Merging and Hierarchical Clustering from an Initially Poisson Distribution

The excursion{set, Press{Schechter mass spectrum for a Poisson distribution of identical particles is derived. For the special case of an initially Poisson distribution the spatial distribution of the Press{Schechter clumps is shown to be Poisson. Thus, the distribution function of particle counts in randomly placed cells is easily obtained from the Press{Schechter multiplicity function. This Poisson Press{Schechter distribution function has the same form as the well-studied gravitational quasi-equilibrium counts-in-cells distribution function which ts the observed galaxy distribution well. The description of merging and hierarchical clustering from an initially Poisson distribution is also formulated and solved. These solutions represent the discrete analogue of those already obtained for an initially Gaussian distribution. In addition, physically motivated arguments are used to provide insight about the structure of the partition function that describes all possible merger histories. From this partition function an expression for the number of progenitor clumps as a function of cluster size is obtained. This, with the knowledge that initially Press{Schechter clumps have a Poisson spatial distribution, is used to calculate the subsequent clustering of these clumps. Thus, the growth of hierarchical clustering on all levels of the hierarchy is quantiied. Comparison of the analytic results with relevant N-body simulations of gravitational clustering shows substantial agreement. A method for extending all these results to describe the growth of clustering from more general, non-Gaussian, compound Poisson distributions is also described. For these compound Poisson processes a scaling relation is obtained that greatly clariies the results of relevant N-body simulations in which particles have a range of masses. This scaling solution and the merger history results are consistent with a simple model of the growth of hierarchical clustering. At early times in this model, clustering on all levels of the hierarchy is well approximated by appropriately renormalized Poisson Press{ Schechter forms.