Diluting the inflationary axion fluctuation by a stronger QCD in the early Universe
aa r X i v : . [ h e p - ph ] M a y CTPU-15-06, KIAS-P15021
Diluting the inflationary axion fluctuation by a stronger QCD in the early Universe
Kiwoon Choi a ∗ , Eung Jin Chun b † , Sang Hui Im a ‡ , Kwang Sik Jeong c § a Center for Theoretical Physics of the Universe, IBS, Daejeon 305-811, Korea b Korea Institute for Advanced Study, Seoul 130-722, Korea c Department of Physics, Pusan National University, Busan, 609-735, Korea
We propose a new mechanism to suppress the axion isocurvature perturbation, while producingthe right amount of axion dark matter, within the framework of supersymmetric axion modelswith the axion scale induced by supersymmetry breaking. The mechanism involves an intermediatephase transition to generate the Higgs µ -parameter, before which the weak scale is comparable tothe axion scale and the resulting stronger QCD yields an axion mass heavier than the Hubble scaleover a certain period. Combined with that the Hubble-induced axion scale during the primordialinflation is well above the intermediate axion scale at present, the stronger QCD in the early Universesuppresses the axion fluctuation to be small enough even when the inflationary Hubble scale saturatesthe current upper bound, while generating an axion misalignment angle of order unity. The non-observation of the neutron EDM requires theCP violating QCD angle to be as tiny as | ¯ θ | < − ,causing the strong CP problem. An appealing solution ofthis puzzle is to introduce a spontaneously broken globalPeccei-Quinn (PQ) symmetry [1]. Then ¯ θ correspondsto the vacuum value of the associated Nambu-Goldstoneboson, the axion, which is determined to be vanishing bythe low energy QCD dynamics [2].An interesting consequence of this solution is that ax-ions can explain the dark matter in our universe. Yet,the prospect for axion dark matter depends on the cos-mological history of the PQ phase transition. A possiblescenario is that the spontaneous PQ breaking occurs af-ter the primordial inflation is over. In such a case, themodel is constrained to have the domain-wall number N DW = 1, where N DW corresponds to the integer-valuedU(1) PQ × SU(3) c × SU(3) c anomaly coefficient. Then ax-ions are produced mainly by the annihilations of ax-ionic strings and domain-walls, which would result in theright amount of axion dark matter for the axion scale f a ∼ × GeV [3]. However it appears to be difficultto realize this scenario within the framework of a funda-mental theory such as string theory, since it requires aPQ symmetry with N DW = 1, as well as a restored PQphase until some moment after the primordial inflation.Another scenario which we will focus on in this paper isthat U(1) PQ is spontaneously broken during the primor-dial inflation and never restored afterwards. Then themodel is not subject to the condition N DW = 1, but isconstrained by the axion isocurvature perturbation [4–6].For instance, from the observed CMB power spectrum, ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] one finds [7], (cid:18) δTT (cid:19) iso ≃ (cid:18) Ω a Ω DM (cid:19) δθθ mis < . × − , (1)where θ mis and δθ denote the average misalignment angleand the angle fluctuation, respectively, for the axion fieldright before the conventional QCD phase transition when m a ( t QCD ) ≈ H ( t QCD ) with a temperature T ( t QCD ) ∼ a Ω DM ≃ . θ (cid:18) f a ( t )10 GeV (cid:19) . , (2)with Ω DM ≈ .
24 being the total dark matter fraction.Here we have assumed that | δθ | ≪ | θ mis | and there isno significant evolution of f a from t QCD to the presenttime t so that f a ( t QCD ) ≈ f a ( t ). In inflationary cos-mology, the primordial quantum fluctuation of the axionfield results in δθ ≡ δθ ( t QCD ) = γδθ ( t I ) = γ H ( t I )2 πf a ( t I ) , (3)where f a ( t I ) and H ( t I ) denote the axion scale and theHubble parameter, respectively, during the primordial in-flation epoch t I , and the factor γ is introduced to takeinto account the evolution of δθ from t I to t QCD . Notethat the inflationary Hubble scale H ( t I ) is bounded bythe tensor-to-scalar ratio of the CMB perturbation as r ≃ . (cid:18) H ( t I )10 GeV (cid:19) < . , (4)and the weak gravity conjecture [8] suggests that genericaxion scales are bounded as f a . O (cid:18) g π M P l (cid:19) , (5)where M P l ≃ . × GeV is the reduced Planck mass.To discuss the implication of the isocurvature con-straint (1), one needs to specify the cosmological evo-lution of the axion scale after the primordial inflation isover. If f a ( t I ) ∼ f a ( t ) as has been assumed in mostof the previous studies, it requires that either H ( t I ) issmaller than its upper bound ∼ GeV by at least fiveorders of magnitude, so that the CMB tensor mode is toosmall to be observable, or δθ should experience a largesuppression after the primordial inflation, which appearsto be difficult to be implemented.The above observation suggests a more attractive sce-nario realizing f a ( t I ) ≫ f a ( t ) [9] in a natural man-ner. Indeed supersymmetric axion models offer a nat-ural scheme to realize such a scenario, generating theaxion scale through the competition between the tachy-onic SUSY breaking mass term and a supersymmetric,but Planck-scale-suppressed higher dimensional term inthe scalar potential [10–13]. One then finds f a ( t ) ∼ p m SUSY M P l ,f a ( t I ) ∼ p H ( t I ) M P l , (6)which explains elegantly the origin of an intermediateaxion scale at present, while giving a Hubble-induced in-flationary axion scale well above the present axion scale,if the supersymmetry (SUSY) breaking mass m SUSY atpresent is around TeV scale. Furthermore, this type ofaxion models can be successfully embedded into stringtheory. Specifically, they can be identified as a low en-ergy limit of string models involving an anomalous U(1) A gauge symmetry with vanishing Fayet-Illiopoulos term[12, 14]. In such string models, the U(1) A gauge bosonis decoupled from the low energy world by receiving aheavy mass M A ∼ g M P l / π through the St¨uckelbergmechanism, while leaving the global part of U(1) A asan unbroken PQ symmetry in the supersymmetric limit.Once SUSY breaking is introduced properly, in both thepresent Universe and the inflationary early Universe, theresidual PQ symmetry can be spontaneously broken togenerate the axion scales as (6).In this paper, we discuss a novel mechanism to sup-press the axion isocurvature perturbation, while produc-ing the right amount of axion dark mater, within theframework of supersymmetric axion models with the ax-ion scales given by (6). The isocurvature constraint (1)and the relic axion density (2) suggest that for H ( t I )near the current upper bound ∼ GeV, the al-lowed amount of axion dark matter is maximal when f a ( t ) ∼ –10 GeV, while f a ( t I ) nearly saturatesthe weak gravity bound (5), e.g. f a ( t I ) ∼ –10 GeV.Interestingly, the axions scales generated by SUSY break-ing as (6) automatically realize such pattern if m SUSY isaround TeV scale. More specifically, for the case f a ( t ) /f a ( t I ) ≈ p m SUSY /H ( t I ) , (7) the isocurvature bound (1) reads off H ( t I )10 GeV < (cid:18) . γ (cid:19) (cid:18) Ω DM Ω a (cid:19) (cid:18) f a ( t )10 GeV (cid:19) . (cid:18) m SUSY (cid:19) , when combined with (2). This implies that a highscale inflation scenario with H ( t I ) ∼ –10 GeV,which would give an observable tensor-to-scalar ratio r = O (0 . .
01) in the CMB perturbation, can be com-patible with the axion dark matter Ω a = Ω DM , if theaxion field fluctuation experiences just a mild suppres-sion after t I , e.g. γ = O (0 . .
01) in (3).To suppress δθ through its cosmological evolution, oneneeds a period with m a ( t ) > H ( t ) well before t QCD . Onthe other hand, usually this is not easy to be realizedbecause the axion mass should be generated mostly bythe QCD anomaly in order for the strong CP problemsolved by the PQ mechanism. (See Refs. [15–18] for analternative possibility.) In the following, we propose asimple scheme to achieve such a cosmological period byhaving a phase of stronger QCD in the early Universe.Our scheme is based on a phase transition at t = t µ ≫ t I , which will be called the µ -transition in the followingas it generates the Higgs µ -parameter through the super-potential term [19], µ ( X ) H u H d ≡ κ X H u H d M P l , (8)where X is a PQ-charged gauge-singlet superfield.Specifically, X ( t ≤ t µ ) = 0 , X ( t > t µ ) ∼ p m SUSY M P l , (9)so that µ ( t ≤ t µ ) = 0 , µ ( t > t µ ) ∼ m SUSY . (10)With this transition, the weak scale experiences an un-usual evolution in a way that the weak scale before the µ -transition is comparable to the axion scale (6), as willbe discussed below.To proceed, let us discuss first the key features of thescheme, and later present an explicit model to realize thewhole ingredients. Including the Hubble-induced contri-bution, the mass of the D -flat Higgs direction H u H d isgenerically given by m φ = c φ H + ξ φ m + 2 | µ | ( φ ≡ H u H d ) , (11)where c φ and ξ φ are model-dependent parameters oforder unity. In our scheme, both c φ and ξ φ are as-sumed to be negative, so m φ < µ -transition.Then φ = √ H u H d is stabilized by the competition be-tween the tachyonic m φ | φ | and a supersymmetric termof O ( | φ | /M P l ) in the scalar potential, which results in f a ( t I ) ∼ φ ( t I ) ∼ p H ( t I ) M P l ,f a ( t µ ) ∼ φ ( t µ ) ∼ p m SUSY M P l . (12)On the other hand, after the µ -transition, m φ > µ ∼ m SUSY . The resulting weak scale and axion scale atpresent are given by φ ( t ) = O (100) GeV ,f a ( t ) ∼ X ( t ) ∼ p m SUSY M P l . (13)A simple consequence of the above evolution of H u H d is that the weak scale is comparable to the axion scalebefore the µ -transition:˜ φ ≡ φ ( t ≤ t µ ) ∼ ˜ f a ≡ f a ( t ≤ t µ ) . (14)This results in a higher QCD scale, i.e. a stronger QCD,and therefore a heavier axion mass which might be evenbigger than the Hubble scale for a certain period. Letus estimate the QCD scale ˜Λ QCD before the µ -transition,which is defined as the scale where the 1-loop QCD cou-pling blows up, as well as the resulting axion mass ˜ m a .For the case with ˜Λ QCD < ˜ m ˜ g ( ˜ m ˜ g ) < − ˜ φ , where ˜ m ˜ g denotes the gluino mass before the µ -transition, we find˜Λ QCD ≈
23 TeV (cid:18) ˜ m ˜ g
30 TeV (cid:19) / × (cid:18) tan β (cid:19) / ˜ φ GeV ! / , (15)where tan β = h H u i / h H d i at present, and ˜ m ˜ g / ˜ g ( ˜ m ˜ g ) ≃ m ˜ g /g ( m ˜ g ) for the gluino mass m ˜ g at present. Here weassume that g ( M GUT ) = ˜ g ( M GUT ) and y q ( M GUT ) =˜ y q ( M GUT ) for the QCD coupling and the quark Yukawacouplings. When the temperature T . ˜Λ QCD , the axionmass during the period of stronger QCD is estimated tobe ˜ m a ≈ ˜Λ / ˜ f a . (16)On the other hand, if ˜ m ˜ g (˜Λ np ) < ˜Λ QCD < − ˜ φ , theresulting QCD scale is estimated as˜Λ QCD ≈
21 TeV (cid:18) tan β (cid:19) / ˜ φ GeV ! / , (17)with the axion mass˜ m a ≈ ˜ m / g ˜Λ / / ˜ f a . (18)Here ˜Λ np denotes the scale where the stronger QCD be-comes nonperturbative, i.e. around ˜ g = 8 π /N c with N c = 3. Note that the axion potential for the axion mass(18) can be obtained by a single insertion of the SUSYbreaking spurion ˜ m ˜ g θ to the nonperturbative superpo-tential W np ∼ ˜Λ induced by the gluino condensation.If the stronger QCD scale ˜Λ QCD is high enough, therecould be a period with ˜ m a ( t ) > H ( t ) well before the conventional QCD phase transition. As is well known,in such a period the axion field experiences a dampedoscillation, with an amplitude ¯ a (averaged over each os-cillation period) evolving as¯ a ∝ R − / ( t ) , (19)where R ( t ) is the scale factor of the expanding universe.Then the spatially averaged vacuum value of the axionfield is settled down at the minimum of the axion poten-tial induced by the stronger QCD, while the axion anglefluctuation is diluted according to δθ = γ H ( t I )2 πf a ( t I ) ≈ (cid:18) T ( t µ ) T ( t i ) (cid:19) / H ( t I )2 πf a ( t I ) , (20)where t = t i denotes the moment when the damped ax-ion oscillation begins, and t = t µ is the moment whenit is over. Note that, after the µ -transition, the weakscale and the QCD scale quickly roll down to the presentvalues, so the axion mass becomes negligible comparedto H ( t ) until t ∼ t QCD when the Universe undergoes theconventional QCD phase transition. Also, the minimumof the axion potential induced by the stronger QCD isgenerically different from the minimum of the axion po-tential at present. As a result, our scheme generates anaxion misalignment angle of order unity: θ mis ≡ (cid:28) a ( t µ ) f a ( t µ ) (cid:29) − (cid:28) a ( t ) f a ( t ) (cid:29) = O (1) , (21)together with an intermediate axion scale at present, sogives rise to Ω a = Ω DM in a natural way.In our case, the damped axion oscillation induced bythe stronger QCD begins at a temperature T ( t i ) ∼ ˜Λ QCD as ˜ m a is highly suppressed by thermal effects for T ≫ ˜Λ QCD . On the other hand, the scalar field X generat-ing µ through (8) is trapped at the origin by thermaleffects until the Universe cools down to a temperature T ( t µ ) ∼ m SUSY . In fact, our scheme involves a varietyof dimensionless parameters which affect the naive esti-mate of the involved scales. We find that there is a largefraction of the natural parameter region where the axionmass ˜ m a ≈ . ˜ f a GeV ! − ˜Λ QCD ! (22)is larger than the Hubble scale H ( t µ ) ≃ . (cid:18) √ V × GeV (cid:19) , (23)over the period t i . t . t µ with a temperature ratio: T ( t µ ) /T ( t i ) = O (10 − –10 − ) . (24)Then the resulting δθ given by (20) can be small enoughto satisfy the isocurvature bound (1) even when H ( t I ) FIG. 1: Upper bound on the inflationary Hubble scale con-sistent with the axion dark matter, Ω a = Ω DM . Here wehave taken m ˜ g = 3 TeV, tan β = 10, and T ( t µ ) = 1 TeV.The shaded region is excluded by the Planck results. Theblack solid line is the constraint in the conventional scenariowith f a ( t I ) = f a ( t ). The magenta lines are for the scenariowith f a ( t I ) /f a ( t ) = p H ( t I ) /m SUSY , but without a strongerQCD. The blue lines are for our scheme which leads to afurther suppression of δθ by the stronger QCD. The SUSYbreaking mass has been taken m SUSY = 1 TeV for the solidlines and 10 TeV for the dotted lines. saturates its upper bound ∼ GeV. Note that during t i . t . t µ , φ ( t ) − φ ( t ) ∼ X ( t ) − X ( t ) ∼ p m SUSY M P l , so the corresponding vacuum energy density V = O ( m f a ( t )). This means that in this period theUniverse is dominated by the vacuum energy density withthe Hubble scale given by (23), which is often called thethermal inflation [20].It should be stressed that in our scheme the ax-ion isocurvature perturbation is suppressed by twosteps. The first suppression is due to f a ( t ) /f a ( t I ) ∼ p m SUSY /H ( t I ) ≪
1, and the second is due to thestronger QCD dynamics before the µ -transition, yield-ing a further suppression by γ ∼ ( m SUSY / ˜Λ QCD ) / . Toillustrate the result, we depict in Fig. 1 the upper boundon the inflationary Hubble scale H ( t I ) resulting from theisocurvature constraint (1) for Ω a = Ω DM . To make acomparison, we depict the results for three distinct cases:( i ) the conventional scenario of f a ( t I ) = f a ( t ) with-out a stronger QCD, ( ii ) a scheme with f a ( t I ) /f a ( t ) ∼ p H ( t I ) /m SUSY , but without a stronger QCD, ( iii ) ourscheme with f a ( t I ) /f a ( t ) ∼ p H ( t I ) /m SUSY and astronger QCD before the µ -transition.Let us now present an explicit model implementing themechanisms discussed above. As a simple example, we consider a model with the following superpotential, W = (MSSM Yukawa terms) + λY ΦΦ c + κ X H u H d M P l + κ XY M P l + κ ( H u H d )( LH u ) M P l , (25)where X and Y are PQ-charged gauge singlets responsi-ble for the µ -transition, L is the MSSM lepton doublet,and Φ + Φ c are U (1) Y -charged exotic matter fields in-troduced to give a thermal mass to Y . Then the scalarpotential for the µ -transition is given by V = m X | X | + m Y | Y | + (cid:18) κ A M P l XY + h . c . (cid:19) + | κ | M P l (cid:0) | Y | + 9 | X | | Y | (cid:1) , (26)where m X = c X H + ξ X m + 4 | µ X | ,m Y = c Y H + ξ Y m + α Y T , (27)for µ X = κ H u H d /M P l . Here c X,Y H are the Hubble-induced masses, ξ X,Y m are the SUSY breakingmasses at zero temperature, and α Y T is the thermalmass of Y induced by the coupling λY ΦΦ c , which is of O ( | λ | T ) for | λY | < T .For simplicity, we will assume that all the dimension-less parameters appearing in the superpotential and theSUSY breaking scalar masses are of order unity. How-ever it should be noted that these parameters can havea variation of O (0 . κ n can have a much wider varia-tion without invoking fine-tuning. This gives us a ratherlarge room to get an enough suppression of the axion an-gle fluctuation δθ through a stronger QCD before the µ -transition. At any rate, assuming that c X,Y > ξ X > ξ Y <
0, the scalar potential (26) indeed yields thedesired µ -transition as X = Y = 0 at t ≤ t µ ,X ∼ Y ∼ p m SUSY M P l at t > t µ , (28)with T ( t µ ) ∼ m SUSY .Now the Higgs and slepton fields can have a nontrivialevolution along the following flat direction: H Td = ( φ d , , L T = ( φ ℓ , ,H Tu = (0 , p | φ d | + | φ ℓ | ) , (29)which satisfies the F and D flat conditions. The relevantterms of the scalar potential of φ d,ℓ are given by V = X m i | φ i | + qX | φ i | (cid:16) Bµφ d + h . c . (cid:17) + (cid:16)X | φ i | (cid:17) (cid:18) κ A φ d φ ℓ M P l + h . c . (cid:19) + | κ | M P l (cid:16)X | φ i | (cid:17) (cid:0) | φ d | + 4 | φ d φ ℓ | + | φ ℓ | (cid:1) , (30)for µ = κ X /M P l , where m φ d = c d H + ξ d m + 2 | µ | ,m φ ℓ = c ℓ H + ξ ℓ m + | µ | . (31)Again, assuming c d,ℓ < ξ d,ℓ <
0, but m φ d ,φ ℓ ( t ) > µ ( t ) ∼ m SUSY , the above scalar potential yields f a ( t I ) ∼ φ d,ℓ ( t I ) ∼ p H ( t I ) M P l ,f a ( t µ ) ∼ φ d,ℓ ( t µ ) ∼ p m SUSY M P l , (32)and φ d ( t ) = O (100) GeV , φ ℓ ( t ) = 0 ,f a ( t ) ∼ X ( t ) ∼ Y ( t ) ∼ p m SUSY M P l . (33)To summarize, under a reasonably plausible assump-tion on the SUSY breaking during the primordial infla-tion and in the present Universe, the model with the su-perpotential (25) can successfully realize the desired cos-mological evolution of the three relevant scales: the axionscale, the weak scale, and the QCD scale as given by (28),(32) and (33). Being generated by SUSY breaking, aninflationary axion scale f a ( t I ) ∼ p H ( t I ) /m SUSY f a ( t )is determined to be well above the present axion scale f a ( t ) ∼ √ m SUSY M P l , and a stronger QCD in the earlyUniverse is realized to yield an enough suppression ofthe axion angle fluctuation even when H ( t I ) saturatesits upper bound. We note that the minimum of the ax-ion potential induced by the stronger QCD depends onarg( κ A ), but not on arg( Bµ ), while the minimum ofthe axion potential at present depends on arg( Bµ ), butnot on arg( κ A ). As a result, the stronger QCD gen-erates an axion misalignment angle θ mis = O (1), so thatthe axion dark matter with Ω a = Ω DM arises naturallyin our scheme.There is a remaining issue which should be addressedto complete our scheme. As we have noticed, the µ -transition is foregone by a late-time thermal inflation.This suggests that the scheme should be accompanied bya late-time baryogenesis operating after the µ -transition.In fact, the model of (25) offers an elegant mechanismto generate the baryon asymmetry through the rollingflat direction LH u [21]. More detailed cosmology of ourscheme, including the leptogenesis by rolling LH u , willbe discussed elsewhere [22]. Acknowledgment
This work was supported by IBS under the projectcode, IBS-R018-D1 [KC and SHI], and by Pusan Na-tional University Research Grant, 2015 [KSJ], and theResearch Fund Program of Research Institute for Ba-sic Sciences, Pusan National University, Korea, 2015,Project No. RIBS-PNU-2015-303 [KSJ]. [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440(1977); Phys. Rev. D , 1791 (1977).[2] For a recent review, see J. E. Kim and G. Carosi, Rev.Mod. Phys. , 557 (2010); A. Ringwald, Phys. DarkUniv. (2012) 116; M. Kawasaki and K. Nakayama, Ann.Rev. Nucl. Part. Sci. , 69 (2013).[3] T. Hiramatsu, M. Kawasaki, K. ’i. Saikawa andT. Sekiguchi, Phys. Rev. D , 105020 (2012) [Erratum-ibid. D , 089902 (2012)]; JCAP , 001 (2013);[4] M. Axenides, R. H. Brandenberger and M. S. Turner,Phys. Lett. B , 178 (1983); D. Seckel andM. S. Turner, Phys. Rev. D , 3178 (1985); A. D. Linde,Phys. Lett. B , 375 (1985); D. H. Lyth, Phys. Lett.B , 408 (1990); M. S. Turner and F. Wilczek, Phys.Rev. Lett. , 5 (1991).[5] P. Fox, A. Pierce and S. D. Thomas, hep-th/0409059;M. Beltran, J. Garcia-Bellido and J. Lesgourgues, Phys.Rev. D , 103507 (2007); M. P. Hertzberg, M. Tegmarkand F. Wilczek, Phys. Rev. D , 083507 (2008);K. J. Mack and P. J. Steinhardt, JCAP , 001 (2011);K. J. Mack, JCAP , 021 (2011).[6] D. J. E. Marsh, D. Grin, R. Hlozek and P. G. Fer-reira, Phys. Rev. Lett. , 011801 (2014); L. Visinelliand P. Gondolo, Phys. Rev. Lett. , 011802 (2014);A. G. Dias, A. C. B. Machado, C. C. Nishi, A. Ringwaldand P. Vaudrevange, JHEP , 037 (2014).[7] P. A. R. Ade et al. [Planck Collaboration], Astron. As-trophys. , A22 (2014).[8] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, JHEP , 060 (2007).[9] A. D. Linde, Phys. Lett. B , 38 (1991).[10] H. Murayama, H. Suzuki and T. Yanagida, Phys. Lett.B , 418 (1992); K. Choi, E. J. Chun and J. E. Kim,Phys. Lett. B , 209 (1997).[11] E. J. Chun, K. Dimopoulos and D. Lyth, Phys. Rev. D , 103510 (2004).[12] K. Choi, K. S. Jeong and M. S. Seo, JHEP , 092(2014).[13] E. J. Chun, Phys. Lett. B , 164 (2014).[14] K. Choi, K. S. Jeong, K. I. Okumura and M. Yamaguchi,JHEP , 049 (2011); G. Honecker and W. Staessens,Fortsch. Phys. , 115 (2014); E. I. Buchbinder, A. Con-stantin and A. Lukas, Phys. Rev. D , no. 4, 046010(2015).[15] K. S. Jeong and F. Takahashi, Phys. Lett. B , 448(2013).[16] T. Higaki, K. S. Jeong and F. Takahashi, Phys. Lett. B , 21 (2014).[17] M. Fairbairn, R. Hogan and D. J. E. Marsh, Phys. Rev.D , no. 2, 023509 (2015).[18] N. Kitajima and F. Takahashi, JCAP , no. 01, 032(2015).[19] J. E. Kim and H. P. Nilles, Phys. Lett. B , 150 (1984).[20] D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. , 201(1995); D. H. Lyth and E. D. Stewart, Phys. Rev. D ,1784 (1996).[21] I. Affleck and M. Dine, Nucl. Phys. B , 361 (1985);M. Dine, L. Randall and S. D. Thomas, Nucl. Phys. B458