From snakes to stars: the statistics of collapsed objects – II. Testing a generic scaling ansatz for hierarchical clustering

We develop a diagrammatic technique to represent the multipoint probability density function 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) shows clear scaling properties in the non‐linear regime. We also show that analytic predictions made using the scaling model for the behaviour of the void probability function (VPF) also match the simulations very well. We generalize the results for the CPDF to the two‐point (bivariate) count probability distribution function (2CPDF), and show that its behaviour in the simulations is also well described by the theoretical model, as is the bivariate void probability function (2VPF). We explore the behaviour of the bias associated with collapsed objects in the 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. Having thus established the validity of the scaling ansatz in various different contexts, we use its consequences to develop a novel technique for correcting finite‐volume effects in the estimation of multipoint statistical quantities from observational data.