 Solution

At fixed conformal time and fixed radial coordinate, the metric gives transverse physical separation $D=a({\tau }_1)r_1\theta$ to first order in the angle. Thus the angular diameter distance isThe coordinate $r_1$ is the transverse comoving radius appearing in the angular metric coefficient. It is not the radial geodesic distance $r_{\mathrm{curv}}\mathrm{arsinh}(r_1/r_{\mathrm{curv}})$; replacing one by the other would omit the curvature dependence of angular projection.