Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 3 / a

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Solution

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 .

This is an order-of-magnitude electroweak-scale estimate under the given approximation. The mass in this formula is the unreduced Planck mass, not the reduced Planck mass.

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