Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S02 / 3 / b / Solution

In the prescribed matter-dominated approximation, neglect curvature, radiation, and the cosmological constant throughout the calculation. The Friedmann equation and ${\rho }_m\propto R^{-3}$ give $\dot{R}\propto R^{-1/2}$, hence $R\propto t^{2/3}$ and $H=2/(3t)$. For $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}=2.269\times 10^{-18}{\mathrm{s}}^{-1}$,The particle horizon is a proper distance on a constant-time slice, not simply $ct$:ThereforeThese are estimates within a spatially flat, matter-only model, rather than the measured age and particle horizon of the actual late-time universe.