Dark energy scenario consistent with GW170817 in theories beyond Horndeski gravity

The Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories up to quartic order are the general scheme of scalar-tensor theories allowing the possibility for realizing the tensor propagation speed $c_t$ equivalent to 1 on the isotropic cosmological background. We propose a dark energy model in which the late-time cosmic acceleration occurs by a simple k-essence Lagrangian analogous to the ghost condensate with cubic and quartic Galileons in the framework of GLPV theories. We show that a wide variety of the variation of the dark energy equation of state $w_{\rm DE}$ including the entry to the region $w_{\rm DE}<-1$ can be realized without violating conditions for the absence of ghosts and Laplacian instabilities. The approach to the tracker equation of state $w_{\rm DE}=-2$ during the matter era, which is disfavored by observational data, can be avoided by the existence of a quadratic k-essence Lagrangian $X^2$. We study the evolution of nonrelativistic matter perturbations for the model $c_t^2=1$ and show that the two quantities $\mu$ and $\Sigma$, which are related to the Newtonian and weak lensing gravitational potentials respectively, are practically equivalent to each other, such that $\mu \simeq \Sigma>1$. For the case in which the deviation of $w_{\rm DE}$ from $-1$ is significant at a later cosmological epoch, the values of $\mu$ and $\Sigma$ tend to be larger at low redshifts. We also find that our dark energy model can be consistent with the bounds on the deviation parameter $\alpha_{\rm H}$ from Horndeski theories arising from the modification of gravitational law inside massive objects.