  To predict the X-ray observables associated to the diffuse baryons in
clusters of galaxies, we develop a new physical approach to model such a hot
intra-cluster plasma. Such approach is based on punctuated equilibria. and
comprises the following blocks: Monte Carlo ``merging histories'' of dark
matter potential wells; the central hydrostatic disposition for the ICP, reset
to a new equilibrium after each merging episode; conditions of shock, or of
closely adiabatic compression at the boundary with the external gas, preheated
by stellar energy feedbacks. We predict the L-T relation, consistent with the
data as for shape and scatter. This we combine with the mass distribution
provided by the hierarchical clustering for different COBE-normalized CDM
models, to predict the z-resolved luminosity functions, the source counts, the
redshift distributions and contribution of the unresolved groups and clusters
to the soft X-ray background. When compared with the recent ROSAT surveys, our
results confirm that the critical cosmology with Standard CDM is ruled out by
its overproduction of local clusters. On account of underproduction, instead,
we rule out open cosmologies except for a narrow range around \Omega_o=0.5;
even there, we find the consistency with the full data base to be hardly
marginal. For \Omega_o=0.3 in flat geometry, we obtain acceptable fits. For the
tilted CDM perturbation spectrum with high baryonic content in the critical
universe, we obtain marginal consistency. Finally, we discuss the effective
limitations of X-ray clusters and groups as cosmological signposts, and their
brighter prospects toward the astrophysics of the ICP and the cosmogony of
large, high-contrast structures.
