Analytical approximations to the low-order statistics of dark matter distributions

Abstract

We show that in a hierarchical clustering model the low-order statistics of the density and the peculiar velocity fields can all be modelled semianalytically for a given cosmology and an initial density perturbation power spectrum $P(k)$. We present such models for the two-point correlation function $\xi(r)$, the amplitude $Q$ of the three-point correlation function, the mean pairwise peculiar velocity $< v_{12}(r)> $, the pairwise peculiar velocity dispersion $< v_{12}^2(r)>$, and the one-point peculiar velocity dispersion $< v_1^2 >$. We test our models against results derived from N-body simulations. These models allow us to understand in detail how these statistics depend on $P(k)$ and cosmological parameters. They can also help to interpret, and maybe correct for, sampling effects when these statistics are estimated from observations. The dependence of the small-scale pairwise peculiar velocity dispersion on rich clusters in the sample, for instance, can be studied quantitatively. There are also significant implications for the reconstruction of the cosmic density field from measurements in redshift space.