(a) The Friedmann-Robertson-Walker metric is given by

$$d s^{2}=-d t^{2}+a^{2}(t)\left[\frac{d r^{2}}{1-k r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

where $k=-1,0,+1$ and $a(t)$ is the scale factor.

For $k=+1$, show that this metric can be written in the form

$$d s^{2}=-d t^{2}+\gamma_{i j} d x^{i} d x^{j}=-d t^{2}+a^{2}(t)\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

Calculate the equatorial circumference $(\theta=\pi / 2)$ of the submanifold defined by constant $t$ and $\chi$.

Calculate the proper volume, defined by $\int \sqrt{\operatorname{det} \gamma} d^{3} x$, of the hypersurface defined by constant $t$.

(b) The Friedmann equations are

$$\begin{aligned} &3\left(\frac{\dot{a}^{2}+k}{a^{2}}\right)-\Lambda=8 \pi \rho, \\ &\frac{2 a \ddot{a}+\dot{a}^{2}+k}{a^{2}}-\Lambda=-8 \pi P, \end{aligned}$$

where $\rho(t)$ is the energy density, $P(t)$ is the pressure, $\Lambda$ is the cosmological constant and dot denotes $d / d t$.

The Einstein static universe has vanishing pressure, $P(t)=0$. Determine $a, k$ and $\Lambda$ as a function of the density $\rho$.

The Einstein static universe with $a=a_{0}$ and $\rho=\rho_{0}$ is perturbed by radiation such that

$$a=a_{0}+\delta a(t), \quad \rho=\rho_{0}+\delta \rho(t), \quad P=\frac{1}{3} \delta \rho(t)$$

where $\delta a \ll a_{0}$ and $\delta \rho \ll \rho_{0}$. Show that the Einstein static universe is unstable to this perturbation.