Sound speed of a P(X, phi) scalar perturbation ($c_s^2$)

On a homogeneous background with $X=\bar{X}$, a $P(X,\varphi )$ perturbation has sound speedThe denominator is its time-kinetic coefficient and $P_X$ is its spatial-gradient coefficient.

For $X={\partial }_{\mu }\varphi {\partial }^{\mu }\varphi /2$ in signature $(+---)$, the derivative part of the quadratic action on a timelike background is $\frac{1}{2}(P_{,X}+2X{P}_{,XX})\delta {\dot{\varphi }}^2-\frac{1}{2}{P}_{,X}{a}^{-2}(\nabla \delta \varphi {)}^2$. Absence of a ghost requires $P_{,X}+2X{P}_{,XX}>0$; positive gradient energy requires $P_{,X}>0$. The squared sound speed is their ratio $P_{,X}/(P_{,X}+2X{P}_{,XX})$.