Clockwork inflation

Article

Reference

Clockwork inflation

KECHAGIAS, Alexandros, RIOTTO, Antonio Walter

Abstract

We investigate the recently proposed clockwork mechanism delivering light degrees of freedom with suppressed interactions and show, with various examples, that it can be efficiently implemented in inflationary scenarios to generate flat inflaton potentials and small density perturbations without fine-tunings. We also study the clockwork graviton in de Sitter and, interestingly, we find that the corresponding clockwork charge is site-dependent. As a consequence, the amount of tensor modes is generically suppressed with respect to the standard cases where the clockwork set-up is not adopted. This point can be made a virtue in resurrecting models of inflation which were supposed to be ruled out because of the excessive amount of tensor modes from inflation.

KECHAGIAS, Alexandros, RIOTTO, Antonio Walter. Clockwork inflation. Physics Letters. B , 2017, vol. 767, p. 73-80

DOI : 10.1016/j.physletb.2017.01.042

Available at:

http://archive-ouverte.unige.ch/unige:91555

Disclaimer: layout of this document may differ from the published version.

1 / 1

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Clockwork inflation

Alex Kehagias

^a,b,∗

, Antonio Riotto

^c

aPhysicsDivision,NationalTechnicalUniversityofAthens,15780ZografouCampus,Athens,Greece bTheoreticalPhysicsDepartment,CERN,CH-1211Geneva23,Switzerland

cDepartmentofTheoreticalPhysicsandCenterforAstroparticlePhysics(CAP),24quaiE.Ansermet,CH-1211Geneva4,Switzerland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received20November2016

Receivedinrevisedform22December2016 Accepted20January2017

Availableonline25January2017 Editor:G.F.Giudice

We investigate therecently proposed clockwork mechanismdelivering lightdegrees offreedom with suppressed interactions and show, with various examples, that it can be efficiently implemented in inflationary scenarios to generate flat inflaton potentials and small density perturbations without fine-tunings. We also study the clockwork graviton in de Sitter and, interestingly,we find that the corresponding clockwork chargeis site-dependent.Asaconsequence, the amountoftensor modes is genericallysuppressedwithrespectto thestandardcaseswhere theclockworkset-upisnot adopted.

Thispointcanbemadeavirtueinresurrectingmodelsofinflationwhichweresupposedtoberuledout becauseoftheexcessiveamountoftensormodesfrominflation.

©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Theclockworkmechanism[1,2] allowstoexplainthepresence oflightdegreesoffreedomwithhighlysuppressedinteractionsin theorieswheretherearenosmallparameterstostartwith.Agen- eraltheoryoftheclockworkmechanismvalidforscalars,fermions, gaugebosons,andgravitonshasbeenrecentlyproposedinRef.[3].

Letusbrieflyshowhowitoperatesforscalarsandconsiderathe- oryendowed withaglobalU(1)^N+1 spontaneouslybrokenatthe scale f.Thedegreesoffreedomatenergiessmallerthan f arethe N+1 Goldstonebosons

π

i

U_i

(

x

) =

^eiπi(x)/f

,

i

=

⁰

, · · · ,

N

.

(1.1) The

π

i fields transform by a phase under the corresponding AbelianfactorU(1)_i.Supposenowthatthelow-energydescription ofthetheoryisdescribedbytheLagrangian

L

= −

^f2 2

N i=0

∂

_μU_i^†

∂

^μUi

+

^m2f2 2

N

−¹ i=0

U^†_iU^q_i+1

+

h

.

c

.

= −

¹ 2

N i=⁰

(∂ π

i

)

²

+

^m2 2

N−¹

i=⁰

( π

i

−

^q

π

i+¹

)

²

+

O

( π

⁴

)

*

Correspondingauthor.

E-mailaddress:[email protected](A. Kehagias).

= −

¹ 2

N i=⁰

(∂ π

i

)

²

+

¹ 2

N i,j=⁰

π

iM²_{πi j}

π

j

,

(1.2)

where the presence of the explicit mass terms breaks softly the symmetryU(1)^N+1downtoasingleU(1).Thesquaremassmatrix isgivenby

M²_π

=

^m2

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

1

−

q 0

· · ·

0

−

^{q 1}

+

^q2

−

^q

· · ·

⁰ 0

−

^{q 1}

+

^q2

· · ·

⁰

.. . .. . .. . . . . .. .

1

+

^q2

−

^q 0 0 0

· · · −

^{q q2}

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

(1.3)

Inthemasseigenstatebasisφi(i=⁰,...,N)

π ₌

O

φ,

O^TM²_πO

=

^diag

(

m²_φ

0

, . . . ,

m²_φ

N

),

(1.4)

where O isarealorthogonalmatrix,theeigenvaluesaregivenby

m²_φ

0

=

⁰

,

m²_φ

k

= λ

_km²

, λ

_k

≡

^q2

+

¹

−

^{2qcos k}

π

N

+

¹

,

k

=

¹

, · · · ,

N

.

(1.5) TheelementsoftherotationmatrixO aregivenby

http://dx.doi.org/10.1016/j.physletb.2017.01.042

0370-2693/©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

74 A. Kehagias, A. Riotto / Physics Letters B 767 (2017) 73–80

O_i0

=

^N0

qⁱ

,

O_ik

=

Nk

qsin ik

π

N

+

¹

−

^sin

(

i

+

¹

)

k

π

N

+

¹

,

i

=

⁰

, ..,

N

;

^k

=

¹

, · · · ,

N

,

(1.6)

and N0

≡

q²

−

¹

q²

−

^q−2N

,

Nk

≡

2

(

N

+

¹

)λ

_k

.

(1.7)

Thekeypointoftheclockwork mechanismisnowthatthemass- lesseigenstateφ0 iscoupledtotherestofthefieldsinthetheory witha coupling which is suppressed by Oi0∼^{q−i. In}particular, iftherestofthedegreesoffreedom inthemattersectorcouples onlytotheN-thpion

π

N,thestateφ0couplestothemwithasup- pressedcouplingscalinglikeq^−N.IfN islargeandq>1,thenthe couplingisefficientlysuppressed.

Inthecaseinwhichthenumberofcopiesisverylarge,ithas been pointed out that there exists also a five-dimensional con- tinuum limit of the clockwork mechanism [3]. It is achieved by introducingadilatonfieldS inafive-dimensionalbraneworldwith thefifthdimensioncompactifiedon S₁/Z2.Thecorrespondingac- tionreads

S

=

d⁴xdy

√

−

g

M³₅

2

R

−

¹

3

∂

MS

∂

^MS

+

4k²e^−23S

−

^e−

1 3S

√

_g

55

[

δ(

y

)

V₀

+ δ(

y

− π

R

)

V_π]

,

(1.8)

where k² characterizes the negative vacuumenergy in the bulk, Ristheradiusofthefifthdimension,M5isthefundamentalscale in the bulk, and V0 and Vπ are tensions on the brane satisfy- ingtherelationV0= −^Vπ= −^4kM35.Thecorrespondingmetricis foundtobe

ds²

=

^e43k|y|

dy²

+ η

_μνdx^μdx^ν

,

(1.9)

with

η

μνtheflatMinkowskimetric.Inthispicturehierarchiesare producedonthey=

π

Rbraneandthediscretesuppressionfactor q^−N isreplacedinthecontinuumwithe^−kπR.

Thegoalofthisnoteistoshowthattheclockworkmechanism canbe adoptedininflationary theoriestoefficientlygenerateflat inflaton potentials sustaining a de Sitter phase as well as small masses and couplings to match the small amount of observed scalar perturbations. We will presentin section 2various exam- plesofsuchabilityfromthefour-dimensionaldiscreteperspective.

Insection 3we willstudythephenomenon ofinflationfromthe five-dimensionalcontinuumperspectiveandshowthattheamount ofclockworkingproducingsmallmasses/couplingsdependsonthe Hubble rate during inflation. Maybe more interestingly, in sec- tion4 weshow that,within theclockwork set-up, theclockwork chargesofthegravitonsaresite-dependentandtheamountoften- sormodesgeneratedduringinflationissuppressedwithrespectto thestandardscenarioduetothefactthattensormodesareintrin- sicallybulkdegreesoffreedom.Section5containsourconclusions.

2. Clockworkinflation:thefour-dimensionaldiscreteperspective

In thissection we show how to exploit the clockwork theory in inflation. The clockwork set-up is suitable to get either small masses(comparedtothefundamentalmassscaleoftheproblem) orsmallcouplings(comparedtocouplingsoforderunity). Thisis exactly what isneeded during inflation in orderto get the right amountofdensityfluctuations.Thecomovingcurvatureperturba- tionζ intheflatgaugeis[4]

ζ =

H

φ ˙

∗

δφ ∼

H

M_pl

√

∗

,

(2.1)

wherethesubscript_∗indicatesthatquantitiesshouldbecomputed attheepochofHubbleradiusexitforthecomovingscalek=^aH, φ is theinflatonfield, H is theHubblerateduring inflation, Mpl is the reduced Planck mass, and one has to remember that ob- servablesscales inourcurrentuniversecorrespondtothelast 60 e-foldsorsobeforetheendofinflation. Dotsindicatedifferentia- tionwithrespecttotimeand

= −

H

˙

H²

^M

2 pl

2

V

V

2

(2.2)

isoneoftheslow-rollparameters.Theobservedperturbationsare matchedif

V^3/2 M_pl³V

∗

⁵

.

3

·

¹⁰⁻⁴

.

(2.3)

2.1. Largefieldmodelsofinflation

Toillustratetheadvantagesoftheclockworkset-upinproduc- ingflatpotentialduringinflation,letusconsidertheclassoflarge field models ofinflation. Thesimplestmodel ofinflation isgiven byalinearpotential

V

(φ) =

^m3

φ.

(2.4)

Of course, this is the potential during inflation and additional termsaresupposedtobethereinordertodescribethedynamics after inflation, mainly reheating. For instance a simple modifica- tionofthepotentialabovewillbeV(φ)=^m3(φ²+^am2)^1/2 where aisaconstantandwhichgivesthepotentialaboveforφ^m,but hastherightminimumatφ=^0.

Weknowthattheslow-rollconditionsareattainedwhenφ Mpl andthatthedensityperturbationsaregivenby[4]

ζ ∼

m

M_pl

3/2

∼

¹⁰⁻⁵

,

(2.5)

for m∼^{1015 GeV M}pl. In the clockwork scenario, we can as- sume that there are N+^{1 copies of the inflaton fields and the} potentialis

V

( π

1

, · · · , π

N

) =

^M21 2

N

−1 i=⁰

( π

i

−

^q

π

i+1

)

²

+

^M3₂

π

N

,

(2.6) where M₂ M₁ (saysmallerbyafactorof10),butbothcloseto thefundamental scale.Thefirstpieceofthepotentialisinvariant underashiftsymmetry

π

i

→ π

i

+

¹

qⁱ

,

(2.7)

whichisbrokenbythelast terminthepotential.Upon diagonal- ization ofthemass matrix(which isnot alteredby the presence ofthelinearterm)andgoing toenergy(tobeidentifiedwiththe HubblerateH)muchsmallerthanM₂,thelightestmasseigenstate stateφ0 willhaveapotential

V

(φ

0

) =

^M23

q^N

φ

0

.

(2.8)

Taking forinstance M2∼^10−1Mpl,q=^{2,weneed N}∼^{20 copies} tomatchtheobservedlevelofperturbations.Weshouldalsonote thatpossibleone-loopcontributionsfromthemattersectortothe

inflatonpotentialaresuppressed,atleastbyafactorofq^−N/16

π

², andthereforesuchcontributionsarefullyundercontrol.

Letusalsostressthat thevacuumexpectationvalue ofthein- flatonplaystheroleoftheorderparameterforthebreakingofthe residualshiftsymmetryoftheclockwork.Therefore,thescale M₂, as we imposed above, needs to be smaller than the clockwork scale M1. In this respect, the implementation of the clockwork mechanismneeds a hierarchy of scales, even though not a fine- tunedone.Thisappliesaswelltotheothermodelsofinflationwe willdiscusslateron.

Another issue the clockwork can be useful for in large field modelsistheoneofsuper-Planckian fieldexcursion.Thiswasal- readynoticedinRef.[2].Ifthescalarperturbationsare ascribable toonlyonescalardegreeoffreedom,then

φ

M_pl

r 2

×

¹⁰⁻²

1/2

,

(2.9)

wherer is theso-called tensor-to-scalarratio.A futuredetection ofgravitationalwavesrequiresingeneralvariationoftheinflaton fieldoftheorderofthePlanckscale[5].Thiswouldposeaprob- lem asslow-roll models ofinflation disregard the possible pres- enceofhigher-orderoperators withpowersof(φ/M_pl).However, super-Planckianfieldexcursionscanbemimickedwhilepreserving theregime ofrenormalizable four-dimensionalfield theorybythe clockwork mechanism. Indeed, the slow-roll of the inflaton field over super-Planckian field values corresponds to a clockwork of phase-rotations of the N+^{1 copies offields}

π

i with a U(1)^N+1 globalsymmetrywhoseeffectivedecayconstantisamplifiedwith respecttotheoriginalonebyafactorq^N.

2.2.Hybridmodelsofinflation

To illustrate the advantage of the clockwork mechanism in termsofefficientlyproducingsmallcouplingsduring inflation,let usconsiderthe hybridmodelofinflation[6,7] withN+^{1 copies} ofthefields

π

i and extrafield

V

( π

1

, · · · , π

N

, ) =

^M2 2

N

−1 i=⁰

( π

i

−

^q

π

i+1

)

²

+

^V0

1

−

M

+

¹

4

λ π

_N²

²

+ · · · .

(2.10) Thedots representone ormoreadditional terms,whichgive the potential a minimum atwhich it vanishes butplay no role dur- ing inflation. By performing the standard diagonalization of the clockworkmechanismforthemasssquaredterminEq.(2.10)and workingatenergiesmuchsmallerthantheHubblerateH,thepo- tential(2.10)forthelightesteigenstateφ0 isreducedto

V

(φ

0

, ) =

^V0

1

−

M

+ λ

0

4

λφ

₀²

²

+ · · · , λ

0

= λ

q^2N

.

(2.11) Forsuitable choicesof theparameters, inflation takes placewith thefield heldatthe instantaneousminimum,leading toa po- tential

V

(φ

0

) =

^V0

1

−

^V0

λ

²M²

φ

₀²

.

(2.12)

Imposingthecondition(2.3)gives[4]

10⁻¹²

λ

0

V₀^1/2M

M³_pl

,

(2.13)

and one could explain a small coupling λ0 with the clockwork mechanismeventhough alltheother massscales inthe problem areoftheorderofthePlanckscale.

2.3. Smallfieldinflation

Insmallfieldmodels ofinflationtheproblemistohaveaflat enoughpotentialclosetotheorigin

V

=

^V0

−

¹

2m²

φ

²

+ · · · .

(2.14) As the spectral index of the scalar perturbations is givenby [4]

1−ⁿ=^2M2_pl^m2/V0∼⁰.04,oneneedsm²∼^{10−1H2tobeinagree-} ment with the observations. To produce a potential suitable for smallfieldinflation,intheclockworkscenarioitisenoughtocou- ple the pion

π

N to fermions charged under some strong group.

Belowthe confinementscalethe lightestmode acquiresa poten- tialoftheformusedinnaturalinflation[8]

V

(φ

0

) =

⁴cos

φ

₀

q^Nf

,

(2.15)

insuchawaythat,expandingaroundthemaximumofthepoten- tialweobtain

V0

=

⁴

,

m²

=

⁴

q^2Nf² (2.16)

andtheconditionm² H² requires

f²

^M

2 pl

q^2N

.

(2.17)

This condition can be easily satisfied if f M_pl even for mod- eratevaluesof N.¹ Furthermore,thenormalizationofthedensity perturbations(2.3)imposes

V₀^1/2

M_pl²

⁵

.

4

·

¹⁰⁻⁴¹

−

ⁿ

2 e^Ne(1−n)/2

φ

_e

M_pl

,

(2.18)

whereNeisthenumberofe-foldstilltheendofinflationandφe isthevalue oftheinflatonfield φ0 wheninflationends.Sincethe typicalscaleforφeisq^Nf^Mpl,oneseesthattheclockworkcan allowsizeable.Inthisset-upinflation willendwhentheslow- rollconditionsare violatedandthe φ0 fieldrolls toits minimum andreheatingwilltakeplaceby couplingtheinflatonfieldto,for instancefermions,throughacoupling∂_μφ0ψ¯

γ

^μ

γ

5ψ.

2.4. Starobinskyinflation

Anotherillustrativeexampleoftheefficiencyoftheclockwork mechanismisprovided bythe so-calledStarobinskymodel ofin- flation[12].FollowingRef.[3],weconsider N copiesoflinearized GRtowhichweaddtheStarobinsky-likeaction

S

=

^M2N 2

d⁴x

√

−

^gNR

(

g_Nμν

) +

¹ 12

α

²

d⁴x

√

−

^gNR²

(

g_Nμν

),

(2.19)

where we have added a quadratic curvature term for the N-th term whosestrength is parametrized by a dimensionlessparam- eter

α

. It is known that in (R+^R2)-theory, there is a massive scalarmodeontopofthegravitonswhichcanbeuncoveredwith theusualmethods,leadingtotheaction

1 Inthisrespect,theclockworkmechanismappliedtonaturalinflationisreminis- centofthealignedmechanism[9–11].However,alignmentdoesrequireahierarchy ofparameterswhichisnaturallyachievedthroughclockwork.

76 A. Kehagias, A. Riotto / Physics Letters B 767 (2017) 73–80

S

=

^M2N 2

d⁴x

√

−

gNR

(

gNμν

)

+

d⁴x

√

−

^gN

−

¹

2

(∂φ

N

)

²

−

³ 4

α

²M⁴_N

1

−

^e−

2 3

φ_N MN

2

.

(2.20) Forthistheory,asshownin Ref.[3],thereis amassless graviton withacorrespondingPlanckmass,whichinthelarge N-limitis

M_pl²

=

^qNM2_N

.

(2.21)

Therefore,theactionforthemassless graviton andthescalarbe- comes

S

=

d⁴x

√

−

^g

M_pl

2 R

−

¹ 2

(∂φ

_N

)

²

−

³

4

α

²M⁴_plq^−4N

1

−

^e−

2 3

φ_{N q}^N Mpl

2

.

(2.22)

Duringinflation φN takeslargevaluesandthedynamicsisdomi- natedbythevacuumenergy

V

=

³

4

α

²M⁴_plq^−4N

.

(2.23)

Inorderto getthe correctnormalization(2.3) one needs q^−2N≈ 10⁻⁵ for

α

_=O(1). For instance, for q=^{2, the moderate value} N =^{8 is required. In addition, the} tensor-to-scalar ratio is r= (12q^−2N/Ne²).Hence, thetensormodesare suppressedbyan ex- trafactorofq^−2N=^{10−5,leavingtonoroomfortensormodesin} theclockworkStarobinskyinflationmodel.

2.5. Theclockworkandthegenerationofperturbationsfromlightfields otherthantheinflaton

Eventhoughtheinflationaryparadigmisbyitselfquiteelegant andsimple,themechanismgivingrisetotheadiabaticcosmologi- calperturbationsisfarfrombeingestablished.Itisfairtosaythat we donot knowatpresentwhatis thesource ofthescalar per- turbations during inflation, theinflaton field itselfor some other field. Thetotal curvatureperturbation ζ might notbe a constant (intime)onsuper-Hubblescalesandchangingonarbitrarilylarge scalesduetoanon-adiabaticpressureperturbationwhichmaybe duetoextrascalardegreesoffreedom.

Forinstance,inthe curvatonmechanism[13,14]the curvature perturbationisgeneratedfromaninitialisocurvatureperturbation associatedtothequantumfluctuationsofalightscalarfield

σ

,the curvaton, whoseenergydensityis not dominantduring inflation.

Thecurvatonisocurvatureperturbationbecomestheadiabaticone oncethecurvatondecaysintoradiationaftertheendofinflation.

During inflation a flat spectrum is produced in the curvaton field

δ σ =

H

2

π

∗

.

(2.24)

After inflation, the curvaton field starts oscillating during some radiation-dominatedera,causingitsenergydensitytoincreaseand convertingtheinitialisocurvatureintocurvatureperturbationζ.

The curvaton mechanism works as long as the curvaton can be quantum mechanically excited during inflation. This requires that the mass m ofthe curvaton field is much smaller than the Hubblerateduringinflation,m H.Thisposesaproblemasnon- renormalizable couplings between the inflaton and the curvaton areexpectedtogenerateO(^H2)correctiontothemasssquaredof

the curvaton. Forinstance,insupergravity onehas correctionsto theKählerpotentialofthecurvatonoftheform

δ

K

⊃

d²

θ φ

^†

φ

M_pl

σ

^†

σ _⊃

O

(

1

)

H²

σ

^†

σ ,

(2.25) where φ is the inflaton field. The clockwork provides a possible solution to thisproblem. Suppose that thereare N+^{1 copiesof} curvatons,letuscallthemagain

π

i withpotential

V

( π

1

, · · · , π

N

) =

^M21 2

N

−¹ i=0

( π

i

−

q

π

i+¹

)

²

.

(2.26)

This potential hasthe usual shiftsymmetry

π

i→

π

i+¹/qⁱ. Like intheconstructionofRef.[2],thisshiftsymmetryisamanifesta- tionofthefactthatthe

π

i’sareindeedpseudoNambu–Goldstone bosonsofaU(1)^N+1 globalsymmetry.Sincegravityisexpectedto breaksuchaglobalsymmetry,oneexpectscorrectionstothemass oftheform

N i,j=⁰

c_{i j}H²

π

i

π

j

=

^cN NH²

π

_N²

+ · · · ,

(2.27)

where the ci j are O(1) coefficients and we have supposed that the vacuum energyis located at the N-th site. Upon diagonaliz- ing themass matrix (2.26) one finds that the lightest eigenstate

σ

0 receivescorrectionstoitsmasssquaredsuppressedatleastby 1/q^N 1,whichisenoughtoobtainalightcurvatonduringinfla- tion.

3. Clockworkinflation:thefive-dimensionalcontinuum perspective

In this section we investigate the clockwork inflationary sce- nario from the continuum limit point of view. We imagine that the vacuumenergydriving inflation is locatedat one ofthe two branesand, asaresult, each fifthdimensional sectionisinflating withconstant Hubblerate. Letusstart withthefive-dimensional action[3,15]

S

=

d⁵x

√

−

^g

M³₅

2

R

−

¹

3

∂

MS

∂

^MS

+

^4k2e−23S

+

Lbulk

+

Sbrane

,

(3.1)

where

Sbrane

= −

a

d⁴x

√

h

M³₅

2

[

K

] +

^e−13SLbrane

.

(3.2)

Lbulkisabulkmatteraction,Lbraneisthebraneactionand[K]= K⁺−K^{−isthejumpofthetraceoftheextrinsiccurvature}

Kμν

=

^hM_μ^hN_ν

∇

MnN

,

(3.3)

definedinterms ofthenormaltothe branesa locatedatposi- tions xa andthe projection tensor h^Mμ.Phenomenologicalaspects ofthistheory havebeenexplored in[15–17].Thefield equations thatfollow fromtheaction(3.1)are

R_{M N}

−

¹

2g_{M N}R

=

¹

M³₅T_{M N}

,

(3.4)

∇

^2S

−

^4k2e−23S

=

⁰

,

(3.5)

supplementedbytheIsraelmatchingconditionsforthebranesa

[

Kμν

]

a

=

¹ M₅³

T_μν^(a)

−

¹

3g_μνT^(a)

a

, [

n^M

∂

MS

]

a

=

³

M₅³

∂

∂

S

e^−13SLbrane

a

,

(3.6)

andTμν^(a) isthefour-dimensionalbraneenergy-momentumtensor.

Wearelookingnowforsolutionsoftheform ds²

=

^e2σ(y)

dy²

+

^ds24

,

S

=

^S

(

y

),

(3.7)

where

ds²₄

=

^gdS_μν^dxμdxν

=

¹ H²

η

²

−

^d

η

²

+

^d

x²

,

(3.8)

isafour-dimensionaldeSittermetric.Theequationsofmotionare then

36

( σ

²

−

^H2

) −

^S2

=

^{12k2e−23S+2σ}

,

(3.9)

σ

− σ

²

+

^H2

+

¹

9S²

=

⁰

,

(3.10)

S

+

³

σ

S

=

^{4k2e−23S+2σ}

.

(3.11) In addition, the Israel matching conditions along the branes a located at y=^ya with tension L⁽_brane^a) =^Va for the metric (3.7) give

σ

a

= −

¹

6M³₅e^σa−13SaVa

,

S

a

= −

¹

2M³₅e^σa−13SaV_a

,

(3.12) where

σ

a=

σ

(ya), Sa=^S(ya).The scalarfield equation(3.11) is notindependentasitisconnectedtotheBianchiidentity.Soonly Eqs.(3.9) and(3.10)are independent.Eliminating S² we getthe systemofequations

3

σ

+

⁹

σ

²

−

^9H2

=

^{4k2e−23S+2σ}

,

(3.13)

σ

− σ

²

+

^H2

+

¹

9S²

=

⁰

.

(3.14)

When H=^{0, the solution to Eqs. (3.13) and (3.14) is the linear} dilatonsolution

σ ₌ σ

0

=

^2k

3 y

,

S

=

^S0

=

^2ky

,

(3.15)

where the boundary condition

σ

0(0)=^S0(0)=^{0 has been as-} sumed.However,thesolutionofEqs.(3.13)and(3.14)fornon-zero Hisnoteasytobefound.Nevertheless,byusingEq.(3.13)wecan expressthescalarS intermsofthewarpfactoras

e^−23S

=

³ 4k²e^−2σ

σ

+

³

σ

²

−

^3H2

.

(3.16)

BytakingthederivativeofEq.(3.14)andcomparingwithEq.(3.13), wecancompletelydecouplethescalar S andwefindthat

σ

sat- isfiesthethirdorderequation

2

σ

3H²

−

³

σ

²

₊

2

σ

₊ σ

²

+

⁴

H²

− σ

²

₊ σ

3H²

−

³

σ

²

₋ σ

²

₌

0

.

(3.17) Althoughwewerenot abletofindanexact solutiontoEq.(3.17) wecantrytofindasolutionperturbativelyinH².Forthis,wemay write

σ _≈ σ

0

+

^H2

σ

1 (3.18)

and treat

σ

1 asa first order perturbation. We then find that

σ

1 satisfies

σ

₁

+

^2k

σ

₁

−

^2k

=

⁰

,

(3.19)

andtherefore,

σ

₁

₌

¹

2y²

+

^C1e^−2ky

+

^C3y

+

^C2

,

(3.20) whereC1,2,3 areintegrationconstants.Inaddition,wefindthat S isgivenby

S

≈

^2ky

+

³ 2H²

y²

+

^5C1e^−2ky

+

^y

2C₃

−

³ k

+

^2C2

−

^3C3 k

+

³

2k²

.

(3.21)

It is straightforward to verify that

σ

₌

σ

0 + ^H2

σ

1 and S in Eq.(3.21)indeedsatisfiestheequations(3.13) and(3.14)to lead- ing orderin H².Withacompactfifth dimensionandtwo branes at y=^{0 and y}=

π

Rwithbraneaction

Sbrane

=

d⁴x πR

0

dy

√

−

^ge−

1 3S

√

g55

[

δ(

y

)

V0

+ δ(

y

− π )

V_π]

,

(3.22) wefindfromEq.(3.12)thatthediscontinuitiesof

σ

,Sshouldsat- isfy

3

[ σ

]

0,πR

= [

S

]

0,πR

.

(3.23) Thesolutionsthatsatisfy(3.23)haveC1=^{0 andifwetakeC}2=⁰ andC3=¹/2ksothat

σ

(0)=^S(0)=^0,weget

σ =

^2k 3

|

^y

| +

¹

2H²

y²

+

¹ k

|

^y

|

,

(3.24)

S

=

^2k

|

^y

| +

³ 2H²

y²

+

¹

k

|

^y

| −

³ ky

.

(3.25)

NotethatifSπ=^S(

π

R),wehavefromEq.(3.25) S_π

=

^2k

π

R

+

³

2H²

k²

π

²R²

−

² k

π

R

,

(3.26)

whichfixes the radionfield R in termsofthe boundaryvalue of thedilaton Saty=

π

R.Thelattercanbedeterminedbyapoten- tialforexampleoftheform

U

(

S

) =

^M25

2

(

S

−

^Sπ

)

²

.

(3.27)

Such apotentialfixes theboundaryvalueofthedilatonandcon- sequently thevalue of R.In addition,thetensions V0 and Vπ in Eq.(3.22)turnouttobe

V0

(

H

) = −

4k

+

^3H2 k

M³₅

,

V_π

(

H

) =

4k

+

^3H2 k

e^{−σ(πR)+S(πR)/3}M₅³

.

(3.28) Atleading orderin H²/k²,theextravacuumenergydrivinginfla- tioninthe y=

π

R brane² isgivenby

2 We are adoptingthe so-called π-framehere, wherethe SM residesinthe y=πRbrane,asopposedtothe0-frameadoptedin[3]forthecontinuumclock- work.Thetwoframesareequivalent,andthereisawell-definedtransformation thatconnectsthetwoframes.

78 A. Kehagias, A. Riotto / Physics Letters B 767 (2017) 73–80

U_π

=

^{e4σ(πR)−S(πR)/3}

(

V_π

(

H

) −

^Vπ

(

0

)) =

^3H2

k e^2kπRM³₅

.

(3.29) Wemaywritethelastequationas

H²

=

^Uπ

3M²_pl

,

(3.30)

whichis justthestandard Friedmannlawonthe braneexpected witha stabilizeddilatontolowest orderin H².In(3.30)we have definedthefour-dimensionalPlanckmassinflatspace–timeas M_pl²

=

^M

3 5

k e^2kπR

,

(3.31)

whichcoincides withthe one foundby dimensional reductionof thefive-dimensionalaction reportedin[3].Ifthe vacuumenergy drivinginflationcomesfromthe y=

π

R brane, thescalarpertur- bations havethe samebehavior at leading orderin theslow-roll parameters as in the four-dimensional case [18] and we expect that no detectable signature remains from the non-trivial geom- etry in the bulk.³ The reason is the following. Differently from thetensormodes,whicharegenuinelyfive-dimensionalfreefields quantizedinthebulk,scalarmetricperturbationsaregeneratedby thebranescalarfieldwhichisquantizedonthebraneandisfour- dimensionalinallregimes.Thecouplingbetweentheinflatonfield andthemetricconcerns thelong-wavelengthlimit forwhichthe metricevolvesasinthefour-dimensionaltheory.Moreover,sucha couplingislocalizedonthebrane,andnosignature remainsfrom thewarpedgeometryinthebulk. Correctionstothisresultare of the order of − ˙^{H R2}=

(H R)² as metric perturbations probe the extra dimension on times scales −^H2/H.˙ The same logic is not validfortensormodes,aswenowproceedtodiscuss.

4. Tensormodesinclockworkinflation

Inthissectionwestudythebehavior ofthetensormodesdur- ingade Sitterstageintheclockwork set-up.Thegoalistoshow thattensormodesare suppressedinthisscenariowithrespectto thestandardcase,thereasonbeingthatthetensormodesfeelthe bulkbyfullstrengthandthat duringthedeSitterstage thelatter ismorewarped.

Westartfromthefive-dimensionalcontinuumperspective.The advantageof usingtensormodesto probe theclockwork mecha- nismis thatthere isadependence onlyonthe geometry,noton themicroscopic models ofinflation andstabilizationof theextra fifthdimension.

4.1. Tensormodes:thefive-dimensionalcontinuumperspective

AsshowninRefs.[18,20],amasslesstensormodeisproduced in braneworld scenarios of inflation with the amplitude H/M_pl. The point isthat the effectivePlanck mass during inflation fora curvedde Sitterbraneworld differsfromthat ofthe flatbrane at lowenergies,duetoadependenceontheHubblerateH.Thus,the amplitudeoftensormodesfrominflation inaclockworkscenario is usually different from a standard period of inflation without clockwork.

Letusindeedconsidertheequation ofmotionof thegraviton field in the continuum clockwork scenario starting from the fac- torized metric(3.7).By decomposing the five-dimensionaltensor modeas

3 Onecan alsocheckthatthefive-dimensionalgeometryis notgravitationally unstabledue tothetachyonicgrowthofthe conformalfluctuations ofthe four- dimensionalmetric[19].

Fig. 1.TheratioofthePlanckianmassduringthedeSitterstageoverthePlanckian masstodayasafunctionofH/k.HerewehavetakenR=2/k.

hM N

(

y

,

x^ρ

) =

e^−32σ(y)hm

(

y

)

Q_μν^(m)

(

x^ρ

),

(4.1) where m is the eigenvalue corresponding to the Kaluza–Klein modes and the transverse-traceless conditions are imposed on Qμν^(m)(xρ),onecanshowthat theeigenvalueproblemcanbewrit- tenas[18,20]

−

D₋D₊h_m

(

y

) =

^m2hm

(

y

),

D±

= ∂

y

∓

³

2

σ

(

y

) = ∂

y

∓

2k

3

θ (

y

) +

^H2 k²

y

+

^2k

θ (

y

)

e^−2ky

.

(4.2) From thisequationone immediatelyreadsoffthepresenceofthe zeromodegravitonwithwavefunction

h0

(

y

) =

^e32σ(y)

.

(4.3)

Furthermore,onecanshowthatthereisamassgapfortheother Kaluza–Klein modesof at least3H²/2 in the mass squared. The key point is that during inflation the effective-four dimensional Planckmassasseenbythemasslesstensormodeisdifferentfrom theoneinflatspacesincethefunction

σ

(y)indeSitter,asshown intheprevioussection,

σ (

y

) =

^2k 3

|

y

| +

¹

2H²

y²

+

¹ k

|

y

|

+

O

H⁴R⁴

(4.4)

ismodifiedfromtheflatcase.Itiseasytoseethatthewarpfactor eσ(y) is convex andan increasing function of H². Therefore, the four-dimensionalMpldefinedas

M^dS_pl

2

=

^2M3₅

πR

0

dy e^3σ

=

^M35 k

e^2kπR

1

+

³

2H²

π

²R²

−

¹

,

(4.5) isan increasing functionof H²,seeFig. 1.Astheamountoften- sor modes is proportional to (H/M_pl^dS)², this means that in the clockwork scenariothe amount oftensor modesis reducedwith respecttotraditionalcase.Isthisanegativepoint?Notnecessarily so. Therearemanymodels ofinflation whichhasbeenruledout bytherecentPlanckdata[21]astheyproduceatoolargeamount oftensormodes.Examplesarethechaoticlargefieldmodelofin- flationλφ⁴ [22]andpower-lawinflation[23].Byembeddingthese