The 1-Point Cluster Distribution Function and its Moments

We derive the 1-point probability density function of the smoothed 3-D Abell-ACO cluster density field and we compare it with that of artificial cluster samples, generated as high peaks of a Gaussian field in such a way that they reproduce the low-order (2- and 3-point) correlation functions and the observed cluster selection functions. We find that both real and simulated {\em pdf}'s are well approximated by a log-normal distribution even when the Gaussian smoothing radius is as large as 40 $h^{-1}$ Mpc. Furthermore the low-order moments of the {\em pdf} are found to obey a relation $\gamma = S_{3} \sigma^4$, with $\gamma$ being the skewness and $S_{3}\approx 1.8$. These results are consistent with clusters being high-peaks of an underlying initial Gaussian density field. A by-product of our analysis is that when we rescale the {\em pdf} cluster moments to those of the QDOT-IRAS galaxies, using linear biasing with $b_{cI}\sim 4.5$ 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 data by Saunders et al. (1991).