(a) Consider a homogeneous and isotropic universe with scale factor $a(t)$ and filled with mass density $\rho(t)$. Show how the conservation of kinetic energy plus gravitational potential energy for a test particle on the edge of a spherical region in this universe can be used to derive the Friedmann equation

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

where $k$ is a constant. State clearly any assumptions you have made.

(b) Now suppose that the universe was filled throughout its history with radiation with equation of state $P=\rho c^{2} / 3$. Using the fluid conservation equation and the definition of the relative density $\Omega$, show that the density of this radiation can be expressed as

$$\rho=\frac{3 H_{0}^{2}}{8 \pi G} \frac{\Omega_{0}}{a^{4}},$$

where $H_{0}$ is the Hubble parameter today and $\Omega_{0}$ is the relative density today $\left(t=t_{0}\right)$ and $a_{0} \equiv a\left(t_{0}\right)=1$ is assumed. Show also that $k c^{2}=H_{0}^{2}\left(\Omega_{0}-1\right)$ and hence rewrite the Friedmann equation $(*)$ as

$$\tag{†} \left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2} \Omega_{0}\left(\frac{1}{a^{4}}-\frac{\beta}{a^{2}}\right)$$

where $\beta \equiv\left(\Omega_{0}-1\right) / \Omega_{0}$.

(c) Now consider a closed model with $k>0$ (or $\Omega>1)$. Rewrite ( $\dagger$ ) using the new time variable $\tau$ defined by

$$\frac{d t}{d \tau}=a$$

Hence, or otherwise, solve $(†)$ to find the parametric solution

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

where $\alpha \equiv H_{0} \sqrt{\left(\Omega_{0}-1\right)} . \quad$ Recall that $\left.\int d x / \sqrt{1-x^{2}}=\sin ^{-1} x .\right]$

Using the solution for $a(\tau)$, find the value of the new time variable $\tau=\tau_{0}$ today and hence deduce that the age of the universe in this model is

$$t_{0}=H_{0}^{-1} \frac{\sqrt{\Omega_{0}}-1}{\Omega_{0}-1}$$