Which particle states are expected to be relativistic and which interacting when the temperature $T$ of the early universe satisfies (i) $10^{10} \mathrm{~K}<T<5 \times 10^{10} \mathrm{~K}$, (ii) $5 \times 10^{9} \mathrm{~K}<T<10^{10} \mathrm{~K}$, (iii) $T<5 \times 10^{9} \mathrm{~K}$ ?

Calculate the total spin weight factor, $g_{*}$, of the relativistic particles and the total spin weight factor, $g_{I}$, of the interacting particles, in each of the three temperature intervals.

What happens when the temperature falls below $5 \times 10^{9} \mathrm{~K} ?$ Calculate the ratio of the temperatures of neutrinos and photons. Find the effective value of $g_{*}$ after the universe cools below this temperature. [Note that the equilibrium entropy density is given by $s=\left(\rho c^{2}+P\right) / T$, where $\rho$ is the density and $P$ is the pressure.]