Unifying framework for scalar-tensor theories of gravity

TL;DR

This work introduces a foliation-based, brane-language framework that unifies a broad class of scalar-tensor theories, including k-essence, Horndeski, EFT of inflation, ghost condensate, and Hořava gravity, by describing dynamical spacelike hypersurfaces coupled to a background. The authors construct a general Lagrangian with kinetic terms built from extrinsic curvature and allow nonlinear dependence on the lapse, enabling a controlled departure from Horndeski while preserving the correct degrees of freedom (two tensor, one scalar) and second-order linear perturbations in a broad sector. A pivotal result is the identification of a new cubic combination of $K_{ab}$ that yields second-order equations for linear perturbations beyond Horndeski, along with explicit degeneracy conditions that recover Horndeski as a special case. The framework also provides explicit mappings between the φ-language and brane-language, enabling covariant formulations and potential nonperturbative applications such as black holes, thereby offering a flexible, covariant platform for exploring beyond-Horndeski theories with robust theoretical consistency.

Abstract

A general framework for effective theories propagating two tensor and one scalar degrees of freedom is investigated. Geometrically, it describes dynamical foliation of spacelike hypersurfaces coupled to a general background, in which the scalar mode encodes the fluctuation of the hypersurfaces. Within this framework, various models in the literature---including $k$-essence, Horndeski theory, the effective field theory of inflation, ghost condensate as well as the Hořava gravity---get unified. Our framework generalizes the Horndeski theory in the sense that, it propagates the correct number of degrees of freedom, although the equations of motion are generally higher order. We also identify new operators beyond the Horndeski theory, which yield second order equations of motion for linear perturbations around an a Friedmann-Robertson-Walker background.