Power spectrum normalization from the local abundance of rich clusters of galaxies

The number density of rich galaxy clusters still provides the most robust way of normalizing the power spectrum of dark matter perturbations on scales relevant to large-scale structure. We revisit this constraint in light of several recent developments: (1) the availability of well-defined samples of local clusters with relatively accurate X-ray temperatures; (2) new theoretical mass functions for dark matter haloes which provide a good fit to large numerical simulations; (3) more accurate mass-temperature relations from larger catalogues of hydrodynamical simulations; (4) the requirement to consider closed as well as open and flat cosmologies to obtain full multi-parameter likelihood constraints for CMB and SNe studies. We present a new sample of clusters drawn from the literature and use this sample to obtain improved results on σ8, the normalization of the matter power spectrum on scales of 8 h Mpc, as a function of the matter density and cosmological constant in a Universe with general curvature. We discuss our differences with previous work, and the remaining major sources of uncertainty. Final results on the normalization, approximately independent of power spectrum shape, can be expressed as constraints on σ at an appropriate cluster normalization scale RCl. We provide fitting formulas for RCl and σ(RCl) for general cosmologies, as well as for σ8 as a function of cosmology and shape parameter Γ. For flat models we find approximately σ8 ≃ (0.495 +0.034 −0.037)Ω −0.60 M for Γ = 0.23, where the error bar is dominated by uncertainty in the mass-temperature relation.