{"id":"5c76f68c-dd7b-4e45-8387-2dcaba2446d1","arxiv_id":"2608.01200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In charged AdS black holes surrounded by perfect fluid dark matter, the fundamental scalar quasinormal mode jumps between two branches at the first-order small/large black hole transition, stays smooth at the critical point, and its frequencies grow with the dark matter parameter.","lead":"This paper computes the damping oscillations, called quasinormal modes, of a charged black hole sitting in dark matter and compares them with the black hole's thermodynamic phase changes. It finds a clear jump in the modes at the first-order small/large black hole transition, no jump at the critical point, and a dark matter parameter that shifts both the transition and the mode frequencies.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The branch-separation and critical-merging conclusions rest entirely on one pseudospectral implementation; without an independent QNM method, the quantitative claims are unverified.","rationale":"I read the paper as a numerical demonstration that the l=0 scalar QNM spectrum tracks the small/large equilibrium branches in a PFDM-charged AdS background. The thermodynamics is internally consistent: the critical-point formulas check out, Eq. (2.6) matches the derivative of the metric function, and the Δr_h estimates in Tables 7-9 follow from Eq. (3.9). The qualitative branch separation is robust to the interpretive caveat that these are equilibrium QNMs rather than a time-dependent transition; the paper carefully limits its claim to the fundamental scalar mode and explicitly leaves gravitational perturbations open in Section 6. What is not secured is the numerical spectrum itself. The absence of an independent method, error bars, or released code means the quantitative content of Figures 2, 4, 7, 8, and 11 is supported only by the authors' implementation. This is precisely the kind of concern that conditional acceptance should require resolving, and it aligns with the reader's request for independent verification of representative frequencies. I do not find a mathematical contradiction in the derivation; the concern is verification rather than soundness of the formalism, so the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":22818,"tokens_out":15291,"duration_ms":141820,"concrete_test":"Recompute the l=0 fundamental QNM for representative states with an independent Leaver continued-fraction code, or a Prony-extracted time-domain integrator: α=0.1, Q=0.75, P/P_c=0.5, T=0.0450 and T=0.0500 (Table 1a); α=0.1, P=P_c, T=0.0654 (Table 10a); and α=0.1, T=0.74 T_c, P=0.00735 (Table 11a). Accept the paper's numbers if Re ω agrees to 1e-3 and Im ω to 1e-3. If any entry differs by more than about 1% in Re ω or 3% in Im ω, or if the small/large branch separation at P/P_c=0.5 shrinks by more than 10%, the central claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a quantitative statement about the fundamental scalar QNM spectrum across the phase transition: separated branches, distinct slopes, a discontinuous jump at T0/P0, and smooth merging at criticality. All of this is produced by a single Chebyshev pseudospectral implementation described in Section 3.1, with no independent method cross-check and no error budget. The only validation reported is that N=120 agrees with N=150 within 1e-6 for the fundamental mode (Section 3.1). Internal resolution convergence does not exclude spurious eigenvalues or a systematic error in the horizon boundary row or the compactification; for QNM computations, an independent method such as continued fraction or time-domain integration is the standard safeguard. The critical-point smoothness claim is additionally delicate: Tables 10 and 11 sample only a few dozen points and no derivative analysis is given, so a non-analytic kink in dω/dT or dω/dP at r_c could go unnoticed. If the pseudospectral implementation has a systematic offset for the large-|Im ω| modes that dominate the large-black-hole branch below coexistence, the claimed separation and slopes could be partly numerical. Therefore the demonstrative conclusion in Section 6 is only as strong as the unverified eigenvalues.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the relationship between thermodynamic phase transitions and fundamental scalar quasinormal mode (QNM) spectra for charged AdS black holes surrounded by perfect fluid dark matter (PFDM). The authors work in the extended phase space, treating the cosmological constant as pressure and the PFDM parameter as an additional thermodynamic variable, and derive the equation of state, critical point, and Gibbs free energy. Using a Chebyshev pseudospectral method, they compute the fundamental l=0 scalar QNMs on equilibrium backgrounds. Below the critical point, along isobaric and isothermal processes, the QNM frequencies of the small and large black hole branches are claimed to form separated trajectories with distinct slopes, with a discontinuous jump at the first-order transition. At the critical point, the QNMs vary smoothly and show no sharp signature. Along the coexistence curve, the separation between branch QNMs shrinks and vanishes at criticality. The paper concludes that the fundamental scalar QNM spectrum is sensitive to the first-order small/large black hole transition and that the PFDM parameter shifts both the thermodynamic and dynamical scales.","tokens_in":22935,"tokens_out":8306,"duration_ms":69739,"significance":"The paper is a straightforward extension of the known QNM-phase transition correspondence to a PFDM background. If the numerical results are reliable, they provide a concrete example of how a dark-matter environment alters both the phase structure and the perturbative response, which could be useful for holographic or observational probes. Strengths include the self-contained thermodynamic derivation (Eqs. (2.8)-(2.13)), the complete data tables for QNM frequencies, and the reported convergence test N=120 vs N=150. However, the central claims are entirely based on a single numerical method with only internal resolution checks, and the 'smoothness' at the critical point is asserted from sparse data. These issues, rather than the physics modeling per se, are the main obstacles to full confidence.","major_comments":[{"comment":"The central branch-separation, slope, and discontinuity claims rest exclusively on the Chebyshev pseudospectral implementation described in §3.1, with no independent method cross-check. The only validation reported is that N=120 agrees with N=150 to 1e-6 for the fundamental mode. Internal resolution convergence does not exclude spurious eigenvalues or systematic boundary errors, which are known risks for pseudospectral QNM codes, especially for large-|Im ω| modes. I request an independent validation: (i) reproduce the α=0 RN-AdS limit and compare with published results (e.g., Ref. [37]); (ii) compute at least a representative subset of the entries in Tables 1 and 2 with a second method (continued fraction or time-domain integration); and (iii) report the numerical uncertainty for the large-black-hole-branch modes. Without this, the claimed discontinuity and distinct slopes could be partly numerical in origin.","section":"§3.1, Tables 1–2, Figs. 2, 4"},{"comment":"The claim that QNM frequencies vary smoothly at the critical point is based on only a few sampled points, with no analysis of derivatives. For example, in Table 10(a) the points near the critical radius r_c ≈ 1.69 are T=0.0654 (r_h=1.623) and T=0.0655 (r_h=1.913), a gap of 0.29 in r_h; a non-analytic kink in ω(r_h) or dω/dT between these points would go unnoticed. To support the 'no sharp signature' conclusion, please compute a denser grid around r_c and explicitly test continuity of dω/dr_h (or dω/dT) across the critical point, or alternatively soften the claim to state that no discontinuous jump was found rather than that the frequencies vary smoothly.","section":"§4, Tables 10–11, Figs. 7–8"}],"minor_comments":[{"comment":"The phrase 'dynamical signature' may overstate the results; the QNMs are computed for a test scalar field on static equilibrium backgrounds, not through a dynamical transition. Please clarify that the observed discontinuity is a spectral branch-structure feature.","section":"Abstract and §6"},{"comment":"The convergence test statement could be strengthened by explicitly stating the comparison for the worst-converging mode (e.g., the large-|Im ω| modes) rather than only the fundamental mode.","section":"§3.1"},{"comment":"The number of decimal places is inconsistent across tables (e.g., 6 decimals in Table 3 vs 5 in Table 1). Please standardize.","section":"Tables 1–6"},{"comment":"It would be helpful to show the α→0 limit explicitly for these expressions (the text states it but does not display the reduction).","section":"Eqs. (2.10)–(2.12)"},{"comment":"The two branches for Re ω appear to overlap over a wide range of T/T_c; the legend and line styles should be made more distinguishable, and the text should quantify the 'separation' rather than relying on visual impression.","section":"Figure 11"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is suitable for the journal's readership. The main weakness is the absence of an independent numerical cross-check for the QNM spectra; however, this is fixable in revision and does not invalidate the thermodynamic derivation. I also note that the paper does not compare its α=0 limit with existing RN-AdS QNM results, which would be a low-cost validation step. Please weigh these points when considering the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, incremental numerical study of scalar QNMs across the small/large black hole transition in charged AdS with perfect fluid dark matter. The headline result, separated QNM branches below the critical point, a discontinuous jump at the coexistence crossing, and smooth behavior at the critical point, matches the pattern known from RN-AdS and its variants. The genuinely new bits are the quantitative trajectories for alpha = 0.1, 0.3, 0.5, and, more interesting, the isothermal decomposition into horizon-radius and pressure effects. Tables 3-6 isolate the two variables and Tables 7-9 show a first-order Taylor split; that is a clean piece of analysis and worth keeping.\n\nThe thermodynamics in Section 2 checks out. The critical quantities in Eqs. (2.10)-(2.12) reduce correctly to RN-AdS when alpha→0, and the tabulated transition temperatures and pressures are consistent with the coexistence curves. The QNM tables are mutually consistent and the pseudospectral implementation is standard, with a reported N=120 vs N=150 agreement at 10^-6. That is a reasonable resolution check.\n\nSoft spots, in proportion. First, the novelty claim is under-hedged. Ref [47], Abbas and Ali, is cited in the reference list but never discussed in the text; by its title it already treats QNMs and phase transitions for the same charged AdS PFDM black hole. The authors should state explicitly what their work adds beyond [47], or the claim that this 'has not been systematically examined' will not survive. This is my main concern and it is fixable.\n\nSecond, all quantitative claims rest on one Chebyshev pseudospectral implementation. No independent method, no code, no error budget. The stress-test note that a systematic offset in the large-|Im w| modes could affect the slopes and the branch separation is a real possibility, though not proven. For a paper whose main message is quantitative (distinct slopes, specific QNM frequencies), an independent cross-check on a handful of representative points, say a continued fraction or time-domain integration, would substantially raise confidence. I would not call it fatal, but it is a clear gap.\n\nThird, the 'dynamical signature' phrasing overstates things a little. These are QNMs of static equilibrium backgrounds, not a time-dependent transition. The authors do acknowledge the step to gravitational perturbations and real observations is open, but the correspondence between the plotted branch jump and a physical ringdown crossing is asserted, not derived. That caveat is worth making prominent.\n\nMinor: tables are long and repetitive; a few figures plus a supplementary dataset would do.\n\nOverall: this paper is for people working on black hole chemistry, PFDM backgrounds, and QNM probes of phase structure. It is a competent extension with a useful new decomposition, but it needs a clear comparison with ref [47] and an independent QNM check before I would trust the quantitative details. Send it to a serious referee; with those two additions it would be a publishable contribution.","headline":"A competent, incremental QNM extension to PFDM-dressed charged AdS black holes with a genuinely useful isothermal decomposition, but the novelty claim needs to be squared with ref [47] and the quantitative QNM results would benefit from an independent method check.","tokens_in":23569,"tokens_out":2719,"would_cite":false,"duration_ms":23100,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Black hole ringdown jumps at the small-large phase transition","keywords":["quasinormal modes","perfect fluid dark matter","charged AdS black hole","thermodynamic phase transition","extended phase space","Van der Waals-like phase transition","Chebyshev pseudospectral method","small/large black hole transition"],"falsifier":"Compute the time-domain ringdown of the same charged-AdS-PFDM background through the first-order transition, allowing the background to evolve or including the gravitational perturbation sector, and check whether the emitted frequency actually jumps between the small- and large-branch quasinormal frequencies at the coexistence crossing; if the time-domain signal interpolates smoothly or selects a different frequency, the claimed branch-jump signature fails. Alternatively, compute higher scalar overtones at $P=P_c$, since the paper predicts no sharp critical signature for the fundamental mode and a sharp feature there would refute the universality of that statement.","tokens_in":2121,"feed_emoji":"🕳️","tokens_out":4170,"duration_ms":86696,"temperature":0.7,"pith_summary":"This paper asks whether a black hole's characteristic ringdown spectrum knows about its thermodynamic phase transitions, and whether an ambient perfect-fluid dark matter environment shifts or destroys that relationship. The authors argue that for a charged anti-de Sitter black hole, the fundamental massless-scalar quasinormal frequencies trace two separated curves below the critical point: one for small black holes and one for large black holes, with the frequency jumping discontinuously when the thermodynamically preferred branch changes. At the critical point the jump disappears and the frequencies vary smoothly, so the fundamental scalar mode carries no sharp second-order signature. The paper presents quasinormal modes as a dynamical complement to the Gibbs free-energy analysis: larger dark-matter intensity raises the critical temperature and pressure, lowers the critical horizon radius, and shifts the quasinormal frequencies to larger values, while the qualitative first-order signature persists.","feed_headline":"Black hole ringdown jumps at the small-large phase transition","feed_subtitle":"Small and large black-hole modes split below the critical point but merge at it, even with dark matter.","key_machinery":"The machinery is the scalar-field effective potential on the PFDM background, $V(r)=f(r)[l(l+1)/r^2+f'(r)/r]$, built from the metric function $f(r)=1-2M/r+Q^2/r^2+8\\pi P r^2/3+(\\alpha/r)\\log(r/|\\alpha|)$, together with a Chebyshev pseudospectral solution of the resulting radial eigenvalue problem for the complex frequency $\\omega$. The same potential drives the qualitative behavior: at fixed pressure, varying $r_h$ reshapes the potential and changes the damping; at fixed $r_h$, varying $P$ (equivalently the AdS radius $L$) rescales the potential and shifts both real and imaginary parts. A first-order Taylor expansion $\\omega(r_h+\\delta r_h,P+\\delta P)$ then separates the two contributions and identifies which thermodynamic variable controls each branch. The thermodynamic side is carried by the equation of state $P=T/(2r_h)-1/(8\\pi r_h^2)+Q^2/(8\\pi r_h^4)-\\alpha/(8\\pi r_h^3)$, whose critical-point conditions determine $r_c$, $T_c$, and $P_c$, together with the Gibbs free energy swallowtail that marks the coexistence curve.","core_discovery":"The central claim is that the fundamental scalar quasinormal-mode spectrum of the charged AdS black hole is sensitive to the first-order small/large black hole phase transition, and that this sensitivity survives in a PFDM background. Concretely, for fixed pressure or fixed temperature below the critical point, the small- and large-black-hole branches occupy clearly separated curves in the complex-frequency plane with different slopes; crossing the coexistence curve changes the preferred branch, so the quasinormal frequency jumps discontinuously. Along an isotherm the evolution is not controlled by the horizon radius alone: a first-order expansion in $\\delta r_h$ and $\\delta P$ shows the small branch is pressure-dominated, while the large branch involves near cancellation in the real part and a horizon-radius-driven increase in damping. At the critical point the same frequencies vary smoothly and monotonically, showing no sharp second-order signal, and along the coexistence curve the two branches' frequencies approach each other as the swallowtail shrinks, merging at criticality. The paper presents this correspondence as the dynamical signature of the phase transition.","pith_inferences":["Editorial inference: if this branch-jump behavior survives for gravitational axial and polar perturbations, quasinormal spectra could become a gravitational-wave-relevant marker of the phase transition, but the paper's test-field scalar computation alone does not establish that.","Editorial inference: the smooth behavior at the critical point suggests the fundamental scalar mode does not directly see the divergence of thermodynamic response functions; computing higher overtones or the pseudospectrum near $P=P_c$ might reveal a distinct imprint of critical slowing down.","Editorial inference: the isothermal branch analysis implies that in a realistic dark-matter halo, frequency shifts from the environment could be mistaken for horizon-radius shifts; disentangling the two requires measuring the pressure/AdS dependence separately, exactly as the paper does analytically.","Editorial inference: the PFDM model is phenomenological, and replacing the logarithmic correction with a particle-physics-based dark matter profile should change both the phase structure and the quasinormal scaling, offering a route to constrain dark matter models through ringdown observations."],"forward_implications":["Below the critical point, the fundamental scalar quasinormal frequency works as a dynamical marker of the small/large black hole transition: an abrupt jump in $\\omega$ accompanies the branch switch in both isobaric and isothermal processes.","Along isotherms, the small-black-hole branch is pressure-dominated while the large-black-hole branch is governed by competing $r_h$ and $P$ effects, so quasinormal measurements could in principle distinguish which thermodynamic quantity drives a perturbation process.","Larger PFDM intensity $\\alpha$ strengthens the first-order signature by enlarging the branch separation and slope difference in the complex-frequency plane, while leaving the smooth behavior at criticality intact.","Along the coexistence curve, the shrinking quasinormal gap between the two branches mirrors the shrinking Gibbs free-energy swallowtail, so the dynamical and thermodynamic descriptions converge together near the critical point.","Because the fundamental scalar mode shows no sharp critical-point signal, any second-order dynamical signature would have to come from other perturbation types or higher overtones, a question the paper leaves open."],"supporting_citations":[{"why":"Establishes the small/large charged AdS black hole phase transition that the paper seeks to detect dynamically.","marker":"[4]"},{"why":"Provides the extended phase-space P-V criticality framework and critical point equations used to locate the transition.","marker":"[5]"},{"why":"Supplies the AdS quasinormal-mode formalism with ingoing-at-horizon and Dirichlet-at-boundary conditions that the pseudospectral implementation follows.","marker":"[19]"},{"why":"Shows that RN-AdS scalar quasinormal frequencies change at the first-order transition, the result this paper extends to the PFDM background.","marker":"[37]"},{"why":"Introduces the coexistence-curve quasinormal-mode analysis that motivates the analogous study here.","marker":"[40]"},{"why":"Defines the perfect fluid dark matter model whose metric correction appears in the function f(r).","marker":"[42]"},{"why":"Provides the quintessential/PFDM solution used for the dark matter contribution to the spacetime metric.","marker":"[43]"},{"why":"Gives the charged AdS black hole in PFDM: the metric, first law, and thermodynamic relations used throughout the paper.","marker":"[44]"},{"why":"Supports the same PFDM phase structure through Lyapunov-exponent probes, used for comparison with the quasinormal-mode picture.","marker":"[50]"},{"why":"Computes quasinormal modes in a PFDM environment, providing the perturbation background for the present calculation.","marker":"[51]"}],"fun_headline_variants":["QNM spectrum marks black hole first-order transition","Small and large black hole ringdowns diverge then merge","Charged AdS quasinormal modes follow phase transition","Dark matter shifts phase boundary and ringdown signature","Ringdown branches split below critical point, merge at it"],"cache_read_input_tokens":25600,"weakest_assumption_plain":"The paper's load-bearing premise is that quasinormal frequencies computed on each equilibrium background separately, for a test scalar field with no backreaction and no dynamical path connecting the branches, can be read as a 'dynamical signature' of what a ringing black hole would actually do when it crosses the first-order transition; if the ringdown is governed by a non-equilibrium or coupled process, the branch discontinuity in the spectrum need not be what a real black hole exhibits.","fun_headline_variants_meta":{"raw":{"variants":["QNM spectrum marks black hole first-order transition","Small and large black hole ringdowns diverge then merge","Charged AdS quasinormal modes follow phase transition","Dark matter shifts phase boundary and ringdown signature","Ringdown branches split below critical point, merge at it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3294,"prompt_tokens":1008,"completion_tokens":2286,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":2207}},"tokens_in":624,"tokens_out":2286,"duration_ms":17283,"temperature":1.0,"reasoning_tokens":2207,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:11:00.783143+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the time-domain ringdown of the same charged-AdS-PFDM background through the first-order transition, allowing the background to evolve or including the gravitational perturbation sector, and check whether the emitted frequency actually jumps between the small- and large-branch quasinormal frequencies at the coexistence crossing; if the time-domain signal interpolates smoothly or selects a different frequency, the claimed branch-jump signature fails. Alternatively, compute higher scalar overtones at $P=P_c$, since the paper predicts no sharp critical signature for the fundamental mode and a sharp feature there would refute the universality of that statement.","supporting_citations":[{"cited_title":"Ali and X.-M","cited_arxiv_id":null,"evidence_quote":"Supports the same PFDM phase structure through Lyapunov-exponent probes, used for comparison with the quasinormal-mode picture."}],"review_version":2}