{"id":"157f9b04-faa8-48ff-b086-1f7f56cbd803","arxiv_id":"2412.07172","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"For a dark-energy-induced dark matter black hole metric, quasinormal mode frequencies decrease in real part and damping as η and λG increase, relative to Schwarzschild.","lead":"This paper computes quasinormal mode frequencies for a black hole spacetime that models dark matter via a dark-energy-induced graviton mass. It claims that the dark matter and shielding parameters lower oscillation frequencies and damping rates, which could serve as a gravitational-wave probe of dark matter.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For M=1/2, λG=2, f(r) in Eq. (2) has no zero for any η≥0, so the central QNM tables are not black hole quasinormal modes; the η=0 row is not the Schwarzschild metric.","rationale":"The reader's weakest assumption is exactly the load-bearing flaw: for the central parameter choice M=1/2, λG=2, the metric function is negative everywhere, so there is no horizon and the system is not a black hole. This invalidates the QNM boundary conditions used throughout. The additional observation that the η=0 row reproduces the exact Schwarzschild frequency even though f=e^{-r/2}-1/r is not Schwarzschild is a second manifestation of the same inconsistency: the numerical pipeline is not actually solving Eqs. (8)-(9) with metric (2). I therefore agree with the reader's REJECT verdict. The test above would settle the issue unambiguously by checking for zeros of f(r) and by recomputing the η=0 row from the stated potential.","tokens_in":13149,"tokens_out":7516,"duration_ms":71341,"concrete_test":"Numerically evaluate f(r) from Eq. (2) with M=1/2, λG=2 for η=0 and η=0.4 on a log-spaced grid r∈[10^{-6},10^3], checking for sign changes; if there are none, the spacetime has no horizon. As a second check, recompute the η=0, l=2 scalar QNM from the actual potential (9) with f=e^{-r/2}-1/r; if the resulting complex frequency differs from the Schwarzschild value 0.967284-0.193532i in Table II, the baseline comparison is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical results (Tables II, IV, VI and Figures 2, 4) fix M=1/2 and λG=2. For these values, Eq. (2) gives f(r)=e^{-2η/r}e^{-r/2}-1/r. Since max_r(r e^{-r/2})=2/e<1, one has e^{-r/2}<1/r for all r>0, and multiplying by e^{-2η/r}≤1 only lowers the left side. Hence f(r)<0 for every r>0 and every η≥0: there is no event horizon and the spacetime is a naked singularity. The tortoise coordinate (7) therefore does not map a horizon to r*→-∞ and infinity to r*→+∞, so the QNM boundary conditions (25)-(26) invoked in the WKB and time-domain calculations are not defined. Moreover, the η=0 row of Table II reports the exact Schwarzschild frequency 0.967284-0.193532i, but the η=0 metric with λG=2 is f(r)=e^{-r/2}-1/r, not Schwarzschild, and it has no horizon. The paper's claim that η=0 degenerates to Schwarzschild would require λG→∞, not λG=2. The claimed reduction relative to Schwarzschild is therefore computed against an inconsistent baseline; the central claim of the paper is not supported by the model as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies quasinormal modes (QNMs) of scalar, electromagnetic, and axial gravitational perturbations of a spherically symmetric dark-energy-induced dark matter spacetime given by Eqs. (1)–(2). It fixes M=1/2 and λG=2, uses EHT M87* shadow constraints to delimit the dark matter parameter η, and computes fundamental QNM frequencies with sixth-order WKB and time-domain/Prony methods (Tables II–VII). The central reported result is that increasing η or λG decreases both the real part of the QNM frequency and the absolute value of its negative imaginary part. However, for the central parameter choice M=1/2, λG=2 the metric function has no zero for any η≥0, so the spacetime has no event horizon and the QNM boundary conditions used in the paper are not defined.","tokens_in":13527,"tokens_out":6176,"duration_ms":65054,"significance":"The intended application is timely: ringdown QNMs are a promising probe of dark-matter environments around supermassive black holes, and the paper applies two standard numerical methods with WKB/time-domain cross-checks. The verification of the Prony extraction against the known Schwarzschild fundamental frequency is a good idea. However, the significance is undermined by a load-bearing geometric error: the spacetime used for Tables II, IV, VI and Figures 2, 4 has no event horizon for λG=2, so the computed frequencies are not black-hole quasinormal modes in the sense required by the stated boundary conditions. In addition, the paper's claim that η=0 degenerates to Schwarzschild is false for λG=2, making the Schwarzschild comparison baseline inconsistent. The physical interpretation and the central claim therefore do not follow from the calculations as presented.","major_comments":[{"comment":"For M=1/2 and λG=2, the metric function is f(r)=e^{-2η/r}e^{-r/2}-1/r. For every η≥0 and r>0, e^{-2η/r}≤1 while the maximum of r e^{-r/2} is 2/e<1, so e^{-r/2}<1/r and hence f(r)<0 for all r>0. Thus f has no zero, no event horizon exists, and the spacetime is a naked singularity. Consequently the tortoise coordinate (7) does not map a horizon to r*→−∞, and the QNM boundary conditions (25)–(26) are not defined. Since all central numerical tables fix λG=2, the computed frequencies are not black-hole quasinormal modes.","section":"Section II, Eq. (2); Tables II–VII"},{"comment":"The claim that η=0 degenerates to the Schwarzschild black hole is incorrect for the parameter values used. With λG=2, the η=0 metric is f(r)=e^{-r/2}-1/r, which has no horizon and is not Schwarzschild; the Schwarzschild metric would require λG→∞. The η=0 rows of Tables II, IV, and VI report the exact Schwarzschild QNM frequencies (e.g., 0.967284−0.193532i for scalar perturbations), but these values cannot be produced by the metric actually employed. The reported percentage shifts relative to Schwarzschild are therefore not meaningful.","section":"Below Eq. (2); Section IV; Tables II, IV, VI"},{"comment":"The effective potentials are not reliably derived as printed. In Eq. (9), the bracket contains terms of different length dimensions: l(l+1)/r^2 has dimension 1/L^2, while terms such as 2M/r^2 have dimension 1/L; the expression also has unbalanced parentheses. For η=0 it does not reduce to the standard scalar/electromagnetic potential f[l(l+1)/r^2+(1−s)2M/r^3]. Similarly, Eq. (18) mixes dimensionless, length, and inverse-length terms, so the axial gravitational potential (23) is not trustworthy. These issues are secondary to the absence of a horizon, but they would need to be corrected in any revision.","section":"Section II, Eqs. (9), (18), and (23)"}],"minor_comments":[{"comment":"There are numerous typographical and language issues, including 'the spherical harmonic function of Hamilton' (should be 'of the spherical harmonics'), inconsistent spacing in section headings, and incomplete sentences in the abstract and introduction.","section":"Throughout"},{"comment":"The horizontal axis labels 'r*/2M' are confusing because the potentials are plotted versus r*, and the scale extends to negative r*; the axis label and tick values should be made consistent with the variable actually used.","section":"Figures 2 and 3"},{"comment":"The discretization formula contains signs and factors that are not fully derived; a reference to the standard null-cone scheme or a brief derivation would improve reproducibility.","section":"Section III.B, Eq. (29)"},{"comment":"The text states that these tables show the effect of λG, but the chosen range includes values for which the metric has no horizon as well as values where a horizon may exist; the physical interpretation of the trend in λG depends on the horizon issue raised above.","section":"Section IV, Tables III, V, VII"}],"recommendation":"reject","confidential_remarks":"The rejection is based on a geometric inconsistency in the background spacetime for the central parameter choice, not on disagreement with the dark-matter interpretation. If the authors can identify a horizon-having parameter region and redo the parameter constraints and QNM computations entirely within that region, a substantially revised manuscript might be considered. As it stands, the central numerical results are not quasinormal modes of a black hole under the stated boundary conditions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on 2412.07172. The paper is the first to compute quasinormal modes for the dark-energy-induced dark matter metric of Pantig [47], and it does so with two independent methods—sixth-order WKB and time-domain Prony—which is more than many papers in this area do. The qualitative trend they report (real frequency and damping both decrease as η and λG grow) is also the kind of behavior you'd expect from a softened potential. So there is real labor here, and the idea of using QNMs to probe this specific spacetime isn't silly on its face.\n\nThat said, the central results are not black hole quasinormal modes. For all the main tables—Tables II, IV, VI, and the corresponding figures—the parameters are M=1/2, λG=2. For those values, the metric function is f(r)=e^{-2η/r}e^{-r/2}-1/r. Even at η=0, e^{-r/2}<1/r for every r>0, so f(r) is negative everywhere and there is no event horizon. The extra factor e^{-2η/r} only pushes it further negative. The spacetime is a naked singularity, not a black hole. The tortoise coordinate in (7) then doesn't map a horizon to r*→-∞, and the boundary conditions (25)-(26) that define QNMs don't exist. The WKB and time-domain codes are solving a scattering problem on a singular spacetime, and the numbers in the tables have no clear physical meaning.\n\nThe paper also misidentifies the η=0 limit. It claims η=0 reduces the metric to Schwarzschild, but that's only true if λG→∞. With λG=2, the η=0 metric is f(r)=e^{-r/2}-1/r, which is not Schwarzschild and has no horizon. The η=0 rows in Tables II, IV, VI, which report the exact Schwarzschild QNM frequencies, are therefore not the correct baseline for comparison.\n\nThere's a secondary issue: Eq. (9) for the effective potential is dimensionally inconsistent as printed and doesn't reduce to the standard scalar potential in the Schwarzschild limit. The axial potential (23) is also hard to trust, given the parent derivation doesn't track dimensions carefully.\n\nNone of this is a matter of taste. The central claim—that this spacetime's QNMs show a measurable dark-matter effect—is unsupported because the spacetime chosen for the main computation isn't a black hole. The paper would need a serious reworking: identify parameter ranges where a horizon actually exists, recompute, and redo the baseline comparison. As written, I wouldn't send it to a referee; it would waste their time. The topic is legitimate, and a corrected version might be worth another look, but this version has a load-bearing flaw.\n\nMy recommendation: desk reject.","headline":"First QNM computation for this dark-matter metric, but the central tables use parameters where no horizon exists, so the numbers are not black hole quasinormal modes.","tokens_in":13963,"tokens_out":3602,"would_cite":false,"duration_ms":34839,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","04.30.-w","95.35.+d"],"model":"deepseek-v4-flash","headline":"The paper claims that dark matter and gravitational shielding measurably lower black hole ringdown frequencies and slow their decay.","keywords":["quasinormal modes","dark matter","dark energy","gravitational shielding","black hole ringdown","WKB approximation","time-domain method","M87*"],"falsifier":"Evaluate $f(r)=e^{-2\\eta/r}e^{-r/2}-1/r$ for $M=1/2$ and $\\lambda_G=2$ with any $\\eta>0$: the function stays negative for all $r>0$, so a horizon search finds no root and the standard quasinormal-mode boundary conditions cannot be imposed; this direct calculation would settle whether the quoted frequencies are legitimate black hole modes.","tokens_in":12995,"feed_emoji":"🕳️","tokens_out":5817,"duration_ms":56139,"temperature":0.7,"pith_summary":"The paper argues that the ringdown gravitational waves from a supermassive black hole like M87* carry a measurable imprint of dark matter and dark energy. It works with a metric in which dark energy gives gravitons an effective mass, producing a Yukawa-like suppression of gravity at a length scale λG and a dark-matter parameter η. By computing quasinormal mode frequencies for scalar, electromagnetic, and axial gravitational perturbations, it finds that increasing η or λG lowers the real oscillation frequency and shrinks the magnitude of the negative imaginary frequency, meaning weaker, longer-lived oscillations. The implication, if correct, is that observed ringdown waveforms could distinguish this dark-matter environment from a bare Schwarzschild black hole and constrain the parameters.","feed_headline":"Dark matter and shielding shift black hole ringdown frequencies","feed_subtitle":"When dark matter and gravitational shielding strengthen, black holes oscillate more slowly and ring down for longer.","key_machinery":"The load-bearing object is the metric function $f(r)=e^{-2\\eta/r}e^{-r/\\lambda_G}-2M/r$, with $\\lambda_G=1/\\sqrt{2\\Lambda}$ acting as a gravitational shielding length and $\\eta$ parametrizing the dark matter distribution. From this $f$, the paper derives three effective potentials — scalar ($s=0$), electromagnetic ($s=1$), and axial gravitational ($s=2$) — and reduces each perturbation to a Schrödinger-like equation in the tortoise coordinate. It then solves those equations with the sixth-order WKB scheme and with a time-domain finite-difference scheme plus Prony extraction, using the $\\eta=0$, $M=1/2$ Schwarzschild mode to calibrate the numerics.","core_discovery":"The central discovery is a parameter-dependent shift in the quasinormal-mode spectrum of the metric $f(r)=e^{-2\\eta/r}e^{-r/\\lambda_G}-2M/r$. For fixed multipole $l=2$, as the dark matter parameter $\\eta$ rises (with $\\lambda_G=2$) or as the gravitational shielding length $\\lambda_G$ rises (with $\\eta=0.4$), the real part of the fundamental mode frequency decreases monotonically and the imaginary part, though negative, decreases in absolute value across all three perturbation types. The $\\eta=0$ limit is taken to reproduce the Schwarzschild quasinormal frequencies, and the same trend appears both in sixth-order WKB approximations and in time-domain Prony extraction, so the paper treats the trend as a robust signature of the model.","pith_inferences":["Beyond the paper's claims: a natural next check is to repeat the calculation for parameter sets where $f(r)$ actually has an event horizon; if the monotonic trends persist there, the signal is physical, and if not, the reported numbers are artifacts of boundary conditions applied to a horizonless spacetime.","The same metric could be tested against echo or shadow signatures; a naked-singularity interpretation would change the expected late-time ringdown entirely.","The connection between $\\lambda_G$ and a cosmological constant suggests the QNM shift could be translated into a bound on $\\Lambda$ using a single ringdown observation."],"forward_implications":["If the model is right, ringdown observations of supermassive black holes should show lower frequencies and longer damping times than Schwarzschild predictions for the same mass.","The M87* 3σ shadow constraints ($0.09978 \\leq \\lambda_G \\leq 6.113$ at $\\eta=0.4$; $0 \\leq \\eta \\leq 0.94$ at $\\lambda_G=2$) define a concrete window in which QNM deviations of order 10–20% in the real part are expected.","Because the imaginary part stays negative for all parameter choices studied, the spacetime is linearly stable against these perturbations within the considered range.","The two independent numerical methods agree closely, so the monotonic frequency shift should be reproducible by other quasinormal-mode codes."],"supporting_citations":[{"why":"Supplies the dark-matter black hole metric $f(r)$ and the M87* parameter ranges used in Table I.","marker":"[47]"},{"why":"Proposes the dark energy-induced graviton mass and Yukawa-like potential that underlies the metric.","marker":"[44]"},{"why":"Introduces the WKB method for black hole scattering that the paper's sixth-order WKB calculation extends.","marker":"[52]"},{"why":"Extends the WKB approximation to higher orders, underpinning the 6th-order formula used here.","marker":"[53]"},{"why":"Completes the higher-order WKB corrections that the paper relies on for the quasinormal frequencies.","marker":"[54]"},{"why":"Provides the sixth-order WKB corrections and the argument that matter perturbations are negligible against the modified background.","marker":"[50]"},{"why":"Gives continued-fraction Schwarzschild quasinormal frequencies used to calibrate the Prony and WKB results.","marker":"[56]"},{"why":"Reviews quasinormal mode theory and establishes that mode frequencies depend only on black hole parameters, motivating the analysis.","marker":"[10]"}],"fun_headline_variants":["Dark matter and shielding make black holes ring slower and longer","Stronger dark matter and shielding slow black hole ringdown","As dark matter and shielding rise, black holes ring down longer","Dark energy-induced dark matter alters black hole oscillation frequencies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that metric (2) with $M=1/2$ and $\\lambda_G=2$ describes a black hole, but in fact for those values the metric function is negative at every radius, so the spacetime has no event horizon; if it is not a black hole, the imposed boundary conditions and the quoted 'black hole' quasinormal modes are not physically applicable.","fun_headline_variants_meta":{"raw":{"variants":["Dark matter and shielding make black holes ring slower and longer","Stronger dark matter and shielding slow black hole ringdown","As dark matter and shielding rise, black holes ring down longer","Dark energy-induced dark matter alters black hole oscillation frequencies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000848,"raw_usage":{"total_tokens":3667,"prompt_tokens":897,"completion_tokens":2770,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":2703}},"tokens_in":513,"tokens_out":2770,"duration_ms":22666,"temperature":1.0,"reasoning_tokens":2703,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:05:00.926397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $f(r)=e^{-2\\eta/r}e^{-r/2}-1/r$ for $M=1/2$ and $\\lambda_G=2$ with any $\\eta>0$: the function stays negative for all $r>0$, so a horizon search finds no root and the standard quasinormal-mode boundary conditions cannot be imposed; this direct calculation would settle whether the quoted frequencies are legitimate black hole modes.","supporting_citations":[],"review_version":1}