 Virial velocity of a spherical-overdensity halo

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.