 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-312/4/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

p^{\overset{a}{^}} = E (1, e)

and

E = ϵ / a

. With degeneracy

g_{ν}

, kinetic theory gives the stress-energy tensor

T_{ν}^{\overset{a}{^} \hat{b}} = g_{ν} \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{p ^{\overset{a}{^}} p ^{\hat{b}}}{E} f, \overset{ρ}{ˉ}_{ν} = \frac{g _{ν}}{a ^{4}} \int \frac{ϵ ^{3} d ϵ d Ω _{e}}{( 2 π ) ^{3}} \overset{ˉ}{f} (ϵ) .

Coordinate components follow by inserting the orthonormal-frame vectors on the indices.

For the perturbation

δ f = - Θ ϵ \overset{ˉ}{f}^{'} (ϵ)

- \int_{0}^{\infty} ϵ^{4} \overset{ˉ}{f}^{'} (ϵ) d ϵ = 4 \int_{0}^{\infty} ϵ^{3} \overset{ˉ}{f} (ϵ) d ϵ,

assuming the boundary term vanishes. Consequently the physical trace-free spatial stress is

π_{ν}^{\hat{i} \hat{j}} = 4 \overset{ρ}{ˉ}_{ν} \int \frac{d Ω _{e}}{4 π} Θ (e^{\hat{i}} e^{\hat{j}} - \frac{1}{3} δ^{\hat{i} \hat{j}}) .

The paper uses the opposite signed source,

Π^{\hat{i} \hat{j}} = - π_{ν}^{\hat{i} \hat{j}}

, consistently with its tensor equation having

- 16 π G a^{2} Π_{ij}

on the right. In that convention the neutrino tensor anisotropic stress is

Π^{\hat{i} \hat{j}} = - 4 \overset{ρ}{ˉ}_{ν} \int \frac{d Ω _{e}}{4 π} Θ (e^{\hat{i}} e^{\hat{j}} - \frac{1}{3} δ^{\hat{i} \hat{j}}) .

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