Weak cosmic growth in coupled dark energy with a Lagrangian formulation

Abstract

We investigate a dark energy scenario in which a canonical scalar field $\varphi$ is coupled to the four velocity $u_c^{\mu }$ of cold dark matter (CDM) through a derivative interaction $u_c^{\mu }{\partial }_{\mu }\varphi$. The coupling is described by an interacting Lagrangian $f(X,Z)$, where $f$ depends on $X=-{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi /2$ and $Z=u_c^{\mu }{\partial }_{\mu }\varphi$. 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.

Comments: 15 pages, 2 figures, published version