Natural chaotic inflation and ultraviolet sensitivity

Nemanja KaloperDepartment of Physics, University of California, Davis, Davis, California 95616, USA

Albion LawrenceMartin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA

(Received 4 May 2014; published 3 July 2014)

If the recent measurement of B-mode polarization by BICEP2 is due to primordial gravitationalwaves, it implies that inflation was driven by energy densities at the grand unified theory scaleMGUT ∼ 2 × 1016 GeV. This favors single-field chaotic inflation models. These models require trans-Planckian excursions of the inflaton, forcing one to address the UV completion of the theory. We usea benchmark 4d effective field theory of axion–four-form inflation to argue that inflation driven by aquadratic potential (with small corrections) is well motivated in the context of high-scale string theorymodels; that it presents an interesting incitement for string model building; and that the dynamics of the UVcompletion can have observable consequences.

DOI: 10.1103/PhysRevD.90.023506 PACS numbers: 98.80.Cq, 11.25.Mj, 14.80.Va

I. INTRODUCTION

The detection of B-mode polarization by the BICEP2experiment [1,2] gives tantalizing evidence that quantumgravity may be directly relevant for observational cosmol-ogy. If the future checks confirm that the BICEP2 B modesare generated by primordial gravitational waves, this willgive support to the idea that these waves are induced byquantum fluctuations of the graviton. Furthermore, simplechaotic inflation models such as V ¼ 1

2m2ϕ2 [3] are in

excellent current agreement with the data. To generate theobserved density fluctuations in the CMB and large scalestructure, chaotic inflation models occur at energy densities≲M4

GUT ∼ ð2 × 1016 GeVÞ4. Any models generating obser-vable primordial gravitational radiation require trans-Planckian excursions of the inflaton in field space of orderΔϕ ∼ 10mpl [4,5] over the course of inflation.1 Therefore,all such models are sensitive to the UV completion at thePlanck scale, requiring a good understanding of quantumgravity. The purpose of this paper is to consider constraintson the UV completion of models dominated by a (possiblydistorted) quadratic potential and present them as anincitement for string model builders.Chaotic inflation is stable against perturbative quantum

corrections, as exemplified as early as [9,10], in response tothe concerns about irrelevant operator contributions raisedin [11]. This in fact follows if one protects the theory with(approximate) shift symmetries, so that all couplings to theinflaton are either very weak or via the derivatives of theinflaton. However, nonperturbative quantum-gravitationaleffects are expected to break such global symmetries,

inducing Oð1Þ coefficients for all Planck-suppressed oper-ators. When ϕ ranges over super-Planckian distances, theseoperators may spoil the small curvature of the inflatonpotential required for slow roll inflation. Another concern isthat in a string compactification, the degrees of freedomcan shift substantially as fields move over super-Planckiandistances; this can also manifest itself in dangerous Planck-suppressed operators. A good UV-complete realizationmust control these operators.2

Another technically natural high-scale model takes asthe inflaton a periodic pseudoscalar (a.k.a. an axion) forwhich the potential is generated by instantons [13–15]: Thetopology of field space protects the shift symmetry of theinflaton. The usual potential is V ∼ Λ4 cosðϕ=fÞ. However,high-scale slow roll inflation requires axion decay con-stants f > ϕ > mpl. Gravitational instantons, such aswormholes [16], and string theoretic instantons [17] typ-ically have actions of order S ∼ ðmpl

f Þp; thus, higher-orderinstantons can spoil slow-roll inflation [17,18].Axion monodromy models control the Planck-

suppressed operators by combining chaotic and naturalinflation. In this scenario, the inflaton is a compact scalarwith periodicity f < mpl. The presence of fluxes or branes“unwraps” the inflaton configuration space [12,19–22],leading to a multivalued potential as in Fig. 1. In whatfollows we will focus on the axion–four-form modelsof [12,21], which we dub “natural chaotic inflation,”as a benchmark theory from which to discuss the generalphenomenon of axion monodromy. We start with theLagrangian density

1We use the reduced Planck mass mpl ¼ 2.4 × 1018 GeV; wetakeMGUT to be the value suggested by supersymmetric couplingunification [6–8].

2For a more complete review and discussion of the issues inthis paragraph, see [12].

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L ¼ 1

2m2

plR −1

2ð∂ϕÞ2 − 1

48F2ð4Þ þ

μ

24ϕ�Fð4Þ; ð1Þ

where Fð4Þ ¼ dAð3Þ is a four-form field strength, with Að3Þa three-form gauge field. The canonical momentumpA123

¼ �Fð4Þ − μϕ is quantized in units of the electriccharge e2; using this, one may write the Hamiltonian as

H ¼ 1

2ðpϕÞ2 þ

1

2ð∇ϕÞ2 þ 1

2ðne2 þ μϕÞ2: ð2Þ

If ϕ → ϕþ f, we must have μf ¼ e2 for consistency. Thequantum number n can be shifted by the nucleation ofmembranes, leading to the multivalued potential for ϕshown in Fig. 1. At tree level, if membrane nucleation issuppressed, one has a model of chaotic inflation with aquadratic potential. The fundamental periodicity of thescalar implies that corrections to V ¼ 1

2μ2ϕ2 take the form

ðV=M4uvÞn, which can be small [12,21]. A danger to the

model remains: Couplings such as μ can depend on moduliwith Planck-suppressed couplings. If the moduli massesare of the order or smaller than the Hubble scale H, theypotentially destabilize inflation.The axion monodromy models of [19,20] turn these bugs

into features, by providing a UV completion in which theoverall effect of higher-order corrections and couplings tomoduli is to flatten the tree-level potential [19,20,23]. Theconstructions in those papers give potentials significantlyflatter than quadratic while still being large-field models,with a significant tensor-to-scalar ratio r. These may beconsistent with the BICEP2 result, given the currentstatistical significance of the results. Further, additionalUV complete models realizing the monodromy mechanism

starting from steeper potentials with larger r are underdevelopment [24]. Here we will focus on whether aquadratic potential generated by integrating the four-formstarting with (1) can be realized with only small correc-tions, using effective field theory as a guide. Our motivationfor selecting this particular class of models is the relativesimplicity of the dynamics which follows; yet we will findthat there is a range of interesting potential signatures. Wewill discuss the relevant dynamical scales arising in variousstring theory scenarios, to argue that existing modelsconsistent with grand unification are at the edge of beingviable in this regard. If viable, they can give a finiteprobability for nonperturbative transitions to occur in earlyepochs of inflation [12], and they can give observablecorrections to the tree-level quadratic potential. In particu-lar, we will show that the natural consequence of a singleUV scale Muv ≳MGUT close to the GUT scale yieldspotentially observable corrections to the scalar and tensorspectra. We present this as motivation for further work onhigh-scale string models.

II. SCALES AND CORRECTIONS

The theory (1) is protected from direct polynomialcontributions to the axion potential of the form ϕn

mn−4pl

by

the periodicity of the scalar [12,21]. Corrections of the form

δL ¼ cnM4n

uvF2ðnþ1Þ ð3Þ

for n ≥ 1 are not forbidden. Upon integrating out thefour-form, these lead to corrections of the form

δV ¼ c0nV tree

�V tree

M4uv

�n: ð4Þ

If Muv ≫ MGUT these are small. They may becomeimportant if Muv is close to but still larger than MGUT.Note that these corrections can be viewed as beinggenerated by integrating out the gauge field A, whose fieldstrength is F ¼ dA. In what follows we will focus mainlyon the signatures of these corrections.The other serious danger to this scenario (and to all

models of high-scale inflation) arises from the coupling ofmoduli to the inflaton. In models with geometric compac-tifications, moduli arise from the metric degrees of freedomwhich have kinetic terms weighted by mpl. Upon writingthese degrees of freedom in terms of canonical scalars ψ ,they appear in potential terms in the form ψ=mpl. We expectthe moduli mass terms to be generated by potential energiesof the form

WðψÞ ¼ M4w

�ψ

mpl

�; ð5Þ

FIG. 1 (color online). Energies as a function of ϕ, for thepotential V ¼ 1

2ðμϕþ qÞ2. The picture repeats itself each time

one shifts ϕ → ϕþ e2=μ≡ ϕþ f.

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where M is the scale of the physics generating the modulipotential. The masses are of the order M2

mod ¼ M4

m2pl. During

inflation, the inflaton potential provides an additionalsource term for the moduli,

Vðϕ;ψÞ ¼ 1

2μ2�

ψ

mpl

�ϕ2 þ 1

2M2

modψ2: ð6Þ

If ∂2ψV ∼ V

m2pl∼H2 ≳M2

mod, the modulus ψ is potentially

destabilized, absent some interesting mechanism; it canalso lead to flattening of the potential as in [23]. This can beavoided if M ≫ MGUT.

3 If M ∼Muv, this will be achieved.The theory may also have nonperturbative instabilities,

generated by the discharge of the fluxes which give riseto the potential V. There are two possibilities. The first isthat the inflaton jumps to a neighboring metastable branchof the multivalued inflaton potential, corresponding to a shiftn → n − 1 in (2). These will be mediated by membranescharged under Að3Þ. As a guide, Ref. [12] found that iffϕ ∼ 0.1mpl, the transitions were suppressed if the chargedmembranes had tension σ > ð0.2VÞ3=4. The second pos-sibility is a jump in some modulus ψ on which μ depends.This is mediated by a nucleation of another membrane. Thespace of possibilities is larger and depends on the jump in μitself. In Ref. [12], we estimated that for Δμ ∼ 0.1μ, thetransitions were suppressed if σ > ð0.8VÞ3=4. Note that if atransition of the former type does occur, one can realize the“unwinding inflation” of [25,26]: There are at least twodirections in the configuration space of (1) which poten-tially support inflation, and it would be worthwhile to studythese directions in a single model.In all these cases, if the “thin wall” approximation holds,

we found that the transition probability is well approxi-mated by the flat space result [27,28]:

P ∼ exp

�−27π2σ4

2ðΔVÞ3�: ð7Þ

For jumps in n,ΔV ∼ ϕf V, while for jumps in μ,ΔV ∼ Δμ

μ V.This means that such processes are increasingly disfavoredat later stages of inflation. However, the probability is largerat larger values of ϕ, implying that the final stage of smoothslow roll with small perturbations is of limited duration, asin unwinding inflation [25,26]. Such transitions couldgenerate interesting signatures early on or, equivalently,on the largest visible scales on the sky.

III. A SAMPLING OF STRING MODELS

A high scale of inflation V ∼M4GUT presents a healthy

incitement to string model builders. Some attractive modelsof axion monodromy inflation, both global compactifica-tions and “local” models describing a patch of putativecompactification, have been constructed in [19,20,22,23].These describe models with a potential flattened away fromthe quadratic minimum.To date there has been no deep investigation of models

realizing the effective field theory described in [12,21],though a schematic version was described in [21].4 We areinterested in the following:

(i) The scale Muv which controls the corrections inEq. (3), usually corresponding to the fundamentalstring or 10=11d Planck scale. Our scenario re-quires Muv ≫ MGUT.

(ii) The scale M which controls the potential for moduliwith generic couplings to the inflation; to avoidexcessive corrections to our tree-level action, we areinterested in whether M ≫ MGUT for these moduli.

The additional challenge is that these models must alsohave a somewhat lower scale μ ∼ 0.1M2

GUT=mpl controllingthe tree-level inflaton potential.In order to gain some intuition, let us consider two

scenarios broadly consistent with grand unification at thescale MGUT, with candidates for realizing our axion–four-form model. A review of the dynamical scales in stringmodels consistent with grand unification (with the aim oflooking for other signatures of high-scale physics in theCMB) can be found in [31].One example is M theory compactified on a manifold of

G2 holonomy, following [32–36]. These models aredescribed by a K3 fibration over a base S3=Zn, whosesize sets the GUT scale. The existence of axion–four-formmonodromy was pointed out in [37]; for an alternatediscussion in the case of toroidal compactifications of Mtheory, which captures the essential physics, see [21,38]. Ifthe GUT theory is realized following [32–36], consistencywith grand unification leads to [31]

m11 ∼MGUT; L−1K3 ∼ 0.1MGUT: ð8Þ

We expect a typical modulus potential term of the form (5)will have M at most equal to m11, up to pure numericalfactors, whileMuv in (3) should also be of this order. In thiscase, our scenario looks tenuous.We can also consider type-IIB theories. We leave explicit

realizations of axion–four-form monodromy for futurework, and simply discuss the dynamical scales of thisclass of models. In typical models consistent with grandunification at MGUT, the realization of the GUT scale will

3As discussed in [12], this may not be sufficient if ψ coupleswith more than gravitational strength, or if the minimum of Vwith respect to ψ is more than a Planck distance away fromψ ¼ 0.

4References [29,30] also present a string construction realizinga quadratic potential, although there is not a complete compacti-fication scenario there.

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depend on the origins of the gauge degrees of freedom. Ifthe gauge group lives on D3-branes, the gauge couplingwill be the string coupling—thus, the latter is weak. Thenatural scale for the GUT symmetry to be broken isthe string scale, in which case M4

uv ∼ V and correctionsof the form (3) are problematic. Furthermore, one is leftwith a Kaluza-Klein scale that is of order 0.1MGUT.Following the discussion in [39], we find that the heaviestmoduli, stabilized by three-form flux, have masses of orderH. If one instead realizes the gauge group on D7-braneswrapping finite-sized four-cycles and setsMGUT to be equalto the Kaluza-Klein scale and the string coupling to beOð1Þ (which is appropriate since the D7-branes source thedilaton), the string scale is of order 2MGUT. If Muv ¼ ms,these models are just at the edge of viability from thestandpoint of our scenario of inflation. Again followingthe discussion in [39], moduli stabilized by closed stringor brane worldvolume fluxes will have masses of orderM2

uv=mpl. However, other moduli are often parametricallylighter, so the inflaton must be shielded from these. We notethat this discussion is consistent with the explicit con-structions given in [40–42], in cases when the stringcoupling is pushed towards the GUT scale.We stress that the examples given here are meant to be an

illustration. Our point is simply that for string theory toproduce a model with a quadratic potential for the inflaton,which is consistent with the existing CMB data, it is naturalto consider compactifications of a fairly high scale, close tothe higher-dimensional Planck scale; furthermore, one mustwork hard to ensure that the moduli do not spoil thescenario. It is not clear that geometric compactifications arethe right arena to work in. Note also that a rescaling ofMuv,M by a factor of a few can make a significant difference. Aswe will see below, a consequence is that such models willfairly easily produce observable corrections of the form (4).It is not clear how much precise quantitative control we cangain over them in the near future. However, we advocatethat the problem of making some moduli heavy andsequestering the inflaton from the remaining moduli, ina class of high-scale models, deserves further scrutiny inand of itself.One should also pay additional attention to how the

inflaton couples to other fields to ensure that reheating can,in principle, proceed smoothly without overproducinglong-lived moduli. Some phenomenological explorationsfrom the bottom up, such as [43], may also provideinteresting points of contact with the high-energy theory.

IV. OBSERVABLE CONSEQUENCES OFCORRECTIONS

The scalar and tensor spectra for single-field slow rollinflation are, respectively,5

PS ¼�H2

2π _ϕ

�2

¼ 1

24π2m4pl

Vϵ

ð9Þ

and

PT ¼ 8

m2pl

�H2π

�2

¼ 2

3π2Vm4

pl

: ð10Þ

The tensor-to-scalar ratio is

r ¼ 16ϵ ¼ 8

m2pl

�_ϕ

H

�2

; ð11Þ

where ϵ ¼ 12m2

plð _ϕHÞ2 is a slow roll parameter. When discus-

sing explicit numbers, we use the following observedvalues as constraints:

ffiffiffiffiffiffiPS

p∼ 5 × 10−5;

r ∼ 0.2þ:07−:05 without dust subtraction;

∼ 0.16þ:06−:05 with dust subtraction; ð12Þ

computed at 50 e-foldings before the end of inflation. Thepower spectrum is based on the COBE result; the tensor-to-scalar ratios come from the BICEP2 result. We willcompute corrections to the tensor and scalar spectrum asa parametric function of the number of e-folds left beforethe end of inflation N (which serves directly as the clock)and the corrections to V; to estimate the size we will use thenumbers above. Future data could change the observedvalue of r, and our choice of N ¼ 50 implies a choice ofreheating history. This will change the detailed numbers wecalculate below, but not our qualitative conclusions.6

First, we review the simple tree-level quadratic potential.The number of e-folds before the end of inflation isN ¼ 1

4ð ϕmplÞ2. The resulting scalar and tensor powers are

PS ¼ μ2

6π2m2pl

N 2; PT ¼ 4μ2

3π2m2pl

N ; ð13Þ

so that r ¼ 8N . The observed value of PS and N ¼ 50 fixes

the inflaton mass,

μ≃ 1.8 × 1013 GeV: ð14Þ

From this it follows that the tensor-to-scalar ratio isr ¼ 0.16, and the energy scale of inflation is V1=4

50 ¼ð2μ2m2

plN 50Þ1=4 ¼ 2.1 × 1016 GeV. At 60 e-folds before

5Here we use the conventions of [44], which differ from thosewe used in [12].

6Note that corrections to N -independent quantities used in[45] are parametrically of the same order as the corrections to thespectrum we find here.

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the end, the scale of inflation is V1=460 ¼ 2.2 × 1016 GeV,

not much different. Thus, the scale of quadratic inflation isvery close to the GUT scaleMGUT ∼ 2 × 1016 GeV. This isconsistent with the BICEP2 result.The principal sources of corrections to this scenario arise

from membrane nucleation, corrections of the form (3), andlight moduli [12,21].7 First, consider nucleation of a bubblewhich changes μ by Δμ ∼ 0.1μ. Suppression of this effect(to preserve large scale homogeneity and isotropy) leads toa bound σ1=3 ≳ 9.8 × 1015 GeV on the membrane tension,very close to the GUT scale. Since ΔV ¼ 2ðΔμ=μÞV in thiscase, the probability will increase in earlier epochs. Howearly that occurs is exquisitely sensitive to the scale settingthe membrane tension, due to the large powers whichappear in (7). For example, if the membrane tension isprecisely M3

GUT, and Δμ ∼ 0.1μ, the nucleation probabilityis of order 1=e when N ∼ 960, V ∼ 3.6 × 1016 GeV,ϕ ∼ 44mpl. A slight decrease in σ1=3 places this epochclose to the observable one. Thus, if the relevant scales areclose to the GUT scale, our Universe may have been theresult of a bubble nucleation in the relatively near past. Thiscan lead to a small negative spatial curvature [46,47].Moreover, the bubble walls themselves can fluctuate,leading to a deformation of the homogeneous slices insidethe bubble. If one assumes that the tension is set by theGUT scale, that small fluctuations are governed by theaction Sbubble ¼ σ

Rd3xð∂XÞ2, and that δð ffiffiffi

σp

XÞ ∼ ffiffiffiffiH

p,

the fluctuations of these homogeneous slices are compa-rable to the fluctuations due to the inflaton. The authors of[48] have argued that this could yield a large scaleasymmetry in the power spectrum, in line with claims thatsuch an asymmetry has been observed in the CMB [49].The corrections (4) to the tree-level potential (2) can

affect the slow roll regime during the observable epoch ofinflation in a significant way if Muv ≳MGUT. As we havediscussed, this occurs for many string models, consistentwith grand unification. Because of the particular nature ofour baseline tree-level quadratic potential, the slow rollcorrections are particularly simple, being parametrized byϵ ∼ 0.01. We will find that the new physics corrections (4)are significantly larger. Since V tree ∼M4

GUT, the correctionscan be comparable to the tree-level potential. For example,the first term in the expansion induces a quartic coupling,c01V tree

V treeM4

uv¼ λ4ϕ

4=4, with the induced coupling

λ4 ¼ 4c01

�μ

Muv

�4

: ð15Þ

Using the mass of the inflaton (14) and takingMuv ≳MGUT ¼ 2 × 1016 GeV, this yields

λ4 ≲ 6.6 × 10−13c01; ð16Þ

where we expect c01 ∼Oð1Þ. A purely quartic inflatonpotential consistent with the observed PS hasλ ∼ 2.8 × 10−13. Such a potential does not fit current valuesof ns and r: It must be subdominant to the quadratic term.However, since λ4 scales as 1=M4

uv, a slight increase inMuvcan place λ in a range consistent with current observations.AsMuv can be within a factor of a few ofMGUT, our theoryis at the edge of respectability [12]. It is therefore importantto examine these corrections carefully: They can correct theamplitude of the scalar density spectrum generated by thetree-level quadratic potential by tens of percents [3].We assume the corrections are subleading, so that the

Taylor expansion for the effective potential whose terms aregiven in (4) is well within its radius of convergence:Upon fitting to (12), this assumption will turn out to beself-consistent. Consider a quadratic potential with a do-

minant subleading term ΔV¼c0nV treeðV treeM4

uvÞn¼ λ2ðnþ1Þ

2ðnþ1Þϕ2ðnþ1Þ,

where λ2ðnþ1Þ ¼ 2ðnþ 1Þc0n μ2ðnþ1ÞMuv

4n ¼ 2ðnþ 1Þcn μ2ðnþ1ÞMGUT

4n and

cn ¼ c0nðMGUTMuv

Þ4n. We compute the corrections to the infla-tionary dynamics (in the slow roll approximation) to thefirst nontrivial order in λ2ðnþ1Þ. We expect that the lowestvalues of n will dominate the corrections to inflationaryobservables. First, if V tree=M4

uv < 1, higher powers aresmall to begin with. Second, corrections from higherpowers die off more quickly during inflation: BetweenN 2 andN 1 e-foldings of inflation, the nth term will reduceby a factor of ∼O½ðN 1=N 2Þnþ1�.Wewish to express physical quantities during inflation in

terms of the number N of e-foldings before the end ofinflation. Since

N ¼Z

ϕ

ϕend

dϕm2

pl

V∂ϕV

; ð17Þ

using V ¼ V tree þ ΔV and expanding in ΔV, we first find acorrection8

N ¼ 1

4

�ϕ

mpl

�2

−nλ2ðnþ1Þ4ðnþ 1Þ2

ϕ2ðnþ1Þ

m2plμ

2; ð18Þ

where we ignore the Oð1Þ ≪ N contribution of ϕend in(17). Inverting (18), we find

7We will put aside for now the interesting possible correctionsarising from periodic modulations of the potential [12,19,20].

8In what follows we will concentrate on monomial correctionsfor simplicity. Since we work to linear order in λ2ðnþ1Þ, morecomplicated cases with corrections involving several differentlow powers can be obtained by superposition, taking care withpossible cross-contamination of different terms by slow rollcorrections.

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1

4

�ϕ

mpl

�2

¼ N þ 22nþ1ncn

ðnþ 1Þ�

μmpl

MGUT2

�2nN nþ1: ð19Þ

At the order we are working in, we can insert the tree-levelexpression for μ into the correction term of (19), to find14ð ϕmplÞ2 ¼ N ½1þ 2 ncn

ðnþ1Þ ðN21Þn�. For the correction to be

subdominant for N < 60, cn < nþ12n ð0.35Þn.

Next, the corrected values of the potential V and the slowroll parameter ϵ in terms of N are9

V ¼ 2μ2m2plN

�1þ 22nþ1ð2nþ 1Þ

nþ 1cn

�μmpl

MGUT2

�2nN n

�

¼�0.45 · 1016 GeVÞ4N

�1þ 2ð2nþ 1Þ

nþ 1cn

�N21

�n�;

ϵ ¼ 1

2N

�1þ 2nð2nþ 1Þ

nþ 1cn

�N21

�n�: ð20Þ

Using Eqs. (9) and (10) we obtain the corrected formulasfor the scalar and tensor spectra,

PS ¼ μ2

6π2m2pl

N 2

�1 −

2ð2nþ 1Þðn − 1Þnþ 1

cn

�N21

�n�;

PT ¼ 4μ2

3π2m2pl

N�1þ 2ð2nþ 1Þ

nþ 1cn

�N21

�n�: ð21Þ

A clear effect of the corrections is to enhance (for cn > 0)the tensor-to-scalar ratio relative to that generated by thetree-level potential, rtree 50 ¼ 0.16,

r ¼ 16ϵ ¼ 8

N

�1þ 2nð2nþ 1Þ

nþ 1cn

�N21

�n�: ð22Þ

We claimed above that we can ignore corrections at highorders in n. If we demand that the corrections be small, wemust have cn<

nþ12nð2nþ1Þð0.32Þn. Since cn ∼ ðMGUT=MuvÞ4n,

this power-law behavior is entirely plausible ifMuv > 1.3MGUT. Corrections then appear as a series incnðN =21Þn ∼ ð0.76Þn even in this worst-case scenario.The corrections at n ∼Oð1Þ can be significant in early

epochs of inflation, and larger than the corrections to slowroll. For a quadratic potential, the latter are multiplicativecorrections of order ϵ, η ∼ 0.01. Let us consider the cases ofn ¼ 1, 2 corresponding to quartic and sextic corrections tothe inflaton potential. In the quartic case, if we fit thepotential to r ¼ 0.2 at N ¼ 50, we find that c1 ∼ 0.035,corresponding to Muv ∼ 2.3MGUT. At the earlier epochN ¼ 60, such a correction gives multiplicative corrections

of order 0.3 to the tree-level expressions for ϵ and V. Notethat the correction to PS vanishes at this order for a quarticcorrection. For a sextic correction, setting r ¼ 0.2 atN ¼ 50 implies c2 ∼ 0.0066. At N ¼ 60, this givesmultiplicative corrections of order 0.36 to ϵ, and of order0.18 to V. In particular, note that ΔPS=PS ∼ −0.18 (if c01 ispositive) at this order.The possible reduction of PS induced by n ≥ 2 terms is

interesting given the claimed observation of the lack ofpower at low multipoles [49]. Obviously, higher ordercorrections can help account for some of it, suppressing thescalar spectrum power by slightly increasing the inflatonrolling. Other mechanisms involving additional light fieldsthat induce a small amount of curvaton-induced perturba-tions have also been proposed (see [50]). In fact, such ascenario might occur in the presence of any additional lightmoduli in our framework, which couple to the inflaton onlyweakly, or not at all.The spectral indices nS ¼ 1þ d lnPS

d ln k ¼ 1 − 1PS

dPSdN and

nT ¼ d lnPTd ln k ¼ − 1

PT

dPTdN are a useful diagnostic of corrections

to V tree. From (21) we find

nS ¼ 1 −2

Nþ 2nð2nþ 1Þðn − 1Þ

21ðnþ 1Þ cn

�N21

�n−1

;

nT ¼ −1

N−2nð2nþ 1Þ21ðnþ 1Þ cn

�N21

�n−1

¼ −2ϵ: ð23Þ

The last of these two equations is the inflationary con-sistency condition [51], nT ¼ −2ϵ ¼ −r=8, which must betrue for single-field slow roll inflation. Further, the sub-leading corrections to nS and nT are related, their ratiobeing fixed by the power of the dominant correction2ðnþ 1Þ. In the case when n ¼ 1 and ΔV is quartic, thecorrections to the spectral indices are scale invariant—thatis, independent of N ∝ k. This should be expectedsince the quartic potential is scale invariant. In this case,the quantitative corrections obey ΔnS ¼ 0, ΔnT ¼−3c1=21≲ 10−2, if we fix the quartic correction bydemanding r ¼ 0.2. A combination of r, nS can be formedwhich isolates corrections to the quadratic potential [45]. Ifpossible, it will also be important to measure nT to test theconsistency condition, as recently stressed in [52].More generally, one should also consider the case where

the subleading corrections to the potential compete directlywith the quadratic potential in the earliest epochs ofinflation, to see their effect on the data. This is a naturalpossibility if, as we stated, the UV completion is near theGUT scale. The quadratic potential eventually dominates,but higher-order corrections can exert more influence onthe largest scales on the sky. A detailed fitting of the data isbeyond the scope of our work here; some attempts fittingpolynomial potentials may be found in [53].The bottom line is that our examples demonstrate that

subleading corrections are observationally relevant (and not

9The factor of 21 and the prefactor in V arises from (14), whichis slightly dependent on N . One may replace N =21 →ð2μmpl=M2

GUTÞ2N in the formulas below for a more generalexpression, but the difference will be down by a factor of ϵ.

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yet ruled out) when the leading potential is quadratic andthe UV cutoff of the low-energy effective theory governinginflation is close to MGUT.

V. SUMMARY—CHAOTIC EVIL INFLATION

The observation of primordial gravitational waves byBICEP2 has ushered a new era in cosmology. The resultsare pointing very strongly to the large field slow roll modelsof inflation [3] as the dynamics that shaped the Universe(although a lively discourse on this is still ongoing; see[45,54–63]). This presents a constructive incitement tomodel building, in particular, in string theory, because largefield models are sensitive to the details of the UVcompletion in which they should be embedded.In this work we have addressed the UV sensitivity of the

specific class of models described in [12,21], which lead toquadratic potentials and are in excellent agreement with thecurrent cosmological data.10 Since such models operate at

the GUT scale and string theories consistent with grandunification have fundamental scales close to the GUT scale,the UV physics may leave imprints in the sky in the form ofsmall corrections to the leading order CMB observables.These signatures can at least be used to constrain the UVphysics. Even more interestingly, future observational testscould use these signatures as benchmarks to search for, inorder to further test large field inflation.We feel that the discussion here only reinforces the

conclusion of [12]: that this class of chaotic inflationmodels lives at the edge of respectability, which is oftenthe most interesting place to be.

ACKNOWLEDGMENTS

We would like to thank Guido D’Amico, Matt Kleban,Andrei Linde, Fernando Quevedo, Eva Silverstein, MartinSloth, Lorenzo Sorbo, and Alexander Westphal for veryuseful discussions. A. L. would like to thank the AspenCenter for Physics for its hospitality during the “NewPerspectives on Thermalization” winter conference, at theonset of this work. N. K. is supported by the DOE GrantNo. DE-FG03-91ER40674. A. L. is supported by the DOEGrant No. DE-SC0009987.

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