From Snakes to Stars, the Statistics of Collapsed Objects - II. Testing a Generic Scaling Ansatz for Hierarchical Clustering

Abstract

We develop a diagrammatic technique to represent the multi-point cumulative probability density function (CPDF) of mass fluctuations in terms of the statistical properties of individual collapsed objects and relate this to other statistical descriptors such as cumulants, cumulant correlators and factorial moments. We use this approach to establish key scaling relations describing various measurable statistical quantities if clustering follows a simple general scaling ansatz, as expected in hierarchical models. We test these detailed predictions against high-resolution numerical simulations. We show that, when appropriate variables are used, the count probability distribution function (CPDF) and void probability distribution function (VPF) shows clear scaling properties in the non-linear regime. Generalising the results to the two-point count probability distribution function (2CPDF), and the bivariate void probability function (2VPF) we find good match with numerical simulations. We explore the behaviour of the bias associated with collapsed objects in limit of large separations, finding that it depends only on the intrinsic scaling parameter associated with collapsed objects and that the bias for two different objects can be expressed as a product of the individual biases of the objects. Finally we use hierarchal ansatz to develop a novel technique for correcting finite-volume effects in the estimation of multi-point statistical quantities from observational data (abridged).