We find that the observed log N - log S relation of X-ray clusters can be reproduced remarkably well with a certain range of values for the fluctuation amplitude \sigma _ { 8 } and the cosmological density parameter \Omega _ { 0 } in cold dark matter ( CDM ) universes .
The 1 \sigma confidence limits on \sigma _ { 8 } in the CDM models with n = 1 and h = 0.7 are expressed as ( 0.54 \pm 0.02 ) \Omega _ { 0 } ^ { -0.35 - 0.82 \Omega _ { 0 } +0.55 \Omega _ { 0 } ^ { 2 } } ( \lambda _ { 0 } = 1 - \Omega _ { 0 } ) and ( 0.54 \pm 0.02 ) \Omega _ { 0 } ^ { -0.28 - 0.91 \Omega _ { 0 } +0.68 \Omega _ { 0 } ^ { 2 } } ( \lambda _ { 0 } = 0 ) , where n is the primordial spectral index , and h and \lambda _ { 0 } are the dimensionless Hubble and cosmological constants .
The errors quoted above indicate the statistical ones from the observed log N - log S only , and the systematic uncertainty from our theoretical modelling of X-ray flux in the best-fit value of \sigma _ { 8 } is about 15 % .
In the case of n = 1 , we find that the CDM models with ( \Omega _ { 0 } , \lambda _ { 0 } ,h, \sigma _ { 8 } ) \simeq ( 0.3 , 0.7 , 0.7 , 1 ) and ( 0.45 , 0 , 0.7 , 0.8 ) simultaneously account for the cluster log N - log S , X-ray temperature functions , and the normalization from the COBE 4 year data .
The derived values assume the observations are without systematic errors , and we discuss in details other theoretical uncertainties which may change the limits on \Omega _ { 0 } and \sigma _ { 8 } from the log N - log S relation .
We have shown the power of this new approach which will become a strong tool as the observations attain more precision .