Matter density perturbations and effective gravitational constant in modified gravity models of dark energy

TL;DR

The paper develops a general perturbation formalism for Lagrangians $f(R,\phi,X)$ on sub-horizon scales to study linear matter growth and gravitational potentials. It derives the effective gravitational constant $G_{\rm eff}$, the anisotropic stress $\eta$, and the lensing parameter $\Sigma$, and applies the framework to both $f(R)$ gravity and scalar-tensor models, including frame transformations between Jordan and Einstein frames. In $f(R)$ gravity, perturbation growth becomes scale-dependent through $m=Rf_{,RR}/f_{,R}$ and the regime $k^2/(a^2R)\,m$, yielding analytic growth exponents for constant-$m$ cases and distinctive ISW signals for evolving potentials. The results provide a path to constrain modified gravity dark-energy scenarios with upcoming large-scale-structure and weak-lensing data, distinguishing them from the concordance $\Lambda$CDM model.

Abstract

We derive the equation of matter density perturbations on sub-horizon scales for a general Lagrangian density f(R, phi, X) that is a function of a Ricci scalar R, a scalar field phi and a kinetic term X=-(nabla phi)^2/2. This is useful to constrain modified gravity dark energy models from observations of large-scale structure and weak lensing. We obtain the solutions for the matter perturbation delta_m as well as the gravitational potential Phi for some analytically solvable models. In a f(R) dark energy model with the Lagrangian density f(R)=alpha R^{1+m}-Lambda, the growth rates of perturbations exhibit notable differences from those in the standard Einstein gravity unless m is very close to 0. In scalar-tensor models with the Lagrangian density f=F(phi)R+2p(phi,X) we relate the models with coupled dark energy scenarios in the Einstein frame and reproduce the equations of perturbations known in the current literature by making a conformal transformation. We also estimate the evolution of perturbations in both Jordan and Einstein frames when the energy fraction of dark energy is constant during the matter-dominated epoch.