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 _ { t } ^ { 2 } from 1 generally leads to the large modification to the propagation speed squared c _ { 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 _ { t } ^ { 2 } away from 1 .
For other general initial conditions , the deviation of c _ { t } ^ { 2 } from 1 gives rise to the large oscillation of \Psi with the frequency related to c _ { s } ^ { 2 } .
In both cases , the model can leave distinct imprints for the observations of CMB and weak lensing .