The APM cluster - galaxy cross - correlation function : Constraints on Ω and galaxy bias

We estimate the cluster-galaxy cross-correlation function (ξcg), from the APM galaxy and galaxy cluster surveys. We obtain estimates both in real space from the inversion of projected statistics and in redshift space using the galaxy and cluster redshift samples. The amplitude of ξcg is found to be almost independent of cluster richness. At large separations, r ∼> 5 h Mpc ( h = H0/100km s , where H0 is the Hubble constant), ξcg has a similar shape to the galaxy-galaxy and cluster-cluster autocorrelation functions. ξcg in redshift space can be related to the real space ξcg by convolution with an appropriate velocity field model. Here we apply a spherical collapse model, which we have tested against N-body simulations, finding that it provides a surprisingly accurate description of the averaged infall velocity of matter into galaxy clusters. We use this model to estimate β (β=Ω/b where b is the linear bias parameter) and find that it tends to overestimate the true result in simulations by only ∼ 10−30%. Application to the APM results yields β = 0.43 with β < 0.87 at 95% confidence. This measure is complementary to the estimates made of the density parameter from larger scale bulk flows and from the virialised regions of clusters on smaller scales. We also compare the APM ξcg and galaxy autocorrelations directly to the mass correlation and clustermass correlations in COBE normalised simulations of popular cosmological models and derive two independent estimates of the galaxy biasing expected as a function of scale. This reveals that both low density and critical density cold dark matter (CDM) models require anti-biasing by a factor ∼ 2 on scales r ≤ 2 hMpc and that the Mixed Dark Matter (MDM) model is consistent with a constant biasing factor on all scales. The critical density CDM model also suffers from the usual deficit of power on large scales (r ∼ > 20 hMpc). We use the velocity fields predicted from the different models to distort the APM real space cross-correlation function. Comparison with the APM redshift space ξcg yields an estimate of the value of Ω 0.6 needed in each model. We find that only the low Ω model is fully consistent with observations, with MDM marginally excluded at the ∼ 2σ level.