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 root f"(0)/f'(0) is larger than the distance d0 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.