Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S2 / 3

At a fixed cosmological time, homogeneous expansion has $\mathbf{v}=H(t)\mathbf{r}$. A comoving galaxy has $\mathbf{r}=R(t)\mathbf{x}$ with constant $\mathbf{x}$, hence $\mathbf{v}=\dot{R}\mathbf{x}=(\dot{R}/R)\mathbf{r}$. Thus $\boxed{H=\dot{R}/R}$. Peculiar velocities must be removed before applying this ideal relation to individual galaxies.

Set the spatial-curvature constant to zero in the stipulated matter-only Friedmann equation to get ${\rho }_c=3H^2/(8\pi G)$. Conservation of pressureless matter with $R(t_0)=1$ gives $\rho ={\rho }_0/R^3$, and thereforeChoosing the Big Bang as $t=0$ yields $R=(t/t_0{)}^{2/3}$ and $H=2/(3t)$. Consequently $\boxed{t_0=2/(3H_0)\simeq 9.3\mathrm{Gyr}}$ for the supplied $H_0$. This is the age of the flat pressureless model without a cosmological constant, not a determination in the current multi-component cosmological model.

The matter rest-energy density is ${\rho }_m{c}^2\simeq 2.25\times 10^{-10}$ J m${}^{-3}$, whereas the given photon density is $8.01\times 10^{-14}$ J m${}^{-3}$. Their present ratio is about $2.8\times 10^3$. Matter scales as $(1+z{)}^3$ and freely redshifting radiation as $(1+z{)}^4$, soThis compares matter specifically with the supplied CMB photons. Adding relativistic neutrinos changes the total radiation equality redshift; it is not included in these data.