Weak cosmic growth in coupled dark energy with a Lagrangian formulation

We investigate a dark energy scenario in which a canonical scalar field ϕ is coupled to the four velocity u_c ^μ of cold dark matter (CDM) through a derivative interaction u_c ^μ∂_μϕ. The coupling is described by an interacting Lagrangian f(X,Z), where f depends on X=−∂^μϕ∂_μϕ/2 and Z=u_c ^μ∂_μϕ. We derive stability conditions of linear scalar perturbations for the wavelength deep inside the Hubble radius and show that the effective CDM sound speed is close to 0 as in the standard uncoupled case, while the scalar-field propagation speed is affected by the interacting term f. Under a quasi-static approximation, we also obtain a general expression of the effective gravitational coupling felt by the CDM perturbation. We study the late-time cosmological dynamics for the coupling f∝X^(2−m)/2Z^m and show that the gravitational coupling weaker than the Newton constant can be naturally realized for m>0 on scales relevant to the growth of large-scale structures. This allows the possibility for alleviating the tension of σ₈ between low- and high-redshift measurements.