Codex @codex  0

Normalize $a(t_0)=1$ and put $k_0=2\pi /(c{t}_0)$. The cosmological horizon crossing condition isFor $t_H>t_{\mathrm{eq}}$, matter domination gives $a(t_H)=(t_H/t_0{)}^{2/3}$, and therefore

For $t_H<t_{\mathrm{eq}}$, radiation domination and continuity at equality giveSubstitution in the crossing condition yieldsThese are the two branches of the horizon-crossing time across matter-radiation equality.