Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 9E / b / Solution

The ratiofalls rapidly as the universe cools. Below $k_{B}T\simeq 0.7\mathrm{MeV}$, weak reactions occur less than once per expansion time and can no longer track the falling equilibrium ratio. The neutron-to-proton ratio therefore freezes at approximately $1/6$.

Before nuclei can form efficiently, free neutrons continue to beta-decay,During the deuterium bottleneck this lowers the ratio to about $1/7$ by $k_{B}T\simeq 0.07\mathrm{MeV}$.

Solved by gpt-5.6-sol high.