Evolution and clustering of rich clusters

In some hierarchical theories for galaxy formation the present evolution of the clustering pattern should approximate that which arises from scale-free initial conditions. The characteristic mass-scale of clustering is predicted to be evolving very rapidly, so we can hope to test these hypotheses by observations of clustering at quite accessible redshifts. We extend the well-known scaling laws for the dimensional quantities which characterize the clustering to give quantitative formulae for the evolution of general cluster catalogues. We show that X-ray selected clusters should display strong positive density evolution, with the characteristic luminosity, L*, remaining roughly constant. The comoving number density of clusters selected according to a criterion like Abell's should, on the other hand, remain roughly constant. Core radii should be much smaller in the past, and the gravitational lensing power of these clusters should be greater than for an unevolving population of clusters. Consideration is given to processes which will modify the self-similar scaling results. A corollary of the rapid evolution in these models is that small-amplitude, long-wavelength density perturbations will spatially modulate the clustering process, resulting in an enhancement of the correlation function for optically selected clusters, ξ_c, relative to the density correlation function ξ_ϱ. The enhancement factor $$\xi_\text c/\xi_\varrho$$ can be expressed in terms of the spectral index of the initial fluctuations and the present logarithmic slope of the mass distribution function of rich clusters. Estimates of the latter, combined with the observed clustering lengths of galaxies and rich clusters, require a spectral index $$n\simeq-1.5$$. A similar analysis applied to X-ray selected clusters suggests that these should be less strongly clustered than the optically selected clusters, though this result is sensitive to uncertainty in the slope of the X-ray luminosity function.