General formulation of cosmological perturbations in scalar-tensor dark
  energy coupled to dark matter
  For a scalar field $\phi$ coupled to cold dark matter (CDM), we provide a
general framework for studying the background and perturbation dynamics on the
isotropic cosmological background. The dark energy sector is described by a
Horndeski Lagrangian with the speed of gravitational waves equivalent to that
of light, whereas CDM is dealt as a perfect fluid characterized by the number
density $n_c$ and four-velocity $u_c^\mu$. For a very general interacting
Lagrangian $f(n_c, \phi, X, Z)$, where $f$ depends on $n_c$, $\phi$,
$X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$, and $Z=u_c^{\mu} \partial_{\mu}
\phi$, we derive the full linear perturbation equations of motion without
fixing any gauge conditions. To realize a vanishing CDM sound speed for the
successful structure formation, the interacting function needs to be of the
form $f=-f_1(\phi, X, Z)n_c+f_2(\phi, X, Z)$. Employing a quasi-static
approximation for the modes deep inside the sound horizon, we obtain analytic
formulas for the effective gravitational couplings of CDM and baryon density
perturbations as well as gravitational and weak lensing potentials. We apply
our general formulas to several interacting theories and show that, in many
cases, the CDM gravitational coupling around the quasi de-Sitter background can
be smaller than the Newton constant $G$ due to a momentum transfer induced by
the $Z$-dependence in $f_2$.
