Present-extrapolated spherical-collapse barrier (${\delta }_c(z)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$)

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.