Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 1 / b / Solution

Since $dt=ad\tau$ and $\dot{\varphi }=a^{-1}{\varphi }^{\prime }$, the interaction isto first order in $\lambda$. The tree-level in-in formalism givesThere are two Wick contractions for each choice of the undifferentiated field at the vertex. With $K=k_1+k_2+k_3$, the stated Bunch-Davies vacuum mode obeysThe two powers of $\tau$ from the differentiated modes cancel $a^2=1/(H^2{\tau }^2)$, and the remaining integral isRemoving the momentum-conserving delta function, the bispectrum from a time-derivative cubic scalar interaction is therefore