  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.
