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arxiv: 2604.24349 · v2 · submitted 2026-04-27 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Scalar, electromagnetic, and Dirac perturbations of regular black holes constituting primordial dark matter

Bekir Can L\"utf\"uo\u{g}lu

Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:15 UTC · model grok-4.3

classification 🌀 gr-qc
keywords regular black holesquasinormal modesDirac-Born-Infeld scalarperturbationsprimordial dark matterringdownWKB method
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The pith

Larger regularity in phantom DBI black holes shifts quasinormal spectra to lower frequencies and damping rates

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

The paper examines massless scalar, electromagnetic, and Dirac perturbations around an exact asymptotically flat regular black hole supported by a phantom Dirac-Born-Infeld scalar. It calculates representative quasinormal frequencies with the Padé-improved WKB method and cross-checks the scalar fundamental mode via time-domain Prony fitting. Larger values of the regularity scale move both the real and imaginary parts of the frequencies downward in a spin-dependent manner, while the quality factor stays nearly constant. The shifts exceed estimated numerical uncertainty. This matters because the black holes are proposed as primordial dark matter candidates whose ringdown signals could appear in gravitational wave data.

Core claim

The exact asymptotically flat regular black hole supported by a phantom Dirac-Born-Infeld scalar produces quasinormal frequencies for massless scalar, electromagnetic, and Dirac perturbations that shift toward smaller oscillation frequencies and damping rates as the regularity scale increases, with the quality factor changing only weakly. The spectral shifts remain well above the estimated numerical uncertainty, showing that the DBI regularity scale leaves a robust spin-dependent imprint on ringdown.

What carries the argument

The effective potentials for scalar, electromagnetic, and Dirac perturbations on the regular metric, with the DBI regularity scale as the parameter that alters the potential shape and thus the quasinormal spectrum, computed via Padé-improved WKB and verified by Prony fitting.

If this is right

  • Ringdown signals from these black holes would exhibit lower oscillation frequencies and slower damping than those of the Schwarzschild solution.
  • The size of the shift depends on perturbation spin, producing distinct signatures for scalar, electromagnetic, and Dirac fields.
  • The nearly constant quality factor implies that the number of observable cycles before damping remains comparable across regularity values.
  • Precise measurements of ringdown in gravitational wave events could in principle reveal or constrain the presence of such regularity scales.

Where Pith is reading between the lines

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

  • Gravitational wave observatories might extract bounds on the DBI regularity parameter from high-signal-to-noise ringdown events if the predicted shifts hold.
  • If the assumed linear stability fails for certain regularity values, these objects would be ruled out as stable dark matter constituents.
  • Applying the same perturbation analysis to other families of regular black holes could test whether the downward frequency shift is a general feature of regularity or tied to the DBI construction.

Load-bearing premise

The regular black hole solution is stable under the linear perturbations considered and the Padé-improved WKB method combined with Prony fitting accurately extracts the fundamental modes without missing significant corrections from the regularity scale.

What would settle it

A time-domain evolution or higher-order calculation on the same regular metric that yields fundamental quasinormal frequencies whose real and imaginary parts do not decrease with increasing regularity scale, or that reveals an instability for large regularity.

Figures

Figures reproduced from arXiv: 2604.24349 by Bekir Can L\"utf\"uo\u{g}lu.

Figure 1
Figure 1. Figure 1: FIG. 1. Effective potentials in the DBI-supported regular view at source ↗
Figure 2
Figure 2. Figure 2: In all three sectors, increasing a lowers the maximum of the barrier and makes the profile slightly broader. This tendency is mild but systematic, and it is precisely the type of deformation that is expected to shift the quasinormal frequencies relative to the Schwarzschild limit. In particular, the DBI regularity scale modifies FIG. 1. Effective potentials in the DBI-supported regular black-hole backgroun… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dependence of the effective potentials on the regu view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Time-domain profile of the massless scalar pertur view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Quality factor view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Time-domain profile of the electromagnetic pertur view at source ↗
read the original abstract

We study massless scalar, electromagnetic, and Dirac perturbations of the exact asymptotically flat regular black hole supported by a phantom Dirac--Born--Infeld scalar. Using the Pad\'e-improved WKB method, with a time-domain Prony check for the scalar fundamental mode, we compute representative quasinormal frequencies and find that larger regularity shifts the spectrum toward smaller oscillation frequencies and damping rates, whereas the quality factor changes only weakly. The spectral shifts remain well above the estimated numerical uncertainty, demonstrating that the DBI regularity scale leaves a robust spin-dependent imprint on ringdown.

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 / 2 minor

Summary. The manuscript computes quasinormal modes for massless scalar, electromagnetic, and Dirac perturbations of an asymptotically flat regular black hole sourced by a phantom Dirac-Born-Infeld scalar. Using the Padé-improved WKB method (with a time-domain Prony validation reported only for the scalar fundamental mode), the authors find that increasing the DBI regularity scale shifts oscillation frequencies and damping rates downward while the quality factor changes only weakly; these shifts are stated to exceed estimated numerical uncertainties and to exhibit a spin-dependent imprint.

Significance. If the numerical accuracy and error estimates hold, the result would indicate that the regularity scale imprints a robust, observable signature on the ringdown spectrum of these black holes, potentially allowing gravitational-wave observations to constrain or rule out such objects as primordial dark matter candidates.

major comments (1)
  1. [Numerical method and results sections] The central claim that spectral shifts exceed numerical uncertainty and constitute a 'robust spin-dependent imprint' rests on the accuracy of the Padé-improved WKB results for all three fields. However, the time-domain Prony check is reported only for the scalar fundamental mode; no convergence tests, residual error estimates, or cross-validation against continued-fraction or direct-integration methods are described for the electromagnetic or Dirac cases, where the modified effective potentials (altered peak location, width, and tails) could introduce O(regularity-scale) corrections not captured by the WKB truncation.
minor comments (2)
  1. [Results] Clarify in the text how the 'estimated numerical uncertainty' is quantified (e.g., variation with WKB order, Prony fitting window, or grid resolution) and provide explicit comparison tables or figures for all three perturbation types.
  2. [Introduction or setup] Ensure that the stability of the background solution under the considered perturbations is explicitly stated or referenced, as assumed in the analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the numerical validation of our quasinormal mode results. We address the concern regarding the scope of the time-domain checks and error estimates below.

read point-by-point responses

  1. Referee: [Numerical method and results sections] The central claim that spectral shifts exceed numerical uncertainty and constitute a 'robust spin-dependent imprint' rests on the accuracy of the Padé-improved WKB results for all three fields. However, the time-domain Prony check is reported only for the scalar fundamental mode; no convergence tests, residual error estimates, or cross-validation against continued-fraction or direct-integration methods are described for the electromagnetic or Dirac cases, where the modified effective potentials (altered peak location, width, and tails) could introduce O(regularity-scale) corrections not captured by the WKB truncation.

    Authors: We agree that extending the validation beyond the scalar fundamental mode would strengthen the robustness of our claims. The Padé-improved WKB method has been widely tested in the literature for black-hole perturbation problems with potentials of comparable shape, and the observed shifts are consistent in sign and magnitude across the three fields. Nevertheless, to address the referee's point directly, in the revised manuscript we will add explicit convergence tests with respect to WKB order for representative electromagnetic and Dirac modes, together with residual error estimates. Where computationally feasible we will also include a limited set of time-domain Prony analyses or direct-integration cross-checks for the lowest-lying modes of each field. These additions will allow us to quantify any potential O(regularity-scale) corrections arising from changes in the effective-potential peak and tails. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical QNM computation on fixed background is independent

full rationale

The paper solves the perturbation equations for scalar, electromagnetic, and Dirac fields on the given regular black-hole metric using Padé-improved WKB (with Prony validation only for the scalar mode). The regularity parameter enters solely through the background metric functions; the computed frequencies and damping rates are outputs of the numerical scheme, not inputs or fitted quantities. No self-citation chain, ansatz smuggling, or redefinition of the regularity scale in terms of the quasinormal spectrum occurs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper relies on the existence of an exact asymptotically flat regular black-hole solution sourced by a phantom Dirac-Born-Infeld scalar field. The perturbation equations are the standard massless wave equations on that fixed background. No new free parameters are introduced beyond the regularity scale already present in the background metric. No invented entities beyond the phantom scalar are postulated in the abstract.

free parameters (1)
  • DBI regularity scale
    The parameter controlling the size of the regular core in the background metric; its value is varied to produce the reported spectral shifts.
axioms (2)
  • domain assumption The background metric is an exact solution of the Einstein equations coupled to the phantom DBI scalar.
    Invoked when the perturbation equations are written on this fixed geometry.
  • standard math Massless scalar, electromagnetic, and Dirac fields propagate on the fixed background without back-reaction.
    Standard linear perturbation assumption in black-hole ringdown calculations.

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Lean theorems connected to this paper

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Scattering of scalar, electromagnetic, and Dirac fields in an asymptotically flat regular black hole supported by primordial dark matter

    gr-qc 2026-05 unverdicted novelty 4.0

    Raising the regularity parameter in this regular black-hole spacetime lowers the single-barrier potentials for all three fields, shifts transmission to lower frequencies, increases absorption cross sections, and produ...

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

    gr-qc 2026-05 unverdicted novelty 3.0

    The primordial dark matter scale suppresses Hawking temperature and spectral energy emission rate relative to Schwarzschild while the heat capacity stays negative and the sparsity parameter receives a small negative c...

Reference graph

Works this paper leans on

120 extracted references · 120 canonical work pages · cited by 2 Pith papers · 30 internal anchors

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    no peak found

    obtained from the Padé-improved WKB approach for representative values of the regularity parameter. As an internal consistency check, we compare the 16th-order WKB results with Padé approximant ˜m = 8 against the 14th-order results with ˜m = 7 . For scalar perturbations with ℓ = 1 and ℓ = 2 at a = 0 .2, the effective potential has no barrier maximum, and i...

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    0.104535 − 0.101138i 0.104552 − 0.100903i 0.163% (b) ℓ = 1 a WKB16 ( ˜m = 8) WKB14 ( ˜m = 7) difference 0.2 no peak found 0.4 0 .291260 − 0.097149i 0.291260 − 0.097149i 0% 0.6 0 .289214 − 0.096526i 0.289214 − 0.096526i 0% 0.8 0 .286438 − 0.095681i 0.286438 − 0.095681i < 10−4%

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    0.283009 − 0.094640i 0.283009 − 0.094640i 0% 1.2 0 .279019 − 0.093429i 0.279019 − 0.093429i 0% 1.4 0 .274561 − 0.092079i 0.274561 − 0.092079i < 10−4% 1.6 0 .269730 − 0.090618i 0.269730 − 0.090618i 0% 1.8 0 .264614 − 0.089073i 0.264614 − 0.089073i 0%

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    0.259297 − 0.087470i 0.259297 − 0.087469i 0.0003% (c) ℓ = 2 a WKB16 ( ˜m = 8) WKB14 ( ˜m = 7) difference 0.2 no peak found 0.4 0 .481021 − 0.096255i 0.481021 − 0.096255i 0% 0.6 0 .477816 − 0.095639i 0.477816 − 0.095639i 0% 0.8 0 .473462 − 0.094802i 0.473462 − 0.094802i 0%

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    0.468076 − 0.093767i 0.468076 − 0.093767i 0% 1.2 0 .461793 − 0.092560i 0.461793 − 0.092560i 0% 1.4 0 .454758 − 0.091209i 0.454758 − 0.091209i 0% 1.6 0 .447113 − 0.089741i 0.447113 − 0.089741i 0% 1.8 0 .438997 − 0.088183i 0.438997 − 0.088183i 0%

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    For the fundamental modes with ℓ ≥ 1 or |κ| ≥ 2, this difference is typi- cally zero in the quoted digits and, when nonzero, re- TABLE II

    0.430536 − 0.086557i 0.430536 − 0.086557i 0% The significance of this shift becomes especially clear once it is compared with the numerical uncertainty es- timated from the difference between the WKB16 and WKB14 Padé-improved results. For the fundamental modes with ℓ ≥ 1 or |κ| ≥ 2, this difference is typi- cally zero in the quoted digits and, when nonzero, ...

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    0.240480 − 0.089460i 0.240480 − 0.089461i 0.0001% 1.2 0 .237329 − 0.088235i 0.237329 − 0.088236i 0.0001% 1.4 0 .233794 − 0.086862i 0.233794 − 0.086862i 0.0001% 1.6 0 .229946 − 0.085367i 0.229946 − 0.085367i 0% 1.8 0 .225851 − 0.083777i 0.225850 − 0.083778i < 10−4%

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    0.221572 − 0.082118i 0.221572 − 0.082118i 0.0001% (b) ℓ = 2 a WKB16 ( ˜m = 8) WKB14 ( ˜m = 7) difference 0.2 0 .456985 − 0.094875i 0.456985 − 0.094875i 0% 0.4 0 .455172 − 0.094491i 0.455172 − 0.094491i 0% 0.6 0 .452208 − 0.093863i 0.452208 − 0.093863i 0% 0.8 0 .448177 − 0.093010i 0.448177 − 0.093010i 0%

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    Even for the higher electromagnetic overtones in Tab

    0.408219 − 0.084559i 0.408219 − 0.084559i 0% mains at the level 10−4%– 10−3%. Even for the higher electromagnetic overtones in Tab. IV, the discrepancy does not exceed 8.82 × 10−3%. The least favorable fun- damental cases are the Dirac mode with |κ| = 1 , where the internal spread reaches about 0.13%, and the scalar monopole, where it is about 0.25%. Yet ...

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    0.177181 − 0.094080i 0.177002 − 0.093919i 0.120% 1.2 0 .174805 − 0.092730i 0.174632 − 0.092619i 0.104% 1.4 0 .172111 − 0.091248i 0.171969 − 0.091159i 0.0860% 1.6 0 .169172 − 0.089648i 0.169066 − 0.089569i 0.0688% 1.8 0 .166052 − 0.087948i 0.165977 − 0.087879i 0.0544%

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    0.162804 − 0.086176i 0.162750 − 0.086116i 0.0438% (b) |κ| = 2 a WKB16 ( ˜m = 8) WKB14 ( ˜m = 7) difference 0.2 0 .379527 − 0.096275i 0.379527 − 0.096275i 0% 0.4 0 .378014 − 0.095887i 0.378014 − 0.095887i 0% 0.6 0 .375540 − 0.095255i 0.375540 − 0.095255i 0% 0.8 0 .372176 − 0.094394i 0.372176 − 0.094394i 0%

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    0.368009 − 0.093328i 0.368010 − 0.093328i 0% 1.2 0 .363143 − 0.092084i 0.363143 − 0.092084i 0% 1.4 0 .357686 − 0.090688i 0.357686 − 0.090688i 0% 1.6 0 .351746 − 0.089170i 0.351746 − 0.089170i 0% 1.8 0 .345429 − 0.087556i 0.345429 − 0.087556i 0%

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    3 shows the waveform for ℓ = 1 , M = 1 , and a = 1 , for which the Prony fit is in excellent agreement with the WKB result listed in Tab

    0.338832 − 0.085870i 0.338832 − 0.085870i 0% scalar fundamental mode, Fig. 3 shows the waveform for ℓ = 1 , M = 1 , and a = 1 , for which the Prony fit is in excellent agreement with the WKB result listed in Tab. I. An analogous electromagnetic profile is shown in Fig. 4 for ℓ = 1 , M = 1 , and a = 2 . In this case the Prony fit gives ω = 0 .221573 − 0.08211...

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    0.422851 − 0.281364i 0.422851 − 0.281364i 0% 1.2 0 .417301 − 0.277582i 0.417301 − 0.277582i 0% 1.4 0 .411071 − 0.273338i 0.411071 − 0.273338i 0% 1.6 0 .404282 − 0.268719i 0.404282 − 0.268719i 0% 1.8 0 .397053 − 0.263805i 0.397053 − 0.263805i 0%

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