Spatially covariant theories of gravity: disformal transformation, cosmological perturbations and the Einstein frame

TL;DR

This paper explores a broad class of spatially covariant gravity theories, analyzing their background evolution, perturbations, and invariance properties under disformal transformations, including implications for primordial spectra and Einstein frame representations.

Contribution

It demonstrates the invariance of primordial spectra under disformal transformations and identifies a family of frames where gravitational wave spectra match the Einstein frame.

Findings

Primordial spectra are invariant under disformal transformations.

A family of frames exists where tensor spectra resemble the Einstein frame.

The structure of the theory remains preserved under disformal transformations.

Abstract

We investigate the cosmological background evolution and perturbations in a general class of spatially covariant theories of gravity, which propagates two tensor modes and one scalar mode. We show that the structure of the theory is preserved under the disformal transformation. We also evaluate the primordial spectra for both the gravitational waves and the curvature perturbation, which are invariant under the disformal transformation. Due to the existence of higher spatial derivatives, the quadratic Lagrangian for the tensor modes itself cannot be transformed to the form in the Einstein frame. Nevertheless, there exists a one-parameter family of frames in which the spectrum of the gravitational waves takes the standard form in the Einstein frame.