The gravitational wave (GW) event GW170817 from a binary neutron star merger together with the electromagnetic counterpart showed that the speed of GWs ct is very close to that of light for the redshift z < 0.009. This places tight constraints on dark energy models constructed in the framework of modified gravitational theories. We review models of the late-time cosmic acceleration in scalar-tensor theories with second-order equations of motion (dubbed Horndeski theories) by paying particular attention to the evolution of dark energy equation of state and observables relevant to the cosmic growth history. We provide a gauge-ready formulation of scalar perturbations in full Horndeski theories and estimate observables associated with the evolution of large-scale structures, cosmic microwave background and weak lensing by employing a so-called quasi-static approximation for the modes deep inside the sound horizon. In light of the recent observational bound of ct, we also classify surviving dark energy models into four classes depending on different structure-formation patterns and discuss how they can be observationally distinguished from each other. In particular, the nonminimally coupled theories in which the scalar field has a coupling with the Ricci scalar R of the form G4()R, including f(R) gravity, can be tightly constrained not only from the cosmic expansion and growth histories but also from the variation of screened gravitational couplings. The cross-correlation of integrated Sachs-Wolfe signal with galaxy distributions can be a key observable for placing bounds on the relative ratio of cubic Galileon density to total dark energy density. The dawn of GW astronomy will open up a new window to constrain nonminimally coupled theories further by the modified luminosity distance of tensor perturbations.
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