{"id":"a5ca3e65-fa4d-4612-b9bd-49aeafc6e195","arxiv_id":"2411.13049","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The optical appearance of a scalar hairy black hole with asymmetric potential is computed, and its shadow size is used to constrain the potential parameters with EHT observations.","lead":"This paper computes what a scalar hairy black hole with an asymmetric potential would look like when surrounded by a thin accretion disk, using a numerical spacetime from earlier work. It then compares the predicted shadow angular diameter to EHT measurements of Sagittarius A* and M87* to narrow down the model parameters.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The EHT constraints violate the paper's own potential-domain condition 0<2ϕ0<ϕ1: for ϕ1=1.0 and 2.0 all quoted ϕ0 intervals in Table I lie outside the valid solution space, so the central exclusion claim is unsupported.","rationale":"The reader's weakest assumption was the identification of the EHT ring angular diameter with the shadow of a static, spherically symmetric, non-rotating metric. That is a legitimate astrophysical caveat, but it is secondary. The more fundamental problem is internal to the paper: the parameter values used for the central claim violate the paper's own stated domain 0<2ϕ0<ϕ1. For ϕ1=1.0, the valid range is ϕ0<0.5, yet Table I claims allowed intervals around 0.89–0.92; for ϕ1=2.0, the valid range is ϕ0<1.0, yet Table I claims 1.38–1.50; for ϕ1=3.0, the upper bounds exceed 1.5. In these regions the potential is non-negative and the WEC-violating mechanism that generates the hairy solution is absent. The paper therefore does not constrain the scalar hairy black hole it set out to study; it constrains parameter values for which the solution likely does not exist. This is an internal inconsistency, not a disagreement with consensus, and it directly undermines the strongest claim: the exclusion of ϕ1=0.5 and the allowed intervals for other ϕ1 values. A recomputation restricted to the valid domain would settle the issue; if no valid point matches EHT, the conclusion changes from 'narrow constraints' to 'no constraint from this model.' The lack of numerical data and absence of propagated uncertainties remain additional concerns, but the domain violation is sufficient by itself to reject the central claim as stated.","tokens_in":18498,"tokens_out":15979,"duration_ms":154772,"concrete_test":"Restrict the parameter scan to 0<ϕ0<ϕ1/2 and recompute the shadow angular diameter in Eq. (37) for ϕ1=1,2,3,5, regenerating Fig. 10 and Table I. If no point in the valid domain overlaps the EHT bands, the quoted constraints and the exclusion claim disappear; report which valid intervals, if any, survive.","verdict_should_be":"REJECT","load_bearing_attack":"Section II introduces the asymmetric potential and states 'Therefore, 0 < 2ϕ0 < ϕ1' immediately after Eq. (2). This is not optional: for ϕ0 > ϕ1/2 the potential is non-negative everywhere, so the WEC-violating hairy solution from Corichi et al. is not the object being studied. Yet the headline constraints in Sec. IV.A and Table I are placed almost entirely outside this domain. For ϕ1=1.0 the valid range is ϕ0<0.5, while Table I gives 0.8967≤ϕ0≤0.9131 (M87*) and 0.89≤ϕ0≤0.92 (Sgr A*). For ϕ1=2.0 the valid range is ϕ0<1.0, while Table I gives 1.39826≤ϕ0≤1.4775 and 1.375≤ϕ0≤1.498. For ϕ1=3.0 the valid range is ϕ0<1.5, but the upper bounds 1.532 and 1.57 exceed it. Figure captions also display invalid points (ϕ0=0.3 for ϕ1=0.5; ϕ0=0.6 for ϕ1=1.0; ϕ0=1.6 for ϕ1=3.0). The claim that ϕ1=0.5 is excluded therefore compares valid small-hair solutions against EHT bounds, while the claimed allowed intervals for ϕ1=1,2,3 are vacuous: they correspond to potentials with no negative minimum, no WEC violation, and hence no valid scalar hair solution. The central constraint result is internally inconsistent.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the optical appearance of static, spherically symmetric scalar hairy black holes with an asymmetric potential, originally constructed by Corichi et al. (2006). Using a Hamiltonian ray-tracing formalism, the authors compute the effective potential, photon sphere, ISCO, shadow radius, specific intensity for spherical accretion, and isoradial curves, redshift maps, and bolometric flux for a thin equatorial accretion disk in the Page-Thorne framework. The predicted angular shadow diameters are compared with EHT measurements of Sgr A* and M87*, leading to claimed constraints on the potential parameter φ0 for several values of φ1 and the claimed exclusion of φ1=0.5.","tokens_in":18829,"tokens_out":10840,"duration_ms":89970,"significance":"The paper treats a nontrivial numerical black hole solution and produces a broad set of standard observables, which is useful if the underlying solution is correctly parameterized. Its most promoted result, however—the EHT-based constraints on φ0—is undermined by the paper's own domain condition, since almost all quoted allowed intervals lie in the region 2φ0>φ1 where the asymmetric potential is not the WEC-violating one described in Eq. (2). If the results are recomputed on the valid parameter domain, the central constraint claim may disappear entirely. The paper's other sections, based on standard formulas, are more robust but still need revisions.","major_comments":[{"comment":"The parameter constraints in Table I violate the domain condition 0<2φ0<φ1 stated in §II immediately after Eq. (2). For φ1=1.0 the valid range is φ0<0.5, yet Table I lists allowed intervals 0.8967≤φ0≤0.9131 (M87*) and 0.89≤φ0≤0.92 (Sgr A*). For φ1=2.0 the valid range is φ0<1.0, yet the table gives 1.39826≤φ0≤1.4775 and 1.375≤φ0≤1.498. For φ1=3.0 the upper bounds 1.532 and 1.57 exceed the valid value 1.5. For these parameters the potential in Eq. (2) is non-negative and does not violate the weak energy condition, so the numerical hairy solution from Ref. [16] is not the object being studied. The associated claims—that φ0 is constrained for φ1=1,2,3 and that φ1=0.5 is excluded—are therefore unsupported. This is a load-bearing inconsistency in the paper's central comparison.","section":"§II and §IV.A/Table I"},{"comment":"The EHT comparison identifies the measured ring angular diameter (51.8±2.3 μas for Sgr A* and 42±3 μas for M87*) with the shadow diameter Θ=2r_shadow/D computed from a static, spherically symmetric, non-rotating metric. Realistic EHT targets are expected to be rotating, and the observed ring is an emission feature rather than the geometric shadow. The paper does not propagate the mass and distance uncertainties into the bounds in Table I, nor does it discuss how spin would affect the shadow size. This assumption should be stated explicitly and its impact on the constraints assessed, at least via a consistency check with the Kerr prediction for the same mass/distance inputs.","section":"§IV.A, Eqs. (36)–(37)"},{"comment":"The metric functions m(r), σ(r), φ(r) are numerical solutions taken from Ref. [17]. The manuscript does not provide the numerical data nor describe the interpolation procedure used to evaluate the metric at arbitrary φ0 between the plotted sample points. The Data Availability Statement says 'This manuscript has no associated data,' which conflicts with the paper's reliance on shared numerical data mentioned in the acknowledgments. Without access to the metric data or the interpolation details, the quantitative results in Table I and Figure 10 cannot be reproduced or checked.","section":"§II and Data Availability"},{"comment":"Several displayed parameter combinations lie outside the allowed domain 0<2φ0<φ1. The caption of Fig. 12 uses (φ1,φ0)=(0.5,0.3), (1.0,0.6), and (3.0,1.6); Table II uses (3.0,2.0). These do not correspond to valid scalar-hairy-black-hole solutions with the asymmetric potential defined in Eq. (2). The physical interpretation of the corresponding curves, images, and fluxes is therefore not well defined.","section":"Captions of Fig. 12 and Table II"}],"minor_comments":[{"comment":"The sentence 'for ϕ1=1.0, the possible values for ϕ0 belong to the interval 0.89 ≤ ϕ0 ≤ 0.92 and ≤ ϕ0 ≤ for M87* and Sgr A*' is incomplete; the second interval is missing its entries.","section":"§IV.A"},{"comment":"Several typos appear throughout: 'introducation' (§IV.B), 'infintiy' (§IV.B.1), 'Hece' (§IV.B.1), 'Schawarzchild' (§V), and 'Hereido' (§II).","section":"Throughout"},{"comment":"In the conclusion, the reference 'Fig. reffig:accretionDiskProfiles' is broken and should be replaced with the actual figure number.","section":"Conclusion"},{"comment":"The Data Availability Statement says 'no observational data related to this article' although EHT data are used; this is at least misleading and should be clarified.","section":"Data Availability Statement"}],"recommendation":"major_revision","confidential_remarks":"The parameter-domain problem is severe and likely changes the headline conclusion; I recommend requesting a revision in which the constraints are recomputed for 0<2φ0<φ1. The paper's other sections are salvageable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the ray-tracing and disk-imaging part is a competent, standard application of Luminet/Page–Thorne machinery to Corichi et al.'s numerical hairy black hole; the geodesic equations, photon sphere, ISCO, shadow radius, specific intensity, redshift factor, and flux formulas all look correct. Second, the central EHT constraint is internally inconsistent: the paper states 0 < 2ϕ0 < ϕ1 in Sec. II, then in Table I quotes allowed intervals for ϕ1 = 1 and 2 that lie entirely outside that domain (e.g., ϕ0 ≈ 0.9 for ϕ1 = 1, where validity requires ϕ0 < 0.5). For ϕ1 = 3 the upper bounds also exceed the limit. So the claimed allowed parameter windows do not correspond to valid hairy solutions. This is not a typo; it is the main observational result.\n\nWhat is genuinely new: this is the first place I know of that computes the optical appearance catalog for this specific asymmetric-potential SHBH solution—shadow size, photon rings, isoradial curves, redshift maps, and bolometric flux images. The work is honest about relying on Chew et al.'s numerical data and cites the relevant literature. The plots are consistent with the stated formalism.\n\nSoft spots, in order of severity. (1) The domain violation above. Because the constraints are computed for parameters where the scalar hair solution does not exist, the exclusion of ϕ1 = 0.5 is the only part of the constraint analysis that survives; and that depends on the additional assumption that the EHT ring angular diameter can be equated with the shadow of a static, spherical, non-spinning metric. Real Sgr A* and M87* are rotating, and the EHT ring is an emission feature, so this identification is an extra modeling leap. (2) Reproducibility: the numerical metric data are not provided, and the interpolation over ϕ0 is not described. The data availability statement says no associated data, which is odd given the numerical input from elsewhere. (3) The quoted ϕ0 intervals have no propagated uncertainties; the EHT errors are used as hard cutoffs.\n\nBottom line: the imaging part is a useful template for this solution and deserves referee time, but the constraint section must be redone inside 0 < 2ϕ0 < ϕ1. If the authors recompute and the results change, the paper could be solid; as written, the central claim is unsupported.","headline":"Competent imaging pipeline for a hairy black hole, but the EHT constraints are computed outside the solution's own validity domain, so the central claim does not stand as written.","tokens_in":19351,"tokens_out":5490,"would_cite":false,"duration_ms":47480,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Scalar hairy black hole shadows match EHT measurements only for narrow parameter windows, and the phi1 = 0.5 branch is excluded.","keywords":["black hole shadow","scalar hairy black hole","asymmetric scalar potential","photon sphere","accretion disk imaging","EHT observations","Sgr A*","M87*"],"falsifier":"Recompute the shadow angular diameter for a rotating version of the same asymmetric-potential scalar hairy black hole and compare with the EHT ring diameters; if spin moves the predicted values for $\\phi_1 = 0.5$ into the observed range, or shifts the $\\phi_0$ windows by more than the measurement uncertainties, the paper's parameter constraints would be overturned.","tokens_in":18284,"feed_emoji":"🕳️","tokens_out":12590,"duration_ms":105247,"temperature":0.7,"pith_summary":"This paper sets out to establish whether a class of black holes that evade the no-hair theorem—static, spherically symmetric solutions of Einstein gravity minimally coupled to a scalar field with an asymmetric potential—can produce the same shadow observations as the supermassive black holes at the centers of M87 and the Milky Way. It computes the photon sphere, the ISCO, the shadow radius, and the thin-disk images for these scalar hairy black holes, and compares the predicted angular diameter of the shadow with the EHT measurements for Sgr A* and M87*. The central claim is that the shadow angular diameter grows with the potential parameter $\\phi_0$, so the observed sizes select narrow windows for $\\phi_0$ for each value of $\\phi_1$, and the case $\\phi_1 = 0.5$ is excluded because its predicted shadows are too small. If this claim is right, shadow imaging becomes a direct observational test of no-hair violations driven by non-positive scalar potentials.","feed_headline":"Hairy black holes match EHT shadows only for narrow parameter windows","feed_subtitle":"The scalar potential branch phi1=0.5 is excluded; for phi1=1.0, phi0 must lie near 0.9.","key_machinery":"The central object is the numerical scalar hairy black hole metric $ds^2 = -N(r)e^{-2\\sigma(r)}dt^2 + N^{-1}(r)dr^2 + r^2(d\\theta^2 + \\sin^2\\theta\\,d\\varphi^2)$, with $N(r) = 1 - 2m(r)/r$ and metric functions $m$, $\\sigma$, and scalar field $\\phi$ obtained by solving the Einstein-Klein-Gordon equations for the asymmetric potential $V(\\phi)$. The argument is carried by the photon-sphere impact parameter $b = r e^{\\sigma(r)}/\\sqrt{N(r)}$ evaluated at the photon sphere radius $r_{\\rm ps}$, where $rN'(r)-2N(r)=0$; this value is the shadow radius $r_{\\rm shadow} = b(r_{\\rm ps})$, and it converts directly into the angular diameter $\\Theta = 2r_{\\rm shadow}/D$ used in the EHT comparison. The same metric, in the Hamiltonian formalism, produces the isoradial curves, redshift factor, and Page-Thorne energy flux that make up the synthetic images.","core_discovery":"On the paper's own terms, a scalar hairy black hole is not just a Schwarzschild black hole in disguise: the asymmetric potential parameters $\\phi_0$ and $\\phi_1$ change the metric functions $N(r)$ and $\\sigma(r)$ near the horizon, which in turn moves the photon sphere and the shadow boundary. The paper's key quantitative result is that for fixed $\\phi_1$ the shadow angular diameter $\\Theta = 2 r_{\\rm shadow}/D$ increases with $\\phi_0$, so matching the EHT values $51.8 \\pm 2.3\\,\\mu$as for Sgr A* and $42 \\pm 3\\,\\mu$as for M87* yields finite allowed intervals. For $\\phi_1 = 1.0$ the allowed range is approximately $0.89 \\leq \\phi_0 \\leq 0.92$; for $\\phi_1 = 2.0$, $3.0$, and $5.0$ the windows are wider, while for $\\phi_1 = 0.5$ the angular diameter never reaches the observed lower bounds. The paper also shows that the intensity of the photon ring weakens and the ring radius grows as $\\phi_0$ increases, and that the Doppler-boosted side of the disk dominates the bolometric flux distribution.","pith_inferences":["Beyond the paper: the constraints are computed for a static, non-rotating metric, whereas Sgr A* and M87* are expected to spin; a rotating version of the same hairy solution could shift the shadow size enough to alter the $\\phi_0$ windows or even rescue the $\\phi_1 = 0.5$ branch.","Beyond the paper: the quoted windows depend on the adopted mass and distance of each black hole, so improved astrometric distance or mass measurements for Sgr A* and M87* would directly rescale the allowed parameter ranges.","Beyond the paper: the predicted dimming of the photon ring as $\\phi_0$ grows is a testable trend, since future higher-resolution interferometric images that measure the ring-to-shadow brightness contrast would discriminate among the allowed parameter windows."],"forward_implications":["Accepting the EHT-shadow identification, the observed angular diameters of Sgr A* and M87* confine the asymmetric-potential parameter $\\phi_0$ to finite windows for $\\phi_1 = 1.0$, $2.0$, $3.0$, and $5.0$.","The branch with $\\phi_1 = 0.5$ is excluded by both EHT measurements, because its shadow angular diameter stays below the observed lower bounds.","Increasing $\\phi_0$ pushes the photon sphere and shadow outward and weakens the bright photon ring, so the same scalar hair that enlarges the shadow also makes it harder to see.","In the thin-disk images, the approaching side of the disk remains blueshifted and carries the bulk of the bolometric flux, with the overall flux magnitude set by $\\phi_0$, $\\phi_1$, and the inclination angle."],"supporting_citations":[{"why":"Constructs the numerical scalar hairy black hole solution with the asymmetric potential used throughout the paper.","marker":"[16]"},{"why":"Supplies the later discussion and numerical data for the same solution that the ray-tracing calculations rely on.","marker":"[17]"},{"why":"Gives the Luminet thin-disk imaging method used to draw isoradial curves and the mushroom-shaped disk images.","marker":"[48]"},{"why":"Reports the M87* angular diameter of 42 ± 3 μas used to set the phi0 constraints.","marker":"[54]"},{"why":"Reports the Sgr A* angular diameter of 51.8 ± 2.3 μas used to set the phi0 constraints.","marker":"[55]"},{"why":"Provides the Page-Thorne steady-disk model used to compute the emitted and observed energy flux.","marker":"[61]"}],"fun_headline_variants":["EHT shadow size narrows scalar hairy black hole parameter space","Asymmetric hairy black holes: only narrow phi0 windows survive EHT","Hairy black hole shadows: EHT data constrains phi0 for given phi1","Shadow size sets narrow parameter windows for scalar hairy black holes","phi1=0.5 hairy black holes excluded by EHT shadow limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The constraint analysis assumes that the ring angular diameter measured for Sgr A* and M87* can be equated with the shadow angular diameter $\\Theta = 2r_{\\rm shadow}/D$ of a static, spherically symmetric, non-rotating scalar hairy black hole.","fun_headline_variants_meta":{"raw":{"variants":["EHT shadow size narrows scalar hairy black hole parameter space","Asymmetric hairy black holes: only narrow phi0 windows survive EHT","Hairy black hole shadows: EHT data constrains phi0 for given phi1","Shadow size sets narrow parameter windows for scalar hairy black holes","phi1=0.5 hairy black holes excluded by EHT shadow limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1880,"prompt_tokens":1055,"completion_tokens":825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":671,"completion_tokens_details":{"reasoning_tokens":730}},"tokens_in":671,"tokens_out":825,"duration_ms":48767,"temperature":1.0,"reasoning_tokens":730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:53:14.912378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the shadow angular diameter for a rotating version of the same asymmetric-potential scalar hairy black hole and compare with the EHT ring diameters; if spin moves the predicted values for $\\phi_1 = 0.5$ into the observed range, or shifts the $\\phi_0$ windows by more than the measurement uncertainties, the paper's parameter constraints would be overturned.","supporting_citations":[],"review_version":1}