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, sigma(g). 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 sigma(8): d log n/dz proportional to -1/sigma(8)(2); low-sigma(8) models evolve exponentially faster than high-sigma(8), models, for a given mass cluster. For example, the number density of Coma-like clusters decreases by a factor of similar to 10(3) from z = 0 to z similar or equal to 0.5 for sigma(8) = 0.5 models, while the decrease is only a factor of similar to 5 for sigma(8) similar or equal to 1. The strong exponential dependence on sigma(8) arises because clusters represent rarer density peaks in low-sigma(8), models. We show that the evolution rate at z less than or similar to 1 is insensitive to the density parameter Omega or to the exact shape of the power spectrum. Cluster evolution therefore provides a powerful constraint on sigma(8). Using available cluster data to z similar to 0.8, we find sigma(8) = 0.83 +/- 0.15. This amplitude implies a bias parameter b similar or equal to sigma(8)(-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, sigma(8) Omega(0.5) similar or equal to 0.5, the cosmological density parameter can be determined: Omega similar or equal to 0.3 +/- 0.1.