Fe b 19 96 CLUSTERING STATISTICS & DYNAMICS

Since the appearance of the classical paper of Lifshitz almost half a century ago, linear stability analysis of cosmological models is textbook knowledge. Until recently, however, little was known about the behavior of higher than linear order terms in the perturbative expansion. These terms become important in the weakly nonlinear regime of gravitational clustering, when the rms mass density contrast is only slightly smaller than unity. In the past, theorists showed little interest in studying this regime, and for a good reason: only a decade ago, it would have been an academic excercise – at scales large enough to probe the weakly nonlinear regime, all measures of clustering were dominated by noise. This is no longer the case with present data. The purpose of this talk is to provide a brief summary of recent advances in weakly nonlinear perturbation theory. We present analytical perturbative results together with results of N-body experiments, conducted to test their accuracy. We compare perturbative predictions with measurements from galaxy surveys. Such comparisons can be used to test the gravitational instability theory and to constrain possible deviations from Gaussian statistics in the initial mass distribution; they can be also used to study the nature of physical processes that govern galaxy formation (“biasing”). We also show how future studies of velocity field statistics can provide a new way to determine the density parameter, Ω. To appear in: Clustering in the Universe, Proc. XXX Moriond Meeting, Les Arcs, 1995, Eds. S. Maurogordato et al. Editions Frontieres, Paris 1 Perturbation theory Tests of theories for the origin of the large scale structure of the universe ultimately depend on a comparison of model predictions with measurable quantities, derived from observations. The statistical measures we will discuss here are the low order N-point correlation functions. They have two clear advantages. First, they can be estimated from galaxy surveys with a reasonable degree of precision and reproducibility (see, e.g. [45, 46, 20, 51], and references therein). Second, correlation functions can be relatively easily related to dynamics (e.g. [45]). Their evolution can be studied by taking moments of the hydrodynamical equations of motion of an expanding self-gravitating pressureless fluid with zero vorticity, which is a good approximation of the real universe after the hydrogen recombination. Low order correlation functions can also be measured from N-body experiments. 1.1 Gravitational instability The full description of gravitational instability is nonlinear. The density contrast δ(x, t) = ρ(x, t) 〈ρ〉 − 1 , (1) the peculiar velocity v and the gravitational potential φ are related by the Euler, Poisson and continuity equations. They can be combined into one expression for the density contrast, δ̈ + 2Hδ̇ − 3 2 ΩHδ = 3 2 ΩHδ + a∇δ · ∇φ+ a∇α∇β[(1 + δ)vαvβ ] . (2) Here x = {xα} and t are, respectively, the comoving spatial coordinates and the cosmological time, the dots represent time derivatives, ∇α = ∂/∂xα , a(t) is the scale factor, H = ȧ/a is the Hubble parameter, and Ω = 8πG〈ρ〉/3H is the density parameter. The brackets 〈. . .〉 denote ensemble averaging and ρ is the mass density. Perturbation theory rests on a conjecture that when the deviations from homogeneity are small, the first few terms of the expansion δ = δ1 + δ2 + δ3 + . . . (3) provide a reasonable approximation of the exact solution of eq. (2). The first, linear term is the well known Lifshitz [39] solution of eq. (2) with the right-hand side set to zero, δ1 = D(t)ε(x) + decaying mode , (4) where D(t) is the standard growing mode (see, e.g., §11 in [45]) and ε is a random field with statistical properties defined by initial conditions. The term δ2 = O(ε ) is the solution of the same equation with quadratic nonlinearities included iteratively by using δ1 as source terms (as in [45], §18). For a vanishing cosmological constant (Λ = 0) and arbitrary Ω, the fastest growing mode in this solution is [35]