{"id":"cf0db591-a712-4e31-9e4e-7484748b0ae5","arxiv_id":"2506.15763","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors analyze thermodynamics, shadow, and quasinormal modes of a Horndeski black hole with a dark matter halo and derive tight EHT bounds that force the new coupling parameters near zero.","lead":"This paper studies a black hole solution in a modified gravity theory (Horndeski) surrounded by dark matter, computing its heat, shadow, and ringdown signals. It uses the shadow size of Sagittarius A* to argue the modified gravity and dark matter parameters must be very small, keeping general relativity nearly unchanged.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The combined Horndeski–PFDM metric in Eq. (8) is never shown to solve the coupled field equations; if it is not a solution, every shadow and QNM result describes a metric outside the claimed theory.","rationale":"The reader's weakest assumption is indeed the most load-bearing: Eq. (8) is a linear superposition of the pure quartic square-root Horndeski solution (Eq. 6, from [5]) and the PFDM lapse (from [27]), with no derivation that the combination solves the coupled scalar–metric–matter equations. This matters because the paper's headline — that EHT Sgr A* shadows force β, η, and b to be tiny — is obtained by computing the shadow of Eq. (8); if Eq. (8) is not a genuine solution, the bound constrains a metric that lives outside the Horndeski–PFDM theory, and the conclusion is void. I agree with the reader here. I also note two additional internal inconsistencies that support rejection but are secondary: (1) the entropy in Eq. (13) is πr_h²/4, which is neither A/4=πr_h² nor consistent with F=M−TS giving Eq. (14); (2) the 'upper bounds' β<−0.5929, η<−0.3263, b<0.01059 in Section V point in the wrong direction given Table I, where the computed shadow radii (3.06–4.09)M all fall below the EHT 1σ interval (4.55–5.22)M, and larger |β| makes the shadow smaller, so the constraint should bound |β| from above, not below. These issues are real but the solution-status question is prior: if the metric fails the field-equation test, the numerical outputs are irrelevant to the theory. The concrete test I propose (direct substitution into the full Horndeski equations) would settle the prior question; it is computationally straightforward and would likely appear in any serious follow-up. A fair reading credits the authors with computing standard diagnostics carefully for their ansatz, and the 6th-order WKB error estimates are a useful check of the method; the failure is the missing link between the action and the metric. Therefore the preprint, as written, does not support its central claim.","tokens_in":21727,"tokens_out":14373,"duration_ms":130157,"concrete_test":"Substitute the metric (5) with f(r) from Eq. (8) into the full Horndeski field equations derived from action (4) plus the PFDM matter action with stress tensor (7), together with a static, spherically symmetric scalar ansatz (first the pure solution's φ′, then a general ansatz). Use a computer algebra system (e.g., xAct) to reduce the tt, rr, angular, and scalar equations to ODEs and check whether they vanish identically for parameter values in Table I. If no scalar profile satisfies both metric and scalar equations, the combined metric is not a solution; if a profile exists, the concern is resolved. As a minimal cross-check, verify whether the pure-Horndeski scalar equation remains satisfied when the (b/r)ln(r/b) term is added to f(r).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central observational claim (Section V) rests entirely on the metric f(r)=1−2M/r − β²/(2ηr²) + (b/r)ln(r/b) in Eq. (8), presented as the quartic square-root Horndeski solution 'immersed in' PFDM. No derivation is given: Eq. (6) is cited to [5] for the pure Horndeski solution, and the PFDM modification is simply asserted via [27], a GR PFDM paper. Adding the PFDM stress-energy tensor T_μ^ν=diag(−ρ,p_r,p_θ,p_ϕ) to the Horndeski equations changes both the metric and the scalar-field equation; the scalar profile that solved the pure case is not shown to survive, and no new profile is supplied. The superposition of lapse functions is not justified in a nonlinear scalar-tensor theory. If the combined metric is not an exact solution of the full field equations, then the thermodynamic quantities (Eqs. 9–14), the photon sphere, shadow radius, the EHT parameter bounds (β<−0.5929, η<−0.3263, b<0.01059), and the 6th-order WKB QNM frequencies in Tables II–IV are all computed for a metric that does not belong to the theory. This is load-bearing because every conclusion in the abstract and Section VII follows from those calculations. In addition, the stated bounds sit uneasily with Table I: for β=−0.5, η=−0.5, b=0.3 the shadow radius is b_crit=3.39M, far below the EHT 1σ window (4.55–5.22)M, and the direction β<−0.5929 indicates large |β|, contradicting the 'small parameters' summary. The decisive issue, however, is the missing solution check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the static spherically symmetric metric f(r)=1-2M/r-β²/(2ηr²)+(b/r)ln(r/b), presented as a quartic square-root Horndeski black hole immersed in perfect-fluid dark matter (PFDM), with parameters β, η, and b. It computes the Hawking temperature, ADM mass, specific heat, entropy, and free energy; analyzes null geodesics and the photon sphere; computes the shadow radius and compares it with Keck/VLTI-based EHT constraints on Sgr A* to obtain 1σ and 2σ bounds on β, η, and b; and calculates scalar quasinormal frequencies using sixth-order WKB approximation. The central claims are that small-horizon black holes are locally stable but never globally preferred, that increasing β and b enlarges the photon sphere and shadow while increasing η mildly shrinks them, and that EHT shadow observations require the Horndeski and PFDM parameters to be very small, leaving general relativity as the effective description near the photon sphere.","tokens_in":22138,"tokens_out":19803,"duration_ms":182538,"significance":"If the combined metric in Eq. (8) were a genuine exact solution of the Horndeski field equations with a PFDM source, the paper would provide a useful connection between modified gravity, dark matter, and EHT observations. The analysis is largely transparent: explicit expressions for thermodynamic quantities, a table of critical impact parameters and photon radii, a table of WKB frequencies, and an external comparison to EHT data with no parameter fitting, so the constraints are not circular. The quantitative shadow and QNM tables are in principle reproducible. However, the significance is fully conditional on the existence of the claimed solution and on resolving the internal inconsistencies in the thermodynamics, in the parameter-bound derivation, and in the reported parameter trends, each of which is problematic in the current text.","major_comments":[{"comment":"The combined metric is asserted without derivation. Equation (6) is cited to [5] for the pure Horndeski solution, and the PFDM term is introduced by invoking [27], a general-relativistic PFDM paper. In a scalar-tensor theory, adding a matter source changes both the metric and the scalar-field equations; the scalar profile that solves the pure Horndeski system is not shown to survive, and no new profile is supplied. The authors need to demonstrate explicitly that f(r) in Eq. (8), together with a specified scalar profile and the PFDM stress tensor (7), satisfies the full Horndeski field equations, or provide a derivation from an existing exact solution. Until this is done, the thermodynamic quantities, the shadow and photon-sphere results in Tables I-V, and the QNM frequencies describe an arbitrary metric rather than the claimed Horndeski-PFDM theory.","section":"II, Eq. (8)"},{"comment":"The entropy and the free energy are mutually inconsistent. Eq. (13) states S_BH = π r_h²/4, but the Helmholtz free energy in Eq. (14) follows from F = M - T S only if S_BH = π r_h², which is also the horizon area/4. With the stated value, Eq. (14) would have a different coefficient of β²/(η r_h), namely -9/32 instead of -3/8. This internal inconsistency affects the thermodynamic stability analysis and the claim that no Hawking-Page transition occurs (Fig. 5 and Section VII). The authors should correct Eq. (13) and verify all subsequent thermodynamic expressions.","section":"III, Eqs. (13)-(14)"},{"comment":"The quoted 1σ bounds β < -0.5929, η < -0.3263, b < 0.01059 are not consistent with the text's conclusion that the parameters are very small, and their inequality directions appear to conflict with the data presented. Table I shows that, with the fixed values used in Fig. 14, the critical impact parameter is far below the EHT 1σ window (4.55M to 5.22M): for example, b_crit = 3.39M for (β,η,b) = (-0.5,-0.5,0.3). The bound β < -0.5929 permits β = -0.7, -0.8, for which the shadow is even smaller, so as written it does not confine the model to the EHT-allowed region. In addition, β²/(2|η|) evaluated at the quoted 1σ limits is about 0.54, not the 0.06 stated near Eq. (45). The authors should display the curves r_s/M versus each parameter over a range that actually enters the EHT band and quote the inequalities in the correct direction.","section":"V.A, Fig. 14 and Table I"},{"comment":"The claimed monotonic dependence of the shadow on the model parameters is opposite to the values in Table I. For fixed β = -0.5, η = -0.5, increasing b from 0.1 to 0.7 decreases b_crit from 4.09145 to 3.06305; for fixed η = -0.5, b = 0.3, increasing |β| from 0.3 to 0.6 decreases b_crit from 3.56247 to 3.26334. The abstract states that increases in the dark matter parameter b enlarge the shadow, and Section VII states that 'increasing |β| or b consistently enlarges the critical impact parameter and photon sphere radius'. These statements are contradicted by the paper's own Table I. The qualitative summary of the geodesic and shadow results must be corrected, or the table and figures must be recomputed.","section":"Table I vs. abstract and Section VII"}],"minor_comments":[{"comment":"The sentence 'where η and β' is incomplete; it should state that they are constants and specify the sign condition required for a real scalar-field solution.","section":"II, Eq. (3)"},{"comment":"The text says the entropy follows the usual Bekenstein-Hawking form, but writes π r_h²/4; the horizon area/4 is π r_h², so the factor must be corrected.","section":"III, Eq. (13)"},{"comment":"The table and the surrounding text use 'η1' and 'β1' where η and β are meant; these should be renamed for clarity.","section":"Table V"},{"comment":"In the first panel of Fig. 3, the legend lists β = -0.5, η = -0.5 twice and does not show a distinct η = -0.4 curve; the legend should be checked.","section":"Fig. 3"},{"comment":"The axis labels in Fig. 14 appear garbled in the compiled text, with missing symbols such as η, β, and b; the figure should be regenerated.","section":"Fig. 14"},{"comment":"The phrase 'upper bounds' is misleading, because β < -0.5929 and η < -0.3263 are lower bounds on |β| and |η|, not upper bounds on the parameters; the wording should be reconciled with the actual inequalities.","section":"V.A"},{"comment":"The conclusion's 'sub-percent departure' statement is inconsistent with the stated β²/(2|η|) ≲ 0.06, which corresponds to a six-percent correction; the numbers should be reconciled.","section":"VII"},{"comment":"References [2] and [21] mix arXiv identifiers and DOIs with journal information; the reference list should be cleaned up.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the unproven combined metric in Eq. (8). I would ask the editor to require the authors to supply either a derivation of the combined Horndeski-PFDM solution or an explicit check of the full field equations and the scalar-field profile. If this cannot be provided, the paper should not proceed, since every thermodynamic and observational statement is built on that metric. The internal inconsistencies in Eqs. (13)-(14), the bound directions in Section V.A, and the contradiction between Table I and the abstract/conclusion also need to be resolved in a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one applies the standard toolkit—thermodynamics, null geodesics, shadows, sixth-order WKB QNMs—to a metric that adds a PFDM term to a known Horndeski black hole. The combination itself appears new, and the qualitative trends (shadow shrinks as |β| or b grow; η acts differently) match intuition. The WKB error table is a nice touch. But the paper does not support its main claim, for two independent reasons.\n\nFirst, the metric in Eq. (8) is asserted. Equation (6) is the pure Horndeski solution, and the PFDM term is borrowed from a GR paper via Eq. (7). Horndeski gravity is nonlinear in the scalar field; adding a matter source changes both the metric and the scalar equation. The paper never shows that the combined lapse function is an exact solution, nor does it cite a derivation. If it is not, every quantity in the paper—entropy, photon sphere, shadow radius, EHT bounds, QNMs—describes a spacetime outside the theory. This is load-bearing, and the stress-test note is right to flag it.\n\nSecond, the observational constraints in Sec. V.A are internally inconsistent. Table I shows that increasing β from -0.6 to -0.3 enlarges the shadow; to reach the EHT 1σ window (4.55–5.22 M), you need smaller |β|, not β < -0.5929. Moreover, with b = 0.3 fixed, the shadow stays near 3.4 M regardless of η, so varying η alone cannot enter that window; the stated η bound is unsupported. The b < 0.01059 bound is plausible, but the other two are not.\n\nThere are smaller issues: Eq. (13) gives S = πr_h²/4, a factor of 4 off from the Bekenstein–Hawking value; Eq. (14) is consistent with the correct entropy, so it is a typo, but it should be fixed.\n\nIn short, the qualitative conclusion—that EHT shadow data force the Horndeski and PFDM parameters to be small—is almost certainly true and aligns with expectations. But the metric is unproven and the specific bounds are wrong. If the authors supply a derivation of the combined metric and recompute the bounds, the paper could be a modest but useful addition. As it stands, I would desk reject, not send to referees; if you do send it out, focus the referee on the solution check and the bound directions.","headline":"Standard toolkit applied to an asserted metric: the combined Horndeski–PFDM spacetime is never shown to solve the field equations, and the EHT bounds contain sign errors that contradict the paper's own tables.","tokens_in":22616,"tokens_out":17884,"would_cite":false,"duration_ms":159867,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Comparing the predicted shadow with the measured Sgr A* image forces the Horndeski and perfect-fluid dark-matter parameters to be tiny, leaving general relativity as the effective description near the photon sphere.","keywords":["Horndeski gravity","black hole shadow","perfect fluid dark matter","quasinormal modes","black hole thermodynamics","null geodesics","Sgr A*","WKB approximation"],"falsifier":"Substitute the combined lapse function $f(r)=1-\\frac{2M}{r}-\\frac{\\beta^2}{2\\eta r^2}+\\frac{b}{r}\\ln(r/b)$ and a scalar-field profile into the Horndeski field equations together with the perfect-fluid dark-matter energy-momentum tensor. If the equations are not satisfied identically for some $(\\beta,\\eta,b)$, the combined metric is not a solution of the theory and the shadow-based bounds do not constrain Horndeski gravity.","tokens_in":21523,"feed_emoji":"🕳️","tokens_out":14223,"duration_ms":132008,"temperature":0.7,"pith_summary":"This paper studies a static black hole in quartic `square-root` Horndeski gravity—a scalar-tensor theory in which the scalar field couples non-minimally to curvature through a square-root kinetic term—immersed in a perfect-fluid dark-matter halo. The model's lapse function carries three extra parameters beyond the mass: $\\beta$, $\\eta$, and $b$. The paper shows how these parameters shift horizon structure, temperature, specific heat, photon orbits, shadow size, and scalar quasinormal ringing. Its main result is that matching the predicted shadow radius to the measured Sgr A* shadow forces $\\beta < -0.5929$, $\\eta < -0.3263$, and $b < 0.01059$ at $1\\sigma$, so the spacetime near the photon sphere is effectively Schwarzschild. A companion thermodynamic result is that small-horizon states are locally stable but globally disfavoured, with no Hawking-Page phase transition.","feed_headline":"Sgr A* shadow keeps Horndeski dark-matter effects below one percent","feed_subtitle":"The model's three extra parameters must be tiny to match the observed shadow of the Galactic-center black hole.","key_machinery":"The load-bearing object is the combined lapse function $f(r)=1-\\frac{2M}{r}-\\frac{\\beta^2}{2\\eta r^2}+\\frac{b}{r}\\ln(r/b)$ for the quartic square-root Horndeski black hole immersed in perfect-fluid dark matter. Everything else flows from this function: the Hawking temperature $T=f'(r_h)/(4\\pi)$, the specific heat $C$, the Helmholtz free energy $F$, the photon-sphere equation $f'(r_p)r_p-2f(r_p)=0$, the shadow radius $r_s=r_p/\\sqrt{f(r_p)}$, and the scalar-field effective potential used in the sixth-order WKB quasinormal-mode calculation. The paper's parameter bounds come from combining the shadow-radius formula with the observed fractional deviation $\\delta$.","core_discovery":"The paper's central claim is that the combined metric $f(r)=1-\\frac{2M}{r}-\\frac{\\beta^2}{2\\eta r^2}+\\frac{b}{r}\\ln(r/b)$ produces observable signatures that current shadow measurements can already bound. Solving the photon-sphere condition $f'(r_p)r_p-2f(r_p)=0$ and using $r_s=r_p/\\sqrt{f(r_p)}$, the paper compares $r_s/M$ with the $1\\sigma$ interval $4.55 \\lesssim r_s/M \\lesssim 5.22$ derived from the fractional shadow deviation $\\delta=-0.060 \\pm 0.065$. This comparison forces $\\beta<-0.5929$, $\\eta<-0.3263$, and $b<0.01059$ at $1\\sigma$ (with looser $2\\sigma$ bounds), implying that any perfect-fluid dark-matter component is negligible within a few gravitational radii of Sgr A* and any Horndeski-induced departure from Schwarzschild is sub-percent. At the same time, the thermodynamic analysis shows the black hole is locally stable for small horizons but globally unstable, with no Hawking-Page transition.","pith_inferences":["A direct field-equation check of the combined ansatz would promote the quoted bounds from constraints on a metric template to constraints on Horndeski gravity itself.","The same shadow-comparison pipeline applied to the M87* image or to future higher-resolution Galactic-center data would tighten the bounds, since each parameter enters the shadow radius monotonically.","A rotating version of the metric would turn the circular shadow into an oval, letting shape measurements break the degeneracy among $\\beta$, $\\eta$, and $b$.","In the eikonal limit the quasinormal frequencies should match the photon-sphere orbital frequency and Lyapunov exponent, so checking that correspondence would connect the WKB table to the geodesic analysis."],"forward_implications":["If the bounds hold, any spherically symmetric dark-matter halo around Sgr A* has negligible gravitational backreaction inside the photon sphere, so current shadow measurements cannot distinguish it from vacuum.","The Horndeski couplings are pinned to a narrow window near the Schwarzschild limit, so future improvements in shadow resolution will either shrink the window further or reveal a deviation.","In this model, black holes with small horizons are locally thermodynamically stable but never globally preferred, and no Hawking-Page phase transition occurs.","Quasinormal-mode oscillation frequencies and damping rates move in opposite directions with $\\beta$ and $\\eta$, while both grow with the dark-matter parameter $b$, giving a potential ringdown signature for modified gravity."],"supporting_citations":[{"why":"Supplies the quartic square-root Horndeski black-hole solution that the paper embeds in dark matter.","marker":"[5]"},{"why":"Provides the perfect-fluid dark-matter energy-momentum tensor and the resulting logarithmic modification to the lapse function.","marker":"[27]"},{"why":"Supplies the dark-matter energy density and pressure profiles used to justify the perfect-fluid source.","marker":"[28]"},{"why":"Supplies the observed shadow of the Galactic-center black hole Sgr A* used as the observational target.","marker":"[29]"},{"why":"Gives the photon-sphere condition and shadow-radius formula used to compute the predicted shadow.","marker":"[30]"},{"why":"Provides the fractional shadow deviation intervals used to set the 1σ and 2σ parameter bounds.","marker":"[31–33]"},{"why":"Supplies the higher-order WKB correction terms used in the sixth-order quasinormal-mode calculation.","marker":"[39–41]"}],"fun_headline_variants":["Sgr A* shadow pins Horndeski dark matter to sub-percent","Galactic center shadow squeezes Horndeski couplings and dark matter","Small shadow demands tiny Horndeski and dark matter parameters","Horndeski black hole shadow matches Sgr A* only for tiny couplings","Shadow test: Horndeski dark matter must be negligible near Sgr A*"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that adding the Horndeski term and the dark-matter term in the metric yields an exact solution of the theory's field equations; if the combined spacetime is not a genuine solution, the thermodynamic, shadow, and ringing results describe a metric the theory does not admit.","fun_headline_variants_meta":{"raw":{"variants":["Sgr A* shadow pins Horndeski dark matter to sub-percent","Galactic center shadow squeezes Horndeski couplings and dark matter","Small shadow demands tiny Horndeski and dark matter parameters","Horndeski black hole shadow matches Sgr A* only for tiny couplings","Shadow test: Horndeski dark matter must be negligible near Sgr A*"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000394,"raw_usage":{"total_tokens":2148,"prompt_tokens":1104,"completion_tokens":1044,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":943}},"tokens_in":720,"tokens_out":1044,"duration_ms":9471,"temperature":1.0,"reasoning_tokens":943,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:33:08.364049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute the combined lapse function $f(r)=1-\\frac{2M}{r}-\\frac{\\beta^2}{2\\eta r^2}+\\frac{b}{r}\\ln(r/b)$ and a scalar-field profile into the Horndeski field equations together with the perfect-fluid dark-matter energy-momentum tensor. If the equations are not satisfied identically for some $(\\beta,\\eta,b)$, the combined metric is not a solution of the theory and the shadow-based bounds do not constrain Horndeski gravity.","supporting_citations":[{"cited_title":"Babichev, C","cited_arxiv_id":null,"evidence_quote":"Supplies the quartic square-root Horndeski black-hole solution that the paper embeds in dark matter."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the perfect-fluid dark-matter energy-momentum tensor and the resulting logarithmic modification to the lapse function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the dark-matter energy density and pressure profiles used to justify the perfect-fluid source."},{"cited_title":"Akiyama, A","cited_arxiv_id":null,"evidence_quote":"Supplies the observed shadow of the Galactic-center black hole Sgr A* used as the observational target."},{"cited_title":"Perlick and O","cited_arxiv_id":null,"evidence_quote":"Gives the photon-sphere condition and shadow-radius formula used to compute the predicted shadow."}],"review_version":2}