 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-3/9b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

In the very early universe, assume species

i

k T ≫ m_{i} c^{2}

and

E (p) ≃ p c

, and has negligible chemical potential,

μ_{i} ≃ 0

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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