Probing the signature of axions through the quasinormal modes of black holes

The advent of gravitational-wave astronomy opened up a new window for probing the physics in strong-gravity regimes [1]. From the merger events of compact binaries, one can constrain not only the masses and charges of black holes (BHs) but also quasinormal modes (QNMs) of damped oscillations. QNMs of the Schwarzschild BH can be modified by the presence of extra degrees of freedom [2, 3, 4, 5, 6, 7], e.g., vector and scalar fields. A simple example is the Reissner-Nordström (RN) BH with an electric charge [8, 9, 10, 11], which arises from the presence of a vector field $A_{\mu}$ in Einstein-Maxwell theory.

Recently, there has been growing interest in understanding properties of the magnetically charged BHs [12]. Such BHs may have primordial origins as a result of the absorption of magnetic monopoles in the early Universe [13, 14, 15, 16, 17]. Since the magnetic BH is not neutralized with ordinary matter in conductive media, it can be a more stable configuration relative to the purely electric BHs [12, 18]. Then, it is worth studying observational signatures of the magnetic monopole carried by BHs. With a given total BH charge and mass, however, it was recently shown that the QNM of the RN BH is the same independent of the mixture between the magnetic and electric charges [19] (see also Refs. [20, 21, 22]). Hence we cannot distinguish between the magnetic and electric RN BHs from the observations of QNMs.

In the presence of an additional scalar field, it is possible to realize nontrivial BH solutions endowed with scalar hairs. The pseudo-scalar axion field $\phi$, which was originally introduced to address the strong CP problem in QCD [23], can be coupled to an electromagnetic field strength tensor $F_{\mu\nu}$ in the form $-(1/4)g_{a\gamma\gamma}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$, where $g_{a\gamma\gamma}$ is a coupling constant and $\tilde{F}^{\mu\nu}$ is a dual of $F^{\mu\nu}$. In string theory, there are also axion-like light particles with a vast range of masses [24]. It is known that there are BHs endowed with the axion hair as well as with the magnetic and electric charges [25, 26, 27]. An important question is whether or not such hairy BHs can be observationally distinguished from the RN BH.

In this letter, we compute the QNMs of hairy BHs in Einstein-Maxwell-axion (EMA) theory in the presence of the axion-photon coupling. We show that, with a given total BH charge and mass, the QNMs are different depending on the ratio between the magnetic and electric charges. This property is in stark contrast with that of the RN BH. Thus, the precise observations of QNMs can allow us to probe the existence of both the magnetic monopole and the axion.

The EMA theory is given by the action

	$\displaystyle{\cal S}$	$\displaystyle=$	$\displaystyle\int{\rm d}^{4}x\sqrt{-g}\left[\frac{M_{\rm Pl}^{2}}{2}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right.$		(1)
			$\displaystyle\left.-\frac{1}{2}m_{\phi}^{2}\phi^{2}-\frac{1}{4}g_{a\gamma\gamma}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}\right]\,,$

where $g$ is the determinant of metric tensor $g_{\mu\nu}$, $M_{\rm Pl}$ is the reduced Planck mass, $R$ is the Ricci scalar, and $m_{\phi}$ is the axion mass. The field strength tensor $F_{\mu\nu}$ is related to the vector field $A_{\mu}$, as $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, and $\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}/(2\sqrt{-g})$ with $\epsilon^{0123}=+1$. The action (1) respects $U(1)$ gauge invariance under the shift $A_{\mu}\to A_{\mu}+\nabla_{\mu}\chi$.

We consider a static and spherically symmetric line element given by

$${\rm d}s^{2}=-f(r){\rm d}t^{2}+h^{-1}(r){\rm d}r^{2}+r^{2}\left({\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}\right)\,,$$

(2)

where $f$ and $h$ are functions of the radial coordinate $r$. The axion and vector-field configurations compatible with this background are $\phi=\phi(r)$ and $A_{\mu}=[A_{0}(r),0,0,-q_{M}\cos\theta]$, where $q_{M}$ is a constant corresponding to the magnetic charge. The axion and the temporal vector component obey the following differential equations

	$\displaystyle\phi^{\prime\prime}+\left(\frac{2}{r}+\frac{f^{\prime}}{2f}+\frac{h^{\prime}}{2h}\right)\!\phi^{\prime}-\frac{m_{\phi}^{2}}{h}\phi-\frac{g_{a\gamma\gamma}q_{M}A_{0}^{\prime}}{r^{2}\sqrt{fh}}=0\,,$		(3)
	$\displaystyle A_{0}^{\prime}=\frac{\sqrt{f}[q_{E}+q_{M}g_{a\gamma\gamma}\phi]}{r^{2}\sqrt{h}}\,,$		(4)

respectively, where a prime represents the derivative with respect to $r$. The integration constant $q_{E}$ in $A_{0}^{\prime}$ corresponds to the electric charge. For $q_{M}\neq 0$, the BH can have a nontrivial axion profile through the coupling with $A_{0}^{\prime}$. The gravitational equations of motion are given by

	$\displaystyle\frac{rh^{\prime}+h-1}{M_{\rm Pl}^{-2}h}+\frac{r^{2}\phi^{\prime 2}}{2}+\frac{r^{2}m_{\phi}^{2}\phi^{2}}{2h}+\frac{q_{M}^{2}}{2h\,r^{2}}+\frac{r^{2}A_{0}^{\prime 2}}{2f}=0\,,$		(5)
	$\displaystyle\Delta\equiv r_{h}\left(\frac{f^{\prime}}{f}-\frac{h^{\prime}}{h}\right)=\frac{r_{h}r\phi^{\prime 2}}{M_{\rm Pl}^{2}}\,,$		(6)

where $r_{h}$ is the outer horizon radius. Around $r=r_{h}$, we expand the metrics and scalar field, as

	$\displaystyle f=\sum_{i=1}f_{i}(r-r_{h})^{i}\,,\qquad h=\sum_{i=1}h_{i}(r-r_{h})^{i}\,,$
	$\displaystyle\phi=\phi_{0}+\sum_{i=1}\phi_{i}(r-r_{h})^{i}\,,$		(7)

where $f_{i}$, $h_{i}$, $\phi_{0}$ and $\phi_{i}$ are constants. For consistency with the background equations, we require that

	$\displaystyle h_{1}$	$\displaystyle=$	$\displaystyle[2M_{\rm Pl}^{2}r_{h}^{2}-q_{E}^{2}-q_{M}^{2}-g_{a\gamma\gamma}q_{M}\phi_{0}(g_{a\gamma\gamma}q_{M}\phi_{0}+2q_{E})$		(8)
			$\displaystyle-m_{\phi}^{2}r_{h}^{4}\phi_{0}^{2}]/(2M_{\rm Pl}^{2}r_{h}^{3})\,,$
	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle[(m_{\phi}^{2}r_{h}^{4}+g_{a\gamma\gamma}^{2}q_{M}^{2})\phi_{0}+g_{a\gamma\gamma}q_{M}q_{E}]/(h_{1}r_{h}^{4})\,.$		(9)

We are interested in hairy BH solutions where $|\phi|$ is a decreasing function of $r$ from the horizon to spatial infinity. Furthermore, to ensure the property $h(r)>0$ for $r>r_{h}$, we require that $h_{1}>0$. Hence, around $r=r_{h}$, these two conditions lead to

$$\phi_{0}\phi_{1}h_{1}r_{h}^{4}=\left(m_{\phi}^{2}r_{h}^{4}+g_{a\gamma\gamma}^{2}q_{M}^{2}\right)\phi_{0}^{2}+g_{a\gamma\gamma}\phi_{0}q_{M}q_{E}<0\,.$$

(10)

Then, it is at least necessary to satisfy the inequality

$$g_{a\gamma\gamma}\phi_{0}q_{M}q_{E}<0\,.$$

(11)

Since this condition is violated for $q_{M}=0$ or $q_{E}=0$, we need the existence of both magnetic and electric charges to realize a nontrivial axion hair. The inequality (11) does not hold for $g_{a\gamma\gamma}=0$ either, so we require the axion-photon coupling $-(1/4)g_{a\gamma\gamma}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$ to realize hairy BH solutions. In other words, the no-hair property of BHs for a canonical scalar field [28, 29] is broken by the appearance of a secondary axion hair through interaction with electromagnetic fields.

Without loss of generality, we will consider the case $\phi_{0}>0$, $q_{M}>0$, $q_{E}>0$, and $g_{a\gamma\gamma}<0$. Because of Eq. (8), combining (10) with $h_{1}>0$ gives

$$\frac{-g_{a\gamma\gamma}q_{M}q_{E}-\sqrt{\cal A}}{q_{M}^{2}g_{a\gamma\gamma}^{2}+m_{\phi}^{2}r_{h}^{4}}<\phi_{0}<\frac{-g_{a\gamma\gamma}q_{M}q_{E}}{q_{M}^{2}g_{a\gamma\gamma}^{2}+m_{\phi}^{2}r_{h}^{4}}\,,$$

(12)

where

$${\cal A}\equiv(2M_{\rm Pl}^{2}r_{h}^{2}-q_{M}^{2})q_{M}^{2}g_{a\gamma\gamma}^{2}+\left(2M_{\rm Pl}^{2}r_{h}^{2}-q_{M}^{2}-q_{E}^{2}\right)m_{\phi}^{2}r_{h}^{4}\,.$$

(13)

If $q_{M}^{2}\geq 2M_{\rm Pl}^{2}r_{h}^{2}$, then ${\cal A}$ is negative. The magnetic charge should be at least in the range $q_{M}^{2}<2M_{\rm Pl}^{2}r_{h}^{2}$ for the existence of hairy BHs with $\phi_{0}\neq 0$. More strictly, so long as the condition

$$q_{M}^{2}+q_{E}^{2}<2M_{\rm Pl}^{2}r_{h}^{2}$$

(14)

is satisfied, we always have ${\cal A}>0$ and hence there is the field value $\phi_{0}$ in the range (12).

We search for the solutions respecting the asymptotic flatness, i.e., $f\to 1$, $h\to 1$, $f^{\prime}\to 0$, and $h^{\prime}\to 0$ as $r\to\infty$. We also impose the boundary condition $\phi(r\to\infty)=0$. In this large-distance regime, Eq. (3) approximately reduces to $\phi^{\prime\prime}+2\phi^{\prime}/r-m_{\phi}^{2}\phi\simeq g_{a\gamma\gamma}q_{M}q_{E}/r^{4}$. The solution to this equation, which respects the boundary condition $\phi(r\to\infty)=0$, can be expressed as

$$\phi(r)\simeq q_{s}\frac{e^{-m_{\phi}r}}{r}-\frac{g_{a\gamma\gamma}q_{M}q_{E}}{m_{\phi}^{2}r^{4}}\,,$$

(15)

where $q_{s}$ is a constant. The first term in Eq. (15) decreases exponentially for $r>m_{\phi}^{-1}$ and hence $\phi(r)\propto r^{-4}$ in this regime. For $m_{\phi}=0$, the large-distance solution is given by $\phi(r)\simeq q_{s}/r+g_{a\gamma\gamma}q_{M}q_{E}/(2r^{2})$. In both cases, the metric approaches that of the RN BH as $r\to\infty$.

To confirm the existence of hairy BH solutions, we numerically solve Eqs. (3)-(6) by imposing the aforementioned boundary conditions around $r=r_{h}$. In Fig. 1, we plot $h$, $\Delta$, $\bar{\phi}=\phi/M_{\rm Pl}$, and $-r_{h}\bar{\phi}^{\prime}(r)$ versus $r/r_{h}$ for $m_{\phi}=0$, $g_{a\gamma\gamma}M_{\rm Pl}=-10$, $q_{M}=0.05M_{\rm Pl}r_{h}$, and $q_{E}=0.5M_{\rm Pl}r_{h}$. In this case, the two conditions (11) and (14) are satisfied, with $-1.827<\phi_{0}/M_{\rm Pl}<1$ from Eq. (12). The axion has a maximum amplitude $\phi_{0}\simeq 0.217899M_{\rm Pl}$ on the horizon and then it decreases toward the asymptotic value $\phi(\infty)=0$ without changing the sign. In Fig. 1, we observe the field dependence $\phi^{\prime}(r)=-q_{s}/r^{2}~{}(<0)$ in the regime $r\gg r_{h}$. Substituting the large-distance solution $\phi(r)=q_{s}/r$ into Eqs. (5) and (6), we obtain $f=1-2M/r+(q_{M}^{2}+q_{E}^{2})/(2M_{\rm Pl}^{2}r^{2})+{\cal O}(r^{-3})$ and $h=f+q_{s}^{2}/(2M_{\rm Pl}^{2}r^{2})+{\cal O}(r^{-3})$, with $\Delta\simeq r_{h}q_{s}^{2}/(M_{\rm Pl}^{2}r^{3})$, where $M$ corresponds to the BH ADM mass. As we see in Fig. 1, the difference between $f^{\prime}/f$ and $h^{\prime}/h$ is most significant around $r=r_{h}$.

For $m_{\phi}\neq 0$ the axion has a growing-mode solution $e^{m_{\phi}r}/r$ manifesting at the distance $r\gtrsim 1/m_{\phi}$, but there should be appropriate boundary conditions respecting the regularities of both infinity and the horizon. We numerically confirm the existence of asymptotically-flat hairy BHs especially in the mass range $m_{\phi}r_{h}\lesssim 1$. For a BH with $r_{h}\simeq 10^{4}$ m, the axion mass corresponding to $m_{\phi}r_{h}\lesssim 1$ is $m_{\phi}\lesssim 10^{-11}$ eV, which includes the case of fuzzy dark matter ($m_{\phi}\simeq 10^{-21}$ eV) [30]. Taking the limit $m_{\phi}\to\infty$ in Eq. (12), the allowed values of $\phi_{0}$ shrink to 0. Hence the hairy BH solution tends to disappear in this massive limit. For the axion mass $m_{\phi}\lesssim 1$ eV, the current limit on the axion-photon coupling is $|g_{a\gamma\gamma}|\lesssim 10^{6}M_{\rm Pl}^{-1}$ [31]. The coupling $g_{a\gamma\gamma}$ used in Fig. 1 is well consistent with such a bound.

In this section, we compute the QNMs of hairy BHs in EMA theory by considering linear perturbations on the background (2). We choose the gauge in which the $\theta$ and $\varphi$ components of $h_{\mu\nu}$ vanish, i.e.,

	$\displaystyle h_{tt}=f(r)H_{0}(t,r)Y_{l}(\theta),\quad h_{tr}=H_{1}(t,r)Y_{l}(\theta),$
	$\displaystyle h_{t\varphi}=-Q(t,r)(\sin\theta)Y_{l,\theta}(\theta),\;\;h_{rr}=h^{-1}(r)H_{2}(t,r)Y_{l}(\theta),$
	$\displaystyle h_{r\theta}=h_{1}(t,r)Y_{l,\theta}(\theta),\quad h_{r\varphi}=-W(t,r)(\sin\theta)Y_{l,\theta}(\theta),$
	$\displaystyle h_{\theta\theta}=0,\quad h_{\varphi\varphi}=0,\quad h_{\theta\varphi}=0,$		(16)

where $Y_{l}(\theta)$’s are the $m=0$ components of spherical harmonics $Y_{lm}(\theta,\varphi)$, and $Y_{l,\theta}(\theta)\equiv{\rm d}Y_{l}(\theta)/{\rm d}\theta$. In Eq. (16), we omit the summation of $Y_{l}(\theta)$ concerning the multipoles $l$. Note that the above gauge choice is different from the Regge-Wheeler-Zerilli gauge [32, 33, 34], but the former can also fix the residual gauge degrees of freedom completely [35, 36, 37, 38]. Since our theory has $U(1)$ gauge symmetry, we can choose a gauge in which the $\theta$ component of the vector-field perturbation $\delta A_{\mu}$ vanishes [38]. Then, we consider the perturbed components of vector and axion fields, as

	$\displaystyle\delta A_{t}=\delta A_{0}(t,r)Y_{l}(\theta),\quad\delta A_{r}=\delta A_{1}(t,r)Y_{l}(\theta),\quad$
	$\displaystyle\delta A_{\theta}=0,\quad\delta A_{\varphi}=-\delta A(t,r)(\sin\theta)Y_{l,\theta}(\theta),$
	$\displaystyle\delta\phi=\delta\phi(t,r)Y_{l}(\theta),$		(17)

respectively. The coupling $-(\sqrt{-g}/4)g_{a\gamma\gamma}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$ in the action (1) does not have contributions from the metric components and depends only linearly on each perturbed field. Moreover, none of the perturbations are coupled to the modes with different values of $l$ or $m$. The background spherical symmetry allows us to set $m=0$ without loss of generality. Indeed, for fixed $l$, the second-order perturbed action does not depend on the values of $m$. Since the perturbations in the odd- and even-parity sectors are mixed for $q_{M}\neq 0$, we must deal with them all at once.

For $l\geq 2$, we expand Eq. (1) up to quadratic order in perturbations. Then, the resulting second-order action contains ten perturbed variables $H_{0}$, $H_{1}$, $H_{2}$, $h_{1}$, $Q$, $W$, $\delta A_{0}$, $\delta A_{1}$, $\delta A$, and $\delta\phi$. The explicit form of the total second-order action ${\cal S}^{(2)}$ is given in Appendix A. Introducing the new fields $\chi_{1}$, $v_{1}$, $\chi_{2}$ defined in Eqs. (27), (28), and (32), respectively, we can express the action in terms of the five dynamical perturbations $\chi_{1}$, $v_{1}$, $\chi_{2}$, $\delta A$, $\delta\phi$, and their $t,r$ derivatives. The process for deriving the reduced second-order action is explained in Appendix B. Here, $v_{1}$ and $\chi_{1}$ are associated with the even- and odd-parity gravitational perturbations, respectively, while $\chi_{2}$ and $\delta A$ arise from the vector-field perturbations in even- and odd-parity sectors, respectively. We also have the dynamical axion perturbation $\delta\phi$.

For the computational simplicity, we make the following field redefinitions

	$\displaystyle\psi_{1}=M_{\rm Pl}rhe^{i\omega t}v_{1},\quad\psi_{2}=M_{\rm Pl}re^{i\omega t}\chi_{1},$
	$\displaystyle\psi_{3}=r^{2}e^{i\omega t}\chi_{2},\quad\psi_{4}=e^{i\omega t}\delta A\,,\quad\psi_{5}=re^{i\omega t}\delta\phi\,,$		(18)

where $\omega$ is an angular frequency. Introducing the tortoise coordinate $r_{*}=\int^{r}{\rm d}\tilde{r}/\sqrt{f(\tilde{r})\,h(\tilde{r})}$, the perturbation equations of motion can be schematically written as

$$\frac{{\rm d}^{2}\psi_{i}}{{\rm d}r_{*}^{2}}+B_{ij}\frac{{\rm d}\psi_{j}}{{\rm d}r_{*}}+C_{ij}\psi_{j}=0\,,\quad i,j\in\{1,\dots,5\},$$

(19)

where the matrices $B_{ij}$ and $C_{ij}$ are background-dependent quantities, and $C_{ij}$ also contain $\omega$.

On the horizon ($r_{*}\to-\infty$) and at spatial infinity ($r_{*}\to+\infty$), Eq. (19) is approximately given by ${\rm d}^{2}\psi_{i}/{{\rm d}r_{*}^{2}}\simeq-\omega^{2}\psi_{i}$. The QNMs are characterized by purely ingoing waves on the horizon and purely outgoing at spatial infinity, and hence

$$\psi_{i}(r_{*}\to-\infty)=A_{i}e^{-i\omega r_{*}}\,,\quad\psi_{i}(r_{*}\to\infty)=B_{i}e^{+i\omega r_{*}}\,,$$

(20)

where $A_{i}$ and $B_{i}$ are constants. For the calculation of QNMs, we will exploit a matrix-valued direct integration method [6] based on higher-order expansions of $\psi_{i}$ both around the horizon and infinity. Using the background solutions (7) expanded in the vicinity of the horizon, we have $r_{*}\simeq(f_{1}h_{1})^{-1/2}\ln(r/r_{h}-1)$ and hence the leading-order solutions to $\psi_{i}$ are proportional to $(r-r_{h})^{-i\omega(f_{1}h_{1})^{-1/2}}$. Then, around $r=r_{h}$, we choose the following ansatz

$$\psi_{i}^{\rm H}=(r-r_{h})^{-i\omega(f_{1}h_{1})^{-1/2}}\sum_{n=0}(\psi_{i}^{\rm H})^{(n)}\,(r-r_{h})^{n}\,,$$

(21)

where $(\psi_{i}^{\rm H})^{(n)}$ is the $n$-th derivative coefficient. To perform this expansion, we need the numerical values of $f_{1}$, $h_{1}$ as well as $r_{h}$. We will find them by numerically solving the background Eqs. (3)-(6) with the boundary conditions (7) expanded up to sufficiently high orders in $i$.

Far away from the horizon where the metric components are given by $f\simeq h=1-2M/r+\mathcal{O}(r^{-2})$, we have $r_{*}\simeq r+2M\ln\,[r/(2M)-1]$ and hence $e^{i\omega r_{*}}\propto e^{i\omega r}r^{2i\omega M}$. At spatial infinity, this leads to the following ansatz

$$\psi_{i}^{\rm I}=e^{i\omega r}r^{2i\omega M}\sum_{n=0}(\psi_{i}^{\rm I})^{(n)}r^{-n}\,.$$

(22)

Solving the perturbation equations order by order in the regime $r\gg r_{h}$, we find that the coefficients $(\psi_{i}^{\rm I})^{(n)}$ with $n\geq 1$ are all functions of $(\psi_{i}^{\rm I})^{(0)}$. Then the space of independent solutions is five, as it is also the case for the expansion (21) around $r=r_{h}$.

In the following, we will focus on the massless axion ($m_{\phi}=0$) and the quadrupole perturbations ($l=2$). The computation of QNMs can be easily extended to the massive axion whose Compton wavelength $m_{\phi}^{-1}$ is much larger than $r_{h}$. For the numerical computation, it is useful to perform the rescalings $r=\bar{r}\,r_{p}$, $q_{M}=\bar{q}_{M}M_{\rm Pl}r_{p}$, and $q_{E}=\bar{q}_{E}M_{\rm Pl}r_{p}$, where $r_{p}$ is a pivot radius. We will first find the value of $r_{h}$ leading to the BH ADM mass $M=1$ and then choose $r_{p}=M=1$. In the following, we will omit the bars from $\bar{r}$, $\bar{q}_{M}$, and $\bar{q}_{E}$ to keep the notation simpler. We also apply this rescaling to the perturbation equations of motion.

We vary the values of $q_{E}$ and $q_{M}$ by keeping the BH mass $M$ and the total charge $q_{T}=\sqrt{q_{E}^{2}+q_{M}^{2}}$ constant. After this, we only have the freedom of choosing five constants on the horizon, $(\psi_{i}^{\rm H})^{(0)}$, and the other five at infinity, namely $(\psi_{i}^{\rm I})^{(0)}$. We can build up ten independent solutions as follows. The first solution is found by integrating the perturbation equations from the vicinity of $r=r_{h}$ up to a value of $r=r_{{\rm mid}}<\infty$ (typically $r_{{\rm mid}}=5$), with the coefficients $(\psi_{1}^{\rm H})^{(0)}=1$ and $(\psi_{j}^{\rm H})^{(0)}=0$ for $j\neq 1$. We repeat this procedure by choosing $(\psi_{2}^{\rm H})^{(0)}=1$ and $(\psi_{j}^{\rm H})^{(0)}=0$ with $j\neq 2$, until we arrive at $i=5$. These solutions and radial derivatives, which are evaluated at $r=r_{{\rm mid}}$, are called $\tilde{\psi}_{i,j}^{\rm H}$ and ${\rm d}\tilde{\psi}_{i,j}^{\rm H}/{\rm d}r$, respectively, where $j$ stands for the nonzero value of $(\psi_{j}^{\rm H})^{(0)}$. In this way, we can build a matrix ${\cal A}$ with the first five columns given by $(\tilde{\psi}_{i,j}^{\rm H},{\rm d}\tilde{\psi}_{i,j}^{\rm H}/{\rm d}r)^{\rm T}$.

We will also find five other independent solutions by integrating from infinity down to $r=r_{{\rm mid}}$. For the boundary conditions, we fix one of the $\psi_{j}^{{\rm I},(0)}$ to 1 and the other four elements to zero. We call these solutions and radial derivatives $\tilde{\psi}_{i,j}^{\rm I}$ and ${\rm d}\tilde{\psi}_{i,j}^{\rm I}/{\rm d}r$, respectively, and again naming by $j$ the nonzero $(\psi_{j}^{\rm I})^{(0)}$. Adding the five columns $(\tilde{\psi}_{i,j}^{\rm I},{\rm d}\tilde{\psi}_{i,j}^{\rm I}/{\rm d}r)^{\rm T}$ to the matrix $\mathcal{A}$, we obtain the $10\times 10$ matrix $\tilde{{\cal A}}$. From the determinant equation $\det\tilde{{\cal A}}=0$, we can obtain the QNM frequency $\omega$.

We consider the two fundamental QNMs that are present in the limit $q_{M}\to 0$.¹¹1If isospectrality is broken, we should have other fundamental frequencies, one from the gravitational side and the other one from the electromagnetic side. In addition, independently of the isospectrality breaking, we should expect to have another frequency coming from the scalar mode, due to its nontrivial hair. This work focuses on the possibility of finding the axion coupled to photons around charged black holes by breaking electric-magnetic duality. The detailed study of the whole spectrum of QNM frequencies will be discussed elsewhere. For $q_{M}=0$, the background solution reduces to the electrically charged RN BH without the axion hair²²2In this limit, we can recognize numerically the spectrum of the QNMs as being gravitational or electromagnetic.. In this case, there are one gravitational and the other electromagnetic QNMs, whose frequencies were computed in Refs. [9, 8, 9]. The study of a possible extra fundamental QNM due to the nontrivial axion profile is left for future work. In fact, in this study, we would like to focus on the crucial property of hairy BHs to distinguish the magnetic charge from the electric one at the level of the gravitational/electromagnetic QNMs, leading to an unequivocal sign for the existence of axions. We choose a configuration with $q_{M}=q_{T}\sin\alpha$ and $q_{E}=q_{T}\cos\alpha$, where the total charge $q_{T}$ is chosen to be $\sqrt{13/50}$. We vary the angle $\alpha$ from 0 to $12/50$, where each solution differs from the previous one by the interval $\Delta\alpha=1/100$. For each value of $\alpha$, we numerically solve the background equations of motion and find the value of $r_{h}$ leading to $M=1$, so that all the BHs have the same mass and total charge.

In Fig. 2, we plot the QNM frequencies $\omega=\omega_{R}+i\omega_{I}$ for the gravitational fundamental mode. In the limit $q_{M}\to 0$ we obtain $\omega M=0.37744-0.08932i$, which coincides with the value of an electrically charged RN BH with $q_{E}=\sqrt{13/50}$. For $q_{M}\neq 0$, both $\omega_{R}$ and $\omega_{I}$ change as a function of $q_{M}$. This property is in stark contrast to the RN BH without the axion-photon coupling, where the QNM is independent of $q_{M}$ for a fixed total charge $q_{T}=\sqrt{q_{E}^{2}+q_{M}^{2}}$ and a mass $M$. The axion-photon coupling breaks this degeneracy of QNMs relevant to electric-magnetic duality [22].

In Fig. 3, we also show the electromagnetic fundamental frequencies as a function of $q_{M}$. In the limit $q_{M}\to 0$, we confirm that the electromagnetic QNM approaches the value $\omega=0.4756-0.09618i$ derived for the RN BH with $q_{E}=\sqrt{13/50}$. For $q_{M}\neq 0$, both the real and imaginary parts of the electromagnetic QNM explicitly depend on the ratio $q_{M}/q_{T}$. We showed this property for a total charge $q_{T}=\sqrt{13/50}$, but it also persists for general nonvanishing values of $q_{T}$. Moreover, the overtones of both gravitational and electromagnetic perturbations are also dependent on $q_{M}/q_{T}$ for fixed values of $q_{T}$ and $M$. Thus, the gravitational-wave observations of QNMs allow us to distinguish between the charged BH with the axion hair and the magnetically (or electrically) charged RN BH.

The magnetically charged BH can be present today as a remnant of the absorption of magnetic monopoles in the early Universe. For a given total charge $q_{T}$ and mass $M$, the QNMs of RN BHs are the same independent of the mixture of magnetic and electric charges. In the presence of the axion coupled to photons, however, we showed that the charged BH with the axion hair breaks this degeneracy. We computed the gravitational and electromagnetic QNMs and found that both QNMs depend on the ratio $q_{M}/q_{T}$ for hairy BH solutions realized by the axion-photon coupling. Hence the upcoming high-precision observations of QNMs offer the possibility for detecting the signatures of both the magnetic charge and the axion.

There are several interesting extensions of our work. First, the computation of QNMs for charged rotating BH solutions with the axion hair [39] is the important next step for placing realistic bounds on our model parameters. Next, the gravitational waveforms emitted during the inspiral phase of charged binary BHs with the axion hair will put further constraints on the theory. Thirdly, the observations of BH shadows such as the Event Horizon Telescope [40] will give upper bounds on the BH charges. Fourth, we leave a detailed study of the isospectrality of QNMs for a future separate work. While isospectrality may be broken, this letter aims to demonstrate the potential for simultaneously finding magnetic charges and axions through the QNMs of BHs. Finally, it will be of interest to study the effect of large magnetic fields on the BH physics near the horizon, e.g., restoration of an electroweak symmetry [12]. These issues are left for future work.

The work of ADF was supported by the Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 20K03969. ST was supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No. 22K03642 and Waseda University Special Research Project No. 2023C-473.