= Solution
A radial null ray obeys $c\,d\tau=dr/\sqrt{1+r^2/r_{\rm curv}^2}$. Hence
$$
c(\tau_0-\tau_*)=r_{\rm curv}\operatorname{arsinh}\frac{r_*}{r_{\rm curv}},\qquad
r_*=r_{\rm curv}\sinh\frac{c(\tau_0-\tau_*)}{r_{\rm curv}}.
$$
For the open matter solution, let $\beta=\sqrt{1-\Omega_m}$. The <Friedmann equation> gives $a'=H_0\sqrt{\Omega_m a+\beta^2a^2}$, and the age in <conformal time> is
$$
\tau_0=\frac1{H_0}\int_0^1\frac{da}{\sqrt{a(\Omega_m+\beta^2a)}}
=\frac2{H_0\beta}\operatorname{arsinh}\frac{\beta}{\sqrt{\Omega_m}}
=\frac2{H_0\beta}\operatorname{artanh}\beta.
$$
Thus
$$
\boxed{\tau_0=\frac1{H_0\beta}\ln\frac{1+\beta}{1-\beta},\qquad
r_*\simeq\frac{c}{H_0\beta}\sinh(2\operatorname{artanh}\beta)
=\frac{2c}{H_0\Omega_m}.}
$$
Here the stated approximation $\tau_*\ll\tau_0$ was used only in the last distance estimate. The flat limit is smooth: $\tau_0\to2/H_0$ and $r_*\to2c/H_0$. Retaining $\tau_*$ in the preceding null-ray formula gives the finite-emission-time correction.
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