The Friedmann equation is

$$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\rho-\frac{k c^{2}}{R^{2} a^{2}}\right)$$

Briefly explain the meaning of $H, \rho, k$ and $R$.

Derive the Raychaudhuri equation,

$$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3 c^{2}}(\rho+3 P),$$

where $P$ is the pressure, stating clearly any results that are required.

Assume that the strong energy condition $\rho+3 P \geqslant 0$ holds. Show that there was necessarily a Big Bang singularity at time $t_{B B}$ such that

$$t_{0}-t_{B B} \leqslant H_{0}^{-1}$$

where $H_{0}=H\left(t_{0}\right)$ and $t_{0}$ is the time today.