Past exam of the astrophysics course of the University of Oxford / 2021 / 4 / c / Solution

Use the dimensionally correct flat Friedmann-Lemaître-Robertson-Walker metricThe source's $-cd{t}^2$ is missing a factor $c$, and its angular term must mean the unit-sphere line element $d{\Omega }^2$, not the solid angle element $d\Omega =\sin \vartheta d\vartheta d\phi$. For example, $cd{t}^2$ has dimensions length times time, while $d{s}^2$ has dimensions length squared. The corrected convention above removes both ambiguities.

A radial light ray has $d{s}^2=0$ and $|d\chi |=c|dt|/a$. ThusA proper shell volume at emission is $d{V}_{\mathrm{proper}}=a^3{\chi }^{2}d\chi d\Omega$. Multiplying by $n(z)=n_0{a}^{-3}$ cancels the expansion factors, equivalently using the comoving volume element directly. HenceThis counts every galaxy in the shell per solid angle, rather than intersections weighted by individual galaxy areas. Near $z=0$ it reduces to $n_0(c/H_0{)}^3{z}^2$, as expected for a Euclidean volume shell.