On the generalized Faber-Jackson relation for galaxy clusters
  The significant deviations among observations and the expectations based on
self-similar scaling model of galaxy clusters, especially up to redshift
$z\lesssim 0.4$, constrain the evolution of the X-ray clusters scaling
relations with the redshift, is claimed that in this redshift range, the data
has a strong influence by selection bias. However, also suggests that some
non-gravitational processes can be responsible for a weak or almost null
evolution, at least to $z\lesssim 0.4$. This almost universality observed in
X-ray galaxy clusters can be understood if we assume that the X-ray emission,
results from thermal bremsstrahlung from a hot diffuse intracluster gas with
temperatures about $10^8$ K. A fraction of it would not be bound to the cluster
and would escape as a wind. This hot wind can warm the local environment, the
thermal bath where the cluster is immersed. This mechanism can put all the
galaxy clusters within thermal baths, with almost the same effective
temperature, independent of the cluster redshift and it can be effective for
clusters with redshifts up to $z\sim 0.4$.
  Debye Gravitational Theory (DGT), allows obtaining a Generalized
Faber-Jackson relation to described the galaxy clusters such as the M-$\sigma$
and M-Tx relations as a function of the bath thermal temperature.
  We show that the DGT prediction to the M-$\sigma$ relation, overlap the fit
on data of an extensive spectroscopic survey of galaxy clusters with
MMT/Hectospec, at 0.1$\leq$ z $\leq$ 0.3. And the DGT predictions to the M-Tx
relation almost overlap the fit on data from Canada France Hawaii Telescope
Lensing Survey and XMM-CFHTLS surveys up to $z\sim 0.47$.
