Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 2 / i

In a spatially flat FLRW metric, a radial null geodesic satisfies $d\chi =cdt/a(t)$. Normalize $a_0=1$. The comoving particle horizon at time $t$ is therefore $r_h=c{\int }_0^{t}d{t}^{\prime }/a(t^{\prime })$, using the initial singularity as the time origin. The cosmological redshift relation $1+z=1/a$ givesFor matter and a cosmological constant in a spatially flat model, the Friedmann equation gives $H(z)=H_0[{\Omega }_{m,0}(1+z{)}^3+{\Omega }_{\Lambda ,0}{]}^{1/2}$. The initial singularity corresponds to $z\to \infty$, so reversing the integration limits yieldsThis is a comoving particle horizon, not a distance to the source currently observed at redshift $z$, and not a future-directed cosmological event horizon. The corresponding proper horizon radius at that epoch is $r_h(z)/(1+z)$.