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 \sigma _ { 8 } , the normalization of the matter power spectrum on scales of 8 h ^ { -1 } 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 \sigma at an appropriate cluster normalization scale R _ { Cl } .
We provide fitting formulas for R _ { Cl } and \sigma ( R _ { Cl } ) for general cosmologies , as well as for \sigma _ { 8 } as a function of cosmology and shape parameter \Gamma .
For flat models we find approximately \sigma _ { 8 } \simeq ( 0.495 ^ { +0.034 } _ { -0.037 } ) \Omega _ { M } ^ { -0.60 } for \Gamma = 0.23 , where the error bar is dominated by uncertainty in the mass-temperature relation .