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In analytical models of structure formation, protohaloes are routinely assumed to be peaks of the smoothed initial density field, with the smoothing filter being spherically symmetric. This works reasonably well to identify a protohalo’s centre of mass, but not its shape. To provide a more realistic description of protohalo boundaries, one must go beyond the spherical picture. We suggest that this can be done by looking for regions of fixed volume, but arbitrary shape, that minimize the enclosed energy. Such regions are delimited by surfaces over which (a slightly modified version of) the gravitational potential is constant. We show that these equipotential surfaces provide an excellent description of protohalo shapes, orientations, and associated torques.

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 (B/T) ratio (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 is 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.