Gravitational resonances in mimetic thick branes

TL;DR

This paper studies gravitational resonances in mimetic thick branes, revealing how the structure and parameters of the branes affect the localization and spectrum of gravitational modes, which could help detect extra dimensions.

Contribution

It introduces new methods to analyze gravitational resonances in both single and double mimetic thick branes, including different definitions of relative probability and their effects on resonance spectra.

Findings

Resonance spectra are similar across different probability definitions.

Distance and thickness of sub-branes influence resonance localization.

New features in resonance behavior are identified based on brane parameters.

Abstract

In this work, we investigate gravitational resonances in both single and double mimetic thick branes, which can provide a new way to detect the extra dimension. For the single brane model, we apply the relative probability proposed in [Phys. Rev. D. 80 (2009) 065019]. For the double brane model, we investigate the resonances quasi-localized on the double brane, on the sub-branes and between the sub-branes, respectively. To investigate the resonances quasi-localized on the double brane, we introduce two different definitions of the relative probability and find that the corresponding mass spectra of gravitational resonances are almost the same. For the gravitational resonances quasi-localized on sub-branes and between the sub-branes, the influence of the distance between the two sub-branes and the thickness of the sub-branes are analyzed and new features are found in both cases.

Tables

Table 1. Table 1: The mass spectrum m n 2 superscript subscript 𝑚 𝑛 2 m_{n}^{2} , relative probability P m ​ a ​ x subscript 𝑃 𝑚 𝑎 𝑥 P_{max} , FWHM Γ Γ \Gamma and life-time τ 𝜏 \tau of the resonances for the single brane model ( 22 ).

$n$	Parity	$m_n^2$	$P_{max}$	$\Gamma$	$\tau$
1	odd	$0.1385$	$0.9902$	$0.002418$	$413.50$
2	even	$0.4609$	$0.9367$	$0.007880$	$126.90$
3	odd	$0.9711$	$0.7609$	$0.019633$	$50.93$
4	even	$1.6428$	$0.4852$	$0.047884$	$20.884$
5	odd	$2.4317$	$0.2817$	$0.107172$	$9.3308$

Table 2. Table 2: The mass spectrum m n 2 superscript subscript 𝑚 𝑛 2 m_{n}^{2} , relative probability P m ​ a ​ x subscript 𝑃 𝑚 𝑎 𝑥 P_{max} , FWHM Γ Γ \Gamma and life-time τ 𝜏 \tau of the resonances quasi-localized on the double brane described by ( 29 ).

Parity	$m_n^2\left(P_1\right)$	$m_n^2\left(P_2\right)$	$\Delta m_n^2$	$\Gamma$	$\tau$	$P_{max}$
even	$0.1606$	$0.1605$	$1\times {10}^{-4}$	$1.248\times {10}^{-4}$	$8015$	0.9978
odd	$0.3907$	$0.3908$	$1\times {10}^{-4}$	$3.760\times {10}^{-4}$	$2660$	0.9982
even	$0.6870$	$0.6872$	$2\times {10}^{-4}$	$9.170\times {10}^{-4}$	$1091$	0.9964
odd	$1.0445$	$1.0445$	$0.0000$	$2.153\times {10}^{-3}$	$464.6$	0.9880
even	$1.4555$	$1.4565$	$1.0\times {10}^{-3}$	$9.597\times {10}^{-3}$	$104.2$	0.02071
odd	$1.9032$	$1.9021$	$1.1\times {10}^{-3}$	$1.305\times {10}^{-2}$	$76.64$	0.7916
even	$2.3424$	$2.3411$	$1.3\times {10}^{-3}$	$4.177\times {10}^{-2}$	$23.94$	0.4164
odd	$2.6949$	$2.6802$	$1.47\times {10}^{-2}$	$0.1384$	$7.223$	0.1762

Table 3. Table 3: The mass spectrum m n 2 superscript subscript 𝑚 𝑛 2 m_{n}^{2} , relative probability P m ​ a ​ x subscript 𝑃 𝑚 𝑎 𝑥 P_{max} , FWHM Γ Γ \Gamma and life-time τ 𝜏 \tau of a part of the resonances quasi-localized on the sub-branes with different brane distance d 𝑑 d for the double brane model ( 29 ).

$d$	Parity	$m_n^2$	$P_{max}$	$\Gamma$	$\tau$
	even	$0.944$	$0.7310$	$0.02726$	$36.68$
1.6	odd	$0.972$	$0.7072$	$0.02763$	$36.20$
	odd	$3.002$	$0.2680$	$0.1227$	$8.149$
	even	$3.345$	$0.2206$	$0.1385$	$7.220$
	odd	$0.943$	$0.7078$	$0.03034$	$32.96$
2.3	even	$0.965$	$0.6955$	$0.03076$	$32.51$
	odd	$2.903$	$0.2507$	$0.06870$	$14.56$
	even	$3.083$	$0.2622$	$0.07261$	$13.77$
	even	$0.963$	$0.6317$	$0.02644$	$37.82$
2.6	odd	$0.942$	$0.6385$	$0.02630$	$38.02$
	odd	$2.945$	$0.1947$	$0.1877$	$5.330$
	even	$3.073$	$0.1945$	$0.1863$	$5.368$

Table 4. Table 4: The mass spectrum m n 2 superscript subscript 𝑚 𝑛 2 m_{n}^{2} , relative probability P m ​ a ​ x subscript 𝑃 𝑚 𝑎 𝑥 P_{max} , FWHM Γ Γ \Gamma and life-time τ 𝜏 \tau of a part of the resonances quasi-localized on sub-branes with different brane thickness b 𝑏 b for the double brane model ( 29 ).

$b$	Parity	$m_n^2$	$P_{max}$	$\Gamma$	$\tau$
	even	$1.726$	$0.444$	$0.06316$	$15.83$
5	odd	$1.628$	$0.347$	$0.09062$	$11.04$
	odd	$0.5744$	$0.835$	$0.01445$	$69.22$
9	even	$0.5794$	$0.867$	$0.01504$	$66.50$
	even	$0.2689$	$0.950$	$0.004628$	$216.1$
13	odd	$0.2683$	$0.952$	$0.004633$	$215.8$

Table 5. Table 5: The mass spectrum m n 2 superscript subscript 𝑚 𝑛 2 m_{n}^{2} , relative probability P m ​ a ​ x subscript 𝑃 𝑚 𝑎 𝑥 P_{max} , FWHM Γ Γ \Gamma and life-time τ 𝜏 \tau of a part of the resonances quasi-localized between the sub-branes with different brane distance for the double brane model ( 29 ).

$d$	Parity	$m_n^2$	$P_{max}$	$\Gamma$	$\tau$
	even	$1.235$	$0.7185$	$0.02655$	$37.66$
1.2	odd	$2.438$	$0.5590$	$0.03195$	$31.30$
	odd	$0.459$	$0.990$	$0.000594$	$1693$
1.5	even	$1.131$	$0.653$	$0.009404$	$106.3$
	even	$0.683$	$0.9765$	$0.001452$	$688.8$
1.8	odd	$0.391$	$0.9979$	$0.0003200$	$3125$

Equations

$$S=\int d^{5}x\sqrt{-g}\left(\frac{R}{2}+\lambda \left[{\partial }^M\varphi {\partial }_N\varphi -U(\varphi )\right]-V(\varphi )\right),$$

$$G_{MN}+2\lambda {\partial }_M\varphi {\partial }_N\varphi -L_{\varphi }{g}_{MN}=0,$$

$$2\lambda {\square }^{(5)}\varphi +2{\nabla }_M\lambda {\nabla }^M\varphi +\lambda \frac{\partial U}{\partial \varphi }+\frac{\partial V}{\partial \varphi }=0,$$

$$g^{MN}{\partial }_M\varphi {\partial }_N\varphi -U(\varphi )=0.$$

$$d{s}^2=a^2(y){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{y}^2,$$

$$\frac{3a^{\prime 2}}{a^2}+\frac{3a^{\prime \prime }}{a}+V(\varphi )+\lambda \left(U(\varphi )-{\varphi }^{\prime 2}\right)$$

$$\frac{6a^{\prime 2}}{a^2}+V(\varphi )+2\lambda \left(U(\varphi )+{\varphi }^{\prime 2}\right)$$

$$\lambda \left(\frac{8a^{\prime }{\varphi }^{\prime }}{a}+2{\varphi }^{\prime \prime }+\frac{\partial U}{\partial \varphi }\right)+2{\lambda }^{\prime }{\varphi }^{\prime }+\frac{\partial V}{\partial \varphi }$$

$${\varphi }^{\prime 2}-U(\varphi )$$

$$d{s}^2=a(z{)}^2[({\eta }_{\mu \nu }+h_{\mu \nu })d{x}^{\mu }d{x}^{\nu }+d{z}^2],$$

$${\square }^{(4)}{\tilde{h}}_{\mu \nu }+{\partial }_z^2{\tilde{h}}_{\mu \nu }-\frac{{\partial }_z^2{a}^{\frac{3}{2}}}{a^{\frac{3}{2}}}{\tilde{h}}_{\mu \nu }=0.$$

$$-{\partial }_z^{2}t(z)+V_{\text{t}}(z)t(z)=m^{2}t(z),$$

$$V_{\text{t}}(z)=\frac{{\partial }_z^2{a}^{\frac{3}{2}}}{a^{\frac{3}{2}}},$$

$$t_0(z)\propto a^{\frac{3}{2}}(z).$$

$$\left(-{\partial }_z+{\partial }_z\ln a^{\frac{3}{2}}\right)\left({\partial }_z+{\partial }_z\ln a^{\frac{3}{2}}\right)t(z)=m_t^{2}t(z).$$

$$d{s}^2=a^2(z)\left[(1+2\psi ){\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+(1+2\Phi )d{z}^2\right],$$

$$-\frac{3}{2}{\partial }_z^2\delta \varphi +\frac{3}{4}\left(\frac{a^2}{{\partial }_z\varphi }\frac{\partial U}{\partial \varphi }+\frac{2{\partial }_z^2\varphi }{{\partial }_z\varphi }-\frac{4{\partial }_{z}a}{a}\right){\partial }_z\delta \varphi$$

$$+\left[\frac{3a{\partial }_{z}a}{{\partial }_z\varphi }\frac{\partial U}{\partial \varphi }+2\lambda ({\partial }_z\varphi {)}^2+\frac{3}{4}{a}^2\left(\frac{{\partial }^{2}U}{\partial {\varphi }^2}-2\frac{\partial U}{\partial \varphi }\frac{{\partial }_z^2\varphi }{({\partial }_z\varphi {)}^2}\right)\right]\delta \varphi =0,$$

$$\Phi =-2\psi ,$$

$$\Phi =\frac{{\partial }_z\delta \varphi }{{\partial }_z\varphi }-\frac{a^2}{2({\partial }_z\varphi {)}^2}\frac{\partial U}{\partial \varphi }\delta \varphi .$$

$$-{\partial }_z^{2}s(z)+V_{\text{s}}(z)s(z)=0,$$

$$V_{\text{s}}(z)=\frac{2({\partial }_{z}a{)}^2-a{\partial }_z^{2}a}{a^2}+\frac{-({\partial }_z^2\varphi {)}^2+2{\partial }_z\varphi {\partial }_z^3\varphi }{4({\partial }_z\varphi {)}^2}.$$

$$a(y)=\text{tanh}[k(y+b)]-\text{tanh}[k(y-b)].$$

$$t_{\text{e}}(0)=1,{\partial }_z{t}_{\text{e}}(0)=0;$$

$$t_{\text{o}}(0)=0,{\partial }_z{t}_{\text{o}}(0)=1.$$

$$P(m^2)=\frac{{\int }_{-z_b}^{z_b}|t(z){|}^{2}dz}{{\int }_{-z_{max}}^{z_{max}}|t(z){|}^{2}dz}.$$

$$a(y)$$

$$P_1$$

$$P_2$$

$$a(y)$$

$$P_3=\{\begin{array}{ll}\frac{{\int }_{z_1}^{z_2}|t(z){|}^{2}dz}{{\int }_{z_m-5(z_2-z_1)}^{z_m+5(z_2-z_1)}|t(z){|}^{2}dz},& z_m\ge 5(z_2-z_1)\\ & \\ \frac{{\int }_{z_1}^{z_2}|t(z){|}^{2}dz}{{\int }_0^{10(z_2-z_1)}|t(z){|}^{2}dz},& z_m<5(z_2-z_1)\end{array}$$

$$P_4$$

$$U(r)\sim G_N\frac{M}{r}\left[1+{\int }_0^{\infty }dm{\text{e}}^{-mr}{t}_m^2(z_0)\right],$$

$$U(r)\sim G_N\frac{M}{r}\left[1+\frac{C}{(kr{)}^2}\right],$$

Full text

1 Introduction

The braneworld scenario has attracted much attention since the renowned Arkani-Hamed-Dimopoulos-Dvali (ADD) model and Randall-Sundrum (RS) model were proposed ArkaniHamed:1998rs ; Randall:1999ee ; Randall:1999vf . It is possible to solve the hierarchy problem and the cosmological constant problem in the braneworld scenario ArkaniHamed:1998rs ; Randall:1999ee ; Randall:1999vf ; Kim:2000mc . Both the ADD model and RS model are thin brane models. Later, various thick brane models were investigated DeWolfe:1999cp ; Gremm:1999pj ; Csaki:2000fc ; Kobayashi:2001jd ; Giovannini:2001fh ; Bazeia:2003aw ; Bazeia:2004dh ; Gremm:2000dj ; Bazeia:2002sd ; Afonso:2006gi ; Dzhunushaliev:2008gk ; Neupane:2010ey and the localization of matter fields on the brane was realized. In the braneworld scenario, our four-dimensional universe is an infinitely thin brane or a domain wall embedded in a higher dimensional space-time. The Standard Model fields are localized on the brane Bajc:1999mh ; Ringeval:2001cq ; Bagger:2004rr ; Liu:2007ku ; Liu:2008wd ; Ghoroku:2001zu ; Kehagias:2000au , while the gravity propagates in all dimensions. In order not to contradict the present observations, the zero mode of the tensor perturbation of gravity should be localized on the brane and recover the four-dimensional Newtonian potential Randall:1999vf ; Csaki:2000fc . In many types of brane models, the extra part of the tensor perturbation satisfies a Schrödinger-like equation, and the effective potential may support resonance modes Csaki:2000fc ; Csaki:2000pp ; Gremm:1999pj ; Cruz:2013uwa ; Xie:2013rka ; Xu:2014jda ; Yu:2015wma . Thus, apart from the localized zero mode, the quasi-localized modes, i.e. the resonance modes may exist and contribute correction to the four-dimensional Newtonian potential Csaki:2000fc ; Csaki:2000pp ; Gremm:1999pj ; Cruz:2013uwa ; Xu:2014jda , which can provide a new way to detect the extra dimension. Therefore, the investigation of gravitational resonances is an important topic in braneworld models. Gravitational resonances also appear in other systems, e.g., the quasi-normal modes outside of black holes Nollert:1999ji . For a recent review, see Ref. Liu:2017gcn .

Since it is widely believed that general relativity should be modified, which is inspired by both the theoretical motivation and cosmological observation data Nollert:1999ji ; Sotiriou:2008rp ; Nojiri:2006ri ; Nojiri:2017ncd , braneworld models in modified gravities were investigated extensively Kastor:2017knv ; Bazeia:2017mjd ; Afonso:2007zz ; Guo:2014bxa ; German:2013sk ; Arias:2002ew ; Zhong:2015pta ; Yang:2012hu ; daSilva:2017jbx ; Yang:2017evd ; Guo:2015qbt ; Guo:2018tpo ; Karam:2018squ . New features such as inner structure of branes and pure geometrical branes were found Arias:2002ew ; Zhong:2015pta ; Yang:2012hu . Recently the mimetic gravity was proposed to solve the dark matter problem Chamseddine:2013kea ; Chamseddine:2014vna . In this theory, the physical metric $g_{\mu\nu}$ is defined in terms of an auxiliary metric $\hat{g}_{\mu\nu}$ and a scalar field $\phi$ as $g_{\mu\nu}=-\hat{g}_{\mu\nu}\hat{g}^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi$. In the framework of mimetic gravity, the geometrical explanation of dark matter was given on galaxy level, cluster level and cosmological evolution and perturbation level Dutta:2017fjw ; Matsumoto:2015wja ; Myrzakulov:2015kda ; Vagnozzi:2017ilo . It is also possible to unify the inflation and late-time acceleration period in this theory Nojiri:2014zqa . For more recent works of mimetic gravity in cosmology see Refs. Odintsov:2015ocy ; Odintsov:2016imq ; Odintsov:2016oyz ; Nojiri:2016vhu ; Odintsov:2018ggm . The Friedmann-Robertson-Walker thin brane was considered and the late time cosmic expansion was explained in the favor of observational data, and the initial time cosmological inflation was also produced Sadeghnezhad:2017hmr . Later, thick branes with inner structure generated by mimetic scalar field were found in Ref. Zhong:2017uhn . The gravitational perturbation was analyzed in detail. It was found that the tensor zero mode is localized on the branes. For specific parameters, the branes split into multi sub-branes, and the effective potential of the tensor perturbation also splits into multi-wells, which is different from the usual braneworld case. Unlike the zero mode of the tensor perturbation which is localized on the sub-brane, the resonance modes can be quasi-localized on or between the sub-branes. This is an important new feature in the mimetic thick brane. Inspired by this we would like to study the resonances of the tensor perturbation in these brane models. We will introduce alternative definitions of relative probability and compare the corresponding mass spectra of gravitational resonances. Then we will analyze how the structure of the brane impacts on the gravitational resonances quasi-localized on different locations of the double brane.

The organization of this paper is as follows. In Sec. 2, we briefly introduce the mimetic thick brane model and the tensor perturbation of gravity. In Sec. 3, we investigate the gravitational resonances in both single and double mimetic brane models. In Sec. 4, we will discuss the contribution of the resonances to the four-dimensional Newtonian potential and give a conclusion.

2 Linear perturbation in a mimetic thick brane model

In this section, we briefly introduce the mimetic thick brane model and the linear perturbation of the brane system, which were given in Ref. Zhong:2017uhn in detail. We take the geometrized units in which the gravitational constant $\kappa^{2}=1$. The action of the five-dimensional mimetic gravity is

[TABLE]

where $\lambda$ is a Lagrange multiplier. Throughout this paper, the indices $M,N\cdots=0,1,2,3,5$ denote the bulk coordinates and $\mu,\nu\cdots=0,1,2,3$ denote the ones on the brane.

In the original mimetic gravity, $U(\phi)=-1$ , and $\phi$ represents the conformal degree of the metric $g_{MN}$ Chamseddine:2013kea . This theory was extended to $U(\phi)<0$ for cosmological application Astashenok2015 . In this work, the mimetic scalar field $\phi=\phi(y)$ is used to generate the thick brane. Therefore, we assume that $U(\phi)=g^{MN}\partial_{M}\phi\partial_{N}\phi>0$ Zhong:2017uhn . This generalization can also provide thick branes with inner structure. The equations of motion can be easily obtained by varying the action with respect to the physical metric $g_{MN}$, the mimetic scalar field $\phi$ and the Lagrange multiplier $\lambda$ Zhong:2017uhn :

[TABLE]

where $L_{\phi}=\lambda\left[g^{MN}\partial_{M}\phi\partial_{N}\phi-U(\phi)\right]-V(\phi)$ is the lagrangian of the mimetic scalar field. Assuming the Minkowski brane metric

[TABLE]

the equations of motion (2)-(4) read

[TABLE]

where the primes denote the derivatives with respect to the extra-dimensional coordinate $y$. Since there are only three independent equations in Eqs. (6)-(9) and five independent variables, we can easily solve $\lambda(y)$, $V(\phi)$ and $U(\phi)$ for any given $a(y)$ and $\phi(y)$. As we will see later, the equation of the tensor perturbation depends only on the warped factor. Therefore, in the next section we will only give the warped factor $a(y)$ and omit the expression of $\phi(y)$, $\lambda(y)$, $V(\phi)$ and $U(\phi)$ for the brane models. When the mimetic scalar $\phi(y)$ is set to be constant, the potentials $V(\phi)$ and $U(\phi)$ and the Lagrange multiplier $\lambda(y)$ also become constants, and the theory turns to general relativity. In this work we will focus on the double brane models, in which $\phi(y)$ can not be a constant.

Next, we consider the linear perturbation of the brane system. It is easy to see that the scalar, vector and tensor modes of the pertubation are decoupled with each other. Thus, we will investigate the tensor and scalar perturbation separately. Redefining the extra-dimensional coordinate $dz=\frac{1}{a(y)}dy$, the perturbed metric in the new coordinate is given by

[TABLE]

where the tensor perturbation $h_{\mu\nu}=h_{\mu\nu}(x^{\mu},z)$ depends on all the coordinates and satisfies the transverse-traceless (TT) condition $\eta^{\mu\nu}\partial_{\mu}h_{\lambda\nu}=0$ and $\eta^{\mu\nu}h_{\mu\nu}=0$. Next we redefine the perturbation as $h_{\mu\nu}=a(z)^{-\frac{3}{2}}\tilde{h}_{\mu\nu}$. After tedious but straightforward derivation, the perturbation of Eq. (2) yields

[TABLE]

Employing the decomposition $\tilde{h}_{\mu\nu}=\epsilon_{\mu\nu}(x^{\gamma})\text{e}^{ip_{\lambda}x^{\lambda}}t(z)$ with the polarization tensor $\epsilon_{\mu\nu}$ satisfying the TT condition $\eta^{\mu\nu}\partial_{\mu}\epsilon_{\lambda\nu}=0$ and $\eta^{\mu\nu}\epsilon_{\mu\nu}=0$, one can obtain the Schrödinger-like equation for the extra part $t(z)$ of the tensor perturbation:

[TABLE]

where the effective potential $V_{\text{t}}(z)$ is given by

[TABLE]

and $m$ is the mass of the tensor perturbation $t(z)$. The zero mode of the tensor perturbation is

[TABLE]

Obviously, the tensor perturbation in mimetic gravity is the same as that in general relativity. Furthermore, Eq. (12) can be factorized as

[TABLE]

The structure in the above equation ($\mathcal{K}\mathcal{K}^{\dagger}$ with $\mathcal{K}=-\partial_{z}+\partial_{z}\ln a^{\frac{3}{2}}$) ensures that the eigenvalues are non-negative and so there is no tensor tachyon mode with $m^{2}<0$. Thus, the brane is stable against the tensor perturbation Zhong:2017uhn . The tensor zero mode is localized around the thick brane embedding in an AdS5 space-time. Nevertheless, the mimetic scalar field generates more types of thick brane, which could lead to new type of potential of the tensor perturbation. Thus, one can expect new phenomena in the resonances of the tensor perturbation.

At last, we turn to the scalar perturbation. The perturbed metric is

[TABLE]

and the perturbed scalar field is $\phi+\delta\phi$. The field equations of the scalar perturbation $\delta\phi$ and the scalar modes $\Phi$ and $\psi$ are

[TABLE]

Redefining the perturbation of the scalar field as $\delta\phi(x^{\mu},z)=\frac{(\partial_{z}\phi)^{\frac{3}{2}}}{a^{2}}s(z)\overline{\delta\phi}(x^{\mu})$, we can obtain the equation of the extra part of the scalar perturbation:

[TABLE]

where the effective potential $V_{\text{s}}(z)$ is given by

[TABLE]

Since there is no term of the form $\Box^{4}\delta\phi$ in Eq. (17), the scalar perturbation does not propagate on the brane. Furthermore, for the following brane models, we will plot the shape of the potential $V_{\text{s}}(z)$ to show that the scalar perturbation is not localized on the brane, and thus does not contribute to the four-dimensional Newton potential. Therefore, though it seems strange that there is no term of the form $\Box^{4}\delta\phi$ in Eq. (17), it does not lead to any problems. Similarly, in the cosmological context, the scalar perturbation of mimetic gravity has also no terms of $\nabla^{2}\delta\phi$ (see Eq. (64) of Ref. Chamseddine:2014vna ) , which implies that the sound speed is identically zero.

3 Gravitational resonance in various thick brane models

In the above section, it was pointed out that the zero mode of the tensor perturbation is localized around the brane embedding in an AdS5 space-time. In this section, we will investigate quasi-localized modes, i.e. the gravitational resonances, in both single and double brane models.

3.1 Gravitational resonances in a single-brane model

First of all, as a simple example, we study a single brane model with the following warped factor Zhong:2017uhn

[TABLE]

The shapes of the warped factor and the corresponding effective potentials (13) and (21) are plotted in Fig. 1. It can be seen that the potential $V_{\text{t}}(z)$ has an obvious double-well with two barriers, which is the main reason leading to resonance KK modes. The potential $V_{\text{s}}(z)$ approaches $0^{-}$ at infinity, therefore the scalar mode is not localized on the brane and does not contribute to the four-dimensional Newton potential.

Due to the complexity of the function $a(y)$ and the coordinate transformation, we can not obtain the analytical expressions of the warp factor $a(z)$ and the effective potential $V_{\text{t}}(z)$. To solve the Schrödinger-like equation (12) for $t(z)$ numerically, we decompose $t(z)$ into an even parity mode $t_{\text{e}}(z)$ and an odd parity mode $t_{\text{o}}(z)$, which are set to satisfy the following boundary conditions:

[TABLE]

To investigate the gravitational resonances, we adopt the concept of the relative probability of the KK mode $t(z)$ with mass $m$, which was defined in Ref. Liu:2009ve ; Almeida:2009jc :

[TABLE]

Here $2z_{b}$ is approximately the width of the thick brane, and $z_{max}=10z_{b}$. Note that there are other methods that can find out KK resonances, such as the transfer matrix method DuZhao2013 ; Landim1105.5573 .

For a given $m^{2}$, the Schrödinger-like equation (12) can be solved numerically for the even parity mode $t_{\text{e}}(z)$ and the odd parity mode $t_{\text{o}}(z)$ with the conditions (23) and (24), respectively. Then the relative probability $P$ corresponding to this $t_{\text{e}}(z)$ or $t_{\text{o}}(z)$ can be obtained. By this means, the relative probability as a function of $m^{2}$ is obtained and plotted in Fig. 2, in which each of the peaks represents a resonance mode.

Furthermore, the corresponding life-time $\tau$ of the resonances can obtained by $\tau=\frac{1}{\Gamma}$, where $\Gamma$ is the full width at half maximum (FWHM) Gregory:2000 ; Almeida:2009jc . The resonances having large life-time can be quasi-localized on the brane for a long time. Therefore, these resonances are approximately four-dimensional gravitons Csaki:2000pp . The mass spectrum, FWHM, and life-times are shown in Table 1. It is shown that the relative probability $P$ and life-time $\tau$ of the resonance modes decrease with the mass square $m^{2}$, while the FWHM $\Gamma$ increases with $m^{2}$. Thus, the behavior of the resonances in the single mimetic brane is similar to that in a single brane model in general relativity Csaki:2000fc ; Xie:2013rka .

The wave functions of the odd and even modes corresponding to the highest peaks in Fig. 2 are plotted in Fig. 3.

3.2 Gravitational resonances quasi-localized on a double-brane

Next we study the gravitational resonances quasi-localized on a double-brane. The warped factor is given by Zhong:2017uhn

[TABLE]

The shapes of the warped factor (26) and the corresponding effective potential (13) are plotted in Fig. 4,

which shows that the effective potential $V_{\text{t}}(z)$ has two sub-wells, and the part between the two sub-wells can also be regarded as a sub-well, and resonances can be quasi-localized on these threes sub-wells. The potential $V_{\text{s}}(z)$ approaches $0^{-}$ at infinity, therefore the scalar mode is not localized on the brane and does not contribute to the four-dimensional Newton potential.

In this subsection, we would like to investigate the resonances quasi-localized on the double brane. Since the effective potential has different structure, we introduce two alternative definitions of the relative probability:

[TABLE]

where $z_{m}=\frac{z_{1}+z_{2}}{2}$, and ($z_{1}$, $z_{2}$) is the $z$-coordinate range of one of the sub-wells (see Fig. 4(c)). Following the same procedure, the relative probabilities $P_{1}$ and $P_{2}$ are plotted in Fig. 5, and the spectra of the resonant modes calculated with the above two definitions of relative probability are listed in Tab. 2. From Tab. 2 we can see that the even and odd parity modes appear alternately. Note that the first even and odd resonance modes are not degenerate. For the two definitions of relative probability, the difference between the spectra $\Delta m_{n}^{2}$ is much less than the mass square $m_{n}^{2}$. Thus, we may draw a conclusion that the two definitions give almost the same spectra of resonance modes. Table 2 shows that the FWHM increases with $m^{2}$, thus the life-time decreases with $m^{2}$. It can be seen that although there is more than one sub-well, the mass spectrum of the resonances is similar to the case of the single brane model in the last subsection. The wave functions of two resonances with mass square $m^{2}=0.1606$ and $m^{2}=0.3907$ are plotted in Fig. 6, which shows that the resonances are indeed quasi-localized on the double brane.

3.3 Gravitational resonances quasi-localized on the sub-branes

From Fig.  4(c) we can see that each sub-brane corresponds to a sub-well, which may support new kinds of gravitational resonances. Therefore, we investigate gravitational resonances quasi-localized on the sub-branes in this subsection. We will analyze the influence of the distance between the two sub-branes and the thickness of the sub-branes. To this end, we consider the following warped factor

[TABLE]

where $2(b+d)$ is approximately the thickness of the brane, and $2d$ is the distance between the two sub-branes in the physical coordinate $y$. The shapes of the warped factor (29) and the corresponding effective potential $V_{\text{t}}(z)$ in this model are similar to those in subsection 3.2. In order to investigate resonance modes which are only quasi-localized on the sub-branes, we define the corresponding relative probability $P_{3}$:

[TABLE]

where $(z_{1}$,$z_{2})$ is the $z$-coordinate range of one of the sub-wells (see Fig. 4(c) as a diagrammatic drawing), and $z_{m}=\frac{z_{1}+z_{2}}{2}$.

Firstly, we fix the parameter $b$ and plot the shape of $P_{3}(m^{2})$ for three values of the parameter $d$ in Fig. 7. The mass spectrum, relative probability, FLHM, and life-time of the resonances in two definitions of relative probability are given in Tab. 3. Note that in Fig. 7 and the figures of the relative probability in the following subsections, only the peaks which satisfy $P>0.1$ and have a FWHM represent the resonances. The relative probability of the resonance modes do not monotonically decrease with the mass square $m^{2}$, and there are only one odd and one even significant resonance modes, which is different from the case in single brane model. As the brane distance $d$ increases, more resonances appear. The mass spectra, FWHMs and life-times of a part of the resonances are shown in Tab. 3. The wave functions $t_{\text{e}}(z)$ and $t_{\text{o}}(z)$ of a part of the resonance modes are shown in Figs. 8-10. For each resonance mode, the amplitude in the sub-wells is larger than the one out of the sub-wells, which shows that the definition (30) of relative probability $P_{3}$ is proper. In Figs. 8-10 the red and blue lines denote even and odd modes, respectively. Nevertheless, Figs. 8(c), 8(d), 9(c), 9(d), 10(c), 10(d) are even modes with respect to the sub-branes, and the others are odd modes. This is crucial to the correction of the four-dimensional Newtonian potential, which will be demonstrated in the next section.

Next, we analyze the influence of the thickness of sub-branes. We fix the parameter $d$ and plot the shapes of the relative probability $P_{3}(m^{2})$ for different values of the brane thickness $b$ in Fig. 11. The mass spectrum, relative probability, FWHM and life-time of a part of the resonances are given in Tab. 4. It is shown that as the sub-brane thickness increases, the mass of the first even and odd modes decrease, while their relative probabilityincrease. Furthermore, for small sub-brane thickness $b=5$, there are only a group of resonances with small relative probability, while for large sub-brane thickness, the resonances with large relative probability appear. The wave functions of the resonances are similar to the ones in Fig.  8(a)-10(d).

Through the above demonstration, it can be seen that the character of the resonances quasi-localized on the sub-branes is quite different from that of the resonances quasi-localized on the double brane and single brane studied before.

3.4 Gravitational resonances quasi-localized between the sub-branes

Figure  4(c) shows that the gravitational resonances could also be quasi-localized between the sub-branes, since sub-well between the sub-branes can support resonances. Therefore, we investigate the gravitational resonances quasi-localized between the sub-branes in this subsection. The warped factor $a(y)$ is also assumed as Eq. (29). The relative probability $P_{4}$ corresponding to the gravitational resonances quasi-localized between the two sub-branes is given by

[TABLE]

where $z_{1}$ is shown in Fig. 4(c). Then we can analyze the influence of the thickness of sub-branes and the distance between the two sub-branes, respectively.

Firstly, we fix the sub-brane thickness and plot the shapes of $P_{3}(m^{2})$ for three values of the sub-brane distance in Fig. 12, and list the information of some resonances in Tab. 5. It can be seen that the number and life-time of the resonances increase with the width $d$ of the middle sub-well, which is similar to the case of a single brane Csaki:2000fc ; Xie:2013rka . While the relative probability of the resonances does not monotonically decrease with the mass square $m^{2}$, which is very different from the case of a single brane Csaki:2000fc ; Xie:2013rka .

Next, we fix the distance $d$ and plot the shapes of $P_{4}(m^{2})$ for different values of the brane thickness $b$ in Fig. 13. It is shown that though the parameter $b$ is related to the sub-wells on the left and right rather than the one in the middle, it has an important impact on the resonance quasi-localized between the sub-branes. With the increasing of the parameter $b$, an even or odd resonance splits into two resonances with small relative probability, and then the two resonances become a single resonance again. While the other resonances are not significantly changed with the parameter $b$, which is different from all the cases above or the case in a single brane model.

4 Conclusion and discussion

In this work, we discussed gravitational resonances quasi-localized on different locations of mimetic branes. Firstly we considered a single brane as an example and used the relative probability in Ref. Liu:2009ve to investigate the gravitational resonances. Then we considered a double brane and investigated resonances quasi-localized on the double brane, on the sub-branes and between the sub-branes, respectively. For the first case, since the effective potential splits into two sub-wells, we introduced two alternative definitions of the relative probability. In each definition, we obtained the spectrum of gravitational resonances and showed that the two spectra are almost the same. We also obtained the FWHM and life-time of the resonances. For the second case, we introduced another definition of relative probability to investigate the gravitational resonances quasi-localized on the sub-branes. The influence of the distance between the two sub-branes and the thickness of the sub-branes was analyzed. We found that more gravitational resonance modes appear with the increasing of the distance between the two sub-branes or the thickness of the sub-branes. Especially, we found some new feature of the resonances. For the third case, we investigated the resonances quasi-localized between the two sub-branes. We showed that the number and life-time of the resonances increase with the distance between the two sub-branes. As the brane thickness increases, an even or odd resonance will split into two resonances with smaller relative probability, and then the two resonances will become a single resonance again.

Finally, we will discuss briefly the contribution of the resonances to the four-dimensional Newtonian potential. For the case that the brane locates at $z=z_{0}$ along the extra dimension, the correction from resonances to the four-dimensional Newtonian potential of a mass point $M$ is given by Csaki:2000fc ; Lykken:1999nb

[TABLE]

where $r$ is the distance to the mass point on the brane, and $t_{m}(z)$ is the normalized wave function of the resonance with mass $m$ and the normalization constant $N_{m}$ is related to $m$. For subsections 3.1, 3.2, and 3.4, $z_{0}=0$, and so only the even modes contribute to the four-dimensional Newtonian potential as $t_{\text{o}}(0)=0$. For subsection 3.3, the resonances are quasi-localized on the sub-branes, therefore $z_{0}=\int^{b+d}_{0}\frac{dy}{a(y)}$, and the resonances shown in Figs. 8(c), 8(d), 9(c), 9(d), 10(c), 10(d) contribute to the four-dimensional Newtonian potential. The resonances which have larger life-time escape to the five-dimension more slowly Csaki:2000pp . On the other hand, it is shown in section 3 that these resonances have larger $t_{m}(z_{0})$. Therefor, the resonances which have larger life-time contribute to the Newtonian potential more than other resonances. The mass spectrum of the resonances show that the mass scale $m$ of the resonances is of about $m\sim k$, where $k$ is the scale on the brane. Usually we assume that $k$ is much less than the Planck scale, therefore $m$ is also much less than the Planck mass. According to the analysis of Refs. Csaki:2000fc ; Lykken:1999nb , the normalization constant $N_{m}$ is decided by the asymptotic behavior of the effective potential $V_{\text{t}}(z)$ in (13) at the boundary of the extra dimension. For the asymptotic AdS5 solutions (22), (26), and (29) considered in this paper, we have $V_{\text{t}}(z)\propto\frac{15}{4z^{2}}$ at $|z|\gg 1/k$ and $t_{m}(0)\sim(m/k)^{1/2}$ for the brane located at $z_{0}=0$, for which the correction to the four-dimensional Newtonian potential is

[TABLE]

where $C$ is a dimensionless constant determined by the structure of the brane. For the case of the double brane, the correction also occurs at the scale of $r\sim 1/k$, but it has a complex form because of the rich structure of the mass spectrum of the resonance KK modes. According the recent test of the gravitational inverse-square law, the usual Newtonian potential still holds down to a length scale at 59$\mu$m TanLuoPRL2016 . Therefore, the thickness of the brane, $1/k$, should be much less than 59$\mu$m.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grants No. 11875151 and No. 11522541) and the Fundamental Research Funds for the Central Universities (Grants No. 531107051196, No. lzujbky-2018-k11, and No. lzujbky-2017-it68).