Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 1 / ii / Solution

The first-order in-in formalism formula at observation time ${\tau }_0$ isFourier transforming the three derivatives giveswhere ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$. Because the four species are distinct, Wick contraction pairs each vertex field with the external field of the same species and introduces no permutation factor. IfthenCombining this with part i gives the requested time integral:The factor $\mathcal{E}$ is a pseudoscalar, so the resulting primordial trispectrum is parity odd and purely imaginary in this momentum-space convention.