Large-Scale Structure of the Universe and Cosmological Perturbation Theory

TL;DR

This paper provides a comprehensive, unified treatment of non-linear cosmological perturbation theory as applied to the growth of large-scale structure. It develops both Eulerian and Lagrangian perspectives, emphasizing the Fourier-space mode coupling with kernels α and β and the recursive construction of density and velocity perturbation kernels F_n and G_n, and it shows how cosmology enters mainly through the linear growth factor D_1 and the growth rate f. The work also clarifies the scale-dependent behavior across Einstein–de Sitter and general cosmologies, derives explicit third-order expressions, and links perturbative results to the spherical-collapse dynamics via ν_n and μ_n, setting a foundation for comparing PT predictions with simulations and observations. It highlights the regimes of validity, the role of shell-crossing, and the need to extend PT to capture non-Gaussian initial conditions and non-linear redshift-space distortions. Overall, the paper establishes PT as a robust, testable framework for predicting the statistics of cosmic fields and guiding interpretation of galaxy surveys and weak lensing data.

Abstract

We review the formalism and applications of non-linear perturbation theory (PT) to understanding the large-scale structure of the Universe. We first discuss the dynamics of gravitational instability, from the linear to the non-linear regime. This includes Eulerian and Lagrangian PT, non-linear approximations, and a brief description of numerical simulation techniques. We then cover the basic statistical tools used in cosmology to describe cosmic fields, such as correlations functions in real and Fourier space, probability distribution functions, cumulants and generating functions. In subsequent sections we review the use of PT to make quantitative predictions about these statistics according to initial conditions, including effects of possible non Gaussianity of the primordial fields. Results are illustrated by detailed comparisons of PT predictions with numerical simulations. The last sections deal with applications to observations. First we review in detail practical estimators of statistics in galaxy catalogs and related errors, including traditional approaches and more recent developments. Then, we consider the effects of the bias between the galaxy distribution and the matter distribution, the treatment of redshift distortions in three-dimensional surveys and of projection effects in angular catalogs, and some applications to weak gravitational lensing. We finally review the current observational situation regarding statistics in galaxy catalogs and what the future generation of galaxy surveys promises to deliver.