Sound from extra dimension: quasinormal modes of thick brane

As the characteristic modes of a dissipative system, quasinormal modes (QNMs) exist in every aspect of our world. These QNMs contain the key features that are characteristics of the physical systems. Studying them would help us to unravel the mysteries of the physical systems. In black hole physics, the QNMs are thought to be able to carry information about black holes and have attracted much attention Berti:2009kk ; Kokkotas:1999bd ; Nollert:1999ji ; Konoplya:2011qq ; Cardoso:2016rao ; Jusufi:2020odz ; Cheung:2021bol , especially after the detection of gravitational waves LIGOScientific:2016aoc . Other physical systems such as leaky resonant cavities, QNMs also play an important role Kristensen:2015qq . So we are curious about what role QNMs might play in the braneworld model.

The braneworld models were originally introduced as a solution to the hierarchical problem between the weak and Plank scales. Among them, the warped extra dimension models proposed by Randall and Sundrum (RS) have attracted a lot of interest Randall:1999ee ; Randall:1999vf . They consist one brane (RS-II model) or two branes (RS-I model) embedded in a five-dimensional anti-de-Sitter spacetime. Since the RS models were proposed, they have been studied in many realms such as particle physics, cosmology, and black hole physics. The applicability has gone far beyond its original scope Shiromizu:1999wj ; Tanaka:2002rb ; Gregory:2008rf ; Jaman:2018ucm ; Adhikari:2020xcg ; Bhattacharya:2021jrn . Combining the RS-II model Randall:1999vf and the domain wall model Akama:1982jy ; Rubakov:1983bb , the thick brane models were developed DeWolfe:1999cp ; Gremm:1999pj ; Csaki:2000fc . It is a smooth extension of the RS-II model. The inclusion of brane thickness gives us new possibilities. Usually, most thick branes are generated by one or more scalar fields, but they can also be generated by a vector or spinor field Dzhunushaliev:2010fqo ; Dzhunushaliev:2011mm ; Geng:2015kvs . In order to recover the physics in our four-dimensional world, the zero modes of various fields should be confined on the brane. In previous literatures, some thick brane models and the localization of various matter fields on the brane were investigated Melfo2006 ; Almeida2009 ; Zhao2010 ; Chumbes2011 ; Liu2011 ; Xie2017 ; Gu2017 ; ZhongYuan2017 ; ZhongYuan2017b ; Zhou2018 ; Chen:2020zzs ; Hendi:2020qkk ; Xie:2021ayr ; Moreira:2021uod ; Xu:2022ori ; Silva:2022pfd ; Xu:2022gth . Besides these zero modes, there are massive Kaluza-Klein (KK) modes which might propagate along extra dimensions. These massive KK modes provide the possibility of detecting extra dimensions. In addition, cosmological thick brane solutions were also investigated Mounaix:2002mm ; Ghassemi:2006qk ; Wu:2010stv .

Quasinormal modes in higher dimensional theories also attract the interest of researchers Chakraborty:2017qve ; Dey:2020pth ; Prasobh:2014zea ; Hashemi:2019jlt ; Chen:2016qii ; Konoplya:2003dd ; Cardoso:2003vt . It is expected that the signatures of extra dimensions can be extracted from QNMs of black holes on the brane. These signals can be used to constrain the extra dimensional models Seahra:2004fg ; Seahra:2006tm ; Chung:2015mna ; Dey:2020lhq ; Banerjee:2021aln ; Mishra:2021waw ; Lin:2022hus . But these researches are mainly focused on the QNMs of black holes on the brane. Does a brane have a characteristic sound? That is, does it have a set of discrete QNMs as characteristic modes of a braneworld model? For the RS-II model, the answer is yes Seahra:2005wk ; Seahra:2005iq . Seahra studied the scattering of KK gravitons in the RS-II model and found that the brane possesses a series of discrete QNMs Seahra:2005wk ; Seahra:2005iq .

Reference Tan:2022uex investigated the evolution of massive modes in the thick brane model and found that the evolution behavior is similar to QNMs. This arouse our interest in QNMs in thick brane models. As far as we know, the QNMs of a thick brane have not been investigated. As the characteristic modes of a brane, it can reflect the structure of the thick brane. On the other hand, since the QNMs dominating the time evolution of some initial fluctuations from the physic system’s equilibrium state, they can be used to verify the stability of the brane Clarkson:2005mg . It is undoubtedly interesting to study the QNMs of a thick brane. We will use semi-analytical and numerical methods to study QNMs in a thick brane model.

The organization of the rest of this paper is as follows. In Sec. II, we review a solution of the thick brane and the linear metric tensor perturbation. Based on this solution, we solve for the QNMs of this thick brane. In Sec. III, we compute the quasinormal frequencies of the thick brane by using semi-analytical methods. We also compare them with the results of numerical evolution. Finally, Sec. IV gives the conclusions and discussions.

In this section, we will briefly review the thick brane solution and its gravitational perturbation. Usually, a thick brane can be generated by a wide variety of matter fields like scalar fields and vector fields. Here we choose a canonical scalar field to generate the brane. The action of this thick brane model is the Einstein-Hilbert action minimally coupled to a canonical scalar field

$\displaystyle S=\int d^{5}x\sqrt{-g}\left(\frac{1}{2\kappa^{2}_{5}}R-\frac{1}{2}g^{MN}\partial_{M}\varphi\partial_{N}\varphi-V(\varphi)\right),$

(1)

where $\kappa_{5}$ is the five-dimensional gravitational constant. Hereafter, capital Latin letters $M,N,\dots=0,1,2,3,5$ label the five-dimensional indices, while Greek letters $\mu,\nu\dots=0,1,2,3$ and Latin letters $i,j\dots=1,2,3$ label the four-dimensional ones and three-dimensional space ones on the brane, respectively. The dynamical field equations are

	$\displaystyle R_{MN}-\frac{1}{2}Rg_{MN}$	$\displaystyle=$	$\displaystyle-g_{MN}\kappa^{2}_{5}\left(\frac{1}{2}\partial^{A}\varphi\partial_{A}\varphi+V(\varphi)\right)$		(2)
			$\displaystyle+\kappa^{2}_{5}\partial_{M}\varphi\partial_{N}\varphi,$
	$\displaystyle g^{MN}\nabla_{M}\nabla_{N}\varphi$	$\displaystyle=$	$\displaystyle\frac{\partial V(\varphi)}{\partial\varphi}.$		(3)

The five-dimensional metric ensuring the four-dimensional Poincaré symmetry is Csaki:2000fc

$$ds^{2}=e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$$

(4)

where $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$ is the four-dimensional Minkowski metric. Now, the specific dynamical equations can be written as

	$\displaystyle 6A^{\prime 2}+3A^{\prime\prime}$	$\displaystyle=$	$\displaystyle-\frac{\kappa^{2}_{5}}{2}\varphi^{\prime 2}-\kappa^{2}_{5}V,$		(5)
	$\displaystyle 6A^{\prime 2}$	$\displaystyle=$	$\displaystyle\frac{\kappa^{2}_{5}}{2}\varphi^{\prime 2}-\kappa^{2}_{5}V,$		(6)
	$\displaystyle\varphi{{}^{\prime\prime}}+4A^{\prime}\varphi^{\prime}$	$\displaystyle=$	$\displaystyle\frac{\partial V}{\partial\varphi},$		(7)

where prime denotes the derivative with respect to the extra dimensional coordinate $y$. By using the first-order formalism, the thick brane solution was investigated in Ref. Gremm:1999pj :

	$\displaystyle A(y)$	$\displaystyle=$	$\displaystyle-b\ln\left(\cosh(ky)\right),$		(8)
	$\displaystyle\varphi(y)$	$\displaystyle=$	$\displaystyle\sqrt{\frac{3b}{\kappa^{2}_{5}}}\arcsin\left(\tanh\left(ky\right)\right),$		(9)
	$\displaystyle V(\varphi)$	$\displaystyle=$	$\displaystyle\frac{3bk^{2}}{4\kappa^{2}_{5}}\left(1-4b+(1+4b)\cos\left(\sqrt{\frac{4\kappa^{2}_{5}}{3b}}\varphi\right)\right).$

Here, $b$ is a dimensionless parameter and $k$ is a parameter with mass dimension one. If we choose $\kappa_{5}=\sqrt{2}$, the above solutions for the scalar field $\varphi$ and potential $V$ are the same to Ref. Gremm:1999pj because $\arcsin\left(\tanh\left(ky\right)\right)=2\arctan\left(\tanh\left(ky/2\right)\right)$³³3There is a typo in the scalar potential $V$ in Ref. Gremm:1999pj , the $-$ in front of $(1+4b)$ should be replaced by $+$.. Besides, the warp factor (8) differs from the one in Ref. Gremm:1999pj a factor $2$ in front of $\cosh(ky)$. It does not matter, because this factor can be absorbed into a new four-dimensional coordinate. Next, we consider the linear transverse-traceless tensor perturbation of the metric. The perturbed metric is given by

$\displaystyle g_{MN}=\left(\begin{array}[]{cc}e^{2A(y)}(\eta_{\mu\nu}+h_{\mu\nu})&0\\ 0&1\\ \end{array}\right),$

(13)

where $h_{\mu\nu}$ satisfies the transverse-traceless condition

$\displaystyle\partial_{\mu}h^{\mu\nu}=0=\eta^{\mu\nu}h_{\mu\nu}.$

(14)

Substituting the perturbed metric (13) into the field equation (2), we obtain the linear equation of the tensor fluctuation:

$\displaystyle\left(e^{-2A}\Box^{(4)}h_{\mu\nu}+h^{\prime\prime}_{\mu\nu}+4A^{\prime}h^{\prime}_{\mu\nu}\right)=0,$

(15)

where $\Box^{(4)}=\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$. Introducing the following coordinate transformation $dz=e^{-A}dy$, the metric (4) can be written as

$$ds^{2}=e^{2A(z)}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}),$$

(16)

and Eq. (15) becomes

$$\left[\partial^{2}_{z}+3(\partial_{z}A)\partial_{z}+\Box^{(4)}\right]h_{\mu\nu}=0.$$

(17)

The perturbation $h_{\mu\nu}$ can be written as Seahra:2005iq

$$h_{\mu\nu}=e^{-\frac{3}{2}A(z)}\Phi(t,z)e^{-ia_{j}x^{j}}\epsilon_{\mu\nu},~{}~{}~{}~{}\epsilon_{\mu\nu}=\text{constant}.$$

(18)

Substituting the above decomposition (18) into Eq. (17), we obtain a one-dimensional wave equation of $\Phi(t,z)$

$$-\partial_{t}^{2}\Phi+\partial_{z}^{2}\Phi-U(z)\Phi-a^{2}\Phi=0,$$

(19)

where

$\displaystyle U(z)=\frac{3}{2}\partial_{z}^{2}A+\frac{9}{4}(\partial_{z}A)^{2}$

(20)

is the effective potential and $a$ is a constant coming from the separation of variables. Furthermore, separability means that the function $\Phi(t,z)$ can be decomposed as

$\displaystyle\Phi(t,z)=e^{-i\omega t}\phi(z).$

(21)

So we can obtain a Schrödinger-like equation of the extra dimensional part $\phi(z)$

$$-\partial_{z}^{2}\phi(z)+U(z)\phi(z)=m^{2}\phi(z),$$

(22)

where $m^{2}=\omega^{2}-a^{2}$ is the mass of the KK modes. Equation (22) supports a bound zero mode $\phi_{0}(z)\propto e^{\frac{3}{2}A(z)}$ which is localized on the brane for the solution (8) with $b>0$, and a series of massive KK modes. Usually, the massive KK modes stay on the brane for a finite time and eventually escape to infinity of the extra dimension. Thus the thick brane is a dissipative system for the massive KK modes. Similar to QNMs in the black hole system, there are also characteristic modes with complex frequencies in the thick brane model. These modes can also reflect the properties of the thick brane model. We will discuss them in the next section.

In this section, we will use some semi-analytical methods to solve the QNMs of the thick brane. As can be seen from the Schrödinger-like equation (22), it is the effective potential $U(z)$ that determines the QNMs. Substituting the thick brane solution (8) into the effective potential (20), we can obtain the specific form of the effective potential. Note that we only consider $b=1$, because the coordinate transformation relation between $y$ and $z$ is analytical for this case. The specific forms of the warp factor $A(z)$, the effective potential $U(z)$, and the zero mode $\phi_{0}(z)$ are given by

	$\displaystyle A(z)$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\ln(k^{2}z^{2}+1),$		(23)
	$\displaystyle U(z)$	$\displaystyle=$	$\displaystyle\frac{3k^{2}\left(5k^{2}z^{2}-2\right)}{4\left(k^{2}z^{2}+1\right)^{2}},$		(24)
	$\displaystyle\phi_{0}(z)$	$\displaystyle=$	$\displaystyle\frac{1}{(1+k^{2}z^{2})^{3/4}}.$		(25)

We plot the effective potential and the zero mode in Fig. 1. It can be seen that the effective potential is volcano-like and $U(z)\rightarrow 0$ when $|z|\rightarrow\infty$. This potential is a smooth extension of the effective potential in the RS-II model. In the thin brane scenario, the general solution of the massive KK modes is the Hankel function. So the QNMs can be analytically obtained by imposing the outgoing boundary condition to the Hankel function Seahra:2005wk . But there is no any analytical solution of massive KK modes for this thick brane. So we use some semi-analytical method to obtain the QNMs of the thick brane. Unlike the case of a black hole, there is a potential well but not a pure barrier for our brane case. Therefore, some methods of solving the QNMs commonly used in black holes, such as the Wentzel-Kramers-Brillouin (WKB) approximation Konoplya:2003ii , cannot solve the QNMs of this thick brane directly. But we notice that the Schrödinger-like equation (22) can be factorized as a super-symmetric form

$$QQ^{\dagger}\phi(z)=m^{2}\phi(z),$$

(26)

where $Q$ and $Q^{\dagger}$ are

$$Q=\partial_{z}+\frac{3}{2}\partial_{z}A,~{}~{}~{}~{}~{}~{}~{}~{}Q^{\dagger}=-\partial_{z}+\frac{3}{2}\partial_{z}A.$$

(27)

The above equation (26) has a corresponding Schrödinger-like equation with the super-symmetric partner potential:

$\displaystyle Q^{\dagger}Q\tilde{\phi}(z)=\left(-\partial_{z}^{2}+U^{dual}(z)\right)\tilde{\phi}(z)=m^{2}\tilde{\phi}(z),$

(28)

where

$\displaystyle U^{\text{dual}}(z)=-\frac{3}{2}\partial_{z}^{2}A+\frac{9}{4}(\partial_{z}A)^{2}=\frac{3k^{2}\left(k^{2}z^{2}+2\right)}{4\left(k^{2}z^{2}+1\right)^{2}}.$

(29)

According to the super-symmetric quantum mechanics, the Schrödinger-like equations (22) and (28) have the same spectrum of massive KK modes Cooper:1994eh . So the effective potential and the super-symmetric partner potential have the same spectrum of QNMs Ge:2018vjq . Plot of the super-symmetric partner potential (29) is shown in Fig. 1(b). We can see that the shape of the dual potential is similar to the effective potentials in the case of the Schwarzschild black hole, for which the QNMs can be solved by the asymptotic iteration method Ciftci:2003As ; ciftci:2005co ; Cho:2011sf and the WKB approximation. Therefore, we can obtain the quasinormal frequencies of the thick brane by using the dual potential (29).

In this paper, we investigated the QNMs of the thick brane model by the semi-analytical and numerical methods. The results obtained by these methods are in good agreement with each other. It shows that there is a zero mode (normal mode) and a series of discrete QNMs in the thick brane model. This is consistent with the results of the RS-II brane Seahra:2005iq . As the characteristic modes of the thick brane, these QNMs play an indispensable role on understanding the structure of the thick brane. This is a new way for the investigation of the gravitational perturbation in thick brane models. It also may provide new ideas for studying thick brane models.

Starting from the solution and the linear metric tensor perturbation given in Ref. Gremm:1999pj , we obtained the wave equation (19) and the Schrödinger-like equation (22). Since the Schrödinger-like equation can be factorized as a super-symmetric form, we can obtain the super-symmetric partner potential which provides the same spectrum of QNMs of the brane. The super-symmetric partner potential is similar to the effective potentials in the case of the Schwarzschild black hole. Some semi-analytical methods can be used to solve the QNMs. In this way, the QNMs of the thick brane were obtained indirectly. We used the asymptotic iteration method and the WKB approximation to solve the QNMs. The results of the two methods agree with each other in the low overtone region, which can be seen from Table 1. To further confirm the above results, we studied the numerical evolution of the wave equation (19). The results show that a zero mode is excited by the incident Gaussian pulse. And the evolution of the odd wave packet reveals the property of the QNMs, which can be seen from Fig. 5. In addition, the frequency extracted from the data is consistent with the frequency of the first QNM obtained using the asymptotic iteration method and the WKB approximation. This enhances the credibility of our results. Finally, we investigated the propagation distance $d_{n}$ of the massive mode on the brane. We found that, for the same $m_{n}$, the distance $d_{n}$ increases with the parameter $a$. If the propagation distance is of the galactic scale, the frequency of the massive mode is extremely high, far beyond the ability of the current detectors. However, the massive mode might play a key role in the early universe. It might be detected as a stochastic gravitational wave background.

Our work could be strengthened in a number of ways. First,we need to develop more methods to calculate higher overtone modes and compare with the asymptotic iteration method. Second, some thick brane models might support long-lived QNMs, which deserve further study. Third, the QNMs of other test fields could be investigated in the future.

We are thankful to J. Chen, C.-C. Zhu for useful discussions. This work was supported by the National Key Research and Development Program of China (Grant No. 2020YFC2201503), the National Natural Science Foundation of China (Grants No. 11875151, No. 12147166, and No. 12047501), the 111 Project under (Grant No. B20063), the China Postdoctoral Science Foundation (Grant No. 2021M701529), and “Lanzhou City’s scientific research funding subsidy to Lanzhou University”.