Show that after nearly all of the positrons were annihilated and the electron number density had nearly leveled off at the proton density, the ratio of the positron number density to the photon number density scaled with temperature as

\[\frac{n_{+}}{n_{\gamma}} \sim\left(\beta m_{e} c^{2}\right)^{3 / 2} \exp \left(-2 \beta m_{e} c^{2}\right)\]