 Einstein-de Sitter virial overdensity

For ideal isolated spherical collapse, energy conservation and the virial theorem give $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$ and ${\rho }_{\mathrm{vir}}=8{\rho }_{\mathrm{ta}}$ if the density-profile shape is unchanged. The cycloid gives $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$ and ${\rho }_{\mathrm{ta}}/\bar{\rho }(t_{\mathrm{ta}})=9{\pi }^2/16$. Since the Einstein-de Sitter universe has $\bar{\rho }\propto t^{-2}$, the overdensity at the nominal collapse time is $32(9{\pi }^2/16)=18{\pi }^2$. Hence ${\rho }_{\mathrm{vir}}=18{\pi }^2{\rho }_0(1+z_{\mathrm{coll}}{)}^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.