Halo correlations in nonlinear cosmic density fields

The question we address in this paper is the determi- nation of the correlation properties of the dark matter halos ap- pearing in cosmic density fields once they underwent a strongly nonlinear evolution induced by gravitational dynamics. A se- ries of previous works have given indications that kind of non- Gaussian features are induced by nonlinear evolution in term of the high-order correlation functions. Assuming such patterns for the matter field, i.e. that the high-order correlation functions behave as products of two-body correlation functions, we de- rive the correlation properties of the halos, that are assumed to represent the correlation properties of galaxies or clusters. The hierarchical pattern originally induced by gravity is shown to be conserved for the halos. The strength of their correlations at any order varies, however, but is found to de- pend only on their internal properties, namely on the parameter x / m=r 3 where m is the mass of the halo, r its size and is the power law index of the two-body correlation function. This internal parameter is seen to be close to the depth of the internal potential well of virialized objects. We were able to derive the explicit form of the generating function of the moments of the halo counts probability distribution function. In particular we show explicitly that, generically, SP(x) ! P P 2 in the rare halo limit. Various illustrations of our general results are presented. As a function of the properties of the underlying matter field, we construct the count probabilities for halos and in particular discuss the halo void probability. We evaluate the dependence of the halo mass function on the environment: within clusters, hierarchical clustering implies the higher masses are favored. These properties solely arise from what is a natural bias (i.e., naturally induced by gravity) between the observed objects and the unseen matter field, and how it manifests itself depending on which selection effects are imposed.