= 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 $4\pi(a_0r)^2$: the astronomical radial coordinate is an areal coordinate, even in curved space. Hence
$$
F=\frac{L}{4\pi(a_0r)^2(1+z)^2},\qquad\boxed{d_L\equiv\sqrt{\frac{L}{4\pi F}}=a_0r(1+z).}
$$
This 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 $a_0r$.
Back to article page
