Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 3 / b

Table of contents
- Solution b

Solution

For one species with two internal states, the relic-neutrino energy density obtained from the frozen Fermi-Dirac distribution is

ρ_{ν} = 2 \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{p ^{2} + m _{ν}^{2}}{e ^{a p / T_{ν}} + 1} = \frac{1}{π ^{2}} \int_{0}^{\infty} \frac{p ^{2} p ^{2} + m _{ν}^{2}}{e ^{a p / T_{ν}} + 1} d p .

In the relativistic limit, set

x = a p / T_{ν}

and use the standard Riemann zeta function integral

\int_{0}^{\infty} \frac{x ^{3}}{e ^{x} + 1} d x = \frac{7 π ^{4}}{120} .

This gives

ρ_{ν} = \frac{7 π ^{2}}{120} (\frac{T _{ν}}{a})^{4} .

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