 Halo virial-velocity conversion between epochs

For a fixed mass under a spherical-overdensity definition, $R_h\propto ({\Delta }_c{H}^2{)}^{-1/3}$ and $v_h^2=G{M}_h/R_h$. Thus equivalent virial mass scales at two epochs have the displayed velocity ratio. With fixed overdensity during matter domination, $v_h\propto (1+z{)}^{1/2}$. This converts an epoch-specific mass–velocity calibration; it does not require a previously collapsed bound halo to physically expand with the later background density.