Evolution of the cluster temperature function is extremely sensitive to the mean matter density of the universe. Current measurements based on cluster temperature surveys indicate that Ω_m≈ 0.3 with a 1 a statistical error ~0.l, but the systematic errors in this method are of comparable size. Many more high-z cluster temperatures will be arriving from Chandra and XMM in the near future. In preparation for future cluster temperature surveys, this paper analyzes the cluster mass-temperature relation, with the intention of identifying and reducing the systematic errors it introduces into measurements of cosmo- logical parameters. We show that the usual derivation of this relation from spherical top-hat collapse is physically inconsistent and propose a more realistic derivation based on a hierarchical merging model that more faithfully reflects the gradual ceasing of cluster evolution in a low-Ω_m . universe. We also analyze the effects of current systematic uncertainties in the M_vir-TX relation and show that they intro- duce a systematic uncertainty of ~0.l in the best-fittingΩ_m Future improvements in the accuracy of the M_vir-Tx relation will most likely come from comparisons of predicted cluster temperature functions with temperature functions derived directly from large-scale structure simulations.
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