Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 1 / ii / Solution

At cubic order, the expansion of $P(X,\varphi )$ produces the schematic operatorsThe derivative self-interactions involving $P_{XX}$ and $P_{XXX}$ are generally largest when $c_s\ll 1$, while coefficients containing explicit $\varphi$ derivatives of a slowly varying $P$ are commonly slow-variation suppressed. The former therefore tend to dominate primordial non-Gaussianity.

The terms contributing to $\phi {\phi }^{\prime 2}$ areThusEquivalently, if the complete time-dependent coefficient is called $\lambda$, then $\lambda (\tau )=a^2{\lambda }_0$.

Treat ${\lambda }_0$ as constant at leading slow variation. To cubic order the corresponding interaction Hamiltonian isFor $q_i=c_s{k}_i$, the mode function above satisfiesWriting $K=k_1+k_2+k_3$, the needed regulated integral isThere are three choices for the undifferentiated leg and two contractions interchanging the differentiated legs. The stated in-in formalism formula therefore giveswithThe powers of $c_s$ cancel for this vertex with the normalization specified in part (i). Reversing the convention for the sign of $H_I$ reverses the displayed overall sign but not the momentum shape of the primordial bispectrum.

Solved by gpt-5.6-sol high.