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- Copyright © IOP Publishing Ltd
- ISSN
- 1126-6708
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- 1029-8479
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- 10.1088/1126-6708/2002/07/051
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Abstract
JHEP07(2002)051 Published by Institute of Physics Publishing for SISSA/ISAS Received: March 21, 2002 Accepted: July 25, 2002 Brane interaction as the origin of in°ation Nicholas Jones, Horace Stoica and S.-H.Henry Tye Laboratory of Elementary Particle Physics, Cornell University Ithaca, NY 14853, USA E-mail: [email protected], [email protected], [email protected] Abstract: We reanalyze brane in°ation with brane-brane interactions at an angle, which include the special case of brane-anti-brane interaction. If nature is described by a stringy realization of the brane world scenario today (with arbitrary compacti¯cation), and if some additional branes were present in the early universe, we ¯nd that an in°ationary epoch is generically quite natural, ending with a big bang when the last branes collide. In an interesting brane in°ationary scenario suggested by generic string model-building, we use the density perturbation observed in the cosmic microwave background and the coupling uni¯cation to ¯nd that the string scale is comparable to the GUT scale. Keywords: D-branes, Tachyon Condensation, Cosmology of Theories beyond the SM, Physics of the Early Universe. ° SISSA/ISAS 2002 http://jhep.sissa.it/archive/papers/jhep072002051 /jhep072002051.pdf JHEP07(2002)051 Contents 1. Introduction 1 2. E®ective action for brane-stacks 5 3. Braneworld at the start of in°ation 7 3.1 Set-up 7 3.2 Brane-anti-brane pair 10 3.3 Branes at angles 10 4. In°ation 13 4.1 General set-up 13 4.2 Hypercubic compacti¯cation 15 4.3 The string scale 18 4.4 Generic compacti¯cation 18 4.5 Reheating and defect production 20 5. Discussions 22 6. Summary and remarks 23 A. Summing over the lattice 24 1. Introduction By now, the in°ationary universe [1] is generally recognized to be the most likely scenario that explains the origin of the big bang. So far, its predictions of the °atness of the uni- verse and the almost scale-invariant power spectrum of the density perturbation that seeds structure formations are in very good agreement with the cosmic microwave background (CMB) observations [2, 3]. In standard in°ationary models [4], the physics lies in the in°aton potential. However, conventional physics (such as the standard electroweak and strong interaction model or grand uni¯ed theories) does not yield an in°aton potential that agrees with observations (such as enough number of e-foldings, the amplitude of the den- sity perturbation etc.). In the past two decades, numerous phenomenological potentials that give the correct in°ationary properties have been proposed, most with little input from fundamental physics. In general, a potential that yields enough e-foldings and the correct magnitude of density perturbation requires some ¯ne-tuning. However, such ¯ne- tuning is in general not preserved by quantum corrections. Recently, the brane in°ationary scenario [5] was proposed, where the in°aton is identi¯ed with an inter-brane separation, { 1 { JHEP07(2002)051 while the in°aton potential emerged from brane interactions that is well-studied in string theory [6]. In particular, exchanges of closed string (bulk) modes between branes (the lowest order of which can be determined by the one-loop partition function of the open string spectrum) determine the form of the in°aton potential during the slow-roll epoch. In°ation ends when the branes collide, heating the universe that starts the big bang. This visualization of the brane dynamics allows one to implement in°ation physics pictorially. The brane in°ation scenario may be realized in a variety of ways [7]{[10]. One possi- bility is the brane-anti-brane interaction and annihilation [7, 8]. Although this potential is too steep for in°ation, a particularly appealing scenario proposed by Burgess etc. [8] shows that a hypercubic compacti¯cation of extra dimensions °attens the potential to yield enough e-foldings for a viable in°ationary epoch. The brane in°ationary scenario for branes at a ¯xed angle is studied by Garcia-Bellido etc. [9], which is a generalization of the brane-anti-brane scenario and is quite natural in string-model building. We re-analyze the brane in°ationary scenario by combining and generalizing these two observations and ¯nd that the result is very robust. For cases where in°ation is generic, we use the CMB data to determine the string scale to be M . M . s GUT From experimental data and assuming MSSM, the gauge couplings of the standard model unify at the GUT scale [11]: ®(GUT ) ' M ' 2£ 10 GeV (1.1) GUT In the brane world scenario, the standard model particles are open string (brane) modes while graviton and other closed string modes are bulk modes. This may be naturally realized in type I (or orientifold) string theories, with string scale M = 1= ®0 and large compacti¯ed dimensions. After compacti¯cation to 4-D spacetime, the Planck mass, M = ¡1=2 18 (8¼G ) = 2:42£ 10 GeV, is given by, via dimensional reduction [12, 13] 8 8 M V M V V t P 2 2 s s g M = = (1.2) s P 6 6 (2¼) ¼ (2¼) ¼ Here V = V V is the 6-D compacti¯cation volume, where the (p ¡ 3) dimensions of the t P stack of Dp-branes we live in are compacti¯ed with volume V . Here g = e is the string t s coupling determined by the dilaton expectation value Á , which is related to the gauge coupling ® via p¡3 ¡(p¡3) g = 2M V (2¼) ®(M ) = 2v ®(M ) (1.3) s t s t s The presence of the factor (2¼) ¼ in eq. (1.2) appears because the 10-D Newton's constant G in type I string theory is given by [13, 6] 2 7 g (2¼) 8¼G = · = (1.4) 2M In general, M ¸ M . Crudely speaking, if M ' M , the gauge couplings are GUT s GUT s uni¯ed at the string scale via conventional (renormalization group) logarithmic running. If M > M , then v > 1 and the gauge couplings will go through a period of power GUT s t running, unifying at the string scale [14]. However, the present experimental data does not determine the value of M . It can take value anywhere between 1 GeV to M [15]. s GUT { 2 { JHEP07(2002)051 INFLATION INFLATION Figure 1: In°ating away brane and bulk Figure 2: In°ating away non-wrapping an- °uctuations. gles between branes. The key data in the cosmic microwave background (CMB) we use are the density perturbation magnitude measured by COBE [2] and its power spectrum index n [3] ¡5 ± = 1:9£ 10 jn¡ 1j < 0:1 (1.5) Our crucial assumptions are that the cosmological constant is negligible after in°ation and that the radions (moduli of the compact dimensions) and the dilaton are stabilized by some unknown physics. This assumption precludes the dynamics of these moduli from playing a role in in°ation. (Otherwise, the picture becomes much more complicated and model-dependent. We shall brie°y discuss how they may play a role in an in°ationary scenario [16, 17].) For two branes at a distance and at angle µ, the parameters coming from string theory are essentially M and 0 < µ · ¼, as well as the integers p ¸ 3 and d , where p + d · 9. ? ? The generic in°ationary picture that emerges in our analysis may be summarized as follows: ² The branes may start out wrinkled and curved, with matter density or defects on them and in the bulk. They may even intersect each other in the uncompacti¯ed directions. Generically, there is a density of strings stretched between branes as well. They look like massive particles to the 4-D observer. Branes with zero, one or two uncompacti¯ed dimensions are defects that look like point-like objects, cosmic strings and domain walls, respectively. Fortunately, in°ation will smooth out the wrinkles, in°ate away the curvature and the matter/defect/stretched-string densities, and red shift the intersections exponentially far away from any generic point on the brane, so one may consider the branes with 3 uncompacti¯ed dimensions to be essentially parallel in the uncompacti¯ed directions, with empty branes and empty bulk. ² Generically the in°aton potential for a brane-anti-brane system is too steep for in°a- tion [7, 8]. Ref. [8] shows that in a hypercubic toroidal compacti¯cation, the images { 3 { JHEP07(2002)051 of the brane also exert forces on the anti-brane. For separations comparable to the compacti¯cation size, the forces tend to cancel. This softens the potential so there may be enough in°ation. If the branes start out far enough apart, there will be a slow-roll period for in°ation before they collide. It is sensible to ask what is the probability P of su±cient in°ation if the brane and the anti-brane are randomly placed inside the compacti¯ed volume. Depending on the number d of large extra dimensions, we ¯nd the probability P » (3%) This means that brane-anti-brane in°ation is not very likely. ² We ¯nd that hypercubic compacti¯cation is not necessary. For the same brane in- teraction and compacti¯cation volume, we ¯nd that P for a generic compacti¯cation is comparable to that for a hypercubic toroidal compacti¯cation. This point is illus- trated with a rectangular torus. ² Ref. [9] argues that branes at angles improves the situation. A Dp-brane di®ers from an anti-Dp-brane only by an opposite Ramond-Ramond (R-R) charge. This charge corresponds to an orientation, so the special case when the angle µ = ¼ corresponds to the brane-anti-brane case. Two branes at generic angle µ collide to form a lower energy brane (while brane-anti-brane annihilates). For two randomly placed branes at an angle, the probability P of su±cient in°ation increases to close to unity as the angle becomes small. Besides the factor of (2¼) ¼ in eq. (1.2) and the inclusion of the compacti¯cation e®ect, our analysis di®ers from ref. [9] in the choice of the vacuum energy during in°ation. Also, we believe it is more natural to choose the rectangular torus (maybe together with a large winding number) to obtain a small µ. In this ¡d case, P / µ . So, for µ . 1=10 and generic compacti¯cation, P is of order unity. This implies that in°ation is quite generic with a relatively small µ. As µ becomes smaller, the compacti¯cation lattice e®ect becomes less important. For small enough ¡2 angle (i.e., µ . 10 ), the compacti¯cation lattice e®ect is negligible. In this case, P(µ) is very close to unity. ² The magnitude of the density perturbation ± (1.5) may be used to ¯x the string scale. Generically, we have M . M In a brane in°ationary scenario suggested s GUT by the analysis and generic orientifold construction, we ¯nd that M » M . It will s GUT be very interesting to see how close M is to M in a more realistic string model. s GUT ² Soon after in°ation ends, the ground state open string modes become tachyonic, which leads to tachyon condensation and brane annihilation/collision, reheating the universe. This is an explicit realization of hybrid in°ation [18]. It is well-known that p-brane collision generically allows the production of lower-dimensional (p ¡ 2k)- branes. Since the in°aton is di®erent from the tachyon modes that are responsible for defect production, the production of defects is generically a serious problem. We argue that the dangerous productions of domain walls and monopole-like defects are actually absent, while the production of the more palatable cosmic strings may happen via the Kibble mechanism. A more careful analysis is clearly needed to determine the cosmic string density. { 4 { JHEP07(2002)051 Generically, bulk modes have gravitational strength couplings and so brane modes are more likely in°aton candidates. In string theory, vacuum expectation values of such scalar brane modes are simply brane separations, which are the in°atons in brane in°ation. Compacti¯cation must take place and it always softens the in°aton potential to improve P for in°ation. Branes at angles is more generic than a brane-anti-brane pair, and a small µ is easily realized if two compacti¯cation sizes have a ratio » µ. These lead us to argue that brane in°ation is probably quite generic. In this sense, brane interaction may be considered as an explanation of the origin of in°ation. The attractive force between two branes is stronger for smaller separation and larger angle µ. If the early universe starts with a number of branes (as will be the case if the universe starts out to be very hot, or if there is a higher-dimensional brane-anti-brane pair or a non-BPS brane), probably the universe goes through multiple in°ationary epochs, with the last collision involving two branes that start out relatively far apart and at small angle. The in°ation we have been discussing is the epoch just before their collision. It is interesting to study this scenario in greater detail. In section 2, we write down the e®ective theory for multi-branes. In section 3, we set up the brane world scenario just before the ¯nal in°ation takes place, for a brane- anti-brane pair and for two branes at a ¯xed angle. The application of this brane pair to in°ation is discussed in section 4, where we ¯nd that two branes at a small angle in a compacti¯ed volume generically generates in°ation. We also discuss the issues of reheating and possible defect productions after in°ation. In section 5, we commend on the bulk modes and the robustness of M . MGUT . The summary and remarks are presented in section 6. Appendix A contains the details of the compacti¯cation lattice e®ect on the in°aton potential by summing over the images of a brane. 2. E®ective action for brane-stacks We write the world-volume action for N parallel BPS Dp-branes and M anti-Dp-branes with d large extra dimensions. This action contains all the salient features of the more general e®ective action which are important in the brane cosmology model: the \slow-roll" potential for the in°aton (brane separation) coming from the combined R-R and NS-NS sectors, multiple and non-commuting scalar ¯elds, and a set of scalar ¯elds which become tachyonic at the end of in°ation (or possibly after the slow roll phase has ended). The e®ective action is expanded about a °at background, with metric ´ =diag(¡1; 1; : : :): ½ · ¸ · ¸ 1 1 1 1 p+1 + I +¹ I + 2 ¡ I ¡¹ I ¡ 2 ~ ~ S = ¡¿ d ¾ tr D Á D Á + (F ) + tr D Á D Á + (F ) + p N M ¹ ¹ 2 4 2 4 h i § §¹ y + tr D TD T + V (Á; Á; T ) : (2.1) ¡(p+1)=2 0 p I=1;:::;d The brane tension is ¿ = ® =g (2¼) , the N brane coordinates, Á , are scalars p s in the adjoint of U(N) written as N £ N matrices in the fundamental representation, Á The e®ects of the curved background due to the branes is taken into account by a doubling of the R-R sector force. { 5 { JHEP07(2002)051 are M £ M scalars in the adjoint of U(M) representing the anti-brane coordinates. T is in § § the (N; M) of U(N)£ (M) [20] and F are the ¯eld strengths of A , the U(N) and U(M) connections on the brane and anti-brane stacks in the fundamental representations. The gauge covariant derivatives are £ ¤ + I I + I D Á = @ Á + A ; Á ; ¹ ¹ 2¼® h i ¡ I I ¡ I ~ ~ ~ D Á = @ Á + A ; Á ; ¹ ¹ 2¼® ¡ ¢ § + ¡ D T = @ T + A T ¡ TA ; ¹ ¹ ¹ 2¼® The potentials are given by ~ ~ ~ ~ V (Á; Á; T ) = V (Á; Á) + V (Á; Á; T ) + V + V (Á; Á); b s T l n ³h ih i´o ¡£ ¤£ ¤¢ I J I J I J I J ~ ~ ~ ~ V = ¡ tr Á ; Á Á ; Á + tr Á ; Á Á ; Á ; b N M 0 2 4(2¼® ) h i I y I I I y I I y ~ ~ ~ V = tr Á TT Á + TÁ Á T ¡ 2Á TÁ T ; s N 0 2 2(2¼® ) y 4 V = ¡ tr (TT ) + O(T ) ; T N 4® 2 0 X ^ ¯g ® ¿ V = (N + M)¡ ; (2.2) l · ¸ (d¡2)=2 nk I I ~ 2 (Á ¡ Á ) nn kk 5 7¡d=2 where ¯ = 2 ¼ ¡((d¡ 2)=2). The potential V is obtained from the non-abelian gener- alization of the Dirac Born-Infeld action [21]. The form of the tachyon scalar interactions, V , can be easily deduced when N 6= M, these being the only quartic interactions allowed. The coe±cients can be deduced by specializing to the N = M = 1 case, and using the fact 2 2 0 that the tachyon quadratic term goes negative for (Á ¡ Á) < 2¼ ® [22]. Alternatively these terms appear as the T-duals of the tachyon gauge-kinetic terms. The tachyon potential, V , can been calculated by various means [23, 24, 25]: ² The result from level truncated cubic string ¯eld theory [23, 24] gives a tachyon mass which agrees with that of the NS open string tachyon, ¡ [6]; this is the mass of 2® the perturbative state which appears in the brane-brane system upon reversal of the R-R charge of one. ² The tachyon potential obtained from boundary string ¯eld theory (BSFT) [25] is not in canonical form, and a ¯eld rede¯nition to its canonical form will involve all components of the string ¯eld. BSFT does however have the advantage that it clearly veri¯es Sen's conjecture [26] that at the minimum of the tachyon potential, the physics is that of the closed string vacuum. ² The T term in V has been calculated at the lowest level in cubic string ¯eld theory 1 4 for N = M = 1 to be + T [23], but there is ambiguity in its form in general, with 2® 4 2 2 both tr(T ) and [tr(T )] terms allowable; BSFT predicts only the former term [25], but it is uncertain whether this holds under the ¯eld rede¯nition. { 6 { JHEP07(2002)051 ² For the purposes here, the T term is most important, since its coe±cient dictates the separation at which T becomes tachyonic, ending in°ation. In generalizing this action to that of branes at angles, the coe±cient of T term should become [6] ¡ , 4¼® where µ is the angle between the brane-stacks; µ = ¼ reverts to the brane anti-brane case above. The e®ects of placing the branes at angles is to lessen the distance between the branes at which T becomes tachyonic. ² The presence of the tachyonic modes originates from the matrix nature of the brane positions. This non-commutative property has interesting cosmological consequences. The non-trivial vacuum structure of the tachyon condensate allows the creation of lower-dimensional branes. Since the tachyons are not the in°aton, lower-dimensional brane production will take place after in°ation, typically via the Kibble mechanism. It is important that this does not re-introduce the old monopole problem. Finally, the potential V takes into account the R-R and NS-NS backgrounds generated I I by the brane and anti-brane stacks [27, 22]. Á is the ln component of Á , and V is a ln I I function of Á ¡ Á only. For d = 2, the Coulomb-like potential is replaced by a log. nn kk This is twice the Chern-Simons brane action, written with the explicit expression for the pull-back of the space-time p + 1 form potential [27] generated by the branes. Only the term from the massless modes in the expansion of the exact potential is necessary for our purposes, to give the long-distance potential between the brane stacks, before the T ¯elds become tachyonic, and in°ation ends. We have doubled the R-R potential to account for the equal force supplied by the NS-NS sector. When we consider branes at angles, the coe±cient of V acquires a factor of tan(µ=2) sin (µ=2) [6, 28], with more complicated angular factors for more than one tilted direction; as we shall see, some brane in°ation scenarios require branes at small angles to achieve su±cient e-foldings. 3. Braneworld at the start of in°ation 3.1 Set-up Realistic string models have 6 of the 9 space dimensions compacti¯ed. Consider Dp- branes in 10-dimensional space-time, where (p ¡ 3) dimensions parallel to the brane are compacti¯ed with volume V and the d dimensions orthogonal to the brane are compacti¯ed with volume V . The remaining 3 spacial dimensions of the Dp-brane are uncompacti¯ed; ¡2 0 they span our observable universe. Let M be the string scale M = ® . So 4 + (p¡ 3) + d = 10 and V = V V . In general, the branes in the early universe that will eventually d k collide after in°ation but before the big bang are di®erent from those present in today's braneworld, so V can be quite di®erent from V in eq. (1.2). To simplify the discussion, k t we shall assume, unless pointed out otherwise, that they largely overlap. 2 2 0 Were we to use the BSFT mass of ¡1=4 ln 2® , the end of the in°ationary phase would be changed little. { 7 { JHEP07(2002)051 In string theory, there is a T-duality symmetry, i.e., physics is invariant under a T- Q Q duality transformation. For toroidal compacti¯cation, we have V = l = 2¼r , where i i i i l is the size of the ith torus/circle and r the corresponding radius, l = 2¼r . If any of i i i i the r is much smaller than the string scale, i.e., M r < 1, the T-dual description is more i s i appropriate: g 1 g ! ; r ! (3.1) s i r M r M i s i In this dual picture, the new r M of the dual T torus is always larger than or equal to i s i unity. Under this duality transformation, the Dirichlet and Neumann boundary conditions of the open strings are interchanged, and so the branes are also mapped to other types of branes. This allows us to consider only the cases where M r ¸ 1, with M r = 1 the s i s i self-dual point. ² Perturbative string theory does not seem to stabilize the dilaton. So we expect dilaton stabilization to arise from non-perturbative stringy e®ects. Very strongly coupled string theory presumably has a weak dual description, where the dilaton is again not stabilized. This leads us to conclude that the string coupling generically is expected to be g 1. To obtain a theory with a weakly coupled sector in the low energy e®ective ¯eld theory (i.e., ®(GUT ) is small), it then seems necessary to have the brane world picture [29]. Let us consider the p > 3 case. In this case, p¡3 g = 2(M r ) ®(r ) = 2v ®(r ) (3.2) s s k k k k p¡3 where v = (M r ) and ®(r ) is the gauge coupling at the scale 1=r . All couplings k s k k k should unify at the string scale. To get a qualitative picture of the impact of a relatively large string coupling, let us take g » 1. So ® has logarithmic running up to the scale 1=r and then power-running between 1=r and M , yielding an k k s ®(M ) » 1. For p = 5, M r » 5 while for p = 7, M r » 5. So ®(r ) is essentially s s k s k k ® . Since the renormalization group °ow ¯ coe±cients (of the standard model GUT gauge couplings) for power-running [14] have almost identical convergence properties as that for the logarithmic running in MSSM and the power running is only over a relatively small energy range, the uni¯cation of the gauge couplings at the string scale is probably assured. ² Of the d dimensions orthogonal to the brane, suppose only d of them are large. To simplify the discussions, let the 6-D compacti¯cation volume V be V = V V V (3.3) k (d¡d ) d¡d where the (d ¡ d )-D volume V is (2¼=M ) (i.e., at the self-dual value). ? (d¡d ) We shall consider d + p · 9. Unless pointed out otherwise, we shall assume all the large extra dimensions to have equal size l = ` or r = r (` = 2¼r ), so the i ? i ? ? ? In general, the radius r here does not necessarily have to be the radius of the ith torus. It is simply a characteristic length scale of the compacti¯ed dimension. In the case of a Z orbifold, the volume is given Q Q Q 1 2 2 ¡1 by v = (2¼R ) ´ (2¼r ) . In general, it stands for the ¯rst KK mode, i.e., at r . i i i i i i i { 8 { JHEP07(2002)051 remaining dimensions orthogonal to the brane have sizes such that M r = 1. For s i branes at angle and p > 3, we shall consider d to range from 2 to 4. Note the e®ect of a larger V is equivalent to an increased d , so the physics will d¡d ? simply interpolate between the values considered. This allows us to use the volumes V = (2¼r ) (transverse to the brane) and v (along the brane) as parameters ? ? k to be determined. Following from dimensional reduction where the 10-D Newton's constant G is given by eq. (1.4), we have 2 2 2 d g M = M (M r ) (3.4) s ? s P s Note that the factor of (2¼) ¼ in eq. (3.4) is missing in ref. [7, 8, 9]. The string coupling is g = e where Á is the dilaton. The BPS Dp-brane has brane tension s D ¿ and R-R charge ¹ : p p p+1 ¿ = ; ¹ = g ¿ (3.5) p p s p (2¼) g while an anti-Dp-brane has the same tension ¿ but opposite R-R charge. These relations are subject to quantum corrections, which we shall ignore for the moment. We shall demand that M r À 1 and d > 1. For instance, this is required within s ? ? string theory if we want to treat the bulk dynamics using only the low-energy e®ective ¯eld theory corresponding to the massless string states. These are also the conditions under which the nonlinear contributions of Einstein's equations are negligible when considering the gravitational ¯eld of a single D-brane, say. Therefore our approximate 4D e®ective ¯eld theory treatment is self-consistent. ² In principle, we can also consider d = 1. However, in this case, the potential between branes becomes con¯ning and the presence of branes will induce a warped geometry in the bulk [30]. As a result of this, the physics is quite di®erent and will not be considered here. ² The standard model of strong and electroweak interactions requires at least 5 Dp- branes. Suppose today's universe is described by a type I or an orientifold string model, which is supersymmetric at scales above the electroweak scale (or non-super- symmetric but with a very small cosmological constant). Suppose the universe starts out with more branes than its vacuum state. This may happen if the early universe is very hot, which probably is a hot gas of di®erent p-branes oriented randomly. Alternatively, the universe may start out with some brane-anti-brane pair or non-BPS branes [19, 26]. This set-up may emerge without a very hot universe. Eventually, the branes will move and collide, until it reaches the brane con¯guration of the vacuum state; that is, all branes except those in the ground state of the string model disappear, via collisions, decays and/or annihilations. Before the last collision, we want to see if in°ation happens, and how generic in°ation is. If the branes that will collide are randomly placed in the compacti¯cation volume, we want to estimate the probability P that su±cient in°ation will take place before collision. The universe { 9 { JHEP07(2002)051 will then reheat to originate the big bang. When this happens, the magnitude of the density perturbation will be used to ¯x the string scale. It is important to see if the density perturbation power spectrum index satis¯es the observational bound. 3.2 Brane-anti-brane pair Suppose that in°ation takes place before the last Dp-brane-anti-Dp-brane pair annihilates. We may also consider the case of a stack of parallel N Dp-branes and a stack of M parallel anti-Dp-branes. The 4-D in°aton potential has a term due to brane tension and a term due to the interactions between the branes. The force between stationary parallel BPS Dp-branes is zero, due to the exact cancellation between the attractive NS-NS and the repulsive Ramond-Ramond (R-R) couplings. On the other hand, the potential between a Dp-brane and an anti-Dp-brane is attractive, since the R-R coupling is also attractive in this case. To simplify the problem, let us take N = M = 1, i.e., a single anti-Dp-brane. In°ation takes place when the last anti-Dp-brane is approaching a Dp-brane and in°ation ends when it annihilates with the brane. The universe is taken to be supersymmetric at this scale, so after annihilation, we expect the vacuum energy to be essentially zero, where the remaining brane tensions are exactly canceled by the presence of orientifold planes. We shall consider the simple case where the remaining branes are sitting at the orientifold ¯xed points (evenly distributed). In this case, the interaction of the anti-Dp-brane with the remaining branes is exactly canceled by that with the orientifold planes. The 4-D e®ective action of a pair of Dp-anti-Dp-branes takes the form: Z µ ¶ ¿ V 4 ¹ ¹ ¡ ' d x jgj (@ y @ y + @ y @ y )¡ V (y) +¢¢¢ (3.6) ¹ b b ¹ a a where the brane is at y and the anti-brane is at y , so the separation is y = y ¡ y . With a a b b 2 9¡p¡d e®ective · =(V V ) where V = (2¼=M ) , the potential for d > 2 is easy to d¡d d¡d s ? ? ? write down: µ ¶ 2 2 9¡p¡d · ¯¿ V p s V (y) = 2¿ V ¡ (3.7) l p d ¡2 y 2¼ where p+1 ¿ = ; (2¼) g ((d ¡ 2)=2) ¡d =2 ¯ = ¼ ¡ d > 2 (3.8) and · is given in eq. (1.4). Here ¯=2 comes from the inverse surface area from Gauss's Law and the extra factor of 2 in ¯ is due to the sum of the NS-NS and the R-R couplings. The potential becomes logarithmic and ¯ = 1=¼ for d = 2. 3.3 Branes at angles String models with branes at angles can be quite realistic [31]. Following ref. [9], let us consider a supersymmetric string model (or a non-supersymmetric model with a zero cosmological constant) where a brane wrapping 1-cycles in a two dimensional torus. Let { 10 { JHEP07(2002)051 the two torus have radii r (or size ` = 2¼r ) and ur , where u < 1. This brane wraps k k k k a straight line in the fn [a] + m [b]g homology class. The energy density of this brane is f f just [6] hq i 2 2 E = ¿ ` n + (um ) (3.9) f 4 f This energy is of course precisely canceled by the presence of other branes and orientifold planes in the model. Suppose this brane results from the collision of two branes in the early universe, each of which wraps a straight line in the fn [a] + m [b]g homology class. Since i i the total (homological) charge is conserved, (m ; n ) = (m + m ; n + n ). The energy f f 1 2 1 2 density of the two-brane system is · ¸ q q 2 2 2 2 E = ¿ ` n + (um ) + n + (um ) : 2 4 k 1 2 1 2 So the energy density before the brane collision, up to an interaction term, is V = E ¡ E 0 2 Let us consider the case where V > 0. Before collision, the branes are at an angle µ = Á ¡ Á , where tan Á = (um =n ) and are separated by a distance in directions orthogonal 1 2 i i i to the torus. When this separation between the branes approaches zero (see ¯gure 3), the collision results in the (m ; n ) brane. In the case where (m ; n ) = (0; 0), with non- f f f f zero (m ; n ), we have a brane-anti-brane pair (with µ = ¼) which annihilate when they i i collide. When (m ; n ) is non-zero, a brane is left behind. If m and n are co-prime, this f f f f brane is at an angle. Such branes appear naturally in many phenomenologcally interesting models [31]. If m and n have a common factor k, m = km ^ , n = kn ^ , we may f f f f f f consider this as k branes in the fn ^ [a] + m ^ [b]g homology class. If either m or n is zero, f f f f presumably the brane (or branes) is parallel to an orientifold plane. In e®ective low-energy ¯eld theory, this is the Higgs mechanism. Suppose there is a U(N) gauge group associated with each brane before collision (i.e., n = n = N). Then there is a U(N) £ U(N) gauge 1 2 symmetry together with a bifundamental scalar, which are open strings stretched between the branes. During collision, the bifundamental scalar develops a vacuum expectation value and spontaneous symmetry breaking takes place: U(N)£ U(N) ! U(N) (3.10) Suppose these two branes are separated by a large distance in the other compacti¯ed directions. Then the e®ective potential is given by [6, 28] 6¡d 2 ¯M sin µ=2 tan µ=2 V (y) = V ¡ : (3.11) d ¡2 4¼y where V / tan µ. For d = 2, the potential has a logarithmic form. Note that the 0 ? potential term vanishes as µ ! 0. In this limit, the branes are parallel and are BPS with respect to each other. In the other limit, as µ ! ¼, the potential blows up due to the { 11 { JHEP07(2002)051 Figure 3: Branes wrapping a torus with sides ` and u` , where u < 1. The two branes ((1,1) and k k (1,0)) are at an angle µ, but they are separated in directions orthogonal to the torus. When that separation approaches zero, the two branes collide to become a single (2,1) brane. tan µ=2 factor. This happens because the brane becomes parallel to the anti-brane and so the strings stretching between them are free to move in that direction. In this case, there is a volume factor to be taken into account. Alternatively, one simply use the potential for the brane-anti-brane system given earlier. Here, we are interested in the small angle case. To get a qualitative feeling of the sce- nario, let us take for example m = 1 and m = 0. Then Á = arctan(um =n ) = 0, 1 2 2 2 2 and Á = arctan(u=n ) ´ µ ¿ 1. This may be achieved with large n or small u, 1 1 1 or a combination of both. Small µ may be realized in a number of ways in string the- ory: ² We may simply take n to be very large, the case considered in ref. [9]. Generically, large n implies the string model today will have a brane with a very high winding number. This is a possibility that deserves further study. ² Suppose the resulting brane that wraps around ` is not among the branes that are responsible for the standard model gauge ¯elds, that is, ` is part of V and not part of V . Then there is hardly any constraint on the value ` and we may take u to be k k very small with n = 1, say. ² Generically, the resulting brane will wrap around ` , so ` is likely to be a part of V k k in eq. (1.2). We may estimate how small u may be. First recall that uM ` ¸ 2¼. On the other hand, ` is bounded by M ` =2¼ . v , so, following eq. (1.3), u ¸ 2®=g . k s k k s ¡2 The smallest we can reasonably have is u ¼ 10 . Since it is quite reasonable for a string model to have n ¸ 1 (to accomodate the standard model, we expect n ¸ 5, 1 f ¡2 which implies n ¸ 4 in our example), µ ' u=n ¼ 10 is not unreasonable. As we 1 1 shall see, µ » 0:1 is quite su±cient. To facilitate comparison with ref. [9], we may consider the d = 4 case. In the small angle approximation we have (taking n = n = 1), 1 2 2 2 ¿ ` tan µ 4 M sin µ=2 tan µ=2 V (y) = ¡ : (3.12) 3 2 4 8¼ y { 12 { JHEP07(2002)051 4. In°ation 4.1 General set-up The in°aton as seen by a 4-D observer µ ¶ 4 ¹ ¡ ' d x jgj @ Ã@ à ¡ V (Ã) +¢¢¢ (4.1) is related to the brane separation y = y ¡ y in eq. (3.6) by: b a r s ¿ V v p k k à = y = yM (4.2) 2g (2¼) ³ ´ p¡3 p¡3 where v = M V = (2¼) . So the e®ective potential for the in°aton becomes, for k k d > 2 and d = 2, respectively, ? ? µ ¶ V (Ã) = A 1¡ d ¡2 µ µ ¶¶ V (Ã) = A 1 + ¸ ln (4.3) where A and ¸ are given by eq. (3.7) for the brane-anti-brane case and by eq. (3.11) for the branes at angle case. For small µ and d = 4, we shall use eq. (3.12). The equation of motion for à is Ä _ à + 3Hà + V = 0 (4.4) where the prime indicates derivative with respect to Ã. For in°ation to take place, there is some choice of à so that the potential V (Ã) satis¯es the slow-roll conditions ² ¿ 1 and j´j ¿ 1, where: µ ¶ 1 V ² = M 2 V ´ = M (4.5) During the slow-roll epoch, à in eq. (4.4) is negligible and the Hubble constant H is given by µ ¶ a_ V H ' = (4.6) a 3M where a(t) is the cosmic scale factor, and we may relate the value of à as a function of the number N of e-foldings before the end of in°ation. The value of à is determined when e end the slow-roll condition breaks down, when either j´j ¼ 1 or ² ¼ 1. Generically, j´j À ² in brane in°ationary scenarios. It takes N e-foldings for à to reach à . e N end Z Z Z t à à end end N dà 1 V N = Hdt = H = dà (4.7) M V t à à N N end { 13 { JHEP07(2002)051 In°ation may also end when the ground state open string mode becomes tachyonic: µ ¶ 2 2 M y M µ 2 s s m = ¡ (4.8) 2¼ 2¼ However, in the scenarios we are considering, the slow-roll condition breaks down before the tachyon mode appears. As we shall see, this has a non-trivial impact on defect productions after in°ation. We know that the density perturbation measured [2] from Cosmic Microwave Back- ¡5 ground (CMB) is ± = 1:9£ 10 . This is the density perturbation at N e-foldings before H e the end of in°ation, where µ ¶ µ ¶ 2 M 1 T s RH N ' 60 + ln + ln (4.9) 16 14 3 10 GeV 3 10 GeV where T is the reheating temperature after in°ation. Here, for d ¸ 2, RH ? · ¸ d ¡ 1 y ? N N = (4.10) d y ? end The power index of the density perturbation n¡ 1 = 2´ ¡ 6² (4.11) For ² ¿ ´, the spectral index n and its variation with respect to the wavenumber k are given by: 2 (d ¡ 1) n¡ 1 = 2´j ¼ ¡ (4.12) d N 0 000 dn V V 2 (d ¡ 1) 2 4 = 2» = 2M ¼ (4.13) N 2 2 d ln k V d N ? e In our case, we have µ ¶ ´ = ¡¯(d ¡ 1)(d ¡ 2) d > 2 ? ? ? µ ¶ 1 ` = ¡ d = 2 (4.14) ¼ y Generically, we need j´j . 1=60 during the early in°ationary stage. Since jy j < ` =2, we i ? see that ´ can never be small enough to satisfy the slow-roll condition. This is noted in ref. [7, 8]. Now let us consider the more general situation of branes at angle. As pointed out in ref. [9], the situation improves. For small µ and n = 1, we have µ ' u=n = u. With 1 1 d +2 2 2 2 d ? ? (2¼) ¼g M = M (M ` )(M u` )(M ` ) , s s s ? s s k k µ ¶ 3µ ` ´ ¼ ¡ d = 4 2¼ y µ ¶ ¼ ¡4¼µ d = 2 (4.15) { 14 { JHEP07(2002)051 To have enough number of e-foldings of slow-roll while the branes are separated, we get a bound on µ by requiring j´j · 1=60 at the beginning of in°ation. However, when jyj » ` =2, the compacti¯cation e®ect is important and eq. (4.15) for ´ is not valid (see the discussion below). For the compacti¯cation e®ect to be negligible so eq. (4.15) is valid, a smaller µ is needed, say ¡2 µ < 10 (4.16) With such a small µ, the probability P of su±cient in°ation approaches unity. A small but not too small value of µ will still increase P and further improves the naturalness of existence of an in°ationary epoch in the brane world. 4.2 Hypercubic compacti¯cation When the brane separation is comparable to the lattice size, we must include the forces exerted by the images of one brane on the other brane. (The net force on a brane due to its own images is exactly zero.) This always tend to soften (or °attens) the potential. Let us ¯rst consider the hypercubic lattice for the brane-anti-brane pair discussed in ref. [8]. ! ¡ ! ¡ Let the brane sit at the origin y = 0 , while we measure the position of the anti-brane from the antipodal point (` =2; ` =2; : : :), at which the net force on the anti-brane is zero: ? ? z = y ¡ ` =2 and de¯ne the corresponding ' = à ¡ à ¤, where à ¤ is the value of à at i i ? i i i i i the antipodal point. (See ¯gure 4.) The potential for the D-dimensional hypercubic lattice is given by: A¡ B ; d > 2 d ¡2 ? i ! ¡ ! ¡ ! ¡ y ¡ r j ij V ( y ) = (4.17) ! ¡ ! ¡ A + B lnj y ¡ r j ; d = 2 ! ¡ where the sum is over all the lattice sites r (positions of the images of the brane), where ! ¡ ! ¡ ! ¡ r is at the origin y = 0 . In order to estimate the potential around the antipodal point, ! ¡ ! ¡ ! ¡ we expand to 4th order in z (or ') around the center of the elementary cell, i.e., z = 0 . z + z Figure 4: Compacti¯cation on hypercubic lattice (solid lines), where one brane sits at the lattice points while the other brane is at z. Here, z (and ') are measured with respect to the antipodal point (the intersection of the dashed lines). { 15 { JHEP07(2002)051 d C D F F¯ 2z =` P(¼) ? ¤ ? ¡3 2 3.94 23.64 15.76 5.0 3.5 % 2£10 ¡4 3 6.22 18.64 37.26 5.9 3.3 % < 10 ¡6 4 15.41 30.82 123.3 6.2 3.1 % » 10 Table 1: Numerical values of the parameters that appear in eq. (4.18) to eq. (4.22). Summing over the images of the other brane for hypercubic lattices (see appendix A for details), we have, for small ', 0 2 31 X X 2 2 4 @ 4 5A V (') ' A 1¡ ¸ D ' ' ¡ C ' (4.18) i j i i6=j where ¸ is related to ¸ by some rescaling, µ ¶ ¡1¡d =2 ¿ V p k ¡2¡d ¸ = ¸` and the values of C and D are given in table 1. As pointed out in ref. [8], the quadratic terms in ' are absent in hypercubic lattices. The in°aton is a multi-component ¯eld, so the in°aton path can be quite complicated. Some sample paths are shown in the ¯gures. To get a qualitative feeling of the potential, let us consider the diagonal path ³ ´ V (') ' A 1¡ ¸F' d (d ¡ 1) ? ? F = D ¡ d C (4.19) For large N , we have µ ¶ ´ = ¡12¸F' = ¡12F¯ N ¼ (4.20) 2j´j The quantum °uctuation in deSitter space H=2¼ yields 3 3 1 V 32A¸FN 2 e ± = ¼ (4.21) 6 2¯ 2 2 75¼ M 75¼ (V ) To have enough time for slow-roll, let us consider the value z¤ at j´j ¼ 1=40, which implies 2z ¼ 3:7% This means the probability P(¼), which is essentially the fraction of the cell volume that yields 60 or more e-foldings for d ¸ 2, µ ¶ 2z P(¼) » » (3:7%) (4.22) { 16 { JHEP07(2002)051 60 efolds limit 0.5 0.5 0.45 0.45 0.4 0.4 Slow-roll 0.35 0.35 ends 0.3 0.3 z /l z /l 0.25 0.25 2 2 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 60 efolds limit 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 z /l 1 z /l Figure 5: The 60 e-folds region for brane Figure 6: The 60 e-folds region for branes at anti-brane interaction. The brane is sitting ¯xed angle µ » 1=6. Three possible in°aton at the upper right corner. There will be 60 paths with enough e-folds are shown. Inter- e-folds or more if the anti-brane starts in the vals along each path acts as a clock. Note region in the lower left corner. that the right path starts out with j´j » 1 and then slows down during the in°ationary epoch. of enough in°ation for an initial randomly placed brane-anti-brane pair is quite small. A more careful estimate of the region with enough in°ation in the d = 2 case is shown in ¯gure 5 (since the lattice has at least a 4-fold symmetry, we need to show only a quarter of a lattice cell in the d = 2 case), while the value of P(¼) is given in table 1. In this case, a ¯ne-tuning on the initial condition seems to be required to obtain enough in°ation. Fortunately, the situation is immensely improved when the branes are at a small µ. Actually, one may argue that, to have a brane-anti-brane pair (that is µ = ¼) involves some sort of a ¯ne-tuning because, apriori, randomly placed branes will be at an angle µ 6= ¼. To get a qualitative feeling of the impact of small µ, let us consider a 4-brane pair with 2 2 n = n = 1 in d = 2, 3 and 4. In this scenario, ' ¼ ¿ ` z =2, and 1 2 ? 4 k µ ¶ ´ ' ¡6µ F¯ (4.23) This is useful since F¯ is relatively insensitive to d . The probability P(µ) for branes at small angle µ is related to that for the brane-anti-brane case via à ! p d µ ¶ 2 5:3% P(µ) » P(¼) » (4.24) µ µ so P(µ) is of order unity for µ . 1=10. In ¯gure 6, we show the case where µ » 1=6 and d = 2. For the diagonal path, the dependence of n of N is essentially the same as in ? e other scenarios where ´ À ², 3 dn 3 n ' 1¡ ' ¡ (4.25) N dln k N Notice that all in°aton paths (see ¯gure 6) tend to move towards the diagonal path. So we expect only at most a slight correction in n and ± for non-diagonal paths. { 17 { JHEP07(2002)051 4.3 The string scale To see the relation between the string scale M and ± , let us consider a particularly s H interesting scenario. A phenomenologically interesting orientifold construction, either su- persymmetric or non-supersymmetric, typically starts with 3 tori. For small µ, we already have 3 sizes: ` and µ` and ` . This ¯xes the 3 torus sizes. Since the brane resulting k k ? from brane collision must wind around ` , we may choose, in eq. (1.1), (1.3), ` = V for k k t 2 6 2 2 2 2 2 2 4-branes, or V = µ` for 5-branes, where (2¼) ¼g M = M (M ` ) (M µ` ) (M ` ) . t s k s k s ? s P s Assuming that ` À ` , we have e®ective d = 2. So using eq. (1.3), eq. (4.21) for the ? k ? 5-brane case becomes µ ¶ 4 3 2g µ F¯N M s s 2 e ± ' (4.26) 3 3 75¼ ® M ¡5 Using ± » 1:9 £ 10 , F¯ ' 5 from table 1, N ' 60, g » 1, ® ' 1=25 and µ » 2®, we H e s have M ' 2£ 10 GeV (4.27) Here, if µ is not small, then the probablity to have enough in°ation, P, will be too small. On the other hand, µ cannot be much smaller than 2®, otherwise V (and so g in eq. (1.3)) t s will be too big. It is precisely this range of µ that allows the compacti¯cation lattice to play a role, so the overall picture is quite consistent with all data. So, within the crude approximations in a toroidal compacti¯cation we have ¡5 ± » 10 () M » M (4.28) H s GUT This means the coupling uni¯cation in MSSM is consistent with brane in°ation in the braneworld. A more precise determination of the relation between ± and M will be H s interesting for phenomenologically realistic compacti¯cations. Note that ` > ` and M µ` ¸ 2¼. If ` À ` , d is e®ectively 2. Otherwise, we may have non-hypercubic s ? ? k k compacti¯cation with d = 3. As we shall see in a moment, the qualitative properties of this case will remain intact. To get a crude estimate of the compacti¯cation sizes, let n + n = 4 + 1 = 5 for the standard model gauge group. Crudely, with µ ' u=n , we have 1 2 1 M ur » 1 and the 3 compacti¯cation radii s k ¡1 ur : r : r ' 1 : u : 400g u (4.29) k k ? For µ » 0:1, we have u » 1=3, M ` » 20 and M ` » 10 . In any case, M . M s k s ? s GUT is much closer to the GUT scale than to the electroweak scale. As long as M is close to ¡1 M , the power-running of the couplings from r to M will not mess up the coupling GUT s uni¯cation [14]. In fact, it is even possible that the coupling uni¯cation may be improved slightly by a small power-running. A more careful analysis will be interesting. 4.4 Generic compacti¯cation For hypercubic lattice, we see that it is easy to obtain a region that have enough in°ation. Since the potential (4.18) has the form V (')¡A / ¡z , it is very °at around the antipodal point z = 0, so it is easy to see that a region around z = 0 will have enough in°ation. i i { 18 { JHEP07(2002)051 60 efolds limit 1.0 0.95 0.9 0.85 0.6 0.8 0.55 0.75 0.5 0.7 0.65 0.45 0.4 0.6 Slow-roll ends 0.35 0.55 0.3 z /l 0.5 z /l 2 0.25 0.45 0.2 0.4 Slow-roll ends 0.15 0.35 0.1 0.3 0.05 0.25 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 z /l 0.15 60 efold limit 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 z /l Figure 7: Rectangular lattice with small Figure 8: Rectangular lattice with asym- asymmetry, a = 1:2. One brane is sitting at metry a = 2. 3 possible trajectories of the the upper right corner. There will be enough in°aton are shown. The paths in the slow- in°ation if the other brane starts inside the roll region are reminiscent to renormalization 60 e-fold region. 3 possible trajectories of the group °ows. in°aton are shown. For generic compacti¯cations, lower powers of z appears in the potential. As a result, the potential is much steeper, so naively one may expect the valid region for enough in°ation to shrink rapidly to zero around the symmetric point where the net force is zero. However, the actual situation is more robust than naively exected. Here, let us illustrate the situation with a rectangle-like lattice. As shown in appendix A, quadratic terms appear in the potential around the symmetric point. Let us consider a 2-dimensional rectangular lattice, with sides ` and a` . In this scenario, the e®ective potential becomes k k ³ h i´ 2 2 2 2 4 4 ^ ^ V (') ' A 1¡ ¸ U(' ¡ ' ) + D' ' ¡ C(' + ' ) +¢¢¢ (4.30) 2 1 1 2 1 2 where U is a function of the ratio a of the two sides of the rectangular lattice. Clearly ^ ^ U ! 0 as a ! 1. In table 2, we show the values U = U=(¿ V ` =2), C and D for p k d = 2. The in°aton is a two-component ¯eld, so the paths for the in°aton can be quite com- plicated. Typical paths are shown in the ¯gures. In ¯gure 7 (for a = 1:2) and ¯gure 8 (for a = 2), one brane is sitting at the upper right corner. If the other brane (at angle µ » 1=6 with respect to the ¯rst one) starts inside the 60 e-fold region around the lower left corner, { 19 { JHEP07(2002)051 a U C D 1.0 0 3.94 23.64 1.2 0.59 2.48 14.93 1.5 0.97 1.08 6.48 2.0 1.03 0.23 1.40 Table 2: Numerical values of the parameters that appear in eq. (4.30). there will be enough in°ation. Comparing ¯gures 7 and 8 to ¯gure 6, each with the same angle µ » 1=6, we see that P is mildly sensitive to the shape (complex structure) of the torus P(µ) P(µ; a) ' (4.31) for a ¸ 1. However, the 60 e-fold region and the in°aton path do depend strongly on the properties of the torus. In rectangular lattices, the quadratic z (i.e., ') terms in the potential dominate during the initial part of the in°aton path. If the asymmetry is large, the quadratic term is dominant over a large region, and the in°aton trajectories are simple, as can be seen in ¯gure 8. When the quartic terms become important they will cause the trajectories to curve towards the corners of the cell. The location of the point where the trajectories turn depends on the asymmetry of the cell. Towards the end of in°ation, when the branes are relatively close, the images become negligible and the potential is Coulombic. By now, it is intuitively clear that the particular hypercubic compacti¯cation is not needed for a reasonable value of P. Generically, the images of one brane exert forces on the other brane, and there is always a point where the forces balance each other so the in°aton potential is °at at that point. Due to the softness of the brane interaction, there is always a region around that point where the potential is °at enough for su±cient in°ation. For initially randomly placed branes at relatively small angle µ . 1=10, which is not hard to arrange for generic compacti¯cations, the probability of su±cient in°ation is close to unity. This is very important, since, in any realistic compacti¯cation in string model-building, a hypercubic lattice is a very special case. 4.5 Reheating and defect production We see that the in°aton and the tachyon modes are charged under the gauge symmetry. In more realistic models, there are chiral ¯elds among the brane modes, which certainly couple to the gauge ¯elds. During reheating after in°ation, we expect the in°aton and the tachyon modes to decay to the gauge bosons and the chiral ¯elds. Since the in°aton and the tachyons are brane modes, we expect this reheating process to be very e±cient. This picture is essentially that of hybrid in°ation. When p-branes collide, tachyons appear and energy is released. Tachyon condensation generically allows the formation of lower-dimensional branes. Following eq. (2.1), (2.2), ¡ ¢ y 4 2 2 V / tr(TT ) M y ¡ 2¼M µ (4.32) s s { 20 { JHEP07(2002)051 so when y is small, the open string ground state T becomes tachyonic and the Higgs mechanism (3.10) takes place. Here, the vacuum structure has a U(N) £ U(N)= U(N) symmetry. Generically, lower-dimensional D(p ¡ 2k)-branes can be formed [20] during the brane collision. Since this happens after in°ation, it is important to check if the old monopole problem returns or not. A naive application of the Kibble mechanism will be disastrous, since that will yield a defect per Hubble volume, resulting in over-abundance of domain walls and monopole-like objects. Fortunately this does not happen, due to the properties of the brane theory. Imagine a string model that describes our world today. The early universe of this model generically contains branes of all types. The higher-dimensional branes collide to produce lower-dimensional branes and branes that are present today. To be speci¯c, con- sider a type IIB orientifold, with D5-branes and orientifold planes. In early universe, there are generically additional D9, D7, D5, D3 and D1-branes with a variety of orientations. (Branes with even p are non-BPS and decay rapidly.) Since the total R-R charges in the compacti¯ed directions must be zero, the branes must appear in sets with zero R-R charges, except for D5-branes, which must appear so the total R-R charge between D5- branes and orientifold planes is zero. For example, if there are only two 9-branes, they must form a pair of D9-anti-D9-branes. Since D9-branes ¯ll the 9-dimensional space, we expect the D9-anti-D9-brane pair to annihilate ¯rst. The resulting tachyon condensation generically produces lower-odd-dimensional branes. Brane on top of each other collides rapidly (unless they are BPS with respect to each other), leaving behind branes separated in the compacti¯ed directions. Consider the last two branes with 3 uncompacti¯ed spatial dimensions that are not in today's string ground state. As they approach each other, the universe is in the in°ationary epoch. Suppose there are other branes with zero, one or two uncompacti¯ed dimensions, which appear as point-like objects, cosmic strings or domain walls, respectively. They are defects in 4-D spacetime. They either annihilate or are in°ated away, resulting in negligible densities. After in°ation ends, the collision of the two branes with 3 uncompacti¯ed spatial dimensions may produce lower-dimensional branes, which appear as defects. A large density of such defects may destroy the nucleosynthesis or even overclose the universe (like the old monopole problem). There are two mechanisms to produce such defects: the Kibble mechanism and thermal production. After in°ation, when the ground state open string mode becomes tachyonic, brane col- lision and tachyon condensation takes place, and lower-dimensional branes will generically appear via the Kibble mechanism. To be speci¯c, consider a Dp-anti-Dp-brane collision, which may result in D(p ¡ 2)-branes and anti-D(p ¡ 2)-branes. At this time, the particle horizon size is typically bigger than the compacti¯cation sizes, 3=2 1 M (2¼) ' À ` (4.33) H M µ so the Kibble mechanism does not happen in the compacti¯ed directions. In the uncom- pacti¯ed directions, the Kibble mechanism can take place, so cosmic string-like defects may be formed: they are D(p ¡ 2)-branes wrapping the same compacti¯ed cycles as the { 21 { JHEP07(2002)051 original p-branes, with one uncompacti¯ed dimension. Similarly, if the Dp-brane collision can produce D(p ¡ 4)-branes, their production will be suppressed since there is less than one Hubble volume in the compacti¯ed directions. This implies that domain walls and monopole-like objects are not produced by the Kib- ble mechanism, while cosmic strings may. Generically, there may be closed and stretched cosmic strings, and they form some sort of a network. Fortunately, in contrast to do- main walls and monopole-like objects, a cosmic string network may be acceptable [32]. A more careful analysis of the production of cosmic strings will be very interesting. This result is somewhat di®erent from that of ref. [33], which does not take into account the compacti¯cation e®ect. Next, let us consider thermal productions. We shall argue that thermal production is negligible. Branes can wrap around the compacti¯cation cycles. The mass of such a brane wrapping a p-dimensional volume V , which appears as a point-like object in 4 dimensional spacetime, is M v M s p s M ¼ ¿ V = ¸ (4.34) p p p 2¼g g s s where v = M V =(2¼) ¸ 1. Since a wrapped brane is charged, while the total charge p p of the defects must be zero, they have to be pair-produced or multi-produced. Assuming e±cient reheating after in°ation, the reheat temperature is given by µ ¶ 1=4 30(2¿ V ) T ¼ (4.35) RH ¼ n dof where n is the number of light degrees of freedom at reheating. Typically n is of the dof dof order of a few hundred. (For less e±cient reheating, the temperature is of course lower.) T ¼ M =3 for brane-anti-brane annihilation. Comparing to M , this temperature is RH s p probably low enough to prevent the production of defects after in°ation. This problem will be totally absent in the collision of branes at angle since the energy released is lower than that from the brane-anti-brane annihilation. Also, if all the compacti¯cation directions are larger than the string scale (by a factor of 2 is enough), the defects will be too heavy to be produced. So the defect production is not a problem in this scenario. 5. Discussions Although we ¯nd that the string scale is quite close to the GUT scale, this does not imply that a low string scale (say M = 10 GeV) is necessarily ruled out. Such a low string scale is possible if the radion grows a lot after in°ation [16, 17]. In this scenario, the model necessarily becomes rather complicated. Keeping the radion mode as a dynamical variable, we can integrate out the d compact- i¯ed dimensions to obtain the low energy e®ective 4-dimensional action " # d© e d(d¡ 1) 4 (4) d© 2 ~ ~ S = d x det g R + e (r©) ¡ V (Ã; ©) +L (g ; Ã) (5.1) sm ¹º 16¼G 16¼G N N { 22 { JHEP07(2002)051 The action (5.1) has the form of a scalar-tensor theory of gravity, written in the Jordan frame. The Jordan-frame potential V is given by ~ ~ d d© (d¡2)© ~ ~ ~ V (Ã; ©) = r e V (©)¡ e + V (Ã; ©) ; (5.2) 0 bulk brane 8¼G r ¡2 where the Ricci scalar of the metric h is 2k r (r being the equilibrium radius of the ab i 0 extra dimensions today) and k is a dimensionless constant that we may neglect. The action may be written in the Einstein frame, " # h i p ^ © © © R 1 1 ¡ ¡2 ¡ 4 2 2 ^ ^ ¹ ¹ ¹ S = d x ¡g^ ¡ (r©) ¡ e (rÃ) ¡ e V (Ã; ©) + S e g^ ;  rest ®¯ rest 16¼G 2 2 (5.3) ¡©=¹ where g^ is the Einstein frame metric, and g = e g^ is the physical, Jordan frame ®¯ ®¯ ®¯ d+2 metric. Here, ¹ = M , and the canonically normalized radion ¯eld © is related to 8d the radius r of the extra dimensions by r = r exp[ ]. The ¯eld à is a brane scalar mode. d¹ The quantity V (Ã; ©) is the Jordan-frame potential for the radion and in°aton (energy per unit proper brane 4-volume). Finally the action S [g ;  ] is the action of the rest ®¯ rest remaining matter ¯elds  , which may be treated as a °uid. rest ¡2©=¹ ¡2©=¹ For generic V (©), the factor e will render the potential e V (©) unsuitable for in°ation. (Such an exponential form is too steep for slow-roll and too shallow for reheating.) However, © may still play a role in an in°ationary universe. An example of this scenario has been considered in ref. [16]. With appropriate V (Ã; ©), let à be the in°aton, and © be frozen during the in°ationary epoch. After in°ation, © grows, resulting in today's value for the Newton's constant, which is substantially bigger than the e®ective Newton's constant during in°ation. This allows the correct value for density perturbation even though the string scale can be much smaller than what we ¯nd earlier. Of course, this scenario is much more complicated and model-dependent than that discussed in this paper. 6. Summary and remarks In brane in°ation, the inter-brane separations play the role of in°atons, while the bulk modes provide the brane interactions that generate the in°aton potential. Such properties are well studied in string theory. By itself, the resulting potential is generically unsuitable for in°ation. Fortunately, any realistic phenomenological string realization requires the compacti¯cation of the extra dimensions. Also, the weak coupling behavior observed in nature probably requires the branes to have extra dimensions (beyond the 3 uncompacti¯ed spatial dimensions) that wrap around some of the compacti¯cation directions. These two properties dramatically improves the brane in°ationary scenario. Without any ¯ne-tuning, the probability of randomly placed branes in the early universe to originate substantial in°ation can easily be of order unity. This allows one to argue that brane interaction provides an explanation of the origin of in°ation. { 23 { JHEP07(2002)051 In a simple brane in°ationary scenario suggested by our analysis, we ¯nd that the amplitude of the density perturbation observed by COBE implies that the string scale is very close to the GUT scale. A more careful analysis in more realistic string models (e.g., orientifolds) can make this relation more precise. We can reverse the analysis: starting with MSSM and coupling uni¯cation, with M = M , we ¯nd that a generic orientifold s GUT model will yield a density perturbation in CMB of the correct magnitude. A number of other issues also deserve further analysis, e.g., the likely braneworld just before in°ation, the reheating and the defect production after in°ation, p-brane-p0-brane interactions, the generation of density perturbation due to multi-component feature of the in°aton, as is likely to be the case in the braneworld. We hope that this work shows that further studies along this direction is worthwhile. We thank Keith Dienes, Sash Sarangi, Ashoke Sen, Gary Shiu and Ira Wasserman for discussions. This research was partially supported by the National Science Foundation. A. Summing over the lattice Here we provide some details to the derivation of the potential via summing over the lattice. The potential for the D-dimensional lattice is given by: 8 P A¡ B ; d > 2 < d ¡2 ? ! ¡ ! ¡ ? j r ¡ r j ! ¡ V ( r ) = (A.1) : P ! ¡ ! ¡ A + B lnj r ¡ r j ; d = 2 i ? where the sum is over all the lattice sites. In order to estimate the potential around the ! ¡ antipodal point we expand to 4th order in z , around the center of the elementary cell. X X 1 1 = (A.2) £ ¤ d d =2 ! ¡ ! ¡ ? d ? ! ¡ ! ¡ 2 2 2 j z ¡ r j i r 1¡ 2 z ¢ r =r + z =r i i i i i ! ¡ ! ¡ 2 2 2 denoting ± = ¡2 z ¢ r =r + z =r and p = d ¡ 2 > 0 we have: i ? i i ³ ´ X X 1 1 p ± p p 1 ± = ¡ + + 1 ¡ p ¡ ¢ ¡ ¢ ¡ ¢ ! ¡ ! ¡ p=2 p=2+1 p=2+2 j z ¡ r j 2 2 2 2 2 2! 2 r r r i i i i i ³ ´³ ´ p p p 1 ± ¡ + 1 + 2 + ¡ ¢ p=2+3 2 2 2 3! ³ ´³ ´³ ´ p p p p 1 ± + + 1 + 2 + 3 +¢¢¢ (A.3) ¡ ¢ p=2+4 2 2 2 2 4! 2 For p = 0 (d = 2), the factor of p in every term should be dropped. Using the expression for ± and grouping the 4th powers of z we obtain: 4 2 ! ¡ ! ¡ X X 1 3z 6z ( z ¢ r ) ! ¡ V ( z ) = ¡p (p + 2) + p (p + 2) (p + 4) ¡ p+4 p+6 4! r r i i i i ! ¡ ! ¡ ( z ¢ r ) ¡ p (p + 2) (p + 4) (p + 6) (A.4) p+8 { 24 { JHEP07(2002)051 ! ¡ As an example, for a two-dimensional square lattice, we express z in terms of the compo- ! ¡ nents, z = (z ; z ) and obtain: 1 2 2 3 ¡ ¢ ¡ ¢ 2 2 1 1 X X ¡ ¢ 48 i + j + 1 6 6 7 4 4 2 2 V (z ; z ) = z + z ¡ ¡ 4 ³ ´ ³ ´ 5 4 1 2 1 2 4 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 2 2 2 2 4! 1 1 1 1 i;j i;j i + + j + i + + j + 2 2 2 2 2 3 ¡ ¢ ¡ ¢ 2 2 1 1 X X 288 i + j + 1 36 6 7 2 2 2 2 ¡ z z ¡ (A.5) 4 ³ ´ ³ ´ 5 1 2 4 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 2 2 2 2 4! 1 1 1 1 i;j i;j i + + j + i + + j + 2 2 2 2 ! ¡ Similarly, in the case of a four dimensional hypercubic lattice, z = (z ; z ; z ; z ), and the 1 2 3 4 potential is: V (z ; z ; z ; z ) = 4 1 2 3 4 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 2 2 2 2 1 1 1 1 ¡ ¢ i + j + +¢¢¢ + k + l + 4 4 2 2 2 2 z +¢¢¢ + z 192 ¡ ³ ´ 1 4 ¡ ¢ ¡ ¢ 2 2 4! 1 1 i;j;k;l i + +¢¢¢ + l + 2 2 ³ ´ ¡ ¢ ¡ ¢ 3 2 2 1 1 i;j;k;l i + +¢¢¢ + l + 2 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 2 2 2 2 1 1 1 1 ¡ ¢ 1 i + j + +¢¢¢ + k + l + 2 2 2 2 2 2 2 2 ¡ z z +¢¢¢ + z z 384 ¡ ³ ´ 1 2 3 4 ¡ ¢ ¡ ¢ 5 4! 2 2 1 1 i;j;k;l i + +¢¢¢ + l + 2 2 ¡ (A.6) ³ ´ ¡ ¢ ¡ ¢ 2 2 1 1 i;j;k;l i + +¢¢¢ + l + 2 2 If the lattice is not hypercubic, the leading term in the expansion is the 2nd order term. 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Journal of High Energy Physics – IOP Publishing
Published: Aug 8, 2002