Scalar-field dark energy nonminimally and kinetically coupled to dark
  matter
  We provide a general framework for studying the dark energy cosmology in
which a scalar field $\phi$ is nonminimally and kinetically coupled to Cold
Dark Matter (CDM). The scalar-graviton sector is described by the action of
Horndeski theories with the speed of gravitational waves equivalent to that of
light, whereas CDM is treated as a perfect fluid given by a Schutz-Sorkin
action. We consider two interacting Lagrangians of the forms $f_1(\phi,X)\rho_c
(n_c)$ and $f_2 (n_c, \phi,X) J_c^{\mu} \partial_{\mu}\phi$, where
$X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$, $\rho_c$ and $n_c$ are the
energy density and number density of CDM respectively, and $J_c^{\mu}$ is a
vector field related to the CDM four velocity. We derive the scalar
perturbation equations of motion without choosing any special gauges and
identify conditions for the absence of ghosts and Laplacian instabilities on
scales deep inside the sound horizon. Applying a quasi-static approximation in
a gauge-invariant manner, we also obtain the effective gravitational couplings
felt by CDM and baryons for the modes relevant to the linear growth of
large-scale structures. In particular, the $n_c$ dependence in the coupling
$f_2$ gives rise to an interesting possibility for realizing the gravitational
coupling with CDM weaker than the Newton gravitational constant $G$.
