Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 9E (Cosmology)

Chemical equilibrium in $n+{\nu }_e\leftrightarrow p+e^{-}$ givesThe negligible lepton chemical potentials therefore imply ${\mu }_n\simeq {\mu }_p$. Nonrelativistic Maxwell--Boltzmann number densities satisfyNeutrons and protons have equal spin degeneracy and nearly equal masses, so their prefactor ratio is approximately one. Hence

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 $k_{B}T\simeq 0.07\mathrm{MeV}$.

Once deuterium survives photodissociation, nuclear reactions rapidly assemble nearly every available neutron into tightly bound helium-4. Protons are more abundant, so neutrons are the limiting constituent. If $r=n_n/n_p=1/7$, thenNeglecting the neutron--proton mass difference,Thus

Increasing $\kappa$ makes ${\Gamma }_W/H$ smaller at every fixed temperature, so freeze-out occurs earlier, at a higher $T$. The equilibrium value $e^{-Q/(k_{B}T)}$ is then closer to one. Although some neutrons subsequently decay, the final neutron fraction is larger, andincreases, approaching unity if the surviving neutron and proton abundances become comparable.

Such a universe begins with less hydrogen and more helium. It has less fuel for long-lived hydrogen-burning stars; stellar evolution would tend to proceed through hotter, faster helium-burning stages, reducing typical stellar lifetimes. The shorter stable-energy window, altered heavy-element production, and lower primordial hydrogen abundance would make familiar water-rich, slowly evolving habitats less likely, although a detailed conclusion would require a stellar-evolution model.