 Solution

The Einstein-de Sitter universe has $a\propto t^{2/3}$ and $H=\dot{a}/a\propto a^{-3/2}$. Normalizing $a_0=1$ therefore gives $H(z)=H_0(1+z{)}^{3/2}$. Differentiating $1+z=1/a$ givesIn the local comoving rest frame, a photon crosses proper path length $d\ell =c|dt|$. Galaxies with proper number density $n(z)$ and geometric cross section ${\sigma }_G=\pi r_G^2$ contribute the mean number of intersections $d\tau =n(z){\sigma }_{G}d\ell$. The infinitesimal probability of an intersection is the same to first order, givingThe radius is a physical radius, and $n(z)$ is a proper number density evaluated at epoch $z$, rather than a density already multiplied by $dz$. For a finite path with overlaps, the integrated mean count $\tau$ need not itself be the probability of at least one intersection.