Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / d / Solution

Adiabatic photon redshifting gives $T\propto a^{-1}$. Using the stipulated temperatures,In the flat dust approximation of part c, $t=t_0{a}^{3/2}$, so $t_{\mathrm{rec}}\simeq (2/3H_0)1000^{-3/2}\simeq 2.75\times 10^5$ years. This approximation omits the significant radiation contribution near the actual recombination epoch.

The elementary travel-distance estimate is $c{t}_{\mathrm{rec}}\simeq 8.4\times 10^4$ pc, whose corresponding comoving scale today is about $84$ Mpc. In an expanding universe, however, that is not the exact proper particle-horizon radius. Integrating the null geodesic givesfor dust. Thus the proper horizon at this epoch is about $0.25$ Mpc and its present comoving size about $250$ Mpc. The factor three accounts for expansion during light propagation. These scales describe the limited causal domain in the simple decelerating model. The near-uniform CMB temperature across much more widely separated last-scattering regions motivates the horizon problem; the question's rough $ct$ estimate and the exact dust horizon should not be conflated.