  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.
