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

The virial radius of a dark-matter halo is usually a specified spherical-overdensity boundary: $M_{\Delta }=(4\pi /3)\Delta {\rho }_{\mathrm{ref}}{r}_{\Delta }^3$. Here ${\rho }_{\mathrm{ref}}$ can be the mean matter density or the mass-equivalent cosmological critical density. The same halo therefore receives different quoted masses and radii under different conventions. This useful comparison radius is not a sharp material edge; the density and orbital population can extend beyond it. The chosen overdensity and reference density must accompany numerical values.

In a spherical-overdensity halo convention, $3M/(4\pi R^3)={\Delta }_f{\rho }_0(1+z_f{)}^3$, so $1+z_f$ follows from the displayed cube root. At fixed measured $M,R,{\rho }_0$, changing the adopted overdensity changes $1+z_f\propto {\Delta }_f^{-1/3}$. It does not change the hydrostatic temperature estimate from the same measured mass and radius. A redshift inferred this way is a model formation epoch, not a direct age measurement.

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