Constraining Omega with cluster evolution

We show that the evolution of the number density of rich clusters of galaxies breaks the degeneracy between Omega (the mass density ratio of the universe) and sigma(8) (the normalization of the power spectrum), sigma(8) Omega(0.5) similar or equal to 0.5, that follows from the observed present-day abundance of rich clusters. The evolution of high-mass (Coma-like) clusters is strong in Omega = 1, low-sigma(8) models (such as the standard biased cold dark matter model with sigma(8) similar or equal to 0.5), where the number density of clusters decreases by a factor of about 10(3) from z = 0 to z similar or equal to 0.5; the same clusters show only mild evolution in low-Omega, high-sigma(8) models, where the decrease is a factor of about 10. This diagnostic provides a most powerful constraint on Omega. Using observations of clusters to z similar or equal to 0.5-1, we find only mild evolution in the observed cluster abundance. We find Omega = 0.3 +/- 0.1 and sigma(8) = 0.85 +/- 0.15 (for Lambda = 0 models; for Omega + Lambda = 1 models, Omega = 0.34 +/- 0.13). These results imply, if confirmed by future surveys, that we live in a low-density, low-bias universe.