Linear spherical-collapse threshold (${\delta }_c$)

In an Einstein-de Sitter universe, extrapolating the linear growing mode to the time when the spherical top-hat solution collapses gives

With linear growth normalized to unity today, a collapse threshold ${\delta }_{\mathrm{sc}}$ at epoch $z$ corresponds to a present-extrapolated initial barrier ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$. Comparing this barrier to present variance is equivalent to comparing ${\delta }_{\mathrm{sc}}$ to variance evolved to the collapse epoch. Mixing the two conventions counts growth twice.