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.
