Approximation Methods for Non–linear Gravitational Clustering

We discuss various analytical approximation methods for following the evolution of cosmological density perturbations into the strong (i.e. nonlinear) clustering regime. We start by giving a thorough treatment of linear gravitational instability in cosmological models and discussing the statistics of primordial density fluctuations produced in various scenarios of structure formation, and the role of non–baryonic dark matter. We critically review various methods for dealing with the non–linear evolution of density inhomogeneities, in the context of theories of the formation of galaxies and large–scale structure. These methods can be classified into five types: (i) simple extrapolations from linear theory, such as the high–peak model and the lognormal model; (ii) dynamical approximations, including the Zel’dovich approximation and its extensions; (iii) non–linear models based on purely geometric considerations, of which the main example is the Voronoi model; (iv) statistical solutions involving scaling arguments, such as the hierarchical closure ansatz for BBGKY, fractal models and the thermodynamic model of Saslaw; (v) numerical techniques based on particles and/or hydrodynamics. We compare the results of full dynamical evolution using particle codes and the various other approximation schemes. To put the models we discuss into perspective, we give a brief review of the observed properties of galaxy clustering and the statistical methods used to quantify it, such as correlation functions, power spectra, topology and spanning trees. To appear in Physics Reports (1995)