1 Simulated observations

1.1 N-body simulation

We have run a 512^{3} particle simulation of a ΛCDM model in a 300h^{−1}Mpc box using the TreePM-SPH code (White et al. [?]) but without any SPH particles. This simulation represents a large cosmological volume, to include a fair sample of rich clusters, while maintaining enough mass resolution to identify galactic mass halos. The cosmological model is chosen to provide a reasonable fit to a wide range of observations with Ω_{m} = 0.3, Ω_{Λ} = 0.7, H_{0} = 100 h kms^{−1}Mpc^{−1} with h = 0.7, Ω_{B}h^{2} = 0.02, n = 1 and σ_{8} = 1 (corresponding to δ_{H} = 5.3 × 10^{−5}). The simulation was started at z = 60 and evolved to the present with the full phase space distribution dumped every 100h^{−1}Mpc from z ≃ 2 to z = 0. The gravitational force softening was of a spline form, with a “Plummer-equivalent” softening length of 20 h^{−1}kpc comoving. The particle mass is 1.7 × 10^{10}h^{−1}M_{⊙} allowing us to find bound halos with masses several times 10^{11}h^{−1}M_{⊙}.

For every output of the simulation we produce a halo catalogue by running a “friends-of-friends” group finder with a linking length b = 0.15. We use this rather than the more canonical b = 0.2 as we find that the larger linking length frequently merges what we would by eye list as separate halos. We keep all groups above 8 particles, which imposes a minimum halo mass of order 10^{11}h^{−1}M_{⊙}. For each identified halo above 10^{13}h^{−1}M_{⊙} we compute, using the 3D distribution of all of the particles in the simulation, the mass (we use M_{200}, the mass enclosed within a radius, r_{200}, within which the mean density is 200 times the critical density at that redshift), velocity dispersion etc.