Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 9B / Solution

For an infinitesimal test mass $\delta m$ on the surface of a homogeneous sphere, its Newtonian mechanical energy isThe spatially flat case corresponds to zero specific energy. Usingand dividing by $\delta m{R}^2/2$ givesSince $R$ is proportional to the scale factor $a$, this is the Newtonian derivation of the flat Friedmann equation

Now take $a(t_0)=1$ and $\rho ={\rho }_0{a}^{-4}$. PutThe contracting branch of the Friedmann equation isIntegrating $ada=-Cdt$ from $t_0$ givesThereforeon the contracting interval, reaching $a=0$ at $t=t_0+1/(2C)$.