Variance, Skewness and Kurtosis: results from the APM Cluster Redshift Survey and Model Predictions

We estimate the variance $\xibar_2$, the skewness $\xibar_3$ and the kurtosis $\xibar_4$ in the distribution of density fluctuations in a complete sample from the APM Cluster Redshift Survey with 339 clusters and a mean depth $ \sim 250\Mpc$. We are able to measure the statistics of fluctuations in spheres of radius $R \simeq 5-80 \Mpc$, with reasonable errorbars. The statistics in the cluster distribution follow the hierarchical pattern $\xibar_J=S_J~\xibar_2^{J-1}$ with $S_J$ roughly constant, $S_3 \simeq 2$ and $S_4 \sim 8$. We analyse the distribution of clusters taken from N-body simulations of different dark matter models. The results are compared with an alternative method of simulating clusters which uses the truncated Zel'dovich approximation. We argue that this alternative method is not reliable enough for making quantitative predictions of $\xibar$. The N-body simulation results follow similar hierarchical relations to the observations, with $S_J$ almost unaffected by redshift distortions from peculiar motions. The standard $\Omega=1$ Cold Dark Matter (CDM) model is inconsistent with either the second, third or fourth order statistics at all scales. However both a hybrid Mixed Dark Matter model and a low density CDM variant agree with the $\xibar_J$ observations.