Ringing of Reissner–Nordström Black Holes with a Non-Abelian Hair in Gravity’s Rainbow

Abstract

In this paper, we consider massless scalar perturbations minimally coupled to gravity in the background spacetime of charged black holes in Yang–Mills theory with gravity’s rainbow modification. We calculate the corresponding quasinormal frequencies by employing the sixth-order Wentzel—Kramers—Brillouin (WKB) approximation for both asymptotically flat and de Sitter (dS) spacetimes. We show that the Yang–Mills modification of the Reissner–Nordström black holes leads to an increase in the real and imaginary parts of frequencies. Furthermore, we find that the perturbations in asymptotically flat spacetime decay faster with more oscillations compared to dS spacetime, and we study the effects of the rainbow functions on the oscillations. Interestingly, we reveal a novel feature of this black hole case study and show that, unlike typical black hole solutions such as Schwarzschild, RN, and Kerr, the higher multipole numbers live longer than the lower ones in both asymptotically flat and dS spacetimes. Furthermore, the reflection and transmission coefficients are explored for Einstein–Maxwell–Yang–Mills black holes, and the results are compared for flat and dS asymptotes.

1. Introduction

Quasinormal modes (QNMs) are generated as the response of black holes to various types of perturbations on their background spacetime, and the QNMs spectrum depends on the black hole’s conserved quantities [1,2,3,4]. The QN modes corresponding to gravitational perturbations could be detected by employing the future space-based gravitational wave detectors, which would allow us to study the properties of black hole spacetimes [5,6].

The QNMs of black holes in general relativity and beyond, undergoing various kinds of perturbations, have been studied extensively so far. For instance, the QN modes have been investigated for black holes in loop quantum gravity [7,8,9,10], quantum-corrected black hole metrics [11,12,13,14], deformed Schwarzschild black holes [15], spinning C-metric [16], and asymptotically safe gravity [17]. In addition, the QN oscillations of black holes in Einstein–Gauss–Bonnet–adS spacetime [18] and dark matter halo [19] have been investigated, and the anomalous decay rate of QNMs of Reissner–Nordström (RN) black holes in asymptotically dS spacetime has been explored [20]. Furthermore, the QN ringing of Dilaton–dS black holes [21], regular black holes [22,23], and Kaluza–Klein black holes [24,25,26] have been studied. Furthermore, black hole perturbations in conformal Weyl gravity and corresponding QNMs have been investigated in [27,28,29,30,31,32,33,34,35].

In addition to Maxwell field coupled to Einstein gravity as an abelian matter source, one can also consider general relativity in the presence of Yang–Mills theory as a non-abelian gauge field. These non-abelian gauge fields and Yang–Mills equations appear in the low-energy limit of some string theory models. Analytic solutions of Einstein gravity coupled to Yang–Mills field have been found in [36], and later, black hole solutions with Wu-Yang monopole have been derived in higher-dimensions [37,38], massive gravity [39], and gravity’s rainbow [40]. According to the no-hair theorem and uniqueness, these black hole solutions led to specific versions of the hairy black holes. It has been shown that the black hole solutions in Einstein–Yang–Mills theory are stable against linear radial perturbations [41,42,43]. In the case of Yang–Mills theory, the QNMs of black holes in Einstein–Yang–Mills massive gravity [39], nonminimal Einstein–Yang–Mills black holes [44,45,46,47,48,49], and higher-dimensional black holes [50,51] have been studied.

On the other hand, doubly special relativity [52] represents a generalization of the standard energy–momentum dispersion relation, and it was shown that this modified dispersion relation can explain the threshold anomalies of ultra-high-energy cosmic rays and TeV photons [53,54,55]. The doubly special relativity has been extended to curved spacetimes, which is known as gravity’s rainbow [56]. Black hole solutions have been derived and analyzed in the context of gravity’s rainbow (for instance, see an incomplete list [40,57,58,59,60,61,62,63,64] and references therein).

The standard cosmological model of the $\mathrm{\Lambda }$-cold dark matter can explain various cosmological phenomena of the current epoch of the cosmos. Therefore, in order to build up a gravitational model consistent with the large-scale structure of the universe, one should incorporate the contribution of dark energy and dark matter to the underlying gravitational theory. In this regard, it was shown that the Hubble law arises from the frequency-shift considerations of geodesic particles circulating the Kerr black hole in asymptotically dS spacetime [65], confirming that adding the cosmological constant term to general relativity is inevitable [66]. Furthermore, some of the differences between the behavior of QNMs spectrum in the asymptotically flat spacetimes and dS spacetimes have been identified so far. For instance, the final stage of the QN ringing dominates by power law tails in asymptotically flat spacetimes [67], whereas in the case of dS spacetimes, it dominates by an exponential decay law [68]. More recently, it has also been demonstrated that the cosmological constant term and the large-scale structure of the cosmos play a crucial role in observing cosmological echoes [35].

In this article, we are going to investigate scalar perturbations in the background geometry of black hole solutions in Einstein–Maxwell–Yang–Mills theory with gravity’s rainbow and in the presence of the cosmological constant, that have been found in [40]. We explore the QNM spectrum in both asymptotically flat and dS spacetimes and study deviations from the standard RN black hole. The outline of this paper is as follows. The next section is devoted to reviewing the black hole solutions, QNMs boundary conditions, and semi-analytical WKB approximation, and also, we compute the radial master wave equation and corresponding effective potential. In Section 3, we calculate the corresponding QN frequencies with the help of sixth-order WKB approximation, explore the effects of the free parameters on the QNM spectrum in both asymptotically flat and dS spacetimes, and find deviations from the standard RN black holes. In Section 4, we study the effects of Wu–Yang magnetic charge, multipole number, and cosmological constant on the reflection and transmission coefficients. Finally, we finish our paper with some concluding remarks.

2. Black Hole Solutions and Massless Scalar Perturbations

The black hole solutions in Einstein–Maxwell–Yang–Mills theory with gravity’s rainbow generalization are described by the following infinitesimal line element [40]

$$d{s}^2=-{\displaystyle \frac{\mathcal{F}\left(r\right)}{f^2\left(\epsilon \right)}}d{t}^2+{\displaystyle \frac{d{r}^2}{g^2\left(\epsilon \right)\mathcal{F}\left(r\right)}}+{\displaystyle \frac{r^2}{g^2\left(\epsilon \right)}}\left(d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2\right),$$

(1)

where $f\left(\epsilon \right)$ and $g\left(\epsilon \right)$ are rainbow functions that depend on energy only, $\epsilon =E/E_P$ is the energy ratio, E is the energy of a test particle, and $E_P$ is the Planck energy. In the infrared limit, where $E<<E_P$, the rainbow functions reduce to [56]

$$\underset{\epsilon \to 0}{lim}f\left(\epsilon \right)=1,\underset{\epsilon \to 0}{lim}g\left(\epsilon \right)=1.$$

(2)

In the Einstein–Maxwell–Yang–Mills system, the metric function $\mathcal{F}\left(r\right)$ depends on the radial coordinate only and has been found to be [40]

$$\mathcal{F}\left(r\right)=1-{\displaystyle \frac{2M}{r}}-{\displaystyle \frac{\mathrm{\Lambda }{r}^2}{3g^2\left(\epsilon \right)}}+{\displaystyle \frac{q^2{f}^2\left(\epsilon \right)}{r^2}}+{\displaystyle \frac{{\nu }^2{g}^2\left(\epsilon \right)}{r^2}}.$$

(3)

In this relation, M and q are two integration constants corresponding, respectively, to the total mass and electric charge of the black hole; $\mathrm{\Lambda }$ is the cosmological constant; and $\nu$ is the magnetic charge. Generally, the metric function (3) has at most four real roots corresponding to the cosmological horizon $r_C$, event horizon $r_H$, Cauchy horizon $r_{-}$, and negative root $r_0$, satisfying $r_0<r_{-}<r_H<r_C$.

It is worth mentioning that $\mathcal{F}\left(r\right)$ reduces to the standard RN black hole solutions in the limit $\mathrm{\Lambda }=0=\nu$ and $f\left(\epsilon \right)=1=g\left(\epsilon \right)$, as it should be. Hence, the set of parameters $\{\mathrm{\Lambda },\nu ,f,g\}$ shows deviations from the RN black hole, each one referring to a distinct type of modification to the Einstein–Maxwell system. The sign of the cosmological constant characterizes the asymptotic behavior of the solutions (1) such that a vanishing $\mathrm{\Lambda }$ corresponds to asymptotically flat spacetime, while $\mathrm{\Lambda }>0$ refers to asymptotically dS spacetimes. Note that we assume $\epsilon \to 0$ in the asymptotic region $r\to \infty$ (or $r\to r_C$) since we want to maintain the asymptotic flatness (or dS) property of the solutions 1.

In order to explore the QNMs spectrum, we first take into account a test massless scalar perturbation minimally coupled to gravity in the background spacetime of black hole solutions (1). Then we employ the semi-analytic method of WKB approximation to calculate the QN frequencies and probe the effects of rainbow functions and magnetic charge on the oscillations and decay rates of scalar perturbations. This would also allow us to compare the black hole solutions with both flat and dS asymptotes.

The evolution of the test scalar field $\mathrm{\Phi }(t,r,\vartheta ,\phi )$ in the background spacetime (1) is governed by the Klein–Gordon equation (see a similar procedure with more details to obtain scalar perturbation equations in the standard Schwarzschild spacetime in [69]):

$${\nabla }_{\mu }{\nabla }^{\mu }\mathrm{\Phi }(t,r,\vartheta ,\phi )=0,$$

(4)

where ${\nabla }_{\mu }$ is the covariant derivative compatible with the metric tensor $g_{\nu \rho }\left(x^{\sigma }\right)$. By taking into account the following decomposition in angular and time variables of the scalar field [69],

$$\mathrm{\Phi }(t,r,\vartheta ,\phi )={\displaystyle \frac{1}{r}}\sum _{l,m}{\psi }_l\left(r\right)Y_{l,m}(\vartheta ,\phi )e^{-i\omega t},$$

(5)

in the Klein–Gordon Equation (4), we find a second-order wave equation for the radial part of perturbations as [69]

$$\left({\partial }_{r_{*}}^2+{\omega }^2\right){\psi }_l\left(r_{*}\right)=V_l\left(r_{*}\right){\psi }_l\left(r_{*}\right),$$

(6)

where l is the multipole number, $r_{*}$ is the tortoise coordinate defined as (see a similar definition for the tortoise coordinate in the standard Schwarzschild spacetime in [70])

$$r_{*}={\displaystyle \frac{f\left(\epsilon \right)}{g\left(\epsilon \right)}}\int {\displaystyle \frac{dr}{\mathcal{F}\left(r\right)}},$$

(7)

and $V_l\left(r_{*}\right)$ is the effective potential given by

$$V_l\left(r\right)={\displaystyle \frac{g^2\left(\epsilon \right)\mathcal{F}\left(r\right)}{f^2\left(\epsilon \right)}}\left({\displaystyle \frac{l(l+1)}{r^2}}+{\displaystyle \frac{{\mathcal{F}}^{\prime }\left(r\right)}{r}}\right).$$

(8)

Note that r is related to $r_{*}$ through Equation (7). The radial coordinate ranges $r\in \left(r_H,\infty \right)$ are for asymptotically flat and $r\in \left(r_H,r_C\right)$ for asymptotically dS spacetime, and therefore, by considering the tortoise coordinate definition given in Equation (7), $r_{*}$ ranges $r_{*}\in (-\infty ,\infty )$ are for both asymptotes.

It is worth mentioning that while a massless scalar field is a simplified probe, it is a standard and effective tool for investigating the fundamental characteristics of black hole spacetimes, such as QNMs and greybody factors. Considering the perturbations of test massless fields allows us to focus on the properties of the black hole background itself, isolating them from the complexities of massive fields. Furthermore, adding mass to the scalar field only shifts the effective potential upward, resulting in a shift in the frequencies while the main physical results remain unchanged. On the other hand, there are some examples of real massless scalar fields from a theoretical perspective, such as the inflaton field, the dilaton field, and moduli fields in string theory and supergravity that can be taken into account as motivations for the present study.

The QNMs are solutions of the wave Equation (6) when we impose the suitable boundary conditions at the asymptotic regions $r_{*}\to \pm \infty$. The boundary conditions require that we have incoming signals at the event horizon and outgoing waves at infinity (at the cosmological horizon in case of dS spacetime), namely [71]

$$\left\{\begin{array}{cc}\psi \left(r_{*}\right)={\mathcal{T}}_l\left(\omega \right)e^{-i\omega r_{*}},& \mathrm{for}{r}_{*}\to -\infty \\ \psi \left(r_{*}\right)=e^{-i\omega r_{*}}+{\mathcal{R}}_l\left(\omega \right)e^{i\omega r_{*}},& \mathrm{for}{r}_{*}\to \infty \end{array}\right.,$$

(9)

where ${\mathcal{T}}_l\left(\omega \right)$ and ${\mathcal{R}}_l\left(\omega \right)$ are the transmission and reflection coefficients, respectively. By applying these boundary conditions to the master wave Equation (6), one obtains a discrete set of eigenvalues ${\omega }_{nl}={\omega }_R-i{\omega }_I$ with the real part ${\omega }_R$ characterizing the oscillations of the perturbed scalar field and the imaginary part ${\omega }_I$ corresponding to its decay rate. In this study, we explore the QNMs of black holes in Einstein–Maxwell–Yang–Mills theory by employing WKB approximation that we briefly explain as follows.

The WKB method is constructed from the matching of WKB expansion of the wave function $\psi \left(r_{*}\right)$ at asymptotic regions with the Taylor expansion near the peak of the effective potential. Hence, the WKB expansion is a powerful approach for the perturbation problems consisting of an effective potential with a single peak that vanishes at asymptotic regions. This method was first applied to the scattering problem around black holes in [71], and then it was extended to higher orders [72,73,74]. The sixth-order WKB formula is given by the following relation:

$$i{\displaystyle \frac{{\omega }^2-Q_0}{\sqrt{-2Q_0^{″}}}}-\sum _{j=2}^6{Q}_j\left(n+{\displaystyle \frac{1}{2}}\right)=n+{\displaystyle \frac{1}{2}},$$

(10)

where $Q_0$ is the maximum of the potential, $Q_0^{″}$ is the corresponding second derivative, and $n=0,1,2,\dots$ is the overtone number. The WKB correction terms $Q_j\left(n+{\displaystyle \frac{1}{2}}\right)$ depend on the value of the potential and its higher derivatives at the maximum. The WKB corrections $Q_j$’s are given up to third order and sixth order in [72,73], respectively. It is worth noting that the QN frequencies calculated from the WKB expansion are not reliable for $n\ge l$. But this method gives quite accurate results in the case of $n<l$ and exact frequencies in the eikonal limit $l\to \infty$. In addition, one should note that the WKB approach can be applied to the effective potentials that form a potential barrier and takes zero values (or small values compared with the height of the barrier) at the asymptotic regions. We shall use this expansion up to the sixth order to calculate the QN frequencies of massless scalar perturbations in the Einstein–Maxwell–Yang–Mills black hole background for $n<l$. We recall that our use of the sixth-order WKB method provides a high degree of accuracy for the QN modes; hence, it is a reliable tool for our investigation.

3. Quasinormal Modes

In this section, we calculate the QNMs in both asymptotically flat and dS spacetimes by employing the WKB expansion through the following two subsections. We also explore the effects of the free parameters on the QNMs spectrum and find deviations from the standard RN black holes.

3.1. Asymptotically Flat Spacetime

In order to explore the effects of rainbow functions and magnetic charge on the perturbations in asymptotically flat spacetime, we calculate the QN frequencies of black hole solutions in Einstein–Maxwell–Yang–Mills theory with gravity’s rainbow modification. For these types of black holes, the profile of the corresponding effective potential (8) is illustrated in the left panel of Figure 1. From this figure, we see that $V_1\left(r\right)$ forms a single peak against incident waves and vanishes at the event horizon $r_H$ and spatial infinity $r\to \infty$, which is also the case for arbitrary values of the multipole number l. Therefore, the WKB approximation is a suitable approach to calculate the QN frequencies. Furthermore, the effective potential is positive definite, which guarantees the dynamical stability of these types of black holes in asymptotically flat spacetime undergoing scalar perturbations for the special choice of free parameters.

The fundamental QNMs for various multipole numbers are presented in

Table 1. The fundamental QN frequencies for various multipole numbers in asymptotically flat spacetime $\mathrm{\Lambda }=0$ for $M=1$ and $q=0.5$. The second row corresponds to the QN frequencies of RN black hole.

ν	f(ε)	g(ε)	ω01	ω02
0	1	1	$0.306551-0.098874i$	$0.505966-0.097949i$
$0.5$	1	1	$0.323537-0.099413i$	$0.533817-0.098580i$
$0.5$	$1.2$	1	$0.277128-0.082690i$	$0.457204-0.082045i$
$0.5$	1	$1.2$	$0.399065-0.119074i$	$0.658374-0.118144i$

. From the second and third rows, we see that adding the Wu–Yang monopole to the RN black holes increases both the real and imaginary parts of the frequencies. Therefore, perturbations in the RN background live longer with fewer oscillations in comparison to the black hole solutions in Einstein–Maxwell–Yang–Mills theory in the presence of gravity’s rainbow.

On the other hand, by comparing the last two rows with the third row, one finds that the rainbow functions $f\left(\epsilon \right)$ and $g\left(\epsilon \right)$ have opposite effects on the QNMs and stability of the black holes. One important novel feature of these types of black holes is that as the multipole number l increases, the imaginary part decreases. This indicates that, unlike typical black hole solutions such as Schwarzschild, RN, and Kerr, the higher multipole numbers live longer than the lower ones.

3.2. Asymptotically dS Spacetime

In order to explore the effects of rainbow functions and magnetic charge on the perturbations in dS spacetime, we calculate the QN frequencies of hairy black hole solutions in Einstein–Maxwell theory with gravity’s rainbow modification and positive cosmological constant $\mathrm{\Lambda }>0$. The profile of the corresponding effective potential (8) is illustrated in the right panel of Figure 1 for these types of black holes. Similar to the asymptotically flat spacetime case, the potential forms a single peak against incident waves and vanishes at the event horizon $r_H$ and cosmological horizon $r_C$, which is also the case for arbitrary values of l. Therefore, the WKB expansion is a suitable approach to calculate the QN frequencies in asymptotically dS spacetime as well. Furthermore, the effective potential is positive definite, which guarantees the dynamical stability of these black holes in dS spacetime undergoing scalar perturbations for the special choice of the free parameters.

The fundamental QNMs for various multipole numbers are presented in

Table 2. The fundamental QN frequencies for various multipole numbers in dS spacetime for $\mathrm{\Lambda }=0.1$, $M=1$, and $q=0.5$. The second row corresponds to the QN frequencies of an RNdS black hole.

ν	f(ε)	g(ε)	ω01	ω02
0	1	1	$0.115348-0.042831i$	$0.205198-0.041584i$
$0.5$	1	1	$0.150169-0.052887i$	$0.263834-0.050955i$
$0.5$	$1.2$	1	$0.139147-0.047360i$	$0.243040-0.045562i$
$0.5$	1	$1.2$	$0.270511-0.089851i$	$0.462468-0.086183i$

. The results show that the QNMs spectrum behaves similarly to the corresponding one in asymptotically flat spacetime when one changes the magnetic charge $\nu$, the multipole index l, and the rainbow functions $f\left(\epsilon \right)$ and $g\left(\epsilon \right)$. In addition, by comparing Table 1 and Table 2, we find that both the real and imaginary parts of the QN frequencies decrease by turning on the cosmological constant and hence by changing the spacetime background from flat to dS. This indicates that the perturbations in dS spacetime live longer with fewer oscillations in comparison to asymptotically flat spacetime.

4. Greybody Factors

Here, we study the behavior of reflection coefficient ${\mathcal{R}}_l\left(\omega \right)$ and transmission coefficient ${\mathcal{T}}_l\left(\omega \right)$ for black hole solutions in Einstein–Maxwell–Yang–Mills gravity’s rainbow as functions of the Wu–Yang magnetic charge $\nu$, cosmological constant $\mathrm{\Lambda }$, and multipole number l. It was shown that black holes can be considered as ordinary thermodynamic systems with a given temperature. The corresponding conjugate parameter is entropy, which is an extensive quantity. Therefore, black holes also emit radiation, and this property is revealed by Hawking [75]. The radiation emission rate from a black hole with frequency $\omega$ is given by [76]

$$\dot{E}=\sum _l{N}_l{\left|{\mathrm{\Gamma }}_l\left(\omega \right)\right|}^2{\displaystyle \frac{\omega }{e^{\beta \omega }\pm 1}}{\displaystyle \frac{d\omega }{2\pi }},$$

(11)

where $N_l$ values are the multiplicities that depend on the spacetime dimension and l. Furthermore, ${\mathrm{\Gamma }}_l\left(\omega \right)$ are the greybody factors observed by a distant detector and specify the dependency of emission rate on the properties of spacetime. Therefore, the free parameters of black hole solutions in gravitational theories beyond Einstein–vacuum gravity have a footprint on the greybody factors ${\mathrm{\Gamma }}_l\left(\omega \right)$. Furthermore, $\beta$ is the inverse of black hole temperature, the plus sign refers to fermionic particles, and the minus sign denotes bosonic species. In this formalism, the curvature of spacetime produced by the black hole plays the role of a potential barrier against the Hawking radiation and the emission rate detected at the asymptotic region decreases. This implies that the greybody factor is unity at the horizon and can be rewritten as [72]

$${\mathrm{\Gamma }}_l\left(\omega \right)={\left|{\mathcal{T}}_l\left(\omega \right)\right|}^2,$$

(12)

in terms of the transmission coefficients ${\mathcal{T}}_l\left(\omega \right)$. One can write ${\mathcal{T}}_l\left(\omega \right)$ in terms of ${\mathcal{R}}_l\left(\omega \right)$ as follows [72]:

$$|{\mathcal{T}}_l{\left(\omega \right)|}^2=1-{\left|{\mathcal{R}}_l\left(\omega \right)\right|}^2,$$

(13)

such that the reflection coefficients ${\mathcal{R}}_l\left(\omega \right)$ have the following explicit form [72]:

$${\mathcal{R}}_l\left(\omega \right)={\displaystyle \frac{1}{\sqrt{1+e^{-\pi (2n+1)i}}}},$$

(14)

where the overtone number n can be calculated by means of (10), and note that n is a function of the frequency $\omega$ through (10). We recall that the relations (12)–(14) are standard equations derived from the WKB approximation for a potential barrier [72] (see also [77]).

Figure 2 and Figure 3 illustrate variation of the reflection (left panels) and transmission (right panels) coefficients as functions of the frequency for various values of Wu–Yang magnetic charge and multipole number. In order to plot these figures, we used a Mathematica package provided in [77] and we fixed the free parameters to $f\left(\epsilon \right)=1=g\left(\epsilon \right)$, $M=1$, and $q=0.5$. From these figures and for small frequencies, one sees that the reflection coefficients are unity, whereas the transmission coefficients vanish. This is because the scalar waves are in the low-energy regime and the signals cannot pass the potential barrier generated by spacetime curvature. On the contrary, as the frequency increases and the scalar waves move to the high energy regime, the signals start to tunnel the potential barrier, and as a result, ${\mathcal{R}}_l\left(\omega \right)$ starts decreasing, whereas ${\mathcal{T}}_l\left(\omega \right)$ starts growing. Finally, highly energetic scalar waves fully pass the potential barrier and the reflection coefficients vanish, whereas the transmission coefficients become equal to one. These figures also demonstrate the effects of $\nu$ and l on ${\mathcal{R}}_l\left(\omega \right)$ and ${\mathcal{T}}_l\left(\omega \right)$. We find that by increasing $\nu$ and l, the tunneling process starts at higher frequencies, and thus, the potential barrier against the incident waves is stronger for higher values of $\nu$ and l. Our calculations show qualitatively similar results for $\nu$ and l in asymptotically dS spacetime, which are not displayed in this article.

Similarly, Figure 4 illustrates how ${\mathcal{R}}_l\left(\omega \right)$ and ${\mathcal{T}}_l\left(\omega \right)$ change in transition from a flat background to a dS one. Interestingly, we see that as we switch the asymptotically flat spacetime to the dS background, the tunneling process starts at lower frequencies, so the potential barrier against the incident waves becomes weaker. This can (partially) be understood by taking into account the anti-gravity nature of the cosmological constant.

5. Outlook and Conclusions

We have explored test scalar field perturbations minimally coupled to the Einstein–Maxwell–Yang–Mills black hole system in the presence of gravity’s rainbow in the asymptotically flat and dS spacetimes. The motivation to incorporate the non-abelian Yang–Mills field, cosmological constant, and gravity’s rainbow into the Einstein-Maxwell system is discussed in the introduction extensively, which demonstrates the novelty of this paper compared to previous literature. We computed the QNMs spectrum of perturbations by employing the sixth-order WKB approximation in order to investigate the effects of the Wu–Yang magnetic monopole, rainbow functions, and cosmological constant on oscillations and relaxation time scales of the test massless scalar field. Then, we studied the reflection and transmission coefficients to probe the magnetic charge and cosmological constant effects.

In the case of asymptotically flat spacetime, we saw that adding the Wu-Yang monopole to the RN black holes increases both the real and imaginary parts of the frequencies. Therefore, perturbations in the RN background live longer with fewer oscillations in comparison to the hairy black hole solutions in Einstein–Maxwell theory in the presence of gravity’s rainbow. In addition, we observed that the rainbow functions have opposite effects on the QNMs and stability of the black holes. Furthermore, it was shown that the imaginary part decreases when the multipole number l increases, which is an important novel feature of this black hole case study. This indicates that, unlike typical black hole solutions such as Schwarzschild, RN, and Kerr, the higher multipole numbers live longer than the lower ones. From these results, we expect to see a similar behavior from the gravitational waves at the ringdown stage that needs further investigation.

On the other hand, in the case of asymptotically dS spacetime, we saw that the QNMs spectrum behaves similarly to the corresponding one in asymptotically flat spacetime when one changes the magnetic charge, the multipole index, and the rainbow functions. In addition, we found that both the real and imaginary parts of the QN frequencies decrease by turning on the cosmological constant, hence changing the spacetime background from flat to dS. This indicates that the perturbations in dS spacetime live longer with fewer oscillations in comparison to asymptotically flat spacetime. As the next step in this direction, we can explore the gravitational perturbations in the background of the Einstein–Maxwell–Yang–Mills black hole system in the presence of gravity’s rainbow in order to study the effects of the cosmological constant, Wu–Yang magnetic charge, and rainbow functions on the gravitational waves at the ringdown stage and compare the results with the present study.

Furthermore, we have investigated the reflection and transmission coefficients as functions of the frequency for various values of Wu–Yang magnetic charge and multipole number. We found that by increasing $\nu$ and l, the tunneling process starts at higher frequencies, and thus, the potential barrier against the incident waves is stronger for higher values of $\nu$ and l. In addition, we have shown that when we switch the asymptotically flat spacetime to the dS background, the tunneling process starts at lower frequencies, so the potential barrier against the incident waves becomes weaker.

Now, we conclude our paper with a couple of suggestions for future work. It would be interesting to employ alternative computational methods to calculate the exact QN frequencies and compare the results with those obtained from the semi-analytic WKB method. On the other hand, it would be nice to consider the gravitational perturbations in the background spacetime of the black hole case study in order to probe the effects of the free parameters on the gravitational QNMs and check if the results of this article hold or depend on the spin of the perturbed field.