= Solution
The <luminosity distance> is defined by $F=L/(4\pi d_L^2)$, where $L$ is the <bolometric luminosity> and $F$ the received <radiative flux>. In the <FRW metric> a sphere through the observer around a comoving source has area $4\pi a_0^2r_e^2$. The <cosmological redshift> reduces each photon's <energy> by $(1+z)^{-1}$, and <cosmological time dilation> reduces their arrival rate by another such factor. Hence
$$
F=\frac{L}{4\pi a_0^2r_e^2(1+z)^2},
\qquad \boxed{d_L=(1+z)a_0r_e=(1+z)r_e.}
$$
The positive radial lookback integral is $\chi_e=c\int_{t_e}^{t_0}dt/a(t)=\int_0^{r_e}dr/\sqrt{1-kr^2}$. The printed time limits are reversed; also the factor $c$ is required unless units $c=1$ are adopted. The areal coordinate $r_e$ and the radial <comoving distance> $\chi_e$ agree only when $k=0$.
For an <Einstein-de Sitter universe>, the <Friedmann equation> gives $H(z)=H_0(1+z)^{3/2}$. Using the <redshift-time relation> in the radial <null geodesic> integral yields
$$
r_e=c\int_0^z\frac{dz'}{H(z')}
=\frac{2c}{H_0}\left[1-(1+z)^{-1/2}\right].
$$
Thus the <Einstein-de Sitter luminosity-distance relation> is
$$
\boxed{d_L(z)=\frac{2c}{H_0}\left[(1+z)-\sqrt{1+z}\right].}
$$
Its small-<redshift> expansion $d_L=(c/H_0)[z+z^2/4+O(z^3)]$ recovers <Hubble's law>.
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