(i) In a homogeneous and isotropic universe, the scalefactor $a(t)$ obeys the Friedmann equation

$$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho,$$

where $\rho(t)$ is the matter density which, together with the pressure $P(t)$, satisfies

$$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+P / c^{2}\right)$$

Use these two equations to derive the Raychaudhuri equation,

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

(ii) Conformal time $\tau$ is defined by taking $d t / d \tau=a$, so that $\dot{a}=a^{\prime} / a \equiv \mathcal{H}$ where primes denote derivatives with respect to $\tau$. For matter obeying the equation of state $P=w \rho c^{2}$, show that the Friedmann and energy conservation equations imply

$$\mathcal{H}^{2}+k c^{2}=\frac{8 \pi G}{3} \rho_{0} a^{-(1+3 w)}$$

where $\rho_{0}=\rho\left(t_{0}\right)$ and we take $a\left(t_{0}\right)=1$ today. Use the Raychaudhuri equation to derive the expression

$$\mathcal{H}^{\prime}+\frac{1}{2}(1+3 w)\left[\mathcal{H}^{2}+k c^{2}\right]=0$$

For a $k c^{2}=1$ closed universe, by solving first for $\mathcal{H}$ (or otherwise), show that the scale factor satisfies

$$a=\alpha(\sin \beta \tau)^{2 /(1+3 w)}$$

where $\alpha, \beta$ are constants. [Hint: You may assume that $\int d x /\left(1+x^{2}\right)=-\cot ^{-1} x+$ const.]

For a closed universe dominated by pressure-free matter $(P=0)$, find the complete parametric solution

$$a=\frac{1}{2} \alpha(1-\cos 2 \beta \tau), \quad t=\frac{\alpha}{4 \beta}(2 \beta \tau-\sin 2 \beta \tau)$$