{"id":"b1df0353-f6d2-437d-a5da-5c231040645b","arxiv_id":"2412.13385","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Sixth-order WKB formulas for wormhole quasinormal modes and grey-body factors are presented, but the grey-body factor part is internally inconsistent.","lead":"The paper derives new analytic WKB formulas for quasinormal modes and grey-body factors of three Morris-Thorne wormhole models, exploiting that the potential peak sits at the throat. The quasinormal mode parts look plausible, but the grey-body factor expressions contain internal inconsistencies and the shadow-correspondence check uses a suspect formula.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Shadow-correspondence check in Sec. V.D uses R_s=b0/Φ(b0), which diverges for the paper's tideless models and is wrong for model 3, leaving the abstract's central claim unverified.","rationale":"The paper's main deliverable is a set of analytic WKB formulas plus the claim that they satisfy the shadow correspondence. The QNM expansions in Secs. V.A-C appear internally plausible: I checked the eikonal limits against the standard parabolic-barrier WKB and they agree for models 1 and 2. The grey-body factor formula (15) also has the correct low- and high-energy limits when combined with the eikonal K. The single point where the argument is not just unproven but demonstrably wrong is the shadow-radius formula in Sec. V.D. A formula that gives an infinite radius for the paper's own first two examples cannot be used to verify a finite QNM correspondence; and for model 3 it disagrees with the value implied by the paper's own eikonal QNM. This directly undermines the abstract's claim. The reader identified exactly this weak point. No other internal inconsistency is as cleanly fatal; the GBF polynomial cancellations I checked are consistent with the eikonal limit. Therefore the concern lands and the reader's rejection is justified. A corrected paper that uses R_s = b0 e^{-Φ(b0)} and shows the equality would remove the objection. If such a correction is supplied, the verdict could be revisited.","tokens_in":10195,"tokens_out":22150,"duration_ms":181409,"concrete_test":"Re-derive the shadow radius for the metric (1) from null geodesics: the critical impact parameter is b_c = r/√(e^{2Φ(r)}) at the throat r=b0, giving R_s = b0 e^{-Φ(b0)}. Compare this with Eq. (22) for the three models: for Φ=0, b0/Φ(b0) diverges while the geodesic result is b0; for Φ=1/r^p, the two differ by a factor e^{2Φ(b0)}/Φ(b0). If the geodesic result is as stated, the paper's Sec. V.D check cannot be correct.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes the eikonal shadow correspondence ω_R = (ℓ+1/2)/R_s. Section V.D asserts this is satisfied and quotes the shadow radius as R_s = b0/Φ(b0) (Eq. 22). For the two tideless models (Φ=0), this gives R_s = ∞, while the paper's own eikonal QNMs (ωb0 = κ for models 1 and 2, up to O(1/κ)) require R_s = b0. For model 3, Φ(r)=1/r^p at b0=1, the paper's formula gives R_s = b0^{p+1}, whereas the eikonal ω_R = eκ (with e = e^{Φ(b0)}) implies R_s = b0/e. The correct null-geodesic impact parameter for this metric is R_s = b0 e^{-Φ(b0)}. Thus the verification step is internally inconsistent and does not support the abstract's claim; the correspondence may be repairable, but as written it is not demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives sixth-order WKB analytic expressions for the quasinormal modes and grey-body factors of three Morris-Thorne wormhole models (Ellis-Bronnikov, a tideless model with b(r)=sqrt(b0 r), and a model with non-zero tidal force). The derivation exploits the claim that the effective potential maximum lies exactly at the throat, which simplifies the WKB expansion. The paper further claims that in the eikonal limit the quasinormal frequencies satisfy the shadow-radius correspondence omega_R = (l+1/2)/R_s. The central claims are the accuracy of the analytic expressions and the eikonal correspondence.","tokens_in":10450,"tokens_out":9945,"duration_ms":86502,"significance":"The idea of using the exact location of the potential maximum at the throat to obtain compact analytic WKB formulas is attractive and could be useful for phenomenological studies of wormhole ringdown and absorption. If correct, the eikonal expressions would provide a simple link between wormhole geometry and observable gravitational-wave signatures. However, the manuscript contains load-bearing inconsistencies in the grey-body factor formulas and in the shadow-radius verification, so the claimed results are not supported as written.","major_comments":[{"comment":"The full WKB relation for K that determines the grey-body factor is inconsistent with the eikonal limit reported in Sec. V.D. Setting Omega=0 in Eq. (19) and keeping the leading term in kappa gives i K approximately -0.943/kappa, i.e. K approximately +i 0.943/kappa, so the transmission probability Gamma_l(0) is approximately 1/2 for large l. In contrast, the eikonal expression K = i(b0^2 Omega^2 - kappa^2)/(2 kappa) gives K = -i kappa/2 at Omega=0, which yields Gamma_l(0) tending to 0. The contradiction shows that the grey-body factor formulas in Eqs. (19) and (21) do not reduce to the claimed eikonal limit and are not reliable for the stated purpose.","section":"V.A, Eq. (19)"},{"comment":"The shadow radius formula R_s = b0/Phi(b0) is incorrect for the models studied. For the tideless models (Phi=0) the denominator vanishes, while the eikonal QNMs in Sec. V.D imply R_s = b0. For model 3, with b0=1 and Phi(r)=1/r^p, the formula gives R_s = 1, whereas the eikonal frequency omega_R = e kappa/b0 requires R_s = b0/e = e^{-1}. The correct impact parameter for null geodesics in this metric is R_s = b0 e^{-Phi(b0)}. Thus the claimed verification of the eikonal QNM-shadow correspondence is unsupported.","section":"V.D, Eq. (22)"}],"minor_comments":[{"comment":"The assertion that the effective potential has its maximum at the throat is used throughout but is not proved for the potentials in Eqs. (9)-(10); a short argument or a citation would make the paper more self-contained.","section":"Sec. II"},{"comment":"The symbol e appears without definition in the eikonal formulas for model 3; it should be introduced explicitly, for example e = exp(Phi(b0)), to avoid confusion with the base of the natural logarithm.","section":"Sec. V.D"},{"comment":"The equation numbers for the definitions of alpha and beta (printed as Eqs. (18) and (19)) collide with the equation numbers used for the grey-body relations in Sec. V.A; the numbering should be corrected.","section":"Sec. V.D"},{"comment":"There are several typos, including 'disccussed' (Sec. I), 'other other method' (Sec. III), 'eseentially' (Sec. IV), 'becuase' (Sec. IV), and 'does require' in Sec. IV which should read 'does not require'.","section":"Throughout"},{"comment":"The abstract claims 'accurate' analytic expressions, but the manuscript provides no numerical comparison with exact or independent results; in light of the inconsistency in Eq. (19), the accuracy claim needs either a correction of the formulas or a benchmark validation.","section":"Abstract and Sec. V"}],"recommendation":"reject","confidential_remarks":"The paper fits the journal's scope, and the approach is not circular: the self-citations are not used as input to the derivation. However, the two load-bearing errors described in the major comments undermine the central claims, and a rejection is recommended."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a standard sixth-order WKB calculation for three Morris-Thorne wormholes, and the new formulas are probably useful, but the paper's advertised eikonal shadow-correspondence check is not actually done—the radius formula used is wrong—and the grey-body factor relation conflicts with its own eikonal limit at Ω=0.\n\nWhat is new: the sixth-order analytic expressions for QNMs and grey-body factors for the Ellis-Bronnikov, Morris-Thorne type 2, and the power-law shape/redshift models. The trick of using the throat as the exact potential maximum is legitimate and gives compact formulas. The QNM series look plausible, and the eikonal real parts are consistent with the geometry.\n\nThe problems are in the verification. Eq. (22) gives the shadow radius as b0/Φ(b0). For the two tideless models Φ=0 this diverges, yet the eikonal QNMs give R_s=b0. For model 3, Φ(b0)=1 (with b0=1), the formula gives R_s=1, while the eikonal ω_R=eκ implies R_s=1/e. So the claimed correspondence is not demonstrated. The correct formula is presumably R_s=b0 e^{-Φ(b0)}, which would make everything consistent, but that is not what is written, and the reader should not have to reverse-engineer it.\n\nSecond: Eq. (19) at Ω=0 gives K ≈ i 0.94/κ, so Γ(0) ≈ 0.5, whereas the eikonal K in Sec. V.D gives K = -iκ/2 and Γ(0) → 0. That is a genuine inconsistency in the grey-body factor part. The GBF formulas might be correct away from Ω=0, but the paper does not check the zero-frequency limit. The third model's GBF is omitted as too lengthy, which is acceptable but leaves that part unverifiable.\n\nCitation pattern is fine; self-citations are to the author's own prior WKB work, not used as input.\n\nWho this is for: people working on wormhole QNMs and wanting ready-made sixth-order expressions. It deserves a serious referee because the core computation is checkable and the flaws are repairable, but as written the central claim fails. Recommendation: send to review, with a request to fix the shadow radius formula and verify the Ω=0 limit of the GBF relations before acceptance.","headline":"Useful sixth-order WKB formulas for wormhole QNMs, but the shadow-correspondence check and grey-body eikonal limit are internally inconsistent as written.","tokens_in":10900,"tokens_out":3450,"would_cite":false,"duration_ms":31066,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.30.Nk","04.50.+h"],"model":"deepseek-v4-flash","headline":"The paper derives analytic sixth-order WKB series for quasinormal frequencies and grey-body factors of three Morris-Thorne wormhole families and claims they satisfy the eikonal shadow-radius correspondence.","keywords":["quasinormal modes","grey-body factors","Morris-Thorne wormholes","WKB approximation","eikonal approximation","wormhole shadow","effective potential","traversable wormholes"],"falsifier":"Integrate the null geodesic equation in the Ellis-Bronnikov metric ($\\Phi=0$, $b(r)=b_0^2/r$) to find the critical impact parameter; if the resulting shadow radius is not $b_0$, then the paper's own eikonal frequency $\\omega_R=(\\ell+1/2)/b_0$ cannot satisfy the shadow correspondence $\\omega_R=(\\ell+1/2)/R_s$, and the claimed extension fails.","tokens_in":10020,"feed_emoji":"🕳️","tokens_out":22498,"duration_ms":182615,"temperature":0.7,"pith_summary":"The paper claims that for a broad class of Morris-Thorne wormholes, the perturbation effective potential has its maximum exactly at the throat, and that this single fact makes the sixth-order WKB method fully analytic. Working from that premise, it derives closed-form expansions, in powers of the inverse multipole number $\\kappa=\\ell+1/2$, for electromagnetic and scalar quasinormal frequencies, together with matching grey-body factors $\\Gamma_\\ell(\\Omega)=1/(1+e^{2\\pi i K})$, for three wormhole families: the Ellis-Bronnikov wormhole, a tideless model with shape function $b(r)=\\sqrt{b_0 r}$, and a family $b(r)=b_0(b_0/r)^q$ with optional redshift function $1/r^p$. In the eikonal limit the frequencies reduce to compact forms whose real parts are proportional to $\\kappa$, and the paper concludes that they satisfy the correspondence $\\omega_R=(\\ell+1/2)/R_s$ with the wormhole shadow radius. If this is right, it provides an analytic bridge between wormhole ringdown, grey-body transmission spectra, and shadow measurements, which could help distinguish wormholes from black holes observationally.","feed_headline":"Wormhole ringdown linked to shadow radius by analytic formulas","feed_subtitle":"Closed-form quasinormal modes and grey-body factors now link wormhole ringdown to shadow size.","key_machinery":"The central object is the effective potential $V(r)$ of the scalar and electromagnetic perturbations. For the Morris-Thorne class considered, its maximum lies exactly at the throat radius $r=b_0$, and this peak location is known exactly rather than found by an expansion. The machinery is the sixth-order WKB formula for quasinormal modes, whose correction terms $\\Lambda_i$ depend on derivatives of $V$ at the peak; because the peak is at $b_0$, all derivatives can be evaluated in closed form, giving the presented series in $\\kappa=\\ell+1/2$. For grey-body factors the same peak data enter the phase $K$ in $\\Gamma_\\ell(\\Omega)=1/(1+e^{2\\pi i K})$, kept unexpanded in inverse multipole number, and the eikonal interpretation is carried by the shadow correspondence $\\omega_R=(\\ell+1/2)/R_s$.","core_discovery":"The paper claims that the throat-peak property of Morris-Thorne wormholes turns the usually numerical WKB calculation into an analytic one. It presents explicit sixth-order expansions for the quasinormal frequencies of scalar and electromagnetic perturbations of the Ellis-Bronnikov wormhole, of the tideless wormhole with $b(r)=\\sqrt{b_0 r}$, and of the family $b(r)=b_0(b_0/r)^q$ with $\\Phi(r)=1/r^p$ (both tideless and redshifted variants). It gives the corresponding grey-body factors through the compact formula $\\Gamma_\\ell(\\Omega)=1/(1+e^{2\\pi i K})$, where $K$ is obtained without an additional inverse-multipole expansion. In the eikonal limit $\\ell\\to\\infty$ the frequencies reduce to simple forms whose real parts are $\\kappa$ divided by a radius, and the paper concludes that these satisfy $\\omega_R=(\\ell+1/2)/R_s$ with $R_s$ the wormhole shadow radius. The intended upshot is that wormhole ringdown and shadow observations probe the same geometric quantity.","pith_inferences":["Beyond the paper: the method's WKB grey-body-factor ansatz excludes superradiant amplification, so extending the calculation to rotating wormholes will need a different approach when the reflection coefficient exceeds unity.","Beyond the paper: the non-tideless eikonal frequency is rescaled by the throat redshift factor; if the shadow correspondence is exact, the effective shadow radius should be the throat radius divided by that factor, a prediction ray tracing could test.","Beyond the paper: the closed-form grey-body factors could be integrated against a thermal spectrum to predict the wormhole's emission signature, which the paper does not do; that would turn the analytic transmission coefficients into an observational template."],"forward_implications":["The sixth-order expansions give immediate closed-form approximations for scalar and electromagnetic ringdown of the three wormhole families, removing the need for numerical integration at high multipoles and low overtones.","The grey-body factors $\\Gamma_\\ell(\\Omega)=1/(1+e^{2\\pi i K})$ become explicit functions of frequency and wormhole parameters, so transmission and reflection spectra can be compared directly with black-hole templates.","The eikonal relation $\\omega_R=(\\ell+1/2)/R_s$ turns a measured high-frequency ringdown into a direct estimate of the wormhole shadow radius, and vice versa, for these spacetimes.","The same throat-peak argument can generate analytic quasinormal modes and grey-body factors for any Morris-Thorne throat where the effective potential peaks at $r=b_0$, making the method a template for other shape and redshift functions.","For the redshifted family the eikonal frequency is multiplied by the throat redshift factor, so the ringdown carries direct information about the lapse function at the throat."],"supporting_citations":[{"why":"It supplies the Morris-Thorne metric ansatz and the throat and traversability conditions used for every model in the paper.","marker":"[1]"},{"why":"It states the eikonal quasinormal-mode and shadow-radius correspondence that the paper claims to verify for the three wormhole families.","marker":"[14]"},{"why":"It provides the wormhole scattering and ringdown setup, including the type-3 shape and redshift model whose frequencies are analyzed.","marker":"[23]"},{"why":"It supplies the grey-body-factor ansatz $\\Gamma_\\ell(\\Omega)=1/(1+e^{2\\pi i K})$ and the distinction between scattering frequencies and quasinormal frequencies.","marker":"[25]"},{"why":"It is the original WKB quasinormal-mode method whose lowest order the paper's sixth-order expansions extend.","marker":"[30]"},{"why":"It provides the higher-order WKB corrections used to build the analytic quasinormal-mode series.","marker":"[31]"},{"why":"It is the sixth-order WKB formula that the paper evaluates at the throat to obtain the closed-form expressions.","marker":"[32]"},{"why":"It supplies the further WKB correction terms that make the sixth-order expansions accurate.","marker":"[33]"},{"why":"It gives earlier analytic quasinormal-mode expressions for wormhole metrics that this paper's throat-peak method extends to asymptotically flat cases.","marker":"[71]"},{"why":"It defines the Ellis-Bronnikov wormhole metric used as the first worked example.","marker":"[72, 73]"}],"fun_headline_variants":["Analytic formulas tie wormhole ringdown to shadow size","Wormhole quasinormal modes solved analytically via WKB","Throat peak yields exact WKB for wormhole modes and shadows","Wormhole ringdown and shadow now linked by closed forms","Eikonal wormhole modes match shadow radius analytically"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the shadow-radius formula used to check the eikonal correspondence is the right one; for the tideless wormholes it gives no finite value, so the claimed agreement depends on correcting or replacing it.","fun_headline_variants_meta":{"raw":{"variants":["Analytic formulas tie wormhole ringdown to shadow size","Wormhole quasinormal modes solved analytically via WKB","Throat peak yields exact WKB for wormhole modes and shadows","Wormhole ringdown and shadow now linked by closed forms","Eikonal wormhole modes match shadow radius analytically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2139,"prompt_tokens":827,"completion_tokens":1312,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":1228}},"tokens_in":443,"tokens_out":1312,"duration_ms":9081,"temperature":1.0,"reasoning_tokens":1228,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:14:54.843042+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate the null geodesic equation in the Ellis-Bronnikov metric ($\\Phi=0$, $b(r)=b_0^2/r$) to find the critical impact parameter; if the resulting shadow radius is not $b_0$, then the paper's own eikonal frequency $\\omega_R=(\\ell+1/2)/b_0$ cannot satisfy the shadow correspondence $\\omega_R=(\\ell+1/2)/R_s$, and the claimed extension fails.","supporting_citations":[],"review_version":1}