Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 15D / b / Solution

The two scalings give $\rho ={\rho }_{r0}{a}^{-4}+{\rho }_{s0}{a}^{-2}$. Flatness and the Friedmann equation give $H^2=8\pi G\rho /3$, with $H_0^2=(8\pi G/3)({\rho }_{r0}+{\rho }_{s0})$. ThusOn the expanding branch, putting the Big Bang at zero, separation givesIts positive square root is the required solution. If $\beta >0$, its early behaviour is $a\propto t^{1/2}$, as for radiation. The values $a=1$ and $a^2=\beta$ giveFor $\beta =0$ there is only the string component, with linear expansion and no radiation era.