On the evolution of the temperature-virial mass relation for clusters of galaxies

X-ray temperature measurements of clusters of galaxies are now reaching to redshifts high enough to constrain Ohm(0). A redshift-dependent relation that maps these X-ray temperatures to the virial masses of clusters is an essential ingredient when one is trying to determine cosmological parameters from cluster evolution. Most such relations assume that clusters form from top-hat perturbations that virialized just before the time at which we are observing them. The smaller Ohm(0) is, the less accurate these relations become. Here we derive a relation between virial mass and cluster temperature that allows for the fact that clusters form gradually and cease forming when the density of the universe drops well below the critical value. We show how sensitively the expected redshift distribution of clusters depends on the mass-temperature relation used and argue that one needs to use a relation that yields no evolution in the low-Ohm(0) limit.