Linear growth of matter density perturbations in f(R,G) theories

TL;DR

This paper derives the behavior of matter density perturbations in general f(R,G) gravity theories, revealing a scale-dependent effective gravitational constant and providing tools to test these models with cosmological observations.

Contribution

It presents the general equations for matter perturbations in f(R,G) theories and identifies conditions for avoiding superluminal modes, advancing the understanding of modified gravity effects.

Findings

Effective gravitational constant scales as distance squared at small scales.

Provides expressions for effective anisotropic stress relevant for observations.

Identifies a class of theories that evade unusual small-scale behavior.

Abstract

We derive the equation of matter density perturbations on sub-horizon scales around a flat Friedmann-Lema\^\i tre-Robertson-Walker background for the general Lagrangian density $f(R,\GB)$ that is a function of a Ricci scalar $R$ and a Gauss-Bonnet term $\GB$. We find that the effective gravitational constant generically scales as distance squared at small distances. The effect of this diminishing of the gravitational constant might be important in the gravitational dynamics of cosmic objects such as galaxies, which can be in principle tested by observations. We also provide the general expressions for the effective anisotropic stress, which is useful to constrain modified gravity models from observations of large-scale structure and weak lensing. We also find that there is a special class of theories which evade this unusual behaviour and that the condition to belong to this special class is exactly the same as the one for not having super-luminal modes with propagation speed proportional to their wavenumber.