Pith. sign in

REVIEW 3 major objections 6 minor 94 references

Homogeneous and inhomogeneous plasma push black-hole shadow size in opposite directions; realistic dark-matter halos barely matter.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 22:52 UTC pith:KCZN5E3T

load-bearing objection Solid Kerr-plasma calculation on an Einasto-halo metric; the homogeneous enlargement is mostly the known refractive Bardeen factor, and realistic DM is negligible. the 3 major comments →

arxiv 2603.11434 v2 pith:KCZN5E3T submitted 2026-03-12 gr-qc astro-ph.HE

Plasma effects on gravitational lensing and shadow observables of a Kerr-like black hole in a dark matter halo

Connor McMillin , Zhichen Guan , Owen Gartlan , Lotus Liu , Leo Rodriguez , Shanshan Rodriguez This is my paper
classification gr-qc astro-ph.HE PACS 11.25.Hf04.60.-m04.70.-s
keywords black-hole shadowplasma lensingKerr-like metricEinasto dark-matter haloEvent Horizon Telescopenull geodesicsenergy emission rate
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.

This paper asks how plasma around a spinning black hole, together with a dark-matter halo, changes the size and shape of the black-hole shadow that instruments such as the Event Horizon Telescope measure. The authors embed a Kerr-like black hole in an Einasto dark-matter halo and then propagate light through two non-magnetized plasma models: a uniform (homogeneous) plasma and a radially varying (inhomogeneous) plasma. They find that raising the plasma density enlarges and distorts the shadow for the homogeneous case, but shrinks and circularizes it for the inhomogeneous case. Dark-matter densities typical of real galaxies leave photon orbits almost unchanged. The same plasma models also reverse the ordering of the black hole’s energy-emission rate. By comparing the predicted shadow radii with the published EHT intervals for M87* and Sgr A*, the authors obtain concrete upper bounds on how strong a homogeneous plasma can be before it exceeds the observations.

Core claim

For a Kerr-like black hole sitting in an Einasto dark-matter halo, increasing plasma density increases both the shadow radius and its deformation when the plasma is homogeneous, but decreases both quantities when the plasma is inhomogeneous. Astrophysically realistic dark-matter densities produce no appreciable shift in the photon trajectories, while the energy-emission rate tracks the same opposite plasma trends.

What carries the argument

The separable plasma-frequency ansatz ω_p² = (f_r(r) + f_θ(θ))/Σ, which keeps the Hamilton–Jacobi equation separable and yields closed-form impact parameters and asymptotic Bardeen celestial coordinates (including a refractive factor 1/√(1−ω_c²/ω₀²) for the homogeneous profile).

Load-bearing premise

The plasma density must be chosen so that the ray equations separate into independent radial and angular pieces; real magnetized, turbulent plasma need not obey that restriction.

What would settle it

A multi-frequency EHT or next-generation shadow measurement that shows the shadow radius either growing or shrinking with frequency in a way that cannot be fit by either of the two plasma models (or that requires dark-matter densities far above the galactic values used here).

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

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

3 major / 6 minor

Summary. The manuscript studies null geodesics and black-hole shadow observables for a Kerr-like metric embedded in an Einasto dark-matter halo, in the presence of non-magnetized pressureless plasma. Two separable plasma profiles (homogeneous and inhomogeneous) are adopted so that the Hamilton–Jacobi equation separates. The authors compute critical photon orbits, deflection angles, Bardeen celestial coordinates (including the homogeneous-plasma refractive factor arising from Δ(r) f_r(r) ∼ r⁴), shadow radius R_s, deformation δ_s, ellipticity K_s, and a heuristic energy-emission rate, and compare geometric critical-curve sizes with EHT-inferred intervals for M87* and Sgr A*. The main reported trends are that realistic Einasto densities leave photon trajectories essentially unchanged, while increasing plasma strength enlarges (shrinks) the shadow for homogeneous (inhomogeneous) plasma, with corresponding effects on emission rate and on allowed plasma-density windows.

Significance. The work is a careful, incremental parameter study in a well-established phenomenological niche (plasma + environmental halo + Kerr-like shadows). Its strengths are (i) explicit recovery of the homogeneous-plasma refractive normalization of the asymptotic Bardeen coordinates, (ii) systematic numerical scans of R_s, δ_s, K_s and emission rate across spin, inclination and plasma strength, and (iii) concrete, falsifiable upper bounds on ω_c²/ω₀² from EHT size intervals for the homogeneous profile. The negative result that astrophysically reasonable Einasto densities do not move the critical curve is useful. The paper does not introduce a new metric or a non-separable plasma model, so its significance is primarily as a controlled comparison of two standard plasma profiles on a previously studied halo spacetime, with direct EHT benchmarks.

major comments (3)
  1. §IV A, Eqs. (20a)–(20b): the closed-form expressions for the critical impact parameters η and χ contain no explicit dependence on f̃_r or f̃_r′, yet the defining conditions R(r_p)=0 and R′(r_p)=0 (Eqs. 18–19) do. Moreover Eq. (20b) is algebraically garbled (identical summands “8 a r_p² Δ + 8 a r_p² Δ”). Because every subsequent shadow, deformation and emission-rate plot is built from these critical parameters, the manuscript must either (i) supply the correct plasma-dependent analytic solution of the R=R′=0 system or (ii) state clearly that η,χ are obtained purely numerically and remove the incorrect closed forms. As written, the central pipeline is not reproducible from the text.
  2. §V A, Eqs. (29a)–(29b) and Figs. 6, 8f, 9c: the dominant growth of R_s with homogeneous-plasma strength is the overall multiplicative factor 1/√(1−ω_c²/ω₀²). Because this factor multiplies both α and β uniformly, pure refractive rescaling leaves shape measures (δ_s, K_s) invariant. Any reported increase of deformation with ω_c²/ω₀² must therefore come from the plasma-induced change of the photon-region boundary (r_p±, η(r_p), χ(r_p)). The text should quantitatively separate the two contributions (e.g., by plotting the un-normalized critical curve in the (η,χ) plane alongside the Bardeen image) so that the claim “homogeneous plasma increases deformation” is not misread as a pure normalization artifact.
  3. §VI, Eqs. (34)–(38): the energy-emission rate is constructed from the geometric shadow area σ_lim=π R_s² and the vacuum surface gravity of the Kerr-like horizons. Plasma modifies photon propagation and, in a consistent treatment, the grey-body factors and the notion of asymptotic energy; the manuscript never discusses whether T_H or the high-frequency absorption cross-section remain valid once ω_p≠0. At minimum the section should be labeled as a purely geometric proxy and the ordering of emission curves should be tied explicitly to the ordering of R_s already shown in Fig. 8, rather than presented as an independent physical prediction.
minor comments (6)
  1. Notation for plasma strength is inconsistent: the text uses ω_c²/ω₀² throughout the figures, yet Eqs. (29a,b) write √(1−ω_c²). Please adopt a single dimensionless ratio everywhere.
  2. PACS codes 11.25.Hf and 04.60.-m are inappropriate for a classical GR/plasma-optics paper; replace with standard codes such as 04.70.-s, 95.30.Sf, 98.62.Sb.
  3. Fig. 1 caption and surrounding text: the statement that ∂r_h/∂a → ∞ at extremality is true for the usual Kerr square-root branch point, but the figure’s “derivative going to infinity” is hard to read; a zoomed inset or a plot of (∂r_h/∂a)⁻¹ would help.
  4. Eq. (4) conversion of ρ_e, r_e into geometric units is given, but the numerical values actually used for Sgr A* versus M87* in §§VII–VIII should be collected in a single table for reproducibility.
  5. Several references appear with incomplete bibliographic data (e.g., Perlick & Tsupko entries missing journal/year in the list). Please clean the bibliography.
  6. Abstract and §V: “asymptotically stable photon sphere” should be “unstable spherical photon orbits” (the critical curve is unstable).

Circularity Check

0 steps flagged

No circularity: geodesic-derived shadows and EHT-constrained plasma bounds are independent of fitted inputs or self-referential definitions.

full rationale

The paper constructs a Kerr-like metric with an Einasto halo (Eqs. 1–3, taken from external literature [73]), imposes the standard separability condition on the plasma frequency (Eq. 10, from Perlick–Tsupko), and solves the resulting Hamilton–Jacobi equation for null geodesics. Shadow observables (Rs, δs, Ks) and the energy-emission rate follow directly from the critical-curve asymptotics (Eqs. 29–33, 36). The homogeneous-plasma refractive factor 1/√(1−ωc^{2}/ω0^{2}) is an explicit leading-order consequence of Δ(r)fr(r)∼r^{4}(1−ωc^{2}/ω0^{2}) when fr∝r^{2}; the paper states this openly rather than presenting it as an independent geometric prediction. Comparison with EHT areal-radius intervals for M87* and Sgr A* (external observational inputs) yields upper bounds on ωc^{2}/ω0^{2}; those bounds are not fitted parameters re-labeled as predictions. No self-citation is load-bearing, no uniqueness theorem is imported from the authors, and no quantity is defined in terms of the result it is claimed to predict. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claim rests on a pre-existing Kerr-like metric with Einasto halo, the standard non-magnetized pressureless plasma Hamiltonian, and the separability ansatz that forces the two plasma profiles. Free parameters are the plasma strength ω_c²/ω_0² (scanned) and the fixed halo parameters taken from galactic estimates; no new entities are invented.

free parameters (3)
  • ω_c²/ω_0² (plasma strength)
    Dimensionless plasma density scanned from 0 to ~1; upper bounds later constrained by EHT size intervals rather than fitted a priori.
  • Einasto halo parameters (ρ_e, r_e, α)
    Fixed to literature values for M87 and Sgr A*; treated as inputs, not fitted to the shadow data.
  • black-hole spin a and observer inclination θ_0
    Scanned or set to literature estimates (a≈0.9 for Sgr A*, θ_0≈17° for M87*); free within observational ranges.
axioms (4)
  • domain assumption Plasma frequency must be of the separable form ω_p² = (f_r(r)+f_θ(θ))/Σ for the Hamilton–Jacobi equation to separate.
    Stated in §III B / Eq. 10; inherited from Perlick & Tsupko and required for analytic treatment.
  • domain assumption Plasma is non-magnetized and pressureless; no back-reaction on the metric.
    Explicitly assumed in §III; standard in the geometric-optics plasma-shadow literature.
  • domain assumption The Kerr-like metric with Einasto halo (Eqs. 2–3) correctly describes the spacetime.
    Taken from Liu et al. (2024) without re-derivation.
  • standard math Geometric-optics limit and refractive-index formulation remain valid near the photon sphere.
    Standard high-frequency approximation used throughout §§III–VI.

pith-pipeline@v1.1.0-grok45 · 24368 in / 2811 out tokens · 23399 ms · 2026-07-14T22:52:35.744593+00:00 · methodology

0 comments
read the original abstract

Plasma surrounding a black hole modifies light propagation and can alter the observed shadow, potentially affecting the interpretation of Event Horizon Telescope data. We study the effects of dark matter and nonmagnetized pressureless plasma on the shadow of a Kerr-like black hole by analyzing null geodesics in both homogeneous and inhomogeneous plasma distributions. For the homogeneous plasma profile, the asymptotic Bardeen coordinates acquire a refractive normalization factor arising from the leading-order coupling between the metric function $\Delta(r)$ and the radial plasma function $f_r(r)\propto r^2$. We show that increasing the black hole spin generally enlarges the shadow radius and increases its deformation, while moving the observer away from the equatorial plane decreases both quantities. For the parameter ranges considered, astrophysically reasonable dark matter densities in this model do not produce appreciable changes in the photon trajectories. Plasma effects, however, are significant: increasing the plasma density increases the shadow radius and deformation for homogeneous plasma, but decreases them for inhomogeneous plasma. The energy emission rate likewise depends strongly on the plasma model, with homogeneous plasma producing a substantially larger rate as the plasma strength increases. As an illustrative benchmark, we compare the resulting geometric critical-curve sizes with EHT-inferred shadow-size intervals for M87* and Sgr A*.

Figures

Figures reproduced from arXiv: 2603.11434 by Connor McMillin, Leo Rodriguez, Lotus Liu, Owen Gartlan, Shanshan Rodriguez, Zhichen Guan.

Figure 1
Figure 1. Figure 1: FIG. 1: Variation of the inner [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Panel (a) shows [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Panel (a) shows the variation of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variations of deflection angle as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Shadow shape for various values of black hole spin [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Shadow shape for various values of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Diagram of the shadow observables with relevant [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Panels (a)-(c) show the variation of shadow radius [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Panels (a)-(c) show the variation of deformation [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: (a) Energy emission rate [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Energy emission rate [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Comparison of shadow radius for various plasma [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

94 extracted references · 58 linked inside Pith

  1. [1]

    E. H. T. Collaboration, Astrophysical Journal Letters875, L1 (2019), arXiv:1906.11238

    Pith/arXiv arXiv 2019

  2. [2]

    E. H. T. Collaboration, Astrophysical Journal Letters930, L12 (2022), arXiv:2311.08697

    Pith/arXiv arXiv 2022

  3. [3]

    J. L. Synge, Monthly Notices of the Royal Astronomical Society131, 463 (1966)

    1966

  4. [5]

    Perlick and O

    V . Perlick and O. Y . Tsupko, Physical Review D90, 104005 (2014), arXiv:1409.4733

    Pith/arXiv arXiv 2014

  5. [6]

    O. Y . Tsupko and G. S. Bisnovatyi-Kogan, Phys. Rev. D87, 124009 (2013)

    2013

  6. [7]

    Perlick, O

    V . Perlick, O. Y . Tsupko, and G. S. Bisnovatyi-Kogan, Physical Review D92, 104031 (2015), arXiv:1507.04217

    Pith/arXiv arXiv 2015

  7. [8]

    Perlick and O

    V . Perlick and O. Y . Tsupko, Physical Review D95, 104003 (2017), arXiv:1702.08768

    Pith/arXiv arXiv 2017

  8. [9]

    G. S. Bisnovatyi-Kogan and O. Y . Tsupko, Grav. Cosmol.15, 20 (2009), arXiv:0809.1021 [astro-ph]

    Pith/arXiv arXiv 2009

  9. [10]

    G. S. Bisnovatyi-Kogan and O. Y . Tsupko, Mon. Not. Roy. Astron. Soc.404, 1790 (2010), arXiv:1006.2321 [astro- ph.CO]

    Pith/arXiv arXiv 2010

  10. [12]

    Rogers, Mon

    A. Rogers, Mon. Not. Roy. Astron. Soc.451, 17 (2015), arXiv:1505.06790 [gr-qc]

    Pith/arXiv arXiv 2015

  11. [13]

    V . S. Morozova, B. J. Ahmedov, and A. A. Tursunov, Astrophys. Space Sci.346, 513 (2013)

    2013

  12. [14]

    Atamurotov and B

    F. Atamurotov and B. Ahmedov, Phys. Rev. D92, 084005 (2015), arXiv:1507.08131 [gr-qc]

    Pith/arXiv arXiv 2015

  13. [15]

    Chowdhuri and A

    A. Chowdhuri and A. Bhattacharyya, Phys. Rev. D104, 064039 (2021)

    2021

  14. [16]

    Yan, Phys

    H. Yan, Phys. Rev. D99, 084050 (2019), arXiv:1903.04382 [gr-qc]

    Pith/arXiv arXiv 2019

  15. [17]

    Zhang, H

    Z. Zhang, H. Yan, M. Guo, and B. Chen, Phys. Rev. D107, 024027 (2023), arXiv:2206.04430 [gr-qc]

    Pith/arXiv arXiv 2023

  16. [18]

    Khodadi, Nucl

    M. Khodadi, Nucl. Phys. B985, 116014 (2022), arXiv:2211.00300 [gr-qc]

    Pith/arXiv arXiv 2022

  17. [19]

    Pahlavon, F

    Y . Pahlavon, F. Atamurotov, K. Jusufi, M. Jamil, and A. Abdujabbarov, Phys. Dark Univ.45, 101543 (2024), arXiv:2406.09431 [gr-qc]

    Pith/arXiv arXiv 2024

  18. [20]

    M. M. Gohain, K. Bhuyan, and P. Chutia, Phys. Dark Univ. 49, 102064 (2025), arXiv:2504.00581 [hep-th]

    arXiv 2025

  19. [21]

    Er and S

    X. Er and S. Mao, Mon. Not. Roy. Astron. Soc.437, 2180 (2014), arXiv:1310.5825 [astro-ph.CO]

    Pith/arXiv arXiv 2014

  20. [22]

    G. S. Bisnovatyi-Kogan and O. Y . Tsupko, Universe3, 57 (2017), arXiv:1905.06615 [gr-qc]

    Pith/arXiv arXiv 2017

  21. [23]

    Liu, C.-K

    C.-Q. Liu, C.-K. Ding, and J.-L. Jing, Chin. Phys. Lett.34, 090401 (2017), arXiv:1610.02128 [gr-qc]

    Pith/arXiv arXiv 2017

  22. [24]

    Crisnejo and E

    G. Crisnejo and E. Gallo, Phys. Rev. D97, 124016 (2018), arXiv:1804.05473 [gr-qc]

    Pith/arXiv arXiv 2018

  23. [25]

    Crisnejo, E

    G. Crisnejo, E. Gallo, and K. Jusufi, Phys. Rev. D100, 104045 (2019), arXiv:1910.02030 [gr-qc]

    Pith/arXiv arXiv 2019

  24. [26]

    Crisnejo, E

    G. Crisnejo, E. Gallo, and A. Rogers, Phys. Rev. D99, 124001 (2019), arXiv:1807.00724 [gr-qc]

    Pith/arXiv arXiv 2019

  25. [27]

    Kumaran and A

    Y . Kumaran and A. Övgün, Turk. J. Phys.45, 247 (2021), arXiv:2111.02805 [gr-qc]

    Pith/arXiv arXiv 2021

  26. [28]

    Atamurotov, A

    F. Atamurotov, A. Abdujabbarov, and J. Rayimbaev, Eur. Phys. J. C81, 118 (2021)

    2021

  27. [29]

    J. Sun, X. Er, and O. Y . Tsupko, Mon. Not. Roy. Astron. Soc. 520, 994 (2023), arXiv:2211.13442 [astro-ph.SR]

    Pith/arXiv arXiv 2023

  28. [30]

    G. S. Bisnovatyi-Kogan and O. Y . Tsupko, Mon. Not. Roy. Astron. Soc.524, 3060 (2023), arXiv:2301.00053 [gr-qc]

    Pith/arXiv arXiv 2023

  29. [31]

    Feleppa, V

    F. Feleppa, V . Bozza, and O. Y . Tsupko, Phys. Rev. D110, 064031 (2024), arXiv:2406.07703 [gr-qc]

    Pith/arXiv arXiv 2024

  30. [32]

    Abdujabbarov, B

    A. Abdujabbarov, B. Toshmatov, Z. Stuchlík, and B. Ahmedov, Int. J. Mod. Phys. D26, 1750051 (2016), arXiv:1512.05206 [gr-qc]

    Pith/arXiv arXiv 2016

  31. [33]

    Huang, Y .-P

    Y . Huang, Y .-P. Dong, and D.-J. Liu, Int. J. Mod. Phys. D27, 1850114 (2018), arXiv:1807.06268 [gr-qc]

    Pith/arXiv arXiv 2018

  32. [34]

    Atamurotov, K

    F. Atamurotov, K. Jusufi, M. Jamil, A. Abdujabbarov, and M. Azreg-Aïnou, Phys. Rev. D104, 064053 (2021), arXiv:2109.08150 [gr-qc]

    Pith/arXiv arXiv 2021

  33. [36]

    F. Ozel, D. Psaltis, and Z. Younsi, Astrophys. J.941, 88 15 (2022), arXiv:2111.01123 [astro-ph.HE]

    Pith/arXiv arXiv 2022

  34. [37]

    A. Das, A. Saha, and S. Gangopadhyay, Class. Quant. Grav. 39, 075005 (2022), arXiv:2110.11704 [gr-qc]

    Pith/arXiv arXiv 2022

  35. [38]

    Badía and E

    J. Badía and E. F. Eiroa, Phys. Rev. D107, 124028 (2023), arXiv:2210.03081 [gr-qc]

    Pith/arXiv arXiv 2023

  36. [39]

    A. Das, A. Saha, and S. Gangopadhyay, Class. Quant. Grav. 40, 015008 (2023), arXiv:2207.06994 [gr-qc]

    Pith/arXiv arXiv 2023

  37. [40]

    Fathi, M

    M. Fathi, M. Olivares, J. R. Villanueva, and N. Cruz, Eur. Phys. J. Plus139, 752 (2024), arXiv:2311.14017 [gr-qc]

    Pith/arXiv arXiv 2024

  38. [41]

    M. A. Raza, M. Zubair, and E. Maqsood, JCAP05, 047 (2024), arXiv:2401.04779 [gr-qc]

    Pith/arXiv arXiv 2024

  39. [42]

    R. Ali, X. Tiecheng, R. Babar, and A. Övgün, Int. J. Theor. Phys.64, 75 (2025), arXiv:2402.07657 [gr-qc]

    Pith/arXiv arXiv 2025

  40. [43]

    R. Ali, X. Tiecheng, M. Awais, and R. Babar, Chin. J. Phys. 94, 416 (2025), arXiv:2502.16273 [gr-qc]

    Pith/arXiv arXiv 2025

  41. [44]

    Yasmin, K

    S. Yasmin, K. Jafarzade, and M. Jamil, Chin. Phys. C49, 065107 (2025), arXiv:2502.18071 [gr-qc]

    Pith/arXiv arXiv 2025

  42. [45]

    Badía and E

    J. Badía and E. F. Eiroa, Phys. Rev. D104, 084055 (2021), arXiv:2106.07601 [gr-qc]

    Pith/arXiv arXiv 2021

  43. [46]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. 912, 35 (2021), arXiv:2105.01804 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  44. [47]

    G. Z. Babar, A. Z. Babar, and F. Atamurotov, Eur. Phys. J. C80, 761 (2020), [Erratum: Eur.Phys.J.C 82, 403 (2022)], arXiv:2008.05845 [gr-qc]

    Pith/arXiv arXiv 2020

  45. [48]

    Kala and J

    S. Kala and J. Singh, Eur. Phys. J. C85, 1047 (2025), arXiv:2507.17280 [gr-qc]

    Pith/arXiv arXiv 2025

  46. [49]

    Kumar, A

    S. Kumar, A. Uniyal, and S. Chakrabarti, Phys. Dark Univ. 44, 101472 (2024), arXiv:2403.12439 [gr-qc]

    Pith/arXiv arXiv 2024

  47. [50]

    Feng, R.-J

    H. Feng, R.-J. Yang, and W.-Q. Chen, Astropart. Phys.166, 103075 (2025), arXiv:2403.18541 [gr-qc]

    Pith/arXiv arXiv 2025

  48. [51]

    Kumar Sahoo and I

    S. Kumar Sahoo and I. Banerjee, JCAP10, 100 (2025), arXiv:2504.09443 [gr-qc]

    arXiv 2025

  49. [52]

    Perlick and O

    V . Perlick and O. Y . Tsupko, Physics Reports947, 1 (2022), arXiv:2105.07101

    Pith/arXiv arXiv 2022

  50. [53]

    V . C. Rubin, N. Thonnard, and W. K. Ford, Jr., Astrophys. J. 238, 471 (1980)

    1980

  51. [54]

    Persic, P

    M. Persic, P. Salucci, and F. Stel, Mon. Not. Roy. Astron. Soc. 281, 27 (1996), arXiv:astro-ph/9506004

    Pith/arXiv arXiv 1996

  52. [55]

    Bertone, D

    G. Bertone, D. Hooper, and J. Silk, Phys. Rept.405, 279 (2005), arXiv:hep-ph/0404175

    Pith/arXiv arXiv 2005

  53. [56]

    Clowe, M

    D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, Astrophys. J. Lett.648, L109 (2006), arXiv:astro-ph/0608407

    Pith/arXiv arXiv 2006

  54. [57]

    Bertone and D

    G. Bertone and D. Hooper, Rev. Mod. Phys.90, 045002 (2018), arXiv:1605.04909 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  55. [58]

    Gondolo and J

    P. Gondolo and J. Silk, Phys. Rev. Lett.83, 1719 (1999)

    1999

  56. [59]

    O. A. Hannuksela, K. C. Y . Ng, and T. G. F. Li, Phys. Rev. D 102, 103022 (2020)

    2020

  57. [60]

    Peirani and J

    S. Peirani and J. A. de Freitas Pacheco, Phys. Rev. D77, 064023 (2008)

    2008

  58. [61]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophysical Journal462, 563 (1996)

    1996

  59. [62]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophysical Journal490, 493 (1997)

    1997

  60. [63]

    Dehnen, Mon

    W. Dehnen, Mon. Not. Roy. Astron. Soc.265, 250 (1993)

    1993

  61. [64]

    M. M. Gohain, P. Phukon, and K. Bhuyan, Phys. Dark Univ. 46, 101683 (2024), arXiv:2407.02872 [gr-qc]

    Pith/arXiv arXiv 2024

  62. [65]

    Al-Badawi, S

    A. Al-Badawi, S. Shaymatov, and Y . Sekhmani, JCAP02, 014 (2025), arXiv:2411.01145 [gr-qc]

    Pith/arXiv arXiv 2025

  63. [66]

    Burkert, Astrophysical Journal Letters447, L25 (1995)

    A. Burkert, Astrophysical Journal Letters447, L25 (1995)

    1995

  64. [68]

    Baes and G

    M. Baes and G. Gentile, Astronomy & Astrophysics665, A136 (2022), arXiv:2208.12321

    Pith/arXiv arXiv 2022

  65. [69]

    Hernquist, Astrophysical Journal356, 359 (1990)

    L. Hernquist, Astrophysical Journal356, 359 (1990)

    1990

  66. [70]

    N. Dai, Y . Gong, Y . Zhao, and T. Jiang, Phys. Rev. D110, 084080 (2024), arXiv:2301.05088 [gr-qc]

    Pith/arXiv arXiv 2024

  67. [71]

    Jusufi, M

    K. Jusufi, M. Jamil, P. Salucci, T. Zhu, and S. Haroon, Phys. Rev. D100, 044012 (2019), arXiv:1905.11803 [physics.gen- ph]

    Pith/arXiv arXiv 2019

  68. [72]

    D. Liu, Y . Yang, Z. Xu, and Z.-W. Long, Eur. Phys. J. C84, 136 (2024), arXiv:2307.13553 [gr-qc]

    Pith/arXiv arXiv 2024

  69. [73]

    Al-Badawi and S

    A. Al-Badawi and S. Shaymatov, Chin. Phys. C49, 055101 (2025), arXiv:2501.15397 [gr-qc]

    Pith/arXiv arXiv 2025

  70. [74]

    Liang, D

    Q.-Q. Liang, D. Liu, and Z.-W. Long, Eur. Phys. J. C85, 1107 (2025), arXiv:2505.15540 [gr-qc]

    Pith/arXiv arXiv 2025

  71. [75]

    Uktamov, S

    U. Uktamov, S. Shaymatov, B. Ahmedov, and C. Yuan, (2025), arXiv:2509.00460 [gr-qc]

    Pith/arXiv arXiv 2025

  72. [76]

    S. V . M. C. B. Xavier, H. C. D. Lima Junior, and L. C. B. Crispino, Phys. Rev. D107, 064040 (2023)

    2023

  73. [77]

    J. Wang, S. Bose, C. S. Frenk, L. Gao, A. Jenkins, V . Springel, and S. D. M. White, Nature585, 39 (2020), arXiv:1911.09720 [astro-ph.CO]

    arXiv 2020

  74. [78]

    J. Wang, S. Bose, C. S. Frenk, L. Gao, A. Jenkins, V . Springel, and S. D. M. White, Nature585, 39 (2020)

    2020

  75. [79]

    Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata5, 87 (1965)

    J. Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata5, 87 (1965)

    1965

  76. [80]

    Perlick and O

    V . Perlick and O. Y . Tsupko,95, 104003 ()

  77. [81]

    A. Das, A. Saha, and S. Gangopadhyay,39, 075005

  78. [82]

    Perlick, O

    V . Perlick, O. Y . Tsupko, and G. S. Bisnovatyi-Kogan,92, 104031

  79. [83]

    Chowdhuri and A

    A. Chowdhuri and A. Bhattacharyya,104, 064039, 2012.12914 [gr-qc]

    Pith/arXiv arXiv 2012

  80. [84]

    C. F. Gammie, J. C. McKinney, and G. Toth, The Astrophysical Journal589, 444 (2003)

    2003

Showing first 80 references.