2008ApJ...672..122E

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 logσ_DM(M,z)=log(1082.9±4.0 km/s)+(0.3361±0.0026)logh(z)M₂₀₀/10¹⁵ M_☉, with h(z) being the dimensionless Hubble parameter. At fixed mass, the velocity dispersion likelihood is nearly lognormal, with scatter σ_lnσ=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 ΛCDM 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 r₅₀₀ is (0.71±0.09) h^–3/2₇₀Ω_b/Ω_m, a value consistent with recent Sunyaev-Zel'dovich observations.