Aligned natural inflation with modulations

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Aligned

natural

inflation

with

modulations

Kiwoon Choi

a

,

Hyungjin Kim

a

,

b

,

∗

a_{CenterforTheoreticalPhysicsoftheUniverse,InstituteforBasicScience(IBS),Daejeon,34051,RepublicofKorea} b_{DepartmentofPhysics,KAIST,Daejeon,305-701,RepublicofKorea}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received7January2016

Receivedinrevisedform20May2016 Accepted30May2016

Availableonline9June2016 Editor:G.F.Giudice

Theweakgravityconjectureappliedforthealignednaturalinflationindicatesthatgenericallytherecan be amodulationoftheinflatonpotential,with aperioddeterminedbysub-Planckianaxion scale.We studytheoscillationsinthe primordialpowerspectruminducedbysuchmodulation,and discussthe resultingobservationalconstraintsonthemodel.

©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Inflationintheearlyuniverseexplainstheflatness,horizon,and entropyproblemsinthestandardbigbangcosmology,while pro-viding a seed of the large scale structure andthe anisotropy in cosmic microwavebackground(CMB) radiation observedtoday. If theenergyscaleofinflationishighenough,thedeSitterquantum fluctuationofspacetimemetriccangiverisetoaprimordialtensor perturbationwhichmightbelargeenoughtobedetectableinthe nearfuture.Ontheotherhand,such highscaleinflationdemands asuper-Planckian excursionoftheinflaton[1],soa scalar poten-tialwhichisflatoverasuper-Planckianrangeoftheinflatonfield. InviewoftheUVsensitivityofscalarpotential,thisisanontrivial conditionrequiredfortheunderlyingtheoryofinflation.

As is well known, a pseudo-Nambu–Goldstone boson

φ

can haveanaturally flatpotentialoverafieldrangecomparabletoits decayconstant f . Thelowenergypotential isprotectedfrom un-knownUV physics underthe simpleassumption that UV physics respects an approximate global symmetry which is non-linearly realized in the low energy limit as

φ/

f

→ φ/

f

+

c, where c is

a real constant. Inthe natural inflation scenario [2,3], inflaton is assumedto be a pseudo-Nambu–Goldstone bosonhaving a sinu-soidalpotentialgeneratedbynon-perturbativedynamics.Thenthe inflationary slow-roll parameters have a size of

O(

M2_Pl

/

f2

)

, and thereforethemodelrequires f



MPl,where

M

Pl



2

.

4

×

1018GeV isthereducedPlanckscale. Althoughitappears tobe technically naturalwithin the framework of effective field theory, there has beena concern that the requiredsuper-Planckian decayconstant may not have a UV completion consistent with quantum

grav-*

Correspondingauthor.

E-mailaddresses:kchoi@ibs.re.kr(K. Choi),hjkim06@kaist.ac.kr(H. Kim).

ity [4]. Also, previous studies on the axion decay constants in stringtheorysuggestthat generically f

<

MPl,atleastinthe per-turbativeregime[5–7].

Nevertheless,onecanengineerthemodeltogeta super-Planck-ian axion decayconstant within the framework of effectivefield theory [8–11], or even in string theory [12–14]. An interesting approach along this direction is the aligned natural inflation [8, 15–24]. In this scheme,initially one starts withmultiple axions, all having a sub-Planckian decay constant. Provided that the ax-ion couplings are aligned to get a specific form of axion poten-tial [8],a helical flat directionwith multiplewindings is formed in the multi-dimensional field space withsub-Planckian volume. Ifthe numberofwindingsislargeenough, thisflatdirectioncan haveasuper-Planckianlength,soresultinaninflatonwith super-Planckianeffectivedecayconstant.

Recentlytherehasbeenarenewedinterestintheimplicationof theweakgravityconjecture(WGC)[25–29]forthealignednatural inflation. TheWGCwas proposedinitially forU

(

1

)

gauge interac-tion [4], implyingthat there shouldexist a chargedparticlewith mass

m and

charge

q satisfying q

/

m

≥

1

/

MPl,sothegravityshould beweakerthanthe

U

(

1

)

gaugeforce.1 Whentranslatedtoaxions, theWGCsuggeststhatthereshouldexistaninstantonwhich cou-plestothecorrespondingaxion withastrengthstrongerthanthe gravity.Thisleadstoanupperboundonthedecayconstantof in-dividualaxion, whichis givenby f

≤

MPl

/

Sins,where Sins isthe Euclideanactionoftheinstanton[4].

1 _{There aretwodifferent versionsofthe WGC, the strongand themild. The} strongWGCrequiresthatthemassandchargeofthelightestchargedparticle sat-isfyq/m≥1/MPl,whileinthemildversiontherequiredparticlewithq/m≥1/MPl doesnothavetobethelightestchargedparticle.Herewearemostlyconcerned withthemildversiongeneralizedtothecaseofmultipleaxions.

http://dx.doi.org/10.1016/j.physletb.2016.05.097

0370-2693/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

It has been noticed [30] that for models with multiple U

(

1

)

gauge interactions, the WGC often leads to a stronger constraint onthecharge-to-massratiosthantheoneobtainedbyconsidering the individual U

(

1

)

separately. The reason is that the WGC ap-pliesforalldirectionsinthemulti-dimensionalcharge space, not justforthechargevectorsoftheindividualparticles.Forthecase ofmultipleaxions,one similarly obtains astronger constrainton theaxioncouplings[27].Onethenfinds[25–27,29]thatthe align-mentmechanism cannot be compatible withthe constraintfrom the WGC if the axion–instanton couplings required by the WGC coincidewiththecouplingsgeneratingtheaxionpotentialthat im-plementsthealignmentmechanism.

Asimplesolutiontothisproblemisthatsomeoftheinstantons requiredbytheWGCdonotparticipateinimplementingthe align-mentmechanism[25]. Sincesome partoftheaxion potentialfor thealignmentmechanismcouldbeinducedbyeitherperturbative effectssuchasfluxorother formsofnonperturbativeeffectssuch ashiddengauginocondensation,thisappearstobearather plau-siblepossibility.Indeed,knownstringtheoreticconstructionofthe alignednaturalinflation involvesan instanton whichis not rele-vantforthealignmentmechanism

[31–35,54]

.Yet,suchinstanton can generatean additional axion potential which doesnot affect thealignment mechanism asit corresponds to a subleading cor-rection, but may give rise to an observable consequence in the precision CMB data. As we will see, in the presence of such in-stanton,thelowenergyinflatonpotentialisgenericallygivenby2

Veff

(φ)

= 

4eff



1

−

cos



φ

feff



+ 

4 mod



1

−

cos



φ

fmod

+ δ



,

(1)

where the first term with feff



MPl corresponds to the domi-nantinflatonpotentialgeneratedbyadynamicsimplementingthe alignment mechanism, while the second term with fmod



MPl and



4_mod

 

4_eff isa subleadingmodulationgeneratedby an in-stantonwhichisrequiredbytheWGC,butdoesnotparticipatein implementingthealignmentmechanism.

Inthispaper, we firstprovide an argumentimplying that the presence of subleading modulation in the inflaton potential is a genericfeatureofthealignednaturalinflationconsistentwiththe WGC.We then studythe observable consequences of such mod-ulation, while focusing on the parameter region favored by the-oretical orphenomenological considerations. Specifically we con-sider the region feff



5MPl to avoid a fine tuning of the initial condition, while being consistent with the CMB data [36], and

fmod



M_2πPl which is suggested by the WGC. Observable conse-quences of modulation in the axion monodromy inflation were studiedextensively in [37–41],where it was noticed that modu-lationcanleadtoanoscillatorybehaviorofthepowerspectrumof theprimordialcurvatureperturbation.Weexaminetheconstraint fromCMBonmodulationsforthecaseofalignednaturalinflation, andfindthattheCMBdatarestricttheamplitudeofmodulationas



4_mod

/

4_eff



O(

10−4–10−6

)

,dependinguponthevalueof fmod. It has been pointed out that modulation may significantly changethepredictedvalueofthetensor-to-scalarratio

r in

natural inflationscenario

[31,54]

.Wefindthatthechangeofthepredicted valueof r due to modulationis minor,e.g.atmostof

O(

10

)

%, if theamplitudeofmodulationiswithin therangecompatiblewith

2 _{Theso-calledmulti-naturalinflationscenario[42,43]assumesthesameformof} inflatonpotential,butwith fmodMPl/2πandeff mod,whoseobservational consequencesaredifferentfromourcasewith fmod<MPl/2πandeff mod.

theobservedCMBdata.3 _{Ontheother hand,includingan}

oscilla-torypartofthecurvaturepowerspectrumindata-fittinganalysis, wefindthat alargerparameterregioninthe

(

ns

,

r

)

planecanbe compatiblewith theCMB datacompared to thecasewithout an oscillatory piece,where

n

s denotesthe spectral indexofthe cur-vature powerspectrum. Thismakes it possible thatthe potential tensionbetweenthe CMBdata andthenaturalinflation scenario is ameliorated underthe assumptionthat there exists a modula-tion ofthe inflatonpotential yielding a proper size ofoscillatory pieceinthecurvaturepowerspectrum.

Thispaperisorganized asfollows.Insection2,we revisit the weak gravityconjecture appliedformodels withmultipleaxions, aswellastheKim–Nilles–Peloso(KNP)alignmentmechanism.We arguethatasmallmodulationoftheinflatonpotentialisageneric featureofthealignednaturalinflationcompatiblewiththeWGC. Insection3,westudytheoscillationsinthecurvaturepower spec-trum induced by modulation, anddiscuss the constraints onthe modelfromtheCMBdata.Section4istheconclusion.

2. WeakgravityconjectureandtheKNPalignment

Inthis section,we revisit theweak gravity conjectureapplied forthealignednaturalinflation, aswell asthealignment mecha-nism.As we willsee,theWGCimpliesthat genericallytherecan beasmallmodulationoftheinflationpotential,withaperiod de-terminedbysub-Planckianaxionscale.

Let us begin with the constraint on the axion couplings for models withmultipleaxions,whichis referred tothe convex hull condition (CHC)[27,30].It requiresfirstthatinthepresence ofN

axions,

φ = (φ

1

, φ

2

, . . . , φ

N

),

thereexistcorresponding(atleast)

N instantons

generating axion-dependentphysicalamplitudesas

A

I

∝

exp



−

SI

+

iq

I

· φ



(

I

=

1

,

2

, . . . ,

N

),

(2)

where SI denotes theEuclideanactionof the I-th instanton,and the axion–instanton couplings

qI are linearly independent from eachother.Itisalwayspossibletoparametrizetheaxion–instanton couplingsas

qI

=



nI1 f1

,

nI2 f2

, . . . ,

nI N fN



,

(3)

where fi (i

=

1

,

2

,

. . . ,

N)

canbe identifiedasthedecayconstant ofthe

i-th

axion, and

n

Ii areinteger-valuedmodelparameters,so theinstantonamplitudesareperiodicundertheaxionshift:

φ

i

→ φ

i

+

2πfi

.

(4)

In thefollowing, we willassume forsimplicitythat all instanton actionshaveacommonvaluebiggerthantheunity,4 _e.g.

SI

=

Sins

>

1

.

By taking an analogy to the case of multiple U

(

1

)

gauge fields, it hasbeen argued that theaxion–instanton couplingsshould be stronger than the gravity in all directions in the N-dimensional

couplingspace.Specifically,onefindsthattheconvexhullspanned by

3 _{Foranalternativescenariowhichcangiverisetoasignificantlysmallerr within} the(aligned)naturalinflation,see[44–46].

4 _UsuallyS

I∝1/g2foracertaincouplingconstant g,andthenastrong–weak

zI

≡

MPl Sins

qI (5)

should contain the N-dimensional unit ball with a center atthe origin.Equivalently,oneneeds

|

qI

| >

Sins

MPl

for all I

,

(6)

andforan

arbitrary unit

vectoru,

|

u

·

qI

| >

Sins

MPl

for some I

.

(7)

Theaboveconvexhullconditionhasanimmediateconsequence onthealignednaturalinflation.Toseethis,letusconsidera sim-pletwoaxionmodelforthealignmentmechanism,whichhasthe followingKNP-typeaxionpotential:

V0

= 

41



1

−

cos



p1

· φ



+ 

4 2



1

−

cos



p2

· φ



,

(8)

wheretheaxioncouplings

pI (I

=

1

,

2)canbeparametrizedas

p1

=



_˜

n11 f1

,

n

˜

12 f2



and p

2

=



_˜

n21 f1

,

n

˜

22 f2



(9) withinteger-valued n

˜

i j. For



14

∼ 

42, which will be assumedin thefollowingdiscussion,theaboveaxionpotentialhasan approx-imatelyflatdirectioninthelimitthat

p1 and

p2 arealignedtobe nearlyparallel: sin

θ

p

=

1

|

p1

||

p2

|

det



p1

p2





1

.

(10)

A particularlyconvenient parametrization ofthis flat directionis providedby

φ

inf

=

1

|

p1

−

p2

|

det



_φ

p1

−

p2



≡ ξ · φ,

(11)

wherethe flatdirectionunit vector

ξ

ischosen tobe orthogonal to

p1

−

p2.Afterintegratingouttheheavyaxion,weareleftwith alightinflaton

φ = φ

inf

ξ

(12)

withaneffectivepotential

V0

= (

41

+ 

42

)



1

−

cos



φ

inf feff



,

(13) where feff

=

1

| ξ ·

p1

|

=

1

| ξ ·

p2

|

=

|

p1

−

p2

|

det

p1

,

p2



T

.

(14)

If

p1 and

p2 are aligned to be nearly parallel, then the inflaton direction

ξ

becomesnearly orthogonal toboth

p1 and

p2, which resultsin feff



MPl although fi are all sub-Planckian.Note that ourparametrizationoftheinflatondirectioncanreceivea correc-tionof

O(

fi

/

feff

)

,whichwouldgiverise toacorrectionof

O(

fi

)

to feff.

Obviously the convex hull condition (7) for u

= ξ

cannot be compatible with feff



MPl in (14), if the axion–instanton cou-plings

{

qI

}

requiredbytheWGCcoincidewiththecouplings

{

pI

}

generatingtheKNP-type axion potential(Fig. 1).Therefore,in or-derforthealignmentmechanismtobecompatiblewiththeWGC, some of

{

qI

}

should not be in

{

pI

}

[27]. In view ofthat at least some partoftheKNP-type axionpotential can beinduced by ei-therperturbative effects,e.g.flux,ornonperturbativeeffectsother thaninstantons,e.g.hiddenquarkorgauginocondensations,there

Fig. 1. TheKNPalignmentmechanismandtheconvexhullcondition.Theaxion

cou-plings p1and p2(bluearrows)arealignedtobenearlyparalleltoproducea super-Planckianeffectivedecayconstant.Thesecouplingsdonotnecessarilycoincidewith theinstantoncouplingsq 1and q2requiredbytheweakgravityconjecture.Herewe assume q1= p1,andthereforeq = q2(blackarrow).(Forinterpretationofthe refer-encestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthis article.)

is no apparent obstacle to satisfyingthis condition. Letq denote

such axion–instanton couplingin

{

qI

}

,whichdoesnotparticipate inthealignmentmechanism. Yet,generically

q can inducean ad-ditionalpieceofaxionpotential



V

= 

4_mod



1

−

cos



q

· φ + δ



(15) with



4_mod

 

4_1,2 asthisadditionalpotentialshouldnotspoilthe alignmentmechanism.NowtheWGCrequiresthattheconvexhull spanned by

{

pI

,

q

}

should contain the unit ball. This meansthat once

{

pI

}

are alignedtoyield 1

/

feff

= | ξ ·

p1

|

= | ξ ·

p2

|



1

/

MPl, thereshouldexista

q satisfying

| ξ ·

q

| >

Sins MPl

.

(16)

Again,onecanintegrateouttheheavyaxiontoderivetheeffective potential ofthelight inflaton.With

φ = φ

inf

ξ

,onefindsthat the totalinflatonpotentialisgivenby

V

=

V0

+ 

V

= 

4 eff



1

−

cos



φ

inf feff



+ 

4 mod



1

−

cos



φ

inf fmod

+ δ



,

(17) where



4_eff

= 

4₁

+ 

4₂

,

feff

=

1

| ξ ·

p1

|

=

1

| ξ ·

p2

|



MPl

,

fmod

=

1

| ξ ·

q

|

<

MPl Sins

.

Theabove considerationsuggeststhatsmallmodulationofthe inflaton potential is a generic feature of the aligned natural in-flation compatible with the WGC. As a specific example for the alignednaturalinflationcompatiblewiththeWGC, onemay con-sidertheaxioncouplings

p1

=



_˜

n f1

,

1 f2



,

p2

=



1 f1

,

0



,

q

=



0

,

1 f2



(18) with f1

∼

f2

,

n

˜



1

,

which corresponds to the axion couplings considered in [34] to generateaKNP-typeaxionpotential.Onethenfinds

Fig. 2. Parameterregionontheplaneof( fmod,δns)with68%(pink)and95%(lightpink)CLlikelihoodwithrespecttothePlanckdataonthetemperatureanisotropyand

low- polarization.Thedashedlinesrepresentthepredictionsfromtheinflatonpotential(28)with(mod/eff)4=10−5,5×10−6,10−6fromthetoptothebottom.The shadedregionintheupperleftcornercorrespondstotheregionthatourperturbativeapproachformodulationbecomesunreliableastheexpansionparameterb isnot smallenough,e.g.b≥0.3 forgrayregionandb≥0.5 fordarkgrayregion.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

ξ =



f1

(

n

˜

−

1

)

f2

,

−

1



+

O



1

˜

n2



,

feff

= ˜

n f2

+

O

(

fi

),

fmod

=

f2

+

O



fi

˜

n2



.

(19)

Itisstraightforwardtogeneralizeourargumenttothecasethat more axions are involved in the alignment mechanism. As was stressedin

[15]

,inthelimitthat

N



1,therecanbeamore vari-etyofwaystoenhancetheeffectiveaxiondecayconstant.For in-stance,onecanevenachieve feff

/

fi

=

O(

eN

)

,whichwouldmakeit possibleto geta super-Planckianeffectivedecayconstantwithout introducinganunreasonablylargenumberoffieldsinthemodel.5

Toimplementthealignmentmechanismwith

φ = (φ

1

,

φ

2

,

. . . ,

φ

N

)

, onecanconsiderapotentialoftheform

V0

=

N



I=1



4_I



1

−

cos



pI

· φ



,

(20) where

pI

=



˜

nI1 f1

,

n

˜

I2 f2

, . . . ,

n

˜

I N fN



(21)

5 _{Recentlythisformofexponentialhierarchybetweentheaxionscaleshasbeen} appliedforthemodelofrelaxion[47–49].

are linearlyindependent fromeachother,buttheyare alignedin such a waythat all

pI lie nearly on an

(

N

−

1

)

-dimensional hy-perplane. Then the potential has a flat direction which can be parametrizedbytheinflatonfield

φ

inf

=

1

N i=1a2i



1/2 det

⎛

⎜

⎜

⎜

⎝

φ

p1

−

p2

..

.

pN−1

−

pN

⎞

⎟

⎟

⎟

⎠

≡ ξ · φ,

(22)

wheretheflatdirectionunitvectorisgivenby

ξ

i

=

ai

N i=1a2i



1/2 with ai

=

det

⎛

⎜

⎜

⎜

⎜

⎜

⎝

˜ n11 f1

· · ·

˜ n1(i−1) fi−1 1 ˜ n1(i+1) fi+1

· · ·

˜ n1N fN ˜ n21 f1

· · ·

˜ n2(i−1) fi−1 1 ˜ n2(i+1) fi+1

· · ·

˜ n2N fN

..

.

..

.

..

.

..

.

..

.

˜ nN1 f1

· · ·

˜ nN(i−1) fi−1 1 ˜ nN(i+1) fi+1

· · ·

˜ nN N fN

⎞

⎟

⎟

⎟

⎟

⎟

⎠

.

Theeffectivedecayconstantofthisinflatonfieldisgivenby

feff

=

1

|

p1

· ξ|

=

1

|

p2

· ξ|

= . . . =

1

|

pN

· ξ|

=

N i=1a2i



1/2

|

det

p1

,

p2

, . . . ,

pN



T

|

,

(23)

which can have a super-Planckian value if

pI are aligned to lie nearlyonan

(

N

−

1

)

dimensionalhyperplane.Notethat

ξ

is nor-malto

pI+1

−

pI (I

=

1

,

2

,

. . . ,

N

−

1),andtherefore

| ξ ·

pI

|

havea commonvalueforall I.

Again,in orderfor feff



MPl tobe compatiblewiththe con-vex hullcondition (7),the axion–instanton couplings

{

qI

}

should notcoincidewiththecouplings

{

pI

}

generatingtheKNP-type ax-ionpotential.Inotherwords,some of

{

qI

}

,whichwillbedenoted as

q, shouldnot belong to

{

pI

}

. Genericallysuch axion–instanton couplingcan inducea subleadingpiece ofaxion potential, taking theformof

(15)

.Also,tobecompatiblewiththeWGC,theconvex hullspanned by

{

pI

,

q

}

should contain theunit ball. Thismeans thatfor

{

pI

}

alignedtogenerate feff



MPl,thereshouldexistan axion–instanton couplingq satisfying

the bound (16). Then, after integratingoutthe

(

N

−

1

)

heavyaxions,theresultinginflaton po-tentialincludes a modulation part as(17).Depending upon how theKNP-typeaxion potential

V

0 isgenerated,i.e.dependingupon theoriginoftheaxioncouplings

{

pI

}

,therecanbemultipleq sat-

isfyingthebound

(16)

,whichwouldresultinmultiplemodulation terms



V

=



q



4q



1

−

cos



ξ ·

q

φ

inf

+ δ

q



=



q



4_q



1

−

cos



kq

φ

inf fmod

+ δ

q



,

(24)

where

k

q areintegersoforderunity,and fmod

<

MPl

/

Sins. Aparticularlyinterestingexampleofthealignmentinvolving

N

axionshasbeenproposedin

[15]

,inwhichtheaxioncouplingsare givenby

⎛

⎜

⎜

⎜

⎝

p1

p2

..

.

pN

⎞

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

1 f1

−

n f2 1 f2

−

n f3

. .

_.

1 fN−1

−

n fN 1 fN

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,

q

=



1 f1

,

0

, . . . ,

0



.

(25)

Thisresultsintheflatdirection

ξ ∝

n n

−

1



1

−

n−N

,

n−1

−

n−N

,

n−2

−

n−N

,

· · · ,

n−(N−1)

−

n−N



,

(26)

withtheeffectivedecayconstants

feff

∼

nN−1fi

,

fmod

∼

fi

,

(27)

whereall fi areassumedtobecomparabletoeachother.

3. Oscillationsinprimordialpowerspectrum

Ourdiscussionintheprevioussectionsuggeststhatsmall mod-ulationoftheinflatonpotential isageneric featureofthealigned natural inflation compatible with the WGC. In this section, we examine the observable consequence ofthis modulation andthe resultingconstraintsonthemodel.

Toproceed,wefirstidentifytheparameterregionofour inter-estfortheinflatonpotentialwithmodulation:

V

=

V0

+ 

V

= 

4 eff



1

−

cos



φ

feff



+ 

4 mod



1

−

cos



φ

fmod

+ δ



.

(28)

To avoid a fine tuning of the initial condition worse than 10%, whileproducingaspectralindexoftheCMBpowerspectrum con-sistent with the observation, we limit the discussion to feff



5

M

Pl.As for themodulation periodicity fmod,the WGCsuggests that fmod



MPl

/

Sins.Quiteoften, onefinds the instanton ampli-tude

∝

e−2πT_{,where 2}

_π

_T

₌

_S

ins

+

i

θ

for an angular axion field

θ

,andtheunderlyingUVtheoryrevealsthestrong–weakcoupling duality under T

→

1

/

T , as well as the discrete shift symmetry:

θ

→ θ +

2

π

[50].One then finds Sins

≥

2

π

, wherethe bound is saturated when the associated coupling

∝

1

/

Sins has a self-dual valuewithrespecttothepresumedstrong–weakcouplingduality. Motivated by these observations,in this section we will concen-trateontheparameterregionwith

feff



5MPl

,

fmod



MPl

2π

.

(29)

Withtheinflatonpotential

(28)

,theequationofmotionisgiven by

¨φ +

3H

˙φ +

V₀

(φ)



1

+



4 mod V₀

(φ)

fmod sin



φ

fmod

+ δ



=

0

.

(30) The effect of modulation can be treated perturbatively if



4

mod

/

V₀

(φ)

fmod



1. Sincewe are interested inthe inflatondynamics around whentheCMBpivotscale

k

∗ exitsthehorizon,we define ourexpansionparameteras[38]

b

≡



4 mod

V₀

(φ

_∗

)

fmod



1

,

(31)

where

φ

_∗ is the inflatonvalue when the CMB scale k_∗ exits the horizon. We thenexpand thesolution oftheinflatonfield

φ

and thecorrespondingslow-rollparametersas

φ

= φ

0

+ φ,

=

0

+ 

,

η

=

η

0

+ 

η

,

(32) where

= −

H

˙

H2

,

η

=

˙

H

,

andthemodulation-inducedcorrections

φ

,



,



η

includeonly theeffectsfirstorderin

b.

Toexaminetheeffects ofmodulationon thecurvature pertur-bation

R

,wechoosethecomovinggaugeforwhich

δ

gi j

=

2a2

R

δ

i j

,

(33)

where

a and

δ

gi j denotethe scalefactorandthe perturbationof spatial metric, respectively.The corresponding evolutionequation ofthecurvatureperturbationisgivenby

R

k

−



2

+

2

+

η

τ



R

k

+

k2

R

k

=

0

,

(34)

where

τ

is the conformal time, d

τ

=

dt

/

a

(

t

)

,and the prime de-notesthederivativewithrespectto

τ

.Thecurvatureperturbation canbeexpandedas

R

k

(

τ

)

=

R

(_k0)

(

τ

)

+

R

(_k0)

(

0

)

gk

(

τ

),

(35) where

R

(0)

₍

_τ

₎

_{denotesthecurvature perturbationintheabsence}

ofmodulations,andthecorrectionfunction

g

k

(

τ

)

satisfies

g_k

−

2

τ

g

Notethat

R

_k(0)

(

0

)

corresponds to the frozen value of

R

_k(0)

(

τ

)

in the superhorizon limit,

−

k

τ

→

0. One then finds gk

(

τ

)

leads to an oscillatory behavior of the curvaturepower spectrum [37,38], whichcanbeparametrizedas

P

R

(

k

)

=

P

_R(0)

(

k

)



1

+ δ

nscos



φ

k fmod

+ β



,

(37)

where

P

_R(0)

(

k

)

is the powerspectrum inthe absence of modula-tions,whichisdescribedwellbythestandardform

P

(0) R

(

k

)

=

A(∗0)



k k_∗



n(s0)−1

.

(38)

Here

δ

ns,

β

, A(∗0),and

n

(s0)areall

k-independent

constants,and

φ

k denotes the inflaton field value when the CMBscale

k exits

the horizon.Then,from

d

φ

k d ln k

= −

√

2 1

−

MPl

,

(39) wefind cos



φ

k 2 feff







k k_∗



M2_Pl 2 f 2_eff cos



φ

∗ 2 feff



.

(40) Itisalsostraightforwardtofind

δ

ns

=

3b



2π γcoth₂π_γ



(

1

+

3₂

γ

2_M2 Pl

/

feff2

)

2

+ (

3γ

)

2

,

(41) where

γ

=

fefffmod M2_Pl tan

φ

_∗ 2 feff

.

(42)

Ourresultagreeswithref.[38]inthelimit

γ



1.Seerefs.

[37,38]

foramoredetaileddiscussionoftheoscillationinthepower spec-trum.

Intheconventionalslowrollinflationscenario,oneusually as-sumes that the curvature power spectrum can be systematically expanded in powers of ln

(

k

/

k_∗

)

over the available CMB scales, whichwouldresultin

P

R

(

k

)

=

A_∗

(

k

/

k_∗

)

ns−1+12αln(k/k∗)+···

_,

₍₄₃₎

where

n

sand

α

are

k-independent

constantssatisfying

α

ln

(

kmax

/

kmin

) <

|

ns

−

1

| 

1

,

(44)

wherekmin

<

k

<

kmax represents therangeoftheobservedCMB scales with ln

(

kmax

/

kmin

)



6–8, and the ellipsis stands for the higher order terms which are assumed to be negligible. In the presenceofmodulations,genericallythecurvaturepowerspectrum cannotbe describedby theaboveform, butrequirestointroduce anoscillatorypieceasin

(37)

.Onemaystillaskinwhichlimitthe oscillatorypiecein

(37)

canbemimickedbytheconventionalform

(43).Toexaminethisquestion,letusexpand

φ

k in

(40)

as

φ

k

= φ

∗

+

d

φ

k d ln k





k_∗ ln



k k_∗



+

1 2 d2

_φ

k d

(

ln k

)

2





k_∗ ln2



k k_∗



+ · · · ,

(45) whichisprovidingawell controlledapproximationfor

φ

kasitis essentiallyanexpansioninpowersof M

2 Pl

f2 eff

ln

(

k

/

k_∗

)



1.Inorderfor theoscillatorypieceinthepowerspectrum

(37)

tobewell mim-icked by the conventional form (43), the sinusoidal functionsof

Fig. 3. 95% CLupperboundon(mod/eff)4 asafunctionoffmodfor feff=5MPl (red),10MPl(blue)and20MPl(green).(Forinterpretationofthereferencestocolor inthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

(φ

k

− φ

∗

)/

fmod needsto bewell approximatedby a simple poly-nomialofln

(

k

/

k_∗

)

withafewterms.Obviouslythisispossibleonly when

φ

kmax

− φ

kmin fmod

≈

√

2_∗MPl fmod ln

(

kmax

/

kmin

) <

1

,

(46)

where

φ

kmax

− φ

kmin corresponds tothe total excursion of

φ

over

the periods when the observable CMB scales exit the horizon. If theaboveconditionissatisfied,thecurvaturepowerspectrumcan beexpandedas

P

R

(

k

)

=

A(_∗0)



k k_∗



n(s0)−1



1

+ δ

ns



cos



φ

_∗ fmod



−

sin



φ

_∗ fmod

 

φ

k

− φ

∗ fmod



−

1 2cos



φ

_∗ fmod

 

φ

k

− φ

∗ fmod



2

+

O



φ

k

− φ

∗ fmod



3



,

(47)

wherewe set

β

=

0 forsimplicity. Comparingthiswith

(43)

,we findthattheoscillatorypiecein

(37)

canbemimickedbythe con-ventionalform

(43)

withthefollowingmatchingconditions:

A_∗

=

A(_∗0)



1

+ δ

nscos

φ

_∗ fmod



,

ns

=

n(s0)

+ δ

ns

√

2∗MPl fmod sin



φ

∗ fmod



,

α

= −δ

ns

√

2_∗MPl fmod

√

2_∗MPl fmod cos



φ

_∗ fmod



−

η

∗ 2 sin



φ

_∗ fmod



.

(48)

However,fortheparameterregion

(29)

ofourinterest,the condi-tion

(46)

isbadlyviolated,sowe needtousetheparametrization

(37)includingtheoscillatorypieceexplicitly,ratherthanusingthe conventionalform

(43)

.

Let us now present the constraints on the model from the CMB data,while taking into account the oscillatory piece in the curvature power spectrum. To this end, we fit the Planck CMB data with the power spectrum (37), and find the likelihood of

Fig. 4. Greenbandsrepresent(n(s0),r)predictedbynaturalinflationwith

modula-tions,i.e.theinflatonpotential(28),whiletheyellowlinesaretheresultsinthe absenceofmodulation.Theredcontoursrepresentthemodel-independent68%and 95%CLrangesof(n(s0),r),whicharecompatiblewiththeobservedCMBdata

fit-tedwiththecurvaturepowerspectrum(37)includinganoscillatorypiece.Theblue contoursaretheresultsofdatafittingintheabsenceofoscillation,i.e.δns=0.(For

interpretationofthereferencestocolorinthisfigurelegend,thereaderisreferred tothewebversionofthisarticle.)

thephenomenologicalparametersincluding

δ

ns and fmod.Forour analysis, we use the

C

osmoMC code [51] with nested sampler

P

olyChord[52].In

Fig. 2

,we show theparameter regions of68% and 95% CLin the plane of( fmod,

δ

ns). Although it dependson thevalues of feff and fmod,the allowed maximalvalue of

δ

ns is around 0

.

1.The shaded region in the upperleft corner of Fig. 2

correspondstotheregionthatourperturbativeapproachfor mod-ulation becomes unreliable as the expansion parameter b is not smallenough.Forthisregion,oneneedstocomputetheprimordial power spectrum numerically because the analytic approximation

(37)isnotreliableanymore.

In

Fig. 3

, we providea 95% CL upperbound on

(

mod

/

eff

)

4 as a function of fmod for three different values of feff6: feff

=

5MPl

,

10MPl

,

20MPl.Fromthis,wefindthattheamplitudeof mod-ulationinthe inflatonpotential isconstrainedas

(

mod

/

eff

)

4



O(

10−4–10−6

)

,dependinguponthevalueofthemodulation peri-odicity fmod.

Aswaspointedoutin

[31]

,modulationsmaymodifythe tensor-to-scalar ratio. In the presence of modulation, the scalar power spectrum is modified by

δ

ns, while the tensor power spectrum remains almost unaffected. Therefore, in our case, the tensor-to-scalarratioatthescale

k is

givenby

r

=

P

t

(

k

)

P

R

(

k

)



160 1

+ δ

nscos



φk fmod

+ β

,

(49)

where

0 corresponds to the slow-roll parameter in theabsence ofmodulation. Consideringthetensor-to-scalar atthe pivotscale

k_∗,it canbe eitherenhanced orsuppressedbecauseofthe oscil-latingbehaviorofthescalarpowerspectrum.In

Fig. 4

,wepresent the tensor-to-scalar ratio predicted by the aligned natural infla-tionwithmodulations,wherethespectral index

n

(s0)isdefinedin

(38)for the curvature power spectrum (37) involvingan oscilla-torypiece.Thegreenbandsshowtheresultswhenanonzero

δ

ns, whichisgivenby (41),is takenintoaccount.Tobe specific,here wechoose

(

mod

/

eff

)

4

=

5

×

10−6and fmod

=

10−3MPl,however

6 _{Herewearefocusingon5M}

Plfeff20MPl.For feff<5MPl,themodelhas adifficultyinproducingacorrectvalueofthespectralindex,andalsorequiresa finetuningoftheinitialcondition.For feff>20MPl,itessentiallycoincideswith thechaoticinflationmodelwithV0=m2φ2.

theresultisnotsensitivetotheseparametersaslongasthey are withintheobservationalbounddepictedin

Fig. 3

.Forcomparison, weprovidealsotheresults(yellowlines)intheabsenceof modu-lation,i.e.

δ

ns

=

0.Becausetheoscillationamplitude

δ

ns islimited to be lessthanabout0

.

1,the changeof

r due

tomodulations is ratherminor,i.e.atmostachangeof

O(

10

)

% comparedtothecase withoutmodulation.

Although the predictedvalue ofr is not significantly affected by modulations, its compatibility with the observed CMB data might be altered. In Fig. 4, we depict also the contours repre-senting the 68% and 95% CL range of

(

n(s0)

,

r

)

compatible with the observed CMB data. Red contours are the results when

δ

ns is allowed to freely vary within the corresponding observational bounds,whilethebluecontoursaretheresultswhen

δ

nsisfrozen to be vanishing. Our results imply that the parameter region of

(

n(s0)

,

r

)

compatible withthe CMB datais enlarged if there were aproperamountofoscillatorypieceinthecurvaturepower spec-trum. Thiscan amelioratethe potentialtension betweenthe nat-ural inflation scenario andthe CMBdata. Forinstance, the point with

r

=

0

.

05 and

n

s(0)

=

0

.

95,whichispredictedbynatural infla-tionwith feff

=

5MPl,isinsidethe95% CLcontourinthepresence of modulation,while itis outsidethe 95% CL contour inthe ab-senceofmodulation.

We finally comment onthe primordial non-Gaussianity. Mod-ulations of the inflaton potential can give rise to a distinct shape of non-Gaussianity, which is known as the resonant

non-Gaussianity[38].Inourcase,itsamplitudeisgivenby

f_{N L}res

=

3b

√

2π

8γ3/2

,

(50)

where

γ

=(

fefffmod

/

M_Pl2

)

tan

(φ

∗

/

2 feff

)

<

1.Thistypeof non-Gauss-ianityrarelyoverlapswiththeothertypesofnon-Gaussianity,such as the local, equilateral, and orthogonal non-Gaussianities [38]. By this reason, we cannot simply apply the constraints on the other typesofnon-Gaussianityforthismodel.Ontheotherhand, the Planckcollaboration has providedan observational constraint on the resonant non-Gaussianity, roughly f_{N L}res



100 for a nar-row range of frequency with

γ

∼

0

.

1 [53]. In our case, we find

f_{N L}res

<

O(

1

)

when

γ

∼

0

.

1 and theamplitudeofmodulation sat-isfiestheboundfromCMB.Thereforethealignednaturalinflation withmodulationssatisfieseasilytheconstraintsontheprimordial non-Gaussianities.

4. Conclusion

In this paper, we have provided an argument implying that a smallmodulation ofthe inflatonpotential is a generic feature of the aligned naturalinflation consistent withthe weak gravity conjecture.Westudiedalsotheobservableconsequencesof modu-lationinthealignednaturalinflationscenarioforthetheoretically orphenomenologicallyfavored parameterregionwith feff



5MPl and fmod



21π MPl.We findthatthe PlanckCMBdataprovides a severe boundon theamplitude ofmodulation.Although modula-tion doesnotcausean appreciablechangeof thetensor-to-scalar ratio

r predicted

bythemodel,itaffectsthecompatibilitybetween theCMBdataandthepredictedvalueof

r,

andthereforecan ame-lioratethepotentialtensionbetweentheCMBdataandthenatural inflationscenario.

Note added While this paper was being finalized, we received ref.[54] discussingthealignednaturalinflation withmodulations within the framework of string theory embedding. Ref. [54] dis-cussesalsotheobservableconsequencesofmodulationsforlower modulationfrequenciesthanours.

Acknowledgements

We thank Chang Sub Shin for initial collaboration and help-ful comments. We thank also Jinn-Ouk Gong, Raphael Flauger, SeokhoonYunandToyokazuSekiguchiforusefuldiscussions. H.K. thanksWonSangChoforhishelpinrunningthecode.Thiswork wassupportedbyIBSundertheprojectcode,IBS-R018-D1.

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