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.
