 Dimensionless spherical-collapse time normalization

For spherical collapse in an Einstein-de Sitter universe, take $y=R/R_{\mathrm{ta}}$ and $\tau =H_{\mathrm{ta}}t$, with ${\Delta }_{\mathrm{ta}}=(3\pi /4{)}^2$. The expanding branch has $\tau ={\Delta }_{\mathrm{ta}}^{-1/2}{\int }_0^y[u/(1-u){]}^{1/2}du$. Its small-radius expansion is $\tau =(8/9\pi )y^{3/2}(1+3y/10+O(y^2))$, and its turnaround time is $2/3$. These constants are linked: using another leading coefficient without changing the time normalization no longer gives a density contrast approaching zero at early times. The collapsing branch is $4/3-{\tau }_{\mathrm{expand}}(y)$.