Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 2

The inverse 3+1 decomposition of spacetime metric givesThe first term is the squared derivative along the hypersurface normal and the second is the spatial-gradient contribution.

At fixed $N$ and $h_{ij}$, varying the shift givesandIntegration by parts in the gravitational term turns the variation of $K_{ij}{K}^{ij}-K^2$ into the spatial divergence of $K_i{}^j-{\delta }_i{}^{j}K$. Since the shift has no time derivative, its Euler-Lagrange equation is the ADM momentum constraint for a P(X, phi) scalar field

In flat gauge in cosmology, the supplied expressions implyto first order. The last two terms cancel after taking ${\partial }_j$, so the geometric part of the momentum constraint is $2H{\partial }_i\delta N$. Its matter part isThereforeand, after discarding a spatially homogeneous lapse mode,

For the homogeneous background, $X={\dot{\bar{\varphi }}}^{,2}/2$. The second Friedmann equation givesDividing by $H{M}_{\mathrm{Pl}}^2\dot{\bar{\varphi }}$ shows that the coefficient found above is

Substitution of $\delta N=C\phi$ and conversion to conformal time turn the interaction intoTreat $C$ and $H$ as constant at leading slow-roll order and use the massless de Sitter modeFor $K=k_1+k_2+k_3$, the in-in formalism time integral at a vertex with leg $i$ undifferentiated is proportional toSumming the three choices of undifferentiated leg gives the primordial bispectrumThe sign follows from the interaction sign displayed in the question and $H_I=-L_{\mathrm{int}}$ at this order.