Distinguishing f(R) theories from general relativity by gravitational lensing effect

TL;DR

This paper investigates whether gravitational lensing can distinguish f(R) gravity theories from general relativity, finding that differences only appear at the third post-Newtonian order under specific conditions and are generally negligible.

Contribution

The study develops a third-order post-Newtonian framework for f(R) theories and identifies conditions under which lensing effects differ from general relativity.

Findings

Differences in lensing appear at third order when certain parameters exceed a threshold.

In axially symmetric spacetimes, f(R) and GR lensing effects are indistinguishable.

Distinguishing f(R) theories via lensing is rare and requires specific conditions.

Abstract

The post-Newtonian formulation of a general class of f(R) theories is set up to 3rd order approximation. It turns out that the information of a specific form of f(R) gravity is encoded in the Yukawa potential, which is contained in the perturbative expansion of the metric components. Although the Yukawa potential is canceled in the 2nd order expression of the effective refraction index of light, detailed analysis shows that the difference of the lensing effect between the f(R) gravity and general relativity does appear at the 3rd order when $\sqrt{f''(0)/f'(0)}$ is larger than the distance $d_0$ to the gravitational source. However, the difference between these two kinds of theories will disappear in the axially symmetric spacetime region. Therefore only in very rare case the f(R) theories are distinguishable from general relativity by gravitational lensing effect at the 3rd order post-Newtonian approximation.