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 $M_{\ast }$ of massive ($M_{\ast }\gtrsim 3\times {10}^{10}{\mathrm{M}}_{\odot }$) 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 ($0\lesssim z\lesssim 2.5$)\mathrm{.}Basedonthefiducialsample,wefindthatthe$M_$-$\sigma_$\mathrm{e}$$relationiswellrepresentedby$\sigma_$\mathrm{e}$\propto M_^{\beta}(1+z)^{\zeta}$,with$\beta\simeq 0.18$independentofredshiftand$\zeta\simeq 0.4$(atgiven$M_$,$\sigma_$\mathrm{e}$$decreasesfordecreasing$z$,forinstancebyafactorof$\approx1.3$from$z=1$to$z=0$)\mathrm{.}Whentheslope$\beta$ 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 $M_{\ast }$-${\sigma }_{\mathrm{e}}$ relation is $\simeq 0.08$ dex in ${\sigma }_{\mathrm{e}}$ at given $M_{\ast }$, independent of redshift. Our results suggest that, on average, the velocity dispersion of $individual$ massive ($M_{\ast }\gtrsim 3\times {10}^{11}{M}_{\odot }$) 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$\beta$varieswithtime\mathrm{.}At$z=2$ETGswith$M_\approx 10^{11},M_\odot$have,onaverage,$\approx1.7$higher$\sigma_$\mathrm{e}$$thanETGsofsimilarstellarmassat$z=0$.