Internal structure of cuscuton Bloch brane
IInternal structure of cuscuton Bloch brane
D. Bazeia, D.A. Ferreira, and M.A. Marques
3, 1 Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Jo˜ao Pessoa, PB, Brazil Unidade Acadˆemica de F´ısica, Universidade Federal de Campina Grande, 58429-900 Campina Grande, PB, Brazil Departamento de Biotecnologia, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, PB, Brazil (Dated: February 16, 2021)This work deals with thick branes in bulk with a single extra dimension modeled by a two-fieldconfiguration. We first consider the inclusion of the cuscuton to also control the dynamics of oneof the fields and investigate how it contributes to change the internal structure of the configurationin three distinct situations, with the standard, the modified and the asymmetric Bloch brane. Theresults show that the branes get a rich internal structure, with the geometry presenting a novelbehavior which is also governed by the parameter that controls the strength of the cuscuton term.We also study the case where the dynamics of one of the two fields is only described by the cuscuton.All the models support analytical solutions which are stable against fluctuations in the metric, andthe main results unveil significant modifications in the warp factor and energy density of the branes.
I. INTRODUCTION
The study of branes in higher dimensional theoriesemerged with great interest as it gives a procedure tounderstand the hierarchy problem; see Refs. [1–4]. Inparticular, the model first proposed in [3] consisted ofa thin brane and, using scalar fields in the modellingof the extra dimension, it was generalized to describethick branes; see, for instance, Refs. [5–15] and referencestherein. Depending on the profile of the scalar fields, thebraneworld scenario may engender distinct features, suchas the presence of internal structure in the energy den-sity of the brane. This occurs in the case of the Blochbrane [16], when one considers a two-field configurationwhich, in the flat spacetime leads to the so-called Blochdomain wall. Over the years, this model has been studiedin several papers in the literature [17–23].In the standard situation, in braneworld models thesector associated to the scalar field appears in the actionas the difference between kinetic and potential terms.However, this is not the only possibility to model the ex-tra dimension in the braneworld scenario, since we canalso consider generalized models. In this situation, it wasshown in Ref. [24] that one can work within a first orderframework for a class of non-canonical models. Amongthe many possibilities, one may consider, for instance,the inclusion of the cuscuton term, which was firstlyconsidered in Refs. [25–27]. In Ref. [26], in particular,the authors investigated it in the context of cosmologyand showed that its presence does not add dynamical de-grees of freedom. Over the years, several papers dealingwith the cuscuton term appeared in the literature [28–36]. In particular, in Ref. [29], it was also considered inthe tachyacoustic cosmology as an alternative to infla-tion; in Refs. [32, 35], extensions of the cuscuton modelwere investigated in the context of dark energy, where theauthors found cosmological solutions that mimic ΛCDMcosmology. The cuscuton term also finds applications inthe braneworld scenario: in Refs. [30, 33] it was shownthat models with the cuscuton term may support stable branes in standard and bimetric theories.A direct motivation of the present study is due to therecent results of Refs. [34–36], where extended cuscutonis investigated. In particular, in the work [34] the authorsdeal with cosmology in cuscuton gravity to find exactsolution describing an accelerated four-dimensional uni-verse with a stable extra dimension. An important issuerelated to thick branes is that, in the standard scenario,the warp factor of the brane has a bell-shaped profilewhich is hard to be modified. This happens even whenone changes the model to accommodate important mod-ifications, as in the case with the inclusion of generalizedterms [37–39] such as F ( R ) gravity [40–43], the Palatiniformalism [44, 45], and the Gauss-Bonnet [46] contribu-tion already studied. These and other possibilities maycontribute to modify the energy density of the brane, butthe warp factor has in general the standard bell-shapedprofile.Motivated by the possibility to study whether the cus-cuton may modify the internal structure of the Blochbrane, in this paper we investigate the braneworld sce-nario described by a two-field model with the inclusionof the cuscuton term associated to one of the two scalarfields. Since the presence of the cuscuton may respondto change the profile of the warp factor in an importantmanner, and since the equations that govern the systemare of second order, in Sec. II we develop a first orderformalism and investigate how the aforementioned mod-ification changes the profile of the Bloch brane in threedistinct situations. For completeness, the pure cuscutoncase is also studied, and the stability of the braneworldscenarios are also investigated in Sec. III. We then endthe work in Sec. IV, where we include some commentsand suggestions of future work. II. THE MODELS
In this work, we investigate scalar fields in
AdS warped geometry with a single extra dimension of in-finite extent. We follow Refs. [3–5] and write the line a r X i v : . [ h e p - t h ] F e b element as ds = e A η µν dx µ dx ν − dy . (1)In the above expression, e A is the warp factor and A = A ( y ) is the warp function, which depends only on theextra dimension y . One also has µ, ν = 0 , , , η µν = diag(+ , − , − , − ). The five-dimensional metrictensor is g ab = diag( e A , − e A , − e A , − e A , − S = (cid:90) dx dy √ g (cid:18) − R + L (cid:19) , (2)where the Lagrange density L has the form L = 12 ∂ a φ∂ a φ + α (cid:112) | ∂ a φ∂ a φ | + 12 ∂ a χ∂ a χ − V ( φ, χ ) . (3)Here, φ and χ denote the scalar fields and the parameter α is non-negative. We highlight the presence of the cus-cuton term here, whose strength is controlled by α . Thecase α = 0 is well known and has been studied previ-ously in Refs. [16–20, 22]; it is an interesting model thatgives rise to an internal structure in the energy density ofthe brane, depending on the specific choise of the poten-tial V ( φ, χ ). Our purpose here is to investigate how theinclusion of the cuscuton term modifies the braneworldconfiguration; as we can see, this effect is controlled bythe parameter α .By varying the action associated to the Lagrange den-sity (3) with respect to the scalar fields and to the metrictensor, one gets1 √ g (cid:18) α (cid:112) | ∂ a φ∂ a φ | ∂ a φ∂ a φ (cid:19) ∂ a ( √ g ∂ a φ )+ α ∂ a (cid:18) (cid:112) | ∂ a φ∂ a φ | ∂ a φ∂ a φ (cid:19) ∂ a φ + V φ = 0 , (4a)1 √ g ∂ a ( √ g ∂ a χ ) + V χ = 0 , (4b) G ab − T ab = 0 , (4c)where V φ = ∂V /∂φ and V χ = ∂V /∂χ . The Einstein ten-sor is calculated standardly, G ab = R ab − g ab R/
2, where R ab is the Ricci tensor and R is the scalar curvature. Theenergy-momentum tensor that appears in the Einstein’sequation (4c) is given by T ab = (cid:18) α (cid:112) | ∂ a φ∂ a φ | ∂ a φ∂ a φ (cid:19) ∂ a φ∂ b φ + ∂ a χ∂ b χ − g ab L . (5)We follow the usual route and consider that the scalarfields are static, depending only on the extra dimension.In this case, Eqs. (4a) and (4b) become φ (cid:48)(cid:48) + 4 A (cid:48) ( φ (cid:48) − α ) − V φ = 0 , (6a) χ (cid:48)(cid:48) + 4 A (cid:48) χ (cid:48) − V χ = 0 , (6b) where the prime represents derivative with respect to y ,i.e., φ (cid:48) = dφ/dy and χ (cid:48) = dχ/dy . Also, the non-vanishingcomponents of Einstein’s equations (4c) are A (cid:48)(cid:48) = − (cid:16) φ (cid:48) − αφ (cid:48) + χ (cid:48) (cid:17) , (7a) A (cid:48) = 16 (cid:16) φ (cid:48) + χ (cid:48) (cid:17) − V, (7b)where A (cid:48) = dA/dy and A (cid:48)(cid:48) = d A/dy .Solving Eqs. (6) and (7) analytically is a hard task,since they present couplings between the functions in-volved in the model and some of them are of secondorder. To simplify the problem, we follow the lines ofRefs. [5, 24] and implement a first order formalism, whicharise for potentials with the form V ( φ, χ ) = 12 ( W φ + α ) + 12 W χ − W , (8)where W = W ( φ, χ ) is in principle an auxiliary functionwhich depends only on the scalar fields. In this case, weobtain the following first order differential equations: φ (cid:48) = W φ + α, χ (cid:48) = W χ , (9)and A (cid:48) = − W. (10)One can show that the above first order equations arecompatible with the equations of motion (6) and theEinstein’s equations (7). Notice that Eq. (9) does notdepend on the warp function, A ( y ). So, one must firstsolve Eqs. (9) and then use the known scalar field solu-tions, φ ( y ) and χ ( y ), to calculate A ( y ) in Eq. (10). Inthis sense, the scalar fields model the geometry and theparameter α which comes from the cuscuton term playsan important role in the process, since it may modify thefield configuration.The energy density is obtained from the T componentof the energy-momentum tensor in Eq. (5); it is given by ρ ( y ) = e A (cid:18) φ (cid:48) − αφ (cid:48) + 12 χ (cid:48) + V (cid:19) . (11)Here we use the first order equations (9) and (10) torewrite the energy density in the form ρ ( y ) = ( e A W ) (cid:48) . (12)Thus, the parameter α which controls the cuscuton termdoes not invalidate the process of writing the energy den-sity as a total derivative. In this sense, adequate choicesof W ( φ, χ ) that allow us to write e A W → y → ±∞ ,lead to models in which the energy of the brane is null,contributing to its stability. A. Cuscuton Bloch Brane
In order to better understand the role of the cuscutonterm in the model, we consider the function that gives riseto the so-called Bloch brane in the model with α = 0, asinvestigated before in Ref. [16]. It has the form W ( φ, χ ) = φ − φ − rφχ , (13)where the real parameter r is such that r ∈ (0 , / V ( φ, χ ) = 12 (cid:104)(cid:0) α − φ − rχ (cid:1) + 4 r φ χ (cid:105) − (cid:18) φ − φ − rφχ (cid:19) . (14)In addition, the first order equations (9) for the scalarfields are φ (cid:48) = 1 + α − φ − rχ , (15a) χ (cid:48) = − rφχ. (15b)These are first order nonlinear differential equations; theyare coupled but one can combine them to obtain the el-liptical orbit φ + r − r χ = 1 + α. (16)So, we can decouple these first order equations and getthe analytical solutions φ ( y ) = √ α tanh (cid:0) r √ α y (cid:1) , (17a) χ ( y ) = (cid:114) (1 − r )(1 + α ) r sech(2 r √ α y ) . (17b)We can see from these expressions that φ ( ±∞ ) →±√ α and χ ( ±∞ ) →
0. Therefore, these solutionsconnect the minima ( φ, χ ) = ( ±√ α,
0) of the poten-tial in Eq. (14). The profile of these solutions can be seenin Figs. 1 and 2.Since the asymptotic behavior of the scalar potentialdefines the five-dimensional cosmological constant, weobtainΛ ≡ V ( φ ± , χ ± ) = −
43 (1 + α ) (cid:20) − (1 + α )3 (cid:21) . (18)where φ ± = φ ( y → ±∞ ) and χ ± = χ ( y → ±∞ ). For α = 2 we have Λ = 0, showing that the bulk is asymp-totically Minkowski. For α (cid:54) = 2 we have Λ <
0, showingthat the bulk is asymptotically
AdS .As we know the profile of the scalar fields in Eq. (17),we combine them with Eq. (10) to get the following warpfunction A ( y ) = 19 r (cid:2) (1 − r )(1 + α ) tanh (cid:0) r √ α y (cid:1) − (2 − α ) ln cosh (cid:0) r √ α y (cid:1) (cid:3) . (19) Note that for α = 0 we get back the warp function of theBloch brane [16], as expected. From now on we will con-sider the case in which the warp factor vanishes asymp-totically. For this, we must restrict the α parameter tovary in the interval [0 , y = 0, the warp function has the following behavior: A (cid:48) (0) = 0 and A (cid:48)(cid:48) (0) = 4 r (1 + α ) [ α − r (1 + α )] /
3. Thismeans that there are two interesting possibilities, whichwe describe below.1. For r ∈ [1 / , / y = 0, for any value of α ∈ [0 , r ∈ (0 , / α ∈ (2 r/ (1 − r ) , y = 0 becomes a minimum and two sym-metric maxima appear in the warp function. Con-sequently, the warp factor is split (see Fig. 2), re-vealing that the cuscuton Bloch brane engendersan internal structure richer than the usual Blochbrane, which only presents a split in the energydensity. When the values of α are outside theaforementioned interval, it has only a maximum at y = 0.The second possibility with r ∈ (0 , /
3) and α ∈ (2 r/ (1 − r ) ,
2) is an interesting novelty, since the warpfactor in this case has two symmetric maxima, and a lo-cal minimum at the center of the brane. This profile maycontribute to change the way the brane entrap fermionsand other matter fields, an issue that deserves furtherinvestigation.
B. Modified cuscuton Bloch brane
In the above model, we have seen that the two pa-rameters r and α contribute importantly for the internalstructure of the Bloch brane. For this reason, in the newmodel to be explored here we add another parameter, s ,with s ∈ [0 ,
1) and use 1 / (1 − s ) which is defined in theinterval [1 , ∞ ). The modification we introduce is simple,we just change W = W ( φ, χ ) which is defined in Eq. (13)by W/ (1 − s ). This changes the previous solutions (17)to the new ones φ ( y ) = (cid:112) α (1 − s ) tanh ( L s y ) , (20a) χ ( y ) = (cid:114) (1 − r )(1+ α (1 − s )) r sech ( L s y ) , (20b)where we have used L s = 2 r (cid:112) α (1 − s )1 − s . (21) - - - y - - y - - y - - y FIG. 1: From top to bottom, we depict the solutions φ ( y )and χ ( y ) in Eq. (17), the warp factor associated to the warpfunction in Eq. (19) and the energy density (11) correspondingto these solutions for r = 0 . α = 0 . , , .
5. The linethickness and color darkness increase as α increases. This makes the cosmological constant (18) to change tothe new formΛ = −
43 1 + α (1 − s )(1 − s ) (cid:18) −
13 (1 + α (1 − s )) (cid:19) . (22)This result shows that for α = 2 / (1 − s ), the above cos-mological constant vanishes, unveiling that the bulk isasymptotically Minkowski. For α (cid:54) = 2 / (1 − s ), it is neg-ative, resulting in a bulk which is asymptotically AdS . - -
10 0 10 20 - y - -
10 0 10 20024 y - - y - - - y FIG. 2: From top to bottom, we depict the solutions φ ( y )and χ ( y ) in Eq. (17), the warp factor associated to the warpfunction in Eq. (19) and the energy density (11) correspondingto these solutions for r = 0 . α = 0 . , . ,
1. The linethickness and color darkness increase as α increases. With the above solutions, we can write the warp func-tion in the form A ( y ) = 19 r (cid:2) (1 − r )(1 + α (1 − s )) tanh ( L s y ) − (2 − α (1 − s )) ln cosh ( L s y ) (cid:3) . (23)To ensure that the warp factor e A ( y ) vanishes asymptot- - - y - - y FIG. 3: From top to bottom, we depict the warp factor asso-ciated to the warp function in Eq. (23) and the energy density(11) corresponding to these solutions for r = 0 . α = 1,with s = 0 . , . , and 0 .
9. The line thickness and color dark-ness increase as s increases. In the case s = 0 .
9, we displayed ρ/ ically, we have to impose that α < / (1 − s ).The above model engenders interesting features, as wecan see from Figs. 3 and 4, where we depict the warpfactor and energy densities for r = 0 . r = 0 .
1, re-spectively. We notice from Eqs. (20) that the parameter s also modifies the field configurations φ ( y ) and χ ( y ), butin Figs. 3 and 4 we focus only on the warp factor andenergy densities of the branes. Also, in both cases we use α = 1 and s = 0 . , . .
9, to show how the increas-ing of s works to modify the profile of the brane. Weremark, in particular, that the increasing of the parame-ter s seems to have an effect similar to the one uncoveredbefore in [47], since it works to shrink the brane into acompact space, in this way teaching us how to build acompact cuscuton Bloch brane. C. Asymmetric cuscuton Bloch brane
We can also consider the possibility to make the Blochbrane asymmetric, using the procedure described beforein Ref. [15]. This implies in the addition of another realparameter c , changing W to W + c . We will implementthis in the Block brane model included in Sec. II A. Thus,we consider W ( φ, χ ) = φ − φ − rφχ + c . (24) - - y - - - y FIG. 4: From top to bottom, we depict the warp factor asso-ciated to the warp function in Eq. (23) and the energy density(11) corresponding to these solutions for r = 0 . α = 1with s = 0 . , . , .
9. The line thickness and color darknessincrease as s increases. In the case s = 0 .
9, we displayed ρ/ The inclusion of the constant c does not change the solu-tions of the scalar field, by it modifies the warp function,which is now given by A ( y ) = 19 r (cid:2) (1 − r )(1 + α ) tanh (cid:0) r √ α y (cid:1) − (2 − α ) ln cosh (cid:0) r √ α y (cid:1) − c r y (cid:3) , (25)and we have to consider α ∈ [0 , c , the cosmologicalconstant is also changed. Here it is written asΛ = − (cid:20) c ± √ α (cid:18) −
13 (1 + α ) (cid:19)(cid:21) . (26)In this case, we also have two distinct possibilities, onefor c ± = ±√ α (cid:18) −
13 (1 + α ) (cid:19) , (27)which leads the bulk asymptotically AdS from one side,and Minkowski from the other side. The other case is for c in between these two positive and negative values, thatis, c ∈ ( c − , c + ). In this case, the brane is also asymmet-ric, but now connecting two distinct asymptotic AdS geometries. Although in the case of c = c + or c = c − ,the model is not capable of accommodating a normal-izable zero mode (see, e.g., Refs. [48, 49] for more onthis issue of quasilocalization of gravity on a brane), we - - y - - - y FIG. 5: From top to bottom, we depict the warp factor asso-ciated to the warp function in Eq. (25) and the energy density(11) corresponding to these solutions for r = 0 . α = 1,with c = 0 . , . , and 0 .
1. The line thickness and colordarkness increase as c increases. still have room to choose c ∈ ( c − , c + ) to build interestingasymmetric cuscuton Block brane scenarios. We then fo-cus on this possibility and in Fig. 5 we depict the warpfactor and energy density to illustrate how the asymme-try induced by the parameter c contributes to make thebrane asymmetric. There we used r = 0 . α = 1 and c = 0 . , .
05 and 0 . D. Pure Cuscuton
The model in Eq. (3) includes the addition of a cus-cuton term for one of the fields into the usual Lagrangedensity for two real scalar fields. One may also investi-gate the braneworld scenario in which the kinetic termassociated to one of the fields is a pure cuscuton, withoutthe standard quadratic term; the Lagrange density hasnow the form L = (cid:112) | ∂ a φ∂ a φ | + 12 ∂ a χ∂ a χ − V ( φ, χ ) . (28)In this case, the equations of motion associated to thescalar fields are 4 A (cid:48) + V φ = 0 , (29a) χ (cid:48)(cid:48) + 4 A (cid:48) χ (cid:48) − V χ = 0 , (29b) and the Einstein’s equations become A (cid:48)(cid:48) = 23 ( φ (cid:48) − χ (cid:48) ) , (30a) A (cid:48) = 13 (cid:18) χ (cid:48) − V (cid:19) . (30b)In this situation, the energy density is given by ρ ( y ) = e A ( − φ (cid:48) + χ (cid:48) / V ). Notice that the equation of mo-tion for the field φ does not present derivatives of φ asin the previous scenario. We were not able to find a firstorder formalism for this model. In this sense, to find so-lutions we first suppose that scalar fields support kinklikesolutions given by φ ( y ) = λ tanh( σy ) , (31a) χ ( y ) = β arctan[tanh( γy )] , (31b)where λ, σ, β and γ are supposed to be positive real num-bers. These solutions have asymptotic behavior φ ( y →±∞ ) = ± λ and χ ( y → ±∞ ) = ± βπ/
4. From thesesolutions, we have verified that the system supports awell-known result for warp function if σ = 1, β = √ λ and γ = 1 /
2; see Ref. [7]. It is given by A ( y ) = λ ln[sech( y )] . (32)Although the profile of the scalar field χ ( y ) is differ-ent, the warp factor and energy density are similar tothe case presented in Fig. 1; for this reason, we donot depict them in this case. In addition, the poten-tial here is V ( y ) = 5 λ sech ( y ) / − λ tanh ( y ) and thefive-dimensional cosmological constant can be written inthe form Λ ≡ V ( y → ±∞ ) = − λ . (33)This reveals that the brane connects two AdS geometry,as expected. III. STABILITY
In order to study the localization of gravity on thebrane, let us consider that the metric (1) is perturbed inthe form ds = e A ( η µν + h µν ) dx µ dx ν − dy , (34)where h µν = h µν ( x µ , y ). In addition, we consider smallperturbations on the scalar fields such that φ → φ + ξ and χ → χ + ζ with ξ = ξ ( x µ , y ) and ζ = ζ ( x µ , y ).It is convenient to rewrite Einstein’s equations in theform R ab = 2 ˜ T ab with ˜ T ab = T ab − g ab T cc , so the µν -components of the linearized Ricci tensor are R (1) µν = e A (cid:18) ∂ y + 2 A (cid:48) ∂ y + A (cid:48)(cid:48) + 4 A (cid:48) (cid:19) h µν − η λρ ( ∂ µ ∂ ν h λρ − ∂ µ ∂ λ h νρ − ∂ ν ∂ λ h µρ )+ 12 η µν e A A (cid:48) ∂ y ( η λρ h λρ ) − (cid:3) h µν , (35)where (cid:3) = η µν ∂ µ ∂ ν and ∂ y = ∂/∂y . Now we use Eq. (5)to obtain the µν -components of the linearized energy-momentum tensor˜ T (1) µν = − e A (cid:20) η µν (cid:18) ξV φ + ζV χ − αξ (cid:48) (cid:19) + h µν (cid:18) V − αφ (cid:48) (cid:19) (cid:21) . (36)Then, with the help of the equations (7), the linearizedequation R (1) µν = 2 ˜ T (1) µν can be written as e A (cid:18) ∂ y + 2 A (cid:48) ∂ y (cid:19) h µν + 12 η µν e A A (cid:48) ∂ y ( η λρ h λρ ) − (cid:3) h µν − η λρ ( ∂ µ ∂ ν h λρ − ∂ µ ∂ λ h νρ − ∂ ν ∂ λ h µρ )= − e A η µν (cid:18) ξV φ + ζV χ − αξ (cid:48) (cid:19) . (37)We have verified that for the model in Eq. (28), the equa-tions are obtained by taking α = 1 in the two previ-ous equations. We then use transverse traceless gauge( ∂ µ h µν = 0 and η µν h µν = h = 0) to decouple the met-ric fluctuations from the scalars, reducing the linearizedequation to ( ∂ y + 4 A (cid:48) ∂ y − e − A (cid:3) ) h µν = 0 . (38)Introducing the z coordinate with the choice dz = e − A ( y ) dy , and defining h µν = e ip · x e − A ( z ) / H µν ( z ), weget the Schr¨odinger-like equation (cid:18) − d dz + U ( z ) (cid:19) H µν ( z ) = p H µν ( z ) , (39)where U ( z ) = 94 A z + 32 A zz . (40)Here A z and A zz correspond to the first and secondderivative of the warp function with respect to the z vari-able. Note that Eq. (39) can be factorized as S † SH µν ( z ) = p H µν ( z ) , (41)with S = d/dz − A z / S † = − d/dz − A z /
2. Thisfactorization forbids the existence of negative eigenval-ues, showing that the system is stable under small per-turbations of the metric. This factorization works for allthe cases investigated before in Sec. II.Furthermore, the zero mode solution ( p = 0) repre-sents the massless graviton and it is obtained by perform-ing SH (0) µν = 0. So, we get H (0) µν = N µν e A ( z ) / , (42)where N µν is a normalization factor. One can verify thatthe zero modes of most of the models investigated inthis paper are normalizable. Hence, the four-dimensionalgravity can be realized on the brane. IV. CONCLUSION
In this paper, we have studied how the inclusion of thecuscuton term modifies the Bloch brane [17], which arisesin a two-field model. The equations of motion and energydensity of the brane were calculated and the stability ofthe gravity sector was investigated. Since the field pro-files were, in principle, calculated through second orderequations with couplings between the involved functions,we have developed a first order formalism for this model.By considering the auxiliary function in Eq. (13),which is associated to the Bloch brane, we have ob-tained the internal structure in the energy density of themodel, which also arises in the absence of the cuscutonterm. Furthermore, we have found a similar feature inthe warp factor. In this case, the parameter that controlsthe strength of the cuscuton term dictates how deep theinternal structure is: as it increases, this feature becomesmore and more apparent. So, in this sense, the internalstructure of the Bloch brane with the presence of the cus-cuton term is richer than the usual one, as it is present inboth the energy density and the warp factor associatedto the brane. We also have also briefly investigated thecase in which the kinetic term of one of the scalar fieldsis the pure cuscuton. Even though we could not obtaina first order formalism, we have shown that it supportsbranes connecting two
AdS geometries.An important result is that the cuscuton modifies boththe geometry and energy density of the brane, as dis-played in Figs. 2, 3, 4 and 5. In this sense, we can think ofinvestigating fermion localization, since the Bloch branehas internal structure and this may make the localizationmore efficient [18]. The localization of matter field canalso be studied in the pure cuscuton model investigatedin Sec. II D; this is of interest since the profile of the warpfunction in the present case, is the same of the model in-vestigated in Ref. [7], so we can compare results fromdifferent procedures; see, e.g., Ref. [50] and referencestherein. The case of the modified cuscuton Bloch branewhich is controlled by the parameter s is of particularimportance, since it may amplify the modification of thewarp factor and energy density or, else, shrink the braneinto a compact space. Moreover, we can add the effectsof s with the asymmetry described by c together, that is,we can consider a modified asymmetric cuscuton Blochbrane, leading us to a novel modified and asymmetricconfiguration, which is also of current interest.Another possibility is to think of considering the modelrecently studied in Ref. [51] in the presence of the cuscu-ton, to see how the parameter α may modify the geome-try and energy density in this novel model in the presenceof Lagrange multiplier. Moreover, we can also suggestthe inclusion of the cuscuton dynamics in the general-ized hybrid metric-Palatini gravity model investigated inthe very recent work [52]. And yet, we can study issuesrelated to asymmetry and acceleration, in particular themechanism to make the brane asymmetric, as explored,for instance, in [14, 15, 53–56], which can also be used todescribe an accelerating four-dimensional universe with astable extra dimension, in which the cuscuton is respon-sible for the accelerating expansion, as recently suggestedin Ref. [34]. These and other open problems are presentlyunder consideration, and we hope to report on them inthe near future. Acknowledgments
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