Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 4 / i / Solution

The comoving particle horizon is the maximum radial distance a photon could have travelled since the initial Big Bang, expressed in comoving coordinates. It describes possible past causal contact, whereas a cosmological event horizon concerns photons that can reach an observer in the future.

For a radial null geodesic in the FLRW metric, $dr/\sqrt{1-k{r}^2}=cdt/a$. Spatial flatness sets $k=0$, so with $a_0=1$,The given Hubble parameter then supplies $a^{2}H=H_0\sqrt{{\Omega }_{r,0}+{\Omega }_{m,0}a+{\Omega }_{\Lambda ,0}{a}^4}$. Since cosmological redshift obeys $a=(1+z{)}^{-1}$,The corresponding proper horizon at that epoch is $a(z)r_p(z)$.