Weak gravitational lensing by compact objects in fourth order gravity

We discuss weak lensing characteristics in the gravitational field of a compact object in the low-energy approximation of fourth order f(R) gravity theory. The particular solution is characterized by a gravitational strength parameter $\sigma$ and a distance scale $r_{c}$ much larger than the Schwarzschild radius. Above $r_{c}$ gravity is strengthened and as a consequence weak lensing features are modified compared to the Schwarzschild case. We find a critical impact parameter (depending upon $r_{c}$) for which the behavior of the deflection angle changes. Using the Virbhadra-Ellis lens equation we improve the computation of the image positions, Einstein ring radii, magnification factors and the magnification ratio. We demonstrate that the magnification ratio as function of image separation obeys a power-law depending on the parameter $\sigma$, with a double degeneracy. No $\sigma \neq 0$ value gives the same power as the one characterizing Schwarzschild black holes. As the magnification ratio and the image separation are the lensing quantities most conveniently determined by direct measurements, future lensing surveys will be able to constrain the parameter $\sigma$ based on this prediction.Comment: 14 pages, 8 figures, published version. Appendix added on the energy conditions for the modifications from GR, interpreted as an effective energy-momentum tenso