List the relativistic species of bosons and fermions from the standard model of particle physics that are present in the early universe when the temperature falls to $1 \mathrm{MeV} / \mathrm{k}_{B}$.

Which of the particles above will be interacting when the temperature is above $1 \mathrm{MeV} / k_{B}$ and between $1 \mathrm{MeV} / k_{B} \geq T \geq 0.51 \mathrm{MeV} / k_{B}$, respectively?

Explain what happens to the populations of particles present when the temperature falls to $0.51 \mathrm{MeV} / k_{B}$.

The entropy density of fermion and boson species with temperature $T$ is $s \propto g_{s} T^{3}$, where $g_{s}$ is the number of relativistic spin degrees of freedom, that is,

$$g_{s}=\sum_{\text {bosons }} g_{i}+\frac{7}{8} \sum_{\text {fermions }} g_{i}$$

Show that when the temperature of the universe falls below $0.51 \mathrm{MeV} / k_{B}$ the ratio of the neutrino and photon temperatures will be given by

$$\frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1 / 3}$$