Unifying framework for scalar-tensor theories of gravity

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 Ho {r} {r}{}ava 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 a Friedmann-Robertson-Walker background.