The generalized Poisson distribution and a model of clustering from Poisson initial conditions

Abstract

This paper presents a new derivation of the generalized Poisson distribution. The derivation is based on the barrier crossing statistics of random walks associated with the Poisson distribution. A simple interpretation of this model in terms of a single server queue is also included. In the astrophysical context, the generalized Poisson distribution is interesting because it provides a good fit to the evolved, Eulerian counts-in-cells distribution measured in numerical simulations of hierarchical clustering from Poisson initial conditions. The new derivation presented here can be used to construct a useful analytic model of the evolution of clustering measured in these simulations. The model is consistent with the assumption that, as the Universe expands and the comoving sizes of regions change as a result of gravitational instability, the number of such expanding and contracting regions is conserved. The model neglects the influence of external tides on the evolution of such regions. Indeed, in the context of this model, the generalized Poisson distribution can be thought of as arising from a simple variant of the well-studied spherical collapse model, in which tidal effects are also neglected. This has the following implication: insofar as the generalized Poisson distribution derived from this model is a reasonable fit to the numerical simulation results, the counts-in-cells statistic must be relatively insensitive to such effects. This may be a consequence of the Poisson initial condition. The model can be understood as a simple generalization of the excursion set model which has recently been used to estimate the number density of collapsed, virialized haloes. The generalization developed here allows one to estimate the evolution of the spatial distribution of these haloes, as well as their number density. For example, it provides a framework within which the halo-halo correlation functions, at any epoch, can be computed analytically. In the model, when haloes first virialize, they are uncorrelated with each other. This is in good agreement with the simulations. Since it allows one to describe the spatial distribution of the haloes and the mass simultaneously, the model allows one to estimate the extent to which these haloes are biased tracers of the underlying matter distribution.