= Einstein-de Sitter virial overdensity
{c}
{title2=$\rho_{\rm vir}/\bar\rho(t_{\rm coll})=18\pi^2$}
For ideal isolated <spherical collapse>, energy conservation and the <virial theorem> give $R_{\rm vir}=R_{\rm ta}/2$ and $\rho_{\rm vir}=8\rho_{\rm ta}$ if the density-profile shape is unchanged. The cycloid gives $t_{\rm coll}=2t_{\rm ta}$ and $\rho_{\rm ta}/\bar\rho(t_{\rm ta})=9\pi^2/16$. 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 $\rho_{\rm vir}=18\pi^2\rho_0(1+z_{\rm coll})^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.
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