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Table of contents
Introduction
I Morphology and dynamics of the large-scale structure
1 Cosmological perturbations and structure formation
1.1 The homogeneous Universe
1.2 Statistical description of cosmological fields
1.2.1 Average and ergodicity
1.2.2 Statistical homogeneity and isotropy
1.2.3 Gaussian and log-normal random fields in cosmostatistics
1.2.4 Correlation functions and power spectra
1.3 Dynamics of gravitational instability
1.3.1 The Vlasov-Poisson system
1.3.2 Fluid dynamics approach, evolution equations in phase space
1.3.3 The single-stream approximation
1.4 Eulerian perturbation theory
1.4.1 Eulerian linear perturbation theory
1.4.2 The growth of fluctuations in linear theory
1.4.3 Eulerian perturbation theory at higher order
1.5 Lagrangian perturbation theory
1.5.1 Lagrangian fluid approach for cold dark matter
1.5.2 The Zel’dovich approximation
1.5.3 Second-order Lagrangian perturbation theory
1.6 Non-linear approximations to gravitational instability
1.6.1 The Zel’dovich approximation as a non-linear approximation
1.6.2 Other velocity potential approximations
1.6.3 The adhesion approximation
2 Numerical diagnostics of Lagrangian perturbation theory
2.1 Correlation functions of the density field
2.1.1 One-point statistics
2.1.2 Two-point statistics
2.1.3 Three-point statistics
2.2 Statistics of the Lagrangian displacement field
2.2.1 Lagrangian ψ versus Eulerian δ: one-point statistics
2.2.2 Perturbative and non-perturbative prescriptions for ψ
2.2.3 Non-linear evolution of ψ and generation of a vector part
2.3 Comparison of structure types in LPT and N-body dynamics
II Bayesian large-scale structure inference
3 Bayesian cosmostatistics
3.1 Introduction: plausible reasoning
3.1.1 On the definition of probability
3.1.2 On parameter determination
3.1.3 Probability theory as extended logic
3.2 Inverse problems and the mechanism of experimental learning
3.2.1 What is Bayesian analysis?
3.2.2 Prior choice
3.3 Bayesian data analysis problems
3.3.1 First level analysis: Bayesian parameter inference
3.3.2 Exploration of the posterior
3.3.3 Second level analysis: Bayesian model comparison
3.4 Markov Chain Monte Carlo techniques for parameter inference
3.4.1 Markov Chains
3.4.2 The Metropolis-Hastings algorithm
3.4.3 Hamiltonian Monte Carlo
4 Physical large-scale structure inference with the BORG algorithm
4.1 The challenge: the curse of dimensionality
4.1.1 Sparse sampling
4.1.2 Shape of high-dimensional pdfs
4.1.3 Algorithms in high dimensions
4.2 The BORG data model
4.2.1 The physical density prior
4.2.2 The large-scale structure likelihood
4.2.3 The posterior distribution
4.2.4 The Γ-distribution for noise sampling
4.3 Sampling procedure and numerical implementation
4.3.1 Calibration of the noise level
4.3.2 Hamiltonian Monte Carlo and equations of motion for the LSS density
4.3.3 The mass matrix
4.3.4 The leapfrog scheme integrator
4.4 Testing BORG
4.4.1 Generating mock observations
4.4.2 Convergence and correlations of the Markov Chain
4.4.3 Large-scale structure inference
4.5 Future extensions of BORG
5 Past and present cosmic structure in the Sloan Digital Sky Survey
5.1 The SDSS galaxy sample
5.2 The BORG SDSS analysis
5.3 Inference results
5.3.1 Inferred 3D density fields
5.3.2 Inference of 3D velocity fields
5.3.3 Inference of LSS formation histories
5.4 Summary and conclusions
III The non-linear regime of structure formation
6 Remapping Lagrangian perturbation theory
6.1 Introduction
6.2 Method
6.2.1 Remapping procedure
6.2.2 Comparison of structure types in LPT and in N-body dynamics
6.2.3 Improvement of the remapping procedure
6.2.4 Remapping function and transfer function
6.3 Statistics of remapped fields
6.3.1 One-point statistics
6.3.2 Two-point statistics
6.3.3 Three-point statistics
6.4 Discussion and conclusion
7 Non-linear filtering of large-scale structure samples
7.1 Introduction
7.1.1 Motivation for non-linear filtering of large-scale structure samples
7.1.2 Filtering in the final conditions
7.1.3 Filtering via constrained simulations
7.2 Fully non-linear filtering with Gadget
7.3 Fast non-linear filtering with COLA
7.3.1 The COLA method
7.3.2 Non-linear BORG-COLA realizations
IV Cosmic web analysis
8 Dark matter voids in the SDSS galaxy survey
8.1 Introduction
8.2 Methodology
8.2.1 Bayesian large-scale structure inference with the BORG algorithm
8.2.2 Generation of data-constrained reconstructions
8.2.3 Void finding and processing
8.2.4 Blackwell-Rao estimators for dark matter void realizations
8.2.5 Void catalogs for comparison of our results
8.3 Properties of dark matter voids
8.3.1 Number function
8.3.2 Ellipticity distribution
8.3.3 Radial density profiles
8.4 Summary and conclusions
9 Bayesian analysis of the dynamic cosmic web in the SDSS galaxy survey
9.1 Introduction
9.2 Methods
9.2.1 Bayesian large-scale structure inference with BORG
9.2.2 Non-linear filtering of samples with COLA
9.2.3 Classification of the cosmic web
9.3 The late-time large-scale structure
9.3.1 Tidal environment
9.3.2 Probabilistic web-type cartography
9.3.3 Volume and mass filling fractions
9.4 The primordial large-scale structure
9.4.1 Tidal environment
9.4.2 Probabilistic web-type cartography
9.4.3 Volume and mass filling fractions
9.5 Evolution of the cosmic web
9.5.1 Evolution of the probabilistic maps
9.5.2 Volume filling fraction
9.5.3 Mass filling fraction
9.6 Summary and Conclusion
10 Cosmic-web type classification using decision theory
10.1 Introduction
10.2 Method
10.3 Maps of structure types in the SDSS
10.4 Conclusions
Summary, Conclusion and Outlook
Appendices
A Complements on Gaussian random fields
A.1 Characteristic function
A.2 General definition of a Gaussian random vector
A.3 Some well-known properties of Gaussian random vectors
A.4 Marginal and conditionals of Gaussian random vectors
B Simulating collisionless dark matter fluids
B.1 Model equations
B.1.1 Model equations in the standard PM code
B.1.2 Model equations with COLA
B.2 Steps and data structures
B.2.1 Main PM steps
B.2.2 Definitions and data structures
B.3 Mesh assignments and interpolations
B.3.1 The mesh assignment function
B.3.2 Low-pass filtering
B.3.3 Common mesh assignment schemes
B.3.4 Interpolation
B.4 Poisson equation and accelerations
B.4.1 Solving the Poisson equation
B.4.2 Computation of the accelerations
B.5 Update of positions and momenta
B.5.1 Time integrators
B.5.2 Kick and Drift operators
B.6 Setting up initial conditions
B.6.1 The initial Gaussian random field
B.6.2 The high-redshift particle realization
C Cosmic structures identification and classification algorithms
C.1 VIDE: the Void IDentification and Examination toolkit
C.1.1 Voronoi Tessellation Density Estimation
C.1.2 The watershed algorithm
C.1.3 Processing and analysis of void catalogs
C.1.4 Radial density profiles
C.2 The T-web
C.2.1 The tidal tensor
C.2.2 Analogy with the Zel’dovich formalism
C.2.3 The T-web: original procedure
C.2.4 Extensions of the T-web
C.2.5 Implementation
C.2.6 Example
Bibliography
