  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.
