Weak gravitational lensing in fourth order gravity

Abstract

For a general class of analytic functions $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}},{R}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}})$ we discuss the gravitational lensing in the Newtonian limit of theory. From the properties of the Gauss-Bonnet invariant it is enough to consider only one curvature invariant between the Ricci tensor and the Riemann tensor. Then, we analyze the dynamics of a photon embedded in a gravitational field of a generic $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}})$ gravity. The metric is time independent and spherically symmetric. The metric potentials are Schwarzschild-like, but there are two additional Yukawa terms linked to derivatives of $f$ with respect to two curvature invariants. Considering first the case of a pointlike lens, and after the one of a generic matter distribution of the lens, we study the deflection angle and the angular position of images. Though the additional Yukawa terms in the gravitational potential modifies dynamics with respect to general relativity, the geodesic trajectory of the photon is unaffected by the modification if we consider only $f(R)$ gravity. We find different results (deflection angle smaller than the angle of general relativity) only due to the introduction of a generic function of the Ricci tensor square. Finally, we can affirm that the lensing phenomena for all $f(R)$ gravities are equal to the ones known for general relativity. We conclude the paper by showing and comparing the deflection angle and position of images for $f(R,{R}_{\ensuremath{\alpha}\ensuremath{\beta}}{R}^{\ensuremath{\alpha}\ensuremath{\beta}})$ gravity with respect to the gravitational lensing of general relativity.