Large-Scale Structure of the Universe and Cosmological ...

5.9 Two-Dimensional Dynamics 115
5.10 The Velocity Divergence PDF 116
5.10.1 The Velocity Divergence Cumulants Hierarchy 117
5.10.2 The Shape of the PDF 119
5.10.3 Comparison with N-Body Simulations 119
5.11 The Velocity-Density Relation 121
5.12 The Two-Point Density PDF 124
5.13 Extended Perturbation Theories 126
6 From Theory to Observations: Estimators and Errors 129
6.1 Introduction 129
6.2 Basic Concepts 132
6.2.1 Cosmic Bias and Cosmic Error 132
6.2.2 The Covariance Matrix 133
6.3 Fair Sample Hypothesis and Local Poisson Approximation 134
6.3.1 The Fair Sample Hypothesis 134
6.3.2 Poisson Realization of a Continuous Field 134
6.4 The Two-Point Correlation Function 137
6.4.1 Estimators 137
6.4.2 Cosmic Bias and Integral Constraint of the LS Estimator 139
6.4.3 Cosmic Error of the LS Estimator 141
6.4.4 The Covariance Matrix 143
6.4.5 Recipes for Error Calculations 144
6.5 The Power Spectrum 144
6.5.1 Simple Estimators 145
6.5.2 Cosmic Bias and Integral Constraint 146
6.5.3 The Cosmic Error 147
6.5.4 The Covariance Matrix 148
6.6 Generalization to Higher-Order Correlation Functions 150
6.7 One-Point Distributions: Counts-in-Cells 152
6.7.1 Definitions: 152
6.7.2 Estimators 153
6.7.3 Error Propagation: Cosmic Bias vs. Cosmic Error 154
6.7.4 Cosmic Error and Cross-Correlations of Factorial Moments 155
6.7.5 Cosmic Error and Cosmic Bias of Cumulants 159
6.8 Multivariate Count-in-Cells 160
6.9 Optimal Weighting 162
6.10 Cosmic Distribution Function and Cross-Correlations 166
6.10.1 Cosmic Distribution Function and Likelihood 166
6.10.2 Cross-Correlations Between Different Statistics 168
6.10.3 Validity of the Gaussian Approximation 169
6.11 Optimal Techniques for Gaussian Random Fields 170
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