The cluster distribution as a test of dark matter models .1. Clustering properties

We present extended simulations of the large-scale distribution of galaxy clusters in several dark matter models, using an optimized version of the truncated Zel'dovich approximation (TZA). In order to test the reliability of our simulations, we compare them with N-body-based cluster simulations. We find that the TZA provides a very accurate description of the cluster distribution as long as fluctuations on the cluster mass-scale are in the mildly non-linear regime (sigma(8) less than or similar to 1). The low computational cost of this simulation technique allows us to run a large ensemble of 50 realizations for each model, so we are able to quantify accurately the effects of cosmic variance. Six different dark matter models are studied in this work: standard CDM (SCDM), tilted CDM (TCDM) with primordial spectral index n = 0.7, cold + hot DM (CHDM) with Omega(hot) = 0.3, low Hubble constant (h = 0.3) CDM (LOWH), and two spatially flat low density CDM models with Omega(0), = 0.2 and Omega(Lambda) = 0.8, having two different normalizations, sigma(8) = 0.8 (Lambda CDM(1)) and sigma(8) = 1.3 (ACDM(2)). We compare the cluster simulations with an extended redshift sample of Abell/ACO clusters, using various statistical measures, such as the integral of the two-point correlation function, J(3), and the probability density function (pdf). We find that the models that best reproduce the clustering of the Abell/ACO cluster sample are the CHDM and the Lambda CDM(1), models. The Lambda CDM(2) model is too strongly clustered, and this clustering is probably overestimated in our simulations as a result of the large sigma(8)-value of this model. All of the other models are ruled out at a high confidence level. The pdfs of all models are well approximated by a lognormal distribution, consistent with similar findings for Abell/ACO clusters. The low-order moments of all the pdfs are found to obey a variance-skewness relation of the form gamma = S-3 sigma(4), with S-3 similar or equal to 1.9, independent of the primordial spectral shape and consistent with observational data. After computing the cluster biasing parameter, b(cl), we estimate the quantity beta(cl) = Omega(0)(0.6)/b(cl) for the different models. Owing to the large observational uncertainties, beta(cl) = 0.20 +/- 0.05, this test does not discriminate strongly between the different models. The scale-independence Of beta(cl) and thus of beta(cl), does, however, suggest that it is probably a reliable procedure to use the linear biasing model to infer the dark matter power spectrum from observational cluster samples. We also note that the abundances of clusters predicted using the Press-Schechter theory provide strong constraints on these models: only the CHDM, LOWH and ACDM, models appear to produce the correct number density of clusters.