Effective gravitational couplings for cosmological perturbations in
  generalized Proca theories
  We consider the finite interactions of the generalized Proca theory including
the sixth-order Lagrangian and derive the full linear perturbation equations of
motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker background in the
presence of a matter perfect fluid. By construction, the propagating degrees of
freedom (besides the matter perfect fluid) are two transverse vector
perturbations, one longitudinal scalar, and two tensor polarizations. The
Lagrangians associated with intrinsic vector modes neither affect the
background equations of motion nor the second-order action of tensor
perturbations, but they do give rise to non-trivial modifications to the
no-ghost condition of vector perturbations and to the propagation speeds of
vector and scalar perturbations. We derive the effective gravitational coupling
$G_{\rm eff}$ with matter density perturbations under a quasi-static
approximation on scales deep inside the sound horizon. We find that the
existence of intrinsic vector modes allows a possibility for reducing $G_{\rm
eff}$. In fact, within the parameter space, $G_{\rm eff}$ can be even smaller
than the Newton gravitational constant $G$ at the late cosmological epoch, with
a peculiar phantom dark energy equation of state (without ghosts). The
modifications to the slip parameter $\eta$ and the evolution of growth rate
$f\sigma_8$ are discussed as well. Thus, dark energy models in the framework of
generalized Proca theories can be observationally distinguished from the
$\Lambda$CDM model according to both cosmic growth and expansion history.
Furthermore, we study the evolution of vector perturbations and show that
outside the vector sound horizon the perturbations are nearly frozen and start
to decay with oscillations after the horizon entry.
