= Solution
For <radiation in cosmology>, $\rho\propto a^{-4}$ by the <cosmological perfect-fluid continuity equation>. The flat <Friedmann equation> gives $a\propto t^{1/2}$, so $a\propto\eta$ in <conformal time>. Hence $1+z=\eta_0/\eta_e$.
The opposite galaxies again have separation $2(\eta_0-\eta_e)$. Direct communication requires this to be at most $\eta_e$, so
$$
\boxed{z_c=\frac12\quad\text{for direct communication}.}
$$
For overlap of their past light cones, the criterion is instead $\eta_e\ge\eta_0/2$, giving $\boxed{z_c=1\text{ for a shared causal past}}$. The two notions are precisely those distinguished in <direct cosmological communication and a shared causal past>.
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