The evolution of groups of galaxies in the Press-Schechter formalism

The Fress-Schechter theory provides a simple analytical description for the evolution of gravitational structure in a hierarchical universe. In this paper, we extend the original theory in order to focus our attention on the subset of regions that will eventually collapse to form clusters of a set mass. The first part of the paper is concerned with obtaining the conditional multiplicity function of groups at an epoch with redshift z, given that they are bound into an object of a particular mass at the present epoch. This is combined with the present distribution of group masses in order to obtain the joint multiplicity function. The difficulty in obtaining these functions lies in determining the cross-corelation between a set of small volumes and the larger volume that contains them. We show that this correlation has a simple form. The formulae we derive are checked for self-consistency, and are compared with the N-body simulations of Efstathiou et al. The numerical results are found to be in extremely good agreement. These multiplicity functions are applied to study the evolutionary histories of groups and clusters. We obtain a number of results relating to the infall of galaxies in small groups into large clusters, and to the density of the more massive groups at early times. In particular, we illustrate our analysis by making an in-depth comparison of the theoretical evolution with the Butcher-Oemler effect observed in rich clusters at moderate redshifts. We find that the growth of the infall rate over these look-back times is not (by itself) sufficient to explain the rapidly increasing fraction of blue galaxies in these clusters, but that a good quantitative fit to the available data can be obtained if allowance is made for the increased star burst activity that is seen in the spectra of the distant blue galaxies.