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: σ ₈ = 0.9 (±10%) for Ω_m = 0.4, increasing slightly to σ₈ = 0.95 for Ω_m = 0.25 and σ₈ = 1.0 for Ω_m = 0.1 (flat cold dark matter 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 Ω_m: σ₈Ω_m ^0.14 = 0.78 ± 0.08. When combined with recent constraints from the present-day cluster mass function, σ₈Ω _m^0.6 = 0.33 ± 0.03, we find σ₈ = 0.98 ± 0.1 and Ω_m = 0.17 ± 0.05. Low-σ ₈ values (≲0.7) are unlikely; they produce an order-of-magnitude fewer massive clusters than observed.