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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 $σ_\mathrm{e}$ 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 $M_*$-$σ_\mathrm{e}$ relation is well represented by $σ_\mathrm{e}\propto M_*^β(1+z)^ζ$, with $β\simeq 0.18$ independent of redshift and $ζ\simeq 0.4$ (at given $M_*$, $σ_\mathrm{e}$ decreases for decreasing $z$, for instance by a factor of $\approx1.3$ from $z=1$ to $z=0$). When the slope $β$ is allowed to evolve, we find it increasing with redshift: $β(z)\simeq 0.16+0.26\log(1+z)$ describes the data as well as constant $β\simeq 0.18$. The intrinsic scatter of the $M_*$-$σ_\mathrm{e}$ relation is $\simeq0.08$ dex in $σ_\mathrm{e}$ 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 ($ζ\simeq 0.5$) and weaker redshift dependence of the slope (${\rm d} β/{\rm d} \log (1+z)\simeq 0.18$) when $β$ varies with time. At $z=2$ ETGs with $M_*\approx 10^{11}\,M_\odot$ have, on average, $\approx1.7$ higher $σ_\mathrm{e}$ than ETGs of similar stellar mass at $z=0$.