VARIANCE, SKEWNESS AND KURTOSIS - RESULTS FROM THE APM CLUSTER REDSHIFT SURVEY AND MODEL PREDICTIONS

We estimate the variance <(xi)over bar>(2), the skewness <(xi)over bar >(3) and the kurtosis <(xi)over bar>(4) in the distribution of density fluctuations in a complete sample from the APM Cluster Redshift Surve y with 339 dusters and a mean depth D similar to 250 h(-1)Mpc.star We are able to measure the statistics of fluctuations in spheres of radiu s R similar or equal to 5-80 h(-1)Mpc, with reasonable error-bars. The statistics in the duster distribution follow the hierarchical pattern <(xi)over bar>(J) = S-J <(xi)over bar>(J-1)(2) with S-J roughly const ant, S-3 similar or equal to 2 and S-4 similar to 8. We analyse the di stribution 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 mak ing quantitative predictions of <(xi)over bar>. The N-body simulation results follow similar hierarchical relations to the observations, wit h S-J almost unaffected by redshift distortions from peculiar motions. The standard Ohm = 1 Cold Dark Matter (CDM) model is inconsistent wit h the second-, third- and fourth-order statistics at all scales. Howev er both a hybrid Mixed Dark Matter model and a low-density CDM variant agree with the <(xi)over bar>(J) observations.