Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 67 / 2 / a / Solution

Normalize the region's reference scale factor to one. Pressureless matter has $\rho ={\rho }_0{a}^{-3}$, so the closed Friedmann equation isThe expanding branch can reach only $a\le a_m$, where $a_m={\Omega }_m/({\Omega }_m-1)$. Substitute $a=a_m(1-\cos \theta )/2$. Since $da/d\theta =a_m\sin \theta /2$ and $\dot{a}=H_0\sqrt{({\Omega }_m-1)(a_m/a-1)}$, cancellation on $0<\theta <\pi$ givesIntegration from the Big Bang establishesThe same parametrization continues through spherical-collapse turnaround at $\theta =\pi$ onto the contracting branch up to $\theta =2\pi$. Its maximum-expansion time is $t_m=\pi {\Omega }_m/[2H_0({\Omega }_m-1{)}^{3/2}]$.