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 ct2 from 1 generally leads to the large modification to the propagation speed squared cs2 of a scalar degree of freedom φ. This problem persists whenever the kinetic energy ρX of the field φ is much smaller than the background energy density ρm, which is the case for most of dark energy models in the asymptotic past. Since the scaling solution characterized by the constant ratio ρX/ρm 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 η=-Φ/Ψ between the two gravitational potentials Ψ and Φ significantly deviates from 1 for ct2 away from 1. For other general initial conditions, the deviation of ct2 from 1 gives rise to the large oscillation of Ψ with the frequency related to cs2. In both cases, the model can leave distinct imprints for the observations of CMB and weak lensing.
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