Clustering of galaxy clusters in cold dark matter universes

We use very large cosmological N-body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model (τCDM) and a flat low density model with a cosmological constant (ΛCDM). Our simulations each use 10⁹ particles to follow the mass distribution within cubes of side 2h^-1Gpc (τCDM) and 3h^-1Gpc (ΛCDM) with a force resolution better than 10^-4 of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from r₀ = 0.4 d_c = 0.4 n_c^-1/3 r₀=12-13 h^-1 Mpc for samples with mean separation b(M) = 1 + (δ_c) / (σ₂(M)) - 1/δ_c, δ_c = 30h^-1Mpc to P_s/P_r = 1 + 2β/3 + β²/5, r₀ = 22-27 h^-1 Mpc for samples with d_c = 100 h^-1 Mpc. The lower value here corresponds to τCDM and the upper to ΛCDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10per cent using the simple model of Sheth, Mo & Tormen. This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the automated plate measurement (APM) cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for ΛCDM. The observed power spectrum (for a cluster sample of mean separation d_c=31h^-1Mpc) lies significantly above the predictions of both models.