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

Abstract

We derive the equation of matter density perturbations on subhorizon scales for a general Lagrangian density $f(R,\ensuremath{\phi},X)$ that is a function of a Ricci scalar $R$, a scalar field $\ensuremath{\phi}$, and a kinetic term $X=\ensuremath{-}(\ensuremath{\nabla}\ensuremath{\phi}{)}^{2}/2$. 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 ${\ensuremath{\delta}}_{m}$ as well as the gravitational potential $\ensuremath{\Phi}$ for some analytically solvable models. In an $f(R)$ dark energy model with the Lagrangian density $f(R)=\ensuremath{\alpha}{R}^{1+m}\ensuremath{-}\ensuremath{\Lambda}$, 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 $f=F(\ensuremath{\phi})R+2p(\ensuremath{\phi},X)$, 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.