Cubic curvature action for a P(X, phi) scalar field

In the limit where scalar field fluctuations dominate metric fluctuations, put $\zeta =-H\delta \varphi /\dot{\bar{\varphi }}$ and neglect derivatives of slowly varying background coefficients. Splitting $\delta X=\delta X_1+\delta X_2$ givesThe cubic terms $P_{XX}\delta X_1\delta X_2+P_{XXX}\delta X_1^3/6$ therefore yieldThe expression before factoring remains well-defined when $P_{XX}=0$. These interactions can generate substantial primordial non-Gaussianity when the Sound speed of a P(X, phi) scalar perturbation is small.

Neglect metric fluctuations and derivatives of $\dot{\bar{\varphi }}/H$, so $\delta X_1=-2\bar{X}\dot{\zeta }/H$ and $\delta X_2=\bar{X}[{\dot{\zeta }}^2-a^{-2}(\nabla \zeta {)}^2]/H^2$. The coefficient of ${\dot{\zeta }}^3$ in the cosmic-time action is $C=-a^3[2{\bar{X}}^2{P}_{,XX}+(4/3){\bar{X}}^3{P}_{,XXX}]/H^3$. This unfactored form is valid even when $P_{,XX}=0$ and a factored ratio would be undefined.