The Amplitude of Mass Fluctuations

We determine the linear amplitude of mass fluctuations in the universe, σ8, from the abundance of massive clusters at redshifts z = 0.5− 0.8. The evolution of massive clusters depends exponentially on the amplitude of mass fluctuations and thus provides a powerful measure of this important cosmological parameter. The relatively high abundance of massive clusters observed at z > 0.5, and the relatively slow evolution of their abundance with time, suggest a high amplitude of mass fluctuations: σ8 = 0.9±10% for Ωm = 0.4, increasing slightly to σ8 = 0.95 for Ωm = 0.25 and σ8 = 1.0 for Ωm = 0.1 (flat CDM models). We use the cluster abundance observed at z = 0.5 − 0.8 to derive a normalization relation from the high-redshift clusters, which is only weakly dependent on Ωm: σ8Ω 0.14 m = 0.78±0.08. When combined with recent constraints from the present-day cluster mass function, σ8Ω 0.6 m = 0.33±0.03, we find σ8 = 0.98±0.1 and Ωm = 0.17±0.05. Low σ8 values (. 0.7) are unlikely; they produce an order of magnitude fewer massive clusters than observed. Subject headings: cosmological parameters — galaxies: clusters: general