The one-point cluster distribution function and its moments

We derive the 1-point probability density function of the smoothed 3-DAbell-ACO cluster density field and we compare it with that of artificialcluster samples, generated as high peaks of a Gaussian field in such a way thatthey reproduce the low-order (2- and 3-point) correlation functions and theobserved cluster selection functions. We find that both real and simulated {\empdf}'s are well approximated by a log-normal distribution even when theGaussian smoothing radius is as large as 40 $h^{-1}$ Mpc. Furthermore thelow-order moments of the {\em pdf} are found to obey a relation $\gamma = S_{3}\sigma^4$, with $\gamma$ being the skewness and $S_{3}\approx 1.8$. Theseresults are consistent with clusters being high-peaks of an underlying initialGaussian density field. A by-product of our analysis is that when we rescalethe {\em pdf} cluster moments to those of the QDOT-IRAS galaxies, using linearbiasing 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 directlyestimated from the QDOT data by Saunders et al. (1991).Comment: LaTex file. Submitted in MNRAS June 1994. Figures upon reques