Noncanonical scalar kinetic stability
= Noncanonical scalar kinetic stability
{title2=$P_{,X}+2XP_{,XX}>0,\quad P_{,X}>0$}
For $X=\partial_\mu\phi\partial^\mu\phi/2$ in signature $(+---)$, the derivative part of the quadratic action on a timelike background is $\tfrac12(P_{,X}+2XP_{,XX})\delta\dot\phi^2-\tfrac12P_{,X}a^{-2}(\nabla\delta\phi)^2$. Absence of a ghost requires $P_{,X}+2XP_{,XX}>0$; positive gradient energy requires $P_{,X}>0$. The squared sound speed is their ratio $P_{,X}/(P_{,X}+2XP_{,XX})$.