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 $f_2$.