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We compute the number counts of clusters of galaxies, the logN-logS relation, in several X-ray and submm bands on the basis of the Press—Schechter theory (Kitayama et al. 1998). We pay particular attention to a set of theoretical models which well reproduce the ROSAT 0.5–2 keV band logN-logS (Ebeling et al. 1997; Rosati et al. 1997), and explore possibilities to further constrain the models from future observations with ASCA and/or at submm bands. The latter is closely related to the European PLANCK mission and the Japanese LMSA project. We exhibit that one can break the degeneracy in an acceptable parameter region on the Ω0–σ8 plane by combining the ROSAT logN-logS and the submm number counts. Models which reproduce the ROSAT band logN-logS will have N(> S) ∼ (150–300)(S/10−12 erg cm−2 s−) −1.3 str−1 at S ≳ 10−12 erg cm−2 s−1 in the ASCA 2–10 keV band, and N(> Sv) ∼ (102–104)(Sv/100 mJy)−1.5 str−1 at Sv ≳ 100m J y in the submm (0.85mm) band. The amplitude of the logN-logS is very sensitive to the model parameters in the submm band. We also compute the redshift evolution of the cluster number counts and compare with that of the X-ray brightest Abell-type clusters (Ebeling et al. 1996). The results, although still preliminary, point to low density (Ω0 ∼ 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 logN-logS.
