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} \simeq 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 CDM model with sigma_{8} \simeq 0.5), where the number density of clusters decreases by a factor of \sim 10^{3} from z = 0 to z \simeq 0.5; the same clusters show only mild evolution in low-Omega, high-sigma_{8} models, where the decrease is a factor of \sim 10. This diagnostic provides a most powerful constraint on Omega. Using observations of clusters to z \simeq 0.5-1, we find only mild evolution in the observed cluster abundance. We find Omega = 0.3 \pm 0.1 and sigma_{8} = 0.85 \pm 0.15 (for Lambda = 0 models; for Omega + Lambda = 1 models, Omega = 0.34 \pm 0.13). These results imply, if confirmed by future surveys, that we live in a low-den sity, low-bias universe.