Horizon-crossing time across matter-radiation equality
= Horizon-crossing time across matter-radiation equality
With $a(t_0)=1$, $k_0=2\pi/(ct_0)$, and $1+z_{\rm eq}=(t_0/t_{\rm eq})^{2/3}$,
$$
\frac{t_H}{t_0}\simeq
\begin{cases}
(k_0/k)^3,&t_H>t_{\rm eq},\\
(1+z_{\rm eq})^{-1/2}(k_0/k)^2,&t_H<t_{\rm eq}.
\end{cases}
$$