Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 1 / a / Solution

A photon loses energy by $1+z=a_0/a_e$ between emission and reception. The interval between successive photon arrivals is also longer by $1+z$, so the received energy per unit time is reduced by $(1+z{)}^2$. An isotropic wavefront at reception has area $4\pi (a_{0}r{)}^2$: the astronomical radial coordinate is an areal coordinate, even in curved space. HenceThis defines the luminosity distance. Photon absorption or conversion would require a further transmission factor; the relation assumes freely propagating conserved photons and bolometric luminosity. The radial cosmological proper distance is generally different from the areal distance $a_{0}r$.