Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 1 / b / 3 / Solution

Set $a(t)=a_0(t/t_0{)}^2$. The remaining future conformal time isA light ray sent to a comoving reflector at separation $\chi$ and immediately returned consumes $2\chi$ of conformal time. The cosmological round-trip communication horizon therefore has supremum physical distanceFinite-time reception requires $d_0<d_{\max }$. Indeed the total travel equation is $2d_0/a_0=(t_0/a_0)(1-t_0/t_r)$, giving $t_r=t_0/(1-2d_0/t_0)$. At the limiting distance the return time tends to infinity. This round-trip radius is half the current physical cosmological event horizon radius.