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Cosmic evolution in DHOST theories with scaling solutions

We study cosmic evolution based on the fixed points in the dynamical analysis of the Degenerate Higher-Order Scalar-Tensor (DHOST) theories. We consider the DHOST theory in which the propagation speed of gravitational waves is equal to the speed of light, the tensor perturbations do not decay to dark energy perturbations, and the scaling solutions exist. The scaling fixed point associated with late time acceleration of universe can be either stable or saddle depending on the parameters of the theory. For some ranges of the parameters, this scaling fixed point and field dominated fixed point can be simultaneously stable. Cosmic evolution will reach either the scaling attractor or the field dominated attractor depending on signs of time derivative of the scalar field in the theory during the matter domination. The density parameter of dark matter can be larger than unity before reaching the scaling attractor if the deviation from the Einstein theory of gravity is too large. For this DHOST theory, stabilities of $ϕ$-matter-dominated epoch ($ϕ$MDE) and field dominated solutions are similar to the coupled dark energy models in Einstein gravity even though gravity is described by different theories. In our consideration, the universe can only evolve from the $ϕ$MDE regime to the field dominated regime. The ghost and gradient instabilities up to linear order in cosmological perturbations have been investigated. There is no gradient instability, while the ghost instability can be avoided for some range of the parameters of the model.