Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise
SSCIPP 15/04UTTG-08-15
Light Scalars and the Cosmos:
Nambu-Goldstone and Otherwise
Nambu Memorial Symposium, University of Chicago, 2016
Michael Dine ( a ) Santa Cruz Institute for Particle Physics andDepartment of Physics, University of California at Santa CruzSanta Cruz CA 95064
Abstract
This talk focuses on the role of light scalars in cosmology, both Nambu Gold-stone bosons and pseudo moduli. The former include QCD axions, which mightconstitute the dark matter, and more general axions, which, under certain condi-tions, might play the role of inflatons, implementing natural inflation . The latterare the actors in (generalized) hybrid inflation. They rather naturally yield largefield inflation, even mimicking chaotic inflation for suitable ranges of parameters. a r X i v : . [ h e p - t h ] A ug Introduction: Homage to Nambu
From my graduate student days, when I first encountered his work on symmetry breakingin the strong interactions, Nambu has been one of my intellectual heroes. This wasreinforced during my years at City College, where I regularly heard stories of Nambufrom Bunji Sakita, who himself was an admirer. While he was somewhat younger thanNambu, Sakita often regaled me with stories of the War years in Japan, involving bothhis experiences and Nambu’s. I finally got to know Nambu in the 1980’s, and all ofmy interactions with him were intellectually stimulating and enhanced by his charm andwit. I remember many conversations at Chicago, but remarks he made at the 1984Argonne meeting on String Theory, which were thoughtful but cautionary, particularlystand out. My final interactions came shortly after his Nobel Prize. Like many, I senthim a congratulatory note, only to receive a ”mailbox is full” reply. About a year later,though, I received the most thoughtful note.In my own work, Nambu’s influence is perhaps heaviest in the areas of string theoryand in the appearance of light scalars in the case of continuous global symmetry breaking.His work with Jona-Lasinio has always been instructive for me in that it takes a modelwhich in detail cannot be taken too seriously, but extracts important, universal features.Some of what I say in this talk I hope can be viewed as a modest effort in this style. Whilemuch of my discussion will center on Nambu-Goldstone bosons, I will also consider somequestions in strong dynamics, where fermionic condensates will play important roles.In his famous work on symmetry breaking, the light scalars were the pions of thestrong interactions. Today, particularly important roles for light scalars arise in cosmol-ogy. Examples include axions as dark matter, but also candidates for the inflaton of slowroll inflation. Indeed, the following statements are often made about inflation:1. The Planck satellite ruled out hybrid inflation2. If tensor fluctuations are observed (requiring Planck scale variation of the inflaton)then inflation is necessarily “natural”, or “chaotic”.3. In the case of “natural” inflation, this must be understood as “monodromy” infla-tion or “aligned axions”.Today I want to push back (gently) on these statements.
We often distinguish two ategories of Inflationary models:1. Large Field Inflation: φ (cid:29) M p . In this rubric, most often one speaks of chaoticinflation or natural inflation and its variant, monodromy inflation. This frameworkpredicts potentially observable gravity waves.1. Small Field Inflation: φ (cid:28) M p . This is often associated with “hybrid inflation”.In this case, there are no observable gravity waves. For small field hybrid inflation,the challenge is to understand n s as reported by Planck ( n s = 0 . ± . . )Indeed, the Planck paper[1] asserts that hybrid inflation is ruled out. Both the small and large field scenarios for inflation face significant theoretical challenges.For the two principle implementations of large field:1. Chaotic inflation[2] is typically modeled with monomial potentials; arguably thisis the definition of chaotic inflation. One requires very small coefficients for themonomial, and this is typically not enforced by any conventional symmetry (oftenapproximate, continuous shift symmetries for non-compact fields are invoked, butwe have little understanding of how such symmetries would arise, say, in stringtheory). The dominance of a particular monomial requires even greater suppressionof a host of terms of the form φ n M n − p for many n . Again, why should this be?2. Natural inflation: natural inflation invokes Nambu-Goldstone bosons as inflatons[3].This is, indeed, the sense in which natural inflation is strictly natural accordingto the criterion of ’t Hooft[4]: such models contain small parameters protectedby approximate symmetries. Axions typically have a periodicity (discrete shiftsymmetry), a → πf a (2.1)where a is a canonically normalized scalar field. So a is compact. Correspondingly,the potential for a is of the form: V ( a ) = Λ cos( af a + δ ) . (2.2)Successful inflation requires that a vary over a range much larger than the Planckmass, i.e. f a (cid:29) M p . This is in the realm of graviational physics. The one frameworkin which we can assess the plausibility of such large f a is string theory, where thisdoesn’t seem to be realized in known constructions[5]. This observation has beenelevated to a principle, the Weak Gravity Conjecture , which makes natural inflation,as originally proposed, seem unlikely[6].In recent years, a plausible generalization of natural inflation has emerged: mon-odromy inflation[7]. Theories of monodromy inflation mimic a large f a by allowing theaxion to vary over a much larger range. This can be understood in terms of a potential: V ( a ) = Λ cos( af a + 2 πkN ) . (2.3)2he 2 π periodicity survives if k → k − a shifts. This phenomenon was arguedto be realized in string theory in [7]; here we’ll see some simple (and in fact familiar)realizations in field theory. We should note that there are related proposals, such as“multi-natural inflation”, which we will not consider here[8]. Here we will focus mainly on hybrid inflation. Hybrid inflation is often understood interms of rather specific (usually supersymmetric) models, but it can be understood ingeneral terms as inflation on a pseudomoduli space. In its favor, as we explain, it issomewhat natural (arguably much moreso than chaotic inflation). But Planck scalecorrections are still important and must be suppressed. There is no obvious symmetryexplanation for this suppression, so there appears an irreducible tuning at the level of10 − or so. One also requires very tiny couplings to account for δρρ . This smallness maybe technically natural, but it is disturbing nevertheless.More generally, there is a challenge in either the small or large field frameworks: towhat extent can one make predictions which would tie to a detailed microscopic picture.We will propose an alternative viewpoint on modeling inflation in this talk, but we won’tgive a completely satisfying answer to this question. Natural Inflation and its Variants
With V ( a ) = Λ cos( θ/f a ) . (3.1)The constraints on f a arise from satisfying the slow roll conditions, for example: η = M p V (cid:48)(cid:48) V (cid:28) f a (cid:29) M p . As we have mentioned, this is difficult to realize in string theory.Silverstein and Westphal[7] suggested an alternative. They noted that in string modelsin the presence of branes, axions can “wind”. They have, as a result, an approximateperiodicity greater than 2 π . Silverstein and Westphal found that the potentials for thesefields were monomial, yielding a form of chaotic inflation.The string constructions are somewhat complicated. A readily understood class ofmodels of monodromy inflation can be exhibited in field theory[9]. Consider an SU(N)supersymmetric gauge theory without chiral matter. In such a theory, there is a gauginocondensate. The idea of non-zero fermion bilinears goes back, of course, to Nambu. Inthe strong interactions, these have been studied phenomenologically since that time, andmore quantitatively in lattice gauge theory. In the last few decades, supersymmetry has3rovided a context in which such condensates can sometimes be computed analytically.In particular, in the SU ( N ) supersymmetric gauge theory[10], (cid:104) λλ (cid:105) = Λ e πikN e iθ/N . (3.3)The phase reflects the spontaneous breaking of a Z N symmetry.If we perturb the theory with a susy-breaking gaugino mass term, m λ λλ , then V ( θ ) = m λ Λ cos( θN + 2 πkN ) . (3.4)So, as we anticipated above, the naive periodicity θ → θ + 2 π is compensated by changingthe branch, k . If we elevate θ to a (pseudo) Nambu-Goldstone boson, θ = a ( x ) /f a , thenif the axion moves slowly, it simply crosses to the other branches. The tunneling ratescales as Γ ∝ e − N (cid:16) Λ mλ (cid:17) . (3.5)so the tunneling rate for changes of k is extremely small[11, 9]. Further observations onthe tunneling rate will appear in [12].For sufficiently large N ( N ∼ − m λ small (not too small, as will be clearfrom [12], we seem to have what we require for successful natural inflation. One mightwonder whether this a particularly plausible story (for example. For example, is suchlarge N in the Swampland ?[13]). Similar questions can be raised about the ingredientsof brane constructions, which one might think roughly dual to these field theories.
Witten, long ago, put forward a particularly interesting proposal for monodromy inQCD[14, 15, 16, 17]. He advocated considering QCD from the viewpoint of the LargeN expansion. In large N , he argued that instanton effects should be exponentially sup-pressed. Instead, he proposed that quantities like correlation functions of F ˜ F , even atzero momentum, should exhibit the same behavior with N as in perturbation theory. Forexample, in pure gauge theory, a correlator (cid:104) (cid:18)(cid:90) d xF ˜ F (cid:19) n (cid:105) ∝ N − n . (3.6)To account for the 2 π periodicity of θ , he suggested that pure QCD, for example, shouldhave N branches, with the θ dependence of the vacuum energy, for example behaving as E ( θ ) = min k ( θ + 2 πk ) . (3.7)It is interesting to revisit these questions in light of our understanding of instantonsand θ in supersymmetric theories, where we have a great deal of theoretical control. This4ontrol extends to the inclusion of small soft breakings; of course, we need to pass to largesoft breakings if we are to understand real QCD. In any case, already examining eqn.3.4, we see exactly the sort of structure anticipated by Witten. In fact, exploring thesetheories, one can also reproduce Witten’s modification of the non-linear lagrangian ofthe pseudogoldstone bosons to include the η (cid:48) . The origin of the branches in each of thesecases is clear: they are associated with the breaking of an approximate Z N symmetry.But said this way, it is natural to ask what happens as one increases, say, m λ ,so that the discrete symmetry is badly broken – and ordinary QCD recovered. Onemight speculate that the N branches would collapse into a small number, and a moreconventional periodicity in θ would be recovered. The problem with this idea is thepresumed suppression of instanton effects.So it is is interesting to revisit these as well. This will be done in [12], but themain point is simple, and not surprising given our experience with SUSY theories. Incases where instanton calculations are reliable, they are not suppressed with N . In fact,they reproduce the counting expected from perturbation theory. So on the one hand,the instanton argument for branches does not hold; on the other hand, where one hascontrol, Witten’s conjectures are realized. In [12], possible behaviors are enumerated,and a set of possible lattice tests will be enumerated.To summarize this section, we have considered the two popular realizations of largefield inflation:1. Chaotic inflation with monimials: Here one has the longstanding puzzle of account-ing for the suppression of an infinite set of operators – polynomials in fields – atlarge fields. These are non-compact fields and it is hard to see how this arises fromconventional symmetries.2. Monodromy inflation has simple realizations in field theory, but these require verylarge gauge groups. It is not clear how plausible this is. Whether similar plau-sibility issues exist for the string constructions would seem a question worthy ofinvestigation.So it is interesting to consider other possibilities for large field inflation. Indeed, stringtheory suggests a simple alternative picture. String theory possesses non-compact fields which, at least in certain regions of the fieldspace, would seem likely to develop very flat potentials. These are the moduli of thetheory. The idea that such fields are candidate inflatons has been around a long time(e.g. [18]). Here we will add some new elements.To justify the existence of a pseudomoduli space, we assume approximate super-symmetry. The non-compact moduli of interest are typically scalar partners of compact5oduli (axions). Together these form the lowest components of superfields, Φ = r + ia .The discrete shift symmetry of the axion, as well as the requirement of sensible behaviorin asymptotic regions of the moduli space (e.g. where the theory is typically weakly cou-pled) allow one to write, for example, the superpotential as a series of terms of the form e − N Φ . This connection between axions and moduli will provide us with a conceptual pegfor our discussion. We will see that the combination of large and small field inflation hasclose parallels to large and small field solutions of strong CP problem. In this section wewill:1. Review (small field) hybrid inflation extracting some general lessons.2. Discuss large and small field solutions of the strong CP problem3. Consider Inflation with non-compact moduli4. Enumerate the Ingredients for successful modular inflation5. Discuss large field excursions in the moduli space Hybrid inflation is often defined in terms of fields and potentials with rather detailed,special features, e.g. a so-called waterfall field[19, 20, 21, 22]. But hybrid inflation canbe characterized in a more conceptual way. Inflation occurs in all such models on apseudomoduli space, in a region where supersymmetry is badly broken (possibly by alarger amount than in the present universe) and the potential is slowly varying[23].Essentially all hybrid models in the literature are small field models; this allowsquite explicit constructions using rules of conventional effective field theory, but it isnot clear that small field inflation is selected by any deeper principle. The simplest(supersymmetric) hybrid model involves two fields, I and φ , with superpotential: W = I ( κφ − µ ) . (4.1) φ is known as the waterfall field.Classically, for large I , the potential is independent of I ; V cl = µ ( φ = 0) . The quantum mechanical corrections control the dynamics of the inflaton: V ( I ) = µ (1 + κ π log( | I | /µ )) . (4.2) κ is constrained to be extremely small in order that the fluctuation spectrum be of thecorrect size; κ is proportional, in fact, to V I , the energy during inflation. The quantumcorrections determine the slow roll parameters. One has: V I = 2 . × − (cid:15) M p (4.3)6 = 0 . × (cid:16) µ GeV (cid:17) = 7 . × × (cid:18) µM P (cid:19) . (4.4) In addition to the quantum corrections we have described, higher dimension operators ofvarious types, even if Planck suppressed, can have dramatic effects. These include:1. Kahler Potential Corrections: One expects corrections to the Kahler potential; wewill assume here that they are Planck suppressed; the constraints are more severeif they are suppressed by some smaller scale. We organize the effective field theoryin powers of I [23]. The quartic term in K , K = αM p I † II † I (4.5)gives too large an η unless α ∼ − . This appears to be one irreducible source offine tuning in this framework.2. Superpotential Corrections: thesel are potentially very problematic. For example,one can write: δW = I n M n − p (4.6)At least the low n terms must be suppressed. This might occur as a result ofdiscrete symmetries.The leading power of I in the superpotential controls the scaleof inflation. For example, N = 4, gives µ ≈ GeV and κ ≈ − . With N = 5,one obtains µ ≈ GeV, and κ ≈ − The scale µ grows slowly with N , reaching10 GeV at N = 7 and 10 GeV for N = 12.This result is interesting from the perspective of understanding (predicting?) thescale of inflation. It is hard to understand a high scale of inflation in this frameworkwithout a rather absurd sort of discrete symmetry. This might be taken as an argumentfor a low scale of inflation. On the other hand, pointing in the opposite direction is κ ,which gets smaller rapidly with V , In addition, achieving n s <
1, consistent with Planck,required a balancing of Kahler and superpotential corrections. Indeed, the abstract of thePlanck theory paper[1] includes the assertion: “the simplest hybrid inflationary models,and monomial potential models of degree n > − level)is required. One also needs a very small dimensionless parameter, progressively smalleras the scale of inflation becomes smaller. Quite likely, any successful model requiressignificant discrete symmetries (or even more severe tunings).7 .2 Generalizing hybrid inflation to large fields: moduli infla-tion So it is clearly interesting to explore the possibility of inflation on (non-compact) modulispaces with fields undergoing variations of order Planck scale or larger. Such modulispaces are quite familiar from string theory. First it is instructive to consider anothersituation where such a small field/large field dichotomy arises: the axion solution to thestrong CP problem. To solve the strong CP problem one must account for an accidental global symmetrywhich is of extremely high quality . Most models designed to obtain a Peccei-Quinnsymmetry can be described as small field models; they are constructed with small axiondecay constant, f a (cid:28) M p , with f a = (cid:104) φ (cid:105) In this case, one can organize the effectivefield theory in powers of φ/M p (again, if higher dimension operators are suppressed by ascale M (cid:28) M p , the objections discussed here to the Peccei-Quinn solution are even moresevere).We can define a notion of Axion Quality [24]. We require Q a ≡ f a m a ∂V∂a = 10 f a ∂V∂a < − . In small field models, if the axion is the phase of φ , the PQ symmetry is the trans-formation φ → e iα φ . This symmetry must be extremely good[25]. One needs to suppress φ N M N − p up to very high N . E.g. Z N , with N >
11 or more, depending on f a . This is notterribly plausible. It involves models of a high degree of complexity, designed to solve aproblem of essentially no consequence (small θ is not, by itself, singled out by anthropicor similar considerations[24, 26]. String theory has long suggested a large field perspective on the axion problem[27]. Stringtheory, as we have stressed, frequently possesses axions. These axions exhibit continuousshift symmetries in some approximation (e.g. perturbatively in the string coupling).Non-perturbatively these symmetries are broken, but usually one has a discrete shiftsymmetry left which is exact: a → a + 2 π. (4.7)We have normalized a to be dimensionless; f a depends on the precise form of the axionkinetic term. The (non-compact) moduli which accompany these axions typically havePlanck scale vev’s. Calling the full chiral axion superfield A = s + ia + . . . , this periodicityimplies that, for large s , in the superpotential the axion appears as e −A . Solving the8trong CP problem then requires suppressing only a small number of possible terms[28,29]. As always, in string theory, one has to understand stabilization of moduli. Morehonestly (thinking of Nambu’s cautionary remarks) in the current state of our knowledge,we can at best conjecture that moduli are stabilized. If string theory is to produce anaxion which can solve the strong CP problem, typically several moduli must be stabilized.Whatever the mechanism, the axion multiplet is special. If the superpotential plays asignificant role in stabilization of the saxion , it is difficult to understand why the axionshould be light. e −A would badly break the PQ symmetry if responsible for saxionstabilization. So the stabilization must result from Kahler potential effects (presumablyconnected with supersymmetry breaking). In perturbative string models the Kahlerpotential is often a function of A + A † . There is no guarantee that would-be correctionsto K which stabilize A do not violate this symmetry substantially, but will take as ahypothesis.For example, as a model, suppose there is some other modulus, T = t + ib , appearingin the superpotential as e − T , where e − T might set the scale for supersymmetry breaking. W ( T ) = Ae − T/b + W (4.8)with small W , leading to T ≈ b log( W ) . (4.9)The potential for s would arise from terms in the supergravity potential: V s = e K (cid:12)(cid:12)(cid:12)(cid:12) ∂K∂ A W (cid:12)(cid:12)(cid:12)(cid:12) g A A ∗ + . . . (4.10)For suitable K ( A , A ∗ ), V might exhibit a minimum as a function of s . If s is, say, twice t at the minimum, e −A is severely suppressed, as is the potential for the (QCD) axion, a . Typical metrics for non-compact moduli fall off as powers of the field for large field.Defining s to be dimension one, g A , A ∗ = C M p /s (4.11)for some constant, C . So large s is far away (a distance of order a M p log( s/M p )) in fieldspace. If, for example, the smallness of e − ( s + ia ) is to account for an axion mass smallenough to solve the strong CP problem, we might require s ∼ M p , corresponding toa distance of order 8 M p from s = M p if C = √ .3 Non-Compact Moduli as Inflatons So the strong CP problem points to Planck scale regions of field space as the arena forphenomenology. This has parallels in inflation. Moduli of the sort required for the axionsolution might also play a role as inflatons. There are some plausible ingredients formoduli as the players in inflation:1. In the present epoch, one or more moduli which are responsible for hierarchicalsupersymmetry breaking.2. In the present epoch, a modulus whose superpotential is highly suppressed, andwhose compact component is the QCD axion. This is not necessary for inflation,but is the essence of a modular (large field) solution to the strong CP problem.3. At an earlier epoch, a stationary point in the effective action with higher scalesupersymmetry breaking then at present and a positive cosmological constant.4. At an earlier epoch, a field with a particularly flat potential which is a candidatefor slow roll inflation.Fields need not play the same role in the inflationary era that they do now. ThePeccei-Quinn symmetry might be badly broken during inflation. Then the axion willbe heavy during this period and isocurvature fluctuations may not be an issue[30]. Insuch a case the initial axion misalignment angle, θ , would be fixed rather than being arandom variable. We know that the scale of inflation is well below M p . So it is plausiblethat even during inflation moduli have large vev’s, e −A , e − T (cid:28)
1, though much smallerthan at present. For example, suppose that there exist a pair of moduli, A , T responsiblefor supersymmetry breaking, and an additional field, I , which will play the role of theinflaton. During inflation, H I ∼ W ∼ e − t (4.12)For typical Kahler potentials, the curvature of the t and i potentials will be of order H I (for i , this is the usual “ η problem”). We will exhibit a model with lower curvaturebelow.A successful model requires a complicated interplay between effects due to the Kahlerpotential and superpotential[31]. • The potential must possess local, supersymmetry breaking minima in A and T , oneof higher, one of lower, energy. The former is the setting for the inflationary phase;the latter for the current, nearly Minkowski, universe. • In the inflationary domain, the potential for I , must be very flat over some range. • In the inflationary domain, the imaginary parts of A and T should have massescomparable to H I (or slightly larger), if the system is to avoid difficulties withisocurvature fluctuations. This would arise if e − s ≈ e − t .10 In the present universe, the imaginary part of A should be quite light and that of I much lighter. • There are additional constraints from the requirement that inflation ends. Forsome value of Re I , the inflationary minimum for T and A must be destabilized(presumably due to Kahler potential couplings of I to A and T ). At this point,the system must transit to another local minimum of the potential, with nearlyvanishing cosmological constant. • The process of transiting from the inflationary region of the moduli space to thepresent day one is subject to serious constraints. Even assuming that there is apath from the inflationary regime to the present one, the system is subject to thewell-known concerns about moduli in the early universe[32, 33]. If they are suffi-ciently massive (as might be expected given current constraints on supersymmetricparticles), they may reheat the universe to nucleosynthesis temperatures, avoidingthe standard cosmological moduli problem. T and A are vulnerable to the moduliovershoot problem[34], for which various solutions have been proposed. r Here we describe a simple model which yields large r and satisfies some of the conditionsenumerated above (because it involves only a single field and we specify the potentialonly in a limited range it cannot satisfy all)[31] For the Kahler potential we take: K = −N log( I + I ∗ ) . (4.13)With I = e φ/ N (4.14)the kinetic term for φ is simply | ∂φ | . V ( φ ) = e −N φ V , (4.15) V being the minimum of the S , T potential.The slow roll parameters are: (cid:15) = 12 N ; η = N = 2 (cid:15). (4.16)Note n − − (cid:15). (4.17)If r = 0 . (cid:15) , n s − .
025 (4.18)on the high end of the range favored by the Planck measurement.This model is discussed in the Planck theory paper which rules it out based on theirmeasurement of n s . Suitable modifications are discussed in [31]. A model with similarfeatures (with cosh rather than exponential potential) has been discussed in[35].11 .3.2 Connection to Chaotic Inflation Chaotic inflation has, for decades, provided a simple model for slow roll inflation, andits prediction of transplanckian field motion and observable gravitational radiation iscompatible with our discussion of large field modular inflation. As we look at the moduliinflation model of the previous section (and more generally moduli models of large fieldinflation), we see, in fact, a realization of the ideas of chaotic inflation[31]. Again, thepotential behaves as V ∼ H I M p e N φ (4.19)The exponent changes, during inflation, by a factor of about 3 /
2. So we can make acrude approximation, expanding the exponent and keeping only a few terms. If we focuson each monomial in the expansion, the coefficient of φ p , in Planck units, is: λ p = 10 − N p p ! . (4.20)where N is the humber of e -folds.We can compare this with the required coefficients of chaotic inflation driven by amonomial potential, φ p . In this case, λ p = 3 × − (2 N p ) p − (4.21)These coefficients are not so different. For example, for p = 1, the moduli coefficient isabout 2 × − , while for the chaotic case it is about four times smaller; the discrepancyis about a factor of two larger for p = 2. So we see that these numbers, which would onehardly expect to be identical, are in a similar ballpark.So moduli inflation provides a rationale for the effective field theories of chaoticinflation. The typical potential is not a monomial, but one has motion on a non-compactfield space, over distances of several M p , with a scale, in Planck units, roughly thatexpected for chaotic inflation. The structure is enforced by supersymmetry and discreteshift symmetries. Physicists are likely a long way from writing down the microscopic model which describesinflation. For the time being the most sensible approach is to consider classes of models,the constraints coming from observations, and possible general features and predictions.Here we have discussed some features of several classes of models:1. Chaotic inflation: we have reviewed why it is puzzling as usually formulated, andhave seen that something like it may arise in frameworks which are more natural.12. Natural inflation in its simplest formulation unlikely.3. Monodromy inflation: has simple field theory realizations, with large amounts ofinflation requiring very large gauge groups; we have speculated about the implica-tions for the plausibility of the mechanism.4. Hybrid inflation: we have stressed that hybrid inflation should be thought of as in-flation on a pseudomoduli space. from this vantage point, large field seems plausible(the simplest forms of small field are ruled out, and even these are highly tuned).Large field hybrid inflation favors higher scales for inflation, and has features whichcan mimic chaotic inflation.In all cases, detailed implementations are challenging; one wants to ask whether thereare any generic features one can extract ( r , non-gaussianity,...) and compare with data.Explaining inflation from an underlying microscopic theory is an extremely chal-lenging problem, quite possibly inaccessible to our current theoretical technologies. Aswe have reviewed, even in so-called small field inflation, it requires control over Planckscale phenomena. Within string theory, this requires understanding of supersymmetrybreaking (whether large or small) and fixing of moduli in the present universe as wellas at much earlier times. It requires an understanding of cosmological singularities, andalmost certainly of something like a landscape.We have stressed a parallel between small/large field inflation and small/large fieldsolutions to the strong CP problem. The existence of moduli in string models is stronglysuggestive of the large field solutions to both problems. The proposal we have put forwardhere is similar to the large field solutions of the strong CP problem.Several moduli likely play a role in inflation in order to achieve the needed degree ofsupersymmetry breaking and slow roll. We have noted that small r is more tuned thanlarge r , giving some weight to the former possibility. We have noted the contrast withsmall field inflation, where extreme tuning to achieve low scale inflation is replaced bythe requirement of an extremely small dimensionless coupling.Returning to the strong CP problem, any would-be Peccei-Quinn symmetry is anaccident, and the accident which holds in the current configuration of the universe neednot hold during inflation; this would resolve the axion isocurvature problem. It wouldimply that θ is not a random variable.The inflationary paradigm is highly successful; the question is whether we can pro-vide some compelling microscopic framework and whether it is testable. In the presentproposal, one does not attempt (at least for now) a detailed microscopic understanding,but considers a class of theories. Within those considered here:1. Higher scales of inflation are preferred2. High scale axions are likely, and the idea of an axiverse gains additional plausability[36].In a more detailed picture, one might hope to connect some lower energy phenomenon,such as supersymmetry breaking, with inflation.13 cknowledgements: I thank my collaborators Patrick Draper, Guido Festuccia, LaurelStephenson-Haskins, and Lorenzo Ubaldi for the many insights they have shared withme. This work was supported in part by the U.S. Department of Energy grant numberDE-FG02-04ER41286.
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