Expand $\delta=\delta^{(1)}+\delta^{(2)}+\delta^{(3)}+\cdots$, where the <standard perturbation theory density kernel> $F_n$ convolves $n$ linear fields. Through fourth order in the Gaussian linear field, the only power-spectrum diagrams are the tree contraction $P_{11}$, the loop joining two $F_2$ vertices $P_{22}$, and the two orderings that join an $F_3$ vertex to a linear leg, $P_{13}+P_{31}=2P_{13}$. Their expressions are
$$
P_{11}(k)=P_L(k),
$$
$$
P_{22}(k)=2\int\frac{d^3q}{(2\pi)^3}
F_2(\mathbf q,\mathbf k-\mathbf q)^2
P_L(q)P_L(|\mathbf k-\mathbf q|),
$$
$$
2P_{13}(k)=6P_L(k)\int\frac{d^3q}{(2\pi)^3}
F_3(\mathbf k,\mathbf q,-\mathbf q)P_L(q).
$$

\Image[../../../paper-312-one-loop-matter-power.svg]
{title=One-loop matter-power-spectrum diagrams and schematic present-day contributions}
{description=The three required contraction topologies are P11, P22, and P13 plus P31. At low wavenumber the one-loop sum approaches the linear spectrum; the loop terms become appreciable around 0.1 inverse megaparsecs times h, and fixed-order perturbation theory eventually fails.}

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-312/3/b/solution