Density perturbations in general modified gravitational theories

TL;DR

The paper develops a comprehensive framework for linear cosmological perturbations in a wide class of modified gravity theories described by the general action with $f(R,\phi,X)$ and a nonlinear self-interaction term $L_c=\xi(\phi)\square\phi(\partial^{\mu}\phi\partial_{\mu}\phi)$. It derives the background and perturbed field equations, then obtains sub-horizon, quasi-static perturbation equations that yield the effective gravitational coupling $G_{\rm eff}$, the gravitational slip $\zeta$, and the lensing potential $\Phi_{\rm eff}$, incorporating a scalar-field mass $M_{\phi}$ and a curvature-induced mass $M_R$. The authors present explicit ghost- and Laplacian-instability conditions for two main classes of theories ($f_{,RR}\neq0$ and $f_{,RR}=f_{,RX}=0$), and illustrate the results with $f(R)$, scalar-tensor, and Brans–Dicke models, including nonlinear self-interactions that realize Vainshtein-type screening and potential GR recovery in high-density or high-curvature regimes. The framework enables systematic viability assessments against observations of large-scale structure, CMB, and weak lensing, and highlights how mass scales and screening mechanisms govern the transition between GR-like and modified-gravity behavior.

Abstract

We derive the equations of linear cosmological perturbations for the general Lagrangian density $f (R,φ, X)/2+L_c$, where $R$ is a Ricci scalar, $φ$ is a scalar field, and $X=-(\nabla φ)^2/2$ is a field kinetic energy. We take into account a nonlinear self-interaction term $L_c$ recently studied in the context of "Galileon" cosmology, which keeps the field equations at second order. Taking into account a scalar-field mass explicitly, the equations of matter density perturbations and gravitational potentials are obtained under a quasi-static approximation on sub-horizon scales. We also derive conditions for the avoidance of ghosts and Laplacian instabilities associated with propagation speeds. Our analysis includes most of modified gravity models of dark energy proposed in literature and thus it is convenient to test the viability of such models from both theoretical and observational points of view.