We compute the number counts of clusters of galaxies , the log N –log S relation , in several X-ray and submm bands on the basis of the Press–Schechter theory .
We pay particular attention to a set of theoretical models which well reproduce the ROSAT 0.5-2 keV band log N –log S , 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 Large Millimeter and Submillimeter Array ( LMSA ) project .
We exhibit that one can break the degeneracy in an acceptable parameter region on the \Omega _ { 0 } - \sigma _ { 8 } plane by combining the ROSAT log N –log S and the submm number counts .
Models which reproduce the ROSAT band log N –log S will have N ( > S ) \sim ( 150 - 300 ) ( S / 10 ^ { -12 } \mbox { erg~ { } cm$ { } ^ { -2 } $~ { } s$ { } ^ { -1 } $ } ) ^ { -1.3 } str ^ { -1 } at S \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 10 ^ { -12 } erg cm ^ { -2 } s ^ { -1 } in the ASCA 2-10 keV band , and N ( > S _ { \nu } ) \sim ( 10 ^ { 2 } -10 ^ { 4 } ) ( S _ { \nu } / 100 \mbox { mJy } ) ^ { -1.5 } \mbox { str } ^ { -1 } at S _ { \nu } \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 100 \mbox { mJy } in the submm ( 0.85mm ) band .
The amplitude of the log N –log S 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 .
The results , although still preliminary , point to low density ( \Omega _ { 0 } \sim 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 .