Field redefinitions in theories beyond Einstein gravity using the
  language of differential forms
  We study the role of field redefinitions in general scalar-tensor theories.
In particular, we first focus on the class of field redefinitions linear in the
spin-2 field and involving derivatives of the spin-0 mode, generically known as
disformal transformations. We start by defining the action of a disformal
transformation in the tangent space. Then, we take advantage of the great
economy of means of the language of differential forms to compute the full
transformation of Horndeski's theory under general disformal transformations.
We obtain that Horndeski's action maps onto itself modulo a reduced set of
non-Horndeski Lagrangians. These new Lagrangians are found to be invariant
under disformal transformation that depend only in the first derivatives of the
scalar. Moreover, these combinations of Lagrangians precisely appear when
expressing in our basis the constraints of the recently proposed Extended
Scalar-Tensor (EST) theories. These results allow us to classify the different
orbits of scalar-tensor theories invariant under particular disformal
transformations, namely the special disformal, kinetic disformal and disformal
Horndeski orbits. In addition, we consider generalizations of this framework.
We find that there are possible well-defined extended disformal transformations
that have not been considered in the literature. However, they generically
cannot link Horndeski theory with EST theories. Finally, we study further
generalizations in which extra fields with different spin are included. These
field redefinitions can be used to connect different gravity theories such as
multi-scalar-tensor theories, generalized Proca theories and bi-gravity. We
discuss how the formalism of differential forms could be useful for future
developments in these lines.
