Constraints on the fluctuation amplitude and density parameter from X-ray cluster number counts

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) universe. The 1 sigma confidence limits on sigma(8) in the CDM models with n = 1 and h = 0.7 are expressed as (0.54 +/- 0.02)Omega(0)(-0.35-0.82 Omega 0 + 0.55 Omega 02) (lambda(0) = 1 - Omega(0)) and (0.54 +/- 0.02)Omega(0)(-0.28-0.91 Omega 0+0.68 Omega 02) (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 statistical errors from the observed log N-log S only; the systematic uncertainty from our theoretical modeling of X-ray flux in the best-fit value of sigma(8) is about 15%. In the case where n = 1, we find that the CDM models with (Omega(0), lambda(0), h, sigma(8)) similar or equal to (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 that the observations are without residual systematic errors, and we discuss in detail other theoretical uncertainties that may change the limits on Omega(0) and sigma(8) from the log N-log S relation. We show the power of this new approach, which will become a strong tool as observations attain more precision.