Drifting Oscillations in Axion Monodromy
Raphael Flauger, Liam McAllister, Eva Silverstein, Alexander Westphal
SSU/ITP-14/29, SLAC-PUB-16165, DESY-14-225
Drifting oscillations in axion monodromy
Raphael Flauger, Liam McAllister, Eva Silverstein, , , and Alexander Westphal Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Physics, Cornell University, Ithaca, NY 14853, USA Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA 94025, USA Kavli Institute for Particle Astrophysics and Cosmology, Stanford, CA 94305, USA Deutsches Elektronen-Synchrotron DESY, Theory Group, D-22603 Hamburg, Germany
Abstract
We study the pattern of oscillations in the primordial power spectrum in axion monodromyinflation, accounting for drifts in the oscillation period that can be important for comparingto cosmological data. In these models the potential energy has a monomial form over asuper-Planckian field range, with superimposed modulations whose size is model-dependent.The amplitude and frequency of the modulations are set by the expectation values of modulifields. We show that during the course of inflation, the diminishing energy density can induceslow adjustments of the moduli, changing the modulations. We provide templates capturingthe effects of drifting moduli, as well as drifts arising in effective field theory models basedon softly broken discrete shift symmetries, and we estimate the precision required to detecta drifting period. A non-drifting template suffices over a wide range of parameters, but forthe highest frequencies of interest, or for sufficiently strong drift, it is necessary to includeparameters characterizing the change in frequency over the e-folds visible in the CMB. Weuse these templates to perform a preliminary search for drifting oscillations in a part of theparameter space in the Planck nominal mission data. a r X i v : . [ h e p - t h ] D ec ontents φ ) . . . . . 28 Introduction: Phenomenology of Axion Monodromy
Cosmic microwave background (CMB) and large-scale structure (LSS) observations are be-coming sensitive enough to constrain broad classes of inflationary models. Two importantsignatures for this purpose are primordial gravitational waves, imprinted in B-mode polar-ization, and the shapes of the power spectrum and non-Gaussian correlators. In this paper,we will be concerned with the search for oscillatory features in the scalar power spectrum.In axion monodromy scenarios [1, 2, 3, 4], a super-Planckian inflaton displacement resultsfrom repeated circuits of a sub-Planckian fundamental period. Gravitational waves are thendetectably large, while the tilt n s can take a range of values determined by the number offields [8] and the reheating scenario. Oscillatory contributions to the scalar power spectrumarise as a direct consequence of the underlying periodicity. However, the oscillatory featureshave a model-dependent amplitude, and may be undetectably small: in particular, the ampli-tude is exponentially suppressed in regimes where the oscillations are generated by instantoneffects. Nonetheless, searching for oscillations is well-motivated, because a discovery wouldimply a very interesting additional structure in the primordial perturbations, and converselya null result can constrain the parameter space in a useful way.A featureless, nearly scale-invariant ΛCDM power spectrum provides a very good fit tothe data, with values of χ / d . o . f typically just a few percent above unity. However, giventhe large number of degrees of freedom in datasets such as Planck , the total excess χ islarge enough that new physics in various forms could be hiding in the data, waiting to bediscovered. Oscillatory features in the power spectrum provide one example, and there mayalso be more general theoretical sources of excess variance, as in the scenarios of [9].Many searches for oscillations that are periodic in the canonical inflaton field have beenperformed [10], with no unambiguous detection. The goal of this paper is to point outthat the symmetry structure and dynamics of axion monodromy allow for — and oftenrequire — slow secular drift of the frequency of oscillations. One key source of this drift isbackreaction of the inflationary sector on other degrees of freedom, such as moduli scalars instring compactifications. Drifting oscillations can also be motivated from low energy effectivefield theory considerations [11], assuming a weakly broken discrete shift symmetry. We willanalyze the conditions under which the drift becomes large enough so that limits set witha constant-frequency oscillatory template are inaccurate, finding that for sufficiently highfrequencies or for sufficiently strong evolution of the period, a drifting-frequency template isrequired.The mechanism underlying monodromy, involving a weakly broken shift symmetry, is This mechanism can be regarded as an ultraviolet completion of large-field chaotic inflation [5], withadditional dynamics including some aspects of natural inflation [6]. As in those scenarios, there are interestinggeneralizations that involve multiple dynamical fields, with the possibility of distinct signatures [7, 8]. The potential in single-field axion monodromy inflation takes the general form V ( φ ) = V + µ − p φ p + Λ( φ ) cos (cid:16) γ + φ/f ( φ ) (cid:17) , (1.1)where V , µ , γ , and p are constants, and we have focused on the large-field regime of thepotential. The variable coupling f , which we will refer to as the axion decay function, reducesto the usual axion decay constant f when f is constant. Examples in the literature includecases with p = 3 , , / , , / f are functions of the modulifields, which can shift over time as the inflationary energy decreases. As we will explain, thiseffect is analogous to the flattening of the potential that arises as a result of shifts of modulivevs during inflation [12, 13]. Indeed, drift and flattening are two natural consequences ofthe adiabatic evolution of massive moduli. However, it is interesting to note that flattening,which determines the power p and the resulting slow-roll parameters, is generally independentof the drift in f : different degrees of freedom may adjust in the two cases, and moreover thefinal value of p depends on the fiducial value p , which is a separate parameter. So the driftin frequency is not determined purely by an expansion in the slow roll parameters of the non-oscillatory potential, even though the latter does make a model-independent contribution tothe drift that is calculable within effective field theory.In this work we will present templates that parameterize key features of ultraviolet-complete examples of axion monodromy inflation, without being so specific as to be tiedto a particular realization. We will provide one specialized template based on the patternof frequency drift deduced from simplified models of axion monodromy, along with a moregeneral Taylor series expansion that captures more general models, including [2, 3]. Weanticipate that these templates will have significant overlap with the predictions of a broadrange of models. This is similar in spirit to the use of simplified models of physics beyondthe standard model in the analysis of LHC data.We will examine oscillations with a drifting frequency in a number of toy models inorder to illustrate the spectrum of possibilities. We will scan over a range of the parametersentering the templates, including the parameter values exhibited by the models. For models Structural constraints analogous to those used in this work can be found in [15]. § § §
4, and the symmetrystructure of axion monodromy, in §
5. In § Planck nominal mission data. Our conclusions appearin § In the single-field version of axion monodromy inflation — or any similar mechanism ex-hibiting a softly broken discrete shift symmetry — one finds a potential for the canonicallynormalized inflation field φ of the form V ( φ ) = V ( φ ) + Λ( φ ) cos[ a ( φ )] . (2.1)Here a ( φ ) is the underlying periodic axion variable, which in general is a nonlinear functionof the canonical inflaton φ . Nontrivial dependence a ( φ ) can have multiple underlying causes,including backreaction of the inflationary energy on compactification moduli, as well as loopeffects derived from the weak explicit breaking of the discrete shift symmetry in V ( φ ). In(2.1) we have also allowed the amplitude Λ to depend on φ , but because amplitude driftis generally less important than frequency drift in the search for oscillatory features, ouranalysis will primarily focus on the drifting frequency encoded in a ( φ ).4 .1 Drift from backreaction on string moduli In string-theoretic models of axion monodromy, the leading contribution to the drift in periodcomes from the coupling of the axion to additional scalar fields σ I known as moduli . Themoduli fields include the size and shape of the extra dimensions of string theory, as wellas the coupling g s . Axion-moduli couplings can appear in both the kinetic and potentialterms, and their effect on the leading potential term V ( φ ) has been studied in [12, 13].These effects can become very complicated in general, but much has been learned from thebasic structures involved in existing moduli stabilization proposals. (In fact, allowing theinflationary potential to participate in moduli stabilization can simplify the latter [13, 15].)The nontrivial dependence a ( φ ) in (2.1) arises from the moduli-dependence of the axiondecay function f ( (cid:54) = constant) in the axion kinetic term. Because the moduli σ I in generaladjust during inflation as a result of their couplings to the axion potential terms, we canwrite (cid:90) d x √− gf [ σ I ( a )] ˙ a = (cid:90) d x √− g ˙ φ , (2.2)with φ the canonical field, given by the solution to dφda = f ( a ). The theory contains sectorsthat are periodic in the axion a ( φ ), leading to the oscillatory term in (2.1).The amplitude of these oscillations is highly model-dependent, and also in general de-pends on the moduli fields and hence on φ . Since there is a regime of parameters in which theleading oscillations arise from instanton effects, these can easily be exponentially suppressedas a function of the natural couplings of the theory. However, in high-scale inflation, thereare limits on how weak these couplings can be, leaving room for a detectable signal, as weexplore in more detail in § The comparison between models with oscillatory power spectra and data is commonly basedon a search for a given set of templates and constraints on their parameters [10]. To assessthe importance of the drifting period for comparison to cosmological data we must thus un-derstand whether a model with drifting period would have led to a detection in these searchesor might have been missed. For simplicity we consider an ideal power spectrum measure-ment for a Gaussian random variable such as the scalar perturbation ζ or the temperatureanisotropies a (cid:96)m . The likelihood for a theoretical power spectrum P k given a measurementˆ P k is L ( P | ˆ P ) ∝ exp (cid:34) − (cid:88) k N k (cid:32) ˆ P k P k + ln P k − N k − N k ln ˆ P k (cid:33)(cid:35) , (2.3)5here N k is the number of modes contributing to P k . The theoretical spectra are typicallytaken to be of the form P k = P k + AδP k , where P k is a smooth spectrum, and δP k is anoscillatory template that will depend on a number of parameters such as the frequency, thephase of the oscillations, etc. We have explicitly introduced one of them, the amplitude A .Oscillatory contributions to the angular power spectrum in the absence of a drift are typicallyconstrained at the few percent level, so we consider A (cid:28) A = (cid:80) k w k ( ˆ P k − P k ) δP k (cid:80) k (cid:48) w k (cid:48) δP k (cid:48) δP k (cid:48) , (2.4)where w k = N k / P k ) , so that the different contributions are inverse-variance weighted,and the variance for small A is ∆ A = 1 (cid:80) k (cid:48) w k (cid:48) δP k (cid:48) δP k (cid:48) . (2.5)Let us now assume that the true power spectrum contains oscillations, but of a shape differentfrom the template δP k , so that (cid:104) ˆ P k (cid:105) = P k + δP true k . (2.6)For this power spectrum, the expected amplitude is (cid:104) ˆ A (cid:105) = (cid:80) k w k δP true k δP k (cid:80) k (cid:48) w k (cid:48) δP k (cid:48) δP k (cid:48) , (2.7)Using equations (2.5) and (2.7), and introducing the notation δP · δP (cid:48) = (cid:80) k w k δP k δP (cid:48) k , wecan write the signal-to-noise ratio for our template as SN = δP · δP true √ δP · δP = δP · δP true √ δP · δP √ δP true · δP true √ δP true · δP true . (2.8)Thus, the signal-to-noise ratio we expect for a given template is the product of the signal-to-noise ratio for the correct template times the overlap, or cosine, between the two templates (cid:18) SN (cid:19) template = cos( δP, δP true ) × (cid:18) SN (cid:19) true with cos( δP , δP ) = δP · δP √ δP · δP √ δP · δP . (2.9)In the approximation in [3], the power spectrum for the potential (2.1) is P ( k ) = P ( k (cid:63) ) (cid:18) kk (cid:63) (cid:19) n s − (1 + δn s ( φ ) cos[ a ( φ k )]) , (2.10)with φ k the value of the scalar field at the time the mode with comoving momentum k exitsthe horizon. For a CMB power spectrum measurement the number of modes contributing6o a given multipole is 2 (cid:96) + 1. To define an inner product on primordial power spectra thatapproximates the inner product on angular power spectra, we choose N k = k . For a powerspectrum of the form (2.10), and ignoring the slow variation of the amplitude, the innerproduct is δP i · δP j ∝ k max (cid:90) k min k dk cos[ a i ( φ k )] cos[ a j ( φ k )] , (2.11)To estimate the level of precision we should require of our templates, let us write a ( φ ) ∼ φ/f ( φ ) and perform a Taylor expansion of f about some point φ in field space.cos[ a ( φ )] = cos (cid:34) φf × (cid:32) φ f dfdφ (cid:12)(cid:12)(cid:12) φ (cid:18) φ − φ φ (cid:19) + 12 φ f d fdφ (cid:12)(cid:12)(cid:12) φ (cid:18) φ − φ φ (cid:19) + . . . (cid:33) − (cid:35) . (2.12)In axion monodromy scenarios in which V ( φ ) is approximately monomial, V ( φ ) ≈ µ − p φ p ,we find that φ is of order 10 M p when the modes observed in the CMB exit the horizon, andthe change in the field is ∆ φ CMB ∼ M p during this time. The expansion (2.12) is thereforeorganized as a power series in | φ − φ | /φ (cid:46) /
10 with coefficients a n = n ! φ n f − d n fdφ n | φ . Wewill find below that based on simple estimates of the effect of backreaction on moduli, thecoefficients a n can easily be of order unity. For example, we will see that there are modelsin which f ( φ ) = f (cid:18) φφ (cid:19) − p f , (2.13)with a power p f of order unity.Under the assumptions made in [3], the oscillatory contribution to the power spectrum isobtained from equation (2.12) by replacing φ by φ k ≈ (cid:112) p ( N (cid:63) − ln k/k (cid:63) ). We will discuss in § k min = 10 − Mpc − , k max = 10 − Mpc − , and k (cid:63) = 0 . − , we find that for afrequency close to the WMAP9 best-fit value f /M p ∼ × − , overlaps as large as 80%can be achieved between the template for constant axion decay function and the templatewith the coefficient of the first correction being of order unity. However, the overlap drops This definition of f ( φ ) will be convenient, even though it differs from the definition obtained from thekinetic term by solving f da = dφ . Although we will focus on the single-field version of axion monodromy for simplicity in the present work,the multifield version of the mechanism [7, 8] involves smaller individual field ranges. f /M p ∼ × − we find overlapsof around 50%, and for the lower end of the range of axion decay constants studied here, f /M p ∼ − , the overlap is further reduced to only a few percent. We conclude that atleast two nontrivial terms in the Taylor expansion (2.12) of the axion decay function arenecessary to achieve an order unity overlap for frequencies f /M p (cid:46) × − . Consequently,previous searches do not cover the entire space of models with drifting frequencies.We should note that there is not always a useful description in terms of a Taylor expan-sion. For example, this expansion breaks down for large p f . This is another regime in whichone cannot rely on a non-drifting template. With such a strong drift in f , one would needto estimate radiative corrections to the slow roll parameters, which we estimate in § V ( φ ), but at some point the oscillations would then become too small to be detectable. The derivation in [3] used above is only accurate to leading order in the slow roll parameters.Given the monomial large-field form of the leading potential V ( φ ) ∼ µ − p φ p , the slow rollparameters η and (cid:15) are of order ( M p /φ ) ∼ − . Let us estimate their importance in thepower spectrum by considering corrections to the argument a ( φ ) of the cosine of the formcos (cid:34) φ k f (cid:32) (cid:88) c n (cid:18) M p φ k (cid:19) n (cid:33)(cid:35) . (2.14)Evaluating the overlaps between templates with coefficients c n of order unity suggests thatboth the n = 1 and n = 2 terms in this expansion are important. This should not come asa surprise. In order to obtain a good approximation to the oscillatory features in the powerspectrum, one must keep correction terms in ∆ a ∼ φ k f (cid:80) c n (cid:16) M p φ k (cid:17) n that can be of order 2 π or bigger.Extending the analytical derivation by two orders in the slow-roll expansion is beyondthe scope of this paper. For our numerical analyses, we will therefore check that a giventemplate agrees with a numerical calculation. To estimate whether the low-energy effective theory is radiatively stable even when theTaylor expansion breaks down, let us consider the case (2.13) with | p f | ∼ O (10). Wechange variables back to the axion a = a ( φ/φ ) p f , with dφ = ˆ f ( a ) da ⇒ ˆ f ( a ) ≡ ˆ f = φ /a (1 + p f ). Starting from a typical axion monodromy potential V ( φ ) ≈ µ − p φ p , we can8rite L = 12 ( ∂φ ) − µ − p φ p − Λ cos (cid:34) γ + φ f (cid:18) φφ (cid:19) p f +1 (cid:35) (2.15)= 12 (cid:18) aa (cid:19) − p f / (1+ p f ) ( ˆ f ( a ) ∂a ) − ˆ µ a p/ ( p f +1) − Λ cos[ a ] , (2.16)where for simplicity we here neglect any drift in the amplitude Λ .Each form of the action makes manifest a different limiting symmetry. As µ → →
0, the first form (2.15) exhibits a continuous shift symmetry in φ . The second form(2.16) makes clear that the oscillatory term by itself respects the discrete shift symmetry a → a + 2 π . These symmetries constrain the form of the corrections.One can estimate corrections using the first form of the action, starting from the one-loopColeman-Weinberg effective potential. Taking into account the leading contributions fromthis, the essential effect of p f is to generalize the expansion in Λ / (4 πf ) to an expansion in( p f + 1)Λ / (4 πf ). We will now explore this using the second form of the action (2.16).Our basic question is whether the strong drift parameterized by p f ∼
10 generates cor-rections that ruin slow roll inflation. To assess this, we can focus on the interactions comingfrom the kinetic plus oscillatory terms, which we will now delineate.In the expansion of the kinetic term, the Lagrangian (2.16) exhibits a sequence of higherdimension operators, with coefficients that depend on our parameter of interest, p f . Ex-panding this (defining a = a + δa ), we find12 ( ∂ [ ˆ f ( a ) δa ]) (cid:32) − ˆ f ( a ) δa (cid:40) f ( a ) a (cid:18) p f p f + 1 (cid:19)(cid:41) + . . . (cid:33) . (2.17)The dimension-one fluctuating field in this Lagrangian is Y ≡ ˆ f ( a ) δa . The higher-dimensionoperators arising from this expansion of the kinetic term are suppressed by the mass scale M ≡ | ˆ f ( a ) a | = φ / | p f | ∼ | p f | M p . (2.18)If for example p f ∼
10, then this suppression scale is still very high, M ∼ M p . In particular,it is much higher than the strong coupling scale ∼ π ˆ f of the low energy theory. We willassume that the low energy theory is effectively cut off at a scale Λ UV (cid:28) ˆ f , and estimate theradiative stability of slow roll inflation against the loop corrections generated by the inter-actions in (2.15), with the loop momenta cut off at Λ UV . It would be interesting to analyzethis in the context of field theoretic (as well as string-theoretic) ultraviolet completions thatdescribe the degrees of freedom at and above the scale 4 π ˆ f .9his latter scale is evident if we expand the oscillatory term:Λ cos( a ) = Λ cos (cid:18) a + Y ˆ f (cid:19) = Λ (cid:88) c j (cid:18) Y ˆ f (cid:19) j . (2.19)However, note that the background solution for Y ranges over a distance in field space muchgreater than the underlying period ˆ f , so one cannot use the last form expanded about Y = 0 for the whole process. In the original cosine form, it is clear that the magnitude ofthe interaction term is bounded by Λ . (Consider the one-loop Coleman-Weinberg potentialTr[log( ∂ + V (cid:48)(cid:48) ( ϕ ))] in scalar field theory: for our sinusoidal potential V , one has V (cid:48)(cid:48) = − (Λ / ˆ f ) cos( a + Y / ˆ f ), which gives smaller corrections than would be estimated from theindividual terms in the series (2.19).)Let us first estimate an upper bound on the corrections to the potential term V ( a ) andtheir effect on the slow roll parameters, assuming for this discussion that the kinetic termis not corrected in an important way. As we just noted, by symmetry the only potentiallyrelevant corrections must use a combination of the interaction vertices from the kineticterm and the oscillatory term. Some corrections contribute to the oscillatory term in thepotential, proportional to cos( a ) and sin( a ) (with mildly drifting amplitude); we find these arecontrollably small. Others are proportional to even powers of cos( a ) and sin( a ), and introducecorrections to the non-oscillatory potential V ( φ ). The corresponding leading contributionto the slow roll parameter (cid:15) V = M p ( V (cid:48) /V ) arises from∆ V (cid:48) = dVdφ = ˆ f ˆ f dVdY ≤ ˆ f ˆ f M Λ ˆ f . (2.20)Here in the second and third factors we introduced a vertex from the kinetic term andtwo from the cosine term, with the understanding that higher loop contributions will besuppressed by additional powers of Λ UV / ˆ f and Λ UV /M . As discussed above, Λ UV is thescale at which the loop is effectively cut off. We write (2.20) as an inequality because theremay be additional cancellations that we are not working out explicitly. The inequality (2.20)translates into ∆ (cid:15) V < (cid:18) Λ V (cid:19) (cid:32) Λ ˆ f min (cid:33) (cid:18) M p M (cid:19) (cid:32) ˆ f ˆ f min (cid:33) (2.21)where we used the fact that dφ/da = ˆ f . To get the most conservative bound, we take ˆ f at itsminimum value, ˆ f min , within the range of field visible in the CMB. The factor ( M p /M ) is oforder 1. The last factor is (cid:46)
1, as the series of higher order terms should resum into a cosine-type dependence in each diagram. Using ˆ f ∝ ( φ/φ ) − p f , we find that the penultimate factoris ∼ (5 / p f (cid:46) /V . 10ext, let us estimate the corrections to the kinetic term. Consider the 1PI 1-loop diagramwith a three-point vertex from the cosine term and a four-point vertex from the kinetic term.This gives a correction to the ( Y /M )( ∂Y ) term in the Lagrangian (with one ∂Y leg fromthe cosine-term vertex and the other Y ( ∂Y ) from the four-point vertex). This diagram isof order (1 /M )(Λ / ˆ f ). So using M ∼ M p , it is down from the leading term of this form ∼ ( Y /M )( ∂Y ) from (2.17) by a factor of Λ /M p ( M p / ˆ f ) . Now since f /M p can range downto 10 − , this correction becomes important only if Λ (cid:38) − M p , which exceeds the existingobservational upper bound (4.9). Again, higher loops will have extra suppression by powersof Λ UV / ˆ f and Λ UV /M . Diagrams with more external legs from the vertices obtained fromthe expansion of the cosine term (2.19) will only contribute terms ∼ Y n ( ∂Y ) over a smallrange (cid:28) ˆ f in Y .As a result, there appears to be a regime of parameters in which the Taylor series tem-plate fails, but the shift-symmetry breaking from the drift is not so large as to producesubstantial corrections to the underlying inflationary slow roll mechanism. This regime isthe most challenging one for the problem of searching for oscillations. It requires knowing thefunctional form of the drift in period, without the benefit of a convergent Taylor expansion.The example we just gave illustrates this possibility. In this section we explore the form of the function a ( φ ) in a few classes of prototypicalexamples of moduli stabilization and axion monodromy inflation in string theory. In § § f . Axions a I arise from higher dimensional analogues of electromagnetic potential fields A ( q ) , A ( q ) = (cid:88) a I ω I ( q ) , (3.1)where the ω I ( q ) are a basis for the cohomology H q ( X ) of the compactification manifold X .There are various duals or analogues of axions under electromagnetic and target space stringdualities, in some of which the axion becomes a brane collective coordinate or a geometricmodulus. In fact, the possibility of monodromy inflation was discovered in the context ofsuch examples [1] (cf. [16]), and interesting mechanisms of that kind continue to appear [17].These duals have similar drifts in their oscillatory periods, so for simplicity we will focus onaxions that descend directly from higher dimensional potential fields.The kinetic term (cid:82) | dA | for A descends to kinetic terms for a , which generally depend on11he moduli fields as indicated in (2.2). The potential descends from gauge-invariant termsin the D -dimensional effective action of the schematic form (cid:90) d D x √− G (cid:88) q | ˜ F q | = (cid:90) d D x √− G (cid:88) q | F q − C q − ∧ H + · · · + F ˜ q B ∧ · · · ∧ B | , (3.2)in terms of potential fields A = { C n , B } and their field strengths dA = { F n +1 , H } . Thecombinations ˜ F q are known as generalized fluxes. The couplings in (3.2) generalize theStueckelberg term (cid:82) | ∂θ − A | in spontaneously broken electromagnetism. In the presence ofgeneric background fluxes, these terms contain direct dependence on A , which descends toa monodromy-unwound potential for a I . Just as in the kinetic terms, these potential termsgenerally depend on the moduli σ I . Correspondingly, the potential energy V ( a ) of the axionfields a I must be incorporated in a consistent way in the metastabilization of the moduli:either V ( a ) must be subdominant to the remainder of the moduli potential V mod , or elsethe forces encoded in V ( a ) must be balanced against additional contributions in V mod . Thelatter case — which may be more generic because it requires less of a hierarchy of scales inthe problem — leads to significant drift in the oscillation period f ( φ ), as well as flatteningof the potential [12, 13]. In this case the role of the axion potential energy in the overallscalar potential is analogous to the contribution of an ordinary flux. These features havebeen reviewed recently in [13, 18].In some circumstances, such as in the early examples [2, 3], the most direct descriptionof the monodromy is that it is induced by a brane’s DBI action. One can use the gravitysolution for the brane, as in AdS/CFT, to relate the brane and flux pictures [12]. We willconsider both cases in our explorations below. We begin by discussing scenarios in which moduli such as the volume V , the string coupling g s , and the axion decay function f are stabilized by power-law effects. By this we mean thatthe forces fixing these fields are powers of V , g s , and f , such as those arising from classicalstress-energy sources (including curvature) in string compactifications. This type of modelwas studied recently in [13], and here we extract some lessons of that work for the questionof drifting oscillations. In other scenarios, such as [19] and [20], moduli stabilization resultsfrom a competition between perturbative and nonperturbative effects. We explore that casein § See [18] and references therein for pedagogical introductions to this approach to moduli stabilization. V = a V x γ a − b V x γ b + c V x γ c , (3.3)with x ∼ x e c x σ x /M p , where σ x is the canonically normalized field and c x a constant. Metasta-bility requires a sufficiently strong intermediate negative term; for example for the simplecase γ a = 2 , γ b = 3 , γ c = 4 we require 1 < acb <
98 (3.4)for a metastable minimum.In other directions, such as those describing flux-stabilization of the relative sizes of dualcycles in the manifold, one finds a simpler two-term structure of the schematic form V = A V y δ A + B V y δ B , (3.5)with again y ∼ y e c y σ y /M p . The prototype for this is a manifold with a product structure M × M with flux threading the two orthogonal directions M and M : in each direction,the flux prevents collapse of the cycle, leading to a potential of the form (3.5) for the ratioof the cycle sizes.In general, the coefficients a V , . . . , B V can depend directly on axions, depending on thedetails of the example. In particular, we can see immediately from the terms (3.2) that therole of fluxes is more generally played by combinations of fluxes and axions. Since at largefield values (or with many axion fields [7]), the axion potential in itself satisfies the slowroll conditions, it can be consistent for the axion-dominated terms to participate in modulistabilization [15, 13].In three-term stabilization, however, there is a significant limitation on the contributionsof axions to the scalar potential. For a single axion that evolves over a range of order 10 M p — implying many circuits of its underlying period — the condition (3.4) can easily fail byvirtue of too large a change in 4 ac/b over this large field range. In two-term stabilization,in contrast, it is straightforward for axions to participate much like fluxes.
Let us next describe an illustrative pair of examples drawn from [13], before attempting todraw some more general lessons that will motivate the oscillatory templates and parameter If many axions each move over a very small range, this problem can be ameliorated. g s and internal volume V with a three-termstructure, and another combination with a leading contribution from the axion b driving in-flation. Let us work from equation (4.8) of [13], giving the effective potential after three-termstabilization of a combination x = g s / V / (with V the volume in string units) U | x = x min ∼ M p (cid:26) C h n V / + C h ( q + q b ) V / + C h ( q + 2 q q b + q b ) (cid:27) , (3.6)where C h is a constant depending on the parameters of the underlying model, n is a quantumnumber corresponding to H flux, and the q i similarly correspond to Ramond-Ramond fluxes,as described in [13]. This potential is valid as long as 2 q q b + q b ≤ O ( q ).The first two terms here stabilize the volume V . In the case where q b (cid:29) q , we obtain b ∝ / V / (3.7)from this. The relation (3.7) has two effects: first, it introduces a linear term in b in thepotential, after substituting (3.7) in (3.6). Second, it changes the relation between b and thecanonically normalized inflaton field φ b because of the V dependence in the b kinetic term:˙ b V / ∝ ˙ b b ⇒ φ b ∝ b / . (3.8)This implies a drift in f given by f ∝ V / ∝ b / ∝ φ / . (3.9)Therefore the oscillatory term in the potential takes the formΛ( φ ) cos (cid:34) γ + φ f (cid:18) φφ (cid:19) p f (cid:35) , p f = − / . (3.10)where φ is the field value at a pivot scale, e.g. at the start of the observable stage of inflation.In the framework [13] in which (3.6) applies, the leading term V ( φ ) is of the form V ( φ ) = µ − p φ p , p = 4 / p = 2 / , (3.11)with p = 4 / q q b dominates in the potential, and p = 2 / q b V / dominates in the potential.The amplitude of oscillations Λ( φ ) is a very model-dependent quantity, one that asemphasized above can easily be undetectably small. In the examples defined by (3.10),(3.11), we can estimate their size as given by worldsheet instanton effects [21] derived from14rocesses in which the string wraps a two-cycle of the internal manifold. Contributions tothe potential from worldsheet instantons are exponentially suppressed in V / , suggesting anamplitude Λ( φ ) = Λ e −V / (cid:16) φφ (cid:17) p Λ , p Λ = − / . (3.12)where V is the volume in string units evaluated at the pivot scale. The amplitude (3.12)is exponentially suppressed in an internal two-cycle volume, something sufficiently model-dependent that it prevents us from obtaining a model-independent signature from the oscil-lations in axion monodromy. But for high-scale inflation, the size of the extra dimensions isnaturally at the inverse GUT scale or below, and the amplitude depends on details such asflux quantum numbers and topology. There is a range of amplitudes of interest, includingboth a regime where the effect is large enough to be detectable as well as a regime where it istoo suppressed to be seen. In section § The examples just discussed motivate a structure for the inflaton potential of the form V ≈ µ − p φ p + Λ e − C (cid:16) φφ (cid:17) p Λ cos (cid:34) γ + φ f (cid:18) φφ (cid:19) p f +1 (cid:35) , (3.13)with powers p, p Λ , p f that may take more general values than those found in the abovescenario. To search for drifting oscillations in cosmological data, we also require an under-standing of the range over which these parameters can vary. The drift in frequency arisingfrom p f is more important to keep track of than the amplitude (related to p Λ , C here) orthe underlying slow-roll potential (related to p ), and we will on a first pass set C = 0and p Λ = 0. The allowed range of axion decay constants f will be discussed in the nextsubsection. To get an idea of the potential range of the parameter p f , let us focus on theexample of an axion b descending from the Neveu-Schwarz two-form potential B that couplesto the fundamental string. In a simple model with one length scale L (in units of the stringtension), the kinetic term (cid:82) | dB | descends to a kinetic term of the form (2.2) with fM p ∼ L , (3.14)as is reviewed for example in [18].As in the above examples, the axion potential energy may contribute to stabilization of L . In a power-law mechanism for moduli stabilization based on the Stueckelberg-like terms153.2), and/or on forces from branes affecting the size L , the result is a relation of the form L = L b p β . (3.15)From this we derive the relation with the canonical field φf db = dφ ⇒ b = ( L (1 − p β ) φ ) / (1 − p β ) . (3.16)This translates into p f = 2 p β − p β , (3.17)and we would like to understand the range of values this parameter might take in moregeneral models realizing axion monodromy inflation.One basic question is whether only one sign of p f is possible. We will show below that bothsigns are possible, with simple mechanisms for each case. In what we will describe next, thetwo signs arise in somewhat different versions of monodromy inflation. As discussed above,although flux-axion couplings can be understood as the source of the monodromy effect foraxions descending from potential fields in string theory, in some cases the most effectivedescription is through a brane action, while in others the description in terms of fluxes issimplest. In the following, we will find scenarios with p f > p f < p f = − / § Brane-induced monodromy
Let us start by noting that power law stabilization with brane-induced monodromy im-mediately gives p f >
0. The simplest way to see this is to work in a duality frame in which wehave a pair of branes moving relative to each other with collective coordinate x = ˜ Lθ arounda circle of size ˜ L √ α (cid:48) , with another brane stretched between them (for example, D4-branesstretching between NS5-branes). This forms a configuration like a windup toy (see e.g. figure1 in [2]). As the toy unwinds, going to smaller values of φ , the circle grows larger because itis less constricted by the brane, and hence the period f grows. Thus f (cid:48) <
0, correspondingto p f > M p ˜ L ˙ θ , and the potential term is of the form θ ˜ LC + C / ˜ L γ , where wehypothesize a second term that stabilizes ˜ L in combination with the brane system. Thisgives an axion decay function f ∝ ˜ L / . We have ˜ L ∝ θ − /γ +1 . Changing to canonicalvariables we find V ∝ φ γ/ (2 γ +1) , f ∝ φ − / (2 γ +1) . (3.18)In other words, this type of toy example gives p f = 2 / (2 γ + 1) > We will likewise find a growing period in the example of axion monodromy inflation induced by anNS5-brane pair in a nonperturbatively stabilized compactification: see § lux-induced monodromy Next, let us explore the behavior of p f in more general models in which fluxes inducemonodromy. A reader interested only in a broad view could skip this discussion, as itmainly serves to motivate a range of values of p f in a way that generalizes the example with p f = − / p β = 1 / b ∼ e φ/M p (up to factors of order unity), anddoes not give rise to an inflationary model of interest for the present discussion. Excludingthis case, there are three possible regimes:(1) p β < − < p f < p β = 0 p f = 0 constant f (3) p β > < p β < / < p f < ∞ p > p β > / p f < − p < V ∝ /φ | p | Here ‘kinetic flattening’ and ‘kinetic steepening’ refer to the effect of the transformation(3.16) on the power p in V ( φ ) = µ − p φ p .The examples in the previous section had p f = − /
3, falling into class (1). More gen-erally, as explained in § φ and b (3.16). This would leadto p <
0, making the potential proportional to a negative power of the inflaton field. Wewill call this possibility ‘kinetic inversion’. This is not viable phenomenologically accordingto current constraints on the tensor to scalar ratio r and the tilt n s of the scalar powerspectrum. This provides another illustration of the importance of taking into account thebackreaction on moduli as a function of the inflaton vev.From the relation (3.15) one might think that kinetic inversion or steepening (i.e., p β >
0, regardless of whether p β < / p β > /
2) might readily arise in some classes ofperturbative stabilization, based on the idea that a larger inflationary energy creates a largerdecompactification tadpole. However, there are two problems with this argument. First, aswe saw in the example (3.6) (3.7), after stabilizing an appropriate combination x = g s L K/ (3.19) Although it is not viable for early universe inflation, this case may be interesting if it can provide a novelmechanism for generating accelerated expansion in string theory.
V ∼ L D − toward smaller values once x = g s /L is stabilized in its minimum. In addition, the relation between b and L is not pure monomialwhen b is a subleading effect in the stabilization, which is an important distinction. Inparticular, in three-term stabilization mechanisms (at least without large relative powers inthe three terms), a single large-field axion cannot dominate, as we explained above.The remaining possibility for phenomenologically viable inflation with kinetic steepeningis case (3a). However, in the single-scale scenarios described here, known mechanisms forperturbative stabilization do not appear to generate case (3a). Recall, as briefly reviewedabove, that fluxes — and their generalizations ˜ F q depending directly on axion fields (3.2)— can stabilize pairs of dual cycles. However, if we were to balance Ramond-Ramondgeneralized flux terms in stabilizing L , there is a simple restriction. Within a generalizedflux, the two-form potential field B acts as a flux of rank 2 on the two-cycle Σ that itthreads. To balance this against some other generalized RR flux term, the latter has to beof lower rank on Σ . Balancing the two terms then gives b n ∼ L n − ∆ , with ∆ ≥
0. Turningthis around, we can write L ∝ b n/ (2 n − ∆) , implying that p β as defined above in (3.15) satisfies p β ≥ /
2. As a result, stabilization based on balancing the energies in generalized fluxes doesnot populate the range 0 < p β < /
2, but could in principle give an example of the sort (3b),i.e. with p <
0, which would be excluded by constraints on the primordial perturbations.Alternatively, let us consider the possibility that the RR generalized flux instead balancesagainst another type of term, for example the term descending from the magnetic flux ofthe internal B field | dB | = H , or the potential energy descending from the curvature ofthe extra dimensions. With some combination (3.19) stabilized in a three-term structure,we would have the scaling Q b n L n ∼ x L n ⇒ p β = 12 − (2˜ n + K ) / n . (3.20)Here we have used the fact that the Ramond-Ramond terms come with a relative factorof g s compared to the H and curvature terms. We know that ˜ n ≥ K <
0. Thus, presently knownconstructions of this sort of flux stabilization do not populate the range 0 < p β < / p f > n + K we have p β < p f = − / p f > p f is p = (1 + p f ) p V , (3.21)where p V is the power computed by incorporating the backreaction of the inflationary en-ergy on the moduli potential, and the corresponding flattening, before taking into accounta nontrivial power from the kinetic term (3.16). In other words, p V is the power that incor-porates flattening from the potential, and p is the final power that also incorporates kineticflattening. If p V is a fixed positive number, then through (3.21), observational limits on theprimordial perturbations imply an upper limit on p f . However, strong potential-flatteningeffects are a logical possibility: p V could in principle be very small to compensate a largevalue of p f . That is, one could have strong potential flattening combined with strong kineticsteepening. It would be interesting to analyze more general axion monodromy scenarios inorder to characterize this possibility. In extreme limits — such as large total dimensionality D — one can obtain large fiducial powers in the higher-dimensional Lagrangian, and it wouldbe worthwhile to characterize the ensuing powers p, p f , p Λ in those cases [15].Let us recap the pattern we have discerned from this exploration. The ‘kinetic inversion’case (3b), p β > /
2, which leads to a monomial with a negative power, is observationallyunviable. The marginal case p β = 1 /
2, which leads to an exponential potential, is alsounviable. The kinetic steepening case 0 < p β < / p V (3.21) appearing in the potential expressed in terms of the basicvariable b . In other words, viability depends on how steep the potential starts out, and howmuch further it gets steepened. However, as just explained, at present we have no examples ofmechanisms that would produce examples with 0 < p β < /
2. Next, examples with p β < p f <
0, arise quite readily in power-law stabilization scenarios, e.g. in theexample of § p f > p f could be used to distinguishthe two classes of constructions described above, or if instead our current exploration is toosimplistic for such distinctions. In this section we will provide simple estimates of the range of frequencies of interest in simplepower law scenarios. At sufficiently high values of f /M p , oscillations become degenerate withparameters determining the overall amplitude and tilt of the power spectrum. From this,one might find interesting effects related to the low- (cid:96) behavior of the power spectrum asstudied previously [10]; however, the statistical significance at this point is low. For stringcompactifications involving single axions, one typically finds f (cid:46) M p , with theoreticallycontrolled examples satisfying f /M p (cid:46) − . In the case of multi-axion mechanisms, suchas [7], larger values of f /M p are possible. 19he lower end of the range of f /M p is of particular interest. One threshold that arises isthe value of f at which a single-field effective field theory description breaks down. Imposingthat the frequency in time of oscillations be less than the unitarity scale ∼ πf leads to alower bound on f /M p of order 10 − . The full theory may continue to produce oscillatoryfeatures beyond this scale, but their functional form becomes more difficult to calculate. Apractical threshold arises from the resolution available in the data.Here we will make a similar rough estimate of the lower end of the range of f /M p fromthe string-theoretic point of view. We will start with the case of single-scale power law sta-bilization, with the axion b derived from the Neveu-Schwarz two-form potential B discussedabove. After making that estimate, we will generalize to scenarios in which the axion mightarise from higher-rank gauge potentials.For large-field axion monodromy inflation we have (cf. eqn (1.38) of [22]) H = 3 × − (cid:114) r . M p ⇒ HM p (cid:38) − . (3.22)In string compactifications from D to 4 dimensions, the four-dimensional Planck mass isgiven by M p ∼ (2 πL √ α (cid:48) ) D − g s κ , (3.23)where L is the size of the extra dimensions in string units, 1 /α (cid:48) is the scale of the stringtension, and g s κ is the higher-dimensional Newton’s constant. For example in D = 10, towhich we will specialize, 2 κ = (2 π ) α (cid:48) .If we go to the boundary of control, imposing only that H √ α (cid:48) ≤ /L we can write fM p ∼ L ≥ H α (cid:48) ∼ L ( H/M p ) × (1 /g s ) . (3.24)Again we may go to the marginal case of g s → bound. Also taking H ∼ − M p , (3.24) gives roughly L (cid:46) ⇒ f (cid:38) − / M p .However, we find a larger window if we allow for the possibility of axions from higher-rankgauge potentials. In that case, we have f /M p ∼ L − r a , where r a is the rank of the potentialfield. The most extreme case would be a potential of rank D −
4. In this case, for D = 10the inequality (3.24) gives f /M p (cid:38) − .Altogether, our top-down theoretical priors are not very informative about the lower endof the range of f /M p . As such, it is well motivated to scan down at least to the unitarityscale, based on EFT reasoning, or even more agnostically to scan down to the resolutionavailable in the data. By conservative here we mean avoiding imposing potentially overly strong theoretical priors that areconsequences of our computational limitations. .2 Nonperturbative stabilization In some cases, the moduli whose adjustments during inflation cause drift in the period maybe stabilized via a balance of nonperturbative and perturbative terms in the potential. Inthis section, we will explore a few such scenarios. They produce a more general functionalform of the drift f ( φ ) than that obtained in (3.13); as such, these scenarios require use ofthe broader template (2.12), keeping more parameters in the Taylor expansion. We will begin by examining a specific realization of axion monodromy in a nonperturbatively-stabilized compactification of type IIB string theory [2]. In this scenario the dimensionlessaxion corresponding to the inflaton is given by a ≡ (cid:90) Σ c C , (3.25)where C is the Ramond-Ramond two-form potential, and Σ c is a homology class correspond-ing to a family of two-cycles. Situating the family Σ c in a warped region and wrapping anNS5-brane and an anti-NS5-brane on well-separated representatives of Σ c introduces a mon-odromy that weakly breaks the shift symmetry of a . The resulting potential is asymptoticallylinear, V = µ φ , with φ = af , where f is given by f M p = g s π c αcc v α V . (3.26)Here v α , α = 1 , . . . h , , are the volumes of two-cycles in the compactification; V is the totalsix-volume; and c αcc are triple intersection numbers. See [3], whose conventions we adopthere, for more detailed background.The structures required for NS5-brane monodromy are compatible with moduli stabiliza-tion in the KKLT [19] scenario, in which K¨ahler moduli are stabilized by nonperturbativesuperpotential terms from Euclidean D3-branes and/or from gaugino condensation on D7-branes. The decay constant is determined by a specific combination of two-cycle volumes c αcc v α , and by the overall volume V . The inflaton potential depends on V via µ = const. × V − . (3.27)This is an example of the universal decompactification instability caused by positive energysources in four dimensions. As a result, V couples strongly to the changing inflationaryenergy, diminishing as inflation proceeds, and we can estimate the shift in f by examiningthe shift in V . 21e will consider a toy model that is a variant of the two-modulus example given in [2].To capture the shift of V , it suffices to work in a simplified model with a single K¨ahlermodulus T . We take W = W + A e − aT , K = − T + ¯ T ) , (3.28)with W = 3 × − , a = π , A = −
1, as in [2]. This theory has a supersymmetric
AdS minimum at σ ≡ Re( T ) ≈
21, and can be uplifted to a metastable de Sitter vacuum throughthe addition of supersymmetry-breaking vacuum energy, e.g. from an anti-D3-brane. Uponfurther including a sufficiently warped NS5-brane/anti-NS5-brane pair, axion monodromyinflation can occur without causing decompactification. However, for sufficiently large in-flationary energy µ φ = V crit , as will arise if the warp factor approaches unity, the quasi-deSitter minimum disappears and decompactification ensues. The maximum drift of the fre-quency of oscillations occurs in a model that is almost unstable to decompactification at thestart of inflation: the shift of V is then the largest possible change compatible with K¨ahlermoduli stabilization.The location σ min of the minimum, as a function of φ , is accurately captured by a second-order Taylor expansion around a point φ in field space, σ (cid:63) ( φ ) ≈ σ min ( φ ) + σ (cid:48) | φ ( φ − φ ) + 12 σ (cid:48)(cid:48) | φ ( φ − φ ) . (3.29)If the volume V is taken to be almost unstable at the start of inflation, and φ is taken to benear the point where the pivot scale exits the horizon, then for the model parameters quotedabove, one finds σ (cid:48) | φ (cid:46) . σ (cid:48)(cid:48) | φ (cid:46) .
1. These are (approximate) upper limits: if therelative sizes of the moduli potential and the inflationary energy are such that V is robustlystabilized at the start of inflation, considerably smaller shifts are possible. Next, expanding f as in (2.12), f ( φ ) /f ≈ f dfdφ (cid:12)(cid:12)(cid:12) φ ( φ − φ ) + 12 f d fdφ (cid:12)(cid:12)(cid:12) φ ( φ − φ ) , (3.30)and using (3.26), we find that for the same assumptions made above, f dfdφ | φ ∼ − and f d fdφ | φ ∼ − . One can then check that for the parameters of this toy model, ∆ f /f (cid:46) . f this effect may be observable.We conclude that in realizations of NS5-brane axion monodromy in KKLT compactifi-cations with realistic parameter values, the shifts of the nonperturbatively-stabilized K¨ahler The shift of the two-cycle whose volume is denoted v + in [2] induces a subleading correction to the shiftin f , for the parameter values we will consider. A K¨ahler moduli space of dimension > h , = 1, as we do here. We now point out that in some classes of nonperturbatively-stabilized compactifications,there is a logarithmic correction to the drift in frequency. Very schematically, in terms ofone relevant modulus L , nonperturbative stabilization can lead to a relation of the form µ V e − L n ∼ µ a a p L γ , (3.31)with µ a / ( µ V L γ ) ≡ (cid:15) a L − γ (cid:28) This translatesinto L n ∼ − log( (cid:15) a a p /L γ ) ∼ − log( (cid:15) a a p ) . (3.32)Combining (3.32) with f = M p L − n f , we find that the sinusoidal dependence is of the formcos (cid:34) b φ (cid:16) − log( (cid:15) a ) − p log( φ/φ ) (cid:17) n f /n (cid:35) , (3.33)where b is a dimensionful constant. Here we have used the fact that at least in a simplesituation with a single overall scale, we have a p /L γ ∼ ( φ/M p ) p .In (3.33) the exponent n f /n is positive, and might be expected to range over a fewdiscrete rational values, roughly of order unity. The presence of the log factor inside thecosine changes the template considerably, leading to a qualitatively different (cid:96) -dependenceof the oscillations.
To explore possible relations among the parameters appearing in the oscillatory templates,it will be instructive to consider additional examples. We will therefore examine oscillationsin a simple N = 1 supergravity theory inspired by those that arise in nonperturbatively-stabilized string compactifications. In this example the moduli-dependence of the axiondecay constant (setting the period of a modulation) will be related to the moduli-dependenceof a dynamically-generated scale (setting the amplitude of the modulation). For example, in KKLT constructions this small parameter is the fine-tuned small value of the classicalflux superpotential W . In supercritical theories one could have large n f and small n for some axions. Whether the moduli potential arises from perturbative or nonperturbative effects will turn out to be oflittle importance in this example, but the fact that the Lagrangian is supersymmetric will be central. N = 1 supergravity theory with a single chiral superfield T = τ + iθ and K¨ahler potential K . First we will discuss the case in which the modulations comefrom a superpotential term, and periodic terms in the K¨ahler potential can be neglected.Modulations in K will be treated below. Taking θ to be the candidate axion-inflaton, wesuppose that K ( T, ¯ T ) = K ( τ ) . (3.34)We take the modulation to arise from a superpotential coupling, W = W + W e − αT , (3.35)where W , W are constants independent of the modulus T , and α ∈ R . To be concrete,suppose that in Planck units, K T ¯ T = τ − β , (3.36)with β ∈ R , up to an overall constant that we will not retain. For β = 1 this supergravitytheory captures key structures arising in flux compactifications of type IIB string theoryon Calabi-Yau orientifolds, with T corresponding to a K¨ahler modulus. The canonically-normalized field is (neglecting τ − θ kinetic mixing) φ = θ τ − β . (3.37)If the nonperturbative term is subleading overall, i.e. if W e − αT (cid:28) W , then the leadingmodulated term is V osc = W W e − ατ cos( αθ ) ≡ Λ ( τ ) cos( ατ β φ ) , (3.38)up to a phase that we omit. So in full we have V osc = Λ (cid:16) τ [ φ ] (cid:17) cos (cid:16) α ( τ [ φ ]) β φ (cid:17) = Λ e − α ( τ [ φ ] − τ ) , cos( α ( τ [ φ ] − τ ) β φ ) , (3.39)where we allow the saxion τ to depend on the axion vev, φ . Notice that the scale andamplitude of the modulations are linked, both depending on τ [ φ ], because the amplitudedetermined by the real part of an instanton action, while the period is determined by theimaginary part of the same action (and, at the same time, by the kinetic term). Linearizingin small shifts of τ , τ [ φ ] = τ + γφ , (3.40)with γ a dimensionful constant, we find V osc = Λ e − αγφ cos (cid:16) αγ β φ β (cid:17) . (3.41)Comparing to the template (3.13), we have p Λ = 1, p f = β .24ext, we turn to modulations arising from instanton corrections to K (this is, for example,a leading source of modulations in the NS5-brane construction of axion monodromy inflationin [2, 3].) Taking as a concrete example (cf. [2]) K = − T + ¯ T + e − T + e − ¯ T ) , (3.42)and neglecting nonperturbative terms in the superpotential, one finds a structure of theschematic form e − τ τ cos (cid:18) φf − e − τ τ sin( φ/f ) (cid:19) , (3.43)or more generally (cid:15) cos (cid:18) φf − (cid:15) sin( φ/f ) (cid:19) . (3.44)We learn that deviations from a pure sinusoidal form can arise if the oscillations in Λ and in f are correlated, e.g. if both originate from a single correction to the K¨ahler potential. However,significant deviations from the sinusoidal template occur only when the modulations are large( (cid:15) ∼
1) and a number of our approximations are breaking down.
To understand whether the relative rigidity of moduli stabilization in the example of § (cid:37) corresponding to Re( T ) is (cid:37) = (cid:114)
34 log( T + ¯ T ) + C (cid:37) , (3.45)with C (cid:37) an integration constant. Comparing to (3.26), we see that f has an exponential de-pendence on (cid:37) . This finding is fairly general: the axion kinetic term typically has exponentialdependence on the canonically-normalized fields that parameterize ‘universal’ fluctuations,such as the overall volume V and the string coupling g s . In supersymmetric compactifi-cations, this is a straightforward consequence of the logarithmic K¨ahler potentials of thesefluctuations, as in (3.28).However, it is worth asking, still in the context of nonperturbative stabilization, whether f could have power-law dependence on a more general modulus ψ that does not parameterizea universal fluctuation. As an example motivated by the Large Volume Scenario [20], considera relative K¨ahler modulus in an example where the K¨ahler potential for two K¨ahler moduli τ b , τ s takes the form K = − V ) = − τ / b − τ / s ) . (3.46)25n the regime τ b (cid:29) τ s , τ s is to good approximation proportional to a power of a canonicalfield: one can expand the logarithmic K¨ahler potential.In nonperturbatively-stabilized compactifications, it is often the case that leading termsin the scalar potential have steep dependence on τ s , so that we have schematically f ∼ τ αs , V ∼ e − τ s , (3.47)for α a constant. It follows that the relative shift of f can be large when τ s (cid:28) δff ∼ ∂ τ s ff ∂ τ s V infl ∂ τ s V mod ∼ ατ s V infl V mod , (3.48)where V infl and V mod denote the inflaton potential and the moduli potential, respectively.We conclude that a shift in τ s that is modest in Planck units can be a large shift relativeto the initial vev, and give rise to a large relative change in f . The dynamics set by theinflationary and moduli potentials can be thought of as determining the shift measured inPlanck units, while the functional form f ( τ s ) (e.g., exponential or power law) determineswhether such a shift leads to a small or large multiplicative change in f . It would beinteresting to identify examples in which the large frequency shifts (3.48) lead to distinctivesignatures. In this section, we will comment on the amplitude Λ ( φ ) of oscillations in the potential (2.1).There is a wide range of possibilities for contributions to the part of the potential periodic in a ( φ ). These contributions are generally exponentially suppressed as a function of the internallength scales in the extra dimensions, so it is important to emphasize that the amplitude ofthe effect is quite possibly unobservably small. In high scale inflation there are significant upper bounds on the sizes of the extra dimensions,leading to reasonable regimes of parameters for which oscillations could be detectable. Wewill now make some simple estimates, in the spirit of the preceding discussion, to illustratethe relation between the amplitude of oscillations and the geometry. As well as helpingto frame our understanding of the motivation for the search for oscillatory features, theseestimates may provide some utility in interpreting the implications of a null result for axionmonodromy and related scenarios. 26s reviewed above, axions can arise from potential fields integrated over topologicallynontrivial submanifolds (‘cycles’) Σ I in the extra dimensions. The spectrum of branes wrap-ping these cycles is periodic under a I → a I + 2 π . The brane masses depend on a I as well ason the volumes v I of the cycles. If a cycle size is relatively small, then as the axion rolls alongits trajectory, new sectors of wrapped branes become light. This can lead to particle/defectproduction, or to radiative corrections, which produce periodic modulations, reflected in theoscillatory term in V ( φ ) (2.1). Similarly, instanton effects generically contribute oscillatoryterms to the potential.Each of these effects is exponentially suppressed as a function of the size of the cyclecontributing the axion. A large oscillatory contribution can arise if the corresponding cycleis stabilized at a small value — even a vanishing value is tractable theoretically undersome circumstances. For a small or vanishing cycle size, one finds a value of f /M p thatis independent of v I . It is therefore well motivated to search for oscillations over the wholerange of values of f /M p for which the data has sufficient resolution.However, the signal may even be detectable for relatively large cycles, given the con-straints on internal dimensions required for large-field inflation. Let us make some estimatesfor the case studied above in § L √ α (cid:48) ≡ √ vα (cid:48) in the internal geometry. Inthis case, for example for an axion from the Neveu-Schwarz two-form potential field B , wehave worldsheet instanton effects that contributeΛ ( φ ) ∼ Λ e − v , (4.1)where for power law stabilization of moduli in a product structure, we can estimate theprefactor as Λ ∼ g s V tot × g s × L M p ∼ V R ∼ V mod . (4.2)In this prefactor, the first factor is the conversion to Einstein frame, the second the standard1 /g s scaling of a classical effect in spacetime, and the factor of L represents the integralover the bosonic zero modes (transverse position) of the worldsheet instanton in the case ofa simple product structure. We should stress that the estimates we are doing here are meantto give a rough idea of the possible scale of the effect, rather than specific predictions ofexplicit models. Here V mod refers to a typical term in the moduli-stabilizing potential, suchas the contribution V R obtained by dimensionally reducing the internal curvature.In this type of scenario, we may also trade v for f /M p via fM p ∼ πL ) ∼ v a . (4.3)This exhibits a simple relation p Λ = p f , but this relation is not general.27ext, we note from previous work [10] that an analysis using current CMB data issensitive to an oscillatory piece roughly of size Λ ( φ ) ∼ − V ( φ ). Putting this togetherwith the previous estimates, we have detectability for e M p /f ≤ Λ V ( φ ) , (4.4)where V is the scale of the inflationary potential energy. Since Λ can be of order the moduli-stabilizing potential barriers, it is at least as large as the inflationary potential V ( φ ). Evenin the marginal case, where they are equal, we see from (4.4) that one requires f /M p ≥ / v is of order 10 in string units. For high-scale inflation, this is areasonable number. And again, given a hierarchy between the inflationary and moduli-barrierscales, the amplitude would be detectable for even larger cycle size despite the exponentialsuppression.Turning this around, a null result at this level would exclude an interesting regime ofmodel parameters, given the bounds on the size of internal cycles arising from the high scaleof inflation. However, it would be difficult to falsify the scenario based on these constraints.In any case, the estimates here provide ample motivation to pursue a search for driftingoscillations over a broad range of parameters. Λ( φ ) In this section, we will make a slight detour to explain a relation between drifting oscillationsand an interesting scenario for the initial conditions (somewhat similar to those explored in[17]) in axion monodromy inflation. An appealing possibility for the precursor to the ≈ e -folds of phenomenological inflation is tunneling from some earlier metastable minimum in theeffective potential. This can be motivated by the structure of the string landscape, althoughdespite interesting efforts there is not anything close to a well-understood framework forcomputing and interpreting physical observables in eternal inflation.With the oscillations studied here, there are regimes of parameters and field values forwhich the oscillations produce local minima in the potential. In order for phenomenologicalslow-roll inflation to proceed, we require that the potential be monotonic during the last 60 e -folds. That is, we require V (cid:48) ( φ ) to dominate at late times. But if the derivative of theoscillatory term in the potential, V (cid:48) osc ( φ ), dominates at earlier times, i.e. further out in φ ,tunneling out of a resulting metastable minimum could provide initial conditions for slowroll inflation. Since the field range of φ is bounded by ultraviolet physics, extending onlyover the range φ < φ UV where the inflationary potential remains consistent with modulistabilization [1, 2, 3, 13], the condition that V (cid:48) osc ( φ ) dominates at large φ leads to a lower bound on the amplitude of oscillations. This bound depends on the parameters p, p f , p Λ .28 .2.1 Rigid case As an initial illustration, consider p Λ = p f = 0 (no drift in amplitude or phase), and V ( φ UV ) ∼ V ( φ ).A convenient parameterization is V (cid:48) = p µ − p φ p − (cid:20) − b p sin (cid:18) φf + γ (cid:19)(cid:21) , b p ≡ Λ p µ − p φ p − f . (4.5)Having a tunneling ‘creation myth’ induced by oscillations requires that the secular term µ − p φ p − dominates V (cid:48) for φ < φ for slow-roll to work, while at some value φ t > φ theoscillatory contribution to V (cid:48) must begin to dominate and create local minima to tunnelfrom. That is, we need b p > φ > φ t and b p < φ < φ . (4.6)Here φ t denotes the point of the (last) tunneling exit, which is in the region where b p crossesfrom b p > b p < φ . That is, b p must grow monotonically with increasing φ , increasing beyond unity around φ ∼ φ t > φ . From the form of b p this immediatelyrequires p < φ UV < ∞ above which the energy densitygrows too large. Hence we need to require φ < φ t ≤ φ UV < ∞ for a tunneling based creationmyth to work. This fact that φ t ≤ φ UV < ∞ immediately enforces a lower bound on Λ.We can analyze this more closely by rewriting the above conditions b p ( φ t ) > φ t ≤ φ UV explicitly as a lower bound on ΛΛ > Λ c ≡ p f µ − p φ p − (cid:18) φ UV φ (cid:19) p − = p V φ (cid:18) V V UV (cid:19) − pp f (4.7)= 3 × − M p p / (cid:18) V V UV (cid:19) − pp fM p . Here we used φ (cid:39) √ p M p and the power spectrum normalization P S = 124 π V M p (cid:15) = 112 π V φ M p p = 2 . × − ⇒ V (cid:39) × − p . (4.8)Observationally, oscillations are bounded in amplitude, giving a constraintΛ < Λ exp ∼ − M p (cid:115) fM p . (4.9)29e now require Λ c < Λ < Λ (4.10)for a model to provide a creation myth consistent with experimental data. This rearrangesinto p / (cid:18) V V UV (cid:19) − pp < × − (cid:115) M p f , (4.11)In turn, this provides an upper bound on p which leads to a constraint on r = 4 p/N e . To geta sense of the numbers in this case, let us allow f /M p to range down to 10 − and estimatefrom the high scale of inflation that V /V UV (cid:38) .
01. This leads to an upper bound r (cid:46) . More generally, however, the additional parameters p f , and especially p Λ , in our driftingtemplates relax this condition. For each value of these parameters that is consistent with atunneling initial condition, there is a lower bound on the amplitude Λ( φ ) , but it will varysignificantly over the range of interest. The power p Λ enters exponentially in (3.13). Overthe range φ < φ < φ UV , this can lead to a large hierarchy between Λ( φ UV ) and Λ( φ ) ,allowing for tunneling initial conditions.We start from the expression for V (cid:48) descending from (3.13), which we write as V (cid:48) = p µ − p φ p − (cid:34) − ˜ b (1) p sin (cid:32) φ f (cid:18) φφ (cid:19) p f + γ (cid:33) − ˜ b (2) p cos (cid:32) φ f (cid:18) φφ (cid:19) p f + γ (cid:33)(cid:35) , (4.12)where ˜ b (1) p = Λ ( φ ) p µ − p φ p − f (1 + p f ) (cid:18) φφ (cid:19) p f , (4.13)˜ b (2) p = Λ ( φ ) p µ − p φ p − f p Λ C f φ (cid:18) φφ (cid:19) p Λ − , (4.14)Λ ( φ ) = M p e − C (cid:16) φφ (cid:17) p Λ . Already at this level, we see that p Λ > b (1) p , ˜ b (2) p → φ > φ . So we look at the parameterranges p f (cid:54) = 0, p Λ ≤ p f ∼ p Λ without a strong hierarchy between them (whichthe power-law cases above seem to support), then ˜ b (2) p (cid:28) ˜ b (1) p due to the extra power of φ /φ and the factor f /φ (cid:28)
1. 30he parameter ranges p f (cid:54) = 0, p Λ ≤ b (2) p as argued above) provide twopossibilities. • case a: p Λ = 0, i.e. only phase drift. In this case the creation myth conditions ˜ b (1) p ( φ t ) > φ t ≤ φ UV boil down to a modification of the rigid case conditionΛ > Λ c ≡ p p f V φ (cid:18) V V UV (cid:19) pf − pp f (4.15)= 3 × − M p p / p f (cid:18) V V UV (cid:19) pf − pp f M p . Consequently, in comparing with the results from the rigid case, p f < p f > f = 10 − M p and V /V UV = 0 .
01. Thenwe get r (cid:46) .
04 (1 + p f ) up to terms logarithmic in 1 + p f with small prefactor. • case b: p Λ <
0, i.e. both amplitude and phase drift. In this case the exponentialdependence in Λ ( φ ) quickly dominates once | p Λ | ≥ O (0 . p f = p Λ = − / p = 4 / φ UV ) Λ( φ ) ∼ exp (cid:32) M p f (cid:34) − (cid:18) φ φ UV (cid:19) / (cid:35)(cid:33) . (4.16)Since f /M p ∼ /L can be small, e.g. of order 10 − − − , this can easily exceed theratio V (cid:48) ( φ UV ) V (cid:48) ( φ ) ∼ φ / φ / UV , (4.17)leading to a monotonic behavior during inflation and a tunneling initial condition.Similar results hold for a significant range of these parameters.The drift in Λ described by p Λ has a limited effect on oscillations in the CMB, becauseof the small range of φ during the observable window. However, we have just seen thatthis drift can have significant impact on tunneling initial condition. Now that we have explored the pattern of drifting oscillations, exploiting the backreactionof the inflationary potential on moduli degrees of freedom, it is worthwhile to summarize31he symmetry structure of axion monodromy inflation. The comments in this section can befound in the existing literature, but we collect them here for completeness, and because ofthe close connection between the discrete shift symmetry, its soft breaking, and the driftingoscillatory features in the resulting power spectrum.There are several ways symmetries come into the physics. First, there are the underlyinggauge symmetries respected by the generalized fluxes ˜ F appearing in the Stueckelberg-liketerms (3.2). As always, a gauge symmetry is a redundancy of description. Nonetheless, thegauge symmetry constrains the form of the action for the fluxes and their couplings to thegauge potential fields A that descend to the axions a I ∼ (cid:82) Σ I A via integration over cyclesΣ I . In the presence of generic fluxes and/or branes, there is direct dependence on the axionsin the four-dimensional effective potential. A shift of a ( φ ) by 2 π can be undone by a shiftof flux quanta, giving an equivalent configuration. But in a given sector of flux quanta— quantum numbers which are locally stable, decaying via nonperturbative effects — thepotential is unwound to a kinematically unbounded field range.If the system is displaced from the minimum of the resulting axion potential, other scalarfields, including moduli corresponding to the volume of the extra dimensions and to thestring coupling g s , feel forces as a result. These tadpoles are not periodic under the shift a → a +2 π . The sourced fields adjust in an energetically favorable way, either by flattening ordestabilizing the potential, as explained for the case of power-law stabilization in § V ( φ ) ∼ µ − p φ p .Given that, there is a sector of particles and defects — those arising from branes wrap-ping the cycle Σ a — that respects a weakly broken discrete shift symmetry. Much like thespacefilling branes that can contribute to the potential as in [2, 3], branes of higher codi-mension also have tensions that depend on the axion a , and that undergo monodromy when a → a +2 π . However, the full spectrum of such branes comes back to itself under a → a +2 π ,up to small corrections inherited from the (generically flattened) potential V ( φ ). Similarly,Euclidean brane configurations, which produce instanton effects, are periodic. These sectorsof the theory are important in the present work, as they generate the oscillatory features.Finally, at low energies there is an effective approximate discrete shift symmetry. Thissymmetry protects against substantial radiative corrections to the potential, as in standardlarge-field inflationary models such as [5]. This structure descends from gauge symmetriesin the underlying string theory, and because of the sub-Planckian underlying periodicities, it As noted above, there are various string-dual fields, including examples from brane motion modes asin [1] or the T-dual ‘windup toy’ picture of [2, 3], which can behave similarly; another interesting class ofexamples can be found in [17].
32s consistent with the expectation that exact continuous global symmetries should be absentin quantum gravity.
We now turn to a search for the models we have discussed in the
Planck nominal missiondata. Our analysis is preliminary and should largely be thought of as a proof of principle,establishing that what we propose is numerically feasible.The analysis divides into two parts: the computation of the primordial power spectrum,and a search for this power spectrum in the data. As we have argued in § § § § Throughout this work, we focus on single-field axion monodromy. We will now specializefurther to the inflaton potential (3.13) motivated by power-law moduli stabilization. Usingthe same approximations as in [3], the power spectrum is∆ R ( k ) = ∆ R (cid:18) kk (cid:63) (cid:19) n s − (cid:32) δn s e − C (cid:16) φkφ (cid:17) p Λ cos (cid:34) φ f (cid:18) φ k φ (cid:19) p f +1 + ∆ ϕ (cid:35)(cid:33) , (6.1)where the amplitude δn s and the phase ∆ ϕ are calculable using the techniques in [3]; φ issome fiducial value for the scalar field to be specified later; and φ k is the value of the scalarfield at the time when the mode with comoving momentum k exits the horizon. At leadingorder in the slow-roll expansion we have φ k M p = (cid:112) p ( N − ln( k/k (cid:63) )) . (6.2)where N = N (cid:63) +( φ end /M p ) / p , and φ end is the value of the scalar field at the end of inflation.In the spirit of our earlier discussions, we will for now neglect the drift in amplitude, setting C = 0. Comparison with a numerical evaluation of the power spectrum shows that theanalytic template (6.1) gives an excellent approximation to the exact power spectrum, aftera small adjustment of the frequency. In other words, an accurate template for the powerspectrum is ∆ R ( k ) = ∆ R (cid:18) kk (cid:63) (cid:19) n s − (cid:32) δn s cos (cid:34) φ ˜ f (cid:18) φ k φ (cid:19) p f +1 + ∆ ϕ (cid:35)(cid:33) , (6.3)with ˜ f higher than the underlying axion decay constant by up to a few percent. This is thefirst template for which we will present results in § n s and the parameters in the potential is well approximated by δn s = 3 b (cid:18) πα (cid:19) / with α = (1 + p f ) φ f N (cid:18) √ pN φ (cid:19) p f , (6.4)and monotonicity parameter b ≡ ˜ b (1) p ( φ ) given by (4.13).Figure 6.1 shows the comparison between the numerical and analytic power spectra fortwo representative choices of parameters. For both we set p = 4 / p f = − /
3. For thefirst example, we set f = 4 × − M p and b = 0 .
01. In this case we find ˜ f = 4 . × − M p .For the second example, we set f = 10 − M p and b = 0 .
05 and find ˜ f = 1 . × − M p .Since f and ˜ f are so close, we will omit the tilde in what follows.It is instructive to expand the argument of the trigonometric function in ln( k/k (cid:63) ). Onefindscos (cid:34) φ f (cid:18) φ k φ (cid:19) p f +1 + ∆ ϕ (cid:35) = cos (cid:34) α (cid:18) ln( k/k (cid:63) ) + 1 − p f N ln ( k/k (cid:63) )+ (1 − p f )(3 − p f )2 N ln ( k/k (cid:63) ) + (1 − p f )(3 − p f )(5 − p f )2 N ln ( k/k (cid:63) ) + . . . (cid:19) + ∆ ˜ ϕ (cid:35) . (6.5)Even for the highest frequencies of interest, the term proportional to the fourth power ofthe logarithm does not significantly modify the phase over the observed seven e-folds, andcan be neglected. However, we should in general keep the terms up to and including thethird power of the logarithm to accurately model the power spectrum. This is consistentwith our findings in § φ k − φ in the Taylor expansion of the axion decay function.We see that the potential (3.13) with p f of order unity leads to an argument of thetrigonometric function that is a rapidly converging series in ln( k/k (cid:63) ). This suggests a naturalphenomenological template of the form∆ R ( k ) = ∆ R (cid:18) kk (cid:63) (cid:19) n s − (cid:32) δn s cos (cid:34) ∆ ϕ + α (cid:32) ln( k/k (cid:63) ) + (cid:88) n =1 c n N n(cid:63) ln n +1 ( k/k (cid:63) ) (cid:33)(cid:35)(cid:33) . (6.6)This is the second template for which we will present results in § c n will berelated to the parameters of the model. These relations can be used to convert bounds on c n into bounds on microphysical parameters.Modulations in the inflaton potential also lead to oscillatory contributions in the powerspectrum of tensor perturbations. However, for f (cid:28) M p , the amplitude of these oscillations35 .001 0.002 0.005 0.010 0.020 0.050 0.100 0.2002. (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) k (cid:64) Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) Mpc (cid:45) (cid:68) P nu m (cid:72) k (cid:76) (cid:144) P a n (cid:72) k (cid:76) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) k (cid:64) Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) Mpc (cid:45) (cid:68) P nu m (cid:72) k (cid:76) (cid:144) P a n (cid:72) k (cid:76) Figure 1: Comparison of a numerical power spectrum and an analytic template after a smallshift in f in the analytic template to maximize the overlap. Top row: C = 0, p f = − , f = 4 × − M p , and b = 0 .
01. Bottom row: C = 0, p f = − , f = 10 − M p , and b = 0 . f /M p ) α (cid:28) h ( k ) ≈ ∆ h (cid:18) kk (cid:63) (cid:19) n t with ∆ h = r ∆ R and n t = − r/ . (6.7)To perform a search, we must specify priors for the various parameters in these templates.The parameters in our first template are n s (or p ), the amplitude δn s , p f , f , and the phase∆ ϕ . Concrete models with p ∈ { , , / , , / } have been explored in the literature, butmore general powers are plausibly realizable. As can be seen from (6.5), there is a degeneracybetween p and the axion decay constant, so that changes in p can be compensated for bychanges in f . So p is only directly constrained by the spectral index and the tensor-to-scalarratio. With current data, the variation in n s and r induced by changes in p only leads toimprovements of the fit of a few. Thus, in an exploratory search for oscillatory patterns in the36ower spectrum, p can be fixed; we will set p = 4 /
3. We will assume instantaneous reheating.For p = 4 / N (cid:63) ≈ . k (cid:63) = 0 .
05 Mpc − . We set φ = 12 . M p andfind φ end = 0 . M p . The best-fit amplitude in the WMAP9 data was large, δn s ≈ . < δn s < .
7. Forthis initial analysis, we consider − / < p f < /
2, but because we do not know of sharptheoretical bounds on the parameters p , p f , we plan to perform scans over a wider range inthe future. For the axion decay constant we scan over 10 − M p < f < − M p , and finallywe take 0 < ∆ ϕ < π . For the second template, we assume that the parameters c n are oforder unity and use − < c n <
2, 1 < α < < ∆ ϕ < π . It should be keptin mind that the single-field effective field theory becomes strongly coupled when α (cid:38) Before we present the results of our search in the
Planck nominal mission data, we will testour methods on simulated data.The introduction of additional parameters into a model is expected to improve the fit.Parameters that enter the model linearly, such as δn s in our case, are expected to improve thefit by ∆ χ = 1 per parameter. Parameters such as the axion decay constant are not of thistype, and the expected improvement is harder to predict. Roughly speaking, for amplitudeslike δn s we only have one trial, and increasing the amplitude too far will lead to a bad fit.For parameters like f , however, an increase will not necessarily eventually lead to a bad fit.Some frequency may fit well, and then as we increase the frequency the fit may become worse,but it may eventually become better again. In other words, we have several independenttrials and should include a trials factor for the look-elsewhere effect. While progress can bemade analytically toward understanding the expected improvements for such parameters, wewill resort to simulations based on a featureless power spectrum to assess the improvementswe should expect for our templates in the absence of a signal. Our simulations includecorrelations between multipoles introduced by the sky masks used for the measurement ofthe spectra in the CAMspec likelihood code. In addition, the simulations include foregroundsconsistent with the foreground model used in the CAMspec likelihood code presented in [25].They do not include instrumental effects, but we assume that instrumental effects are wellenough controlled to be unimportant for the statistical question we are interested in. Wehave performed searches in five simulations for the template (6.3), varying only the powerspectrum parameters. The largest improvements found in the five runs range from ∆ χ ≈ Ongoing work to analyze the powers arising in axion monodromy inflation more systematically may yieldmore interesting bounds on, or relations among, the parameters.
37o ∆ χ ≈
19. If these were drawn from a Gaussian distribution, we would be led to expectan improvement of ∆ χ = 16 . ∼ .
5. We know from the firstreference in [10] that the distribution is not quite Gaussian, and more substantial simulationsare required in the event of a potential detection to assess the significance. However, thesenumbers do provide a useful guide, and improvements of this magnitude should not beconsidered significant.We now demonstrate that our methods allow recovering a signal from simulated databased on an oscillatory power spectrum. As before, the simulations include effects of skymasks and foregrounds, but do not include any instrumental effects. We have performedsimulations for a signal of the form of template (6.3) with p f = − / δn s = 0 .
02 to δn s = 0 . f = 2 × − M p , f = 4 × − M p , and f = 10 − M p . We find that the signal is recovered in all simulationsthat lead to an improvement larger than that expected in the absence of a signal, but with anamplitude δn s that is slightly biased towards larger values. For our preliminary search thisis of no concern, but would have to be addressed in the event of a detection. The results areshown in Figure 2 for the simulations with amplitudes δn s = 0 . δn s = 0 . δn s = 0 . δn s = 0 .
05 for axion decay constant f = 10 − M p and p f = − /
3. The four simulationsare based on the same random seed, so the only difference is the amplitude of the signal. Thesize and color of the circles represents the improvement. Larger circles represent larger valuesof ∆ χ . For δn s = 0 .
02 the improvement corresponding to the input parameters is smallerthan for other parameter choices that fit random scatter in the data. As the amplitude isincreased to δn s = 0 .
05 the improvement grows to ∆ χ ≈
25 and becomes significantly largerthan the improvement expected in the absence of a signal. As expected from our discussionof overlaps, p f is not well constrained and ∆ χ is approximately constant along the line ofconstant α = ω/H as defined in equation (6.4). We now turn to the results of a preliminary search for oscillatory contributions to the powerspectrum with drifting periods in the
Planck nominal mission data using the data set re-ferred to as
Planck +WP [25]. In previous searches for logarithmic oscillations or oscillationsmotivated by the linear axion monodromy model, with negligible drift of the axion decayconstant, in the
Planck nominal mission data using the public likelihood code, the improve-ment in fit was exactly as expected statistically (see figure 6 of the first reference in [10]).That is, the fit improved by ∆ χ ≈
10 over the best-fit featureless power spectrum, justas expected based on simulations that include the look-elsewhere effect. In this work, weadd parameters characterizing the drift in frequency, and would like to understand if theyimprove the fit in a significant way. 38igure 2: Results for a search for oscillations in simulated data. The simulations are basedon the template (6.3) with p f = − / f = 10 − M p , and amplitudes δn s ranging from δn s = 0 .
02 to δn s = 0 .
05, as indicated by the labels. Points included in the plot lead toan improvement over the best-fit with δn s = 0 of ∆ χ ≥
4. The sizes of the dots and theircolor indicate the amount of improvement. The orange cross indicates the input values of f and p f , and the red line represents the corresponding value of α = ω/H as defined inequation (6.4). As the amplitude δn s increases the significance of the improvements grows.39n summary, we find that the fit improves by ∆ χ ≈
18 for both templates (6.3) and (6.5).Based on the simulations discussed in section § χ ≈
18. At the best-fit point, the axion decay constant is f ≈ − M p , p f = 0 .
2, and the amplitude is δn s ≈ .
27. As expected, the improvement compared toΛCDM is better when the ΛCDM or nuisance parameters are varied, but only slightly.Points that lead to an improvement of ∆ χ ≥ χ ≈
10 corresponds to a frequency ω/H = 28 . Planck collaboration [26]. As was pointedout in the first reference of [10], some of the improvement can be attributed to a range ofmultipoles around (cid:96) = 800 and is seen in all frequencies. However, a significant contributionto the improvement originates from the 217 GHz data in the region around (cid:96) = 1800. Thisrange of multipoles of the 217 GHz data is known to be affected by interference between the4K cooler and the read-out electronics [27, 25]. This disfavors a cosmological interpretation,but the agreement between analyses provides a consistency check for our search. The factthat the improvement is nearly constant along the line of constant ω/H shows that the datais not sensitive to the ln ( k/k (cid:63) ) term for this relatively low frequency. At smaller f or larger ω/H the improvement is no longer uniform along a line of constant ω/H , so that the dataprefers a certain amount of drift.In addition, we have performed a search based on the likelihood described in [27]. This isbased on survey cross-spectra and as a consequence is essentially insensitive to the cooler linebut may be subject to other systematics such as changes over the course of six months. Inthis case, we find an improvement of ∆ χ ≈
14 relative to the unmodulated power spectrum.The fraction of sky is slightly different between the two analyses and we have not performedsimulations for our search tailored to this likelihood, but we expect improvements comparableto those seen in the simulations for the CAMspec likelihood. The best-fit frequency in thiscase is f = 2 . × − M p with p f = 0 .
05 and an amplitude of δn s = 0 .
15. The resultsare shown in the right panel of Figure 3. The solid lines shown in the figure indicate thevalues of ω/H that led to improvements in the search based on the CAMspec likelihood.40igure 3: Points that lead to an improvement over ΛCDM of ∆ χ ≥ Planck likelihood [25], and the rightpanel shows the results for the likelihood discussed in [27]. The sizes of the dots and theircolor indicate the amount of improvement. Larger blue points represent larger values of∆ χ . The solid lines indicate constant α = ω/H . The red solid line represents ω/H = 28 . Planck collaboration [26]. A significant contribution tothe improvement derives from the region around (cid:96) = 1800 in the 217 GHz data. This rangeof multipoles of the 217 GHz data is known to be affected by interference between the 4Kcooler and the read-out electronics [27, 25].As expected, the improvement at ω/H ≈ . ω/H ≈
60 and ω/H ≈ f = 4 . × − M p and is the frequency that led to the improvement of ∆ χ ≈
20 in theWMAP9 data. The improvement seen in both likelihoods is ∆ χ (cid:46) − / < p f < / ω/H = 28 . ω/H ≈
60 and ω/H ≈
210 are seen in both likelihoods, but with ∆ χ (cid:46) c for the frequencies that lead to the most notable41mprovements to the fit, there is no dependence on the parameter c for ω/H < Planck nominal mission data at ω/H ≈
210 (∆ χ ≈
10) and the WMAP9 data (∆ χ ≈ Planck nominal mission data once the multipoles included in the analysis are restrictedto the range in which both experiments are cosmic variance limited. While the best-fitparameters in this case approach each other, the improvement in the analysis based on theCAMspec likelihood remains ∆ χ ≈
10. A preliminary analysis [28] based on WMAP9 W-band data as well as
Planck nominal mission 100 GHz survey spectra performed with thesame masks and approximations to the covariance matrix led to an improvement ∆ χ ≈ Planck analysis in both datasets. As the sky fraction is increased,the improvement increases and approaches ∆ χ ≈
20 for both the W-band and 100 GHz data.This indicates that a more thorough analysis is needed. Finally, the range of parameterssearched here was limited and a careful search should be carried out for a wider range ofparameters and including the drift in the amplitude.Figure 4: Marginalized posterior in the δn s − f plane for the template (6.3), for the runin which only the power spectrum parameters are varied. The left panel shows the resultsfor the public Planck likelihood, and the right panel shows the results for the likelihooddescribed in [27]. 42igure 5: Results for the second template, (6.6). As before, points included in the plot leadto an improvement over ΛCDM of ∆ χ ≥
4, and the sizes of the dots and their color indicatethe amount of improvement. The orange dot indicates the best-fit point. The top panelsshows the results for the public
Planck likelihood, and the bottom panels show the results forthe likelihood described in [27]. While especially for higher frequency there is a dependenceon c , indicating that a drift in frequency leads to a better fit, the dependence on c is weak.43 Conclusions
We have shown that the symmetry structure of axion monodromy inflation allows oscillatoryterms in the scalar potential whose amplitude and frequency can drift over the course of in-flation, as a consequence of moduli vevs that shift in response to the diminishing inflationaryenergy. Incorporating the drifts in frequency can be critical in searching for oscillatory signa-tures in the scalar power spectrum. We presented templates that encode the drifts that occurin a broad range of axion monodromy scenarios in string theory and in effective field theory.The parameter ranges we propose for these templates include the values found in modelssuch as [2, 3, 13] but extend well beyond this regime, including ranges in which the driftis strong enough to preclude a useful Taylor series expansion of the axion decay function,but weak enough for radiative stability. The range of frequencies and drift parameters con-sistent with an effective field theory description includes regions where the drift parametersare essential in comparison to cosmological data, although for a wide range of parametersprevious analyses without drift parameters suffice. It would be interesting to pursue thetheory of oscillations at higher frequencies, beyond the scale at which effective field theorybreaks down, and to characterize the full range of parameters realized in string-theoreticaxion monodromy.Using the templates proposed here, we performed a preliminary search for drifting-frequency oscillations in the
Planck nominal mission data, in order to demonstrate that suchan analysis is numerically feasible. A complete analysis that explores the full parameterspace is an important problem for the future.
Acknowledgements
We are very grateful to Hiranya Peiris for extensive discussions and correspondence, andto Richard Easther for early discussions of the role of the drift in f . We would also like tothank Thomas Bachlechner, Dick Bond, Fran¸cois Bouchet, Xi Dong, Olivier Dor´e, George Ef-stathiou, Daniel Green, Guy Gur-Ari, Ren´ee Hloˇzek, Enrico Pajer, Leonardo Senatore, DavidSpergel, Gonzalo Torroba, and Herman Verlinde for useful discussions. The work of L.M. wassupported by NSF grant PHY-0757868 and by a Simons Fellowship. The work of E.S. wassupported in part by the National Science Foundation under grant PHY-0756174 and NSFPHY11-25915 and by the Department of Energy under contract DE-AC03-76SF00515. Theresearch of A.W. was supported by the Impuls und Vernetzungsfond of the Helmholtz Asso-ciation of German Research Centres under grant HZ-NG-603. We thank the Aspen Centerfor Physics and the Princeton Center for Theoretical Science for hospitality during the courseof this work. 44 eferences [1] E. 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