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Gravitational microlensing in modified gravity theories: Inverse-square theorem

Microlensing studies are usually based on the lens equation that is valid only to the first order in the gravitational constant G and lens mass M. We consider corrections to the conventional lens equation in terms of differentiable functions, so that they can express not only the second-order effects of GM in general relativity but also modified gravity theories. As a generalization of Ebina et al. (Prog. Theor. Phys. 104 (2000) 1317), we show that, provided that the spacetime is static, spherically symmetric and asymptotically flat, the total amplification by microlensing remains unchanged at the linear order of the correction to the deflection angle, if and only if the correction takes a particular form as the inverse square of the impact parameter, whereas the magnification factor for each image is corrected. It is concluded that the light curve shape by microlensing is inevitably changed and will thus allow us to probe modified gravity, unless a modification to the deflection angle takes the particular form. No systematic deviation in microlensing observations has been reported. For instance, therefore, the Yukawa-type correction is constrained as the characteristic length > $10^{14}$ m.