Geometrical approach to strong gravitational lensing in $f(R)$ gravity

We present a framework for the study of lensing in spherically symmetric spacetimes within the context of $f(R)$ gravity. Equations for the propagation of null geodesics, together with an expression for the bending angle, are derived for any $f(R)$ theory and then applied to an exact spherically symmetric solution of $R^n$ gravity. We find that for this case more bending is expected for $R^n$ gravity theories in comparison to general relativity and is dependent on the value of $n$ and the value of the distance of closest approach of the incident null geodesic.

Figure 1

Plot of the bending angle $\alpha$ compared to the bending angle in general relativity against $n$ corresponding to different values of $r_1$ (distance from the source), with $r_0=100$ (the distance of closest approach) and $r_2=30000$. The result shows that the bending angle value is dependent on $r_1$.

Figure 2

Plot of the bending angle $\alpha$ compared to the bending angle in general relativity against $n$ corresponding to different values or $r_0$ (the distance of closest approach). $r_1=25000$ and $r_2=10000$.

Figure 3

The basic geometric setup for a gravitational lens system with corresponding angles and angular diameters.

Figure 4

The variation of dimensionless angular luminosity distance as functions of redshift, for different values $n$.

Figure 5

Plot of the ratio of the mass of the lensing galaxy to its GR mass against $n$.

Figure 6

The image positions of the Einstein radius against $n$. Two images of the background source are obtained as in the classical GR case which is recovered at the value of $n=1$. A second ring also forms for $1.07<n<1.16$.

Figure 7

Plot of deviation of the Einstein angle $\theta$ from the classical GR value against $n$. The Einstein angle decreases with respect to the classical angle.