The metric for a homogenous isotropic universe, in comoving coordinates, can be written as

$$d s^{2}=-d t^{2}+a^{2}\left\{d r^{2}+f^{2}\left[d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right]\right\}$$

where $a=a(t)$ and $f=f(r)$ are some functions.

Write down expressions for the Hubble parameter $H$ and the deceleration parameter $q$ in terms of $a(\eta)$ and $h \equiv d \log a / d \eta$, where $\eta$ is conformal time, defined by $d \eta=a^{-1} d t$.

The universe is composed of a perfect fluid of density $\rho$ and pressure $p=(\gamma-1) \rho$, where $\gamma$ is a constant. Defining $\Omega=\rho / \rho_{c}$, where $\rho_{c}=3 H^{2} / 8 \pi G$, show that

$$\frac{k}{h^{2}}=\Omega-1, \quad q=\alpha \Omega, \quad \frac{d \Omega}{d \eta}=2 q h(\Omega-1)$$

where $k$ is the curvature parameter $(k=+1,0$ or $-1)$ and $\alpha \equiv \frac{1}{2}(3 \gamma-2)$. Hence deduce that

$$\frac{d \Omega}{d a}=\frac{2 \alpha}{a} \Omega(\Omega-1)$$

and

$$\Omega=\frac{1}{1-A a^{2 \alpha}}$$

where $A$ is a constant. Given that $A=\frac{k}{2 G M}$, sketch curves of $\Omega$ against $a$ in the case when $\gamma>2 / 3$.

[You may assume an Einstein equation, for the given metric, in the form

$$\frac{h^{2}}{a^{2}}+\frac{k}{a^{2}}=\frac{8}{3} \pi G \rho$$

and the energy conservation equation

$$\left.\frac{d \rho}{d t}+3 H(\rho+p)=0 .\right]$$