{"id":"f1df3fcb-f613-42e4-a4b0-cbd404a3ab32","arxiv_id":"2412.13775","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"In F(R) gravity, the black hole shadow shift is controlled by a free perturbing function, so matching the M87* and Sgr A* shadow sizes reduces to fitting that function.","lead":"This paper derives a perturbative framework for static black holes in F(R) gravity and computes how the photon sphere and shadow radius shift from the Schwarzschild values. It concludes that the Event Horizon Telescope constraints on M87* and Sgr A* can be satisfied, but only by tuning a free metric perturbation function.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The EHT constraints bound a free seed function ε ν1(3M), not a parameter of F(R) gravity; without a concrete action or a global asymptotically flat extension, the claim that F(R) shadows 'easily pass' EHT is unsupported.","rationale":"The reader's weakest-assumption diagnosis is correct and is the most load-bearing issue. The paper's own summary states that the observational constraints are applied to the combination -3√3 ε ν1(3M), and the text explicitly describes the freedom as 'by tuning the parameters.' Since ν1 is introduced as an arbitrary seed perturbation in Eq. (27) and never derived from a specific F(R) action, the EHT comparison is not a test of F(R) gravity; it is a restatement of how the shadow radius responds to a chosen metric deformation. This is not merely a matter of interpretation: for any genuine F(R) model, the metric perturbation is determined by the theory, and the paper supplies no such model. In addition, the solution is constructed only near the photon sphere, and the paper does not demonstrate that the perturbed metric can be extended to an asymptotically flat spacetime where the shadow formula (41) applies. The claim that F(R) gravity 'may easily pass' the EHT constraints therefore overreaches the derivation. I agree with the reader's verdict and do not see a reason to change it: the technical ODE may be a useful contribution, but the headline claim is unsupported. The proposed concrete test would settle whether a simple, well-defined F(R) action actually admits the kind of shadow-shifting solutions the paper postulates.","tokens_in":10612,"tokens_out":27850,"duration_ms":227902,"concrete_test":"Test with an explicit F(R) model, e.g. F(R)=R+αR^2. Linearize the full field equations around Schwarzschild with ν=ν0+ε ν1, λ=λ0+ε λ1, and the consistency relation f_R=2αR1, where R1 is the first-order Ricci scalar computed from ν1 and λ1. Impose horizon regularity at r=2M and asymptotic flatness, and solve for ν1. If the only regular asymptotically flat solution is ν1≡0, then the free-ν1 construction of §IIB1 does not correspond to any black hole of this natural F(R) class. A complementary local check: take a concrete seed such as ν1=(r-3M)^2, compute f_R and λ1 from Eqs. (32)-(35), then compute R1 and verify whether f_R is proportional to R1 and whether the full metric satisfies Eq. (2) with a single F(R); if not, the local reconstruction is internally inconsistent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conclusion is not supported by the derivation. In Eqs. (26)-(27) the metric perturbation ν1 and the F(R) perturbation f_R are introduced as independent O(ε) quantities, and Eq. (29) is then solved locally for f_R in terms of an arbitrarily chosen ν1. Consequently Eq. (54) states only that the shadow radius shifts by -3√3 M ε ν1(3M), and the inequalities (55)-(56) are restrictions on the choice of the free seed function ν1, not on any specified F(R) action. The paper never constructs a concrete F(R) theory for which ν1 is the corresponding solution, nor does it explain why the free choice of ν1 should be regarded as a property of F(R) gravity rather than as input data. Moreover, the solution is obtained only locally near r=3M; no proof is given that the local expressions (32)-(35) extend to a global, horizon-regular, asymptotically flat black hole. Because the shadow formula r_sh = r e^{-ν(r)}|_{r=r_ph} assumes an asymptotic observer, the absence of an asymptotically flat extension is a genuine gap. In a real F(R) model the metric perturbation is fixed by the action through the field equations, so the freedom to tune ε ν1(3M) to match M87* and Sgr A* is not a prediction of F(R) gravity. The ODE (29) may be a useful technical step, but it does not establish the stated claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates photon-sphere and black-hole-shadow radii in F(R) gravity. For a general static spherically symmetric metric (5) the authors derive, in Eq. (15), a third-order ODE for F_R(r); they then specialize to a perturbative setup in which the metric is a small deviation from Schwarzschild and F_R=1+(ε/2)f_R. Solving the linearized Eq. (29) locally near r=3M, they obtain f_R and λ_1 as expansions around the photon-sphere radius. Using the standard null-geodesic potential W(r), they find the shadow radius shift r_sh=3√3 M(1-ε ν_1(3M)), Eq. (54). The EHT constraints on M87* and Sgr A* are then translated into bounds on the combination δ=-3√3 ε ν_1(3M), Eqs. (55)-(56), and the paper concludes that F(R)-gravity black-hole shadows may easily pass EHT constraints.","tokens_in":10951,"tokens_out":4411,"duration_ms":41665,"significance":"If the central claim were established, the paper would show that a wide class of F(R) gravity models can evade EHT shadow-radius constraints, and the perturbative framework for deriving the third-order F_R equation could be a useful technical tool for modified-gravity shadow calculations. The leading-order shadow formula is a straightforward consequence of the null-geodesic calculation once the perturbative ansatz is accepted. However, the main conclusion is not supported by the presented derivation: the quantity constrained by EHT is an arbitrary seed function ν_1, not a parameter of a specified F(R) action, and no global asymptotically flat solution is constructed. The paper is therefore better viewed as a consistency relation for a free metric perturbation than as a prediction of F(R) gravity.","major_comments":[{"comment":"The EHT constraints are imposed on ε ν_1(3M), where ν_1 is a freely chosen seed function introduced in Eq. (27). The paper explicitly states that 'we did not specify ν_1 as long as Cν_1 is finite and does not vanish.' Thus Eqs. (55) and (56) restrict the choice of ν_1, not any parameter of a concrete F(R) action. The conclusion 'by tuning the parameters, the models can satisfy the constraints' is therefore circular with respect to the claim that F(R) gravity predicts shadow sizes consistent with EHT; a genuine test would require fixing an action F(R), deriving ν_1 from the resulting field equations, and then comparing the predicted ε ν_1(3M) with the bounds.","section":"II B 1 and III D, Eqs. (29)-(35), (55)-(56)"},{"comment":"The derivation of Eq. (15) is not shown. The text states that substituting Eq. (13) into Eq. (14) yields Eq. (15), but the algebra is nontrivial and no intermediate steps or independent check are provided. Since Eq. (29) and all subsequent shadow results depend on Eq. (15), the derivation should be supplied in an appendix or at least sketched in sufficient detail for the reader to verify the reduction.","section":"II, Eq. (15)"},{"comment":"The solution for f_R and λ_1 is only a local Taylor expansion around r=3M. No argument establishes that these local expressions extend to a global, horizon-regular, asymptotically flat solution. The shadow formula Eq. (41) is evaluated at the photon sphere but refers the radius to an asymptotic observer; without asymptotic flatness the interpretation of r_sh as the observed shadow size is not justified. The paper neither constructs the global metric nor invokes any existence or asymptotic-flatness result that would warrant the local-to-global step.","section":"II B 1, Eqs. (32)-(35)"},{"comment":"The final sentence, 'Hence, it is proved that BH shadows in F(R) gravity may easily pass the Event Horizon Telescope constraints,' overstates the result. What is proved, given the assumptions, is that for any sufficiently regular seed ν_1 one can locally solve for f_R near r=3M and obtain a shadow shift proportional to ν_1(3M). Since ν_1 is not derived from a specified F(R) model and no global solution is shown, the statement about F(R) gravity as a theory is not established by the manuscript.","section":"IV, Summary and Conclusion"}],"minor_comments":[{"comment":"The displayed formula for N has ambiguous fraction formatting; the denominator should be written explicitly, e.g. N = exp(-∫ dr1 [F_R''/(F_R/r1 + (1/2)F_R')]), to avoid confusion.","section":"II, Eq. (13)"},{"comment":"The symbol E is used for both the conserved energy and the Lagrangian, and the sentence 'E = L vanishes identically E = L = 0' is confusing; a distinct symbol for the Lagrangian would make the null-geodesic argument clearer.","section":"III A, Eqs. (43)-(44)"},{"comment":"The statement that 'the expression in (41) is general' is too strong; Eq. (54) is derived under the specific local perturbative ansatz of Section II B 1 and the assumption Cν_1 ≠ 0.","section":"III C, after Eq. (54)"},{"comment":"The notation α_±^3 = -10/27 ± 90/27 and the subsequent numerical values appear correct but are displayed in a compressed way; writing the cube roots explicitly would improve readability.","section":"II A, Eqs. (21)-(22)"}],"recommendation":"reject","confidential_remarks":"The paper contains a useful perturbative scheme and the leading-order shadow formula is a legitimate geometric calculation, but the advertised conclusion about EHT constraints is not supported by the derivation because the constrained quantity is an arbitrary seed function rather than a parameter of a specified F(R) theory. The local and non-global nature of the solution compounds the problem. These are load-bearing gaps that cannot be fixed by minor revisions, hence my recommendation to reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the third-order ODE (15) for F_R in a spherically symmetric static background, and the systematic perturbative expansion around Schwarzschild that leads to the shadow shift formula (54). Those steps are competently done, and the paper is honest that the observational constraints are satisfied \"by tuning the parameters.\" If you work on modified-gravity black holes, the ODE is a plausible starting point for constructing perturbative F(R) solutions.\n\nThe soft spots are in the interpretation, and they are load-bearing. The shadow shift is controlled entirely by the free combination ε ν1(3M), where ν1 is an unconstrained metric perturbation. The paper never constructs a concrete F(R) action for which ν1 is the actual solution; it only shows that for any ν1 with a finite nonzero Cν1, a local f_R exists near r=3M. So the \"prediction\" is really a constraint on a seed function, not on F(R) gravity. The absence of a global, asymptotically flat extension is a further gap: the shadow formula assumes an asymptotic observer, and no argument is given that the local expansion extends cleanly to infinity. And, as the reader notes, the Schwarzschild shadow already falls inside the quoted M87* and Sgr A* ranges, so consistency with those numbers is not informative.\n\nThe derivation of Eq. (15) is not shown in detail, which makes independent verification harder, but the equation is plausible and the final shift formula follows from standard geodesic arguments. The paper is not incoherent; it just overreaches. The technical machinery could be useful, but the advertised conclusion — that F(R) shadows \"easily pass\" EHT constraints — is unsupported in the present form.\n\nWho gets value from this? Someone looking for the ODE as a tool, or a referee who wants to push the authors to either construct a concrete model or reframe the paper as formalism with no astrophysical claim. I would not cite the EHT consistency result, but I would accept the paper for peer review because the technical core is new and worth the referees' time to force a proper framing.","headline":"Novel ODE for F(R) black hole perturbations, but the EHT 'prediction' reduces to tuning a free metric seed; the technical core deserves review, the conclusion does not.","tokens_in":11484,"tokens_out":2512,"would_cite":false,"duration_ms":24750,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83D05"],"pacs":["04.50.Kd","04.70.-s"],"model":"deepseek-v4-flash","headline":"F(R) gravity can shift black hole shadows into the observed EHT ranges.","keywords":["F(R) gravity","black hole shadow","photon sphere","Event Horizon Telescope","Schwarzschild perturbation","modified gravity","M87*","Sgr A*"],"falsifier":"For a specified F(R) model, solve the linearized field equation (29) (or the full equation (15)) to compute $\\nu_1(r)$, then evaluate $-3\\sqrt{3}\\,\\varepsilon\\nu_1(3M)$ and check it against the inequalities (55) and (56); a value outside the relevant window shows that model is incompatible with the corresponding EHT shadow observation.","tokens_in":10345,"feed_emoji":"🕳️","tokens_out":11919,"duration_ms":96875,"temperature":0.7,"pith_summary":"This paper shows that in F(R) gravity, a modified-gravity theory whose Lagrangian is a general function of the Ricci scalar, spherically symmetric vacuum black holes can be built as small perturbations around the Schwarzschild solution. The construction leaves the metric perturbation $\\nu_1(r)$ free, subject only to a finiteness condition at the photon-sphere radius $r=3M$. For any such perturbation the photon sphere moves to $3M(1+\\varepsilon M\\nu_1'(3M))$ while the shadow radius becomes $3\\sqrt{3}M(1-\\varepsilon\\nu_1(3M))$, so all observational content is carried by the single combination $\\varepsilon\\nu_1(3M)$. Tuning that combination puts the shadow inside the reported M87* range $r_{\\mathrm{sh}}/M\\simeq 5.5\\pm 0.8$ and the Sgr A* range $4.21\\lesssim r_{\\mathrm{sh}}/M\\lesssim 5.56$. The upshot is that F(R) gravity as a class is not excluded by current shadow measurements, in contrast to the simplest mimetic-gravity model discussed in the paper's introduction.","feed_headline":"F(R) gravity can match the Event Horizon Telescope shadows","feed_subtitle":"A perturbative reconstruction tunes the shadow radius to fit both M87* and Sgr A* observations.","key_machinery":"The load-bearing object is the third-order differential equation (15) that the F(R) field equations impose on $F_R(r)$ for a given static spherically symmetric metric function $\\nu(r)$, together with its perturbative reduction (29) near the photon sphere. The reconstruction pipeline—solve (15) for $F_R(r)$, use (13) to obtain $\\lambda(r)$, compute the scalar curvature $R(r)$, invert to $r(R)$, and integrate $F_R(R)$ to recover $F(R)$—converts an assumed geometry into a concrete F(R) theory. The observable side is the shadow formula $r_{\\mathrm{sh}}=r e^{-\\nu(r)}|_{r=r_{\\mathrm{ph}}}$, which in the perturbative branch collapses to $r_{\\mathrm{sh}}=3\\sqrt{3}M(1-\\varepsilon\\nu_1(3M))$; the photon-sphere shift itself drops out of this expression.","core_discovery":"The paper's central claim is that a perturbative reconstruction of F(R) from an assumed geometry near the photon sphere is enough to make black hole shadows compatible with observations. Writing $F_R=1+\\frac12\\varepsilon f_R$ and $\\nu=\\nu_0+\\varepsilon\\nu_1$, the vacuum field equations reduce to an inhomogeneous linear third-order equation (29) for $f_R(r)$. In the region $r\\sim 3M$, a regular branch of solutions exists whenever $C_{\\nu_1}\\equiv(4M\\nu_1'''-\\frac{32}{3}\\nu_1''+\\frac{8}{3M}\\nu_1')|_{r=3M}$ is finite and nonzero; along that branch the shadow radius is $r_{\\mathrm{sh}}=3\\sqrt{3}M(1-\\varepsilon\\nu_1(3M))$. Because $\\nu_1(3M)$ is not fixed by the reconstruction, the M87* and Sgr A* bounds reduce to simple inequalities on the combination $-3\\sqrt{3}\\,\\varepsilon\\nu_1(3M)$, and parameter choices satisfying them exist. The paper therefore claims that F(R)-gravity black holes can easily pass the Event Horizon Telescope constraints while remaining perturbatively close to Schwarzschild.","pith_inferences":["The paper leaves open which explicit F(R) action realizes a given $\\nu_1$; committing to a concrete model fixes $\\nu_1$ and may remove the tuning freedom used to satisfy the inequalities.","The same perturbative scheme should transfer to other metric-modified gravity theories: whenever the field equations linearize to a scalar equation for a metric perturbation near the photon sphere, EHT constraints reduce to a one-parameter test.","The 'easily pass' conclusion is an existence statement about parameter choices, not a prediction for generic F(R) models; the method supplies a diagnostic for classifying individual models by their value of $\\varepsilon\\nu_1(3M)$.","A single shadow measurement constrains only the combination $\\varepsilon\\nu_1(3M)$, not the full radial dependence of the deviation; distinguishing among F(R) models will require additional observations or a theoretical prior on $\\nu_1$."],"forward_implications":["F(R) gravity remains viable at horizon scales: the current M87* and Sgr A* shadow measurements do not exclude the theory class.","Any metric perturbation $\\nu_1$ with $C_{\\nu_1}$ finite and nonzero at $r=3M$ determines a locally reconstructed F(R) whose shadow radius is controlled by $\\varepsilon\\nu_1(3M)$ alone.","The constraints become explicit numerical targets for model building: $-0.5<-3\\sqrt{3}\\,\\varepsilon\\nu_1(3M)<1.1$ for M87* and $-1.01\\lesssim -3\\sqrt{3}\\,\\varepsilon\\nu_1(3M)\\lesssim 0.36$ for Sgr A*.","A shift in the photon-sphere radius at first order does not feed into the shadow radius, because $d(r/\\sqrt{1-2M/r})/dr$ vanishes at $r=3M$; the resulting first-order shadow formula depends only on $\\nu_1(3M)$.","The non-perturbative $\\nu=0$ power-law branch yields no finite photon sphere, so not every F(R) vacuum solution can describe observed shadows."],"supporting_citations":[{"why":"Supplies the EHT observation of the M87* shadow, the observational basis for the constraints.","marker":"[1]"},{"why":"The prior mimetic-gravity exclusion that frames the paper's motivation: shadow measurements can rule out modified theories.","marker":"[2]"},{"why":"Formalism for photon spheres and shadows, used for the radius formulas (41)-(46).","marker":"[6-8]"},{"why":"Foundational F(R) gravity references defining the action and vacuum field equations.","marker":"[9-11]"},{"why":"M87* shadow-radius bound with r_sh/M approximately 5.5 plus or minus 0.8, converted into inequality (55).","marker":"[23]"},{"why":"Sgr A* shadow-radius bound 4.21 to 5.56, converted into inequality (56).","marker":"[24]"}],"fun_headline_variants":["F(R) gravity shadows fit Event Horizon Telescope data","Perturbative F(R) matches M87* and Sgr A* shadows","F(R) black holes pass shadow tests with ease","Shadow radii in F(R) gravity align with EHT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the metric perturbation $\\nu_1(r)$ may be chosen freely and then realized by some local F(R) theory; the paper never derives $\\nu_1$ from a specified action, so the tuning freedom that fits the shadow data could disappear for a concrete model.","fun_headline_variants_meta":{"raw":{"variants":["F(R) gravity shadows fit Event Horizon Telescope data","Perturbative F(R) matches M87* and Sgr A* shadows","F(R) black holes pass shadow tests with ease","Shadow radii in F(R) gravity align with EHT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1451,"prompt_tokens":973,"completion_tokens":478,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":407}},"tokens_in":589,"tokens_out":478,"duration_ms":4274,"temperature":1.0,"reasoning_tokens":407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:49:13.525748+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a specified F(R) model, solve the linearized field equation (29) (or the full equation (15)) to compute $\\nu_1(r)$, then evaluate $-3\\sqrt{3}\\,\\varepsilon\\nu_1(3M)$ and check it against the inequalities (55) and (56); a value outside the relevant window shows that model is incompatible with the corresponding EHT shadow observation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the EHT observation of the M87* shadow, the observational basis for the constraints."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The prior mimetic-gravity exclusion that frames the paper's motivation: shadow measurements can rule out modified theories."}],"review_version":1}