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 c_T and the effective gravitational constant for scalar perturbations G_eff, are characterized by two functions of an auxiliary variable. We study the phenomenology of several models all having c_T=1. We obtain a scenario with c_T=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.

We study the phenomenology of a class of minimally modified gravity theories called $$f(\mathcal {H})$$f ( H ) theories, in which the usual general relativistic Hamiltonian constraint is replaced by a free function of it. After reviewing the construction of the theory and a consistent matter coupling, we analyze the dynamics of cosmology at the levels of both background and perturbations, and present a concrete example of the theory with a 3-parameter family of the function f. Finally, we compare this example model to Planck data as well as some later-time probes, showing that such a realization of $$f(\mathcal {H})$$f ( H ) theories fits the data significantly better than the standard $$\Lambda $$Λ CDM model, in particular by modifying gravity at intermediate redshifts, $$z\simeq 743$$z ≃ 743 .

Working directly with a general Hamiltonian for the spacetime metric with the $3+1$ decomposition and keeping only the spatial covariance, we investigate the possibility of reducing the number of degrees of freedom by introducing an auxiliary constraint. The auxiliary constraint is considered as part of the definition of the theory. Through a general Hamiltonian analysis, we find the conditions for the Hamiltonian as well as for the auxiliary constraint, under which the theory propagates two tensorial degrees of freedom only. The class of theories satisfying these conditions can be viewed as a new construction for the type-II minimally modified gravity theories, which propagate the same degrees of freedom of but are not equivalent to general relativity in the vacuum. We also illustrate our formalism by a concrete example and derive the dispersion relation for the gravitational waves, which can be constrained by observations.

Following the path of minimalism in alternative theories of gravity, we construct the “Minimal Theory of Bigravity” (MTBG), a theory of two interacting spin-2 fields that propagates only four local degrees of freedom instead of the usual seven ones and that allows for the same homogeneous and isotropic cosmological solutions as in Hassan-Rosen bigravity (HRBG). Starting from a precursor theory that propagates six local degrees of freedom, we carefully choose additional constraints to eliminate two of them to construct the theory. Investigating the cosmology of MTBG, we find that it accommodates two different branches of homogeneous and isotropic background solutions, equivalent on-shell to the two branches that are present in HRBG. Those branches in MTBG differ however from the HRBG ones at the perturbative level, are both perfectly healthy and do not exhibit strong coupling issues nor ghost instabilities. In the so-called self-accelerating branch, characterized by the presence of an effective cosmological constant, the scalar and vector sectors are the same as in General Relativity (GR). In the so-called normal branch, the scalar sector exhibits non-trivial phenomenology, while its vector sector remains the same as in GR. In both branches, the tensor sector exhibits the usual HRBG features: an effective mass term and oscillations of the gravitons. Therefore MTBG provides a stable nonlinear completion of the cosmology in HRBG.

In this paper, we show that any solution of general relativity (GR) that can be rendered spatially flat by a coordinate change is also a solution of the self-accelerating branch of the minimal theory of massive gravity (MTMG), with or without matter. We then for the first time obtain black hole and star solutions in a theory of massive gravity that agree with the corresponding solutions in GR and that are free from strong coupling issues. This in particular implies that the parametrized post-Newtonian parameters ${\beta }^{\mathrm{PPN}}$ and ${\gamma }^{\mathrm{PPN}}$ are unity, as in GR. We further show how these solutions can be embedded in a cosmological setting. While cosmological scales have already been considered in previous works, this is the first study of the phenomenology at shorter scales of the self-accelerating branch of MTMG.