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

Understand

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.