Cosmological Constraints from a 31 GHz Sky Survey with the Sunyaev-Zel'dovich Array

We present the results of a 6.1 square degree survey for clusters of galaxies via their Sunyaev- Zel'dovich (SZ) effect at 31 GHz. From late 2005 to mid 2007 the Sunyaev-Zel'dovich Array (SZA) observed four fields of roughly 1.5 square degrees each. One of the fields shows evidence for significant diffuse Galactic emission, and we therefore restrict our analysis to the remaining 4.4 square degrees. We estimate the cluster detectability for the survey using mock observations of simulations of clusters of galaxies; and determine that, at intermediate redshifts (z ~ 0.8), the survey is 50% complete to a limiting mass (M200 rho mean) of ~ 6.0 x 10^14M_{solar}, with the mass limit decreasing at higher redshifts. We detect no clusters at a significance greater than 5 times the RMS noise level in the maps, and place an upper limit on \sigma_8, the amplitude of mass density fluctuations on a scale of 8h^-1 Mpc, of 0.84 + 0.07 at 95% confidence, where the uncertainty reflects calibration and systematic effects. This result is consistent with estimates from other cluster surveys and CMB anisotropy experiments.