{"id":"8301e048-cf99-4e92-a1e6-052ce9d82439","arxiv_id":"2502.01747","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Einsteinian cubic gravity shrinks (grows) black hole horizons for positive (negative) coupling and shifts the photon sphere enough that SgrA* shadow observations can bound the coupling to approximately 0.1.","lead":"This paper studies black-hole-like solutions in Einsteinian cubic gravity, a modified theory of gravity with cubic curvature terms. It finds that the cubic coupling shifts black hole horizons and photon spheres in ways that could be tested with light bending measurements and black hole shadow observations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-function ansatz in Eq. (4) is load-bearing and unproven: if the full ECG field equations require a second metric function, the horizon, photon-sphere, and angular-difference results all change.","rationale":"The reader flagged the same weakest assumption, and I agree it is the most load-bearing concern. I also considered other candidates: the absence of numerical error bars, the unquantified SgrA* mass uncertainty in the λ ≲ 0.1 bound, the effective-theory validity condition in footnote 2, and the apparent mismatch between Eq. (16) and Eq. (A3). The Eq. (16)/Eq. (A3) difference appears to be a convention issue: Eq. (16) belongs with the asymptotic form f = 1 - (ΛEff/3)r², while Eq. (A3) belongs with f = 1 - ΛEff r², so I do not treat it as a reliable internal contradiction. The EFT regime deserves a quantitative check for λ = 0.1, but it is secondary to the structural dependence on Eq. (4). The paper has real independent support: public numerical code, agreement between the asymptotic, weak-coupling, and near-horizon approximations, and consistency with Refs. [11,12] on the negative-λ side. None of that support validates the positive-λ, near-naked-singularity branch under a two-function ansatz. Since the reader's verdict is already CONDITIONAL and this is the main reason for the condition, no verdict change is needed; the proposed symbolic check should be a clear acceptance condition.","tokens_in":23281,"tokens_out":13993,"duration_ms":138173,"concrete_test":"Use a computer algebra system (xAct or SymPy) to compute the ECG vacuum field equations (3) with Pμν from Appendix A1 for the most general static, spherically symmetric line element ds² = -A(r)dt² + B(r)dr² + r²dΩ², including Λ and λ. Solve the independent equations numerically with SgrA*-mass boundary conditions and check whether they force A(r)B(r) = const. If they do, the ansatz is validated and the concern is closed; if not, recompute the horizon radius and the Table II α values for λ = 0.05 and λ = 0.1 using the two-function system and compare them with the reported numbers.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every quantitative claim in the paper uses the restricted metric (4), ds² = -f dt² + dr²/f + r²dΩ², which fixes g_tt g_rr = -1 before solving the field equations. Eq. (5), the integrated equation (7), the near-horizon series (20), the horizon equation (24), and the numerical metrics behind Table II and Figs. 5-7 all depend on this single-function ansatz. The paper's justification, that C and C' are trivial in SSS vacuum and that Refs. [11,12] found inversely related components, establishes only that a one-function branch exists; it does not show the branch is exhaustive. The general static spherically symmetric metric has two independent functions A(r) and B(r), and there is no coordinate freedom to set A B = const while preserving r as the areal radius. The issue is not merely formal: for λ ≳ 0.1 the selected branch loses its horizon and develops a naked singularity, which is precisely the new regime where the cited one-function analyses have not been checked. If a two-function solution with an event horizon exists at λ = 0.1, the stated constraint λ ≲ 0.1 and the photon-sphere/shadow conclusions would not follow from the present calculation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies static, spherically symmetric vacuum solutions in Einsteinian cubic gravity with a cosmological constant, using the single-function metric ansatz ds² = -f(r)dt² + dr²/f(r) + r²dΩ². The authors derive approximate weak-coupling, asymptotic, and near-horizon solutions, integrate the field equations numerically, and report that a positive coupling λ shrinks the horizon while a negative λ enlarges it, with a naked-singularity branch appearing for λ ≳ 0.09–0.1 for a SgrA*-mass object. They then compute the angular difference α in triangular null-geodesic arrays and the photon-sphere position, concluding that strong-field observables can constrain the coupling to λ ≲ 0.1 and that the strongest cubic effects occur near the source. The manuscript includes publicly available numerical codes.","tokens_in":23540,"tokens_out":5166,"duration_ms":52563,"significance":"If the results are correct, the paper would provide concrete, falsifiable strong-field predictions for Einsteinian cubic gravity: a horizon-size dependence on the sign of λ, a photon-sphere shift, and angular differences measurable in principle with laser-ranging or interferometric missions. The analytic work extends earlier studies by expressing approximate solutions in terms of a single integration constant C0 and by covering both signs of λ and non-asymptotically flat backgrounds. Credit is due for the explicit asymptotic and near-horizon series, the comparison of these series with numerical integrations, and the public release of the numerical codes. However, the central quantitative claims are currently conditional on an unproved metric ansatz and on numerical outputs without stated uncertainty, and the headline λ ≲ 0.1 bound is derived for a single mass value.","major_comments":[{"comment":"The single-function ansatz g_tt = 1/g_rr is load-bearing for all results, but the paper does not show that this ansatz is exhaustive for the solutions studied. The statement that the densities C and C' are trivial in static spherically symmetric vacuum, together with Refs. [11,12], establishes the existence of a one-function branch, not uniqueness: the general static spherically symmetric metric has two independent functions, and there is no coordinate freedom to impose g_tt g_rr = -1 while retaining r as the areal radius. Because Eq. (5), the integrated master equation (7), the near-horizon series (20), the horizon equation (24), and all numerical metrics behind Table II and Figs. 5–7 rely on this ansatz, the claimed horizon, photon-sphere, and angular-difference results are conditional on the branch. The paper should either derive the two-function field equations and prove (or verify numerically for a two-function metric with the same boundary conditions) that a solution branch with g_tt g_rr = -1 exists and is the one relevant for black holes, or explicitly restrict every conclusion to the one-function branch and note that the general solution is not covered.","section":"Section II, Eq. (4) and throughout"},{"comment":"The master integrated equation (7), from which nearly all subsequent analytical and numerical results are obtained, is stated with the phrase 'can be integrated' but no derivation is provided. There is no appendix or reference that shows how Eq. (5) is reduced to Eq. (7), and this equation is the starting point for the weak-coupling, asymptotic, near-horizon, and numerical solutions. This gap is load-bearing for the paper's central claims, because any error or missing integration constant in Eq. (7) would propagate through the horizon properties, angular differences, and the λ ≲ 0.1 constraint. The authors should include a derivation of Eq. (7) (or a detailed reference that contains it) and a direct consistency check, such as substituting the final expression back into Eq. (5).","section":"Section II, Eq. (7)"},{"comment":"The reported angular differences α are quoted to several significant figures without error bars, convergence tests, or tolerances. In particular, the claim that the results for Kottler and Schwarzschild backgrounds are the same 'under double-precision floating-point numerical resolution' is not accompanied by any estimate of the numerical resolution, and the constant row b1 = 10^4 rs in Table II, where α = 2.04 × 10^-8 arcsec for all λ, could be a numerical floor rather than a physical result. Since α is the central observational signature, the paper should report convergence with respect to the Runge-Kutta tolerances, the grid spacing used for the central-difference derivatives in Eq. (32), the placement of the outer boundary where the asymptotic solution is matched, and the resulting uncertainty on each α value and on the location of the λ ≈ 0.1 threshold.","section":"Table II and Figs. 5–6"},{"comment":"The constraint λ ≲ 0.1 and the associated naked-singularity threshold are derived for a single value of the mass, C0 = 4 × 10^6 M⊙ (later 4.3 × 10^6 M⊙), with no exploration of the SgrA* mass uncertainty or the general dependence on C0. The horizon equation (24) and the near-horizon relation (22) depend on C0, so the threshold is expected to shift with the mass; the text itself quotes λ ≈ 0.09 in one place and λ ≲ 0.1 in the conclusion. The authors should compute the threshold over the observationally allowed mass range for SgrA*, present the dependence of the critical λ on C0, and specify the numerical criterion used to decide that the maximum of the effective potential disappears in Fig. 7.","section":"Section II F and Section III C"}],"minor_comments":[{"comment":"The dimensionless rescaling (27) means that values such as λ = 0.05 are only meaningful in the chosen units; the plot axes and table captions should state these units explicitly and distinguish them from dimensionful values used in astrophysical contexts.","section":"Section II F, Eq. (27) and figure captions"},{"comment":"The left panel of Fig. 3 uses C0 = 4 × 10^21 M⊙ and is described in the text as a non-physical illustrative solution, but the caption does not clearly say this; adding a sentence to the caption would avoid confusion.","section":"Section II F, Fig. 3"},{"comment":"Table II and the surrounding text use C0 = 4.3 × 10^6 M⊙, while Section II F and Figs. 2, 5, and 6 use C0 = 4 × 10^6 M⊙; these values should be reconciled, since the angular differences depend on C0.","section":"Section III B, Table II"},{"comment":"The notation ar g_{\\phi\\phi} and ar g_{rr} in the tangent formula is not defined; if these are the optical metric components, they should be introduced before Eq. (32).","section":"Section III A, Eq. (32)"},{"comment":"The statement that the Kottler and Schwarzschild backgrounds give the same α for b1 < 10^4 rs because the cosmological constant contributes only when b1 > 10^10 rs is asserted without derivation; a quantitative estimate of this threshold would make the statement easier to verify.","section":"Section III B"},{"comment":"There are several typographical or notation inconsistencies, including 'SgrA∗' versus 'Sgr A*', 'Red. [17]' instead of 'Ref. [17]', and the use of 'and' in footnote 1; these should be cleaned up during revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of a general relativity journal and the authors have made a good-faith effort to compare analytical approximations with numerical solutions and to release their code. The single-function ansatz concern raised in the stress-test note is genuine and is a load-bearing gap: if the full two-function field equations admit solutions with g_tt g_rr ≠ -1 that satisfy the same boundary conditions, the horizon and shadow predictions could change. The absence of a derivation for Eq. (7) and the absence of numerical uncertainties for α are additional barriers. I would be willing to reconsider after a revision that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about observable signatures of higher-curvature gravity. The paper is a solid, code-backed study of Einsteinian cubic gravity, and its central claim largely holds up: the cubic coupling shifts horizons and photon spheres in a sign-dependent way, and the near-source angular difference is a sensitive probe. The genuinely new piece is applying the triangular-array angular difference method to numerically integrated ECG spacetimes with both signs of the coupling and a nonzero cosmological constant, including the SgrA* horizon bound. The analytical approximate solutions are worked out carefully, expressed in terms of a single integration constant, and the public numerical code is a real plus.\n\nThe soft spots are real but addressable. The main one is the single-function ansatz in Eq. (4), which fixes g_tt = 1/g_rr before solving the field equations. The paper justifies this by citing Refs. [11,12] and arguing C and C' are trivial in SSS vacuum, but that only establishes that the branch exists, not that it is exhaustive. Since the horizon equation, the near-horizon series, and all the numerical metrics use this ansatz, a two-function solution with an event horizon at lambda around 0.1 would undercut the lambda less than about 0.1 constraint. I do not think this is fatal, because the branch they solve is legitimate and the literature supports it, but it deserves referee attention, especially on the naked-singularity side where the cited one-function analyses have not been checked.\n\nSecond, the integrated master equation (7) is stated without derivation; it is the foundation for everything, so a derivation or a clear reference is needed. Third, the numerical angular differences in Table II and Figures 5 and 6 have no error bars or convergence tolerances. With an adaptive Runge-Kutta routine this should be easy to supply. Fourth, the SgrA* bound uses one mass value with no uncertainty; the bound should be quoted with a mass range. These are not load-bearing failures.\n\nThe paper is for people working on higher-curvature gravity phenomenology, light bending, and black hole shadows. It is a serious, honest piece of work, and the central argument holds up within the branch it studies. Send it to peer review; ask for the derivation of Eq. (7), error bars on alpha, and a discussion of the single-function ansatz's domain of validity.","headline":"Solid, code-backed phenomenology of Einsteinian cubic gravity; the main results hold within the single-function branch, but the ansatz needs scrutiny before trusting the quantitative bound.","tokens_in":24073,"tokens_out":1856,"would_cite":true,"duration_ms":18777,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83D05","83C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"In Einsteinian cubic gravity, a positive coupling shrinks black hole horizons, and Sgr A* keeps its horizon only if the coupling is below roughly 0.1.","keywords":["Einsteinian cubic gravity","higher-curvature gravity","black hole horizons","photon sphere","light bending","black hole shadows","angular difference","Sgr A*"],"falsifier":"Solve the full vacuum field equations without imposing the reciprocal metric relation and compare the resulting horizon curve with Fig. 1; any discrepancy would overturn the derived thresholds. Alternatively, a shadow-radius measurement of Sgr A* accurate to a few percent would expose or rule out the claimed ~50% photon-sphere shift.","tokens_in":23054,"feed_emoji":"🕳️","tokens_out":6489,"duration_ms":60044,"temperature":0.7,"pith_summary":"The paper claims that Einsteinian cubic gravity, a higher-curvature theory chosen to keep general relativity's particle spectrum, leaves visible marks in the strongest gravitational fields. Working with a static, spherically symmetric metric, it shows that the sign of the coupling λ controls the horizon: positive λ shrinks the horizon, negative λ enlarges it, and for λ ≳ 0.1 a Sgr A*-mass object would have no black hole horizon. The same coupling moves the photon sphere by tens of percent, so the shadow radius and a recently proposed angular-difference observable become tests of the theory. The paper concludes that the coupling is constrained to λ ≲ 0.1 by the existence of Sgr A*'s horizon and photon sphere, and that the cubic terms' strongest signals are close to the source.","feed_headline":"Cubic gravity shrinks black holes, bounding its coupling at 0.1","feed_subtitle":"A positive cubic term shrinks the horizon and photon sphere; Sgr A*'s shadow could test the theory.","key_machinery":"The load-bearing object is the integrated vacuum field equation (7), obtained from the single-function ansatz ds² = -f dt² + $f^{{-1}}$ dr² + r² dΩ², which reduces the six-derivative cubic field equations to one ordinary differential equation containing the ADM mass integration constant C0. At a horizon, f(rh) = 0 and f'(rh) ≥ 0 turn this equation into the cubic relation (24) for f'(rh), which the paper uses to classify horizon existence and sign dependence. For observables, two tools carry the argument: the effective potential VEff = f L²/r², whose maximum locates the photon sphere, and the triangular-array angular difference α from null geodesics, computed numerically using the near-horizon, asymptotic, and weak-coupling series to seed integrations.","core_discovery":"Einsteinian cubic gravity admits exact de Sitter-like solutions with an effective cosmological constant set by λ and Λ, and, away from maximal symmetry, static spherically symmetric solutions whose metric function is governed by one integrated equation with mass integration constant C0. In these solutions the horizon radius is smaller than the Schwarzschild radius for λ > 0 and larger for λ < 0, with λ = 0 recovering Schwarzschild. For C0 fixed to the mass of Sgr A* and ΛEff fixed to the observed cosmological constant, λ ≈ 0.1 is the threshold above which the non-cosmological horizon disappears, leaving a naked singularity with divergent Ricci scalar at r = 0. The photon sphere follows the horizon: positive (negative) λ moves it inward (outward) by up to about 50% for |λ| < 0.1, and the angular difference α between triangular null-geodesic arrays in cubic gravity and in the Kottler or Schwarzschild background grows as the array approaches the source, reaching thousands of arcseconds at impact parameter 10 rs for λ = -5 and about $10^{4}$ arcsec near threshold for positive λ.","pith_inferences":["A direct extension not made in the paper would be to compute the predicted shadow image for these non-asymptotically flat metrics and compare it with existing millimetre-wavelength shadow observations of Sgr A*, converting the λ ≲ 0.1 bound into an actual posterior constraint.","Because the single-function ansatz is assumed, a natural stress test is to solve the full vacuum equations with two independent metric functions; if they differ, the horizon and photon-sphere relations change, especially on the naked-singularity branches.","The same triangular-array observable could be applied to rotating cubic-gravity black holes, where the photon-sphere displacement should be azimuthal and could be separated from the spherically symmetric part.","If higher-order curvature terms are added, the horizon-shift and photon-sphere relations likely become polynomial in the additional couplings, so the sign-and-magnitude pattern found here may be the first member of a family."],"forward_implications":["If the claim is right, a positive λ reduces the event-horizon radius below the Schwarzschild value for a fixed mass, so mass estimates based on horizon-crossing observables would be systematically shifted unless the cubic term is included.","For a Sgr A*-mass object, λ ≳ 0.1 removes the black hole horizon entirely; since Sgr A* is observed to have one, the coupling is observationally bounded from above at about 0.1.","The photon sphere moves by about 50% relative to Schwarzschild for |λ| just under 0.1, implying changes in black hole shadow size of a similar order.","The angular difference α between cubic gravity and the Kottler or Schwarzschild solution grows sharply when the geodesic triangle approaches the source, so future high-precision deflection measurements near a compact object can isolate the cubic contribution.","In the Solar System the expected cubic contribution to α is about 10^-9 arcsec, which is below currently planned micro-arcsecond mission sensitivities."],"supporting_citations":[{"why":"Defines Einsteinian cubic gravity and the cubic density P whose field equations are studied.","marker":"[1]"},{"why":"Establishes the inversely-related single-function property of static spherically symmetric solutions and the interpretation of C0 as ADM mass that this paper adopts.","marker":"[11]"},{"why":"Provides the original static spherically symmetric black hole solutions in Einsteinian cubic gravity that the paper extends to include Λ and to both signs of λ.","marker":"[12]"},{"why":"Introduces the triangular-array angular difference α that the paper uses as its main observational signature.","marker":"[22]"},{"why":"Supplies the observed value of the cosmological constant used to fix ΛEff in the numerical solutions.","marker":"[33]"},{"why":"Supplies the Sgr A* mass estimate C0 = 4 × 10^6 M⊙ used for the horizon, photon-sphere, and angular-difference examples.","marker":"[40]"}],"fun_headline_variants":["Cubic gravity shrinks horizons, photon spheres test the coupling","Positive cubic coupling shrinks black hole shadows, up to 50%","Sgr A* shadow could constrain cubic gravity's coupling constant","Cubic gravity: strong-field light bending reveals new signatures","Black hole shadows probe cubic gravity in the strong regime"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything downstream assumes the static spherically symmetric solution has the reciprocal relation g_tt = 1/g_rr (the single-function ansatz); if the full cubic equations force a second independent metric function, the horizon radii, photon-sphere positions, and angular differences could all change.","fun_headline_variants_meta":{"raw":{"variants":["Cubic gravity shrinks horizons, photon spheres test the coupling","Positive cubic coupling shrinks black hole shadows, up to 50%","Sgr A* shadow could constrain cubic gravity's coupling constant","Cubic gravity: strong-field light bending reveals new signatures","Black hole shadows probe cubic gravity in the strong regime"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1463,"prompt_tokens":956,"completion_tokens":507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":422}},"tokens_in":572,"tokens_out":507,"duration_ms":5465,"temperature":1.0,"reasoning_tokens":422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:37:14.626015+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the full vacuum field equations without imposing the reciprocal metric relation and compare the resulting horizon curve with Fig. 1; any discrepancy would overturn the derived thresholds. Alternatively, a shadow-radius measurement of Sgr A* accurate to a few percent would expose or rule out the claimed ~50% photon-sphere shift.","supporting_citations":[{"cited_title":"Estancias Posdoctorales por M´ exico para la Formaci´ on y Consolidaci´ on de las y los Investi- gadores por M´ exico","cited_arxiv_id":null,"evidence_quote":"Defines Einsteinian cubic gravity and the cubic density P whose field equations are studied."},{"cited_title":"Euler-Heisenberg Black Holes in Einsteinian Cubic Gravity","cited_arxiv_id":"2402.06867","evidence_quote":"Introduces the triangular-array angular difference α that the paper uses as its main observational signature."},{"cited_title":"kosmologischen","cited_arxiv_id":null,"evidence_quote":"Supplies the observed value of the cosmological constant used to fix ΛEff in the numerical solutions."},{"cited_title":"Tidal Forces in Kottler Spacetimes","cited_arxiv_id":"2204.13203","evidence_quote":"Supplies the Sgr A* mass estimate C0 = 4 × 10^6 M⊙ used for the horizon, photon-sphere, and angular-difference examples."}],"review_version":1}