For symmetrized <standard perturbation theory density kernels>, the two one-loop contractions are
$$
\boxed{P_{22}(k)=2\int\frac{d^3q}{(2\pi)^3}
\left[F_2(\mathbf q,\mathbf k-\mathbf q)\right]^2
P(q)P(|\mathbf k-\mathbf q|)},
$$
and
$$
\boxed{P_{13}(k)=3P(k)\int\frac{d^3q}{(2\pi)^3}
F_3(\mathbf k,\mathbf q,-\mathbf q)P(q)}.
$$
The factors $2$ and $3$ count the relevant <Wick contractions>. Since the cross-correlation occurs in both orders, the <one-loop matter power spectrum> is
$$
P_{\rm 1-loop}(k)=a^2P(k)+a^4[P_{22}(k)+2P_{13}(k)].
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 4 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-312/4/ii/solution