Non-minimal Einstein–Yang–Mills black holes: fundamental quasinormal mode and grey-body factors versus outburst of overtones

Recently, an exact black hole solution in non-minimal Einstein–Yang–Mills theory was obtained, and its quasinormal modes were subsequently analyzed using the JWKB approximation (Gogoi and Ponglertsakul in Eur. Phys. J. C 84:652, 2024. https://doi.org/10.1140/epjc/s10052-024-12946-9. arXiv:2402.06186 [gr-qc]). However, we demonstrate that this analysis lacks sufficient accuracy when studying modes with ℓ≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \le n$$\end{document}, where ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} is the multipole number and n is the overtone number. To address this, we compute the quasinormal frequencies using the precise Leaver method. Our results show that while the fundamental mode deviates only slightly from the Schwarzschild value, the first few overtones exhibit significantly larger deviations, with the discrepancy growing rapidly with the overtone number. Moreover, beginning with the second overtone, we observe a striking phenomenon: the real part of the frequency tends to zero quickly as the non-minimal coupling constant increases. This combination of spectral stability in the fundamental mode and high sensitivity of the overtones suggests that the Yang–Mills contribution primarily deforms the metric in the near-horizon region, while the geometry quickly transitions back to a Schwarzschild-like form at larger radii. In addition, we compute the grey-body factors and confirm that they represent a more stable characteristic of the geometry, exhibiting a correspondence with quasinormal modes already at the first multipole. The Yang–Mills coupling enhances the grey-body factors, further increasing the transmission probability.