 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/9d/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Before electron-positron annihilation, the electromagnetic plasma has

g_{* s} = 2 + \frac{7}{8} (2 + 2) = \frac{11}{2},

whereas afterward only the two photon polarizations remain, so

g_{* s} = 2

. Entropy conservation in the decoupled electromagnetic sector gives

\frac{11}{2} T_{before}^{3} a^{3} = 2 T_{γ}^{3} a^{3} .

Neutrinos receive none of this entropy and continue cooling as

T_{ν} \propto a^{- 1}

. Therefore

\frac{T _{ν}}{T _{γ}} = (\frac{4}{11})^{1/3} .

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