The cosmic evolution of the stellar mass$-$velocity dispersion relation
  of early-type galaxies
  We study the evolution of the observed correlation between central stellar
velocity dispersion $\sigma_\mathrm{e}$ and stellar mass $M_*$ of massive
($M_*\gtrsim 3\times 10^{10}\,\mathrm{M_\odot}$) early-type galaxies (ETGs) out
to redshift $z\approx 2.5$, exploiting a Bayesian hierarchical inference
formalism. Collecting ETGs from state-of-the-art literature samples, we build a
$fiducial$ sample ($0\lesssim z\lesssim 1$), which is obtained with homogeneous
selection criteria, but also a less homogeneous $extended$ sample ($0\lesssim
z\lesssim 2.5$). Based on the fiducial sample, we find that the
$M_*$-$\sigma_\mathrm{e}$ relation is well represented by
$\sigma_\mathrm{e}\propto M_*^{\beta}(1+z)^{\zeta}$, with $\beta\simeq 0.18$
independent of redshift and $\zeta\simeq 0.4$ (at given $M_*$,
$\sigma_\mathrm{e}$ decreases for decreasing $z$, for instance by a factor of
$\approx1.3$ from $z=1$ to $z=0$). When the slope $\beta$ is allowed to evolve,
we find it increasing with redshift: $\beta(z)\simeq 0.16+0.26\log(1+z)$
describes the data as well as constant $\beta\simeq 0.18$. The intrinsic
scatter of the $M_*$-$\sigma_\mathrm{e}$ relation is $\simeq0.08$ dex in
$\sigma_\mathrm{e}$ at given $M_*$, independent of redshift. Our results
suggest that, on average, the velocity dispersion of $individual$ massive
($M_*\gtrsim 3\times 10^{11}\,M_\odot$) ETGs decreases with time while they
evolve from $z\approx 1$ to $z\approx 0$. The analysis of the extended sample
leads to results similar to that of the fiducial sample, with slightly stronger
redshift dependence of the normalisation ($\zeta\simeq 0.5$) and weaker
redshift dependence of the slope (${\rm d} \beta/{\rm d} \log (1+z)\simeq
0.18$) when $\beta$ varies with time. At $z=2$ ETGs with $M_*\approx
10^{11}\,M_\odot$ have, on average, $\approx1.7$ higher $\sigma_\mathrm{e}$
than ETGs of similar stellar mass at $z=0$.
