Derivative-cubic coefficient of a noncanonical scalar ($C=-\frac{a^3}{H^3}(2{\bar{X}}^2{P}_{,XX}+\frac{4}{3}{\bar{X}}^3{P}_{,XXX})$)

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.