Virial scaling of massive dark matter halos: Why clusters prefer a high normalization cosmology

We present a precise estimate of the bulk virial scaling relation of halos formed via hierarchical clustering in an ensemble of simulated cold dark matter cosmologies. The result is insensitive to cosmological parameters; the presence of a trace, dissipationless gas component; and numerical resolution down to a limit of ∼ 1000 particles. The dark matter velocity dispersion scales with total mass as log[σ_DM(M, z)] = log(1082.9 ± 4.0 km s^_1) + (0.3361 ± 0.0026)log [h(z)M₂₀₀ /10 ¹⁵ M_⊙], with h(z) being the dimensionless Hubble parameter. At fixed mass, the velocity dispersion likelihood is nearly lognormal, with scatter σ_{ln σ} = 0.0426 ± 0.015, except for a tail with higher dispersions containing 10% of the population that are merger transients. We combine this relation with the halo mass function in ACDM models and show that a low normalization condition, S₈ = σ₈(Ω_m/0.3)^0.35 = 0.69, favored by recent WMAP and SDSS analysis requires that galaxy and gas-specific energies in rich clusters be 50% larger than that of the underlying dark matter. Such large energetic biases are in conflict with the current generation of direct simulations of cluster formation. A higher normalization, S₈ = 0.80, alleviates this tension and implies that the hot gas fraction within r ₅₀₀ is (0.71 ± 0.09) h₇₀^-3/2 Ω_b/Ω_m, a value consistent with recent Sunyaev-Zel'dovich observations.