Before <electron-positron annihilation in cosmology>, photons and electron-positron pairs share a temperature. Their effective entropy degrees of freedom are
$$
g_{*s}^{\rm before}=2+\frac78(2+2)=\frac{11}{2}.
$$
After annihilation the electromagnetic plasma contains only the two photon polarizations, so $g_{*s}^{\rm after}=2$. The already decoupled neutrinos receive none of this entropy and simply cool as $T_\nu\propto a^{-1}$. Applying <cosmological entropy conservation> separately to the coupled electromagnetic plasma gives
$$
g_{*s}^{\rm before}T_{\rm before}^3a_{
m before}^3
=g_{*s}^{\rm after}T^3a^3,
$$
whereas $T_\nu^3a^3=T_{\rm before}^3a_{
m before}^3$. Division yields the <Cosmic neutrino background> temperature
$$
\boxed{\frac{T_\nu}{T}=\left(\frac4{11}\right)^{1/3}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-310/2/a/solution