Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 9D (Cosmology)

Since $p\propto L^{-1}=V^{-1/3}$, $dp/dV=-p/(3V)$. At fixed occupations, $dE=\int {\mathcal{E}}^{\prime }(p)(dp/dV)\bar{n}(p),dp,dV$. Equating this to $-P,dV$ yields $P=(3V{)}^{-1}\int p{\mathcal{E}}^{\prime }(p)\bar{n}(p),dp$.

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 $T\propto a^{-1}$.

Here $\mathcal{E}=m{c}^2+p^2/(2m)+\cdots$, so $p{\mathcal{E}}^{\prime }=p^2/m$. Consequently $P=(3V{)}^{-1}\int p^2/m,\bar{n},dp=2U/(3V)$. For an adiabatic monatomic gas, $u\propto a^{-5}$ and $n\propto a^{-3}$, hence $T\propto u/n\propto a^{-2}$.