Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 1 / a / 2 / 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 ${\eta }_e$, soFor 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.