Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 2 / ii

Expand

δ = a δ_{1} + a^{2} δ_{2} + a^{3} δ_{3} + \dots

. For a Gaussian linear density field, odd linear correlators vanish and Wick theorem reduces the fourth-order terms to products of the linear power spectrum. Hence

P_{1 - loop} (k) = a^{2} P (k) + a^{4} [P_{22} (k) + 2 P_{13} (k)] .

(1)

The two contractions of the two quadratic fields give

P_{22} (k) = 2 \int \frac{d ^{3} q}{( 2 π ) ^{3}} F_{2} (q, k - q)^{2} P (q) P (∣ k - q ∣) .

(2)

The three choices for which argument of

F_{3}

carries the external momentum give

P_{13} (k) = 3 P (k) \int \frac{d ^{3} q}{( 2 π ) ^{3}} F_{3} (k, q, - q) P (q) .

(3)

Thus the contribution conventionally called

P_{13} + P_{31}

2 P_{13} = 6 P (k) \int F_{3} P

. These are the two one-loop diagrams of the one-loop matter power spectrum.

Solved by gpt-5.6-sol high.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 2 / ii

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