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