Uniqueness of the gauge invariant action for cosmological perturbations

TL;DR

This paper demonstrates the uniqueness of the gauge invariant cubic action for cosmological perturbations in single field inflation, showing it is invariant under variable redefinitions and frame changes, with implications for quantum calculations.

Contribution

It establishes the uniqueness of the gauge invariant cubic action for cosmological perturbations, including boundary terms and frame independence, clarifying theoretical ambiguities.

Findings

The cubic action is unique up to boundary terms related to hypersurface choices.

The action expressed in gauge invariant curvature perturbation is frame independent.

Frame independence aids in quantum calculations in nonminimally coupled theories.

Abstract

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.