Breaking the degeneracy among regular black holes with gravitational lensing

Abstract

We examine parameter degeneracies in Culetu, Bardeen and Hayward regular black holes across lensing, shadow and quasinormal mode regimes. Our analysis reveals that while Einstein ring data yield extremely loose constraints, with the regularization parameter $q$ exceeding $\mathcal{O}(10^3)$, they fail to improve the parameter estimation when combined with strong lensing observables. In contrast, the Event Horizon Telescope observations provide remarkably tight limits: $0 \leq q < 0.0466 <0.0847$ for Culetu, $0 \leq q < 0.5115 <0.6682$ for Bardeen and $0 \leq q < 1.0258 <1.1881$ for Hayward, which shows that the strong field regime alone dominates the available parameter space. Despite these bounds, leading order geometric observables remain highly degenerate, which masks the microscopic details of non-singular cores. To break this ``macroscopic universality,'' we identify high order signatures, such as the Lyapunov exponent and subleading time delays, as sensitive probes of near horizon curvature. Crucially, we discover that the brightness hierarchy of accretion induced intensity profiles undergoes a fundamental inversion when transitioning from lensing dominated static flows to dynamics dominated infalling flows. These results demonstrate that high resolution temporal and intensity profiles are essential for distinguishing between regular black hole geometries.

Figures (9)

• Figure 1: The angular radius of Einstein ring $\theta_E$ as functions of parameter $q$ and the $\chi^2$ test results for Culetu, Bardeen and Hayward black holes. In left one, the data from galaxy ESO325-G004 are $\theta_E^{\rm obs}=2.85^{+0.55}_{-0.25} as$Smith:2005pqGao:2024ejs. The black dashed line denotes best value. The dotted lines indicate $1\sigma$ and $2\sigma$ ($\theta_E=2.35 as$) confidence levels, while the gray and yellow shaded regions represent corresponding uncertainties in $\theta_E$. To provide a clearer visualization of constrained regions, we have restricted the display range of $\theta_E$ by excluding its upper values. In the right one, the constraints are derived from strong gravitational lensing data. The observed shadow angular radius for M87* and Sgr A* are $21 \pm 1.5\,\mu\text{as}$ and $24.35 \pm 3.5\,\mu\text{as}$, respectively. The $x$-axis utilizes a logarithmic scale to capture high resolution details of the deviation parameter $q$. Given that the Schwarzschild limit ($q = 0$) is the best fit result, we employ a practical lower limit of $10^{-4}$ to circumvent the singularity of the logarithmic axis at zero. This serves as a numerical proxy to represent the regime where $q \geq 0$, allowing for a continuous visualization from near Schwarzschild conditions to more extreme deviations.
• Figure 2: Evolution of photon sphere radius $x_m(r_m/M)$ and critical impact parameter $b_m(R_{sh}/R_s)$ as a function of parameter $q$. Following the strong lensing constraints established in Table \ref{['sigmas']}, the region for $q$ is defined as $10^{-4} < q < q_{2\sigma}$ with the lower limit effectively capturing the $q=0$ case.
• Figure 3: Same as Fig. \ref{['rmb']}, but for the angular observable $\theta_{d}/\mu as(\theta_{\infty}/\mu as)$, shadow area $\tilde{A}/M^2$, time delay leading order term $\Delta T^0_{n,m}$ and QNM frequency $\Omega_m$.
• Figure 4: Shadow characteristics for the Culetu/Bardeen/Hayward black hole models. The shadows cast in the celestial plane $(X/M, Y/M)$ are illustrated for the $1\sigma$ and $2\sigma$ constraints on the deviation parameter $q$. The outermost boundary (gray line) represents the standard Schwarzschild shadow ($q=0$). The intermediate boundaries correspond to the $1\sigma$ constraints, where the shadows for the Culetu (blue), Bardeen (cyan) and Hayward (orange) models are effectively degenerate and overlap. Similarly, the innermost boundaries represent the $2\sigma$ constraints for the Culetu (purple), Bardeen (green) and Hayward (red) models which also show significant overlap.
• Figure 5: Same as Fig. \ref{['rmb']}, but for the lensing coefficients $\bar{a}$ and $\bar{b}$.