\sigma}_{B)}^{\\ \\ \\, j} \\ .\n\\eea\n\n\\noindent Note that in a tetrad basis contracting with $\\tilde{h}_{AB}$\ncommutes with $\\pate{}$, but not with $\\patl{}$.\nWe denote indices on the space orthogonal to $n^A$\nand $e^A$ with small Latin letters from the beginning of the\nalphabet, $e^a\\equiv \\tilde{h}^a_{\\ B} e^B$, and specialise the choice\nof basis by taking $t_{\\a}^{\\ 3}=e_\\a$, so that $e^A=\\delta^{A3}, e_A=\\delta_{A3}$.\nPutting together \\re{shear1} and \\re{shear2}, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{patsigma2}\n  \\pate \\tilde{\\sigma}_{ab} &=&  - {E^{(n)}}^{-1} \\tilde{\\theta} \\tilde{\\sigma}_{ab} + {E^{(n)}}^{-1} C_{ab} + ( 2 \\Omega_{c(a} +  2 \\tilde\\epsilon_{d[c} N^{d}_{\\ (a]} + \\tilde\\epsilon_{c(a} N^{3}_{\\ 3} ) \\tilde{\\sigma}_{b)}^{\\ \\, c} \\ .\n\\eea\n\nLike equation \\re{eeq} for the components of the deflection,\n\\re{patsigma2} is an ordinary differential equation for the two\nindependent components of $\\tilde{\\sigma}_{ab}$. However, we cannot\nstraightforwardly apply Picard's theorem. First, \\re{patsigma2}\ncontains the unknown $\\tilde{\\theta}$.\nEven if we include the equation \\re{Raynull} given below\nto obtain a closed first order system of equations, Picard's theorem does\nnot apply, because it assumes that the variables remain bounded,\nwhereas the initial condition for the area expansion rate is\n$\\tilde{\\theta}(\\eta_0)=-\\infty$ at the observer.\n\nNevertheless, the reasoning about cancellations due the lack\nof preferred directions still holds, because the solution\ndepends on the source term $C_{ab}$ only via an integral.\nThis is transparent with the change of variable\n$\\tilde{\\sigma}_{ab}\\equiv\\tilde\\Sigma_{ab}+\\int\\mathrm{d}\\eta {E^{(n)}}^{-1} C_{ab}$.\nWe now argue as before that due to statistical homogeneity\nand isotropy $C_{ab}$ contributes dominantly via its trace,\nwhich is zero, so $\\tilde{\\sigma}_{ab}\\approx\\tilde\\Sigma_{ab}$.\nThis eliminates the source term in \\re{patsigma2}.\nGiven the initial condition $\\tilde\\Sigma_{ab}(\\eta_0)=0$,\nwe obtain $\\tilde{\\sigma}_{ab}\\approx\\tilde\\Sigma_{ab}\\approx0$.\nAs with the deflection, this argumentation needs to\nbe made more rigorous, and the amplitude of the small\nshear that is generated should be calculated.\n\n\\paragraph{The angular diameter distance.}\n\nApplying the derivative $\\patl{}$ to $\\tilde{\\theta}$, we\nobtain the evolution equation for the area expansion rate\n(see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Raynull}\n  \\patl\\tilde{\\theta} &=& - G_{AB} k^A k^B - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2 \\nonumber \\\\\n  &=& - 8\\piG_{\\mathrm{N}} T_{AB} k^A k^B - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2  \\nonumber \\\\\n  &=& - 8\\piG_{\\mathrm{N}} \\big( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B \\big) {E^{(n)}}^2 - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2 \\ ,\n\\eea\n\n\\noindent where we have used the Einstein equation\n\\re{Einstein} and applied the decomposition \\re{emdecn}\nof the energy-momentum tensor.\nAs discussed in \\sec{sec:geom}, we could equally\nregard \\re{emdecn} as the decomposition of the\nEinstein tensor (or, in the present context, the\nRicci tensor, as the trace does not contribute to \\re{Raynull}).\n\nUsing \\re{DA}, we obtain from \\re{Raynull} the evolution\nequation for the angular diameter distance:\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAeq}\n  \\frac{\\mathrm{d}^2 D_A}{\\mathrm{d}\\l^2} &=& - \\left[ 4 \\piG_{\\mathrm{N}} \\big( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B \\big) {E^{(n)}}^2 + \\tilde{\\sigma}^2 \\right] D_A \\ .\n\\eea\n\nThe solution again depends on the source functions only via\nan integral. This is transparent with the change of variable\n$\\tilde{\\theta}\\equiv\\tilde\\Theta - 8\\piG_{\\mathrm{N}} \\int\\mathrm{d}\\l ( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B ) {E^{(n)}}^2$.\nWe can write\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{muint}\n  && \\int\\mathrm{d}\\l ( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B ) {E^{(n)}}^2 \\nonumber \\\\\n  && \\approx E^{(n)}(\\eta_0) \\int\\mathrm{d} t a^{-1} \\av{ \\Gamma ( \\rho^{(n)} + p^{(n)} ) } \\nonumber \\\\\n  && = E^{(n)}(\\eta_0) \\int\\mathrm{d} t a^{-1} \\av{ \\Gamma ( \\rho^{(u)} + p^{(u)} )+ \\frac{4}{3} \\Gamma F } \\ ,\n\\eea\n\n\\noindent where \n$F\\equiv v^2 (\\rho^{(u)} + p^{(u)}) + 2 \\gamma^{-1} q^{(u)}_A v^A + \\pi^{(u)}_{AB} v^A v^B\\simeq 2 q^{(u)}_A v^A $.\nOn the second line we have taken into account\n$\\mathrm{d}\\l={E^{(n)}}^{-1}\\mathrm{d}\\eta$ and the approximate scaling\n$E^{(n)}\\approx E^{(n)}(\\eta_0) a^{-1}$.\nWe have also again applied the reasoning that statistical\nhomogeneity and isotropy together with slow evolution implies\nthat the contributions of $q^{(n)}_A e^A$ and $\\pi^{(n)}_{AB} e^A e^B$\nare suppressed, and that the dominant\ncontribution of $\\rho^{(n)} + p^{(n)}$ comes from the average,\nFinally, we have written $\\rho^{(n)} + p^{(n)}$ in terms of $u^\\a$\nframe quantities using \\re{h}, \\re{n}, \\re{emdecn} and \\re{emdecu}.\nDropping the null shear, the solution $\\tilde{\\theta}$ to \\re{Raynull}\ndepends only on the quantity \\re{muint} and\n$\\l=\\int\\mathrm{d}\\eta {E^{(n)}}^{-1} \\approx E^{(n)}(\\eta_0)^{-1}\\int\\mathrm{d} t a\\av{\\Gamma}$,\nwhere we have assumed that $\\Gamma$ has a statistically\nhomogeneous and isotropic distribution and varies slowly, so\nthat the integral is dominated by the average value.\nBecause both \\re{muint} and $\\l$ depend approximately only\non $t$ (and $E^{(n)}(\\eta_0)$), so does $\\tilde{\\theta}$.\nWriting \\re{Raynull} as an integral equation, dropping\nsubdominant parts which depend on position, taking the\ntime derivative and expressing the equation in terms\nof the angular diameter distance, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAbareq}\n  H \\partial_{\\bar{z}} \\left[ \\av{\\Gamma}^{-1} (1+\\bar{z})^2 H \\partial_{\\bar{z}} \\bar{D}_A \\right] &=& - 4\\piG_{\\mathrm{N}} \\av{ \\Gamma (\\rho^{(u)} + p^{(u)}) + \\frac{4}{3} \\Gamma F } \\bar{D}_A \\nonumber \\\\\n  &\\simeq& - 4\\piG_{\\mathrm{N}} \\av{ \\Gamma (\\rho^{(u)} + p^{(u)}) + \\frac{8}{3} \\Gamma q^{(u)}_A v^A } \\bar{D}_A \\ ,\n\\eea\n\n\\noindent where the notation $\\bar{D}_A(t)$ refers to the dominant\npart of the angular diameter distance with the corrections to the\nmean dropped, and we have used the relation\n$\\partial_t\\bar{D}_A=-(1+\\bar{z}) H \\partial_{\\bar{z}}\\bar{D}_A$, with $H\\equiv\\pat_t{a}\/a$.\nThe quantity $\\bar{D}_A$ has a similar physical interpretation\nas $\\bar{z}$: if we take any two points on $\\mathcal{N}(t)$ and $\\mathcal{N}(t_0)$\nconnected by a null geodesic (or several of them), the angular\ndiameter distance along the geodesic(s) is $\\bar{D}_A$\nplus small corrections.\nAs noted in \\cite{Rasanen:2008b}, while $\\bar{D}_A(\\bar{z})$\nis well-defined, there does not exist a function $D_A(z)$ even along\na single geodesic, because the redshift is in general\nnot monotonic along the null geodesic\\footnote{In\n\\cite{Rasanen:2008b} it was incorrectly claimed that when\nthe factor in the parenthesis on the right-hand side of\n\\re{DAeq} is positive, $D_A$ would be monotonic, because\nthe initial condition (at the observer) for $\\patl{D_A}$\nis negative. However, because $\\l$ decreases\nalong the null geodesic away from the observer, this\nonly implies that $\\patl{D_A}$ has at most one zero,\nso $D_A$ is separately monotonic on at most two sections\nof the null geodesic, as is well known from FRW spacetimes.\nIn the present case, the sign of the factor in the parenthesis\nis not determined, and the number of zeros of $\\patl{D_A}$ is not limited.}.\n\nApart from $\\Gamma$ and $F$, the equation \\re{DAbareq}\nfor the mean distance in terms of $\\bar{z}$ is the same as in\nFRW spacetimes.\nHowever, the relation between $\\av{\\rho^{(u)} + p^{(u)}}$ and\n$\\bar{z}$ is different than in the FRW case, as we discuss\nin the next section (see also \\cite{Rasanen:2008b}).\nAs with the redshift and the\ndeflection, it is important to make the arguments\nabout the null shear and the angular diameter distance\nmore rigorous by evaluating the variation around the\nmean. Observationally, the angular diameter distance\nis known not to vary much with direction \\cite{peakvar}.\n\n\\section{The average expansion rate and the scale factor} \\label{sec:av}\n\n\\subsection{The average expansion rate} \\label{sec:avexp}\n\n\\paragraph{Defining the average.}\n\nWe have started with light propagation, which involves\nthe observed quantities directly, and have been led to\nconsider averages. The average of a scalar $f$ on $\\mathcal{N}$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{av}\n  \\av{f}(t) \\equiv \\frac{ \\int \\epsilon f }{ \\int \\epsilon } \\ ,\n\\eea\n\n\\noindent where $t$ is constant on $\\mathcal{N}$.\nRecall that in general, $t$ is not a proper time.\nIn particular, it is neither the proper time\nassociated with $n^\\a$ nor the proper time measured\nby the observers. The commutation rule between averaging\nand taking a derivative with respect to $t$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{comm}\n  \\partial_t\\av{f} = \\av{\\partial_t f} + \\av{\\Gamma\\theta f} - \\av{f} \\av{\\Gamma\\theta}\n\\eea\n\nThe scale factor $a$ was defined in \\re{a} to give the mean\nredshift. From the definition it follows that $a^3$ is\nproportional to the volume of $\\mathcal{N}$. The average expansion\nrate of interest is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{H}\n  3 \\H &=& \\av{\\Gamma\\theta} \\nonumber \\\\\n  &=& \\av{ \\Gamma \\gamma^{-1} \\theta^{(u)} - \\hat{\\nabla}_\\a ( \\Gamma v^\\a ) - \\Gamma \\gamma^2 (\\dot{v}_\\a v^\\a + v^\\a v^\\b \\hat{\\nabla}_\\a v_\\b) } \\nonumber \\\\\n  &\\simeq& \\av{ \\Gamma \\theta^{(u)} - \\hat{\\nabla}_\\a ( \\Gamma v^\\a ) - \\Gamma \\dot{v}_\\a v^\\a } \\ ,\n\\eea\n\n\\noindent where we have used \\re{reltheta} and the relation\n$\\dot{n}_\\a=\\Gamma^{-1}\\hat{\\nabla}_\\a\\Gamma$.\nIn the irrotational ideal fluid case, the scale factor\nwas originally defined as the volume of the hypersurface\northogonal to $u^\\a$, or equivalently by using the\naverage of $\\theta^{(u)}$, with the lapse function\nincluded \\cite{Buchert:1999, Buchert:2001}.\nWe start from light propagation, and while $u^\\a$\nis the relevant velocity field for the redshift, the\nsymmetry of the spacetime selects $\\Gamma \\nabla_\\a n^\\a=\\Gamma\\theta$\nas the relevant local expansion rate.\nLocally $\\theta$ can be very different from $\\theta^{(u)}$, even for\nsmall $v$, because the derivatives of $v^\\a$ can be large.\nHowever, using Gauss' theorem \\cite{Wald:1984} (page 433)\nthe total derivative in \\re{H} can be converted into a surface\nintegral which describes the flux of $\\Gamma v^\\a$ through the boundary.\nIf the distribution is statistically homogeneous and isotropic,\nthere should be an equal flux through the surface in both\ndirections, so the integral vanishes (up to statistical\nfluctuations).\nTherefore, the difference between the average quantities\n$\\av{\\Gamma\\theta}$ and $\\av{\\Gamma\\theta^{(u)}}$ is suppressed\nby $v$.\n\n\\subsection{Evolution equations for the scale factor} \\label{sec:evo}\n\n\\paragraph{The average equations.}\n\nWe now write down the evolution equations for\nthe scale factor $a$, or equivalently for the\naverage expansion rate.\nThese generalise the Buchert equations derived\nfor irrotational dust \\cite{Buchert:1999}.\nIn \\cite{Buchert:2001} the average equations were\nwritten down in the irrotational ideal fluid case\nusing the ADM formalism, assuming that the\naveraging hypersurface is orthogonal to the fluid flow\\footnote{These\naverage equations, apart from the conservation law\nof the energy-momentum tensor, were written down in the context of\ngeneral irrotational matter content in \\cite{Behrend}.\nNote that the perturbative calculation in \\cite{Behrend} is incorrect,\nbecause the averages of both first order terms and intrinsic second order\nterms taken in the perturbed spacetime are neglected in comparison\nwith the averages of squares of first order terms. In fact,\nall these terms are of the same order, and the distinction between\nthem is gauge-dependent \\cite{Kolb:2004}.}.\nIn \\cite{Larena:2009} the average equations were derived\n(also in the ADM formalism) for an ideal fluid\nincluding rotation, taking the expansion rate to be\n$h^{\\a\\b}\\nabla_\\b u_\\a=\\theta^{(u)}+n_\\a \\dot u^\\a$,\nand keeping the hypersurface of averaging arbitrary.\n(The formalism was applied to second order perturbation\ntheory in \\cite{Clarkson:2009a}, with the hypersurface\nfixed by the condition $H_{\\a\\b}=0$.\nAveraging in second order perturbation theory was also\nconsidered in \\cite{Brown:2009}, with different hypersurfaces\nfixed by coordinate conditions.)\nIn \\cite{Gasperini:2009}, the average equations were\nderived for an ideal fluid in the covariant formalism,\nwith an arbitrary averaging hypersurface.\nWe consider general matter content, an arbitrary\nhypersurface of averaging and use the covariant formalism.\n\nCombining the Einstein equation \\re{Einstein}\nwith the Bianchi and Ricci identities for $n^\\a$,\nthe evolution equations can be conveniently written in\nterms of the decompositions \\re{gradn} and \\re{emdecn}\nand the electric and magnetic components of the Weyl tensor\n\\cite{Ellis:1998c, Clarkson:2000, Tsagas:2007}.\nWe are only interested in the three scalar equations\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n   \\label{Rayloc} \\dot{\\theta} + \\frac{1}{3} \\theta^2 &=& - 4\\piG_{\\mathrm{N}} ( \\rho^{(n)} + 3 p^{(n)} ) - 2 \\sigma^2 + \\dot{n}_\\a \\dot{n}^\\a + \\hat{\\nabla}_\\a \\dot{n}^\\a \\\\\n  \\label{Hamloc} \\frac{1}{3} \\theta^2 &=& 8 \\pi G_{\\mathrm{N}} \\rho^{(n)} - \\frac{1}{2} {^{(3)}R} + \\sigma^2 \\\\\n  \\label{consloc} \\dot{\\rho}^{(n)} + \\theta ( \\rho^{(n)} + p^{(n)} ) &=& - \\hat{\\nabla}_\\a q^{(n)\\a} - 2 \\dot{n}_\\a q^{(n)\\a} - \\sigma_{\\a\\b} \\pi^{(n)\\a\\b} \\ .\n\\eea\n\n\\noindent where ${^{(3)}R}$ is the spatial curvature of $\\mathcal{N}$.\nIf $\\mathcal{N}$ is a hypersurface of constant proper time,\nthen $\\dot{n}^\\a=0$, and the equations differ from\nthe irrotational dust case only by the pressure term in the\nRaychaudhuri equation \\re{Rayloc} and the non-dust terms in\nthe conservation law \\re{consloc}. In terms of the Hamiltonian\nconstraint \\re{Hamloc}, the only difference is the different\nevolution of the energy density. We keep $\\dot{n}^\\a$ arbitrary.\nChanging to derivatives with respect to $t$, averaging,\napplying the commutation rule \\re{comm} and using\nthe relations $\\av{\\Gamma\\theta}=3\\pat_t{a}\/a$ and\n$\\dot{n}_\\a=\\Gamma^{-1}\\hat{\\nabla}_\\a\\Gamma$, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayavn} && 3 \\frac{\\pat_t^2{a}}{a} = - 4\\piG_{\\mathrm{N}} \\av{\\rho^{(n)} + 3 p^{(n)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\av{\\hat{\\nabla}_\\a \\dot{n}^{\\a}} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } \\\\\n  \\label{Hamavn} && 3 \\frac{(\\pat_t a)^2}{a^2} = 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(n)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} \\\\\n  \\label{consavn} && \\partial_t \\av{\\rho^{(n)}} + 3 \\H \\av{\\rho^{(n)} + p^{(n)}} = - \\av{\\Gamma\\theta p^{(n)}} + \\av{\\Gamma\\theta} \\av{p^{(n)}} - \\av{\\Gamma \\dot{n}_\\a q^{(n)\\a} + \\Gamma \\sigma_{\\a\\b} \\pi^{(n)\\a\\b}} \\nonumber \\\\\n  && \\quad - \\av{\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})} \\ ,\n\\eea\n\n\\noindent where\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\mathcal{Q} &\\equiv& \\frac{2}{3}\\left( \\av{{\\Gamma\\theta}^2} - \\av{\\Gamma\\theta}^2 \\right) - 2 \\av{\\sigma^2} \\ .\n\\eea\n\n\\noindent In \\cite{Buchert:2001, Behrend, Larena:2009},\nthe equivalent of the factors of $\\Gamma$ were put inside the\naverages of the terms which appear on the right-hand side of\n\\re{Rayloc} and \\re{Hamloc}, rather than the left-hand side.\nInserting \\re{Rayloc} into the last term of \\re{Rayavn} and\n\\re{Hamloc} into the last term of \\re{Hamavn} would recover\nthat form of the equations.\nHowever, the present convention keeps the $\\hat{\\nabla}_\\a\\dot{n}^\\a$\nterm and the last term of \\re{consavn} as total derivatives,\nwhich we can neglect.\n\nBecause the backreaction variable $\\mathcal{Q}$ is a statistical\nquantity which characterises the distribution of the\nspatial geometry, it is appropriate to give it in\nterms of $\\theta$ and $\\sigma$, which are related to $n^\\a$.\nHowever, it seems more appropriate to express the energy-momentum tensor\nin terms of the decomposition with respect to $u^\\a$ from the point of\nview of estimating the magnitude of the different terms.\nUsing \\re{h}, \\re{n}, \\re{emdecn} and \\re{emdecu}, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayavu} && 3 \\frac{\\pat_t^2{a}}{a} = - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\av{\\hat{\\nabla}_\\a \\dot{n}^{\\a}} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } - 8 \\piG_{\\mathrm{N}} \\av{F} \\\\\n  \\label{Hamavu} && 3 \\frac{(\\pat_t a)^2}{a^2} = 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} + 8 \\piG_{\\mathrm{N}} \\av{F} \\\\\n  \\label{consavu} && \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)} + p^{(u)}} = - \\av{\\Gamma\\theta ( p^{(u)} + \\frac{1}{3} F ) } + \\av{\\Gamma\\theta} \\av{p^{(u)} + \\frac{1}{3} F} \\nonumber \\\\\n  && \\quad - \\av{\\gamma \\Gamma \\dot{n}_\\a q^{(u)\\a} + \\gamma^2 \\Gamma\n    (\\rho^{(u)}+p^{(u)}) \\dot{n}_\\a v^\\a + \\gamma \\Gamma \\dot{n}_\\a\n    q^{(u)}_\\b v^\\a v^\\b + \\Gamma \\dot{n}_\\a \\pi^{(u)\\a\\b} v_\\b } \\nonumber \\\\ \n  && \\quad - \\av{ \\Gamma \\sigma_{\\a\\b} \\pi^{(u)\\a\\b} + \\gamma^2 \\Gamma (\\rho^{(u)}+p^{(u)}) \\sigma_{\\a\\b} v^\\a v^\\b + 2 \\gamma \\Gamma \\sigma_{\\a\\b} q^{(u)\\a} v^\\b } \\nonumber \\\\\n  && \\quad - \\partial_t \\av{F} - 4 \\H \\av{F} - \\av{\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})} \\ ,\n\\eea\n\n\\noindent with $F=v^2 (\\rho^{(u)} + p^{(u)}) + 2 \\gamma^{-1}\nq^{(u)}_\\a v^\\a + \\pi^{(u)}_{\\a\\b} v^\\a v^\\b$ as before.\nWe have not written $\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})$ in terms of\n$u^\\a$ frame quantities, because it is suppressed due to\nstatistical homogeneity and isotropy, like $\\hat{\\nabla}_\\a \\dot{n}^\\a$.\nVorticity does not appear in the above equations, because\n$n^\\a$ is hypersurface-orthogonal by construction.\nWere we to decompose $\\nabla_\\b n_\\a$ with respect\nto the $u^\\a$ frame, the vorticity of $u^\\a$ would\n(to leading order in $v$) emerge from\n$\\hat{\\nabla}_\\a \\dot{n}^\\a$ and ${^{(3)}R}$ \\re{Rayloc} and \\re{Hamloc}.\n(For the definition of ${^{(3)}R}$ for velocity fields which are not\nhypersurface-orthogonal, see \\cite{Ellis:1990}.)\nIn particular, the leading order contribution of the $u^\\a$\nframe vorticity to the average Raychaudhuri equation \\re{Rayavn}\nvanishes due to statistical homogeneity and isotropy, because it\nis contained in the boundary term $\\av{\\hat{\\nabla}_\\a \\dot{n}^\\a}$.\n(In Newtonian gravity, the vorticity combines with $\\mathcal{Q}$ to give\na boundary term, so backreaction vanishes for periodic\nboundary conditions and for statistical homogeneity and\nisotropy \\cite{Buchert:1995}.)\n\nDropping the boundary terms $\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})$ and\n$\\hat{\\nabla}_\\a \\dot{n}^\\a$ as well as all terms higher than first\norder in $v$, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayapp} && 3 \\frac{\\pat_t^2{a}}{a} \\simeq - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } - 16 \\piG_{\\mathrm{N}} \\av{q^{(u)}_\\a v^\\a} \\\\\n  \\label{Hamapp} && 3 \\frac{(\\pat_t a)^2}{a^2} \\simeq 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} + 16 \\piG_{\\mathrm{N}} \\av{q^{(u)}_\\a v^\\a} \\\\\n  \\label{consapp} && \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)} + p^{(u)}} \\simeq - \\av{\\Gamma\\theta \\big( p^{(u)}+\\frac{2}{3}q^{(u)}_\\a v^\\a \\big)} + \\av{\\Gamma\\theta} \\av{ p^{(u)} + \\frac{2}{3} q^{(u)}_\\a v^\\a } \\nonumber \\\\\n  && \\quad - \\av{ \\Gamma \\dot{n}_\\a q^{(u)\\a} + \\Gamma (\\rho^{(u)}+p^{(u)}) \\dot{n}_\\a v^\\a + \\Gamma \\dot{n}_\\a v_\\b \\pi^{(u)\\a\\b} } - \\av{ \\Gamma\\sigma_{\\a\\b} \\pi^{(u)\\a\\b} + 2 \\Gamma \\sigma_{\\a\\b} q^{(u)\\a} v^{\\b} } \\nonumber \\\\\n  && \\quad - 2 \\partial_t \\av{q^{(u)}_\\a v^\\a} - 8 \\H \\av{q^{(u)}_\\a v^\\a} \\ .\n\\eea\n\n\\paragraph{The dust approximation.}\n\nOne reason for deriving the general equations\n\\re{Rayavu}--\\re{consavu} is to take into account\ndeviations from the approximation of treating\nthe matter as dust in the late universe.\nThe importance of the different terms depends\non the matter model, and cannot be determined from\ngeneral arguments. However, it is possible\nto say what would would be necessary for the\nnon-dust terms to have a significant effect.\nFor the $\\dot{n}^\\a$ terms to be important in \\re{Rayapp}--\\re{consapp},\n$\\dot{n}_\\a \\dot{n}^\\a$ would have to be of the order of the\nsquare of the expansion rate in a large fraction of space (contrary\nto what was argued in \\cite{Tsagas:2009}; see also \\cite{Coley}),\nor the contraction of the energy flux and $\\dot{n}^\\a$ would\nhave to be of the order of the product of the average energy\ndensity and the average expansion rate.\nIn order for the pressure or the anisotropic stress\nto be important, they would have to be on average of the same order\nof magnitude as the average energy density.\nFor the time dilation to be important, the spatially\nvarying part of $\\Gamma$ would have to be of order one\nin a fraction of space which is of order one.\nIf the matter content is a gas of Standard Model particles\nplus cold or warm dark matter, and structures evolve from\nsmall adiabatic perturbations with a nearly scale-invariant\nspectrum, it seems unlikely that any of these conditions would\nbe satisfied in the late universe when radiation pressure can be neglected.\n\nLet us assume that the matter is approximately dust in\nthe $u^\\a$ frame, i.e. that $p^{(u)}, q^{(u)\\a}, \\pi^{(u)}_{\\a\\b}$\nand $A^\\a$ are small\\footnote{From the equations\nof motion it follows that $A^\\a$ is zero if\n$p^{(u)}, q^{(u)\\a}$ and $\\pi^{(u)}_{\\a\\b}$ are zero, and $A^\\a$ is small if\n$p^{(u)}, q^{(u)\\a}, \\pi^{(u)}_{\\a\\b}$ as well as $\\hat{\\nabla}_\\a p^{(u)}$ and\n$\\hat{\\nabla}^\\b \\pi^{(u)}_{\\a\\b}$ are small.},\nand the deviation of $\\Gamma$ from unity (and the\ntime derivative of the deviation) is small,\n$\\Gamma\\equiv1-\\delta\\Gamma$, with $|\\delta\\Gamma|\\ll1$.\nWhen we drop all squares of small terms (whether\nthey are non-dust terms or $v^\\a$), the\nequations \\re{Rayapp}--\\re{consapp} simplify to\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayav} 3 \\frac{\\pat_t^2{a}}{a} &\\simeq& - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\mathcal{Q} \\nonumber \\\\\n  && + \\av{ \\dot{v}_\\a \\dot{v}^\\a + \\frac{2}{3} \\delta\\Gamma \\theta^2 + 2 \\delta\\Gamma\\partial_t\\theta + \\theta \\partial_t\\delta\\Gamma } \\\\\n  \\label{Hamav} 3 \\frac{(\\pat_t a)^2}{a^2} &\\simeq& 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{2}{3} \\av{ \\delta\\Gamma \\theta^2} \\\\\n  \\label{consav} \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)}} &\\simeq& - \\av{\\theta p^{(u)}} + \\av{ \\dot{v}_\\a q^{(u)\\a} + \\rho^{(u)} \\dot{v}_\\a v^\\a } - \\av{\\sigma^{(n)}_{\\a\\b} \\pi^{(u)\\a\\b} } \\ ,\n\\eea\n\n\\noindent where we have used the fact that\n$\\dot{n}^\\a\\simeq A^\\a-\\dot{v}^\\a$ plus corrections of order $v$.\nEquations \\re{Rayav}--\\re{consav} give the leading\ncorrections to the treatment of matter as irrotational dust,\ncompared to the original Buchert equations \\cite{Buchert:1999}.\nIn particular, they cover the case when the matter can be\nlocally treated as dust, but has rotation, and $n^\\a$ is orthogonal\nto the hypersurface of constant proper time of observers\ncomoving with the dust. Then $\\dot{v}^\\a$ is of order $v$,\nand we can choose $\\Gamma=1$, so the difference between\n$n^\\a$ and $u^\\a$ arises only from vorticity.\nWe see that vorticity alone has (to leading order in $v$)\nno effect on the averages, because the dominant contribution\ncomes from the total derivative $\\hat{\\nabla}_\\a \\dot{n}^\\a$. \nIt does change the relation of the scale factor $a$\nto the geometry, because $a$ will be defined with a\ndifferent vector field, but not the relation of $a$ to the redshift.\n\nFor irrotational dust, the Raychaudhuri equation \\re{Rayloc}\ncan be integrated as an inequality before averaging to obtain\nthe bound $Ht\\leq1$ \\cite{Wald:1984} (page 220),\n\\cite{Nakamura:1995, Rasanen:2005}.\nWhen rotation or non-dust terms are important,\nthere is no such local inequality.\nIndeed, having $\\dot{\\theta}+\\frac{1}{3}\\theta^2>0$ locally\nis required in order for collapsing regions to stabilise.\nFrom a physical point of view, we would still expect\nto recover $Ht\\leq1$ unless there is sustained acceleration\nin a significant fraction of space, but the conditions for this\nderived from \\re{Rayloc} involve combinations of spatial\naverages and integrals over time, and are not entirely transparent.\n\nNote that in order for the approximation of treating the matter\nas dust on average to hold, it is only necessary that the contribution of\nthe non-dust terms to the averages is smaller than that of the average\nenergy density. It is not required that the energy-momentum\ntensor of matter could be locally approximated as dust everywhere.\nIn fact, deviations from the irrotational dust behaviour\nare necessarily important on small scales.\nAs the local Raychaudhuri equation \\re{Rayloc}\nshows, in order to stabilise structures, a large\n$\\dot{n}^\\a$ or its gradient is needed.\nThis can correspond to $u^\\a$ frame vorticity\nas with rotating baryonic structures,\nor the acceleration can be generated by\nanisotropic stress (or a pressure gradient or energy flux)\nas in the case of dark matter \\cite{Pueblas:2008}.\nFor dark matter, the dust approximation is\nlocally invalid in structure formation due to\nmultistreaming \\cite{Buchert:2005}.\nNevertheless, as long as the volume occupied\nby regions where such terms are important is small, their\ncontribution to the average is not important.\nIn Newtonian calculations, this is certainly the case \\cite{Pueblas:2008}.\n\nApproximating the matter content as dust on average does not\nimply  viewing the matter as infinitesimal grains.\nFor example, the issue of what the ``particles''\nof the dust fluid are is sometimes raised, and whether one\ncan consistently ``renormalise'' the scale of the\ndescription as larger stable structures form\n\\cite{Rasanen:2006b, Rasanen:2008a}.\nHowever, the dust approximation is properly understood\nas the statement that when considered on large scales,\nthe energy density dominates over the pressure, the energy\nflux, the anisotropic stress, and their gradients.\n\nIt has been argued that because of gradients of\nspatial curvature, clocks in different regions\nrun at different rates, and that this effect is important\nfor cosmology but neglected in the dust approximation \\cite{Wiltshire}.\nAny such effects are accounted for in the present analysis,\nto the extent they are part of general relativity, and\nnot outside of the geometrical optics approximation.\nIf the $n^\\a$ frame is non-geodesic, different points on the\nhypersurface of statistical homogeneity and isotropy indeed\nhave different values of the $n^\\a$ frame proper time\n(though this cannot be understood as being due to spatial curvature\ngradients), which in turn is close to the proper time\nmeasured by the observers if $v$ is small.\nFor this time dilation to be significant, the acceleration\n$\\dot{n}^\\a$ would have to be of the order of the average\nexpansion rate in a significant fraction of space.\nThis in turn requires that either the motion of the observers\nis very non-geodesic (large $A^\\a$) or the acceleration\nbetween the two frames is significant (large $\\dot{v}^\\a$);\nthe latter possibility however has to contend with the fact\nthat the velocity between the frames cannot become large,\nas this would violate the small anisotropy observed in the CMB.\nIn order to generate such large accelerations, the non-dust terms\nin the energy-momentum tensor would have to be significant in a large\nfraction of space. This would likely have important cosmological\neffects apart from the time dilation.\nNote that it follows from statistical homogeneity and isotropy\nthat the spatial difference in the CMB temperature\nbetween different regions is small, in contrast to\nthe arguments made in \\cite{Wiltshire}.\nThis issue can be observationally probed with the\nblackbody shape of the CMB spectrum \\cite{Fixsen:1996, blackbody}.\n\n\\section{Discussion}\n\n\\subsection{Modelling issues}\n\n\\paragraph{Discreteness.}\n\nWhile we have kept the energy-momentum tensor generic,\nthe arguments about light propagation in \\sec{sec:light}\ncontain the implicit assumption that matter is so\nfinely distributed in space that it can be treated\nas a continuous distribution which light rays sample.\nThe redshift, the angular diameter distance and other quantities\nrelated to light propagation depend on the spacetime geometry\nonly via an integral along the null geodesic. If the matter\nconsists of discrete clumps whose size and number density is\nso small that a typical light ray will never encounter matter,\nthe energy-momentum tensor along the light ray is zero, regardless\nof the average energy density (or pressure or other components).\nFor example, the integrand in \\re{muint} vanishes identically,\nand our arguments about cancellation between regions of low\nand high density do not apply.\n\nThe approximation of discrete matter as a continuous fluid\nhas been studied from first principles for the dynamics of\nmatter in Newtonian gravity \\cite{discrete}.\nHowever, the effect on light propagation has been looked\nat mostly from the perspective of adding perturbations to\na FRW metric. With a statistically homogeneous and isotropic\ndistribution and small structures, it is then not surprising\nthat the deviation of the expansion rate from the FRW case\nis small \\cite{Rasanen:2008a}.\nOne exception is \\cite{Clifton}, where the effect of discreteness\nlight propagation was considered in a lattice model without any FRW\napproximation.\n\nIf the light travels in vacuum, and we assume statistical\nhomogeneity and isotropy, the mean angular diameter distance\nis given by \\re{DAbareq} with zero on the right-hand side.\nThe equation can be integrated to yield\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAempty}\n  \\bar{D}_A = \\int_0^{\\bar{z}} \\mathrm{d} z \\frac{\\bar{\\Gamma(z)}}{(1+z)^2 \\bar{H}(z)} \\ ,\n\\eea\n\n\\noindent where $\\bar{H}$ and $\\bar{\\Gamma}$ are the mean expansion rate\nand the mean time dilation along the null geodesic,\nwhich in general do not coincide with the spatial averages.\nBecause the null geodesic samples only vacuum, the expansion rate along\nthe null geodesic is larger than the average\nexpansion rate (assuming that the matter satisfies the\nstrong energy condition). The equations \\re{Rayloc} and \\re{Hamloc}\nshow that if the acceleration $\\dot{n}^\\a$ and the shear\n$\\sigma_{\\a\\b}$ could be neglected, the expansion rate sampled\nby the light ray would be the same as in an empty FRW universe.\nIf the time dilation could be neglected, the angular diameter\ndistance would then correspond to the 'coasting universe'\n(or Minkowski space, for non-expanding regions).\nHowever, it is probably not reasonable to neglect\nthe shear $\\sigma_{\\a\\b}$. For example, the existence of both expanding\nand non-expanding regions means that there is a gradient\nin the expansion rate, which implies non-zero shear.\nEvaluating the expansion rate along a null geodesic,\nand the angular diameter distance, is thus reduced to the\nquestion of realistically modelling the shear scalar (and the\nacceleration $\\dot{n}^\\a$) along the geodesic.\n\nDiscussing light propagation in terms of null geodesics assumes\nthe validity of the geometrical optics approximation.\nGeometrical optics is in turn based on modelling\nlight as local plane waves, which requires the wavelength\nof the light to be much smaller than both the curvature scale\nand the scale over which the amplitude, wavelength and\npolarisation of the light change significantly. If the fraction of the\nvolume occupied by matter is so small that light rays\nnever come close to the matter particles, this is satisfied.\nHowever, if the light passes through small discrete\nregions where the energy-momentum tensor is non-zero,\nthe situation is very different from the geometrical optics limit.\n(We are here concerned only with gravitational\ninteractions, and are not taking into account gauge\ninteractions between photons and matter.)\nIn evaluating the validity of the continuous fluid\napproximation, the extension of the wave-packet of the\nmatter particles should therefore be taken into account.\nThe treatment of the photon waves should also be more\ndetailed, instead of simply treating their transverse\nwidth as zero (as in the null geodesic picture) or\ninfinite (as implicit in the plane wave approximation).\nThe effect of discreteness on light propagation is not\nobvious, and should be studied in a realistic model.\n\n\\paragraph{The relevance of averages.}\n\nThe equations \\re{Rayavn}--\\re{consavn} or\n\\re{Rayavu}--\\re{consavu} generalise\nthe average equations for the irrotational\ndust case derived by Buchert \\cite{Buchert:1999}\nto arbitrary matter content and rotation,\nand an arbitrary averaging hypersurface.\nThe scale factor has been defined to give the mean\nredshift in the case that the difference between the\nobserver frame and the frame of statistical homogeneity\nand isotropy is small, and we have assumed statistical\nhomogeneity and isotropy and slow evolution, which are\nrequired to have a meaningful notion of mean redshift.\nThe equations \\re{Rayavn}--\\re{consavn} or\n\\re{Rayavu}--\\re{consavu} are of course valid\nwithout any symmetry assumptions. However,\nthe quantity $a^3$ is of limited use in interpreting observations\nunless the space is statistically homogeneous and isotropic,\nso that the change in the total volume of the spatial hypersurface\nis the dominant effect for light propagation.\n\nThe system of equations \\re{Rayavn}--\\re{consavn}\nor \\re{Rayavu}--\\re{consavu} is derived from the\nscalar part of the full Einstein equation, which is\nnot closed, because it is coupled to the vector and tensor parts.\nAs the sum of two tensors at different spacetime\npoints is not a tensor, the average of a tensor (or vector)\nin curved spacetime is not well-defined, so it is sometimes\nsaid that the rest of the evolution equations cannot be averaged.\nHowever, we can write the evolution equations\nin terms of components, and average these.\nThe problem is not lack of covariance\\footnote{One can even average\nthe vector and tensor part of the equations covariantly\nby first projecting with a vector field such as $v^\\a$\nor $\\hat{\\nabla}_\\a\\theta$. The Einstein equation can be\nexpressed in full generality in scalar form\nby using a projection \\cite{Padmanabhan:2004}.},\nbut the fact that products of variables become\nindependent correlation terms, so the number of unknowns\nincreases when taking the average, and the set of average\nequations does not close.\n\nMethods for covariantly averaging tensors on curved spacetime\nhave been suggested, including the macroscopic gravity\nformalism \\cite{Zalaletdinov, Paranjape}, the Ricci flow \\cite{Carfora},\na statistical averaging formalism \\cite{Debbasch},\na procedure which relies on a specific choice of tetrads \\cite{Behrend:2008},\nand the proposal of \\cite{Korzynski:2009} which is more\na way to rewrite the tensors than average them.\nHowever, the relevant issue is not the mathematical definition\nof averages in some covariant formalism which is an extension\nof general relativity or in a statistical ensemble of spacetimes.\nRather, we want to determine the impact of structures in\nthe spacetime which actually describes the universe we\nobserve, with the dynamics determined by the Einstein equation\nand the local equations for light propagation.\nAverages are useful insofar as they provide an approximate\ndescription of observed quantities in this complex system.\nThe relevant averaging procedure, and the hypersurface\nof averaging, emerges from considering observations,\nand cannot be determined on abstract mathematical grounds. \n\nIt bears emphasising that taking the average on a different\nhypersurface would correspond to considering a different velocity\nfield, and this is a physical choice. This issue is\nseparate from the question of gauge-invariance, i.e. dependence of\nunphysical quantities on the chosen coordinate system.\nThe Buchert equations were originally derived using the ADM\nformalism \\cite{Buchert:1999}, where the distinction between\nchoice of velocity field and choice of coordinates is\nnot entirely transparent. However, the problem can be considered\ncompletely covariantly, without introducing coordinates\n\\cite{Tsagas:2007, Rasanen:2008a, Rasanen:2008b}.\nThe averages depend on the choice of the averaging hypersurface\n\\cite{Geshnizjani, Rasanen:2004}, but not on the coordinate\nsystem \\cite{Kolb:2004}\\footnote{In \\cite{Brown:2009},\nchoice of the averaging hypersurface and the coordinate system\nwas conflated. This mixes up defining a quantity of interest\nusing physical criteria and using different coordinates to\ndescribe the same physics.}.\nThe relevant velocity field is singled out as that of the\nobserver by the redshift, and the relevant averaging hypersurface\nis given by the symmetry properties of the spacetime.\n\n\\paragraph{Deviation from the FRW universe.}\n\nIf backreaction is important for the average\nexpansion rate, i.e. if $\\mathcal{Q}$ contributes significantly\nto \\re{Rayavn} and \\re{Rayavu}, there is no ``FRW background''\nthat would emerge on large scales \\cite{Rasanen:2006b, Rasanen:2008a}.\n(Note that $\\mathcal{Q}$ being small does not guarantee that the spacetime\ncan be described by the FRW metric.)\nThe FRW metric describes a universe that is exactly\nhomogeneous and isotropic, not a universe where there\nare large non-linearities with a statistically homogeneous\nand isotropic distribution.\n\nWhile the deviation of the average expansion rate from the\nFRW equations could be attributed to an effective matter\ncomponent in a FRW universe, this is not the case for\nother observables. For the shear scalar, this is obvious,\nbecause it is zero in FRW models, but generally positive.\nAs a less trivial example, the spatial curvature \nin the FRW case is fixed to be proportional to $a^{-2}$, while the\naverage spatial curvature in an inhomogeneous and\/or\nanisotropic space can evolve non-trivially \\cite{Rasanen:2007}.\nIn fact, if the\nmatter can be treated as dust, the effect of backreaction\nis encoded in the non-trivial evolution of the spatial curvature\n\\cite{Buchert:1999, Rasanen:2005, Rasanen:2006b}. (The\ndifference between backreaction in general relativity\nand Newtonian gravity can also be understood in terms\nof the spatial curvature \\cite{Rasanen:2008a}.)\nFor this reason, calling the effect of backreaction a\nchange of background as in\n\\cite{Kolb:2009, Clarkson:2009a, Clarkson:2009b}\nis misleading. A FRW model which reproduces the\naverage expansion rate of a clumpy model is better called\na ``fitting model'' or something similar. The metric associated\nwith it cannot be used to calculate quantities other than\nthe one specifically fitted for, and usual perturbation theory\naround it does not make sense.\n\nIn particular, if backreaction is important,\nthe relation between the expansion rate\nand the angular diameter distance given by \\re{DAbareq}\n(assuming that discreteness is not important) is\ndifferent from the FRW case \\cite{Rasanen:2008b}\\footnote{In contrast,\nthe relation $D_L=(1+z)^2 D_A$ for the luminosity distance $D_L$\nis universal \\cite{Ellis:1971}, \\cite{Schneider:1992} (page 111),\n\\cite{Etherington:1933}.}.\nEven though either the change of the expansion rate or the change\nof the distance due to backreaction can be reproduced in a FRW\nmodel by introducing extra sources of matter or changing the\nEinstein equation, it is not possible to do both\nat the same time, since FRW models cannot mimic\nthe correlation \\re{DAbareq}.\nIn a clumpy space, if the dust approximation holds, the distance\nis uniquely determined by the function $H(\\bar{z})$ and\nthe matter density today \\cite{Rasanen:2008b}.\n(Note that fitting the distance observations may not\nnecessarily require accelerating expansion, because the relation\nbetween $H$ and $D_A$ is different from the FRW case.)\nAnalogously, in a FRW universe with general matter content,\nthe distance is determined by $H(z)$ and the spatial curvature\ntoday \\cite{Clarkson:2007b}.\nIn LTB models, the relationship is different from\neither of these cases \\cite{February:2009}.\n\nThis prediction for both FRW models and backreaction\ncan be tested with independent observations of distance\nand the expansion rate \\cite{Avgoustidis:2009}.\nThe deviation of the backreaction case from the FRW\nconsistency condition is related to the\ndifference of the average expansion rate from the FRW\ncase with vacuum energy and dust \\cite{Rasanen:2008b}.\nThere are relatively good constraints on the distance scale as\na function of redshift from type Ia supernova observations\n\\cite{FRWexp}, but measurements of the expansion rate\nusing the ages of passively evolving galaxies are less\nprecise \\cite{ages}. The expansion rate at different\nredshifts also enters into the radial mode of the baryon\nacoustic oscillations, and a measurement\nwas reported in \\cite{BAO2} (see also\n\\cite{SylosLabini:2009, MiraldaEscude:2009}).\nWith better observations of the expansion rate, it\nwill be possible to more tightly test the statement that\nthe universe is described by a FRW metric,\nindependent of the possible existence of exotic matter\nor modified gravity \\cite{Clarkson:2007b, Shafieloo:2009}\\footnote{Note\nthat this is not a test of the Copernican principle.\nThe statement that our position in the universe is\nnot untypical is different from the statement that the\nmetric is FRW. In fact, the Copernican principle says\nnothing about the metric.}.\nSimilarly, backreaction can be tested without having a\nprediction for the average expansion rate.\nIn the case of backreaction, it helps that there are\nindependent observational constraints on the matter density today,\nwhile the only way to determine the spatial curvature of a\nFRW universe is to make independent measurements\nof the expansion rate and distance, and use the FRW relation\nbetween them. (In particular, the CMB has no model-independent\nsensitivity to the spatial curvature.)\n\n\\subsection{Conclusions} \\label{sec:disc}\n\n\\paragraph{Summary.}\n\nIn \\cite{Rasanen:2008b} it was argued that light\npropagation can be approximately described in terms\nof the overall geometry (meaning the average expansion rate\nand other average scalar quantities) in statistically homogeneous\nand isotropic dust universes where the distribution\nevolves slowly compared to the time it takes for\nlight to cross the homogeneity scale.\nThe calculation was incomplete because it\nwas simply assumed that the light deflection is small,\nand there was no result for the amplitude of the image shear.\nIt was also assumed that the matter is dust and irrotational,\nwhile it is known that such a description does not locally\nhold everywhere.\n\nWe have now considered general matter content,\nwith a general observer velocity and a general\nhypersurface of statistical homogeneity and isotropy,\nwith a slowly evolving distribution.\nFrom these assumptions about the spacetime symmetry,\nwe find that the propagation of typical light rays\nover distances longer than the homogeneity scale\ncan to leading order be treated in terms of a\nfew average quantities, as in the irrotational dust case.\nThe redshift is given by the average volume\nexpansion rate of the hypersurface of statistical\nhomogeneity and isotropy, assuming that the velocity difference\nbetween the frame of statistical homogeneity and\nisotropy and the observer frame is non-relativistic.\nThe relevant averaging hypersurface is selected by the\nsymmetry of the situation as the one of statistical\nhomogeneity and isotropy, while the relevant\nvelocity is that of the observers, because it\ngives the observed redshift.\nThe angular diameter distance is to leading order determined\nby the averages of the expansion rate, time dilation,\nenergy density and pressure.\nThe light deflection and the image shear are small.\n\nWe have also written down the generalisation of the\nBuchert equations \\cite{Buchert:1999} for the\nevolution of the average expansion rate to general\nmatter content and averaging hypersurface.\nProvided that the difference between frame\nof statistical homogeneity and isotropy and the\nobserver frame is small, and that pressure,\nenergy flux and anisotropic stress are not significant\nin a large fraction of space, we recover the Buchert\nequations for dust.\n\n\\paragraph{Outlook.}\n\nIf backreaction has a large effect on the average\nexpansion rate, the relation between\nthe expansion rate and the angular diameter distance\nis different from the FRW case, and the difference\ngrows with the deviation of the expansion rate\nfrom that of the FRW model with dust and vacuum energy.\nThis is a distinct prediction of backreaction, which\ncannot be mimicked by any FRW model \\cite{Rasanen:2008b}.\n\nOur arguments are qualitative, and \nshould be followed up by a more rigorous quantitative study.\nIn particular, evaluating the deviations from the mean is\nnecessary to study lensing and the low multipoles of the CMB.\nThe equations we have written in the covariant formalism\ncontain all general relativistic effects and are exact, except\nfor the geometrical optics approximation.\nThe study of light propagation in a general\nspacetime is reduced to the system of coupled ordinary differential\nequations \\re{Eder}, \\re{eeq}, \\re{patsigma2} and \\re{Raynull}.\nTo obtain a solution, we do not have to know\nthe global geometry, it is only necessary to specify\nthe distribution along the null geodesic. This approach was\nused in the perturbative case in \\cite{Kainulainen:2009}.\nThis formulation is well-suited to slowly evolving statistically\nhomogeneous and isotropic universes, where the solution is\nexpected to depend only on the statistical properties of the\ndistribution. In particular, because in the real universe the\ndistribution originates from small almost Gaussian fluctuations, its\nstatistics are even at late times determined by the\ninitial power spectrum, processed by gravity.\nThe effect of the discreteness of the matter content of the\nuniverse on light propagation should be clarified.\nFinally, even though the relation between the expansion rate\nand the distance is already a prediction which can be checked,\nit remains of central importance to derive the average expansion\nrate from the statistics of structure formation \\cite{Rasanen:2008a}\nin a reliable manner to allow easier and more comprehensive\ncomparison with observations.\n\n\\acknowledgments\n\nI thank Thomas Buchert, Timothy Clifton, Ruth Durrer, Pedro\nFerreira and David Wiltshire for discussions and disagreements, \nand Thomas Buchert also for comments about the manuscript,\nand the Helsinki Institute of Physics and the Tuorla Observatory\nfor hospitality.\n\nThis paper is dedicated to Heidi Hopeamets\\\"{a}.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect:1}\n\nLet $\\Omega$ and $D$ be bounded Lipschitz domains in $\\mathbb{R}^n$, $n\\geq 2$, such that $D\\Subset\\Omega$ and $\\Omega\\backslash\\overline{D}$ is connected. Let $\\eta\\in L^\\infty (D)$ be such that $\\eta(\\mathbf{x})>\\eta_0\\in\\mathbb{R}_+$, $\\mathbf{x}=(x_j)_{j=1}^n\\in \\Omega$, and $|\\eta(\\mathbf{x})-1|>0$, $\\mathbf{x}\\in\\mathrm{neigh}(\\partial D)\\cap D$, where $\\mathrm{neigh}(\\partial D)$ denotes an open neighbourhood of $\\partial D$. Set\n \\begin{equation}\\label{form}\n \\gamma(\\mathbf{x})=1+(\\eta-1)\\chi_D(\\mathbf{x}),\\: \\mathbf{x}\\in \\Omega.\n \\end{equation} \n Introduce \n \\[\n H_0^{-1\/2}(\\partial \\Omega):=\\{f\\in H^{-1\/2}(\\partial\\Omega); \\int_{\\partial\\Omega} f\\, ds=0\\},\n \\]\nand consider the following elliptic PDE problem for $u\\in H^1(\\Omega)$,\n\\begin{equation}\\label{EQ_Calderon}\n\\begin{cases}\n\\displaystyle{\\mathrm{div}(\\gamma\\nabla u)=0}\\qquad & \\mbox{in}\\ \\ \\Omega,\\medskip\\\\\n\\displaystyle{\\frac{\\partial u}{\\partial\\nu}=g\\in H_0^{-1\/2}(\\partial\\Omega)} & \\mbox{on}\\ \\ \\partial\\Omega,\n\\end{cases}\n\\end{equation} \nwhere $\\nu\\in\\mathbb{S}^{n-1}$ is the exterior unit normal vector to $\\partial\\Omega$.\nAssociated with \\eqref{EQ_Calderon}, we introduce the following Neumann-to-Dirichlet (DtN) map, $\\Lambda_\\gamma: H_0^{-1\/2}(\\partial\\Omega)\\mapsto H^{1\/2}(\\partial\\Omega)$, \n\\begin{equation}\\label{eq:NtD}\n\\Lambda_\\gamma(g)=u|_{\\partial\\Omega},\n\\end{equation}\nwhere $u\\in H^1(\\Omega)$ is the solution to \\eqref{EQ_Calderon}. For a given $g\\in H_0^{-1\/2}(\\partial\\Omega)$, we are concerned with the inverse inclusion problem of determining $D$ by the pair of boundary data $(g, \\Lambda_\\gamma(g))$, independent of $\\eta$. That is, \n\\begin{equation}\\label{eq:ic1}\n(g, \\Lambda_\\gamma(g))\\rightarrow \\partial D,\\quad \\mbox{independent of $\\eta$}. \n\\end{equation}\n\n\nThe inverse inclusion problem constitutes a particular set of sub-problems for the celebrated Calder\\'on's inverse problem. The Calder\\'on problem is concerned with the recovery of the conductivity of a body by the associated boundary electric current and flux measurements. Mathematically, it can formulated as determining $\\gamma$ by knowledge of $\\Lambda_\\gamma$ associated with the PDE system \\eqref{EQ_Calderon}, where $\\gamma\\in L^\\infty(\\Omega)$ is not necessarily of the specific form \\eqref{form}. Calder\\'on's inverse problem arises from industrial applications of practical importance including electrical impedance tomography (EIT) in medical imaging and geophysical exploration. The problem was first posed and studied by A. P. Calder\\'on \\cite{CALDERON2006} and has a profound impact on the mathematical developments of inverse problems. On the other hand, in many physical and engineering applications such as detecting defects in a composite medium, one is concerned more about recovering the geometric shape of an inhomogeneous inclusion embedded in a homogeneous conductive body. This naturally leads to the inverse inclusion problem \\eqref{eq:ic1}. In the practical point of view, many reconstruction methods have been developed in the literature for the inverse inclusion problem. They include the monotonicity-based method \\cite{HU}, the factorization method \\cite{GH}, the enclosure method \\cite{I}, and the method of using the generalized polarization tensors deduced from the layer potential techniques \\cite{ammari2004reconstruction}.\n\nIt is noted that knowing $\\Lambda_\\gamma$ is equivalent to knowing $(g, \\Lambda_\\gamma(g))\\in H_0^{-1\/2}(\\partial\\Omega)$ $\\times H^{1\/2}(\\partial\\Omega)$ for any $g\\in H_0^{-1\/2}(\\partial\\Omega)$, which corresponds to infinitely many boundary measurements in the practical scenario. In order to recover a generic $\\gamma(x)$, it is necessary to make use of infinitely \nmany boundary measurements. In fact, one can see that the cardinality of the unknown $\\gamma(x)$ is $n$, whereas the cardinality of the given data encoded into $\\Lambda_\\gamma$ is $2(n-1)$. Here, by cardinality, we mean the number of independent variables in a given quantity. In such a case, one has $2(n-1)\\geq n$ for $n\\geq 2$, which means that the inverse problem in recovering a generic $\\gamma(x)$ is at least formally determined (indeed, it is formally determined when $n=2$, and over-determined when $n\\geq 3$). Nevertheless, for the inverse inclusion problem \\eqref{eq:ic1}, the cardinality of the unknown $\\partial D$ is $n-1$, and for a fixed $g$, the cardinality of the boundary measurement data $(g, \\Lambda_\\gamma(g))$ is also $n-1$. That is, the inverse inclusion problem \\eqref{eq:ic1} is formally determined with a single measurement. Hence, it is unobjectionable to expect that one can establish the unique identifiability result for \\eqref{eq:ic1} in a generic scenario.  However, this problem constitutes a long-standing open problem in the literature. \n\nIn \\cite{beretta2019lipschitz,AMR,ammari2017identification,moi2,moi3}, the uniqueness and reconstruction issues for \\eqref{eq:ic1} were studied under multiple measurements. There are also several works for \\eqref{eq:ic1} by a single boundary measurement, provided that the inclusion $D$ is a-priori known to belong to certain specific geometrical classes. If $D$ is within the radial geometry, uniqueness and stability results were established in \\cite{fabes1999inverse,moi1,Kang_ball_R3} by a single measurement. In the case that $D$ is a convex polygon in $\\mathbb{R}^2$ or a convex polyhedron in $\\mathbb{R}^3$, the uniqueness results are respectively derived in \\cite{friedman1989uniqueness} and \\cite{barcelo1994inverse}. The reconstruction of an insulating curvilinear polygonal inclusion was considered in \\cite{CHL}. In a recent paper \\cite{StabilityPolygon}, we proved a logarithmic type stability in determining polygonal inclusions. In those studies mentioned above by a single measurement, it is a technical requirement that the content of the inclusion has to be uniform; that is, the conductivity $\\eta$ of the inclusion in \\eqref{eq:ic1} is a positive constant. Moreover, in all of the aforementioned literature except \\cite{StabilityPolygon}, full boundary measurements are required. Here, by full boundary measurement, we mean that the measurement dataset $(\\psi, \\Lambda_\\gamma(\\psi))$ is given over the whole boundary $\\partial\\Omega$. For comparison, the following partial boundary measurement was used in \\cite{StabilityPolygon}: $(\\psi, \\gamma\\partial_\\nu u|_{\\Gamma_0})$ with $\\mathrm{supp}(\\psi)\\subset\\Gamma_0$, where $\\psi=u|_{\\partial\\Omega}$ and $\\Gamma_0\\Subset\\partial\\Omega$ is a proper subset. We would like to mention the partial-data Calder\\'on problem constitutes another challenging topic in the field of inverse problems (cf. \\cite{CSU,IUY}). Nevertheless, it is pointed out that a mild condition was imposed for the study in \\cite{StabilityPolygon} which depends on the a-priori knowledge of the underlying inclusion as well as the corresponding boundary input. \n\nThe mathematical argument in \\cite{StabilityPolygon} is of a localized feature, which is based on carefully studying the singular behaviors (in the phase space) of the solution to the conductivity problem \\eqref{EQ_Calderon} around a corner point on the polygonal inclusion. In this paper, we show that the corner singularity in \\cite{StabilityPolygon} can be relaxed to be a certain high-curvature condition. Indeed, the corner singularity can be regarded as having an extrinsic curvature being infinity. Our argument in tackling the singular behaviors of the solution to \\eqref{EQ_Calderon} around an admissible high-curvature point on the boundary of the conductive inclusion is mainly motivated by a recent article \\cite{Liu_curve}. However, the study in \\cite{Liu_curve} mainly deals with high-curvatures occurring on the support of a parameter $q$, which is the coefficient for the lower-order term of an elliptic partial differential operator, namely $-\\Delta+q$. In the current study, the high-curvatures enter into the coefficient of the leading-order term, namely $\\gamma$ associated with $\\nabla(\\gamma\\nabla u)$. It is pointed out that in \\cite{StabilityPolygon}, quantitative stability estimates are established in determining polygonal inclusions in two dimensions, whereas in this paper, we are mainly concerned with the qualitative unique identifiability issue in any dimension $n\\geq 2$. Finally, we would also like to mention in passing some recent related works \\cite{ACL,B1,B2,BLa,BLb,BL17,BL172,BLX,CX,CDL,CDL2,LX} on characterizing the geometric singularities in the coefficients of certain partial differential operators and their implications to the related inverse inclusion problems. \n\nThe rest of the paper is organized as follows. In Section 2, we derive some auxiliary results. Section 3 is devoted to the main results for the inverse inclusion problem. In the Appendix, we present some discussion about a generic condition in the main theorem. \n\n\\section{Some auxiliary results}\n\nIn this section, we derive several auxiliary results that shall be of critical importance for establishing the unique determination results in Section 3. By following the treatment in \\cite{Liu_curve}, we first introduce an important geometric notion for our study. \n\n\\begin{definition}\\label{def:k}\nLet $K,L,M,\\delta$ be positive constants and $D$ be a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$. A point $\\mathbf{p}\\in \\partial D$ is said to be an admissible $K$-curvature point with parameters $L,M,\\delta$ if the following conditions are fulfilled.\n\\begin{enumerate}\n\\item Up to a rigid motion, the point $\\mathbf{p}$ is the origin $\\mathbf{x}=\\mathbf{0}$ and $\\mathbf{e}_n:=(0,\\cdots,0,1)$ is the interior unit normal vector to $\\partial D$ at $\\mathbf{0}$.\n\\item Set $b=\\sqrt{M}\/K$ and $h=1\/K$. There is a $\\mathcal{C}^{2,1}$ function $w: B(\\mathbf{0},b)\\rightarrow \\mathbb{R}_+\\cup \\{0\\}$ with $B(\\mathbf{0},b)\\subset \\mathbb{R}^{n-1}$ such that if \n\\begin{equation}\nD_{b,h}=B(\\mathbf{0},b)\\times(-h,h)\\cap D,\n\\end{equation}\nthen\n\\begin{equation}\nD_{b,h}=\\{\\mathbf{x}\\in \\mathbb{R}^n; |\\mathbf{x}'|<b, -h<x_n<h, w(\\mathbf{x}')<x_n<h\\},\n\\end{equation}\nwhere $\\mathbf{x}=(\\mathbf{x}',x_n)$ with $\\mathbf{x}'\\in \\mathbb{R}^{n-1}$. Here and also in what follows, $B(\\mathbf{x}, b)$ signifies a ball centered at $\\mathbf{x}$ and of radius $b$. \n\\item The function $w$ in (2) satisfies\n\\begin{equation}\nw(\\mathbf{x}')=K|\\mathbf{x}'|^2+\\mathcal{O}(|\\mathbf{x}'|^3),\\qquad \\mathbf{x}'\\in B(\\mathbf{0}',b).\n\\end{equation}\n\\item $M\\geq 1$ and there are $0<K_-\\leq K\\leq K_+<\\infty$ such that \n\\[ K_-|\\mathbf{x}'|^2\\leq w(\\mathbf{x}')\\leq K_+|\\mathbf{x}'|^2,\\qquad |\\mathbf{x}'|<b, \\]\n\\[M^{-1}\\leq \\frac{K_\\pm}{K}\\leq M, \\qquad K_+-K_-\\leq LK^{1-\\delta}.\\]\n\\item The intersection $V=\\overline{D_{b,h}}\\cap \\mathbb{R}^{n-1}\\times\\{h\\}$ is a Lipschitz domain.\n\\end{enumerate}\n\\end{definition}\n\nIn what follows, we suppose that $\\partial D$ possesses an admissible $K$-curvature point $\\mathbf{p}$. We shall work within an Euclidean system of coordinates, which is transformed from the cardinal one after a rigid motion such that $\\mathbf{p}$ is the origin. We assume that in this coordinate system, the boundary $\\partial D$ in a neighborhood of $\\mathbf{p}$ can be represented by the equation $x_n=w(\\mathbf{x}')$, where $w(\\mathbf{x}')$ is the function in Definition~\\ref{def:k} satisfying $w(\\mathbf{0}')=0, \\nabla_{\\mathbf{x}'} w(\\mathbf{0}')=\\mathbf{0}'$. We denote by $U_{b,h}:=B(\\mathbf{0}',b)\\times (-h, h)$ the cylinder centered at $\\mathbf{p}$, and by $D_{b,h}:=U_{b,h}\\cap D$ the neighborhood of $\\mathbf{p}$, and by $\\partial S :=\\partial D_{b,h}\\setminus \\partial D$ the complementary part of its surface.\n\nWe next study the local behaviors of the solution $u$ to \\eqref{EQ_Calderon}. We recall that $\\eta$ in \\eqref{form} is a positive constant and $\\eta\\neq 1$. Nevertheless, we shall remark the case that $\\eta$ is a variable function in Section 3. It is straightforward to show that \\eqref{EQ_Calderon} is equivalent to the following transmission problem with $u_i:=u|_{D}$ and $u_e:=u|_{\\Omega\\backslash\\overline{D}}$: \n\\begin{equation}\\label{EQ2}\n\\begin{cases}\n     \\triangle u_e =0 & \\text{in}\\ \\ \\Omega\\setminus\\overline{D}, \\\\\n     \\triangle u_i =0 & \\text{in}\\ \\ D,\\\\\n     u_i=u_e & \\text{on}\\ \\ \\partial D,\\\\\n     \\eta\\partial_\\nu u_i=\\partial_\\nu u_e & \\text{on}\\ \\ \\partial D,\\\\\n     \\partial_\\nu u_e=g & \\text{on}\\ \\ \\partial \\Omega. \n\\end{cases}  \n\\end{equation}\nIn principle, we shall show that $u_e$ in \\eqref{EQ2} cannot be harmonically extended across the admissible $K$-curvature point $\\mathbf{p}\\in\\partial D$, provided $K$ is sufficiently large. It is easily seen that the harmonic extension across $\\mathbf{p}$ is equivalent to the analytic extension of $u_e$ across $\\mathbf{p}$. Then using this non-extension result together with an absurdity argument, we can establish the unique identifiability results for the inverse problem \\eqref{eq:ic1}. To that end, throughout the rest of this section, we assume on the contrary that the function $u_e$ admits a harmonic extension into $D_{b, h}$, which is still denoted by $u_e$. That is, when $u_e|_{D_{b,h}}$ is involved in what follows, it is actually referred to the harmonic extension mentioned above. Under such a situation, we shall derive several key properties of the solution $u\\in H^1(U_{b,h})$ locally around the point $\\mathbf{p}\\in\\partial D$, which shall be used for the absurdity argument in Section~\\ref{sect:3} for the inverse inclusion problem. \n\n\\begin{proposition}\\label{TIntId}\nLet $u_0\\in H^1_{loc}(U_{b,h})$ be harmonic in $U_{b,h}$. Then,\n\\begin{equation}\\label{IntId}\n(\\eta-1)\\int_{D_{b,h}}\\nabla u\\cdot \\nabla u_0 \\, d\\mathbf{x}=\\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)u_0-(u_i-u_e)\\partial_\\nu u_0\\, ds,\n\\end{equation}\nwhere $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}\\in H^1(U_{b,h})$ is the solution to (\\ref{EQ2}). \n\\end{proposition}\n\\begin{proof}\nThe integral identity \\eqref{IntId} can be directly verified by using Green's formula and the transmission conditions across $\\partial D$ of $u$.\n\\end{proof}\n\\begin{proposition}\\label{prop:regularity}\nUnder the assumption that $u_e$ can be harmonically extended into $D_{b,h}$, one has $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ and $u_e\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ for some $0<\\alpha \\leq 1$ in two and three dimensions. \n\\end{proposition}\n\\begin{proof}\n Clearly, from \\eqref{EQ2}, one sees that $u_i\\in H^1(D_{b, h})$ satisfies $\\Delta u_i=0$ in $D_{b, h}$, and \n $$\n u_i=u_e\\quad \\mbox{and}\\quad \\partial_\\nu u_i=\\eta^{-1}\\partial_\\nu u_e\\quad\\mbox{on}\\ \\ {\\partial D_{b, h}\\cap \\partial D}. \n $$   \n According to Definition~\\ref{def:k}, we know that ${\\partial D_{b, h}\\cap \\partial D}$ is $\\mathcal{C}^{2,1}$. Hence, by the regularity estimate up the boundary for elliptic PDEs (cf. \\cite[Theorem 4.18]{McL}), we have that $u_i\\in H^3(D_{b, h})$. Finally, by the Sobolev embedding, we readily see $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ for some $0<\\alpha \\leq 1$ in two and three dimensions. $u_i\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ is obvious since we assume that $u_e$ allows a harmonic extension into $D_{b,h}$. \n\\end{proof}\n\n\\begin{remark}\nThe H\\\"older regularity of $u^i$ up to interface $\\partial\\Omega$ around $\\mathbf{p}$ is an important ingredient in our subsequent argument. In Proposition~\\ref{prop:regularity}, we derive the $H^3$-smoothness, which yields the desired H\\\"older regularity by the Sobolev embedding in two and three dimensions. For general dimensions greater than 3, one can follow a completely similar argument to derive the $\\mathcal{C}^{1,\\alpha}$ regularity of $u^i$ in $\\overline{D_{b,h}}$, but subject to requiring that ${\\partial D_{b, h}\\cap \\partial D}$ is $\\mathcal{C}^{r+1,1}$, where $r:=\\lceil{\\frac n 2-1+\\alpha}\\rceil$. That is, one needs to modify $\\mathcal{C}^{2,1}$ in item (2) in Definition~\\ref{def:k} to be $\\mathcal{C}^{r+1,1}$ for the general dimension case in order to have Proposition~\\ref{prop:regularity}. However, only two and three dimensions are physically meaningful, and hence we mainly focus on the two and three dimensional cases. Nevertheless, it is emphasized that all our subsequent arguments can be extended to higher dimensions after some necessary but slight modifications.  \n\\end{remark}\n\n\n\\begin{theorem}\\label{estimationT}\nLet $u_0\\in H^1_{loc}(U_{b,h})$ be a harmonic function, known as the CGO (Complex Geometric Optics) solution, constructed in the from\n\\begin{equation}\\label{CGO}\nu_0=\\exp(\\mathbf{\\xi}\\cdot \\mathbf{x}),\n\\end{equation}\n with the parameter \n \\begin{equation}\\label{eq:rho}\n \\xi=\\mathrm{i}\\tau \\hat{\\mathbf{v}}-\\tau \\mathbf{e}_n\\in\\mathbb{C}^n, \\ \\tau\\in\\mathbb{R}_+,\n \\end{equation}\n where\n \\begin{equation}\\label{eq:pv1}\n \\hat{\\mathbf{v}}:=\\begin{cases}\n \\displaystyle{\\frac{\\nabla u_i(\\mathbf{p})-(\\nabla u_i(\\mathbf{p})\\cdot \\mathbf{e}_n)\\mathbf{e}_n}{\\big|\\nabla u_i(\\mathbf{p})-(\\nabla u_i(\\mathbf{p})\\cdot \\mathbf{e}_n)\\mathbf{e}_n\\big|}} \\ \\ \\ &\\mbox{if\\ \\ $\\nabla u_i(\\mathbf{p})\\nparallel \\mathbf{e}_n$,}\\medskip\\\\\n \\mathbf{e}_1:=(1,0,\\ldots,0)\\ \\ &\\mbox{if\\ \\ $\\nabla u_i(\\mathbf{p})\\parallel \\mathbf{e}_n$}. \n \\end{cases}\n \\end{equation}\n Then it holds that\n \\begin{equation}\\label{estimation}\n \\begin{split}\n &C_{n,\\eta,\\alpha}|\\nabla u_i(\\mathbf{p})|\\\\\n \\leq&  \\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(1+(\\tau h)^{(n-1)\/2})e^{\\tau(\\frac{1}{4K}-h)}\\\\\n &+\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}\\left((\\frac{K}{K_-})^{\\frac{n-1}{2}}-(\\frac{K}{K_+})^{\\frac{n-1}{2}}\\right)e^{\\frac{\\tau}{4K}}\\\\\n&+\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(h+K^{-1}_-)^{\\alpha\/2}h^{(n+1+\\alpha)\/2}(K\/K_-)^{(n-1)\/2}\\tau^{3\/2}e^{\\frac{\\tau}{4K}}\\\\\n&+(\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}+\\Vert u_e \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})})h^{\\alpha+(n-1)\/2}(K\/K_-)^{(n-1)\/2}\\\\\n&\\quad \\times (1+\\tau h)\\tau^{(n-1)\/2}e^{\\tau(\\frac{1}{4K}-h)},\n \\end{split}\n \\end{equation}\nwhere $C_{n,\\eta,\\alpha}$ is a positive constant depending on $n, \\eta$ and $\\alpha$. \n\\end{theorem}\n\\begin{proof}\nIt is directly verified that $u_0$ constructed in \\eqref{CGO}--\\eqref{eq:pv1} is a harmonic function. Next, we apply the constructed $u_0$ to the integral identity \\eqref{IntId}. \nBy straightforward calculations, we split the integral at the left hand side of (\\ref{IntId}) into the following identity,\n\\begin{eqnarray}\\label{IntId2}\n\\nabla u_i(\\mathbf{p})\\cdot \\xi \\int_{x_n>K|\\mathbf{x}'|^2}e^{\\xi\\cdot \\mathbf{x}} dx=\\nabla u_i(\\mathbf{p})\\cdot \\xi \\int_{x_n>\\max(h, K|\\mathbf{x}'|^2)}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x} \\nonumber\\\\\n+\\nabla u_i(\\mathbf{p})\\cdot \\xi \\left( \\int_{K|\\mathbf{x}'|^2<x_n<h}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_{\\mathbf{x}}-\\int_{D_{b,h}} e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x} \\right) \\nonumber\\\\\n+\\int_{D_{b,h}} \\xi\\cdot(\\nabla u_i(\\mathbf{x})-\\nabla u_i(\\mathbf{p})) e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}\\nonumber \\\\\n+\\frac{1}{\\eta-1}\\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)e^{\\xi\\cdot \\mathbf{x}}-(u_i-u_e)\\partial_\\nu e^{\\xi\\cdot \\mathbf{x}}\\, ds.\n\\end{eqnarray}\nFor notational convenience, we rewrite the integral identity (\\ref{IntId2}) in the following form,\n\\begin{equation}\\label{eq:eee1}\n\\nabla u_i(\\mathbf{p})\\cdot \\xi I_0=\\nabla u_i(\\mathbf{p})\\cdot \\xi I_1+\\nabla u_i(\\mathbf{p})\\cdot \\xi I_2+ I_3+\\frac{1}{\\eta-1}I_4.\n\\end{equation}\nwhere $I_j$, $j=1,\\ldots,4$, are respectively defined as\n\\begin{equation}\\label{eq:I}\n\\begin{split}\nI_0=& \\int_{x_n>K|\\mathbf{x}'|^2}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_1=& \\int_{x_n>\\max(h,K|\\mathbf{x}'|^2)}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_2=& \\int_{K|\\mathbf{x}'|^2<x_n<h}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}-\\int_{D_{b,h}} e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_3=& \\int_{D_{b,h}} \\xi\\cdot(\\nabla u_i(\\mathbf{x})-\\nabla u_i(\\mathbf{p})) e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}, \\\\\nI_4=& \\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)e^{\\xi\\cdot \\mathbf{x}}-(u_i-u_e)\\partial_\\nu e^{\\xi\\cdot \\mathbf{x}}\\, ds.\n\\end{split}\n\\end{equation}\nUsing Lemmas 2.17--2.20 in \\cite{Liu_curve} we have the following estimates of the integrals $I_0,I_1,I_2,I_3,$:\n\\begin{equation}\\label{eq:E1}\n\\begin{split}\nI_0=& \\frac{1}{-\\xi_n}\\left(\\frac{\\pi}{-\\xi_n K}\\right)^{(n-1)\/2}\\exp\\left( -\\frac{\\xi'\\cdot\\xi'}{4\\xi_n K} \\right),\\\\\n|I_1|\\leq & C_n \\frac{1+(\\tau h)^{\\frac{n-1}{2}}}{\\tau^{\\frac{n+1}{2}}K^{\\frac{n-1}{2}}}e^{-\\tau h},\\\\ \n|I_2|\\leq & C_n \\left(K_-^{-\\frac{n-1}{2}}-K_+^{-\\frac{n-1}{2}}\\right)\\tau^{-\\frac{n+1}{2}},\\\\ \n|I_3|\\leq & C_{n,\\alpha}\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(h+K_-^{-1})^{\\alpha\/2}h^{(n+\\alpha+1)\/2}K_-^{-(n-1)\/2},\n\\end{split}\n\\end{equation}\nwhere $\\xi=(\\xi_j)_{j=1}^n$ and $\\xi'=(\\xi_j)_{j=1}^{n-1}$. \nBy following a similar argument to the proof of Proposition 2.21 in \\cite{Liu_curve}, the last integral $I_4$ can be estimated as follows,\n\\begin{equation}\\label{eq:I4}\n|I_4|\\leq C_{n,\\alpha}h^{\\alpha+(n-1)\/2}K_-^{-(n-1)\/2}(1+\\tau h)e^{-\\tau h}(\\eta\\Vert u_i\\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}+\\Vert u_e\\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}).\n\\end{equation}\nFinally, by applying the estimates in \\eqref{eq:E1} and \\eqref{eq:I4} to \\eqref{eq:eee1}, together with grouping similar terms, one can arrive at the estimate (\\ref{estimation}).\n\nThe proof is complete. \n\\end{proof}\n\n\\section{Unique determination results for Calder\\'on's inverse inclusion problem}\\label{sect:3}\n\nIn this section, we consider the inverse problem \\eqref{eq:ic1}. In principle, we aim to establish a local unique determination result as follows. Let $(D, \\eta)$ and $(\\widetilde D, \\widetilde \\eta)$ be two conductive inclusions supported in $\\Omega$ as described in \\eqref{form}. Let $\\Gamma_0\\Subset\\partial\\Omega$ be an open subset and let $g\\in H_0^{-1\/2}(\\partial\\Omega)$ with $\\mathrm{supp}(g)\\subset\\Gamma_0$. If\n\\begin{equation}\\label{eq:u1}\n\\Lambda_{\\gamma}(g)=\\Lambda_{\\widetilde{\\gamma}}(g)\\quad\\mbox{on}\\ \\ \\Gamma_0, \n\\end{equation}\nwhere $\\gamma$ and $\\widetilde\\gamma$ are given in \\eqref{form}, respectively associated with $\\eta$ and $\\widetilde\\eta$, then in a certain generic scenario as shall be detailed in what follows, $D\\Delta\\widetilde D:=(D\\backslash\\overline{\\widetilde{D}})\\cup(\\widetilde{D}\\backslash\\overline{D})$ cannot possess a $K$-curvature point as described in Definition~\\ref{def:k} with $K$ sufficiently large, which lies on $\\partial \\Sigma$. Here, $\\Sigma$ is the connected component of $\\Omega\\backslash\\overline{D\\cup\\widetilde{D}}$ that connects to $\\partial\\Omega$. In the simpler geometric setup with $D$ and $\\widetilde{D}$ both being convex, this means $D\\Delta\\widetilde{D}$ cannot possess a $K$-curvature point with $K$ sufficiently large. To establish the local uniqueness results, we shall make use of the contradiction argument by assuming that $D\\Delta\\widetilde D$ possesses an admissible $K$-curvature point. Let $\\mathbf{p}$ signify the aforementioned admissible $K$-curvature point. Without loss of generality, we assume that $\\mathbf{p}\\in \\partial D\\cap \\partial \\Sigma$; see Fig.\\ref{dispo_inverse} for a schematic illustration. Let $u$ and $\\widetilde u$ be the electric potential fields in \\eqref{EQ2} associated with $\\gamma$ and $\\widetilde\\gamma$, respectively. By virtue of \\eqref{eq:u1}, we readily see that \n\\begin{equation}\\label{eq:u2}\nu=\\widetilde u\\quad\\mbox{and}\\quad \\partial_\\nu u=\\partial_\\nu \\widetilde u\\quad\\mbox{on}\\ \\ \\Gamma_0. \n\\end{equation}\nNoting that $\\Delta u=\\Delta \\widetilde u=0$ in $\\Sigma$, we thus have from the unique continuation principle that $u=\\widetilde u$ in $\\Sigma$. The contradiction shall be established from the quantitative properties of the solution $u\\in H^1(U_{b,h})$ locally around the point $\\mathbf{p}\\in\\partial D$. It is recalled that according to our discussion below \\eqref{EQ2}, $u=u_i\\chi_{D_{b,h}}+u_e\\chi_{U_{b,h}\\backslash\\overline{D_{b,h}}}$ in $U_{b,h}$. By taking $b$ and $h$ sufficiently small, we can assume that $U_{b,h}$ stays away from $\\widetilde D$ with a positive distance.  Since $u_e=\\widetilde{u}$ in ${U_{b,h}\\backslash\\overline{D_{b,h}}}$, and $\\widetilde{u}$ is obviously analytic in $U_{b,h}$, we clearly have that $u_e$ can be harmonically extended into $D_{b,h}$, which is exactly $\\widetilde u$. \n\nTo establish the uniqueness result mentioned above, we start with introducing an admissibility condition on the input $g\\in H_0^{-1\/2}(\\partial\\Omega)$ in \\eqref{EQ2} for our inverse problem study.  \n\n\\begin{figure}\n\\includegraphics[scale=0.6]{dispo_inverse} \n\\caption{A $K$-curvature point and its neighborhood}\n\\label{dispo_inverse}\n\\end{figure}\n\\begin{proposition}\\label{inf_grad}\nLet $u$ be the solution to \\eqref{EQ_Calderon}. Then there exists $g\\in H_0^{-1\/2}(\\partial\\Omega)$ such that for all $\\mathbf{x}\\in \\Omega$,\n\\begin{equation}\\label{eq:inf_grad}\n\\lim_{r\\rightarrow +0}\\frac{\\int_{B_r(\\mathbf{x})} |\\nabla u(\\mathbf{z})|d\\mathbf{z}}{|B_r(\\mathbf{x})|}\\geq m_g,\n\\end{equation}\nwhere $m_g>0$ is a positive constant independent of the conductivity function $\\gamma$. Here and also in what follows, $B_r(\\mathbf{x})$ signifies a ball centered at $\\mathbf{x}$ with radius $r$.\n\\end{proposition}\n\n\\begin{remark}\\label{rem:input}\nIn fact, the statement of Proposition \\ref{inf_grad} in the 2-dimensional case is true if the input data $g$ has only one local maximum and only one local minimum on $\\partial \\Omega$ (see e.g. \\cite{gradu1,gradu2}). Through out the rest of the paper, we assume that the condition \\eqref{eq:inf_grad} holds true in our study. In the context of the inverse inclusion problem, it means that the boundary input should be properly chosen. \n\\end{remark}\n\n\n\n\\subsection{Local behaviours of the solution $u$ near the high curvature point $\\mathbf{p}$}\nIn Theorem \\ref{estimationT}, we obtain an estimate of $|\\nabla u_i (\\mathbf{p})|$ in terms of the $K$-curvature parameters and the $\\mathcal{C}^{1,\\alpha}$-norm of $u_{i}$. In this subsection, we further refine the estimate \\eqref{estimation}. Specifically, we shall need an estimates of $\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}$ and $\\Vert u_e \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}$ appeared in \\eqref{estimation} in terms of the geometric parameters of the a-priori parameters. It is recalled that according to our discussion at the beginning of this section, $u_e=\\widetilde u$ in $U_{b,h}$, with $\\widetilde u$ satisfying \\eqref{EQ2} associated with $(\\widetilde D, \\widetilde\\eta)$ that arises from the contradiction argument in what follows. It is noted again that $\\widetilde u$ is harmonic in $U_{b,h}$. \n\nBy Proposition~\\ref{prop:regularity}, we know that $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ and $u_e\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ for some $0<\\alpha\\leq 1$. However, the H\\\"{o}lder norms of the solution $u$ on each side intricately depend on the geometric shape, and particularly depend on the local geometric parameter $K$ of the admissible $K$-curvature point $\\mathbf{p}$. The following lemmas give an estimate of these H\\\"{o}lder norms in terms of the interface's local curvature \\cite{DEF}.\n\n\\begin{lemma}\\label{lemma_DEF}\nLet $Q_R$ be a cube in $\\mathbb{R}^n$ of side length $R$, centered at $\\mathbf{p}\\in\\partial\\Omega$. We denote the two sub-domains of $Q_R$ lying on the two sides of $\\partial D$ by $Q^+_R:=Q_R\\cap \\{x_n>w(\\mathbf{x}')\\}$ and $Q^-_R:=Q_R\\cap \\{x_n<w(\\mathbf{x}')\\}$, respectively. We consider the conductivity equation,\n\\begin{equation}\\label{eq_DEF}\n\\mathrm{div} [(1+(\\eta-1)\\chi_\\pm)\\nabla u]=0 \\ \\ \\text{in}\\ \\ Q_R,\n\\end{equation}\nwhere $\\chi_\\pm(\\mathbf{x})=1$ in $Q^+_R$ and $\\chi_\\pm(\\mathbf{x})=0$ in $Q^-_R$. Then there exist positive constants $\\alpha\\in (0, 1),\\mu,C$ independent of $u$ and $w$ such that\n\\begin{equation}\\label{holder_estimate}\n\\Vert \\nabla u\\Vert_{\\mathcal{C}^\\alpha(Q^\\pm_{R\/4})}\\leq C(1+\\Vert \\nabla w \\Vert_{\\mathcal{C}^1(Q_R)})^\\mu \\Vert \\nabla u\\Vert_{L^2(Q_R)}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{remark}\\label{rem:n1}\nIn the context of our study, we let $R_0>0$ be such that $U_{b,h} \\Subset Q_{R_0}\\Subset \\Omega$, then Lemma~\\ref{lemma_DEF} implies that there exist $C_{n,\\eta,R_0}$ and $\\mu$ which depend only on the a-priori data such that\n\\begin{equation}\\label{holder_estimate_K}\n\\Vert \\nabla u\\Vert_{\\mathcal{C}^\\alpha(Q^\\pm_{R_0\/4})}\\leq C_{n,\\eta,R_0} K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}.\n\\end{equation}\n\\end{remark}\n\nClearly, \\eqref{holder_estimate_K} gives the estimates of $\\|\\nabla u_i\\|_{C^\\alpha(\\overline{D_{b,h}})}$ and $\\|\\nabla u_e\\|_{C^\\alpha(\\overline{U_{b,h}}\\backslash{D_{b,h}})}$. We next further derive the estimates of $\\|u_i\\|_{C(\\overline{D_{b,h}})}$ and $\\|u_e\\|_{C(\\overline{U_{b,h}}\\backslash {D_{b,h}})}$ in terms of the local curvature of the admissible $K$-curvature point, which then yield the desired estimates of $\\|u_i\\|_{C^{1,\\alpha}(\\overline{D_{b,h}})}$ and $\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}}\\backslash {D_{b,h}})}$.\n\n\\begin{lemma}\\label{lem:holder}\nLet $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}\\in H^1(\\Omega)$ be the solution to (\\ref{EQ2}). Suppose that $D\\Subset B(\\mathbf{x}_0, r_0)\\Subset\\Omega$, where $B(\\mathbf{x}_0, r_0)$ is a given ball. Furthermore, it is assumed that for any point $\\mathbf{x}\\in B(\\mathbf{x}_0, r_0)\\backslash {D}$, there exists a line segment within $\\Omega$ which connects $\\mathbf{x}$ to a point $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0,r_0)}$. Then one has that \n\\begin{equation}\\label{eq:estn1}\n\\|u_i\\|_{C^{1,\\alpha}(\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:estn2}\n\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}}\\backslash {D_{b,h}} )}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere the positive constant $C$ depends on generic constant in the estimate \\eqref{holder_estimate_K} as well as $\\Omega$ and $B(\\mathbf{x}_0, r_0)$, but independent of $K$. \n\\end{lemma}\n\n\\begin{proof}\nWe first prove \\eqref{eq:estn2}, and by virtue of \\eqref{holder_estimate_K}, we see that\n\\begin{equation}\\label{eq:estn25}\n\\|\\nabla u_e\\|_{C^\\alpha(U_{b,h}\\backslash\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}. \n\\end{equation}\nand hence it suffices for us to show that \n\\begin{equation}\\label{eq:estn3}\n\\|u_e\\|_{C(U_{b,h}\\backslash\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}. \n\\end{equation}\n\nBy the standard elliptic PDE estimate, we know $u_e\\in C(\\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)})$, and for any $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)}$, \n\\begin{equation}\\label{eq:estn4}\n|u_e(\\mathbf{x}')|\\leq C_1\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere $C_1$ is a positive constant depending only on $\\eta, \\Omega$ and $B(\\mathbf{x}_0, r_0)$, but independent of $K$. For any $\\mathbf{x}\\in \\overline{U_{b, h}}\\backslash{D_{b,h}}$, we let $\\mathbf{x}'\\in \\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)}$ and ${l}(\\mathbf{x},\\mathbf{x}')$ be the line segment connecting $\\mathbf{x}$ and $\\mathbf{x}'$ such that $l(\\mathbf{x},\\mathbf{x}')\\Subset\\Omega$. Then by the intermediate value theorem, we have\n\\begin{equation}\\label{eq:estn5}\nu_e(\\mathbf{x})-u_e(\\mathbf{x}')=\\nabla u_e(\\xi)\\cdot(\\mathbf{x}-\\mathbf{x}'),\n\\end{equation}\nwhere $\\xi\\in l(\\mathbf{x}, \\mathbf{x}')$. By combining \\eqref{eq:estn25} and \\eqref{eq:estn4}, we readily have from \\eqref{eq:estn5} that\n\\begin{equation}\\label{eq:est6}\n|u_e(\\mathbf{x})|\\leq C_2(1+K^\\mu)\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}\\leq 2C_2 K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere we have made use of the facts that $|\\mathbf{x}-\\mathbf{x}'|\\leq \\mathrm{diam}(\\Omega)$, and without loss of generality that $K^\\mu\\geq 1$. \\eqref{eq:est6} clearly implies \\eqref{eq:estn3}, which in combination with \\eqref{eq:estn25} immediately yields \\eqref{eq:estn2}. \n\n\\eqref{eq:estn1} can be proved by following a completely similar argument. Indeed, for any $\\mathbf{x}\\in D_{b, h}$, one can take $\\mathbf{x}'\\in \\partial D\\cap U_{b,h}$, and then it holds that\n\\begin{equation}\\label{eq:estn7}\nu_i(\\mathbf{x})-u_e(\\mathbf{x}')=u_i(\\mathbf{x})-u_i(\\mathbf{x}')=\\nabla u_i(\\xi)\\cdot(\\mathbf{x}-\\mathbf{x}'),\n\\end{equation}\nwhere $\\xi\\in l(\\mathbf{x}, \\mathbf{x}')\\subset\\overline{D_{b, h}}$. Finally, by combining Remark~\\ref{rem:n1}, \\eqref{eq:est6} and \\eqref{eq:estn7}, one can show \\eqref{eq:estn1}. \n\nThe proof is complete.  \n\n\\end{proof}\n\n\\begin{remark}\\label{rem:n2}\nThe geometric condition in Lemma~\\ref{lem:holder}, namely for any point $\\mathbf{x}\\in B(\\mathbf{x}_0, r_0)\\backslash {D}$, there exists a line segment within $\\Omega$ which connects $\\mathbf{x}$ to a point $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0,r_0)}$, can be easily fulfilled if $D$ is convex or star-shaped. \n\\end{remark}\n\nAs mentioned earlier, we assume that $u_e$ can be harmonically extended into $D_{b,h}$, which is still denoted by $u_e$. That is, $u_e$ is harmonic in $U_{b,h}$ and hence is real analytic in $U_{b,h}$. Then for $b,h\\in\\mathbb{R}_+$ sufficiently small, we can have from \\eqref{eq:estn2} that\n\\begin{equation}\\label{eq:estn8}\n\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere $C$ depends on the same a-priori data as those in \\eqref{eq:estn2}. Since throughout the paper, our argument is localized around an admission $K$-curvature point $\\mathbf{p}$ (cf. Theorems~\\ref{estimationT} and \\ref{decayT}). Hence, in what follows, we shall always assume that \\eqref{eq:estn8} holds true. By combining \\eqref{estimation} of Theorem~\\ref{estimationT} and Lemma~\\ref{lem:holder}, one can derive the following theorem.\n\n\\begin{theorem}\\label{decayT}\nLet $u\\in H^1(\\Omega)$ be the solution to \\eqref{EQ2} and $\\mathbf{p}\\in\\partial D$ be an admissible $K$-curvature point. Suppose that \\eqref{eq:estn1} and \\eqref{eq:estn2} hold. We further suppose that the exponent $\\mu$ in \\eqref{eq:estn1} and \\eqref{eq:estn2} (or equivalently in (\\ref{holder_estimate_K})) satisfies\n\\begin{equation}\\label{eq:cond1}\n\\mu <\\frac{\\min(1,\\delta)}{2}, \n\\end{equation}\nwhere $\\delta$ is the a-priori parameter associated to $\\mathbf{p}$ (cf. Definition~\\ref{def:k}). \nThen it holds that\n\\begin{equation}\\label{decay}\n|\\nabla u_i(\\mathbf{p})|\\leq \\mathcal{E}  \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(\\alpha,\\delta)\/2},\n\\end{equation}\nwhere $\\mathcal{E}$ depends on the same a-priori data as those in \\eqref{eq:estn1}--\\eqref{eq:estn2} as well as $\\alpha$ and $L, M$ in Definition~\\ref{def:k} , but independent of $g$ and $K$. \n\\end{theorem}\n\\begin{proof}\nBy plugging the estimates \\eqref{eq:estn1} and \\eqref{eq:estn8} into \\eqref{estimation} of Theorem \\ref{estimationT}, we can obtain\n\\begin{equation}\\label{estimation_2}\n\\begin{split}\n& C |\\nabla u_i(\\mathbf{p})|\\leq K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(1+(\\tau h)^{(n-1)\/2})e^{\\tau(\\frac{1}{4K}-h)} \\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\left((\\frac{K}{K_-})^{\\frac{n-1}{2}}-(\\frac{K}{K_+})^{\\frac{n-1}{2}}\\right)e^{\\frac{\\tau}{4K}}\\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(h+K^{-1}_-)^{\\alpha\/2}h^{(n+1+\\alpha)\/2}(K\/K_-)^{(n-1)\/2}\\tau^{3\/2}e^{\\frac{\\tau}{4K}} \\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}h^{\\alpha+(n-1)\/2}(K\/K_-)^{(n-1)\/2}\\\\\n& \\quad \\times (1+\\tau h)\\tau^{(n-1)\/2}e^{\\tau(\\frac{1}{4K}-h)}.\n \\end{split}\n \\end{equation}\n It is noted that in \\eqref{estimation_2}, we have absorbed the generic constant involved into the right-hand side term, and it depends on the a-priori data as stated in the theorem, which should be clear in the context. \n \nNext, by following a similar argument to the proof of Proposition 2.22 in \\cite{Liu_curve}, one can show that there exists $C_{n,L,M}>0$ such that \n \\begin{equation}\n \\left|\\left(\\frac{K}{K_-}\\right)^{\\frac{n-1}{2}}-\\left(\\frac{K}{K_+}\\right)^{\\frac{n-1}{2}}\\right|\\leq C_{n,L,M}K^{-\\delta}.\n \\end{equation}\n Using $h=1\/K$ and $b=\\sqrt{M}\/K$ in the definition of the $K$-curvature point in Definition~\\ref{def:k}, the estimate (\\ref{estimation_2}) further yields\n \\begin{eqnarray}\\label{estimation_3}\n& &C|\\nabla u_i(\\mathbf{p})|\\leq (1+(\\tau \/K)^{(n-1)\/2})K^\\mu e^{-\\frac{3\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\nonumber \\\\\n& &+K^{\\mu-\\delta}e^{\\frac{\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}+K^{\\mu-(n+2\\alpha+1)\/2} \\tau^{3\/2}e^{\\frac{\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\nonumber \\\\\n& &+K^{\\mu-\\alpha-(n-1)\/2} (1+\\tau \/K)\\tau^{(n-1)\/2}e^{-\\frac{3\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}.\n \\end{eqnarray}\nChoosing $\\tau=4K\\ln K^\\rho$ for some $\\rho>0$ and dividing by $\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}$, the left-hand side of (\\ref{estimation_3}) can be estimated by\n\\begin{equation}\\label{estimation_4}\n(\\ln K)^{(n-1)\/2}K^{\\mu-3\\rho}+K^{\\mu-\\delta+\\rho}+(\\ln K)^{3\/2}K^{\\mu+1-n\/2-\\alpha+\\rho}+(\\ln K)^{(n+1)\/2}K^{\\mu-\\alpha-3\\rho}.\n\\end{equation}\nBy setting $\\rho=\\min(\\alpha,\\delta)\/2$, each of the terms in (\\ref{estimation_4}) can be estimated by\n\\begin{equation}\nC(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(\\alpha,\\delta)\/2},\n\\end{equation}\nand thus the claim of this theorem follows.\n\nThe proof is complete. \n\\end{proof}\n\n\\begin{remark}\\label{rem:nn2}\nIn Appendix, we shall present two examples to numerically verify that the condition~\\eqref{eq:cond1} can be fulfilled in generic scenarios of practical interest.\n\\end{remark}\n\n\\subsection{Local uniqueness result}\n\nWe are in a position to present the main local uniqueness result for the inverse inclusion problem \\eqref{eq:ic1}. \n\n\\begin{theorem}\\label{mainT}\nLet $(D, \\eta)$ and $(\\widetilde D, \\widetilde\\eta)$ be two conductive inclusions in $\\Omega$, and $u, \\widetilde u$ be the solutions to \\eqref{EQ2} associated respectively to $(D, \\eta)$ and $(D, \\widetilde\\eta)$. Suppose that  \n\\begin{enumerate}\n\\item $\\partial D$ and $\\partial\\widetilde D$ are of class $C^{2, 1}$; \n\n\\item $D$ and $\\widetilde D$ satisfy the geometric condition in Lemma~\\ref{lem:holder};\n\n\\item $G:=\\Omega\\setminus \\overline{(D\\cup \\widetilde{D})}$ is connected;\n\n\\item the condition~\\eqref{eq:cond1} is fulfilled for $u\/\\widetilde u$; \n\n\\item Proposition \\ref{inf_grad} holds for $u$ and $\\widetilde{u}$. \n\\end{enumerate}\nLet $d_0\\in\\mathbb{R}_+$ and $\\Gamma_0\\subset\\partial\\Omega$. If $u=\\widetilde u$ on $\\Gamma_0$, then $D\\Delta\\widetilde{D}=(D\\setminus \\widetilde{D})\\cup (\\widetilde{D}\\setminus D)$ cannot possess an admissible $K$-curvature point $\\mathbf{p}$ such that \n\\begin{equation}\\label{eq:cc1}\n\\max\\{\\mathrm{dist}(\\mathbf{p}, \\partial D), \\mathrm{dist}(\\mathbf{p}, \\partial\\widetilde D)> d_0, \n\\end{equation}\nand $K\\geq K_0$, where $K_0\\in\\mathbb{R}_+$ is sufficiently large and depends on the a-priori parameters of $\\mathbf{p}$ in Definition~\\ref{def:k} as well as $\\Omega, d_0, g, \\eta, \\widetilde\\eta$ and $B(\\mathbf{x}_0, r_0)$ in Lemma~\\ref{lem:holder}. \n\\end{theorem}\n\n\\begin{proof}\nBy absurdity, we assume without loss of generality that there exists an admissible $K$-curvature point $\\mathbf{p}\\in \\partial D \\cap \\partial G$, such that \n$B(\\mathbf{p},d_0)\\Subset\\Omega\\backslash\\overline{\\widetilde{D}}$. We next show that as $K\\rightarrow+\\infty$, $|\\nabla u(\\mathbf{p})|\\rightarrow 0$, which yields a contraction the assumption (5) in the statement of the theorem. \n\n\nFirst, let us consider the function $u-\\widetilde{u}$. This function is harmonic in $G$. Moreover, $u-\\widetilde{u}$ and $\\partial_\\nu (u-\\widetilde{u})$ vanish on $\\Gamma_0$. It follows from the unique continuation property that $u=\\widetilde{u}$ in $G$.\n\nIt is recalled that $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}$. Since $u=\\widetilde u$ in $G$ and both $u$ and $\\widetilde u$ are harmonic in $G$, we see that $u_e$ admits a harmonic extension $D\\cap B(\\mathbf{p}, d_0)$, which is actually $\\widetilde u$. Hence, Theorem~\\ref{decayT} applies and we immediately obtain from \\eqref{decay}\n that\n\\begin{equation}\\label{eq:nn2}\n|\\nabla u(\\mathbf{p})|\\leq \\mathcal{ET}  \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(1,\\delta)\/2}.\n\\end{equation}\nClearly, the right-hand side of the above estimate tends to zero as $K\\rightarrow +\\infty$. Therefore, we can choose $K_0$ such that when $K>K_0$, $|\\nabla u(\\mathbf{p})| < m_g$ which contradicts to the assumption (5) stated in the theorem. \n\nThe proof is complete. \n\\end{proof}\n\n\\begin{remark}\nIt is remarked that the assumptions (2) and (3) in Theorem~\\ref{mainT} can be fulfilled if $D$ and $\\widetilde D$ are convex; see Remark~\\ref{rem:n2}. Nevertheless, they might be more general than being convex in Theorem~\\ref{mainT}. Moreover, since our argument is localized around the admission $K$-curvature point, one may consider a even more general geometric situation where there are multiple conductive inclusions. In such a case, one may also relax the requirement that $G:=\\Omega\\setminus \\overline{(D\\cup \\widetilde{D})}$ is connected and replace $G$ to be the connected component of $\\Omega\\backslash\\overline{D\\cup\\widetilde D}$ that connects to $\\partial\\Omega$. \n\\end{remark}\n\n\\begin{remark}\nAs discussion in Remark~\\ref{rem:input}, the condition~(5) in Theorem~\\ref{mainT} can be fulfilled by choosing a suitable input $g$. As mentioned in Remark~\\ref{rem:nn2}, we shall show in the Appendix that the condition~(4) can be fulfilled in generic scenarios. \n\\end{remark}\n\n\n\\begin{remark}\nSince our argument is localized around an admissible $K$-curvature point, it is sufficient for us to require that the conductivity parameter $\\eta$ is constant in a small neighbourhood of the admissible $K$-curvature, and it can be an $L^\\infty$ variable function in the rest part of the inclusion $D$. Our unique recovery result in Theorem~\\ref{mainT} can be extended to such a case by straightforwardly modifying the relevant arguments. Finally, we would like to point out if sufficient a-priori information is available about the underlying inclusion, the local uniqueness result in Theorem~\\ref{mainT} also implies a certain global uniqueness result, and we shall explore more along this direction in our future study. \n\\end{remark}\n\n\n\n\\section*{Acknowledgment}\nThe work of H Liu was supported by the startup grant from City University of Hong Kong, Hong Kong RGC General Research Funds, 12302017, 12301218 and 12302919.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe aim of this paper is to find a part of a John decomposition\non which a given nontrivial operator is invertible in a certain sense, \nand to apply this to the study of contact points of convex bodies. \n\nJohn decomposition of identity is by now a classical tool in the local \ntheory of Banach spaces.\nSuppose $X  =  ({\\bf R}^n, \\| \\cdot \\|)$ is a Banach space whose \nellipsoid of maximal volume contained in $B(X)$ \ncoincides with the unit euclidean ball. \nThe John decomposition of the identity operator on $X$ is\n\\begin{equation}                                                             \\label{Johndecomp}\n  {\\it id}  =  \\sum_{j = 1}^m  x_j \\otimes x_j,\n\\end{equation}\nwhere $x_j \/ \\|x_j\\|_X$ are some contact points of the surfaces\nof $B(X)$ and the unit euclidean ball.\nThis celebrated theorem of F.~John has been used extensively \nover past 30 years. Recently it was interpreted as an isotropic\ncondition \\cite{G-M}, and was generalized to a non-convex case\nin \\cite{G-P-T}.\n\nIt is important to know good parts of John decomposition, \nas it provides a good set of contact points of $X$, \nwhich can be useful for understanding the geometrical structure of $X$,\nsee \\cite{R2}.\nIn the present paper we find a part of a decomposition (\\ref{Johndecomp})\nwhich preserves the orthogonal structure under action of a given \nlinear operator $T$.\nMore precisely, if $\\|T\\|_{2 \\rightarrow 2}  =  1$\nthen there exists a subset of indices $\\sigma$\nof cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\nthe system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\nto an orthogonal basis in Hilbert space.\nThis result is non-trivial even for the coordinate decomposition\n${\\it id}  =  \\sum e_j \\otimes e_j$.\nIn this case it generalizes the principle of restricted invertibility\nproved by J.~Bourgain and L.~Tzafriri \\cite{B-Tz}. \nThey considered only operators $T$ for which all norms $\\|T e_j\\|_2$ \nare well bounded below and proved the principle with some fixed $0 < \\varepsilon < 1$.\n\n\n$T$ being an orthogonal projection, we derive a new lemma of \nDvoretzky-Rogers type. \nSuppose $P$ is an orthogonal projection in $X$ with ${\\rm rank} P = k$.\nThen for any $\\kappa  <  k$\nthere are contact points $x_1, \\ldots, x_\\kappa$\nsuch that setting $z_j  =  P x_j \/ \\|P x_j\\|_2$\nwe have\n\n\\begin{itemize}\n\\item{the system $(z_j)$ is $C(\\kappa \/ k)$-equivalent in $l_2$-norm \n    to the canonical basis of $l_2^\\kappa$;}\n\\item{$\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}$  for all $j$.}\n\\end{itemize}\n\n\\noindent To put the result in other words, \nthe orthonormal system in $Z$ guaranteed by the classical \nDvoretzky-Rogers Lemma is essentially the normalized projections \nof contact points of $X$.\nMoreover, the result holds for selfadjoint operators as well \nas for projections, the Hilbert-Schmidt norm substituting ${\\rm rank} P$.\nFor a general operator $T$ the best lower bound for $\\|T x_j\\|_X$ \nis equivalent to $\\frac{1}{n} |{\\rm trace} T|$.\n\n$T$ being the identity operator, we obtain a set of \n$k  >  (1 - \\varepsilon) n$ contact points of $X$\nwhich is $C(\\varepsilon)$-equivalent in $l_2$-norm to the canonical \nbasis in $l_2^k$.\nThis settles an isomorphic version of a problem of \nN.~Tomczak-Jaegermann (\\cite{T-J}, p.127),\nand confirms the feeling that contact points are always \ndistributed fairly uniformly on the surface of the maximal volume ellipsoid\n(see \\cite{B1}).\nBesides, this yields the proportional Dvoretzky-Rogers factorization\n(with constant $C(\\varepsilon)  =  \\varepsilon^{c \\log \\varepsilon}$, \nwhich is however not the best known estimate).\n\nThe Dvoretzky-Rogers Lemma is proved useful in study \nof subspaces of $X$ well isomorphic to $l_\\infty^k$.\nThe use of the refined Dvoretzky-Rogers lemma above\nimproves a \"Gaussian\" version of Alon-Milman-Talagrand Theorem \nabout $l_\\infty^k$-subspaces of $X$.\nLet $P$ be an orthogonal projection in $X$ with ${\\rm rank} P = k$.\nThen there exists a subspace $Z  \\subset X$ \nwhich is $M$-isomorphic to $l_\\infty^m$ \nfor $m  \\ge  c k \/ \\sqrt{n}$\nand $M  =  c \\sqrt{\\frac{n}{k}} \\ell(P)$.\nThe subspace $Z$ is canonically spanned by the projections\nof $m$ contact points $x_j$.\nMoreover, the norm on $Z$ is $M$-equivalent to \n$|||z|||  =  \\max_{j \\le m} | \\< z , x_j \\> |$.\nThis improves the estimates obtained by \nM.~Rudelson in \\cite{R1}, and also provides information\nabout the position of $Z$ in $X$.\nBesides, this yields a refinement of M.~Rudelson's \nresult about $l_\\infty^k$-subspaces in spaces\nwith large volume ratio.\nIf ${\\rm vr}(X)  \\ge  a \\sqrt{n}$ then $X$ has a subspace $Z$ \nof dimension $m  \\ge  C_1(a) \\sqrt{n}$\nwhich is $C_2(a) \\log{n}$-isomorphic to $l_\\infty^m$.\n\nThe extraction results about John decompositions\ncan be reformulated in the language of frames. \nSuppose we are given a tight frame $(x_j)$ in Hilbert space $H$,\nand a norm-one linear operator $T : H  \\rightarrow  H$. \nThen there is a subsequence \n$(T x_j)_{j \\in \\sigma}$ with $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ \nwhich is $C(\\varepsilon)$-equivalent to an orthogonal basis in Hilbert space.\nThis theorem again can be interpreted as an extension of\nthe invertibility principle. \nIt also generalizes results of P.~Casazza \\cite{C2} and \nthe author \\cite{V}, who worked with the identity operator \n$T = {\\it id}$.\n\nThe rest of the paper is organized as follows. \nIn \\S \\ref{SecPrelim} we recall some basic tools used later. \nThe extraction result about John decompositions, as\nwell as some modifications, is proved in \\S \\ref{SecMain}.\nIts relation to the principle of restricted invertibility\nand infinite-dimensional analogs are discussed in \\S \\ref{SecInvert}.\nDvoretzky-Rogers type lemmas are derived from these results\nin \\S \\ref{SecDR}.\nThey help to understand structure of the set of contact points. \nApplications to $l_\\infty^k$-subspaces of a finite dimensional \nspace are given in \\S \\ref{SecCube}.\nFinally, in \\S \\ref{SecFrames} we discuss a relation of \nthese results to the theory of frames in Hilbert space. \n\nI am grateful to M.~Rudelson for many important discussions.\nThe research would not be accomplished without help and \nencouragement of my wife Lilya.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}                                                \\label{SecPrelim}\n\n\nWe denote by $c, c_1, c_2$ absolute constants, \nand by $C(t)$, $C_1(t)$, $C_2(t)$ constants which depend on the\nparameter $t$ only. The values of these constants may differ\nfrom line to line.\nThe canonical vectors in ${\\bf R}^n$ are denoted by $e_j$.\n\nA sequence of vectors $(x_j)$ in a Banach space is called \n{\\em $K$-Hilbertian} if \n$\\| \\sum a_j x_j \\|  \\le  K ( \\sum |a_j|^2 )^{1\/2}$\nfor any finite set of scalars $(a_j)$.\nSimilarly, $(x_j)$ is called \n{\\em $K$-Besselian} if \n$K \\| \\sum a_j x_j \\|  \\ge  ( \\sum |a_j|^2 )^{1\/2}$\nfor any finite set of scalars $(a_j)$.\nSuppose we are given two sequences $(x_j)$ and $(y_j)$ \nin Banach spaces $X$ and $Y$ respectively.\nThe sequences $(x_j)$ and $(y_j)$ are called {\\em $K$-equivalent}\nif there exist constants $K_1$ and $K_2$ with $K_1 K_2  \\le  K$\nsuch that for any finite sequence of scalars $(a_j)$\n$$\nK_1^{-1} \\| \\sum a_j y_j \\|_Y\n\\le  \\| \\sum a_j x_j \\|_X\n\\le  K_2 \\| \\sum a_j y_j \\|_Y.\n$$\nIn other words, the linear operator \n$T : {\\rm span}(x_j)  \\rightarrow  {\\rm span}(y_j)$ defined as\n$T x_j  =  y_j$ for all $j$\nis a $K$-isomorphism: $\\|T\\| \\|T^{-1}\\|  \\le  K$.\n\nHere and in the next section we work in a Hilbert space $H$\nwhose scalar product is denoted by $\\< \\cdot, \\cdot \\> $, \nand the norm by $\\| \\cdot \\|$.\nFirst we observe that the Hilbert-Schmidt norm of an operator on $H$\ncan be computed on the elements\nof certain decompositions of identity.\n\n\\begin{lemma}                                                             \\label{hs}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space $H$, \n  and $T : H \\rightarrow H$ be a linear operator. \n  Then \n  $$\n  \\|T\\|_{\\rm HS}^2  =  \\sum \\|T x_j\\|^2.\n  $$\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\  \nIt is enough to write \n$$\n\\sum T x_j  \\otimes  T x_j\n=  T T^*\n= \\sum T e_j  \\otimes  T e_j\n$$\nand to take traces.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\\noindent As an immediate consequence we have \n\n\\begin{corollary}                                                             \\label{squares}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space $H$. \n  Then \n  $$\n  \\sum \\|x_j\\|^2  =  \\dim H.\n  $$\n\\end{corollary}\n\n\n\\begin{lemma}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space.\n  Then the system $(x_j)$ is $1$-Hilbertian. \n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nNotice that for every vector $x$\n$$\n\\|x\\|^2  =  \\langle x, x \\rangle  \n  =  \\Big\\langle \\sum \\langle x_j, x \\rangle x_j, x \\Big\\rangle\n  =  \\sum | \\langle x_j, x \\rangle |^2.\n$$\nThus $\\| \\sum x_j \\otimes e_j \\| = 1$, \nand by duality $\\| \\sum e_j \\otimes x_j \\| = 1$.\nThis yields that $(x_j)$ is $1$-Hilbertian.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nThe starting point of this paper is the \nprinciple of restricted invertibility proved \nby J.~Bourgain and L.~Tzafriri \\cite{B-Tz}.\n\n\\begin{theorem} (J.~Bourgain, L.~Tzafriri).                          \\label{BTz}  \n  Let $T$ be a linear operator in $l_2^n$\n  for which $\\|T e_j\\| = 1$, $j = 1, \\ldots, n$. \n  Then there exists a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \n  of cardinality $|\\sigma|  \\ge  c_1 n \/ \\|T\\|^2$ such that\n  $$\n  \\Big\\| \\sum_{j \\in \\sigma} a_j T e_j \\Big\\|\n    \\ge  c \\Big( \\sum_{j \\in \\sigma} |a_j|^2 \\Big)^{1\/2}\n  $$\n  for any choice of scalars $(a_j)$.\n\\end{theorem}\n\n\nThe invertibility principle will be used together with\nthe following restriction theorem. It can easily be recovered \nfrom A.~Kashin's and L.~Tzafriri's paper \\cite{K-Tz}, \nsee the proofs of Theorem 1 and Corollary 2 there. \n\n\\begin{theorem} (A.~Kashin, L.~Tzafriri).                                  \\label{Lunin}\n  Let $A$ be a norm-one linear operator in $l_2^m$.\n  Fix a number $\\lambda$ with $1\/m  \\le  \\lambda  \\le 1$.\n  Then there exists a subset $\\nu  \\subset \\{1, \\ldots, m\\}$\n  of cardinality $|\\nu|  \\ge  \\lambda m \/ 4$\n  such that \n  $$\n  \\| P_\\nu A \\|  \\le  c \\left( \\sqrt{\\lambda} + \\frac{\\|A\\|_{\\rm HS}}{\\sqrt{m}} \\right).\n  $$\nHere $P_\\nu$ denotes the coordinate projection onto ${\\bf R}^\\nu$.\n\\end{theorem}\n\n\nNow we introduce an elementary procedure of splitting a sequence.\nGiven a sequence $(x_j)$ in $H$, let $(y_k)$ be any sequence of vectors in $H$ \nsuch that for every $j = 1, 2, \\ldots$\n\n\\begin{itemize}\n\\item{the vectors $y_k$, $k \\in \\sigma_j$, are multiples of the vector $x_j$;}\n\\item{$\\sum_{k \\in \\sigma_j} \\|y_k\\|^2  =  \\|x_j\\|^2$.}\n\\end{itemize}\n\n\\noindent Then we say that $(y_k)$ is the {\\em splitted sequence} $(x_j)$.\nSplitting allows us to make the norms of the vectors almost equal. Still the key \nproperty of a sequence, being $h$-Hilbertian, is preserved by splitting.\n\n\n\n\n\\section{The main result}                                               \\label{SecMain}\n\nIn this section we prove an extraction theorem which is a core of the paper.\n\n\\begin{theorem}                                                          \\label{main}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on $l_2^n$,\n  and $T$ be a norm-one linear operator. \n  Then for any $\\varepsilon > 0$ there exists a set of indices $\\sigma$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  \n  (i) the system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\n      to an orthogonal basis in $l_2^\\sigma$;\n\n  (ii) $\\|T x_j\\|  \\ge  c \\sqrt{\\varepsilon} \\frac{\\|T\\|_{\\rm HS}}{\\sqrt{n}} \\|x_j\\|$\n      for all $j \\in \\sigma$.\n\\end{theorem} \n\n\\noindent {\\bf Proof.}\\ \\ \nPut $h  =  \\|T\\|_{\\rm HS}^2$.\nBy an approximation one can assume that the system $(x_j)$ is finite,\nso we enumerate it as $(x_j)_{j \\le m}$.\nDenote $y_j  =  T x_j$ for all $j$. \nSplitting the system $(x_j)_{j \\le m}$ we can assume that \n$$\n0.9 \\sqrt{ \\frac{h}{m} }  \\le  \\|y_j\\|  \\le  1.1 \\sqrt{ \\frac{h}{m} }\n\\ \\ \\ \\ \\mbox{for all $j \\le m$.}\n$$\nLet $\\delta  =  \\varepsilon \/ 3$.\nConsider the set of indices satisfying (ii), i.e.\n$$\n\\tau  =  \\Big\\{ j \\le m :  \n          \\|y_j\\|  \\ge  0.9 \\sqrt{\\delta} \\sqrt{\\frac{h}{n}} \\|x_j\\|  \\Big\\}.\n$$\nWe claim that \n\\begin{equation}                                            \\label{sizeoftau}\n  |\\tau|  \\ge  (1 - \\delta) m.\n\\end{equation}\nIndeed, by Corollary \\ref{squares}\n\\begin{eqnarray*}\nn  =  \\sum_{j \\le m} \\|x_j\\|^2  \n    &\\ge&  \\sum_{j \\in \\tau^c} \\|x_j\\|^2                                     \\\\\n    &\\ge&  \\sum_{j \\in \\tau^c} \n                  \\left( \\frac{1}{0.9 \\sqrt{\\delta}} \\sqrt{\\frac{n}{h}} \n                  \\|y_j\\| \\right)^2                                        \\\\\n    &\\ge&  |\\tau^c| \\left( \\frac{1}{0.9 \\sqrt{\\delta}} \\sqrt{\\frac{n}{h}} \n                           0.9 \\sqrt{\\frac{h}{m}} \\right)^2                \n      =    |\\tau^c| \\cdot \\frac{n}{\\delta m}.\n\\end{eqnarray*}\nThus $|\\tau^c|  \\le  \\delta m$, which proves (\\ref{sizeoftau}). \n\nNow we have to find a further subset $\\sigma  \\subset \\tau$ \nof cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) h$,\nsuch that the system $(y_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\nto an orthogonal basis in $l_2^\\sigma$.\nThe set $\\sigma$ will be constructed by successive iterations. \nOn the first step $\\sigma = \\emptyset$.\nOn each successive step, the remainder $h - |\\sigma|$ \nwill be reduced in a fixed proportion.\nSo, it is enough to prove the following.\n\n\\begin{lemma}\n  Let $\\sigma  \\subset \\tau$ with $|\\sigma|  <  (1 - \\varepsilon)h$ be given, \n  and suppose the system $(y_j)_{j \\in \\sigma}$ is $K$-equivalent to an \n  orthogonal basis in Hilbert space.\n  Then $\\sigma$ can be extended in $\\tau$ to a subset $\\sigma_1$ so that \n  \n  (a) the system $(y_j)_{j \\in \\sigma_1}$ is $C(K, \\varepsilon)$-equivalent to an\n      orthogonal basis in Hilbert space;\n\n  (b) for some absolute constant $\\alpha < 1$ \n      $$\n      h - |\\sigma_1|  \\le  \\alpha (h - |\\sigma|).                          \\label{remainder}\n      $$\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nLet $P$ be the orthonormal projection in $l_2^n$\nonto $l_2^n  \\ominus  {\\rm span}(y_j)_{j \\in \\sigma}$\n(at the first step $P = {\\it id}$).\nFirst observe that by Lemma \\ref{hs}\n\\begin{eqnarray*}\n\\sum_{j \\le m} \\|P y_j\\|^2\n  & = &  \\sum_{j \\le m} \\|P T x_j\\|^2\n    =    \\|PT\\|_{\\rm HS}^2                                          \\\\\n  & = &  h  -  \\| ({\\it id} - P) T \\|_{\\rm HS}^2                         \\\\\n  &\\ge&  h  -  \\|{\\it id} - P\\|_{\\rm HS}^2 \\|T\\|^2                       \\\\\n  & = &  h  -  |\\sigma|.\n\\end{eqnarray*}\nUsing (\\ref{sizeoftau}), we get\n\\begin{eqnarray*}\n\\sum_{j  \\in  \\tau \\setminus \\sigma} \\|P y_j\\|^2  \n  & = &  \\sum_{j \\in \\tau} \\|P y_j\\|^2                                       \\\\\n  &\\ge&  \\sum_{j \\le m}  \\|P y_j\\|^2  \n            -  |\\tau^c| \\cdot \\Big( 1.1 \\sqrt{h \/ m} \\Big)^2                 \\\\\n  &\\ge&  h - |\\sigma| - 1.21 \\delta h                                              \\\\\n  &\\ge&  (1 - 2 \\delta) h - |\\sigma|   =:  h_0.\n\\end{eqnarray*}\nNote that $h_0$ is comparable with $h$. \nIndeed, since $|\\sigma|  <  (1 - \\varepsilon) h  =  (1 - 3 \\delta) h$, we have\n\\begin{equation}\n  h_0  \\ge  \\delta h.                                              \\label{hzeroh}\n\\end{equation}\nNow we can split the system $(y_j)_{j  \\in  \\tau \\setminus \\sigma}$\nso that the resulting system $(y'_j)_{j \\le M}$ satisfies\n\\begin{equation}\n  \\|P y'_j\\|  \\ge  0.9 \\sqrt{\\frac{h_0}{M}}                   \n     \\ \\ \\ \\ \\mbox{for all $j \\le M$}.                                \\label{Pydash}\n\\end{equation}\n\nWe are going to apply Kashin-Tzafriri's extraction result, Theorem \\ref{Lunin}.\nIn the dual setting it can be reformulated as follows.\n\n\\begin{itemize}\n\\item{Let $(x_j)_{j \\le m}$ be a $1$-Hilbertian system in a Hilbert space, \n  and put $\\sum\\|x_j\\|^2  =  h$. \n  Fix a number $\\lambda$ with $1\/m  \\le  \\lambda  \\le 1$.\n  Then there exists a subset $\\nu  \\subset \\{1, \\ldots, m\\}$\n  of cardinality $|\\nu|  \\ge  \\lambda m \/ 4$\n  such that setting \n  $K  =  \\Big( \\sqrt{\\lambda} + \\sqrt{h \/ m} \\Big)^{-1}$,\n  $$\n  \\mbox{the system $(K x_j)_{j \\in \\sigma}$ is $c$-Hilbertian}.\n  $$}\n\\end{itemize}\n\n\\noindent Apply this to the system $(y'_j)_{j \\le M}$ \nwhich is $1$-Hilbertian and \n$$\n\\sum_{j \\le M} \\|y'_j\\|^2  \n  \\le  \\sum_{j \\le m} \\|y_j\\|^2  \n   =   h.\n$$\nWith $\\lambda  =  4 h \/ M$, we obtain a subset \n$\\nu  \\subset \\{1, \\ldots, M\\}$\nof cardinality $|\\nu|  \\ge  h$\nsuch that \n\\begin{equation}                                                    \\label{ishilb}\n\\mbox{the system $\\Big( \\sqrt{\\frac{M}{h}} y'_j \\Big)_{j \\in \\nu}$ \n      is $c$-Hilbertian}\n\\end{equation}\n(notice that we could make $M$ large enough to have $\\lambda  \\le  1$\nas required in Kashin-Tzafriri's Theorem).\nTherefore, the system \n$\\Big( \\sqrt{\\frac{M}{h}} P y'_j \\Big)_{j \\in \\nu}$ \nis $c$-Hilbertian, too.\nMoreover, by (\\ref{Pydash}) and (\\ref{hzeroh})\n$$\n\\Big\\| \\sqrt{\\frac{M}{h}} P y'_j \\Big\\|   \n  \\ge  0.9 \\sqrt{\\frac{h_0}{h}}\n  \\ge  0.9 \\sqrt{\\delta}  \n  \\ \\ \\ \\ \\mbox{for all $j \\in \\nu$}.\n$$\n\nAt this point we use the original invertibility principle\nof J.~Bourgain and L.~Tzafriri \\cite{B-Tz}, \nwhich can be reformulated as follows.\n\n\\begin{itemize}\n\\item{Let $(x_j)_{j \\le n}$ be an $H$-Hilbertian system in $l_2$ \n      and $\\|x_j\\|  \\ge  \\alpha$ for all $j$.\n      Then there exists a subset $\\rho  \\subset \\{1, \\ldots, n\\}$\n      of cardinality $|\\rho|  \\ge  c (\\alpha \/ H)^2 n$ such that\n      the system $(x_j)_{j \\in \\rho}$ is $(c_1 \/ \\alpha)$-Besselian.}\n\\end{itemize}\n\n\\noindent Apply this to the system \n$\\Big( \\sqrt{\\frac{M}{h}} P y'_j \\Big)_{j \\in \\nu}$.\nThere exists a subset $\\rho'  \\subset  \\nu$ of cardinality \n$|\\rho'|  \\ge  c (0.9 \\sqrt{\\delta})^2 h  =  c \\delta h$\nsuch that\n\\begin{equation}                                               \\label{isbess}\n\\mbox{the system $(\\sqrt{\\frac{M}{h}} P y'_j)_{j \\in \\rho'}$\n      is $c_1 \/ \\sqrt{\\delta}$-Besselian.}\n\\end{equation}\nRecall that each vector $y'_j$ with $j \\in \\rho'$ \nis a multiple of some vector $y_{k(j)}$ \nwith $k(j)  \\in  \\tau \\setminus \\sigma$.\nBy (\\ref{isbess}), these vectors $y_{k(j)}$ must be linearly independent. \nIn particular, the correspondence $j \\mapsto k(j)$ is one-to-one.\nConsider the subset $\\rho  \\subset  \\tau \\setminus \\sigma$\nconsisting of vectors\n$$\n\\rho  =  \\{ k(j) : j \\in \\rho' \\}.\n$$\nNow put $\\sigma_1  =  \\sigma \\cup \\rho$.\n\nWe see that (b) is satisfied with $\\alpha  =  1 - c \\delta \/ 2$:\n$$\nh - |\\sigma_1|  \\le  h - |\\sigma| - c \\delta h  \\le  \\alpha (h - |\\sigma|).\n$$\n\nThe reason why (a) holds is that \nthe system $(y_j)_{j \\in \\rho}$ is well equivalent to an\northogonal basis, \nand the spans of $(y_j)_{j \\in \\sigma}$ and $(y_j)_{j \\in \\rho}$\nare well disjointed.\nTo implement this idea, note that by the preseding observations\n(\\ref{ishilb}) and (\\ref{isbess}) yield that there exist constants \n$(\\lambda_j)_{j \\in \\rho}$ such that\n\n\\begin{itemize}\n\\item{the system $(\\lambda_j   y_j)_{j \\in \\rho}$ is $c$-Hilbertian,}\n\\item{the system $(\\lambda_j P y_j)_{j \\in \\rho}$ \n                               is $c_1 \/ \\sqrt{\\delta}$-Besselian.}\n\\end{itemize}\n\n\\noindent Now it is easy to complete the proof.\nFix any scalars $(a_j)_{j \\in \\rho}$.\nThen defining $\\lambda_j  =  y_j \/ \\|y_j\\|$ for $j \\in \\sigma$\nwe have\n\\begin{eqnarray*}\n\\Big( \\sum_{j \\in \\sigma \\cup \\rho} |a_j|^2 \\Big)^{1\/2} \n  & = &  \\Big( \\sum_{j \\in \\sigma}  |a_j|^2 \n             + \\sum_{j \\in \\rho}|a_j|^2 \\Big)^{1\/2}                   \\\\\n  &\\le&  (K + c_1 \/ \\sqrt{\\delta}) \\left( \n         \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j   y_j \\Big\\|^2\n       + \\Big\\| \\sum_{j \\in \\rho} a_j \\lambda_j P y_j \\Big\\|^2\n                               \\right)^{1\/2}                            \\\\\n  & = &  (K + c_1 \/ \\sqrt{\\delta}) \n         \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j   y_j \n             +  \\sum_{j \\in \\rho} a_j \\lambda_j P y_j \\Big\\|       \n            \\\\ \\mbox{by orthogonality}                       \\\\\n  &\\le&  (K + c_1 \/ \\sqrt{\\delta}) \n         \\Big\\| \\sum_{j \\in \\sigma \\cup \\rho} a_j \\lambda_j y_j \\Big\\|,\n\\end{eqnarray*}\nbecause $P y_j = y_j$ for $j \\in \\sigma$.\nThis shows that the system $(\\lambda_j y_j)_{j \\in \\sigma_1}$ \nis $(K + c_1 \/ \\sqrt{\\delta})$-Besselian.\nNext, \n\\begin{eqnarray*}\n\\Big\\| \\sum_{j \\in \\sigma_1} a_j \\lambda_j y_j \\Big\\|\n  &\\le&  \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j y_j \\Big\\|\n       + \\Big\\| \\sum_{j \\in \\rho} a_j \\lambda_j y_j \\Big\\|               \\\\\n  &\\le&  K \\Big( \\sum_{j \\in \\sigma}   |a_j|^2 \\Big)^{1\/2}        \n       + c \\Big( \\sum_{j \\in \\rho} |a_j|^2 \\Big)^{1\/2}              \\\\\n  &\\le&  \\sqrt{2} (K + c) \n           \\Big( \\sum_{j \\in \\sigma \\cup \\rho} |a_j|^2 \\Big)^{1\/2},\n\\end{eqnarray*}\nso the system $(\\lambda_j y_j)_{j \\in \\sigma_1}$ is $\\sqrt{2}(K + c)$-Hilbertian as well. \nThis establishes (a) of the Lemma\nand completes the proof of Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nLet us rewrite Theorem \\ref{main} in a different form,\nwhich is useful in applications like Dvoretzky-Rogers type lemmas.\n\n\\begin{theorem}                                                            \\label{mainc}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on $l_2^n$,\n  and $T$ be a norm-one linear operator, $\\|T\\|_{\\rm HS}^2  =  h$.\n  Then for any integer $\\kappa  <  h$ there exists a set of indices $\\sigma$\n  with $|\\sigma| = \\kappa$ such that\n  \n  (i) the system $(T x_j)_{j \\in \\sigma}$ is $C(\\kappa \/ h)$-equivalent\n      to an orthogonal basis in $l_2^\\sigma$;\n\n  (ii) $\\|T x_j\\|  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}} \\|x_j\\|$\n      for all $j \\in \\sigma$.\n\\end{theorem} \n\n\n\\noindent It is sometimes more useful to get a lower bound \nfor $\\< x_j, T x_j \\> $ rather than for $\\|T x_j\\|$.\n\n\\begin{proposition}                                                   \\label{mainscalar}\n  In Theorem \\ref{main} statement (ii) can be replaced by \n  \n  (ii') $| \\< x_j, T x_j \\> |  \n          \\ge  c \\varepsilon \\frac{|{\\rm trace} T|}{n} \\|x_j\\|^2$\n          for $j \\in \\sigma$.\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\ \nNotice that $\\sum  \\< x_j, T x_j \\>  =  {\\rm trace} T$.\nTherefore, splitting our system $(x_j)_{j \\le m}$\nwe can assume that\n\\begin{equation}                                                       \\label{tracem}\n  | \\< x_j, T x_j \\> |  \n    \\ge  0.9 \\frac{|{\\rm trace} T|}{m}\n    \\ \\ \\ \\ \\mbox{for all $j$}.\n\\end{equation}\n\nLet us examine the proof of Theorem \\ref{main}.\nThe set $\\tau$ was responsible for the lower bound of $\\|T x_j\\|$.\nSo, we replace $\\tau$ by \n$$\n\\tau'  =  \\Big\\{ j \\le m :\n       | \\< x_j, T x_j \\> |  \n       \\ge  (\\varepsilon \/ 5) \\frac{|{\\rm trace} T|}{n} \\|x_j\\|^2\n        \\Big\\},\n$$\nand all we have to check is\n\\begin{equation}                                                   \\label{tdash}\n  |\\tau'|  \\ge  (1 - \\varepsilon\/3) m.\n\\end{equation}\nBy (\\ref{tracem})\n$$\n\\tau'  \\supset  \\rho  \n  =  \\Big\\{ j \\le m : \\|x_j\\|^2  \\le  (3 \/ \\varepsilon) \\frac{n}{m} \\Big\\}.\n$$\nSince $\\sum_{j \\le m} \\|x_j\\|^2  =  n$, \nwe have $|\\rho|  \\ge  (1 - \\varepsilon\/3) m$.\nThis verifies (\\ref{tdash}) and allows us \nto finish the proof as in Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\\section{Principle of restricted invertibility}                    \\label{SecInvert}\n\nOur first application stems from the look at Theorem \\ref{main}\nas an extension of the \"principle of restricted invertibility\",\nTheorem \\ref{BTz}, proved by J.~Bourgain and L.~Tzafriri.\nIndeed, for the coordinate decomposition \n$id = \\sum e_j \\otimes e_j$ we get\n\n\\begin{corollary}                                                \\label{BTzgeneralized}\n  Let $T$ be a norm-one linear operator in $l_2$.\n  Then for any $\\varepsilon > 0$ there exists a subset $\\sigma \\subset \\{1, 2, \\ldots\\}$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  the sequence $(T e_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent to an orthogonal\n  basis in Hilbert space.\n\\end{corollary}\n\nThis theorem generalizes the invertibility principle in two ways. \nFirst, instead of requiring that {\\em all} norms $\\|T e_j\\|$ \nbe large, we can assume only largeness of their average, \nwhich is the Hilbert-Schmidt norm of $T$.\nThis makes the result independent of the dimension $n$ of the space,\n$\\|T\\|_{\\rm HS}^2$ being a natural substitute for the dimension $n$.\n\nThe second improvement is that we obtain the subset $\\sigma$ with \nthe largest possible cardinality.\nCorollary \\ref{BTzgeneralized} allows us to get \n$|\\sigma|  \\ge  (1 - \\varepsilon) n \/ \\|T\\|^2$ in the original invertibility\nprinciple for any $0 < \\varepsilon < 1$\n(while the Bourgain-Tzafriri's argument proves only the \nexistence of such $\\varepsilon$).\nIn some applications one really needs almost full percentage;\nparticularly, this is important in estimating the distance to the \ncube, see \\cite{Sz-T}.\n\nNotice that the infinite-dimensional analogs of Theorem \\ref{main}\nand Corollary~\\ref{BTzgeneralized} hold, too.\n\n\\begin{proposition}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$ be a decomposition \n  on a Hilbert space, and $T$ be a linear operator which is \n  not Hilbert-Schmidt. \n  Then for any $\\varepsilon > 0$ there exists an infinite subset $\\sigma$\n  such that the sequence $(T x_j)_{j \\in \\sigma}$ \n  is $(1 + \\varepsilon)$-equivalent to an orthogonal basis in Hilbert space. \n\\end{proposition}\n  \n\\noindent {\\bf Proof.}\\ \\ \nThe subset $\\sigma$ is constructed by a standard induction argument, \nmodulo the following claim:\n\n\\begin{itemize}\n\\item{For any finite dimensional subspace $E$\n$$\n\\sup_j {\\rm dist} (T x_j \/ \\|T x_j\\|, E)  =  1.\n$$\n}\n\\end{itemize}\n\nAssume the contrary. \nDenoting the orthogonal projection in $l_2$ onto $E$ by $P$, we would have \n$$\n\\inf_j  \\Big\\| P (T x_j \/ \\|T x_j\\|) \\Big\\|  =  \\delta  >  0,\n$$\nthat is \n$$\n\\|P T x_j\\|  \\ge  \\delta \\|T x_j\\| \n  \\ \\ \\ \\ \\mbox{for all $j$.}\n$$\nBy Lemma \\ref{hs}, this yields\n$$\n\\|PT\\|_{\\rm HS}  \\ge  \\delta \\|T\\|_{\\rm HS}  =  \\infty.\n$$\nBut the operator $PT$ has finite rank, \nthus $\\|PT\\|_{\\rm HS}$ must be finite. \nThis contradiction completes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nAn infinite dimensional analogue of Bourgain-Tzafriri's Theorem \\ref{BTz}\nsays that, given a linear operator $T$ in $l_2$ with \n$\\|T e_j\\|  =  1$, $j = 1, 2, \\ldots$, \nthere exists a subset $\\sigma$ of $\\{1, 2, \\ldots \\}$\nwith upper density $\\overline{\\rm dens}\\; \\sigma  \\ge  c \/ \\|T\\|^2$, \nsuch that the sequence $(T e_j)_{j \\in \\sigma}$\nis $c$-Besselian \\cite{B-Tz}.\n\nAs we loose the normalizing condition $\\|T e_j\\| = 1$, nothing can be said\nin general about the density of $\\sigma$.\nIndeed, let $(y_k)$ be the result of a splitting of a canonical basis\nin $l_2$, so that the sets $\\sigma_j$ from the definition of splitting\nsatisfy $|\\sigma_j|  \\rightarrow  \\infty$ as $j  \\rightarrow  \\infty$.\nThere exist a norm-one operator $T$ in $l_2$ such that \n$\\|T\\|_{\\rm HS}  =  \\infty$ and\n$T e_k  =  y_k$, $k = 1, 2, \\ldots$.\nHowever, each term $y_k$ in the sequence $(y_k)$ is repeated $|\\sigma_j|$\ntimes. Therefore, the upper density of any subset $\\sigma$ satisfying\nthe conclusion of Corollary \\ref{BTzgeneralized} must be zero.\n\nSimilarly, in some cases $\\sigma$ must be a sparse set with respect \nto the dimension.\nMore precisely, in general the sequence $(T x_j)_{j \\in \\sigma}$ \nspans a subspace of infinite codimension.\nThis follows easily from a result of P.~Casazza and O.~Christensen\ndiscussed in Section \\ref{SecFrames}.\n\n\n\n\n\n\n\n\\section{Contact points and Dvoretzky-Rogers lemmas}                       \\label{SecDR}\n\nHere we apply Theorem \\ref{main} to John decompositions. \nLet $X = ({\\bf R}^n, \\| \\cdot\\|)$ be a Banach space. \nThe ellipsoid of maximal volume contained in $B(X)$ is unique \nand it is called {\\em maximal volume ellipsoid} of $X$.\nSuppose the euclidean structure on $X$ is chosen so that\nthe maximal volume ellipsoid coincides with the unit euclidean ball $D_n$.\nLet us write a John decomposition on $X$:\n\\begin{equation}                                               \\label{Johndecomposition}\n  {\\it id}  =  \\sum_{j = 1}^m  x_j \\otimes x_j,\n\\end{equation}\nwhere $x_j \/ \\|x_j\\|_X$ are some contact points of $B(X)$ with the John's \nellipsoid (see \\cite{T-J}, \\S 15.3).\nJohn decompositions can be considered as a subclass\nin the class of all decompositions of type ${\\it id}  =  \\sum x_j \\otimes x_j$.\nConversely, each decomposition (\\ref{Johndecomposition})\nis a John decomposition for a suitable Banach space \n$X = ({\\bf R}^n, \\| \\cdot\\|)$, whose maximal volume ellipsoid is the unit\neuclidean ball $D_n$.\nActually, the norm on $X$ is given by \n$\\|x\\|_X  =  \\max_{j \\le m} | \\< x, \\frac{x_j}{\\|x_j\\|_X} \\> |$.\nThis result goes back to F.~John \\cite{J}, \nalthough other proofs were found recently by \nK.~Ball \\cite{B2} and A.~Giannopoulos and V.~Milman \\cite{G-M}.\nTherefore, working with contact points instead of decompositions\n${\\it id}  =  \\sum x_j \\otimes x_j$ we do not lose generality.\n\nRecasting Theorem \\ref{main} in this light, we have\n\n\\begin{corollary} \n  Let $X$ be an $n$-dimensional Banach space whose maximal volume ellipsoid\n  is the unit euclidean ball. \n  Let $T$ be a linear operator with $\\|T\\|_{2 \\rightarrow 2}  \\le  1$.\n  Then for any $\\varepsilon > 0$ there are contact points $x_1, \\ldots, x_k$\n  with $k  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$\n  such that the system $(Tx_j)_{j \\le k}$ is $C(\\varepsilon)$-equivalent \n  in $l_2$-norm to the canonical basis of $l_2^k$.\n\\end{corollary}\n\n\\noindent Moreover, the norms $\\|T x_j\\|_2$ are well bounded below:\n$$\n\\|T x_j\\|  \\ge  C_1(\\varepsilon) \\frac{\\|T\\|_{\\rm HS}}{\\sqrt{n}}\n\\ \\ \\ \\ \\mbox{for all $j \\le k$.}\n$$\n\n$T$ being an orthogonal projection, this result leads us to \na new Dvoretzky-Rogers type lemma , which we will discuss now. \nSuppose $X$ is an $n$-dimensional Banach space whose John's \nellipsoid is the unit euclidean ball. \nLet $Z$ be a $k$-dimensional subspace of $X$.\nDvoretzky-Rogers Lemma states that, given a positive integer\n$\\kappa < k$,\nthere is an orthonormal system $(z_j)_{j  \\le  \\kappa}$ in $Z$ such that\n$$\n\\|z_j\\|_X  \\ge  \\sqrt{\\frac{k - \\kappa + 1}{n}}\n  \\ \\ \\ \\ \\mbox{for all $j  \\le  \\kappa$}.\n$$\nLet us sketch the proof. By induction, it is enough to find {\\em one}\nvector $z$ in $Z$ such that $\\|z\\|_2 = 1$ and \n$\\|z\\|_X  \\ge  \\sqrt{\\frac{k}{n}}$\n(then substitute $Z$ by $Z  \\ominus  {\\rm span}(z)$ and repeat the argument).\nBy duality, this is equivalent to finding a functional \n$x^*  \\in  B(X^*)$ with $\\|P x^*\\|_2  \\ge  \\sqrt{\\frac{k}{n}}$,\nwhere $P$ is the orthogonal projection onto $Z$.\nLet ${\\it id}_{X^*}  =  \\sum \\lambda_j x_j^* \\otimes x_j^*$\nbe a John decomposition in $X^*$, that is $\\sum \\lambda_j  =  n$\nand $x_j^*$ are contact points of $B(X^*)$.\nThen $P  =  \\sum \\lambda_j P x_j^* \\otimes P x_j^*$.\nTaking the trace, we get $k  =  \\sum \\lambda_j \\|P x_j^*\\|_2^2$.\nSo, there is a $j$ such that $\\|P x_j^*\\|_2^2  \\ge  \\sqrt{\\frac{k}{n}}$.\nThis completes the proof.\n\nHowever, this argument, as well as other known proofs of the \nDvoretzky-Rogers Lemma, only establish the existence of the vectors $z_j$.\nIn contrast to that, the argument based on Theorem \\ref{main}\nprovides information about their position.\n\n\\begin{theorem}                                                           \\label{DR}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $P$ be an orthogonal projection, ${\\rm rank} P = k$.\n  Then for any positive integer $\\kappa  <  k$\n  there are contact points $x_1, \\ldots, x_\\kappa$\n  such that setting $z_j  =  P x_j \/ \\|P x_j\\|_2$\n  we have\n  \n  (i)  the system $(z_j)$ is $C(\\kappa \/ k)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}$\n       for all $j$.\n\\end{theorem}\n\n\n\\noindent {\\bf Proof.}\\ \\ \nNote that $\\|P\\|_{\\rm HS}  =  \\sqrt{k}$ and apply Theorem \\ref{mainc}\nto a John decomposition on $X$. \nThis gives us contact points $x_1, \\ldots, x_\\kappa$ such that\n(i) is satisfied, and\n$$\n\\|P x_j\\|_2  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nIt remains to note that for every $j \\le m$\n$$\n\\|P x_j\\|_X  \\ge  \\frac{\\< P x_j, x_j \\> }{\\|x_j\\|_{X^*}}\n  =  \\< P x_j, x_j \\>  =  \\|P x_j\\|_2^2,\n$$\nhence\n$$\n\\|z_j\\|_X  \\ge  \\|P x_j\\|_2\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nA natural question is whether Theorem \\ref{DR} \ncan be extended for arbitrary operator $T$, \n$\\|T\\|_{2 \\rightarrow 2} = 1$, \nwith $k$ substituted by $\\|T\\|_{\\rm HS}^2$. \nThe answer is negative. \nIndeed, let $n  =  2^m$ for a positive integer $m$,\nand denote by $W_m$ the Walsh $n \\times n$ matrix.\nConsider the operator $T  =  n^{-1\/2} W_m$\nacting in the space $X  =  l_\\infty^n$.\nAll contact points $(x_j)$ of $X$ are simply the\ncoordinate vectors. However, denoting $z_j = T x_j \/ \\|T x_j\\|_2$\nwe have for any $j$\n$$\n\\|z_j\\|_X  =  \\|T x_j\\|_X \/ \\|T x_j\\|_2  =  n^{-1\/2}.\n$$\nThis shows that (ii) in Theorem \\ref{DR} would fail \nfor the operator $T$.\n\nStill, a lower bound for $\\| \\cdot \\|_X$-norm exists\nand is equivalent to $\\frac{1}{n} |{\\rm trace}{T}|$.\n\n\\begin{proposition}\n  Let $X$ be an $n$-dimensional Banach space\n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $T$ be an operator with $\\|T\\|_{2 \\rightarrow 2} \\le 1$.\n  Then for any $\\varepsilon > 0$\n  there are $k  >  (1 - \\varepsilon) n$ contact points $x_1, \\ldots, x_\\kappa$\n  such that \n  \n  (i)  the system $(T x_j)$ is $C(\\varepsilon)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|T x_j\\|_X  \\ge  c \\varepsilon \\frac{|{\\rm trace} T|}{n}$.\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\ \nThe argument is similar to that of Theorem \\ref{DR};  \none only uses Proposition \\ref{mainscalar}\ninstead of Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nThe estimate in (ii) is essentially sharp. \nIndeed, for any positive integers $k \\le n$\none can construct an orthogonal projection $P$\nin ${\\bf R}^n$ with ${\\rm rank} P = k$, and such that\n$$\n\\|P e_j\\|_2  =  \\sqrt{\\frac{k}{n}}\n\\ \\ \\ \\ \\mbox{for all $j \\le n$}.\n$$\nNotice that \n$\\|P\\|_{1 \\rightarrow 2}       =  \\sqrt{\\frac{k}{n}}$, \ntherefore \n$\\|P\\|_{2 \\rightarrow \\infty}  =  \\sqrt{\\frac{k}{n}}$.\nThus \n$\\|P\\|_{1 \\rightarrow \\infty}  \\le  \\frac{k}{n}$.\nNow consider the space $X  =  l_\\infty^n$\nand the operator $T = P$ on it.\nThe only contact points $(x_j)$ of $X$ are the coordinate \nvectors $(e_j)$. Then for all $j$\n$$\n\\|T x_j\\|_X  =  \\|P e_j\\|_\\infty  \n  \\le  \\frac{k}{n}  \n   =  \\frac{|{\\rm trace} T|}{n}.\n$$\nThis shows that the lower bound in (ii) is essentially sharp.\n\nFinally, there is a class of operators for which \nTheorem \\ref{DR} itself can be extended: selfadjoint operators.\nSo, the desired general result for arbitrary operator can be obtained\nif we allow a suitable rotation of $(z_j)$.\nThere is always a unitary operator (coming from the polar\ndecomposition of $T$) which sends the vectors $(z_j)$\nto vectors of a good $\\| \\cdot \\|_X$-norm.\n\n\\begin{theorem}                                                    \n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $T$ be an operator with $\\|T\\|_{2 \\rightarrow 2} \\le 1$, \n  and put $\\|T\\|_{\\rm HS}^2  =  h$.\n  Then for any positive integer $\\kappa  <  h$\n  there are contact points $x_1, \\ldots, x_\\kappa$\n  such that setting $z_j  =  |T| x_j \/ \\|T x_j\\|_2$\n  we have\n  \n  (i)  the system $(z_j)$ is $C(\\kappa \/ h)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}}$\n       for all $j$.\n\\end{theorem}\n\n\\noindent {\\bf Proof.}\\ \\ \nLet $T = U |T|$ be the polar decomposition of $T$, \nwhere $|T| = (T^* T)^{1\/2}$ is a positive selfadjoint operator\nand $U$ is a partial isometry on $l_2^n$.\nApply Theorem \\ref{mainc} to the operator $|T|$\nand a John decomposition on $X$. \nAs before, this gives us contact points $x_1, \\ldots, x_\\kappa$ \nsuch that (i) is satisfied, and\n$$\n\\Big\\| |T| x_j \\Big\\|_2  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}}\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nFrom diagonalization of $|T|$ it follows that\n$\\Big\\| |T|^{1\/2} \\Big\\|_{2 \\rightarrow 2}  =  \\|T\\|_{2 \\rightarrow 2}^{1\/2}  \\le  1$.  \nTherefore we can bound for every $j \\le m$\n\\begin{eqnarray*}\n\\Big\\| |T| x_j \\Big\\|_X  \n &\\ge&  \\frac{\\< |T| x_j, x_j \\> }{\\|x_j\\|_{X^*}}\n   =    \\Big\\| |T|^{1\/2} x_j \\Big\\|_2^2                \\\\\n &\\ge&  \\Big\\| |T| x_j \\Big\\|_2^2\n   =    \\Big\\| T x_j \\Big\\|_2^2.\n\\end{eqnarray*}\nHence\n$$\n\\|z_j\\|_X  \\ge  \\|T x_j\\|_2\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\nNow we turn to a particular case when $T$ is the identity operator, \nwhich also happens to be interesting. We clearly have \n\n\\begin{corollary}                                            \\label{contact}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Then for any $\\varepsilon > 0$ there is a set of $k  >  (1 - \\varepsilon) n$\n  contact points which is $C(\\varepsilon)$-equivalent to \n  the canonical vector basis of $l_2^k$.\n\\end{corollary}\n\n\\noindent This result is related to another variant of the \nclassical Dvoretzky-Rogers Lemma, which establishes the existence\nof contact points whose distance to a certain orthonormal basis\nis controlled (\\cite{T-J}, Theorem 15.7).\nMore precisely, there exist contact points $x_1, \\ldots, x_n$\nand an orthonormal basis $h_1, \\ldots, h_n$ such that\n$$\n\\|x_j - h_j\\|_2  \\le  2 \\Big(1 - \\sqrt{\\frac{n - i + 1}{n}} \\Big)\n\\ \\ \\ \\ \\mbox{for $i  \\le  n$}.\n$$\nHowever, this estimate is to crude to assure that a fixed \nproportion of the system $(x_j)_{j \\le n}$ is equivalent \nin $l_2$-norm to an orthonormal system. Such an isomorphism \ncan be established only for $c \\sqrt{n}$ contact points. \n\nUsing this argument, it is proved that for $k = [\\sqrt{n} \/ 4]$\nthere exist orthonormal vectors \n$x_1, \\ldots, x_k$ in $(E, \\|\\cdot\\|_2)$ on which all three norms \n$\\|\\cdot\\|$, $\\|\\cdot\\|_2$, and $\\|\\cdot\\|_*$ \ndiffer by the factor $2$ at most (\\cite{T-J}, p.127). \nIt has been an open problem to make $k$ proportional to $n$.\nBy Corollary \\ref{contact} we actually have $k  \\ge  (1 - \\varepsilon)n$\nand make all three norms $\\|\\cdot\\|$, $\\|\\cdot\\|_2$, and $\\|\\cdot\\|_*$\nequal to $1$ on our sequence (paying however in exact orthogonality).\n\nBy duality, Corollary \\ref{contact} holds also for the ellipsoid of minimal \nvolume containing $B(X)$ instead of the maximal volume ellipsoid. \nThis variant of Corollary \\ref{contact} yields also a proportional \nDvoretzky-Rogers factorization result from \\cite{B-Sz}. \nNamely, given an $n$-dimensional Banach space $X$ and $\\varepsilon > 0$, \nthere is a $k > (1 - \\varepsilon) n$ such that the formal identity \n${\\it id} : l_2^k \\rightarrow l_\\infty^k$\ncan be written as ${\\it id} = \\alpha \\cdot \\beta$ for some \n$\\beta : l_2^k \\rightarrow X$, $\\alpha : X \\rightarrow l_\\infty^k$, \nwith $\\|\\alpha\\| \\|\\beta\\|  \\le  C(\\varepsilon)$.\nThis can be obtained by duality from the factorization of the identity\non the contact points, ${\\it id} : l_1^k  \\rightarrow  X  \\rightarrow  l_2^k$, \nguaranteed by Corollary \\ref{contact}.\n\nA few comments about the dependence $C(\\varepsilon)$ in Theorem \\ref{main}.\nIt is a challenge to find the correct asympthotics. \nIndeed, by an argument of S.~Szarek and M.~Talagrand \\cite{Sz-T}\nthe proportional Dvoretzky-Rogers factorization above\nyields a non-trivial estimate on the distance from $X$ to $l_\\infty^n$ --\na well known and hard problem in the local theory of normed spaces.\nThe factorization constant $C(\\varepsilon)$ lies in the heart of the computation\nof this distance.\n\nThe proof of Theorem \\ref{main} guarantees that\n$C(\\varepsilon)  \\le  \\varepsilon^{c \\log \\varepsilon}$.\nHowever, for the Dvoretzky-Rogers factorization constant \n$C_{\\rm DR} (\\varepsilon)$ much better bounds are found \\cite{G}: \n$C_{\\rm DR} (\\varepsilon)  \\le  c \\varepsilon^{-1}$. \nSince $C_{\\rm DR} (\\varepsilon)  \\le  C(\\varepsilon)$ and \n$C_{\\rm DR} (\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon \\rightarrow 0$  (see \\cite{Sz-T}), \nwe necessarily have $C(\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon \\rightarrow 0$.\n\nHowever, $C(\\varepsilon)  \\rightarrow  1$ as $\\varepsilon \\rightarrow 1$.\nThis follows directly from \n\n\\begin{lemma}\n  Let a normalized sequence $(x_j)_{j \\le n}$ in Hilbert space \n  be $M$-Hilbertian, and $\\varepsilon > 0$.\n  Then there is a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$\n  of cardinality $|\\sigma|  \\ge  C(M, \\varepsilon) n$ such that\n  the system $(x_j)_{j \\in \\sigma}$ is $(1 + \\varepsilon)$-equivalent\n  to the canonical basis of $l_2^\\sigma$.\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nWe can assume that the given Hilbert space is $l_2^n$.\nDefine a linear operator $T : l_2^n  \\rightarrow  l_2^n$ by \n$$\nT e_j  =  x_j  \n\\ \\ \\ \\ \\mbox{for $j \\le n$.}\n$$\nLet $A = T^*T - {\\it id}$.\nThen the matrix of $A$ has zeros on the diagonal and \n$\\|A\\|  \\le  M^2 + 1$.\nNow, by a theorem of J.~Bourgain and L.~Tzafriri\n(\\cite{B-Tz} Theorem 1.6, see also \\cite{K-Tz})\nthere is a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \nof cardinality $|\\sigma|  \\ge  C(M, \\varepsilon)$ such that\n$$\n\\| P_\\sigma A P_\\sigma \\|  <  \\varepsilon.\n$$\nThis shows that for any sequence of scalars $(a_j)_{j \\le n}$\n$$\n\\Big| \\Big\\langle\n(T^*T - {\\it id}) \\sum_{j \\in \\sigma} a_j e_j , \\sum_{j \\in \\sigma} a_j e_j\n\\Big\\rangle \\Big|\n<  \\varepsilon,\n$$\nhence\n$$\n\\Big| \\Big\\langle\n\\sum_{j \\in \\sigma} a_j x_j , \\sum_{j \\in \\sigma} a_j x_j\n\\Big\\rangle \n  -  \\sum_{j \\in s} |a_j|^2\n\\Big|\n<  \\varepsilon.\n$$\nThis clearly finishes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\n\n\n\n\\section{Embeddings of the cube}                                     \\label{SecCube}\n\nIn this section we apply Theorem \\ref{main} to the study \nof embeddings of $l_\\infty^k$ into finite dimensional spaces. \nN.~Alon and V.~Milman proved that if a given normalized sequence $(x_j)$ \nin a Banach space $X$ has small Rademacher average ${\\bf E} \\| \\sum \\varepsilon_j x_j\\|_X$, \nthen it must contain a large subsequence well equivalent \nto the canonical basis of $l_\\infty^k$.\nLater on, M.~Talagrand \\cite{T} improved this result \nand simplified the argument.\n\n\\begin{theorem} (M.~Talagrand).  \\                                   \\label{Talagrand}\n  Suppose we are given vectors $(x_j)_{j \\le n}$ in a Banach space $X$\n  with $\\|x_j\\|_X  \\ge  1$.\n  Set $M = {\\bf E} \\| \\sum \\varepsilon_j x_j \\|_X$ \n  and $\\omega = \\sup \\Big\\{ \\sum |x^*(x_j)| : x^* \\in B(X^*) \\Big\\}$.\n  Then there exists a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \n  of cardinality $|\\sigma|  \\ge  c n \/ \\omega$ such that\n  $$\n  \\frac{1}{2} \\max_{j \\in \\sigma} |a_j|\n    \\le  \\Big\\| \\sum_{j \\in \\sigma} a_j x_j \\Big\\|_X\n    \\le   4 M \\max_{j \\in \\sigma} |a_j|\n  $$  \n  for any choice of scalars $(a_j)$.\n\\end{theorem}\n\nA few years earlier, M.~Rudelson obtained in \\cite{R1} \na \"Gaussian\" version of this theorem.\nRecall that the $\\ell$-norm of an operator $u : l_2^n  \\rightarrow  X$\nis defined as \n$$\n\\ell(u)^2  =  \\int \\|ux\\|_X^2 \\, d \\gamma_n(x),\n$$\nwhere $\\gamma_n$ is the canonical Gaussian measure on ${\\bf R}^n$.\nWe will sometimes write $\\ell(X)$ instead of $\\ell({\\it id}_X)$.\n\nSuppose $X$ is an $n$-dimensional Banach space whose maximal volume ellipsoid\nis the unit euclidean ball.\nLet $P$ be an orthogonal projection in $X$ \nand set $k  =  {\\rm rank} P$, $a  =  k \/ n$.\nThe result of M.~Rudelson states that there is a subspace \n$Z  \\subset  P(X)$  of dimension $m  \\ge  C_1(a) \\frac{\\sqrt{n}}{\\ell(P)}$\nwhich is $C_2(a) \\ell(P)$-isomorphic to $l_\\infty^m$.\n\nWe will remove $\\ell(P)$ from the estimate on the dimension.\nFurther, it will be shown that $Z$ is canonically spanned by the projections\nof some contact points of $X$.\nIn particular, the norm on $Z$ is well equivalent to \n$\\max_{j \\le m} | \\< z , x_j \\> |$,\nwhere $x_j$ are contact points.\nThis yields automatically that $Z$ is well complemented \nby the orthogonal projection. \nMoreover, the dependence on $a$ will be improved.\n\n\\begin{theorem}                                                    \\label{GT}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is $B(l_2^n)$.\n  Let $P$ be an orthogonal projection, ${\\rm rank} P = k$.\n  Then there are contact points $(x_j)_{j  \\le  m}$\n  with $m  \\ge  c_1 k \/ \\sqrt{n}$ such that \n  $$\n  \\max_{j \\le m} | \\< x , x_j \\> |    \\le   \\|x\\|_X    \n  \\le   c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< x , x_j \\> |\n  $$\n  for every $x$ in $Z  =  {\\rm span}(P x_j)_{j \\le m}$.\n\\end{theorem}\n\nA particular case $k = n$ is also interesting: \nwe get a sequence of $m  \\ge  c_1 \\sqrt{n}$ contact points \nwhich is $c \\ell(X)$-equivalent to the canonical vector basis \nof $l_\\infty^m$.\nLet us prove this latter fact separately. \nBy Corollary \\ref{contact}, there exists a set of contact points \n$(x_j)_{j \\le m'}$, $m'  \\ge  n \/ 2$, \nwhich is $c$-equivalent in $l_2$-norm \nto the canonical basis of $l_2^{m'}$.\nLet $(g_j)$ be independent standard Gaussian random variables.\nTo apply Talagrand's Theorem \\ref{Talagrand}, we bound\n$$\nM   =   {\\bf E} \\| \\sum \\varepsilon_j x_j \\|_X\n   \\le  c {\\bf E} \\| \\sum g_j x_j \\|_X\n   \\le  c c_2 {\\bf E} \\| \\sum g_j e_j \\|_X\n$$\nby the ideal property of the $\\ell$-norm, see Lemma \\ref{Slepian} below. \nFurther, for any finite system of scalars $(a_j)$\n\\begin{eqnarray*}\n\\| \\sum a_j x_j \\|_X   \n  &\\le&   \\| \\sum a_j x_j \\|_2                     \\\\\n  &\\le& c_2 \\Big( \\sum |a_j|^2 \\Big)^{1\/2}          \n   \\le  c_2 \\sqrt{n} \\max_j |a_j|.\n\\end{eqnarray*}\nThus\n$$\n\\omega  =  \\sup \\Big\\{ \\sum |x^*(x_j)| : x^* \\in B(X^*) \\Big\\}\n   \\le c_2 \\sqrt{n}.\n$$\nNow Talagrand's Theorem \\ref{Talagrand} finishes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nThe proof of Theorem \\ref{GT} is longer, but the main idea remains \nto combine results of Section \\ref{SecDR} with Talagrand's theorem. \nFirst, we need to know what vectors canonically span a large\nsubspace $Z$ well isomorphic to $l_\\infty^m$.\nThey happen to be multiples of some contact points $(x_j)_{j \\le m}$.\nMore precisely, we have\n\n\\begin{proposition}                                                           \\label{spancube}\n  Under the assumptions of Theorem \\ref{GT}, the system \n  $\\Big( \\frac{Px_j}{\\|Px_j\\|_2} \\Big)_{j \\le m}$\n  is $c \\sqrt{\\frac{n}{k}} \\ell(P)$-equivalent \n  to the canonical basis of $l_\\infty^m$.\n\\end{proposition}\n\nTo prove this, let $(x_j)_{j \\le k'}$, $k'  \\ge  k \/ 2$,\nbe the contact points provided by Theorem \\ref{DR}.\nPut \n$$\nz_j  =  \\alpha \\sqrt{\\frac{n}{k}} \\frac{Px_j}{\\|Px_j\\|_2}\n$$\nfor an appropriate $\\alpha > 0$ and all $j$.\nThen \n\n(i)  the system $(z_j)_{j \\le k'}$ is $(c \\sqrt{\\frac{n}{k}})$-equivalent\n     in $l_2$-norm to the canonical basis of $l_2^{k'}$;\n\n(ii) $\\|z_j\\|  \\ge  1$ for all $j$.\n\n\\noindent For a future reference note that the proof of \nTheorem \\ref{DR} gives also \n\\begin{equation}                                                                 \\label{Pxj}\n  \\|Px_j\\|_2  \\ge  c_1 \\sqrt{\\frac{n}{k}}.\n\\end{equation}\n\nTo apply Talagrand's Theorem \\ref{Talagrand} to the system $(z_j)_{j \\le k'}$,\nrecall the ideal property of the $\\ell$-norm (cf. \\cite{T-J}, \\S 12)\n\n\\begin{lemma}                                                                   \\label{Slepian}\n  For any two linear operators \n  $A : l_2^n  \\rightarrow  X$ and $B : l_2^n  \\rightarrow  l_2^n$\n  $$\n  \\ell(AB)  \\le  \\|B\\| \\ell(A).\n  $$\n\\end{lemma}\nLet $(h_j)_{j \\le k'}$ be an orthonormal basis in ${\\rm span}(z_j)$.\nNow we bound \n\\begin{eqnarray*}\nM  & = &    {\\bf E} \\| \\sum \\varepsilon_j z_j \\|_X  \n    \\le   c {\\bf E} \\| \\sum  g_j z_j \\|_X                          \\\\\n   &\\le&  c \\sqrt{\\frac{n}{k}} {\\bf E} \\| \\sum g_j h_j \\|_X         \n          \\ \\ \\ \\ \\mbox{by (i) and Lemma \\ref{Slepian}}       \\\\\n   & = &  c \\sqrt{\\frac{n}{k}} \\ell(P(X))\n     =    c \\sqrt{\\frac{n}{k}} \\ell(P).\n\\end{eqnarray*}\nNoting (ii) above, we apply Talagrand's Theorem \\ref{Talagrand}.\nThis finishes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nTo obtain Theorem \\ref{GT}, Talagrand's Theorem will be used more delicately.\nIts proof in \\cite{T} gives the following additional property.\n\n\\begin{lemma}                                                                 \\label{Talagrandplus}\n  In the situation of Theorem \\ref{GT}, suppose $(x_j^*)_{j \\le n}$\n  are functionals in $X^*$ such that \n  $$\n  x_j^* (x_j)  \\ge  1  \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \n  \\|x_j^*\\|_{X^*}  =  1      \\ \\ \\ \\ \\mbox{for all $j \\le n$.}\n  $$\n  Then the subset $\\sigma$ can be found so that \n  $$\n  \\sum_{j  \\in  \\sigma \\setminus \\{i\\}} |x_i^* (x_j)|  \\le  1\/2\n  \\ \\ \\ \\ \\mbox{for all $i \\in \\sigma$}.\n  $$\n\\end{lemma}\n\n\nLet us turn again to the proof of Proposition \\ref{spancube}.\nWe applied Talagrand's Theorem to the system $(x_j)_{j \\le k'}$.\nNote that by (\\ref{Pxj})\n\\begin{eqnarray*}\n\\< x_j, z_j \\>  & = &  \\alpha \\sqrt{\\frac{n}{k}} \\frac{\\< x_j, Px_j \\> }{\\|Px_j\\|_2}  \\\\\n  & = &   \\alpha \\sqrt{\\frac{n}{k}} \\|Px_j\\|_2\n   \\ge  1   \\ \\ \\ \\ \\mbox{for all $j$.}\n\\end{eqnarray*}\nTherefore we obtain a subset $\\sigma  \\subset \\{1, \\ldots, k'\\}$\nof cardinality $|\\sigma|  \\ge  c_1 k \/ \\sqrt{n}$ such that\n\\begin{itemize}\n  \\item{the system $(z_j)_{j \\in \\sigma}$ \n        is $c \\sqrt{\\frac{n}{k}} \\ell(P)$-equivalent to the \n        canonical basis of $l_\\infty^\\sigma$;}\n  \\item{$\\sum_{j  \\in  \\sigma \\setminus \\{i\\}} |\\< x_i , z_j\\> |  \\le  1\/2$\n        for all $j \\in \\sigma$.} \n\\end{itemize}\nNow fix an $x  =  \\sum_{j \\in \\sigma} a_j z_j$,\nand let $i \\in \\sigma$ be such that $|a_i|  =  \\max_{j \\in \\sigma} |a_j|$.\nThen\n\\begin{eqnarray*}\n\\max_{j \\in \\sigma} |\\< x, x_j \\> |  \n  &\\ge&  |\\< x_i, x \\> |\n    =    \\Big| \\sum_{j \\in \\sigma} a_j \\< x_i, z_j \\> \\Big|          \\\\\n  &\\ge&  |a_j| \\Big( |\\< x_i, z_i \\> |  \n           -  \\sum_{j  \\in  \\sigma \\setminus \\{i\\}} | \\< x_i, z_j \\> | \\Big) \\\\\n  &\\ge&  \\frac{1}{2} |a_i|\n    =    \\frac{1}{2} \\max_{j \\in \\sigma} |a_j|                    \\\\\n  &\\ge&  \\Big( c \\sqrt{\\frac{n}{k}} \\ell(P) \\Big)^{-1} \\|x\\|_X.\n\\end{eqnarray*}\nThis proves the second inequality in Theorem \\ref{GT}, \nwhile the first one is obvious.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nTheorem \\ref{GT} yields also that $Z$ is well complemented by \nthe orthogonal projection. \n\n\\begin{proposition}\n  In the situation of Theorem \\ref{GT}, \n  let $P_Z$ be the orthogonal projection in $X$ onto $Z$. Then \n  $$\n  \\|P_Z\\|  \\le  c \\sqrt{\\frac{n}{k}} \\ell(P).\n  $$\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\  \nFor any $x \\in X$\n\\begin{eqnarray*}\n\\|P_Z x\\|_X  \n  &\\le&  c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< P_Z x, x_j \\> | \\\\\n  & = &  c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< x, x_j \\> |     \\\\\n  &\\le&  c \\sqrt{\\frac{n}{k}} \\ell(P) \\|x\\|_X.\n\\end{eqnarray*}\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nAnother consequence of Theorem \\ref{GT} is a refined isomorphic \ncharacterization of spaces with large volume ratio.\nRecall that the volume ratio of an $n$-dimensional Banach space $X$\nis defined as\n$$\n{\\rm vr}(X)  =  \\min \\left( \\frac{{\\rm Vol}(B_X)}{{\\rm Vol}(E)} \\right)^{1\/n}\n$$\nover all ellipsoids $E$ contained in $B_X$, see \\cite{L-M}, \\cite{P}.\nThe maximal value of ${\\rm vr}(X)$ among all $n$-dimensional spaces \nis of order $\\sqrt{n}$, \nand the only space with maximal volume ratio is $l_\\infty^n$\n(K.~Ball \\cite{B1}).\nLater on, M.~Rudelson proved in \\cite{R1} that\nif ${\\rm vr}(X)$ is proportional to the maximal volume ratio \nthen $X$ has a subspace isomorphic to $l_\\infty^m$ with \nthe isomorphism constant of order $\\log n$, \nand such that \n$m  \\sim  \\frac{\\sqrt{n}}{\\log n}$.\nUsing Theorem \\ref{GT} in the M.~Rudelson's proof of this\nresult removes the parasitic factor $\\log n$ from the estimate \non the dimension. \n\n\\begin{theorem}\n  Let $a > 0$, and $X$ be an $n$-dimensional Banach space. \n  If ${\\rm vr}(X)  \\ge  a \\sqrt{n}$ \n  then there exists a subspace $Z$ of dimension \n  $m  \\ge  C_1(a) \\sqrt{n}$ which is $C_2(a) \\log n$-isomorphic \n  to $l_\\infty^m$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\section{Frames}                                                     \\label{SecFrames}\n\nThe notion of frame goes back to the work of R.~Duffin and A.~Schaeffer\non nonharmonic Fourier series \\cite{D-S}.\nA sequence $(x_j)$ in a Hilbert space $H$ is called a {\\em frame} \nif there exist positive numbers $A$ and $B$ such that\n$$\nA \\|x\\|^2  \\le  \\sum_j | \\langle x, x_j \\rangle |^2  \\le  B \\|x\\|^2\n\\ \\ \\ \\ \\mbox{for $x \\in H$.}\n$$\nThe number $(B\/A)^{1\/2}$ is called a {\\em constant} of the frame.\nWe call $(x_j)$ a {\\em tight frame} if $A = B = 1$.\nFor introduction to the theory of frames, its relation to wavelets and signal \nprocessing, see \\cite{B-W}.\nGeometric structure of frames is studied extensively in recent years, see\n\\cite{Ho}, \\cite{A}, \\cite{C-C1}, \\cite{C-C2}, \\cite{C2}, \\cite{V}.\n\nIt is known by now that finite dimensional frames are essentially the \nsame object as John decompositions. \nIn the equivalent theory, it is sufficient to work only with tight frames,\nbecause every frame with constant $M$ is $M$-equivalent to a tight frame\n(cf. e.g. \\cite{Ho}).\nFurther, one has the following equivalence between frames and \nJohn decompositions:\n$$\n\\mbox{ $(x_j)$ is a tight frame in $H$ }   \n  \\iff   {\\it id}_H  =  \\sum_j x_j \\otimes x_j.\n$$\nThis observation allows to interpret the results of \nSections \\S \\ref{SecMain} and \\S \\ref{SecInvert} as statements about frames.\nTheorem \\ref{main} yields:\n\n\\begin{corollary}                                                    \\label{frame}\n  Let $(x_j)$ be a tight frame in Hilbert space $H$,\n  and $T$ be a norm-one linear operator in $H$. \n  Then for any $\\varepsilon > 0$ there exists a subset of indices $\\sigma$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  the system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\n  to an orthogonal basis in Hilbert space.\n\\end{corollary}\nFor operators $T$ which are not Hilbert-Schmidt this means \nthat the subset $\\sigma$ is infinite.\n\nClearly, Corollary \\ref{frame} itself generalizes the invertibility\nprinciple of J.~Bourgain and L.~Tzafriri.\nWhen applied to the identity operator, it yields that every tight \nframe in $l_2^n$ has a subset of length $(1 - \\varepsilon) n$ \nwhich is $C(\\varepsilon)$-equivalent to an orthogonal basis in Hilbert space.\nThis result was proved in \\cite{V} as a generalization of \nP.~Casazza's theorem \\cite{C2}.\n\nNotice that one necessarily has $C(\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon  \\rightarrow  1$,\nas explained in Section \\S \\ref{SecDR}.\nAn infinite dimensional analog of this phenomenon holds, too.\nA frame may not in general contain a complete subsequence equivalent\nto an orthogonal basis. The counterexample was found \nby P.~Casazza and O.~Christensen in \\cite{C-C2}, see also \\cite{V}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere is now substantial evidence that environments of intermediate density,\nsuch as galaxy groups, play an important role in the transformation of field\ngalaxies into the passively evolving populations found in galaxy clusters.\nSeveral studies have found that group populations already exhibit reduced\nstar formation rates (SFR; Lewis et al.~2002; Gomez et al.~2003) and high\nearly-type fractions similar to those observed in denser environments\n(Zabludoff \\& Mulchaey 1998; Jeltema et al.~2007).  While the physical\nmechanisms responsible for the preprocessing of galaxies in the group regime\nare still heavily debated, several recent studies have reported an\noverdensity of X-ray luminous Active Galactic Nuclei (AGN) on the outskirts\nof clusters and within the substructure surrounding unrelaxed systems (D'Elia\net al.~2004; Cappelluti et al. 2005; Kocevski et al.~2009a,2009b; Gilmour et\nal.~2009). These observations suggest that increased nuclear activity may be\ntriggered in such environments and that AGN-driven outflows may play a role\nin suppressing star formation within galaxies during cluster assembly.\nIndeed the increased dynamical friction within groups and their low relative\nvelocity dispersions make them conducive to galaxy interactions which can\ntrigger such activity (Hickson 1997; Canalizo \\& Stockton 2001) and recent\nhydrodynamical simulations suggest that merger-triggered AGN feedback can\nhave a profound effect on the gas content and star formation activity of\ntheir host galaxies (Hopkins et al.~2007; Somerville et al.~2008). \n\nSince the mass density of virialized structures increases with redshift,\nmergers are expected to play an even greater role in the group environment in\nthe past.  Therefore, if galaxy interactions and subsequent AGN feedback are\ndriving a significant portion of the preprocessing found in intermediate\ndensity environments, we may expect to find an overdensity of AGN in high\nredshift groups in the early stages of hierarchical cluster formation.  In\nthis Letter we report on \\emph{Chandra} observations of one such system, the\nCl0023+0423 (hereafter Cl0023) four-way group merger at $z=0.84$ (Lubin et\nal.~2009a).  The Cl0023 structure consists of four interacting galaxy groups\nwhich simulations suggest are the direct progenitors of a future massive\ncluster.  As such, the system provides a unique look at the level\nof processing and evolution already underway in the group environment prior\nto cluster assembly. \n\nTo search for evidence of triggered nuclear activity within the Cl0023\nstructure, we present the number counts of X-ray point sources detected in a\nfield covering the entire system, as well as a cross-correlation of these\nsources with our extensive spectroscopic database.  Surprisingly, we find\nno evidence for an overdensity of X-ray detected point sources in the\ndirection of the Cl0023 groups.  We discuss the implications of this finding\non the role of AGN feedback in regulating galaxy evolution in such\nstructures. We also examine possible explanations for the lack of increased\nnuclear activity in the system.  Throughout this Letter we assume a\n$\\Lambda$CDM cosmology with $\\Omega_{m} = 0.3$, $\\Omega_{\\Lambda} = 0.7$, and\n$H_{0} = 70$ $h_{70}$ km s$^{-1}$ Mpc$^{-1}$. \n\n\n\\begin{figure*}\n\\epsscale{1.1}\n\\plotone{.\/f1.astroph.eps}\n\\caption{(\\emph{left}) Adaptively smoothed density map of\n  color-selected red galaxies in the Cl0023 field.  Three density peaks that\n  correspond with four spectroscopically confirmed galaxy groups are marked.\n  Adapted from Lubin et al.~(2009a).  (\\emph{right}) Adaptively smoothed, ACIS-I image of the Cl0023 field\n  in the soft X-ray band (0.5-2 keV). \\label{fig-densmap}}\n\\end{figure*}\n\n\\vspace{0.5in}\n\\section{The Cl0023+0423 System}\n\nOriginally detected in the cluster survey of Gunn, Hoessel \\& Oke (1986),\nthe Cl0023 system consists of four galaxy groups separated by roughly 3000\nkm s$^{-1}$ in radial velocity (Lubin et al.~2009a).  Two of the constituent\ngroups have measured velocity dispersions of 428 and 497 km s$^{-1}$, while\nthe second, poorer pair, have dispersions of 206 and 293 km s$^{-1}$ (Lubin et\nal.~2009a).  $N$-body simulations suggest that the groups are likely bound and in\nthe process of forming a massive cluster within the next $\\sim1$ Gyr, which,\nbased on virial mass estimates of the individual groups, will have a final\nmass of $\\sim5\\times10^{14}$ M$_{\\odot}$ (Lubin, Postman \\& Oke 1998). \n\nDetails of our optical observations of the Cl0023 field are presented in\nLubin et al.~(2009a).  In short, the system was imaged with the Sloan Digital\nSky Survey (SDSS) $r'i'z'$ filters using the Large Format Camera (LFC; Simcoe\net al.~2000) on the Palomar 5-m telescope and follow-up spectroscopy carried\nout with the Deep Imaging Multi-object Spectrograph (DEIMOS; Faber et\nal.~2003) on the Keck 10-m telescopes.  Our spectroscopic observations\nyielded 423 extragalactic redshifts in the Cl0023 field and an additional 73\ngalaxy redshifts were incorporated from the spectroscopic survey of Oke,\nPostman \\& Lubin (1998). The combined catalog contains redshifts for 134\ngalaxies in the Cl0023 structure with $0.820<z<0.856$.   An adaptively smoothed\ndensity map of color-selected red galaxies in the Cl0023 field constructed\nfrom our ground-based optical imaging of the system is shown in Figure\n\\ref{fig-densmap}, while the redshift distribution of the system is shown in\nFigure \\ref{fig-zhist}.  Of the three density peaks visible in Figure\n\\ref{fig-densmap}, concentrations A and C consist of galaxies belonging to the\nredshift peaks at $z=0.839$ and $z=0.845$, respectively, while concentration\nB is a superposition of two groups along the line of sight at $z=0.828$ and\n$z=0.845$.  The galaxies within the highest redshift peak at $z\\sim0.864$ are\nnot centrally concentrated but rather extend across the entire region,\nsuggesting a sheet of galaxies in the near background.\n\n\n\\section{X-ray Observations} \n\\label{sect-data}\n\nObservations of the Cl0023 group system were carried out with {\\it Chandra's}\nAdvanced CCD Imaging Spectrometer (ACIS; Garmire et al.~2003) on 2007 August\n30 (obsID 7914).  The observation consists of a single 49.4 ks pointing of\nthe $16\\farcm9\\times16\\farcm9$ ACIS-I array, with the aimpoint located at\n$\\alpha_{2000} = 00^{\\rm h}23^{\\rm m}50.9^{\\rm s}$, $\\delta_{2000} =\n+04^{\\circ}22^{\\prime}55^{\\prime\\prime}$.  The dataset was reprocessed and\nanalyzed using standard CIAO 3.3 software tools\\footnote{Available through\n  the {\\it Chandra} X-ray Center at http:\/\/cxc.harvard.edu\/} in an identical\nmanner to the procedure detailed in Kocevski et al.~(2009a).  Images for use\nin object detection were created from the level 2 event lists with a 0.492\n$^{\\prime\\prime}$ pixel$^{-1}$ binning and corresponding spectrally-weighted\nexposure maps were constructed to account for vignetting.  An adaptively\nsmoothed, exposure corrected image of the Cl0023 field in the soft band is\nshown in Figure \\ref{fig-densmap}.\n\n\\begin{figure}\n\\epsscale{1.125}\n\\plotone{.\/f2.astroph.eps}\n\\caption{Redshift distribution in the Cl0023 field.  Shaded regions\n  correspond to galaxies within the Cl0023 structure.\\label{fig-zhist}}\n\\end{figure}\n\n\\begin{figure*}\n\\epsscale{1.125}\n\\plotone{.\/f3.astroph.eps}\n\\caption{Combined cumulative source number counts versus flux for the\n  Cl0023 group merger system in the soft (0.5-2 keV;\n  \\emph{left}) and hard band (2-10 keV; \\emph{right}).  The shaded region denotes a\n  $1\\sigma$ variance in the number counts.  Only sources detected above the\n  $3\\sigma$ level are included.  The results of an \\emph{XMM-Newton} survey of the COSMOS\n  field (Cappelluti et al.~2007) and that of a 130 ks \\emph{Chandra} observation\n  of the CDFN and CDFS are shown for comparison (solid, dashed and\n  dashed-dotted lines, respectively). The vertical dotted line represents the flux at which our\n  sky coverage dropped to 20\\% of the full ACIS-I field of view. \\label{fig-lognlogs}}\n\\end{figure*}\n\nWe searched for point sources using the wavelet-based {\\tt wavdetect}\nprocedure in CIAO, employing the standard $\\sqrt2$$^i$ series of wavelet\npixel scales, with $i=0-16$.   We adopted a minimum exposure threshold of\n20\\% relative to the exposure at the aimpoint of the observation and a\nthreshold significance for spurious detections of $10^{-6}$.  Object\ndetection was carried out on the unvignetting-corrected images and source\nproperties, including count rates and detection significances, were\ndetermined with follow-up aperture photometry on the vignetting-corrected\nimages.  A total of 151 sources were found by {\\tt  wavdetect}, of which 91\nhad detection significances greater than $3\\sigma$ in at least one of the\n0.5-2 keV (soft), 2-8 keV (hard) and 0.5-8 keV (full) bands. \n\nSource fluxes in the soft and hard bands were determined by normalizing a\npower-law spectral model to the net count rate measured for each source.\nWe assumed a photon index of $\\gamma = 1.4$ for the power-law\nmodel\\footnote{As this is the slope of the X-ray background (Tozzi et\n  al.~2001; Kushino et al.~2002)} and a\nGalactic neutral hydrogen column density of $2.66\\times10^{20}$ cm$^{-2}$\n(Dickey \\& Lockman 1990). Full-band fluxes were determined by summing the\nflux in the soft and hard bands.   Finally, rest frame X-ray luminosities\nwere calculated for sources matched to galaxies with measured redshifts (see\n\\S\\ref{sect-opt-matching}) using the luminosity distance equation and a\n$(1+z)^{\\gamma - 2}$ $k$-correction appropriate for a power-law spectrum:\n\n\\begin{equation}\n   L_{\\rm X} = 4\\pi d^{2}_{L} f_{\\rm X} (1+z)^{\\gamma - 2},\n\\end{equation}\n\n\\noindent where $d_{L}$ is the luminosity distance and $f_{\\rm X}$ is the 0.5-8 keV X-ray\nflux.\n\n\n\n\n\\section{Point Source Number Counts}\n\nTo calculate the cumulative number counts, $N(>S)$, of X-ray sources in\nthe field of the Cl0023 system, we employed the method described by Gioia et\nal.~(1990):\n\n\\begin{equation}\n   N(>S) = \\sum_{i=1}^{N} \\frac{1}{\\Omega_{i}} {\\rm deg}^{-2}.\n\\end{equation}\n\n\\noindent Here $N$ is the total number of detected point sources and\n$\\Omega_{i}$ is the sky area in square degrees sampled by the detector down\nto the flux of the $i$th source.  The variance of the number counts was in turn calculated as\n\n\\begin{equation}\n   \\sigma_{i}^{2} = \\sum_{i=1}^{N} \\left(\\frac{1}{\\Omega_{i}}\\right)^{2}.\n\\end{equation}\n\nIn order to determine $\\Omega_{i}$ we constructed a flux limit map using the\nmethod employed by Kocevski et al.~(2009a). First, all point sources detected\nby {\\tt wavdetect} were replaced with an estimate of the local background\nwith the CIAO tool {\\tt dmfilth} and the resulting images binned to a pixel\nscale of $32^{\\prime\\prime}$ pixel$^{-1}$ to produce a coarse background map.  This\nmap is used to determine the flux limit, $S_{\\rm lim}$, for a $3\\sigma$ point\nsource detection in any one pixel.  We can then calculate $\\Omega_{i}$ by\nsumming the sky area covered by all pixels with $S_{\\rm lim}$ equal to or\ngreater than the flux of the $i$th source.   \n\nThe resulting cumulative source number counts for the Cl0023 group system in\nthe 0.5-2 keV (left panel) and 2-10 keV (right panel) bands are shown in\nFigure \\ref{fig-lognlogs}.  The latter was chosen to ease comparison with\nprevious studies and obtained by extrapolating our 2-8 keV fluxes to 10 keV.\nAlso shown are the cumulative number counts measured in the COSMOS field\n(Scoville et al.~2007) and the {\\it Chandra} Deep Field South and North (CDFS\nand CDFN; Rosati et al.~2002; Brandt et al.~2001).   The COSMOS results are\nthose of Cappelluti et al.~(2007) converted to a spectral index of $\\gamma = 1.4$, while the CDFS and CDFN counts are the\nresults of our own reanalysis of single ACIS-I pointings in each\nfield\\footnote{Observation ID numbers 581 and 2232.}. \n\n\nIn both the soft and hard bands, we find that the number of sources detected\nin the Cl0023 field is statistically consistent ($<1\\sigma$ deviation) with\nthe source counts observed in the reference blank fields.  In the hard band,\nthis agreement extends over roughly the entire sampled flux range, while in\nthe soft band, we find an underdensity of bright sources relative to the\nCOSMOS field, in agreement with the underdensity of soft sources previously\nreported at these fluxes in the CDFS (Yang et al.~2003).  Our measured source\ncounts suggest that there is no increased nuclear activity in the Cl0023 system\ndetectable at X-ray wavelengths above our $3\\sigma$ flux limit, which amounts\nto $2.9\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ (0.5-8 keV).  At the median\nredshift of the Cl0023 system, this corresponds to a rest-frame 0.5-8 keV\nluminosity of $6.9\\times10^{42}$ $h_{70}^{-2}$ erg s$^{-1}$, making our\nobservations sensitive to moderate luminosity Seyferts and QSOs in the\ncomplex. \n\nTo parameterize the number counts, we fit the unbinned soft- and hard-band\ncounts in the Cl0023 field with a power-law model of the form $N(>S) =\nk(S\/S_{0})^{-\\alpha}$ using the maximum likelihood method of Murdoch et\nal.~(1973).  Our best-fit slopes, $\\alpha$, to the faint- and bright-end\nnumber counts in the soft band are $\\alpha_{\\rm faint} = 0.57\\pm0.12$ and\n$\\alpha_{\\rm bright} = 1.61\\pm0.30$, with a break in the distribution at\nroughly $S\\simeq3\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$.  In the hard band we\nonly fit to the bright-end counts as we do not sample the faint-end\npopulation sufficiently to obtain a separate fit.  Our best-fit slope above\n$S\\simeq8\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ is $\\alpha_{\\rm bright} =\n1.69\\pm0.31$.  These slopes are in good agreement (within the errors) with\nprevious studies of the \\emph{Chandra} Deep Fields, which found Euclidean\nslopes at the bright-end and $\\alpha_{\\rm faint} = 0.63\\pm0.13$,\n$0.67\\pm0.14$ in the soft band for the CDFS and CDFN, respectively (Brandt et\nal.~2001; Rosati et al.~2002).  \n\n\n\n\\section{Optical Source Matching }\n\\label{sect-opt-matching}\n\n\\begin{center}\n\\tabletypesize{\\scriptsize}\n\\begin{deluxetable}{cccrrr}\n\\tablewidth{0pt}\n\\tablecaption{Properties of X-ray Detected Galaxies in the Cl0023 Field with\n  Measured Redshifts\\label{tab-prop}}\n\\tablecolumns{6}\n\\tablehead{\\colhead{RA} & \\colhead{Dec} &\n           \\colhead{} &  \\colhead{Net} &  \\colhead{$F_{\\rm x}$$^{\\dagger}$} & \\colhead{$L_{\\rm x}$$^{\\ddagger}$}  \\\\  \n           \\colhead{(J2000)} &  \\colhead{(J2000)} &  \n           \\colhead{$z$} & \\colhead{Counts} & \\colhead{($\\times10^{-15}$)} & \\colhead{($\\times10^{42}$)}} \n\\startdata\n00:23:47.5   &  04:21:17.6  &  1.487  &  7.7  &  1.49  & 12.02  \\\\   \n00:23:43.3   &  04:18:06.3  &  0.169  &  6.4  &  2.94  &  0.21  \\\\    \n00:23:56.3   &  04:17:60.0  &  0.683  &  3.8  &  1.14  &  1.71  \\\\   \n00:23:55.2   &  04:25:20.2  &  1.091  & 13.5  &  4.77  & 19.86  \\\\   \n00:24:02.5   &  04:22:12.9  &  0.442  & 39.5  & 11.29  &  6.46  \\\\    \n00:23:51.1   &  04:27:19.8  &  0.113  & 19.7  &  5.34  &  0.17  \\\\   \n00:23:52.2   &  04:25:53.7  &  0.682  &  5.7  &  0.69  &  1.03  \\\\   \n00:23:48.9   &  04:21:23.7  &  0.745  & 43.0  &  9.53  & 17.23  \\\\   \n00:23:58.5   &  04:24:51.1  &  1.336  & 10.5  &  3.76  & 24.13  \\\\   \n\\vspace*{-0.075in}\n\\enddata\n\\tablecomments{All X-ray properties measured in the 0.5-8 keV band; $^{\\dagger}$ In units of erg\n  s$^{-1}$ cm$^{-2}$;  $^{\\ddagger}$ In units of $h_{70}^{-2}$ erg s$^{-1}$.} \n\\end{deluxetable}\n\\end{center}\n\\vspace{-0.2in}\n\nDespite the absence of a clear overdensity of X-ray sources in the Cl0023\nfield, we searched for AGN within the Cl0023 structure by matching our X-ray\nsource list to our preexisting spectroscopic catalog.  To perform this cross-correlation,\nwe determined the positional uncertainty associated with each X-ray source\nusing the empirical relationship of Kim et al.~(2007), who find that\ncentroiding errors increase exponentially with off-axis angle from the\naimpoint of the observation and decrease as the source counts increase with a\npower-law form.  To determine the reliability of a given match, we employed a\nmaximum likelihood technique described by Sutherland \\& Saunders (1992) and\nmore recently implemented by Kocevski et al.~(2009a).  The method gauges the\nlikelihood that a given optical object is matched to an X-ray source by\ncomparing the probability of finding a genuine counterpart with the\npositional offset and magnitude of the optical candidate relative to that of\nfinding a similar object by chance.  We refer the reader to Kocevski et\nal.~(2009a) for details.  \n\nUsing this technique we have matched a total of nine X-ray sources to\ngalaxies with measured redshifts in our spectroscopic catalog.  These\ngalaxies cover a broad range in redshift ($0.442 < z < 1.487$) and there is no\nevidence for a concentration near the redshift of the Cl0023 system.  In fact,\nwe find no X-ray point sources matched to the 134 galaxies spectroscopically\nassociated with the four groups in the Cl0023 structure.  \nHowever, we should note that we did not specifically target X-ray\nsources with our spectroscopic observations, but instead simply cross-correlated\ntheir positions with out existing spectroscopic database.  In future DEIMOS\nobservations of the system we plan to have dedicated masks for X-ray and\nradio detected AGN in order to both increase our spectroscopic completeness\nof X-ray sources and to determine if the lack of AGN currently observed in the Cl0023 groups holds.\nThe coordinates, redshifts, and X-ray properties of the nine galaxies\ncurrently matched to X-ray point sources in the Cl0023 field are listed in Table 1.  \n\n\n\n\n\\section{Discussion and Conclusions}\n\nUsing \\emph{Chandra} imaging of the Cl0023 complex, we have searched for\nevidence of triggered nuclear activity within a dynamically active system of\nfour galaxy groups in the early stages of cluster formation.   Both the\nredshift distribution and cumulative number counts of X-ray point sources in\nthe Cl0023 field reveal little evidence to suggest that the system contains X-ray\nluminous AGN in excess to what is observed in the field population.  \nThese results are at odds with previous reports of source excesses on the outskirts\nof dynamically unrelaxed clusters at high redshift.  They also appear to challange \nthe notion that AGN-driven outflows play a significant role in the preprocessing \nobserved in galaxy groups and environments of moderate overdensity relative to the field.\nIf preprocessing is underway in the Cl0023 system, our\nobservations suggest that powerful (quasar mode) nuclear activity is not the\npredominant mechanism quenching star formation and driving the evolution of\nCl0023 galaxies.   Of course we cannot rule out a population of\nlow-luminosity AGN powering ``radio mode'' feedback (Croton et al.~2006) in\nthe Cl0023 complex as our observations are only sensitive to moderate\nluminosity Seyferts and QSOs.  We are currently analyzing \\emph{Very Large Array} (\\emph{VLA}) 20-cm\nobservations of Cl0023 to search for such a population and expect to present\na full radio study of the system in a forthcoming paper (L.~M.~Lubin et al.~2009b,\nin preparation). \n\nOur current findings are in stark contrast to the overdensity of AGN recently detected in\nsimilar \\emph{Chandra} observations of the Cl1604 supercluster at $z=0.9$,\nwhere we find a population of Seyferts associated with an unrelaxed cluster and two rich groups \n(Kocevski et al.~2009a, 2009b).  However, the galaxy populations of these groups\ndiffer in significant ways from those of the Cl0023 system.  The Cl1604\ngroups tend to have higher velocity dispersions and more evolved galaxy\npopulations than the Cl0023 groups, as indicated by their average SFRs and\nmorphological fractions (Gal et al.~2008; Lubin et al.~2009a).  Previous\nobservations of Cl0023 galaxies found them to be predominately late-type\nsystems (75\\%; Lubin et al.~1998) with substantial amounts of ongoing star\nformation\\footnote{This is consistent with the galaxy properties of high\n  redshift groups with similar velocity dispersions (e.g.~Poggianti et\n  al.~2006)} (Postman, Lubin, Oke 1998; Lubin et al.~2009a), whereas the hosts\nof the Cl1604 AGN tend be bulge-dominated, post-starburst galaxies which show\nsigns of recent or ongoing galaxy interactions.   Therefore, while Cl0023\ncontains galaxies which have the gas necessary to fuel nuclear activity, it apparently\nlacks the bulge-dominated and massive early-type hosts in which powerful AGN\nhave been shown to reside (Kauffmann et al.~2003). \n\nA likely explanation for the absence of luminous AGN in the Cl0023 groups is\nthat the system lacks galaxies with sufficiently massive nuclear black holes\nrequired to power such activity.  It has previously been shown that the\nbulge-dominated S0 population in clusters and groups builds up over time at\nthe expense of the spiral population and that this morphological evolution is\nmore pronounced in lower mass systems (Poggianti et al.~2009).  There is also\nevidence that these galaxies are typically more massive than their suspected\nprogenitors (Dressler et al.~2009), suggesting they experience growth in\ntheir stellar bulges while in overdense environments, possibly via a\ncentrally concentrated burst of star formation (Dressler et al.~1999). \nGiven the correlation between bulge mass and central black hole mass\n(Gebhardt et al.~2000), we would expect similar growth in galactic nuclei\nover the same period.  Therefore, if disruptive AGN-driven\noutflows play a role in quenching star formation in groups, as has been\nsuggested, it may only become an important factor in the preprocessing of\ngalaxy populations during a later stage in the evolution of such groups and\nstructures, when sufficiently massive galaxies (and nuclear black holes) have\nbuilt up, but prior to hydrodynamical processes within clusters stripping\nthem of their gas reservoirs.   \n\nFurther observations of a larger sample of systems in the early\nstages of cluster formation, with a variety of velocity dispersions and\nmorphological fractions, will be required to test this scenario. \nIn the mean time, we are planning additional spectroscopic follow-up of the\nCl0023 groups targeting the radio bright population as well as the\nremaining X-ray point sources that currently lack redshifts.  \nThis will give us a greater spectroscopic completeness of X-ray luminous AGN in\nthe Cl0023 field, which will enable us to test our current findings and should allow us to better discern\nthe prevalence of powerful nuclear activity during cluster formation.\n\n\n\n\n\\acknowledgments\nThis work is supported by the Chandra General Observing Program under award\nnumber G07-8126X.  The spectrographic data used herein were obtained at the\nW.M. Keck Observatory, which is operated as a scientific partnership among\nthe California Institute of Technology, the University of California and the\nNational Aeronautics and Space Administration. The Observatory was made\npossible by the generous financial support of the W.M. Keck Foundation.\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}