The Universe Dynamics from Topological Considerations
TThe Universe Dynamics from Topological Considerations
Miguel A. Garc´ıa-Aspeitia ∗† and Tonatiuh Matos ∗‡ Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del I.P.N. Apdo. Post. 14-740 07000, D.F., M´exico (Dated: October 31, 2018)We explore the possibility that the dynamics of the universe can be reproduced choosing appropri-ately the initial global topology of the Universe. In this work we start with two concentric sphericalthree-dimensional branes S , with radius a < a immersed in a five-dimensional space-time. Thenovel feature of this model is that in the interior brane there exist only spin-zero fundamental fields(scalar fields), while in the exterior one there exist only spin-one fundamental interactions. Asusual, the bulk of the universe is dominated by gravitational interactions. In this model, like in theEkpyrotic one, the Big Bang is consequence of the collision of the branes and causes the existence ofthe particles predicted by the standard model in the exterior brane (our universe). The scalar fieldson the interior brane interact with the spin-one fields on the exterior one only through gravitation,they induce the effect of Scalar Field Dark Matter with an ultra-light mass on the exterior one. Wediscuss two different regimes where the energy density and the brane tension are compared, withthe aim to obtain the observed dynamics of the universe after the collision of the branes. PACS numbers: 04.50.+h
I. INTRODUCTION.
In the last part of the twentieth century, cosmology be-came a precision science getting observational data withan accuracy comparable with the data obtained throughthe standard model. The last data arising from observa-tions confirm that our universe is accelerating due to theexistence of some unknown type of energy and also re-quires the existence of an unknown field that permeatesthe universe and dictates the formation and evolution ofthe structures at large scales. Both dark components ofthe universe are not predicted by the standard model orby the general theory of relativity either. This opens thepossibility to extend these theories to limits beyond thecurrent ones and to formulate new paradigms that pre-dict new physics, under the condition that in the appro-priated limits these new theories reproduce the currentstandard observations. So far the best accepted candi-date to be the dark energy of the universe is the cos-mological constant Λ. It is well-known that observationsof the Cosmic Microwave Background Radiation (CMB)and galaxies surveys fit very well with a small value of Λ.The main problem is to relate Λ with some physical phe-nomena. There are many interpretations for doing so, forexample, the most simple one is that the Einstein equa-tions have two free constants, the gravitational one G and the cosmological constant Λ. Here the cosmologicalconstant contains only a geometrical interpretation. Wefix Λ in the same way as we have fixed G , namely, usingcosmological observations. The problem here is that wehave now a theory with two coupling constants, givingrise to the interpretation that the Einstein theory could ∗ † Electronic address: agarca@fis.cinvestav.mx ‡ Electronic address: tmatos@fis.cinvestav.mx not be the most fundamental theory for gravitation thatcan exist. Other example is to relate the cosmologicalconstant with the vacuum energy of the universe. Herethe problem is that the values derived from any particlephysics models is many orders of magnitude too big (orvanishes in some SUSY cases) compared with the one ob-tained using astronomical observations [6]. From this in-terpretation follows an incompatibility between the the-ory of general relativity and the particle physics modelsthat we have to face on at this moment. There are someother proposals where the space-time contains subspacesembedded in a higher dimensional one, for example thebranes theory [2], the DGP model [10], Ekpyrotic [13],[3] or cyclic universe [29], [23] in which it is proposed theexistence of a four dimensional manifold embedded in afive dimensional bulk. It is important to mention thatin the last two models [13], [3] and [29], [23] the branesmove through the bulk and collide giving origin to theBig Bang. The main goal of this work is to explore thepossibility that the dynamics of the universe follows fromits initial global topology. For doing so there are differentways, but the idea here is the following:1. We start with two three-dimensional branes embed-ded in a 5-dimensional space-time, where the fivedimensional Einstein equations with cosmologicalconstant are fulfilled. For facility we will supposethat the branes are three dimensional concentricspheres S (FIG. 1) but this result can be gener-alized to the other two homogeneous topologies R and H . The initial conditions are such that theinterior brane has the scale factor a and the exte-rior brane (where we live) has the scale factor a ,with a < a (FIG. 1). For simplicity, each branehave a common center.2. We assume that matter is confined on the branes,the matter content of the interior brane is a selfinteracting scalar field (spin zero field) and of the a r X i v : . [ g r- q c ] M a y FIG. 1: Schematic representation of the interior and exte-rior branes and the different values of the 5D cosmologicalconstant Λ i i = 1 , , σ . exterior brane is the one of the standard model in-teractions (spin one fields).With these hypothesis we show that1. The scalar field on the interior brane acts as aninflaton, expanding this brane and provoking thecollision with the exterior one, which was in ther-mal equilibrium before the collision. It follows abig bang scenario like in the Ekpyrotic models.2. It is possible to choose the free parameters of themodel such that after the collision, both branes ex-pand together, (see equations (34) and (35)).3. The fluctuations induced by the collision in thescalar field provoke the potential wells that give riseto the structure formation in the exterior brane,acting as the seeds for the galaxies formation onthe brane where we live, via gravitational interac-tion. After the collision the physics of the exteriorbrane is close related with the Ekpyrotic and Cyclicmodels [29], [28].The main difference of this toy model with previousones is the hypothesis of the initial conditions and thematter content of the branes. For example, after thecollision the exterior brane behaves in a very similar wayas in the model presented in [19]. The problem of thislast model is the reheating period, where it was not verynatural to reheat the Universe. In the model presentedhere the reheating epoch is not necessary because theheat of the Universe is consequence of the collision of thebranes, as in the Ekpyrotic model [29], [28].It is important to mention that we let some open ques-tions and use the results of previous works (see for exam-ple [28], [29], [3], [23], [25]), the main goal of this workis to show that there exist the possibility that the initial topology of the Universe could reproduce its observeddynamics.This work is organized as follows. In section II wepresent the model in detail and write the main field equa-tions in section III. In section IV we present the differentpossible scenarios and regimes of the model. In section Vwe present the dynamics of the universe with this modeland in section VI we give some conclusions and remarks. II. THE MODEL.
In this section we write the most important featureof the model in a general way. Suppose two concentric3-dimensional branes embedded in a 5-dimensional bulk,(we use natural units c = (cid:125) = 1). The shape of the actionto model this physical structure is given by S = (cid:90) dX (cid:112) − g (5) m (cid:0) R (5) + Λ (cid:1) − (cid:88) ± (cid:90) ± dx (cid:112) − g ± (cid:16) m K ± + £ ± (cid:17) , (1)being ± the exterior or interior region of the brane respec-tively, g (5) is the determinant of the five-dimensional (5D)metric and g the determinant of the four-dimensional(4D) one, m (5) is the 5D Planck mass, R (5) is the 5Dscalar curvature, K is the extrinsic curvature and Λ isthe 5D cosmological constant, £ + is the Lagrangian ofthe fields content in the exterior brane (spin zero) and £ − is the Lagrangian of the fields content in the interiorbrane (spin one). The bulk can be described with thenatural coordinates in the form [12] ds = − A ( a ) ± dt ± + 1 A ( a ) ± da + a (cid:2) dχ + sin ( χ ) (cid:0) dθ + sin θdϕ (cid:1)(cid:3) . (2)Now we solve the 5D Einstein equations R (5) AB = Λ ± g (5) AB with ( A, B = 1 , · · · ,
5) for the most general A ( a ) ± function, we obtain A ( a ) ± = 1 − M ± m a − Λ ± a , (3)where M ± m is the mass parameter of the AdS-S (5) blackhole [23] and Λ ± is the 5D cosmological constant in thebulk.On the other hand, if we consider the location of thetwo 3-brane by t = t ( τ ), a = a ( τ ) parametrized by thetime τ in the brane, we have the induced four dimensionalmetric given by ds i = − dτ + a i ( τ ) (cid:2) dχ + sin ( χ ) (cid:0) dθ + sin ( θ ) dϕ (cid:1)(cid:3) , (4)where the subindex i = 1 , a < a . The functions t ( τ ) and a ( τ ) arethen constrained by u ν u ν = g µν u µ u ν = − A ( a i ) ˙ t + A ( a i ) − ˙ a i = − , (5)being u µ = (cid:0) ˙ t, ˙ a i , , (cid:1) the brane velocity. The dots rep-resent here differentiation with respect to τ . We definethe unit normal vector on the two branes n ± µ such that n ± µ u µ = 0 and n ± µ n ± µ = 1, their components are givenby n ± µ = ± (cid:16) ˙ a i , − A ( a i ) − ( A ( a i ) + ˙ a i ) , , (cid:17) . (6)Following [28], [24], [1] we can construct the extrinsiccurvature using the equation K µν = − g λµ ∇ λ n ν . (7)The no null components of K µν can be written as K ± tt = − (cid:16) ¨ a i + ∂A ( a i ) ± ∂a i (cid:17) ( ˙ a i + A ( a i ) ± ) , (8) K ± χχ = K ± θθ = K ± ϕϕ = (cid:0) ˙ a i + A ( a i ) ± (cid:1) a i . (9)With the equations of motion of the brane, it follows [1],[28], [24] [ K ] g µν − [ K µν ] = k T µν , (10) ∇ ν T νµ = − [ T µn ] , (11)where k = πm , [ K µν ] = K + µν − K − µν and T µν is theenergy-momentum tensor. Using equation (9) we find (cid:0) ˙ a i + A ( a i ) − (cid:1) − (cid:0) ˙ a i + A ( a i ) + (cid:1) = a i k ρ i , (12)where we are supposing that the matter content of thebranes can be represented as perfect fluids with density ρ i and presion P i [1], [28], [24].The equation (11) represents the energy-momentumconservation on the brane, given by ddt (cid:0) ρ i a i (cid:1) + P i ddt (cid:0) a i (cid:1) = 0 . (13)We will use the equation (12) to give a physical interpre-tation of the model. III. THE MODIFIED FRIEDMANNEQUATIONS.
In this section we write the field equations of themodel. First observe that we have to distinguish be-tween three vacuum regions, the first one is located in-side the interior brane, we call it region I, the second oneis located between the branes, region II, and the thirdvacuum region surrounds the exterior brane, region III,(see FIG. 1). Now consider the following analogy with anelectromagnetic system. Imagine two charged conduct-ing parallel plates with opposite charges, we know thatin the region in between the plates, there exist an electricfield induced by the charges on the plates, while in thesurrounded region, the electric field vanishes. If we nowsuppose that the branes are the plates and the electriccharge is the gravitational field, the analogy tell us thatit should exist an extreme difference between the expec-tation value for the vacuum in the region in between thebranes and zero in the exterior region, having in mindthat in the linear regime, the electromagnetic and thegravitational fields behave in the same way.Thus, for the region inside of the interior branes (regionI), equation (3) becomes A ( a ) − = 1 − Λ a , (14)since there does not exist a gravitational potential insideof the interior brane. But in the region in between thebranes (region II), the equation (3) can be written as A ( a ) + = 1 − M m a − Λ a , (15)where M is the mass of the interior brane. If we substi-tute the equations (14) and (15) into equation (12), wehave ˙ a a + 1 a = k ρ
36 + Λ +Λ
12 + M m a + (Λ − Λ ) ρ k + 9 M m k ρ a + 3 M (Λ − Λ )2 m a k ρ . (16)On the other hand, inside of the exterior brane (regionII again) we have A ( a ) − = 1 − M m a − Λ a . (17)Finally, outside of the exterior brane (region III), thetotal mass is M = M + M . Therefore we have A ( a ) + = 1 − M + M ) m a − Λ a . (18)Again, we substitute the equations (17) and (18) into theequation (12), to obtain˙ a a + 1 a = k ρ
36 + Λ +Λ
12 + 2 M + M m a + (Λ − Λ ) ρ k + 9 M m k ρ a + 3 M (Λ − Λ )2 m a k ρ . (19)Expressions (16) and (19) are the Friedman equationson the interior and exterior branes respectively. In bothequations, (16)) and (19), the third, fourth, fifth andsixth terms are present only if the assumption of Z -symmetry is dropped out. Both equations, (16) and (19),are in agreement with Ida et. al [12], except for the third,fourth, fifth and sixth terms which are characteristic ofthe proposed model. IV. THE SCENARIOS
According to the brane world scenario, it is natural toassume that the matter component consists of the branetension σ and the ordinary fields ρ sf, and ρ m , such thatthe brane densities are given by ρ = ρ sf + σ, P = p sf − σ, (20) ρ = ρ m + σ, P = p m − σ, (21)where ρ m and p m denote the energy density and the pres-sure of the matter with spin one interactions before thecollision. After the collision we can interpret ρ m and p m as radiation in the early universe (high energy regime)and as baryons plus radiation plus neutrinos, etc., in thelate universe (low energy regime). ρ sf and p sf denotethe energy density and the pressure of the self interact-ing scalar field respectively and σ is the constant branetension [12].Now we assume two scenarios on the branes, the firstone corresponds to very big densities and the second onewhen they are very small in comparison with the tension σ , that is ρ i (cid:29) σ and ρ i (cid:28) σ , i = sf, m . A. High Energy Limit in the Early Universe.
First we analyse the branes when ρ i (cid:29) σ , i = sf, m .This scenario corresponds to the universe before the col-lision. We assume the next fine tuning conditions onthe interior brane Λ = Λ (5) , Λ = − λ (5) , with k = 36 κ σ where λ (5) ∼ (10 GeV ) is of the orderof magnitude of the quantum fluctuations of the vacuum predicted by the standard model [6]. We replace the pre-vious conditions and equation (20) in equation (16) toobtain˙ a a + 1 a = κ ρ sf (cid:16) ρ sf σ (cid:17) + (4) Λ Mm a (cid:34) − σ (Λ (5) + λ (5) )2 κ σρ sf (cid:0) ρ sf σ (cid:1) + σ κ (cid:35) + 3 M m a κ σ σρ sf (cid:0) ρ sf σ (cid:1) +2 σ , (22)with (4) Λ = κ σ (5) − λ (5) )+ σ ( λ (5) +Λ (5) ) κ σρ sf (cid:0) ρ sf σ (cid:1) +2 σ κ . (23)This is because in the high energy limit it follows that ρ sf ∼ σρ sf (cid:16) ρ sf σ (cid:17) + σ , For simplicity we assume the ansatz M = − M = − M with non physical interpretation. Then, imposing ρ sf (cid:29) σ and proposing κ σ ≈ λ (5) we obtain˙ a a + 1 a ≈ κ ρ sf λ (5) + (4) Λ Mm a , (24)with (4) Λ = − λ (5) . (25)In the same way, for the exterior brane we can set thenext fine tuning condition Λ = Λ (5) . Again, if one re-place the previous conditions and equation (21) in equa-tion (19), we obtain˙ a a + 1 a = κ ρ m (cid:16) ρ m σ (cid:17) + (4) Λ Mm a (cid:34) − σ (Λ (5) + λ (5) )2 κ σρ m (cid:0) ρ m σ (cid:1) + σ κ (cid:35) + 3 M m a κ σ σρ m (cid:0) ρ m σ (cid:1) + 2 σ , (26)with (4) Λ = κ σ (5) − λ (5) )+ σ ( λ (5) +Λ (5) ) κ σρ m (cid:0) ρ m σ (cid:1) +2 σ κ . (27)Again, we have set the limit condition ρ m ∼ σρ m (cid:16) ρ m σ (cid:17) + σ , Thus, in the limit where ρ m (cid:29) σ and κ σ ≈ λ (5) , weobtain ˙ a a + 1 a ≈ κ ρ m λ (5) + (4) Λ − Mm a , (28)with (4) Λ = − λ (5) . (29)The relations (24) and (28) are the Friedmann equationsfor the early universe of the model. Observe that thedensity ρ sf appears quadratic in these equations, thus,the brane containing the scalar field inflates, even witha very small scalar field mass and collides with the ex-terior one, where the matter content is only of spin oneinteractions and the brane expands without inflation. B. Low Energy Limit in the Late Universe.
In the same way we analyse the regime ρ i (cid:28) σ , i = sf, m for the late universe using the same conditions asin the previous regime. We obtain the next two relationsfor the branes 1 and 2.˙ a a + 1 a = κ ρ sf (cid:16) ρ sf σ (cid:17) + (4) Λ Mm a (cid:34) − σ (Λ (5) + λ (5) )2 κ σρ sf (cid:0) ρ sf σ (cid:1) + σ κ (cid:35) + 3 M m a κ σ σρ sf (cid:0) ρ sf σ (cid:1) +2 σ , (30)˙ a a + 1 a = κ ρ m (cid:16) ρ m σ (cid:17) + (4) Λ Mm a (cid:34) − σ (Λ (5) + λ (5) )2 κ σρ m (cid:0) ρ m σ (cid:1) + σ κ (cid:35) + 3 M m a κ σ σρ m (cid:0) ρ m σ (cid:1) + 2 σ , with (4) Λ = κ σ (5) − λ (5) )+ σ ( λ (5) +Λ (5) ) κ σρ sf (cid:0) ρ sf σ (cid:1) +2 σ κ , (32) (4) Λ = κ σ (5) − λ (5) )+ σ ( λ (5) +Λ (5) ) κ σρ m (cid:0) ρ m σ (cid:1) +2 σ κ . (33)In the same way as before, this is because ρ i =2 σρ i (cid:0) ρ i σ (cid:1) + σ , ( i = sf, m ). Then, imposing the condi-tion ρ i (cid:28) σ , i = sf, m and again κ σ ≈ λ (5) , we obtain˙ a a + 1 a ≈ κ ρ sf + (4) Λ M m λ (5) a , (34)˙ a a + 1 a ≈ κ ρ m + (4) Λ M m λ (5) a , (35)where now we find the remarkable result that (4) Λ , = Λ (5) . Thus, we can fix the cosmological constant Λ (5) ∼ (10 − GeV ) such that (4) Λ , have the value of the clas-sical observed 4-dimensional cosmological constant [6].The relations (34) and (35) are the Friedmann equationsfor the late time universe and give the dynamics of thebranes. From the similitude of equations (34) and (35)we observe that both branes could have a very similardynamics in the late time universe. We conclude thatwe can fix the free parameter of the model in order thatboth branes expand together. V. THE DYNAMICSA. The Inflation in a Spherical Geometry.
Now we can follow the dynamics of the branes. Themain idea of the model we are dealing with here isnot to fix the initial conditions a (0) = something , a (0) = something else, etc. Instead of that we set theinitial topology of the model as follows. Suppose thereexist two concentric S branes, in the interior one with avery small radius lives a scalar field; in the exterior branelive the particles of spin one. The matter contained in theexterior brane has a state equation p = ( γ − ρ , whichdensity evolve as ˙ ρ + 3 ˙ aa γρ = 0 with the equation (13).During the early universe the expansion of the brane fol-lows equation (28), thus the exterior brane evolves as˙ a + 1 = κ ρ a γ − λ (5) + Λ a − Mm a , (36)or, if we derive this expression with respect to t , we arriveat 2 ¨ a = − (6 γ − κ ρ a γ − λ (5) + 2 Λ a + 2 Mm a . (37)If during this period the exterior brane is dominated bymatter, the first term goes like ∼ a − , if this brane isdominated by radiation, this term goes like ∼ a − . In anycase, during this period the first term dominates over theother two, indicating that this brane starts with a verybig negative acceleration but decreasing its accelerationvery fast.On the other brane the situation is very different dueto the fact that the interior brane is permeated with ascalar field, its evolution is governed by equation (24).During the epoch when the density is enough big, suchthat the two last terms on the right hand side of equations(24) can be neglected, in the slow-roll approximation thisequation reduces to H (cid:39) (cid:32) π m (cid:33) V (cid:20) V σ (cid:21) , (38)˙Φ (cid:39) − V (cid:48) H . (39)where V is the scalar field potential. Then, using the twoslow-roll parameters (cid:15) (cid:39) κ (cid:18) V (cid:48) V (cid:19) V /σ (2 +
V /σ ) , (40) η ≡ m π (cid:18) V (cid:48)(cid:48) V (cid:19) (cid:20) σ σ + V (cid:21) , (41)if, for example, the scalar field potential is an exponentialone V = V exp( − ακ (4) φ ), (cid:15) is given by [8] (cid:15) (cid:39) α σV . (42)During this period V /σ >> (cid:15) < (cid:15) ∼
1, this is, till the scalar field po-tential reaches the value V end ∼ α σ . If the potential is V = m φ φ , the slow-roll parameter reads (cid:15) ∼ κ φ V /σ (2 +
V /σ ) = m P l π φ V /σ (2 +
V /σ ) , that means that if V >> σ , inflation ends when the scalarfield reaches the value φ ∼ π σV m P l . For both poten-tials, the interior brane inflates and collide with the exte-rior one, heating the branes very fast. The heating of thebranes essentially depends on the interaction constantbetween the scalar field and the matter and, of course,on the speed of the collision. Inflation in the interiorbrane also grows the quantum fluctuations of the scalarfield, in such a way that after inflation ends, the inte-rior brane contains a spectrum of semiclassical potentialwells which become the seeds for the structure formationin the exterior brane. Between these two regimes, duringthe branes collision, the physics of the system is just likein the Ekpyrotic model [3]. During the collision, quan-tum gravitational effects due to the interactions of bothspace-times take place. From this moment, both branesfollow the dynamics of equations (34) and (35), indicat-ing that both branes expand with the same dynamics.The scalar field riches the minimum of its potential im-plying that, from this period on, the scalar field potentialhas a behaviour like ∼ φ . The scalar field in the inte-rior brane induces potential wells into the exterior onevia gravitational interaction, which evolve as cold darkmatter (see [16]). B. The Scalar Field as Dark Matter.
As we have explained before, the scalar field confinedon the interior brane produces potential wells along thewhole brane, caused by the extreme growing of the scalarfield fluctuations during inflation. On the exterior brane,matter feels this potential wells as an extra gravitationalforce, but it cannot detect the scalar field in any otherway. In order to fit cosmological and galactic obser-vations, the scalar field mass measured in the exteriorbrane must be ultra-light, such that m sfdm ∼ − eV.This ultra-light mass causes, of course, a problem of hi-erarchy, however this problem can be solved using theprescription of Randall-Sundrum [2] as follows.The scalar field part of the action on the interior branereads S interior = (cid:90) d x £ sf = (cid:90) d x √− g (cid:2) g µν φ ,µ φ ,ν + m sf φ (cid:3) . (43)Using normal Gaussian coordinates for the AdS metric ds = e − ky η µν dx µ dx ν + dy , action (43) translated intothe exterior brane reads S exterior = (cid:90) d x (cid:2) η µν φ ,µ φ ,ν + m sfdm φ (cid:3) , (44)with the identification e ξy φ −→ φ and e − ξy m sf = m sfdm , where ξ = Λ6 M . As long as the two branesexpand together, y remains constant after the collision.Thus, with this prescription it is enough that ξy (cid:118) m sf ∼ × M (5) [30]. That means, we obtain the required ultra-light massof the scalar field on the exterior brane through a scalarfield on the interior one. Furthermore, this ultra-lightmass sets a minimum distance between the branes.With this mass, that means, with only one free pa-rameter, the scalar field on the exterior brane behavesexactly in the same way as Scalar Field Dark Matter(SFDM) [15] with the following important features:The ultra-light scalar field mass ( m SF DM ∼ − eV)fits:1. The evolution of the cosmological densities [16].2. The rotation curves of galaxies [31] and the centraldensity profile of LSB galaxies [5],3. With this mass, the critical mass of collapse for areal scalar field is just 10 M (cid:12) , i.e., the one obe-served in galaxies haloes [4].4. The central density profile of the dark matter is flat[5].5. The scalar field has a natural cut off, thus the sub-structure in clusters of galaxies is avoided natu-rally. With a scalar field mass of m φ ∼ − eVthe amount of substructure is compatible with theobserved one [17].6. SFDM forms galaxies earlier than the cold darkmatter model, because they form Bose-EinsteinCondensates at a critical temperature T c >> TeV.So, if SFDM is right, we have to see big galaxies atbig redshifts.
VI. CONCLUDING REMARKS
The question we are facing on here is the following; isit possible that the universe has the dynamics we observebecause of its global topology? This question makes senseif we start from the 4D general Einstein equations invacuum, R µν − / g µν R − Λ g µν = 0. This equation iscompletely geometrical, there is no matter content in it.The Friedman equation for this case is˙ a a + ka − κ , where k = 0 , ± S or H topology respectively.Or, in a more convenient form for our goal here, we canwrite it as 12 ˙ a − κ a = − k, which can be seen as a (1 /
2) ˙ a + V = E dynamical equa-tion. We observe that even when there is no matter, thespace-time posses a dynamics. Even if there is no cosmo-logical constant Λ = 0, it is enough that k (cid:54) = 0 to have adynamical universe. In other words, it is sufficient thatat the beginning, the global topology is fixed to have aspecific dynamics.In this work we consider a special global topology ofthe universe, namely, two concentric spherical branes andsome content of matter with different interactions in eachregion of the universe: fundamental interactions of spinzero confined in the interior brane, fundamental interac-tions of spin one confined in the exterior brane (the branethat will lead to the standard model) and gravitation inthe bulk. We find that the spin zero interactions inflatethe interior brane, making this brane to collide with theexterior one due to the concentric topology, we assumethe branes heat because of a small interactions constantbetween the spin zero and the spin one particles.After the collision, both branes expand together asshown by equations (34) and (35). The scalar field fluc-tuations build potential wells which are seen as SFDM inthe exterior brane via the RS prescription, with the ideashowed in the equations (43), (44). However, in the exte-rior brane we expect that the collision provokes the exis-tence of particles with fractional spin and spin one, givingthe zoo of particles predicted by the standard model likein the Ekpyrotic or Cyclic models [29]. This idea is veryinteresting, but so far there does not exist mathematicalevidence to back it up. It is important to remark thatduring the collision it is impossible to analyse all the in-teractions with the classical theory of general relativitydue to the quantum-gravitational effects and the interac-tions between the fields that permeates both branes.Thus, in this model the universe expands as we see it,but the particles of the standard model can not inter-act with the scalar field particles because they are con-fined in the interior brane and the standard model onesare confined in the exterior one. The five-dimensionalcosmological constant Λ is pure geometrical and can beseen as a second free constant of this toy model. Theextreme difference between the expectation values in theinterior and exterior branes can be explained using ananalogy with electromagnetism. If the fields in the 5Dbulk produced by the branes, behaves like a electromag-netic field, remember that for a weak field, linearised Ein-stein equations are analogous to the electromagnetic one,then there exist a huge expectation value inside of the re-gion between the two branes which is associated with λ ,while in the exterior region the expectation value is zero.The benefit of working with spherical branes is the nat-ural acceleration predicted by this kind of models wherethe acceleration starts at redshift z > . VII. ACKNOWLEDGEMENT
We want to acknowledge the enlightening conversationwith Roy Maartens during his visit in Leon Guanaju- ato, M´exico and many helpful discussions with RubenCordero, Abdel P´erez Lorenzana and Luis Ure˜na. Thiswork was partially supported by CONACyT M´exico, un-der grants 49865-F, 216536/219673 and I0101/131/07 C-234/07, Instituto Avanzado de Cosmologia (IAC) collab-oration. [1] Cordero Ruben and Vilenkin Alexander., et al ., Phys.Rev. D , 083519. arXiv.hep-th/0107175[2] Randall Lisa and Sundrum Raman., et al ., Phys. Rev.Lett. , 149-160 (2008), arXiv:astro-ph/0303455[6] Carroll Sean M. Living Reviews in Relativity, (1999),arXiv:astro-ph/0004075v2[7] Coley Alan A. Proceedings of ERE-99, arXiv:gr-qc/9910074v1[8] Copeland Edmund J, Liddle Andrew R. and LidseyJames E et al ., Phys. Rev. D , 023539 (2002).[9] Dodelson Scott. Academic Press, Copyright 2003, Else-vier[10] Dvali G., Gabadadze G., Porrati M. et al ., Phys. Lett. B et al Phys. Lett. B et al ., Phys. Rev. D :123522 (2001)[14] Knop R. A, et al ., Astrophys. J.
102 (2003) arXiv:astro-ph/0309368; A. G. Riess, et al ., Astrophys. J.
665 (2004) 665 arXiv: astro-ph/0402512[15] Matos Tonatiuh and Siddhartha Guzm´an F. et al ., Class. Quant. Grav. , L9-l16 (2000), arXiv: gr-qc /9810028[16] Matos Tonatiuh, Vazquez Alberto and Maga˜na J.A. et al MNRAS., , 13957 (2009) arXiv:0806.0683[17] Matos Tonatiuh and Ure˜na Luis A., et al ., Phys Rev. D , 063506 (2001), arXiv:astro-ph/ 0006024[18] Langlois D., Maartens R., Sasaki M., Wands D. et al .,Phys. Rev. D , 084009 (2001), arXiv:hep-th/0012044[19] Liddle A. and Ure˜a-L´opez L., et al ., Phys. Rev. Lett. , 161301 (2006).[20] Lidsey James, Matos Tonatiuh and Ure˜na Luis A. et al .,Phys Rev. D , 023514 (2002), arXiv:astro-ph/ 0111292[21] Maartens Roy., et al Living Rev. Rel.7:7, (2004),arXiv:gr-qc/0312059v2[22] Maartens R., Wands D., Bassett B.A. and Heard I.P.C. et al
Phys. Rev. D , 041302 (2000).[23] Mac Fadden P., PhD thesis. arXiv:hep-th/0612008v2[24] Maeda, Mizuno, Torii., et al ., Phys. Rev. D , 024033(2003)[25] Niz Quevedo Gustavo., PhD thesis, (2006)[26] Lorenzana Abdel., et al ., J.Phys.Conf.Ser. et al Phys. Rev. D et al ., JCAP0706