= Direct cosmological communication and a shared causal past
{title2=$d\le\eta\text{ versus }d\le2\eta$}
In a spatially flat <Friedmann-Lemaître-Robertson-Walker metric> with initial <conformal time> zero, two comoving worldlines at separation $d$ can exchange a one-way light signal by time $\eta$ only if $d\le\eta$. Their past light cones overlap if $d\le2\eta$: a common event halfway between them can influence both. These are different statements. For opposite emission events observed at conformal time $\eta_0$, their separation is $2(\eta_0-\eta_e)$, so the respective thresholds are $\eta_e\ge2\eta_0/3$ and $\eta_e\ge\eta_0/2$.
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