 Virial radius of a dark-matter halo

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.