The contents of a spatially homogeneous and isotropic universe are modelled as a finite mass $M$ of pressureless material whose radius $r(t)$ evolves from some constant reference radius $r_{0}$ in proportion to the time-dependent scale factor $a(t)$, with

$$r(t)=a(t) r_{0}$$

(i) Show that this motion leads to expansion governed by Hubble's Law. If this universe is expanding, explain why there will be a shift in the frequency of radiation between its emission from a distant object and subsequent reception by an observer. Define the redshift $z$ of the observed object in terms of the values of the scale factor $a(t)$ at the times of emission and reception.

(ii) The expanding universal mass $M$ is given a small rotational perturbation, with angular velocity $\omega$, and its angular momentum is subsequently conserved. If deviations from spherical expansion can be neglected, show that its linear rotational velocity will fall as $V \propto a^{-n}$, where you should determine the value of $n$. Show that this perturbation will become increasingly insignificant compared to the expansion velocity as the universe expands if $a \propto t^{2 / 3}$.

(iii) A distant cloud of intermingled hydrogen (H) atoms and carbon monoxide (CO) molecules has its redshift determined simultaneously in two ways: by detecting $21 \mathrm{~cm}$ radiation from atomic hydrogen and by detecting radiation from rotational transitions in CO molecules. The ratio of the $21 \mathrm{~cm}$ atomic transition frequency to the CO rotational transition frequency is proportional to $\alpha^{2}$, where $\alpha$ is the fine structure constant. It is suggested that there may be a small difference in the value of the constant $\alpha$ between the times of emission and reception of the radiation from the cloud.

Show that the difference in the redshift values for the cloud, $\Delta z=z_{C O}-z_{21}$, determined separately by observations of the $\mathrm{H}$ and $\mathrm{CO}$ transitions, is related to $\delta \alpha=$ $\alpha_{r}-\alpha_{e}$, the difference in $\alpha$ values at the times of reception and emission, by

$$\Delta z=2\left(\frac{\delta \alpha}{\alpha_{r}}\right)\left(1+z_{C O}\right)$$

(iv) The universe today contains $30 \%$ of its total density in the form of pressureless matter and $70 \%$ in the form of a dark energy with constant redshift-independent density. If these are the only two significant constituents of the universe, show that their densities were equal when the scale factor of the universe was approximately equal to $75 \%$ of its present value.