Toward a unified view of agnostic parametrizations for deformed black holes

TL;DR

The paper addresses the challenge of comparing model‑independent parametrizations for deformed black holes by constructing explicit transformation maps among Johannsen–Psaltis (JP), Rezzolla–Zhidenko (RZ), and Effective Metric Description (EMD) frameworks. It analyzes how a finite set of deformation coefficients in each scheme can reproduce the quasi‑normal mode spectrum in the eikonal limit for representative BH geometries, emphasizing near‑horizon and large‑distance behaviours. Through careful mappings and perturbative expansions, the authors demonstrate a one‑to‑one correspondence between the parameter sets and show that, to leading order, the three parametrizations yield identical predictions for QNM observables across Hayward, Bardeen, and Simpson–Visser II models; no single framework universally outperforms the others. The results support a democratic view of parametrizations as complementary tools, enabling cross‑checks and targeted refinements for specific observables, and pave the way for applying the approach to more general spacetimes and quantum‑inspired corrections in BH physics.

Abstract

A variety of robust and effective descriptions have been devised to extract model-independent information about the fundamental properties of black holes from observational data when searching for deviations from general relativity. In this work, we construct explicit transformation maps establishing the equivalence among three relevant parametrizations for different spacetime patches: Johannsen-Psaltis, Rezzolla-Zhidenko, and Effective Metric Description. We then select representative black hole geometries to determine the minimal number of parameters required within each scheme to reproduce the associated quasi-normal mode spectra with a prescribed degree of accuracy. Our analysis shows that, for the given observables, a finite set of coefficients suffices to attain the desired precision in the three frameworks. Finally, we emphasize how the individual strengths of these effective descriptions can be exploited to probe complementary aspects of black hole physics.

Figures (2)

• Figure 1: Comparison of the convergence to the linear coefficient (top row) and quadratic coefficient (bottom row) for the Bardeen BH model, varying the number of the related parameters. The horizontal dashed black line is the value computed from the model, reported in Table \ref{['tab:BH-models']}. For the near-horizon EMD with four parameters, two Padé approximants were used: (2,2) and (1,3), the latter being more accurate.
• Figure 2: Coefficients $\gamma_{2k}$ (left) and $\delta_{2k}$ (right) for different orders of the Padé approximation $(N,3)$.