Weak cosmic growth in coupled dark energy with a Lagrangian formulation

We investigate a dark energy scenario in which a canonical scalar field $\phi$ is coupled to the four velocity $u_{c}^{\mu}$ of cold dark matter (CDM) through a derivative interaction $u_{c}^{\mu} \partial_{\mu} \phi$. The coupling is described by an interacting Lagrangian $f(X, Z)$, where $f$ depends on $X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$ and $Z=u_{c}^{\mu} \partial_{\mu} \phi$. 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 $f \propto X^{(2-m)/2}Z^m$ 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 $\sigma_8$ between low- and high-redshift measurements.