Disformal invariance of curvature perturbation

TL;DR

This paper investigates the invariance properties of the curvature perturbation under disformal transformations, showing superhorizon invariance in Horndeski-related theories and analyzing nonlinear and degrees of freedom aspects.

Contribution

It demonstrates the conditions under which the curvature perturbation remains invariant under disformal transformations in Horndeski theories and explores nonlinear and degrees of freedom considerations.

Findings

Curvature perturbation is invariant on superhorizon scales for Horndeski-related theories.

Disformal transformations preserve nonlinear curvature perturbation in attractor regimes.

The degrees of freedom in disformally related theories are analyzed and clarified.

Abstract

We show that under a general disformal transformation the linear comoving curvature perturbation is not identically invariant, but is invariant on superhorizon scales for any theory that is disformally related to Horndeski's theory. The difference between disformally related curvature perturbations is found to be given in terms of the comoving density perturbation associated with a single canonical scalar field. In General Relativity it is well-known that this quantity vanishes on superhorizon scales through the Poisson equation that is obtained on combining the Hamiltonian and momentum constraints, and we confirm that a similar result holds for any theory that is disformally related to Horndeski's scalar-tensor theory so long as the invertibility condition for the disformal transformation is satisfied. We also consider the curvature perturbation at full nonlinear order in the unitary gauge, and find that it is invariant under a general disformal transformation if we assume that an attractor regime has been reached. Finally, we also discuss the counting of degrees of freedom in theories disformally related to Horndeski's.