The Force of Gravity from a Lagrangian containing Inverse Powers of the Ricci Scalar

TL;DR

The work studies a class of modified gravity theories with $f(R) = -\mu^{2(p+1)} R^{-p}$ to explain late-time acceleration without dark energy. It analyzes the gravitational response to a diffuse spherical source in a locally de Sitter background by solving the trace equation and matching interior and exterior Ricci-scalar perturbations for $p=1$. The key result is a linearly growing exterior gravitational force, $h'(y) \propto y$, which greatly exceeds the GR prediction with a cosmological constant, signaling strong phenomenological tension. The authors discuss potential remedies, such as adding an $R^2$ term, which can stabilize one aspect but may spoil cosmology or require fine-tuning, and note that brane-world scenarios like Dvali–Gabadadze–Shifman (DGS) can avoid both the interior instability and the long-range growth.

Abstract

We determine the gravitational response to a diffuse source, in a locally de Sitter background, of a class of theories which modify the Einstein-Hilbert action by adding a term proportional to an inverse power of the Ricci scalar. We find a linearly growing force which is not phenomenologically acceptable.