First-order formalism and thick branes in mimetic gravity
aa r X i v : . [ g r- q c ] F e b First-order formalism and thick branes in mimetic gravity
Qun-Ying Xie a,b , Qi-Ming Fu c , Tao-Tao Sui b , Li Zhao b , Yi Zhong d, ∗ a School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, P. R. China b Lanzhou Center for Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, P. R. China c Institute of Physics, Shaanxi University of Technology, Hanzhong 723000, P. R. China d Hunan Provincial Key Laboratory of High-Energy Scale Physics and Applications, Hunan University, Changsha 410082, P. R. China
Abstract
In this paper, we investigate thick branes generated by a scalar field in mimetic gravity theory. By introducing twoauxiliary super-potentials, we transform the second-order field equations of the system into a set of first-order equa-tions. With this first-order formalism, several types of analytical thick brane solutions are obtained. Then, tensor andscalar perturbations are analysed. We find that both kinds of perturbations are stable. The e ff ective potentials for thetensor and scalar perturbations are dual to each other. The tensor zero mode can be localized on the brane while thescalar zero mode cannot. Thus, the four-dimensional Newtonian potential can be recovered on the brane. Keywords: T hick brane , Mimetic graivty
1. Introduction
Modified gravity theories have obtained great devel-opment and performance in the study of some unsolvedproblems in general relativity such as the dark energyproblem, the dark matter problem, the singularity prob-lem, etc. By isolating the conformal degree of freedomof general relativity, Chamseddine and Mukhanov pro-posed a theory called mimetic gravity [1]. This theorywas studied from the view of variational principle [2]. Itwas shown that after introducing the scalar potential thisconformal degree of freedom becomes dynamical andcan mimic cold dark matter [1, 3] or dark energy [4–7]and can resolve the singularity problem [8] and the cos-mic coincidence problem [9]. The extension of mimeticgravity was used to investigate the inflationary solu-tion [10]. In Ref. [11], the authors proposed mimeticEinstein-Cartan gravity and proved that torsion is a non-propagating field in this mimetic gravity. Mimetic grav-ity theory was also extended to Horava-like theory andapplied to galactic rotation curves [12]. It was also ap-plied to other gravity theories such as f ( R ) gravity [13–19], Horndeski gravity [20–22] and Gauss-Bonnet grav-ity [23, 24]. ∗ The corresponding author.
Email addresses: [email protected] (Qun-Ying Xie), [email protected] (Qi-Ming Fu), [email protected] (Tao-Tao Sui), [email protected] (Li Zhao), [email protected] (Yi Zhong )
On the other hand, in order to solve the gauge hi-erarchy problem and the cosmological constant prob-lem, Randall and Sundrum (RS) proposed that our four-dimensional world could be a brane embedded in five-dimensional space-time [25]. With the warped extra di-mension, it was further found that the size of extra di-mension can be infinitely large without conflicting withNewtonian gravitational law [26]. This charming ideahas attracted substantial researches in particle physics,cosmology, gravity theory, and other related fields [27–36].Recently, one of the interesting works appeared inRef. [37], which applied mimetic gravity theory to thethin RSII brane model [26]. It was shown that themimetic scalar field can mimic the dark sectors on thebrane and explain the late time cosmic expansion in thefavor of observational data, and it has the capability toexplain initial time cosmological inflation [37]. Later,other related topics about thick branes in mimetic grav-ity were studied in Refs. [38–41]. Thick branes withthe inner structure and the stability of the perturbationswere first investigated in Ref. [40]. Besides, it is knownthat first-order formalism is a very powerful tool to ob-tain analytical brane solutions [42–44]. With this for-malism, the second-order coupled field equations can bewritten as a set of first-order ones by introducing one ormore auxiliary super-potentials. Very recently, Bazeiaet al. used first-order formalism to find brane solutions
Preprint submitted to Physics Letters B February 23, 2021 n mimetic gravity [41, 45]. In this paper, we would liketo investigate thick branes in mimetic gravity with thehelp of first-order formalism by including two super-potentials. In order to show the systematicness and ef-fectiveness of the first-order formalism on finding ana-lytical brane solutions, we will utilize polynomial, pe-riod, and mixed super-potentials. We will also investi-gate the stabilities of the tensor and scalar perturbationsas well as the relationship between the localizations ofthe tensor and scalar zero modes.The paper is organized as follows. In Sec. 2, we give areview of the thick brane model and reduce the second-order field equations to the first-order ones by introduc-ing two auxiliary super-potentials. In Sec. 3, we obtainthree types of analytical brane solutions by consider-ing di ff erent forms of super-potentials. In Sec. 4 andSec. 5, we focus on tensor and scalar perturbations, re-spectively. Finally, the conclusion and discussion aregiven in Sec. 6.
2. First-order formalism for thick brane models
We consider thick branes generated by a scalar fieldin five-dimensional mimetic gravity. The correspondingaction is given by S = Z d xdy √− g (cid:18) R + L φ (cid:19) , (1)where R is the five-dimensional scalar curvature, the La-grangian of the mimetic scalar field φ is L φ = λ h g MN ∂ M φ∂ N φ − U ( φ ) i − V ( φ ) . (2)Here λ represents the Lagrange multiplier, U ( φ ) and V ( φ ) are two potentials. In this paper, x M and x µ de-note respectively the five-dimensional bulk coordinatesand the four-dimensional brane ones, where the indices M , N , · · · = , , , , µ, ν, · · · = , , , g MN , the scalar field φ , and the La-grange multiplier λ lead to the following field equations G MN + λ∂ M φ∂ N φ − L φ g MN = , (3)2 λ (cid:3) (5) φ + ∇ M λ ∇ M φ + λ ∂ U ∂φ + ∂ V ∂φ = , (4) g MN ∂ M φ∂ N φ − U ( φ ) = . (5)Here the five-dimensional d’Alembert operator is de-fined as (cid:3) (5) = g MN ∇ M ∇ N .The line-element ansatz for a flat brane is given by ds = e A ( y ) η µν dx µ dx ν + dy (6) = a ( y ) η µν dx µ dx ν + dy . (7) Using the line-element (6), and considering that thescalar field is static and depends only on the extra-dimensional coordinate, we can get the followingsecond-order nonlinear coupled di ff erential field equa-tions 6 A ′ + λ (cid:16) U ( φ ) + φ ′ (cid:17) + V ( φ ) = , (8)6 A ′ + A ′′ + λ (cid:16) U ( φ ) − φ ′ (cid:17) + V ( φ ) = , (9) λ A ′ φ ′ + φ ′′ + ∂ U ∂φ ! + λ ′ φ ′ + ∂ V ∂φ = , (10) φ ′ − U ( φ ) = . (11)Here, the primes denote derivatives with respect to theextra-dimensional coordinate y . It can be seen that itis not easy to analytically solve the above second-orderfield equations directly. However, one can reduce themto first-order field equations by introducing two auxil-iary super-potentials [41] Q = Q ( φ ) , W = W ( φ ) , (12)and providing the potential V ( φ ) = Q φ W φ − W , (13)where Q φ = dQd φ and W φ = dWd φ . The resulting first-orderfield equations can be written as A ′ = − W ( φ ) , (14) φ ′ = Q φ , (15) U ( φ ) = Q φ , (16) λ ( φ ) = − W φ Q φ . (17)These equations would be helpful to give analyticalbrane solutions. One can see from (16) that the scalarpotential U ( φ ) is related with the super-potential Q ( φ ),which is a function of the scalar field φ . The other po-tential V ( φ ) is determined by the two super-potentials Q and W (see Eq. (13)).Note that the first-order equations (14)-(16) can bedivided into two groups. One is Eq. (14) related with A and W , the other is Eqs. (15) and (16) related with φ, Q , U . Therefore, these equations can be solvedwith di ff erent approaches by giving di ff erent combina-tions from ( A , W ) and ( φ, Q , U ). For example, we canchoose W ( φ ) and Q ( φ ), or W ( φ ) and φ ( y ).The energy density of the brane is given by ρ ( y ) = T MN u M u N , (18)where u M = ( u , , , ,
0) is the velocity of a static ob-server. From the condition of the velocity g MN u M u N =
1, we have u = e − A . Thus, the energy density can bewritten as ρ ( y ) = V ( φ ( y )) , (19)which shows that the potential V ( φ ) and the profile ofthe scalar φ ( y ) determine the distribution of the thickbrane along the extra dimension.On the other hand, in order to localize gravity on thebrane, the warp factor e A ( y ) should tend to zero rapidlyenough as y → ±∞ , such that the condition R e A dy < ∞ is satisfied. This will be derived in section 4. Usually,we consider the solutions with e A ( y ) | y →±∞ → e − k | y | ,which corresponds to the branes embedded in an AdSspacetime with a negative cosmological constant. Itshould be pointed out that the contribution of the cos-mological constant has been included in the energy den-sity (19) for this case. Therefore, the net energy densityshould be ρ ( y ) = V ( φ ( y )) − Λ . (20)
3. The thick brane solutions
Next, we will take focus on finding some specific an-alytical solutions of the thick brane model by solvingthe first-order equations (14)-(17). Our main motivationhere is to show the systematicness and e ff ectiveness ofthe first-order formalism, which is also called the super-potential method [34, 35, 46]. First, supposing that one of the super-potential Q hasthe following polynomial form considered in Ref. [41]: Q ( φ ) = k v φ − φ v ! , (21)and solving Eq. (15), we can easily get the solution ofthe scalar field φφ ( y ) = v tanh( ky ) . (22)The solution of the potential U can be read directly fromEq. (16) as U ( φ ) = k v (cid:16) φ − v (cid:17) . (23)The other super-potential W is chosen as W ( φ ) = knv φ, (24) where n is a non-vanishing parameter. Then, fromEq. (14), we obtain a simple solution for the warp factor A ( y ) = ln sech n ( ky ) . (25)The other functions are given by λ ( y ) = − n v cosh ( ky ) , (26) V ( φ ) = nk v h v − (2 n + φ i . (27)The energy density of the brane including the cosmo-logical constant is ρ ( Λ ) = V = k n (cid:16) − (2 n +
1) tanh ( ky ) (cid:17) . (28)From the solution (25), we can calculate the cosmolog-ical constant and hence the net energy density ρ ( y ) = ρ ( Λ ) − Λ = k n (2 n + ( ky ) . (29)The shapes of the warp factor a ( y ) = e A ( y ) and the energydensity ρ ( y ) are plotted in Fig. 1, which shows that theparameter n a ff ects the warp factor and the energy den-sity. With the increase of n , the warp factor becomesnarrower while the energy density becomes larger andnarrower. The maximum of the energy density is givenby ρ max = k n (2 n +
1) for n > n < − /
2. It isobvious that the parameter v does not a ff ect the warpfactor and the energy density, it only a ff ects the ampli-tude of the scalar field φ and hence the localization of abulk fermion Ψ when one introduces the Yukawa cou-pling η ¯ Ψ φ Ψ [47]. In fact, v is the vacuum expectationvalue of the scalar potential U ( φ ) given in (23). - - ky e A (a) The warp factor - - ky Ρ (b) The energy densityFigure 1: The shapes of the warp factor a ( y ) and the energy density ρ ( y ) of the first brane model. The parameter n is set as n = , , Next, we also consider the same polynomial super-potential Q as in previous subsection Q ( φ ) = k v φ − φ v ! . (30)3herefore, we will get the same φ and U ( φ ) as solutionI: φ ( y ) = v tanh( ky ) , (31) U ( φ ) = k v ( φ − v ) . (32)But di ff erent from last model, we fix W ( φ ) = Q ( φ ),which results in the constant solution of the Lagrangemultiplier λ ( y ) = − W φ Q φ = − . (33)Then we get a di ff erent form of the warp factor via Eq.(14) A ( y ) = v
18 sech ( ky ) + v ky ) . (34)The scalar potential V has the following φ form V ( φ ) = k v (cid:16) φ − v (cid:17) − k v (cid:16) φ − v (cid:17) φ . (35)In this case, the energy density of the brane is ρ ( y ) = k v (cid:16) + v + + v )cosh(2 ky ) (cid:17) × sech ( ky ) . (36)In this model, the vacuum expectation value v of thescalar potential U ( φ ) has an explicit e ff ect on the warpfactor and the energy density. The shapes of the scalarfield, the warp factor, and the energy density do notchange with non-vanishing v . Note that A ( | y | → ∞ ) → v (cid:16) e − k | y | − k | y | (cid:17) → − v k | y | . (37)Therefore, the five-dimensional spacetime is alsoasymptotic AdS. Next, we try to construct another form of brane solu-tion by giving a period super-potential, such as W ( φ ) = kn sin p (cid:16) φ v (cid:17) . (38)At the same time, the warp factor A ( y ) is assumed to be A ( y ) = ln sech n ( ky ) . (39)Then the scalar field can be solved from Eq. (14) as φ ( y ) = v arcsin[tanh p ( ky )] , (40) which is a kink and a double kink for p = p = n + n , respectively. And fromEq. (15) we know that the super-potential Q is Q ( φ ) = kp v (cid:2) F ( p , φ ) + F ( − p , φ ) (cid:3) . (41)where F ( p , φ ) = ( − p sec p (cid:16) φ v (cid:17) × F p , p ; 1 + p ; sec (cid:16) φ v (cid:17)! . Here F is the hypergeometric function. The Lagrangemultiplier and the two potentials can also be solved: λ ( y ) = n cot (cid:16) φ v (cid:17) sin p (cid:16) φ v (cid:17) p v (cid:20) sin p (cid:16) φ v (cid:17) − (cid:21) , (42) V ( φ ) = k n (cid:20) − (1 + n ) sin p (cid:16) φ v (cid:17)(cid:21) , (43) U ( φ ) = k p v sech (cid:16) φ v (cid:17) (44) × (cid:20) sin + p (cid:16) φ v (cid:17) − sin − p (cid:16) φ v (cid:17)(cid:21) . (45)The energy density in this case is given by ρ ( y ) = k n (2 n + ( ky ) , (46)which is the same as Eq. (29) for the first model. Infact, from the definition of the energy density ρ = T MN u M u N = (1 / G MN u M u N , we know that the energydensity will have the same configuration for the samewarp factor. Here, the warp factors in this model andthe first model have the same form, and hence so do theenergy densities. Similarly, for the period super-potentials Q ( φ ) = W ( φ ) = kv sin (cid:16) φ v (cid:17) , (47)we can obtain the following solution A ( y ) = v (cid:2) sech( ky ) (cid:3) , (48) φ ( y ) = v arctan (cid:2) sinh( ky ) (cid:3) , (49) λ ( y ) = − , (50) V ( φ ) = k v cos (cid:18) φ v (cid:19) − v sin (cid:18) φ v (cid:19)! , (51) U ( φ ) = k v cos φ v , (52) ρ ( y ) = k v (2 v + ( ky ) . (53)4 .2.3. Solution V Next, we consider di ff erent period super-potentials W ( φ ) and Q ( φ ): W ( φ ) = kn tan (cid:16) φ v (cid:17) , (54) Q ( φ ) = kv sin (cid:16) φ v (cid:17) , (55)The warp factor and the scalar field will have the fol-lowing explicit forms A ( y ) = ln (cid:2) sech n ( ky ) (cid:3) , (56) φ ( y ) = v arctan[tanh( ky )] . (57)The other functions are given by λ ( y ) = − n v sec[2 arctan tanh( ky )] , × (1 + tanh ( ky )) , (58) V ( φ ) = k n sec (cid:18) φ v (cid:19) " (1 + n ) cos φ v ! − n , (59) U ( φ ) = k v cos (cid:16) φ v (cid:17) . (60) Finally, we would like to generate the brane model bygiving the pair of super-potentials Q ( φ ) and W ( φ ) as themixed of polynomial and period, for example Q ( φ ) = W ( φ ) = kv h φ + v sin (cid:16) φ v (cid:17)i , (61)which results in the constant Lagrange multiplier λ ( y ) = − . (62)Then governed by Eq. (15), the scalar field φ is deter-mined as φ ( y ) = v arctan( ky ) , (63)from which one can see that the asymptotic behavior of φ is lim y →±∞ φ ( y ) = π . The potentials U and V are U ( φ ) = k v cos (cid:16) φ v (cid:17) , (64) V ( φ ) = k v " (cid:16) φ v (cid:17) − (cid:16) φ + v sin (cid:16) φ v (cid:17)(cid:17) . (65)The warp factor and the energy density read as A ( y ) = − v ky arctan( ky ) , (66) ρ ( y ) = k v " (cid:0) k y + (cid:1) + π v − v (cid:16) ky ) + sin (2 arctan( ky )) (cid:17) . (67) The asymptotic behavior of A ( y ) is A ( | y | → ±∞ →− π v k | y | /
4. Tensor perturbation
In this section, we consider the linear tensor fluctu-ation of the metric around the background. Followingthe previous research works in Refs. [33, 40, 41], weperform the following coordinate transformation dz = e − A ( y ) dy , (68)to get a conformally flat metric ds = e A ( z ) ( η µν dx µ dx ν + dz ) . (69)To simplify the fluctuations of the metric around thebackground, we only consider the transverse and trace-less part of the metric fluctuation, i.e., we consider thefollowing tensor perturbation of the metric: ds = (cid:16) e A ( z ) η µν + ˆ h µν ( x , z ) (cid:17) dx µ dx ν + e A ( z ) dz = e A ( z ) h(cid:16) η µν + h µν ( x , z ) (cid:17) dx µ dx ν + dz i . (70)Here h µν is the tensor perturbation of the metric, andit satisfies the transverse and traceless conditions [34]: h µµ = ∂ ν h µν =
0. The non-vanishing part of the pertur-bation of Einstein tensor in Eq. (3) is the µν components(since h = δ G =
0) and it reads δ G µν = − (cid:3) (4) h µν + (6 A ′ + A ′′ ) e A h µν − A ′ e A h ′ µν − e A h ′′ µν , (71)where the four-dimensional d’Alembertian is defined as (cid:3) (4) ≡ η µν ∂ µ ∂ ν . Using Eq. (10), we get the perturbationequation − (cid:3) (4) h µν − A ′ e A h ′ µν − e A h ′′ µν = . (72)Considering Eq. (68), we rewrite the above equation un-der the coordinate z as (cid:3) (4) h µν + A ˙ h µν + ¨ h µν = , (73)where the dot represents the derivative with respect tothe coordinate z . By performing the following decom-position h µν ( x , z ) = ε µν e ikx h ( z ) e − A , (74)Eq. (73) leads to a kind of Schr¨odinger equation (cid:16) − ∂ z + V T ( z ) (cid:17) h ( z ) = m h ( z ) , (75)5here k = − m with m the four-dimensional mass of agraviton KK mode, and the e ff ective potential given by V T ( z ) =
32 ¨ A ( z ) +
94 ˙ A ( z ) . (76)The e ff ective potential of the tensor perturbation underthe physical coordinate y is V T ( z ( y )) = e A ( y ) A ′′ ( y ) + A ′ ( y ) ! . (77)The zero mode of the tensor perturbation reads as h ( z ) = e A ( z ) / c + c Z e − A ( z ) dz ! . (78)This general form of the zero mode was first found in f ( R )-brane model in Ref. [48]. One can see that thee ff ective potentials and the zero mode of the tensor per-turbation are only determined by the warp factor A ( y ).It is easy to show that the tensor zero mode (78) can belocalized on the brane with the choice of c = n > v , n and v increase, the potential wells in Figs. 2 and 3become narrower and deeper, respectively. - - ky - V T (cid:144) k (a) - - ky h (b)Figure 2: The e ff ective potential V T and the non-normalized zeromode of the tensor perturbation for brane models I, III and V. Theparameter is set as n = n = n = It is easy to verify that the zero modes for the abovebrane models are localized around the brane. So thefour-dimensional Newtonian potential can be realizedon the brane. There is no tensor tachyon mode, thus thebrane is stable against the tensor perturbation. - - ky - V T (cid:144) k (a) - - ky h (b)Figure 3: The e ff ective potential V T and the non-normalized zeromode of the tensor perturbation for brane model II. The parameteris set as v = v = . v =
5. Scalar perturbation
At last, we come to the scalar perturbation in this sec-tion. The perturbed metric is given by ds = e A ( z ) h (1 + ψ ) η µν dx µ dx ν + (1 + Φ ) dz i . (79)The scalar perturbation equations can be derived as e A U φφ − U φ ¨ φ ˙ φ + A ˙ φ U φ − e − A λ ˙ φ ! δφ + e A ˙ φ U φ + ¨ φ ˙ φ − A ! ˙ δφ − ¨ δφ = , (80)and Φ = ˙ δφ ˙ φ − e A φ U φ δφ, (81) Φ = − ψ. (82)It can be seen that there is only one degree of freedomfor the scalar perturbation. Finally, by using the back-ground equations (3)-(5), we can replace the potential U and λ in Eq. (80) with the functions A and φ and obtainthe final form of the scalar perturbation δφ : A ¨ φ ˙ φ − A − φ ˙ φ + ... φ ˙ φ ! δφ + φ ˙ φ − A ! ˙ δφ − ¨ δφ = . (83)Redefining δφ ( x µ , z ) = δφ ( x µ ) s ( z ) A − / ( z ) ˙ φ ( z ) , (84)with the four-dimensional part of δφ satisfying (cid:3) (4) δφ ( x µ ) =
0, we get the perturbation equation forthe scalar degree of freedom s ( z ) from Eq. (83): (cid:16) − ∂ z + V S ( z ) (cid:17) s ( z ) = , (85)6here the e ff ective potential is given by V S ( z ) = −
32 ¨ A ( z ) +
94 ˙ A ( z ) (86)in the z coordinate or V S ( z ( y )) = − e A A ′′ ( y ) + A ′ ( y ) ! (87)in the y coordinate. Note that the corresponding e ff ec-tive potential of (86) in Ref. [40] is not right since thereis an error in the matter equation (45) in that paper (theterm + λ ( ∂ z φ ) should be − λ ( ∂ z φ ) ).Comparing (86) with (76), one can see that the e ff ec-tive potential of the scalar perturbation V S is dual to thatof the tensor perturbation V T : V S = V T ( A → − A ), andEqs. (85) and (75) can be rewritten as P † P s ( z ) = , (88) PP † h ( z ) = , (89)where P = ∂ z + ˙ A . The above two equations ensurethat both the scalar and tensor perturbations are stable.The zero mode can be obtained by replacing A → − A from the solution of the tensor zero mode (78): s ( z ) = e − A ( z ) / c + c Z e A ( z ) dz ! . (90)This will lead to the conclusion that only one of the ten-sor and scalar zero modes can be localized on the brane.The e ff ective potential (87) for solution I is given by V S = k n sech ( ky ) (cid:18) + n − n cosh(2 ky ) (cid:19) , (91)which is shown in Fig. 4. From this figure, it can beseen that, the value of the e ff ective potential at z = n . This can be checkedfrom the expression V S (0) = k n . The potential has twovery shallow wells, and approaches 0 − when y → ±∞ . - - ky V S (cid:144) k Figure 4: The e ff ective potential V S ( z ( y )) for solution I. The parameteris set as n = n = n = For the brane solution I in (25) with n >
0, it is easyto show that the zero mode (90) of the scalar perturba-tion cannot be localized on the brane. Thus, there is no additional fifth force coming from the scalar perturba-tion. For other brane solutions, we have also the sameconclusion.
6. Conclusion
In this work, we investigated the super-potentialmethod with which the second-order equations can bereduced to the first-order ones for thick brane mod-els in modified gravity with Lagrange multiplier. Themain step of this method is to introduce a pair of aux-iliary super-potentials, i.e., W ( φ ) and Q ( φ ). With thesetwo super-potentials, the field equations are rewritten asEqs. (13)-(17). Then we try to use the method to finda series of analytical brane solutions via some poly-nomial super-potentials, period super-potentials, andmixed super-potentials. The warp factor has the sameshape and the same asymptotic behavior at the bound-ary of the extra dimension for all those solutions, and allof these branes are embedded in five-dimensional AdSspacetime. The scalar field φ is a double kink for (40)with odd integer p ≥ ff ect thelocalization properties of fermions on the brane throughthe Yukawa coupling η ¯ Ψ φ Ψ [47].We also considered the tensor and scalar perturba-tions of the brane system. It was shown that bothequations of motion of the perturbations can be trans-formed into Schr¨odinger-like equations. Furthermore,these equations can be recast as the forms of (88) and(89), which show that both the perturbations are stable.The e ff ective potential of the tensor perturbation is dualto that of the scalar perturbation. Therefore, only oneof the tensor and scalar zero modes can be localizedon the brane. For all of our brane solutions, the ten-sor zero mode can be localized on the brane while thescalar zero mode cannot. Thus, the four-dimensionalNewtonian potential can be recovered on the brane andthere is no additional fifth force contradicting with theexperiments. Acknowledgement
This work was supported by the National Natural Sci-ence Foundation of China (Grant No. 11875151, No.12047501, and No. 11705070). Yi Zhong was sup-ported by the Fundamental Research Funds for the Cen-tral Universities (Grant No. 531107051196).7 eferences [1] A. H. Chamseddine and V. Mukhanov,
Mimetic Dark Matter , JHEP (2013)135, [ arXiv:1308.5410 ].[2] A. Golovnev, On the recently proposed Mimetic Dark Matter , Phys. Lett.
B 728 (2014) 39-40, [ arXiv:1310.2790 ].[3] A. H. Chamseddine, V. Mukhanov and A. Vikman,
Cosmology with Mimetic Matter , JCAP arXiv:1403.3961 ].[4] A. Casalino, M. Rinaldi, L. Sebastiani, and S. Vagnozzi,
Mimicking dark matter and dark energy in a mimetic modelcompatible with GW170817 , Phys. Dark Univ.
22 (2018) 108,[ arXiv:1803.02620 ].[5] J. Matsumoto,
Unified description of dark energy and dark mat-ter in mimetic matter model , [ arXiv:1610.07847 ].[6] S. Nojiri, S. D. Odintsov and V. K. Oikonomou,
Unimodular-Mimetic Cosmology , Class. Quant. Grav.
33 (2016) 125017,[ arXiv:1601.07057 ].[7] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou,
ViableMimetic Completion of Unified Inflation-Dark Energy Evolu-tion in Modified Gravity , Phys. Rev.
D 94 (2016) 10, 104050[ arXiv:1608.07806 ].[8] S. Brahma, A. Golovnev, and D. H. Yeoma,
On singularity-resolution in mimetic gravity , Phys. Lett.
B 782 (2018) 280,[ arXiv:1803.03955 ].[9] J. Dutta, W. Khyllep, E. N. Saridakis, N. Tamanini,and S. Vagnozzi,
Cosmological dynamics of mimetic gravity , JCAP arXiv:1711.07290 ].[10] S. A. H. Mansoori, A. Talebian, and H. Firouzjahi,
MimeticInflation , [ arXiv:2010.13495 ].[11] F. Izaurieta, P. Medina, N. Merino, P. Salgado, and O. Val-divia,
Mimetic Einstein-Cartan-Kibble-Sciama (ECKS) gravity ,[ arXiv:2007.07226 ].[12] L. Sebastiani, S. Vagnozzi and R. Myrza-kulov, Mimetic gravity: a review of recent devel-opments and applications to cosmology and astro-physics , Adv. High Energy Phys. arXiv:1612.08661 ].[13] K. Nozari, and N. Sadeghnezhad,
Braneworld mimetic f(R)gravity , Int. J. Geom. Meth. Mod. Phys.
16 (2019) 03, 1950042,[ arXiv:1612.08661 ].[14] D. Momeni, R. Myrzakulov and E. G¨udekli,
Cosmolog-ical viable mimetic f ( R ) and f ( R , T ) theories via Noethersymmetry , Int. J. Geom. Meth. Mod. Phys.
12 (2015) 1550101,[ arXiv:1502.00977 ].[15] S. Nojiri and S. D. Odintsov,
Mimetic f(R)gravity: Inflation, dark energy and bounce , Modern Physics Letters A
29 (dec, 2014) 1450211,[ arXiv:1408.3561 ].[16] S. D. Odintsov and V. K. Oikonomou,
Mimetic F ( R ) in-flation confronted with Planck and BICEP2 / Keck Array data , Astrophys. Space Sci.
361 (2016) 174, [ arXiv:1512.09275 ].[17] S. D. Odintsov and V. K. Oikonomou,
Viable MimeticF ( R ) Gravity Compatible with Planck Observations , Annals Phys.
363 (2015) 503–514, [ arXiv:1508.07488 ].[18] S. D. Odintsov and V. K. Oikonomou,
Accelerat-ing cosmologies and the phase structure of F(R) gravitywith Lagrange multiplier constraints: A mimetic approach , Phys. Rev.
D 93 (2016) 023517, [ arXiv:1511.04559 ].[19] V. K. Oikonomou,
Reissner-Nordstr¨om Anti-de SitterBlack Holes in Mimetic F ( R ) Gravity , Universe arXiv:1511.09117 ].[20] F. Arroja, T. Okumura, N. Bartolo, P. Karmakar, and S.Matarrese,
Large-scale structure in mimetic Horndeski gravity , JCAP arXiv:1708.01850 ]. [21] F. Arroja, N. Bartolo, P. Karmakar, and S. Matar-rese,
Cosmological perturbations in mimetic Horndeski gravity , JCAP
04 (2016) 042, [ arXiv:1512.09374 ].[22] G. Cognola, R. Myrzakulov, L. Sebastiani, S. Vagnozziand S. Zerbini,
Class. Quant. Grav.
33 (2016) 225014 . arXiv:1601.00102 .[23] S. Capozziello, A. N. Makarenko, and S. D. Odintsov,
Gauss-Bonnet dark energy by Lagrange multipliers , Phys. Rev.
D 87 (2013) 084037, [ arXiv:1302.0093 ].[24] A. V. Astashenok, S. D. Odintsov and V. K.Oikonomou,
Modified Gauss-Bonnet gravity with theLagrange multiplier constraint as mimetic theory , Class. Quant. Grav.
32 (2015) 185007, [ arXiv:1504.04861 ].[25] L. Randall and R. Sundrum,
A Large mass hierarchy froma small extra dimension , Phys. Rev. Lett.
83 (1999) 3370–3373,[ arXiv:hep-ph/9905221 ].[26] L. Randall and R. Sundrum,
An Alternative to com-pactification , Phys. Rev. Lett.
83 (1999) 4690–4693,[ arXiv:hep-th/9906064 ].[27] H. Davoudiasl, J. L. Hewett and T. G. Rizzo,
Bulk gauge fieldsin the Randall-Sundrum model , Phys. Lett.
B 473 (2000) 43–49,[ arXiv:hep-ph/9911262 ].[28] T. Shiromizu, K. i. Maeda and M. Sasaki,
The Einsteinequation on the 3-brane world , Phys. Rev.
D 62 (2000) 024012,[ arXiv:gr-qc/9910076 ].[29] T. Gherghetta and A. Pomarol,
Bulk fields and supersym-metry in a slice of AdS , Nucl. Phys.
B 586 (2000) 141–162,[ arXiv:hep-ph/0003129 ].[30] T. G. Rizzo,
Introduction to Extra Dimensions , AIP Conf. Proc. arXiv:1003.1698 ].[31] K. Yang, Y.-X. Liu, Y. Zhong, X.-L. Du and S.-W. Wei,
Grav-ity localization and mass hierarchy in scalar-tensor branes , Phys. Rev.
D 86 (2012) 127502, [ arXiv:1212.2735 ].[32] K. Agashe, A. Azatov, Y. Cui, L. Randall and M. Son,
Warped Dipole Completed, with a Tower of Higgs Bosons , JHEP
06 (2015) 196, [ arXiv:1412.6468 ].[33] C. Csaki, J. Erlich, T. J. Hollowood and Y.Shirman,
Universal aspects of gravity localizedon thick branes , Nucl. Phys.
B 581 (2000) 309–338,[ arXiv:hep-th/0001033 ].[34] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A.Karch,
Modeling the fifth-dimension with scalars and gravity , Phys. Rev.
D 62 (2000) 046008, [ arXiv:hep-th/9909134 ].[35] M. Gremm,
Four-dimensional gravity on a thickdomain wall , Phys. Lett.
B 478 (2000) 434–438,[ arXiv:hep-th/9912060 ].[36] Y.-X. Liu,
Introduction to Extra Dimensions and ThickBraneworlds , arXiv:1707.08541 .[37] N. Sadeghnezhad and K. Nozari, Braneworld Mimetic Cosmol-ogy , Phys. Lett.
B 769 (2017) 134–140, [ arXiv:1703.06269 ].[38] Y. Zhong, Y. Zhong, Y.-P. Zhang, W.-D. Guo, Y.-X. Liu,
Gravitational resonances in mimetic thick branes , JHEP
04 (2019) 154, [ arXiv:1812.06453 ].[39] Q. Xiang, Y. Zhong, Q.-Y. Xie, L. zhao,
Flat andbent branes with inner structure in two-field mimetic gravity ,[ arXiv:2011.10266 ].[40] Y. Zhong, Y. Zhong, Y.-P. Zhang, Y.-X. Liu, Thick branes withinner structure in mimetic gravity , Eur. Phys. J.
C 78 (2018) 45,[ arXiv:1711.09413 ].[41] D. Bazeia, D. A. Ferreira, D. C. Moreira,
First order formalismfor thick branes in modified gravity with Lagrange multiplier , EPL
129 (2020) 11004, [ arXiv:2002.00229 ].[42] V.I. Afonso, D. Bazeia, and L. Losano,
First-order for-malism for bent brane , Phys.Lett.
B 634 (2006) 526-530,[ arXiv:hep-th/0601069 ].
43] B. Janssen, P. Smyth, T. V. Riet, and B. Vercnocke,
A first-order formalism for timelike and spacelike brane solutions , JHEP
007 (2008) 0804, [ arXiv:0712.2808 ].[44] R. Menezes,
First Order Formalism for Thick Branes inModified Teleparallel Gravity , Phys. Rev.
D 89 (2014) 125007,[ arXiv:1403.5587 ].[45] D. Bazeia, D. A. Ferreira, F. S. N. Lobo, and J. L. Rosa,
Novelmodified gravity braneworld configurations with a Lagrangemultiplier , arXiv:2011.06240 .[46] D. Bazeia and A. R. Gomes, Bloch brane , JHEP
05 (2004) 012, [ arXiv:hep-th/0403141 ].[47] Y.-X. Liu, J. Yang, Z.-H. Zhao, C.-E. Fu, and Y.-S. Duan,
Fermion Localization and Resonances on A de Sitter ThickBrane , Phys. Rev.
D 80 (2009) 065019, [ arXiv:0904.1785 ].[48] Z.-Q. Cui, Z.-C. Lin, J.-J. Wan, Y.-X. Liu, and L. Zhao,
Ten-sor Perturbations and Thick Branes in Higher-dimensional f ( R ) Gravity , JHEP
12 (2020) 130, [ arXiv:2009.00512 ].].