Cosmological event horizon (${\chi }_{\mathrm{eh}}(t)={\int }_t^{\infty }\frac{d{t}^{\prime }}{a(t^{\prime })}$)

For a comoving observer in a spatially flat Friedmann-Lemaître-Robertson-Walker metric, a finite future conformal time interval gives an event horizon at the displayed comoving radius. Its physical radius at time $t$ is $a(t){\chi }_{\mathrm{eh}}(t)$.

A light signal sent from a comoving observer to a comoving reflector at separation $\chi$, and immediately returned, consumes conformal time $2\chi$. The supremum of the initial physical reflector distances allowing a return is therefore half the cosmological event horizon radius. Equality generally requires infinite return time; finite-time reception requires a strictly smaller distance. For a cosmological domain wall network with $a\propto t^2$, the supremum is $t/2=1/H$.

For two comoving observers in an Einstein-de Sitter universe, light observed at redshift $z$ today fixes their separation through the past null geodesic. A newly sent signal needs the same comoving distance for every one-way leg. Since $a=(t/t_0{)}^{2/3}$, each leg adds a fixed conformal time interval, producing the displayed reception time after $n$ legs when replies are immediate. At $z=1$ the first round-trip wait is $[(3-\sqrt{2}{)}^3-1]t_0$. There is no finite count limit because future conformal time diverges, though infinitely many legs require infinite cosmic time.