THE CLUSTERING OF GALAXY CLUSTERS - SYNTHETIC DISTRIBUTIONS AND THE CORRELATION-FUNCTION AMPLITUDE

We present a discussion of the relative amplitudes of the galaxy-galax y and cluster-cluster correlation functions in terms of specific analy tic models for the galaxy clustering process, To this end we examine t he relative merits of various cluster finding algorithms: minimal span ning tree, friends-of-friends and wavelet decomposition Then we look a t specific models of the galaxy distribution: two variants of the Sone ira-Peebles model and the multifractal distribution of Jones et al. Th ese models are interesting because they reproduce various aspects of t he observed galaxy distribution. Clusters of various richness can be d efined by imposing a scale-independent threshold on the galaxy density field of from cluster-finding algorithms. The clustering models yield substantially different behaviours for xi(cc) as a function of cluste r richness. The simplest variant of the Soneira-Peebles model in which the number of levels in the hierarchy is fixed fails to reproduce a r ichness-dependent xi(cc). Varying the number of levels randomly as sug gested by Soneira & Peebles to fix the scaling relation between the tw o- and three-point galaxy correlation functions does produce the requi red richness dependence, The multifractal model also has a manifest ri chness dependence of the cluster correlation function amplitude, but t he slope of the function varies systematically with richness. Whether or not this is in accord with the data is still an open question. An i nteresting result emerges from this investigation: the position of the first zero-crossing of xi(cc) as a function of cluster richness would be a sensitive discriminator of clustering models if it could be meas ured. The cluster correlation function contains information about the growth of structure in the Universe that cannot be simply obtained fro m the underlying galaxy distribution alone.