The velocity function of dark matter halos at r=20 kpc: Remarkably little evolution since z ~ 4

We investigate the evolution in the dark matter halo circular velocity function, measured at a fixed physical radius of 20 kpc (v_20), which is likely to be a good proxy for galaxy circular velocity, in the Millennium-II simulation. We find that the v_20 function evolves remarkably little since z ~ 4. We analyze the histories of the main progenitors of halos, and we find that the dark matter distribution within the central 20 kpc of massive halos has been in place since early times. This provides evidence for the inside-out growth of haloes. The constancy of the central circular velocity of halos may offer a natural explanation for the observational finding that the galaxy circular velocity is an excellent predictor of various galaxy properties. Our results also indicate that we can expect a significant number of galaxies with high circular velocities already at z=4 (more than one per 10^6 (Mpc/h)^3 with circular velocities in excess of 450 km/s, and more than one per 10^4.5 (Mpc/h)^3 with circular velocities in excess of 350 km/s). Finally, adding baryonic mass and using a simple model for halo adiabatic contraction, we find remarkable agreement with the velocity dispersion functions inferred observationally by Bezanson et al. (2011) up to z ~ 1 and down to about 220 km/s.