  We investigate the minimal theory of massive gravity (MTMG) recently
introduced. After reviewing the original construction based on its Hamiltonian
in the vielbein formalism, we reformulate it in terms of its Lagrangian in both
the vielbein and the metric formalisms. It then becomes obvious that, unlike
previous attempts in the literature of Lorentz-violating massive gravity, not
only the potential but also the kinetic structure of the action is modified
from the de Rham-Gabadadze-Tolley (dRGT) massive gravity theory. We confirm
that the number of physical degrees of freedom in MTMG is two at fully
nonlinear level. This proves the absence of various possible pathologies such
as superluminality, acausality and strong coupling. Afterwards, we discuss the
phenomenology of MTMG in the presence of a dust fluid. We find that on a flat
homogeneous and isotropic background we have two branches. One of them
(self-accelerating branch) naturally leads to acceleration without the genuine
cosmological constant or dark energy. For this branch both the scalar and the
vector modes behave exactly as in general relativity (GR). The phenomenology of
this branch differs from GR in the tensor modes sector, as the tensor modes
acquire a non-zero mass. Hence, MTMG serves as a stable nonlinear completion of
the self-accelerating cosmological solution found originally in dRGT theory.
The other branch (normal branch) has a dynamics which depends on the
time-dependent fiducial metric. For the normal branch, the scalar mode sector,
even though as in GR only one scalar mode is present (due to the dust fluid),
differs from the one in GR, and, in general, structure formation will follow a
different phenomenology. The tensor modes will be massive, whereas the vector
modes, for both branches, will have the same phenomenology as in GR.
