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A black hole embedded in a Hernquist dark-matter halo has a shadow radius that grows with the halo compactness C, and its ringdown frequencies are redshifted by the factor 3√3 M_BH / b_c = 1 - C + C²/6; current shadow-imaging data bound the

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For black holes embedded in Hernquist dark matter halos, the shadow radius and quasinormal mode frequencies are redshifted by a factor 1 - C + C^2/6 in the halo compactness C, with EHT observations implying C <= 0.092.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A solid but modest extension of the known environmental QNM redshift, with a clean shadow bound and an over-stated validity range for the new C^2/6 formula. the 2 major comments →

arxiv 2509.04001 v2 pith:PQFRD3UU submitted 2025-09-04 gr-qc

Shadow and Quasi-Normal Modes of Schwarzschild-Hernquist Black Hole

Xing-Hui Feng , Guang-Yu Zhang This is my paper
classification gr-qc PACS 04.70.-s95.35.+d04.30.-w
keywords Schwarzschild-Hernquist black holedark matter haloEinstein clusterblack hole shadowquasi-normal modeseikonal correspondencecompactness boundEvent Horizon Telescope
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper studies a black hole sitting at the center of a Hernquist dark-matter halo, using an exact solution of Einstein's equations built from an Einstein cluster with zero radial pressure. It claims that two observable signals are controlled mainly by the halo compactness C = M_DM/a_0: the shadow radius grows with C, and the gravitational-wave ringdown (quasi-normal mode) frequencies are redshifted relative to vacuum Schwarzschild. The central result is a simple scaling law—the ringdown frequency ratio equals 3√3 M_BH/b_c, which expands as 1 - C + C²/6 + O(C³) for C ≤ 0.3. Taking the Event Horizon Telescope's shadow measurements at their roughly 10% precision, the paper converts the enlarged shadow into an upper bound C ≤ 0.092. This matters because future ringdown observations of galactic black holes must account for this environmental redshift, and because the bound constrains how concentrated dark matter can be around a supermassive black hole.

Core claim

The paper's central claim is that for the Schwarzschild-Hernquist black hole, the strong-field observables are governed to leading order by a single dimensionless number, the halo compactness C = M_DM/a_0. In the astrophysically relevant limit of small mass ratio ε = M_BH/M_DM, the critical impact parameter of the photon sphere—which determines the shadow radius seen by a distant observer—grows with C, and the quasi-normal mode frequencies are redshifted by the inverse of that same impact parameter: ω(C, ε)/ω(0,0) = 3√3 M_BH / b_c = 1 - C + C²/6 + O(C³). The paper computes axial gravitational QNMs by three independent methods (matrix, pseudospectral, and sixth-order WKB), confirms the redshi

What carries the argument

The load-bearing objects are the compactness parameter C = M_DM/a_0 and the critical impact parameter b_c of the photon sphere. For null geodesics the effective potential is V_L = f(r)/r², and the photon sphere follows from r - 3m(r) = 0; b_c = r_ph/√f(r_ph) then fixes both the shadow radius and, through the eikonal light-ring/QNM correspondence, the ringdown frequency scale. The argument expands b_c in powers of C, inserts it into the eikonal correspondence b_c ω = κ + c_1/κ + ..., and obtains the redshift formula (5.1); the same expansion is checked against direct numerical QNM calculations at l=2 and l=3.

Load-bearing premise

The redshift formula (5.1) assumes the eikonal (large-multipole) relation between ringdown frequency and photon-sphere impact parameter stays accurate at the low multipoles l=2 and l=3 for compactness up to C≈0.3, where the paper checks it only for a small set of parameter values.

What would settle it

Run an independent time-domain evolution code for the axial l=2 fundamental mode of the Schwarzschild-Hernquist black hole at, say, C=0.5 and ε=0.01, and compare Re[ω]/Re[ω_Schwarzschild] with 1-C+C²/6. If the difference exceeds the numerical error (a few tenths of a percent), the claimed C≤0.3 validity range of the redshift formula fails. Observationally, a future shadow measurement of the Galactic center black hole with a few-percent precision would tighten or contradict the C≤0.092 bound: a shadow diameter above 5.716 M_BH would require either a larger compactness or a different halo model.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the bound C ≤ 0.092 holds, current shadow images of M87* and the Galactic center black hole cannot distinguish a Schwarzschild-Hernquist black hole from vacuum Schwarzschild, since the shadow shift stays within the observational error.
  • A future claim of a shadow excess at the roughly 10% level could be interpreted as a halo-compactness measurement rather than a deviation from general relativity.
  • The ringdown redshift is controlled by b_c, so measuring a galactic black hole's shadow radius and its ringdown frequency together would test the eikonal relation and the halo model simultaneously.
  • Dark-matter halos with compactness near the allowed range produce enough frequency redshift to act as a systematic error in ringdown-based tests of general relativity, so those tests must marginalize over environmental compactness.
  • The extension to C ~ O(1) makes the calculation relevant for dense dark-matter spikes or compact halos, where lower-order redshift formulas fail.

Where Pith is reading between the lines

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

  • Editorial inference: if the same eikonal redshift structure persists for other Einstein-cluster halo profiles, then the ringdown spectrum would encode an integrated compactness rather than the detailed shape of the density profile, making different halo models hard to tell apart by ringdown alone.
  • Editorial inference: the paper checks the low-multipole validity of the redshift formula for only a few parameter combinations (ε = 0.1, C = 0.1–0.3); a systematic scan over (C, ε) with an independent time-domain code would show whether the formula can serve as a template for actual ringdown observations, which are dominated by l = 2.
  • Editorial inference: a next-generation very-long-baseline interferometer with shadow precision of a few percent would tighten the bound on C by an order of magnitude, or, if the measured shadow came out larger than 5.716 M_BH, would point to a more compact halo or a breakdown of the Hernquist Einstein-cluster model.
  • Editorial inference: the same b_c-based redshift factor should also appear in the damping times of the ringdown, so gravitational-wave detectors with good signal-to-noise ratio on the ringdown of a galactic black hole could constrain halo compactness independently of the shadow measurement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies a spherically symmetric black hole surrounded by a Hernquist-type dark matter halo, the Schwarzschild-Hernquist solution of Cardoso et al. (2021). It computes the photon sphere and shadow radius as functions of halo compactness C and mass ratio ε, giving small-C analytic expansions. It then computes axial gravitational quasi-normal modes (QNMs) with pseudospectral, matrix, and 6th-order WKB methods, and proposes a redshift formula ω(C,ε)/ω(0,0)=1−C+C²/6+O(C³) in the eikonal and ε→0 limits. Using the EHT shadow-size uncertainty of about 10%, it derives an upper bound C≤0.092.

Significance. If the main claims hold, the paper provides a simple, potentially observable mapping from halo compactness to shadow radius and QNM frequency shift, including a concrete astrophysical bound from EHT observations. The numerical work is carefully cross-validated: pseudospectral and matrix methods agree to six digits, and WKB matches numerical results for large ℓ. The analytic expansions for the photon sphere and impact parameter are useful and reduce correctly to known Schwarzschild limits. The shadow bound is a direct, falsifiable prediction. However, the QNM redshift formula is derived in the eikonal limit, and its accuracy for low multipoles is not quantified; this limits the strength of the claims about observable ringdown frequencies.

major comments (2)
  1. [Section 5, Eq. (5.1)] The redshift formula is obtained from the eikonal limit (4.31) together with the small-C expansion of the critical impact parameter, dropping all finite-ℓ terms (c1/κ, c3/κ³, ...). At the observationally relevant ℓ=2 mode, these terms are not negligible: for C=0.3, Table 1 gives the 6th-order eikonal Reω=0.278094 versus the numerical value 0.276139 (~0.7% error), while Eq. (5.1) predicts a ratio ω(C)/ω(0)=0.715 compared with the numerical ratio 0.739 (~3% error). The paper states that Eq. (5.1) is "sufficiently accurate up to C≤0.3" without providing an error budget or clearly stating that it is an eikonal-limit prediction. Since the abstract and Section 6 present this as the redshift of "the QNMs" relative to Schwarzschild, the statement can be misread as applying to the ℓ=2 ringdown spectrum. Please either explicitly identify Eq. (5.1) as an eikonal-limit result with a quantitative err
  2. [Abstract and Section 6] The paper claims to calculate axial gravitational QNMs "up to C∼O(1)", but Tables 1 and 2 only present results for C≤0.3, and Figure 3 stops at C=0.5. No numerical data are shown in the claimed O(1) range. Given the authors' own statement that the eikonal approximation "works bad when C>0.3," the claim of coverage up to O(1) is not supported by the evidence shown. Please either provide the missing results (e.g., a table for C=0.5, 0.7, 1.0) or amend the claim to reflect the actually presented range.
minor comments (5)
  1. [Abstract and Section 5] The word "fit" is misleading: Eq. (5.1) is derived analytically from the eikonal limit and the impact-parameter expansion, not fitted to numerical data. Please rephrase, e.g., "we derive" or "we obtain".
  2. [Eq. (2.8)] The formula for Υ is typeset ambiguously: "arctan r + a0 − MDM√MDMξ" should have parentheses around the arctangent argument. This is a readability issue.
  3. [Eqs. (2.11) and (3.2)] The symbol C is used both for the compactness and as the constant term in the cubic equation (3.2). This can confuse the reader; consider renaming one of them, e.g., using C₀ or a different letter for the cubic coefficient.
  4. [Throughout] Several typos: "Noth" should be "Note", "alow" should be "allow", "paseudospectral" should be "pseudospectral", "quantity" should be "quantify", "Schwarschild" should be "Schwarzschild".
  5. [Figure 3 caption] The bottom panels show ratios of QNMs to Schwarzschild for ε=10⁻¹, 10⁻², 10⁻³, but the caption does not specify the multipole ℓ and overtone n for these panels. The top caption indicates ℓ=2 and distinguishes n=0 and n=1 by marker shapes; please clarify that the bottom panels follow the same convention.

Circularity Check

0 steps flagged

No significant circularity: Eq. (5.1) is an analytic eikonal-limit result using the independently computed impact parameter b_c; numerical QNMs serve as validation, not as fit inputs.

full rationale

The central chain is self-contained and non-circular. The photon-sphere and critical impact parameter b_c are computed analytically from the metric of Ref. [1]: the small-compactness expansion is given in Eq. (3.5), b_c = 3√3 M_BH [1 + C + (5-18ε)/6 C^2 + O(C^3)], and the ε→0 expression is Eq. (3.6), b_c = 3√3 e^Γ(C) M_BH + O(ε). The QNM redshift formula (5.1), ω(C,ε)/ω(0,0) = 3√3 M_BH/b_c = 1 - C + C^2/6 + O(C^3), is obtained from the standard eikonal light-ring/QNM correspondence, Eq. (4.31), by keeping the leading term b_cω = κ and normalizing to Schwarzschild. The coefficients -1 and 1/6 come from expanding b_c, not from fitting the numerically computed QNMs. The abstract's wording 'fit the redshift' and the Discussion's phrase 'fitting formula' are loose language; no parameter is adjusted to the numerical QNM data. The numerical tables (Tables 1 and 2) and Figure 3 are consistency checks between pseudospectral, matrix, 6th-order WKB, and eikonal approximations, not inputs that determine Eq. (5.1). The EHT bound C ≤ 0.092 is derived directly from the analytic b_c expression (3.6) and the assumed 10% uncertainty (Eq. (6.1)); it does not depend on the QNM calculation. The only self-citation, Ref. [37], is cited alongside Refs. [16,23] for the standard axial master wave equation and is not load-bearing. No uniqueness theorem, ansatz, or fitted parameter is imported from the authors' prior work. The finite-l accuracy caveat for l=2 (the 6th-order eikonal deviates from the numerical value by ~0.7% at C=0.3, and Eq. (5.1) is a leading-eikonal truncation) is a correctness/validity concern, not a circularity: the formula is derived rather than fitted, and the shadow bound remains independent of that approximation.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central results depend on the Hernquist/Einstein-cluster spacetime from Ref. [1] and on the eikonal correspondence. No new entities are introduced, and no parameters are fitted to data; the compactness C is a model variable constrained by observations.

axioms (5)
  • standard math Einstein field equations G_mu nu = 8 pi T_mu nu in general relativity
    Used to construct the spacetime from the stress-energy tensor (Sec. 2).
  • domain assumption Einstein cluster ansatz with vanishing radial pressure: T^mu nu = diag(-rho, 0, P_t, P_t)
    Borrowed from Ref. [1]; defines the matter model for the halo (Eq. 2.2).
  • domain assumption Hernquist-type density profile rho = M_DM(a0+2M_BH)(1-2M_BH/r)/(2 pi r (r+a0)^3)
    Specific dark matter halo profile imported from Ref. [1] (Eq. 2.5).
  • standard math Eikonal (large-l) correspondence between QNMs and photon sphere, omega ~ (l+1/2)/b_c - i(n+1/2)lambda, and its higher-order corrections (4.31)
    Used to derive the redshift formula (5.1); validity at low l is the paper's weakest assumption.
  • domain assumption EHT shadow radius measurement error is about 10%
    Used in Sec. 6 to derive the bound C<=0.092; the paper does not propagate actual EHT uncertainties.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Shadow and Quasi-Normal Modes of Schwarzschild-Hernquist Black Hole." pith.science (2026). https://pith.science/paper/PQFRD3UU

@misc{pith2026250904001,
  author       = {Pith},
  title        = {Pith review of: Shadow and Quasi-Normal Modes of Schwarzschild-Hernquist Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQFRD3UU}},
  note         = {Machine review of arXiv:2509.04001}
}
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abstract

In this paper we study the shadow and quasi-normal modes (QNMs) of a black hole (BH) surrounded by a dark matter halo with Hernquist-type density distribution, which was reported in Ref. \cite{Cardoso:2021wlq}. In astrophysical scenarios, we find that the shadow radius enlarges as the compactness of halo increases. Therefore, we obtain an upper bound for the compactness ${\cal C}\le0.092$ with the Event Horizon Telescope (EHT) observations. We calculate axial gravitational QNMs of the galactic BH up to ${\cal C}\sim{\cal O}(1)$, and fit the redshift relative to Schwarzschild QNMs up to second order in the compactness (for ${\cal C}\le0.3)$. These highly redshifted QNMs, resulting from large compactness, are key to modeling the dark matter halo.

Figures

Figures reproduced from arXiv: 2509.04001 by Guang-Yu Zhang, Xing-Hui Feng.

Figure 1
Figure 1. Figure 1: The top panel in the left column is the region (blue) for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The critical impact parameter bc as a function of the compactness C when the mass ratio ϵ → 0. compactness C in the extremal mass ratio ϵ → 0 in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Top panel: Relative percentage difference between the ℓ = 2 QNMs with ϵ = 0.1 computed with the pseudospectral or matrix method and the 6th order WKB approximation (left), the 6th order eikonal limit (right). Filled (empty) markers correspond to the real (imaginary) parts, and circle (square) correspond to the n = 0 fundmental (n = 1 first overtone) QNMs. Bottom panel: Ratio between QNMs of Schwarzschild-H… view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

49 extracted references · 11 canonical work pages · cited by 4 Pith papers · 5 internal anchors

  1. [1]

    Black holes in galaxies: Environmental impact on gravitational-wave generation and propagation,

    V. Cardoso, K. Destounis, F. Duque, R. P. Macedo and A. Maselli, “Black holes in galaxies: Environmental impact on gravitational-wave generation and propagation,” Phys. Rev. D 105, no.6, L061501 (2022) [arXiv:2109.00005 [gr-qc]]

    Pith/arXiv arXiv 2022

  2. [2]

    The mathematical theory of black holes

    S. Chandrasekhar, “The mathematical theory of black holes” (Oxford University Press, New York, 1992)

    work page 1992

  3. [3]

    A cool accre- tion disk around the Galactic Centre black hole,

    E. M. Murchikova, E. S. Phinney, A. Pancoast, and R. D. Blandford, “A cool accre- tion disk around the Galactic Centre black hole,” Nature (London) 570, 83 (2019) [arXiv:1906.08289 [astro-ph.GA]]

    Pith/arXiv arXiv 2019

  4. [4]

    Probing Accretion Physics with Gravitational Waves,

    L. Speri, A. Antonelli, L. Sberna, S. Babak, E. Barausse, J. R. Gair and M. L. Katz, “Probing Accretion Physics with Gravitational Waves,” Phys. Rev. X 13, no.2, 021035 (2023) [arXiv:2207.10086 [gr-qc]]. 14

    Pith/arXiv arXiv 2023

  5. [5]

    The Structure of cold dark matter halos,

    J. F. Navarro, C. S. Frenk and S. D. M. White, “The Structure of cold dark matter halos,” Astrophys. J. 462, 563-575 (1996) [arXiv:astro-ph/9508025 [astro-ph]]

    Pith/arXiv arXiv 1996

  6. [6]

    The Dark Matter Distribution in Disk Galaxies

    A. Borriello and P. Salucci, “The Dark matter distribution in disk galaxies,” Mon. Not. Roy. Astron. Soc. 323, 285 (2001) [arXiv:astro-ph/0001082 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  7. [7]

    How far do they go? The outer structure of dark matter halos

    F. Prada, A. A. Klypin, E. Simonneau, J. Betancort-Rijo, S. Patiri, S. Gottlober and M. A. Sanchez-Conde, “How far do they go? The Outer structure of dark matter halos,” Astrophys. J. 645, 1001-1011 (2006) [arXiv:astro-ph/0506432 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    Into the lair: gravitational-wave signatures of dark matter,

    C. F. B. Macedo, P. Pani, V. Cardoso and L. C. B. Crispino, “Into the lair: gravitational-wave signatures of dark matter,” Astrophys. J. 774, 48 (2013) [arXiv:1302.2646 [gr-qc]]

    Pith/arXiv arXiv 2013

  9. [9]

    Environmental effects in gravitational-wave physics: Tidal deformability of black holes immersed in matter,

    V. Cardoso and F. Duque, “Environmental effects in gravitational-wave physics: Tidal deformability of black holes immersed in matter,” Phys. Rev. D 101, no.6, 064028 (2020) [arXiv:1912.07616 [gr-qc]]

    Pith/arXiv arXiv 2020

  10. [10]

    Extreme mass ratio inspirals into black holes surrounded by matter,

    L. Polcar, G. Lukes-Gerakopoulos and V. Witzany, “Extreme mass ratio inspirals into black holes surrounded by matter,” Phys. Rev. D 106, no.4, 044069 (2022) [arXiv:2205.08516 [gr-qc]]

    Pith/arXiv arXiv 2022

  11. [11]

    Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A,

    S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. “Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A,” Class. Quant. Grav. 40, no.16, 165007 (2023) [arXiv:2205.07787 [gr-qc]]

    Pith/arXiv arXiv 2023

  12. [12]

    Influence of dark matter equation of state on the axial gravitational ringing of supermassive black holes,

    Y. Zhao, B. Sun, Z. Cao, K. Lin and W. L. Qian, “Influence of dark matter equation of state on the axial gravitational ringing of supermassive black holes,” Phys. Rev. D 109, no.4, 044031 (2024) [arXiv:2308.15371 [gr-qc]]

    Pith/arXiv arXiv 2024

  13. [13]

    Black holes surrounded by Einstein clusters as models of dark matter fluid,

    K. Jusufi, “Black holes surrounded by Einstein clusters as models of dark matter fluid,” Eur. Phys. J. C 83, no.2, 103 (2023) [arXiv:2202.00010 [gr-qc]]

    Pith/arXiv arXiv 2023

  14. [14]

    Solutions of the Einstein Equations for a Black Hole Surrounded by a Galactic Halo,

    R. A. Konoplya and A. Zhidenko, “Solutions of the Einstein Equations for a Black Hole Surrounded by a Galactic Halo,” Astrophys. J.933, no.2, 166 (2022) [arXiv:2202.02205 [gr-qc]]. 15

    Pith/arXiv arXiv 2022

  15. [15]

    Who knows what dark matter lurks in the heart of M87: The shadow knows, and so does the ringdown

    R. G. Daghigh and G. Kunstatter, “Effect of dark matter on galactic black hole ringdown waveforms and shadows,” Phys. Rev. D 109, no.8, 083004 (2024) [arXiv:2308.15682 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  16. [16]

    Black holes surrounded by generic dark matter profiles: Appearance and gravitational-wave emission,

    E. Figueiredo, A. Maselli and V. Cardoso, “Black holes surrounded by generic dark matter profiles: Appearance and gravitational-wave emission,” Phys. Rev. D 107, no.10, 104033 (2023) [arXiv:2303.08183 [gr-qc]]

    Pith/arXiv arXiv 2023

  17. [17]

    Black holes surrounded by generic matter distributions: Polar perturbations and energy flux,

    N. Speeney, E. Berti, V. Cardoso and A. Maselli, “Black holes surrounded by generic matter distributions: Polar perturbations and energy flux,” Phys. Rev. D 109, no.8, 084068 (2024) [arXiv:2401.00932 [gr-qc]]

    Pith/arXiv arXiv 2024

  18. [18]

    Analytical models of supermassive black holes in galaxies surrounded by dark matter halos,

    Z. Shen, A. Wang, Y. Gong and S. Yin, “Analytical models of supermassive black holes in galaxies surrounded by dark matter halos,” Phys. Lett. B 855, 138797 (2024) [arXiv:2311.12259 [gr-qc]]

    Pith/arXiv arXiv 2024

  19. [19]

    Inner radius and energy conditions of dark mat- ter halos surrounding Schwarzschild black holes,

    Z. Shen, A. Wang and S. Yin, “Inner radius and energy conditions of dark mat- ter halos surrounding Schwarzschild black holes,” Phys. Lett. B 862, 139300 (2025) [arXiv:2408.05417 [gr-qc]]

    Pith/arXiv arXiv 2025

  20. [20]

    Black hole in the Dekel-Zhao dark matter profile,

    A. ¨Ovg¨ un and R. C. Pantig, “Black hole in the Dekel-Zhao dark matter profile,” Phys. Lett. B 864, 139398 (2025) [arXiv:2501.12559 [gr-qc]]

    Pith/arXiv arXiv 2025

  21. [21]

    Spinning black holes in astrophysical environ- ments,

    P. G. S. Fernandes and V. Cardoso, “Spinning black holes in astrophysical environ- ments,” [arXiv:2507.04389 [gr-qc]]

    arXiv

  22. [22]

    Black holes in galactic centers: Quasinormal ringing, grey-body factors and Unruh temperature,

    R. A. Konoplya, “Black holes in galactic centers: Quasinormal ringing, grey-body factors and Unruh temperature,” Phys. Lett. B 823, 136734 (2021) [arXiv:2109.01640 [gr-qc]]

    Pith/arXiv arXiv 2021

  23. [23]

    Gravitational Waves from Extreme-Mass-Ratio Systems in Astrophysical Environments,

    V. Cardoso, K. Destounis, F. Duque, R. Panosso Macedo and A. Maselli, “Gravitational Waves from Extreme-Mass-Ratio Systems in Astrophysical Environments,” Phys. Rev. Lett. 129, no.24, 241103 (2022) [arXiv:2210.01133 [gr-qc]]

    Pith/arXiv arXiv 2022

  24. [24]

    Shadows of black holes with dark matter halo,

    S. V. M. C. B. Xavier, H. C. D. Lima, Junior. and L. C. B. Crispino, “Shadows of black holes with dark matter halo,” Phys. Rev. D107, no.6, 064040 (2023) [arXiv:2303.17666 [gr-qc]]

    Pith/arXiv arXiv 2023

  25. [25]

    Shadow bound of black holes with dark matter halo,

    Y. S. Myung, “Shadow bound of black holes with dark matter halo,” [arXiv:2402.03606 [gr-qc]]. 16

    Pith/arXiv arXiv

  26. [26]

    Optical appearance of black holes surrounded by a dark matter halo,

    C. F. B. Macedo, J. L. Rosa and D. Rubiera-Garcia, “Optical appearance of black holes surrounded by a dark matter halo,” JCAP 07, 046 (2024) [arXiv:2402.13047 [gr-qc]]

    Pith/arXiv arXiv 2024

  27. [27]

    Superradiance of charged black holes embedded in dark matter halos,

    A. Mollicone and K. Destounis, “Superradiance of charged black holes embedded in dark matter halos,” Phys. Rev. D 111, no.2, 024017 (2025) [arXiv:2410.11952 [gr-qc]]

    Pith/arXiv arXiv 2025

  28. [28]

    Black Hole Spectroscopy in Environments: Detectability Prospects,

    T. F. M. Spieksma, V. Cardoso, G. Carullo, M. Della Rocca and F. Duque, “Black Hole Spectroscopy in Environments: Detectability Prospects,” Phys. Rev. Lett. 134, no.8, 081402 (2025) [arXiv:2409.05950 [gr-qc]]

    Pith/arXiv arXiv 2025

  29. [29]

    Quasinormal modes of black holes embedded in halos of matter,

    L. Pezzella, K. Destounis, A. Maselli and V. Cardoso, “Quasinormal modes of black holes embedded in halos of matter,” Phys. Rev. D 111, no.6, 064026 (2025) [arXiv:2412.18651 [gr-qc]]

    Pith/arXiv arXiv 2025

  30. [30]

    Tidal Love numbers and quasinormal modes of the Schwarzschild-Hernquist black hole,

    S. Chakraborty, G. Comp` ere and L. Machet, “Tidal Love numbers and quasinormal modes of the Schwarzschild-Hernquist black hole,” Phys. Rev. D 112, no.2, 024015 (2025) [arXiv:2412.14831 [gr-qc]]

    Pith/arXiv arXiv 2025

  31. [31]

    Effect of generic dark matter halo on transonic accretion onto galactic black holes,

    A. Chowdhury, G. Sen, S. Chakrabarti and S. Das, “Effect of generic dark matter halo on transonic accretion onto galactic black holes,” [arXiv:2503.08528 [gr-qc]]

    work page arXiv

  32. [32]

    Extreme mass ratio inspirals in dark matter halos: dynamics and distinguishability of halo models,

    S. Gliorio, E. Berti, A. Maselli and N. Speeney, “Extreme mass ratio inspirals in dark matter halos: dynamics and distinguishability of halo models,” [arXiv:2503.16649 [gr- qc]]

    arXiv

  33. [33]

    A multi-parameter expansion for the evolution of asymmetric binaries in astrophysical environments,

    S. Datta and A. Maselli, “A multi-parameter expansion for the evolution of asymmetric binaries in astrophysical environments,” [arXiv:2507.04471 [gr-qc]]

    Pith/arXiv arXiv

  34. [34]

    Gravitational-wave tails and memory effect for mergers in astrophysical environments,

    Q. Alnasheet, V. Cardoso, F. Duque and R. Panosso Macedo, “Gravitational-wave tails and memory effect for mergers in astrophysical environments,” Phys. Rev. D 112, no.4, 044066 (2025) [arXiv:2508.20238 [gr-qc]]

    Pith/arXiv arXiv 2025

  35. [35]

    Lensing by black holes within as- trophysical environments,

    G. Kouniatalis, A. G. Suvorov and K. Destounis, “Lensing by black holes within as- trophysical environments,” [arXiv:2508.19333 [gr-qc]]

    arXiv

  36. [36]

    Chaos and Carter: Extreme-mass-ratio systems of relativistic rotating black holes in astrophysical environments,

    K. Destounis and P. G. S. Fernandes, “Chaos and Carter: Extreme-mass-ratio systems of relativistic rotating black holes in astrophysical environments,” [arXiv:2508.20191 [gr-qc]]. 17

    arXiv

  37. [37]

    Axial gravitational perturbations of slowly rotating com- pact objects in general relativity and beyond,

    X. H. Feng and J. Peng, “Axial gravitational perturbations of slowly rotating com- pact objects in general relativity and beyond,” Phys. Rev. D 110, no.6, 6 (2024) [arXiv:2404.16437 [gr-qc]]

    Pith/arXiv arXiv 2024

  38. [38]

    Stability of a Schwarzschild singularity,

    T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. 108, 1063-1069 (1957)

    work page 1957

  39. [39]

    Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes,

    A. Jansen, “Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes,” Eur. Phys. J. Plus 132, no.12, 546 (2017) [arXiv:1709.09178 [gr-qc]]

    Pith/arXiv arXiv 2017

  40. [40]

    A non grid-based interpolation scheme for the eigenvalue problem

    K. Lin and W. L. Qian, “A non grid-based interpolation scheme for the eigenvalue problem,” arXiv:1609.05948 [math.NA]

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    A Matrix Method for Quasinormal Modes: Schwarzschild Black Holes in Asymptotically Flat and (Anti-) de Sitter Spacetimes

    K. Lin and W. L. Qian, “A Matrix Method for Quasinormal Modes: Schwarzschild Black Holes in Asymptotically Flat and (Anti-) de Sitter Spacetimes,” Class. Quant. Grav. 34, no.9, 095004 (2017) [arXiv:1610.08135 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  42. [42]

    Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations,

    R. A. Konoplya, A. Zhidenko and A. F. Zinhailo, “Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations,” Class. Quant. Grav. 36, 155002 (2019) [arXiv:1904.10333 [gr-qc]]

    Pith/arXiv arXiv 2019

  43. [43]

    Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering,

    S. Iyer and C. M. Will, “Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering,” Phys. Rev. D 35, 3621 (1987)

    work page 1987

  44. [44]

    Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach,

    R. A. Konoplya, “Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach,” Phys. Rev. D 68, 024018 (2003) [arXiv:gr- qc/0303052 [gr-qc]]

    work page arXiv 2003

  45. [45]

    Quasinormal modes of the Schwarzschild black hole and higher order WKB approach,

    R. A. Konoplya, “Quasinormal modes of the Schwarzschild black hole and higher order WKB approach,” J. Phys. Stud. 8, 93-100 (2004)

    work page 2004

  46. [46]

    Quasinormal modes of black holes. The improved semi- analytic approach,

    J. Matyjasek and M. Opala, “Quasinormal modes of black holes. The improved semi- analytic approach,” Phys. Rev. D 96, no.2, 024011 (2017) [arXiv:1704.00361 [gr-qc]]

    Pith/arXiv arXiv 2017

  47. [47]

    Geodesic stability, Lyapunov exponents and quasinormal modes,

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, “Geodesic stability, Lyapunov exponents and quasinormal modes,” Phys. Rev. D 79, no.6, 064016 (2009) [arXiv:0812.1806 [hep-th]]. 18

    Pith/arXiv arXiv 2009

  48. [48]

    Correspondence of eikonal quasinormal modes and unstable fundamental photon orbits for a Kerr-Newman black hole,

    P. C. Li, T. C. Lee, M. Guo and B. Chen, “Correspondence of eikonal quasinormal modes and unstable fundamental photon orbits for a Kerr-Newman black hole,” Phys. Rev. D 104, no.8, 084044 (2021) [arXiv:2105.14268 [gr-qc]]

    Pith/arXiv arXiv 2021

  49. [49]

    Analytic expressions for quasinormal modes and grey-body factors in the eikonal limit and beyond,

    R. A. Konoplya and A. Zhidenko, “Analytic expressions for quasinormal modes and grey-body factors in the eikonal limit and beyond,” Class. Quant. Grav. 40, no.24, 245005 (2023) [arXiv:2309.02560 [gr-qc]]. 19

    Pith/arXiv arXiv 2023

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.