{"id":"5a9f4e3d-8dd9-40c3-9946-e0af4e875194","arxiv_id":"2411.15757","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"For a black hole in F(R)-ModMax gravity, shadow and lensing calculations compared with M87* data suggest f_R0<-1 for AdS backgrounds and f_R0>-1 for dS backgrounds, modulo a mass-normalization ambiguity.","lead":"Researchers computed how the shadow, radiation, and light bending of a black hole in a modified gravity model change with the model's parameters. They report that M87* data prefer a negative F(R) parameter for AdS black holes, but the comparison has a units ambiguity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The EHT comparison is built on an unstated mass normalization: with the standard identification m0=2M, the plotted shadow diameters of 3-6.5 should be roughly 10.4 in M87* mass units, so the claimed fR0<-1 preference may be a factor-of-two artifact.","rationale":"The reader's weakest assumption was exactly the unresolved relation between m0 and the astrophysical mass M, noting that the plotted d_sh values (roughly 4-6) sit at roughly half of d_M87* ~ 11 and that either the plotted quantity is a radius, m0 is not 2M, or the EHT comparison is off by a factor of two. My stress test confirms that this is the most load-bearing issue: the central claim is the EHT parameter preference, and that preference is obtained by comparing the shadow size to Eq. (3.18). The metric's GR limit unambiguously fixes m0 = 2M, so the logical default is that physical masses in the plots are m0/2. Under that normalization, a plain Schwarzschild shadow already matches M87* within 1 sigma, which would invalidate the conclusion that fR0 must be below -1 for AdS consistency. I therefore agree with the reader's conditional verdict: the optical formalism is standard and the calculation is likely reproducible, but the headline constraint cannot be accepted until the mass normalization is stated and the plots are redone with dmetric expressed in units of M_ADM. A secondary concern, the fR0<-1 region having 1+fR0<0 and hence the scalar ghost/instability of F(R) gravity, would further weaken the physical interpretation even if the normalization issue were resolved, but the normalization ambiguity is the more direct and decisive threat to the central claim.","tokens_in":16077,"tokens_out":10419,"duration_ms":98873,"concrete_test":"Recompute the shadow diameter dmetric = 2 r_sh for the parameters of Fig. 5(c) (e.g. q = 0.7, gamma = 1, R0 = -0.1, fR0 = -1.05) using Eqs. (3.10)-(3.11), and for fR0 = -1.05, -0.5, 0, 0.5, with the ADM mass relation M_ADM = m0/2. Overlay these values on the EHT band (9.5-12.5 for 1 sigma, 8-14 for 2 sigma) of Eq. (3.18). In particular, check the Schwarzschild limit (q=0, gamma=0, R0=0, fR0=0) with m0=2M: if the code gives dmetric/M = 6*sqrt(3) ~ 10.4, the normalization is correct; if it gives 5.196, the paper has used M=m0 and the reported fR0 constraints are factor-of-two artifacts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new result is the EHT-based preference fR0<-1 for AdS and fR0>-1 for dS, obtained by comparing the model shadow diameter d_sh with d_M87* = D theta / M ~ 11.0 +/- 1.5 (Eq. 3.18). The metric (Eq. 2.13) is h(r) = 1 - m0/r - R0 r^2/12 + q^2 e^{-gamma}/((1+f_R0) r^2), and in the GR limit (fR0=0, gamma=0, R0=4Lambda) this is RN-(A)dS with m0 = 2M. With m0=2M, the Schwarzschild limit of Eqs. (3.10)-(3.11) gives r_sh = 3*sqrt(3) M and dmetric = 2 r_sh = 6*sqrt(3) M ~ 10.4 M, which is already inside the EHT 1-sigma band. The paper's Fig. 4 colorbars instead show d_sh in the range 3-6.5, which corresponds to r_sh computed with m0=1 and then interpreted as M=1, i.e. to dividing by half the astrophysical mass. The text never states whether m0 = 2M or m0 = M, so the comparison in Figs. 5 and 6 is ambiguous. If the plotted quantity is d_sh with m0=M, all physical lengths in units of M should be doubled before comparison with Eq. (3.18); then the allowed parameter regions and the sign preference for fR0 could shift substantially, possibly making fR0=0 (GR-like) consistent and removing the headline constraint. A compounding internal inconsistency is that Eq. (3.14) gives X = -r_sh sqrt(1 + R0 r_sh^2/12), whose magnitude is not r_sh, so the celestial-coordinate circle does not have radius r_sh as assumed elsewhere in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the optical properties of a static, spherically symmetric black hole solution in F(R)-ModMax gravity, with metric function given by Eq. (2.13). It computes the photon-sphere radius and shadow radius (Eqs. 3.10-3.11), plots shadow boundaries and their parameter dependence, compares the shadow diameter with EHT M87* constraints, and separately derives the energy emission rate and the weak-field deflection angle. The headline claim is that the EHT data favor f_R0 < -1 for AdS black holes and f_R0 > -1 for dS black holes, so that astrophysical observations can fix the sign of the F(R) parameter.","tokens_in":16486,"tokens_out":5627,"duration_ms":46779,"significance":"If the EHT comparison were performed with an unambiguous mass normalization, the claimed sign preference for f_R0 would be an interesting observational handle on F(R)-ModMax gravity. The paper has clear strengths: it works with an explicit exact solution, derives analytic expressions for the photon-sphere radius, shadow radius, energy emission rate, and deflection angle, and compares a parameter scan against published EHT numbers. However, the central constraint is currently not trustworthy because the relation between the metric parameter m0 and the astrophysical mass M is never stated, and the plotted shadow diameters appear to be off by a factor of two relative to the standard m0 = 2M convention. In addition, the f_R0 < -1 region, which drives the headline result, lies on the far side of a pole in the metric and is not physically justified in the paper. The significance is therefore conditional on a corrected reanalysis.","major_comments":[{"comment":"The relation between the metric integration constant m0 and the astrophysical mass M used in the EHT comparison is never stated. In the GR limit (f_R0 = 0, gamma = 0, R0 = 4 Lambda -> 0, q = 0), the metric (2.13) reduces to Schwarzschild only if m0 = 2M; then Eqs. (3.10)-(3.11) give r_sh = 3 sqrt(3) M and shadow diameter d_metric = 6 sqrt(3) M ~ 10.4, which is consistent with Eq. (3.18), d_M87* = 11.0 +/- 1.5. Yet the colorbars in Figs. 4(a)-4(c) show d_sh values in the range 3 to 6.5, which is what one obtains if m0 = M, i.e., if all lengths are measured in units of half the astrophysical mass. Because the paper never states which convention is used, the comparison in Figs. 5 and 6 is ambiguous. If m0 = M is the intended convention, all plotted shadow diameters must be doubled before comparison with Eq. (3.18), and the allowed regions and the claimed f_R0 preference could shift substantially. The authors must state the normalization explicitly, recompute the contour plots under the standard identification m0 = 2M, and verify whether the f_R0 < -1 (AdS) and f_R0 > -1 (dS) conclusions survive.","section":"III B, Eq. (3.18), Figs. 4-6, Eq. (2.13)"},{"comment":"The metric (2.13) contains (1 + f_R0) in the denominator of the electric-charge term, so f_R0 = -1 is a singular point at which the charge contribution diverges. For f_R0 < -1, the coefficient of q^2 e^{-gamma}/r^2 changes sign, turning the usual repulsive electromagnetic contribution into an attractive one. The paper scans through f_R0 = -1 in Figs. 5(c) and 6(c) and claims consistency with EHT data for f_R0 < -1 without discussing this pole or the sign flip. This is load-bearing because the headline result is precisely the f_R0 < -1 region; the authors need to justify that a physically admissible F(R)-ModMax solution exists on both sides of the pole, including the sign of the effective charge term and the behavior of the field equations (2.4)-(2.7) at f_R0 = -1.","section":"II, Eq. (2.13); III B, Figs. 5(c), 6(c)"},{"comment":"The celestial coordinates are given as X = -r_sh sqrt(1 + R0 r_sh^2/12) and Y = 0, which is a single point, not the boundary of a circular shadow. For a spherically symmetric and static spacetime, the shadow boundary should be a circle, e.g., X = r_sh cos(phi), Y = r_sh sin(phi), or an equivalent angular parametrization. The figures in Fig. 3 clearly plot closed circular curves, so the formula actually used is not the one written in Eq. (3.14). This needs to be corrected and the definition of r_sh clarified for the non-asymptotically flat AdS/dS backgrounds; otherwise the shadow shapes and the derived d_sh values are not properly defined.","section":"III A, Eq. (3.14); Fig. 3"}],"minor_comments":[{"comment":"Panels (c) and (d) of Fig. 1 have identical captions, although the plotted ranges of R0 differ; the caption should distinguish the two panels.","section":"Fig. 1 caption"},{"comment":"The energy emission rate formula is written as d^2E/(dtdomega) = 2 pi^2 omega^3 r_sh^2 / (exp(omega/T) - 1), but the displayed text lacks parentheses around omega/T in the exponential; please fix the typographical ambiguity.","section":"III C, Eq. (3.20)"},{"comment":"The sentence 'Employing Eqs. (4.13) and (4.14), we obtain the correct deflection angle' is self-referential because Eq. (4.14) is the deflection angle itself; the intended reference is probably Eqs. (4.12) and (4.13).","section":"IV, around Eq. (4.14)"},{"comment":"The discussion of Fig. 8 says 'the opposite behavior is observed in dS spacetime (see Fig. 8(e))', but Fig. 8(e) shows negative R0 (AdS); the second mention of dS should be AdS.","section":"IV, Fig. 8 text"},{"comment":"References [51] and [78] are the same paper (Zhu, Wu, Jamil, Jusufi, Phys. Rev. D 100, 044055 (2019)) and should be consolidated; also 'ModMox' appears in Section II and should read 'ModMax'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central EHT constraint is currently built on an unstated mass normalization and on a parameter region (f_R0 < -1) that sits beyond a pole in the metric. I recommend that the editor request a reanalysis with the explicit identification m0 = 2M, a discussion of the f_R0 = -1 singularity, and a correction of the celestial-coordinate formula before the paper can be considered for publication. The paper is within the journal's scope, and the authors' effort to confront the model with EHT data is commendable, but the headline claim is not yet supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nHere's my read on arXiv:2411.15757. The paper has a genuinely useful optical analysis of a known BH solution in F(R)-ModMax gravity, but the central claim — that EHT data prefer f_R0 < -1 for AdS and f_R0 > -1 for dS — is built on an unstated mass normalization that looks off by a factor of two. The stress-test note is correct: with the standard identification m0 = 2M, the plotted shadow diameters of 3–6.5 should roughly double to 6–13 in units of M. That puts the GR-like case (f_R0 = 0) right inside the EHT 1σ band, so the preference for f_R0 < -1 is likely an artifact.\n\nWhat the paper does well: the photon sphere, shadow, energy emission, and Gauss-Bonnet deflection calculations are standard and mostly transparent. The qualitative trends (charge shrinks the shadow, γ and f_R0 enlarge it, etc.) match earlier work. The specific EHT comparison for this model is new relative to the cited literature, and the paper is honest about using a known solution from [31].\n\nThe soft spots: first, the mass normalization is never stated. Equation (2.13) gives m0 as an integration constant, but the plots and EHT comparison implicitly assume m0 = M. That is a factor-of-two error in the shadow diameter relative to the astrophysical mass. Second, Eq. (3.14) for the celestial coordinate X is not consistent with a circular shadow — it gives a radius that depends on R0, yet Fig. 3 shows circles. This looks like a typo, but it signals sloppiness. Third, the preferred region f_R0 < -1 is generally considered unphysical in F(R) gravity because it flips the sign of the effective gravitational coupling; the paper does not address this.\n\nMy verdict: the optical analysis is worth reading, and the paper deserves a serious referee, but the EHT constraint needs major rework. The authors should state the m0–M relation, rerun the parameter scans with the correct normalization, and discuss the stability of f_R0 < -1. If they do, the headline constraint may well disappear. I'd use this as a cautionary example of normalization traps in shadow comparisons.","headline":"The EHT constraint on F(R)-ModMax parameters is undermined by an unstated factor-of-two mass normalization, so the headline f_R0 sign preference is likely an artifact, though the optical analysis is careful.","tokens_in":17050,"tokens_out":9797,"would_cite":false,"duration_ms":78229,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that the shadow of a black hole in F(R)-ModMax gravity matches the Event Horizon Telescope's M87* measurement only when the F(R) parameter f_R0 lies below -1 for AdS spacetime or above -1 for dS spacetime, making the…","keywords":["black hole shadow","F(R) gravity","ModMax electrodynamics","M87*","Event Horizon Telescope","gravitational lensing","deflection angle","energy emission rate"],"falsifier":"Compute the shadow diameter from Eqs. (3.10) and (3.11) with $q=0$, $\\gamma=0$, and $R_0=0$, and check whether the diameter $2r_{sh}$ reproduces the Schwarzschild value $6\\sqrt{3}M \\approx 10.39M$; a plot whose $d_{sh}$ colorbar runs from about 3 to 6 while the target band is $11.0 \\pm 1.5$ indicates that the plotted quantity is a radius rather than a diameter, and recomputing with the correct diameter would settle whether the $f_{R_0} < -1$ AdS conclusion survives.","tokens_in":15841,"feed_emoji":"🕳️","tokens_out":11002,"duration_ms":84909,"temperature":0.7,"pith_summary":"This paper builds a static, spherically symmetric black hole solution in a modified gravity that combines $F(R)$ gravity with ModMax nonlinear electrodynamics, and asks whether its predicted shadow could be the one imaged at the center of M87*. The central argument is that the $F(R)$ parameter $f_{R_0}$ controls the shadow diameter so strongly that matching the Event Horizon Telescope's measured value $d_{M87*} \\approx 11.0 \\pm 1.5$ singles out specific parameter ranges: $f_{R_0} < -1$ in anti-de Sitter (negatively curved) backgrounds and $f_{R_0} > -1$ in de Sitter (positively curved) backgrounds. If this is right, a single astrophysical image can be used to fix the sign and rough size of a modified-gravity parameter. The paper also reports how the same parameters affect the Hawking energy emission rate and the gravitational deflection angle of light.","feed_headline":"M87* shadow size constrains modified-gravity parameter","feed_subtitle":"If right, the EHT image fixes f_R0 below -1 in AdS and above -1 in dS backgrounds.","key_machinery":"The central object is the charged $F(R)$-ModMax black hole, whose metric function is $h(r)=1 - m_0/r - R_0 r^2/12 + q^2 e^{-\\gamma}/((1+f_{R_0}) r^2)$, with $R_0$ the constant scalar curvature, $\\gamma$ the ModMax coupling, $q$ the electric charge, and $f_{R_0}$ the derivative of the $F(R)$ correction evaluated at $R_0$. The argument runs through the photon effective potential: the unstable circular photon orbit follows from $V_{eff}=0$ and $V'_{eff}=0$, yielding $r_{ph} = (3/4)\\left(m_0 + \\sqrt{m_0^2 - 32 q^2/(9 e^{\\gamma}(1+f_{R_0}))}\\right)$, and the shadow radius is $r_{sh} = r_{ph}/\\sqrt{A(r_{ph})}$. The comparison with M87* uses the EHT diameter $d_{M87*} = D\\theta/M \\approx 11.0 \\pm 1.5$, and the paper maps this number, through these formulas, into allowed regions for $(q, \\gamma, R_0, f_{R_0})$.","core_discovery":"On the paper's own terms, the discovery is that the shadow of the $F(R)$-ModMax black hole is a quantitative filter: imposing the EHT constraint $d_{sh} = d_{M87*} \\approx 11.0 \\pm 1.5$ selects $f_{R_0} < -1$ when $R_0 < 0$ (AdS) and $f_{R_0} > -1$ when $R_0 > 0$ (dS). This follows from the geodesic computation of the photon sphere and shadow radius for the metric $h(r) = 1 - m_0/r - R_0 r^2/12 + q^2 e^{-\\gamma}/((1+f_{R_0}) r^2)$, together with the second check that the Schwarzschild shadow deviation $\\delta$ stays inside the EHT bound $-0.18 < \\delta < 0.16$. The paper concludes that $f_{R_0}$, not the charge or the ModMax parameter, is the decisive quantity for consistency with the M87* image.","pith_inferences":["The paper never states the relation between $m_0$ and the astrophysical mass $M$ used in the EHT comparison; if the standard Schwarzschild identification $m_0=2M$ is intended rather than $m_0=M$, the plotted shadow diameters would double and the favored $f_{R_0}$ ranges could shift.","The same constraint pipeline could be applied to Sgr A*, whose EHT shadow diameter is measured separately; agreement with the M87*-derived $f_{R_0}$ ranges would strengthen the claim, while disagreement would suggest the model needs another parameter.","Because the spacetime is spherically symmetric, the model predicts a perfectly circular shadow; a future precision measurement of shadow circularity would directly test the static-solution assumption.","The deflection-angle formula depends on $R_0$ and $f_{R_0}$ at finite impact parameter, so galaxy-scale strong-lensing observations could provide a shadow-free test of the same parameter ranges."],"forward_implications":["If the central claim holds, the M87* shadow measurement becomes a direct bound on $f_{R_0}$: $f_{R_0} < -1$ for AdS-type backgrounds and $f_{R_0} > -1$ for dS-type backgrounds, so the sign of $f_{R_0}+1$ is tied to the sign of the cosmological curvature.","Because the admissible-region plots (Figs. 5 and 6) show that a range of $(q, \\gamma, R_0, f_{R_0})$ values satisfies the 1$\\sigma$ and 2$\\sigma$ EHT bounds, the observation constrains combinations of parameters rather than a single parameter.","For the same black hole parameters, the energy emission rate is lower for larger $q$ and higher for larger $\\gamma$ or $f_{R_0}$, implying that charged $F(R)$-ModMax black holes evaporate more slowly while the ModMax and $F(R)$ corrections speed evaporation up.","In the weak-field lensing calculation, the deflection angle increases with $q$ and $f_{R_0}$ and decreases with $\\gamma$, providing an independent optical signature that lensing observations could test."],"supporting_citations":[{"why":"supplies the $F(R)$-ModMax action and the charged black hole solution $h(r)$ from which all shadow and lensing calculations start.","marker":"[31]"},{"why":"introduces the ModMax electrodynamics Lagrangian whose parameter $\\gamma$ enters the metric through the $e^{-\\gamma}$ charge term.","marker":"[25]"},{"why":"provides the M87* angular diameter, mass, and distance used to define the target shadow size.","marker":"[82]"},{"why":"states the EHT-derived shadow diameter $d_{M87*} \\approx 11.0 \\pm 1.5$ in units of mass that the model is compared against.","marker":"[83]"},{"why":"gives the bound on the Schwarzschild shadow deviation $\\delta$ used as a second consistency check.","marker":"[84]"},{"why":"earlier study of black hole shadows in $F(R)$ gravity that this paper extends to the $F(R)$-ModMax case.","marker":"[72]"}],"fun_headline_variants":["M87* shadow size fixes sign of f_R0 in modified gravity","EHT image of M87* constrains f_R0 for AdS and dS black holes","Black hole shadow in F(R)-ModMax matches EHT only for specific f_R0","f_R0 range pinned by M87* shadow in modified gravity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole EHT comparison rests on an unstated identification between the metric parameter $m_0$ and the astrophysical mass $M$ used in the plots; if that identification is off by the factor that separates $m_0$ from $2M$, the shadow diameters would shift enough to move the reported $f_{R_0}$ boundaries.","fun_headline_variants_meta":{"raw":{"variants":["M87* shadow size fixes sign of f_R0 in modified gravity","EHT image of M87* constrains f_R0 for AdS and dS black holes","Black hole shadow in F(R)-ModMax matches EHT only for specific f_R0","f_R0 range pinned by M87* shadow in modified gravity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000347,"raw_usage":{"total_tokens":1917,"prompt_tokens":982,"completion_tokens":935,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":847}},"tokens_in":598,"tokens_out":935,"duration_ms":8646,"temperature":1.0,"reasoning_tokens":847,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:58:44.295391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the shadow diameter from Eqs. (3.10) and (3.11) with $q=0$, $\\gamma=0$, and $R_0=0$, and check whether the diameter $2r_{sh}$ reproduces the Schwarzschild value $6\\sqrt{3}M \\approx 10.39M$; a plot whose $d_{sh}$ colorbar runs from about 3 to 6 while the target band is $11.0 \\pm 1.5$ indicates that the plotted quantity is a radius rather than a diameter, and recomputing with the correct diameter would settle whether the $f_{R_0} < -1$ AdS conclusion survives.","supporting_citations":[{"cited_title":"Bandos, K","cited_arxiv_id":null,"evidence_quote":"introduces the ModMax electrodynamics Lagrangian whose parameter $\\gamma$ enters the metric through the $e^{-\\gamma}$ charge term."},{"cited_title":"Kumar, A","cited_arxiv_id":null,"evidence_quote":"provides the M87* angular diameter, mass, and distance used to define the target shadow size."},{"cited_title":"Akiyama, et al., EHT Collaboration, Astrophys","cited_arxiv_id":null,"evidence_quote":"states the EHT-derived shadow diameter $d_{M87*} \\approx 11.0 \\pm 1.5$ in units of mass that the model is compared against."},{"cited_title":"Akiyama, et al., Astrophys","cited_arxiv_id":null,"evidence_quote":"gives the bound on the Schwarzschild shadow deviation $\\delta$ used as a second consistency check."},{"cited_title":"Dastan, R","cited_arxiv_id":null,"evidence_quote":"earlier study of black hole shadows in $F(R)$ gravity that this paper extends to the $F(R)$-ModMax case."}],"review_version":1}