Constraints on Q0 and cluster evolution using the ROSAT log N–log S relation

We examine the likelihoods of different cosmological models and cluster evolutionary histories by comparing semi-analytical predictions of X-ray cluster number counts with observational data from the ROSAT satellite. We model cluster abundance as a function of mass and redshift using a Press–Schechter distribution, and assume that the temperature TðM; zÞ and bolometric luminosity LXðM; zÞ scale as power laws in mass and epoch, in order to construct expected counts as a function of X-ray flux. The LX 1 M scaling is fixed using the local luminosity function, while the degree of evolution in the X-ray luminosity with redshift LX ~ ð1 þ zÞ s is left open, with s an interesting free parameter which we investigate. We examine open and flat cosmologies with initial, scale-free fluctuation spectra having indices n 1⁄4 0, 11 and 12. An independent constraint arising from the slope of the luminosity– temperature relation strongly favours the n 1⁄4 12 spectrum. The expected counts demonstrate a strong dependence on Q0 and s, with lesser dependence on l0 and n. Comparison with the observed counts reveals a ‘ridge’ of acceptable models in the Q0 1 s plane, roughly following the relation s , 6Q0 and spanning low-density models with a small degree of evolution to Q 1⁄4 1 models with strong evolution. Models with moderate evolution are revealed to have a strong lower limit of Q0 * 0:3, and low-evolution models imply that Q0 < 1 at a very high confidence level. We suggest observational tests for breaking the degeneracy along this ridge, and discuss implications for evolutionary histories of the intracluster medium.