Recall that the horizon distance at a time $t$ is given by
$d_H(t) = R(t) \int_0^t \frac{dt'}{R(t')}$.
Suppose that the scale factor $R(t)$ follows
$R(t) = R(t_{mr}) \times \begin{cases} \left(\frac{t}{t_{mr}}\right)^{1/2} & 0 < t \le t_{mr},\\ \left(\frac{t}{t_{mr}}\right)^{2/3} & t > t_{mr},\end{cases}$
where $t_{mr} \approx 50\,\,000\,\text{y}$ is the time of matter-radiation equality. Show that the horizon distance at
the time of last scattering, $t_{ls} \approx 380\,\,000\,\text{y}$ is
$d_H(t_{ls}) = 3t_{ls} - t_{ls}^{2/3} t_{mr}^{1/3}$.
