Determining the Amplitude of Mass Fluctuations in the Universe

We present a method for determining the rms mass fluctuations on 8 h −1 Mpc scale, σ 8. 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 σ 8 : d log n/dz ∝ −1/σ 2 8 ; low σ 8 models evolve exponentially faster than high σ 8 models, for a given mass cluster. For example, the number density of Coma-like clusters decreases by a factor of ∼ 10 3 from z = 0 to z ≃ 0.5 for σ 8 =0.5 models, while the decrease is only a factor of ∼ 5 for σ 8 ≃ 1. The strong exponential dependence on σ 8 arises because clusters represent rarer density peaks in low σ 8 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 σ 8. Using available cluster data to z ∼ 0.8, we find σ 8 = 0.83 ± 0.15. This amplitude implies a bias parameter b ≃ σ −1 8 = 1.2 ± 0.2, i.e., a nearly unbiased universe with mass approximately tracing light on large scales. subject headings : galaxies : clusters – galaxies : evolution – galaxies : formation – cos-mology : theory – large scale structure of universe