Virial velocity of a spherical-overdensity halo ($v_{\mathrm{vir}}=H(z)R_h\sqrt{{\Delta }_v/2}$)

For halo mean density ${\Delta }_v$ times the critical density at its epoch, the circular virial velocity obeys $v_{\mathrm{vir}}^2=G{M}_h/R_h=({\Delta }_v/2)H^2{R}_h^2$. Thus $v_{\mathrm{vir}}=H{R}_h\sqrt{{\Delta }_v/2}$. Specifying whether overdensity is relative to mean matter or critical density is necessary for numerical inference.

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.