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arxiv: 2605.27242 · v1 · pith:OWR2VNYKnew · submitted 2026-05-26 · 🌀 gr-qc · astro-ph.CO· astro-ph.HE

Strong-lensing degeneracies of black holes embedded in self-interacting scalar field dark matter halos

Mohsen Fathi , Gabriel G\'omez This is my paper

Pith reviewed 2026-06-29 15:41 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COastro-ph.HE
keywords strong gravitational lensingblack holesscalar field dark matterdark matter halostime delaysEinstein clusterM87*Sgr A*
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The pith

Black holes embedded in self-interacting scalar field dark matter halos produce strong-lensing observables that differ from the pure Schwarzschild case by at most one part in a thousand.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper numerically reconstructs the spacetime of black holes surrounded by self-interacting scalar field dark matter halos using the Einstein cluster formalism and compares the resulting photon trajectories and lensing observables to the Schwarzschild metric and to NFW halos. It computes the photon sphere radius, critical impact parameter, relativistic Einstein ring radii, image positions, separations, magnifications, and time delays, focusing on the supermassive black holes M87* and Sgr A*. The calculations show that all considered halo models shift these quantities by O(10^{-3}) or less relative to the isolated black hole, producing a strong observational degeneracy. Time delays between relativistic images exhibit the largest and most systematic halo-induced corrections, and these corrections become more pronounced for higher black hole masses.

Core claim

The spacetime geometries of black holes embedded in self-interacting scalar field dark matter halos, reconstructed numerically via the Einstein cluster formalism, yield photon spheres and critical impact parameters that differ from the Schwarzschild metric by O(10^{-3}) or smaller. Consequently, strong-lensing observables including relativistic Einstein rings, image separations, magnifications, and especially time delays between relativistic images exhibit small but systematic deviations, with time delays providing the clearest amplification for supermassive black holes such as M87* and Sgr A*.

What carries the argument

Numerical reconstruction of the spacetime metric through the Einstein cluster formalism, which determines the effective potential for photon trajectories in the combined black hole and halo system.

If this is right

  • Relativistic image positions, separations, and magnifications remain within 0.1 percent of their Schwarzschild values for all examined halo configurations.
  • Time delays between relativistic images display the largest relative corrections and scale with black hole mass, becoming more detectable for objects like M87*.
  • NFW-type halos produce deviations of comparable magnitude to the self-interacting scalar field models.
  • Standard strong-lensing observables stay highly robust against the presence of these dark matter distributions around the black hole.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-precision timing observations could distinguish among dark matter halo models around black holes where positional and magnification data cannot.
  • Existing Event Horizon Telescope imaging of M87* and Sgr A* is unlikely to resolve the reported halo-induced shifts.
  • The same numerical reconstruction approach could be applied to other dark matter density profiles to identify which halo properties most influence time-delay signals.

Load-bearing premise

The spacetime geometry for black holes embedded in self-interacting scalar field dark matter halos can be accurately reconstructed numerically through the Einstein cluster formalism, allowing reliable computation of photon trajectories and lensing observables.

What would settle it

A precision measurement of the time delay between the first two relativistic images of M87* that lies outside the narrow range predicted by the halo models (differing from the Schwarzschild value by more than a few parts in 10^3) would falsify the claim of strong observational degeneracy at the reported level.

Figures

Figures reproduced from arXiv: 2605.27242 by Gabriel G\'omez, Mohsen Fathi.

Figure 1
Figure 1. Figure 1: FIG. 1. Halo density profiles for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The panels show (a) the profile of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Radial profile of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Shadow radius for the different halo configurations con [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The values for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The outermost RERs for M87* (a,b) and Sgr A* (c,d), as black holes with DM halo, for all discussed cases of halo configurations. The [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Deviations of the finite-order relativistic image positions, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Separation between the first two finite-order relativistic images, [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Relative magnification deviation [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Halo-induced correction to the time delay between the first two relativistic images. Left: absolute deviation [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

In this paper, we explore the strong gravitational lensing properties of black holes embedded in self-interacting scalar field dark matter halos, together with NFW-type configurations for comparison. The corresponding spacetime geometry is reconstructed numerically through the Einstein cluster formalism, allowing us to study how the surrounding dark matter distribution affects the propagation of photons near the black hole. We first analyze the effective function governing photon trajectories and calculate the corresponding photon sphere radius and critical impact parameter. We then investigate different strong-lensing observables, including relativistic Einstein rings, finite-order image positions, image separations, magnifications, and time delays, with particular attention to the supermassive black holes M87* and Sgr A*. Our results show that the considered halo configurations produce only small deviations with respect to the Schwarzschild case, typically at the level of $\mathcal{O}(10^{-3})$ or smaller, leading to a strong observational degeneracy among the models. Nevertheless, small but systematic differences remain present, especially in the time delay between relativistic images, which provides the clearest amplification of the halo-induced corrections for very massive black holes. These results suggest that, although standard strong-lensing observables remain highly robust against the considered halo environments, time-domain signatures may offer a more promising way to probe the effect of dark matter surrounding black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript numerically reconstructs the spacetime metric for black holes embedded in self-interacting scalar field dark matter halos (and NFW profiles for comparison) via the Einstein cluster formalism. It computes the effective potential for photon trajectories, the photon-sphere radius, critical impact parameter, and strong-lensing observables including relativistic Einstein rings, finite-order image positions, separations, magnifications, and time delays, with explicit application to M87* and Sgr A*. The central claim is that halo-induced deviations from the Schwarzschild case remain at the level of O(10^{-3}) or smaller for most observables, producing strong observational degeneracy, while time delays between relativistic images exhibit systematic differences that are amplified for very massive black holes.

Significance. If the numerical results hold, the work establishes that standard strong-lensing observables are robust against the considered dark-matter halo environments, while identifying time delays as a potentially more sensitive probe. The numerical reconstruction approach via the Einstein cluster formalism is a positive feature, enabling study of metrics without closed-form solutions and yielding falsifiable predictions for time-domain signatures around M87* and Sgr A*.

major comments (1)
  1. [Numerical reconstruction (abstract and methods)] The section on numerical reconstruction of the metric (invoked in the abstract and used to obtain all reported observables) provides no details on the integrator, step-size control, convergence tests against the Schwarzschild limit or known halo solutions, or residual error estimates on the photon-sphere radius, critical impact parameter, or integrated time-delay quantities. This validation is load-bearing for the central claim, as the reported O(10^{-3}) deviations and systematic time-delay differences cannot be distinguished from numerical artifacts without such checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Dear Editor, We thank the referee for their careful and constructive review of our manuscript. The single major comment identifies a genuine omission in the presentation of our numerical methods. We address it below and will revise the manuscript accordingly to strengthen the validation of our results.

read point-by-point responses

  1. Referee: [Numerical reconstruction (abstract and methods)] The section on numerical reconstruction of the metric (invoked in the abstract and used to obtain all reported observables) provides no details on the integrator, step-size control, convergence tests against the Schwarzschild limit or known halo solutions, or residual error estimates on the photon-sphere radius, critical impact parameter, or integrated time-delay quantities. This validation is load-bearing for the central claim, as the reported O(10^{-3}) deviations and systematic time-delay differences cannot be distinguished from numerical artifacts without such checks.

    Authors: We agree that the current manuscript does not provide sufficient documentation of the numerical reconstruction procedure. In the revised version we will add a dedicated subsection in the methods that specifies: the integrator employed and its implementation; the step-size control and convergence criteria; explicit convergence tests recovering the Schwarzschild limit to machine precision and, where analytic halo solutions exist, matching those profiles; and quantitative residual error bounds on the photon-sphere radius, critical impact parameter, and integrated time-delay observables. These additions will demonstrate that the reported O(10^{-3}) deviations and the systematic time-delay differences are not numerical artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: metric reconstruction and observable computation are independent

full rationale

The paper takes external density profiles as input, reconstructs the metric numerically via the Einstein cluster formalism (integrating the Einstein equations for the given halo), and then computes photon trajectories and lensing observables directly from the resulting geometry. No step fits parameters to the output observables and renames them as predictions, invokes self-citations for uniqueness theorems, or defines quantities in terms of each other. The reported O(10^{-3}) deviations are direct numerical outputs rather than tautological re-expressions of the inputs, rendering the derivation chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, new entities, or detailed axioms beyond the stated numerical method; halo models are treated as standard inputs from the field.

axioms (1)
  • domain assumption The Einstein cluster formalism can be applied to numerically reconstruct the spacetime geometry of a black hole embedded in a self-interacting scalar field dark matter halo.
    Directly invoked in the abstract as the method for obtaining the metric used in all subsequent lensing calculations.

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