Consider a finite sphere of zero-pressure material of uniform density $\rho(t)$ which expands with radius $r(t)=a(t) r_{0}$, where $r_{0}$ is an arbitary constant, due to the evolution of the expansion scale factor $a(t)$. The sphere has constant total mass $M$ and its radius satisfies

$$\ddot{r}=-\frac{d \Phi}{d r}$$

where

$$\Phi(r)=-\frac{G M}{r}-\frac{1}{6} \Lambda r^{2} c^{2},$$

with $\Lambda$ constant. Show that the scale factor obeys the equation

$$\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{K c^{2}}{a^{2}}+\frac{1}{3} \Lambda c^{2},$$

where $K$ is a constant. Explain why the sign, but not the magnitude, of $K$ is important. Find exact solutions of this equation for $a(t)$ when

(i) $K=\Lambda=0$ and $\rho(t) \neq 0$,

(ii) $\rho=K=0$ and $\Lambda>0$,

(iii) $\rho=\Lambda=0$ and $K \neq 0$.

Which two of the solutions (i)-(iii) are relevant for describing the evolution of the universe after the radiation-dominated era?