After the positrons were annihilated, the energy density of the universe was dominated by the photons and the neutrinos. Show that the energy density in that era was given by \(u_{\text {total }}=\left[1+(21 / 8)(4 / 11)^{4 / 3}\right] u_{\gamma}\). Next, use the Hubble expansion relation (9.1.1), the temperature scaling relation (9.1.3), and the energy density after the electron-positron annihilation to show that the photon temperature as a function of time was \(T(t) \approx 10^{10} \mathrm{~K} \sqrt{\frac{1.78 \mathrm{~s}}{t}}\). This relation held from \(t \approx 100 \mathrm{~s}\) until \(t \approx 200,000\) years, when the energy density due to baryonic and cold dark matter began to dominate.