 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-55/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

The new neutrino species undergoes thermal decoupling in cosmology when its interaction rate becomes comparable to the Hubble parameter. Equating the given rates gives

G_{A}^{2} T_{D}^{5} = \frac{1.66 g _{*} T _{D}^{2}}{m _{Pl}}, T_{D} = (\frac{1.66 g _{*}}{G _{A}^{2} m _{Pl}})^{1/3} .

With

g_{*} ≃ 100

G_{A} = 1 0^{- 12} GeV^{- 2}

and

m_{Pl} ≃ 1 0^{19} GeV

T_{D} ≃ (1.66 \times 1 0^{6})^{1/3} GeV ≃ 1.2 \times 1 0^{2} GeV .

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