The horizon at recombination

Adapted from IOAA 2008 (Bandung), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The background radiation has almost the same temperature in every direction. Assume that light given out at recombination, at $T_{\text{r}}\approx 3000$ K and $t_{\text{r}}\approx 300000$ years, reaches us only now, at $T_0\approx 3$ K and $t_0\approx 1.5\times {10}^{10}$ years. Let the scale factor $S$ be 1 today, with $T\propto 1/S$. The Universe was ruled by radiation, with $S\propto t^{1/2}$, from the end of inflation at $t={10}^{-32}$ s until recombination, and by matter, with $S\propto t^{2/3}$, after that.

(a) Estimate the horizon distance at recombination, as an angle on the sky today. Here the horizon is the largest separation of two points in the background that could have exchanged light by the time it was given out. [40 points]

(b) Two points of the background are now seen 5° apart. Could they have exchanged light? Answer yes or no, with a reason. [20 points]

(c) Estimate the size of the observable Universe at the end of inflation. [40 points]

Answer(a) About 1.15°. (b) No, since 5° is more than the horizon angle. (c) About 3.4 m. The problem's model is rough. It takes the horizon as $ct$ and the distance as $c{t}_0$, and its two ways of scaling back, by temperature (1000) and by time (1360), do not quite agree. Our model gives a horizon of about 1.2°, close to the problem's answer.