Disformal invariance of second order tensor-scalar theories: framing the Horndeski action
Dario Bettoni, Stefano Liberati
TL;DR
The paper analyzes Horndeski gravity under disformal transformations, identifying conditions under which second-order field equations are preserved and frames that render the theory Einstein-like. It shows that a restricted class of disformal maps (field-dependent only for A,B) preserves the Horndeski structure and renormalizes coefficient functions, while a broader kinetic dependence generically spoils it. The authors define a network of equivalent frames (Jordan, Einstein, Galileon, Disformal) connected by disformal maps and field rescalings, and they identify when an Einstein frame is achievable, often requiring elimination of non-minimal couplings. They also discuss general disformal frames without an Einstein frame and provide concrete forms that realize disformal equivalence, with implications for unifying diverse scalar-tensor models in cosmology.
Abstract
The Horndeski action is the most general one involving a metric and a scalar field that leads to second order field equations in four dimensions. Being the natural extension of the well known Scalar-Tensor theories, its structure and properties are worth analysing along the experience accumulated in the latter context. Here we argue that disformal transformations play, for the Horndeski theory, a similar role to that of conformal transformations for Scalar-Tensor theories a l`a Brans-Dicke. We identify the most general transformation preserving second order field equations and discuss the issue of viable frames for this kind of theories, in particular the possibility to cast the action in the so called Einstein frame. Finally, we investigate the physical equivalence of such frames and their reciprocal relationship.