Observable Chiral Gravitational Waves from Inflation in String Theory

Evan McDonough1, ∗ and Stephon Alexander1, †

1Department of Physics, Brown University, Providence, RI, USA. 02903

We consider gravitational wave production during inflation in type IIB string theory, and thepossibility of observable gravitational waves in small field inflation. We show that the gauge fieldexcitations on a set of coincident D7 branes, itself critical for moduli stabilization and hence intrinsicto inflation in string theory, coupled with axion fields from bulk fluxes, can act as a spectator sectorduring inflation. This results in a large production of chiral gravitational waves, even for relativelysmall values of the axion-gauge field coupling. We extend this to include a monodromy for the axion,and demonstrate that in both cases an observable level of gravitational waves is produced in smallfield inflation in string theory, with a spectrum that is maximally chiral. Finally, we demonstrate theconsistency with moduli stabilization and with arbitrary (large or small field) inflationary dynamicsof the host model, considering as an explicit example Kahler Moduli Inflation.

I. INTRODUCTION

Observable primordial gravitational waves from infla-tionary cosmology are typically associated with large fieldmodels, as required by the Lyth bound [1]. Unfortu-nately, these models generically suffer from the η-problem[2], motivating an analysis from a UV complete frame-work, such as string theory. However, even in string the-ory, large field inflation faces severe challenges, notablyfrom back-reaction on string theory moduli [3–5] and theWeak Gravity Conjecture [6].

Fortunately, intrinsic to string theory realizations ofinflation is the existence of much more structure thensimply an inflaton and its potential energy, e.g. [7–9] (fora comprehensive review see [5]). All constructions re-quire that the additional moduli, potentially numberingin the hundreds, be stabilized. We interpret this not asa drawback or complication, but a feature of inflation instring theory, which may lead to distinct observationalsignatures.

With this goal in mind, in this work we consider theproduction of gravitational waves from the dynamics ofthese additional fields, and find that an observable level of(chiral) gravitational waves can generically arise in bothlarge and small field models of inflation in string theory.

More concretely, the stabilization of string theory mod-uli is most often achieved (e.g. [10–12]) via a combina-tion of fluxes and gaugino condensation. The latter iscaptured by a simple superpotential,

W = e−2πN fD7 , (1)

where N is the rank of the SU(N) gauge theory on world-

volume of N coincident D7 branes, and fD7 is the gaugekinetic function. This generates exponential potentialsfor volume moduli and oscillatory potentials for axions,

accompanied by couplings RefD7F2 and ImfD7FF of the

moduli to the SU(N) gauge field. This complex system of

∗ evan mcdonough@brown.edu† stephon alexander@brown.edu

interacting axions and non-Abelian gauge fields is presentin all string inflation scenarios, and could in principle beevolving during inflation. Similarly, the possibility that astring theory axion could be slowly-rolling but not itselfbe the driver of inflation was first considered in [13] (seealso [14]).

In [15] it was shown that the dynamics of precisely sucha ‘spectator sector’, i.e. a set of fields decoupled fromthe inflaton and subdominant in energy density, couldlead to significant production of gravitational waves oncosmological scales. This construction evades the Lythbound [1], raising the interesting possibility that observ-able gravitational waves may not require large field in-flation [16]. Moreover, this occurs without a significantproduction of scalar perturbations, in contrast with othermechanisms for gravitational wave production in stringtheory [17].

In this paper we realize variants of [15] via the inter-acting system of a C2 axion and the SU(N) gauge fieldson a set of D7-branes. Worldvolume fluxes on an internaltwo-cycle generate a C2 dependence of the gauge kineticfunction, giving rise to an oscillatory potential for theaxion and an axion-gauge field coupling that is enhancedby a factor of N . We extend this to include a monodromypotential for the axion [18, 19], and show these scenarioscan be consistently realized in the large volume scenario(LVS) [11, 12], without backreaction, and independent ofthe details of inflation.

We demonstrate that gravitational waves are copiouslyproduced, though for a periodic axion potential the pro-duction generically (i.e. barring extremely large N) oc-curs only for the first few e-folds of inflation. With the in-clusion of a monodromy potential, the gravitational waveproduction is long lasting, occurring for tens of e-folds.

Provided that the CMB pivot scale exited the horizonwhile gravitational waves are being produced, the resultof both cases is an amplification of the tensor-to-scalarratio, and a net chirality on large scales. This leads to antensor-to-scalar ratio of r ' 10−3−10−2 in small field in-flation, observable by next generation Cosmic MicrowaveBackground B-mode polarization experiments [20], alongwith parity violating CMB cross correlations [21, 22], of

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which the observational prospects have been studied in[23, 24]. This is particularly interesting given the role ofchiral gravitational waves in models of leptogenesis [25–29] and dark matter [30].

Finally, as a concrete and complete model, we embedthis into Kahler Moduli Inflation [31], with moduli sta-bilization occurring as in the LVS. This inflation modelpredicts r ≤ 10−10, and we show that with the inclusionof the spectator sector, this can be raised to an observablelevel.

The structure of this paper is as follows: we first re-view the mechanism for production of gravitational wavesput forward in [15], and its generalization from SU(2) toSU(N). In Section III we discuss the realization of thisvia C2 axions and gaugino condensation on branes car-rying worldvolume flux, and in Section IV we considerthe effect of a monodromy potential for the axion. InSection V we implement this in the context of the LargeVolume Scenario, and demonstrate the consistency withmoduli stabilization and arbitrary inflationary dynamics.In section VI we consider an example realization in thecontext of Kahler Moduli Inflation. We close in sectionVII with a discussion of directions for future work.

II. GRAVITATIONAL WAVES FROMAXION-GAUGE FIELD DYNAMICS

A. The DFF Model: Spectator SU(2)

In this work we study the production of gravitationalwaves in slight variants of the model [15]. This is a spec-tator version of the inflationary model [32, 33] (see also[34]), which in turn builds upon [35, 36]. The modelswe study have the nice feature that while the inflationmodel [32, 33] is ruled out on observational grounds [33],the spectator models inherit the scalar perturbation pre-dictions (e.g. ns) of their ‘host’ model, which can be ar-bitrarily chosen.

This can be succinctly described the matter La-grangian,

L = Lφ[φ] + LGW [χ, F ], (2)

where Lφ is the inflaton and its potential, while LGW isthe axion-gauge field system responsible for the produc-tion of gravitational waves. The field χ slowly rolls downits potential during inflation, and the fixed sign of χ re-sults in a gravitational wave spectrum that is maximallychiral.

In [29] it was shown that the chiral gravitational wavesin a variant of this scenario can lead to successful baryo-genesis, and in [30] that model was used to constructa new class of dark matter models, dubbed the DarkBaryon Superfluid. The observational predictions of [15]were further refined in [37–39], where it was shown thata large tensor non-Gaussianity can be produced in thesemodels, without an accompanying production of scalarnon-Gaussianity (see also Appendix A). Similar phenom-ena can also arise in the case that the gauge field is

Abelian, given suitable values of the model parameters[40].

The Lagrangian for the gravitational wave sector in[15] is given by,

LGW = −1

2(∂χ)

2 − U(χ)− 1

4F aµνF

aµν +λχ

4fF aµν F

aµν ,

(3)where F aµν is the non-Abelian field strength with couplingg. During inflation the dynamics of the universe are pre-dominantly determined by the inflaton φ, with the axionand gauge field vastly subdominant in energy density.Moreover, as in [32, 33], the gauge field Aµ is chosen totake the isotropic configuration,

〈Aa0〉 = 0 , 〈Aai 〉 = δai a(t)Q(t) . (4)

The equations of motion for the axion-gauge field systemare then given by,

χ+ 3Hχ+ U,χ = −3gλ

fQ2(Q+HQ

), (5)

Q+ 3HQ+(H + 2H2

)Q+ 2g2Q3 =

gλ

fχQ2 . (6)

The potential for χ and initial conditions for χ and Qare chosen that they are both slowly-rolling during in-flation. There is an attractor solution wherein Q and χnon-trivially balance one another, which occurs provided

λQ

f 1, (7)

and

λQ

f· gQH 1. (8)

This imposes constraints on the parameters of the model,as we will come back to shortly.

The slowly rolling attractor solution is given by [41],

Q '[−f

3gλHU,χ

]1/3

, χ ' 2fg

λQ. (9)

These equations accurately describe the evolution of thespectator axion-gauge field system during inflation pro-vided that the parameters g, λ, f, µ are such that theconditions (7) (8) are satisfied.

B. Gravitational Waves

The color-spatial fluctuations in the gauge field δAµacan be decomposed into two scalars (e.g. δQδia), two vec-tors (e.g. ∂iMa), and two tensor polarizations (δAia =δa

jtij). The tensor fluctuations tij are amplified dueto the dynamics of χ, and this sources gravitationalwaves at linear order in perturbations, via the combi-nation 〈A〉δA ∼ Qtij . For a detailed discussion seee.g. [29, 33, 41, 42].

3

The equations of motion for the coupled metric andgauge field tensor fluctuations are given by [41]

∂2xtR,L +

[1 +

2mQξ

x2∓ mQ + ξ

x

]tR,L ' 0, (10)

∂2xψR,L +

(1− 2

x2

)ψR,L ' SψRL, (11)

where x ≡ k/aH, ψR,L ≡ aMPlhij/2 is the canonicallynormalized gravitational wave mode function, and tR,L ≡aMPltij/2 is the gauge field tensor mode function. Theparameters ξ and mQ are defined as

ξ ≡ λχ

2fH, mQ ≡

gQ

H, (12)

which are positive during the slow-roll phase of χ. Thisleads to an exponential growth of a single handedness oft, with solution [41],

tR =1√2keπ2 (mQ+ξ)Wα,β

(−2ik

aH

). (13)

There is no corresponding growth of scalar fluctuationsprovided that mQ >

√2 [15].

This growth of tR is converted to a growth of ψR viathe source term on the right-hand-side of (11),

SψR,L ≡ (2√εE/x)∂xtR,L+(2

√εB/x

2)(mQ∓x)tR,L, (14)

where εE and εB are the slow-roll parameters of the elec-tric and magnetic energy densities in the gauge field,

εE ≡(Q2 +HQ)2

H2M2Pl

, εB ≡g2Q4

H2M2Pl

. (15)

After solving for the ψ using the Green’s functionmethod, the resulting sourced tensor power spectrum isgiven by

P sh(k) ' 2

π2

H2

M2Pl

· εBe3.62mQ |k=aH , (16)

where mQ and εB are evaluated at k = aH. The tensor-to-scalar ratio is then given by a sum of the single-fieldand sourced tensor fluctuations,

r ' rinfl ·(1 + εBe

3.6mQ)|k=k∗ , (17)

where rinfl is the single-field inflation prediction for rand k∗ is the CMB pivot scale.

C. Generalization to SU(N)

The gauge group of interest in the present work is notsimply SU(2), but SU(N), where N is the number of co-incident D7-branes. In the original moduli stabilizationproposal of KKLT [10], gaugino condensation occurs on a

stack of N = 20π D7-branes, and we will consider similarvalues of N here.

This introduces two important effects: (1) a rescal-ing of the string theory coupling λ, which we will see indetail in Section III, and (2) the possibility of multipleexcited independent subgroups of SU(N). In this sub-section we focus on the latter, proceeding along the linesof Appendix A of [29] and [43].

The production of gravitational waves in [15] is intrin-sically dependent on the group SU(2), and hence onemust first split the SU(N) into disjoint SU(2) subgroups,of which the maximum number is N/2 mod 2.

While in this work we consider only a single SU(2) sub-group to be excited, in principle one could consider a setofN SU(2) subgroups with non-vanishing field strengths.The equations of motion take a simple form if the SU(2)subgroups have a common field strength (which can beachieved e.g. by initializing with a common initial condi-tion), in which case the background equations of motionare equivalent to the single SU(2) case (5) (6) with thereplacement [29]

g →√N g , Q→ Q√

N. (18)

Under this replacement, and ignoring the string theo-retic rescaling of λ, the value of mQ is unmodified whileεB → εB/N . On the other hand, the gravitational waveequation of motion now has N statistically uncorrelatedsources. The net effect is that the tensor to scalar ratiois unchanged from the single SU(2) case,

r → rinfl ·(

1 +N · εBNe3.6mQ

)= rinfl ·

(1 + εBe

3.6mQ).

(19)With this in mind, in this work we consider for simplicitythe case that a single SU(2) subgroup is excited, that is,we take N = 1.

We now consider the values of the other parametersdescribing the model.

D. Constraints on Model Parameters for aConsistent Realization

In order for the phenomena studied here to arise gener-ically, there should exist a large region of parameter spacethat (i) realizes a self-consistent cosmological evolution ofthe axion and gauge field as a spectator sector, (ii) givesphenomenologically interesting results, and (iii) is selfconsistent from the perspective of the underlying stringcompactification.

To satisfy (i) we require that the slow-roll attractorexists, which requires (7) and (8) be satisfied. The first ofthese can re-phrased as a constraint on f/H as a functionof λ and mQ:

f

H λmQ

g. (20)

4

Interestingly, the slow-roll solution does not necessarilyrequire λ 1.

In order to have a sizeable amplification of the gravi-tational wave power spectrum, and hence satisfy (ii), itis required that

mQ > 1, (21)

which in conjunction with (20) implies that (8) is satis-fied.

On the other hand, the time duration of the amplifi-cation can be approximated as the duration of slow-rollevolution. The slow-roll solution for χ can be written as,

1

f

dχ

dN=

2mQ

λ. (22)

In the absence of a monodromy potential, the amount ofslow-roll evolution of χ is bounded by ∆χ ' πf , and inthis case the number of e-folds of production is given by,

NGW .πλ

2mQ. (23)

Hence, efficient gravitational wave production (mQ > 1)will occur for more then a single e-fold for only for λ > 1.This is modified in the presence of a monodromy poten-tial, as we study in section IV.

Regarding (iii), string theory examples generically leadto small values of the axion-gauge field coupling λ [5].This is an obstacle to driving inflation using solely thisconfiguration, as the number of e-folds of inflation isgiven in that case by [32]

Ninfl .3

5λ, (24)

independent of f . In contrast, the spectator axion westudy here need not slowly-roll for the full 60 e-folds ofinflation, and thus the model can accommodate muchsmaller values of λ. We will see this in more detail insections II-VI.

As an example, consider tuning µ to make mQ = 5,and fix λ = 20. In this case, the production lasts forNGW ' 4 e-folds of inflation. If in addition g = 10−2

and Hinf = 10−7MPl the constraint on f reads:

f 10−3MPl, (25)

which allows for a large range of f . More generally, themodel building challenge for these models, assuming asmall λ, is to engineer a small enough f to allow for slowroll dynamics. This is in stark contrast to the modelbuilding challenge of Natural Inflation [35, 36], whereone must engineer a super-Planckian decay constant (forexample, by alignment [44, 45]), in conflict with the WeakGravity Conjecture [6].

From this we conclude that there is a large parameterspace of consistent and phenomenologically interestingmodels of an axion-gauge field spectator sector, and wewill now proceed to study their realization in string the-ory.

III. C2 AXIONS AND GAUGINOCONDENSATION ON MAGNETIZED BRANES

The C2 axion was considered as a candidate for naturalinflation in [46] and [47]. Here we use these constructionsto build the C2 axion as a spectator field, whose interac-tions with the gauge fields on a stack of D7 branes leadsto an amplification of gravitational waves on large scales.

A. Salient Points

1. Oscillatory Potential : Gaugino condensationon D7 branes carrying worldvolume flux generates an os-cillatory potential for the C2 axion c of the form [46]

V = Λ4

[1− cos

(2π

NMc

)](26)

Λ4 ' |W0|AV2N

τe−aτ , (27)

where M is the number of units of flux on the branes,τ is the 4-cycle volume modulus, and a,A are theparameters of the non-perturbative superpotential. Weprovide further details on this in section V.

2. Decay Constant : As shown in equation (20), theslow-roll attractor can be realized via a suitably smalldecay constant f . This naturally arises in warped fluxcompactifications due to the ‘warping down’ of the decayconstant [48, 49]. For example, in the Klebanov-Strasslergeometry [50] the axion decay constant is given by

f ' e−4π3

KMgs f, (28)

where M and K are the number of RR and NS fluxquanta respectively, and f is the decay constant in theunwarped compactification. For K ' M and at weakstring coupling gs 1, this leads to a large suppressionof the decay constant.

In a compactification with multiple warped throats,the decay constant is given by [48],

f ' f√∑i c−2i h−2

i

, (29)

where the ci are a set of O(1) constants, and hi is thewarp factor at the tip of the ith warped throat, which for

Klebanov-Strassler is simply hi = e− 4π

3

KiMigs . It follows

that the axion decay constant is effectively set by thelongest throat in the compactification [48].

This generates a large λ/f , necessary for the slowroll evolution of the axion gauge field system, while notaffecting the gauge coupling of the stack of D7’s, whichwe take to be localized far from the longest throat in thecompactification.

5

0 2 4 6 8 10Ne0

1

2

3

4

Log10 Ph/Phinfl

0 2 4 6 8 10 12Ne0.0

0.2

0.4

0.6

0.8

1.0

1.2

HR-L

FIG. 1. Spectator C2 with a periodic potential, for a model with rinfl = 10−5 and with coupling λ = 10 (left curves) and λ = 50(right curves). [Left Panel]: Amplification of the tensor power spectrum Ph(k) for modes that exit the horizon in the first10 e-folds of inflation. The red dashed lines corresponds to a scale-invariant spectrum with r = .01 and r = 0.1 respectively.[Right Panel]: The net helicity fraction of the gravitational wave power spectrum .

3. Coupling to Gauge Fields: As per equation(23), the number of e-folds of slow-roll evolution is deter-mined by the size of λ, independent of f .

Details of the couplings of the various string theoryaxions can be found in e.g. [51]. The coupling of C2

arises from Chern-Simons interaction on a D-brane car-rying worldvolume flux,∫

C2 ∧ F2 ∧ F2 ∧ F2 =

∫d4x McFF , (30)

where M is defined as

M =1

2πα′

∫Σ2

F2. (31)

The coupling to the canonically normalized gauge field isg2M , where g2 = 1/τ is the gauge coupling. When thiscoupling is defined as a ratio to the decay constant, λ/4fwith f read off from the oscillatory potential, one finds1

λ =Ng2

2π=

N

2πτ, (32)

independent of the warping down of f or the number ofunits of flux M . Sizeable values of λ can result if the C2

axion wraps a small cycle in the compactification, e.g. forτ = 2, one can achieve λ = 5 by N = 62.

The allowed values for N , and hence λ, can be probedby demanding consistency with tadpole cancellation,which in turn relates N to the topology of the internalspace, as discussed in [52]. Ref. [52] found manifolds al-lowing for N = O(103), and with this in mind here weconsider N = O(1)× 102.

1 Carefully accounting for the (1/4) factor convention [51].

The global structure of the compactification is furtherconstrained by demanding weak coupling to the stan-dard model U(1). which is necessary to avoid the over-production of scalar non-Gaussianities known to arise inAbelian models (see Appendix A for a detailed discus-sion). This is easily achieved if the standard model isrealized on a large cycle τS.M. 1, which we will assumeto be the case here.

B. The Model and Gravitational Wave Signal

Putting the puzzle pieces together, we arrive at thecoupled axion gauge field system

L = −1

2(∂χ)

2 − µ4 cos

(χ

f

)− 1

4F 2 +

λχ

4fF F , (33)

with the parameters specified by:

µ4 =|W0|AV2N

τe−2πτ/N , f =fN

2πM

λ =N

2πτ, g2 =

1

τ(34)

where f is the warped-down axion decay constant, andf is defined as the factor appearing inside the cosine po-tential.

We numerically solve for the evolution of the χ,Qsystem in an inflationary background, and from this com-pute the gravitational wave signal. The amplification ofthe tensor power spectrum, for a host inflation modelwith rinf = 10−5 (Hinf ' 6× 10−7MPl), is given in theleft panel of Figure 1, where the two black curves corre-spond to λ = 10 and λ = 50. The other parameters arechosen as,

µ = 8.4× 10−6MPl, 1.2× 10−5MPl ,

6

g2 =1

2, f = 10−12MPl , χ∗ =

πf

40, (35)

where the two values of µ correspond to the λ = 10 and50 cases respectively, and χ∗ is the initial value of χ.Achieving these values of µ generically requires A 1,which we will see is also required by moduli stabilization.

For λ = 10, corresponding to N ' 126, the amplifica-tion of gravitational waves occurs for roughly an e-fold ofinflation, and hence a small range of k around the CMBpivot scale. On the other hand, for λ = 50, correspondingto N ' 628, the amplification lasts for ' 5 e-folds.

In both cases the amplification tracks the evolutionof mQ ∝ Q ∝ sin1/3(χ/f), leading to a peaked struc-ture, with the peak corresponding to the moment whenχ = πf/2. For larger values of λ, the amplification lastslonger, scaling roughly linearly with λ, while maintainingthe same peaked structure. This implies that both thetensor-to-scalar ratio r and the tensor tilt nt depend sen-sitively on the time at which the CMB pivot scale exitedthe horizon, as observed previously in [16].

The chirality of gravitational waves is also substan-tially impacted. One can define the helicity of the gravi-tational wave power spectrum as

HR−L(k) =PhR − PhLPhR + PhL

, (36)

where PhL,R refers to the tensor power spectrum of theleft,right tensor modes. The gravitational waves pro-duced by the non-Abelian gauge fields are purely right-handed, leading to a sizeable helicity on large scales. Thisis shown in the right panel of Figure 1. As discussed in[21, 22], this can be probed observationally by parity vi-olating CMB cross-correlations EB and TB.

The hallmark of this scenario is the short duration ofgravitational wave production, occurring for O(1)- O(10)e-folds of inflation. This is can be modified by allowingfor a large hierarchy between N and τ , or by a mon-odromy for the axion, which we now proceed to study.

IV. C2 AXION MONODROMY

An alternative construction is to include a monodromypotential [18, 19] for the C2 axion. This monodromypotential arises e.g. from the DBI action of an NS5 brane,leading to a potential of the form,

Vmono = µ4

√(χcf

)+

(χ

f

)2

, (37)

where µ is given by [19]

µ4 ≡ f · ε

g2s(2π)5α′2

, (38)

where ε encodes a warp factor dependence, and the valueof χc dictates the transition from a linear to quadratic

potential. As in [19], we consider the case that χ∗ χc,where the potential is approximated as V ' µ4χ/f .

As suggested in [53], axion monodromy can be straight-forwardly accommodated into supergravity, and modulistabilization, in the framework of F-term Axion Mon-odromy [54]. As a simple example, following [55], onecan realize this via the spontaneous symmetry breakingof a nilpotent superfield [56–58], via

δK =1

V(G,G)· SS , δW = MS, (39)

with M a constant and S satisfying the nilpotency con-straint S2 = 0. In [59] this was dubbed ‘D3 induced ge-ometric inflation’, but it applies more generally to braneand anti-brane constructions that spontaneously breaksupersymmetry [60].

The resulting correction to the scalar potential is de-coupled from the other pieces, due to the nilpotency ofS. The new term takes a simple form,

δV = |DSW |2 = M2V(G,G). (40)

The monodromy potential (37) is then realized by the

replacement χ = (G−G)/√

2i in (37).The monodromy potential allows the axion to roll over

a large distance in field space. This occurs providedthat the monodromy potential dominates over the non-perturbative oscillatory potential, which is indeed thecase when τs is stabilized at larger values. This quali-tatively changes the gravitational wave signal, which isnow amplified for a large number of e-folds, and this sub-stantially softens the constraint on λ.

The χ,Q system can again be numerically solved,and the resulting tensor power spectrum is shown in theleft panel of Figure 2, where we take λ = 2 and assume ahost inflation model with r = 10−6. The initial conditionfor χ is taken to be χ∗ = −200f , and we take µ = 1.861×10−6MPl.

This leads to an amplification of the tensor power spec-trum by a factor of 104, lifting the tensor-to-scalar ratioto r = 10−2. The helicity fraction is shown in the rightpanel of Figure 2, where again we see that the gravita-tional wave power spectrum is maximally helical on largescales.

In contrast with the case studied in Section III, thegravitational wave production lasts for a large numberof e-folds, even for λ = O(1). Additionally, the peakedstructure of the amplification is no longer present andthe gravitational wave spectrum on large scales is indis-tinguishable from large field inflation with rinfl = 10−2.

V. REALIZATION IN THE LARGE VOLUMESCENARIO

We now come to the consistency with moduli stabiliza-tion. Before studying the C2 axion and D7-branes with

7

0 10 20 30 40Ne0

1

2

3

4

5

Log10 Ph/Phinfl

0 10 20 30 40Ne0.0

0.2

0.4

0.6

0.8

1.0

1.2

HR-L

FIG. 2. Spectator C2 with a monodromy potential, with rinfl = 10−6 and with λ = 2. [Left Panel]: Amplification of the tensorpower spectrum Ph(k) for modes that exit the horizon in the first 40 e-folds of inflation. The red dashed line corresponds to ascale-invariant spectrum with r = 10−2. [Right Panel]: The net helicity fraction of the gravitational wave power spectrum.

worldvolume fluxes, we first study generalities of mod-uli stabilization and the possibility of gravitational waveproduction via interactions of the other axions of stringtheory.

A. Moduli Stabilization and Axions in the LargeVolume Scenario

String theory flux compactifications [61, 62] genericallylead to many axions in the four-dimensional effective fieldtheory [51]. However, not all of these, and indeed thevast majority of these, cannot realize the mechanism of[15], either because they are too heavy or because theirdynamics would lead to destabilization of the internalspace.

In this work we focus on the large volume scenariofor moduli stabilization [11, 12]. The complex structuremoduli and axio-dilaton are stabilized by the flux inducedsuperpotential,

W0 =

∫(F3 − SH3) ∧ Ω, (41)

where S = 1/gs + iC0. This provides the “universal ax-ion” C0 with a mass that is generically of similar size tothat of the dilaton, and hence stabilization of the dila-ton prevents C0 from having a parametrically small mass(m2

C0 H2 m2

gs), though some possible exceptions tothis have been studied in [63]. This prevents C0 fromslowly-rolling during inflation, and thus the C0 axion isnot a suitable candidate for the spectator axion of [15].

We now turn to the Kahler moduli. We consider aninternal manifold is of the ‘Swiss Cheese’ type, with onelarge bounding 4-cycle of size τb and many smaller 4-cycles of size τi. The volume is given by

V = α

τ3/2b −

h1,1∑i=2

λiτ3/2i

, (42)

where α and λi are model-dependent positive constants.Moduli stabilization is achieved by a combination of α′

corrections, appearing in the Kahler potential as [64]

K = −2 log

(V +

ξ

2

), ξ = − χ ζ(3)

2(2π)3g3/2s

, (43)

and by introducing a non-perturbative superpotential forone or more of the small cycles. The superpotential isgiven by

W = W0 +

h1,1∑i=2

Aie−aiTi . (44)

The scalar potential can be expanded in a series of 1/V,to give

V =1

V

h1,1∑i=2

8a2i

3λiA2i

√τie−2aiτi

+1

V2

h1,1∑i=2

4aiτie−aiτiW0 cos(aiθi)

+1

V3

3|W0|2ξ4

+ δVup, (45)

where δVup is an ‘uplift’ potential, responsible for makingthe post-inflation local minimum de Sitter 2.

The axions θi each have an oscillatory potential, whichone might hope could lead to the axion-spectator be-haviour necessary for gravitational waves. However,

2 The uplift term is conjectured to arise e.g. from anti-D3 branes[10] or D7 worldvolume fluxes [65], though this has been sub-ject to considerable debate e.g. [58, 66–68]. This term will notdirectly enter into the current analysis, which is focused on in-flation, and hence we do not study it further.

8

the stabilization of τi requires that the second term in(45) be negative and hence that the θi be stabilized atcos(aiθi) = −1 [69] . In contrast, in the example cosmolo-gies considered by DFF [15] the axion spectator beginsits cosmological evolution at cos(χ/f) = 0, in which casethe second term in (45) vanishes and the correspondingτi will runaway to infinity, decompactifying the internalspace. Hence, as is the case for C0, none of the θi aresuitable candidates to be the axion spectator of [15].

With the θi stabilized at cos(aiθi) = −1 moduli sta-bilization can proceed. Provided that one cycle is para-metrically smaller then the others, τs τi, such that theτs term dominates the superpotential, then one can ana-lytically find stabilized solutions by varying with respectto V and τs. The result is

〈V〉 =3λs√τsW0e

asτs

4aA, 〈τs〉 =

(ξ

2λs

)2/3

. (46)

The remaining moduli τi can be similarly fixed via theirnon-perturbative superpotentials, or else undergo infla-tionary dynamics.

Finally, we note that the τb axion θb also cannot realize[15]. The gauge kinetic function of the C4 axions couplingto gauge fields is given by [51]

fD7 = T. (47)

And hence the coupling of θb to canonically normalizedgauge fields on a stack of D7 branes wrapping the τb cycle,rescaled by the argument of the oscillatory potential, isgiven by λ = g2N/(2π) = N/(2πτb), as in equation (32).Since τb ' V2/3 is required to be large, sizeable valuesof λ require extremely large values of N , and hence anextremely large number of condensing branes.

B. Dynamics of C2 axion

We now come to the C2 axion. An important fea-ture of the large volume scenario is that volume stabi-lization does not involve the τi and hence is decoupledfrom the inflationary dynamics, up to perturbative cor-rections. We will demonstrate that the same is true oncedynamics are included for a C2 axion, thus allowing forarbitrary inflationary dynamics to be added in to themodel.

In the presence of worldvolume fluxes, the D7 gaugekinetic function is modified to include the two-form fieldsas [46, 47]3

fD7 = T + f iGi +1

2gsf ifi, (48)

3 We do not consider the possibility of multiple windings, whichhave the effect of rescaling fD7 [46].

where f i are the F2 flux quanta and Gi contain the two-form fields,

Gα = Sbα + icα, (49)

where again S = e−φ + iC0 and

cα =

∫Σ2α

C2 , bα =

∫Σ2α

B2. (50)

Considering only a single C2 axion, the Kahler potentialis given by [46, 47]

K = −2 log[τ

3/2b − λs(τs + γcτ

2G)3/2

], (51)

where τG ≡ G + G, and γc = c+−−/gs, with c+−− thetriple-intersection number of the even and odd cycles.This breaks the shift symmetry of the B2 axion, whileleaving that of C2 intact.

The D7 world-volume fluxes introduce further sub-tleties, notably in the analysis of anomaly cancellationand of D-terms, which have been studied in detail in thecontext of D3/D7 inflation [70]. Anomaly cancellationrequires additional fluxes and brane sources, which ismost easily studied in M-theory [70], where IIB world-volume fluxes are described by localized contributions tothe bulk G4 flux [71, 72]. On the IIB side, the inducedD-term takes the form [73, 74],

D =α′tα2V

καbc(bb − f b

)Wc, (52)

where tα are the two-cycle volumes, καbc are the tripleintersection numbers, andWc is the wrapping number ofthe brane. This leads to additional stabilization of B2,and again leaves C2 unaffected.

A potential for C2 can be generated by gaugino con-densation on the small cycle τs, as studied in section III.Here we consider that this occurs by either the condensa-tion of a product group or else of two independent branestacks. The superpotential is then given by a sum of twonon-perturbative terms4,

W = W0 +Ase−asTs +Ace

−ac(Ts+MG), (53)

with M the units of F2 flux. Moreover, we have absorbed

the fifi term in fD7 into the definition of Ac, and will

consider the magnetized gaugino condensate to be a sub-dominant contribution to the superpotential,

Ac As. (54)

4 We emphasize that both non-perturbative terms must necessar-ily arise from gaugino condensation, and not instantons. Ananalysis of the latter would require a sum over all instanton con-figurations, which would generically introduce a G dependencein the exponent of the first term.

9

0 10 20 30 40Ne0

2

4

6

8

10

Log10 Ph/Phinfl

FIG. 3. Kahler Moduli Inflation with a spectator C2. The peaked curve is an oscillatory potential with λ = 50, while the flatcurve is the monodromy potential with λ = 2. Dashed lines indicate (from bottom to top), r = 10−3, 10−2 and 10−1.

which is the case if fifi is large. This could also arise

if e.g. Ac has a dependence on τb as e−abτb , or couldeffectively be the case if ac as.

The scalar potential is given by, setting ac = as andγc = 1 for simplicity, and with the θi and B2 axionsstabilized,

V =1

V8a2s

3λsA2s

√τse−2asτs

[1 +

AcAs

cos(acMc)

]− 1

V24asAsτse

−asτsW0

[1− Ac

2Ascos(acMc)

]+

1

V3

3|W0|2ξ4

+ δVup +O(

(AcAs

)2

). (55)

Comparing with (45) we see that c-induced correctionsto the scalar potential scale as Ac/As, and hence modulistabilization of τs,V proceeds unchanged provided thatAc As.

It follows from this that the dynamics of C2 will notdestabilize the internal space, nor generate any potentialfor the τi. The latter implies that the C2 spectator sce-nario is consistent with arbitrary inflationary dynamicsfor the τi moduli.

Finally, in the stabilized minimum the first two linesin (55) are of the same order of magnitude, and the oscil-latory potential for c is precisely of the form anticipatedin section III.

VI. EXAMPLE HOST INFLATION MODEL:KAHLER MODULI INFLATION

As a concrete model example, we now consider KahlerModuli Inflation [31]. This inflation scenario arises from

supplementing (53) with a superpotential for a Kahlermodulus τφ, of the form Aφe

−aφTφ . The resulting modelis of the DFF form (2), with inflationary potential givenby

V (φ) ' V0

[1− α

(φ

MPl

)4/3

e−β(φ/MPl)4/3

], (56)

where φ ≡√

4λφ3V τ

3/4φ is the canonically normalized infla-

ton, and the parameters are V0 'W 20 /V3, α ' V5/3, and

β ' V2/3. Typical values of the volume in this scenarioare 105− 107, while 104 can achieved by a suitably smallW0 [69].

The precise observational predictions for Kahler Mod-uli Inflation have been studied by [75], and are given by

ns ' 1− 2

∆N∗,

r ' 4

81β3/2N2∗

log5/2

(24

√β

2α∆N∗

), (57)

where ∆N∗ is the number of e-folds before the end ofinflation when the CMB pivot scale exited the horizon.For typical values of the parameters, this evaluates to[31]

0.960 ≤ ns ≤ 0.967 , r ≤ 10−10. (58)

This value of the tensor-to-scalar ratio is not obervablein the foreseeable future. The corresponding energy scale

of inflation is V1/4inf ' 1013 − 1014 GeV, and the field

excursion is well within the small field regime.As an example of the amplification of r due to gauge

field production, we consider a model at the upper bound,

10

r = 10−10. We consider both an oscillatory potentialwith

λ = 50 , g2 =1

2, f = 10−12MPl , µ = 5.89× 10−8MPl,

corresponding to τs = 2 and N = 628, and a monodromypotential with

λ = 2 , g2 =1

2, f = 10−15MPl , µ = 2.48× 10−8MPl,

corresponding to τs = 2 and N = 13.The resulting tensor power spectra are shown in Fig-

ure 3. As in the example in section III, the oscillatorypotential case exhibits a peaked structure, and in this ex-ample rises above that of a scale-invariant spectrum withr = 10−3 for roughly 3 e-folds of inflation. The observedtensor-to-scalar ratio r, and also the tensor tilt nt, thusdepend sensitively on the time at which the pivot scaleleft the horizon. This is not the case for the monodromypotential, which exhibits an amplification to r = 10−3

for a large range of Ne.Thus we see that both spectator scenarios can real-

ize an observably large tensor-to-scalar ratio in KahlerModuli Inflation.

VII. CONCLUSION

In this work we have studied the gravitational waveproduction due to excitations of non-Abelian gauge fieldson D7 branes, the presence of which is intrinsic to stringinflation scenarios, due to the necessity of moduli stabi-lization. Worldvolume fluxes on the branes lead to anoscillatory potential for an axion, and an axionic cou-pling to the gauge fields. This leads to a realizations ofthe spectator scenario [15], and is easily extended to ainclude a monodromy potential for the axion.

The former case leads to a production of gravita-tional waves at the beginning of inflation, lasting fora few e-folds of expansion. The tensor mode powerspectrum is greatly amplified during this time, and thetensor-to-scalar ratio can be lifted to an observable level,r ' 10−3 − 10−2 [20]. The latter case leads to sustainedproduction of gravitational waves, and a large amplifica-tion of r even for λ = O(1).

This indicates that an observable level of gravitationalwaves is possible in small field inflation in string theory,once the full structure of model realizations is taken intoaccount. Moreover, the resulting gravitational wave spec-trum is maximally chiral, distinguishing it from othersources of gravitational wave production. To furtherquantify and constrain the observational signatures willrequire a full CMB polarization analysis, which we leaveto future work.

We have demonstrated that this can incorporated intothe Large Volume Scenario, and argued that there is neg-ligible backreaction on moduli stabilization, or on the in-flationary dynamics of the host model. However, a more

detailed analysis is certainly warranted, especially giventhe backreaction issues known to affect monodromy mod-els [3, 4], and recent results concerning the cosmologicalbackreaction of gravitational waves [76]. Related to this,in order to fully characterize the model predictions it isnecessary to perform a scan of the self-consistent param-eter space, both of the parameters describing the stringcompactification and of the those describing the cosmo-logical model.

Finally, we mention the connection of this work tothe dark matter scenario [30]. That work connected theinflationary production of chiral gravitational waves tothe simultaneous generation of dark and visible particle-antiparticle asymmetries, and via the condensed matterphysics of gauge theories [77], to a model of superfluiddark matter. The present work is a first step in the stringtheory realization of that scenario.

ACKNOWLEDGMENTS

The authors thank Keshav Dasgupta, Jerome Quintin,Ryo Namba, and Edward Wilson-Ewing, for insightfulcomments and suggestions. EM is supported in part bythe National Science and Engineering Research Councilof Canada via a PDF fellowship.

Appendix A: Scalar Non-Gaussianities

As shown in [15], the spectator sector considered hereleads to negligible induced scalar perturbations, andhence no change to the power spectrum. However,the contribution of the spectator sector to scalar non-Gaussianities requires close attention. This was recentlycomputed in [39], and confirmed to be small. Here wewill put their results in context.

There are two contributions to non-Gaussianities inaxion models:

1. Direct production of highly non-Gaussian inflatonperturbations.

2. Non-Gaussianity of the inflaton-sourced curvatureperturbation induced by new interaction vertices.

To understand these, it useful to first review non-Gaussianities in the simple case of axion inflation coupledto an Abelian gauge field [78, 79]. In that case, the dom-inant effect is the direct production of non-Gaussian per-turbations, occuring at second-order in perturbation the-ory via inverse decay AiA

i → δϕ of gauge fluctuations Aito inflaton fluctuations δϕ. The associated gravitationalwave production is similarly given by hij ' AiAj . Sinceboth these effects occur at second-order in Ai, achiev-ing an obersvable tensor-to-scalar is incompatible withsatisfying observable bounds on non-Gaussianities .

If the axion is instead taken to be a spectator χ, theinverse decay AiA

i → δχ produces entropy perturbations

11

δχ, whose contribution to the curvature perturbation issuppressed by ρχ/H (see e.g. [80]). In this way, the effect1. can be neutralized. However, effect 2. is still present,and the coupling

√−gχF F introduces an interaction ver-

tex [81],

Lint =λχ

4fHζF F . (A1)

The presence of this term can be seen by performing atime-translation t → t − δϕ/ϕ to the uniform ϕ gauge(δϕ = 0), where ϕ the inflaton, which simultaneouslytransforms δχ as δχ→ δχ+ (χ/H)ζ [81].

The non-Gaussianities induced by (A1) can be com-puted via a Feynman diagram with a loop of gauge fieldsAi and three external lines of ζ. The result is that thecurvature perturbation three-point function 〈ζζζ〉 inher-its the exponential growth of Ai, and is again generallyin conflict with observations, unless λ 1 (in which caseno interesting phenomenology occurs), or else if χ is non-zero only for a few e-folds of inflation [40].

We now come to the model studied in this paper, wherea spectator axion is coupled to non-Abelian gauge fieldsin the isotropic configuration. The suppression of effect1. is even more pronounced then in the Abelian spectatorcase, since the gravitational waves are produced at linearorder in fluctuations hij ' 〈Ai〉δAj , while the inversedecay is second order. The scalar sector is complicatedby the presence of genuinely scalar perturbations 〈Ai〉 =Q → Q + δQ, but as shown in [15], this system is free

from instabilities provided mQ >√

2.

Effect 2. has been calculated for the model studiedhere in Appendix C of [39]. There it was shown thatdemanding perturbativity of the curvature perturbation(i.e. that the one-loop correction to the power spectrumis subdominant to the tree-level contribution) ensuresthat the induced non-Gaussianity is well within obser-vational bounds, and that both of these conditions canbe satisfied for a broad range of parameter space (includ-ing that studied here).

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