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 theories 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, the accelerated scaling point and the field dominated point can be simultaneously stable. Cosmic evolution will reach the accelerated scaling point if the time derivative of the scalar field in the theory is positive during the matter domination. If the time derivative of the scalar field is negative during the matter domination, the background universe will evolve towards the field dominated point. The density parameter of the matter can be larger than unity before reaching the scaling fixed point if the deviation from the Einstein theory of gravity is too large and the initial conditions for the dynamical variables during the matter domination are significantly different from the accelerated scaling point. The stabilities of $\phi$MDE fixed point are similar to the coupled dark energy models. In our consideration, the universe can only evolve from the $\phi$MDE to the field dominated point.Comment: 19 pages, 3 figures, 1 tabl