Introductory Overview of Eulerian and Lagrangian Perturbation Theories

TL;DR

This paper provides an introductory overview of Eulerian and Lagrangian perturbation theories in cosmology, covering their derivations, solutions, corrections, and applications to gravitational instability and structure formation.

Contribution

It offers a comprehensive review of both Eulerian and Lagrangian perturbation approaches, including derivations, solutions, and applications, with some new insights into higher order corrections and their uses.

Findings

Derivation of fluid equations from microscopic descriptions.

Linear solutions and higher order corrections for density and velocity fields.

Applications in modeling the evolution of density PDFs and redshift space distortions.

Abstract

These lectures notes give an introduction to the fast developing area of research dealing with perturbative descriptions of the gravitational instability in an expanding universe. I just sketch the outlines of some proofs, and many important contributions (and references) were left out. Many untouched aspects are reviewed in \cite{sah_col} which also contains a useful reference list. In the notes dedicated to the Eulerian approach (\S\ref{sec:fluid} - \S\ref{sec:further}), I derive the fluid equations starting from a microscopic description, I give their linear solution and some higher order corrections, and describe a number of applications concerning the evolution of the one point probability density function for the density contrast field and for the divergence of the corresponding velocity field. In the notes dedicated to the Lagrangian point of view (\S\ref{sec:zeldo} - \S\ref{sec:lag_app}), I recall Zeldovich Approximation and its higher orders corrections, and present some applications, like their use in describing the statistical effect of the real space-redshift mapping, or as approximate description of the full non-linear dynamics.