The effective field theory of inflation/dark energy and the Horndeski
  theory
  The effective field theory (EFT) of cosmological perturbations is a useful
framework to deal with the low-energy degrees of freedom present for inflation
and dark energy. We review the EFT for modified gravitational theories by
starting from the most general action in unitary gauge that involves the lapse
function and the three-dimensional geometric scalar quantities appearing in the
Arnowitt-Deser-Misner (ADM) formalism. Expanding the action up to quadratic
order in the perturbations and imposing conditions for the elimination of
spatial derivatives higher than second order, we obtain the Lagrangian of
curvature perturbations and gravitational waves with a single scalar degree of
freedom. The resulting second-order Lagrangian is exploited for computing the
scalar and tensor power spectra generated during inflation. We also show that
the most general scalar-tensor theory with second-order equations of
motion-Horndeski theory-belongs to the action of our general EFT framework and
that the background equations of motion in Horndeski theory can be conveniently
expressed in terms of three EFT parameters. Finally we study the equations of
matter density perturbations and the effective gravitational coupling for dark
energy models based on Horndeski theory, to confront the models with the
observations of large-scale structures and weak lensing.
