  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 Omega_M ~ 0.3 with a 1-sigma statistical
error ~0.1, 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 analyses the cluster mass-temperature relation, with the intention of
identifying and reducing the systematic errors it introduces into measurements
of cosmological 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-Omega_M universe. We
also analyze the effects of current systematic uncertainties in the M-T
relation and show that they introduce a systematic uncertainty of ~0.1 in the
best-fitting Omega_M. Future improvements in the accuracy of the M-T relation
will most likely come from comparisons of predicted cluster temperature
functions with temperature functions derived directly from large-scale
structure simulations.
