Successfully combining SUGRA hybrid inflation and moduli stabilisation
aa r X i v : . [ h e p - t h ] A p r arXiv:0801.2116DESY 08-003 Successfully combining SUGRA hybrid inflation andmoduli stabilisation
S C Davis and M Postma , Service de Physique Th´eorique, Orme des Merisiers, CEA/Saclay, 91191Gif-sur-Yvette Cedex, France DESY, Notkestraße 85, 22607 Hamburg, Germany Nikhef, Kruislaan 409, 1098 SJ Amsterdam, The NetherlandsE-mail: [email protected] , [email protected] Abstract.
Inflation and moduli stabilisation mechanisms work well independently,and many string-motivated supergravity models have been proposed for them. Howevera complete theory will contain both, and there will be (gravitational) interactionsbetween the two sectors. These give corrections to the inflaton potential, whichgenerically ruin inflation. This holds true even for fine-tuned moduli stabilisationschemes. Following a suggestion by [1], we show that a viable combined model can beobtained if it is the K¨ahler functions ( G = K + ln | W | ) of the two sectors that areadded, rather than the superpotentials (as is usually done). Interaction between thetwo sectors does still impose some restrictions on the moduli stabilisation mechanism,which are derived. Significantly, we find that the (post-inflation) moduli stabilisationscale no longer needs to be above the inflationary energy scale. Keywords: inflation, cosmology of theories beyond the SM
1. Introduction
Many attempts have been made to implement inflation in extensions of the standardmodel, although to date there is still no model that is truly convincing. Supersymmetric(SUSY) theories appear to be more promising. They include numerous moduli fields,i.e. scalar fields which in the supersymmetric limit have an exactly flat potential, as isrequired for slow-roll inflation. Any one of these moduli fields could play the role ofthe inflaton field. As a concrete example we will consider F -term hybrid inflation inthis work. In the SUSY limit it has a flat direction, but when extended to includegravity the situation is less rosy. The large energy density during inflation breaksSUSY spontaneously, and supergravity (SUGRA) effects lift the flatness of the modulipotential. This is the infamous η -problem [2, 3].Furthermore, the particular form of the SUGRA potential means that all other, non-inflationary sectors of the full theory will couple to the inflation sector. The couplingwill be small, in the models we consider it is only of gravitational strength, but it can uccessfully combining SUGRA hybrid inflation and moduli stabilisation N = 1 SUGRA F -term hybrid inflation. All the other, non-inflationarymoduli fields must be fixed during inflation, and so a full SUGRA theory must includeadditional physics to do this. For this we will consider KKLT-like [4] and KL-like [5]moduli stabilisation schemes. As we will see, the moduli sector gives rise to additional— and quite generically fatal — corrections to the inflaton potential. This raises thequestions of whether the original SUSY hybrid inflation model can actually be embeddedin a full, realistic theory, and if so, are its original predictions valid? For the answerto both these questions to be yes, the coupling between the two sectors must somehowbe minimal, so that neither the moduli corrections to the inflation potential, nor theinflaton corrections to the moduli stabilisation potential ruin the model. As we willshow, it is possible, but non-trivial, to achieve this.There are of course many other models of inflation, which offer alternativeapproaches to the issue of moduli-inflation coupling. For example, in modular inflationmodels the modulus field itself is the inflaton [6]. In a sense, the coupling is maximal— nevertheless successful (fine-tuned) models have been constructed [7]. In braneinflation models the inflaton potential arises from brane interactions, and dependsexplicitly on the volume modulus. Stabilising the modulus field then inevitably givesa curvature correction to the inflation potential [8]. However explicit examples havebeen constructed where, for fine-tuned parameters, the corrections to η cancel to a highdegree, allowing inflation [9, 10]. In contrast to the above models, our strategy is todecouple the inflation and modulus sectors as much as possible. One advantage of thisis that it also allows us to decouple the scale of inflation from the gravitino mass scale.At the cost of tuning, it is then possible to have the gravitino in the phenomenologicallyfavoured TeV range without the need for low scale inflation.The η -problem is a common feature of SUGRA inflation models. To illustrateit, consider a canonically normalised inflaton field with K = | φ | . The inflationarypotential is of the form V ∼ e K V ∗ ∼ V ∗ (1 + | φ | + · · · ), with V ∗ the nearly constantenergy density driving inflation. It follows that the slow-roll parameter η = V ′′ /V isof order unity, and slow-roll inflation does not occur. To avoid this conclusion one canfine-tune the model such that the coefficient of the | φ | -term in the potential cancels.More elegantly perhaps, one can try to achieve the same using a symmetry. An exampleof the latter approach is the (accidental) Heisenberg symmetry of the K¨ahler potential in D -term hybrid inflation [11]. In this paper we avoid the above η -problem by using a shiftsymmetry for the inflaton, φ → φ + a , which leaves the K¨ahler potential invariant [6, 12].Since the inflaton field Re( φ ) no longer appears explicitly in the K¨ahler potential, thelarge mass corrections to the inflaton field are avoided.However, the shift symmetry does not kill all the corrections to the inflatonpotential. In the presence of moduli fields η - (and ǫ -) problems appear again. As aconcrete example, consider the case of a single modulus field T . If moduli fields are uccessfully combining SUGRA hybrid inflation and moduli stabilisation m T > H ∗ , with H ∗ being the Hubble constant during inflation [5].Now since the moduli stabilisation mechanism breaks SUSY, there are soft correctionsto the inflaton potential, typically of O( m / H ∗ ). The flatness of the inflaton potentialis lost unless the gravitino mass is sufficiently small m / < H ∗ . The problem with thisis that one cannot tune the gravitino mass arbitrarily: in a generic, KKLT-like potential m / ∼ m T , and a small gravitino mass is at odds with keeping the modulus fixed. It istherefore difficult to embed inflation in such a scheme.A solution to the above moduli problem put forward by Kallosh and Linde(henceforth denoted by KL) [13] is to fine-tune the modulus potential so that m / ≪ m T . Then if the Hubble constant during inflation is between these two mass scales, themodulus remains fixed while the soft corrections to the inflaton mass are small. Sucha set-up has the additional advantage that the gravitino mass can be in the TeV rangewithout the need for low scale inflation. KL gave an explicit realisation of this idea usinga racetrack potential for the modulus. All problems then appear to be solved, but this isdeceiving. Although the moduli corrections are small after inflation thanks to the fine-tuning in the KL set-up, this is not necessarily true during inflation. During inflationthe modulus field T is slightly displaced from its post-inflationary minimum, disruptingthe minute fine-tuning of the potential, with potentially serious consequences. Indeed,as we will show, in F -term hybrid inflation the effects of the modulus displacement aresubstantial, resulting in η ≈ − F -term hybrid inflation, which serves to illustrate allthe observations made above. It is a multi-field model of inflation, consisting of theinflaton field, and two oppositely charged waterfall fields which are responsible for endinginflation. When combined with a KKLT modulus sector, the corrections to both theinflaton and the waterfall field potentials are large. Although the mass correction to theinflaton can be protected by a shift symmetry, this is not the case for the waterfall fields,and as a result there is generally no graceful exit from inflation. Tuning the modulussector, as in the KL set-up, can reduce these corrections to a harmless size. Howeverall of this is under the assumption that the modulus T is fixed during inflation. Takingthe modulus dynamics into account we find that even in the fine-tuned KL-stabilisationscheme the corrections are not harmless after all. On the contrary, they prevent inflationfrom working.In all previous studies of the effect of the moduli sector on inflation [14, 15, 16, 17],the K¨ahler and superpotentials of the modulus and inflaton sectors were simply added toget the combined theory, i.e. take W total = W inf + W mod to get the full superpotential. Inthis paper we instead multiply the superpotentials: W total = W inf W mod , as proposed byAch´ucarro and Sousa [1]. As we will show, this greatly reduces the moduli corrections. uccessfully combining SUGRA hybrid inflation and moduli stabilisation F -term hybrid inflation combined with KL, or even KKLT, in this way can give aviable inflation model. Although multiplying superpotentials may sound odd at first, itis natural in a supergravity formulation in terms of the K¨ahler function G = K +ln | W | .Any supersymmetric theory only depends on the K¨ahler- and superpotential throughthe combination G , suggesting that it is the only significant quantity. Adding theK¨ahler functions of the two sectors is equivalent to adding their K¨ahler potentials andmultiplying their superpotentials.Adding K¨ahler functions has the nice property that a SUSY critical point of themodulus sector is automatically a SUSY critical point of the full theory as well [1, 18]— this feature is at the heart of the reduced moduli corrections. In the limit of a smallgravitino mass, all the corrections to the inflaton potential are small, including thosedue to the dynamics of the modulus field during inflation. The resulting inflationarymodel thus gives similar inflationary predictions to the usual F -term hybrid inflation inthe absence of a modulus sector. Although there are still some constraints on the modelparameters, we want to stress that successful inflation is achieved without the needfor fine-tuning — this is in contrast to most other combined inflaton-moduli models.A notable feature of the model is that it is possible for the vacuum modulus mass tobe smaller than the Hubble scale during inflation, without the modulus running off toinfinity.This paper is organised as follows. In the next section we provide the relevantbackground material. We start with a short review of standard F -term hybrid inflation,both in a SUSY and SUGRA theories. This is followed by a concise discussion of modulistabilisation in KKLT- and KL-style schemes. In section 3 we discuss the resulting modelwhen the two sectors are combined by adding superpotentials. As we will see, even inthe fine-tuned KL set-up this does not give a working model. In section 4 we combinethe modulus and inflaton sectors by their multiplying superpotentials, or equivalentlyby adding their K¨ahler-functions. The modulus corrections to the inflaton potentialnow are under control, and for a certain range of parameters we get successful inflation.The parameter range for which the standard F -term hybrid inflation predictions applyis determined in section 5. We end with some concluding remarks.Throughout this article we will work in units with M pl = 1 / √ πG N = 1.
2. Background F -term hybrid inflation The superpotential for standard SUSY F -term hybrid inflation is [19, 20] W inf = λφ ( φ + φ − − v ) . (1)with φ the singlet inflaton field, and φ ± the waterfall fields with charges ± U (1) symmetry. We can make λ real by an overall phase rotation of the superpotential,whereas the phase of v can be absorbed in the waterfall fields. This is the conventionwe will use throughout this paper. In particular, in sections 3 and 4 where we combine uccessfully combining SUGRA hybrid inflation and moduli stabilisation V inf = λ | φ | (cid:0) | φ + | + | φ − | (cid:1) + λ (cid:12)(cid:12) φ + φ − − v (cid:12)(cid:12) + V D . (2)Vanishing of the D -term potential enforces | φ + | = | φ − | . Inflation takes place for | φ | > v ,during which the waterfall fields sit at the origin φ ± = 0. The potential then reduces toa constant energy density V inf = V ∗ ≡ λ v , (3)which drives inflation. The inflaton potential is flat at tree level, but quantumcorrections generate a slope for the inflaton field. The one-loop potential is given by theColeman-Weinberg formula [21, 22] V loop = 132 π Str M Λ + 164 π Str M (cid:18) log M Λ − (cid:19) , (4)with the supertrace defined as Str f ( M ) = f ( M (boson) ) − f ( M (fermion) ), and Λ is the cut-offscale. During inflation SUSY is broken and the masses of the waterfall field and theirsuperpartners are split m ± = λ ( | φ | ± v ) , ˜ m ± = λ | φ | , (5)giving a non-zero contribution to the logarithmic term in V loop . Inflation ends when theinflaton drops below the critical value | φ | = v , and one combination of the waterfall fieldsbecomes tachyonic. During the phase transition ending inflation the U (1) symmetry getsbroken and cosmic strings form according to the Kibble mechanism [23, 24].The predictions for the CMB power spectrum and spectral index are P = V π ǫ , n s = 1 − d ln P ( N ) dN ≈ η − ǫ , (6)evaluated at N = N ∗ ∼
60, where N = − log a is the number of e -folds before the end ofinflation. The slow-roll parameters are ǫ = (1 /
2) ( V ′ /V ) and η = V ′′ /V , with primesdenoting differentiation with respect to the canonically normalised real inflaton field ϕ ,which for the above model is ϕ = √ | φ | . The COBE normalisation [25] for the powerspectrum is P ≈ × − , and WMAP3 results [26] give n s ≈ . ± .
02. We notehowever that if cosmic strings give a minor contribution to the power spectrum, largervalues of the spectral index are favoured [27].We can get approximate analytical expressions in two limiting cases. For largecouplings λ & . × − inflation takes place for large field values ϕ ≫ v , and thepotential including loop corrections approximates to V inf ≈ V ∗ (cid:20) λ π log λϕ √ (cid:21) . (7)It follows that N e -folds before the end of inflation, the inflaton field is ϕ ≈ λ √ N / (2 π ).The prediction for the power spectrum is P ≈ N ∗ v /
75, which when normalised tothe COBE scale gives v ≈ . × − . The spectral index is n s ≈ − /N ∗ ≈ . λ . . × − , inflation takes place for inflaton values close uccessfully combining SUGRA hybrid inflation and moduli stabilisation ϕ ∗ ≈ ϕ end ≈ √ v . Fitting the power spectrum to the COBEnormalisation now gives v = 5 . × − [ λ / (7 . × − )] / , and an approximately scaleinvariant spectrum n s ≈ v < − — 10 − , which implies λ < − — 10 − [30, 31]. However there are ways to avoid cosmic string production,or at least relax the bound [32]. In any case, the precise inflationary predictions andthe issue of cosmic strings is not the main point of this paper. Even if ruled out byfuture data, F -term hybrid inflation still serves as a useful toy model to study theeffects of a moduli sector on inflation. In particular it provides an explicit example forwhich multiplying superpotentials, instead of adding them, helps to keep the modulicorrections under control. F -term hybrid inflation Generically when an inflaton model is extended to include supergravity corrections thepotential develops a large curvature, resulting in a slow-roll parameter η ∼ F -term hybrid inflation with a canonically normalisedinflaton field this curvature correction miraculously vanishes [33]. However, when higherorder corrections to the the K¨ahler potential are taken into account, or when a modulussector is included, this accidental cancellation is destroyed, and the η -problem reappears.It can be solved by introducing a shift symmetry for the inflaton field into the inflationaryK¨ahler potential [6, 12] K inf = − ( φ − ¯ φ ) | φ + | + | φ − | . (8)The canonically normalised inflaton, which is now ϕ = √ φ ) (rather than | φ | ), doesnot appear explicitly in the K¨ahler.However, the SUGRA model with K¨ahler (8) and superpotential (1) still does notwork. The reason is that the mass of the axion field a = √ φ ) is tachyonic: m a = − λ v . This problem is solved if we include an extra no-scale modulus field T in the model. Explicitly, take K = − T + ¯ T ) + K inf and W inf = λ φ ( φ + φ − − v ) . (9)The modulus field T can arise in string theory as the breathing mode of compactifiedextra dimensions; we will discuss it in more detail in the next subsection. In the limitthat T is fixed we recover (3) with v = v , and λ = λ (2 Re T ) − / the rescaled coupling.The mass of the axion field is now positive definite m a = 2 λ v (3 + 2 φ ). The massesof the waterfall fields are also altered m ± = λ [ φ + v (1 + φ ) ± v (1 + 2 φ )] , ˜ m ± = λ | φ | . (10)Since v ≪ v term is negligbly small. For λ . . φ .
1, and theother correction is also small. The waterfield masses then reduce to the global SUSYresults (5), and the model approaches the SUSY limit. uccessfully combining SUGRA hybrid inflation and moduli stabilisation T . The full theory must include additionalpotential terms, which break SUSY and are expected to give corrections to the effectiveinflaton potential. This is actually part of a wider issue, namely that inflation doesnot exist in isolation — it is part of a full theory containing other very high energyphysics (such a stabilisation mechanisms for moduli fields like T ). Given the restrictiveform of SUGRA theories, interaction between different sectors is unavoidable (gravitycouples to everything). As we will see in later sections, this can be catastrophic for manyapparently good theories, and leads to severe restrictions on others. Before discussingthe moduli corrections to inflation, we will first review moduli stabilisation in the KKLTand KL set-ups. KKLT devised an explicit method for constructing dS or Minkowksi vacua in stringtheory [4]. In their set-up all moduli fields are fixed by fluxes [34], except for the volumemodulus T which is stabilised by the superpotential W KKLT = W + A e − aT , K = − T + ¯ T ] , (11)where W comes from fluxes, and the non-perturbative exponential term from gauginocondensation or alternatively from instanton effects. For a general SUGRA theory, the F -term potential is V F = e K (cid:16) K I ¯ J D I W ¯ D ¯ J ¯ W − |W| (cid:17) (12)with D I W = W ,I + K I W . The minimum of the above superpotential (11) is SUSYpreserving and AdS. However, we require a Minkowski or dS vacuum with a smallcosmological constant to desribe our universe. This can be obtained by adding anuplifting term, which then gives a minumum in which SUSY is broken. In the originalKKLT paper an anti-D-brane was used for uplifting. Alternatively a D -term can beused [35] although additional meson fields are required to implement this [36]. D -term uplifting has the advantage that the full theory can still be described by SUGRA,whereas the KKLT uplifting term breaks SUSY explicitly. In this paper we assume anylifting term takes the form V lift ∝ K T Re f ( T ) , (13)where f ( T ) ∝ T , or is a constant. This gives the correct form for the KKLT lifting V lift ∝ (Re T ) − n with n = 2 ,
3. The D -term will also include the meson fields, although V lift is qualatively the same (at least for the analysis of this paper).Alternatively one can introduce an uplifting F -term sector, such as anO’Raifeartaigh [37] or ISS [38] sector. An explicit example of this is the O’KKLTmodel [13], in which a minimal O’Raifeartaigh sector is added to (11). In this paper we uccessfully combining SUGRA hybrid inflation and moduli stabilisation K = − (cid:20) T + ¯ T − K O ′ (cid:21) , W = W KKLT + W O ′ (14)with K O ′ = S ¯ S − ( S ¯ S ) Λ s , W O ′ = − µ S . (15)The O’Raifeartaigh sector breaks SUSY and lifts the AdS vacuum to Minkowski. Thereis then no need for a separate non- F lifting term in the theory.The resulting stabilisation potential V mod = V F + V lift has only one scale m T ∼ m / .The Minkowski minimum is separated from T = ∞ by a barrier of height V max ∼ m T .The barrier needs to be higher than the inflationary scale, otherwise the moduli willroll off to infinity and the internal space will be decompactified, which gives the bound H ∗ < m / on the inflationary scale [5].KL devised a moduli stabilisation scheme that circumvents the above bound onthe Hubble scale during inflation [13]. Instead of the KKLT superpotential they use amodified racetrack superpotential W KL = W + A e − aT + B e − bT . (16)The extra parameters in the superpotential allow us to tune W ,T = W = 0, giving ametastable SUSY Minkowski vacuum without the need for a lifting term. As it stands,the model has m / = 0. This can be avoided by slightly perturbing the Minkowskisolution to obtain an AdS minimum V ∼ − m / ≪ m T , which is then uplifted to aSUSY breaking Minkowski vacuum. Uplifting can be done with a small KKLT liftingterm, or alternatively by adding an uplifting F -term sector (15), as was used in section3 of [13]. If the SUSY-breaking scale is small, we have T W ,T ∼ W ∼ m / T / and thegravitino mass is far smaller than the modulus mass scale, which is typically set by W in the superpotential. It is then possible to have m / ≪ H ∗ ≪ V max ∼ m T , whichopens the possibility of having inflation with fixed moduli but small soft correctionsto the inflaton potential. Note that such a scenario cannot be implemented with anuplifting D -term. In this case gauge symmetry implies that the Minkowski solution W ,T = W = 0 is obtained along a flat direction in the meson-modulus field space. Asa result, after perturbing the solution and uplifting to Minkowski, only one modulusmass eigenstate is large. The other is only O( m / ), and so the barrier height along thepreviously flat direction is also small V max ∼ m / , even when the modulus mass is large m T ≫ m / .The above model (14) uses a slightly different K to [13], although it has similarproperties. We have chosen the above K¨ahler to simplify the analytical expressions. Butwe want to emphasise that the exact way the modulus potential and the O’Raifeartaighsection are combined does not significantly affect inflation. For that matter, the upliftingsector does not have to be O’Raifeartaigh either, but can be some other F -term SUSYbreaking sector such as the ISS model. The differences in the resulting potential will uccessfully combining SUGRA hybrid inflation and moduli stabilisation m / ), and as long as m / ≪ H ∗ such differences are irrelevant duringinflation. As we will see in section 4, whether the uplifting is F -term or not can make amajor difference. For the case where the modulus and inflaton sector are combined byadding their respective K¨ahler functions it is the difference between a viable model andno model at all.
3. Combining inflation and moduli stabilisation by addition
The usual way to combine the models of the previous sections is to add the respectivesuperpotentials W = W + W inf . Here W is the modulus superpotential, eitherKKLT (11) or KL (16), possibily including an F -term O’Raifeartaigh lifting sector.For the K¨ahler potential we consider the simplest possibility K = − X ] + K inf , (17)with X = T + ¯ T − K O ′ . (18)If uplifting is achieved via an anti-D-brane or D -term, W O ′ and K O ′ are simply set tozero. To verify that the qualitative results are independent of the exact form of theK¨ahler, we also consider the more general expression K = − (cid:20) X − X α K inf (cid:21) . (19)For α = 0 this gives a fully no-scale K¨ahler potential: K a K a ¯ b K ¯ b = 3 with a, b runningover both moduli and inflaton fields.Slow-roll inflation with a scale invariant spectrum of perturbations requires ǫ, η ≪ φ ± becoming tachyonic, and thereis no exit from inflation. Alternatively, if the corrections are large and tachyonic thesystem ends up in the wrong vacuum. Furthermore, the axion mass has to be positivedefinite during inflation, which is not automatic. For the moment we work in theapproximation that the moduli are fixed at the minimum T = T during inflation. Atthe end of this section we will drop this assumption, and analyse its implications.For either choice of K¨ahler we find there are corrections to the slope of theinflationary potential [14]. For (17), the full F -term potential for the combined theoryis V F = e K inf V F + V lift + e K | ∂ i W inf + K i ( W inf + W ) | + V mix , (20)which is roughly the sum of the potential for the inflation and moduli sectors (withsome rescaling), and the additional mixing terms V mix = 2 e K Re[( K I ¯ J D I WK ¯ J − W ) ¯ W inf )] + e K ( K I ¯ J K I K ¯ J − | W inf | . (21) uccessfully combining SUGRA hybrid inflation and moduli stabilisation i runs over the inflation sector fields, while I, J run over the moduli sectorfields. During inflation all K i = 0, and the SUGRA K i W corrections to SUSY inflationvanish. Furthermore, for a no-scale moduli K¨ahler (17) the second term of V mix isidentically zero. The K¨ahler potential (19) gives rise to similar mixing terms.Much of the moduli interaction effectively re-scales the inflationary parameters, andso it is convenient to introduce λ = λ X α/ , v = v X − α , V ∗ = 3 H ∗ = λ v , ϕ = √ X α − Re φ . (22)These apply to the general K¨ahler (19), and also to (17) if α is set to 1. In both casesthe inflationary potential reduces to V inf = V ∗ + V mod + √ W ,T ) √ X λv ϕ (23)with ϕ the canonically normalised inflaton. The inflaton independent modulus potentialis V mod ( T ) = V F + V lift . We see that a nearly flat inflaton potential, with ǫ ≪
1, requires V mix ∝ Re W ,T to be small. This can be achieved either be making |W ,T | small (whichis the case for the two-scale KL-style stabilisation), or by having W ,T imaginary, i.e.having a phase difference between the inflation and moduli superpotentials.We also need to check that the corrections to the masses of the waterfall fields donot radically change the ending of inflation, and that the axion a = √ φ remainsstable. We introduce the mass scales m = W X / , m ′ = W ,T √ X , M = √ X W ,T T . (24)Up to small O(e K inf ) corrections | m | ≈ m / is the gravitino mass after inflation, and ina KL-style scheme |M| ≈ m T the modulus mass. For KKLT we still have |M| ∼ m T .For the K¨ahler (17) with canonically normalised inflaton sector fields the masses of theaxion and waterfall fields are m a = 2 λ v (3 + 2 φ ) + 2 V F + 4 | m | − m − m ′ ] λv φ , (25) m ± = λ φ ± λ v (cid:12)(cid:12)(cid:12)(cid:12) − m − m ′ λv φ + 2 φ (cid:12)(cid:12)(cid:12)(cid:12) + V F + | m | + λ v (1+ φ )+2 λv Re[ m ′ − m ] φ . (26)For a one-scale KKLT-like moduli sector m, m ′ ∼ m / ∼ m T . The requirement that themoduli remain fixed during inflation, i.e. H ∗ < V max ∼ m T , implies that the O( m, m ′ )moduli sector corrections to m ± dominate, preventing a gracefull exit from inflation. Afurther problem for models which use a D-brane or D -term lifting term V lift is that theaxion and waterfall masses recieve large tachyonic contributions from the moduli sector F -term potential ∝ V F ∼ − m / . For F -term lifting V F = 0 in the Minkowski vacuumafter inflation, and so the contribution of V F during inflation is small.In principle, all these problems can be avoided with sufficient fine-tuning, althoughthe single mass scale superpotential (11) does not contain enough parameters. Hencewe must switch to a two-scale KL moduli stabilisation scheme, which is tuned so that W = W ,T ≈ m, m ′ , V F ≈
0. The moduli corrections to the waterfall (26) uccessfully combining SUGRA hybrid inflation and moduli stabilisation T was fixed at the minimum of V mod . However, no field istruly fixed at a constant value during inflation, and in particular the modulus minimumwill shift slightly during inflation. Taking the dynamics of the modulus into account,we will now show that it produces siginificant curvature corrections to the potential,and consequently gives too large a value for η [14]. To do so we Taylor expand thepotential (23) in δT = T − T , with as before T the modulus value that minimises the post -inflationary potential: V inf = V ∗ ( T ) + V mod ( T ) + 2 Re W ,T ( T ) X λ v φ + δV inf + O (cid:0) | δT | , λ v φ | δT | , V ∗ | δT | (cid:1) (27)where δV inf = V mod ,T ¯ T δT δT + Re[ V mod ,T T δT ]+ 2 [ X Re( W ,T T δT ) − W ,T ) Re( δT )] λ v X φ (28)gives the leading order corrections to V inf from the variation of T . Now for KL |M| ≫ | m | , | m ′ | , hence this reduces to δV inf ≈ |M| X | δT | + 3 √ λv ϕX Re[ M δT ] . (29)Minimising with respect to δT we find δTX ≈ − λv ϕ √ M (30)which is small (as expected). However when this is substituted back into the abovepotential, it produces a large negative inflaton mass δV inf ≈ − V ∗ ϕ . (31)The η -problem rears its head again: η = V ,ϕϕ /V ≈ −
3. For KL without the SUSYbreaking O’Raifeartaigh sector the above expressions are exact, while an uplifting sector— O’Raifeartaigh or otherwise — gives rise to small O( m / ) corrections (both due tothe above δT expression, as well as the displacement of e.g. the O’Raifeartaigh field δS ). The large slow-roll parameter rules out F -term hybrid inflation with KL modulistabilisation. The reason for the large corrections, even in the fine-tuned KL set-up isthat although W ∼ W ,T ≈ |W ,T T | = 3 X V mod ,T ¯ T + O( M m / ) is not. Inthe Minkowski vacuum after inflation the potential is fine-tuned so that m / ≪ m T ,but during inflation, due to the small displacement of the modulus field, this tuning isdisrupted, and corrections are large.For the more general K¨ahler (19) the inflaton potential is still given by (23). Thewaterfall masses take the form m ± = λ ϕ ± λ v (cid:12)(cid:12)(cid:12)(cid:12) α ) m ′ − αm √ λv ϕ + αϕ (cid:12)(cid:12)(cid:12)(cid:12) + 2 + α V F + 2(1 − α )3 V lift uccessfully combining SUGRA hybrid inflation and moduli stabilisation α | m | + √ λv α ) m ′ − αm ] ϕ + λ v (cid:18)
23 + αϕ (cid:19) . (32)In general, the model will have all the same problems as that arising from the simplerK¨ahler (17), and one-scale KKLT-style moduli stabilisation superpotentials are ruledout. It is interesting to note that for a no-scale α = 0 model most of the corrections to m ± cancel (compare with the D -term inflation model proposed in [16]). In particular,all the m , V F and V lift corrections disappear. It would seem that we then only need toimpose the single fine-tuning m ′ ≈
0, to obtain a viable inflation model. Unfortunatelythe KKLT superpotential (11) does not have enough freedom to do this, and viableinflation is not obtained. Furthermore, the above discussion does not take into accountthe varation of T during inflation. The above analysis of δT also applies for the moregeneral K¨ahler (19), and so it too is ruled out.To conclude, F -term hybrid inflation does not work for either KKLT- or KL-stylemoduli stabilisation, no matter what the form the K¨ahler takes, at least if we combinethe inflation and modulus sector by adding superpotentials. In fact, if more exponentialterms are added to the moduli stabilisation superpotential, its first three derivatives areappropriately tuned, and the K¨ahler is carefully choosen, the moduli dynamics couldbe different to those used to get (31). A viable model of inflation could concievably beconstructed, although it is hard to justify all the fine-tuning. Furthermore, there is noguarantee that additional problems will not arise as a result of this tuning. We will notconsider such as set-up here, and will instead turn to a much more elegant solution.
4. Combining inflation and moduli stabilisation by multiplication
The inflaton and modulus sectors can also be combined by multiplying theirsuperpotentials. Although due to its unfamiliarity this seems strange at first, we arguethat from a supergravity point of view it is a rather natural thing to do. Multiplyingsuperpotentials greatly reduces the mixing between sectors [1, 18]. Indeed, as we willdiscuss in this section F -term hybrid inflation combined in this way with KL or even aKKLT moduli sector gives a viable inflation model.The supergravity formulation in terms of K and W is redundant, as a K¨ahlertransformation leaves the theory invariant. Instead the theory can be formulated interms of single K¨ahler invariant function G = K + ln | W | , which is known as theK¨ahler function. The kinetic terms and F -term potential are then given in terms of G only. This suggests that the K¨ahler function is a more “fundamental” or “natural”quantity to consider. Hence when combining sectors, it may be argued that one shouldadd their respective K¨ahler functions, which corresponds to adding K¨ahler potentialsand multiplying superpotentials.For the combined theory we then take G = G mod + G inf . The reduced inflaton-moduli interactions are a result of the following property. Consider a SUSY critical point T = T of the modulus sector ∂ T G mod ( T ) = 0, which corresponds to a SUSY extremumof the moduli potential. It can easily be shown that this is then a SUSY critical point uccessfully combining SUGRA hybrid inflation and moduli stabilisation ∂ T G ( T ) = 0 [1, 18]. This is exactly what we want, as itimplies that the modulus minimum is not shifted during inflation. The δT correctionsto the potential, which were fatal when adding superpotentials, are then absent. Ofcourse, with SUSY broken in the modulus sector the minimum of the modulus potentialis not exactly in a critical point. But in the KL-like set-up the deviations away from theSUSY critical point are small, of the order of the small gravitino mass. Consequently weexpect the modulus field to be nearly constant during inflation, and the correspondingcorrection to the potential to be suppressed by the smallness of the gravitino mass. Aswe will see, this is indeed the case.One disadvantage of the K¨ahler function formulation of SUGRA is that it is illdefined whenever W = 0. This presents a problem for F -term hybrid inflation, as theinflationary superpotential (1) is zero after inflation. To solve this problem we “correct”the superpotential by adding a constant W inf = λ φ ( φ + φ − − v ) − C . (33)Here we will assume that C is real and positive, although generalisation of the analysis toinclude a phase is straightforward. The constant C is of course irrelevant in the IR globalSUSY limit, whereas in the UV regime it makes the model well behaved. Similarly, forthe modulus potential we cannot take the supersymmetric KL limit, a finite amountof SUSY breaking (explicitly provided in (16) by an O’Raifeartaigh sector) is required.The effective superpotential of the model with the modulus included is now W = W W inf . (34)For the K¨ahler potential we still use (17) with canonically normalised inflaton fields.To test the dependence of the results on the exact form of the K¨ahler we also give theresults for the general expression (19).For the minimal K¨ahler (17) the potential that follows from (33),(34) is V = e K inf | W inf | V F + e K |W| e K inf | ∂ i W inf + K i W inf | + V lift . (35)As advertised, the mixing between the inflaton and modulus sector is drastically reducedcompared to the case of adding superpotentials (21). The main effect is just a re-scalingof the potential. We define the re-scaled quantities λ = λ |W| X α/ , v = v X − α , V mod = C V F + V lift , ϕ = √ X α − Re φ . (36) V ∗ = 3 H ∗ = λ v is then the rescaled inflationary potential driving inflation, while V mod is the full rescaled modulus stabilsation potential after inflation. The field ϕ is the real,canonically normalised, inflaton field. As before, the expressions for (17) correspond to α = 1. We also define the mass scales m = C |W| X / , M = C √ X |W ,T T | , (37) uccessfully combining SUGRA hybrid inflation and moduli stabilisation V inf = V ∗ + V mod (cid:18) λv ϕ √ m (cid:19) − λv ϕ √ m (cid:18) λv ϕ √ m (cid:19) V lift . (38)We see that if a seperate lifting term is present (either an anti-D-brane or a D -term),its potential V lift ∼ m / gives a large negative contribution to η . This holds for boththe KKLT and KL superpotential, and so all our moduli stabilisation scenarios withnon- F lifting terms are incompatible with F -term hybrid inflation. In the remainder ofthis section will thus focus on the case of F -term lifting with V lift = 0.In the limit that the modulus remains fixed during inflation V mod = 0 for F -termlifting, and there are no corrections to the inflaton potential at all. This is in sharpcontrast to the potential obtained when adding superpotentials (23). Although themodulus is not truly fixed during inflation, we will see below that the corrections to thisassumption are small.In multiplying the superpotentials, our intention was to reduce the effect of themoduli sector on inflation. We see from (38) that a beneficial side effect of this is thatthe inflaton enhances the moduli stabilisation. In particular the barrier height for themoduli stabilisation potential is now V max ∼ M √ H ∗ ϕ √ m ! . (39)Hence we expect the moduli to remain near their minimum during inflation if M ≫ H ∗ (as is usually assumed), or if ( M /m ) ϕ ≫
1. Since ϕ > ϕ end ∼ v , the moduli should bestable thoughout inflation if either( a ) M ≫ H ∗ or ( b ) M ≫ mv & × m . (40)Significantly, the second possibility does not depend on the Hubble constant duringinflation, and so having H ∗ > M is not a problem. The H ∗ < M bound was a majormotivation for the KL scenario, and its removal suggests that a two-scale, KL-stylemoduli sector is no longer needed. However, while the bound (40b) is easily satisfiedfor KL, it cannot be satisfied by KKLT. Hence it seems that a two-scale KL-like modulisector is needed after all, although not necessarily for the reasons that were originallyenvisaged.For the simplest K¨ahler (17) the waterfall field masses are m ± = λ φ ± λ v (cid:18) mλv φ + 2 φ (cid:19) + ( m + λv φ ) + λ v . (41)In the limit m ≈ m / ≪ λv ϕ (42)the moduli corrections are subdominant, and inflation ends as in usual hybrid inflation.From the COBE normalisation it follows that v ≪ v corrections can be uccessfully combining SUGRA hybrid inflation and moduli stabilisation m ∼ M , it is difficultto satisfy both of the above bounds (40), (42) simultaneously, and most vlaues of M are ruled out. For smaller values of λ (for which ϕ ∗ ≪
1) there is a small window ofparameter space H ∗ ≪ M ≪ H ∗ /ϕ ∗ where inflation will be viable. For a two-scaleKL-style scenario there is more room to satisfy the bounds (40), (42), but at the costof fine-tuning the potential.For the more general K¨ahler (19) the waterfall masses are instead m ± = λ ϕ ± λ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) α + (1 − α ) X W ,T W (cid:21) " ϕ + √ mλv ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + α (cid:18) m + λv ϕ √ (cid:19) + 2 λ v , (43)For α = 1 there are additional corrections to the watefall fields proportional to W ,T .These are expected to be of the same size as the other corrections. Hence KKLT-stylemodels are again mostly ruled out, except for a small range of M .We now turn to the behaviour of the moduli fields during inflation. We saw abovehow a lower bound on M arises from the requirement that V max ≫ V ∗ . In fact, a strongerbound on M comes from the inflationary corrections to the moduli sector masses. Therespective masses of the real and imaginary parts of T , and their fermionic superpartnersare m T ≈ ˜ m T + M m V ∗ , m T ≈ ˜ m T − M m V ∗ , ˜ m T ≈ M (cid:18) λv ϕ √ m (cid:19) (44)up to O( m ) corrections. To get the above expressions we have used that |W ,T T | =3 X V mod ,T ¯ T + O( M m ) in the KL set-up; it should also be remembered that the rescaledcoupling λ is modulus dependent. We have assumed, for simplicity, that W and itsderivatives all have the same phase. The masses (44) for KKLT will have differentcoefficents, but will be qualitatively similar. Requiring that Im T is not tachyonic implieseither ( a ) M & H ∗ m or ( b ) M & mv & × m . (45)For large enough M , (a) is satisfied by KL- and KKLT-style moduli sectors, and can inboth cases be combined with (42). The other range (b) is easily satisfied for KL, butnot for KKLT.Finally, we need to check that taking the modulus fixed during inflation, as assumedabove, is a good approximation. As we saw in section 3, the modulus dynamicsdestroys inflation even for the fine-tuned KL set-up when the modulus and inflationsuperpotentials are added. For a model with multiplied superpotentials, this problem isavoided. We will assume that W and all its dervatives have the same phases. Expanding,much as before, around the minumum of V mod , we take T = T + δT R + i δT I . Minimisingthe resulting potential, we find δT I = 0 and δT R X ≈ − V ∗ m T (cid:18) X D T WW + 1 − α (cid:19) (46) uccessfully combining SUGRA hybrid inflation and moduli stabilisation − δV inf V ∗ ≈ V ∗ m T (cid:18) D T WW + 1 − αX (cid:19) . min (cid:18) H ∗ M , m M ϕ , m M (cid:19) . (47)This is just a small correction to the inflationary potential (38) provided that either M ≫ H ∗ , or M ≫ m . At least one of these conditions is satisfied if we require that T is not tachyonic during inflation (45).To summarise, combining the two bounds (42) and (45) gives p M T m / ≫ H ∗ ≫ m / ϕ ∗ , (48)or alternatively m / ≪ H ∗ ϕ ∗ , M T v , (49)where M T ≈ M is the mass of T after inflation. Either of the above bounds canbe satisfied by a KL-style scenario without additional fine-tuning. KKLT-style modelscan also satisfy bound (48) and give a viable model of inflation for a limited range of M . These conclusions also apply for the more generic, α -dependent K¨ahler (19). Inboth KKLT and KL moduli stabilisation potentials, if either of the above bounds issatisfied, then the modulus does not vary significantly during inflation. Hence with onlya moderate degree of tuning, inflation can be successfully combined with a modulussector when their respective superpotentials are multiplied.
5. Inflationary predictions
Having investigated the effects of the moduli stabilisation sector on the tree level inflatonpotential, we will now determine the moduli corrections to the one-loop potential. Theinflaton slope and curvature, which determine the power spectrum and the spectralindex, are dominated by the one-loop contribution. This is given explicitly by theColeman-Weinberg formula (4). V loop receives contributions from the non-degenerateboson and fermion pairs, which in our model are not only the waterfall fields, but alsothe modulus field T (we will ignore any other fields for simplicity). Since the masses are ϕ -dependent, their contribution to the loop potential will generate a non-trivial potentialfor the inflaton field. In the limit that the slope and curvature of the inflaton potentialis dominated by the waterfall field contribution to the loop potential, the inflationarypredictions are the same as for the global SUSY model discussed in subsection 2.1.We will then have a working model of inflation. In this section we will determine thecorresponding parameter space. More precise bounds could be obtained by comparisonwith the WMAP data, although the results will be sensitive to the details of the modulisuperpotential. Here, we will content ourselves with order of magnitude bounds. Likethe conclusions of the previous section, our results will apply to the simple K¨ahler (17),and to the more generic one (19) for any choice of α .We start by calculating the loop potential. In the limit that the gravitino mass issmall and the bound (42) is satisfied, the expressions for the waterfall masses approach uccessfully combining SUGRA hybrid inflation and moduli stabilisation ϕ < λ . .
5, wherethe results are manifestly cut-off independent, we retrieve the global SUSY results (5).The loop potential due to the waterfall fields is given by the familiar expression [20] V ( φ )loop = λ V ∗ π (cid:20) (cid:18) λ v x Λ (cid:19) + ( x + 1) ln(1 + x − ) + ( x − ln(1 − x − ) − (cid:21) (50)with x = ϕ / (2 v ). Inflation takes place for x > x → V ( T )loop = V ∗ M π m (cid:20) (cid:18) V ∗ M z Λ m (cid:19) + ( z + 1) ln(1 + z − ) + ( z − ln(1 − z − ) − (cid:21) (51)with z = ˜ m T mV ∗ M = M m (cid:18) mλv + ϕ √ (cid:19) . (52)The loop potential gives a negligible contribution to the total energy density duringinflation V ∗ , but it is the dominant contribution to the slow-roll parameters ǫ and η .Hence to see whether it is the waterfall or the modulus contribution to the potentialwhich dominates the inflationary dynamics, we have to compare their derivatives. Inaddition we need to satisfy the upper bound on m (42), so that neglecting O( m ) terms isa good approximation. Requiring that the axion is non-tachyonic during inflation givesa further, lower bound on the modulus mass scale M (45). Finally, we note that bothKKLT and KL moduli stabilisation potentials have m . M , which restricts the allowedparameter space. If the above constraints are satisfied, then the modulus automaticallyremains fixed during inflation, and its dynamics do not produce further constraints.We expect to retrieve standard hybrid inflation results in the limit that the masssplitting between the modulus field and its superpartners is small, as this sets the overallscale of the modulus loop potential. In this limit z ≫
1. The ϕ -dependence only enters V ( T )loop via ˜ m T , and we find it convenient to write˜ m T = M (1 + δ m ) , with δ m = λv ϕ √ m . (53)The modulus loop effects are suppressed in the limit δ m →
0. As it turns out the δ m → z ≫ z -limit is (cid:16) V ( T )loop (cid:17) ′∗ ≈ λ v M √ π m δ m ) . (54)This is to be compared with the equivalent expression for the waterfall field potential. λ & − In the large coupling regime, λ > . × − , we can approximate (50) by the large x result (7) and lim x ≫ (cid:16) V ( φ )loop (cid:17) ′∗ ≈ λ v π √ N ∗ , (55) uccessfully combining SUGRA hybrid inflation and moduli stabilisation H a L - - - - log m - - - - log M III III IV H a L - - - - log m - - - - log M H b L - - - - log m - - - - log M III IVIII H b L - - - - log m - - - - log M Figure 1.
Parameter space in { log ( m ) , log ( M ) } for (a) λ = 0 . λ = 10 − .In the white region the model reduces to SUSY hybrid inflation. Regions I-IV areexcluded, because I: the modulus mass dominates the 1-loop potential, II: the gravitinomass is too large, III: the modulus is tachyonic during inflation, and IV: the moduluspotential property m . M is not satisfied. The dashed lines correspond to H ∗ = M where we used ϕ ∗ ≈ λ √ N ∗ / (2 π ). This dominates over (54) for M < √ πm (1 + δ m ) √ N ∗ λ v ≈ ( . × m λ − , δ m ≪ . × m , δ m ≫ v ≈ × − and N ∗ = 60. Small m < . × − λ corresponds to thelarge δ m > M > λ v m (1 + δ m ) ≈ ( . × − m − λ , δ m ≪ . mλ − , δ m ≫ m ± (42), which gives m < . × − .The parameter space in the { log ( m ) , log ( M ) } -plane is shown for λ = 0 . F -term hybrid inflation. Thisis in sharp contrast to a combined model in which the superpotentials are summed: aswe saw in section 3, inflation fails in this case.In all of parameter space z ≫
1, and our analytic results are valid. In region I theloop potential is dominated by the modulus contribution (56); when this becomes toolarge inflation is ruined. In region II the bound (42) on the gravitino mass is violated,and moduli corrections are too large for successful inflation. Region III is excludedas it gives a tachyonic axion (45). Except for very near the border with region IIIthe η -parameter is dominated by the waterfall field contribution to the loop potential.Finally, region IV bounds m . M which is a property of both KKLT and KL-stylemoduli sectors. Viable, KKLT-style models correspond to the upper-left edge of regionIV. Since this class of models has only one mass scale M ∼ m , it corresponds to a linein the plotted, two-dimensional parameter space. The fact that ϕ ∗ < uccessfully combining SUGRA hybrid inflation and moduli stabilisation M (whichincreases in size as coupling λ is reduced). The two-scale KL model works throughoutthe white region of parameter space in the plot.In the δ m ≫ M , as can be seen from (39). This allows for thepossibility of having m < M < H ∗ , yet with the modulus fixed during inflation. For λ = 0 . H ∗ ≈ − . The dashed lines in figure 1 correspondto M = H ∗ ; we see that indeed M < H ∗ is realised in large part of parameter space,contrary to naive expectations. λ . − We can apply the same analysis for the small coupling regime λ < . × − . In thiscase ϕ ∗ ≈ √ v and v = 5 . × − [ λ / (7 . × − )] / . In the small x → x → (cid:16) V ( φ )loop (cid:17) ′ ≈ λ v log(2)4 √ π (58)which is to be compared with (54). The waterfall field contribution dominates theone-loop potential for M < δ m ) m λv ≈ ( . × m λ − / , δ m ≪ . × m λ − / , δ m ≫ m < . × − λ corresponds to the large δ m > m < . × − λ / from (42) and M > ( . × − m − λ / , δ m ≪ . × mλ − / , δ m ≫ λ = 10 − are shown in figure 1b. We see that for smallercouplings the modulus stabilisation scale needs to be larger than the Hubble scale duringinflation. E.g. for λ = 10 − the inflationary scale is H ∗ ≈ − , and M > H ∗ in allof parameter space for successful inflation. This contrasts with the situation for largercouplings, as we saw in the previous subsection.
6. Conclusions
The flatness of the inflationary potential in SUGRA models is typically spoilt bycorrections coming from supersymmetry breaking. Ironically enough, the vacuum energywhich drives inflation breaks SUSY spontaneously, and so gives soft corrections to theinflaton; this is the well-known η -problem. Introducing a shift symmetry for the inflatonwill protect the inflation sector from itself, and remove the problem. However there willstill be corrections coming from other sectors of the full theory, which can also disruptinflation. In this paper we studied the effects of a moduli stabilisation sector on a F -term SUGRA hybrid inflation model. uccessfully combining SUGRA hybrid inflation and moduli stabilisation m T ∼ m / , as well as a fine-tuned two-scale KL-likeset-up with m T ≫ m / . In the KKLT set-up, requiring the modulus to be fixed duringinflation raises the scale of the modulus potential, and as a result the soft correctionsto both the inflaton slope and the waterfall field masses are too large for inflation totake place. This problem is circumvented in the KL set-up where the gravitino mass,and consequently the corrections to the inflationary potential, can be tuned arbitrarilysmall.One would be inclined to conclude that KL moduli stabilisation can be combinedalmost effortlessly with inflation. But this is not true. The above conclusions onlyhold in the limit that the modulus field remains fixed during inflation. Although thisseems like a good approximation, as the displacement of the modulus minimum duringinflation is indeed small, the correction to the flat inflaton potential is nevertheless large.In fact, it gives η ≈ −
3, and thus no slow-roll inflation. This analysis shows that itis important to take the dynamics of all fields during inflation into account, otherwisecrucial effects may be missed.We have proposed a way to solve all of the above problems, and successfully combine F -term hybrid inflation with moduli stabilisation. The idea is to combine the modulusand inflaton sectors not by adding their respective superpotentials, as is usually done,but by adding their respective K¨ahler functions G = K +ln | W | instead. Adding K¨ahlerfunctions corresponds to adding K¨ahler potentials and multiplying superpotentials. Thisway of combining sectors greatly reduces their interactions. In particular, for the caseof combining inflation with a modulus sector, it greatly reduces the displacement ofthe modulus during inflation. Consequently the correction to the inflationary potentialis harmlessly small. For the fine-tuned two-scale KL set-up, or for a one-scale KKLTset-up with a fine-tuned mass scale, the corrections to the inflaton slope and waterfallmasses are small as well. Hence we indeed succeeded in constructing a successful modelof inflation in the presence of moduli.Even when multiplying superpotentials, there are still some constraints on themoduli sector parameters for viable inflation. The graviton mass should be smallenough to suppress the moduli corrections during inflation. The modulus mass needsto be heavy and non-tachyonic during inflation to remain stabilised. Finally the looppotential should be dominated by the contribution of the waterfall fields rather thanby the modulus contribution. Nevertheless, there is still a large region of gravitinoand modulus mass scales for which inflation works, and the inflationary predictions arenearly indistinguishable from the global SUSY model in the absence of moduli fields. Acknowledgments
We are both grateful to K. Sousa and particularly A. Ach´ucarro for useful discussions,inspiration and for the suggestion that multiplication of superpotentials could be naturaland helpful. We also thank N. Bevis, C. Burgess and J. Rocher for a useful comments. uccessfully combining SUGRA hybrid inflation and moduli stabilisation
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