{"id":"ffdd7b6d-60cd-41c8-b19f-09f6b363442b","arxiv_id":"2505.16883","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Test-field QNM spectra of BCL black holes deviate strongly from physical axial gravitational QNM spectra at high overtones, while fundamental modes remain close.","lead":"The authors computed the test-field (scalar and gravitational-wave) quasinormal-mode frequencies of the BCL black hole in scalar-tensor gravity and compared them with the physical gravitational perturbations computed earlier. They find the fundamental modes nearly agree but the higher overtones and asymptotic spectra differ strongly, cautioning against using test-field QNMs as proxies for physical black-hole ringdown in modified gravity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's main asymptotic conclusion is conditional on the external [47] physical QNM spectrum; the paper's own WKB check covers only low-lying l=2 modes, so an error or normalization mismatch in that external spectrum could make the claimed test-field/physical difference an artifact.","rationale":"The reader's weakest assumption and the most load-bearing concern coincide: the central comparison depends on trusting the external physical QNM data of [47]. The paper's own test-field results have independent support—the continued-fraction computation, the WKB cross-check, the monodromy formula (74), and the near-horizon sl2 gap all point to a consistent test-field spectrum. However, the physical side of the comparison is not independently verified for the modes that matter most: the high overtones that determine whether the physical asymptote is truly non-vertical and whether no imaginary-axis crossing occurs. The WKB check in Appendix A only probes low-lying l=2 modes and is not accurate enough to validate the asymptotic behavior. A normalization mismatch in Eq. (105) would systematically shift the physical frequencies and could erase the claimed difference, while a finite-mode fitting artifact in [47] could produce a spurious non-vertical slope. These are concrete, addressable possibilities, not accusations; the appropriate response is to require an independent recomputation of the physical spectrum before fully accepting the qualitative asymptotic claim. Since the reader already issued a CONDITIONAL verdict, the stress-test does not move the verdict; it reinforces the condition.","tokens_in":35167,"tokens_out":7833,"duration_ms":70252,"concrete_test":"Independently recompute the physical axial QNM spectrum for l=2 at r_- = 0.1, 0.3, 0.5, pushing to overtones with Im(omega) ≤ -20i, by reimplementing the matrix continued-fraction method of [47] from the first-order system (94)-(102), or by direct time-domain evolution of the c=1-normalized potential (106). Then check whether (i) the fundamental modes still match the test-field values to the reported ~1-2%, and (ii) the high-overtone branch keeps a non-vertical asymptote with no imaginary-axis crossing. If [47]'s spectrum is reproduced in detail, the concern is refuted; if the slope flattens or a crossing appears at higher n, the central claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the comparison in Sec. V.C between the new test-field spectrum and the physical spectrum from [47]. The authors explicitly write that the comparison holds 'as long as the previous numerical analysis of the BCL BH [47] can be trusted', but they do not recompute the physical spectrum beyond a third-order WKB check of a few l=2 modes (Appendix A, Table II). That WKB check is not decisive: for n=3 the WKB value already differs from the matrix-continued-fraction value of [47] by about 7%. The qualitative headline—imaginary-axis crossing and vertical asymptote for test-field modes versus no crossing and a non-vertical asymptote for physical modes—is therefore only as solid as the completeness and normalization of [47]. In particular, the r-dependent speed normalization (Eqs. 104-106) must be exactly the one used to define the physical QNMs; if the [47] frequencies were quoted in the original tortoise coordinate rather than the c=1 rescaled one, the comparison would shift systematically. Moreover, the asymptotic regime is precisely where numerical methods are hardest to control, so the fitted non-vertical slope of the physical spectrum could be an artifact of a finite number of modes. This does not undermine the paper's internal test-field results, but it is the weakest link in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes test-field (spin-0 and spin-2) quasi-normal mode (QNM) spectra for the BCL black hole in scalar-tensor theories, using Leaver's continued fraction method, the monodromy technique, near-horizon sl2 symmetry, and WKB cross-checks. It derives an asymptotic formula (4πRω = log 2 − i(2n+1)π, with R = r_+^2/(r_+ + r_-)) for highly damped test-field modes. The central comparison is between the spin-2 test-field spectrum and the physical axial gravitational QNM spectrum taken from Roussille et al. [47]. The paper reports that the fundamental modes are close (0.78% difference at r_- = 0.1, 2.5% at r_- = 0.5), but that the test-field spectrum crosses the imaginary axis and has a vertical asymptote, whereas the physical spectrum shows no crossing and a non-vertical asymptote, concluding that test-field spin-2 QNMs differ substantially from physical gravitational QNMs for highly damped modes.","tokens_in":35476,"tokens_out":28228,"duration_ms":189741,"significance":"If the comparison is valid, the paper provides a concrete, well-studied example in which test-field perturbations fail to reproduce physical gravitational perturbation spectra for a black hole in modified gravity, with the deviation growing in the highly damped regime. This is relevant for black hole spectroscopy and for effective quantum-gravity models that lack explicit modified Einstein equations. The monodromy result (74) is a new analytical prediction, and the test-field computations are cross-checked by multiple methods (WKB, Leaver, monodromy, near-horizon symmetry). However, the headline comparison relies on external data from [47] in precisely the regime where numerical control is hardest, and there is an internal inconsistency in the quoted physical effective potential. These issues must be addressed before the central claim can be regarded as fully supported.","major_comments":[{"comment":"There is an inconsistency between the displayed physical effective potential (106) and its quoted large-r asymptotic (108). Equation (106) writes the prefactor as (1 + 2 r_- r_+/r^2)^3, while the text immediately below says the factor is (1 + 2 r_- r_+/r^2)^{-3}; neither choice reproduces the 1/r^4 coefficient in Eq. (108). For example, with r_+ = 1, r_- = 0.5 and λ = 2, expanding Eq. (106) at large r gives a 1/r^4 coefficient of +7.25 for exponent +3 and −4.75 for exponent −3, whereas Eq. (108) gives −7.75. Because Vphys is the basis for the WKB checks in Appendix A and for the comparison with [47], this inconsistency must be resolved and the corrected potential used to re-verify the numerical results.","section":"V.B, Eqs. (106) and (108)"},{"comment":"The central comparison of test-field and physical QNM spectra rests entirely on the physical spectrum from [47]. The independent WKB check in Table II covers only l = 2 for a few low overtones (where WKB is known to be inaccurate for n ~ l, as the authors note) and l = 10, 100 for the first few overtones; it does not probe the highly damped regime where the claimed qualitative differences (crossing of the imaginary axis vs no crossing, vertical vs non-vertical asymptote) occur. The authors should either compute the physical QNM spectrum from the corrected Vphys with the same continued fraction method used on the test-field side, or provide a quantitative convergence and completeness analysis of the [47] data. Without this, the headline claim is conditional on external numerical results in exactly the asymptotic regime where such results are most difficult to control.","section":"V.C and Appendix A"},{"comment":"The comparison presupposes that the rescaled tortoise coordinate d\\tilde{x} = dx/c(r) defined in Eq. (105) is exactly the normalization used by [47] to define their physical QNM frequencies. If [47] quoted frequencies in the unrescaled tortoise coordinate, the physical spectrum would be shifted systematically and the comparison would be invalid. The paper should state explicitly how the [47] frequencies are normalized and, ideally, verify this by reproducing a few of their low-lying modes from the corrected Vphys with the authors' own continued fraction implementation.","section":"V.B, Eq. (105)"}],"minor_comments":[{"comment":"The statement that 'The limited data we have did not allow us to infer the coefficient B, so we only focused on the constant one A' is unclear: if B is not fitted, the fit reduces to a constant A, and the reported error bars (obtained by varying the number of removed low-lying modes) should be described accordingly; please clarify the exact fitting procedure.","section":"IV.C"},{"comment":"Individual QNM frequencies are plotted without numerical error bars; a convergence study with respect to the continued-fraction truncation order (e.g., varying the number of retained terms) would strengthen the spectral plots and support the claim of 'good accuracy'.","section":"IV.B, Figs. 4-8"},{"comment":"The fitted relation 1/a = −0.14(±0.01)r− + 0.04(±0.01)r2− for the physical asymptote slope is presented without details on how it was obtained from [47]'s data; please provide the underlying data or a clear reference to the specific plot in [47].","section":"V.C"},{"comment":"There are several typographical issues, including 'obsctable' in the Discussion and the reference to 'table 9' for a figure; please proofread the manuscript.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and relevant question, and the test-field computations are methodologically sound with multiple cross-checks. The main risk is the reliance on the external [47] data for the physical spectrum, which the paper does not independently reproduce in the high-overtone regime; the WKB check is too weak to validate the asymptotic comparison. The inconsistency in the quoted physical potential (Eq. (106) vs Eq. (108)) is a technical but fixable issue. If the authors can provide an independent continued-fraction computation of the physical QNMs from the corrected Vphys, or otherwise convincingly validate the [47] data in the relevant regime, the paper would be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper delivers the first test-field QNM spectra for the BCL black hole and derives a neat asymptotic formula, 4πRω = log(2) − i(2n+1)π. It also makes a plausible case that test-field spin-2 QNMs track the physical axial gravitational QNMs for the fundamental mode (sub-percent to ~2.5% difference) but diverge substantially at high damping, with the test-field spectrum crossing the imaginary axis and becoming vertical while the physical spectrum does neither.\n\nWhat is genuinely good: the internal test-field results are multiply cross-checked. The monodromy prediction for the imaginary gap matches their Leaver continued-fraction data to 0.02–0.6% across r−, which is strong quantitative support. The near-horizon sl2 argument independently reproduces the same gap, and the WKB checks for low-lying modes are there. The paper is also honest about its numerical limits and explicitly flags that the physical comparison holds only \"as long as the previous numerical analysis of the BCL BH [47] can be trusted.\"\n\nThe soft spots are real but addressable. The central contrast — vertical versus non-vertical asymptote, presence versus absence of imaginary-axis crossing — is taken from [47]'s published physical QNM data. The authors did not recompute that spectrum; their WKB check of [47] covers only l=2 low overtones, and for n=3 the WKB value already differs from [47] by about 7%. So the headline is conditional, exactly as the authors say, but the condition is load-bearing. Second, Eq. (54) writes the near-zero potential as 4/(9x²) for both spins, yet the stated Bessel index for s=0 corresponds to −2/(9x²); the final determinant happens to be the same, so the formula survives, but the displayed equation is wrong for s=0. Third, the Gaussian reduction branch choice that selects the ζn → −1/4 branch is supported only numerically, not proved. Finally, no error bars on individual frequencies and no code/data release, which matters for a numerical paper.\n\nOverall, this is a serious and careful paper. The internal test-field results and the analytical asymptotics stand on their own, and the comparison with physical QNMs is plausible and honestly hedged. I would send it to a good referee; the referee should push on the physical-spectrum normalization and ask for data/code, but the paper deserves referee time.","headline":"Solid test-field QNM computation for the BCL black hole with a clean log(2) monodromy formula, but the headline contrast with physical QNMs is only as strong as the external [47] data it leans on.","tokens_in":35992,"tokens_out":3249,"would_cite":true,"duration_ms":27406,"reading_group":"maybe","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":"For the BCL scalar-tensor black hole, spin-2 test-field quasi-normal modes diverge from physical gravitational modes at high damping while the fundamental modes stay close.","keywords":["quasi-normal modes","scalar-tensor theories","BCL black hole","test-field perturbations","Teukolsky equation","Leaver continued fraction","monodromy method","black hole spectroscopy"],"falsifier":"Independently recompute the physical axial QNM spectrum of the BCL black hole from the first-order system [46,76] using a code that reaches $\\mathrm{Im}(\\omega)$ beyond roughly $-10$ and check the asymptote's slope and whether any mode crosses the imaginary axis; if the physical spectrum becomes vertical or crosses, the claimed contrast with test-field modes fails.","tokens_in":34969,"feed_emoji":"🕳️","tokens_out":8043,"duration_ms":62121,"temperature":0.7,"pith_summary":"The paper asks whether test-field perturbations—waves that propagate on a fixed black-hole background without back-reaction—can stand in for the true gravitational perturbations of a black hole in a modified theory of gravity. It answers this for the BCL solution of quadratic shift-symmetric Horndeski scalar-tensor theories, where both the physical axial gravitational quasi-normal modes and the metric are already known. The authors compute the spin-0 and spin-2 test-field quasi-normal spectra with Leaver's continued-fraction method and analytic monodromy techniques, then compare the spin-2 test-field spectrum with the physical one. They find that the fundamental modes agree to within about one to three percent, but that the higher overtones differ sharply: the test-field spectrum keeps the Schwarzschild-like shape with an imaginary-axis crossing and a vertical asymptote, while the physical spectrum has neither. This matters because test-field calculations are the only option for many effective black-hole models that lack explicit modified Einstein equations, and the result shows such calculations can be systematically misleading outside the low-damping sector.","feed_headline":"Test-field QNMs stray far from physical ones at high damping","feed_subtitle":"On the BCL scalar-tensor black hole, fundamental modes agree to 1-3%, but higher overtones diverge sharply.","key_machinery":"The argument runs on the BCL metric, $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\\Omega^2$ with $f(r) = (1 - r_+/r)(1 + r_-/r)$, where $r_-$ is a single deformation parameter away from Schwarzschild. Both test-field and physical perturbations are reduced to Schr\\\"odinger-like equations, $d^2\\Psi/dx^2 + [\\omega^2 - V(r)]\\Psi = 0$, with effective potentials that agree to third order at infinity but differ near the horizon. The numerical engine is Leaver's continued-fraction method, here applied to a five-term recursion reduced by Gaussian elimination; the analytic engine is the monodromy technique, which tracks solutions around the complex-plane singularities and yields the asymptotic formula $4\\pi R\\omega = \\log 2 - i(2n+1)\\pi$, plus the near-horizon $sl_2$ symmetry that reproduces the equally spaced imaginary gap $\\Delta_{\\rm Im} = 1/(2R)$. To make the physical and test-field problems directly comparable, the propagation speed of the physical gravitational perturbations is normalized to $c = 1$ by the rescaled tortoise coordinate $d\\tilde{x} = (1/c(r))dx$ with $c(r) = r/\\sqrt{r^2 + 2r_- r_+}$.","core_discovery":"The paper's central claim is that, on the BCL black hole of quadratic shift-symmetric Horndeski theories, the spin-2 test-field quasi-normal spectrum is not a reliable proxy for the physical axial gravitational spectrum outside the low-damping sector. The fundamental modes stay close—the test-field value differs from the physical one by about 0.78% at $r_- = 0.1$ and 2.5% at $r_- = 0.5$—but the overtones diverge systematically. The test-field spectrum crosses the imaginary axis and approaches a vertical asymptote whose first-order analytic law is $4\\pi R\\omega = \\log 2 - i(2n+1)\\pi$ with $R = r_+^2/(r_+ + r_-)$, while the physical spectrum shows no imaginary-axis crossing and a non-vertical asymptote that the prior numerical analysis [47] parametrizes by $\\mathrm{Im}(\\omega) = a\\,\\mathrm{Re}(\\omega) + b$. As the authors phrase it, as long as that prior numerical analysis can be trusted, the test-field spin-2 QNMs are substantially different from the physical gravitational ones, especially for highly damped modes.","pith_inferences":["If this pattern is generic, spin-2 test-field QNMs should only be trusted for the fundamental and lowest overtones of effective black-hole models that lack explicit Einstein equations; high-overtone predictions from such models would carry an unquantified systematic error.","The monodromy discontinuity ($\\log 2$ for BCL versus $\\log 3$ for Schwarzschild as $r_- \\to 0$) suggests a boundary-layer regime at small $r_-$; a next-to-leading-order monodromy computation could explain the observed bouncing of the real part with $r_-$.","A monodromy-style analysis of the physical perturbation equations, which the authors did not perform, would test whether the non-vertical asymptote is a genuine feature of the scalar-tensor dynamics or an artifact of the numerical scheme in [47].","The eikonal-limit WKB comparison hints that real parts of physical and test-field QNMs converge for large $l$ while imaginary parts do not; an analytic proof of this $l \\to \\infty$ split could isolate which potential terms control damping."],"forward_implications":["For the BCL black hole, the fundamental spin-2 test-field QNM tracks the physical one within 1–3% for $r_-$ between 0.1 and 0.5, so test fields remain useful for the dominant ringdown mode.","At higher overtones the two spectra part ways: the test-field spectrum has a vertical asymptote and an imaginary-axis crossing, while the physical spectrum has neither.","The asymptotic imaginary gap of the test-field spectrum is $\\Delta_{\\rm Im} = 1/(2R)$ with $R = r_+^2/(r_+ + r_-)$, matching the monodromy prediction to sub-percent accuracy for the imaginary part.","The physical asymptote is instead non-vertical, with $1/a \\approx -0.14\\,r_- + 0.04\\,r_-^2$ from the prior data [47], which a future analytical treatment would need to explain."],"supporting_citations":[{"why":"Supplies the BCL black-hole solution whose perturbations are the object of study.","marker":"[29]"},{"why":"Derives the linear perturbation equations for the BCL solution, providing the physical perturbation setup.","marker":"[46]"},{"why":"Provides the numerical physical axial QNM spectrum and the matrix continued fraction data that the comparison rests on.","marker":"[47]"},{"why":"Introduces Leaver's continued fraction method used to compute test-field QNM frequencies.","marker":"[48]"},{"why":"Gives the Teukolsky equations and effective potentials for massless spin-0 and spin-2 test fields on spherically symmetric metrics.","marker":"[70]"},{"why":"Establishes the monodromy technique and the Schwarzschild asymptotic QNM formula that the authors adapt to the BCL case.","marker":"[57]"},{"why":"Provides the near-horizon $sl_2$ symmetry used to derive the equally spaced imaginary gap for high overtones.","marker":"[61]"},{"why":"Derives the physical effective potential $V_{\\rm phys}$ and the first-order matrix system for axial perturbations used in the comparison.","marker":"[76]"}],"fun_headline_variants":["Test-field QNMs betray physical spectrum at high overtones","Spin-2 test-field QNMs diverge sharply at high damping","For BCL black holes, test-field QNMs fail for overtones","High-damping QNMs: test-field vs physical starkly differ","Test-field QNMs unreliable for gravitational overtones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The comparison's headline result rests on trusting the prior numerical data [47] for the physical axial QNMs and on the speed-normalizing rescaling of the tortoise coordinate, so if that data or that normalization is wrong, the claimed test-field/physical difference could be an artifact.","fun_headline_variants_meta":{"raw":{"variants":["Test-field QNMs betray physical spectrum at high overtones","Spin-2 test-field QNMs diverge sharply at high damping","For BCL black holes, test-field QNMs fail for overtones","High-damping QNMs: test-field vs physical starkly differ","Test-field QNMs unreliable for gravitational overtones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3274,"prompt_tokens":1035,"completion_tokens":2239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":2148}},"tokens_in":651,"tokens_out":2239,"duration_ms":14093,"temperature":1.0,"reasoning_tokens":2148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:53:25.688261+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently recompute the physical axial QNM spectrum of the BCL black hole from the first-order system [46,76] using a code that reaches $\\mathrm{Im}(\\omega)$ beyond roughly $-10$ and check the asymptote's slope and whether any mode crosses the imaginary axis; if the physical spectrum becomes vertical or crosses, the claimed contrast with test-field modes fails.","supporting_citations":[{"cited_title":"As a complement we computed the physical QNM frequencies using the WKB method applied to Vphys for l = 10 and l = 100","cited_arxiv_id":null,"evidence_quote":"Provides the numerical physical axial QNM spectrum and the matrix continued fraction data that the comparison rests on."},{"cited_title":"Quasinormal modes of Schwarzschild black holes: The determination of quasinormal frequencies with very large imaginary parts,","cited_arxiv_id":null,"evidence_quote":"Derives the physical effective potential $V_{\\rm phys}$ and the first-order matrix system for axial perturbations used in the comparison."}],"review_version":1}