{"id":"3e279ccd-2912-4d01-b2b3-2cfd09e6a6b3","arxiv_id":"2507.07196","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For the Bonanno-Reuter quantum-corrected black hole, gravitational quasinormal modes and grey-body factors deviate from Schwarzschild only at small masses and small interpolation parameter, recovering the classical result for large mass and large gamma.","lead":"This paper computes the gravitational wave ringing frequencies and escape probabilities for a quantum-corrected black hole model from asymptotic safety, across different values of the model's interpolation parameter. It finds that the model matches the classical Schwarzschild prediction except for very small masses, where quantum corrections leave a detectable signature in the ringing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed 'large-γ' convergence to Schwarzschild is contradicted by the paper's own Eq. (3) and Tables 1–2: increasing γ at fixed M moves QNM frequencies away from Schwarzschild, and for large enough γ the horizon disappears.","rationale":"The reader's weakest assumption is the borrowed effective-fluid potential (Eq. 11). That is a legitimate concern, and if the potential is wrong all quantitative results fail. However, the most load-bearing, internally checkable problem is the paper's own claim about the γ-limit. Eq. (3) makes the geometry tend to flat space, not Schwarzschild, as γ→∞ for fixed M, and within the black-hole range the tables show increasing γ moves the QNM frequencies away from the Schwarzschild values. This undermines the central conclusion as literally stated, independent of any debate about the perturbation potential. The large-M part of the claim is well supported: for M=10 all tabulated frequencies are within sub-percent of Schwarzschild, and the WKB6-vs-WKB7 differences plus one time-domain check provide reasonable numerics. But the γ-limit clause should be removed or replaced by a correct statement that increasing γ (up to the horizon bound) enhances the quantum imprint, while the Schwarzschild limit is recovered for M→∞ (equivalently, γ≪M²). A revision that corrects this interpretation and re-checks the γ=9/2 tables would preserve the useful numerical content; hence CONDITIONAL rather than outright rejection.","tokens_in":125,"tokens_out":31643,"duration_ms":969817,"concrete_test":"Recompute the fundamental ℓ=2 axial QNM for M=5 at γ=0.1, γ=1, γ=4.5, and a value just below γ_cr(M) determined from f(r)=0, using both the WKB6+Padé code and an independent direct-integration or continued-fraction method. Also evaluate f(r) and V(r) at the potential peak. If Re(ω) increases and |Im(ω)| decreases monotonically as γ grows while the Schwarzschild value is approached only for γ→0 within the horizon-allowed range, the 'large-γ convergence' statement in the abstract and conclusions is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim has two limits: 'large M or large γ.' The large-M limit is supported. The large-γ limit is not. In Eq. (3), with G0=1, G(r)=r³/[r³+ω̃(r+γM)], so for fixed M, γ→∞ gives G(r)→0 and f(r)→1—flat space, not Schwarzschild. Beyond a finite γ_cr(M) the spacetime has no horizon at all, so the limit is not in the black-hole parameter range. Inside the allowed range, increasing γ raises f at the Regge–Wheeler peak (r≈3M), hence raises V(r) in Eq. (11). The paper's own tables contradict its §4.4 statement that 'increasing γ tends to suppress the deviation from the classical limit': for M=5, ℓ=2, γ=0.1 gives ω=0.075714−0.017632i, γ=1 gives 0.076034−0.017481i, while Schwarzschild is 0.07473−0.01779i. The larger-γ value is farther from Schwarzschild in both real and imaginary parts. The same pattern holds at M=4 and M=10. Moreover, Tables 2 and 3 are identical for M≥4, suggesting the γ=9/2 data were not independently recomputed. Thus the 'large-γ convergence' clause of the central claim is not merely unproven—it is contradicted by the paper's own equations and numbers.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper computes the axial gravitational quasinormal-mode (QNM) frequencies and grey-body factors of the Bonanno–Reuter (BR) renormalization-group-improved black hole, with the RG parameter ω̃ fixed and the interpolation parameter γ treated as free. The QNMs are obtained with the sixth- and seventh-order WKB method with Padé approximants, checked by one time-domain integration, while the grey-body factors are computed with the sixth-order WKB transmission formula and with the recently proposed QNM–grey-body correspondence. The results are tabulated and plotted as functions of the mass M and γ, and the paper's central claim is that the Schwarzschild limit is recovered for large M or large γ, while at fixed mass the quantum-corrected black hole has longer-lived QNMs and smaller grey-body factors.","tokens_in":12988,"tokens_out":17378,"duration_ms":186541,"significance":"If the results held as stated, the paper would provide a useful reference for how asymptotic-safety-inspired corrections to Schwarzschild modify ringdown frequencies and Hawking-radiation transmission spectra, with deviations confined to the small-mass regime. The authors deserve credit for several concrete internal consistency checks: the WKB6/WKB7 order comparison (Tables 1–4, typically δ ≲ 0.01%), the time-domain/Prony cross-check for one parameter set (Fig. 6, agreement better than 0.005%), and the comparison of two grey-body factor prescriptions (Figs. 7–8). However, the physical input is an effective potential borrowed from an anisotropic-fluid construction whose validity for the BR metric is not established, and, more seriously, the claimed large-γ Schwarzschild limit is contradicted by the paper's own equations and tables, while the γ=9/2 tables coincide with the γ=1 tables at every overlapping entry. The niche is well populated (refs. 10–13, 32, 35–37), so the incremental contribution, once corrected, is modest but suitable for a specialist venue.","major_comments":[{"comment":"The claim that quasinormal frequencies converge to Schwarzschild 'for large masses or large values of γ' is not supported and is contradicted by the paper's own input. In Eq. (3), at fixed M, γ→∞ gives G(r)→0 and f(r)→1, i.e., flat space, not Schwarzschild; moreover, for γ above a critical value the spacetime has no horizon (as the paper itself notes in §2 and Fig. 1), so the large-γ limit exists only outside the black-hole parameter range. Within the allowed range, the tables show the opposite trend: at M=5, ℓ=2, γ=0.1 gives ω=0.075714−0.017632i and γ=1 gives ω=0.076034−0.017481i, both moving away from the Schwarzschild value (Mω≈0.37367−0.08896i, i.e., ω≈0.07473−0.01779i at M=5). The §4.4 sentence 'increasing γ tends to suppress the deviation from the classical limit' and the statement that γ≳5 leads to frequencies indistinguishable from Schwarzschild for M≳5 are likewise inconsistent with Tables 1–3: at M=5, γ=9/2 still differs from Schwarzschild by about 1.7% in Re ω, two orders of magnitude above the WKB-order uncertainty δ≈0.007%. The abstract and Conclusions should be revised to state the correct limit structure: Schwarzschild is recovered for large M at fixed γ, while increasing γ at fixed M enhances the deviation and eventually removes the horizon.","section":"§2, Eq. (3); §4.4; §6"},{"comment":"The γ=9/2 entries in Table 3 coincide exactly with the γ=1 entries in Table 2 at every overlapping mass (ℓ=2 and ℓ=3; M=3.5, 4, 4.5, 5, 10), and Table 3 adds only the intermediate masses M=3.6–3.9. This cannot be a physical saturation effect: at the barrier peak r≈3M the lapse functions differ by several percent for these parameters (e.g., at M=4, f≈0.348 for γ=1 versus f≈0.361 for γ=9/2), so the QNM frequencies should differ at the 10⁻³–10⁻² level, well above the digits displayed. The authors should recompute and present the actual γ=9/2 results, and clarify the provenance of the γ=10 curves in Figs. 4–5, since this data is the only support for the claimed γ-dependence of the spectrum.","section":"Tables 2 and 3; §4.4"},{"comment":"The effective potential—the single physical input for all QNM and grey-body results—is taken from the anisotropic-fluid construction of refs. [14,16] without a derivation for the BR metric, which is not presented as a solution of the Einstein equations for any specified matter content. The paper explicitly calls this a 'practical workaround' and justifies it only by the assertion that it 'reliably captures the leading quantum corrections' for small deviations from Schwarzschild. This is a load-bearing assumption: if Eq. (11) is not the correct axial gravitational perturbation potential of the RG-improved geometry, then Tables 1–4 and Figs. 6–8 do not describe gravitational perturbations of the BR black hole. The authors should either (i) derive the axial perturbation equations for the improved metric, specifying the effective matter and its perturbation properties, or (ii) clearly reframe the paper as an analysis of the Regge–Wheeler-type equation (11) associated with the effective geometry, as done in refs. [14,16], and state the attendant limitation prominently.","section":"§3, Eq. (11)"},{"comment":"The agreement between the two grey-body factor computations does not by itself validate the transmission coefficients, because both methods are WKB-type approximations built from the same potential V(r): the WKB formula (20) uses V₀ and V₀′′ at the barrier, while the QNM-correspondence formula (22) uses the WKB-Padé quasinormal frequencies of that same potential. Sub-percent agreement therefore tests the internal consistency of two approximations to the same scattering problem and cannot rule out a common systematic error from Eq. (11) or from the WKB barrier treatment. An independent check—direct numerical integration of the scattering problem (18) for real ω—should be added before the grey-body factor plots are presented as quantitative results.","section":"§5, Eqs. (20)–(22)"}],"minor_comments":[{"comment":"The text refers to 'the QNM–grey-body correspondence (equation (23))', but the correspondence formula is Eq. (22); the equation number should be corrected.","section":"§5 (near Eq. (22))"},{"comment":"The sentence 'the quantum corrected black hole of the same mass as its classical counterpart has longer lived modes with slightly smaller oscillations rate' is contradicted by Tables 1–3, which show Re ω slightly larger than the Schwarzschild value at the same mass (e.g., M=5, ℓ=2, γ=0.1: 0.075714 versus ≈0.07473); the wording should be aligned with the tabulated values.","section":"§4.4"},{"comment":"The axis labels of Figs. 2–3 appear garbled ('V/LParen1r/Star/RParen1', 'r/Star') and the captions contain spacing artifacts ('γ = 0 .1'); the figures should be regenerated with standard notation so that the potential plots are legible.","section":"Figs. 2–3"},{"comment":"The captions read 'of the gravitational perturbations the Bonanno–Reuter black hole' and omit the word 'of'; each table caption should also state the fixed γ value explicitly.","section":"Tables 2–4"},{"comment":"Since the central claim is convergence to Schwarzschild, the authors should tabulate the corresponding Schwarzschild fundamental frequencies explicitly (e.g., Mω≈0.37367−0.08896i for ℓ=2, n=0, axial) so that statements such as 'indistinguishable from Schwarzschild' have a stated numerical reference.","section":"§4.4; Tables 1–4"}],"recommendation":"major_revision","confidential_remarks":"The entry-for-entry identity of Tables 2 and 3 at every overlapping mass strongly suggests that the γ=9/2 n=0 data were copied from the γ=1 calculation rather than independently recomputed; the editor should ask the authors to regenerate and verify these values before publication. The 'large γ' convergence claim is internally inconsistent with Eq. (3) and the tables, and must be corrected in the abstract, §4.4, and the Conclusions. The reference list is heavily self-referential (refs. 4, 21–24, 28–29), and the novelty statement should be checked against refs. 10–13 and 32, 35–37, several of which address closely related models. The Data Availability statement ('Not applicable') is unfortunate for a numerical paper whose tables are central to the claims; providing data files or code would strengthen reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: read this as a numerical scan, not as a demonstration about asymptotic safety. The large-M part of the central claim holds up: as M grows, the RG corrections become subleading and the QNMs approach Schwarzschild. The large-gamma part does not. Eq. (3) with G0=1 gives f(r)->1 as gamma->infinity, which is flat space, not Schwarzschild, and beyond a finite gamma_cr(M) there is no horizon. Inside the allowed range, the paper's own tables contradict the statement that increasing gamma suppresses the deviation: for M=5, ell=2, gamma=0.1 gives omega=0.075714-0.017632i, gamma=1 gives 0.076034-0.017481i, while Schwarzschild is 0.07473-0.01779i. Both the real and imaginary parts are farther from Schwarzschild at larger gamma. So the abstract and Conclusions overstate what the calculation actually shows.\n\nWhat the paper does well: it is the first gravitational axial QNM and grey-body scan for this specific Bonanno-Reuter metric, and the numerical execution looks careful. The WKB6/Pade and WKB7/Pade results agree to well below a percent, and the one time-domain Prony extraction matches the WKB frequency. Those internal checks are real. The grey-body comparison between WKB and the QNM-correspondence also agrees, but that is not an independent validation: both methods are built on the same effective potential, so agreement mainly checks the WKB formula against an approximate eikonal mapping.\n\nThe main soft spot is Eq. (11). The axial potential is borrowed from an effective anisotropic-fluid construction in refs [14,16], and the paper calls this a workaround but never justifies why that Regge-Wheeler-like equation describes the gravitational perturbations of the Bonanno-Reuter spacetime, which is not an exact Einstein solution. If that potential is wrong, every frequency and grey-body factor shifts. This is a load-bearing assumption. I also noticed that Tables 2 and 3 are identical for M>=3.5 or 4 onward; either the gamma=9/2 data were not independently recomputed or the effect is below the printed precision, and the paper should say which. The citation pattern is fine: earlier asymptotically safe black-hole QNM papers, including their own review, are cited, and the new element is specifically the Bonanno-Reuter metric plus the parameter scan.\n\nWho is this for? Someone benchmarking quasinormal modes of regular or asymptotically safe black holes. It is not an observational claim. I would not cite it in its current form because the gamma-dependence conclusion is wrong and the potential issue is unresolved. But a serious referee could get a corrected version: rewrite the large-gamma claim, justify or remove the borrowed potential, and recompute or explain the duplicated tables. It deserves peer review, not silent desk rejection.","headline":"The large-M Schwarzschild recovery in this Bonanno-Reuter QNM scan is plausible, but the paper's matching claim about large gamma is contradicted by its own metric and tables, and the borrowed gravitational potential needs more justification.","tokens_in":13496,"tokens_out":4024,"would_cite":false,"duration_ms":48553,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C45"],"pacs":["04.70.-s","04.30.-w","04.60.-m"],"model":"deepseek-v4-flash","headline":"Gravitational quasinormal frequencies of the Bonanno–Reuter black hole converge to Schwarzschild for large mass or gamma, while small quantum-corrected black holes ring longer and emit less.","keywords":["quasinormal modes","grey-body factors","Bonanno–Reuter black hole","regular black holes","asymptotic safety","axial gravitational perturbations","WKB method","ringdown"],"falsifier":"Derive axial gravitational perturbations of the Bonanno–Reuter metric directly from the linearized field equations with the full effective stress-energy tensor, or from the RG-improved action without the anisotropic-fluid analogy, and compare the fundamental $\\ell=2$ frequency at $M=1.66$, $\\gamma=0.1$ with the paper's Table 1 value $\\omega\\approx 0.263295-0.042414i$; a mismatch larger than the roughly $0.01\\%$ WKB error would refute the quantitative claim.","tokens_in":12477,"feed_emoji":"🕳️","tokens_out":10894,"duration_ms":105057,"temperature":0.7,"pith_summary":"This paper computes the gravitational quasinormal-mode frequencies and grey-body factors of the Bonanno–Reuter regular black hole, a quantum-corrected Schwarzschild-like spacetime arising in the asymptotic-safety scenario. It claims that as the black hole mass $M$ or the interpolation parameter $\\gamma$ grows, both the real and imaginary parts of the fundamental quasinormal frequencies smoothly approach their classical Schwarzschild values, while significant deviations appear for small masses. The central physical result is that a quantum-corrected black hole of the same mass as a Schwarzschild one oscillates slightly more slowly and decays more slowly, and emits radiation with smaller grey-body factors. A sympathetic reader would care because these are concrete, in-principle observable signatures of quantum gravity in ringdown and Hawking radiation.","feed_headline":"Quantum-corrected black holes ring longer than Schwarzschild","feed_subtitle":"Quasinormal frequencies hit Schwarzschild values for large M or gamma; deviations remain only at small masses.","key_machinery":"The load-bearing object is the effective axial-gravitational potential of Eq. (11), obtained by modeling the quantum-corrected background as an effective anisotropic fluid so that standard Regge–Wheeler gauge perturbation theory can be applied even though the Bonanno–Reuter metric is not an exact Einstein solution. Around that potential, the paper builds a tortoise-coordinate wave equation and extracts frequencies with the sixth-order WKB formula with Padé approximants, cross-checks with characteristic time-domain integration, and computes grey-body factors both by WKB transmission and by the eikonal correspondence between grey-body factors and the fundamental quasinormal frequency. The height and width of the potential barrier transfer the geometry's quantum corrections into the ringdown frequencies and the transmission probabilities.","core_discovery":"In the paper's own terms: gravitational axial perturbations of the Bonanno–Reuter metric obey a Regge–Wheeler-like wave equation with the effective potential $V(r)=f(r)[2g(r)/r^2 - (f(r)g(r))'/(2r f(r)) + (\\ell+2)(\\ell-1)/r^2]$, and solving that equation with sixth-order WKB plus Padé approximants, confirmed by time-domain integration, yields quasinormal frequencies and grey-body factors that interpolate between quantum-corrected and classical behavior. The Schwarzschild limit is recovered for large $M$ or large $\\gamma$; for fixed mass, larger $\\gamma$ suppresses the deviation. For equal masses, the quantum-corrected black hole has longer-lived quasinormal modes (smaller $|\\operatorname{Im}\\omega|$) and smaller grey-body factors than Schwarzschild. The deviations are one to two orders of magnitude larger than the WKB error estimate, so the paper treats them as genuine physical effects rather than numerical artifacts.","pith_inferences":["If the anisotropic-fluid effective potential does not reproduce the true axial gravitational perturbations of the RG-improved geometry, the specific numbers in the tables would shift; the qualitative direction, longer-lived modes and suppressed transmission, is likely to survive because it follows from the barrier being lower and wider in the quantum regime.","A direct derivation of the axial master equation from the full effective action of the quantum-gravity scenario, without the anisotropic-fluid workaround, would settle whether the quantitative spectrum is correct.","The same machinery could be applied to electromagnetic and scalar perturbations and to higher overtones, which are typically more sensitive to near-horizon geometry and could show a stronger quantum imprint than the fundamental mode.","Translating these results into an observational test would require an estimate of whether the small-mass quantum regime is ever populated and whether its modified ringdown or suppressed emission is within reach of future gravitational-wave or primordial-black-hole searches; the paper does not make that estimate."],"forward_implications":["For small masses near the critical horizon limit (for example $M=1.66$ at $\\gamma=0.1$), the quasinormal frequencies deviate from Schwarzschild by far more than the WKB error, so ringdown of quantum relics would carry a measurable quantum signature.","For large masses, or $\\gamma\\gtrsim 5$ with $M\\gtrsim 5$, the predicted frequencies are indistinguishable from Schwarzschild within numerical error, so classical gravitational-wave tests in the astrophysical mass range are unaffected.","Smaller grey-body factors for equal-mass quantum-corrected black holes mean Hawking radiation is suppressed relative to Schwarzschild, which would slow evaporation and lengthen the lifetime of the remnant.","The agreement between WKB and quasinormal-mode-correspondence grey-body factors, within about $1\\%$ even for $\\ell=2$, supports using the eikonal correspondence at moderately low multipoles in this class of spacetimes.","Since the WKB error is one to two orders of magnitude smaller than the quantum-induced shift, the tabulated frequencies provide a concrete benchmark for future full time-domain or nonlinear evolutions."],"supporting_citations":[{"why":"Supplies the renormalization-group running of Newton's constant that motivates the quantum-corrected metric.","marker":"[8]"},{"why":"Provides the Bonanno–Reuter RG-improved metric that is the background spacetime under study.","marker":"[9]"},{"why":"Supplies the effective anisotropic-fluid interpretation and the axial perturbation setup from which the effective potential is derived.","marker":"[14]"},{"why":"Defines the Regge–Wheeler gauge used for the axial metric perturbations.","marker":"[15]"},{"why":"Co-sources the effective axial-gravitational potential formula (Eq. 11) used for the wave equation.","marker":"[16]"},{"why":"Provides the higher-order WKB formula whose sixth order is used to compute quasinormal frequencies and grey-body factors.","marker":"[19]"},{"why":"Introduces the Padé resummation that improves the WKB accuracy.","marker":"[20]"},{"why":"Supplies the characteristic time-domain integration scheme used to independently confirm the WKB frequencies.","marker":"[25]"},{"why":"Proposes the eikonal correspondence between grey-body factors and the fundamental quasinormal mode, used as a second method for grey-body factors.","marker":"[27]"}],"fun_headline_variants":["Quantum black holes ring longer than Schwarzschild","Small quantum black holes deviate from classical ringdown","Quantum corrections prolong ringdown and suppress grey-body radiation","Longer ringdown for quantum black holes at small mass","Grey-body factors drop for quantum-corrected black holes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Bonanno–Reuter metric is not an exact solution of Einstein's equations, so the paper replaces the true gravitational perturbations with perturbations of an effective fluid model; if that replacement is wrong, every computed quasinormal frequency and grey-body factor is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Quantum black holes ring longer than Schwarzschild","Small quantum black holes deviate from classical ringdown","Quantum corrections prolong ringdown and suppress grey-body radiation","Longer ringdown for quantum black holes at small mass","Grey-body factors drop for quantum-corrected black holes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000838,"raw_usage":{"total_tokens":3658,"prompt_tokens":954,"completion_tokens":2704,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":2628}},"tokens_in":570,"tokens_out":2704,"duration_ms":20317,"temperature":1.0,"reasoning_tokens":2628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:47:12.612895+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Derive axial gravitational perturbations of the Bonanno–Reuter metric directly from the linearized field equations with the full effective stress-energy tensor, or from the RG-improved action without the anisotropic-fluid analogy, and compare the fundamental $\\ell=2$ frequency at $M=1.66$, $\\gamma=0.1$ with the paper's Table 1 value $\\omega\\approx 0.263295-0.042414i$; a mismatch larger than the roughly $0.01\\%$ WKB error would refute the quantitative claim.","supporting_citations":[{"cited_title":"The Asymptotic Safety Scenario in Quantum Gravity","cited_arxiv_id":null,"evidence_quote":"Supplies the renormalization-group running of Newton's constant that motivates the quantum-corrected metric."},{"cited_title":"Renormalization group improved black hole space-times","cited_arxiv_id":null,"evidence_quote":"Provides the Bonanno–Reuter RG-improved metric that is the background spacetime under study."},{"cited_title":"A consistent model of non-singular Schwarzschild black hole in loop quantum gravity and its quasinormal modes","cited_arxiv_id":null,"evidence_quote":"Supplies the effective anisotropic-fluid interpretation and the axial perturbation setup from which the effective potential is derived."},{"cited_title":"Stability of a Schwarzschild singularity","cited_arxiv_id":null,"evidence_quote":"Defines the Regge–Wheeler gauge used for the axial metric perturbations."},{"cited_title":"Probing the effective quantum gravity via quasinormal modes and shadows of black holes","cited_arxiv_id":null,"evidence_quote":"Co-sources the effective axial-gravitational potential formula (Eq. 11) used for the wave equation."},{"cited_title":"Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach","cited_arxiv_id":null,"evidence_quote":"Provides the higher-order WKB formula whose sixth order is used to compute quasinormal frequencies and grey-body factors."},{"cited_title":"Quasinormal modes of black holes","cited_arxiv_id":null,"evidence_quote":"Introduces the Padé resummation that improves the WKB accuracy."},{"cited_title":"Late time behavior of stellar collapse and explosions: 1","cited_arxiv_id":null,"evidence_quote":"Supplies the characteristic time-domain integration scheme used to independently confirm the WKB frequencies."},{"cited_title":"Correspondence between grey-body factors and quasinormal modes","cited_arxiv_id":null,"evidence_quote":"Proposes the eikonal correspondence between grey-body factors and the fundamental quasinormal mode, used as a second method for grey-body factors."}],"review_version":1}