Global adiabaticity and non-Gaussianity consistency condition

In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation. However, the conservation of $\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\R_c$ dominates over the other (usually decaying) mode. In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $\delta P_{nad}\equiv\delta P-c_w^2\delta\rho$ where $c_w^2=\dot P/\dot\rho$, usually becomes also negligible on superhorizon scales. Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity. This is in fact not true. In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\R_c$ on super-horizon scales. In this paper, we consider models that satisfy $\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation. A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$. In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor. Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition.