Determining the Amplitude of Mass Fluctuations in the Universe

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Abstract
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 z0.5 for σ8=0.5 models, while the decrease is only a factor of ~5 for σ81. The strong exponential dependence on σ8 arises because clusters represent rarer density peaks in low-σ8 models. We show that the evolution rate at z1 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 bσ−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, σ8Ω0.50.5, the cosmological density parameter can be determined: Ω0.3±0.1.
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