Late-time cosmological evolution in degenerate higher-order scalar-tensor models

We study the late cosmological evolution, from the nonrelativistic matter dominated era to the dark energy era, in modified gravity models described by degenerate higher-order scalar-tensor (DHOST) theories. They represent the most general scalar-tensor theories propagating a single scalar degree of freedom and include Horndeski and beyond Horndeski theories. We provide the homogeneous evolution equations for any quadratic DHOST theory, without restricting ourselves to theories where the speed of gravitational waves coincides with that of light since the present constraints apply to wavelengths much smaller than cosmological scales. To illustrate the potential richness of the cosmological background evolution in these theories, we consider a simple family of shift-symmetric models, characterized by three parameters and compute the evolution of dark energy and of its equation of state. We also identify the regions in parameter space where the models are perturbatively stable.