{"id":"0b8012df-5f7e-4c15-8bf4-2aa5f10ec157","arxiv_id":"2504.21460","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper computes scalar quasinormal modes of rotating Bardeen and Hayward black holes and finds they follow the eikonal null-geodesic relations, with Bardeen deviations up to about 19% from Kerr.","lead":"This paper calculates the vibration patterns, called quasinormal modes, of a scalar field around two spinning 'regular' black holes that have no inner singularity. The patterns stay within tens of percent of standard Kerr black holes, so ringdown measurements alone would struggle to tell the regular models from Kerr.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Bardeen counterrotating eikonal verification rests on a duplicated table: Tables III and IV are identical, so Table IX cannot support the claimed connection.","rationale":"The reader's verdict is CONDITIONAL, and my read does not move it. The most decisive flaw is the duplicated Bardeen Lyapunov tables, because Table IX's claimed one-thousandth-level verification for counterrotating Bardeen modes uses Table IV directly. Since the table values are identical to the corotating case, the counterrotating verification is unsupported unless a sign symmetry of Eq. (20) is demonstrated, which the paper does not do. This is a demonstrated internal inconsistency rather than a speculative numerical risk, and it touches one of the two model branches named in the abstract's central claim. The missing N-convergence study for the matrix method is also a real concern and should be addressed in revision, especially near the Bardeen parameter limit g* ~ 0.744, but it is a risk about possible inaccuracy; the identical tables are a concrete error. The Hayward results and the corotating Bardeen results may survive a correction, so rejection is not warranted, but the counterrotating Bardeen verification must be redone and the affected statements in Sections 5 and 6 re-evaluated.","tokens_in":17840,"tokens_out":6806,"duration_ms":74486,"concrete_test":"Recompute Table IV with the lower sign in Eq. (18), d = a - sqrt(r_c^4/(2 f_b(r_c) - r_c f_b'(r_c))), using the same (a,g*) grid and the same r_c solutions of Eqs. (15)-(16), then recompute the delta columns of Table IX with these corrected lambda values. If the new entries differ from Table III (as expected from the sign-dependent terms in Eq. (20)), the published counterrotating Bardeen verification is based on erroneous Lyapunov exponents and must be redone; if they coincide numerically, the duplication is physically explained and the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central claim is that the eikonal QNM/null-geodesic connection is explicitly verified for both rotating regular black holes. In Section 5 that verification is presented separately for corotating and counterrotating branches: Tables VI/VII for Hayward and Tables VIII/IX for Bardeen. The Bardeen counterrotating branch, Table IX, uses the Lyapunov exponents reported in Table IV. However, Table IV is numerically identical to the corotating Table III for every listed (a,g*). This cannot be a symmetry of Eq. (20): the two signs in Eq. (18) give d = a ± sqrt(r_c^4/(2 f(r_c) - r_c f'(r_c))), and Eq. (20) depends on d through d^2 and through the denominator term 2a(a-d) r_c f(r_c), so changing the sign changes lambda^2 unless an algebraic cancellation is proved; none is shown. The Hayward tables I and II display the expected corotating/counterrotating split, confirming that the Bardeen duplication is not generic behavior. Since Table IX is the only support for the counterrotating Bardeen part of the eikonal connection, that branch of the central claim is not established by the paper as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies massless scalar quasinormal modes (QNMs) of rotating Hayward and Bardeen regular black holes, computing the fundamental and first-overtone frequencies with a third-order WKB method and a matrix method. It also computes corotating and counterrotating Lyapunov exponents of equatorial null circular geodesics, checks the eikonal connection between QNM imaginary parts and Lyapunov exponents, and compares shadow radii obtained from n=1 QNM real parts with those from closed photon orbits. The central claims are that Hayward QNMs differ from Kerr at the percent level, Bardeen QNMs at the ten-percent level, and that the eikonal QNM/null-geodesic correspondence, including for first overtones, is explicitly verified for both families.","tokens_in":18022,"tokens_out":4576,"duration_ms":48740,"significance":"If the numerical results are correct, the matrix-method frequencies for these regular black holes are a useful new dataset, and the paper provides a concrete check of the eikonal QNM/geodesic correspondence beyond Kerr. The validation of the matrix method against Leaver's continued-fraction results for Kerr to roughly 1e-6 is a genuine strength, as is the explicit presentation of corotating and counterrotating Lyapunov exponents for both models. However, the duplicated Bardeen tables and the absence of a convergence study for the regular cases mean that part of the central verification, especially for counterrotating Bardeen modes, is not established as written.","major_comments":[{"comment":"Table IV, labeled \"Counterrotating Lyapunov exponents\" for the Bardeen black hole, is numerically identical to Table III, the corotating table, for every listed (a, g*). This cannot be a symmetry of Eq. (20). With d = a ± s where s = sqrt(r_c^4/(2 f(r_c) - r_c f'(r_c))), Eq. (20) contains d^2 = a^2 + s^2 ± 2 a s and the denominator term 2 a (a-d) r_c f(r_c) = ∓ 2 a s r_c f(r_c), so the two branches generally give different λ^2; no cancellation is proved. The Hayward Tables I and II show the expected corotating/counterrotating split, confirming that the Bardeen duplication is not generic behavior. Since Table IX, the only support for the counterrotating Bardeen eikonal check, is built directly on Table IV, that branch of the central claim is not supported by the paper as written. The authors must recompute Table IV and revise Table IX accordingly.","section":"Section 2, Tables III and IV"},{"comment":"No convergence study is presented for the matrix method in the regular black hole cases. Section 3.2.2 states that N = 20 grid points are used, and Table V validates the method only for the l = 1 fundamental Kerr mode against Leaver. The paper's quantitative claims, however, include n = 1 overtones, l up to 12, and Bardeen parameters close to the extremal value g* = 0.744 where the n = 1 real part reverses its monotonic trend. A shift of the numerical roots with N could change the percent-level comparisons in Section 5. The authors should add a convergence check (for example, N = 20, 25, 30 for representative Hayward and Bardeen cases) to establish that the reported frequencies are converged.","section":"Section 3.2.2 and Section 4"},{"comment":"The conclusion states that \"the massless scalar QNMs of the two rotating regular black holes have at most percent-level increase in comparison with the Kerr black hole case,\" but Section 4.2 reports that the Bardeen real part with g* = 0.7 increases by about 13% and the imaginary part by about 19%, and the abstract itself quotes a \"ten-percent-level increase\" for Bardeen. This inconsistency must be corrected; as written, the conclusion misstates the main quantitative result for the Bardeen model.","section":"Section 6 versus Section 4.2"},{"comment":"The verification of the real-part/shadow-radius connection for n = 1 QNMs is presented only as a visual comparison in Fig. 11, with no numerical errors or deviations reported. Given that the abstract claims the connection is \"explicitly verified,\" a quantitative measure — for example, the maximum or average relative difference between the two methods for each m value — should be provided, rather than relying on visual agreement.","section":"Section 5, Figure 11"}],"minor_comments":[{"comment":"There are typographical errors: \"Layapunov\" in the Introduction should be \"Lyapunov,\" and \"matirx\" in Section 3.2.2 should be \"matrix.\"","section":"Introduction and Section 3.2"},{"comment":"Equation (23) appears garbled: the denominator contains \"csc^2 σ\" where csc^2 θ is presumably intended, and the notation is hard to follow. Please rewrite this equation and the surrounding definitions clearly.","section":"Equation (23)"},{"comment":"Reference [69] duplicates reference [58] (the same Iyer and Will paper appears twice with different formatting). Please consolidate the bibliography.","section":"References"},{"comment":"The caption of Fig. 11 has grammatical issues (\"are show in lines\" / \"are show in dots\") and should be rewritten.","section":"Caption of Figure 11"}],"recommendation":"major_revision","confidential_remarks":"The duplicated Tables III and IV strike me as a copy-paste error rather than a conceptual failure, but the effect is load-bearing: Table IX, and therefore the counterrotating Bardeen part of the eikonal verification, must be redone with correct counterrotating Lyapunov exponents. I would also insist on a convergence study for the matrix method in the regular cases before acceptance, since the paper's central dataset depends on it. The paper fits the journal's scope, but these issues require a substantive revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this paper supplies a genuinely useful numerical dataset: fundamental and n=1 scalar QNM frequencies for rotating Hayward and Bardeen black holes, with corotating and counterrotating Lyapunov exponents and a shadow-radius check for the n=1 modes. Second, Tables III and IV, which are supposed to list the Bardeen corotating and counterrotating Lyapunov exponents, are numerically identical. That is not a symmetry of their Eq. (20); the Hayward Tables I and II show the expected split, so the Bardeen duplication is almost certainly a copy-paste error. Table IX uses the counterrotating values from Table IV, so the Bardeen counterrotating branch of their central eikonal verification is not supported as written.\n\nWhat the paper does well: the matrix method is benchmarked against Leaver's continued fraction results for Kerr to about one part in a million, which is real evidence that the QNM frequencies are reliable. The Hayward corotating and counterrotating checks pass at the one-thousandth level, and the Bardeen corotating check passes except near the highest g* values, where the deviation grows to a few tenths of a percent. The n=1 overtone results and the shadow-radius check using n=1 QNMs are new, even if the framework is a routine extension of ref. [57]. The plots and tables are clear, and the authors are honest about the WKB method's limitations.\n\nSoft spots, in proportion: the duplicated Tables III/IV is the load-bearing flaw, but it is local. The counterrotating Bardeen verification in Table IX should be redone with correct numbers. Also, the abstract and Section 4.2 state that Bardeen QNMs show ten-percent-level increase over Kerr, while the conclusion says only \"percent-level increase.\" That is an internal contradiction that needs to be fixed. A smaller concern: the matrix method uses N=20 with no convergence study for the regular cases. Given the Kerr benchmark and the eikonal agreement where the tables are not duplicated, I would call that minor, but a referee should ask for a check at one or two parameter points.\n\nWho is this for: anyone computing scalar QNMs in rotating regular spacetimes or testing the eikonal QNM-geodesic correspondence. It does not overturn anything, but it fills in numbers that were missing. A serious referee can confirm the fix in a few hours. I would send it to peer review and require the authors to correct the Bardeen tables, rerun Table IX, and align the conclusion with the rest of the paper.","headline":"Useful numerical QNM dataset with one glaring table duplication that breaks the Bardeen counterrotating check; fixable, not fatal.","tokens_in":18608,"tokens_out":1486,"would_cite":true,"duration_ms":17508,"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":"The massless scalar quasinormal modes of rotating Hayward and Bardeen black holes follow the eikonal null-geodesic correspondence, with Hayward modes 1–4 percent above Kerr and Bardeen modes up to 13–19 percent above Kerr.","keywords":["quasinormal modes","rotating regular black holes","Hayward black hole","Bardeen black hole","Lyapunov exponents","null geodesics","black hole shadow","eikonal approximation"],"falsifier":"Compute the Bardeen n=1 modes at a=0.2, g*=0.7 and g*=0.744 with N=40 and N=80 grid points, or with an independent continued-fraction or time-domain code, and recompute the counterrotating Bardeen Lyapunov exponents from the lower sign in Eq. (18), since the printed corotating and counterrotating tables are identical. If the real part no longer reverses or the counterrotating exponents differ from the printed values, the claimed overtone behavior and the counterrotating eikonal verification collapse.","tokens_in":17591,"feed_emoji":"🕳️","tokens_out":6892,"duration_ms":69595,"temperature":0.7,"pith_summary":"This paper argues that the massless scalar quasinormal modes of rotating Hayward and Bardeen regular black holes remain close to Kerr's, with the deviation parameter g moving Hayward frequencies and decay rates by roughly one to four percent and g* moving Bardeen's by up to about thirteen to nineteen percent. It further claims that the usual eikonal correspondence—imaginary parts set by the Lyapunov exponent of unstable equatorial null geodesics, real parts tied to shadow radius—holds for the fundamental and first-overtone modes of both families. If correct, the paper supplies a concrete dataset of scalar ringing frequencies for two popular singularity-free black hole models and sharpens which observable deviations could distinguish them from Kerr.","feed_headline":"Bardeen black holes ring up to 19% above Kerr","feed_subtitle":"Hayward modes stay within 5%, and both follow the null-geodesic correspondence.","key_machinery":"The central objects are the rotating Hayward and Bardeen metrics, obtained from the Kerr-like metric ansatz with f(r) replaced by f_h(r)=M $r^{4}$/($r^{3}$+$g^{3}$) and f_b(r)=M $r^{4}$/($r^{2}$+g_*^2)^{3/2}. The argument runs on two tools: the null-geodesic effective potential V_r, whose second derivative at the unstable circular orbit yields the Lyapunov exponent λ via Eq. (20), and the eikonal QNM correspondence Im(ω)=λ(n+1/2) together with the shadow-radius relation Re(ω)≈(l+1/2)/R_s. The frequencies are computed with a third-order WKB expansion and an N=20 grid matrix method, with the matrix method validated against Kerr and used for the main tables.","core_discovery":"For massless scalar perturbations, the fundamental (n=0) and first-overtone (n=1) quasinormal frequencies of rotating Hayward black holes increase by only percent-level amounts as g grows (at g=0.7, real part +1.9%, imaginary +4.1% for l=1, m=1; for n=1, +2.8% and +3.7%), while rotating Bardeen black holes show ten-percent-level shifts (at g*=0.7, fundamental real part +13%, imaginary +19%; n=1 real +16%, imaginary +19%). Bardeen n=1 real parts are non-monotonic in g*, rising to a maximum near g*=0.744 and then decreasing slightly, whereas imaginary parts keep increasing. Tables of corotating and counterrotating equatorial Lyapunov exponents satisfy Im(omega)=lambda(n+1/2) at the one-thousandth level for m=±l eikonal modes, and shadow radii reconstructed from n=1 real parts agree with closed photon orbit calculations.","pith_inferences":["If the percent-level closeness to Kerr survives for gravitational perturbations—which the paper does not compute—regular rotating black holes would be nearly degenerate with Kerr in ringdown observables, making the high-g* Bardeen case the most promising target for distinguishing regularity.","A time-domain evolution code, not used in the paper, could independently check the matrix-method frequencies near the Bardeen turnaround; if it confirms the non-monotonicity, the reversal becomes a sharper probe of g* than the absolute shift.","The eikonal correspondence verified here suggests that for lower multipoles, deviations from the geodesic prediction encode finite-l corrections and could be fitted to constrain g and g* simultaneously with spin.","The identical printed corotating and counterrotating Lyapunov tables for Bardeen look like an error; recalculating the counterrotating branch from the lower sign in Eq. (18) would either confirm an unexpected symmetry or remove the counterrotating verification in Table IX."],"forward_implications":["Scalar ringdown observations of a rotating Hayward black hole would look almost Kerr-like, with deviations below five percent in frequency and decay rate for allowed g.","For rotating Bardeen black holes, deviations grow to roughly 13–19 percent at the upper end of g*, making high-g* Bardeen more distinguishable from Kerr by scalar QNM measurements.","The first overtone obeys the eikonal geodesic correspondence just as the fundamental mode does, so overtone ratios can be used to test the photon-ring interpretation.","The non-monotonic dependence of the Bardeen n=1 real part on g* offers a potential signature of the regularizing charge, if the trend is confirmed.","Shadow radii inferred from n=1 QNM real parts match closed photon orbits, extending the shadow-ring connection to overtones."],"supporting_citations":[{"why":"Supplies the continued-fraction reference values used as the Kerr benchmark that validates the matrix and WKB codes in Table V.","marker":"[70]"},{"why":"Supplies the Lyapunov-exponent formula for equatorial null circular geodesics and the eikonal QNM/geodesic correspondence.","marker":"[43]"},{"why":"Provides the shadow-radius-from-QNM formula and the closed-photon-orbit quantization used to test the real-part connection.","marker":"[57]"},{"why":"Provides the rotating Hayward and Bardeen metrics that the whole calculation is built on.","marker":"[47]"},{"why":"Introduces the matrix method for Schwarzschild quasinormal modes, extended here to rotating regular black holes.","marker":"[62]"},{"why":"Extends the matrix method to Kerr black holes, the basis for the rotating regular calculations.","marker":"[63]"},{"why":"Gives the small-aω expansion of the angular separation constant used in the WKB computation.","marker":"[68]"},{"why":"Supplies the third-order WKB formula used as the second numerical method.","marker":"[58]"}],"fun_headline_variants":["Bardeen rings 19% louder than Kerr, Hayward stays quiet","Rotating regular holes: Bardeen deviates 19%, Hayward 4%","QNMs of regular rotators: Hayward near Kerr, Bardeen up to 19%","Null geodesics match ringing of Bardeen and Hayward black holes","Bardeen black hole rings 19% off Kerr, Hayward less than 5%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The matrix method with N=20 grid points is assumed to give converged quasinormal frequencies for all reported parameter values, including Bardeen near g*=0.744 where the n=1 real part reverses its monotonic trend; the paper benchmarks the method against Kerr but gives no convergence study for the regular cases.","fun_headline_variants_meta":{"raw":{"variants":["Bardeen rings 19% louder than Kerr, Hayward stays quiet","Rotating regular holes: Bardeen deviates 19%, Hayward 4%","QNMs of regular rotators: Hayward near Kerr, Bardeen up to 19%","Null geodesics match ringing of Bardeen and Hayward black holes","Bardeen black hole rings 19% off Kerr, Hayward less than 5%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1544,"prompt_tokens":987,"completion_tokens":557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":457}},"tokens_in":603,"tokens_out":557,"duration_ms":5078,"temperature":1.0,"reasoning_tokens":457,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:03:13.301405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Bardeen n=1 modes at a=0.2, g*=0.7 and g*=0.744 with N=40 and N=80 grid points, or with an independent continued-fraction or time-domain code, and recompute the counterrotating Bardeen Lyapunov exponents from the lower sign in Eq. (18), since the printed corotating and counterrotating tables are identical. If the real part no longer reverses or the counterrotating exponents differ from the printed values, the claimed overtone behavior and the counterrotating eikonal verification collapse.","supporting_citations":[{"cited_title":"Geodesic stability and quasinormal modes of non-commutative Schwarzschild black hole employing Lyapunov exponent","cited_arxiv_id":"2204.09006","evidence_quote":"Introduces the matrix method for Schwarzschild quasinormal modes, extended here to rotating regular black holes."},{"cited_title":"Stability Analysis of Geodesics and Quasinormal Modes of a Dual Stringy Black Hole Via Lyapunov Exponents","cited_arxiv_id":"2108.05772","evidence_quote":"Extends the matrix method to Kerr black holes, the basis for the rotating regular calculations."}],"review_version":1}