Quasinormal modes in effective loop quantum gravity and consequences for isospectrality

We investigate the quasinormal mode spectra of axial and polar perturbations about the effective Ashtekar, Olmedo, and Singh black hole geometry within the hybrid approach to loop quantum gravity. We also compare our results with those previously obtained by del-Corral and Olmedo for the dressed metric approach. Starting from mode equations that are straightforward effective counterparts of the classical equations, we compute these spectra by means of a high-order WKB method with Pad\'e resummation. The violation of isospectrality between axial and polar perturbations that was claimed to exist in the dressed metric approach persists with the hybrid quantization, with deviations of the same order of magnitude as in the former effective case, although slightly larger for the hybrid prescription. In both situations, the isospectrality breaking decreases with the cubic root of the squared black hole mass in Planck units, consistently recovering the classical Schwarzschild limit. In addition, we perform a careful assessment of the applicability of the WKB approximation to the present effective analysis, confirming the parameter regions where the method remains reliable and highlighting specific mode cases for which the approximation breaks down.