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: dlogn/dz~-1/σ²₈ 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~=σ^-1₈=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.