Spherically Symmetric Thick Branes Cosmological Evolution
SSpherically Symmetric Thick Branes Cosmological Evolution
A. E. Bernardini ∗ Departamento de F´ısica, Universidade Federal de S˜ao Carlos,PO Box 676, 13565-905, S˜ao Carlos, SP, Brazil
R. T. Cavalcanti † Centro de Ciˆencias Naturais e Humanas,Universidade Federal do ABC, 09210-580, Santo Andr´e, SP, Brazil
Rold˜ao da Rocha ‡ Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao,Universidade Federal do ABC, 09210-580, Santo Andr´e, SP, Brazil andInternational School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy. (Dated: October 16, 2018)Spherically symmetric time-dependent solutions for the 5D system of a scalar field canoni-cally coupled to gravity are obtained and identified as an extension of recent results obtainedby Ahmed, Grzadkowskia and Wudkab [1]. The corresponding cosmology of models with reg-ularized branes generated by such a 5D scalar field scenario is also investigated. It has beenshown that the anisotropic evolution of the warp factor and consequently the Hubble likeparameter are both driven by the radial coordinate on the brane, which leads to an emergentthick brane-world scenario with spherically symmetric time dependent warp factor. Mean-while, the separability of variables depending on fifth dimension, y , which is exhibited by theequations of motion, allows one to recover the extra dimensional profiles obtained in Ref. [1],namely the extra dimensional part of the scale (warp) factor and the scalar field dependenceon y . Therefore, our results are mainly concerned with the time dependence of a sphericallysymmetric warp factor. Besides evincing possibilities for obtaining asymmetric stable brane-world scenarios, the extra dimensional profiles here obtained can also be reduced to thoseones investigated in [1]. PACS numbers: 11.25.-w, 04.50.-h, 04.50.Gh ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ g r- q c ] N ov Brane-world models are a straightforward 5D phenomenological realization of the Hoˇrava-Witten supergravity solutions [2], where the hidden brane is placed at infinity and the moduli effects from compact extra dimensions are neglected [3]. Once introduced in the context of an ef-fective theory of supergravity on domain walls [4], brane-world scenarios are supported by seminalresults [5–8] which are relevant in realizing 4D gravity on a domain wall in 5D space-time [7–9].Brane-world cosmology has also been investigated in several suitable contexts. Classes of exactsolutions with a constant 5D radius on a cosmologically evolving brane were provided in [10],allowing unconventional cosmological equations with the matter content of the brane dominatingthat of the bulk. This framework is in full compliance to standard cosmology, as the present valuesof the Hubble parameter and of the cosmological background radiation temperature fits theirrespective values at the time of nucleosynthesis. Moreover, brane-world cosmology in thin braneshas been studied for any equation of state describing the matter in the brane, where standardcosmological evolution can be obtained after an early non-conventional phase in typical Randall-Sundrum [11] scenarios, where the brane tension compensates the bulk cosmological constant. Theaccelerated Universe could be the result of the gravitational leakage into extra dimensions on Hubbledistances rather than the consequence of non-zero cosmological constant [12]. Some attempts ofdevising the Friedmann law on the brane have involved a dark radiation term due to the bulkWeyl tensor, which depends linearly on the brane energy densities. For any equation of state onthe brane, the radiation was shown to evolve such as to generate conventional radiation-dominatedcosmology, consistent with nucleosynthesis [13].Subsequent to the brane-world cosmology on thin branes, the thick brane-world paradigm hasexhibited a fine structure [1, 14, 15] that supports the above discussed phenomenology. In spiteof their success, thick brane-world models do encompass neither anisotropy on the brane nor theimportant framework of asymmetric branes, as well as spherically symmetric thick brane worlds.Even if anisotropic brane-worlds have been comprehensively investigated, there still are severalreasons to depart from the standard isotropic models, in particular focusing on spherically sym-metric time-dependent warp factors, in the thick brane-world scenario. As an example, in BianchiI brane-world cosmology, for scalar fields with a large kinetic term, the initial expansion of theUniverse is quasi-isotropic. The Universe grows anisotropically during an intermediate transientregime and the anisotropy finally disappears during the inflationary expansion [16]. In addition,anisotropic brane-worlds are realized in the context of exact solutions of the gravitational fieldequations in the generalized Randall-Sundrum model for an anisotropic brane with Bianchi typeI and V geometry, with perfect fluid and scalar fields as matter sources. Under the assumptionof a conformally flat bulk with vanishing Weyl tensor for a cosmological fluid obeying a linearbarotropic equation of state, the general solution of the field equations was expressed as an exactparametric form for both Bianchi type I and V spacetimes [17]. The dynamics of the correspondinganisotropy in such Bianchi type I and V cosmological scenarios has also been investigated in thecontext of Randall-Sundrum brane-worlds [18], given that a Randall-Sundrum brane-world can bemimicked without the assumption of spatial isotropy, by means of an homogeneous and anisotropicKasner type solution of the Einstein-AdS equations in the bulk [19]. Other interesting anisotropicbrane-world models have been studied in Refs. [20, 21].On the other hand, observations of the CMB tell one that the Universe is isotropic with agreat accuracy [22]. The natural framework to approach this highly isotropic Universe impliesinto assuming that the Universe setup from a highly anisotropic state, and thus a dynamicalmechanism gets rid of almost all its anisotropy. Inflation mechanisms [23, 24] are the most promisingcandidates for explaining such a behavior. In these lines, the simplest generalization of FRWcosmologies are the Bianchi cosmologies, as they provide anisotropic but homogeneous cosmologies,where the central point of discussion is if the Universe can isotropize without additionally fine-tuning the parameters of the model. The isotropization of Bianchi I brane-world cosmologies hasbeen investigated, from several points of view [25, 26]. It has been shown, for instance, that alarge initial anisotropy does not suppress inflation in a Bianchi I brane-world [16]. Otherwise,considering negative values of dark radiation in Bianchi I models leads to interesting solutions forwhich the Universe can both collapse or isotropize [18, 26, 27]. It is required that isotropizationshould be accompanied by a phase of accelerated expansion in order to be a good candidate toexplain the results that indicate the current speeding up of the observable Universe [28]. Thislatter observational fact is approached from two directions: modifying the gravitational sector [29]or introducing dark energy [30]. From this point of view, a model in which a dark energy componentlives in a Bianchi brane-world combines both approaches.The study of the dynamics of a scalar field with an arbitrary potential trapped in brane-worldmodel can be further performed [1, 15, 31, 32]. Homogeneous and anisotropic Bianchi I branes filledalso with a perfect fluid are the mostly approached models. In particular, by taking into accountthe effect of a positive dark radiation term on the brane [34], the effect of the projection of the 5DWeyl tensor onto the brane in the form of a negative dark radiation term is considered [33].All the above-mentioned reasons motivate the investigation of both spherically symmetric andanisotropic brane models. It is indeed worthwhile to emphasize that cosmological solutions of thegravitational field equations in the generalized Randall-Sundrum model for an anisotropic branewere obtained, with Bianchi I geometry and with perfect fluid as matter sources described by ascalar field [35]. The solution admits an inflationary era and, at a later epoch, the anisotropy of theUniverse washes out. Two classes of cosmological scenarios are involved, regarding universes thatevolve from a singularity and without singularity [36]. Moreover, by using a metric-based formalismto treat cosmological perturbations [37, 38], the connection between anisotropic stress on the braneand brane bending are discussed in [39].Our main aim here is to provide an initial approach for spherically symmetric thick branecosmology. By exploring the framework of isotropic thick branes [1, 14, 15], one can realize thatthe separability of the warp factor is fundamental in order to explicitly describe the time-dependentsolutions. It is noway obvious that, for spherically symmetric thick brane-worlds, the warp factorsto be considered in this paper – that are dependent on time, extra dimension, and radial coordinateon the brane – should be separable in the context of solving the equations of motion. Likewise, itsuggests that it might be hopeful to find time-dependent soliton solutions leading to non-separableforms of the warp factor [40–42]. Separable solutions are normally discussed in the framework ofthin brane-world models that are rather unnatural in case of thick defects, since the brane thickness∆ must fulfill the limits 2 . × − m (cid:46) ∆ (cid:46) µ m [43], having thus a minimal thickness [44].In fact, thick brane cosmology has been widely discussed in [40–42, 48], further regarding othertype of warp factors [49–53] and tachyonic solutions, with a decaying warp factor that enableslocalization of 4D gravity as well as other matter fields [54]. Some applications in the thin branelimit have been provided, e. g., in [45–47].Departing from a general 5D spherically symmetric warped spacetime, our purpose is to solve thecoupled system of Einstein equations and the equations of motion for a scalar field. The procedureintroduced in the following results into an explicit formula for both the extra dimension-dependentpart of the warp factor and the spherically symmetric time-dependent component. The warp factorsfor flat, closed, and open spacetimes are obtained and discussed, and the properties of Hubble typeparameter are also investigated. Our analysis results into deploying the fundamentals of thick brane-world cosmology with time dependent spherically symmetric warp factors, exclusively departingfrom the Einstein equations.To provide a generalization of the successful achievements on brane-world cosmology in the thinbrane paradigm [10, 11] as well as in the thick brane scenario [1, 15], one considers 5D spacetimesfor which the metric assumes the following form: ds = a ( t, r, y ) g µν dx µ dx ν + dy , (1)where x µ denotes a chart of 4D coordinates on the brane, whereas g µν is the metric given by g µν dx µ dx ν = − dt + (cid:18) dr − kr + r d Ω (cid:19) , where d Ω stands for the usual area element of the 2-sphere and k denotes the curvature parameterassuming the values −
1, 0 and 1, leading respectively to an open, a flat or a closed Universe. Thefunction a ( t, r, y ) is the conformal scale factor extraordinarily depending upon the radial coordinate r on the brane, also referred as a warp factor due to the extra dimension y in (1). The 4D solutionsare sourced by the bulk scalar field.The action for scalar field in the presence of 5D gravity is given by S = (cid:90) d x √− g (cid:18) −
12 g MN ∇ M φ ∇ N φ − V ( φ ) + 2 M R (cid:19) , (2)where g denotes the 5D metric, M is the Planck mass of the fundamental 5D theory and R denotesthe 5D Ricci scalar. For the above prescribed scenario, one assumes that the scalar field, φ , dependsexclusively on time and upon the extra coordinate, y , and V ( φ ) is the scalar field potential.The Einstein equations and the equation of motion for φ resulting from the above action (2)are provided by ∇ φ − dVdφ = 0 , (3) R MN −
12 g MN R = 14 M T MN , (4)where ∇ is the 5D Laplacian operator, and the energy-momentum tensor, T MN , for the scalarfield φ ( t, y ) reads T MN = − g MN (cid:18)
12 ( ∇ φ ) + V ( φ ) (cid:19) + ∇ M φ ∇ N φ . In particular, the energy-density ( T ) is implied by φ ( y ) and localized near y = 0. Moreover, theequation of motion for the scalar field is expressed by φ (cid:48)(cid:48) − a ¨ φ + 4 a (cid:48) a φ (cid:48) − aa ˙ φ = dVdφ , where one denotes ∂f∂t = ˙ f , ∂f∂r = ¯ f , and ∂f∂y = f (cid:48) , for any scalar function f hereupon. By assuminga static scalar field scenario, the components of the Einstein tensor are given by the followingexpressions: G = a (cid:26) (cid:20) ˙ a a − (cid:18) a (cid:48)(cid:48) a + a (cid:48) a (cid:19) + ka (cid:21) + (1 − kr ) (cid:20) ¯ a a − aa − aa r (cid:21) + 2¯ aa r (cid:27) ,G = − g (cid:26)(cid:20) a (cid:18) aa − ˙ a a (cid:19) − (cid:18) a (cid:48)(cid:48) a + a (cid:48) a (cid:19) + ka (cid:21) − (1 − kr ) (cid:20) a a + 4¯ aa r (cid:21)(cid:27) ,G = − g (cid:26)(cid:20) a (cid:18) aa − ˙ a a (cid:19) − (cid:18) a (cid:48)(cid:48) a + a (cid:48) a (cid:19) + ka (cid:21) − (1 − kr ) (cid:20) aa − ¯ a a + 4¯ aa r (cid:21) + 2¯ aa r (cid:27) ,G = g G /g ,G = 3 (cid:18) a (cid:48) a − ¨ aa − ka (cid:19) + 3(1 − kr ) (cid:18) ¯¯ aa + 3¯ aa r (cid:19) − aa r ,G = 2 (cid:18) a ˙ aa − ˙¯ aa (cid:19) ,G = 3 (cid:18) ˙ aa (cid:48) a − ˙ a (cid:48) a (cid:19) ,G = 3 (cid:18) ¯ aa (cid:48) a − ¯ a (cid:48) a (cid:19) . The Einstein equations G MN = T MN , when 4 M ∗ = 1, can be used to find the form of the warpfactor. By separating the variables a ( t, r, y ) = A ( t, r ) B ( y ), the Einstein equation G = T = 0yields 0 = 2 ¯ AA − ˙¯ A ˙ A = ∂ r ln A − ∂ r ln ˙ A ⇒ ln A ˙ A = T ( t ) ⇔ ˙ A = A e − T , (5)or 0 = 2 ˙ AA − ˙¯ A ¯ A = ∂ t ln A ¯ A ⇒ ln A ¯ A = R ( r ) ⇔ ¯ A = A e − R , (6)implying that ˙ A = ¯ Ae R − T . (7)The expressions ˙ A = A e − T and ¯ A = A e − R follow from (5), (6) and (7), and they imply that˙¯ A = = 2 A e − ( T + R ) , ¨ A = = A e − T (cid:16) Ae − T − ˙ T (cid:17) , (8)¯¯ A = A e − R (cid:0) Ae − R − ¯ R (cid:1) . (9)One of the off-diagonal Einstein equations G = T = ˙ φφ (cid:48) yields˙ φφ (cid:48) = 3 (cid:32) ˙ AB (cid:48) AB − ˙ AB (cid:48) AB (cid:33) = 0 ⇒ ˙ φ = 0 , that means that the scalar field φ is time independent.Hence the components of energy-momentum tensor become T µν = − g µν (cid:18) φ (cid:48) + V ( φ ) (cid:19) , T = 12 φ (cid:48) − V ( φ ) . The explicit form of the components of the Einstein equation for the metric ansatz (1) can bewritten for k = 0 , ±
1, by denoting the spatial curvature of the 4D homogeneous and isotropicspace-time for Minkowski, de Sitter and anti-de Sitter space, respectively. The diagonal components(remembering that the component equals ) are respectively expressed as:12 φ (cid:48) + V ( φ ) = 1 B (cid:40) − (cid:0) BB (cid:48) (cid:1) (cid:48) + 1 A (cid:34) (1 − kr ) (cid:32) ¯ A A − AA − AAr (cid:33) +2 ¯
AAr +3 ˙ A A +3 k (cid:35)(cid:41) , φ (cid:48) + V ( φ ) = 1 B (cid:40) − (cid:0) BB (cid:48) (cid:1) (cid:48) + 1 A (cid:34) (1 − kr ) (cid:18) A A +4 ¯ AAr (cid:19) +2 ¨ AA − ˙ A A + k (cid:35)(cid:41) , φ (cid:48) + V ( φ ) = 1 B (cid:40) − (cid:0) BB (cid:48) (cid:1) (cid:48) + 1 A (cid:34) (1 − kr ) (cid:32) AA − ¯ A A +4 ¯ AAr (cid:33) +2 ¨ AA − ˙ A A + k +2 ¯ AAr (cid:35)(cid:41) , φ (cid:48) − V ( φ ) = − B (cid:40) − B (cid:48) + 3 A (cid:34) ( kr − (cid:32) ¯¯ AA +3 ¯ AAr (cid:33) + ¨ AA + k + ¯ AAr (cid:35)(cid:41) . (10)Therefore, the equations for φ or y can be expressed as φ (cid:48)(cid:48) ( y ) + 4 a (cid:48) a φ (cid:48) ( y ) = dVdφ ,B ( y ) (cid:20) (cid:18) B (cid:48)(cid:48) ( y ) B ( y ) + B (cid:48) ( y ) B ( y ) (cid:19) + 12 φ (cid:48) ( y ) + V ( φ ) (cid:21) = c ,B ( y ) (cid:20) B (cid:48) ( y ) B ( y ) − φ (cid:48) ( y ) + V ( φ ) (cid:21) = c , (11)for c and c separation constants, whereas the ones for t and r are summarized respectively by˙ A ( t, r ) = A ( t, r ) e − T ( t ) , ¯ A ( t, r ) = A ( t, r ) e − R ( r ) , (12) : 1 A (cid:34) (1 − kr ) (cid:32) ¯ A A − AA − AAr (cid:33) + 2 ¯
AAr (cid:35) + 3 ˙ A A + 3 kA = c , (13) : 1 A (cid:20) (1 − kr ) (cid:18) A A + 4 ¯ AAr (cid:19)(cid:21) + 2 ¨ AA − (cid:32) ˙ AA (cid:33) + kA = c , (14) = : 1 A (cid:34) (1 − kr ) (cid:32) AA − ¯ A A + 4 ¯ AAr (cid:33) + 2 ¯
AAr (cid:35) + 2 ¨ AA − ˙ A A + kA = c , (15) : 3 A (cid:34) ( kr − (cid:32) ¯¯ AA + 3 ¯ AAr (cid:33) + ¯
AAr (cid:35) + 3 ¨ AA + 3 kA = c . (16)The role of the bulk scalar field is to provide the cosmological constant on a brane as is clear fromEqs.(12-15). By imposing c = Λ one has analogous cosmological implications for suitable limits,where the warp factor has no dependence on r , as in the thick brane cosmology with isotropic warpfactor [1, 15]. In an isotropic thick brane-world the condition c = 2 c = Λ holds [1]. Nevertheless,in this scenario such two constants restrict further the form of the function A ( r, t ), when the aboveequations are used, by the following relationship: − (cid:18) kr − c r (cid:19) −
12 (1 − kr ) / c Ar + 6 √ − kr c Ar = 2 c − c . (17)Note that this consistency equation is trivial if the 4D scale factor is independent of the radialcoordinate, as in [1]. Now, by computing the difference of (14) and (15), one obtains the equation(1 − kr ) ¯ R − r = 0 which has solution R ( r ) = ln c r √ − kr , (18)where c is a constant of integration. Moreover, the solution for Eq. (12) is provided by (hereonone shall notice the index k in order to denote the dependence on k = 0 , ± A k ( t, r ) = c c Y k ( t ) + f k ( r ) , (19)where Y k ( t ) is a constant of integration with respect to the r coordinate, f k ( r ) ≡ ln √ − kr + 1 r − (cid:112) − kr , (20)and A k ( t, r ) depends on k = 0 , ±
1. It implies hence that˙ A k ( t, r ) A k ( t, r ) = − ˙ Y k ( t ) = e − T ( t ) . (21)We can simplify Einstein equations using (6), (8), (9), and (21) to make Eq.(13) — that corre-sponds to the component of the Einstein equations — to read:˙ Y k = − kY k + w k ( r ) Y k + z k ( r ) , (22)where w k ( r ) and z k ( r ) are respectively given by the following expressions: w k ( r ) = − (cid:20) − kr ) e − R (cid:18) R − r (cid:19) + 2 e − R r + 6 kf k ( r ) c (cid:21) ,z k ( r ) = − (cid:26) (1 − kr ) e − R (cid:20) − e − R + 2 f k ( r ) c (cid:18) R − r (cid:19)(cid:21) + 2 f k ( r ) c e − R r + 3 k f k ( r ) c − c (cid:27) . The solutions for Minkowski, anti-de Sitter and de Sitter spacetimes are respectively providedby: Y ±− ( t ) = 14 e ± ( t ∓ α − ) (cid:20)(cid:16) e ∓ ( t ∓ α − ) − w − ( r ) (cid:17) − z − ( r ) (cid:21) , (23) Y ± ( t ) = α ± (cid:112) z ( r ) t, (24) Y ± ( t ) = 12 (cid:20) w ( r ) ± (cid:113) w ( r ) + 4 z ( r ) sin( t + | α | ) (cid:21) . (25)respectively for k = − , ,
1. The constant parameters α , α ± are constants of integration.When f k ( r ) = 0, then A k ( t, r ) = 1 /Y k ( t ), and one has the results from [1] for thick branecosmology, with c = 2 c = Λ, where Λ denotes the 4D cosmological constant. For the caseexplicitly provided by Eq. (19), A k ( t, r ) indeed does not depend on the r coordinate. Firstly, it isevident that A k ( t, r ) = 0 when r → ∞ , as f k ( r ) diverges in this case. However, it is not properlythe useful case here. For k = 0, A k ( t, r ) is independent of r when f k ( r ) = 0, namely, when r = 2 /e .Moreover, when k = 1, f k ( r ) = 0 for r = 1, and hence A k ( t, r ) in Eq. (19) has no dependence onthe variable r . Finally, A k ( t, r ) is merely a function of the cosmic time t for k = − r solvesthe algebraic equation √ r +1 r = exp (cid:16) √ r (cid:17) , which is r ≈ . a ( t ) ∼ sech( t + α − ) , k = − < / ( t + α ) , k = 0 (Λ > t + α ) , k = +1 (Λ >
0) (26)For k = 0 , t = − α a + ( n + 1 / πk , ( a = 0 , n ∈ Z [1].It further implies that in this particular case the Hubble parameter reads H ( t ) = tanh( t + α − ) , k = − < / ( t + α ) , k = 0 (Λ > t + α ) , k = +1 (Λ >
0) (27)for the appropriate limits above analyzed where f k ( r ) = 0.Once the component of the Einstein equations is considered, one can further analyze the component. Eq. (14) thus reads(1 − kr ) e − R (cid:18) e − R + 4 c r f k ( r ) (cid:19) + kc f k ( r ) + kY + (cid:20) (1 − kr ) e − R r + 2 kc f k ( r ) (cid:21) Y − f k ( r ) c ¨ Y − Y ¨ Y + 3 ˙ Y = c . Y (cid:18) Y + f k ( r ) c (cid:19) = 3 ˙ Y + kY + u ( r ) Y + v ( r ) , (28)where u k ( r ) and v k ( r ) are respectively given by: u k ( r ) = 4 r (1 − kr ) e − R + 2 kc f k ( r ) ,v k ( r ) = u k ( r ) f k ( r ) c + 3(1 − kr ) e − R − kc f k ( r ) − c . Moreover, the and components of Einstein equations are provided by Eq. (15), yielding2 ¨ Y (cid:18) Y + f k ( r ) c (cid:19) = 3 ˙ Y + kY + m ( r ) Y + n ( r ) , (29)where m ( r ) = e − R (cid:20) − kr −
1) ¯ R + 6 r − kr (cid:21) + 2 k f k ( r ) c ,n ( r ) = m ( r ) f k ( r ) c + 3 e − R (1 − kr ) − k f k ( r ) c − c . Analogously, the component of Einstein equations Eq. (16) reads¨ Y (cid:18) Y + f k ( r ) c (cid:19) = 2 ˙ Y + kY + g ( r ) Y + h ( r ) , (30)where g ( r ) = e − R (cid:20) ( kr − (cid:18) r − ¯ R (cid:19) + 1 (cid:21) + k f k ( r ) c ,h ( r ) = g ( r ) f k ( r ) c − k f k ( r ) c − (cid:18) − ( kr ) c r (cid:19) − c . Eqs. (28), (29) and (30) can be reduced to first order EDOs. By defining a new variable X = ˙ Y ,Eqs. (28) and (29) can be written as2 X dXdY ( Y + α ) = 3 X + kY + bY + c, (31)where α = f k ( r ) /c , for Eqs. (28) and (29), by identifying respectively b to u ( r ) and m ( r ), and c to v ( r ) and n ( r ). Solutions are provided by˙ Y = k ( Y + α ) − kY − bY − kα − αb − c, (32)where k is a constant of integration. Note that when k = 0 the above equation has exactly thesame form of Eq. (22), thus has the same kind of solutions.1Moreover Eq. (29) can be recast analogously as X dXdY ( Y + α ) = 2 X + kY + g ( r ) Y + h ( r ) , and reduced to a first order EDO in a similar way, giving˙ Y = k ( Y + α ) − kY −
23 ( αk + g ( r )) Y −
13 ( α k + αg ( r ) + 3 h ( r )) . (33)In general, the above obtained equations do not exhibit analytical solutions. However when k = 0the same form of the Eq. (22) is again achieved.Given the form of the above equations, the constant parameter k fixes the initial accelerationassociated to the scale factor of the universe. One can compare it to the expected data for thedynamics of the scale factor and set k according to the initial conditions.In the set of Figs. 1-3 and Figs. 4-6 one respectively depicts the form for the warp factor A k ( t, r )for k = 0 , ±
1, and the associated Hubble like parameter, calculated from the respective warp factor.2 r t A (cid:72) r,t (cid:76) FIG. 1:
Graphic of the warp factor A − ( t, r ) in (19), for c = 2. r t (cid:45) A (cid:72) r,t (cid:76) FIG. 2:
Graphic of the warp factor A ( t, r ) in (19), for c = 2. r t (cid:45) A (cid:72) r,t (cid:76) FIG. 3:
Graphic of the warp factor A ( t, r ) in (19), for c = 2. r t (cid:45) A (cid:72) r,t (cid:76) FIG. 4:
Graphic of the warp factor A − ( t, r ) in (19), for c = 0 . r t (cid:45) A (cid:72) r,t (cid:76) FIG. 5:
Graphic of the warp factor A ( t, r ) in (19), for c = 0 . r t A (cid:72) r,t (cid:76) FIG. 6:
Graphic of the warp factor A ( t, r ) in (19), for c = 0 . The above two sets of plots indicate a dependence on the integration parameter c , that is amultiple of the brane cosmological constant in an isotropic thick brane-world. Instead, here the3constant c is related to the cosmological constant c = Λ by Eq. (17), and the spherical symmetryof the warp factor sets in. This explains the dependence of the A k ( t, r ) on the parameter c .When the constant of integration c in (17) is assumed to be equal to 2, Fig. 1 illustrates amonotonically increasing scale factor both radially and temporally. The larger is the radial positionon the brane the steeper is the time dependence is. Fig. 2 presents a range of singularity that attainslower values for the radial coordinate as time elapses. Fig. 3 illustrates a scale factor that increasesin the range presented therein. However such an increment is smoother as the cosmic time elapses.When c = 0 .
1, Fig. 4 depicts a time-independent singularity for a fixed value r ≈ .
663 for thescale factor of a closed Universe, whereas the singularity evinced in Fig. 2 is smoother in Fig. 5.Finally, Fig. 6 shows a similar profile as that one in Fig. 3, instead the radial increment is planer.Hereupon the Hubble like parameter can also be depicted for k = 0 , ±
1. Their profiles are stilldependent on the constant c in (17), nonetheless the range of H k ( t, r ) changes slightly for differentvalues of the c . For the sake of completeness, and in order to match the results from Figs. 1-6,one the Hubble like parameter for c = 0 . r t H (cid:72) r,t (cid:76) FIG. 7:
Graphic of the Hubble like pa-rameter H − ( t, r ) = ˙ A − ( t, r ) /A − ( t, r )in (19), for c = 0 . r t H (cid:72) r,t (cid:76) FIG. 8:
Graphic of the Hubble likeparameter H ( t, r ) = ˙ A ( t, r ) /A ( t, r ) in(19), for c = 0 . r t H (cid:72) r,t (cid:76) FIG. 9:
Graphic of the Hubble like pa-rameter H ( t, r ) in (19), for c = 0 . k = 1,the Hubble like parameter increases monotonically, being steepest for lower values of the radialcoordinate.The y -dependent part of the solutions that are determined by Eqs. (10). By defining ¯ B ( y ) ≡ e b ( y ) , it can be written as 3 B (cid:48)(cid:48) + 32 Λ e − b = − φ (cid:48) , (34)6 B (cid:48) − e − b = 12 φ (cid:48) − V ( φ ) , (35) φ (cid:48)(cid:48) + 4 B (cid:48) φ (cid:48) − dVdφ = 0 . (36)As such equations are the same as those ones obtained in [1], our results for the extra-dimensionalprofiles are likewise similar here. When one assumes a value for the function B ( y ), thus Eqs. (34)-(36) determine φ ( y ) and V ( φ ), or vice-versa [55–58]. For instance, the warp factor B ( y ) =ln[sinh( βy )] can be adopted in [1], for β usually assumed as a constant parameter, in order tohave A ( y ) ∝ e −| y | when y → + ∞ , recovering thus the Randall-Sundrum model. For small (large)values of y , | y | (cid:46) β − ( | y | (cid:38) β − ), the kink solutions are given respectively by φ small ( y ) = 2 √ βy/ ,φ large ( y ) = (cid:114) − β sinh( βy ) , (37)These results can be turned into asymmetric thick brane-world scenarios, generated after addinga constant to the superpotential associated to the scalar field. Asymmetric branes can be generatedirrespective of the potential being symmetric or asymmetric, and the sine-Gordon-type model in thiscontext can be shown to have a stable graviton zero mode, despite the presence of an asymmetricvolcano potential [59]. Indeed, the superpotential method described in [1] can be further extendedwhen one proposes the sine-Gordon-type model determined by the superpotential W c ( φ ) = 2 (cid:114)
32 sin (cid:32)(cid:114) φ (cid:33) + c, that is obtained by the standard one, by shifting it by a constant parameter c such that | c | ≤ √ φ (cid:48) = 12 W φ , (38) A (cid:48) = − W ( φ ) , (39)5were obtained [59]: φ ( y ) = (cid:114)
32 arcsin(tanh( y )) , (40) A c ( y ) = − ln[sech( y )] − cy, (41)where φ ( y ) is the standard solution of the sine-Gordon model, for c = 0. The Schr¨odinger likeequation with a quantum mechanical potential in conformal coordinates have already been studiedin [59] in order to derive an stable graviton asymmetric zero model . The asymmetry induced inthe thick brane-world scenario is also phenomenologically constrained to be consonant with theAdS bulk curvature and with the experimental and theoretical limits of the brane thickness [44].The constants c and c = Λ in (11) and (13)-(16) are severely constrained, besides Eq. (17), bythe fine tuning, relating the 4D and 5D cosmological constants, and the brane tension. To end up,it is worthwhile to emphasize that for each k , the functions w k ( r ) and z k ( r ) in Eq.(22) constrainthe range of the variable r , and should be used to comply the model to observational data. Thesame principle must be applied in the other differential equations for Y k ( t ). These constraints shallbe addressed in a forthcoming publication [60].To finalize, further ways to analyze the system is to include the radial dependence in the bulkscalar field that supports the radial dependence in the 4D scale factor. However, in this case, it isno longer possible to solve the equations analytically. Acknowledgements
A. E. B. would like to thank for the financial support from the Brazilian Agencies FAPESP(grant 08/50671-0) and CNPq (grant 300809/2013-1) R. T. C. thanks to UFABC and CAPES forfinancial support. R. da R. is grateful to SISSA for the hospitality, to CNPq grants No. 473326/2013-2 and No. 303027/2012-6 and is also