 Solution

In the printed convention $\rho$ is energy density, since the gravitational source is $8\pi G\rho /(3c^2)$. For nonrelativistic pressureless matter, ${\rho }_m={\rho }_{m,0}(1+z{)}^3$. Define the cosmological density parameters byThe Friedmann equation then givesLet $n_c$ be the constant comoving number density of absorbers, and let ${\sigma }_p$ be their fixed proper interception area. Their proper number density is $n_p(z)=n_c(1+z{)}^3$, with $a_0=1$. The redshift-time relation gives a local proper photon-path incrementFor an infinitesimal path, the expected number of interceptions, and to first order the probability of an interception, is $d\mathcal{N}=n_p{\sigma }_{p}d\ell$. Thus absorber incidence and comoving number density implyTwo factors survive: proper density brings three powers of $1+z$, whereas proper path length brings the inverse of one power. A fixed comoving area instead of the assumed fixed proper area would change the result. The formula is an incidence rate, not a probability bounded by one over a finite redshift interval; multiple interceptions can occur.