Cosmological disformal transformations to the Einstein frame and gravitational couplings with matter perturbations

TL;DR

This paper explores how disformal transformations to the Einstein frame affect gravitational couplings and matter perturbations in scalar-tensor theories, providing a framework for understanding modifications to gravity in cosmology.

Contribution

It demonstrates the impact of disformal transformations on gravitational couplings and matter perturbations, and clarifies the correspondence between Jordan and Einstein frames in scalar-tensor theories.

Findings

Effective gravitational potential includes GR and modification-induced couplings.

No anisotropic stress in Horndeski class due to gravitational de-mixing.

Proposes a dark energy model including Brans-Dicke and variable tensor speed theories.

Abstract

The disformal transformation of metric $g_{\mu \nu} \to \Omega^2 (\phi)g_{\mu \nu}+\Gamma(\phi,X) \partial_{\mu}\phi \partial_{\nu}\phi$, where $\phi$ is a scalar field with the kinetic energy $X= \partial_{\mu}\phi \partial^{\mu}\phi/2$, preserves the Lagrangian structure of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories (which is the minimum extension of Horndeski theories). In the presence of matter, this transformation gives rise to a kinetic-type coupling between the scalar field $\phi$ and matter. We consider the Einstein frame in which the second-order action of tensor perturbations on the isotropic cosmological background is of the same form as that in General Relativity and study the role of couplings at the levels of both background and linear perturbations. We show that the effective gravitational potential felt by matter perturbations in the Einstein frame can be conveniently expressed in terms of the sum of a General Relativistic contribution and couplings induced by the modification of gravity. For the theories in which the transformed action belongs to a class of Horndeski theories, there is no anisotropic stress between two gravitational potentials in the Einstein frame due to a gravitational de-mixing. We propose a concrete dark energy model encompassing Brans-Dicke theories as well as theories with the tensor propagation speed $c_{\rm t}$ different from 1. We clarify the correspondence between physical quantities in the Jordan/Einstein frames and study the evolution of gravitational potentials and matter perturbations from the matter-dominated epoch to today in both analytic and numerical approaches.