We provide a general framework for studying the dark energy cosmology in which a scalar field φ 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 f1(φ,X)ρc(nc) and f2(nc,φ,X)Jcμ∂μφ, where X=-∂μφ∂μφ/2, ρc and nc are the energy density and number density of CDM respectively, and Jcμ 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 quasistatic 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 nc dependence in the coupling f2 gives rise to an interesting possibility for realizing the gravitational coupling with CDM weaker than the Newton gravitational constant G.
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