aa r X i v : . [ h e p - ph ] M a y hep-th/yymmnnnSCIPP 11/03 Studies in Small Field Inflation
Michael Dine and Lawrence Pack
Santa Cruz Institute for Particle Physics andDepartment of Physics, Santa Cruz CA 95064
Abstract
We explore some issues in slow roll inflation in situations where field excursions are small com-pared to M p . We argue that for small field inflation, minimizing fine tuning requires low energysupersymmetry and a tightly constrained structure. Hybrid inflation is almost an inevitable out-come. The resulting theory can be described in terms of a supersymmetric low energy effectiveaction and inflation completely characterized in terms of a small number of parameters. Demandingslow roll inflation significantly constrains these parameters. In this context, the generic level of finetuning can be described as a function of the number of light fields, there is an upper bound on thescale of inflation, and an (almost) universal prediction for the spectral index. Models of this typeneed not suffer from a cosmological moduli problem. Introduction
There is good evidence that the universe underwent a period of inflation early in its history. Yet itis probably fair to say that there do not exist completely reliable, calculable microscopic theories ofinflation. Slow roll inflation provides a simple phenomenology; many, if not most, microscopic scenariosfor inflation, involving branes, extra dimensions and the like, admit such a description. Indeed, slowroll inflation makes clear why it is nearly impossible, at present, to formulate a compelling microscopictheory. Planck scale effects are necessarily important, and this requires a full understanding of issueslike dynamics of moduli and supersymmetry breaking, even within a consistent theory of gravity (i.e. astring model). Even as a phenomenology, there are a number of models for slow roll inflation. We followthe review of Lyth[1] in dividing these into “large field”, “medium field” and “small field” types, wherelarge, medium or small here refers to field variations much larger than, comparable, or much smallerthan the Planck mass.Almost by definition, large or medium field inflation is difficult to describe in a systematic fashion,without a complete theory of quantum gravity. Small field inflation, however, is another matter. Hereone should be able to characterize inflaton in terms of a low energy effective action for some numberof light fields, with a limited set of relevant parameters. , This is the goal of the present paper. Wewill see that with some very mild assumptions about genericity, we can characterize small field inflationquite simply:1. The effective theory should exhibit an approximate (global) supersymmetry. Otherwise, the theoryis extremely tuned.2. The effective theory should obey a discrete R symmetry. This accounts for smallness of thesuperpotential during inflation, the absence of terms in the effective action which would spoilinflation, and leads to an approximate R symmetry which accounts for supersymmetry breaking.While many models of inflation posit a continuous R symmetry, such symmetries almost certainlydon’t exist in consistent theories of quantum gravity.3. Supersymmetry is spontaneously broken in the effective theory which describes inflation. Thisbreaking is not related in any simple way to the breaking of supersymmetry in the universe atpresent and the scale is far higher than the TeV scale.4. The (approximate) goldstino may or may not lie in a multiplet with the inflaton.5. The effective theory exhibits an approximate, continuous R symmetry, as an accidental (but typi-cal) consequence of the discrete R symmetry. Expansions of this type have been considered by various authors; an early discusison appears in [2]. Here we are not using “relevant” in the conventional renormalization group sense; but instead referring to theirrelevance to inflation; the correspondence to the usual terminology will be clear shortly.
2. The continuous R symmetry is not exact; terms allowed by the discrete symmetry break thecontinuous global symmetry and spoil inflation, unless the inflationary scale (the square of theGoldstino decay constant) is sufficiently small. In other words, there is an upper bound on thescale of inflation.7. If the requirement above is satisfied, there are further requirements on the Kahler potential in orderto obtain slow roll inflation with adequate e -foldings. This sets an irreducible minimum amountof fine tuning necessary to achieve acceptable inflation. This tuning grows in severity with thenumber of Hubble mass fields.8. In order that inflation ends, the inflaton must couple to other light degrees of freedom, or musthave appreciable self-couplings in the final ground state. The coupling to this extra field, or theself couplings, are fixed by δρρ and the inflationary scale. In the case of an extra field, the resultingstructure is necessarily what is called “hybrid inflation”[3, 4, 5, 2, 6]. In the latter, which we willcall “R breaking inflation” (RBI), further fine tuning is required. In either case, the spectral indexis less than one.Many of these points have been made before, but perhaps not in the systematic fashion discussedhere; in particular, the inevitability of this structure for small field inflation does not seem to be ap-preciated. Similarly, most models of hybrid inflation invoke supersymmetry and R symmetries[6, 4, 5,2, 7, 8, 9, 10]. However, the approximate supersymmetry of the effective action has not been stressed;more important, the R symmetries have generally been taken to be continuous, and the consequences ofdiscreteness, particularly the upper bound on the scale of inflation, have not been considered before, toour knowledge. The restrictions on the Kahler potential have been noted in early work[4, 6], but thenseem frequently ignored; their role in determining the irreducible level of fine tuning and the possiblenumber of e -foldings, particularly sharp in light of the observations about R symmetry, does not seemto have been appreciated.Given the role of supersymmetry in small field inflation, it is natural to investigate embedding thisstructure into theories of low energy breaking. This has the potential to expose connections betweenscales of inflation and scales of low energy physics, as we discuss. Ideas concerning metastable, dynamicalsupersymmetry breaking raise the prospect of sharpening these connections further.In the following sections, we elaborate on each of these points. In section 2, we explain the re-quirement for supersymmetry and the structure of the effective action. In sections 3,4, we relate theinflationary observables to the parameters of the effective action. In section 6, we determine how thelevel of fine tuning depends on the number of fields. In section 7, we obtain an upper bound on the scaleof inflation from interactions which violate the continuous R symmetry. Section 8 discusses the problemof reheating. In section 9, we consider possible alternative structures which might result from relaxingour assumptions; non-hybrid models can arise but require some additional fine tuning and/or degreesof freedom. In sections 10,11, we extend the inflationary models to generate low energy supersymmetry3reaking, in the latter section dynamically. In our concluding section, we briefly consider large fieldinflation and provide an assessment. If we impose some fine tuning or genericity constraints, we can characterize small field inflation quitesimply. First we require a field which is light compared to the Hubble constant during inflation, H .Scalar fields with mass of order H are already unnatural, unless the theory, at least at these scales, issupersymmetric . As we will review, scalar fields with mass much smaller than H , even with supersym-metry, are unnatural. So having one field light compared to H represents a fine tuning; it would seemunlikely that there is more than one such field. For H ≪ M p , we can write a supergravity effectiveaction in an approximately flat space. More precisely, given our assumption that the inflaton is smallcompared to M p , the system is described, approximately, by a globally supersymmetric effective actionwhich exhibits spontaneous supersymmetry breaking. This may seem obvious, but it is perhaps worthpointing out that it follows from the assumption of small field excursions. Consider, for simplicity, asingle light field, S . For fields small compared to M p , we can write: K ≈ K + S † S + α ( S † S ) M p + . . . (1)Similarly, we can expand W in a power series in S . W = W + µ S + m S + λ S + . . . . (2)Because of the small field assumption, W cannot dominate during inflation, so W < H M p . (3)Then µ ∼ H M p . The slow roll conditions then imply W ≪ µ M p ; m ≪ µ (cid:18) µM p (cid:19) ; λ ≪ µ M p . (4)There are also constraints on the Kahler potential parameter α which we will discuss shortly. Formulatedin this way, the fermionic component of S is the Goldstino, and its scalar component is the inflaton. Thepossibility that there is another chiral field, whose scalar component is the inflaton will be consideredfurther when we discuss the Kahler potential constraints.The absence (smallness) of terms W , S , S is most readily accounted for if the theory possessesan R symmetry. A constant in the superpotential can only be forbidden by an R symmetry. Powersof S might be accounted for if S carries a charge, and µ is a spurion for the corresponding symmetry. Ordinary dynamical symmetry breaking is problematic, since one needs a vast mismatch between the associated energyscale and the mass scale of the excitations; Goldstone excitations, as in “natural” inflation, require decay constants muchgreater than the Planck scale. R symmetry for (meta-)stable supersymmetry breaking. Because we do not expect continuousglobal symmetries, we will assume that the underlying R symmetry is discrete (and we will, in general,take it to be Z N , N > R symmetry seems necessary for successful inflation.At the end of inflation, supersymmetry must be restored, and the cosmological constant vanish.One might try to model this without adding additional degrees of freedom. Once the R symmetryis understood as a discrete symmetry, one expects higher order terms in the S superpotential, andsupersymmetric vacua at large fields. We will see that understanding inflation in terms of flow towardssuch a minimum is possible, but adds additional complications. So we will first add additional degreesof freedom coupled to S . For the moment we will suppose that there is one such field, φ . Any additionallight fields coupled to S are likely to be quite light at the end of inflation. φ gains mass by combiningwith S ; if φ were, say, in a non-trivial representation of a non-abelian symmetry, only one componentcould gain mass through this coupling; we will discuss this issue further later. We are led, then, to write W = S ( κφ − µ ) + non − renormalizable terms . (5)This system has a supersymmetric minimum at κφ = µ , S = 0. Classically, however, it has a modulispace with | κS | > | κµ | . (6)This pseudomoduli space will be lifted both by radiative corrections in κ , quantum and Planck-supressedterms in the Kahler potential, and the non-renormalizable terms in the superpotential. As we will see,all are necessarily relevant if inflation occurs in the model. Inflation takes place on this pseudomodulispace; φ is effectively pinned at zero during inflation.The R symmetry now explains the absence of additional dangerous couplings, such as φ . Perhapsmost strikingly, though, it forbids a constant in the superpotential, guaranteeing that at the end ofinflation, when supersymmetry is (nearly) restored, the vacuum energy (nearly) vanishes. We willexplore, in section 9, non-hybrid models in which the superpotential must be suitably tuned.Three points should be noted:1. The assumption that the symmetry is discrete means couplings like S N +1 M N − p are permitted, and, aswe will soon see, they significantly constraint inflation.2. There are additional conditions, as we will shortly enumerate, on the Kahler potential in orderthat one obtain inflation with an adequate number of e -foldings. These constitute at least one finetuning needed to obtain an inflaton with mass small compared to the Hubble scale.3. This structure is not unique; the inflaton need not lie in a supermultiplet with the Goldstino. Ifthere are several multiplets with non-zero R charges, it is possible to tune parameters so that the5calar component of one of these other multiplets is light, while the partner of the Goldstino isheavy. As an example, one can contemplate another field, I . If I couples simply to φ , this isproblematic; inflation does not end. So it is necessary to introduce a field φ ′ , and take for thesuperpotential W = S ( κφ − µ ) + λIφφ ′ + . . . (7)Note that at the minimum of the potential (assumed to be at S = I = φ ′ = 0), all fields aremassive.4. When we consider the constraints arising from points (1) and (2) above, and the values of thecosmological parameters, we will see that the basic hybrid model of eqn. 5 does not producesuitable inflation unless the N of the Z N is very large. The model of eqn. 7, on the other hand, does produce successful inflation for suitable values of the parameters and modest N .5. We will see that these structures can be embedded in a model of low energy supersymmetrybreaking. This is not required, but would seem elegant and economical. In this section, we assume supersymmetry at scales above the Hubble constant of inflation, and thepresence of a discrete Z N R-symmetry. We focus, for now, on the single field model. Our considerationswill generalize immediately to the multi-field case. For slow roll, it is crucial that the curvature of the S potential, during inflation, be smaller than the Hubble constant, V ′′ ≪ µ M p . (8)This is a strong condition, and as we will describe, requires tuning the parameters of the potential for S . An even stronger condition arises from the requirement that the field actually flows towards theminimum at the origin.We have already argued that an R symmetry is a necessary ingredient in successful small-fieldinflation. Because we do not expect continuous global symmetries in nature, the R symmetry must bediscrete; we will take it to be Z N . So the superpotential has the form of eqn. 5, but with additionalterms, which we assume to be Planck suppressed : W = S ( κφ − µ ) + W R ; W R = λ N + 1) S N +1 M N − p + O (cid:18) S N +1 M N − p (cid:19) . (9)We have called the S N +1 term W R because it breaks the would-be continuous R symmetry. Such terms in inflationary models have been considered in [12], where they also play an important role; the scalesassumed there, and the detailed picture of inflation, are quite different, though they resemble some of our discussion insection 9. S much smaller than the Hubble constantand successful inflation. Expanding in powers of S , and exploiting the assumption of R symmetry, ittakes the form K = S † S + φ † φ − α M p ( S † S ) + . . . . (10)We are assuming that, apart from µ , the Planck scale is the only relevant scale; if this is not the case, thefine-tuning problems we discuss below will be more severe. With this assumption, the neglected termswill not be important when the fields are small. We have assumed that other physics is controlled bythe Planck scale.For large values of the fields, the supergravity contributions to the potential, arising from thequartic terms in the Kahler potential, dominate the potential for S . Most importantly, they give rise toa quadratic term[4, 6]: V SUGRA = αµ S † SM p . (11)For lower values of S , the quantum corrections arising from integrating out φ , can be important. Inparticular, in the regime where | κ S | ≫ | κµ | (i.e. on the pseudo moduli space, eqn. 6), these correctionsare easily computed[5, 6, 10]: δK quant ( S, S † ) = κ π S † S log( S † S ) . (12)This Kahler potential is appropriate to the description of the theory at scales below κS , and at timeswhen S is slowly varying. The corresponding quantum correction to the potential is: V quant = κ π µ log( S † S ) (13)The quantum corrections will dominate on the pseudo moduli space if the scale, S quant , belowwhich the quantum contribution to the potential are larger than the supergravity contribution, lies onthe moduli space. S quant is obtained by comparing the second derivative of V SUGRA with that of V quant : | S quant | = 12 α κ π M p . (14)The structure we have outlined above constitutes what is usually called supersymmetric hybridinflation [4, 5, 2, 6, 7, 8, 9, 10], but with the modification W R . Again we see that with the assumption ofsmall field excursions (compared to M p ), and some modest assumptions about naturalness, hybrid infla-tion is almost inevitable. After further studies of this structure, we will subject the various assumptionsto closer scrutiny, and ask whether some may be relaxed.So far we have neglected W R . This term creates an additional potential on the pseudo moduli space(indeed it is not sensible to speak of such a moduli space in anything but an approximate sense, even7eglecting supersymmetry breaking). The leading correction to the potential behaves as λµ S N M N − p . (15)For sufficiently large field, this overwhelms both V SUGRA and V quant . The potential, in this regime, isnot flat enough to inflate. As we will see, this constrains µ . In this section, we attempt to implement inflation in the single field model. We will encounter difficulties,finding that the model is not compatible with facts of cosmology except for very large N . But the modelwill be illustrative and is readily modified to accommodate astrophysical observations.Let us first suppose that W R is sufficiently suppressed that it can be ignored during inflation. Wewill quantify this in the next section. Our interest is in inflation on the pseudo moduli space. The S potential arises from the Kahler potential. The slow roll conditions are: η = V ′′ V M p ≪ ǫ = 12 (cid:18) V ′ V (cid:19) M p ≪ . (16)If V SUGRA dominates, both conditions are satisfied if α ≪
1, and if | S | ≪ M p ; these are minimalconditions for successful inflation in any case.If there is a region where V quant dominates, i.e. if quantum corrections dominate over the super-gravity potential before reaching the “waterfall regime” ( | κtS | ≫ vertµ | ), then inflation may end before S reaches the waterfall regime; η ≈ S f = κ π M p . (17)Alternatively, inflation may end when one enters the waterfall region, S wf = 1 κ µ (18)We will see now that the requirement that κ be suitable to lead to sufficient inflation yields S qu ≪ S wf (19)So inflation, if it occurs at all, takes place in the supergravity regime. This is problematic, if nothingelse, from the point of view of the spectral index, n s , n s = 1 + 2 η. (20)In the supergravity regime, this is greater than one, which appears inconsistent with results fromWMAP. We should stress that this is an issue in the particular class of hybrid inflation models considered here; ”inflectionpoint models” and possibly other small field models can accommodate a “blue” spectrum[13]. In the present context, wewill shortly see that the spectrum is blue as a result of quantum effects. S as real (this implies no loss of generality provided slow roll is valid andfor small fields, i.e. when W R is negligible), and define σ = √ S . The number of e -foldings in thesupergravity regime is: N = Z σ i σ dσ VV ′ M p (21)where σ is the larger of σ quant , σ wf . This yields: N = 12 α log σ i σ quant , (22)The total number of e -foldings in the would-be quantum regime is N quant = 14 α . (23) σ i , the initial value of the field, is not yet constrained by any of our considerations. For now we see thatsignificant inflation requires that α is small. Generically, this is a tuning, of order N (possibly moduloa logarithm). This is irreducible .If there are 60 or so e -foldings in the quantum regime, we can determine the values of the slow rollparameters purely in terms of known quantities. δρρ is determined in terms of µ and κ : V / /V ′ = 5 . × − M p . (24)This expression determines µ in terms of κ (or vice versa): κ = 0 . × (cid:16) µ GeV (cid:17) = 7 . × × (cid:18) µM p (cid:19) . (25)Assuming 60 e-foldings of inflation in the quantum regime, we have, then, for S
60 e-foldings beforethe end of inflation, S = κ π √ M p . (26)Substituting our result for κ , the condition that one not have already entered the waterfall regime is: S S wf = κ √ π (cid:18) M p µ (cid:19) = 4 × (cid:18) µM p (cid:19) (27)The requirement that this be much greater than one yields: µM p ≫ − . (28)This is a rather high scale.We will see in section 7 that there is an upper bound on the scale of inflation, µ (depending on N ).Only for very large N or small λ are scales as large as those of eqn. 36 achievable.9 Inflation in the Two Field Model
The difficulty we have encountered in the single field model can be resolved by invoking the model ofeqn. 7, with two Hubble-mass fields. In this model, as we noted, the inflation is not the partner of theGoldstino. The quantum potential is now: δK quant ( S, S † , I, I † ) = κ π S † S log( I † I ) . (29)The condition on κ required to obtain a suitable fluctuation spectrum is essentially as before, with S replaced by I ; similarly for the formula for the number of e -foldings. But the condition that I qu ≫ I wf is now much different, since I wf = κµ λ (30)and λ can be of order one. Indeed, in this model, inflation ends when I is sufficiently small that η ≈ η and n s areuniversal, and ǫ is small. η = − N ; n s = 1 + 2 η. (31)So one expects, quite generally, that if inflation occurs in the quantum regime, n s ≈ .
98. Again, wenote that if inflation occurs in the supergravity regime, one predicts n s >
1, which appears to be ruledout by current CMBR observations.
So far, we have taken the point of view that in a supersymmetric framework, fields light on the scale H are natural. However, tuning is required to have at least one field much lighter than H . Here we notethat the degree of tuning grows with the number of light fields which can mix with the inflaton (withthe same quantum numbers as the inflaton).In the case of a single field, we have seen that successful inflation requires that the parameter α be less than N , the number of e -foldings. In the context of a landscape (or just standard ’t Hooft’iannotions of tuning) this corresponds to a tuning of order N . More fields require more tuning. In particular,suppose we have a set of singlets, S i , i = 1 , . . . N , all with the same R charge as the superpotential.Then, by a field redefinition, we can take, for the linear term in the superpotential, W = µ S . (32)We can take the Kahler potential to be: K = S † i S i + 1 M p [ α ( S † S ) + (cid:16) α i S † S S † S i + c . c . (cid:17) + 1 M p (cid:16) α ij S † S S † i S j + c . c . (cid:17) . (33)10ur assumption in considering tuning is that all of the α parameters are naturally of order one. Inorder that S , say, have mass of order 1 / N , it is necessary that α be of this order, while the real andimaginary parts of α i be of order √N . This suggests a fine tuning of order N N . Indeed, this indicatesthat fine tuning is minimized if the inflaton does indeed lie in a multiplet with the gravitino.In a somewhat different context (D-brane inflation), an analysis of the fine tuning required to obtaininflation with multiple fields has appeared in [14]. There, an effective action was written for a set offields, and coefficients in the lagrangian chosen at random. The models there are different in a number ofrespects – there is no low energy supersymmetry, terms are included to higher orders in fields – yet thescalings observed in a Monte Carlo analysis are similar. We see that in the framework of supersymmetriceffective lagrangians, these estimates are very simple. W R In the presence of W R , the system has supersymmetric minima, satisfying S N = 2 µ M N − p λ φ = 0 . (34)At large S , the potential includes terms δV R = 2 λµ S N M N − p + c . c . (35)If these terms dominate, the system will be driven towards the supersymmetric minimum. So if we insistthat the system is driven to the R symmetric stationary point, we must require that these terms aresmall, and this in turn places limits on the scale µ (or, through equation 25, the coupling κ ), as well as σ i . The analysis of the previous section goes through provided that σ i > σ quant , and that V ′′ R ( σ i ) ≪ V ′′ quant ( σ i ). The real constraint on the underlying model comes from the requirement that V ′′ R ( σ quant ) ≪ V ′′ quant ( σ quant ). This translates into a restriction on µ , or equivalently κ : (cid:16) µ GeV (cid:17) N − ≪ . N − α N/ λN ( N − × − (36)For N = 4 −
6, this yields for the maximal scale: N = 4 : µ ≈ . × ( α × N = 5 : µ ≈ . × ( α × / GeV (38) N = 6 : µ ≈ . × ( α × / GeV (39)Even for larger N , the scale is not extremely large; e.g. for N = 12, it is only of order 8 × .For N = 3, there is no choice of µ for which W R does not dominate. One can try to resolve thisby including higher order terms in the Kahler potential and considering supergravity corrections to the11 otential of the form: δV = βµ | S | M p . (40)However, in this case, all of the activity occurs, for β ∼
1, for S ∼ √ αM p , which does not seem consistentwith the idea of small field inflation, unless α is tuned to be extremely small. Alternatively, the coefficientof the operator appearing in W R might be very small. Calling W R = λ N + 1) S N +1 M N − p , (41)in the case N = 3, we require λ ≪ − α / . (42)Indeed, we could consider, for any N , the possibility that µ is larger than implied by eqn. 36, and λ issmall. The general condition is: λ ≪ (cid:18) µM p (cid:19) − N α N/ . × − ) N − N ( N − . (43)Thus λ has to be extremely small for reasonable values of N . Small µ is arguably more plausible.In the two-field model, one has similar constraints on the inflationary scale. A coupling like that of W R , with S N +1 replaced by SI N , for example, has essentially identical effects.Note that we now have now enumerated three types of constraints/tunings:1. α must be small, comparable, up to a logarithmic factor, to 1 / N , one over the number of e -foldings.2. µ and κ must be small, in order that the superpotential corrections not drive S to a large fieldregime.3. The initial conditions for S are constrained. S must lie in a range small enough that, at least fora time, the quantum potential is dominant, and large enough that inflation can take place. So far, we have not discussed the quantum numbers and couplings of φ . These are important since it isthe lifetime of φ which determines the reheating temperature. For the scales of interest here, φ = κ − / µ (44)and this can readily be of order grand unified scales. So many hybrid inflation models take φ to be, say, inthe adjoint representation of some grand unified group. If S is a singlet, this is potentially problematic,12ince there may be additional light states (in the final vacuum); these can spoil unification and lead toother difficulties. A very simple possibility is to take φ to be a gauge singlet, and couple φ to somecharged fields; schematically δW = λφ ¯55 . (45)In the vacuum, the masses of ¯5, 5, are of order m Q = λφ . So the φ lifetime is of orderΓ = (cid:16) α s π (cid:17) λ π m φ m Q . (46)This can be rewritten, using φ = κ − / µ , and the relation between κ and µ :Γ = 14 π (cid:16) α s π (cid:17) κ / µ (47) ≈ × (cid:16) µ (cid:17) GeV . For µ = 10 GeV, this corresponds to Γ = 3 × − GeV, or a reheat temperature of order 10 . GeV.Larger µ leads to higher reheat temperatures. Such temperatures are clearly interesting from the pointof view of the gravitino problem and other cosmological issues. So far, we have insisted on an unbroken, discrete R symmetry at the end of inflation. But we mightrelax this. For example, consider a model with superpotential W = µ S − λ N + 1) S N +1 M N − p . (48)Here we might try to arrange that the field, during inflation, rolls towards the supersymmetric minimumat S N = 2 M N − p µ λ . (49)As we will see, the conditions for slow roll inflation can be satisfied for a range of initial field values,and adequate e-foldings and fluctuation spectrum obtained. Inflation ends as the field moves towardsthe minimum. It is important, however, that the energy not be negative at the end of inflation, and thisrequires a small constant in the superpotential. This constant breaks the discrete R symmetry. Whilesmall, this term is not a small perturbation. In particular, it gives a large contribution to V ′′ at thepoint where V ′′ ∼ H . This issue has been discussed in [15], where possible additional tunings and/oradditional degrees of freedom which might permit suitable inflation have been considered.Even if one builds a successful model, when inflation ends, S oscillates about S . It is importantthat there be a mechanism to dissipate the energy of oscillation. This does require coupling to additionalfields. For example, adding again a 5 and ¯5 of SU (5): δW = κS ¯55 , (50)13 can decay to gauge boson pairs. Note, however, that for sufficiently small S , the quantum contributionsto the potential will dominate over those we have considered up to now. For initial configurations inthis regime, the field will flow towards the origin. This places a lower limit on σ i , which depends on κ . Overall, then, the inflationary scenario we have outlined in the earlier sections seems less tuned andsimpler than the non-hybrid scenarios.
10 Supersymmetry Breaking
The basic structure of the hybrid inflation superpotential is reminiscent of O’Raifeartaigh models. It isinteresting to include additional degrees of freedom so that S is part of a sector responsible for (observed)supersymmetry breaking. For small S , in addition to the field φ , we can include another field, X , with R charge two and superpotential: W = S ( κφ − µ ) + mXφ. (51)Assuming that the scale of inflation is large compared to the scale of supersymmetry breaking leads toconsideration of the limit | m | ≪ | κµ | . In this limit, the vacuum state has φ ≈ κ − / µ , X = S = 0,and F X ≈ κ − / mµ. (52)By choosing m , we can arrange F X as we please. We might worry that X is light, and relativelylong-lived. Classically, the X mass vanishes. Quantum mechanically, it is of order m X = κ π m m S F † X F X . (53)Calling m = ǫκµ (we require ǫ < φ have a vev), and noting m s = κµ , F X = ǫ / µ ,we have m X = ǫ π µ . (54)So ǫ can be quite small, with X still in the TeV range.In the model as it stands, the R symmetry is unbroken by loop effects. This can be avoided throughadditional, “retrofitted” couplings[16], or through models like that of [17].
11 Hybrid Inflation and Dynamical Supersymmetry Breaking
It would be appealing if the scales appearing in our models of inflation (and supersymmetry breaking)could be understood dynamically. The simplest implementation of (metastable) dynamical supersym-metry breaking is through “retrofitting”[18, 16, 19]. In particular, we want to generate the scales µ m , of eqn. 51 dynamically. In these references, the scale µ was of order Λ M p , where Λ is thescale of some underlying, supersymmetry preserving but R symmetry breaking dynamics. In the presentcase, however, this would lead to too large a value of W , the value of the superpotential at the end ofinflation. Instead, we follow [16, 19], and consider theories with order parameters of dimension one aswell as dimension three. These are singlet fields, which we will denote by Φ , h Φ i ∼ Λ. The Φ fields havemass of order Λ. We replace the superpotential of eqn. 51 by W = S ( κφ − λ Φ ) + λ ′ Φ M p φY. (55) λ need not be particularly small in order not to appreciably perturb the Φ dynamics, i.e. to satisfy therequirement that Φ is massive compared to S , Φ. Then at scales well below Λ, the theory is that of eqn.51. < W > ∼ Λ ∼ µ at the minimum of the potential, so the supergravity corrections to the potentialare negligible at low energies. In fact, we have F Y ≈ mφ = Λ κ − / /M p (56)
12 Conclusions
So we have seen that small field inflation is likely to require supersymmetry, and that conventionalnotions of naturalness also lead to the inevitable requirement of an R symmetry. This leaves two classesof models: hybrid and RBI. In the former, we have seen that the requirement that the R symmetry bediscrete places an upper bound on the scale of inflation, which makes observations of tensor modes inthe CMB extremely unlikely. Also inevitably, n s <
1, typically about 0 .
98. In the RBI case, there arealso constraints on scales, and one requires some sort of soft breaking of the R symmetry, describable,perhaps, by spurion fields.We have seen that there is a simple effective field theory description of these types of inflation, andthat one can use the language of global supersymmetry, perturbed slightly (but in critically importantways) by coupling to supergravity. In this framework, the simplest models have an inflaton which liesin a supermultiplet with the gravitino, but this is not necessary. Indeed, we have seen that once oneconsiders higher dimension operators, there is an upper bound on the energy scale of inflation, and thatmodels with at least one additional field more readily lead to successful inflation. One could go furtherthan we have here in applying the language of [20] to this problem.We have made much of the naturalness of supersymmetry for the problem of small field inflation,and one might wonder whether, in, say, an anthropic landscape framework, these considerations wouldbe relevant. Needless to say, it is hard to make a definitive statement; it could be that supersymmetricstates are very rare in the landscape, and that this overwhelms any tuning considerations (just as forthe Higgs[21]). But we can ask the question a different way. Given the assumption of a supersymmetriclandscape, could it be that the requirement of inflation accounts for a little (or not so little) hierarchy?E.g. small κ and correspondingly small µ might be disfavored by landscape distributions; depending on15orrelations between m (in eqn. 51) and µ , one might be driven to larger scales of supersymmetry thanmight be expected from naive naturalness considerations. This issue will be discussed elsewhere[ ? ].We have not discussed the problem of initial conditions at any length. Certainly there are constraintson the initial values of the fields, their velocities, and the degree of homogeneity required. These issueslook different in different contexts, and we will leave them for further study. One striking feature ofthis framework is the nature of the cosmological moduli problem. In the models discussed here, theminimum of the potential is a point with an approximate R symmetry. Moduli which are charged underthe symmetry naturally sit near the origin, and are not particularly light. Neutral fields (such as thefield X in the retrofitted models) naturally sit near the origin as a result of the accidental, continuous R symmetry. There seems to be no moduli problem in this context. This issues will be explored furtherelsewhere.Finally, as we noted at the beginning, almost by definition it is hard to make general statementsabout large field inflation. One could attempt this as a limit of the analysis we described above, but thenthe effective lagrangian has many more “relevant” parameters, and one can probably, at best, simplystate constraints consistent with observations on combinations of these. It is hard to see how predictionscan emerge without a detailed microscopic understanding of the underlying gravity (supergravity) theory. Acknowledgements:
Conversations with Willy Fischler, Dan Green, and Liam McAllister helped clar-ify our thinking about many issues here. Correspondence with David Lyth provided valuable perspective.M. Dine thanks Stanford University and the Stanford Institute for Theoretical Physics for hospitalityduring the course of this project. He thanks the Theory Group at the University of California, Berkeleyand the Theory Group at Lawrence Berkeley National Laboratory for their hospitality as well. We thankthe authors of ref. [15], and especially S. Mooij and M. Postma, for conversations and pointing out anerror in our original discussion of the non-hybrid models of section 9. This work was supported in partby the U.S. Department of Energy.
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