Cosmological and Astrophysical Limits on Brane Fluctuations

PHYSICAL REVIEW D 68, 103505 ͑2003͒

Cosmological and astrophysical limits on brane ﬂuctuations

J. A. R. Cembranos, A. Dobado, and A. L. Maroto Departamento de Fı´sica Teo´rica, Universidad Complutense de Madrid, 28040 Madrid, Spain ͑Received 11 July 2003; published 17 November 2003͒ We consider a general brane-world model parametrized by the brane tension scale f and the branon mass M. For a low tension compared to the fundamental gravitational scale, we calculate the relic branon abundance and its contribution to the cosmological dark matter. We compare this result with the current observational limits on the total and hot dark matter energy densities and derive the corresponding bounds on f and M. Using the nucleosynthesis bounds on the number of relativistic species, we also set a limit on the number of light branons in terms of the brane tension. Finally, we estimate the bounds coming from the energy loss rate in supernovae explosions due to massive branon emission.

DOI: 10.1103/PhysRevD.68.103505 PACS number͑s͒: 95.35.ϩd, 11.10.Kk, 11.25.Ϫw

I. INTRODUCTION nario ͑BWS͒, which is becoming one of the most popular extensions of the SM. In these models, the standard model The increasing observational precision is making cosmol- particles are bound to live on a three-dimensional brane em- ogy a useful tool in probing certain properties of particle bedded in a higher dimensional (Dϭ4ϩN) space-time, physics theories beyond the standard model ͑SM͒. The limits whereas gravity is able to propagate in the whole bulk space. on the neutrino masses and the number of neutrino families The fundamental scale of gravity in D dimensions M D can are probably the most clear examples ͓1,2͔. In some cases, be lower than the Planck scale M P . In the original proposal the cosmological bounds are complementary to those ob- in ͓3͔, the main aim was to address the hierarchy problem, tained from colliders experiments, and therefore the combi- and for that reason the value of M D was taken around the nation of both allows us to restrict the parameter space of a electroweak scale. However more recently, brane cosmology theory in a more efficient way. models have been proposed in which M D has to be much There are two main ways in which cosmology can help. larger than the TeV ͓4,5͔. In this work we will consider a On one hand we have the relative abundances of the light general BWS with arbitrary fundamental scale M D . elements, which is one of the most precise predictions of the The existence of extra dimensions is responsible for the standard cosmological model. Indeed, the calculations are appearance of new fields on the brane. On one hand, we have very sensitive to certain cosmological parameters and thus, the tower of Kaluza-Klein modes of fields propagating in the for instance, the production of 4He increases with the rate of bulk space, i.e. the gravitons. On the other, since the brane expansion of the universe H. A successful nucleosynthesis has a finite tension f 4, its fluctuations will be parametrized requires that H should not deviate from its standard value by some ␲␣ fields called branons. These fields, in the case in more than around 10% during that epoch. Since H depends which translational invariance in the bulk space is an exact on the effective number of relativistic degrees of freedom symmetry, can be understood as the massless Goldstone geff(T) at a given temperature T, the above constraint trans- bosons arising from the spontaneous breaking of that sym- ϳ ͓ ͔ lates into a bound on geff(Tnuc), where Tnuc 1 MeV. metry induced by the presence of the brane 6,7 . However, Apart from the number of particle species, cosmology also in the most general case, translational invariance will be ex- sets a limit on their energy density. For particles which are plicitly broken and therefore we expect branons to be mas- nonrelativistic at present, there exists an upper bound given sive fields. ⍀ ϭ Ϯ ͓ ͔ by the measured dark matter density M 0.23 0.08 at the It has been shown 8 that when branons are properly 95% C.L. ͓1͔. taken into account, the coupling of the SM particles to On the other hand, stars also provide useful information any bulk field is exponentially suppressed by a factor ͓Ϫ 2 2 ␲2 4 ͔ on theories containing light and weakly interacting particles. exp MKKMD/(8 f ) , where M KK is the mass of the cor- The extreme opacity of ordinary matter to photons means responding KK mode. As a consequence, if the tension scale that it takes a very long time for a photon produced in the f is much smaller than the fundamental scale M D , i.e. Ӷ center of the star to reach its surface. Indeed, this fact ex- f M D , the KK modes decouple from the SM particles. plains the longevity of stars. However, particles like neutri- Therefore, for flexible enough branes, the only relevant de- nos or axions, still can be produced abundantly in nuclear grees of freedom at low energies in the BWS are the SM reactions in the core of the star, but since they are weakly particles and branons. interacting, the rate at which they can carry away energy can The phenomenological implications of KK gravitons for be much larger. This was particularly evident in the 1987A colliders physics, cosmology and astrophysics have been supernova explosion, in which most of the energy was re- studied in a series of papers ͑see ͓9͔ and references therein͒, leased in the form of neutrinos. Again, this fact can be used and the corresponding limits on M D and/or the number of to set limits on the mass and couplings of the new particles. extra dimensions N have been obtained. In the case of bran- In this paper we will study the constraints that cosmology ons, the potential signatures in colliders have been studied in and astrophysics impose on the so called brane-world sce- the massless case in ͓10͔ and in the massive one in ͓11͔.

Limits from supernovae and modifications of Newton’s law assume that B is a homogeneous space, so that brane fluc- at small distances in the massless case were obtained in ͓12͔. tuations can be written in terms of properly normalized co- ␣ ␣ Moreover in ͓13͔ the interesting possibility that massive bra- ordinates in the extra space: ␲ (x)ϭ f 2Y (x), ␣ϭ1,...,N. nons could account for the observed dark matter of the uni- The induced metric on the brane in its ground state is simply verse was studied in detail. The main aim of the paper is to given by the four-dimensional components of the bulk space analyze the cosmological and astrophysical limits on the metric, i.e. g␮␯ϭ˜g␮␯ϭG␮␯ . However, when brane excita- BWS through the effects due to massive branons. To that end tions are present, the induced metric is given by we will be assuming that the evolution of the universe is M N ␯Y G x,Y͑x͒ץ ␮Yץg␮␯ϭ standard up to a temperature around f. Indeed, this is the case MN„ … of realistic brane cosmology models in five dimensions ͓4͔. m n ␯Y ˜gЈ Y͑x͒ . ͑2͒ץ ␮Yץϭ˜g␮␯ x,Y͑x͒ Ϫ Also more recently, six-dimensional models have been pro- „ … mn„ … posed in which the Friedmann equation has the standard The contribution of branons to the induced metric is then form for arbitrary temperature ͓5͔. obtained expanding Eq. ͑2͒ around the ground state The paper is organized as follows. In Sec. II we give a ͓7,14,11͔: brief introduction to the dynamics of massive branons. Sec- tion III contains a summary of the main steps used in the 1 ␣ ␤ ␯␲ץ ␮␲ץ␤␣␦ g␮␯ϭ˜g␮␯Ϫ standard calculations of relic abundances generated by the f 4 freeze-out phenomenon in an expanding universe. In Sec. IV, we give our results for the thermal averages of branon anni- 1 2 ␣ ␤ ϩ ˜g␮␯M ␣␤␲ ␲ ϩ ... . ͑3͒ hilation cross sections into SM particles. Section V is de- 4 f 4 voted to the study of the limits on the branon mass and on the brane tension scale from the cosmological dark matter Branons are the mass eigenstates of the brane fluctuations in abundances. In Sec. VI, after reviewing the limits imposed the extra-space directions. The branon mass matrix M ␣␤ is by nucleosynthesis on the number of relativistic species, we determined by the metric properties of the bulk space and, in apply them to the case of branons. Section VII contains the the absence of a general model for the bulk dynamics, we calculation of the rate of energy loss from a supernova core will consider its elements as free parameters ͑for an explicit in the form of branons and the corresponding limits on f and construction see ͓15͔͒. Therefore, branons are massless only ͓ ͔ M. Finally, Sec. VIII includes the main conclusions of the in highly symmetric cases 7,14,11 . →ϱ paper. In an Appendix we have included the explicit formu- Since in the limit in which gravity decouples M D , ͓ ͔ las for the creation and annihilation cross sections of branons branon fields still survive 16 , branon effects can be studied pairs. independently of gravity. The mechanism responsible for the creation of the brane is in principle unknown, and therefore we will assume that the brane dynamics can be described by II. THE BRANON FIELD a low-energy effective action derived from the Nambu-Goto In this section we will briefly review the main properties action ͓7͔. Also, branon couplings to the SM fields can be of massive brane fluctuations ͑see ͓7,14,11͔ for a more de- obtained from the SM action defined on a curved background tailed description͒. We will consider a single-brane model in given by the induced metric ͑2͒, and expanding in branon ␲ large extra dimensions. Our four-dimensional space-time M 4 fields. Thus, the complete action, up to second order in is embedded in a D-dimensional bulk space which, for sim- fields, contains the SM terms, the kinetic term for the bran- ϭ ϫ plicity, we will assume to be of the form M D M 4 B. The ons and the interaction terms between the SM particles and B space is a given N-dimensional compact manifold, so that the branons: ϭ ϩ D 4 N. The brane lies along M 4 and we neglect its contribution to the bulk gravitational field. The coordinates pa- ϭ ͵ 4 ͱ ͓Ϫ 4ϩL ͑ ͔͒ ␮ SB d x g f SM g␮␯ m M rametrizing the points in M D will be denoted by (x ,y ), 4 where the different indices run as ␮ϭ0,1,2,3 and m ϭ1,2,...,N. The bulk space M is endowed with a metric 4 ͱ 4 D ϭ ͵ ˜ͫ Ϫ ϩL ͑˜ ␮␯͒ ϩ d x g f SM g tensor which we will denote by GMN , with signature ( , M 4 Ϫ,Ϫ ...Ϫ,Ϫ). For simplicity, we will consider the following ansatz: 1 ␮␯ ␣ ␤ ␯␲ץ ␮␲ץ␤␣␦ ϩ ˜g 2 ˜g␮␯͑x,y͒ 0 G ϭͩ ͪ . ͑1͒ 1 1 MN Ϫ˜Ј ͑ ͒ 2 ␣ ␤ ␣ ␤ ␯␲ץ ␮␲ץ␤␣␦gmn y Ϫ M ␣␤␲ ␲ ϩ ͑4 0 2 8 f 4 The position of the brane in the bulk can be parametrized as M ␮ m Y ϭ x ,Y (x) , with Mϭ0,...,3ϩN and where we have 2 ␣ ␤ ␮␯ „ … ϪM ␲ ␲ ˜g␮␯͒T ͑˜g␮␯͒ͬϩ ... ͑4͒ chosen the bulk coordinates so that the first four are identi- ␣␤ SM fied with the space-time brane coordinates x␮. We assume m ␮␯ ˜ the brane to be created at a certain point in B, i.e. Y (x) where TSM(g␮␯) is the conserved energy-momentum tensor ϭ m Y 0 which corresponds to its ground state. We will also of the standard model evaluated in the background metric.

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␦L ber. Notice that we are considering only the lowest order ␮␯ ϭϪͩ˜ ␮␯L ϩ SMͪ ͑ ͒ Lagrangian and assuming that all the branons have the same TSM g SM 2 . 5 ␦˜g␮␯ mass. This implies that each branon species evolves independently. Therefore in the following we will drop the ␣ index. It is interesting to note that under a parity transformation The thermal average ͗␴ v͘ of the total annihilation cross on the brane, the branon field changes sign if the number of A section times the relative velocity is given by spatial dimensions of the brane is odd, whereas it remains unchanged for even dimensions. Accordingly, branons on a 3 3 ͑ ͒ 3-brane are pseudoscalar particles. This implies that if we 1 d p1 d p2 w s ͗␴ v͘ϭ ͵ f ͑E ͒f ͑E ͒ ͑8͒ want to preserve parity on the brane, terms in the effective A 2 ͑ ␲͒3 ͑ ␲͒3 1 2 E E neq 2 2 1 2 Lagrangian with an odd number of branons would be forbid- den. where The quadratic expression in Eq. ͑4͒ is valid for any internal B space, regardless of the particular form of the metric ␴ 2 s A 4M ˜gЈ . In fact the low-energy effective Lagrangian is model w͑s͒ϭE E ␴ v ϭ ͱ1Ϫ . ͑9͒ mn 1 2 A rel 2 s independent and is parametrized only by the number of branon fields, their masses and the brane tension. The depen- The Mandelstam variable s can be written in terms of the dence on the geometry of the extra dimensions will appear at components of the four momenta of the two branons p and higher orders. These effective couplings thus provide the 1 ϭ ϩ 2ϭ 2ϩ Ϫ͉ ជ ͉͉ជ ͉ ␪ necessary tools to compute cross sections and expected rates p2 as s (p1 p2) 2(M E1E2 p1 p2 cos ). Assum- of events involving branons in terms of f and the branons ing vanishing chemical potential, the branon distribution masses only. functions are From the previous expression, we see that since branons interact by pairs with the SM particles, they are necessarily 1 ͑ ͒ϭ ͑ ͒ stable. In addition, their couplings are suppressed by the f E 10 eE/Tϩa brane tension f 4, which means that they could be weakly interacting, and finally, according to our previous discussion, ϭ in general, they are expected to be massive. As a conse- with a 0 for the Maxwell-Boltzmann distribution and aϭϪ1 for the Bose-Einstein distribution. In the case of non- quence their freeze-out temperature can be relatively high, Ӷ which implies that their relic abundances can be cosmologi- relativistic relics T 3M, the Maxwell-Boltzmann distribu- cally important. tion is a good approximation and we will use it for simplicity instead of the Bose-Einstein distribution. Finally, the equilibrium abundance is given by III. RELIC BRANON ABUNDANCES

In order to calculate the thermal relic branon abundance, d3p we will use the standard techniques given in ͓17,18͔ in two n ϭ ͵ f ͑E͒. ͑11͒ eq ͑ ␲͒3 limiting cases, either relativistic ͑hot͒ or nonrelativistic 2 ͑cold͒ branons at decoupling. In this section we will review the basic steps of the calculation method. From Eq. ͑4͒, the thermal average will include, to leading ␣ The evolution of the number density n␣ of branons ␲ , order, annihilations into all the SM particle-antiparticle pairs. ␣ϭ1,...,N with N the number of different types of branons, If the universe temperature is above the QCD phase transi- Ͼ interacting with SM particles in an expanding universe is tion (T Tc), we consider annihilations into quark-antiquark Ͻ given by the Boltzmann equation, and gluons pairs. If T Tc we include annihilations into light hadrons. For the sake of definiteness we will take a critical dn␣ Ӎ 2 eq 2 temperature Tc 170 MeV and a Higgs boson mass mH ϭϪ3Hn␣Ϫ͗␴ v͓͘n Ϫ͑n ͒ ͔ ͑6͒ dt A ␣ ␣ Ӎ125 GeV, although the final results are not very sensitive to the concrete value of these parameters. where In order to solve the Boltzmann equation we introduce the new variables. xϭM/T and Yϭn/s with s the universe en- ␴ ϭ ␴͑␲␣␲␣→ ͒ ͑ ͒ tropy density. We will assume that the total entropy of the A ͚ X 7 3 X universe is conserved, i.e. Sϭa sϭconst, where a is the scale factor of the universe and we will make use of the is the total annihilation cross section of branons into SM Friedmann equation particles X summed over final states. The Ϫ3Hn␣ term, with H the Hubble parameter, takes into account the dilution of 8␲ the number density due to the universe expansion. These are 2ϭ ␳ ͑ ͒ H 2 12 the only terms which could change the number density of 3M P branons. In fact, since branons are stable they do not decay into other particles and since they interact always by pairs where the energy density in a radiation dominated universe is the conversions like ␲␣X→␲␣Y do not change their num- given by

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2 ⌫ ␲ which can be solved explicitly for T f , expanding A(T f ) for ␳ϭg ͑T͒ T4. ͑13͒ ӷ eff 30 T f M/3. Once we know x f , the relic abundance today Ӎ ͑ ͒ ͓Y ϱ Y(x f )͔ is given by Eq. 16 . From this expression we In a similar way, the entropy density reads can obtain the current number density of branons and the corresponding energy density which is given by 2␲2 sϭh ͑T͒ T3 ͑14͒ 1 M eff 45 ⍀ 2ϭ ϫ Ϫ2 ͑ ͒ Brh 7.83 10 ͒ . 19 heff͑x f eV where geff(T) and heff(T) denote the effective number of relativistic degrees of freedom contributing to the energy The calculation of the decoupling temperature in the case density and the entropy density respectively at temperature T of cold branons is more involved. The well-known result is (T being the temperature of the photon background͒. Notice given by Ͼ Ӎ that for T MeV we have heff geff . Using these expres- ϩ ͒ ␴ sions we get 0.038c͑c 2 M PM͗ Av͘ x ϭlnͩ ͪ ͑20͒ f 1/2 1/2 ␲ 2 1/2 geffx f dY M P heffM ϭϪͩ ͪ ͗␴ v͑͘Y 2ϪY 2 ͒ ͑15͒ dx 45 1/2 2 A eq geffx where cӍ0.5 is obtained from the numerical solution of the Boltzmann equation. This equation can be solved iteratively. where we have ignored the possible derivative terms The corresponding energy fraction reads dheff /dT. ϱ Ϫ The qualitative behavior of the solution of this equation x c 1 goes as follows: if the annihilation rate defined as ⌫ ⍀ 2ϭ ϫ Ϫ11 Ϫ2 f ͩ ͚ n Ϫn ͪ A Brh 8.77 10 GeV 1/2 x f ϭ ␴ nϭ0 nϩ1 neq͗ Av͘ is larger than the expansion rate of the universe geff Ӎ ͑ ͒ H at a given x, then Y(x) Y eq(x), i.e., the branon abun- 21 dance follows the equilibrium abundances. However, since ⌫ ͗␴ ͘ Ϫ1 A decreases with the temperature, it eventually becomes where we have expanded Av in powers of x as ϭ similar to H at some point x x f . From that time on branons are decoupled from the rest of matter or radiation in the ϱ ␴ ϭ Ϫn ͑ ͒ universe and its abundance remains frozen, i.e. Y(x) ͗ Av͘ ͚ cnx . 22 Ӎ у ͑ ͒ nϭ0 Y eq(x f ) for x x f . For relativistic hot particles, the equilibrium abundance reads ϰ ␴ Notice that in general, Y ϱ 1/͗ Av͘, i.e. the weaker the cross section the larger the relic abundance. This is the ex- 45␨͑3͒ 1 Y ͑x͒ϭ , ͑xӶ3͒ ͑16͒ pected result, since, as commented before the sooner the de- eq 4 ͑ ͒ 2␲ heff x coupling occurs, the larger the relic abundance, and decoupling occurs earlier as we decrease the cross section. whereas for cold relics Therefore the cosmological bounds work in the opposite way as compared to those coming from colliders. Thus, a bound 1/2 ⍀ ϽO 45 ␲ 1 such as Br (1) translates into a lower limit for the cross Y ͑x͒ϭ ͩ ͪ x3/2 eϪx, ͑xӷ3͒. ͑17͒ eq 4 ͑ ͒ sections and not into an upper limit as those obtained from 2␲ 8 heff x nonobservation in colliders. In the following we apply the previous formalism to ob- We see that for hot branons the equilibrium abundance is not tain the relic abundance of branons ⍀ h2, both when they very sensitive to the value of x. In the case of cold relics Br are relativistic and nonrelativistic at decoupling. For that pur- however, Y eq decreases exponentially with the temperature, ␴ pose we need to evaluate the thermal averages ͗ Av͘ for the which implies that the sooner the decoupling occurs the Ϯ annihilation of branons into photons, massive W and Z larger the relic abundance. gauge bosons, three massless neutrinos, charged leptons, Let us first consider the simple case of hot branons. Since quarks and gluons ͑or light hadrons͒ and a real scalar Higgs its equilibrium abundance depends on x only through f boson field, in terms of the brane tension f and the branon h (x ), the relic abundance is not very sensitive to the eff f mass M. exact time of decoupling. In this case, in order to calculate ϭ the decoupling temperature T f M/x f , it is a good approxi- ⌫ ϭ mation to use the condition A H. From the explicit ex- IV. BRANON ANNIHILATION CROSS SECTIONS: pression of the Hubble parameter in a radiation dominated THERMAL AVERAGE universe we have We give the results for the different channels contributing 2 to the thermal average of the annihilation cross section T f ␴ ͑ ͒ϭ 1/2 ͑ ͒ ϭ⌫ ͑ ͒ ͑ ͒ ͗ Av͘ of branons into SM particles. The explicit production H T f 1.67geff T f A T f 18 M P and annihilation cross section can be found in the Appendix.

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For cold relics, we have expanded the expressions for each 2 1 m␺ particle species in powers of 1/x as follows: ϭ 2 2 ͑ 2Ϫ 2 ͒ͱ Ϫ c1 M m␺ 67M 31m␺ 1 192␲2 f 8 M 2 ͑25͒ 1 1 ␴ ϭ ϩ ϩ ϩO Ϫ3͒ ͑ ͒ ͗ Av͘ c0 c1 c2 ͑x . 23 x x2 1 M 2 6 4 2 c ϭ ͑17 408M ϩ13 331M m␺ 2 2 8 2 2 7680␲ f M Ϫm␺ In the hot branons case, we give the results for the different contributions to the decay rate ⌫ ϭn ͗␴ ͘, where we 2 A eq Av m␺ Ϫ 2 4 ϩ 6 ͒ͱ Ϫ ͑ ͒ have considered the ultrarelativistic limit for the branons, i.e. 46 606M m␺ 18 927m␺ 1 . 26 2 Mϭ0. For massive SM particles, the ﬁnal expressions can- M not be given in closed form. Therefore, in this section, in Notice that this expansion is not valid near SM particles order to show the high-temperature behavior, we only give thresholds, i.e. for branon masses close to some SM particle the results for fermions, gauge bosons and scalars in the limit mass. In addition, since the c coefﬁcient is different from in which their masses vanish. Also in this case, we have used 0 zero, annihilation will mainly take place through the s-wave. the Bose-Einstein form as the equilibrium distribution. x™3 (Hot) A. Dirac fermions For massless fermions, we obtain xš3 (Cold)

2 9 9 1 m␺ 8␲ T ϭ 2 2 ͑ 2Ϫ 2 ͒ͱ Ϫ ͑ ͒ ⌫Diracϭ ϩO͑ ͒ ͑ ͒ c0 M m␺ M m␺ 1 24 A x . 27 16␲2 f 8 M 2 297 675␨͑3͒f 8

B. Massive gauge ﬁeld xš3 (Cold)

m2 2ͱ Ϫ Z ͑ 4Ϫ 2 2 ϩ 4 ͒ M 1 4M 4M mZ 3mZ M 2 ϭ c0 64f 8␲2

m2 2ͱ Ϫ Z ͑ 6Ϫ 4 2 ϩ 2 4 Ϫ 6 ͒ M 1 364M 584M mZ 349M mZ 93mZ M 2 c ϭ 1 8͑ 2Ϫ 2 ͒␲2 768f M mZ

2 mZ M 2ͱ1Ϫ M 2 c ϭ ͑415 756M 8Ϫ755 844M 6m2 ϩ356 541M 4m4 Ϫ76 294M 2m6 ϩ56 781m8 ͒. 2 8͑ 2Ϫ 2 ͒2␲2 Z Z Z Z 30 720f M mZ ͑28͒ Again this expansion is not valid near SM particles thresholds, and the leading term corresponds to the s-wave.

x™3 (Hot) ӷ In the limit where T mZ , one can obtain the following expression: 8␲9T9 ⌫Z ϭ ϩO͑ ͒ ͑ ͒ A x . 29 99 225␨͑3͒f 8

C. Massless gauge ﬁeld xš3 (Cold) ϭ ͑ ͒ c0 0 30

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ϭ c1 0

68M 6 ϭ c2 . 15f 8␲2 ϭ ϭ In this case, c0 c1 0 and the leading term corresponds to d-wave annihilation.

x™3 (Hot)

16␲9T9 ⌫␥ ϭ ϩO͑ ͒ ͑ ͒ A x . 31 297 675␨͑3͒f 8

D. Complex scalar ﬁeld xš3 (Cold)

2 m⌽ 2͑ 2ϩ 2͒2ͱ Ϫ M 2M m⌽ 1 M 2 ϭ ͑ ͒ c0 32 32f 8␲2

2 m⌽ 2͑ 2ϩ 2͒ͱ Ϫ ͑ 4Ϫ 2 2Ϫ 4͒ M 2M m⌽ 1 182M 115M m⌽ 31m⌽ M 2 c ϭ 1 8 2 2 2 384f ͑M Ϫm⌽ ͒␲

2 m⌽ M 2ͱ1Ϫ M 2 8 6 2 4 4 2 6 8 c ϭ ͑92 164M Ϫ123 556M m⌽ ϩ12 269M m⌽ ϩ9754M m⌽ ϩ6309m⌽ ͒. 2 8 2 2 2 2 5120f ͑M Ϫm⌽ ͒ ␲

We ﬁnd the same problems near SM particle thresholds. The dominant contribution is the s-wave.

x™3 (Hot) V. COSMOLOGICAL BOUNDS FROM THE DARK MATTER ENERGY DENSITY For a massless scalar, one can obtain the following expression: For cold branons, once we know the cn coefficients for the total cross section, we can compute the freeze-out value ␲9 9 ⌽ 16 T ͑ ͒ ⌫ ϭ ϩO͑x͒. ͑33͒ x f from Eq. 20 and the relic contribution to the energy A 8 ⍀ 2 ͑ ͒ 297 675␨͑3͒f density of the universe Brh from Eq. 21 in terms of f and M. Imposing the observational limit on the total dark matter For real scalar fields, the above results should be divided by energy density from the Wilkinson Microwave Anisotropy two. ͑ ͒ ⍀ 2Ͻ Ϫ Probe WMAP : Brh 0.129 0.095 at the 95% C.L. Notice that for conformal matter the leading contribution which corresponds to ⍀ ϭ0.23Ϯ0.08 and hϭ0.79Ϫ0.65 is the d-wave. This explains why in the massless limit for M ͓1͔, we obtain the exclusion plots in Figs. 1 and 6. fermions, the leading contribution is no longer the s-wave, ϭ but the d-wave, whereas for massless scalars or taking the Notice that in Fig. 6 we have plotted the x f 3 curve, → which limits the range of validity of the cold relic approxi- mZ 0 limit for massive gauge bosons, the s-and p-waves survive. mation. Therefore, the excluded region is that between the Concerning the validity of the above results, in order to two curves. It is also important to note that for those values avoid the mentioned problems of the Taylor expansion near of the parameters on the solid line, branons would constitute SM thresholds, we have taken branon masses sufficiently all the dark matter in the universe. separated from SM particle masses where the usual treatment For hot branons, we have computed numerically the total ⌫ ͑ ͒ is adequate ͓17,18͔. Such treatment is known to introduce annihilation rate into SM particles A . Using Eq. 18 ,we errors of the order of 10% in the relic abundances. In addi- can find the freeze-out temperature T f in terms of the brane tion, coannihilation effects are absent in this case since there tension scale f. Approximately, the relation between the loga- are no slightly heavier particle which eventually could decay rithms of these quantities is linear: log10( f /1 GeV) Ӎ ϩ into the lightest branon. (7/8)log10(T f /1 GeV) 2.8. This expression is almost in

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FIG. 3. Exclusion plot for hot relics in models with N ϭ1,2,3,7 branons. For a given N, the curve corresponds to the hot ⍀ 2ϭ FIG. 1. Exclusion plot from cold relics abundance. The thick dark matter limit Brh 0.0076, therefore the region above such a ⍀ 2ϭ Ϫ ϭ ϭ lines correspond to Brh 0.129 0.095 for N 1 and N 7. curve is excluded by hot branon overproduction. ⍀ 2 Therefore the areas above the curves corresponding to Brh Ͼ0.129 are excluded. The striped region contains the curves corresponding to 1ϽNϽ7. Thus, this jump is due to the sudden growth in the number of effective degrees of freedom when passing from the hadronic dependent of the number of branons. From the numerical to the quark-gluon plasma phase. The exclusion areas depend ⍀ 2 values of T f in terms of f, it is possible to obtain Brh from on the number of branon species and we have plotted them Eq. ͑19͒. In this case we have considered two kinds of limits. for Nϭ1,2,3,7 in Figs. 2 and 3. On one hand there are those coming from the total dark The validity of the previous limits requires that branons ⍀ 2Ͻ Ϫ matter of the universe Brh 0.129 0.095 in Fig. 2. On were relativistic particles at freeze-out. Therefore we require Ӷ the other hand, more constraining limits on the hot dark mat- x f 3, which implies that the bounds do not work for f Ͻ Ϫ4 ϭ ter energy density can be derived from a combined analysis 10 GeV. The curve x f 3 in the hot relic case is also of the data from WMAP, CBI, ACBAR, 2dF and Lyman ␣ plotted in Fig. 6. ͓ ͔ ⍀ 2Ͻ 1 . The bound reads Brh 0.0076 at the 95% C.L. and it As commented on in the Introduction, in all the previous Ӷ is obtained thanks to the fact that hot dark matter is able to calculations we are assuming, apart from f M D , that the cluster on large scales but free-streaming reduces the power evolution of the universe is standard up to a temperature on small scales, changing the shape of the matter power around f. In fact, the effective Lagrangian ͑4͒ is only valid at spectrum. In Fig. 3 we have plotted the corresponding limits low energies relative to f and therefore, it is this scale that Ϫ in the f M plane. Notice the abrupt jump around f ﬁxes the range of validity of the results. We have checked Ӎ ͑ ͒ 60 GeV in Figs. 2 and 3 and also in Fig. 4 . This f value that our calculations are consistent with these assumptions Ӎ corresponds to a decoupling temperature of T 170 MeV since the decoupling temperatures are always smaller than f which is the assumed value for the QCD phase transition. in the allowed regions in Fig. 6.

30 N 25 EXCLUDED

10 UNEXCLUDED

-2 0 2 4 6 10 10 10 10 10 FIG. 2. Exclusion plot for hot relics in models with N f (GeV) ϭ1,2,3,7 branons. For a given N, the shaded area corresponds to the ⍀ 2ϭ Ϫ total dark matter limit Brh 0.129 0.095, therefore the region FIG. 4. Restrictions from nucleosynthesis on the number of above such an area is excluded by branon overproduction. massless branon species N as a function of f for ⌬N␯ϭ1.

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VI. BOUNDS FROM NUCLEOSYNTHESIS eq ͑ ͒ 3 geff T f ,B T ϭ ͑38͒ As commented on above, one of the most successful pre- geq ͑T͒ T3 dictions of the standard cosmological model is the relative eff B abundance of the light elements. The calculated abundances where T f ,B is the branon freeze-out temperature and we have are very sensitive to certain cosmological parameters, in par- used the fact that for particles in equilibrium with the pho- ticular it has been shown that the production of 4He in- eq ϭ eq ͑ ͒ tons geff heff . Using Eq. 35 , we can set the following creases with an increasing rate of expansion H. From Eq. limit on the number of massless branon species N: ͑18͒ we see that the Hubble parameter depends on the effec- eq 4/3 tive number of relativistic degrees of freedom geff . Usually, 7 T 4 g ͑T ͒ ⌬ у ͩ B ͪ ϭ eff nuc ͑ ͒ this number is parametrized in terms of the effective number N␯ N Nͩ ͪ . 39 eq 4 Tnuc ͑ ͒ of neutrino species N␯ϭ3ϩ⌬N␯ as geff T f ,B If branons decouple after nucleosynthesis we get the direct SM new 7 g ͑TϳMeV͒ϭg ϩg р10.75ϩ ⌬N␯ ͑34͒ limit eff eff eff 4 7 ϳ Nр ⌬N␯ . ͑40͒ where T MeV corresponds to the universe temperature dur- 4 ing nucleosynthesis. In the SM with three massless neutrino families, we have gSM(TϳMeV)ϭ10.75 corresponding to eq ϭ eff If they decouple before, we have geff(Tnuc) 10.75, as seen the photon field, the three neutrinos and the electron field. In before. Accordingly, we can rewrite the bound as Eq. ͑34͒, in order to avoid deviations of the predicted abun- 4/3 dances from observations, the conservative limit ⌬N␯ϭ1 for 7 g ͑T ͒ р ⌬ ͩ eff f ,B ͪ ͑ ͒ ͓ ͔ N N␯ . 41 the contribution from new physics is usually imposed 17 , 4 10.75 i.e. there could be only one new type of light neutrino. Including branons, the number of relativistic degrees of Taking ⌬N␯ϭ1, the relation between the freeze-out tem- freedom at a given temperature T is given by perature T f ,B and the brane tension scale f that we have obtained for hot relics can be used to get limits on the number 4 TB of branons N ͑see Fig. 4͒. Thus, for f Ͻ10 GeV, we get N g ͑T͒ϭgSM͑T͒ϩNͩ ͪ ͑35͒ eff eff T р1. This result is obtained using Eq. ͑40͒ for f Ͻ3 GeV ͑ Շ ͑ ͒ which corresponds to T f ,B 1 MeV) or Eq. 41 otherwise. SM However the limits are less restrictive in the range f Ӎ10 where geff(T) is the contribution from the SM particles, TB denotes the temperature of the cosmic branon brackground Ϫ60 GeV. In this case we get Nр3. Above the QCD phase and we are assuming that there are no additional new par- transition which, as commented before, corresponds to f ticles. If branons are not decoupled at a temperature T then Ӎ60 GeV, the bound rises so much that the restrictions are ϭ SM տ TB T, i.e. they have the same temperature as the photons. very weak. Notice that we are taking geff(T 300 GeV) On the other hand, if they are already decoupled then its ϭ106.75, i.e. we only include the minimal standard model temperature will be in general lower than that of the photons. matter content with a Higgs boson doublet and three mass- In order to calculate it, we use the fact that the universe less neutrinos. expansion is adiabatic. Let us write Concerning the value of ⌬N␯ , more constraining analysis using only BBN suggests ⌬N␯ϭ0.5 ͓19͔. However, using 3 ϭ ϩ0.4 ͑ TB also WMAP results, the constraints are N␯ 2.6Ϫ0.3 95% h ͑T͒ϭhSM͑T͒ϩNͩ ͪ ͑36͒ ͒ eff eff T C.L. , which are only marginally consistent with the LEP measurements of the number of neutrino families ͓2͔. Such values would severely constrain the number of any new light where hSM(T) includes only the contributions from SM par- eff particles present during nucleosynthesis. However, it has ticles. If at some time between branon freeze-out and nucleo- been suggested that these results could be potentially af- synthesis, some other particle species becomes nonrelativis- fected by systematic errors in the BBN predictions of the tic while still in thermal equilibrium with the photon primordial abundances ͓2,20͔. background, then its entropy is transferred to the photons, but not to the branons which are already decoupled. Thus, the entropy transfer increases the photon temperature relative VII. BOUNDS FROM SUPERNOVA SN1987A to the branon temperature. The total entropy of particles in Important astrophysical bounds on the brane tension scale equilibrium with the photons remains constant i.e., can be obtained from the energy loss in supernovae ͓12͔. Such energy loss is carried away essentially by light par- eq 3 3ϭ ͑ ͒ heffa T const 37 ticles, i.e. photons and neutrinos if we restrict ourselves to SM particles only. However, if the branon mass is low eq and since the number of relativistic degrees of freedom heff enough, we expect branons to carry some fraction of the has decreased, then T should increase with respect to TB . energy, whose importance will depend on their couplings to Thus, we find the SM particles. In this section, we study the constraints on

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ϱ ϱ f and M imposed by the cooling process of the neutron star in 1Ϫ2M 2/(E E ) ϭ ͵ ͵ ͵ 1 2 ͑ ͒͑ ϩ ͒ supernovae explosions. We will perform an analysis of the QBr dE1 dE2 d cos E1 E2 2 Ϫ energy emission rate from the supernova core, similar to that 0 M /E1 1 done by Kugo and Yoshioka for massless branons ͓12͔. The Ϫ ͒Ϫ 2 5/2 Ϫ ͒3/2 N͕2E1E2͓2E1E2͑1 cos 4M ͔͖ ͑1 cos aim is to extend their study to arbitrary M and compare with ϫ . 5 8 (E Ϫ␮)/T (E ϩ␮)/T the colliders and cosmological bounds. In their work, they ͑2␲͒ 7680f ͑1ϩe 1 ͒͑1ϩe 2 ͒ consider the channel corresponding to electron-positron pair ͑43͒ annihilation. Although this contribution could be subdomi- In fact, it is possible to calculate analytically the angular nant, it will allow us to get an order of magnitude estimation integral, whereas the integral over the two energies has been of the branon effect. performed numerically. The corresponding constraints de- If branons are produced in the core, they can be scattered pend on the supernova temperature (TSN) and the number of or absorbed again depending on their couplings to the SM branons (N). In Fig. 5 we show the limits on f and M for particles. Only if the branon mean free path L inside the T ϭ30,50,70 MeV and Nϭ1. ͓ ϳO ͔ SN neutron star is larger than the star size R (10) Km It is interesting to note that for a branon mass of the order could they escape and carry the energy away. For a massive of the GeV, the restrictions on the brane tension scale disap- ӷ ϳ ␲ 8 2 4 branon (M TSN), we have L (8 f )/(M TSNne), where ϭ pear even for TSN 70 MeV due to the limitations in the ne is the electron number density in the star. Therefore the mean free path discussed above. restrictions we will obtain will be valid typically only for f տ5 GeV. These restrictions appear due to the fact that the VIII. CONCLUSIONS emitted energy in the form of branons could spoil the agree- ment between the predictions for the neutrino ﬂuxes from In this work we have studied the limits that cosmology supernova 1987A and the observations in the Kamiokande II and astrophysics impose on the brane-world scenario through ͓21͔ and IMB ͓22͔ detectors. Branons could shorten the du- the effects of massive branons. Using the effective low- ration of the neutrino signal if the energy loss rate per unit energy Lagrangian for massive branons interacting with the Ϫ time and volume is Qտ5ϫ10 30 GeV5. In particular, the SM ﬁelds, we have computed the annihilation cross sections contribution of the mentioned channel to the volume emis- of branon pairs into SM particles. From the solutions of the sivity has the form Boltzmann equation in an expanding universe, we have studied the freeze-out mechanism for branons and obtained their 2 3 thermal relic abundances both for the cold and hot cases. d ki Q ͑ f ,M ͒ϵ ͵ ͟ ͭ 2 f ͮ Comparing the results with the recent observational limits on Br i ϭ ͑ ␲͒3 i 1 2 2Ei the total and hot dark matter energy densities, we have obtained exclusion plots in the f ϪM plane. Such plots are ϫ͑E ϩE ͒2s␴ ϩ Ϫ→␲␲͑s, f ,M ͒ ͑42͒ 1 2 e e compared with the limits coming from collider experiments and show that there are essentially two allowed regions in where i refers to the electron ͑1͒ and positron ͑2͒ particles, Fig. 6: one with low branon masses and large brane tensions whose masses can be neglected in the supernova core. The ͑weak couplings͒ corresponding to hot branons, and a second chemical potential in the Fermi-Dirac distribution function ͑ ͒ Ϫ␮ region with large masses and low tensions strong couplings ϭ (Ei /T /T)ϩ ␮ϳ ␲2 1/3 f i 1/(e 1) can be estimated as (3 ne) in which branons behave as cold relics. In addition, there is with the number density of electrons: ne an intermediate region where f is comparable to M, which is ϳ ϫ Ϫ3 3 1.4 10 GeV . With these assumptions, QBr is given by precisely the region studied in ͓13͔, and where branons could account for the measured cosmological dark matter. Using the nucleosynthesis limits on the number of relativ- f (GeV) 50 istic species, we set a bound on the number of light branons TSN = 70 MeV in terms of the brane tension. We see that if branons de- 40 couple after the QCD phase transition corresponding to f UNEXCLUDED Ͻ60 GeV, the limits can be rather stringent Nр3, whereas 30 TSN = 50 MeV they become very weak otherwise. Finally, we have analyzed the possibility that massive bra- EXCLUDED 20 nons could contribute to the cooling of a supernova core. After estimating the energy loss rate, we again get some 10 limits on the f and M parameters which are compared to the previous ones. It is shown that they are not competitive with TSN = 30 MeV 0.5 1 those coming from LEP-II. M (GeV) In conclusion, cosmology imposes limits on the BWS which are complementary to those coming from collider ex- FIG. 5. Exclusion regions from supernovae cooling by branons periments and astrophysics. Although the combination of ϭ ϭ for TSN 10,50,70 MeV and N 1. The solid lines come from the both bounds excludes an important region of the parameters limits on the volume emissivity, and the dashed lines are the L space, still there are brane-world model which could be com- ϭ10 Km limits on the branon mean free path. patible with observations. Future hadronic colliders such as

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7 10 son ﬁeld or the neutral pion, the only change comes from the branon annihilation cross section that should be divided by EXCLUDED 0.0076 2 two. 3 2 h > 0.129 10 h< 2. Fermions EXCLUDED SN1987A -1 ϩ Ϫ 10 ␴ : ␺ ͑p ͒,␺ ͑p ͒→␲͑p ͒,␲͑p ͒ (GeV) 3 1 2 3 4 f

2 h < 0.095 N -5 EXCLUDED ␴ ϭ ͱ͑ Ϫ 2͒͑ Ϫ 2 ͒ͫ͑ Ϫ 2͒2 10 3 s 4M s 4m␺ s 4M LEP-II 30 720f 8␲

2 -9 2m␺ 10 ϩ ͑ 4Ϫ 2 ϩ 2͒ͬ ͑ ͒ -9 -5 -1 3 7 23M 14M s 3s . A3 10 10 10 10 10 s M (GeV) ␴ ␲ ͒ ␲ ͒→␺ϩ ͒ ␺Ϫ ͒ 4 : ͑p1 , ͑p2 ͑p3 , ͑p4 FIG. 6. Combined exclusion regions in a model with a single branon from total and hot dark matter, LEP-II single photon events 2 3 1 ͑sϪ4m␺͒ ͓11͔, and supernovae cooling. The solid line on the right corre- ␴ ϭ ͱ ͫ͑ Ϫ 2͒2 4 s 4M sponds to the cold dark matter limit. The two solid lines on the left 3840Nf8␲ ͑sϪ4M 2͒ correspond to hot dark matter: the thicker one comes from the total 2 2 ⍀ ϭ Ϫ 2m␺ dark matter range Brh 0.129 0.095, whereas the thin one is 4 2 2 ⍀ 2ϭ ϩ ͑23M Ϫ14M sϩ3s ͒ͬ. ͑A4͒ the hot dark matter limit Brh 0.0076. The two dashed lines s ϭ ͑ ͒ ͑ ͒ correspond to x f 3 for hot upper line and cold lower line dark matter. These results are valid for a Dirac fermion of mass m␺ . LHC, Tevatron-II or the planned linear electron-positron col- For massless Weyl fermions, we should multiply the branon liders, and the possibility of detecting dark matter branons production cross section by two, whereas the annihilation directly ͓13͔ or indirectly will allow us to explore a wider cross section should be divided by the same factor of two. region of the parameter space. Work is in progress in these directions. 3. Photons ␴ ␥͑ ͒ ␥͑ ͒→␲͑ ͒ ␲͑ ͒ APPENDIX: 5 : p1 , p2 p3 , p4 BRANON PRODUCTION AND ANNIHILATION 2 CROSS SECTIONS WITH SM PARTICLES N 4M ␴ ϭ ͱ1Ϫ s͑sϪ4M 2͒2. ͑A5͒ 5 8␲ s In this section we present the branon production and an- 7680f nihilation cross sections in processes involving SM particles. ␴ : ␲͑p ͒,␲͑p ͒→␥͑p ͒,␥͑p ͒ The results are presented with all the internal degrees of 6 1 2 3 4 freedom summed for the ﬁnal particles and averaged for the 1 s͑sϪ4M 2͒2 initial ones. We have used the Feynman rules given in ͓11͔, ␴ ϭ ͑ ͒ 6 . A6 where N is the number of branons. 1920Nf8␲ 4M 2 ͱ1Ϫ s 1. Scalars

␴ ⌽† ͒ ⌽ ͒→␲ ͒ ␲ ͒ 4. Z 1 : ͑p1 , ͑p2 ͑p3 , ͑p4 ␴ : Z͑p ͒,Z͑p ͒→␲͑p ͒,␲͑p ͒ N ͑sϪ4M 2͒ 7 1 2 3 4 ␴ ϭ ͱ ͓Ϫ ͑ 2 ϩ ͒͑ Ϫ 2͒2 s 8m⌽ s s 4M 1 8 2 7680f ␲s ͑sϪ4m⌽͒ N sϪ4M 2 ␴ ϭ ͱ ͓ ͑ 2 ϩ ͒͑ Ϫ 2͒2 7 3s 8M s s 4M 2 2 4 2 2 8␲ Ϫ 2 Z ϩ͑2m⌽ϩs͒ ͑23M Ϫ14M sϩ3s ͔͒. ͑A1͒ 69 120f s s 4M Z ␴ ␲ ͒ ␲ ͒→⌽† ͒ ⌽ ͒ ϩ͑12M 4 ϩ4sM2 ϩs2͒͑23M 4Ϫ14M 2sϩ3s2͔͒. ͑A7͒ 2 : ͑p1 , ͑p2 ͑p3 , ͑p4 Z Z

2 1 ͑sϪ4m⌽͒ ␴ : ␲͑p ͒,␲͑p ͒→Z͑p ͒,Z͑p ͒ ␴ ϭ ͱ ͓Ϫ ͑ 2 ϩ ͒͑ Ϫ 2͒2 8 1 2 3 4 2 s 8m⌽ s s 4M 3840Nf8␲s ͑sϪ4M 2͒ 1 sϪ4M 2 ϩ͑ 2 ϩ ͒2͑ 4Ϫ 2 ϩ 2͔͒ ͑ ͒ ␴ ϭ ͱ Z͓ ͑ 2 ϩ ͒͑ Ϫ 2͒2 2m⌽ s 23M 14M s 3s . A2 8 3s 8M Z s s 4M 7680Nf8␲s sϪ4M 2 The scalar results are given for a complex scalar such as ϩ͑ 4 ϩ 2 ϩ 2͒͑ 4Ϫ 2 ϩ 2͔͒ ͑ ͒ charged pions or kaons. For a real scalar like the Higgs bo- 12M Z 4sMZ s 23M 14M s 3s . A8

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5. WÁ 6. Gluons ␴ Ϯ͑ ͒ ϯ͑ ͒→␲͑ ͒ ␲͑ ͒ 9 : W p1 ,W p2 p3 , p4 ␴ ͒ ͒→␲ ͒ ␲ ͒ 11 : g͑p1 ,g͑p2 ͑p3 , ͑p4

N sϪ4M 2 ␴ ϭ ͱ ͓ ͑ 2 ϩ ͒͑ Ϫ 2͒2 9 3s 8M W s s 4M 69 120f 8␲s sϪ4M 2 N 4M 2 W ␴ ϭ ͱ1Ϫ s͑sϪ4M 2͒2. ͑A11͒ 11 8␲ s ϩ͑ 4 ϩ 2 ϩ 2͒͑ 4Ϫ 2 ϩ 2͔͒ ͑ ͒ 61 440f 12M W 4sMW s 23M 14M s 3s . A9

␴ ␲͑ ͒ ␲͑ ͒→ Ϯ͑ ͒ ϯ͑ ͒ 10 : p1 , p2 W p3 ,W p4 ␴ ␲ ͒ ␲ ͒→ ͒ ͒ 12 : ͑p1 , ͑p2 g͑p3 ,g͑p4

2 sϪ4M 2 ␴ ϭ ͱ W͓ ͑ 2 ϩ ͒͑ Ϫ 2͒2 10 3s 8M W s s 4M 7680Nf8␲s sϪ4M 2 1 s͑sϪ4M 2͒2 ␴ ϭ ͑ ͒ 12 . A12 ϩ͑ 4 ϩ 2 ϩ 2͒͑ 4Ϫ 2 ϩ 2͔͒ 8␲ 2 12M W 4sMW s 23M 14M s 3s . 240Nf 4M ͱ1Ϫ ͑A10͒ s

͓1͔ D.N. Spergel et al., Astrophys. J., Suppl. Ser. 148, 175 ͑2003͒. ͓11͔ J. Alcaraz, J.A.R. Cembranos, A. Dobado, and A.L. Maroto, ͓2͔ V. Barger, J.P. Kneller, H.S. Lee, D. Marfatia, and G. Steig- Phys. Rev. D 67, 075010 ͑2003͒. man, Phys. Lett. B 566,8͑2003͒; A. Pierce and H. Murayama, ͓12͔ T. Kugo and K. Yoshioka, Nucl. Phys. B594, 301 ͑2001͒. ͓ ͔ hep-ph/0302131; S. Hannestad, J. Cosmol. Astropart. Phys. 05, 13 J.A.R. Cembranos, A. Dobado, and A.L. Maroto, Phys. Rev. ͑ ͒ 004 ͑2003͒. Lett. 90, 241301 2003 . ͓14͔ J.A.R. Cembranos, A. Dobado, and A.L. Maroto, Phys. Rev. D ͓3͔ N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 65, 026005 ͑2002͒. 429, 263 ͑1998͒; Phys. Rev. D 59, 086004 ͑1999͒; I. Antonia- ͓15͔ A.A. Andrianov, V.A. Andrianov, P. Giacconi, and R. Soldati, dis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. J. High Energy Phys. 07, 063 ͑2003͒. ͑ ͒ Lett. B 436, 257 1998 . ͓16͔ R. Contino, L. Pilo, R. Rattazzi, and A. Strumia, J. High En- ͓4͔ D. Langlois, in Proceedings of YITP Workshop: Braneworld: ergy Phys. 06, 005 ͑2001͒. Dynamics of Space-time Boundary, Kyoto, Japan, 2002, ͓17͔ E.W. Kolb and M.S. Turner, The Early Universe ͑Addison- hep-th/0209261. Wesley, Reading, MA, 1990͒ ͓5͔ T. Multamaki and I. Vilja, Phys. Lett. B 559,1͑2003͒. ͓18͔ M. Srednicki, R. Watkins, and K.A. Olive, Nucl. Phys. B310, ͓6͔ R. Sundrum, Phys. Rev. D 59, 085009 ͑1999͒. 693 ͑1988͒; P. Gondolo and G. Gelmini, ibid. B360, 145 ͓7͔ A. Dobado and A.L. Maroto, Nucl. Phys. B592, 203 ͑2001͒. ͑1991͒. ͓8͔ M. Bando, T. Kugo, T. Noguchi, and K. Yoshioka, Phys. Rev. ͓19͔ K.N. Abazajian, Astropart. Phys. 19, 303 ͑2003͒. Lett. 83, 3601 ͑1999͒. ͓20͔ R.H. Cyburt, B.D. Fields, and K.A. Olive, Phys. Lett. B 567, ͓9͔ J. Hewett and M. Spiropulu, Annu. Rev. Nucl. Part. Sci. 52, 227 ͑2003͒. 397 ͑2002͒. ͓21͔ K. Hirata et al., Phys. Rev. Lett. 58, 1490 ͑1987͒. ͓10͔ P. Creminelli and A. Strumia, Nucl. Phys. B596, 125 ͑2001͒. ͓22͔ R.M. Bionta et al., Phys. Rev. Lett. 58, 1494 ͑1987͒.

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