A supersymmetric inflationary stage dominated by an $F$-term has the problem
that the flatness of the potential is spoiled by supergravity corrections, that
is the slow-roll parameter $\eta$ gets contributions of order unity. We show
that in $F$-term inflationary models based on strings there is natural way of
obtaining small values of $\eta$. This happens in models of hybrid inflation
based on orbifold constructions, in which a modulus $T$ field is responsible
for the large value of the potential during inflation, and a second field
$\phi$ with appropriate modular weight is responsible for the roll-over. We
illustrate the mechanism with a model in which the inflaton potential is
provided by gaugino condensation, leading to succesful inflation.Comment: 9 pages, LaTe

A Riotto

Brignole

Casas

Ferrara

G.B Gelmini

J.A Casas

Kaplunovsky

Physics Letters B

English

arXiv

A supersymmetric inflationary stage dominated by an $F$-term has the problem that the flatness of the potential is spoiled by supergravity corrections, that is the slow-roll parameter $\eta$ gets contributions of order unity. We show that in $F$-term inflationary models based on strings there is natural way of obtaining small values of $\eta$. This happens in models of hybrid inflation based on orbifold constructions, in which a modulus $T$ field is responsible for the large value of the potential during inflation, and a second field illustrate the mechanism with a model in which the inflaton potential is provided by gaugino condensation, leading to succesful inflation.A supersymmetric inflationary stage dominated by an $F$-term has the problem that the flatness of the potential is spoiled by supergravity corrections, that is the slow-roll parameter $\eta$ gets contributions of order unity. We show that in $F$-term inflationary models based on strings there is natural way of obtaining small values of $\eta$. This happens in models of hybrid inflation based on orbifold constructions, in which a modulus $T$ field is responsible for the large value of the potential during inflation, and a second field $\phi$ with appropriate modular weight is responsible for the roll-over. We illustrate the mechanism with a model in which the inflaton potential is provided by gaugino condensation, leading to succesful inflation.A supersymmetric inflationary stage dominated by an F -term has the problem that the flatness of the potential is spoiled by supergravity corrections, that is the slow-roll parameter η gets contributions of order unity. We show that in F -term inflationary models based on strings there is natural way of obtaining small values of η . This happens in models of hybrid inflation based on orbifold constructions, in which a modulus T field is responsible for the large value of the potential during inflation, and a second field φ with appropriate modular weight is responsible for the roll-over. We illustrate the mechanism with a model in which the inflaton potential is provided by gaugino condensation, leading to successful inflation

Casas, J.A.

Gelmini, G.B.

Riotto, A.

CERN Document Server

CERN-TH/99-84IEM-FT-178/98IFT-UAM/CSIC-99-11F -term inflation in Superstring TheoriesJ.A. Casas 1,2,3,, G.B. Gelmini 4,y and A. Riotto 3,z1 Instituto de Estructura de la materia, CSIC, Serrano 23, 28006 Madrid, Spain2 Instituto de Fsica Teorica, Univ. Autonoma de Madrid, 28049 Madrid, Spain3 Theory Division, CERN, CH-1211 Geneva 23, Switzerland4 Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547AbstractA supersymmetric inflationary stage dominated by an F -term has the problemthat the flatness of the potential is spoiled by supergravity corrections, that isthe slow-roll parameter η gets contributions of order unity. We show that inF -term inflationary models based on strings there is natural way of obtainingsmall values of η. This happens in models of hybrid inflation based on orbifoldconstructions, in which a modulus T eld is responsible for the large value ofthe potential during inflation, and a second eld φ with appropriate modularweight is responsible for the roll-over. We illustrate the mechanism with a modelin which the inflaton potential is provided by gaugino condensation, leading tosuccesful inflation.March, 1999E-mail: casas@mail.cern.chyE-mail: gelmini@physics.ucla.eduzEmail: riotto@nxth04.cern.chThe flatness and the horizon problems of the standard big bang cosmology are elegantlysolved if during the evolution of the early Universe the energy density happens to be dom-inated by some form of vacuum energy and comoving scales grow quasi-exponentially [1].An inflationary stage is also required to dilute any undesirable topological defects left asremnants after some phase transition taking place at early epochs. The vacuum energydriving inflation is generally assumed to be associated to some scalar eld φ, the inflaton,which is initially displaced from the minimum of its potential. As a by-product, quantumfluctuations of the inflaton eld may be the seeds for the generation of structure. The levelsof density and temperature fluctuations observed in the present Universe, δρ/ρ  10−5,require the inflaton potential to be extremely flat. Actually, for inflation to be viable, thereare two parameters which must be small with respect to unity to insure sucient slow rollin the inflationary potential V . These are the so called  = 12M2Pl(V0/V )2 and η = M2PlV00/Vparameters. Here MPl = 2.4  1018 GeV is the reduced Planck mass. The value of η inparticular is constrained by the observational bound in the spectral index jn − 1j < 0.2.Since in an inflationary scenario n = 1 − 6 + 2η, this requires jηj < 0.1 (the contributionproportional to  is usually negligible ). Note that η is essentially given by the eectiveinflaton mass during inflation.The required extreme flatness of the inflaton potential is the main reason why inflationis more natural in the context of supersymmetric (SUSY) theories. Flat directions arefrequent in supersymmetric theories, and they are stable under radiative corrections, aslong as SUSY is not broken. However, the previous argument does not hold if SUSY isbroken, which necessarily occurs during inflation, when there is a vacuum energy densityhV iin  H2M2Pl > 0. Thus, one generically expects eective soft terms, in particular softscalar masses, of order H , spoiling the desired flatness of the potential.The supergravity potential V consists of two pieces | the so-called D-term and F -term.The latter is given byVF = eG(Gj¯Kj¯lGl − 3M4Pl)= F l¯Kjl¯Fj − 3eGM4Pl . (1)Here G = KM−2Pl + ln(jW j2M−6Pl ), where W is the superpotential, K l¯j is the inverse of theKa¨hler metric Kjl¯  ∂K/∂φj∂ φl, φj are the (scalar components) of the chiral supereldsand F j = eG/2K k¯jGk¯M3Pl are the corresponding auxiliary elds. In the following, we willassume that during inflation the potential is dominated by the F -term, i.e. V ’ VF , whichis the so-called F{inflation. Notice that the F -term depends only upon two objects | the1superpotential W , an holomorphic function of the scalar elds, and the Ka¨hler potential K,a real function. During inflationhV iin  V0 = 3H2M2Pl, (2)which implies that some F elds are dierent form zero, thus breaking supersymmetry.It is easy to check that this breaking of SUSY generically gives a sizeable soft mass-squared to the inflaton. Using a basis where the elds have canonical normalization at theorigin (Kij¯ = δij at φi = 0), a generic scalar mass m2i = ∂2V/∂φi∂φi is given bym2i = M−2Pl V0 + eK∑nmKnmii WnWm −M−2Pl eK jWij2 + eK∑njWnij2 +    . (3)The contribution of the rst term to η  M−2Pl m2φV −1 is precisely 1. For generic models, thecontribution of the second term to η is also O(1). So, even if in the global SUSY limit theinflaton is not responsible for the SUSY breaking (Wφ = 0), remaining as a flat directionduring inflation (Wnφ = 0), we see from eq.(3) that the mass-squared of the inflaton isexpected to be O(M−2Pl V ), and thus η = O(1). Therefore, it is generally dicult to naturallyimplement a slow-roll inflation in the context of supergravity (recall we need jηj < 0.1).Solutions to the η-problem already exist in the literature [4]. Among them, D-terminflation seems to be a promising one [5]. In this paper, however, we show that in F -terminflationary models based on (weakly coupled) strings there is a natural way of obtainingsmall values of η. This happens in models of hybrid inflation [6] based on orbifold construc-tions, in which a modulus T eld is responsible for the large value of the potential V duringinflation, and a second eld φ with modular weight nφ = −3 is responsible for the rollover[7].String models of F -term inflation scenarios present special characteristics because theKa¨hler potential, K, which plays a crucial role, is greatly constrained. In orbifold construc-tions, the tree{level Ka¨hler potential is given by [8]K = − log(S + S)− 3 log(T + T ) +∑j(T + T )nj jφjj2 +O( jφjj4M−2Pl)) . (4)Here S is the dilaton and T denotes generically the moduli elds, all in units of MPl, φjare the chiral elds and nj the corresponding modular weights. The latter depend on thetype of orbifold considered and the twisted sector to which the eld belongs. The possiblevalues of nj for ZN orbifolds are nj = −1,−2,−3,−4,−5. The discrete character of nj will2play a relevant role later on. Eq.(4) is written with the usual simplication of considering asingle \overall modulus" T . hSi and hT i are expected to have determined values of O(1) inPlanck units at low energy. This comes from the fact that hSi and hT i have precise physicalmeanings. Namely, hSi is the value of the unied gauge coupling constant and hT i is thesquared radius of the compactied space, both in Planck units.From eq.(4) it is easy to understand that T is a good candidate for providing the vacuumenergy during inflation (i.e. F T 6= 0). The reason is the coupling between φ and T in thequadratic term (/ jφj2) of K. Notice that, without this coupling, if φ corresponds to a flatdirection in global SUSY breaking (Wφ = Wnφ = 0), then m2φ = V0M−2Pl (see eq.(3)) andthus η = 1.If SUSY is broken by the T (and/or S) elds, the corresponding soft terms, in particularmass terms, for φ are straightforwardly extracted from Eqs.(1) and (4). This was preciselythe sort of scenario considered in Refs. [9,10]. Although the motivation of these workswas dierent, namely to study the form of the soft breaking terms at low{energy withm3/2 = O(1 TeV), their results are applicable here. The only dierence here is that thescale of the breaking is much higher and the non-vanishing cosmological constant, V0 > 0,plays a major role. In particular, the eective φ mass squared, m2φ, of a chiral eld φ withmodular weight nφ, for the canonically normalized eld (Kφφ¯)1/2φ, is given by [10]m2φ = m23/2{(3 + nφ cos2 θ)C2 − 2}, (5)where the eective gravitino mass during inflation, m3/2, is dened bym23/2 = eGM2Pl = eK/M2Pl jW j2M−4Pl . (6)(since V0 = Fl¯Kjl¯Fj − 3m23/2M2Pl, unless there is some ne{tuning, m23/2 is at mostO(V0)M−2Pl ); tan2 θ = (KSS¯/KT T¯ )∣∣∣F S/F T ∣∣∣2 and C2 = 1 + [V0/(3M2Plm23/2)]. Notice thatC2 > 1.From (5) we conclude the following. If cos2 θ = 0, the case of S{driven inflation, thenm2φ = m23/2+V0M−2Pl . Hence, η = O(1) as expected from previous arguments. Let us examinenow the more promising case of T -driven inflation, where cos2 θ = 1. Note that in this caseV0 = KT T¯ jF T j2 − 3m23/2M2Pl = 3H2M2Pl. If it happens that KT T¯ jF T j2 = O(m23/2)M2Pl, whatmeans C2 = O(1), then still one obtains the usual result of m2φ = O(m23/2) = O(H2), andthus η = O(1). However, it may perfectly happen thatKT T¯ jF T j2  m23/2M2Pl. (7)3Then V0  O(m23/2M2Pl) and C2  1. Then Eq. (5) simplies tom2φ ’13V0M−2Pl (3 + nφ) = H2(3 + nφ). (8)Hence, for nφ = −3 we get m2φ  O(H2) and thus η  1, as desiredy.Therefore, we have shown that the possibility of a very small mass m2φ can occur in Tdriven F -term inflation, if V0  O(m23/2M2Pl) (namely C2  1) and the modular weight φis nφ = −3. Notice that this possibility appears with no ne tunning at all, thanks to thediscrete character of nφ, thus avoiding the slow-roll η-problem. In this scenario the energydensity of the vacuum is provided by the modulus T and the slow-roll eld is the φ eld. Weare envisaging, therefore, an hybrid inflationary scenario where the dynamics of the systemwill make the energy density of the vacuum disappear for some critical value of the eld φ.To get these nice results we have used the form of the Ka¨hler potential of eq.(4), whichcontains two main approximations: (i) it is a tree-level expression, and (ii) it is leadingorder in the expansion in powers of jφj2.Concerning approximation (i), perturbative corrections to eq.(4) are known at one-looplevel [8] and are small, so they do not aect any of the results presented here. More precisely,at one loop the correction consists in replacing in Eq. (4)S + S ! Y = S + S − δGS8pi2ln(T + T ), (9)where δGS is the Green-Schwarz parameter, which is negative and jδGSj  O(10). With thisreplacement, Eq. (5) becomesm2φ = m23/23 + nφ(1− δGS24pi2Y)−1cos2 θC2 − 2 . (10)In the case we identied above, in which cos2 θ = 1, C2  1 and nφ = −3, from (10) weobtainm2φ ’ −H2δGS8pi2Y’ −H2 δGS32pi2, (11)where we have used that Y is the inverse of the unied coupling constant g, namely Y ’2/g2, so we expect hY i ’ 4. Eq.(11) shows that m2φ < 0.1 H2, which leads to valuesyTo get this result we have assumed eq.(7), which is equivalent to jWT j  9(T+T¯ )2 jW j2. This iseasily fulfilled for slightly large values of T . E.g. if W / [η(T )]−6, as dictated by target-spacemodular invariance, then jWT j  52 jW j2 and jηj < 0.1 for ReT > 2.5.4of η small enough to allow for the implementation of inflation. On the other hand, non-perturbative corrections are very poorly known (see e.g. Ref. [11] for an analysis of theirpossible phenomenological signicance).Concerning approximation (ii), the Ka¨hler potential will generically contain terms of theformK = K0 + Kφφ¯jφj2 + λ4jφj4 + λ6jφj6 + ..., (12)where the rst two terms are the ones explicited in eq.(4), and one expects λn = O(M2−nPl ).These extra terms modify the potential to beV = V0(1 + κ41jφj4 + κ62jφj6 + ...), (13)where the coecients κi are all of the order of M−1Pl . For φ < 0.1MPl the jφj4, jφj6, ... termsbecome negligible (this is a common situation and, as we will see, it is the case of theparticular model we present below). For example, keeping only the jφj4 term above, andignoring the phase of φ, the  and η parameters are given by =8M2Plκ81φ6(1 + κ4φ4)2, η =12M2Plκ21φ21 + κ21φ4. (14)Then, if φ < 0.1MPl we get  ’ 8κ61φ6 < 10−5, η ’ 12(φ/MPl)2 < 0.1.It is amusing to note that the exact (tree-level) form of K(φ, φ) is not known, but hasbeen conjectured [12]. For our case of interest, nφ = −3, it simply becomesK = − log(S + S)− log[(T + T )3 − jφj2]. (15)Then, assuming V0  M2Plm23/2 (i.e. C2  1), it turns out that all the jφj4, jφj6, ... contribu-tions in (13) are exactly vanishing!We have discussed a natural way to solve the slow-roll problem (i.e. to get jηj  1 duringinflation). However, without knowing the T -dependent potential we cannot know the valueof the eld φ at the end of inflation φe. Neither we can calculate its value 60 inflation-foldsbefore the end, φ60. This is important to get more denite results. Next we show a modelbased on gaugino condensation in which the previous mechanism to get jηj  1 is naturallyimplemented, allowing for further predictions.Suppose the hidden sector of the theory has an SU(Nc) gauge group with Nf flavours.Then the gaugino condensation eects generate a non-perturbative superpotential Wnp ofthe form [13]5Wnp = detM1/Nc f(T ). (16)HereM is the eective \quark" mass matrix and f(T )  e−8pi2/Ncg2 [η(T )]α, where α = O(1).Suppose that M has an invariant contribution and a eld-dependent contribution, i.e.M = M − λAq , (17)where M is a constant mass term (typically M = O(MPl)), A is a scalar eld, λ is a couplingconstant and q is a numerical exponent (q = 1 or q = 2 is okay for our purposes). The A eldcan also have perturbative interactions with the inflaton φ (i.e. a eld with nφ = −3), inparticular  A2φr, where r is an undetermined exponent. Then the relevant superpotentialgets the formWnp = [M − λAq]Nf/Nc f(T ) + λ0A2φr , (18)where λ0 is a coupling constant. As we will see, we need r  2 for our model to work.It is easy to check that the supergravity potential, VSUGRA, derived from eq.(18) has aglobal minimum atλAq = M φ = 0 . (19)Notice that at this point detM = 0 and V = 0, and supersymmetry is unbroken sinceW = WT = WA = Wφ = 0 . (20)On the other hand, for large hφi the A eld gets an eective large mass and, hence, theeective minimum of VSUGRA is atA ’ 0 . (21)So, the φ potential in this region is simply given byV (φ) = KT T¯ jF T j2  V0 , (22)which is flat, leading to an inflationary process. Actually, the potential is not exactly flat.As explained above, perturbative contributions to the Ka¨hler potential give to φ a positivetiny mass m2φ ’ H2j δGS j32pi2 (see eqs.(9{11)). So, the inflaton φ rolls slowly towards the origin.Inflation ends when the eective A mass becomes unimportant. This occurs for a value ofφ = φe given by6φe ’ V 1/2r0 MPl1−2/r . (23)For φ < φe, φ and A move quickly towards the true (global) minimum and inflation ends.The value of φ 60 inflation-folds before the end, φ60, is given byφ60 ’ ejδGS j/16φe ’ 2φe . (24)This is crucial to evaluate the primordial spectrum of scalar perturbationsPξ = κ63(2pi)2V 360(V 060)2. (25)According to observations Pξ ’ (5 10−5)2, requiringV1/40 =[2 10−5] r2(r−1) MPl . (26)More precisely, for r  2 this translates into5 1013GeV  V 1/40  1016GeV , (27)where the lower bound corresponds to the simplest case r = 2. We nally note thatthese numbers for V0 are perfectly natural in our model. More precisely, V0 is given byeqs.(22,18,21), so typically is V1/40  e−4pi2/Ncg2O(MPl). Thus, the suppression needed tobring V1/40 into the range of eq.(27) is provided by Nc  8, a perfectly reasonable num-ber. It is also important to keep in mind that the gaugino condensate we have used forinflation does not play any role for low-energy SUSY breaking, as it becomes vanishing atthe nal (global) minimum. Hence, the magnitud of W (and thus of Nc) during inflationis not constrained by any low-energy SUSY breaking requirement, which would demandW = O(1 TeV )M2Pl.In conclusion, we have shown how the slow-roll problem, which generically aects all theinflationary models based on supergravity when the vacuum is dominated by an F -term, isnaturally solved in the context of SUGRA theories coming form superstrings. In particular,this occurs if the vacuum energy during inflation is dominated by a modulus (T ) F -termand the role of the inflaton is played by any eld with the appropriate modular weight,namely n = −3. In this case, the inflaton mass is exactly vanishing at tree-level, and getsnon-dangerous contributions at higher radiative levels. Inflation takes place in a hibrid-likemanner. We have explicitely shown that it is easy to construct models implementing thisidea successfully. In particular we have presented a model based on the vacuum energy7created by a gaugino condensate in the hidden sector, coupled to the inflaton eld in asuitable way.AcknowledgementsThis research was supported in part by the CICYT (contract AEN95-0195) and theEuropean Union (contract CHRX-CT92-0004) (JAC), and by the US Department of Energyunder grant DE-FG03-91ER40662 Task C (GG).8REFERENCES[1] A.H. Guth, Phys. Rev. D23, 347 (1981). See also A.D. Linde, Particle Physics andInflationary Cosmology, (Harwood Academic, New York, 1990).[2] M. Grisaru, W. Siegel nd M. Rocek, Nucl. Phys. B159, 429 (1979).[3] M. Dine, W. Fischler and D. Nemeschansky, Phys. Lett. 136B, 169 (1984); G. D.Coughlan, R. Holman, P. Ramond and G. G. Ross, Phys. Lett. 140B, 44 (1984); E. J.Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, Phys. Rev. D 49,6410 (1994).[4] D. H. Lyth and A. 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F-term inflation in Superstring Theories

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F-term inflation in superstring theories

F-term inflation in Superstring Theories

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