Gravitational Lensing Effect from The Revised Deser-Woodard Nonlocal Gravity

TL;DR

This paper analyzes gravitational lensing by a static spherically symmetric black hole in the revised Deser-Woodard nonlocal gravity. By deriving the weak-field deflection with a leading nonlocal correction $\alpha(b) \approx \left(4 + \tfrac{2\zeta}{3}\right)\tfrac{1}{b}$ and applying Bozza's strong-field formalism to obtain $\alpha(b) = -a \ln\left(\tfrac{b}{b_m}-1\right) + u$, the authors show that corrections scale linearly with the dimensionless coupling $\zeta$ and are exponentially suppressed by the exponent parameter $n$. They compute lensing observables such as $\theta_{\infty}$, the angular separation $s$, and the flux ratio $\mu$, finding that photon lensing preserves scale-invariance at fixed time similarly to GR and conformal gravity, while massive-particle lensing can exhibit scale-variance. The work highlights the crucial role of $n$ in determining deviations from Schwarzschild behavior and suggests a pathway to constrain revised D-W gravity with future astronomical lensing observations, while noting limitations due to the chosen static ansatz and the need for dynamic cosmological coupling.

Abstract

We investigate the gravitational lensing effects of a static spherically symmetric black hole (BH) within the framework of the revised Deser-Woodard (D-W) nonlocal gravity. By analyzing the deflection angle in both the weak and strong field limits, we derive several distinguishing features of the model. In the weak field limit, we report a leading-order correction to the deflection angle directly attributed to the non-local nature of the theory. In the strong field limit, we find that the lensing corrections are almost linearly dependent on the coupling parameter $ζ$ while being exponentially suppressed by the exponent parameter $n$. Furthermore, the gravitational lensing effect in the revised D-W model at a given time shares similar scale-invariant behavior to General Relativity and conformal gravity, offering a potential pathway to distinguish it from other alternatives using astronomical observations.

Figures (3)

• Figure 1: Comparison between results obtained via different methods ($n=2,\ \zeta=0.5$): The numerical results obtained by directly performing the integration (\ref{['dangle']}) (Black line). Strong field limit (Blue dashed line). Weak field limit (Orange dashed line).
• Figure 2: The main results of our work: (a) The photon rings of models of different parameters, with the innermost one being the Schwarzschild BH. (b) Corrections to the strong field limit for $\zeta=0.5,\ n=2$ compared to the Schwarzschild BH. (c) Relations between the relative difference $R_{\alpha}$ of deflection angle and the coupling parameter $\zeta$, with $n=2$ (d) Relations between the relative difference $R_{\alpha}$ of deflection angle and $n$, with $\zeta=1$. (e) Relations between the relative difference $R_{\mathrm{Obs}}$ of observables and the coupling parameter $\zeta$, with $n=2$. (f) Log-Log Continuation of (e) to the regime where $\zeta$ is extremely small.
• Figure 3: (a) The contour plot of the dependence of $|R_{\theta}|$ on both $\zeta$ and $n$. (b) The 3D plot of the dependence of $R_{obs}$ for all three observables on both $\zeta$ and $n$.