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

4.1.1 Emergence of Non-Gaussianity 51
4.1.2 Power Spectrum Evolution in Linear PT 54
4.1.3 The Bispectrum induced by Gravity 55
4.1.4 The Three-Point Correlation Function 57
4.2 The Transition to the Non-Linear Regime: “Loop Corrections” 58
4.2.1 One-Loop PT and Previrialization 58
4.2.2 The One-Loop Power Spectrum 60
4.2.3 The One-Loop Bispectrum 64
4.3 The Power Spectrum in the Zel’dovich Approximation 67
4.4 Non-Gaussian Initial Conditions 68
4.4.1 General Results 68
4.4.2 χ 2 Initial Conditions 69
4.5 The Strongly Non-Linear Regime 72
4.5.1 The Self-Similar Solution 72
4.5.2 Stable Clustering 73
4.5.3 Scale Invariance 76
4.5.4 The Non-Linear Evolution of Two-Point Statistics 78
4.5.5 The Hierarchical Models 79
4.5.6 Hyperextended Perturbation Theory 82
5 From Dynamics to Statistics: The Local Cosmic Fields 84
5.1 The Density Field Third Moment: Skewness 84
5.1.1 The Unsmoothed Case 84
5.1.2 The Smoothed Case 85
5.1.3 Physical Interpretation of Smoothing 86
5.1.4 Dependence of the Skewness on Cosmological Parameters 88
5.1.5 The Skewness of the Local Velocity Divergence 89
5.2 The Fourth-Order Density Cumulant: Kurtosis 89
5.3 Results for Gaussian Smoothing Filters 90
5.4 The Density Cumulants Hierarchy 91
5.4.1 The Unsmoothed Density Cumulant Generating Function 92
5.4.2 Geometrical Properties of Smoothing in Lagrangian Space 94
5.4.3 Lagrangian to Eulerian Space Mapping: Smoothed Case 96
5.5 One-Loop Corrections to One-Point Moments 99
5.5.1 Exact Results 99
5.5.2 The Spherical Collapse Model Approximation 100
5.6 Evolution from Non-Gaussian Initial Conditions 101
5.7 Transients from Initial Conditions 106
5.8 The Density PDF 111
5.8.1 Reconstruction of the PDF from the Generating Function 111
5.8.2 Dependence on Cosmological Parameters 113
5.8.3 The PDF in the Zel’dovich Approximation 114
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