Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 53 / 1 / ii / Solution

For $0<\Omega <1$, choose the Big Bang to occur at $\tau =t=0$ and retain the expanding solution. The first integral found above isUse $a=(\Omega /b){\sinh }^{2}u$. Substitution gives $u^{\prime }=H_0\sqrt{b}/2=\alpha /2$, so $u=\alpha \tau /2$. This gives the flat matter-coasting-fluid Friedmann solution:Integrate $dt/d\tau =a$, with the same zero of time:As a check, $a^{\prime }=(\Omega \alpha /2b)\sinh (\alpha \tau )$, and the identity ${\sinh }^{2}v=(\cosh v-1)(\cosh v+1)$ verifies $a^{\prime 2}=H_0^2(\Omega a+b{a}^2)$. Also $t^{\prime }=a$ exactly. At early times $a\simeq \Omega H_0^2{\tau }^2/4$ and $t\simeq \Omega H_0^2{\tau }^3/12$, reproducing the matter-dominated relation $a\propto t^{2/3}$.

The endpoint $\Omega =1$ is obtained by taking the smooth limit: $a=H_0^2{\tau }^2/4$ and $t=H_0^2{\tau }^3/12$. At $\Omega =0$ the coasting cosmic-string universe instead has $a=H_{0}t$ and $\mathcal{H}=H_0$. Its Big Bang is at $\tau =-\infty$, so a finite conformal-time origin at the bang is no longer available; choosing $a=1$ at $\tau =0$ gives $a=e^{H_0\tau }$ and $t=H_0^{-1}{e}^{H_0\tau }$. The physical-age limit remains regular.