Observational constraints on nonlocal black holes via gravitational lensing
Rocco D'Agostino, Vittorio De Falco
Abstract
In this paper, we study the gravitational lensing around the static and spherically symmetric DD black holes, which we recently derived as perturbations of the Schwarzschild geometry within the revised Deser-Woodard theory of nonlocal gravity. We first present general analytical expressions for the deflection angle in both weak- and strong-deflection limits, explicitly relating them to the nonlocal corrections to Schwarzschild spacetime. Subsequently, we analyze lensing observables, such as the post-Newtonian effects and the black hole shadow, to constrain the DD black hole parameter space using current observational bounds. Finally, we perform a joint statistical analysis based on the Fisher information matrix, combining these findings with our previously obtained constraints from quasinormal modes. Our results indicate consistency with general relativity at the level. This work provides a first assessment of the DD parameter space and offers new insights to probe deviations from Einstein's gravity in view of future larger datasets.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper studies gravitational lensing around "DD black holes" — static, spherically symmetric solutions derived as perturbations of Schwarzschild geometry within the revised Deser-Woodard theory of nonlocal gravity. The main contributions are: (1) analytical expressions for the deflection angle in both weak-deflection limit (WDL) and strong-deflection limit (SDL), explicitly parameterized by the nonlocal correction parameters (ξ, k); (2) application of these results to constrain the DD BH parameter space using current astrophysical observations (PPN constraints from GRAVITY, EHT shadow measurements, and previously obtained QNM bounds); and (3) a joint Fisher information matrix analysis yielding ξ = 0.044 ± 0.039 and k = 2.51 ± 0.64, consistent with GR at the 1.13σ level.
The paper positions itself as a refinement and extension of Ref. [49] (Li & Zhang, 2026), which performed a first assessment of gravitational lensing for DD BHs. The authors claim to clarify analytical derivations and, critically, to connect these to observational data — a step not taken in the prior lensing study.
Methodological Rigor
The analytical framework is standard and well-established. The WDL expansion follows the classical approach of series expanding the integrand in powers of 1/r₀ and then converting to the impact parameter b. The SDL treatment follows the Bozza (2002) formalism, decomposing the integral into divergent and regular parts and extracting the universal logarithmic divergence near the photon sphere. Both are correctly applied to the DD metric, and the results reduce to Schwarzschild in the ξ → 0 limit, providing an important consistency check.
The observational analysis, however, has several notable limitations:
1. PPN constraint: The authors adopt γ = 1.00 ± 0.01 from GRAVITY observations of the S2 star. While reasonable, this is a relatively loose bound compared to Cassini (|γ−1| ≲ 10⁻⁵), and the authors justify excluding Cassini on the grounds that Solar System probes are "far from the gravitational environment surrounding BHs." This is physically motivated but somewhat selective — the PPN parameter γ is theory-dependent and its value should be universal if the theory is correct.
2. Shadow constraints: The conversion of EHT angular diameters to dimensionless impact parameters yields large uncertainties (bm,M87* = 5.50 ± 0.98 and bm,SgrA* = 5.33 ± 1.30), dominated by mass uncertainties. The resulting constraints are therefore quite weak.
3. QNM constraints: The QNM bounds from Ref. [46] are incorporated assuming symmetric Gaussian distributions derived from requiring ≤10% deviation from Schwarzschild predictions. This is somewhat ad hoc — the "10% deviation" threshold is a modeling choice rather than a statistically derived constraint.
4. Fisher matrix approach: While computationally efficient, the Fisher matrix approximation assumes Gaussianity of the likelihood, which may not hold given the bounded nature of the parameters (ξ > 0, k > 1) and the small number of observational inputs. The authors acknowledge this limitation and defer a full Bayesian analysis to future work.
Potential Impact
The paper contributes to the growing literature on testing modified gravity theories with black hole observations. The DD BH framework provides a concrete, physically motivated alternative to phenomenological parameterizations of BH spacetimes (e.g., the Rezzolla-Zhidenko or Johannsen metrics). By connecting nonlocal gravity corrections to observable quantities in both electromagnetic and gravitational-wave channels, this work demonstrates a multi-messenger approach to constraining beyond-GR physics.
However, the practical impact is currently limited by the weakness of available constraints. The 1.13σ consistency with GR is neither a strong confirmation nor an interesting tension — it essentially says the data are insufficiently precise to meaningfully constrain the model. The constraints on ξ are consistent with zero at barely more than 1σ, meaning the nonlocal parameter is essentially unconstrained.
The framework could become more impactful with next-generation EHT observations (ngEHT), LISA gravitational wave detections of extreme mass-ratio inspirals, and improved stellar-orbit monitoring at the Galactic Center.
Timeliness & Relevance
The paper is timely in that it addresses an active research area — testing GR with BH observations from EHT and GRAVITY — and connects it to a theoretically motivated modified gravity framework. The revised Deser-Woodard model is relatively well-studied in cosmological contexts, and extending it to strong-field BH physics is a natural and worthwhile direction. The paper also builds systematically on the authors' own prior work developing the DD BH solutions, their perturbation theory, and QNM analysis.
Strengths
Limitations
Overall Assessment
This is a competent and well-executed study that applies standard gravitational lensing techniques to a specific modified gravity BH solution and derives first observational constraints. It represents useful groundwork for future, more precise tests but currently lacks the observational leverage to make strong physical statements. The 1.13σ consistency with GR is unsurprising and not particularly constraining.
Generated May 13, 2026
Comparison History (38)
Paper 2 has higher impact potential: it targets testable deviations from GR using gravitational lensing, shadows, and combines with quasinormal-mode constraints—an approach aligned with rapidly growing observational capabilities (EHT, strong-lensing surveys, GW astronomy). It provides analytic weak/strong deflection formulas and a joint statistical framework (Fisher) for near-term forecasts, enabling broad cross-field relevance (gravity theory, astrophysics, cosmology). Paper 1 is a phenomenological cosmology fit with modest novelty (Chaplygin/spinor + anisotropy) and limited discriminating power since shear is consistent with zero and improvements over ΛCDM are incremental.
Paper 2 likely has higher impact due to broader applicability and timeliness: it develops a full, gauge-invariant cosmological perturbation framework for an extended vector-tensor theory and connects directly to structure formation and multiple large-scale observational probes (CMB, LSS, weak lensing) across redshifts. This can influence both theoretical model-building and data analyses. Paper 1 is more specialized (lensing/shadow constraints for a particular nonlocal black hole solution) and its near-GR consistency may limit downstream impact, though it is methodologically solid.
Paper 2 likely has higher impact due to stronger timeliness and real-world applicability: it derives lensing/shadow observables and performs statistical constraints (including a Fisher-matrix joint analysis with quasinormal-mode results) directly tied to current and near-future astrophysical datasets testing gravity. This makes it broadly relevant across gravitational theory, observational astrophysics, and data analysis. Paper 1 is more niche (higher-dimensional EYM/GB quantum singularity regularity via HM criterion), with limited direct observational pathways and narrower cross-field uptake, though conceptually interesting.
Paper 2 combines multiple contemporary observational probes, including gravitational lensing, black hole shadows, and quasinormal modes, using a rigorous joint statistical framework. Its integration of modern multi-messenger constraints to test a specific modified gravity theory gives it broader relevance and higher potential impact compared to Paper 1, which primarily relies on standard geodesic analyses and weak-field solar system tests.
Paper 2 likely has higher impact due to clearer, near-term observational applicability: it derives lensing/shadow observables, confronts them with current bounds, and combines constraints with quasinormal-mode results via a Fisher analysis, making it directly relevant to ongoing and upcoming black-hole datasets (EHT, precision lensing, GW). This breadth (gravity theory + astrophysical tests + data-driven constraints) increases cross-field impact and timeliness. Paper 1 advances theoretical understanding of scalarization phase structure (including rotation and linear couplings) but is more specialized and less immediately tied to specific observational constraints.
Paper 1 offers a broader and more rigorous approach by combining multiple cleaner observables (gravitational lensing, black hole shadows, and quasinormal modes) to constrain deviations from General Relativity. Its joint statistical analysis provides concrete limits on fundamental physics. Paper 2 focuses on QPOs, which, while valuable, suffer from high model dependency and astrophysical uncertainties regarding accretion flows, making Paper 1's findings more robust and likely to yield a higher scientific impact in tests of gravity.
Paper 1 provides a fundamental, innovative geometric explanation for the decoupling of perturbations in Kerr spacetimes, a foundational result in general relativity. This deep theoretical insight broadly impacts black hole perturbation theory and gravitational wave physics. Paper 2, while observationally relevant, focuses on constraining a specific, somewhat niche modified gravity theory, limiting its broader fundamental impact compared to Paper 1.
Paper 2 likely has higher impact: it connects a modified-gravity black-hole solution to multiple observable channels (weak/strong lensing, shadow, plus joint Fisher analysis with quasinormal-mode constraints), yielding quantitative bounds and near-term relevance for current/future datasets. This gives clearer real-world applicability, broader cross-field reach (theory, astrophysical observations, statistics), and stronger methodological scope. Paper 1 is more niche and perturbative (specific wormhole in R=0, vanishing Love number only to linear order in a regularization parameter), with less immediate observational leverage.
Paper 1 addresses a fundamental question in mathematical physics—weakly turbulent dynamics and nonlinear stability of Schwarzschild-AdS black holes—with rigorous mathematical results (norm inflation for quasilinear wave equations) that apply broadly to backgrounds with stable trapping. This connects to the deep open problem of AdS instability and introduces novel techniques with cross-disciplinary relevance (mathematical GR, PDE theory, turbulence). Paper 2, while timely and observationally relevant, is more incremental—constraining a specific nonlocal gravity model using standard lensing/shadow analyses with results consistent with GR, limiting its transformative potential.
Paper 1 likely has higher impact due to strong novelty (new dynamical regime in ultra-relativistic BH encounters), high methodological rigor (state-of-the-art numerical relativity at γ≈5.1), and broadly relevant implications for strong-field GR, waveform modeling, and high-energy gravitational dynamics. The reported irregular, prolonged emission and very high radiated energy fraction (>65%) could reshape expectations for extreme encounters. Paper 2 is timely and application-oriented (constraints on nonlocal gravity via lensing), but is more incremental/parameter-constraining and depends on model-specific assumptions, with results largely consistent with GR.
Paper 1 addresses a critical computational bottleneck in gravitational-wave astronomy—parameter estimation for binary neutron star signals—by introducing a novel compression strategy combining simulation-based inference with relative binning. This has immediate practical applications for current and future GW detectors (e.g., next-generation detectors with even longer BNS signals). The methodological innovation of integrating heterodyning-based summary statistics with neural posterior estimation is broadly applicable and timely given the increasing detection rate. Paper 2, while rigorous, tests a specific nonlocal gravity model and finds consistency with GR, offering more incremental contributions to the modified gravity landscape.
Paper 2 presents original scientific research, providing new analytical expressions and statistical constraints on nonlocal gravity theories using observational data. In contrast, Paper 1 is a memorial article summarizing past historical contributions. Original research with testable physical constraints inherently carries a much higher potential for direct scientific impact than biographical pieces.
Paper 2 has broader impact: it combines theoretical predictions with observational constraints using multiple independent probes (gravitational lensing, shadows, quasinormal modes) and statistical analysis, directly connecting modified gravity theory to current and future observations. It addresses the timely topic of testing GR deviations with real data (EHT, etc.) and provides a methodology applicable to other modified gravity theories. Paper 1, while technically interesting, addresses a narrower theoretical question about mass inflation in a specific class of gravity theories, with less immediate observational relevance.
Paper 1 provides direct observational constraints on a specific modified gravity theory using multiple complementary methods (weak/strong lensing, shadows, quasinormal modes) with joint statistical analysis against real observational data (EHT). It offers concrete, falsifiable predictions and quantitative parameter constraints. Paper 2, while creative in connecting Tsallis statistics to black hole thermodynamics and photon-sphere observables, remains more theoretical and speculative, with the optical-thermodynamic analogy being qualitative rather than producing testable predictions against current data. Paper 1's methodological rigor and direct contact with observations give it broader and more immediate impact.
Paper 1 likely has higher impact: it connects a specific nonlocal-gravity black hole model to multiple observational probes (weak/strong lensing, shadow, quasinormal modes) and performs a joint Fisher-matrix constraint study, making it timely with current/future gravitational-wave and EHT-era datasets. Its real-world applicability (parameter constraints, test of GR) and breadth (modified gravity, astrophysics, cosmology, data analysis) exceed Paper 2’s more specialized numerical study of TTMs in an analogue-gravity draining bathtub model, which is narrower in applications and broader-field relevance.
Paper 1 likely has higher impact due to strong timeliness and direct operational relevance for current gravitational-wave astronomy. Its hierarchical Bayesian framework with quantile compression and recycled-likelihood approximations provides a broadly applicable, methodologically rigorous tool for characterizing non-Gaussian noise, improving detector sensitivity and inference reliability, and has immediate uptake potential by LIGO-Virgo-KAGRA analyses. It also demonstrates real-data application and addresses high-profile event validation. Paper 2 is valuable but more niche: constraints on a specific nonlocal-gravity black hole model via lensing are less immediately actionable and depend on future datasets for decisive impact.
Paper 2 likely has higher impact due to direct, timely connection to observations (lensing, shadow, joint constraints with quasinormal modes) and a clear pathway to falsifiable tests with near-future datasets. Its methodology (analytic weak/strong lensing limits plus Fisher-matrix forecasting and existing bounds) is comparatively concrete and broadly relevant across gravity theory, astrophysics, and data analysis. Paper 1 is more speculative and model-dependent (semiclassical/Madelung treatment of interior collapse and singularities), with less immediate observational leverage and narrower near-term applicability.
Paper 1 likely has higher impact due to stronger timeliness and direct observational relevance: it connects a specific modified-gravity model (nonlocal Deser–Woodard) to measurable lensing observables (weak/strong deflection, shadow) and combines constraints with quasinormal-mode information via a Fisher analysis. This yields quantitative bounds and an explicit consistency level with GR, aligning with active EHT/LIGO–Virgo–KAGRA/large-survey programs. Paper 2 is conceptually elegant but more foundational and harder to test experimentally, with narrower near-term applicability.
Paper 1 has broader scientific impact because it connects theoretical nonlocal gravity modifications to current observational data (gravitational lensing, black hole shadows, quasinormal modes), providing concrete constraints on deviations from general relativity. It employs a rigorous joint statistical analysis framework applicable to future observations. Paper 2, while technically interesting, addresses a more niche topic (Nambu-Goto string energy extraction from Kerr black holes) with limited observational relevance, shorter simulation timescales, and narrower applicability. Paper 1's direct connection to observational astronomy and tests of gravity theories gives it greater potential for cross-disciplinary impact.
Paper 2 likely has higher impact due to broader methodological and cross-field relevance: it develops a systematic large-c expansion framework in ADM variables bridging Galilean/Carrollian limits, derives actions/equations to NNLO, and constructs spin/quadrupole/mixed solutions with clear utility for modeling rotating compact objects. This advances nonrelativistic gravity as an effective-theory tool applicable to GR, waveform modeling, and compact-object astrophysics. Paper 1 is timely and observationally motivated but is more specialized (a specific nonlocal-gravity black-hole model) and its constraints are limited by current data, reducing near-term influence.