Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 2 / b

Table of contents
- Solution b

Solution

Photons are bosons with two polarizations and temperature

T

, so

ρ_{γ} = (π^{2} /30) 2 T^{4}

. Each relativistic neutrino species includes a neutrino and antineutrino helicity state, is fermionic, and has temperature

T_{ν} = (4/11)^{1/3} T

. Consequently

ρ_{ν} = \frac{π ^{2}}{30} \frac{7}{8} 2 N_{eff} T_{ν}^{4} .

Adding the two contributions gives the defining expression for the effective number of neutrino species:

ρ_{r} = \frac{π ^{2}}{30} [2 + \frac{7}{8} \times 2 \times (\frac{4}{11})^{4/3} N_{eff}] T^{4} .

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