arxiv: 2605.04923 · v1 · submitted 2026-05-06 · 🌀 gr-qc · hep-th

Recognition: unknown

Hawking Temperature, Sparsity and Energy Emission Rate of Regular Black Holes Supported by Primordial Dark Matter

Faizuddin Ahmed , Ahmad Al-Badawi , Edilberto O. Silva

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:01 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords regular black holesHawking temperatureprimordial dark matterDBI scalar fieldenergy emission ratethermodynamic stabilitysparsity parameterADM mass normalization

0 comments

The pith

Primordial dark matter modeled by a DBI scalar reduces Hawking temperature and emission rate in regular black holes relative to the Schwarzschild case.

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

The paper examines thermodynamic and radiative properties of regular black holes whose source is primordial dark matter, treated as a DBI scalar field. It computes the Hawking temperature, entropy from the first law, heat capacity, sparsity of the flux, and spectral energy emission rate, with focus on how the regularity scale affects these quantities and on recovery of the Schwarzschild limit. Using the normalization in which the integration constant equals the ADM mass, the calculations show that the dark-matter scale lowers both temperature and emission rate. Heat capacity stays negative on the physical branch, so local instability persists. The sparsity parameter receives a small negative correction in the perturbative regime, implying modestly less intermittent radiation.

Core claim

With the metric normalized so that M is the ADM mass and f(r) = 1 - 2M/r + O(r^{-3}), the PDM scale α suppresses the Hawking temperature and the spectral energy emission rate relative to Schwarzschild black holes of equal mass. The fixed-α heat capacity remains negative along the physical branch. Within the adopted effective-area prescription, the geometrical sparsity parameter receives a negative leading correction for α ≪ 2M, which reduces the intermittency of the Hawking flux. The analysis distinguishes the near-horizon geometrical estimate from the shadow-based high-energy absorption cross-section employed in the emission-rate calculation.

What carries the argument

The regularity scale parameter α arising from the DBI scalar field that sources the regular metric, which modifies the near-horizon geometry and thereby alters the surface gravity and emission properties.

If this is right

Hawking temperature decreases with increasing α at fixed ADM mass.
Spectral energy emission rate is suppressed relative to the Schwarzschild case.
Heat capacity remains negative, preserving local thermodynamic instability in the canonical ensemble.
Sparsity parameter receives a negative correction, implying slightly lower intermittency of the flux.
Emission-rate calculations require separating near-horizon geometry from the shadow-based absorption cross-section.

Where Pith is reading between the lines

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

Longer evaporation timescales would follow if the suppression of temperature and emission rate holds in realistic astrophysical environments containing dark matter.
Shadow observations or gravitational-wave ringdown signals could provide independent bounds on α that would then predict specific deviations in any detectable Hawking radiation.
The same DBI modeling approach might be applied to other regular black-hole families to test whether the temperature suppression is generic once dark-matter sources are included.

Load-bearing premise

Primordial dark matter is effectively described by a DBI scalar field that produces a specific regular metric whose perturbative expansion around the Schwarzschild solution is controlled by α, together with the choice of effective-area prescription for the sparsity parameter.

What would settle it

A direct or indirect measurement of Hawking radiation from a stellar-mass black hole candidate whose temperature or emission spectrum matches the unsuppressed Schwarzschild value rather than the reduced value predicted once α is fixed by independent constraints on the regularity scale.

Figures

Figures reproduced from arXiv: 2605.04923 by Ahmad Al-Badawi, Edilberto O. Silva, Faizuddin Ahmed.

**Figure 1.** Figure 1: shows the behavior of the lapse function as a function of the dimensionless coordinate r/M for different values of α/M. The Schwarzschild curve is recovered when α = 0. For nonzero α, the regularity scale shifts the horizon location while preserving the asymptotically flat behavior at large distances. The mass function is displayed in view at source ↗

**Figure 3.** Figure 3: FIG. 3. Hawking temperature. Panel (a) shows the exact dimensionless temperature view at source ↗

**Figure 4.** Figure 4: FIG. 4. Fixed- view at source ↗

**Figure 5.** Figure 5: FIG. 5. Shadow-related quantities. Panel (a) shows view at source ↗

**Figure 6.** Figure 6: FIG. 6. Geometrical sparsity ratio. Panel (a) shows the exact ratio view at source ↗

**Figure 7.** Figure 7: FIG. 7. Spectral energy emission rate as a function of view at source ↗

read the original abstract

In this paper, we investigate the thermodynamic and radiative properties of a regular black hole sourced by primordial dark matter (PDM), modeled effectively through a Dirac--Born--Infeld (DBI) scalar field. We compute the Hawking temperature, the entropy obtained from the first law at fixed PDM scale, the specific heat capacity, the sparsity parameter of the Hawking flux, and the spectral energy emission rate. Particular attention is devoted to the role played by the regularity scale parameter \(\alpha\) and to the recovery of the Schwarzschild limit. Using the normalization in which the integration constant \(M\) is the ADM mass and \(f(r)=1-2M/r+\mathcal{O}(r^{-3})\), we find that the PDM scale suppresses the Hawking temperature and the spectral energy emission rate relative to the Schwarzschild case. The fixed-\(\alpha\) heat capacity remains negative along the physical branch, indicating the persistence of local thermodynamic instability in the canonical ensemble. Moreover, within the effective-area prescription adopted here, the geometrical sparsity parameter receives a negative leading correction in the perturbative regime \(\alpha\ll 2M\), implying a slight reduction of the intermittency of the Hawking flux. We also distinguish between the near-horizon geometrical estimate and the shadow-based high-energy absorption cross-section used in the emission rate.

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.

Desk Editor's Note private letter to a colleague

This paper computes standard thermodynamic and radiative quantities for one specific DBI-sourced regular black hole and reports perturbative suppressions from the dark-matter scale, with no internal contradictions in the derivations.

read the letter

The main takeaway is that in this regular black hole metric normalized so the integration constant equals the ADM mass, the primordial dark matter scale lowers the Hawking temperature and the spectral energy emission rate relative to Schwarzschild while the fixed-alpha heat capacity stays negative. The sparsity parameter picks up a small negative correction under the effective-area prescription they adopt, and they separate the near-horizon and shadow-based absorption cross sections in the emission-rate formula. All of this is done in the perturbative regime where the regularity scale is much smaller than the mass. These explicit numbers for temperature, capacity, sparsity, and rate are new for this particular DBI-supported metric. The calculations follow the usual surface-gravity definition for temperature and the first law at fixed alpha, and the normalization condition is applied correctly without forcing the outcomes. The stress-test note confirms the steps are standard once the metric is given and that no hidden inconsistency appears in the full text. The paper states its modeling assumptions up front and recovers the Schwarzschild limit as alpha goes to zero. The soft spots are modest and expected for this kind of work. Everything rests on the choice of DBI scalar to represent primordial dark matter and on the perturbative expansion, so the reported suppressions hold only inside those limits. No new framework or general theorem is offered, just concrete corrections for one effective model. Higher-order terms or a different source for the dark matter could alter the size or sign of the effects. This is the sort of paper that specialists in regular black holes and modified thermodynamics will want to see for the specific numbers, but it will not change how most people think about Hawking radiation or stability. A reader already working on extra scales in black-hole metrics gets the most value. I would send it to peer review because the computations are reproducible from the given metric and the claims are stated clearly enough to be checked.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the Hawking temperature (via surface gravity), entropy (via first law at fixed PDM scale α), heat capacity C_α, sparsity parameter η, and spectral energy emission rate for a regular black hole sourced by primordial dark matter modeled by a DBI scalar field. With the metric normalized so M is the ADM mass and f(r)=1-2M/r+O(r^{-3}), it reports that the PDM scale suppresses T_H and the emission rate relative to Schwarzschild, that fixed-α heat capacity remains negative on the physical branch, and that η receives a negative leading correction in the α≪2M regime (implying reduced intermittency); near-horizon and shadow-based absorption cross-sections are distinguished.

Significance. If the results hold, the work supplies explicit perturbative expressions showing how a regularity scale tied to a DBI dark-matter model modifies standard black-hole thermodynamics and radiation, with the persistence of negative heat capacity and the sign of the sparsity correction being the main concrete outcomes. Credit is due for the standard derivations, the explicit normalization condition, the two absorption-cross-section prescriptions, and the recovery of the Schwarzschild limit. Significance remains moderate because the findings are tied to the effective DBI modeling choice and the perturbative regime; they do not constitute a parameter-free or model-independent prediction.

minor comments (3)

The explicit form of the metric function f(r) (including the precise O(r^{-3}) term) should be stated in §2 or §3 so that the surface-gravity and first-law calculations can be reproduced without reference to external literature.
The validity range of the α≪2M expansion should be quantified (e.g., by showing the size of the next term in the temperature or emission-rate expressions) to support the claim that the reported suppressions are robust.
The effective-area prescription adopted for the sparsity parameter η is presented as a modeling choice; a brief comparison with the alternative (shadow-based) prescription already used for the emission rate would clarify why the negative correction is tied to one choice rather than the other.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. We appreciate the positive assessment of the derivations, the normalization condition, and the recovery of the Schwarzschild limit. Since no specific major comments were raised beyond the overall summary and significance evaluation, we provide a brief response to the key points noted in the report.

read point-by-point responses

Referee: Significance remains moderate because the findings are tied to the effective DBI modeling choice and the perturbative regime; they do not constitute a parameter-free or model-independent prediction.

Authors: We agree that the results are specific to the DBI scalar-field effective description of primordial dark matter and to the perturbative regime α ≪ 2M. The manuscript does not claim model-independent or parameter-free predictions; it explicitly presents the corrections for this setup and recovers the Schwarzschild case. We have added a clarifying sentence in the introduction and conclusions to emphasize the model dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper computes Hawking temperature from surface gravity, entropy via the first law at fixed alpha, heat capacity as its derivative, sparsity parameter, and spectral emission rate using standard formulas applied to the given DBI-sourced metric with the explicit normalization f(r) = 1 - 2M/r + O(r^{-3}) where M is the ADM mass. These steps follow directly from the metric function and stated modeling choices (effective-area prescription, perturbative regime alpha << 2M, two absorption cross-section options) without any reduction to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims are independent explicit calculations once the metric ansatz is adopted; no uniqueness theorems or ansatze are smuggled via prior self-citation, and the normalization is a standard boundary condition rather than a tautology forcing the reported suppressions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claims rest on an effective DBI scalar-field description of primordial dark matter, a static spherically symmetric metric ansatz, the first law applied at fixed PDM scale, and the perturbative limit alpha much less than 2M; these are introduced without independent derivation inside the paper.

free parameters (2)

regularity scale alpha
Length scale introduced to remove the central singularity; treated as a small perturbative parameter alpha << 2M
PDM scale parameter
Scale controlling the strength of the primordial dark matter source in the DBI effective model

axioms (2)

domain assumption The spacetime is static and spherically symmetric with asymptotic flatness
Standard assumption for black-hole thermodynamics calculations invoked to define the metric function f(r)
domain assumption The first law of black-hole thermodynamics holds when the PDM scale is held fixed
Used to obtain the entropy from the temperature and mass

invented entities (1)

Regular black hole sourced by primordial dark matter via DBI scalar field no independent evidence
purpose: To provide a singularity-free black-hole solution whose matter content is identified with early-universe dark matter
The model is postulated as an effective description; no independent observational or theoretical evidence for this exact sourcing is supplied inside the paper

pith-pipeline@v0.9.0 · 5548 in / 1733 out tokens · 54490 ms · 2026-05-08T16:01:05.399385+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 7 canonical work pages · 3 internal anchors

[1]

J. M. Bardeen, Non-singular general-relativistic gravita- tional collapse, inProceedings of the International Con- ference GR5(Tbilisi, U.S.S.R., 1968) p. 174

1968
[2]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett.80, 5056 (1998)

1998
[3]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, New regular black hole solution from nonlinear electrodynamics, Phys. Lett. B 464, 25 (1999)

1999
[4]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, The bardeen model as a nonlinear magnetic monopole, Phys. Lett. B493, 149 (2000)

2000
[5]

Ayon-Beato and A

E. Ayon-Beato and A. Garcia, Four-parametric regular black hole solution, Gen. Relativ. Gravit.37, 635 (2005)

2005
[6]

S. A. Hayward, Formation and evaporation of nonsingu- lar black holes, Phys. Rev. Lett.96, 031103 (2006)

2006
[7]

Dymnikova, Vacuum nonsingular black hole, Gen

I. Dymnikova, Vacuum nonsingular black hole, Gen. Rel- ativ. Gravit.24, 235 (1992)

1992
[8]

Fan and X

Z.-Y. Fan and X. Wang, Construction of regular black holes in general relativity, Phys. Rev. D94, 124027 (2016)

2016
[9]

Balart and E

L. Balart and E. C. Vagenas, Regular black holes with a nonlinear electrodynamics source, Phys. Rev. D90, 124045 (2014)

2014
[10]

Burinskii and S

A. Burinskii and S. R. Hildebrandt, New type of regu- lar black holes and particlelike solutions from nonlinear electrodynamics, Phys. Rev. D65, 104017 (2002)

2002
[11]

Culetu,On a regular charged black hole with a nonlinear electric source,Int

H. Culetu, On a regular charged black hole with a nonlin- ear electric source, Int. J. Theor. Phys.54, 2855 (2015), arXiv:1408.3334 [gr-qc]

work page arXiv 2015
[12]

Culetu, Nonsingular black hole with a nonlinear elec- tric source, Int

H. Culetu, Nonsingular black hole with a nonlinear elec- tric source, Int. J. Mod. Phys. D24, 1542001 (2015)

2015
[13]

Ma, Magnetically charged regular black hole in a model of nonlinear electrodynamics, Ann

M.-S. Ma, Magnetically charged regular black hole in a model of nonlinear electrodynamics, Ann. Phys.362, 529 (2015), arXiv:1509.05580 [gr-qc]

work page arXiv 2015
[14]

K. A. Bronnikov, Regular magnetic black holes and monopoles from nonlinear electrodynamics, Phys. Rev. D63, 044005 (2001)

2001
[15]

Ma, Magnetically charged regular black hole in a model of nonlinear electrodynamics, Annals of Physics 362, 529 (2015)

M.-S. Ma, Magnetically charged regular black hole in a model of nonlinear electrodynamics, Annals of Physics 362, 529 (2015)

2015
[16]

Simpson and M

A. Simpson and M. Visser, Black-bounce to traversable wormhole, JCAP2019(02), 042
[17]

Vertogradov and A

V. Vertogradov and A. Övgün, Exact regular black hole solutions with de sitter cores and hagedorn fluid, Class. Quantum Grav.42, 025024 (2025)

2025
[18]

H. W. Hu, C. Lan, and Y. G. Miao, A regular black hole as the final state of evolution of a singular black hole, Eur. Phys. J. C83, 1047 (2023)

2023
[19]

K.A.BronnikovandJ.C.Fabris,Regularphantomblack holes, Phys. Rev. Lett.96, 251101 (2006)

2006
[20]

R. A. Konoplya and A. Zhidenko, Dark matter halo as a source of regular black-hole geometries, Phys. Rev. D 113, 043011 (2026)

2026
[21]

Parvez and S

T. Parvez and S. Shankaranarayanan, Exact, nonsingu- lar black holes from a phantom dbi field as primordial dark matter, Phys. Rev. D 10.1103/118s-qpgy (2026), ac- cepted 15 April 2026

work page doi:10.1103/118s-qpgy 2026
[22]

S. W. Hawking, Particle creation by black holes, Com- mun. Math. Phys.43, 199 (1975)

1975
[23]

F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, The hawking cascade and its sparsity, Class. Quantum Grav. 33, 115003 (2016)

2016
[24]

Hod, The hawking cascades of gravitons from higher- dimensionalschwarzschildblackholes,Phys.Lett.B756, 133 (2016)

S. Hod, The hawking cascades of gravitons from higher- dimensionalschwarzschildblackholes,Phys.Lett.B756, 133 (2016)

2016
[25]

Hod, The hawking evaporation process of rapidly- rotating black holes: an almost continuous cascade of gravitons, Eur

S. Hod, The hawking evaporation process of rapidly- rotating black holes: an almost continuous cascade of gravitons, Eur. Phys. J C75, 329 (2015)

2015
[26]

Y.-P. Hu, F. Pan, and X.-M. Wu, The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box, Phys. Lett. B772, 553 (2017)

2017
[27]

Schuster,Black Hole Evaporation: Sparsity in Ana- logue and General Relativistic Space-Times, Ph.D

S. Schuster,Black Hole Evaporation: Sparsity in Ana- logue and General Relativistic Space-Times, Ph.D. the- sis, Victoria University of Wellington, Wellington, New Zealand (2018), arXiv:1901.05648 [gr-qc]

work page arXiv 2018
[28]

Visser, F

M. Visser, F. Gray, S. Schuster, and A. Van-Brunt, Spar- sity of the hawking flux, inThe Fourteenth Marcel Gross- mann Meeting(World Scientific, 2017) pp. 1724–1729

2017
[29]

B. C. Lütfüoğlu, (2026), arXiv:2604.24349 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[30]

Massive scalar quasinormal modes of an asymptotically flat regular black hole supported by a phantom Dirac--Born--Infeld field

M. Skvortsova, (2026), arXiv:2604.25471 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[31]

S. V. Bolokhov, (2026), arXiv:2605.03137 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[32]

J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys.31, 161 (1973)

1973
[33]

R. M. Wald,Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics(University of Chicago Press, 1994)

1994
[34]

Wei and Y.-X

S.-W. Wei and Y.-X. Liu, Observing the shadow of einstein-maxwell-dilaton-axion black hole, JCAP2013 (11), 063
[35]

Sanchez, Absorption and emission spectra of a schwarzschild black hole, Phys

N. Sanchez, Absorption and emission spectra of a schwarzschild black hole, Phys. Rev. D18, 1030 (1978)

1978
[36]

Décanini, G

Y. Décanini, G. Esposito-Farèse, and A. Folacci, Univer- sality of high-energy absorption cross sections for black 11 holes, Phys. Rev. D83, 044032 (2011)

2011
[37]

Ditta, X

A. Ditta, X. Tiecheng, G. Mustafa, and et al., Thermal stability with emission energy and joule–thomson expan- sion of regular btz-like black hole, Eur. Phys. J. C82, 756 (2022)

2022
[38]

Ditta, X

A. Ditta, X. Tiecheng, R. Ali, and G. Mustafa, Thermal stability and tunneling radiation in van der waals black hole, Nucl. Phys. B994, 116287 (2023)

2023
[39]

Javed, G

F. Javed, G. Mustafa, S. Mumtaz, and F. Atamuro- tov, Thermal analysis with emission energy of perturbed black hole in f(q) gravity, Nucl. Phys. B990, 116180 (2023)

2023
[40]

Javed, A

F. Javed, A. Waseem, P. Channuie, G. Mustafa, T. Muhammad, and E. Güdekli, Particle dynamics and joule–thomson expansion of phantom anti-de sitter black hole stability and thermal fluctuations in massive gravity, Phys. Dark Univ.47, 101766 (2025)

2025
[41]

Mahmood, Impact of gravitational collapse ex- hibiting loop quantum black holes on thermodynamical featuresandweakgravitationallensing,Phys.DarkUniv

A.Ditta, A.Bouzenada, G.Mustafa, F.Javed, F.Afandi, and A. Mahmood, Impact of gravitational collapse ex- hibiting loop quantum black holes on thermodynamical featuresandweakgravitationallensing,Phys.DarkUniv. 47, 101818 (2025)

2025