THE ONE-POINT CLUSTER DISTRIBUTION FUNCTION AND ITS MOMENTS

We derive the one-point probability density function (pdf) of the smoo thed Abell-ACO cluster density field and we compare it with that of ar tificial cluster samples, generated as high peaks of a Gaussian field in such a way that they reproduce the low-order (two- and three-point) correlation functions and the observed cluster selection functions. W e find that both real and simulated pdfs are well approximated by a lo g-normal distribution, even when the Gaussian smoothing radius is as l arge as 40 h-1 Mpc. Furthermore, the low-order moments of the pdf are found to obey a relation gamma almost-equal-to 1.8 (+/- 0.2)sigma4, wi th gamma being the skewness. Since clusters have not had enough time t o depart significantly from their original birth-place positions, thes e results are consistent with the theory that clusters are high peaks of an underlying initial Gaussian density field. A by-product of our a nalysis is that when we rescale the pdf cluster moments to those of th e QDT-IRAS galaxies, using linear biasing with b(cI) approximately 4.5 and for the common smoothing radius of 20 h-1 Mpc, we find them to be significantly smaller than those directly estimated from the QDOT dat a by Saunders et al.