{"id":"547c700c-5ce3-467d-b0b9-269271a4bd47","arxiv_id":"2510.19857","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The shadow angular radius of a Schwarzschild black hole is set to 3√3 GM/(c² D_A(z)), re-expressing a known local formula with the fitted cosmological distance.","lead":"The paper proposes that the apparent angular size of a black hole shadow at redshift z is the Schwarzschild photon-sphere impact parameter divided by the cosmological angular-diameter distance D_A(z). It is a readable teaching note, but the claimed derivation from the McVittie and Kottler metrics is never actually carried out.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9) is introduced by substituting the Schwarzschild impact parameter b_c into the Euclidean relation α=R/D_A, without geodesic derivation; this rigid-ruler assumption is the load-bearing step and is not established for McVittie/Kottler spacetimes.","rationale":"The reader's verdict is REJECT, and my stress-test identifies the same load-bearing concern: Eq. (9) is an assertion, not a derivation. The paper's stated goal is to 'derive' a relation from McVittie/Kottler metrics, but no such derivation is provided. The substitution of b_c into the standard angular-diameter-distance relation is a dimensional guess that happens to reproduce the Schwarzschild limit, but there is no argument that the expanding background leaves the critical impact parameter intact. The existing literature (Refs. [6,7]) already contains exact shadow calculations in these spacetimes, and the paper's failure to engage with them is a serious omission. If Eq. (9) is meant as a leading-order approximation, the paper should state the expansion parameter and the regime of validity; absent that, the formula is unsupported. The numerical results only evaluate a known function, so they do not provide independent evidence. Given that the central claim is both central and unsubstantiated, a rejection is warranted. My proposed concrete test would settle whether Eq. (9) matches exact results; if it does in the appropriate limit, the paper might be rehabilitated as a pedagogical estimate, but as written it lacks the necessary justification. Therefore I agree with the reader's verdict and recommend no change.","tokens_in":6195,"tokens_out":4955,"duration_ms":47585,"concrete_test":"Compare Eq. (9) with the exact shadow angular radius in Schwarzschild–de Sitter (Kottler) spacetime from Perlick et al. (2018), Ref. [6], using the same mass M and cosmological constant Λ, with the observer placed at the comoving radius corresponding to a redshift z (e.g., z=1, 3, 5) in the flat ΛCDM background of Eq. (10). If the exact result differs from 3√3GM/(c² D_A(z)) by more than a negligible amount (e.g., >0.1%), then Eq. (9) is not a valid leading-order estimate and the central claim fails. An alternative check: repeat the exact McVittie shadow calculation of Tsupko & Bisnovatyi-Kogan (2020) and compare with Eq. (9).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central formula, Eq. (9), is not derived from the McVittie or Kottler null-geodesic structure. In §II.B, the text states that for any object of proper radius R the apparent angle is α = arcsin(R/D_A(z)) (Eq. 8), then simply substitutes R = b_c = 3√3GM/c² to obtain Eq. (9). No geodesic computation in the McVittie or Kottler metric appears. This is not a perturbative step; it is an ansatz that assumes the shadow's physical size is the flat-space impact parameter and that the angular diameter distance of the background FLRW metric remains valid without modification. For Schwarzschild–de Sitter (Kottler), the photon-sphere impact parameter depends on Λ (see Perlick et al. 2018, Ref. [6]), so the effective 'proper radius' is not b_c. For McVittie, the time-dependent background further complicates the definition of a critical impact parameter. The paper's own cited Ref. [7] contains an exact analytic calculation of the shadow in McVittie, but the present work neither uses nor compares to it. Thus the quantitative results are unverified; the numerical evaluation is merely an integration of the known D_A(z) multiplied by a constant, so the claimed 'redshift dependence' is inherited from the background, not a new prediction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a simple analytic formula, Eq. (9), for the apparent angular radius of a Schwarzschild black hole shadow in an expanding universe: α_shadow(z) = arcsin[3√3 GM/c^2 / D_A(z)], where D_A(z) is the FLRW angular-diameter distance. The derivation is based on the McVittie and Kottler metrics, but the actual calculation only substitutes the Schwarzschild impact parameter b_c into the standard angular-size relation for an object of proper radius R. The paper then expands D_A(z) at small z, numerically evaluates the formula for various masses and cosmologies, and concludes that cosmological expansion has a negligible effect for local sources but a conceptually interesting effect at high redshift.","tokens_in":6509,"tokens_out":5259,"duration_ms":48097,"significance":"If Eq. (9) were established, the paper would provide a simple pedagogical bridge between strong-field black hole physics and cosmology. The manuscript is clearly organized and the numerical integration is straightforward and reproducible. However, the central relation is not derived from the null-geodesic structure of the McVittie or Kottler spacetimes; it is an ansatz that reduces the problem to the background distance-redshift relation. The cited exact calculations by Perlick, Tsupko, and Bisnovatyi-Kogan (Refs. [6,7]) show that the shadow impact parameter is modified in these spacetimes, and the present work neither uses nor compares against them. Consequently, the main claim that cosmic expansion influences the shadow in the manner described is unverified, and the significance is therefore limited to a heuristic, pedagogically framed estimate.","major_comments":[{"comment":"The central formula is introduced by substituting R = b_c into the FLRW angular-size relation Eq. (8). No null-geodesic computation in the McVittie metric is performed. In the static Schwarzschild case, the shadow angle follows from the bending of null geodesics and the critical impact parameter; in McVittie/Kottler, the photon-sphere impact parameter is modified (see Refs. [6,7]). Identifying the shadow's 'proper radius' with the flat-space b_c is an unproven ansatz, not a derivation. The paper should either derive Eq. (9) from the McVittie geodesic equation or state explicitly that it is an approximation and quantify its error.","section":"II.B, Eq. (9)"},{"comment":"The small-z expansion and the numerical 'verification' check the same assumed relation. Eq. (12) is just the Taylor expansion of D_A(z) inserted into Eq. (9), and the numerical evaluation in Eq. (15) is the same expression. The agreement between the two does not verify the physics; it merely confirms the numerical integration is internally consistent. The redshift dependence of α is inherited from the background D_A(z), which is an input, not a new prediction about shadow optics.","section":"II.C, Eq. (12) and III.B"},{"comment":"The manuscript cites Refs. [6,7] but does not use or compare with their exact results. For Kottler, the shadow depends on Λ and on the observer's position relative to the cosmological horizon; for McVittie, Tsupko and Bisnovatyi-Kogan derived an analytic expression with additional dependence on the Hubble rate. These corrections are precisely the 'influence of cosmic expansion' the paper claims to estimate. Without showing that those corrections are negligible in the regime considered, Eq. (9) cannot be validated even as an approximation.","section":"II.B and III.D"}],"minor_comments":[{"comment":"Typo: 'for larger o' should presumably be 'for large r_o'.","section":"II.A, Eq. (5)"},{"comment":"The dashed portions labeled 'R > D_A(z)' correspond to physically unrealizable cases for the masses discussed; for any astrophysical black hole, b_c << D_A(z) at all redshifts. The figure caption should state which masses produce the dashed curves and why this regime is included.","section":"III.C, Fig. 1"},{"comment":"The factor '206265×10^6 rad arcsin(...)' is dimensionally unclear; it should read '206265×10^6 μas/rad × arcsin(...)' or similar.","section":"III.A, Eq. (15)"},{"comment":"Reference [14] is incomplete: it lists 'Perlick, V., Oleg Yu. Tsupko' without a title, journal volume, or page numbers in the reference list.","section":"References"},{"comment":"The paper uses α, θ_sh, and α_shadow interchangeably for the shadow angular radius; please define one symbol and use it consistently.","section":"General notation"}],"recommendation":"reject","confidential_remarks":"The manuscript reads as a pedagogical exercise rather than a research contribution with a derived result. The central formula is an unproven heuristic, and the numerical 'verification' is circular. The authors cite the exact papers that solve the problem more rigorously but do not engage with them, which is a significant omission. This does not meet the standard for publication in a serious journal as a research paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know: this paper has no new physics, no new derivation, and no new result. Eq. (9) is the standard shadow angle for a Schwarzschild black hole, θ ≈ 3√3 GM/(c² D_A(z)), with D_A(z) the usual cosmological angular-diameter distance. The entire paper is an evaluation of that formula for a few masses and redshifts.\n\nWhat it does well: it is clearly written and pedagogically transparent. The steps from the Schwarzschild photon sphere to the critical impact parameter are correct, and the small-z expansion is a textbook Taylor series. For a student who has never seen the relation between local strong-field optics and cosmological distances, this is a readable summary. The numerical values for Sgr A* and M87* are reproduced correctly, and the caveat that cosmological corrections are below 10^-10 μas for observed sources is honest.\n\nThe soft spot is load-bearing: the paper claims to derive the shadow size in expanding universes from the McVittie and Kottler metrics, but no geodesic computation appears. Section II.B simply asserts that any object of proper radius R subtends angle α = arcsin(R/D_A(z)), then substitutes R = b_c. That is an ansatz, not a derivation. For Schwarzschild–de Sitter the critical impact parameter depends on Λ, and for McVittie the time-dependent background changes the definition of the photon sphere. The paper's own cited Refs. [6] and [7] already contain exact analytic shadow calculations in those spacetimes; the present work neither uses nor compares to them. The numerical 'verification' in Section III.B merely checks the small-z expansion against Eq. (15), which is the same assumed formula, so it verifies nothing independent. The redshift dependence of the shadow is inherited from the fitted D_A(z), not predicted.\n\nThe paper is not wrong in its final formula as a leading-order estimate for distant observers. It is just not a research contribution. The abstract overstates the derivation, and the discussion claims conceptual insight that is already standard in Refs. [6,7,14].\n\nWho is this for? An undergraduate or graduate student looking for a compact statement of why cosmological expansion hardly affects observed shadows. As a pedagogical note in a teaching journal it could be acceptable. As a gr-qc research paper it is a desk reject: it adds no derivation, no method, and no benchmark beyond what the cited literature already provides.\n\nMy recommendation: do not send this to a serious referee for a research journal. If the authors want to publish it, they should reframe it explicitly as a pedagogical review and acknowledge that Eq. (9) is an approximation known from the existing analytical shadow literature.","headline":"A clean but essentially content-free substitution of the Schwarzschild impact parameter into the standard angular-diameter-distance formula; cites the literature that already does this properly.","tokens_in":7031,"tokens_out":1648,"would_cite":false,"duration_ms":17234,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C10","83F05"],"pacs":["04.70.-s","98.80.-k","04.20.-q"],"model":"deepseek-v4-flash","headline":"The paper claims the shadow angle of any non-rotating black hole in an expanding universe is simply the inverse sine of its photon-sphere radius over the angular-diameter distance.","keywords":["black hole shadow","cosmological expansion","angular diameter distance","photon sphere","McVittie metric","Schwarzschild spacetime","FLRW cosmology","Lambda-CDM"],"falsifier":"A full null-geodesic ray-tracing calculation in the McVittie metric (or the Kottler metric) that computes the exact apparent shadow boundary would settle whether Eq. (9) holds; if the true boundary differs from b_c/D_A(z) by a term of order H_0*b_c/c or larger, the formula is only an approximation. Observationally, a high-redshift black hole with independently measured mass and distance whose shadow angle disagrees with Eq. (9) beyond measurement uncertainty would falsify it.","tokens_in":6040,"feed_emoji":"🕳️","tokens_out":9231,"duration_ms":74238,"temperature":0.7,"pith_summary":"The paper tries to establish a one-line formula for how cosmic expansion changes the apparent shadow size of a non-rotating black hole: the angular radius is the inverse sine of the local photon-sphere impact parameter divided by the standard angular-diameter distance to the source. All cosmological dependence enters through that distance, so the shadow becomes a probe of the distance-redshift relation rather than a purely local quantity. The authors motivate the formula by embedding Schwarzschild in an expanding background and then evaluate it numerically for flat Lambda-CDM, including a small-redshift expansion that shows the shadow angle falls roughly as 1/z with corrections set by the deceleration parameter. A sympathetic reader would care because this reduces a would-be ray-tracing problem in a dynamic spacetime to a known cosmological integral, making the expansion's influence on shadows analytically tractable and pedagogically transparent.","feed_headline":"Black hole shadow size reduces to one cosmic distance formula","feed_subtitle":"Expansion enters only through angular-diameter distance, so shadows are static nearby and shift at high redshift.","key_machinery":"The central object is the critical impact parameter b_c = 3*sqrt(3)*G*M/c^2 of the Schwarzschild photon sphere, the boundary between photons that fall into the black hole and those that escape. The load-bearing step is to substitute this b_c for the generic proper radius R in the cosmological relation alpha = arcsin(R/D_A(z)), yielding Eq. (9). The McVittie metric provides the conceptual bridge: it describes a point mass embedded in an expanding FLRW background and reduces to Schwarzschild locally and FLRW at large distances, which the paper uses to justify replacing the static-observer distance with D_A(z). All the cosmological work in the formula is done by the distance integral D_A(z), wh","core_discovery":"The paper's central claim is Eq. (9): alpha_shadow(z) = arcsin(3*sqrt(3)*G*M/c^2 / D_A(z)), where D_A(z) is the standard angular-diameter distance in the background cosmology. It asserts that the apparent angular radius of a Schwarzschild black hole's shadow in an expanding universe is obtained by taking the static critical impact parameter b_c = 3*sqrt(3)*G*M/c^2, the radius of the photon-sphere boundary for captured versus escaping light, and treating it as a rigid proper size at the source location. In flat Lambda-CDM, D_A(z) is the usual integral over H_0, Omega_m, and Omega_Lambda, so the formula carries all expansion effects through the distance-redshift relation. The paper then derive","pith_inferences":["If Eq. (9) is taken as a quantitative prediction rather than an estimate, its main unstated risk is the rigid-ruler substitution; a natural next step would be a full null-geodesic ray tracing in the McVittie or Kottler metric to compute the true shadow boundary and measure the error in b_c/D_A(z).","The same construction should extend to rotating black holes: replacing b_c with the appropriate spin-dependent critical impact parameter would give a closed-form, cosmology-dependent shadow size for Kerr-de Sitter black holes, a direction the paper only gestures at.","The non-monotonic behaviour of alpha(z) means a measured shadow size does not uniquely determine a black hole's redshift on its own; using shadows as cosmological probes would require independent distance or mass information to resolve the degeneracy.","Since D_A(z) depends on H_0, Eq. (9) could be inverted to estimate H_0 from a shadow angle plus a mass measurement, but the microarcsecond scales involved make this a long-term prospect rather than an immediate test."],"forward_implications":["For the Milky Way's central black hole at redshift roughly 10^-6, the cosmological correction to the shadow radius is below one part in 10^10, so existing shadow measurements are unaffected by expansion.","For hypothetical black holes at redshift 1 to 5, the apparent shadow angle falls sharply; a 10^5 solar-mass black hole would subtend less than 10^-3 microarcseconds, below foreseeable interferometric resolution.","Because D_A(z) peaks near z about 1.6 in flat Lambda-CDM, a fixed-mass black hole appears slightly larger beyond that redshift; shadow size is non-monotonic in z.","Different background cosmologies, such as matter-dominated, de Sitter, and Lambda-CDM, yield systematically different shadow sizes at the same redshift, so shadow angles in principle encode H_0 and the matter and dark-energy densities.","In the static limit the formula reduces to the familiar Schwarzschild shadow result, making Eq. (9) an interpolation between local strong-field gravity and large-scale expansion."],"fun_headline_variants":["Black hole shadow size set by angular diameter distance","Expansion only shifts black hole shadows through distance","One distance formula ties shadows to cosmic expansion","Shadow size depends on distance in expanding universe","Cosmic expansion enters black hole shadows via distance"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes the shadow behaves as a rigid standard ruler of proper radius 3*sqrt(3)*G*M/c^2 placed at the black hole's location, so its observed angle is just that radius divided by the unmodified FLRW angular-diameter distance; this substitution is stated between Eqs. (8) and (9) rather than derived from the McVittie or Kottler null-geodesic equations.","fun_headline_variants_meta":{"raw":{"variants":["Black hole shadow size set by angular diameter distance","Expansion only shifts black hole shadows through distance","One distance formula ties shadows to cosmic expansion","Shadow size depends on distance in expanding universe","Cosmic expansion enters black hole shadows via distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1704,"prompt_tokens":750,"completion_tokens":954,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":885}},"tokens_in":494,"tokens_out":954,"duration_ms":8385,"temperature":1.0,"reasoning_tokens":885,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:45:47.628415+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A full null-geodesic ray-tracing calculation in the McVittie metric (or the Kottler metric) that computes the exact apparent shadow boundary would settle whether Eq. (9) holds; if the true boundary differs from b_c/D_A(z) by a term of order H_0*b_c/c or larger, the formula is only an approximation. Observationally, a high-redshift black hole with independently measured mass and distance whose shadow angle disagrees with Eq. (9) beyond measurement uncertainty would falsify it.","supporting_citations":[],"review_version":1}