The Sunyaev-Zel'dovich Effect and the Value of Omega

We consider the Sunyaev-Zel'dovich (SZ) effect as a probe of Omega_o: Using a self-consistent modeling of X-ray clusters, we examine the dependence of both the mean Compton y parameter and the SZ source counts on Omega_o. These quantities increase with decreasing Omega_o due to the earlier epoch of structure formation in low-density cosmogonies; the results depend only on the quantity of gas heated to the virial temperature of collapsed objects and are independent of the spatial distribution of the gas in the potential wells. Specifically, we compare two models which reproduce the present-day abundance of clusters - a biased, critical universe and an unbiased, open model with Omega_o=0.3. We find that the mean y parameter approaches the current FIRAS limit of y<2.5 10^{-5} for the open model, demonstrating the importance of improving this limit on spectral distortions, and that the SZ source counts and corresponding redshift distribution differ significantly between the two cosmogonies; millimeter surveys covering a large area should thus provide interesting constraints on the density parameter of the Universe and on the evolution of the heated gas fraction in virialized objects.