Measuring \Omega_0 from the Entropy Evolution of Clusters

In this paper, we have extended the entropy-driven model of cluster evolution developed by Bower (1997) in order to be able to predict the evolution of galaxy clusters for a range of cosmological scenarios. We have applied this model to recent measurements of the evolution of the L_x-T normalisation and X-ray luminosity function in order to place constraints on cosmological parameters. We find that these measurements alone do not select a particular cosmological frame-work. An additional constraint is required on the effective slope of the power spectrum to break the degeneracy that exists between this and the background cosmology. We have therefore included a theoretical calculation of the \Omega_0 dependence on the power spectrum, based on the cold dark matter paradigm, which infers \Omega_0<0.55 (0.1<\Omega_0<0.7 for \Omega_0+\Lambda_0=1), at the 95% confidence level. Alternatively, an independent measurement of the slope of the power spectrum from galaxy clustering requires \Omega_0<0.6 (\Omega_0<0.65 for \Omega_0+\Lambda_0=1), again to 95% confidence. The rate of entropy evolution is insensitive to the values of \Omega_0 considered, although is sensitive to changes in the distribution of the intracluster medium.