Virial radius of a dark-matter halo ($r_v$)

The virial radius of a dark-matter halo is conventionally the radius within which the mean density is a chosen overdensity $v$ times a reference critical or matter density.

If a halo retains the mean density acquired at collapse, $3M/(4\pi R^3)={\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ estimates its formation redshift. In the non-dissipative spherical-collapse model during matter domination, ${\Delta }_{\mathrm{vir}}=18{\pi }^2$. Subsequent mass growth, profile evolution and the reference-density convention limit this inference.

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.

If the virial radius is defined relative to critical density ${\rho }_c$, the virial mass is