(a) Introduce the concept of comoving co-ordinates in a homogeneous and isotropic universe and explain how the velocity of a galaxy is determined by the scale factor $a$. Express the Hubble parameter $H_{0}$ today in terms of the scale factor.

(b) The Raychaudhuri equation states that the acceleration of the universe is determined by the mass density $\rho$ and the pressure $P$ as

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

Now assume that the matter constituents of the universe satisfy $\rho+3 P / c^{2} \geqslant 0$. In this case explain clearly why the Hubble time $H_{0}^{-1}$ sets an upper limit on the age of the universe; equivalently, that the scale factor must vanish $\left(a\left(t_{i}\right)=0\right)$ at some time $t_{i}<t_{0}$ with $t_{0}-t_{i} \leqslant H_{0}^{-1}$.

The observed Hubble time is $H_{0}^{-1}=1 \times 10^{10}$ years. Discuss two reasons why the above upper limit does not seem to apply to our universe.