Phenomenology in type-I minimally modified gravity

We study cosmology in a class of minimally modified gravity (MMG) with two local gravitational degrees of freedom. We classify modified gravity theories into type-I and type-II: theories of type-I have an Einstein frame and can be recast by change of variables as general relativity (GR) with a non-minimal matter coupling, while theories of type-II have no Einstein frame. Considering a canonical transformation of the lapse, the 3-dimensional induced metric and their conjugate momenta we generate type-I MMG. We then show that phenomenological deviations from GR, such as the speed of gravitational waves cT and the effective gravitational constant for scalar perturbations Geff, are characterized by two functions of an auxiliary variable. We study the phenomenology of several models all having cT=1. We obtain a scenario with cT=1 in which the effective equation-of-state parameter of dark energy is different from −1 even though the cosmic acceleration is caused by a bare cosmological constant, and we find that it is possible to reconstruct the theory on choosing a selected time-evolution for the effective dark energy component.