Effective field theory approach to modified gravity including Horndeski theory and Ho\v{r}ava-Lifshitz gravity

We review the effective field theory of modified gravity in which the Lagrangian involves three dimensional geometric quantities appearing in the 3+1 decomposition of space-time. On the flat isotropic cosmological background we expand a general action up to second order in the perturbations of geometric scalars, by taking into account spatial derivatives higher than two. Our analysis covers a wide range of gravitational theories-- including Horndeski theory/its recent generalizations and the projectable/non-projectable versions of Ho\v{r}ava-Lifshitz gravity. We derive the equations of motion for linear cosmological perturbations and apply them to the calculations of inflationary power spectra as well as the dark energy dynamics in Galileon theories. We also show that our general results conveniently recover stability conditions of Ho\v{r}ava-Lifshitz gravity already derived in the literature.