Near-extremal asymptotics and strong cosmic censorship for black holes immersed in a Chaplygin-like dark fluid

Hao-Peng Yan; Xiang-Qian Li; Xiao-Jun Yue; Zeng-Yi Zhang

REVIEW

Near-extremal asymptotics and strong cosmic censorship for black holes immersed in a Chaplygin-like dark fluid

Hao-Peng Yan, Zeng-Yi Zhang, Xiang-Qian Li, Xiao-Jun Yue

General Relativity and Quantum Cosmology · Submitted 1 Sep 2026 · 56 references

Pith's verdict: Solid analytic-numerical study of SCC in a neutral matter-supported black hole, but the spectral gap scan is not exhaustive and Eq. (36) misstates the pure-dS reference for l=1, which should be resolved before settling the phase diagram.

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arXiv ID2609.00980 (version 1)

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CitationHao-Peng Yan, Xiang-Qian Li, Xiao-Jun Yue, Zeng-Yi Zhang. "Near-extremal asymptotics and strong cosmic censorship for black holes immersed in a Chaplygin-like dark fluid." arXiv:2609.00980 (2026). https://pith.science/paper/LLE32LWS

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Editor's note

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This paper does a nice job on a well-defined problem. The analytic parametrization of the degenerate-horizon boundaries and the closed-form near-extremal scalar spectrum are clean and reusable. The numerical QNM calculations are cross-checked with direct integration and WKB, and the comparison with the PFDM case is a welcome sanity check. The CDF-specific phase diagram — the dS-to-PS-to-dS controller sequence and the nonmonotonic normalized width — is new and worth knowing.

Two issues deserve attention. First, the spectral gap is defined as the minimum over only three candidate families (low-ℓ dS, eikonal PS, fundamental NE). The paper follows the standard Cardoso et al. strategy, but it never argues that no other branch — a higher finite-ℓ PS mode, an intermediate-ℓ dS overtone, or a different NE overtone — could lie below this envelope. The central quantitative claims (qcrit, the exchange locations, the maximum width) all inherit this gap. The authors should either add an exhaustive scan over a mesh of ℓ values (say ℓ up to 30) and several overtones, or state explicitly that the three-family ansatz is an assumption and note the uncertainty it introduces.

Second, Eq. (36) gives the pure-dS ℓ=1 massless scalar frequency as -i sqrt(sqrt(B)/3). The standard result for a minimal massless scalar in pure dS is -i(ℓ+3)κ_c, which for ℓ=1 is -4i sqrt(sqrt(B)/3). The paper uses this reference for continuation (Appendix B). If the reference is off by a factor of four, the branch identification could be affected. The numerical dS modes shown in Fig. 3 may well be correct — they were computed and labeled with the full CDF problem — but the paper needs to correct Eq. (36) and clarify how the dS branch was identified independently of that reference. This is a load-bearing mistake for the claimed dS-to-PS-to-dS sequence.

Overall, the paper is a solid within-subfield contribution for readers working on strong cosmic censorship in matter-supported or modified-gravity black holes. It deserves peer review, but the above points should be addressed before the phase diagram is considered settled. I would recommend asking for a revised version with a corrected Eq. (36), a clearer spectral-gap justification, and preferably a release of the numerical data for reproducibility.

Referee report

Significance

If the central claims hold, the paper provides the first SCC analysis for a neutral, matter-supported black hole with an asymptotically de Sitter horizon but no independent cosmological constant—a scenario in which the asymptotic decay scale and the strong-field geometry are coupled through the same matter sector. The exact degenerate-horizon parametrization and the analytic near-extremal spectrum (Eqs. 24–31) are clean and testable. The result that SCC violation is confined to a narrow near-extremal strip, and that the controller sequence is nonmonotonic, extends the phenomenology established for RNdS and PFDM black holes to a new background. The paper is self-contained, the numerical QNM results are cross-checked, and the analytic cold-limit predictions are verified in Fig. 2.

Major objections (2)

[Eq. (36), Sec. V B, Appendix B] Equation (36) gives the pure-de Sitter reference frequency for ℓ=1 as ω_dS^(0) = -i sqrt(sqrt(B)/3), i.e., -iH with H = sqrt(Λ_eff/3). For a minimally coupled massless scalar on pure de Sitter, the correct QNM frequency is ω = -i (ℓ+3) H = -4iH for ℓ=1. The factor ℓ+3 is missing. Since Appendix B states that the dS branches are identified by continuation from pure-de Sitter reference frequencies on the imaginary axis, a wrong reference could lead to misidentification of which numerical mode is the dS branch, especially in the small-M^2 sqrt(B) regime where the CDF geometry is close to pure de Sitter. The authors must correct the reference and demonstrate that the dS branch used in the SCC phase diagram is correctly identified; otherwise the dS-controlled segments of the phase boundary (including the controller-exchange locations) are not reliable.
[Sec. V C, Figs. 5 and 6] The spectral gap α = inf_{ℓ,n} (-Im ω_{ℓn}) is constructed as the lower envelope of only three candidate families: the lowest de Sitter mode (ℓ=0 or ℓ=1), the eikonal photon-sphere mode (ℓ→∞, represented by ℓ=500), and the fundamental near-extremal branch (ℓ=0, n=0). The paper does not prove that no other QNM branch or overtone can produce a smaller value of −Im ω. In particular, dS overtones (n≥1) and dS modes with ℓ≥2, higher PS overtones (n≥1) at moderate ℓ, and NE overtones (ℓ>0, n≥1) are not systematically scanned. While the authors compare ℓ=10 and ℓ=20 PS data to the eikonal limit, and while the fundamental NE mode is expected to be the slowest NE mode (since β_NE,ℓn = n + h_ℓ, with n≥0 and h_ℓ≥1), the dS sector requires stronger justification. Damping rates of dS modes are known to increase with ℓ and n for standard potentials, but this property should be demonstrated or referenced for the CDF geometry. The location of q_crit(B) and the reported controller assignments depend on this envelope being correct; an unexamined branch could shift the threshold. Please provide a more thorough discussion or additional numerical checks (e.g., select a few representative points and compute the lowest-lying eigenvalues for a wider range of ℓ and n) to support the claim that the envelope is indeed the infimum.

Minor objections (6)

[Eq. (36)] As noted above, the factor (ℓ+3) is missing in the pure-de Sitter reference. Even if this factor is not used quantitatively beyond setting the scale, the equation as written is incorrect; please correct it or clarify that only the parametric scaling is meant.
[Sec. IV, Eq. (30)] The limit ω_NE,ℓn / κ_{-} → -i (n + h_ℓ(B)) is given without explicit discussion of the normalization of surface gravities. Since κ_{+}/κ_{-} → 1 in the cold limit, either horizon can be used, but the text should state this equivalence explicitly to avoid confusion.
[Fig. 3 caption] The caption states that 'the figure is not intended as an exhaustive enumeration of all QNM overtones.' This is appropriate, but it would be helpful to specify the number of overtones shown (e.g., n=0,1,2 for each family) to aid reproducibility.
[Table I] At q/q_max=0.995, M^2 sqrt(B)=0.04, the direct-integration entry for ℓ=20 shows β=0.805723, compared to pseudospectral β=0.809974 and WKB β=0.809903. The deviation of ≈0.5% from the other two methods is noted but not discussed. Please add a brief comment on the expected accuracy or possible causes.
[Sec. V D and Appendix C] The PFDM comparison is informative. It would be strengthened by including a brief discussion of whether the PFDM cold-limit estimate (Eq. C6) is monotonic for all Λ, or whether the nonmonotonicity in PFDM might also appear in a different parameter range. This is not required, but would improve the comparative section.
[References] Ref. [40] (Bécar et al.) is listed as 'JCAP 06, 061' without year (presumably 2024). Please verify the year and volume/page numbers in the compilation to ensure completeness.

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The paper

The paper claims

Massless scalar perturbations can violate strong cosmic censorship in a narrow near-extremal region for black holes immersed in a Chaplygin-like dark fluid.

The reading

This paper investigates strong cosmic censorship (SCC) for massless scalar perturbations of neutral black holes embedded in a Chaplygin-like dark fluid (CDF). The CDF produces an asymptotically de Sitter exterior while supporting an inner Cauchy horizon, making the geometry a neutral analogue of Reissner-Nordström–de Sitter. The authors derive an exact parametrization of the degenerate horizon boundaries, which gives analytic control over the near-extremal geometry and the near-extremal quasinormal mode spectrum. They then compute the spectral gap – the slowest damped mode – by comparing three families: de Sitter, photon-sphere, and near-extremal modes. The central finding is that the Christodoulou SCC criterion (β > 1/2) is satisfied only in a narrow strip adjacent to extremality, with a nonmonotonic width that peaks at about 1.47% of the allowed parameter range.

What carries the argument

The central object is the exact parametric representation of the Nariai and cold degenerate-horizon boundaries, given by Eqs. (8)–(10): r_d(x) = 6M/(2 + x tanh x), q_d(x) = r_d(x) tanh x, B_d(x) = $sech^{2}$ x / $r_d^{4}$(x), with x ∈ [0,∞). This parametrization reduces the two-boundary problem to a single variable and makes the near-extremal expansion fully analytic, yielding closed forms for the horizon-splitting coefficient C_r(B) and Cauchy-horizon surface-gravity coefficient K(B). Combined with the near-horizon AdS₂ description, it gives the near-extremal scalar spectrum ω_{NE,ℓn} → −i κ_− (n + h_ℓ(B)) with h_ℓ explicitly known. These analytic results anchor the global numerical competition among de Sitter, photon-sphere, and near-extremal families that defines the SCC phase boundary.

Extended reading notes

Core claim

The paper claims that for linear massless scalar fields on CDF black holes, the spectral gap ratio β = α/κ_− can exceed 1/2 in a finite near-extremal interval q_crit(B) < q < q_max(B). The threshold q_crit(B) is determined by the lowest of the lowest de Sitter mode, the eikonal photon-sphere mode, and the fundamental near-extremal mode. As the effective cosmological scale $M^{2}$√B increases, the threshold-controlling mode switches from de Sitter to photon-sphere near $M^{2}$√B ≈ 0.0139, and back to de Sitter near 0.1288. The normalized width Δ̃q = 1 − q_crit/q_max reaches a maximum of about 0.0147 at $M^{2}$√B ≈ 0.0857, within the photon-sphere-controlled segment. The near-extremal analytic results show that the fundamental ℓ = 0 mode approaches β = 1, while higher ℓ modes approach β = h_ℓ(B), where h_ℓ is the conformal weight of the near-horizon AdS₂ geometry. The paper thus demonstrates that potential SCC violation can occur in charge-neutral matter-supported de Sitter black holes, extending earlier results from charged and perfect-fluid dark matter backgrounds.

Load-bearing premise

The spectral gap is assumed to be the minimum of only three candidate families: the lowest de Sitter mode, the eikonal photon-sphere mode, and the fundamental near-extremal mode; the paper does not prove that no other mode (e.g., a higher finite-ℓ overtone or a different branch) decays more slowly.

Editorial extensions

If this is right

If correct, SCC can be violated in neutral matter-supported black holes, not just in charged ones, generalising the phenomenon.
The analytic cold-limit expressions provide a template for studying SCC in other matter models with an inner horizon.
The nonmonotonic width of the violation region suggests a subtle interplay between the asymptotic cosmological scale and the strong-field geometry, which may be testable in other three-horizon spacetimes.
The dS→PS→dS controller sequence indicates that the dominant decay channel is fragile and depends on the relative balance of horizon scales.
The fundamental near-extremal mode recovering β = 1 in the cold limit confirms that the Cauchy horizon regularity is maximally challenged exactly at extremality, but the violation region is displaced slightly away.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper’s claim depends on the assumption that the spectral gap is attained by one of only three mode families (lowest dS, eikonal PS, fundamental NE), but a slower-decaying overtone or a finite-ℓ mode outside these families could shift the threshold.
The quantitative width maximum (~1.5%) implies that any astrophysical realization would require extremely fine-tuned parameters, making observational signatures of SCC violation unlikely in realistic CDF black holes.
The cold-limit width estimate Δbq_PS ≈ (λ_{e,ph}/K)² could be used as a proxy to predict the shape of the violation region in other matter models without full numerical scans.

Original abstract

We investigate strong cosmic censorship (SCC) for massless scalar perturbations of electrically neutral black holes immersed in a Chaplygin-like dark fluid (CDF). The matter distribution produces an asymptotically de Sitter exterior while supporting an inner Cauchy horizon. We derive a closed-form parametrization of the extremal and Nariai horizon boundaries. This analytic control yields explicit near-extremal asymptotics for the horizon splitting and Cauchy-horizon surface gravity, together with an analytic expression for the leading near-extremal scalar quasinormal spectrum. The fundamental near-extremal damping rate approaches the Cauchy-horizon surface gravity, whereas higher angular multipoles retain explicit dependence on the CDF extremal geometry. Combining these analytic results with global quasinormal-mode calculations, we determine the spectral gap and assess the Christodoulou formulation of SCC throughout the three-horizon domain. Potential SCC violation is confined to a narrow region near extremality. As the effective cosmological scale increases, the lower boundary of this region is controlled successively by de Sitter, photon-sphere, and de Sitter modes, while its normalized width varies nonmonotonically.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate strong cosmic censorship (SCC) for massless scalar perturbations of electrically neutral black holes immersed in a Chaplygin-like dark fluid (CDF). The matter distribution produces an asymptotically de Sitter exterior while supporting an inner Cauchy horizon. We derive a closed-form parametrization of the extremal and Nariai horizon boundaries. This analytic control yields explicit near-extremal asymptotics for the horizon splitting and Cauchy-horizon surface gravity, together with an analytic expression for the leading near-extremal scalar quasinormal spectrum. The fundamental near-extremal damping rate approaches the Cauchy-horizon surface gravity, whereas higher angular multipoles retain explicit dependence on the CDF extremal geometry. Combining these analytic results with global quasinormal-mode calculations, we determine the spectral gap and assess the Christodoulou formulation of SCC throughout the three-horizon domain. Potential SCC violation is confined to a narrow region near extremality. As the effective cosmological scale increases, the lower boundary of this region is controlled successively by de Sitter, photon-sphere, and de Sitter modes, while its normalized width varies nonmonotonically.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate strong cosmic censorship (SCC) for massless scalar perturbations of electrically neutral black holes immersed in a Chaplygin-like dark fluid (CDF).	contribution 0.97	0.93	0.04	asserted 0.71
s1	The matter distribution produces an asymptotically de Sitter exterior while supporting an inner Cauchy horizon.	background 0.94	0.19	0.03	asserted 1.00
s2	We derive a closed-form parametrization of the extremal and Nariai horizon boundaries.	contribution 0.63	0.96	0.04	asserted 1.00
s3	This analytic control yields explicit near-extremal asymptotics for the horizon splitting and Cauchy-horizon surface gravity, together with an analytic expression for the leading near-extremal scalar quasinormal spectrum.	result 0.80	0.96	0.09	asserted 1.00
s4	The fundamental near-extremal damping rate approaches the Cauchy-horizon surface gravity, whereas higher angular multipoles retain explicit dependence on the CDF extremal geometry.	result 1.00	0.91	0.09	asserted 1.00
s5	Combining these analytic results with global quasinormal-mode calculations, we determine the spectral gap and assess the Christodoulou formulation of SCC throughout the three-horizon domain.	method 0.50	0.96	0.17	asserted 1.00
s6	Potential SCC violation is confined to a narrow region near extremality.	result 0.98	0.90	0.57	hedged possibility 0.54
s7	As the effective cosmological scale increases, the lower boundary of this region is controlled successively by de Sitter, photon-sphere, and de Sitter modes, while its normalized width varies nonmonotonically.	result 0.98	0.90	0.14	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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sources read

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not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Geometry and three-horizon parameter domain of the CDF black hole. (a) Representative lapse function for

**Figure 2.** FIG. 2. Near-extremal scaling at fixed

**Figure 4.** FIG. 4. Lowest-lying damping ratios

**Figure 5.** FIG. 5. Massless-scalar SCC phase diagram for

**Figure 6.** FIG. 6. Normalized width of the SCC-violation region, ∆

Figures reproduced from arXiv:2609.00980 by the authors.

References

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Reduction of the degenerate-horizon conditions For the anisotropic CDF source one may write Tµ ν = diag(−ρ,−ρ,p t,pt), p t = ρ 2− 3B 2ρ,(A1) where the second relation follows from the angularly av- eraged equation of state p =−B/ρ. The independent Einstein equations are f+rf ′−1 r2 =−ρ,(A2) 2f′ +rf′′ 2r = ρ 2− 3B 2ρ.(A3) Letr d be a degenerate positive ro...

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The near-horizon spectrum therefore obeys the scaled cold-limit relation ωNE,ℓn κ− −→−i n+h ℓ(B) ,(30) where n = 0, 1, 2,... is the overtone number, consistent with the general near-horizon description of zero-damped modes in nearly extremal cosmological black holes [ 32, 46, 47]. Since κ+/κ−→ 1, either horizon surface gravity gives the same leading scale...

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Nariai, extremal, and triple-horizon branches The sign of f′′(rd) gives an analytic classification of the two degenerate branches. Evaluating Eq. (A3) on a degenerate horizon and using Eqs. (A5) and (A7) gives f′′(rd) = 3u2 d−2 r2 d .(A13) Thus u2 d < 2/3 gives f′′(rd) < 0, so the double root is a local maximum of f and corresponds to the merger r+ = rc (...

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Relation to previous horizon analysis Ref. [41] analyzed the global shape of the same CDF metric, proved that f(r) can have at most three positive roots, and derived the critical relation terminating the three-horizon domain. Their variables are related to those used here by BF = B 9, Q= q 3.(A18) Accordingly, Eq. (A17) becomes BFQ4 = 4 39,(A19) and q⋆/3≃...

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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Reduction of the degenerate-horizon conditions For the anisotropic CDF source one may write Tµ ν = diag(−ρ,−ρ,p t,pt), p t = ρ 2− 3B 2ρ,(A1) where the second relation follows from the angularly av- eraged equation of state p =−B/ρ. The independent Einstein equations are f+rf ′−1 r2 =−ρ,(A2) 2f′ +rf′′ 2r = ρ 2− 3B 2ρ.(A3) Letr d be a degenerate positive ro...

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The near-horizon spectrum therefore obeys the scaled cold-limit relation ωNE,ℓn κ− −→−i n+h ℓ(B) ,(30) where n = 0, 1, 2,... is the overtone number, consistent with the general near-horizon description of zero-damped modes in nearly extremal cosmological black holes [ 32, 46, 47]. Since κ+/κ−→ 1, either horizon surface gravity gives the same leading scale...

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Nariai, extremal, and triple-horizon branches The sign of f′′(rd) gives an analytic classification of the two degenerate branches. Evaluating Eq. (A3) on a degenerate horizon and using Eqs. (A5) and (A7) gives f′′(rd) = 3u2 d−2 r2 d .(A13) Thus u2 d < 2/3 gives f′′(rd) < 0, so the double root is a local maximum of f and corresponds to the merger r+ = rc (...

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Relation to previous horizon analysis Ref. [41] analyzed the global shape of the same CDF metric, proved that f(r) can have at most three positive roots, and derived the critical relation terminating the three-horizon domain. Their variables are related to those used here by BF = B 9, Q= q 3.(A18) Accordingly, Eq. (A17) becomes BFQ4 = 4 39,(A19) and q⋆/3≃...

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