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, sigma8. 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 sigma8: d log n/dz ~ -1/sigma8^2; low sigma8 models evolve exponentially faster than high sigma8 models, for a given mass cluster. For example, the number density of Coma-like clusters decreases by a factor of ~10^3$from z = 0 to z ~ 0.5 for sigma8=0.5 models, while the decrease is only a factor of ~5 for sigma8 ~ 1. The strong exponential dependence on sigma8 arises because clusters represent rarer density peaks in low sigma8 models. We show that the evolution rate at z < 1 is insensitive to the density parameter$\Omega$ or to the exact shape of the power spectrum. Cluster evolution therefore provides a powerful constraint on sigma8. Using available cluster data to z ~ 0.8, we find sigma8 = 0.83 +/- 0.15. This amplitude implies a bias parameter b ~ 1/sigma8 = 1.2 +/- 0.2, i.e., a nearly unbiased universe with mass approximately tracing light on large scales.