  In second order perturbation theory different definitions are known of gauge
invariant perturbations in single field inflationary models. Consequently the
corresponding gauge invariant cubic actions do not have the same form. Here we
show that the cubic action for one choice of gauge invariant variables is
unique in the following sense: the action for any other, non-linearly related
variable can be brought to the same bulk action, plus additional boundary
terms. These boundary terms correspond to the choice of hypersurface and
generate extra, disconnected contributions to the bispectrum. We also discuss
uniqueness of the action with respect to conformal frames. When expressed in
terms of the gauge invariant curvature perturbation on uniform field
hypersurfaces the action for cosmological perturbations has a unique form,
independent of the original Einstein or Jordan frame. Crucial is that the gauge
invariant comoving curvature perturbation is frame independent, which makes it
extremely helpful in showing the quantum equivalence of the two frames, and
therefore in calculating quantum effects in nonminimally coupled theories such
as Higss inflation.
