Two-point correlation for rich clusters of galaxies.

Rich clusters of galaxies are observed to be strongly correlated at distances where the correlations of individual galaxies appear to be negligible. ' For example, the two-point correlation function for Abell clusters (richness R & 1) is unity at about 25k ' Mpc, where h is the Hubble constant and Mpc denotes megaparsecs, while the two-point correlation function for galaxies is unity at about 5h ' Mpc. In addition, the correlations of Abell clusters increase with richness. Kaiser has suggested that the origin of this enhancement is essentially statistical. 2 On the assumption that rich clusters of galaxies arise wherever suitably averaged (filtered) Gaussian primordial mass-density fluctuations are above a high threshold, the n-point correlation functions of rich clusters can be expressed in terms of the two-point correlation function for the filtered mass-density fluctuations. 2 3 The resulting richcluster two-point correlation function is enhanced over that of the filtered mass-density fluctuations, and, furthermore, this enhancement gets larger as the threshold is increased. This model for rich clusters, while reasonable for a universe dominated by cold, dark matter where hierarchical clustering occurs, would not be acceptable if rich clusters arose from the fragmentation of larger objects, as would occur if the universe were dominated by hot, dark matter. Demanding that the filtered mass-density fluctuations exceed a given threshold is a constraint on their distribution that does not involve any derivatives. Consequently, in this model, zeros of the two-point correlation function for rich clusters must coincide with those of the two-point correlation function for the filtered mass-density fluctuations. Theoretical prejudices favor an 0 =1 universe with primordial massdensity fluctuations that have a Zel'dovich power spectrum. With cold, dark matter, the power spectrum of the mass-density fluctuations falls off at large wave numbers, k, as ln(k)/k3 because fluctuations that entered the horizon in the radiation-dominated era only grow logarithmically prior to matter domination. In linear perturbation theory, this results in a (filtered) two-point correlation function for the mass-density fluctuations that crosses zero at about 17h Mpc. If the filtered mass-density fluctuation field is above the rich-cluster threshold at one point, it is likely to be above at a neighboring point. In general, one does want to associate separate rich clusters with such points. A simple modification of Kaiser's original model which removes this difficulty, in a physically reasonable fashion, is to associate rich clusters only with local maxima (peaks) of the filtered mass-density fluctuations that are above the threshold. ~ 5 Because the peak constraint on the probability distribution for the filtered mass-density fluctuations involves derivatives of the mass-density field, zeros of the rich-cluster two-point correlation function no longer coincide with those of the filtered mass-density fluctuations. In removing close pairs of rich clusters, the peak condition introduces an anticorrelation. Hence, it is expected that the rich-cluster two-point correlation function will now cross zero before the two-point correlation of the filtered mass-density fluctuations. In the limit of large separation, the two-point correlation function of peaks is identical to the correlation function of all points above a high threshold; in this limit, these cases differ only in their densities, which are divided out of the dimensionless correlation functions. The purpose of this Letter is to study the richcluster two-point correlation function in the model where rich clusters occur at high peaks of suitably averaged Gaussian primordial mass-density fluctuations. Particular attention will be paid to the distance out to which there is a significant correlation for rich clusters when the primordial mass-density fluctuations have a Zel'dovich power spectrum. 6 Our results may present a problem for this model since the rich-cluster two-point correlation function is observed to be significant at quite large distances. For Gaussian statistics, the probability that the filtered primordial mass-density fluctuations take the value e(x) is