The sharper constraint belongs to BH-II
Two effective quantum-corrected black-hole models produce sharply different limits in the paper's comparison with Event Horizon Telescope measurements. For BH-II, the reported upper bounds on the correction amplitude range from 0.3M to 1.0M for M87* and from 0.4M to 1.0M for Sgr A*, depending on the exponent s. For BH-I, the corresponding ranges are 1.2M to 5.8M and 1.7M to 8.3M. Here M is the model's black-hole mass. The authors summarize BH-II as roughly two to four times more tightly constrained than BH-I.
The work is a theoretical modeling analysis, not an empirical sample study. It examines two effective metric families, BH-I and BH-II, using analytic proofs, perturbative expansions and numerical parameter scans, then compares their predicted shadow sizes with quoted EHT measurements. The modeled exponent s is greater than one-half for BH-I and nonnegative for BH-II.
Different horizons, different photon-sphere boundaries
BH-I retains an inner Cauchy horizon, while its outer horizon stays at twice the black-hole mass for every allowed value of the two parameters. The analysis reports no extremal solution within that domain.
BH-II has a critical horizon boundary instead. On the horizon-containing side, the model has two horizons; above the boundary, the paper reports no horizon and what it calls a naked singularity. The authors restrict their physical analysis to the region with horizons. The photon-sphere boundary lies above the horizon boundary across the nonnegative s range, so the photon sphere exists over a broader parameter region in the equations than the horizon does.
A photon sphere is the calculated light-orbit location used in the shadow analysis. For BH-I, the paper finds a unique physical photon sphere throughout its stated exponent range. At s = 1, its inverse-radius coordinate returns to one-third, corresponding to a photon-sphere radius of 3M. The cancellation does not remove the shadow's parameter dependence: the shadow radius still changes and decreases as the amplitude grows relative to M.
The BH-I results distinguish the photon-sphere radius from the shadow radius. The tabulated photon-sphere radius varies non-monotonically with s, while the shadow radius changes monotonically with s. BH-II shows a different pattern: both radii increase monotonically with s and approach their Schwarzschild values from below. At the representative amplitude ξ = 0.8M, the reported results list no photon sphere when s is 0.75 or lower. The paper's abstract contains a conflicting summary of the BH-I s-dependence; the detailed results report the pattern described here.
A shadow alone cannot settle the question
Small-correction calculations reinforce the special role of s = 1. In BH-I, the first-order shift of the photon sphere carries a coefficient of 1 minus s and vanishes there, while the BH-II correction is always negative. At ξ = 0.1M, the reported expansions match the exact numerical results within 0.5 percent.
Both photon spheres are analytically unstable across the physical parameter spaces. The paper uses a Lyapunov exponent, a quantity that tracks the strength of this instability, and reports different deviations in the two models: they are typically enhanced for BH-I at small s and suppressed for BH-II. That contrast is the proposed route to information beyond the shadow size.
A single shadow-radius measurement cannot identify both the correction amplitude and the exponent. In a BH-I example, a shadow radius of about 5.19M is compatible with both the Schwarzschild case, ξ = 0, and a quantum-corrected point near ξ = 0.16M with s = 0.6. The paper proposes adding a Lyapunov-exponent measurement to break that degeneracy, but no joint observational measurement is presented.
The telescope comparison is illustrative
For the EHT comparison, the paper quotes a shadow diameter of 42 ± 3 microarcseconds for M87* and 48.7 ± 7 microarcseconds for Sgr A*. It treats the two measurements as independent windows on the amplitude and exponent. The resulting bounds are presented as illustrative upper limits, and BH-II's values are partly limited by whether a horizon exists.
Those caveats matter because the calculation tests effective metric families against quoted measurements. The study does not directly measure the Lyapunov exponent, so the second observable remains a proposal for breaking the degeneracy rather than a result already demonstrated.
The work is an arXiv preprint, arXiv:2608.28186v1, dated 28 August 2026, with no journal publication, DOI, PMID or PMCID reported in the supplied metadata. Its conclusions are limited to the stated effective metrics and parameter domains. The authors' proposed next test is a joint reading of the shadow radius and instability signal, aimed at telling the two model families apart.
Paper data and sources
Original title: Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general $μ$-scheme
Authors: Yu Han, Meng Liu, Yongzhuang Li
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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