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We use a description based on differential forms to systematically explorethe space of scalar-tensor theories of gravity. Within this formalism, wepropose a basis for the scalar sector at the lowest order in derivatives of thefield and in any number of dimensions. This minimal basis is used to constructa finite and closed set of Lagrangians describing general scalar-tensortheories invariant under Local Lorentz Transformations in a pseudo-Riemannianmanifold, which contains ten physically distinct elements in four spacetimedimensions. Subsequently, we compute their corresponding equations of motionand find which combinations are at most second order in derivatives in four aswell as arbitrary number of dimensions. By studying the possible exact forms(total derivatives) and algebraic relations between the basis components, wediscover that there are only four Lagrangian combinations producing secondorder equations, which can be associated with Horndeski's theory. In thisprocess, we identify a new second order Lagrangian, named kinetic Gauss-Bonnet,that was not previously considered in the literature. However, we show that itsdynamics is already contained in Horndeski's theory. Finally, we provide a fullclassification of the relations between different second order theories. Thisallows us to clarify, for instance, the connection between differentcovariantizations of Galileons theory. In conclusion, our formulation affordsgreat computational simplicity with a systematic structure. As a first step wefocus on theories with second order equations of motion. However, this newformalism aims to facilitate advances towards unveiling the most generalscalar-tensor theories.

Towards the most general scalar-tensor theories of gravity: a unified approach in the language of differential forms