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

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