Scalar Perturbations Research Articles

Stability of Cauchy horizon in Einstein–Power–Maxwell–de Sitter black holes

We investigated the Strong Cosmic Censorship (SCC) conjecture in Einstein–Power–Maxwell–de Sitter (EPMdS) black holes by analyzing the quasinormal modes (QNMs) of a massless neutral scalar field perturbation. Using the pseudospectral method, we calculated the QNM frequencies across various cosmological constants and nonlinear electromagnetic parameters. Our results show that for black holes far from extremality, the lowest QNM frequency -Im(ω)/κ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ ext {Im} (\\omega )/\\kappa _-$$\\end{document} remains below 1/2. However, as the black holes approach extremality, the lowest mode’s -Im(ω)/κ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ ext {Im} (\\omega )/\\kappa _-$$\\end{document} consistently exceeds 1/2, leading to a violation of SCC. Moreover, with a fixed cosmological constant, as the nonlinearity parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} increases, the interval of SCC violation under the charge extremality ratio narrows. Considering that k=1/2+1/α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k = 1/2 + 1/\\alpha $$\\end{document} is inversely proportional to α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, our results indicate that increasing the order k of the non-linear electromagnetic field can effectively mitigate SCC violations.

Meso-inflationary Peccei–Quinn Symmetry Breaking with Nonminimal Coupling

We study a realization of the inflationary scenario where the Peccei–Quinn (PQ) symmetry is spontaneously broken during inflation, facilitated by its nonminimal coupling to gravity. This results in, effectively, a two-field inflation: the early stage is driven by an inflaton field with the PQ symmetry intact, and the later stage is driven by the PQ scalar after its effective mass becomes tachyonic, causing destabilization from the origin. The nonminimal coupling serves the dual purpose of restoring the PQ symmetry during early inflation and flattening the PQ potential posttachyonic shift, allowing for continued slow-roll. We analyze the inflationary background solutions and scalar perturbations, which are amplified at small scales via significant isocurvature perturbations generated near the symmetry-breaking epoch. These perturbations lead to second-order gravitational waves, detectable by next-generation space-based experiments.

Reheating Constraints and the H0 Tension in Quintessential Inflation

In this work, we focus on two important aspects of modern cosmology: reheating and Hubble constant tension within the framework of a unified cosmic theory, namely the quintessential inflation connecting the early inflationary era and late-time cosmic acceleration. In the context of reheating, we use instant preheating and gravitational reheating, two viable reheating mechanisms when the evolution of the universe is not affected by an oscillating regime. After obtaining the reheating temperature, we analyze the number of e-folds and establish its relationship with the reheating temperature. This allows us to connect, for different quintessential inflation models (in particular for models coming from super-symmetric theories such as α-attractors), the reheating temperature with the spectral index of scalar perturbations, thereby enabling us to constrain its values. In the second part of this article, we explore various alternatives to address the H0 tension. From our perspective, this tension suggests that the simple Λ-Cold Dark Matter model, used as the baseline by the Planck team, needs to be refined in order to reconcile its results with the late-time measurements of the Hubble constant. Initially, we establish that quintessential inflation alone cannot mitigate the Hubble tension by solely deviating from the concordance model at low redshifts. The introduction of a phantom fluid, capable of increasing the Hubble rate at the present time, becomes a crucial element in alleviating the Hubble tension, resulting in a deviation from the Λ-Cold Dark Matter model only at low redshifts. On a different note, by utilizing quintessential inflation as a source of early dark energy, thereby diminishing the physical size of the sound horizon close to the baryon–photon decoupling redshift, we observe a reduction in the Hubble tension. This alternative avenue, which has the same effect of a cosmological constant changing its scale close to the recombination, sheds light on the nuanced interplay between the quintessential inflation and the Hubble tension, offering a distinct perspective on addressing this cosmological challenge.

Resilience of DBI screened objects and their ladder symmetries

Scalar field theories with a shift symmetry come equipped with the K-mouflage (or kinetic screening) mechanism that suppresses the scalar interaction between massive objects below a certain distance, the screening radius. In this work, we study the linear response of the scalar field distribution around a screened (point-like) object subject to a long range external scalar field perturbation for the Dirac-Born-Infeld theory. We find that, for regular boundary conditions at the position of the particle, some multipoles have vanishing response for a lacunar series of the multipole order ℓ for any dimension. Some multipoles also exhibit a logarithmic running when the number of spatial dimensions is even. We construct a ladder operator structure, with its associated ladder symmetries, formed by two sets of ladders that are related to the properties of the linear response and the existence of conserved charges. Our results exhibit a remarkable resemblance with the Love numbers properties of black holes in General Relativity, although some intriguing differences subsist.

Transient dynamics of quasinormal mode sums

Quasinormal modes of spacetimes with event horizons are typically governed by a non-normal operator. This gives rise to spectral instabilities, a topic of recent interest in the black hole pseudospectrum programme. In this work we show that non-normality leads to the existence of arbitrarily long-lived sums of short-lived quasinormal modes, corresponding to localising packets of energy near the future horizon. There exist sums of M quasinormal modes whose lifetimes scale as log M. This transient behaviour results from large cancellations between non-orthogonal quasinormal modes. We provide simple closed-form examples for a massive scalar field in the static patch of dSd+1 and the BTZ black hole. We also provide numerical examples for scalar perturbations of Schwarzschild-AdSd+1, and gravitational perturbations of Schwarzschild in asymptotically flat spacetime, using hyperboloidal foliations. The existence of these perturbations is linked to certain properties of black hole pseudospectra. We comment on implications for thermalisation times in holographic plasmas.

Mass ladder operators and quasinormal modes of the static BTZ-like black hole in the Einstein-bumblebee gravity

Abstract We investigate the mass ladder operators for the static BTZ-like black hole in the Einstein-bumblebee gravity, and probe the quasinormal frequencies of the mapped modes by the mass ladder operators for a scalar perturbation under the Dirichlet and Neumann boundary conditions. It is found that the mass ladder operators depend on the Lorentz symmetry breaking parameter, and the imaginary parts of the frequencies shifted by the mass ladder operators increase with the increase of the Lorentz symmetry breaking parameter under two boundary conditions. It should be noted that, under the Neumann boundary condition, the mapped modes caused by the mass ladder operator $D_{0,k_+}$ are unstable. Moreover, the mass ladder operators do not change the Breitenlohner-Freedman bound for the scalar modes, just as in the usual BTZ black hole. These results could help us to further understand the conformal symmetry and Lorentz symmetry breaking in the Einstein-bumblebee gravity.

A unified spectral approach for quasinormal modes of Morris–Thorne wormholes

Abstract In this paper, we undertake a comprehensive examination of quasinormal modes (QNMs) linked to Morris–Thorne, also known as Bronnikov–Ellis wormholes, delving into scalar, electromagnetic, and gravitational perturbations using the spectral method. Our research corrects inaccuracies previously reported in the literature and addresses areas where the Wentzel–Kramers–Brillouin (WKB) approximation proves inadequate. Moreover, we introduce and evaluate a novel spectral technique designed to consolidate recent advancements in formulating QNM boundary conditions at both the wormhole throat and space-like infinity. This innovative approach bridges critical gaps in existing methodologies and enhances the accuracy and applicability of QNM analysis in the study of wormhole physics.

Correlated scalar perturbations and gravitational waves from axion inflation

Abstract The scalar and tensor fluctuations generated during inflation can be correlated, if arising from the same underlying mechanism. In this paper we investigate such correlation in the model of axion inflation, where the rolling inflaton produces quanta of a U(1) gauge field which, in turn, source scalar and tensor fluctuations. We compute the primordial correlator of the curvature perturbation, ζ, with the gravitational energy density, Ω GW , at frequencies probed by gravitational wave detectors. This two-point function receives two contributions: one arising from the correlation of gravitational waves with the scalar perturbations generated by the standard mechanism of amplification of vacuum fluctuations, and the other coming from the correlation of gravitational waves with the scalar perturbations sourced by the gauge field. Our analysis shows that the former effect is generally dominant. For typical values of the parameters, the correlator, normalized by the amplitude of ζ and by the fractional energy in gravitational waves at interferometer frequencies, turns out to be of the order of 10-4 ÷ 10-2.

Induced gravitational waves: the effect of first order tensor perturbations

Scalar induced gravitational waves contribute to the cosmological gravitational wave background. They can be related to the primordial density power spectrum produced towards the end of inflation and therefore are a convenient new tool to constrain models of inflation. These waves are sourced by terms quadratic in perturbations and hence appear at second order in cosmological perturbation theory. While the focus of research so far was on purely scalar source terms we also study the effect of including first order tensor perturbations as an additional source. This gives rise to two additional source terms: a term quadratic in the tensor perturbations and a cross term involving mixed scalar and tensor perturbations. We present full analytical expressions for the spectral density of these new source terms and discuss their general behaviour. To illustrate the generation mechanism we study two toy models containing a peak on small scales. For these models we show that the scalar-tensor contribution becomes non-negligible compared to the scalar-scalar contribution on smaller scales. We also consider implications for future gravitational wave surveys.

Evolution of coupled scalar perturbations through smooth reheating. Part I. Dissipative regime

If the inflaton is a heavy scalar field, it may equilibrate slower thansome other degrees of freedom, e.g. non-Abelian gauge bosons. In this case,perturbations in the inflaton field and in a thermalplasma coexist from a given moment onwards. We derive a gauge-invariant setof three coupled equations governing the time evolution of such a system.Despite singular coefficients,a reliable numerical solution can be obtained for a longtime period, starting from phase oscillations inside the Hubble horizon,and extending until acoustic oscillations in a radiation-dominated universe.Benchmarks are illustrated from a “weak regime”,where perturbations have a quantum-mechanical origin but get dissipatedby interactions with the plasma. Among applications of our formalismcould be inhomogeneity-induced nucleations inpost-inflationary phase transitions, andthe production of scalar-induced gravitational waves.

Scalar Quasi-Normal Modes of a loop quantum black hole

We compute the Quasi-Normal Mode (QNM) frequencies for scalar perturbations for modified Schwarzschild black holes in Loop Quantum Gravity. We study the singularity-free polymerized metric characterized by two parameters encoding loop quantum effects: the minimal area gap a 0 and the polymeric deformation parameter P. We perform numerical computations using Leaver's continued fraction method and compare our results to other semi-analytical methods and existing literature. We study the effects on the QNM spectrum of variation of both deformation parameters and systematically compare to the standard Schwarzschild case. In particular we find that the scalar fundamental mode is modified from the third decimal for values of P in accordance with the most recent astrophysical constraints. We also show that qualitative differences arise for highly damped modes: on the one hand, a new crossing of the imaginary axis occurs for high values of a 0 and, on the other hand, increasing P produces a positive shift of the real part and an increase of the spacing in imaginary part between modes.

Dark energy with a shift-symmetric scalar field: Obstacles, loophole hunting and dead ends

We discuss the possibility of a scalar field being the fundamental description of dark energy. We focus on shift-symmetric scalar-tensor theories since this symmetry potentially avoids some fine-tuning problems. We also restrict attention to theories satisfying that the propagation speed of gravitational waves is equal to the speed of light. These considerations lead us to investigate shift-symmetric Kinetic Gravity Braiding theories. Analysing the stability of scalar linear perturbations, we discuss the conditions that seems to be necessary to describe (super) accelerated cosmic expansion without introducing instabilities. However, it has been previously established that the linearised analysis does not guarantee the stability of this non-canonical scalar theory, as potentially dangerous interactions between dark energy fluctuations and tensor perturbations (essentially gravitational waves) appear at a higher order in perturbation theory. Indeed, although we shall point out that the standard proof of absence of dark energy stable braiding models due to this interaction has a possible way-out, we find general arguments suggesting that there are no dark energy stable solutions that can exploit this loophole. Thus, we discuss future research directions for finding viable fundamental descriptions of dark energy. We also provide a dictionary between the covariant version of the theory and that of the Effective Field Theory approach, explicitly computing the parameters in the latter formalism in terms of the functions appearing in the covariant version, and its derivatives. To the best of our knowledge, this is the first time these expressions are explicitly obtained up-to arbitrary order in perturbation theory.

Primordial black holes and scalar-induced gravitational waves in radiative hybrid inflation

We study the possibility that primordial black holes (PBHs) can be formed from large curvature perturbations generated during the waterfall phase transition due to the effects of one-loop radiative corrections of Yukawa couplings between the inflaton and a dark fermion in a non-supersymmetric hybrid inflationary model. We obtain a spectral index ns\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_s$$\\end{document}, and a tensor-to-scalar ratio r, consistent with the current Planck data. Our findings show that the abundance of PBHs is correlated to the dark fermion mass mN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_N$$\\end{document} and peak in the GW spectrum. We identify parameter space where PBHs can be the entire dark matter (DM) candidate of the universe or a fraction of it. Our predictions are consistent with any existing constraints of PBH from microlensing, BBN, CMB, etc. Moreover, the scenario is also testable via induced gravitational waves (GWs) from first-order scalar perturbations detectable in future observatories such as LISA and ET. For instance, with inflaton mass m∼2×1012\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m \\sim 2\ imes 10^{12}$$\\end{document} GeV, mN∼5.4×1015\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_N \\sim 5.4\ imes 10^{15}$$\\end{document} GeV, we obtain PBHs of around 10-13M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-13}\\, M_\\odot $$\\end{document} mass that can explain the entire abundance of DM and predict GWs with amplitude ΩGWh2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _\ extrm{GW}h^2$$\\end{document}∼10-9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 10^{-9}$$\\end{document} with peak frequency f∼\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim $$\\end{document} 0.1 Hz in LISA. By explicitly estimating fine-tuning we show that the model has very mild tuning. We discuss successful reheating at the end of the inflationary phase via the conversion of the waterfall field into standard model (SM) particles. We also briefly speculate a scenario where the dark fermion can be a possible heavy right-handed neutrino (RHN) which is responsible for generating the SM neutrino masses via the seesaw mechanism. The RHN can be produced due to waterfall field decay and its subsequent decay may also explain the observed baryon asymmetry in the universe via leptogenesis. We find the reheat temperature TR≲5×109\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_R\\lesssim 5\ imes 10^9$$\\end{document} GeV that explains the matter-anti-matter asymmetry of the universe.

Quasinormal modes of the Mannheim–Kazanas black holes

Abstract A spherically symmetric black hole solution in the conformal Weyl gravity was found by Mannheim and Kazanas in 1988. While the quasinormal modes (QNMs) of these black holes have been considered in a few works, here we complement these studies by considering in detail the regime of vanishing cosmological constant and negative values of the Weyl parameter for which we find quasinormal frequencies for scalar, Dirac and electromagnetic perturbations with the time-domain integration and WKB methods. In particular, we derive the compact and remarkably accurate analytic formula for the frequencies in the form of expansion in terms of the inverse multipole number. Comparison with the time-domain integration shows that the 6th order WKB method with the Padé approximants is quite accurate, unless the black hole is in the near extreme state.

Quantum instability of the Cauchy horizon in a charged de-Sitter spacetime with dark matter

The strong cosmic censorship conjecture (SCCC) requires that spacetime cannot be extended beyond the Cauchy horizon. This ensures the predictability of spacetime. In this paper, we investigate the SCCC for a spherically symmetric charged de-Sitter black hole surrounded by dark matter using classical and quantum scalar fields. At the classical level, we analyze the behavior of scalar waves near the Cauchy horizon using the method developed by Hintz and Vasy. We find a relationship between the Sobolev regularity of scalar waves and the spectral gap of quasinormal modes. In the nearly extremal region, this may lead to a violation of the SCCC. At the quantum level, we first provide a proof of the renormalizability of the quantum scalar field in dark-matter black holes. Using numerical methods, we then demonstrate that the renormalized quantum stress-energy tensor for any Hadamard state exhibits quadratic divergence near the Cauchy horizon in the nearly extremal region. The quadratic divergence of the renormalized quantum stress-energy tensor is sufficient to convert the Cauchy horizon into a singularity. Thus, the SCCC is preserved by quantum effects. Since the quadratic divergence is more singular than the behavior of classical scalar field perturbations near the Cauchy horizon, it means there is a region where physics is dominated by quantum effects. We study the influence of dark matter on quantum effects in this region and we find there is a monotonic relationship between the dark matter and the strength of quantum effects. The numerical results show that the quantum effects will become stronger as dark matter increases.

Improved model of large-field inflation with primordial black hole production in Starobinsky-like supergravity

Abstract A viable model of large-field (chaotic) inflation with efficient production of primordial black holes is proposed in Starobinsky-like (modified) supergravity leading to the ‘no-scale-type’ Kähler potential and the Wess-Zumino-type (‘renormalizable’) superpotential. The cosmological tilts are in good (within 1σ) agreement with Planck measurements of the cosmic microwave background radiation. In addition, the power spectrum of scalar perturbations has a large peak at smaller scales, which leads to a production of primordial black holes from gravitational collapse of large perturbations with the masses about 1017 g. The masses are beyond the Hawking (black hole) evaporation limit of 1015 g, so that those primordial black holes may be viewed as viable candidates for a significant part or the whole of the current dark matter. The parameters of the superpotential were fine-tuned for those purposes, while the cubic term in the superpotential is essential whereas the quadratic term should vanish. The vacuum after inflation (relevant to reheating) is Minkowskian. The energy density fraction of the gravitational waves induced by the production of primordial black holes and their frequency were also calculated in the second order with respect to perturbations.

Analytical QNMs of fields of various spin in the Hayward spacetime

By employing an expansion in terms of the inverse multipole number, we derive analytic expressions for the quasinormal modes (QNMs) of scalar, Dirac and Maxwell perturbations in the Hayward black hole (BH) background. The metric has three interpretations: as a model for a radiating BH, as a quantum-corrected BH owing to the running gravitational coupling in the Asymptotically Safe Gravity, and as a BH solution in the Effective Field Theory. We show that the obtained compact analytical formulas approximate QNMs with remarkable accuracy for ℓ > 0.

Scalar perturbations from inflation in the presence of gauge fields

We study how Abelian-gauge-field production during inflation affects scalar perturbations in the case when the gauge field interacts with the inflaton directly (by means of generic kinetic and axial couplings) and via gravity. The homogeneous background solution is defined by self-consistently taking into account the backreaction of the gauge field on the evolution of the inflaton and the scale factor. For the perturbations on top of this background, all possible scalar contributions coming from the inflaton, the metric, and the gauge field are considered. We derive a second-order differential equation for the curvature perturbation, ζ, capturing the impact of the gauge field, both on the background dynamics and on the evolution of scalar perturbations. The latter is described by a source term in the ζ equation, which is quadratic in the gauge-field operators and leads to non-Gaussianities in the curvature perturbations. We derive general expressions for the induced scalar power spectrum and bispectrum. Finally, we apply our formalism to the well-known case of axion inflation without backreaction. Numerical results show that, in this example, the effect of including metric perturbations is small for values of the gauge-field production parameter ξ>3. This is in agreement with the previous results in the literature. However, in the region of smaller values, ξ≲2, our new results exhibit order-of-unity deviations when compared to previous results. Published by the American Physical Society 2024

Selecting energy–momentum trace dependent gravity theories with LSS

We study scalar cosmological perturbations in f(R,T) modified gravity theories, where T represents the trace of the energy–momentum tensor. We provide detailed equations for the matter energy density contrast. We solve them numerically to facilitate a comparison with available large-scale structure (LSS) formation observational data on fσ8, while also addressing the S8 tension. We identify f(R,T) models that either lead to growth enhancement or suppression. Since recent results in the literature indicate a preference for the latter feature, this type of analysis is quite useful for selecting viable modifications of gravity. The studied class of such f(R,T) models are either ruled out or severely restricted. After selecting the surviving f(R,T) models we show how they deal with the S8 tension.

Quasinormal Frequencies of Fields with Various Spin in the Quantum Oppenheimer–Snyder Model of Black Holes

AbstractA recent development involves an intriguing model of a quantum‐corrected black hole, established through the application of the quantum Oppenheimer–Snyder model within loop quantum cosmology [Lewandowski et al., Phys. Rev. Lett. (2023) 130, 101501]. Employing both time‐domain integration and the Wentzel–Kramers–Brillouin (WKB) approach, the quasinormal frequencies for scalar, electromagnetic, and neutrino perturbations in these quantum‐corrected black holes are computed. This analysis reveals that while the real oscillation frequencies undergo only minor adjustments due to the quantum parameter, the damping rate experiences a significant decrease as a result of its influence. The author also deduced the analytic formula for quasinormal frequencies in the eikonal limit and showed that the correspondence between the null geodesics and eikonal quasinormal modes holds in this case.