Galton-Watson branching processes and the growth of gravitational clustering

The Press-Schechter description of gravitational clustering from an initially Poisson distribution is shown to be equivalent to the well-studied Galton-Watson branching process. This correspondence is used to provide a detailed description of the evolution of hierarchical clustering, including a complete description of the merger history tree. The relation to branching process epidemic models means that the Press-Schechter description can also be understood using the formalism developed in the study of queues. The queueing theory formalism, also, is used to provide a complete description of the merger history of any given Press-Schechter clump. In particular, an analytic expression for the merger history of any given Poisson Press-Schechter clump is obtained. This expression allows one to calculate the partition function of merger history trees. It obeys an interesting scaling-relation; the partition function for a given pair of initial and final epochs is the same as that for certain other pairs of initial and final epochs. The distribution function of counts in randomly placed cells, as a function of time, is also obtained using the branching process and queueing theory descriptions. Thus, the Press-Schechter description of the gravitational evolution of clustering from an initially Poisson distribution is now complete. All these interrelations show why the Press-Schechter approach works well in a statistical sense, but cannot provide a detailed description of the dynamics of the clustering particles themselves. One way to extend these results to more general Gaussian initial conditions is discussed.