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 $\sigma \ne 0$ 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.

Figure 1

The dependence of correction parameter $\sigma$ in the potential upon the exponent $n$.

Figure 2

The ratio of the power-law $f(R)$ gravitational potential and the Newtonian gravitational potential as a function of $\sigma \in [0,1)$ and of the logarithm of the distance $r\ge r_c$, normalized to $r_S$ (the Schwarzschild radius of the lens). We have chosen a typical bulge mass of $m_{\mathrm{bulge}}\approx {10}^{10}{M}_{\odot }$ and typical bulge radius $r_c\approx r_{\mathrm{bulge}}\approx 3240\mathrm{pc}$.

Figure 3

The plots show the $\delta (\sigma )$ dependence in radians for different values of $b/r_c$, in units $2Gm/c^2{r}_c=1$, with the choice $r_c=3240\mathrm{pc}$. The respective values of $b/r_c$ are from top to bottom 1 (dashed), 2 (solid), 4 (diamond) on panel (a) and 10 (cross), 100 (circle), 1000 (dotted) on panel (b). The critical behavior appears at $b/r_c=2$.

Figure 4

The lensing geometry (see text for discussion of the symbols). Positive angles are represented with counterclockwise directed arcs.

Figure 5

(a) The difference between the Einstein angles ${\theta }_{VE}$, obtained from Eq. (18) and ${\theta }_0$, obtained from Eq. (17), as function of the parameters $\sigma$ and $\overline{\epsilon }$. (b) The difference between the positive apparent angles ${\theta }_{VE}$ and ${\theta }_{DS}$ [obtained from Eq. (19)] of the images of a source located at $\beta =1$. While the two surfaces have the same shape it should be noted that the vertical scales differ by 3 orders of magnitude. We set the distances $D_l=1\mathrm{Mpc}$ and $D_{ls}=2\mathrm{Mpc}$.

Figure 6

The image positions $\theta$ as function of $\overline{\epsilon }$ and $\beta$ for $\sigma =0.25$ (a) and $\sigma =0.75$ (b), for the distances $D_l=1\mathrm{Mpc}$ and $D_{ls}=2\mathrm{Mpc}$. The angle $\beta$ is varied up to 0.0015 rad, similarly as on Fig. 4(b) of Ref. 36. With decreasing $\beta$, the image separations shrink accordingly. At $\beta =0$ the angle $\theta$ represents the angular radius of the Einstein ring. As we expect the $\beta =0$ sections of the surfaces are symmetric with respect to the plane $\theta =0$.

Figure 7

The image separations and magnifications as functions of $\beta /{\theta }_E$ for $\sigma =0.25$ (a) and $\sigma =0.75$ (b). We fixed $\overline{\epsilon }=3.375\times {10}^{-9}$, while the distances $D_l=1\mathrm{Mpc}$ and $D_{ls}=2\mathrm{Mpc}$ were chosen for the plots. The upper and lower solid curves represent the primary and secondary image magnification factors, respectively; their ratio is the dashed curve; and the dotted curve is the normalized image separation.

Figure 8

The ratio of the magnification factor of the primary and secondary images as function of the image separation (normalized to the Einstein angle ${\theta }_E^{\sigma =0}$, characterizing the $\sigma =0$ case), on log-log scale for a series of $\sigma$ values (left panel). There is a double degeneracy in $\sigma$ of the power $\kappa$, represented as a function of $\sigma$ (right panel). The plot refers to $\overline{\epsilon }=3.375\times {10}^{-9}$ and distances $D_l=1\mathrm{Mpc}$, $D_{ls}=2\mathrm{Mpc}$.