Analysis of the Yukawa gravitational potential in $f(R)$ gravity I: semiclassical periastron advance

TL;DR

This paper investigates how modifications to gravity via $f(R)$ theories affect planetary orbit precession, deriving constraints on the strength of the resulting fifth force from observational data.

Contribution

It provides a semiclassical analysis of the Yukawa-like potential in $f(R)$ gravity and derives bounds on the fifth force parameters from planetary precession measurements.

Findings

The modified potential leads to Keplerian orbits under certain conditions.

The fifth force causes measurable orbital precession.

Constraints on the fifth force strength are established as $[2.70-6.70]\times10^{-9}$ with a scale of about 5000 AU.

Abstract

The {\it concordance} cosmological model has been successfully tested throughout the last decades. Despite its successes, the fundamental nature of dark matter and dark energy is still unknown. Modifications of the gravitational action have been proposed as an alternative to these dark components. The straightforward modification of gravity is to generalize the action to a function, $f(R)$, of the scalar curvature. Thus one is able to describe the emergence and the evolution of the Large Scale Structure without any additional (unknown) dark component. In the weak field limit of the $f(R)$-gravity, a modified Newtonian gravitational potential arises. This gravitational potential accounts for an extra force, generally called fifth force, that produces a precession of the orbital motion even in the classic mechanical approach. We have shown that the orbits in the modified potential can be written as Keplerian orbits under some conditions on the strength and scale length of this extra force. Nevertheless, we have also shown that this extra term gives rise to the precession of the orbit. Thus, comparing our prediction with the measurements of the precession of some planetary motions, we have found that the strength of the fifth force must be in the range $[2.70-6.70]\times10^{-9}$ whit the characteristic scale length to fixed to the fiducial values of $\sim 5000$ AU.