Abstract
The observed evolution of the galaxy cluster X-ray integral temperature distribution function between $z=0.05$ and $z=0.32$ is used in an attempt to constrain the value of the density parameter, $\Omega_{0}$, for both open and spatially-flat universes. We estimate the overall uncertainty in the determination of both the observed and the predicted galaxy cluster X-ray integral temperature distribution functions at $z=0.32$ by carrying out Monte Carlo simulations, where we take into careful consideration all the most important sources of possible error. We include the effect of the formation epoch on the relation between virial mass and X-ray temperature, improving on the assumption that clusters form at the observed redshift which leads to an {\em overestimate} of $\Omega_0$. We conclude that at present both the observational data and the theoretical modelling carry sufficiently large associated uncertainties to prevent an unambiguous determination of $\Omega_{0}$. We find that values of $\Omega_{0}$ around 0.75 are most favoured, with $\Omega_{0}<0.3$ excluded with at least 90 per cent confidence. In particular, the $\Omega_{0}=1$ hypothesis is found to be still viable as far as this dataset is concerned. As a by-product, we also use the revised data on the abundance of galaxy clusters at $z=0.05$ to update the constraint on $\sigma_8$ given by Viana & Liddle 1996, finding slightly lower values than before.
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