13.1 | The CMB horizon and the end of inflation

Measurements of the cosmic microwave background (CMB) show practically the same temperature at every point in the sky. Assume that light emitted at recombination ($T_r\approx3000\,\mathrm K$, $t_r\approx300000$ years) is only reaching us now ($T_0\approx3\,\mathrm K$, $t_0\approx1.5\times10^{10}$ years).

Define the scale factor by $S_0=S(t_0)=1$ and $S(t<t_0)<1$. In this problem's simplified model, radiation domination lasts from the end of inflation at $t_i=10^{-32}\,\mathrm s$ to recombination, with $S\propto t^{1/2}$. Matter domination begins at recombination, with $S\propto t^{2/3}$.

(a) Estimate the horizon distance at recombination, including its angular size today. Assume $T\propto1/S$. The angular horizon is the maximum separation between two points on the CMB imprint that could have communicated when the radiation was emitted.

(b) Two points on the CMB imprint are observed at an angular separation of $5^\circ$. Could they have communicated using photons by recombination? Answer YES or NO with a mathematical reason.

(c) Estimate the size of the region corresponding to our observable Universe at the end of inflation.