The clustering of galaxy clusters: synthetic distributions and the correlation function amplitude
Abstract
We present a discussion of the relative amplitudes of the galaxy–galaxy and cluster–cluster correlation functions in terms of specific analytic models for the galaxy clustering process. To this end we examine the relative merits of various cluster finding algorithms: minimal spanning tree, friends-of-friends and wavelet decomposition. Then we look at specific models of the galaxy distribution: two variants of the Soneira–Peebles model and the multifractal distribution of Jones et al. These models are interesting because they reproduce various aspects of the observed galaxy distribution. Clusters of various richness can be defined by imposing a scale-independent threshold on the galaxy density field or from cluster-finding algorithms. The clustering models yield substantially different behaviours for ξcc as a function of cluster richness. The simplest variant of the Soneira–Peebles model in which the number of levels in the hierarchy is fixed fails to reproduce a richness-dependent ξcc. Varying the number of levels randomly as suggested by Soneira & Peebles to fix the scaling relation between the two- and three-point galaxy correlation functions does produce the required richness dependence. The multifractal model also has a manifest richness dependence of the cluster correlation function amplitude, but the slope of the function varies systematically with richness. Whether or not this is in accord with the data is still an open question. An interesting result emerges from this investigation: the position of the first zero-crossing of ξcc as a function of cluster richness would be a sensitive discriminator of clustering models if it could be measured. The cluster correlation function contains information about the growth of structure in the Universe that cannot be simply obtained from the underlying galaxy distribution alone.