Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations

TL;DR

The paper derives the full linear perturbation equations for the most general scalar-tensor theories with second-order field equations (Horndeski framework) including non-relativistic matter, and computes the effective gravitational coupling $G_{ m eff}$ and the lensing-related potential $\Phi_{\rm eff}$ under a quasi-static approximation on sub-horizon scales. It provides explicit expressions for $G_{ m eff}$ and the anisotropic stress parameter $\eta$ in terms of the model functions and the scalar mass $M$, and applies the results to several dark energy models (e.g., $f(R)$, Brans-Dicke, kinetic gravity braiding, covariant Galileon, and field-derivative couplings to the Einstein tensor). These results enable robust tests of modified gravity using large-scale structure, weak lensing, and CMB observations, and clarify the conditions under which the quasi-static approximation is valid. The work also clarifies how different Horndeski models imprint distinct signatures on matter growth and light propagation, providing a framework to discriminate between competing theories with upcoming surveys.

Abstract

In the Horndeski's most general scalar-tensor theories the equations of scalar density perturbations are derived in the presence of non-relativistic matter minimally coupled to gravity. Under a quasi-static approximation on sub-horizon scales we obtain the effective gravitational coupling $G_{eff}$ associated with the growth rate of matter perturbations as well as the effective gravitational potential $Φ_{eff}$ relevant to the deviation of light rays. We then apply our formulas to a number of modified gravitational models of dark energy--such as those based on f(R) theories, Brans-Dicke theories, kinetic gravity braidings, covariant Galileons, and field derivative couplings with the Einstein tensor. Our results are useful to test the large-distance modification of gravity from the future high-precision observations of large-scale structure, weak lensing, and cosmic microwave background.