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

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 (i.e. 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 (or kinetic braiding) theory, one has to invoke the gravitational-field equations to show the conservation.