Observational signatures of the theories beyond Horndeski

Abstract: In the approach of the effective field theory of modified gravity, we derive the equations of motion for linear perturbations in the presence of a barotropic perfect fluid on the flat isotropic cosmological background. In a simple version of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, which is the minimum extension of Horndeski theories, we show that a slight deviation of the tensor propagation speed squared $c_{\rm t}^2$ from 1 generally leads to the large modification to the propagation speed squared $c_{\rm s}^2$ of a scalar degree of freedom $\phi$. This problem persists whenever the kinetic energy $\rho_X$ of the field $\phi$ is much smaller than the background energy density $\rho_m$, which is the case for most of dark energy models in the asymptotic past. Since the scaling solution characterized by the constant ratio $\rho_X/\rho_m$ is one way out for avoiding such a problem, we study the evolution of perturbations for a scaling dark energy model in the framework of GLPV theories in the Jordan frame. Provided the oscillating mode of scalar perturbations is fine-tuned so that it is initially suppressed, the anisotropic parameter $\eta=-\Phi/\Psi$ between the two gravitational potentials $\Psi$ and $\Phi$ significantly deviates from 1 for $c_{\rm t}^2$ away from 1. For other general initial conditions, the deviation of $c_{\rm t}^2$ from 1 gives rise to the large oscillation of $\Psi$ with the frequency related to $c_{\rm s}^2$. In both cases, the model can leave distinct imprints for the observations of CMB and weak lensing.