From snakes to stars: the statistics of collapsed objects -- I.Lower order clustering properties

Abstract

The highly non-linear regime of gravitational clustering is characterized by the presence of scale-invariance in the hierarchy of many-body correlation functions. Although the exact nature of this correlation hierarchy can only be obtained by solving the full set of BBGKY equations, useful insights can be obtained by investigating the consequences of a generic scaling ansatz. Extending earlier studies by Bernardeau & Schaeffer, we calculate the detailed consequences of such scaling for the implied behaviour of a number of statistical descriptors, including some new ones, developed to provide useful diagnostics of scale-invariance. We generalize the two-point cumulant correlators (now familiar in the literature) to a hierarchy of multipoint cumulant correlators (MCCs) and introduce the concept of reduced cumulant correlators (RCCs) and their related generating functions. The description of these quantities in diagrammatic form is particularly attractive. We show that every new vertex of the tree representation of higher order correlations has its own reduced cumulant associated with it, and, in the limit of large separations, MCCs of arbitrary order can be expressed in terms of RCCs of the same and lower order. The generating functions for these RCCs are related to the generating functions of the underlying tree vertices for matter distribution. Relating the generating functions of RCCs to the statistics of collapsed objects suggests a scaling ansatz of a very general form for the many-body correlation functions which, in turn, induces a similar hierarchy for the correlation functions of overdense regions. In this vein, we compute the lower order SN parameters and two-point cumulant correlators CNM for overdense regions and study how they vary as a function of the initial power spectrum of primordial density fluctuations. These are especially important results because they are model-independent, at least within the class of models considered here. We also show that our results match those obtained by the extended Press–Schechter formalism (which is based on entirely different arguments) in the limit of large mass.