Measuring the Cosmic Equation of State with Galaxy Clusters in the DEEP2 Redshift Survey

The abundances of groups and clusters of galaxies are expected to have changed substantially since high redshift, with the strength of this evolution dependent upon fundamental cosmological parameters. Upcoming large redshift surveys of distant galaxies will make it possible to measure these quantities at z~1; when combined with the results of local redshift surveys currently underway, the evolution of cluster abundances may be determined. The DEEP2 Redshift Survey, planned to begin in Spring 2002, is particularly well-suited for this work because of the high spectroscopic resolution to be used; velocity dispersions of groups will be readily measurable. In this paper, we determine the constraints on dark energy models that counts of DEEP2 clusters should provide. We find that comparing cosmological models using the simultaneous distribution of clusters in both velocity dispersion and redshift yields significantly stronger constraints than the redshift distribution alone. The method can be made more powerful by employing a value of the fluctuation amplitude sigma_8 determined with upcoming techniques (external to DEEP2) that have no Omega_m degeneracy. The equation-of-state parameter for dark energy models, w=P/rho, can then be measured to ~0.1 from observations of clusters alone.