 Solution

Normalize today's scale factor to $a_0=1$ and use $c=1$. A source of bolometric luminosity $L$ emits photons whose energies arrive smaller by $1+z$, and whose arrival intervals are longer by $1+z$. Their wavefront has physical area $4\pi a_0^2{r}_1^2$. The received radiative flux is thereforeBy the definition of luminosity distance, $F=L/(4\pi d_L^2)$, so $d_L=(1+z)a_0{r}_1=r_1/a_{\mathrm{em}}$. The final equality assumes the stated normalization of the scale factor; it is not a formula invariant under an arbitrary rescaling of $a_0$.

A radial null geodesic of the FRW metric obeys $dt/a=dr/\sqrt{1-K{r}^2}$. Using $1+z=1/a$ and $dz/dt=-(1+z)H(z)$ gives the radial comoving distanceFor $K>0$ this last integral is $\arcsin (\sqrt{K}{r}_1)/\sqrt{K}$, continued along the radial coordinate when necessary; for $K<0$ it is $\mathrm{arsinh}(\sqrt{|K|}{r}_1)/\sqrt{|K|}$. Inverting, with ${\Omega }_K=-K/H_0^2$, provesHere $S_K$ is sine for positive spatial curvature of an FLRW universe, hyperbolic sine for negative spatial curvature of an FLRW universe, and its argument for zero spatial curvature of an FLRW universe. The flat case is the continuous limit $d_L=(1+z)\chi$, not a literal division by zero. The original PDF has $H_0\sqrt{|{\Omega }_K|}$ in the denominator and $H=\dot{a}/a$; the converted TeX loses these details.