The amplitude of mass fluctuations

We determine the linear amplitude of mass fluctuations in the universe, sigma(8), from the abundance of massive clusters at redshifts z = 0.5-0.8. The evolution of massive clusters depends exponentially on the amplitude of mass fluctuations and thus provides a powerful measure of this important cosmological parameter. The relatively high abundance of massive clusters observed at z > 0.5 and the relatively slow evolution of their abundance with time suggest a high amplitude of mass fluctuations: sigma(8) = 0.9 (+/-10%) for Omega(m) = 0.4, increasing slightly to sigma(8) = 0.95 for Omega(m) = 0.25 and sigma(8) = 1.0 for Omega(m) = 0.1 (flat cold dark matter models). We use the cluster abundance observed at z = 0.5-0.8 to derive a normalization relation from the high-redshift clusters, which is only weakly dependent on Omega(m): sigma(8)Omega(m)(0.14) = 0.78 +/- 0.08. When combined with recent constraints from the present-day cluster mass function, sigma(8)Omega(m)(0.6) = 0.33 +/- 0.03, we find sigma(8) = 0.98 +/- 0.1 and Omega(m) = 0.17 +/- 0.05. Low-sigma(8) values (less than or similar to0.7) are unlikely; they produce an order-of-magnitude fewer massive clusters than observed.