The strong interaction exhibits asymptotic freedom at high energies justifying treating the quarks an gluons as noninteracting. The effective number of species is much larger than during the time near \(t=1 \mathrm{~s}\) due to
the muons, quarks and gluons.

\(u_{\gamma}=2 \frac{u_{\gamma}}{2}\)

photons

\(u_{e^{-}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{e^{+}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) electrons/positrons

\(u_{u_{e}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\bar{u}_{e}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) electron neutrinos/antineutrinos

\(u_{u_{\mu}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\bar{u}_{\mu}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) muon neutrinos/antineutrinos

\(u_{u_{\tau}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\bar{u}_{\tau}}=\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) tau neutrinos/antineutrinos

\(u_{\mu^{-}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\mu^{+}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2}\) muons/antimuons

\(u_{u}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\bar{u}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) up quarks/antiquarks

\(u_{d}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad u_{\bar{d}}=2\left(\frac{7}{8} \right) \frac{u_{\gamma}}{2} \quad\) down quarks/antiquarks

\(u_{g}=(8) 2 \frac{u_{\gamma}}{2} \quad\) gluons

The result is \(u=(149 / 8) u_{\gamma}\). Proceeding as in problem 9.1 we get

\[
T(t)=10^{10} \mathrm{~K} \sqrt{\frac{0.53 \mathrm{~s}}{T}} .
\]

Therefore at \(k T=300 \mathrm{MeV}\left(T \simeq 3.5 \times 10^{12} \mathrm{~K} \right)\), the age of the universe was about \(4 \times 10^{-6} \mathrm{~s}\).