Observational signatures of the theories beyond Horndeski
  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.
