Dark energy in Horndeski theories after GW170817: A review

Abstract: The gravitational-wave event GW170817 from a binary neutron star merger together with the electromagnetic counterpart showed that the speed of gravitational waves $c_t$ 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 $c_t$, 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 $\phi$ has a coupling with the Ricci scalar $R$ of the form $G_4(\phi) R$, 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 gravitational-wave astronomy will open up a new window to constrain nonminimally coupled theories further by the modified luminosity distance of tensor perturbations.