GaltonWatson branching processes and the growth of gravitational clustering
Abstract
The PressSchechter description of gravitational clustering from an initially Poisson distribution is shown to be equivalent to the well studied GaltonWatson 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 PressSchechter 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 PressSchechter clump. In particular, an analytic expression for the merger history of any given Poisson PressSchechter 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 PressSchechter description of the gravitational evolution of clustering from an initially Poisson distribution is now complete. All these interrelations show why the PressSchechter 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.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 August 1996
 DOI:
 10.1093/mnras/281.4.1277
 arXiv:
 arXiv:astroph/9602112
 Bibcode:
 1996MNRAS.281.1277S
 Keywords:

 GALAXIES: CLUSTERS: GENERAL;
 GALAXIES: EVOLUTION;
 GALAXIES: FORMATION;
 COSMOLOGY: THEORY;
 DARK MATTER;
 Astrophysics
 EPrint:
 13 pages, uuencoded, gzipped, postscript, submitted to MN