M. Plionis et R. Valdarnini, THE ONE-POINT CLUSTER DISTRIBUTION FUNCTION AND ITS MOMENTS, Monthly Notices of the Royal Astronomical Society, 272(4), 1995, pp. 869-877
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.