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

The post-Newtonian formulation of a general class of f ( R ) theories is set up in a third-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 second-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 third order when 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 in a third-order post-Newtonian approximation.