Highly damped Quasi-Normal Modes of a Loop Quantum Black Hole

TL;DR

This work investigates highly damped Quasi-Normal Modes of a Loop Quantum Gravity inspired black hole (Modesto BH) in the simplified regime $a_0=0$, focusing on the polymeric deformation parameter $P$ as the primary source of deviation from Schwarzschild. It combines two complementary methods—the analytical monodromy technique and the numerical continued fraction (Leaver) method—to compute asymptotic QNM spectra for spin-0 and spin-2 test-field perturbations, revealing oscillatory patterns in both the real part of the frequencies and the imaginary spacing as $P$ and the angular momentum $\ell$ vary. The monodromy approach yields explicit asymptotic conditions that depend on the horizons through $r_\pm$ and on $\nu=\sqrt{4\lambda+8s+1}/6$, while Leaver’s method provides cross-checks and practical convergence for less-damped modes; together they show that increasing $P$ drives significant deviations from Schwarzschild, including a transition toward larger imaginary gaps and, at large $P$, real-part oscillations and potential purely damped branches. The results illuminate how quantum-gravity motivated modifications alter the deep, highly damped QNM regime and illustrate the strengths and limitations of both analytical and numerical techniques in probing the complex-plane spectrum of modified black holes.

Abstract

We compute asymptotic Quasi-Normal Mode (QNM) frequencies -- i.e. frequencies with a very large Imaginary part -- of a Loop Quantum Gravity inspired Black Hole. The deformations from the Schwarzschild Black Hole are encoded via two parameters: the minimal area gap $a_0$ and the polymeric deformation parameter $P$. In this study, we focus on the effect of the latter one, $P$, on the highly-damped part of QNM spectra. We consider both spin 0 and spin 2 test-field perturbations on the Black Hole as proper gravitational perturbations cannot be performed on an effective model. We use an analytical method of computation of QNMs referred to as the monodromy technique, which allows us to compute the asymptotic behaviour of QNMs. We found interesting oscillating behaviour in both the Real part and the Imaginary part of the QNMs, where the oscillation period varies with the polymeric deformation parameter $P$. We compare these analytical predictions to numerical results obtained thanks to the Continued fraction method. Even though the latter does not converge for QNMs with a very large Imaginary part, the numerical results are in rather good agreement with the monodromy prediction.

Figures (15)

• Figure 1: Evolution of the effective potential describing spin 0 and 2 test-field perturbations on the Modesto BH with $a_0=0$. The polymeric deformation parameter $P$ spans between 0 and 0.8. The case $P=0$ corresponds to the Schwarzschild BH.
• Figure 2: Contour for the calculation of QNM frequencies in the complex $r$ plane. The different colour regions are separated by the associated Stokes lines and the dark blue region corresponds to Re$(x)>0$.
• Figure 3: (a) Plot in the complex plane of the QNMs values predicted by the monodromy calculation, for scalar perturbations ($s=0$). $P=0$ corresponds to the well-known Schwarzschild case and the other spectra correspond to small deformations. (b) Plot of the gap in imaginary part $\Delta$Im between two successive QNMs.
• Figure 4: Plots in the complex plane of the QNMs values predicted by the monodromy calculation, for scalar perturbations ($s=0$) and $\ell=0,2,3,5,6,8,...$.
• Figure 5: Plots on the complex plane of the evolution of the imaginary gap $\Delta$Im$(\omega_n)$ in terms of the mode number $n$ for scalar modes computed via the monodromy technique. Five different values of $P$ are represented, from 0.1 to 0.9.