We examine the likelihoods of different cosmological models and cluster evolutionary histories by comparing semi-analytical predictions of X-ray cluster number counts to observational data from the ROSAT satellite. We model cluster abundance as a function of mass and redshift using a Press-Schechter distribution, and assume the temperature T(M,z) and bolometric luminosity L_X(M,z) scale as power laws in mass and epoch, in order to construct expected counts as a function of X-ray flux. The L_X-M scaling is fixed using the local luminosity function while the degree of evolution in the X-ray luminosity with redshift L_X propto (1+z)^s is left open, with s an interesting free parameter which we investigate. We examine open and flat cosmologies with initial, scale-free fluctuation spectra having indices n = 0, -1 and -2. An independent constraint arising from the slope of the luminosity-temperature relation strongly favors the n = -2 spectrum. The expected counts demonstrate a strong dependence on Omega_0 and s, with lesser dependence on lambda_0 and n. Comparison with the observed counts reveals a ridge of acceptable models in the Omega_0 - s plane, roughly following the relation s = 6 Omega_0 and spanning low-density models with a small degree of evolution to Omega = 1 models with strong evolution. Models with moderate evolution are revealed to have a strong lower limit of Omega_0 gtrsim 0.3, and low-evolution models imply that Omega_0 < 1 at a very high confidence level. We suggest observational tests for breaking the degeneracy along this ridge, and discuss implications for evolutionary histories of the intracluster medium.
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