 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/9d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

For equal internal degeneracy, the number density of an isotropic ultrarelativistic species is proportional to

\int_{0}^{\infty} \frac{p ^{2} d p}{e ^{βp c} \mp 1} .

With

x = βp c

\frac{1}{e ^{x} + 1} = \frac{1}{e ^{x} - 1} - \frac{2}{e ^{2 x} - 1} .

For

s > 0

\int_{0}^{\infty} \frac{x ^{s} d x}{e ^{x} + 1} = (1 - 2^{- s}) \int_{0}^{\infty} \frac{x ^{s} d x}{e ^{x} - 1} .

s = 2

, so

n_{f} = (1 - 2^{- 2}) n_{b} = 3 n_{b} /4

. The energy density contains one further factor

E = p c

, so it has

s = 3

and

ρ_{f} = (1 - 2^{- 3}) ρ_{b} = \frac{7}{8} ρ_{b} .

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