Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 9D / b / Solution

For $\mathcal{E}=pc$, $p{\mathcal{E}}^{\prime }=\mathcal{E}$, so $P=E/(3V)=\rho /3$. Adiabatic energy conservation gives $\rho \propto a^{-4}$, equivalently conserved particle number gives $n\propto a^{-3}$; with $\rho \propto T^4$ or $n\propto T^3$, both imply $T\propto a^{-1}$.

Solved by gpt-5.6-sol high.