Generalized disformal invariance of cosmological perturbations with second-order field derivatives

TL;DR

This paper studies how cosmological perturbations transform under a generalized disformal transformation involving second-order derivatives of a scalar field, identifying conditions for invariance on superhorizon scales.

Contribution

It extends disformal invariance analysis to include second-order derivatives of the scalar field, revealing conditions for invariance of curvature and tensor perturbations.

Findings

Curvature perturbation invariance under disformal transformation when entropy perturbation vanishes.

Tensor perturbations are disformally invariant if they are conserved over time.

Superhorizon scale invariance depends on specific conditions of the perturbations.

Abstract

We investigate how the comoving curvature and tensor perturbations are transformed under the generalized disformal transformation with the second-order covariant derivatives of the scalar field, where the free functions depend on the fundamental elements constructed with the covariant derivatives of the scalar field with at most the quadratic order of the second-order covariant derivatives. Our analysis reveals that on the superhorizon scales the difference between the comoving curvature perturbations in the original and new frames is given by the combination of the time derivative of the comoving curvature perturbation, the intrinsic entropy perturbation of the scalar field, and its time derivative in the original frame. Thus, in the case that on the superhorizon scales (1) the intrinsic entropy perturbation and its time derivative vanish and (2) the comoving curvature perturbation in the original frame, ${\cal R}_c$, is conserved, the comoving curvature perturbation becomes invariant under the disformal transformation on the superhorizon scales. We also show that the tensor perturbations are also disformally invariant, in the case that the tensor perturbations in the original frame are conserved with time.