Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / c / Solution

For pressureless matter, ${\rho }_m{a}^3={\rho }_{m,0}$ when $a_0=1$. Setthen ${\dot{a}}^2=D/a-k$. Substitution of $a=A(1-\cos \theta )$ makes this identity hold whenIndeed $dt/d\theta =B(1-\cos \theta )=Ba/A$, and henceafter setting $t=0$ at $\theta =0$. This is the closed matter-dominated Friedmann solution.