Cosmological round-trip communication horizon ($d_{\mathrm{round}}(t)=\frac{a(t)}{2}{\int }_t^{\infty }\frac{d{t}^{\prime }}{a(t^{\prime })}$)

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 $t/2=1/H$.