Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 9B / a

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- Solution a

Solution

In the very early universe, assume species

i

is in thermal equilibrium, is ultrarelativistic so that

k T ≫ m_{i} c^{2}

and

E (p) ≃ p c

, and has negligible chemical potential,

μ_{i} ≃ 0

. Its occupation factor is then the Bose-Einstein distribution for a boson or the Fermi-Dirac distribution for a fermion.

Substitute

E = p c

and set

y = p c / (k T)

. Since

p = (k T / c) y

and

d p = (k T / c) d y

,

ϵ_{i} = \frac{4 π g _{i}}{h ^{3}} \int_{0}^{\infty} \frac{p c p ^{2} d p}{e ^{p c / (k T)} \mp 1} = \frac{4 π g _{i}}{( h c ) ^{3}} (k T)^{4} \int_{0}^{\infty} \frac{y ^{3}}{e ^{y} \mp 1} d y .

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