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- Copyright © IOP Publishing Ltd
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- 1029-8479
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- 10.1088/1126-6708/2002/01/036
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Abstract
JHEP01(2002)036 Received: January 17, 2002, Accepted: January 24, 2002 HYPER VERSION In°ationary scenarios from branes at angles Juan Garc¶³a-Bellido, Raul ¶ Rabad¶an and Frederic Zamora Theory Division CERN, CH-1211 Genµeve 23, Switzerland E-mail: [email protected], [email protected], [email protected] Abstract: We describe a simple mechanism that can lead to in°ation within string-based brane-world scenarios. The idea is to start from a supersymmetric con¯guration with two parallel static Dp-branes, and slightly break the supersymmetry conditions to produce a very °at potential for the ¯eld that parametrises the distance between the branes, i.e. the in°aton ¯eld. This breaking can be achieved in various ways: by slight relative rota- tions of the branes with small angles, by considering small relative velocities between the branes, etc. If the breaking parameter is su±ciently small, a large number of e-folds can be produced within the D-brane, for small changes of the con¯guration in the compacti¯ed directions. Such a process is local, i.e. it does not depend very strongly on the compact- i¯cation space nor on the initial conditions. Moreover, the breaking induces a very small velocity and acceleration, which ensures very small slow-roll parameters and thus an almost scale invariant spectrum of metric °uctuations, responsible for the observed temperature anisotropies in the microwave background. In°ation ends as in hybrid in°ation, triggered by the negative curvature of the string tachyon potential. In this paper we elaborate on one of the simplest examples: two almost parallel D4-branes in a °at compacti¯ed space. Keywords: D-branes, Supersymmetry Breaking, Cosmology of Theories beyond the SM, Physics of the Early Universe. JHEP01(2002)036 Contents 1. Introduction 1 2. Description of the model 2 3. Supergravity description at long distances 7 4. The in°ationary scenario 9 4.1 Phenomenological constraints 9 4.2 The model of brane in°ation 12 4.3 Geometrical interpretation of brane in°ation parameters 14 5. E®ective ¯eld theory description at short distances 15 6. Preheating and reheating 18 7. Conclusions 19 A. Probe brane e®ective action 20 B. Velocity-dependent corrections to the in°aton potential 23 C. Discussion on compact potentials 24 1. Introduction In°ation is a paradigm in search of a model [1]. It has been for several years the aim of particle physicists to construct models of in°ation based on supersymmetry (in search of su±ciently °at potentials) and string theory (in hope of a description within quantum gravity) [2]. It is thus worthwhile the exploration of the various possibilities present within string theory, to come up with cosmological models consistent with observations. In this respect, intersecting brane systems have very interesting features; see, among others, refs. [3, 4]. For instance, they provide constructions that are very close to the standard model, even with the same spectrum of particles [5]{[10]. In general these are non-supersymmetric, but there are also some supersymmetric constructions [9]. Within string theory, the interaction between the branes arises due to the exchange of mass- less closed string modes. When supersymmetry is preserved, the Neveu-Schwarz-Neveu- Schwarz (NS-NS) and Ramond-Ramond (R-R) charges cancel and there is no net force between the branes. Recently, the proposals of refs. [12]{[19] have derived some of the in°ationary properties from concrete (non-supersymmetric) brane con¯gurations. In this paper we will show that in°ation is a very general feature for non-supersymmetric con¯gurations which are not far { 1 { JHEP01(2002)036 away from the supersymmetric one. The idea is to break slightly the supersymmetric con¯guration so one can smoothly turn on an interaction between the branes. In°ation occurs as the branes are attracted to each other, but a tachyonic instability develops when the branes are at short distances compared with the string scale. To our knowledge, this property of ending in°ation like in hybrid models through the open string tachyon was ¯rst proposed in ref. [14]. This is a signal that a more stable con¯guration with the same R-R total charge is available, which triggers the end of in°ation. There are many ways in which this general idea can be implemented: by a slight rota- tion of the branes intersecting at small angles (equivalent to adding some magnetic °uxes in the T-dual picture), by considering small relative velocities between the branes, etc. In this paper we elaborate on one of the simplest examples: a pair of D4-branes intersecting at a small angle in some compact directions. The interaction can be made arbitrarily weak by choosing the appropriate angle to be su±ciently small. The brane-antibrane system is the extreme supersymmetry-breaking case, where the angle is maximised such that the orientation for one brane is opposite to the other. The interaction is so strong in this case, that in°ation seems hard to realise. One should take very particular initial conditions on the system for in°ation to proceed. On the other hand, if the supersymmetry breaking parameter is small, a huge number of e-folds are available within a small change of the internal con¯guration of the system, due to the almost °atness of the potential; in this way the initial conditions do not play an important role. Thus, we ¯nd that in°ation appears very naturally in systems that are not so far from supersymmetry preserving ones. In this paper we will mainly focus on one of the simplest realisations of intersecting branes and extract the ¯rst consequences for in°ation. In section 2 we will describe the system and its decay, due to the tachyon instability, to a supersymmetric brane. We devote section 3 to derive the e®ective action for the in°aton at large distances with respect to the string length scale; we present two di®erent (although equivalent) ways to obtain the action. In section 4 we explain how in°ation is produced in this model, indicating explicitly the conditions that a generic model should satisfy for producing a successful cosmological scenario. After that, in section 5, we discuss the behaviour of the system for inter-brane distances of the order of the string scale by using the description of the low energy e®ective ¯eld theory living on the branes. In section 6 we discuss very brie°y the conditions that give rise to an e±cient reheating within our world brane and derive the reheating temperature. The last section is devoted to the conclusions. The paper comes with three complementary appendices, where concrete computations are done for the in°aton's e®ective action (appendices A and B) and the transverse space compacti¯cation e®ects (appendix C). 2. Description of the model 3;1 6 6 Consider type-IIA string theory on R £ T , with T a (squared) six torus. Let us put 3;1 two D4-branes expanding 3 + 1 world-volume dimensions in R , with their fourth spatial dimension wrapping some given 1-cycles of T . In ¯gure 1 we have drawn a concrete con- ¯guration. The same happens in refs. [15, 17, 18]. { 2 { JHEP01(2002)036 X X X X X 2 4 8 ‘ 6 X X X X X 3 5 7 9 Figure 1: This ¯gure represents two D4-branes at an angle µ. The usual 4-dimensional spacetime 0 3 is along directions fx ; : : : ; x g. The branes are located at particular points on the compact four 4 7 8 directions fx ; : : : ; x g. Finally, they wrap di®erent cycles on the last two compact directions, x and x , and intersect at a given angle µ. If both branes are wrapped on the same cycle and with the same orientation, we have a completely parallel con¯guration that preserves sixteen supercharges, i.e. N = 4 from the 4-dimensional point of view. If they wrap the same cycle but with opposite orientations, we have a brane-antibrane con¯guration, where supersymmetry is completely broken at the string scale [20]. But for a generic con¯guration with topologically di®erent cycles, there is a non-zero relative angle, let us call it µ, in the range 0 · µ · ¼, and the squared supersymmetry-breaking mass scale becomes proportional to 2µ=(2¼® ), for small angles. The case we are considering can be understood as an intermediate case between the supersymmetric parallel branes (µ = 0) and the extreme brane-antibrane pair (µ = ¼), with the angle playing the role of the smooth supersymmetry breaking parameter in units of the string length. Notice that this con¯guration does not satisfy the R-R tadpole conditions [21]. These conditions state that the sum of the homology cycles where the D-branes wrap must add up to zero. In our case these conditions do not play an important role, since we can take a brane, with the opposite total charge, far away in the transverse directions. This brane will act as an expectator during the dynamical evolution of the other two-branes. We will come back to this issue of R-R tadpole cancellation with an expectator brane when discussing reheating. Also, since the con¯guration is non supersymmetric, there are uncancelled NS- NS tadpoles that should be taken into account [22]{[25]. They act as a potential for the internal metric of the manifold, e.g. the complex structure of the last two torus in the 8-9 plane, and for the dilaton. Along this paper we will consider that the evolution of these closed string modes is much slower than the evolution of the open string modes. Each brane is located at a given point in the two planes determined by the compact 4 7 directions fx ; : : : ; x g; let us call these points y , i = 1; 2. In the supersymmetric (µ = 0) con¯guration there is no force between the branes and they remain at rest. When µ 6= 0, a non-zero interacting force develops, attracting the branes towards each other. From the open string point of view, this force is due to a one-loop exchange of open strings between the two-branes. The coordinate distance between the two-branes in the compact space, 2 2 y = jy ¡ y j , plays the role of the in°aton ¯eld, whose vacuum energy drives in°ation. 1 2 See for example [10], where NS-NS tadpoles are analysed in the context of intersecting brane models. { 3 { JHEP01(2002)036 0 2 Field 2¼® m ¡ 2¼® 1 scalar 3µ 2 massive fermions 2µ 3 scalars µ 1 massive gauge ¯eld µ 2 massive fermions (massless for y = 0) 0 1 scalar (tachyonic for y = 0) ¡µ Table 1: The mass spectrum of the N = 4 supermultiplet. Table 1 gives the spectrum for the lightest open string states between the branes. The open string spectrum can be obtained from the quantization of open strings with ends attached to the D-branes at an angle, or in the T-dual picture by considering a magnetic °ux on the brane. Besides the usual contribution to the mass from the y 6= 0 expectation value, there is a non-trivial splitting for the whole N = 4 supermultiplet, proportional to the angle µ. It is interesting to point out the origin of this splitting from the point of view of the ¯eld theory living on the brane. We start from two D4-branes, one wrapping the cycle 8 9 (0; 1) and the other the cycle (1; N) of the squared torus in fx ; x g. If we T-dualise in the x direction, we obtain a 6-dimensional N = 2, U(N + 1) Super Yang-Mills theory living on the D5-branes. The distance between the branes triggers a Super-Higgs mechanism of U(N + 1) ! U(N)£ U(1). The non-trivial homological charge due to the wrapping in the x direction gives one unit of magnetic °ux through the dual torus, for the U(1) ½ U(N) factor, F = HI , where we have N N 1 1 µ H = = = : 0 0 0 2¼NR R 2¼® N 2¼® The magnetic ¯eld H couples to the corresponding o®-diagonal °uctuations of the N + 1 adjoint matrices representing the charged particles with respect to this abelian ¯eld. One can compute their mass spectrum on the 4-dimensional reduced ¯eld theory by ¯nding the eigen-states of the operator [11] 2 2 2 M = (p ¡ iA ) + (p ¡ iA ) + 2HS ; (2.1) 8 8 9 9 89 with S the spin operator on the 8-9 toric plane. Since the magnetic ¯eld is constant, the operator M gives a spectrum of Landau levels, with a spin-dependent splitting for the 6-dimensional N = 2 vector multiplet at each level, see ¯gure 2. We have the eigenstates m = (2n + 1)H + 2HS ; n = 0; 1; 2; : : : : (2.2) H;n The 6-dimensional N = 2 vector multiplet contains four scalars, two opposite chiral fermions and one massless vector. A chiral fermion in six dimensions becomes a Dirac fermion in four dimensions. From (2.2), its two Weyl components get splitted depending on the sign of S = §1=2. Out of the four physical degrees of freedom of the massless vector, only those two with the spin in the 8-9 plane, with S = §1, are splitted. From the 4-dimensional point of view, they are the two scalars of the N = 2 vector multiplet. { 4 { JHEP01(2002)036 + - 2 ψ + 2 ψ 4 ϕ + 2 A + - 2 ψ + 2 ψ θ/2 Figure 2: Splitting of the mass spectrum of the N = 4 Yang-Mills supermultiplet when the angles are non vanishing or, in the T-dual picture, degenerancy splitting for the ¯rst Landau level. à are the chiral fermions and Á are the scalars. With respect the remaining four scalars, which belong to the 4D N = 2 hypermultiplet, one of them is mixed with the two vectorial degrees of freedom with spin S = §1 to give a 4D massive spin 1 ¯eld. As one can check, after equating the angle, µ = 1=N, the lowest Landau level reproduces the spectrum of the lightest open strings given in table 1. Apart from this low-energy supermultiplet, there are copies of these particles at higher 2 0 levels, separated by a mass gap ¢m = 2µ=(2¼® ); these states were called gonions in ref. [7]. In the T-dual picture, where an angle corresponds to a magnetic °ux perpendicular to the brane, all these states correspond to higher Landau levels [3, 11], see eq. (2.2). In the next section we will describe their connection to the string states. Note that, for each Landau level, not only the supertrace of the squared masses cancel, but also the sum of their masses up to the sixth power: F 2n (¡1) m = 0 ; n = 1; 2; 3 : (2.3) This is because supersymmetry is spontaneously broken by the non-zero magnetic °ux in the T-dual 6-dimensional Super Yang-Mills theory. Notice that the ¯rst scalar will be tachyonic if the distance between the two-branes 2 0 is smaller than y = 2¼® µ, i.e. the two-brane system becomes unstable. It can minimise its volume (and therefore its energy) by decaying to a single brane with the same R-R charges as the other two but with lower volume, see ¯gure 3. Sen's conjecture [26] relates the di®erence between the energy of the initial and ¯nal states with the change of the tachyonic potential: ¢V = T (V + V ¡ V ) ; (2.4) tachyon 4 1 2 f p+1 ¡1 p 0 ¡1=2 where T = M g =(2¼) is the tension of the branes, with M = (® ) the string p s s mass and g the string coupling; V are the brane world-volumes, which take into account s i the possible wrappings around the compacti¯ed space. To visualise how this process happens let us take a simple toy model. Consider two- branes wrapping 1-cycles in a two-dimensional squared torus of radius R. Each of these branes wraps a straight line in the fn [a] + m [b]g homology class. The energy density of i i the two-brane system is just [25] · ¸ q q 2 2 2 2 E = T R n + m + n + m : 0 4 1 1 2 2 { 5 { JHEP01(2002)036 Figure 3: Decay of a two brane system on a T . These two-branes have a total (homological) charge f(n +n )[a]+(m +m )[b]g. Consider 1 2 1 2 for example two cycles with angles Á = arctan(m =n ) ´ µ and Á = arctan(m =n ) = 0, 1 1 1 2 2 2 so that the angle between the two-branes is µ = jÁ ¡ Á j ¿ 1. In this case we can write 1 2 the initial energy density as µ ¶ n tan µ E ' 2T L 1 + +O(µ ) ; (2.5) 0 4 2(n + n ) 1 2 where 2L = R(n +n ) is the length of the brane wrapped around the two cycles. However, 1 2 there is a brane con¯guration that has the same charges but less energy, see ¯gure 3. This is a brane wrapping a straight line in the f(n + n )[a] + (m + m )[b]g homology class. In 1 2 1 2 this case, the energy density of the system is: µ ¶ 2 2 n tan µ 1 4 2 2 E = T R (n + n ) + (m + m ) ' 2T L 1 + +O(µ ) : (2.6) f 4 1 2 1 2 4 2(n + n ) 1 2 From Sen's conjecture (2.4), we have ¢V = E ¡ E ' T R tan µ ; (2.7) tachyon 0 f 4 where ¹ = n n =(n + n ) is the \reduced" winding number of the two-branes. It is 1 2 1 2 interesting to note that in the case of a brane-brane system (µ = 0) we obtain ¢V = tachyon 0, as expected, while in the case of brane-antibrane (where supersymmetry is broken at the string scale) we have ¢V = 2T L, since E = 0. In our case, since we are tachyon 4 f breaking supersymmetry only slightly (µ ¿ 1), the energy di®erence is proportional to 4 0 2 M » (µ=2¼® ) . In the small angle approximation we have susy µ ¶ tan µ M L µ ¢V ' 2T L ' : (2.8) tachyon 4 2 0 8 16¼ g 2¼® This corresponds to the energy di®erence between the false vacuum with E = 2T L, and 0 4 the ¯nal true vacuum at the minimum of the tachyon potential. Note that the argument can be straightforwardly generalised to an arbitrary number of branes. In higher dimensions, i.e. branes wrapping higher-dimensional manifolds that intersect at some points, the stability can be analysed in a similar way [27, 28, 29]. The absence of tachyons depends on the angles of the system at the intersecting point. In these cases, branes can intersect without preserving any supersymmetry and still being non tachyonic. { 6 { JHEP01(2002)036 3. Supergravity description at long distances When the two-branes are at a distance much larger than the string scale, i.e. y À l , the e®ective action for the in°aton ¯eld y can be computed from the exchange of massless closed string modes. In this section, we will present two di®erent (although equivalent) ways to obtain the in°aton e®ective action. One can compute the closed string tree-level interaction between two D-branes by going to the open string dual channel. In that case, the interaction potential corresponds to the one-loop vacuum amplitude for the open strings [25, 30], X P dt t 2 2 0 ¡2 ¡ (y +m 2¼R ) i i i 0 i 2¼® V (y ; µ) = ¡V (8¼ ® t) e Z(µ; t) ; (3.1) i 4 µ (iµt=2¼; it) Z(µ; t) = ; (3.2) iµ (iµt=¼; it)´ (it) where ´(¿) is the Dedekind eta function, and µ (º; ¿) are the Jacobi elliptic functions [25]. Here m are the winding modes of the strings on the fX ; i = 4; : : : ; 7g transverse direc- i i tions, i.e. the Dirichlet-Dirichlet ones, see ¯gure 1. The R are the radii of these compact dimensions and the y are the distances between the branes in each of these compact di- mensions. The prefactor V is the regularised volume of our 4 Minkowski coordinates. To get the vacuum energy density one should take the ratio over the 4-volume, V (y )=V . i 4 Let us explain brie°y how this potential arises. The ¯rst factor comes from the inte- gration over momenta in the non-compact dimensions. The sum over the integer numbers m comes from the winding modes in the compact transverse directions. The µ and ´ i 11 elliptic functions come from the bosonic and fermionic string oscillators of the world sheet, see ref. [25]. Notice also that the potential (3.1) is not invariant under rotations in the compact coordinates. This is due to the toroidal compacti¯cation. We will see that in the limit where the compacti¯cation scale is much greater than the distance y between branes, the potential becomes invariant under the group of rotations SO(4) in these coordinates. For simplicity, we will consider a particular type of toroidal compacti¯cation, although the details are not very important for our model. The brane knows about the shape of the compacti¯cation space through the winding modes. When the distance between the D-branes, y , is smaller than the compacti¯cation radii R , the sum over the winding modes i i can be approximated by: P 2 t 2 y ¡ (y +m 2¼R ) ¡t i i i 0 0 2¼® 2¼® e ¡! e ; (3.3) 2 2 where y = y , i.e. these winding modes are so massive that they decouple from the i i low energy modes or, in other words, it costs a lot of energy to wind a string around the compact space. To illustrate this point let us consider a two-dimensional compacti¯ed model. The potential is represented in ¯gure 4. Close enough to the branes the potential recovers the expected rotational invariance. The dynamics of the branes close to these points is described very accurately by the non-compact potential. We analyse this approximation in greater detail in appendix C. { 7 { JHEP01(2002)036 –2 –2 –4 –4 –6 –8 –6 –10 –12 –8 0 0 1.5 –10 1.5 1 0.5 0.5 0.5 1 y1 –12 y2 1 1 0.5 y2 y1 1.5 1.5 2 2 Figure 4: Two views of the e®ective potential for the in°aton ¯eld in a two-dimensional torus transverse to the branes. The periods of the lattice are normalised to 1 in both directions. Notice how the rotational symmetry is recovered close to the brane. For distances much larger than the string scale y À l , the terms that contribute most to the integral are those that appear in the limit t ! 0. In that limit, the partition 3 2 function (3.2) becomes Z(µ; t) ! 4 t sin µ=2 tan µ=2, and the potential, for l ¿ y ¿ s i 2¼R , can be approximated by: sin µ=2 tan µ=2 V (y) = 2T L¡ : (3.4) 3 0 2 8¼ ® y This potential has the expected form, with the right power of the distance, as could be deduced from Gauss law, i.e. from the exchange of massless ¯elds in d = 4 transverse dimensions. We have plotted this potential in ¯gure 5 for the case of µ = ¼=12. The brane-antibrane system of refs. [13, 14] is the extreme case of two-branes with opposite orientations. The brane and antibrane attract each other with a force that depends on the distance [20]: V (y) = 2T L¡ ; (3.5) d ¡2 where B is a positive constant of order one in string units, and d = 9¡ p is the number of transverse dimensions to the branes. In the brane-antibrane system, due to the strong force between them, one has to chose a special location for the branes in order to have enough number of e-folds. In our case, as we will show, the number of e-folds is not so sensitive to the location of the branes, as long as the angle is su±ciently small. An alternative, but equivalent, way to obtain the in°aton e®ective action at large distances is to start in the closed string picture from the beginning and consider a probe brane moving in the supergravity background created by another BPS p-brane solution. The details of the computation for general Dp-branes are given in the appendix A. { 8 { JHEP01(2002)036 0.00002 0.00001 -0.00001 -0.00002 1 2 3 4 5 6 Figure 5: The attractive in°aton potential between two D4-branes. The red line corresponds to the string-derived potential (3.4); in blue is the Real part of the Coleman-Weinberg ¯eld-theory limit of the same potential (5.2), and in green is the Imaginary part, which is non zero only when 2 0 the tachyon condenses, at y < 2¼® µ. We have chosen here µ = ¼=12. The potential V (y) and the distance y are both in units of ® = 1. The resulting e®ective action can be organised in a perturbative expansion of small brane velocities and small supersymmetry breaking angle µ. The validity of both pertur- bative expansions are related, since a small supersymmetry breaking induces a small brane velocity. Indeed, from the appendix B, we can see that the O(v ) terms are negligible 0 2 within the supergravity regime ® =y ¿ 1. Here we just present the ¯rst µ-dependent corrections to the brane probe e®ective action: Z · µ ¶ 1 g ¼ sin µ 4 2 S = ¡(T L) d x ¡g 1 + j@yj ¡ probe 4 2 2(M L)(M y) s s µ ¶¸ g ¼ tan µ ¡ 2 1¡ ; (3.6) 8(M L)(M y) s s whose potential coincides precisely with eq. (3.4), in the small angle approximation. 4. The in°ationary scenario In this section we will describe in some detail the constraints that a generic model of in°ation in the brane, and its subsequent cosmological evolution, should satisfy in order to agree with observations. We will then use the brane in°ation model described above to obtain phenomenological constraints on the parameters of the model. 4.1 Phenomenological constraints We will give here the most relevant contraints that should be satis¯ed in any scenario of string brane in°ation and its subsequent cosmology. For reviews see [1, 2, 31]. 1. In°ation should be possible, i.e. the energy density of the universe should be such that the scale factor accelerates: a=a Ä > 0, or, in terms of the rate of expansion H = a=a _ , we should have ² = ¡H=H < 1. { 9 { V(y) - V 0 JHEP01(2002)036 2. Su±cient number of e-folds during in°ation, N = ln(a =a ), in order to solve ini end the horizon and °atness problems. This constraint depends on the scale of in°ation. In order for the universe to be essentially °at, with = 1:0 § 0:1, we require ( ¡ 1)= = exp(¡2N)(T =T ) (1 + z ) ' 0:1, which implies 0 0 rh eq eq µ ¶ rh N ¸ 54 + ln : (4.1) 10 GeV For the case of GUT-scale in°ation, one needs N ¸ 60, while for EW-scale in°ation, N ¸ 34, where the number of e-folds is computable in terms of the variations in the scalar ¯eld that drives in°ation, Z Z 1 V (Á) dÁ N = Hdt = ; (4.2) 2 0 M V (Á) Áe ¡1=2 18 with M = (8¼G) = 2:4 £ 10 GeV the Planck mass. 3. Amplitude and tilt of scalar density perturbations and the induced temperature aniso- tropies of the microwave background. Quantum °uctuations during in°ation leave the horizon and imprint classical curvature perturbations on the metric, which later (during radiation and matter eras) enter inside our causal horizon, giving rise through gravitational collapse to the large scale structure and the observed temperature anisotropies of the cosmic microwave background (CMB). The slow-roll parameters are de¯ned as µ ¶ 2 0 M V (Á) ² = ¿ 1 ; (4.3) 2 V (Á) V (Á) ´ = M ¿ 1 : (4.4) V (Á) The density contrast (± = ±½=½) and the scalar tilt at horizon satisfy [32, 33] 3=2 2 1 V (Á) 1=2 ¡5 ± = P = = 1:91 £ 10 ; 90% c.l. ; (4.5) 0 3 5 V (Á)M 5¼ 3 @ lnP (k) n¡ 1 = ' 2´ ¡ 6² ; jn¡ 1j < 0:10 ; 90% c.l. : (4.6) @ ln k 4. Amplitude and tilt of gravitational waves. Not only scalar curvature perturbations are produced, but also transverse traceless tensor °uctuations (gravitational waves), with amplitude and tilt: 2 H 1=2 ¡5 P = < 10 ; CMB bound; (4.7) ¼ M @ lnP (k) n = ' ¡2² ¿ 1 ; (not yet observed) . (4.8) @ ln k { 10 { JHEP01(2002)036 5. Graceful exit. One must end in°ation and enter the radiation dominated era. This typically occurs by the end of slow-roll, like in the usual chaotic in°ation models [1], when ² = 1, and not when ´ = 1, as incorrectly stated in the literature. One end end should then compute the number of e-folds from this time backwards, to see if there are su±cient e-folds to solve the horizon and °atness problems. A di®erent way to end in°ation is through its coupling to a tachyonic (Higgs) ¯eld, where the spontaneous symmetry breaking triggers the abrupt end of in°ation, like in hybrid in°ation [34]. This allows for ² ¿ 1. end 6. Reheating before primordial nucleosynthesis. Reheating is the most di±cult part in model building since we don't know to what the in°aton couples to. Eventually one hopes everything will thermalise and the hot Big Bang will start. One thing we know for sure is that the universe must have reheated before primordial nucleosynthesis (T > 1 MeV), otherwise the light element abundances would be in con°ict with rh observations. However, since the scale of in°ation is not yet determined observation- ally, we are allowed to consider reheating the universe just above a few MeV. 7. The matter-antimatter asymmetry of the universe. The universe is asymmetric with respect to baryon number, and the decays of the in°aton ¯eld typically conserve Baryon number, so it remains a mystery how the baryon asymmetry of the universe came about. According to Sakharov, we need B-, C- and CP-violating interactions out of equilibrium. The ¯rst three occur in the Electroweak theory, but we would need to reheat the universe above 100 GeV, which may be too demanding, unless the fundamental Planck scale (in this case the string scale) is relatively high. 8. The di®use gamma ray background constraints. If reheating occurs by emission of massless states to the bulk as well as into the brane, one must be sure that the bulk gravitons do not reheat at too high a temperature, because their energy does not redshift inside the large compact dimensions (contrary to our (3+1)-dimensional ¡4 world, where radiation redshifts with the scale factor like a ), and they could interact again with our (presently cold) brane world and inject energy in the form of gamma rays, in con°ict with present bounds from observations of the di®use gamma ray background [35]. 9. Model dependent constraints. In the case of brane in°ation models with large extra dimensions one may prefer that the low energy e®ective ¯eld theory remains (3 + 1)-dimensional (otherwise the cosmological evolution in the brane has to take into account the evolution of the extra dimensions). In that case, the Hubble scale should be much larger than the compacti¯cation scale, H R ¿ 1. This does not impose any serious constraint, in general. Also, in order to prevent fundamental couplings from evolving during or after in°ation we require that the moduli ¯elds of the compacti¯ed space be ¯xed. The observed amplitud of temperature anisotropies in the CMB only gives a relation between the scale and the slope of the in°aton potential. We do not provide however any stabilisation mechanism. { 11 { JHEP01(2002)036 tachyon y y Figure 6: Two D4-branes attract each other to decay in the last step to a bound state. The in°ation process will take place when the two-branes are far away if the angle between the two- branes is small enough. Note that we have chosen here µ = ¼=2 for pictorical purposes. In fact the branes are at an angle µ ¿ 1, so we would expect the ¯nal brane to be wrapped around the corresponding cycle. 4.2 The model of brane in°ation We will consider here a concrete brane model based on two D4-branes separated by a distance y, and intersecting at a small angle µ, which attract each other due to the soft supersymmetry breaking induced by this angle, see ¯gure 6. As previously mentioned, we consider that all the closed string moduli are frozen out. We are also ignoring the e®ect of the relative velocity of the branes, which would introduce a correction to the potential proportional to the fourth power of the speed. We have computed these corrections in appendices A and B, and we have con¯rmed that they are negligible for our model. They can be computed to all order in ® from the one-loop open string channel. At long distances, only the massless closed string modes contribute to these corrections, i.e. the supergravity approach is reliable. At short distances, only high speed e®ects are important, and they are bigger if the angle between the two-branes is small. But the process of in°ation we are considering occurs at very low velocity and long distances. As we will see below, the velocity is related to the slow-roll parameter ², which in our case is very small. The corrections are therefore negligible. These e®ects are very important when the distance between the two-branes is of the order of the string scale, then the speed contributions are the dominant ones. The 4-dimensional e®ective action can be written as · ¸ p 1 1 4 2 2 S = d x ¡g M R¡ T L(@ y) + V (y) ; (4.9) 4 ¹ 2 2 2 2 6 ¡2 where M = M (2¼R M ) g is the 4-dimensional Planck mass, and R is related to the s s compacti¯cation volume as V = (2¼R) . The potential (3.4) for the canonically normalised 1=2 in°aton ¯eld, à = (T L) y, is given by µ ¶ 4 s V (Ã) = M 1¡ ¯ ; (4.10) { 12 { JHEP01(2002)036 4 ¡4 4 ¡1 where M ´ 2T L = 2(2¼) M (M L)g , and the parameter ¯ is a function of the angle 4 s s s µ between the branes, 4 sin µ=2 tan µ=2 ¯ = ; (4.11) (4¼) which is expected to be very small in order to get a su±ciently °at (slow-roll) potential. Let us calculate the derivatives of the potential, 0 3 V (Ã) M M M ' 2¯ ; V (Ã) M à 00 2 4 V (Ã) M M 2 s M ' ¡6¯ ; (4.12) 2 4 V (Ã) M à and the number of e-folds (4.2) · ¸ 2 4 4 2 4 1 M à à 1 M à s s end N = ¡ ' ; (4.13) 2 4 4 2 4 8¯ M M M 8¯ M M s s s P P in terms of which we can write the slow-roll parameters (4.3) and (4.4), and the scalar tilt (4.6) 1 à 3 ² ' ; ´ ' ¡ ; 32N M 4N n ' 1¡ = 0:974 ; (4.14) 2N which is well within the present bounds from CMB anisotropies, for N = 54. The amplitude of scalar metric perturbations (4.5) also gives a constraint on the model parameters, 3=4 ¡1=4 1=2 N 2 g (M L) s s ¡5 ± = p = 1:91 £ 10 ; (4.15) 1=4 9=2 ¯ (2¼R M ) 5 3¼ which implies 1=2 g (M L) s s 1=4 3 ¯ = 3:27 £ 10 ; (4.16) 9=2 (2¼R M ) and thus 1=2 à (M Lg ) ¤ s s 1=4 3=2 ¡1=2 4 = (8¯N) (2¼R M ) g = 1:48 £ 10 ; (4.17) M (2¼R M ) s s where the asterisc denotes the time when the present horizon-scale perturbation crossed the Hubble scale during in°ation, 54 e-folds before the end on in°ation. In terms of the distance between branes, it becomes y 5:88 £ 10 g ¤ s = ; l (2¼R M ) s s y 5:88 £ 10 g ¤ s = ; (4.18) 2¼R (2¼R M ) so a compacti¯cation radius of order 2¼R M = 60 gives, for g = 0:1, i.e. in the weak s s { 13 { JHEP01(2002)036 string coupling regime, 2¼R l < 2:3 l = y = ¿ 2¼R s s ¤ 1=4 ¡6 1=2 and ¯ = 7:4 £ 10 (M L) , which could be made somewhat larger by chosing a large wrapping length L of the brane around the cycle in the compacti¯ed space, e.g. 1=4 ¡3 LM » 200, or ¯ » 10 , in which case the angle for supersymmetry breaking is ¡3 µ = 2 £ 10 . This small angle ensures that in°ation will end, triggered by the tachyon ¯eld, when y · y = 0:1 l . This value of the compacti¯cation radius, 2¼R M = 60, gives c s s a string scale, a Hubble rate and a scale of compacti¯cation ¡3 12 M = M (2¼R M ) g ' 9£ 10 GeV ; s P s s ¡1 1=4 13 M = (2M L g ) ' 1£ 10 GeV ; 2¼ H = p ' 3£ 10 GeV ; 3M ¡1 12 R = 2£ 10 GeV : (4.19) Let us now study the production of gravitational waves with amplitude (4.7), ¡1 1=2 4 (M Lg ) 1=2 s ¡10 P = = 6£ 10 ; (4.20) (2¼R M ) 3(2¼) which is well below the present bound (4.7). Finally we may ask whether our approximation of using a 4-dimensional e®ective theory ¡1 is correct. For that we need to have the Hubble radius, H , of the 4D theory during in°ation much larger than the compacti¯ed dimensions, ¡1 1=2 2 (M Lg ) s ¡5 H R = = 1:5£ 10 ; (4.21) 3 2 (2¼) (2¼R M ) so we are indeed safely within an e®ective 4D theory. 4.3 Geometrical interpretation of brane in°ation parameters Here we will give a geometrical interpretation of the number of e-folds and the slow-roll parameters in our model. The epsilon parameter (4.3) is in fact the relative squared velocity 2 4 2 2 4 2 (v = y_) of the branes in the compact dimensions. Since à = M v =2 and 3H = M =M , we have _ _ H à 3 ² = ¡ = = v : (4.22) 2 2 2 H 2M H 4 The number of e-folds (4.2) can be seen to be proportional to the distance between the branes in the compacti¯ed space, Z Z dà H N = p = dy : (4.23) M 2² { 14 { JHEP01(2002)036 Finally, the eta parameter (4.4) is the acceleration of the branes with respect to each other due to an attractive po- 2¡d tential of the type V (y) / y , coming from Gauss law in d transverse dimensions, d ¡ 1 ´ = ¡ ; (4.24) d N which only depends on the dimensionality of the compact space d . Note that the spectral tilt of the scalar perturba- tions (4.6) therefore depends on both the velocity and accel- eration within the compact space, and is very small in our Figure 7: The dilaton ¯eld y is interpreted as the dis- model of spontaneous supersymmetry breaking, which makes tance between the two-branes. the branes approach eachother very slowly, driving in°ation Quantum °uctuations of this and giving rise to a scale invariant spectrum of °uctuations. ¯eld will give rise upon col- In fact, the induced metric °uctuations in our (3 + 1)- lision to density perturba- dimensional universe can be understood as arising from the tions on comoving hypersur- fact that, due to quantum °uctuations in the approaching faces. These °uctuations will D4-branes, in°ation does not end at the same time in all be later observed as temper- ature anisotropies in the mi- points of our 3-dimensional space, see ¯gure 7, and the gauge crowave background. invariant curvature perturbation on comoving hypersurfaces, R = ±N = H ±y =v, is non vanishing, being much later k k k responsible for the observed spectrum of temperature anisotropies in the microwave back- ground [32, 33]. 5. E®ective ¯eld theory description at short distances It is important to make the matching between the superstring theory at large distances and the supersymmetric quantum ¯eld theory in the brane, whose dynamics will be important for the reheating of the universe after in°ation. For that purpose, note that we can write the partition function (3.2) in terms of in¯nite products, à ! 1 1 4 m 4 m ¡1 4 X Y (z ¡ 1) (1¡ q z) (1¡ q z ) 2n Z(µ; t) = z ; m 6 m 2 m ¡2 z (1¡ q ) (1¡ q z )(1¡ q z ) n=0 m=1 ¡2¼t ¡µt q = e ; z = e : (5.1) 4 ¡1 2 3 The ¯rst factor, (1¡z) =z = z ¡4+6z¡4z +z , gives precisely the lowest lying N = 4 supermultiplet, including the tachyon, see table 1 and ¯gure 2, with the correct multiplicity 2 0 and (bosonic/fermionic) sign. Together with the exponential factor, exp(¡t y =2¼® ), in eq. (3.1), it gives the masses for the one-loop potential ¡1 dt 0 2 F ¡2¼® t m V = (¡1) e 1¡loop 2 0 2 3 (8¼ ® ) t F 4 2 = (¡1) m log m ; (5.2) i i 64¼ { 15 { JHEP01(2002)036 0 2 Field 2¼® m ¡ 2¼® 1 scalar 7µ 2 massive fermions 6µ 3 scalars 5µ 2 massive gauge ¯elds 5µ 4 massive fermions 4µ 4 scalars 3µ 2 massive gauge ¯elds 3µ 4 massive fermions 2µ 3 scalars µ 2 massive gauge ¯elds µ 2 massive fermions (massless for y = 0) 0 1 scalar (tachyonic for y = 0) ¡µ Table 2: The mass spectrum of the lowest 3 Landau levels, which fall into N = 4 supermultiplets. which corresponds to the Coleman-Weinberg potential for the low energy e®ective ¯eld F 2n theory. The fact that (¡1) m = 0; (n = 1; 2; 3), ensures that (5.2) is ¯nite, as it i i should be, since supersymmetry is being spontaneously broken. 1 2n We can then consider the next series of states in (5.1). The factor z = 1 + n=0 2 4 6 z + z + z + ¢¢¢ corresponds to the Landau levels induced, in the dual picture, by the supersymmetry breaking °ux associated to the angle µ. They give, at any order N, a F 2n supermultiplet with (¡1) m = 0; (n = 1; 2; 3), so they still provide a ¯nite one-loop i i potential (5.2). For a given supersymmetry-breaking angle µ, one should include in the low energy e®ective theory the whole tower of Landau levels up to N = 1=µ. For instance, the 4 2 4 ¡1 2 3 spectrum for µ = ¼=2 is derived from (1¡ z) =z (1 + z + z ) = z ¡ 4 + 7z ¡ 8z + 8z ¡ 4 5 6 7 8z + 7z ¡ 4z + z , with masses given by table 2. Finally, one could include also the ¯rst low-lying string states, whose masses are de- termined by the expansion of the in¯nite products in (5.1), à ! · ¸ 4 4 4 2 4 (z ¡ 1) (1¡ z) (1¡ z) (1 + 7z + z ) 2n 2 4 Z(µ; t) = z 1 + q + q +¢¢¢ : (5.3) 2 4 z z z n=0 Their structure still comes in N = 4 supermultiplets, so they again give a ¯nite Coleman- Weinberg potential (5.2). We will use the whole tower of Landau levels in the e®ective ¯eld theory to connect the one-loop potential at short distances, determined by the Coleman-Weinberg potential (5.2), with the full string theory one-loop potential coming from exchanges of the massless string modes at large distances, responsible for in°ation. This connection will be essential for the latter stage of preheating and reheating, because it will provide the low energy e®ective masses, and the couplings between the in°aton ¯eld y and the e®ective ¯elds living on the D4-brane. { 16 { JHEP01(2002)036 The potential (5.2) is ¯nite if there are no massless or tachyonic ¯elds. When the tachyon appears, i.e. at distances smaller tham y , there is an exponentially divergent amplitude for m < 0. A possible strategy to attach physical meaning to this divergence is to analytically continue the potential (3.1) in the complex y-plane. After the continuation, there is a logarithmic branch point at y = y . In this way, we get rid of the divergence and the potential develops a non-vanishing imaginary part for y < y , which signals the instability of the vacuum. We have plotted in ¯gure 5 the attractive potential V (y) between two D4-branes at an 2 0 angle µ = ¼=12. The large distance behaviour y À ® is determined from the supergravity amplitude (3.4), in red, while the short distance potential is obtained from the Coleman- Weinberg potential corresponding to the lowest-lying e®ective ¯elds, which fall into N = 4 supermultiplets. The real part of the Coleman-Weinberg potential is drawn in blue in ¯gure 5, while the imaginary part is in green. In the ¯eld theory limit (q ! 0 and z 6= 0) we can consider the in¯nite tower of Landau 4 2n¡1 levels, while ignoring the string levels, i.e. Z(µ; t) ' (z ¡ 1) z . The ¯nite low n=0 energy e®ective potential up to Landau level N can be written as 2 2 2 4 2 2 2 2 V (y; µ) = (y ¡ µ) ln(y ¡ µ)¡ 4y ln y + 7(y + µ) ln(y + µ) ¡ 32¼ 2N n 2 2 2 ¡ 8 (¡1) (y + nµ) ln(y + nµ) + n=2 2 2 2 + 7(y + (2N + 1)µ) ln(y + (2N + 1)µ)¡ 2 2 2 ¡ 4(y + (2N + 2)µ) ln(y + (2N + 2)µ) + 2 2 2 + (y + (2N + 3)µ) ln(y + (2N + 3)µ) ; (5.4) 0 0 where y stands for y=(2¼® ) and µ for µ=(2¼® ). From this expression we can compute what is the value of the potential at the distance y = 0; we expect the tachyon to give an imaginary contribution to the vacuum energy, which could be interpreted as the rate of decay of the false vacuum towards the minimum of the tachyon potential. On the other hand, the real part can be summed over, 2N n 2 2 V (0; µ) = i¼ ¡ 8 (¡1) n ln(n) + 7(2N + 1) ln(2N + 1) ¡ 32¼ n=2 µ ¶ 2 2 ¡ 4(2N + 2) ln(2N + 2) + (2N + 3) ln(2N + 3) : (5.5) 2¼® The in¯nite sum converges in the limit N ! 1 to: µ ¶ i¼ ¡ 1:70638 µ V (0; µ) = : (5.6) 2 0 32¼ 2¼® This quantity corresponds to the di®erence between the false vacuum energy at large dis- tances between the branes, E = 2T L, and the height of the tachyon potential at zero 0 p { 17 { JHEP01(2002)036 distance (y = 0), and should be compared with ¢V in eq. (2.8), the di®erence be- tachyon tween E and the ¯nal energy density in the tachyonic vacuum E . Both are proportional 0 f to µ , as expected, since at y = 0 the one-loop potential (5.2) in the ¯eld theory limit q ! 0 can be written as Z µ ¶ Z 1 1 1 1 X X ¡1 dt (1¡ z) µ du 2n u ¡u 4 ¡2nu z = ¡ e (1¡ e ) e : (5.7) 2 0 2 3 2 0 3 (8¼ ® ) t z 8¼ ® u 0 0 n=0 n=0 What we have done to obtain (5.6) is to regularise this integral, for instance introducing a mass cut-o®, to make it absolutely convergent and then commute it with the sum. The fact that the one-loop e®ective action is ¯nite for each Landau level allows us to send the cut-o® to in¯nity such that only the series (5.5) remains. We have to compare the two quantities: µ ¶ 1:70638 µ ¢V = ; 2 0 32¼ 2¼® µ ¶ ¡1 T 2M Lg µ ¢V = ; (5.8) 2 0 32¼ 2¼® Figure 8: A sketch of the in°aton-tachyon potential V (T; y). in the small angle approximation. In order to ensure that The °at region corresponds to the the tachyon minimum is a global minimum for the low- in°ationary regime. energy e®ective theory, we need M L > 0:8352 g , which s s is easy to accommodate within the model. A sketch of the potential in both in°aton and tachyon directions is shown in ¯gure 8. The dotted line indicates where in°ation ends and the tachyon instability sets in. Note that after both branes collide, a vacuum energy density remains, which, for small angles, is of the same order as the original V . The cancellation of this energy density depends on the expectator branes/orientifolds necessary to cancel the R-R tadpoles and is irrelevant for the period of in°ation, but should be taken into account at reheating. 6. Preheating and reheating In this section we will brie°y describe reheating after in°ation, i.e. the mechanism by which the in°aton potential energy density gets converted into a thermal bath at a given temperature. The details of reheating lie somewhat out of the scope of the present paper. We will give here a succint account of what should be expected and leave for the next paper a more detailed description. In°ation ends like in hybrid in°ation, still in the slow-roll regime, when the string- tachyon becomes massless and the tachyon symmetry is broken immediately after. In ref. [36] it was shown that this typically occurs very fast, within a time scale of order the ¡1 0 ¡1=2 inverse curvature of the tachyon potential, t » m ´ (µ=2¼® ) . From the point of view of the low energy e®ective ¯eld theory description, this is seen as the decay rate (per unit time and unit volume) or the imaginary part of the one-loop energy density (5.6). { 18 { JHEP01(2002)036 Assuming that the false vacuum energy E ' 2T L = M of the two-branes is all of 0 p it eventually converted into radiation, we can compute the reheating temperature of the universe as µ ¶ 1=4 T ' M = 2:2 £ 10 GeV ; (6.1) rh ¼ g where we have taken g » 10 for the number of relativistic degrees of freedom at reheating, and we have neglected the energy lost in the expansion of the universe from the end of in°ation to the time of reheating, since the rate of expansion at the end of in°ation H » M = 3M is negligible compared with m . P T The actual process of reheating is probably very complicated and there is always the possibility that some ¯elds may have their occupation numbers increased exponentially due to parametric resonance [37] or tachyonic preheating [36]. Moreover, a signi¯cant fraction of the initial potential energy may be released in the form of gravitational waves, which will go both to the bulk and into the brane. Fortunately, since the fundamental gravitational scale, M , in this model is large enough compared with all the other scales, the coupling of those bulk graviton modes to the brane is suppressed, so we do not expect any danger with the di®use gamma ray background [35], but a detailed study remains to be done. 7. Conclusions In this paper, we have analysed the realisation of in°ationary models arising from the dynamics of D-branes departing only slightly from a supersymmetric con¯guration. The in°aton ¯eld is realised in these models as the inter-brane distance within a compact space transverse to the branes. As a ¯rst example, we obtained the e®ective interac- tion potential for the in°aton ¯eld in the case of two almost parallel D4-branes. Due to the small supersymmetry breaking, the potential is almost °at, and therefore satis- ¯es the slow-roll conditions. It is also interesting to point out the geometrical inter- pretation for various other cosmological parameters, such as the number of e-folds, or the slow-roll parameters ² and ´, as well as the quantum °uctuations that give rise to CMB anisotropies. We have analysed the period of in°ation in detail within the supergravity regime. Both D-branes attract each other with a small velocity at distances much larger than the string scale, but still much smaller than the compacti¯cation scale. In this way, the process involved is essentially local, without much dependence on the type of compacti¯cation. We ¯nd that a su±cient number of e-folds to solve the °atness and horizon problems can easily be accommodated within the model. For a concrete compacti¯cation radius in units of the string scale, 2¼R = 60 l , we ¯nd a mass scale M » M » 10 GeV. Moreover, the s s ¡1 12 radius of compacti¯cation turns out to be R » 10 GeV. An account of the reheating mechanism after in°ation remains to be studied in detail, and in particular the ratio of energy which is radiated to the bulk versus that into the brane. 7 5 The bulk graviton coupling is suppressed by powers of M , not M , which is good, since M =M » 10 P s P s in our model. { 19 { JHEP01(2002)036 A nice feature of the model is the fact that the tachyonic instability inherent to the brane model triggers the end of in°ation. Since the D4-branes wrap di®erent non-trivial homology cycles, a single supersymmetric D4-brane remains after both branes collide. We analysed the brane interaction at short distances by taking the e®ective ¯eld theory description on the brane. We computed the Coleman-Weinberg potential and obtained a good matching with the supergravity potential. It remains to study the reheating involved in this regime. We believe that since the supersymmetry breaking on the brane ¯eld theory is small, maybe it will be possible to study the interactions of the tachyon with other ¯elds by standard supersymmetric ¯eld theory methods. There are a series of important issues which should be analysed [38]: i) The ¯rst issue is the cancellation of the R-R tadpoles. Since the branes move in a compact space, the total R-R charge should vanish. We can put another D4-brane, wrapping the ¯nal cycle with opposite orientation, or an orientifold plane with exactly the same opposite R-R charge, far away from the two-branes driving in°ation, such that during the period of in°ation and the reheating of the universe, this extra brane is an expectator where the R-R °ux can end. ii) The second issue is the moduli stabilisation. There is a non-supersymmetric back-reaction on the bulk, as well as non-zero NS-NS tadpoles, that produces a non-trivial temporal evolution for the closed string moduli, such us the dilaton and the compacti¯cation radii. The idea is that the time scale involved for this process is much larger than the time involved during in°ation and reheating. One can compute the tadpoles by taking the square root of the annulus amplitude. In general terms, the tadpoles are proportional to V =V , k ? where V and V are the volumes of the compact spaces parallel and perpendicular to the k ? brane, respectively. Therefore, for large enough compacti¯cation radii, the tadpoles can be neglected. This is exactly our original con¯guration at the beginning of in°ation, so we hope to being able to freeze the moduli, at least during the in°ationary period. Finally, note that from the general expression of the tadpole potential we can expect that the universe will naturally evolve towards decreasing both the string coupling and the parallel volume (to minimise the brane energy), while increasing the transverse volume. However, in order to precisely quantify this possibility, one would have to solve the cosmo- logical equations of motion for this higher-dimensional model. We will leave this discussion for our forthcoming publication [38]. A. Probe brane e®ective action Following the conventions of ref. [25], the type-II supergravity solution for an extremal ¹ ^ Dp-brane extended in the directions x , for ¹^ = 0; 1; : : : ; p, is 2 ¡1=2 2 1=2 2 ds = H (x )ds + H (x )dx ; (A.1) ? ? string p p+1 p ? 3¡p e = H (x ) ; (A.2) ¡1 A = 1¡ H (x ) ; (A.3) 01:::p ? with all the other supergravity ¯elds vanishing. The function H is harmonic in the 9¡ p transverse coordinates x , I = p + 1; : : : ; 9, with the boundary condition H ! 1 for jx j = x x ! 1, in order to recover the °at Minkowski spacetime for the metric (A.1). ? I { 20 { JHEP01(2002)036 Considering the rotational invariance in the transverse space, we have c g N p s H (jx j) = 1 + ; (A.4) p ? 7¡p (M jx j) s ? µ ¶ 7¡ p 5¡p c = (2 ¼) ¡ : (A.5) Our case corresponds to p = 4 and N = 1 (number of background branes), but to stress the generality of this technique, we shall keep p < 7 and N arbitrary. The e®ective action for a probe Dp-brane moving in this background is given by S = S + S ; p-brane BI WZ p+1 ¡Á M N S = ¡T d » e ¡ det(G (X)@ X @ X ) ; BI p MN ¹ º Dp S = ¡T A : (A.6) WZ p 5 Dp p p+1 We take the con¯guration for the probe rotated in the fx ; x g plane: ¹ ¹ X = » ; ¹ = 0; 1; : : : ; p¡ 1 ; p p X = » cos µ ; p+1 p X = » sin µ ; I I ¹ X = y (» ) ; I = p + 2; : : : ; 9 : (A.7) Plugging into (A.6), we obtain q q p p ¡1 ¹ ¹ I S = ¡T cos µ d x d» ¡g H 1 + H tan µ det(± + H @ y @ y ) ; (A.8) BI p p p º I p p ¡1 S = ¡T cos µ d x d» ¡g (1¡ H ) : (A.9) WZ p There are two conditions in order to have that the brane probe action (A.6) is a valid approximation for the e®ective action of the in°aton ¯eld. The ¯rst is that the supersymmetry breaking mass scale is small with respect the string and Plank scales, to aboid relevant supersymmetry-breaking back-reaction e®ects on the background. In our model, supersymmetry is spontaneously broken by the rotated brane con¯guration (A.7). For su±ciently small angle, we can take the non-supersymmetric back-reaction as a sub- leading e®ect, at least in the region of small curvature. This brings us to the second condition, which is to use the action (A.6) only in the valid supergravity regime. It corresponds to large inter-brane distances with respect to the string length, i.e. M jx j À 1, in order to neglect the massive stringy e®ects. Later on, we s ? will see that small angle allows us to treat the in°ationary period within the supergravity regime, such that both conditions are satis¯ed. For the analysis of preheating, when M jx j » 1 and the tachyon is turned on, massive closed string states become relevant and s ? one should follow a di®erent strategy to describe the relevant degrees of freedom. We leave the quantitative analysis of preheating for the future [38]. { 21 { JHEP01(2002)036 ¡1 Notice that the two above mentioned conditions, µ=¼ and (M jx j) much smaller than one, correspond to the two natural expansion parameters for the probe action (A.6). We can perform an expansion for the ¯rst square-root factor in (A.8) of the form p p cos µ 1 + H tan µ = 1 + I = 1 + ¡ +O(I ) ; (A.10) p p 2 8 where we have de¯ned c g N p s I = sin µ ¿ 1 : (A.11) 7¡p (M jx j) For the second square-root in (A.8), we can expand with respect small velocities, or ¹ I what is the same, in the gradient K = @ y @ y , of the form º º I µ ¶ 2 2 3 1 + H K = 1 + H tr K + (tr K) ¡ tr(K ) +O(K ) : (A.12) p p 4 2 The result is that we can organise the kinetic and potential terms for the in°aton in a perturbative expansion in µ=¼ and inverse powers of (M jx j): S = d x ¡g (L (y)¡ V (y)) ; (A.13) p-brane K 1;p¡1 with the kinetic terms given by µ ¶ 1 1 p p 2 p L (y) = ¡T d» 1 + I (y; » )¡ I (y; » ) +¢¢¢ £ K p p 2 8 µ µ ¶ ¶ 1 H (y; » ) 1 2 2 £ tr K(y) + (tr K(y)) ¡ tr(K (y)) +¢¢¢ ; (A.14) 2 4 2 and the potential terms given by µ ¶ 1 1 p 2 p 4 V (y) = T cos µ d» 1 + tan µ ¡ H (y; » ) tan µ +¢¢¢ : (A.15) p p 2 8 Since the brane probe is rotated with respect the brane background, the harmonic function H (and consequently I also) depends on the world-volume coordinate » through the p p 2 2 combination jx j = y + » sin µ. This longitudinal Dp-brane direction is wrapping p p+1 a non-trivial cycle in the toric plane fx ; x g. If we consider a squared torus with size 2 2 4¼ R , then the lenght of the cycle is L = 2¼R = sin µ. Notice that we can also include the corrections due to a ¯nite compacti¯cation scale in the remaining transverse coordinates, X for I = p + 2; : : : ; 9, by just replacing the expression of the harmonic function H in (A.4) by its corresponding Kaluza-Klein expression. We can look at the appendix C for a discussion on this. Again, they will be small if we can keep our analysis for an inter-brane distance smaller than the compacti¯cation scale. As an illustrative example, let us compute the ¯rst non-supersymmetric corrections to the e®ective action of the in°aton. Form (A.14), the ¯rst correction to the kinetic energy comes from the integral Z Z 1 1 p¡7 ¡ ¢ 1 sin µc g N p s p p 2 2 2 d» I (y; » ) = dx M y + x 2 2M 0 0 c g N ¼ sin µ p+1 s = : (A.16) 6¡p (M y) 2M s s { 22 { JHEP01(2002)036 In this case we have for the kinetic term µ ¶ T L ¼ sin µ c g N p p+1 s L = ¡ 1 + +¢¢¢ j@yj ; (A.17) 6¡p 2 2M L (M y) s s and for the potential energy, µ ¶ ¼ tan µ c g N p+1 s V (y) = T L 1¡ +¢¢¢ ; (A.18) 6¡p 8M L (M y) s s For p = 4, N = 1 and small angles µ ¿ ¼, it gives exactly half of eq. (3.4). This should be expected, since we have only considered one probe brane, and we have not introduced, for instance, the rest energy of the other. B. Velocity-dependent corrections to the in°aton potential The corrections due to the speed of the branes can be obtained very easily from the one- loop open string amplitude (or equivalently by the tree level exchange of closed strings) by considering two angles. The ¯rst angle is our µ, the second one can be taken to be imaginary and corresponds to the hyperbolic tangent of the speed. This procedure is described in de- tail in [25]. The potential at long distances has a dependence with the velocity of the form: ¡1 (cos µ ¡ °) V (y; v) = ; (B.1) 3 0 2 16¼ ® r ° sin µ 2 2 2 2 where r = y + ¿ v is the distance between the two-branes and ° is the relativistic fac- tor: ° = 1= 1¡ v . The same expresion can be obtained from the probe brane e®ective 0 2 action (A.13) at leading order in ® =y . Notice also that this potential is invariant v ! ¡v, as expected from time reversal. One can expand the above expression in the low speed limit to get the corrections to the static potential from the motion of the branes: 2 2 2 4 2 (cos µ ¡ °) (cos µ ¡ 1) v v (cos µ ¡ 3) = + sin µ ¡ +O(v ) : (B.2) ° sin µ sin µ 2 8 sin µ The ¯rst term is just the static interaction: ¡1 (cos µ ¡ 1) ¡1 µ µ V (y; 0) = = sin tan : (B.3) 3 0 2 3 0 2 16¼ ® r sin µ 8¼ ® r 2 2 The second term is the correction to the in°aton kinetic term: ¡1 V (y; v) = sin µ : (B.4) 3 0 2 32¼ ® r This correction vanishes in the supersymmetric case when the two-branes become paral- lel [25]. The fourth order correction is: ¡1 (3¡ cos µ) V (y; v) = : (B.5) 3 0 2 128¼ ® r sin µ { 23 { JHEP01(2002)036 Since our description of the in°ationary process occurs at very low velocities we can estimate the order of magnitude of the correction due to these terms. It can be easily estimated by considering the highest speed v associated with the slow-roll parameter ² during the in°ationary process. One may worry that at low angles the sine factor in the denominator could give big corrections. Fortunately, when substituting the speed v into the above formulae, one gets negligible corrections to our results. For completeness, we can take the ultra-relativistic limit v ! 1. If we denote ± = 1¡v and expanding the above potential one gets: (cos µ ¡ °) 1 = p ¡ 2 cot µ +O( ±) : (B.6) ° sin µ sin µ 2± Notice that the interaction becomes stronger at high speeds. For a discussion about these limits, see ref. [25]. C. Discussion on compact potentials Within this model we are assuming that we are close enough to one of the D-branes that the e®ects of the other branes, as well as the e®ect of the winding modes around the compact space, are negligible. This means that the process we are considering is a local one, not depending very strongly on the details of the compacti¯cation and initial conditions. We will justify here this approximation. To evaluate the importance of this approximation, we consider that the Dirichlet direc- tions are compacti¯ed on a torus. In order to make a more general discussion let us consider D Dirichlet dimensions in a torus T . These directions appear in the cylinder partition function as a sum over winding modes, see eq. (3.2). Commuting the integral with the sum one can express the potential as a sum over the images of the brane on the torus. Thus one obtains at distances bigger than the string length an e®ective potential of the form: ¡k V (y ) = ; (C.1) P D¡2 [ (y + L n ) ] i i i i where k is a constant that will depend on the angles, °uxes, etc. When the system is supersymmetric the k vanish, i.e. there are no interactions. This formula is valid if the number of Dirichlet torroidal dimensions is di®erent from two. In the case D = 2 the integral produces a logarithmic potential. It is easy to see that the long distance behaviour is divergent and one should try to regularise the above expression. This divergence can be undertood very easily. It is due to the propagation of massless closed strings ¯elds at long distances, i.e. an IR divergence. From the formulae below one can see that it is related to the NS-NS uncancelled tadpoles. If these tadpoles are not present one expects this divergence to disappear. That is what happens in the supersymmetric case, where the constant k is zero, and in the non-compact case, where the images are not present. The computation can be done very easily by expressing the above sum as an integral: X P ¡k dt D¡2 ¡t (y +L n ) i i i 2 i V (y ) = t e ; (C.2) D¡2 ¡( ) 2 n { 24 { JHEP01(2002)036 which can be expressed, through the change of variables t = ¡¼=x, in terms of Elliptic functions as D¡2 Z µ ¶ ¡k ¼ y ix V (y ) = dx µ ; ; (C.3) i 3 D¡2 L L ¡( )V i T i 2 i=1 where V = L is the volume of the torus. 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Journal
Journal of High Energy Physics – IOP Publishing
Published: Feb 7, 2002