The richness dependence of galaxy cluster correlations: Results from a redshift survey of rich APM clusters

We analyse the spatial clustering properties of a new catalogue of very rich galaxy clusters selected from the APM Galaxy Survey. These clusters are of comparable richness and space density to Abell Richness Class $\geq 1$ clusters, but selected using an objective algorithm from a catalogue demonstrably free of artificial inhomogeneities. Evaluation of the two-point correlation function $\xi_{cc}(r)$ for the full sample and for richer subsamples reveals that the correlation amplitude is consistent with that measured for lower richness APM clusters and X-ray selected clusters. We apply a maxmimum likelihood estimator to find the best fitting slope and amplitude of a power law fit to $\xi_{cc}(r)$, and to estimate the correlation length $r_{0}$ (the value of $r$ at which $\xi_{cc}(r)$ is equal to unity). For clusters with a mean space density of $1.6\times 10^{-6}\hmpccc$ (equivalent to the space density of Abell Richness $\geq 2$ clusters), we find $r_{0}=21.3^{+11.1}_{-9.3} \hmpc$ (95% confidence limits). This is consistent with the weak richness dependence of $\xi_{cc}(r)$ expected in Gaussian models of structure formation. In particular, the amplitude of $\xi_{cc}(r)$ at all richnesses matches that of $\xi_{cc}(r)$ for clusters selected in N-Body simulations of a low density Cold Dark Matter model