Determining the amplitude of mass fluctuations in the universe

We present a method for determining the amplitude of mass fluctuations on 8 h^-1 Mpc scale, σ₈. The method utilizes the rate of evolution of the abundance of rich clusters of galaxies. Using the Press-Schechter approximation, we show that the cluster abundance evolution is a strong function of σ₈: d log nldz ∝ -1/σ₈²; low-σ₈ models evolve exponentially faster than high-σ₈ models, for a given mass cluster. For example, the number density of Coma-like clusters decreases by a factor of ∼10³ from z = 0 to z ≃ 0.5 for σ₈ = 0.5 models, while the decrease is only a factor of ∼5 for σ₈ ≃ 1. The strong exponential dependence on σ₈ arises because clusters represent rarer density peaks in low-σ₈ models. We show that the evolution rate at z ∝ 1 is insensitive to the density parameter Ω or to the exact shape of the power spectrum. Cluster evolution therefore provides a powerful constraint on σ₈. Using available cluster data to z ∼ 0.8, we find σ₈ = 0.83 ± 0.15. This amplitude implies a bias parameter b ∼ σ₈^-1 = 1.2 ± 0.2, i.e., a nearly unbiased universe with mass approximately tracing light on large scales. When combined with the present-day cluster abundance normalization, σ₈Ω^0.5 ≃ 0.5, the cosmological density parameter can be determined: Ω ≃ 0.3 ± 0.1.