In this talk we discuss a few relevant aspects of heterotic M-theory. These
are the stabilization of the two relevant moduli (the length of the eleventh
segment (pi rho) and the volume of the internal six manifold (V)) in models
where supersymmetry is broken by multiple gaugino condensation and
non-perturbative corrections to the Kahler potential; the existence of almost
flat directions in the scalar potential; the possibility of lifting them, and
their role in constructing a viable model of inflation. Finally, we review the
status of the moduli problem within these models. TComment: LaTeX, 8 pages, 4 eps figures, uses ws-p8-50x6-00.cls; invited talk
  at the International Workshop on Particle Physics and the Early Universe
  (COSMO-99), Trieste, Italy, September 27 - October 2, 199

Barreiro, T.

de Carlos, B.

English

arXiv

T. BARREIRO

B. DE CARLOS

Crossref

COSMOLOGICAL ASPECTS OF HETEROTIC M-THEORY

In this talk we discuss a few relevant aspects of heterotic M-theory. These are the stabilization of the two relevant moduli (the length of the eleventh segment (pi rho) and the volume of the internal six manifold (V)) in models where supersymmetry is broken by multiple gaugino condensation and non-perturbative corrections to the Kahler potential; the existence of almost flat directions in the scalar potential; the possibility of lifting them, and their role in constructing a viable model of inflation. Finally, we review the status of the moduli problem within these models. TIn this talk we discuss a few relevant aspects of heterotic M-theory. These are the stabilization of the two relevant moduli (the length of the eleventh segment (pi rho) and the volume of the internal six manifold (V)) in models where supersymmetry is broken by multiple gaugino condensation and non-perturbative corrections to the Kahler potential; the existence of almost flat directions in the scalar potential; the possibility of lifting them, and their role in constructing a viable model of inflation. Finally, we review the status of the moduli problem within these models. TIn this talk we discuss a few relevant aspects of heterotic M-theory. These are the stabilization of the two relevant moduli (the length of the eleventh segment (πρ) and the volume of the internal six manifold (V)) in models where supersymmetry is broken by multiple gaugino condensation and non-perturbative corrections to the Kähler potential; the existence of almost flat directions in the scalar potential; the possibility of lifting them, and their role in constructing a viable model of inflation. Finally, we review the status of the moduli problem within these models

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arXiv:hep-ph/0001199v1  19 Jan 2000Cosmological Aspects of HeteroticM-theoryT. BarreiroaCentre for Theoretical Physics, University of SussexFalmer, Brighton BN1 9QJ, UKB. de CarlosbTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandIn this talk we discuss a few relevant aspects of heterotic M-theory. Theseare the stabilisation of the two relevant moduli (the length of the eleventhsegment (piρ) and the volume of the internal six manifold (V )) in modelswhere supersymmetry is broken by multiple gaugino condensation and non-perturbative corrections to the Ka¨hler potential; the existence of almost flatdirections in the scalar potential; the possibility of lifting them, and their rolein constructing a viable model of inflation. Finally, we review the status ofthe moduli problem within these models.Talk given at theInternational Workshop on Particle Physics and the EarlyUniverse (COSMO-99)Trieste, Italy, September 27th - October 2nd, 1999(to be published in the proceedings)CERN-TH/2000-023SUSX-TH/00-001January 2000ae-mail address: mppg6@pcss.maps.susx.ac.ukbe-mail address: Beatriz.de.Carlos@cern.chCOSMOLOGICAL ASPECTS OF HETEROTIC M-THEORYT. BARREIROCentre for Theoretical Physics, University of SussexFalmer, Brighton BN1 9QJ, UKE-mail: mppg6@pcss.maps.susx.ac.ukB. DE CARLOSTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandE-mail: Beatriz.de.Carlos@cern.chIn this talk we discuss a few relevant aspects of heterotic M-theory. These are thestabilization of the two relevant moduli (the length of the eleventh segment (piρ)and the volume of the internal six manifold (V )) in models where supersymmetryis broken by multiple gaugino condensation and non-perturbative corrections tothe Ka¨hler potential; the existence of almost flat directions in the scalar potential;the possibility of lifting them, and their role in constructing a viable model ofinflation. Finally, we review the status of the moduli problem within these models.1 IntroductionAs it was well established by Horˇava and Witten 1, the strong coupling limitof the E8×E8 heterotic string theory can be described by d = 11 supergravity(SUGRA). This theory can be compactified on a manifold with boundaries,usually expressed as X × S1/Z2, where X is the d = 6 Calabi–Yau manifoldand S1/Z2 is the so-called eleventh segment. Altogether the picture we getis that of two walls (so-called hidden and observable) that interact throughgravity.The relevant parameters in this theory are V , the volume of the d = 6manifold, and piρ, the length of the eleventh segment. Among the attractivefeatures of this theory, it is worth mentioning the fact that gauge couplingunification is now entirely natural 2, and the scale at which the gauge couplingsmeet, 1016 GeV, is reconciled with the d = 4 Planck scale, MP ∼ 1018 GeV.For this to happen, the process of compactification must occur in the order d =11 → d = 5 → d = 4. In other words, in ordet to fit the phenomenologicallypreferred values for αGUT, MGUT and MP, given byαGUT = (4pi)2/3κ4/3〈V 〉−1 ,MGUT = 〈V 〉−1/6 , (1)MP = κ−1√piρV ,we need 〈piρ〉 ∼ (4 × 1015 GeV)−1 and 〈V 〉 ∼ (3 × 1016 GeV)−6. κ2 is the2d = 11 gravitational coupling and its corresponding Planck scale is given byM11 = κ−2/9.Within this framework we are interested in studying phenomena such assupersymmetry (SUSY) breaking and the cosmological evolution of the modulifields, which occur at scales below the compactification one. Therefore we shallconcentrate on the d = 4 effective SUGRA theory from now on.2 Moduli stabilizationWe shall discuss the stabilization of moduli as a previous step to address thequestion of their cosmological evolution. In order to do that, let us noticethat the values of V and piρ are determined by the chiral superfields S (thedilaton) and T (the modulus). More precisely the real parts of these chiralfields are given bySR ∼ κ4/3V ∼ O(α−1GUT) , (2)TR ∼ κ−2/3piρV 1/3 ∼ O(α−1GUT) .Note that, in the weakly coupled heterotic case, SR ∼ O(α−1GUT) ∼ O(20), thesame as here, whereas TR ∼ O(1) (always in MP units).To study the dynamical behaviour of these fields, and whether they ac-quire the desired vacuum expectacion values (VEVs), we analyse the scalarpotentialV = eK{(Wi +KiW )(Kji )−1(W¯ j +KjW¯ )− 3|W |2}, (3)where W (S, T ) is the superpotential and K(SR, TR) is the Ka¨hler potential.The sub (super) indices indicate derivatives of the functions with respect tothe (conjugate) fields. Together with the gauge kinetic functions, fa, W andK determine the SUGRA Lagrangian.To be more precise, we shall assume a particular source for SUSY break-ing, which is gaugino condensation in the hidden wall 3. This is, so far,the most promising mechanism for breaking SUSY at the right scale in or-der to give rise to an acceptable phenomenology. It assumes the existence ofa strong-type interaction in the hidden sector of the theory which, below acertain scale Λ, triggers the formation of gaugino condensates, λλ, and thebreakdown of SUSY. More precisely,W ∼ 〈λλ〉 ∼N∑i=1Cie−αifi , (4)3where the sum runs over all the condensing groups in the hidden sector(GH = G1×G2×. . .×GN ), αi are proportional to the 1-loop β-function coeffi-cients associated to each condensing group, Ci are also related to each group’scharacteristics and fi are the already-mentioned gauge kinetic functions. Inparticular, in the hidden sector fi = (S − niT/2)/4pi, with ni being modeldependent coefficients and, in the observable wall, fobs = (S + nobsT/2)/4pi.The Ka¨hler potential is given by the expressionK = K0 +Knp , (5)where K0 = − ln(2SR) − 3 ln(2TR) is the tree level piece and Knp standsfor M-theoretic non-perturbative effects. For the latter we will use a specificansatz 4.Now that we have defined all the functions we need to proceed with theanalysis of the scalar potential, Eq. (3). We have first of all confirmed theresults of Choi et al. 5 (obtained with a slightly different ansatz for Knp),namely that for one condensate only the weakly coupled minimum exists.With two condensates (W = C1e−α1f1 + C2e−α2f2), however, things becomemore interesting. It is possible to find plenty of examples of condensing groupsfor which both moduli are fixed at the desired VEVs and SUSY is broken atthe right scale (i.e. the gravitino mass, m3/2, is of order 1 TeV). In fact, inorder to understand the vacuum structure a bit better, it is convenient toredefine the fields S and T as follows:• The combinationsΦ−r ≡ SR −n1α1 − n2α22(α1 − α2)TR , (6)Φ−i ≡ SI −n1α1 − n2α22(α1 − α2)TI =4pi2kα1 − α2are fixed by the interplay between condensates, very much in the same way asthe racetrack mechanism worked in the weakly coupled heterotic case (note,from the second equation, that the condensates at the minimum, i. e. for kodd, are in opposite phase).• The orthogonal combinations are potentially flatΦ+r ≡n1α1 − n2α22(α1 − α2)SR + TR , (7)Φ+i ≡n1α1 − n2α22(α1 − α2)SI + TI .In fact we can easily check that these combinations of the fields do not appearin the superpotential. Therefore the potential flatness of Φ+r will be lifted by4−20020−1−0.500.51x 10−602468(a)Φ+iΦ−i  − 1.2V − V0x 1040−20020−1−0.500.51x 10−60510(b)Φ+iΦ−i  − 1.2V − V0x 10−40Figure 1. Scalar potential for (a) two condensates SU(3)M=0×SU(4)M=8 with n1 = n2 = 1,and (b) an additional third condensate SU(2)M=0 with n3 = 0.5. V0 is the value of thepotential at the minimum.the presence of the Ka¨hler potential, which depends on SR, TR, whereas Φ+iremains totally flat. This is a completely new feature associated to M-theorymodels, which depends entirely on the structure of the gauge kinetic functionsfa.Let us proceed to analyse cases with more than two condensates. Therethe definitions of Φ−r,i and Φ+r,i can be easily generalized to many condensates,and it is also possible to show that the flat direction will only remain so ifand only if all the ni coefficients that enter the definition of the gauge kineticfunctions, fi, are the same. This is shown in Fig. 1 where we plot the scalarpotential as a function of Φ±i , for Φ±r fixed, in the case of two (Fig. 1a) andthree (Fig. 1b) condensates. As we can see, the flat direction of Fig. 1a islifted by the presence of a third condensate with n3 6= n1 = n2. This opensup new scenarios from the cosmological point of view, as we are about to see.3 Cosmological Evolution and Moduli ProblemLet us then study the possible cosmological role of the Φ±r,i fields.• Φ−r has an exponential-type potential. In fact it is the analogous of thedilaton SR in the weakly coupled heterotic string and will therefore suffer fromthe same problems 6. The potential is too steep just before the minimum andthe field tends to roll past it towards infinity. It can be, however, stabilizedin the presence of a dominating background 7 but, in any case, it is totallyunsuitable as an inflaton.• Φ−i has a steep sinusoidal potential, analogous to that of SI in the50 5 10 15 20 25 30 35280300320340360380400420440460ΝΦ+i(a)0 10 20 30 40 50 60 70250300350400450500550ΝΦ+i(b)Figure 2. Evolution of Φ+ias a function of N (number of e-folds) for the three condensatesof Figure 1b, with n3 = 0.995, and two different initial conditions. The real fields are fixedto their minimum values.weakly coupled heterotic case. Again, this field is not a good candidate foran inflaton.On the other hand, the fields Φ+r,i have more promising potentials, as itwas pointed out above. It is then worth studying their evolution equations inthe presence of an expanding UniverseKSS¯(S¨ + 3HS˙) +KSSS¯S˙2 +∂V∂S¯= 0 , (8)KTT¯ (T¨ + 3HT˙ ) +KTTT¯ T˙2 +∂V∂T¯= 0 ,where H2 = 13KSS¯S˙˙¯S + 13KTT¯ T˙˙¯T + V3is the Hubble constant. It is crucialto note here that both S and T have non-minimal kinetic terms, i.e. Lkin =KSS¯DµSDµS¯+KTT¯DµTDµT¯ . This is what introduces the new terms in theevolution equations (8).Let us start with the Φ+r direction. It is easy to show that the potentialalong this direction behaves as an inverse power-law, i.e. V (Φ+r ) = A/(Φ+r )n,at least along the slope to the left of the minimum, which is the relevantone in terms of the evolution. By solving the evolution equations we havechecked that the field inflates for a few e-folds before reaching its minimum.These kinds of inflationary scenarios are denoted as ‘intermediate’ inflationin the literature 8. An important thing to notice is that, if the field Φ+r werecanonically normalized, then this inverse power-law potential would have givenenough e-folds of inflation.Finally we turn to the remaining direction to be analysed, namely Φ+i : aswe had said before, in the two-condensate case this is a totally flat direction,6however when we introduce a third condensate with n3 6= n1 = n2, thingschange. It can be shown that the potential in this latter case is a sinusoidalone,V = V0[1 + cos(aΦ+i + b)] , (9)where a and b are functions of n3. It is easy to see that the closer n3 is to n1and n2 the flatter the potential is. Therefore we have a way of controlling theflatness of the potential, which can lead us to successful examples of inflation.A couple of them are shown in Fig. 2, where we plot Φ+i as a function of thenumber of e-folds N . It is clear then that a combination of the imaginaryparts of the dilaton S and modulus T fields, what we have denoted as Φ+i ,can be a suitable inflaton in the context of heterotic M-theory. These areexamples of the so-called natural inflation 9.Let us finish this section by discussing very briefly the status of the moduliproblem within these models. This arises when we have very weakly interact-ing particles with VEVs of the order of MP and light masses, of the order ofm3/2. If these relics decay, they must do it before nucleosynthesis in order notto ruin its predictions. This imposes a lower bound on their masses of ∼ 10TeV. On the other hand, if these particles are stable, their oscillations shouldnot overclose the Universe. This sets an upper bound of ∼ 10−24 eV on theirmasses. In the weakly coupled heterotic string case, where all moduli masseswere of order 1 TeV, these bounds were obviously difficult to fulfil 10.We have calculated these masses for the present models. In general, weobtain that, along the Φ−r,i directions, the corresponding masses are of theorder of 103 m3/2, well above the lower bound for decaying particles; alongΦ+r they are of the order of 10 m3/2, which may be just enough to save thebound, and for Φ+i they are very small and dependent on how close n3 is to n1,n2. For example, for SU(3)M=0 × SU(4)M=8 × SU(2)M=0 with n1 = n2 = 1we find mΦ+i∼ 1.2 × 10−3 m3/2 for n3 = 0.98 and mΦ+i∼ 1.5 × 10−4 m3/2for n3 = 0.995. There is therefore a conflict with the upper bound for stableparticles, and the particle excess should be washed away with a period ofthermal inflation 11.4 ConclusionsWe have studied the dynamics of the two typical M-theory moduli, namely piρand V in terms of S, T . In the presence of non-perturbative corrections to theKa¨hler potential, and gaugino condensation as the source of SUSY breaking,the scalar potential presents very interesting features.7• It depends on S, T essentially through fa = S − naT/2, motivating aredefinition of fields to Φ−r,i and Φ+r,i.• Φ−r,i behave similarly to the dilaton of weakly coupled heterotic stringtheory: unsuitable as inflatons.• Φ+r would be flat in the absence of K: it has an inverse power-lawpotential, which gives very little intermediate inflation.• Φ+i is totally flat for two condensates and can have a sinusoidal potentialfor three or more: very promising inflaton (many e-folds of natural inflation).• There might be a moduli problem associated to this almost flat direction.These lighter moduli can be diluted with a small period of weak scale inflation.AcknowledgmentsWe thank the organizers for their hospitality. The work of TB is supportedby PPARC.References1. P. Horˇava, E. Witten, Nucl. Phys. B 460, 506 (1996) and Nucl. Phys.B 475, 94 (1996).2. E. Witten, Nucl. Phys. B 471, 135 (1996); T. Banks, M. Dine, Nucl.Phys. B 479, 173 (1996).3. P. Horˇava, Phys. Rev. D 54, 7561 (1996); Z. Lalak, S. Thomas, Nucl.Phys. B 515, 55 (1998); A. Lukas, B.A. Ovrut, D. Waldram, Phys. Rev.D 57, 7529 (1998) and JHEP 9904, 009 (1999).4. T. Barreiro, B. de Carlos, E.J. Copeland, Phys. Rev. D 57, 7354 (1998).5. K. Choi, H.B. Kim, H.D. Kim, Mod. Phys. Lett. A 14, 125 (1999).6. R. Brustein, P.J. Steinhardt Phys. Lett. B 302, 93 (1993).7. T. Barreiro, B. de Carlos, E.J. Copeland, Phys. Rev. D 58, 083513(1998).8. A.G. Muslimov, Class. Quant. Grav. 7, 231 (1990); J.D. Barrow, Phys.Lett. B 235, 40 (1990).9. K. Freese, J.A. Frieman, A.V. Olinto, Phys. Rev. Lett. 65, 3233 (1990).10. B. de Carlos, J.A. Casas, F. Quevedo, E. Roulet, Phys. Lett. B 318,447 (1993); T. Banks, D.B. Kaplan, A.E. Nelson, Phys. Rev. D 49, 779(1994).11. D.H. Lyth, E.D. Stewart, Phys. Rev. Lett. 75, 201 (1995) and Phys.Rev. 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Cosmological Aspects of Heterotic M-theory

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