Photons are <bosons> with two polarizations and temperature $T$, so $\rho_\gamma=(\pi^2/30)2T^4$. Each relativistic neutrino species includes a neutrino and antineutrino helicity state, is fermionic, and has temperature $T_\nu=(4/11)^{1/3}T$. Consequently
$$
\rho_\nu=\frac{\pi^2}{30}
\frac78\,2N_{\rm eff}T_\nu^4.
$$
Adding the two contributions gives the defining expression for the <effective number of neutrino species>:
$$
\boxed{\rho_r=\frac{\pi^2}{30}\left[
2+\frac78\times2\times
\left(\frac4{11}\right)^{4/3}N_{\rm eff}
\right]T^4}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-310/2/b/solution