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

We estimate the cluster-galaxy cross-correlation function (xi(cg)), 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 xi(cg) is found to be almost independent of cluster richness. At large separations, r greater than or similar to 5 h(-1) Mpc (h = H-0/100 km s(-1) Mpc(-1), where H-0 is the Hubble constant), xi(cg) has a similar shape to the galaxy-galaxy and cluster-cluster autocorrelation functions. xi(cg) in redshift space can be related to the real-space xi(cg) 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 an accurate description of the averaged infall velocity of matter into galaxy clusters. We use this model to estimate beta (beta = Omega(0.6)/b, where b is the linear bias parameter), and find that it tends to overestimate the true result in simulations by only similar to 10-30 per cent. Application to the APM results yields beta = 0.46 with beta < 0.73 at 95 per cent confidence. This measure is complementary to the estimates made of the density parameter from larger scale bulk flows and from the virialized regions of clusters on smaller scales. We also compare the APM xi(cg) and galaxy autocorrelations directly with the mass correlation and cluster-mass correlations in COBE-normalized simulations of popular cosmological models, and derive two independent estimates of the galaxy biasing expected as a function of scale. This analysis reveals that both low-density and critical-density COBE-normalized cold dark matter (CDM) models require anti-biasing by a factor similar to 2 on scales r less than or equal to 2 h(-1) Mpc, 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 greater than or similar to 20 h(-1) Mpc), 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 xi(cg) yields an estimate of the value of Omega(0.6) needed in each model. We find that only the low-Omega model is fully consistent with observations, with MDM marginally excluded at the similar to 2 sigma level.