(i) Introducing the concept of a co-moving distance co-ordinate, explain briefly how the velocity of a galaxy in an isotropic and homogeneous universe is determined by the scale factor $a(t)$. How is the scale factor related to the Hubble constant $H_{0}$ ?

A homogeneous and isotropic universe has an energy density $\rho(t) c^{2}$ and a pressure $P(t)$. Use the relation $d E=-P d V$ to derive the "fluid equation"

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

where the overdot indicates differentiation with respect to time, $t$. Given that $a(t)$ satisfies the "acceleration equation"

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

show that the quantity

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

is time-independent.

The pressure $P$ is related to $\rho$ by the "equation of state"

$$P=\sigma \rho c^{2}, \quad|\sigma|<1 .$$

Given that $a\left(t_{0}\right)=1$, find $a(t)$ for $k=0$, and hence show that $a(0)=0$.

(ii) What is meant by the expression "the Hubble time"?

Assuming that $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now (of the current cosmological era), obtain a formula for the radius $R_{0}$ of the observable universe.

Given that

$$a(t)=\left(\frac{t}{t_{0}}\right)^{\alpha}$$

for constant $\alpha$, find the values of $\alpha$ for which $R_{0}$ is finite. Given that $R_{0}$ is finite, show that the age of the universe is less than the Hubble time. Explain briefly, and qualitatively, why this result is to be expected as long as

$$\rho+3 \frac{P}{c^{2}}>0 .$$

Part II