LAUSR

Velocity dispersion (σ) is a key driver for galaxy structure and evolution. We here present a comprehensive semi-empirical approach to compute σ via detailed Jeans modelling assuming both a constant and scale-dependent mass-to-light ratio M*/L. We compare with a large sample of local galaxies from MaNGA and find that both models can reproduce the Faber-Jackson (FJ) relation and the weak dependence of σ on bulge-to-total ratio B/T (for B/T ≳ 0.25). The dynamical-to-stellar mass ratio within R ≲ Re can be fully accounted for by a gradient in M*/L. We then build velocity dispersion evolutionary tracks σap[M*, z] (within an aperture) along the main progenitor dark matter haloes assigning stellar masses, effective radii and Sérsic  indices via a variety of abundance matching and empirically motivated relations. We find: 1) clear evidence for downsizing in σap[M*, z] along the progenitor tracks; 2) at fixed stellar mass σ∝(1 + z)0.2 − 0.3 depending on the presence or not of a gradient in M*/L. We extract σap[M*, z] from the TNG50 hydrodynamic simulation and find very similar results to our models with constant M*/L. The increasing dark matter fraction within Re tends to flatten the σap[M*, z] along the progenitors at z ≳ 1 in constant M*/L models, while σap[M*, z] have a steeper evolution in the presence of a stellar gradient. We then show that a combination of mergers and gas accretion are likely responsible for the constant or increasing σap[M*, z] with time. Finally, our σap[M*, z] are consistent with a nearly constant and steep Mbh − σ relation at z ≲ 2, with black hole masses derived from the LX − M* relation.