{"id":"f4c8f94a-8b53-43d8-985f-65d907e2a2e1","arxiv_id":"2412.17073","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The first fully non-perturbative quasinormal mode spectrum (polar, axial, scalar) for rapidly rotating Einstein-Gauss-Bonnet-dilaton black holes is computed and shown to deviate from Kerr, with mode ordering depending on coupling and spin.","lead":"This paper computes the vibration frequencies, the quasinormal modes, of rapidly spinning black holes in a modified theory of gravity with a dilaton field. These modes control the gravitational wave 'ringdown' signal, so the results give a concrete prediction that future detectors could use to test Einstein's theory.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scalar-led mode ordering at strong coupling is decided in the region where the authors concede numerical accuracy is lost, and no EGBd-specific convergence test is provided.","rationale":"The reader's conditional verdict is driven by the absence of a convergence or error study for the EGBd eigenvalue problem, and my reading identifies the same load-bearing weakness: the paper's most striking physical conclusion, that the dominant mode can switch to the scalar sector and even to an octupole mode, is delivered by data points at the boundary of the domain of existence, where the authors themselves report reduced accuracy. The j=0.8 positive-frequency case is the clearest instance: the scalar l=3 mode becomes the longest lived only at the last tabulated coupling, after a sharp drop in its imaginary part, and the margin over the polar l=2 mode is comparable to the kind of error one expects from an unconverged spectral truncation near a critical boundary. The paper does validate the pipeline on Kerr and against second-order perturbative results, but that validation is limited: the perturbative comparison in Fig. 5 covers only polar-led modes, while the scalar-led modes--which drive the headline claim--are checked only in the xi=0 Kerr limit. The absence of scalar-mode validation in the strong-coupling regime compounds the convergence concern rather than resolving it. A targeted resolution study at the disputed points, ideally supplemented by an independent time-domain computation for one scalar-dominated case, would decide whether the ordering reversal is physical. Until then, the conditional verdict is appropriate: the general phenomenon of scalar modes becoming competitive near the boundary is plausible and partially supported at j=0.2 and 0.4, but the full mode-order phenomenology, especially octupole dominance at j=0.8, is not yet established. My concern therefore does not change the reader's verdict.","tokens_in":25842,"tokens_out":5570,"duration_ms":52329,"concrete_test":"Recompute the j=0.8, Mz=2, positive-frequency fundamental modes at xi=0.10926 and 0.11221 using at least double the spectral resolution in both x and y (e.g., Nx=Ny=30 or 40) and with the same extended-precision eigenvalue solver; repeat at j=0.2, xi=0.17185 for the l=2 scalar and polar modes. If the scalar l=3 omega_I at j=0.8 shifts by more than the roughly 0.006 gap to polar l=2, or if the least-damped mode label changes between resolutions, the claimed ordering reversal is not established. A complementary check is to compute the j=0.2 scalar l=2 mode with a time-domain code, since that case is outside the accuracy disclaimer.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim--that the dominant ringdown mode switches to the scalar-led sector with increasing GB coupling--rests on data taken at the edge of the domain of existence. Section 4 states that for j=0.6 and 0.8 'our calculations lose accuracy towards the boundary of the domain of existence.' Yet the most dramatic ordering reversal, the scalar l=3 mode becoming the least damped positive-frequency mode at j=0.8, occurs at the final tabulated point xi=0.11221 (Tables 13 and 16): its M*omega_I jumps from -0.06379 at xi=0.10926 to -0.05660, overtaking polar l=2 (M*omega_I=-0.06271) by only about 0.006. This is precisely the regime disclaimed. The independent checks are the Kerr limit (xi=0) and a perturbative comparison (Fig. 5) that covers polar-led modes only; scalar-led modes in the strong-coupling regime are validated neither against the static limit nor against an independent solver. A spectral method can produce spurious eigenvalues near a boundary of existence, and without a resolution study for the EGBd system itself these last-point crossings cannot be distinguished from numerical artifacts. If the j=0.8 scalar-octupole dominance is dropped, the qualitative statement that scalar modes dominate near the boundary still has support at j=0.2 and 0.4, but the stronger conclusion about octupole dominance and the full mode-order phenomenology is not secure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a numerical spectral method for computing the quasinormal mode (QNM) spectrum of rapidly rotating Einstein-Gauss-Bonnet-dilaton (EGBd) black holes without perturbative expansions in either the Gauss-Bonnet coupling or the angular momentum. The authors first obtain stationary, axially symmetric background solutions with the FIDISOL/CADSOL package, then linearize the metric and dilaton equations using a seven-function perturbation ansatz (polar, axial, and scalar sectors), apply ingoing/outgoing boundary conditions at the horizon and infinity, and solve the resulting quadratic eigenvalue problem with a Chebyshev-Legendre spectral decomposition. They report fundamental l=2- and l=3-led modes with azimuthal number Mz=2 for angular momenta j=0.2, 0.4, 0.6, 0.8 and both signs of the real frequency. The results are validated against the Kerr limit and against second-order perturbative results for small coupling and rotation, and they exhibit isospectrality breaking and a reordering of modes with increasing coupling, including a claimed dominance of scalar-led modes near the boundary of the domain of existence.","tokens_in":26156,"tokens_out":4832,"duration_ms":43925,"significance":"If the results are correct, this is a significant advance: it provides the first fully non-perturbative QNM spectrum for rapidly rotating black holes in a well-motivated modified gravity theory, with direct implications for ringdown-based tests of gravity. The paper's strengths include a detailed description of the background construction and perturbation method, extensive numerical tables, and genuine external checks against Kerr and second-order perturbative results in the weak-coupling regime. However, the central phenomenological claim that the dominant ringdown mode changes to the scalar-led sector—and in particular that the l=3 scalar mode can dominate at high angular momentum—rests on data in a region where the authors themselves state that accuracy is degraded, and no EGBd-specific convergence or error study is provided. As presented, the qualitative scalar-dominance trend is plausible, but the stronger quantitative claims are not yet fully established.","major_comments":[{"comment":"The claimed octupole dominance at j=0.8 is decided by the last tabulated point xi=0.11221: the scalar l=3 mode's M*omega_I changes from -0.06379 at xi=0.10926 to -0.05660, overtaking the polar l=2 mode (M*omega_I=-0.06271) by only about 0.006. This is precisely the regime covered by the sentence in Section 4: 'for j=0.6 and 0.8 our calculations lose accuracy towards the boundary of the domain of existence.' Since a spectral method can produce spurious eigenvalues near a boundary, the crossing at the final tabulated point cannot be distinguished from a numerical artifact without a convergence or resolution study. The conclusion in Section 5 that 'the octupole modes may also dominate over the quadrupole modes for large coupling' is therefore not secure at j=0.8.","section":"Section 4, Tables 13 and 16"},{"comment":"No convergence tests or error estimates are reported for the EGBd eigenvalue problem itself. The Kerr limit (xi=0) checks the method only in the decoupled case, and the perturbative comparison in Fig. 5 explicitly covers polar-led modes only (the text notes 'the axial modes were not given in there'). The scalar-led and axial-led strong-coupling modes are therefore validated neither against an independent solver nor against the static EGBd limit. To support the central mode-ordering claim, the paper should provide, at minimum for the j=0.6 and j=0.8 cases, a convergence study in Nx and Ny for the fundamental polar, axial, and scalar modes, and report the resulting errors on M*omega_R and M*omega_I, or state explicitly which tabulated points meet a chosen accuracy threshold.","section":"Section 3.4, Eqs. (93)-(102)"},{"comment":"The scalar-led modes in the strong-coupling regime are not checked against the static (j=0) EGBd QNMs of Refs. [38,41] or any other independent non-Kerr limit. Such a comparison would directly test the scalar sector and the axial/polar splitting at finite coupling, where no perturbative anchor exists. If this check cannot be performed, the claim that 'scalar modes tend to dominate' toward the boundary should be restricted to the j=0.2 and 0.4 data, where the trend is visible before the accuracy caveat becomes severe, and the stronger j=0.8 octupole-ordering statement should be deferred until the numerical accuracy is demonstrated.","section":"Section 4 and Fig. 4"}],"minor_comments":[{"comment":"The classification of modes as polar-led, axial-led, or scalar-led is based on visual inspection of the parity of Re(T~), Re(h0~), and Re(Phi~) at a single point (j=0.6, xi=0.11). Because rotation and coupling mix multipoles, and the mixing increases with xi, a quantitative criterion (for instance, the relative magnitude of the dominant spectral coefficients) would make the mode labels reproducible and less subjective.","section":"Section 3.1 and Fig. 3"},{"comment":"The text states that for j=0.6 and 0.8 the calculations lose accuracy towards the boundary of the domain of existence and that the authors 'refrained from showing their values there,' yet Tables 9-16 do list values at the largest couplings. Please clarify which entries are considered unreliable and mark them in the tables (for example, with parentheses or italics).","section":"Section 4 and Appendix"},{"comment":"The dotted vertical lines are said to represent the maximal value of the scaled coupling constant for which background solutions exist, but the caption does not explain how those values are obtained; a reference to Fig. 1b or to the boundary curves would help the reader.","section":"Fig. 4 caption"},{"comment":"The quadratic eigenvalue problem is solved 'with the same numerical methods described previously [49],' but the present paper does not specify the linearization procedure, matrix sizes, or tolerance settings used for the EGBd runs; adding one sentence with these details would improve reproducibility.","section":"Section 3.4, Eq. (102)"},{"comment":"The tables report four or five significant digits without error bars; if convergence information is obtained, including error estimates in the tables would make the numerical content more useful to future comparisons.","section":"Appendix tables"}],"recommendation":"major_revision","confidential_remarks":"The paper is a methods paper with a strong phenomenological conclusion. The main obstacle is not the method itself but the absence of a convergence study for the EGBd system in the regime where the most striking ordering change is claimed. If the authors can provide resolution tests and mark or remove the unreliable boundary-region points, the central qualitative conclusion about scalar dominance may survive, but the quantitative octupole-dominance statement at j=0.8 needs direct support. I would not reject the paper on the present evidence, but I cannot recommend acceptance without the convergence data."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the paper in one breath: it numerically solves the coupled metric-scalar perturbation equations on fully non-perturbative rotating EGBd backgrounds and produces l=2,3 fundamental QNM frequencies for Mz=2, positive and negative real parts. That genuinely goes beyond earlier static or perturbative work, and the spectral pipeline was already validated on Kerr. The agreement with the Kerr limit and with the second-order perturbative results at small spin/coupling is real and gives the method credibility. The physical conclusion—that the dominant ringdown mode can switch to the scalar-led sector as the GB coupling grows—is plausible and, importantly, supported at j=0.2 and 0.4, which are well inside the trustworthy region.\n\nThe main soft spot is the one the authors flag themselves: for j=0.6 and 0.8 they state accuracy is lost toward the boundary of the domain of existence. The most dramatic claim, scalar octupole dominance at j=0.8, is decided at the final tabulated point xi=0.11221, where M omega_I jumps from -0.06379 to -0.05660 and overtakes the polar l=2 value by about 0.006. That is precisely in the disclaimed regime. Without a convergence study in Nx, Ny for the EGBd system itself, a spectral artifact near the boundary cannot be excluded. The validation against Kerr doesn't cover this; neither does the perturbative comparison, which shows polar-led modes only. I'd want the authors to release the spectral resolution data or run an independent check before banking on the octupole ordering.\n\nOther minor points: the stated 10^-5 accuracy is for background functions, not eigenvalues; no code or parameter files are provided; and the tables report five-to-six digits without error bars. Still, the paper is transparent about these issues, and the qualitative scalar-dominance phenomenon does not collapse even if the j=0.8 octupole crossing is discarded.\n\nThis is a useful paper for anyone working on ringdown predictions in Einstein-scalar-Gauss-Bonnet theories and on spectral methods for coupled perturbation systems. It deserves a serious referee, not a desk rejection. The referee should ask for convergence tests at fixed j for the EGBd system, a statement of which table entries are considered reliable at j=0.6 and 0.8, and preferably a released dataset or code. With that, the central claim is very likely to survive.","headline":"First non-perturbative rotating EGBd QNM spectrum, with a credible method and honest caveats, but the headline mode-ordering reversal at j=0.8 sits exactly where the authors concede accuracy loss.","tokens_in":26639,"tokens_out":3603,"would_cite":true,"duration_ms":33323,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C35","83D05"],"pacs":["04.70.-s","04.30.-w","04.50.Kd"],"model":"deepseek-v4-flash","headline":"This paper computes the quasinormal-mode spectrum of rapidly rotating Einstein-Gauss-Bonnet-dilaton black holes non-perturbatively and finds that the dominant ringdown mode can change with the coupling and spin.","keywords":["quasinormal modes","Einstein-Gauss-Bonnet-dilaton theory","rotating black holes","ringdown","spectral methods","isospectrality breaking","Gauss-Bonnet coupling","scalar-led modes"],"falsifier":"Repeat the eigenvalue calculation for the fundamental l=2 scalar-led mode at j=0.2 and ξ near the upper end of the domain (around 0.17) with doubled spectral resolution (larger Nx and Ny) and with an independent time-domain evolution of the linearized EGBd equations; if the sharp drop in MωI that makes the scalar mode longest-lived near the domain boundary shifts or disappears as the resolution changes, the claimed mode-order change is a numerical artifact rather than a property of the theory.","tokens_in":25643,"feed_emoji":"🌌","tokens_out":9112,"duration_ms":75291,"temperature":0.7,"pith_summary":"This paper reports a fully non-perturbative computation of the quasinormal-mode spectrum of rapidly rotating black holes in Einstein-Gauss-Bonnet-dilaton (EGBd) theory, a modified gravity in which a scalar dilaton couples to the Gauss-Bonnet invariant. The authors first construct numerically exact rotating EGBd black hole backgrounds, then linearize the coupled metric-dilaton equations and solve them with a spectral method. The central result is that the Kerr degeneracy between polar-led and axial-led modes is broken, and that as the Gauss-Bonnet coupling grows the ordering of the fundamental l=2 and l=3 modes changes: near the boundary of the region where black hole solutions exist, scalar-led modes can become the longest-lived. The dominant ringdown mode therefore depends on the coupling of the theory and on the spin of the final black hole, which matters for using gravitational-wave ringdown observations to constrain such theories.","feed_headline":"Ringdown's longest-lived mode can switch with Gauss-Bonnet coupling","feed_subtitle":"Non-perturbative spectra show scalar-led modes overtaking metric modes near the domain's edge, reshuffling ringdown.","key_machinery":"The load-bearing machinery is the spectral reduction of the coupled perturbation equations. The seven perturbation functions — the metric functions H1, T, N, L, the axial functions h0, h1, and the dilaton perturbation Φ1 — are expanded in Chebyshev polynomials in the compactified radial coordinate x and in associated Legendre functions of y=cosθ, with the outgoing and ingoing boundary behavior factored out. Evaluating the PDEs and boundary conditions on a Gauss-Lobatto and uniform grid converts the system into a quadratic eigenvalue problem (M0 + M1 ω + M2 ω²) C = 0, whose eigenvalues are the complex quasinormal frequencies. The same scheme is validated against Kerr and against second-order-in-spin, sixth-order-in-coupling perturbative results.","core_discovery":"The central claim is that the quasinormal modes of rapidly rotating EGBd black holes can be extracted without treating either the Gauss-Bonnet coupling or the angular momentum as small parameters. Working with numerically exact stationary axisymmetric backgrounds, the authors linearize the metric and dilaton field equations, impose purely outgoing waves at infinity and purely ingoing waves at the horizon, and expand the seven perturbation functions in Chebyshev and Legendre series, reducing the problem to a quadratic eigenvalue problem. For azimuthal number Mz=2, l=2- and l=3-led fundamental modes, they find that polar-led and axial-led modes split once the coupling is turned on, and that the imaginary parts — the damping times — develop a strong coupling dependence. Toward the boundary of the domain of existence the scalar-led modes cross the metric-led modes and become the longest lived, so the ordering of modes, and hence which mode dominates the ringdown, changes with both the coupling and the angular momentum.","pith_inferences":["(Editorial inference) The crossing pattern suggests that the ringdown of an EGBd black hole could show a sharp transition in the longest-lived mode as a function of spin or coupling; a search that follows the longest-lived mode continuously would be a sharper test than comparing isolated frequencies.","(Editorial inference) The same spectral method should carry over to other scalar-tensor theories, such as shift-symmetric Einstein-Gauss-Bonnet gravity, and the tendency of scalar-led modes to dominate near the domain boundary is likely a general feature; this is a testable prediction for those theories.","(Editorial inference) The paper computes the fundamental modes only; overtone spectra could change which mode actually dominates a matched-filter ringdown search at early times, so the dominance claim is a statement about fundamentals, not the full signal.","(Editorial inference) A practical next step would be to compute excitation coefficients from binary merger initial data, since spectral dominance alone does not determine how strongly each mode is rung up in a real event."],"forward_implications":["Isospectrality is generically broken in EGBd: once the Gauss-Bonnet coupling is nonzero, polar-led and axial-led modes split, and the splitting grows with the coupling.","Near the boundary of the domain of existence, scalar-led modes can be the longest lived, so the dominant ringdown mode is not simply the Kerr quadrupole mode; ringdown analyses that assume Kerr isospectrality would misidentify the mode.","Perturbative expansions in the spin and the coupling deviate increasingly from the exact spectrum at large angular momentum and large coupling, so strong-coupling ringdown predictions require the full numerical treatment.","The domain of existence of rotating EGBd black holes, including solutions that slightly exceed the Kerr bound j=1, is charted by the background construction, showing where perturbative methods break down.","Precision ringdown observations could in principle place bounds on the Gauss-Bonnet coupling by comparing the measured frequencies and damping times with these non-perturbative spectra."],"supporting_citations":[{"why":"Provides the spectral method and its validation on Kerr black holes, which the EGBd calculation extends.","marker":"[49]"},{"why":"Supplies the second-order-in-spin, sixth-order-in-coupling perturbative quasinormal modes used as a comparison in Fig. 5.","marker":"[43]"},{"why":"Constructs the rotating EGBd black hole solutions that serve as the backgrounds for the perturbations.","marker":"[31]"},{"why":"Gives the quasi-isotropic metric parametrization and the domain of existence of rotating EGBd black holes.","marker":"[37]"},{"why":"Earlier static EGBd quasinormal mode results showing broken isospectrality, which the rotating calculation extends.","marker":"[38]"},{"why":"The companion letter reporting the exact rotating EGBd backgrounds and first non-perturbative mode results.","marker":"[44]"}],"fun_headline_variants":["Rotating black holes in EGBd: mode ordering flips near domain edge","Scalar modes overtake metric modes in rotating EGBd ringdown","Ringdown mode switch: coupling can change dominant quasinormal mode","Gauss-Bonnet coupling flips which mode leads black hole ringdown","EGBd rotating black holes: scalar modes can dominate ringdown"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the spectral expansion of the seven coupled perturbation functions is converged for the EGBd backgrounds, so the complex frequencies it returns are true quasinormal modes; the paper validates the pipeline on Kerr and against weak-coupling perturbative results, but it reports no dedicated convergence or error study for the EGBd eigenvalue problem itself, and it states that for j=0.6 and 0.8 the calculations lose accuracy toward the boundary of the domain of existence.","fun_headline_variants_meta":{"raw":{"variants":["Rotating black holes in EGBd: mode ordering flips near domain edge","Scalar modes overtake metric modes in rotating EGBd ringdown","Ringdown mode switch: coupling can change dominant quasinormal mode","Gauss-Bonnet coupling flips which mode leads black hole ringdown","EGBd rotating black holes: scalar modes can dominate ringdown"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3280,"prompt_tokens":878,"completion_tokens":2402,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2305}},"tokens_in":494,"tokens_out":2402,"duration_ms":16350,"temperature":1.0,"reasoning_tokens":2305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:48:58.798432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the eigenvalue calculation for the fundamental l=2 scalar-led mode at j=0.2 and ξ near the upper end of the domain (around 0.17) with doubled spectral resolution (larger Nx and Ny) and with an independent time-domain evolution of the linearized EGBd equations; if the sharp drop in MωI that makes the scalar mode longest-lived near the domain boundary shifts or disappears as the resolution changes, the claimed mode-order change is a numerical artifact rather than a property of the theory.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the spectral method and its validation on Kerr black holes, which the EGBd calculation extends."},{"cited_title":"Pierini and L","cited_arxiv_id":null,"evidence_quote":"Supplies the second-order-in-spin, sixth-order-in-coupling perturbative quasinormal modes used as a comparison in Fig. 5."},{"cited_title":"Kleihaus, J","cited_arxiv_id":null,"evidence_quote":"Gives the quasi-isotropic metric parametrization and the domain of existence of rotating EGBd black holes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier static EGBd quasinormal mode results showing broken isospectrality, which the rotating calculation extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The companion letter reporting the exact rotating EGBd backgrounds and first non-perturbative mode results."}],"review_version":1}