Lensing by black holes within astrophysical environments

TL;DR

This work addresses how astrophysical halos around black holes alter strong-field gravitational lensing. By employing an exact GR solution for a static, spherically symmetric BH embedded in a Hernquist halo, the authors compute deflection angles and image magnifications beyond perturbative treatments. They find that the halo induces a characteristic bump in both deflection and magnification, shifts the light-ring outward, and broadens the regime of strong lensing; these effects depend primarily on the halo compactness ${M/a_0}$ and can mimic the inference of a heavier central mass if environmental effects are neglected. The analysis is extended to gravitational-wave echoes, showing that halos can enhance and delay echoes via multiple images, with practical implications for GW data analysis and the interpretation of potential beyond-GR signatures.

Abstract

Astrophysical black holes are likely to be surrounded by various forms of matter in the form of disks or halos. While a number of studies have examined the impact of an environment on the lensing of light or gravitational waves from cosmological sources, these have, thus far, been carried out in either a Newtonian or post-Newtonian framework where the environment is superimposed on the black-hole spacetime. By using an exact solution in general relativity describing a black hole embedded within a realistic halo of Hernquist matter distribution, we study deflection angles and image amplification in a fully relativistic setup. It is shown that large ``bumps'', that also arise at the Newtonian and post-Newtonian levels, track the transition scale set by the halo parameters that control the strong-lensing upturn and can significantly adjust the inferences made for either the source or lens in various contexts. As an application, we consider ``echoes'' of gravitational waves, sourced by astrophysical lenses rather than being intrinsic to the compact object that produces the signal.

Figures (4)

• Figure 1: Total bending angle $\hat{\alpha}(r_0)$ (arcsec) versus photon periastron, $r_0$. Left: Compactness $M/a_0$ is varied by changing $a_0$ at fixed $(M_{\rm BH},M)=(10^6 M_\odot,\,10^4 M_{\rm BH})$; curves show $a_0/M\in\{10,50,10^2,10^4\}$ alongside the Schwarzschild baseline (see figure legends). Increasing compactness boosts bending across all $r_0$ and shifts the strong-lensing upturn outward (larger $r_0$), reflecting the outward shift of the light ring and larger capture impact parameter. Right: Halo mass is varied instead at fixed $a_0/M=10^2$; curves show $M/M_{\rm BH}\in\{1,10^2,10^4\}$ plus Schwarzschild. Larger $M$ similarly enhances bending and advances the transition into the strong-lensing regime; overall amplitude is chiefly controlled by the compactness $M/a_0$.
• Figure 2: Similar to Fig. \ref{['deflections_plot']}, though instead showing combined parameter variations and degeneracies. Changing $M_{\rm BH}$ primarily rescales the horizontal axis (via $r_{\rm Sch}\!\propto\!M_{\rm BH}$), whereas compactness $M/a_0$ sets the overall amplitude and sharpness of the near-light-ring upturn. The figure highlights a degeneracy between central mass and environmental compactness if only $\hat{\alpha}(r_0)$ is used, suggesting joint constraints from weak- and strong-field portions of the curves.
• Figure 3: Total bending angle $\hat{\alpha}(r_0)$ (arcsec) versus photon periastron, $r_0$. Here, we use the full metric \ref{['eq:g_tt']}, the Newtonian ($1/c^2$) and the PN expansion of $f(r)$, through the Eq. \ref{['eq:PN_expansion']}. Top panel: Compactness is set to $M/a_0=10^{-1}$ (left), with fixed $(M_{\rm BH},M)=(10^6 M_\odot,\,10^4 M_{\rm BH})$; curves show the total bending angle of the Newtonian and PN metric with respect to the full solution. On the right panel the comparison is the same but with $M/a_0=10^{-2}$. Bottom panel: The relative difference of total bending angles of the full GR solution and the Newtonian, PN approximations of the metric. It is obvious that smaller compactness leads to an order of magnitude less error.
• Figure 4: Amplification factors, $\mu$, as a function of source offset, $\beta$. Left: Compactness $M/a_0$ varied at fixed $(M_{\rm BH},M)=(10^6 M_\odot,\,10^4 M_{\rm BH})$ with $a_0/M\in\{10,10^2,10^4,10^5\}$, with the Schwarzschild case shown for comparison (see figure legends). Right: Halo mass varied at fixed $a_0/M=10^2$ with $M/M_{\rm BH}\in\{1,10^2,10^4\}$ plus Schwarzschild. Denser or more massive halos systematically raise $\mu$ at fixed $\beta$ and extend the domain of strong magnification to larger angular offsets; in the diffuse/low-mass limit the curves converge to the vacuum result.