Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 3 / b / Solution

Write $\Omega ={\Omega }_{M0}$, ${\beta }^2=H_0^2(1-\Omega )$ and $\mathcal{H}=a^{\prime }/a=\dot{a}$. The Friedmann equation isThe Raychaudhuri equation has only the matter component on its active-gravity side, soCombining these expressions provesThis is the conformal Riccati equation for matter and a coasting fluid. The prime denotes conformal time, whereas $H_0$ is the physical present Hubble parameter.