Approximation methods for non-linear gravitational clustering

Abstract

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.

5 Dark Matter and Dark Energy

“…CDM-type spectra have the following approximate form of the transfer function [165,194,166] T (k) = 1 + Ak 2 log (1 + Bk)…”

“…(The precise location of the turn-around and the amplitude of P (k) depend upon specific details of the cosmological model, see for instance [16].) The 'standard' cold dark matter (SCDM) paradigm, which assumed that Ω CDM = 1, was introduced during the early 1980's at roughly the same time when HDM was perceived to be in trouble (see [101,96,50,166] for references to earlier work on this subject). Although SCDM was very successful in explaining a host of observational details, it was clear already a decade ago, that the processed power spectrum of SCDM lacked sufficient power on large scales and so fell short of explaining the angular two point correlation function for galaxies on scales ∼ 50 Mpc [60].…”

Initial conditions for imperfect dark matter

“…That condition is obeyed with a high accuracy well before the matter/radiation equality. In particular, this guarantees that IDM is indistinguishable from CDM at the level of cosmic microwave background measurements 6 .…”

Gravitational instability via the Schrödinger equation

“…Even the simplest of these (Lagrangian) approaches, the Zeldovich approximation (Zeldovich 1970), has proved highly successful in comparison against full N-body calculations in describing the quantitative morphology of the clustering pattern (Coles et al 1993). For a review of both these approaches, see Sahni and Coles (1995).…”