  We compute the third order gauge invariant action for scalar-graviton
interactions in the Jordan frame. We demonstrate that the gauge invariant
action for scalar and tensor perturbations on one physical hypersurface only
differs from that on another physical hypersurface via terms proportional to
the equation of motion and boundary terms, such that the evolution of
non-Gaussianity may be called unique. Moreover, we demonstrate that the gauge
invariant curvature perturbation and graviton on uniform field hypersurfaces in
the Jordan frame are equal to their counterparts in the Einstein frame. These
frame independent perturbations are therefore particularly useful in relating
results in different frames at the perturbative level. On the other hand, the
field perturbation and graviton on uniform curvature hypersurfaces in the
Jordan and Einstein frame are non-linearly related, as are their corresponding
actions and $n$-point functions.
