In the limit where <scalar field> fluctuations dominate metric fluctuations, put $\zeta=-H\delta\phi/\dot{\bar\phi}$ and neglect derivatives of slowly varying background coefficients. Splitting $\delta X=\delta X_1+\delta X_2$ gives
$$
\delta X_1=-\frac{2\bar X}{H}\dot\zeta,\qquad
\delta X_2=\frac{\bar X}{H^2}\left(\dot\zeta^2-a^{-2}(\partial_i\zeta)^2\right).
$$
The cubic terms $P_{XX}\delta X_1\delta X_2+P_{XXX}\delta X_1^3/6$ therefore yield
$$
S_3=\int dt\,d^3x\,\frac{a^3\epsilon(1-c_s^2)}{Hc_s^2}
\left[a^{-2}\dot\zeta(\partial_i\zeta)^2+\mathcal A\dot\zeta^3\right],
\qquad
\mathcal A=-1-\frac{2\bar X P_{XXX}}{3P_{XX}}.
$$
The 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.


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

ID: cubic-curvature-action-for-a-p-x-phi-scalar-field