Cosmological implications of number counts of clusters of galaxies: log N log S in X-ray and submm bands

We have computed the number counts of clusters of galaxies, the log N-logS relations, in X-ray and submm bands based on the Press-Schechter theory. We paid particular attention to a set of theoretical models which well reproduce the ROSAT soft X-ray logN-logS, and explored possibilities to further constrain the models from future observations in hard X-ray and submm bands. The former is closely related to the ASCA, AXAF, and Astro-E missions, while the latter to the PLANCK and LMSA projects. We have shown that one can break the degeneracy in an acceptable parameter region on the Omega(0)-sigma g plane by combining the X-ray and submm number counts. Models which reproduce the ROSAT 0.5-2 keV band logN-logS will have N(> S) similar to (150-300)(S/10(-12) erg cm(-2) s(-1))(-1.3) str(-1) at S greater than or similar to 10(-12) erg cm(-2) s(-1) in the 2-10 keV band, and N(> S-upsilon) similar to (10(2)-10(4))(S-upsilon/100 mJy)(-1.5) str(-1) at S upsilon greater than or similar to 100 mJy in the submm (0.85 mm) band. The amplitude of logN-logS is very sensitive to the model parameters in the submm band. We also computed the redshift evolution of the cluster number counts and compared it with that of the X-ray brightest Abell-type clusters. The results, although still preliminary, point to low density (Omega(0) similar to 0.3) universes. The contribution of clusters to the X-ray and submm background radiations is shown to be insignificant in any model compatible with the ROSAT log N-log S relation.