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

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.

Solved by gpt-5.6-sol high.