P(X, phi) scalar field theory ($P(X,\varphi )$ theory)

A $P(X,\varphi )$ scalar theory has Lagrangian density $P$ depending on the field and $X=-g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi /2$. It includes canonical scalar theory and models with noncanonical kinetic interactions.

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.