Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 1 / ii / Solution

Write $y=a/a_0$. Conservation of pressureless matter gives ${\rho }_m={\rho }_{m,0}{y}^{-3}$. The Friedmann equation, retaining spatial curvature, isSince $\dot{y}=yH$ and ${\Omega }_{m,0}=2q_0$, multiplication by $y^2$ givesIn the curvature convention $H^2=8\pi G{\rho }_m/3-k{c}^2/a^2$, this also fixes $-k{c}^2/(a_0^2{H}_0^2)=1-2q_0$.