Gravitational Lensing and Image Distortion by Buchdahl Inspired Metric in $\mathcal{R}^2$ Gravity

TL;DR

This paper analyzes gravitational lensing effects caused by a Buchdahl inspired metric in $ ext{R}^2$ gravity, deriving deflection angles in strong and weak limits, and constraining model parameters using observations of supermassive black holes and Einstein rings.

Contribution

It provides analytical expressions for light deflection angles in both limits within $ ext{R}^2$ gravity and applies these to real astrophysical objects to constrain the model parameter $ ilde{k}$.

Findings

Derived deflection angles in strong and weak limits for the Buchdahl inspired metric.

Constrained the Buchdahl parameter $ ilde{k}$ using observational data from Sgr A*, M87*, and Einstein rings.

Showed that the deflection angle reduces to Schwarzschild case under specific conditions.

Abstract

We investigate gravitational lensing by \textit{special} Buchdahl inspired metric with the Buchdahl parameter $\tilde{k}$. In strong deflection limit, we derive the deflection angle analytically for the light rays that diverge as photons approach the photon sphere. These are then used in order to compute the angular image positions modeling supermassive black holes, Sgr A* and M87* as lenses. The Einstein rings for the outermost relativistic images are also depicted here alongside observational constraints on $\tilde{k}$ by the Einstein radius and lens mass. Constraints on $\tilde{k}$ are obtained modelling black holes ( Sgr A* and M87*) and Canarias Einstein ring. In weak deflection limit, the analytic expression of deflection angle of the subject asymptotically flat metric in $\mathcal{R}^2$ gravity is determined using the Gauss Bonnet theorem. Considering M87* as a lens, weak deflection angle is used to study the image magnification and image distortion for primary and secondary images. It is shown that image distortion satisfies the hypothesis of Virbhadra. Moreover, it is seen that our general expression of deflection angle reduces, as a special case, to the deflection angle of Schwarzschild metric in both weak and strong deflection limits.