Conservation of the nonlinear curvature perturbation in generic single-field inflation

TL;DR

This work shows that the nonlinear curvature perturbation on comoving slices remains conserved on superhorizon scales for a wide class of single-field inflation models, including k-essence and Galileon theories, provided the scalar field dynamics reach an attractor regime where $\partial_\tau \phi = f(\phi)$. By employing a gradient-expansion framework and a general Lagrangian $W(X,\phi) - G(X,\phi) \Box \phi$, the authors derive conditions under which the scalar field equation contains only first time derivatives of the metric, and demonstrate how Einstein equations are needed to handle second-time-derivative terms in the Galileon case. The key result is that on uniform-$\phi$ slicing the nonlinear curvature perturbation $\psi_c$ is time-independent, i.e., $\psi_c(t,x^k) = \psi_c(t_i,x^k)$, implying that horizon-crossing statistics can be predicted from horizon-exit data without solving full dynamics. This extends the separate-universe intuition to a broad class of single-field models and clarifies when gravitational dynamics are essential for the conservation of curvature perturbations.

Abstract

It is known that the curvature perturbation on uniform energy density (or comoving or uniform Hubble) slices on superhorizon scales is conserved to full nonlinear order if the pressure is only a function of the energy density (ie, if the perturbation is purely adiabatic), independent of the gravitational theory. Here we explicitly show that the same conservation holds for a universe dominated by a single scalar field provided that the field is in an attractor regime, for a very general class of scalar field theories. However, we also show that if the scalar field equation contains a second time derivative of the metric, as in the case of the Galileon theory, one has to invoke the gravitational field equations to show the conservation.