A theoretical study of the mass temperature relation for clusters of   galaxies
  I derive the mass-temperature relation and its time evolution for clusters of galaxies in different cosmologies by means of two different models. The first one is a modification and improvement of a model by Del Popolo & Gambera(1999), namely based upon a modification of the top-hat model in order to take account of angular momentum acquisition by protostructures and of an external pressure term in the virial theorem. The second one is based on the merging-halo formalism of Lacey & Cole (1993), accounting for the fact that massive clusters accrete matter quasi-continuously, and is an improvement of a model proposed by Voit (2000). The final result is that, in both models, the M-T relation shows a break at T \sim 3-4 keV. The behavior of the M-T relation is as usual, M \propto T^{3/2}, at the high mass end, and M \propto T^{\gamma}, with a value of \gamma>3/2 depending on the chosen cosmology. The evolution of the M-T relation, for a given M_{\rm vir}, is more modest both in flat and open universes in comparison to previous estimate found in literature, even more modest than what found by V2000. Moreover the time evolution is more rapid in models with L=0 than in models in which the angular momentum acquisition by protostructures is taken into account (L \neq 0). The effect of a non-zero cosmological constant is that of slightly increasing the evolution of the M-T relation with respect to open models with L \neq 0. The evolution is more rapid for larger values (in absolute value) of the spectral index, n. The comparison of the mass-temperature relation with the data by Finoguenov, Reiprich & Bohringer (2001), shows that the FRB data is able to rule out very low Omega_0 models (<0.3), particularly in the open case, and that better fit are obtained by LambdaCDM models and by CDM models with Omega_0>0.3. 
A. Del Popolo (1,2,3) ((1) Dipartimento di Matematica, Universita'   Statale di Bergamo, Bergamo, ITALY, (2) Feza Gursey Ins...