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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 $t/2=1/H$.

A fixed comoving volume has physical volume proportional to $a^3$. The stretched domain walls within it have area proportional to $a^2$, and their energy per physical area is constant. The cosmological domain wall network therefore has $\rho \propto a^{-1}$, equivalently a barotropic equation of state with $w=-2/3$.

The flat Friedmann equation becomes ${\dot{a}}^2=Ca$ for a positive constant $C$. On the expanding branch, integration gives $a=(C/4)(t-t_B{)}^2$. Measuring age from the initial time $t_B=0$, $H=\dot{a}/a=2/t$. Thus

The scale factor of the cosmological domain wall network is $a\propto t^2$, so $\ddot{a}>0$: the expansion accelerates. Equivalently the Friedmann acceleration equation has $\rho +3P=\rho (1+3w)=-\rho <0$ for $w=-2/3$.

For a constant equation-of-state parameter $w=P/\rho$, the cosmological perfect-fluid continuity equation integrates to $\rho \propto a^{-3(1+w)}$. Comparing the given powers yieldsThis constant-equation-of-state density scaling describes a cosmic string network and a cosmological domain wall network. An intuitive check is that the energy in a fixed comoving cell grows in proportion to the total string length, $a$, or total wall area, $a^2$, while the physical cell volume grows as $a^3$. Thus the densities scale as $a^{-2}$ and $a^{-1}$ respectively. These effective equations of state use the noninteracting, statistically isotropic network assumptions of the problem.