PHYSICAL REVIEW D 67, 046006 ͑2003͒

3-form induced potentials, stabilization, and running moduli

Andrew R. Frey* Department of Physics, University of California, Santa Barbara, California 93106

Anupam Mazumdar† CHEP, McGill University, 3600 University Road, Montre´al, Quebec, Canada H3A 2T8 ͑Received 8 November 2002; published 27 February 2003͒ We study the potential induced by imaginary self-dual 3-forms in compactifications of theory and the cosmological evolution associated with it. The potential contains exponentials of the volume moduli of the compactification, and we demonstrate that the exponential form of the potential leads to a power law for the scale factor of the universe. This power law does not support accelerated expansion. We explain this result in terms of and comment on corrections to the potential that could lead to inflation or quintes- sence.

DOI: 10.1103/PhysRevD.67.046006 PACS number͑s͒: 11.25.Mj, 04.65.ϩe, 98.80.Cq

I. INTRODUCTION so these models have the phenomenology of the Randall- Sundrum models ͓17–19͔. The warp factor depends on the If we believe that ͑or M theory͒ is the fun- position of D3- ͑and planes͒ on the com- damental description of interactions in our Universe, then we pact space and also determines the 5-form field strength. The are obviously forced to place the basic processes of cosmol- condition ͑1͒ gives rise to a potential for many of the light ogy into a string theoretic framework. Important steps have scalars, including the dilaton generically, which vanishes at been made in this direction by examining four dimensional the classical minimum and furthermore has no preferred models for potentials that could support the compactification volume. We will be interested in the behav- early phase of the accelerated expansion of the Universe, ior of these systems above the minimum, and the 4D metric known as inflation, which solves some of the outstanding will generalize ␩␮␯→g␮␯ . problems of the hot big bang cosmology ͓1͔. See, for recent For simplicity, we will mainly consider the case where the ͓ ͔ 6 ͓ examples, 2,3 . Other work has identified string theory mod- internal manifold is a T /Z2 orientifold, as described in 20– els in which D- physics leads to inflation ͓3–6͔.1 At the 22͔͑or in dual forms in ͓16,23͔͒. We take the torus coordi- mӍ mϩ ␲ same time, however, it has proven challenging to incorporate nates to have square periodicities, x x 2 ls , so that cosmological acceleration into string theory backgrounds be- the geometric structure is encoded in the metric. On this cause they tend to relax to supersymmetric vacua ͓7,8͔.In torus, the 3-form components must satisfy the Dirac quanti- this paper, we ask whether a stringy potential generated by zation conditions higher dimensional magnetic fields can give rise to acceler- ated expansion. We restrict our analysis to the classical po- 1 1 ϭ ϭ ෈ tential of supergravity. Hmnp ␲ hmnp , Fmnp ␲ f mnp , hmnp , f mnp Z. 2 ls 2 ls We study a class of exact solutions to type IIB supergrav- ͑3͒ ity that have a vacuum state ͓denoted by superscript (0)] with 3-form magnetic fluxes that satisfy a self-duality rela- Boundary conditions at the orientifold planes give large tion Kaluza-Klein masses to many fields ͑including the metric ͒ components g␮m , for example , and the remaining theory is Ϫ⌽(0) Nϭ Ã(0)͑FϪC(0)H͒ϭe H ͑1͒ described by an effective 4D gauged 4 supergravity with 6 completely or partially broken supersymmetry via the super- Higgs effect ͓24–29͔. ͑ ͒ on the compact space, which should be Calabi-Yau CY In the following section, we discuss the dimensional re- ͓ ͔ space 9,10 . These vacua were described in some detail in duction of the type IIB superstring in toroidal compactifica- ͓ ͔ ͓ ͔ 11 and in dual versions in 12–16 . The metric is of tions with self-dual 3-form flux, ignoring the warp factor, ‘‘warped product’’ form, paying particular attention to the potential for a subset of the light scalars. Next, in Sec. III, we find the cosmological evo- (0)2ϭ A␩ ␮ ␯ϩ ϪA m n ͑ ͒ ds e ␮␯dx dx e gmndy dy , 2 lution driven by our potential based on known inflationary models; we find that our potentials do not lead to an accel- erating universe. Finally, in Sec. IV, we comment on the *Electronic address: [email protected] generalization of our results to more complicated models, †Electronic address: [email protected] compare our results to other models that do lead to inflation, 1As did ͓47͔, which found inflation in a small region of moduli and discuss corrections to our potential that might or might space. not lead to inflation.

0556-2821/2003/67͑4͒/046006͑7͒/$20.0067 046006-1 ©2003 The American Physical Society ANDREW R. FREY AND ANUPAM MAZUMDAR PHYSICAL REVIEW D 67, 046006 ͑2003͒

II. DIMENSIONAL REDUCTION AND POTENTIAL M 2 1 ␮␥mnץ ␥ ץ ϭ P ͵ 4 ͱϪ ͫ ϩ S d x g R ␮ kin 16␲ E E 4 mn Here we will review the dimensional reduction of 10D type IIB supergravity in compactifications with imaginary 1 1 mn ␮ qp m ␮ n ␣ ץ ␣␮ץ ␥ ϩ ␥ ␥ D␮␤ D ␤ Ϫ self-dual 3-form flux on the internal manifold. For simplicity 4 mp nq 2 mn I I and specificity, we will concentrate on the toroidal compac- ͓ ͔ 1 1 ͬ ␮ץ ץ␮⌽Ϫ 2⌽ץ⌽ ץ tifications of 21,22 , extending our analysis to more general Ϫ ␮ e ␮C C , cases in Sec. IV. We will ignore the warp factor, which as- 2 2 sumes that the compactification radius is large compared to the string scale.2 1 M 2 ϭ . ͑6͒ P ␲2 2 8 ls A. Kinetic terms Here, M is the Planck mass, and we are using a coset space We will start with the kinetic terms, mostly following the P covariant derivative analysis of ͓22͔, using the Nϭ4 SO(6,22) ϫSU(1,1)/SO(6)ϫSO(22)ϫU(1) language because we 1 are studying configurations away from the at mn mn m n n m ␮␣ ͒ ͑7͒ץ ␣␮␣ Ϫץ ␣͑ ␮␤ ϩץD␮␤ ϵ the bottom of the potential. Our main purpose is to identify 2 I I I I the physical interpretation of the canonically normalized sca- lars, so we will skip the algebraic details. which arises from the magnetic coupling of the D3-branes to As was shown in ͓22͔, the moduli must be tensor densities ␤; this is the dimensionally reduced action for the heterotic in order to avoid double trace terms in the action, theory of ͓30,32͔, as one might expect. In deriving the ac- tion, one needs the identity ⌬ ⌬ ␥mnϭ Ϫ⌽ mn ␤mnϭ ⑀mnpqrs ͒ ͑ ⌬ ץ ⌽Ϫ ץ mnϭ␥ ץ ␥ϭϪ ␥ ץe g , C pqrs , ␥mn 2 2ϫ4! ␮ mn mn ␮ 6 ␮ 4 ␮ln . 8

It is easiest to study the cosmology of canonically normal- ⌬ϵͱ ͑ ͒ ized scalars; so we will break down the geometric moduli. det gmn, 4 For simplicity we will consider only the factorized case T6 ϭ(T2)3. We can then parametrize the metric on an indi- 3 ␣mϭ m ␲ vidual 2-torus ͓say, the (4Ϫ7) torus͔ as along with the D-brane positions I XI /2 ls and the 10D dilaton-axion. For the purpose of cosmology, we want to Ϫ␨ ␨ 2 ␨ work in the 4D Einstein frame ͑note that this is different than e ϩe d Ϫe d ␥mnϭe2␴ͫ ͬ. ͑9͒ in ͓22͔ because we are allowing the dilaton to vary͒ Ϫe␨de␨

Here, ␴ gives the overall size of the T2, ␨ gives the relative ⌬ E Ϫ2⌽ length of the two sides, and d controls the angle between the g␮␯ϭ e g␮␯ . ͑5͒ 2 two directions of periodicity. Then the ␥ kinetic term be- comes

From stringy dualities, it can be seen that the moduli defini- M 2 3 1 ␮␨ץ ␨ ץ ␮␴ ϩץ ␴ ץ ͫ ϭϪ P ͵ 4 ͱϪ ͒ ͑ tions 4 correspond to the geometric moduli gmn ,Bmn in a Skin ␲ d x gE ͚ 2 ␮ i i ␮ i i toroidal heterotic compactification, and the metric ͑5͒ is the 16 iϭ1 2 ͓ ͔ 4D ‘‘canonical metric’’ 30,31 in the heterotic description 1 ͒ ͑ ͬ ␮ץ ץ ϩ 22͔.4͓ ␮di di . 10 The kinetic action obtained from dimensional reduction of 2 type IIB supergravity ͑SUGRA͒ and the D3-brane action is then For canonical normalization, the coefficient of the kinetic terms should simply be Ϫ1/2, so a further rescaling is nec- essary. 2Actually, because the warp factor A scales like RϪ4 ͓14,22͔, the radius need only be a few times the string scale for our approxima- B. Potential tion to be reasonable. 3If the D-branes are coincident, the index I labels the adjoint rep- The scalar potential comes from dimensional reduction of resentation of U(N); the kinetic terms remain the same ͓29͔. the background 3-form terms in the type IIB action. After 4Strictly speaking, these are only the heterotic dual variables with converting to our variables, the potential for the bulk modes vanishing fluxes; see ͓16͔. is, in generality,

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2 4 M P M ϭ ͑ ␥ ͒␥mq␥nr␥ ps͓ ⌽͑ Ϫ ͒ ͑ P 2 Ϫ2͚ ␴ V det mn e F CH mnp F V ϭ h e i i ͭ coshͩ ⌽Ϫ͚ ␨ ͪ ϫ ␲ 0 i 4! 32 4͑8␲͒3 i Ϫ ͒ ϩ Ϫ⌽ ͑ ͒ CH qrs e HmnpHqrs͔ 11 1 ⌽ϩ͚ ␨ Ϫ ␨ ϩ e i i͓C2Ϫ2Cd d d ϩd2d2d2ϩe 2 3d2d2 2 1 2 3 1 2 3 1 2 along with an additional term that subtracts off the vacuum 5 Ϫ ␨ Ϫ ␨ Ϫ ␨ Ϫ ␨ Ϫ ␨ Ϫ ␨ ϩ 2 2 2 2ϩ 2 1 2 2ϩ 2 2 2 3 2ϩ 2 1 2 3 2 energy. This potential was derived from dimensional reduc- e d1d3 e d2d3 e d1 e d2 tion in ͓11,21͔, from gauged supergravity in ͓24,28͔, and from the superpotential of ͓12͔. One feature to note in this Ϫ ␨ Ϫ ␨ ϩe 2 1 2 2d2͔Ϫ1ͮ ͑14͒ potential is that it always has ͑at least͒ three flat directions at 3 the minimum, corresponding to the radii of factorization T6 ϭT2ϫT2ϫT2. Also, the ␤ moduli do not enter into the po- using again Eq. ͑13͒. It is straightforward but tedious to tential, although some become Goldstone bosons via the su- show that this potential is positive definite, and the only ex- ͓ ͔ ⌽Ϫ͚ ␨ ϭ ϭ ϭ per Higgs effect 22,24–26 . tremum is at i i C di 0. As this case is nonsuper- For cosmological purposes, we will need to have a more symmetric, quantum mechanical corrections should lift the explicit form of the potential in hand. Since there are 23 flat directions. scalars ␥mn,⌽,C, writing the full potential for a given set of On the other end of the supersymmetry spectrum are the 3-form fluxes would be prohibitively complicated, but we Nϭ3 models of ͓22͔, which fix the dilaton as well as all the ͑ can write down a few simple examples and focus on the complex structure. If we ignore C,di set them to a vanishing universal aspects. vacuum value͒, we find a potential The simplest case is to take the three T2 to be square, so ␴ that the geometric moduli are ␥44ϭ␥77ϭe2 1, etc., with all 4 M P Ϫ ͚ ␴ others vanishing. Then, above a vacuum that satisfies Eq. ͑1͒, V ϭ h2e 2 i i͓cosh͑⌽Ϫ␨ Ϫ␨ Ϫ␨ ͒ϩcosh͑⌽Ϫ␨ 3 3 1 2 3 1 we can calculate the potential ͑8␲͒ ϩ␨ ϩ␨ ͒ϩ ⌽ϩ␨ Ϫ␨ ϩ␨ ͒ϩ ⌽ϩ␨ ϩ␨ 2 3 cosh͑ 1 2 3 cosh͑ 1 2 4 M P Ϫ ͚ ␴ Ϫ⌽(0) ϭ 2 2 i iͫ ͑⌽Ϫ⌽(0)͒ Ϫ␨ ͒Ϫ ͔ ͑ ͒ Vdil h e e cosh 3 4 . 15 4͑8␲͒3 1 This again has the same cosh structure for the dilaton; the ϩ e⌽͑CϪC(0)͒2Ϫ1ͬ, ͑12͒ only difference is a factor of 4 due to the number of compo- 2 nents of flux in the background. Including the non-Abelian coupling for the D3-brane sca- 1 lars ␣m introduces new terms in the potential ͑see ͓29͔ for a h2ϭ h h ␦mq␦nr␦ ps. ͑13͒ I 6 mnp qrs supersymmetry based approach͒. In the absence of fluxes and even in the ground state, this potential is monotonic and simply forces the ␣m to commute. Otherwise, the branes pick This potential was written explicitly in SU(1,1) notation in I ͓ ͔ up a 5-brane dipole moment and become noncommuting, as 28 and is valid for any 3-form background. The most im- ͓ ͔ portant feature of this potential is that there is a vanishing discussed in 35 . Writing the brane positions as U(N) ma- trices, the potential is vacuum energy, and, further, the radial moduli ␴ feels a po- tential only when the dilaton-axion system is excited. Since this is the simplest potential to write down, it will be our ϭ ␲ 4 ͫ ␲ ⌽␥ ␥ ͓͑␣m ␣n͔͓␣q ␣ p͔͒ Vb 2 M 2 e mp nqtr , , primary focus in Sec. III. It is very interesting to note that the P cosmology of this potential for the dilaton-axion has been i discussed earlier in ͓3,33,34͔ from SUGRA. Importantly, ϩ ␥ ͒1/2 ⌽ Ϫ⌽ ϪÃ ͑det mn e e h 6͑ f though, their models did not include the radial moduli or the 12 „ negative term that subtracts off the cosmological constant. Adding the complex structure is more complicated and Ϫ ͒ ͑␣m␣n␣ p͒ͬ ͑ ͒ Ch …mnptr . 16 more model-dependent. The simplest possible case, for ex- ample, f ϭϪh , is nongeneric in that Eq. ͑1͒ is satisfied 456 789 ϭϪ at ⌽Ϫ͚ ␨ ϭCϭd ϭ0, so the ␨ give extra moduli com- To illustrate this potential, we take f 456 h789 as before, set i i i i ϭ ϭ␨ ϭ ␣4,5,6ϰ ␣7,8,9ϭ␳ 1,2,3 pared to other background fluxes ͑at the classical level͒. C di i 0, and consider IN and t ⌽Ϫ͚ ␨ with ti a representation of SU(2). Then However, we still have i i fixed by a cosh potential with a polynomial in C,di : ⌽ Ϫ ␴ Ϫ ␴ Ϫ ␴ Ϫ ␴ Ϫ ␴ Ϫ ␴ ϭ ␲ 4 ͫ ␲ ͑ 2 1 2 2ϩ 2 1 2 3ϩ 2 2 2 3͒␳4 Vb 2 M P 16 e e e e 5This comes from the D3/O3 tension, which must cancel the vacuum potential for string tadpole conditions to be satisfied to h789 Ϫ ␴ ⌽ Ϫ⌽ ϩ e 2͚ ie ͑e Ϫ1͒␳3ͬ. ͑17͒ i leading order in ls . 2

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There are actually more terms in this potential as required by ␲ 8 1 1 Ϫ͚ ␣ ␴ supersymmetry; these are just the lowest order terms that H2ϭ ͫ ⌽˙ 2ϩ ͚ ␴˙ 2ϩe i i iV͑⌽͒ͬ. ͑21͒ 2 2 2 i appear in the D-brane action given by ͓35͔. For example, the 3M P i underlying Nϭ4 supersymmetry gives a ␳6 term,6 and there is also a ␳2 term from gravitational back reaction that has The Hubble expansion is given by a˙ /a; an overdot denotes been calculated using supersymmetry in one case ͑see ͓36͔͒; derivative with respect to physical time and prime denotes ␣m differentiation with respect to ⌽. in any event, there is a local maximum in the I direction. Like the bulk potential, this potential has exponential prefac- Note that depending upon the slopes of the fields along tors from the ␴ moduli, and if the bulk scalars are away from their classical trajectories the dilaton can roll slowly com- ͓Ϫ2͚ ␴ ͔ factor. pared to the moduli, in which case we might be able to solve their minimum, there is the same exp i i 8 The key point to take from this discussion of the potential the moduli equations exactly. With this simple assumption ͑ ͒ ͑ ͒ ⌽˙ Ӷ␴˙ ⌽ is the exponential prefactor that appears in all terms, whether we first consider Eqs. 20 , 21 with i , and V( ) ϳ bulk or brane modes. V0, the latter condition is true if the dilaton time varying VEV changes slowly. Much stronger condition can be laid on the kinetic terms for the moduli and dilaton if we assume III. COSMOLOGICAL EVOLUTION In this section we seek the cosmological evolution of the M VЈ͑⌽͒ ␴˙ ӷ P ⌽˙ ͑ ͒ ϭ i . 22 dilaton and the moduli fields in a flat d 4 dimensional 2ͱ2␲␣ V͑⌽͒ space time background. However, for the purpose of illustra- i tion it is prudent that we consider a toy model which illus- The above equation can be derived from Eqs. ͑19͒,͑20͒ by trates the behavior of the potentials Vdil , V0 and V3 de- ⌽¨ Ӷ ⌽˙ ␴¨ Ӷ ␴˙ ⌽˙ Ӷ␴˙ assuming 3H , i 3H i and , which is equiva- scribed in the earlier section. lent to slow-roll conditions. Ϫ͚ ␣ ␴ Now we are interested in solving the moduli field evolu- Ϸ i i i ͑⌽͒ ͑ ͒ V e V . 18 tion without imposing slow roll conditions on them. We ar- gue that there exists an attractor region with a power law Let us also assume that the above potential has a global solution a(t)ϰt p, which from Eq. ͑21͒, dimensionally satis- ⌽ ⌽ ⌽ 2 Ϫ2 Ϫ͚ ␣ ␴ minimum 0 determined by V( ). At 0 the potential van- fies H ϰt ϰe i i iV(⌽). Hence we write ⌽ ␴ ishes. In the above, mimics the dilaton and i play the role of moduli with various coefficients ␣ determines the i ␣ ␴ ki slope of the potential. For generality we have assumed that e i iϭ , ͑23͒ ci there are i number of moduli. In our original potential all the t ␣ ϭ ͱ␲ ͑ ͒ slopes are fixed at i 4 /M P with normalized scalars , n ͑ ͒ see Eq. 14 . We will model Vb by slightly different poten- ϭ ͑ ͒ ͚ ci 2, 24 tial. iϭ1 For the sake of simplicity and generality in Eq. ͑14͒,we ␨ do not assume any form for di and i at the moment. It is where ki are dimensional and ci are dimensionless constants interesting to note that the potential Eq. ͑18͒ is quite ad- respectively. Equation ͑23͒, coupled with the equations of equate to determine the cosmological evolution if they domi- motion Eq. ͑20͒ results in ⌽ nate the energy density, which is fixed by the value V( )in n ⌽ ϰ 4 7 our case. Further note that V( ) (M P) . Therefore, given ͑ Ϫ ͒ ϭ␣2 ͑⌽͒ ͑ ͒ ␴ ϳ 3p 1 ci i V ͟ kk , 25 generic initial conditions for all the moduli i M P in the kϭ1 dimensionally reduced action, we hope that the rolling moduli could lead to the expansion of the universe. In order from which we find, using Eq. ͑24͒ and Eq. ͑20͒, to see this clearly, one must obtain the equations of motion n for both dilaton and moduli if coupled to the gravity in a 2͑3pϪ1͒ ͑⌽͒ ϭ Robertson-Walker space-time metric with an expansion fac- V ͟ kk n , kϭ1 ␣2 tor a(t), where t represents the physical time. The equations ͚ i of motion are in the Einstein frame iϭ1

Ϫ͚ ␣ ␴ c 2 4␣2 ⌽¨ ϩ3H⌽˙ ϩe i i iVЈ͑⌽͒ϭ0, ͑19͒ ͩ i ͪ ϭ i ͑ ͒ ␣ n 2 . 26 i ͩ ␣2 ͪ Ϫ͚ ␣ ␴ ͚ k ␴¨ ϩ ␴˙ Ϫ␣ i i i ⌽͒ϭ ͑ ͒ ϭ i 3H i ie V͑ 0, 20 k 1

6We thank S. Ferrara for discussions on this point. 8We are obviously assuming a priori that the dilaton is moving 7 4 Strictly speaking potential energy ought to be less than (M P) in very slowly which may or may not be the case. Nevertheless, our order to make sense of field theoretic description of the expanding scenario shall be able to discern some of the aspects of the actual Universe. dynamics, such as inflationary or noninflationary.

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͑ ͒ ⌽˙ Ӷ␴˙ one in order to be consistent with the Hubble equation ͑21͒. When substituted into Eq. 21 with i , we obtain the key result without using any slow roll condition for the In this case, we can find the exact solution for the dilaton moduli where the exponent of the scale factor a(t)ϰt p goes 2 as Ϫ␩ 1 1 1 ⌽͑t͒ϰa ; ␩ϭ ͫͩ 3Ϫ ͪ Ϫͱͩ 3Ϫ ͪ Ϫ4cͬ. 2 p p 16␲ 1 ͑33͒ ϭ ͑ ͒ p n . 27 M 2 P ␣2 Unlike the dilaton, the moduli have no minimum, and ͚ j jϭ1 they face the usual run away moduli problem. Note that once dilaton reaches its minimum the potential Eq. ͑18͒ vanishes, We also note that the scaling solution for the moduli fields and so the effective potential for the moduli. However, once ␴ can be found quickly as follows for any two moduli, i and the expansion of the universe driven by the dynamics for the ␴ ⌽ k : moduli comes to an end, the dilaton settles down at 0, then the moduli still continue to evolve accordingly ␴˙ 2 ␣ 2 i i ͩ ͪ ϭͩ ͪ . ͑28͒ ␴˙ ␣ d k k ␴˙ a͑t͒3 ϭ0, ͑34͒ dt „ i … The above equation ensures the late time attractor behavior for all the moduli in our case, which has a similarity to the provided there is some source of energy-momentum tensor assisted inflation discussed in ͓37,38͔. From Eqs. ͑23͒,͑24͒, supporting the expansion of the universe. The moduli can we can also write indeed come to rest at some finite value. So far we have been concentrating upon the toy model c with the potential Eq. ͑18͒. Nevertheless, the situation re- ␴ ϭ␴ ͑ ͒Ϫ i ͑ ͒ i i 0 ␣ ln t, 29 mains unchanged for the type of potentials we are interested i in; see Eqs. ͑12͒,͑14͒,͑15͒. Note that the dynamical behavior ␴ of the moduli will remain unchanged, but the dilaton may where i(0) is a constant depending on the initial condi- tions. roll slow or fast depending upon the actual slope of the di- Inflationary solutions exist provided pϾ1, which can be laton potential. By inspecting the potentials we find the cor- ␣ ␣ ϭ ͱ␲ attained in our case only when the slopes i are small responding slope of the moduli, i.e. i 4 /M P , and n enough, or in other words the moduli should have suffi- ϭ3. Therefore, the moduli driven expansion of the universe ciently shallower slope. The power law solution also applies leads to to any p in the range 0ϽpϽ1, where the expansion is non- inflationary. 1 pϭ Ͻ1; a͑t͒ϰt1/3. ͑35͒ Note that so far we have neglected the dynamics of the 3 dilaton. In spite of rolling down slowly, ⌽ eventually comes down to the bottom of the potential. So, the prime question is The expansion is noninflationary and will not solve any of ⌽ how fast does it roll down to its minimum 0. This will the outstanding problems of the big bang cosmology. Never- again depend on the exact slope of the potential for V(⌽). theless, this expansion which is slower than either radiation Nevertheless, if we demand that the dilaton is indeed rolling dominated or matter dominated epoch could be the precursor down slowly such as ⌽¨ Ӷ3H⌽˙ , then we can mimic the or end stage of inflation in this particular model. slow-roll regime for the dilaton, and the situation mimics Now, we briefly comment on bulk potential derived in Eq. that of soft inflation studied in Refs. ͓39–41͔, ͑17͒. Note, even if the dilaton is settled down the minimum with eϪ⌽ϭ1, the moduli fields still contribute to the poten- ⌽͒ϭ ⌽ ͒Ϫ ͑ ͒ f ͑ f ͑ 0 p ln t, 30 tial. It would then be interesting to note whether we get any expansion of the universe from the moduli driven potential. where Further note that the structure of the potential is quite differ- ent from Eq. ͑18͒. The potential rather follows ͑taking ␳ to 8␲ V͑⌽͒ be slowly rolling and ␳Ӷ1) f ͑⌽͒ϵ ͵ d⌽ . ͑31͒ 2 Ј͑⌽͒ M P V n m ϭ ␲2 4 ␳4 ͩ ␣ ␴ ͪ ͑ ͒ Vb 32 M P ͚ exp ͚ sj j . 36 Here the subscript 0 indicates the initial value. sϭ1 jϭ1 Ϫ␣ ␴ With aϰt p and e i iV(⌽)ϰH2, we can then param- etrize the dilaton equation of motion by This kind of potential has also been solved exactly without using slow-roll conditions ͓38͔. Of course with the possibil- 2 ␣ ϭ ⌽¨ ϩ3H⌽˙ ϭϪcH ⌽, ͑32͒ ity of some of sj 0 for some combination of s, j. Our case Eq. ͑17͒ exemplifies with s, jϭ1,2,3. For Eq. ͑36͒, again we ͚m ␣ ␴ ϰ 2 where c is a constant factor which determines the unknown demand that exp( jϭ1 sj j) 1/t . The late time attractor so- shape parameter of V(⌽), which ought to be smaller than lution for the moduli fields can be established with ͓38͔

046006-5 ANDREW R. FREY AND ANUPAM MAZUMDAR PHYSICAL REVIEW D 67, 046006 ͑2003͒

n 2 This sort of argument based on supersymmetry is readily ␣ q 2 ͚ qjB generalized to the Calabi-Yau models with 3-form fluxes that ␴˙ ϭ j q 1 ͓ ͔ ͩ ͪ ϭ . ͑37͒ were studied in 11 . Indeed, the form of the bulk mode ␴˙ ͩ n ͪ ͑ ͒ potential 11 is identical, although the complex structure l ␣ r ͚ rlB rϭ1 decomposition of the metric will differ from case to case. The key thing to note is that the overall scale of the internal ϵ ͚m ␣ ␣ T ␴ ϭ␴ In the above equation B ( jϭ1 sj qj)COF , where T stands manifold is always a modulus, as if we set 1,2,3 . In fact, for transpose and COF stands for the cofactor, and Bs it works out so that the exponential prefactor gives the same ϵ͚n ϳ 1/3 qϭ1Bsq is the sum of elements in row s. The power law a t evolution. The potential for brane modes should also solution a(t)ϰt p can be found to be ͓38͔ be similar, at least for small non-Abelian parts of the brane coordinates. Considering a more complicated CY compacti- n n fication is not the route to an accelerating universe. Again, ͚ ͚ Bsq this seems to be a feature of the broken supersymmetry. 16␲ s q pϭ , ͑38͒ We should contrast this case to other work that does find 2 det A M P inflationary physics in supergravity. In the 1980s, Refs. ͓43,44͔ found no-scale supergravities with inflation, but they ϭ͚m ␣ ␣ where Asq jϭ1 sj qj . specified the potential to give slow-roll inflation. The free- ␣ ͑ ͒ Now, we can read sj from Eq. 17 . After little calcula- ␣ dom to insist on inflation does not exist here. More recently, tion with the normalized sj , we obtain the value of p from other gauged supergravities have been found that can give at Eq. ͑38͒ least a give few e-foldings of inflation ͓2,3,33,34͔, but these 3 do not yet have a known embedding in string theory. These pϭ Ӷ1. ͑39͒ gauged supergravities are not of the no-scale type and have a 16 cosmological constant. Also, ͓3,4,6͔ describe inflation based ͓ ͔ Again we find that there is no accelerated expansion. The on the motion of branes in a warp factor. In fact, 3,6 use a assisted inflation in all these cases provides expansion but background very similar to the one considered here but in- could not be used to solve inflation or even late time accel- clude the warp factor. eration during the matter dominated era. In all our examples There is clearly, then, some hope for finding acceleration we found that the moduli trajectories follow the late time in compactifications with 3-form magnetic fields, and it is attractor towards the supersymmetric vacuum. Finally, a possible to think of other methods than D3-brane motion. word upon supersymmetry breaking in the observable sector, For example, the warp factor can modify the potential, al- which will induce mass ϳ1 TeV to the moduli and dilaton in though it does not seem likely to change the basic features. gravity mediation. Unless the moduli amplitude is damped Another possibility is that the small volume region of moduli considerably, the large amplitude oscillations of the moduli space, where supergravity breaks down, has a different form field will eventually be a cause for worry ͑through particle of the potential. It has been argued that some type IIB com- production͒. The late time moduli domination may lead to pactifications with flux with one T2 shrinking are dual to the infamous moduli problem ͓42͔. heterotic compactifications with intrinsically stringy mono- dromies ͓16,45͔, so it is conceivable that inflation could oc- IV. DISCUSSION cur in such a compactification with a decelerating end stage described by our model. In this section, we would like to comment on the conclu- Finally, there are many possible corrections associated sion that we cannot get power-law inflation ͑or quintessence͒ with supersymmetry breaking. It is known that there should from the 3-form induced potential. The reason seems related be stringy corrections to the potential in nonsupersymmetric to comments in ͓7,8͔; exponential potentials consistent with cases and that these would break the no-scale structure, giv- the constraints of supersymmetry are generically too steep. ͑ ͓͒ ͔ Our results, then, are consistent with a generalization to ing the radial modulus mass at least in the CY case 46 , many fields of the work of ͓7,8͔ that a system cannot simul- and there should also be supergravity loop corrections. It taneously relax to a supersymmetric minimum and cause would be very difficult to compute this potential, but it seems cosmological acceleration. Even though the models consid- likely that the potential could have a local maximum for the ered here do not necessarily preserve supersymmetry, they compactification radius, allowing for inflation. There are also are all classically of ‘‘no-scale’’ structure, meaning that they potentials from corrections, given by wrapped Eu- all have vanishing cosmological constant and no potential for clidean D3-branes ͓22͔. Since the instanton action is propor- the radial moduli. So even the nonsupersymmetric vacua tional to the volume of the cycle it wraps, it would actually have characteristics of supersymmetric cases. Furthermore, generate a potential like the exponential of an exponential. the potential arises from the supergravity Ward identity This type of potential could very possibly be shallow enough ͓24,28͔, which means it suffers from the same kind of con- to support inflation, although we have not investigated this straints imposed by the arguments of ͓7,8͔. Heuristically, the point. vacua of our system give Minkowski spacetime, which is In summary, we have examined the cosmology induced static, and there is no way to accelerate into a static state. by 3-form fluxes in type IIB superstring compactifications

046006-6 3-FORM INDUCED POTENTIALS, DILATON . . . PHYSICAL REVIEW D 67, 046006 ͑2003͒ and concluded that the classical bulk action does not lead to ACKNOWLEDGMENTS inflation or quintessence because the potential contains ex- ponential factors that are too steep, much as in ͓7,8͔. How- The authors are thankful to Cliff Burgess, Ed Copeland, ever, we have noted loopholes in our analysis which could Sergio Ferrara, and for helpful discussions allow accelerating cosmologies. We leave the exploration of and feedback. The work of A.F. is supported by National those loopholes for future work. Science Foundation grant PHY97-22022.

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