Dark energy survivals in massive gravity after GW170817: SO(3) invariant
  The recent detection of the gravitational wave signal GW170817 together with
an electromagnetic counterpart GRB 170817A from the merger of two neutron stars
puts a stringent bound on the tensor propagation speed. This constraint can be
automatically satisfied in the framework of massive gravity. In this work we
consider a general $SO(3)$-invariant massive gravity with five propagating
degrees of freedom and derive the conditions for the absence of ghosts and
Laplacian instabilities in the presence of a matter perfect fluid on the flat
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) cosmological background. The
graviton potential containing the dependence of three-dimensional metrics and a
fiducial metric coupled to a temporal scalar field gives rise to a scenario of
the late-time cosmic acceleration in which the dark energy equation of state
$w_{\rm DE}$ is equivalent to $-1$ or varies in time. We find that the
deviation from the value $w_{\rm DE}=-1$ provides important contributions to
the quantities associated with the stability conditions of tensor, vector, and
scalar perturbations. In concrete models, we study the dynamics of dark energy
arising from the graviton potential and show that there exist viable parameter
spaces in which neither ghosts nor Laplacian instabilities are present for both
$w_{\rm DE}>-1$ and $w_{\rm DE}<-1$. We also generally obtain the effective
gravitational coupling $G_{\rm eff}$ with non-relativistic matter as well as
the gravitational slip parameter $\eta_s$ associated with the observations of
large-scale structures and weak lensing. We show that, apart from a specific
case, the two quantities $G_{\rm eff}$ and $\eta_s$ are similar to those in
general relativity for scalar perturbations deep inside the sound horizon.
