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 nonminimal 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 G eff , 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.

The constant-roll inflation with small positive value of the constant-roll parameter β ≡φ Hφ = const. has been known to produce a slightly red-tilted curvature power spectrum compatible with the current observational constraints. In this work, we shed light on the constant-roll inflation with negative β and investigate how a stage of constant-roll inflation may realize the growth in the primordial curvature power spectrum necessary to produce a peaked spectrum of primordial black hole abundance. We first review the behavior of constant-roll models in the range of parameters − 3 2 < β < 0, which allows for a constant-roll attractor stage generating a blue-tilted curvature power spectrum without superhorizon growth. As a concrete realization, we consider a potential with two slow-roll stages, separated by the constant-roll stage, in a way that satisfies the current constraints on the power spectrum and the primordial black hole abundance. The model can produce primordial black holes as all dark matter, LIGO-Virgo events, or OGLE microlensing events. Due to the range of different scalar tilts allowed by the constant-roll potential, this construction is particularly robust and testable by future observations.

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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 .

The minimal theory of quasidilaton massive gravity allows for a stable self-accelerating de Sitter solution in a wide range of parameters. On the other hand, in order for the theory to be compatible with local gravity tests, the fifth force due to the quasidilaton scalar needs to be screened at local scales. The present paper thus extends the theory by inclusion of a cubic Horndeski term in a way that (i) respects the quasidilaton global symmetry, that (ii) maintains the physical degrees of freedom in the theory being three, that (iii) can accommodate the Vainshtein screening mechanism and that still (iv) allows for a stable self-accelerating de Sitter solution. After adding the Horndeski term (and a k-essence type nonlinear kinetic term as well) to the precursor action, we switch to the Hamiltonian language and find a complete set of independent constraints. We then construct the minimal theory with three physical degrees of freedom by carefully adding a pair of constraints to the total Hamiltonian of the precursor theory. Switching back to the Lagrangian language, we study cosmological solutions and their stability in the minimal theory. In particular, we show that a self-accelerating de Sitter solution is stable for a wide range of parameters. Furthermore, as in the minimal theory of massive gravity, the propagation speed of the massive gravitational waves in the high momentum limit precisely agrees with the speed of light.