Asymptotically Schwarzschild solutions in $f(R)$ extension of general relativity

TL;DR

This paper develops a method to construct $f(R)$ gravity models compatible with solar system tests by perturbatively deriving corrections around Schwarzschild solutions, providing a bottom-up approach to constrain $f(R)$ theories.

Contribution

It introduces a perturbative approach to derive $f(R)$ models consistent with local tests without prior assumptions on $f$, aiding in constraining cosmological models.

Findings

Derived a class of $f(R)$ functions compatible with solar system tests.

Provided a method to constrain $f(R)$ models using local observational data.

Highlighted the importance of low curvature corrections in $f(R)$ gravity.

Abstract

We address the question of how to build a class of $f(R)$ extensions of General Relativity which are compatible with solar system experiments, without making any preliminary assumption on the properties of $f$. The aim is reached by perturbatively solving the modified Einstein equations around a Schwarzschild background and retrieving a posteriori the corresponding $f(R)$. This turns out to be non analytical in $R=0$ and should be intended as the leading correction to the Einstein-Hilbert action in the low curvature limit. The parameters characterizing the $f(R)$ class are then set by constraining the corrections to four different local tests with the observations. The result is a class of $f(R)$ theories built up from a purely bottom-up approach and compatible with the local tests. At a more general level, this result can help constraining exact $f(R)$ models working in Cosmology, since it provides the correct local limit. Further developments and possible extensions of the approach to Cosmology are also discussed.