Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of the secular growth in Euclidean vacuum

0.58 MB·English

by  Takahiro Tanaka

Strong restrictiononinflationary vacua from the localgaugeinvarianceII:
Infrared regularity and absenceofthe seculargrowth inEuclidean vacuum
Takahiro Tanaka1∗ and Yuko Urakawa2†
1 Yukawa Institute for Theoretical Physics, Kyoto university, Kyoto, 606-8502, Japan
2 Departament de F´ısica Fonamental i Institut de Cie`ncies del Cosmos,
Universitat de Barcelona, Mart´ı i Franque`s 1, 08028 Barcelona, Spain
(Dated:January15,2013)
Weinvestigatetheinitialstateoftheinflationaryuniverse.Inourrecentpublications,weshowedthatrequest-
ingthegaugeinvarianceinthelocalobservableuniversetotheinitialstateguaranteestheinfrared(IR)regularity
ofloopcorrectionsinageneralsingleclockinflation.Followingthisstudy,inthispaper,weshowthatchoosing
theEuclideanvacuumensuresthegaugeinvarianceinthelocaluniverseandhencetheIRregularityofloopcor-
rections. Ithasbeensuggestedthatloopcorrectionstoinflationaryperturbationsmayyieldtheseculargrowth,
3
whichcanleadtothebreakdownoftheperturbativeanalysisinanextremelylongterminflation. Theabsence
1
oftheseculargrowthhasbeenclaimedbypickinguponlytheIRcontributions,whichwethinkisincomplete
0
becausethenon-IRmodeswhicharecomparabletoorsmallerthantheHubblescalepotentiallycancontribute
2
totheseculargrowth. Weprovetheabsenceoftheseculargrowthwithoutneglectingthesenon-IRmodestoa

n certainorderintheperturbativeexpansion. Wealsodiscusshowtheregularityofthen-pointfunctionsforthe
a genuinelygaugeinvariantvariableconstrainstheinitialstatesoftheinflationaryuniverse. Theseresultsapply
J inafullygeneralsinglefieldmodelofinflation.

4
1

]
h
I. INTRODUCTION
t
-
p
e A. MotivationandthecurrentstatusofIRissues
h
[
Initial states of the observable universe. How our universe began? This is one of the biggest question in cosmology.

1 Theobservationofthecosmicmicrowavebackgroundtellsus aboutthecosmologicalperturbationatthelastscattering,which
v realistic scenarios of the early universe should explain. If inflation took place preceding the big-bang nucleosynthesis, the
8 quantum fluctuation of the inflaton can generate the seed of the cosmological fluctuation which is consistent with the scale-
8 invariantspectrumatlargescales. Therefore,inthecontextoftheinflationaryscenario,whichiscurrentlythemostsuccessful
0
scenario of the early universe, the period of inflation is the earliest part of our observable universe. In this series of papers,
3
we pursue the question; “What can we claim aboutthe initial quantum state of the observableuniverseif we require that the
.
1 theoreticalpredictionshouldbestableagainsttheinfrared(IR)loopcontribution?”Theadiabaticvacuumiswidelyacceptedto
0 bethemostnaturalvacuumatleastforafreefieldtheory,sinceitmimicsthevacuumoftheflatspacetimeintheultraviolet(UV)
3
limit. However,inanumberofpublications[1–23],ithasbeensuggestedthattheadiabaticvacuummaynotbestableagainst
1
theIRcontributionsinthepresenceofnon-linearinteractions.
:
v Non-localityoftheactionandIRdivergenceproblem.Whenweassumethatthefreefieldhasthescaleinvariantspectrum
i
X intheIRlimit,anaiveconsiderationcaneasilyleadtotheIRdivergenceduetoloopcorrections. HereweillustratehowtheIR
divergencecanappearfromtheloopcorrectionsofthecurvatureperturbationinsinglefieldmodels. Choosingthetimeslicing
r
a onwhichtheinflatonfieldishomogeneous,wecanexpresstheactionintermsoftheuniquedynamicaldegreesoffreedomζ,
the curvature perturbation, and the Lagrangemultipliers N and N , the lapse function and the shift vector. The Hamiltonian
i
andthemomentumconstraintequationsrelatethedynamicalvariableζ tothemultipliersN andN . Asisexplicitlyshownin
i
variouspapers,forinstanceinRef.[24–26],theseconstraintequationsareelliptic-typeequations,andschematicallywrittenas
∂2N =f[ζ], ∂2N =f [ζ], (1.1)
i i
where ∂2 denotes the spatial Laplacian. By requesting the regularity at the spatial infinity, the boundaryconditions of these
elliptic-typeequationsareuniquelyfixed. SubstitutingtheexpressionsofN andN intotheaction,weobtain
i
S = d4xL[ζ, N, N ]= d4xL[ζ, ∂−2f[ζ], ∂−2f [ζ]], (1.2)
i i
Z Z
∗Electronicaddress:tanaka˙at˙yukawa.kyoto-u.ac.jp
†Electronicaddress:yurakawa˙at˙ffn.ub.es
2
andhencetheevolutionofζ isdescribedbytheabovenon-localaction. HeretheinverseLaplacian∂−2 isusuallysupposedto
bedefinedasmultiplyingtheinverseoftheeigenvalueoftheLaplacianoperatorbyusingtheharmonicdecomposition. When
weevaluatetheloopcorrectionstothen-pointfunctionsexpandingthemintermsoftheinteractionpicturefieldζ ,weneedto
I
evaluatetheexpectationvaluessuchas
ζ2 , ζ ∂−2ζ , . (1.3)
h Ii h I Ii ···
Insertingthescaleinvariantspectruminto ζ2 leadstothelogarithmicdivergenceas ζ2 d3k/k3. Thesecondexpression
h Ii h Ii∝
of Eq. (1.3), which may arise as a consequenceof the operation of ∂−2, is more singularas ζ ∂−2ζ d3k/k5, which
I I
Rh i ∝
diverges quadratically. The presence of non-local interactions enhances the long range correlations, and hence the singular
behaviourintheIR.WhenweintroducetheIRcutoff,sayattheHubblescaleataparticulartime t , thevarRiance ζ2 shows
0 h Ii
the logarithmicseculargrowthas ζ2 aH dk/k loga/a where a andH , respectively,denotethe scale factorand
the Hubble scale at t = t . If thehIIRid∝iverag0eHn0ceexists∼, the loop0correction0s, whic0h are suppressed by an extra powerof the
0
amplitude of the powerspectrum (H/M R)2, may dominate in case inflation continuessufficiently long, leading to the break
pl
downofperturbation.
ThedilatationsymmetryasanecessaryingredientforIRregularity. TheregularizationoftheIRcontributionshasbeen
discussedinanumberofpublications[25–38].Theimportantaspectindiscussingthelongwavelengthmodeofζisthedilatation
symmetryofthesystem. Asisexpectedfromthefactthatthespatialmetricisgivenintheforma2e2ζdx2,aconstantshiftof
thedynamicalvariableζ canbeabsorbedbytheoverallrescalingofthespatialcoordinates. Hence,theactionforζ preserves
thedilatationsymmetry:
xi e−sxi, ζ(t, x) ζ(t, e−sx) s, (1.4)
→ → −
where s is a constant parameter. (There are a number of literatures where this dilatation symmetry is addressed. See for
instance, Refs. [39, 40]andthe referencestherein.) Onemaynaivelyexpectthatwe canabsorbthe IR divergentcontribution
ofζ usingthisconstantshift. Asanexample,wesettheparameterstothespatialaverageofthecurvatureperturbationwithin
the Hubble patch at t , ζ¯(t ), where the size of the Hubble patch in comovingcoordinatesis given by 1/(a H ). Then, the
0 0 0 0
logarithmicallydivergenttwo-pointfunction ζ2 seems to be replaced with (ζ ζ¯ )2 aH dk/k, which is finite but
still growslogarithmicallyin time. One mayhthIinik that if the system is descrhibeId−in sIuchia∝waay0Hth0at the symmetryunderthe
R
timedependentdilatationtransformationismanifest,settings(t)tothetimedependentspatialaverageintheHubblepatch,the
logarithmicgrowthof ζ¯(t)mightbeeliminated. However,thereducedactionwrittenin termsofζ (1.2)doesnotpreservethe
invarianceunderthedilatationtransformationwiththetimedependentparameters(t). Forexample,intherecentliterature[40],
the authors showed that when we consider the whole universe with the infinite spatial volume, the dilatation transformation
shouldbetimeindependenttopreservetheactioninvariant. Inaddition,thetwo-pointfunctionwith∂−2cannotberegularized
by considering the dilatation symmetry alone. This quick consideration tells us that the presence of the dilatation symmetry
ofthe system mayplay animportantrole in theregularizationof the IR contributionsbutis notsufficienttoguaranteethe IR
regularityandtheabsenceoftheseculargrowth.
Residualgaugedegreesoffreedominthelocaluniverse. Amissingpieceintheabovediscussionistopaycarefulattention
towhatarethequantitieswecanactuallyobserve. Sinceour observableregionisalimitedportionofthewholeuniverse,the
observablefluctuationsmustbecomposedoflocalquantities. Furthermore,astheinformationthatwecanaccessislimitedto
ourobservableregion,thereisnoreasontorequesttheregularityatthespatialinfinityinsolvingtheellipticconstraintequations
(1.1). Then, there arise degrees of freedom in choosing the boundary conditions of Eqs. (1.1). The degrees of freedom in
solutionsofN andN canbeunderstoodasthedegreesoffreedominchoosingcoordinates. AsweshowedinRefs.[25,26],
i
theseresidualcoordinatetransformationsareexpressedintermsofhomogeneoussolutionstotheLaplaceequationas
xi xi s(t)xi si (t)xj1 xjm + , (1.5)
→ − − j1···jm ··· ···
m=1
X
where sij1···jm(t) are symmetric traceless tensors, which satisfy δjj′sij1···j···j′···jm(t) = 0. Here, we abbreviated the non-
linear terms in the coordinate transformation. Note that this coordinate transformationsinclude the dilatation transformation
with the time dependentfunctions(t). Since the transformationsin Eq. (1.5) are nothingbut coordinatetransformations, the
diffeomorphicinvariantactionS = d4xL[ζ, N, N ]shouldpreservethesymmetryunderthesetransformations. Thus,when
i
weconsideronlythelocalobservableregion,whichisapotionofthewholeuniverse,wefindaninfinitenumberofcoordinate
R
transformationswhich keep the action invariant. Considering the dilatation transformation in the whole universe is subtle in
the sense that the transformationdivergesat the spatial infinity, even if the parameter s is very small. By contrast, restricted
tothelocalregion,themagnitudeofthecoordinatetransformationsinEq.(1.5)iskeptperturbativelysmall. Inthispaper,we
refer to the local observable (spacetime) region as O. The size of the observable region on each time slicing is supposed to
beof order1/a(t)H(t)atleastin thefarpastsincethe pastlightconeasymptotestothatsize. We shouldnotethat, oncewe
inserttheexpressionsofN andN intotheactiontoobtaintheactionforthecurvatureperturbationζ,thesymmetryunderthe
i
3
residualcoordinatestransformationislost,becausespecificboundaryconditionsarechosenforN andN infixingcoordinates.
i
Toemphasizethedistinctionbetweenthecoordinatetransformationsassociatedwiththechangeoftheboundaryconditionsand
theusualgaugetransformation,whichkeepstheactioninvariant,wedenotetheformerbythegaugetransformationintheitalic
font.
Removingtheresidualgaugedegreesoffreedom.Onewaytorealizetheinvarianceunderthegaugetransformationisfixing
thegaugeconditionscompletely. Theresidualgaugedegreesoffreedomintroducedabovecanbealsoremovedbyemploying
additionalgaugeconditions,i.e.,byfixingtheboundaryconditionsof N andN attheboundaryofthelocalregion O. Then,
i
wenaturallyexpectthattheIRregularitymaybeexplicitlyshownbyperformingthequantizationinthislocalregion,sincethe
wavelengthsthatfitwithinthislocalregionOwillbeboundedbythesizeoftheregion. Althoughthequantizationinthelocal
regionisaninterestingapproach,itisnotso clearhowtoperformthe quantizationafterremovingtheresidual gaugedegrees
offreedom. Oneofthedifficultiesisthateventhetranslationsymmetryofthequantumstatecannotbeeasilyguaranteedinthe
localsystem,sinceitismanifestlybrokenbyintroducingtheboundaryconditionatafinitedistance. (Seealsothediscussionin
Ref.[27]).
As an alternativeway, in Ref. [28], we first set the initial state consideringthe whole universe, and then we performedthe
residualgaugetransformation(1.5)tofixthecoordinatessothattheIRcontributionsareabsorbed.Throughthetransformation
withs(t)=ζ¯(t),thecurvatureperturbationistransmittedas
ζ(t, x) ζ(t, e−ζ¯(t)x) ζ¯(t)=ζ(t, x) ζ¯(t)+O(ζ2). (1.6)
→ − −
Here,ζ(x)istheoriginalcurvatureperturbationdefinedinthewholeuniverseanditsspatialaverageoverthewholeuniverseis
setto0asintheconventionalcosmologicalperturbationtheory. Bycontrast,ζ(t, e−ζ¯(t)x) ζ¯(t)isthecurvatureperturbation
relevant to the local universe, and its spatial average over the local region Σ O is set−to 0, where Σ is a time constant
t t
∩
surface. In Ref. [28] we consideredthe fluctuation of the inflaton, using the flat gauge, but the same discussion follows also
for the curvature perturbation ζ. In the recent publication by Senatore and Zaldarriaga [38], the same degrees of freedom in
choosingcoordinatesareusedinaslightlydifferentwayto absorbtheIRdivergentcontributions. Ifthenon-lineartermsinthe
residualgaugetransformationattheinitialtime(1.6)didnotyieldIRdivergentcontributions,thediscussioninRef.[28]would
have provedthe absence of IR divergencein general. What was shown there is that once the field operatorafter the residual
gauge transformationis guaranteedto be regularat the initial time, its succeedingevolution doesnot produceIR divergence.
Theheartoftheproofisthatζ (x)isreplacedwith ζ (x) ζ¯ (t)intheexpansionofthecompositeoperatorsintermsofthe
I I I
−
interactionpicturefield,aftertheresidualgaugetransformation,andhencetheIRcontributionsfromζ (x)arealwayscanceled
I
bythosefromζ¯ (t). However,thenon-linearpartofthetransformationattheinitialtimecontainsζ¯(t)whoseIRcontributions
I
logarithmicallydiverge.ThelessonisthatitisnotstraightforwardtoreformulatethewayofquantizationsothattheIRdivergent
contributionsthereinare allabsorbedbythe residual gaugetransformation. (Theabsorptionof theIR modesofthe curvature
perturbationwasintendedinotherframeworkssuchas δN formalism[31,32]andthesemi-classicalapproach[33]. )
Theseculargrowth. TheappearanceofIRdivergenceduetotheresidualgaugetransformationmentionedabovemightbe
evadedbysendingtheinitialtimetothepastinfinity.ThisisbecausethesizeofthelocalregionΣ Oincomovingcoordinates
t
∩
becomes infinitely large in this limit, making the discrepancy between the average in the local region and that in the global
universesmallerandsmaller.Thenitmightbeeffectivelyunnecessarytoperformtheresidualgaugetransformationattheinitial
time,althoughthisstatementisnotveryrigorous.Weshouldnotethatwhenwesendtheinitialtimetothepastinfinity,itistoo
naivetoneglectthenon-IRmodeswhicharecomparabletoorshorterthantheHubblelengthscale,becauseallthemodeswere
much shorterthan the Hubble length scale in the distant past. This makes the issue regardingthe secular growth much more
complicated. Forinstance,onceweincludethecontributionsfromthenon-IRmodes,wecannotusetheconservationofζk in
thelimitk/aH 1,wherekisthecomovingwavenumberoftheexternalleg,relyingonthelongwavelengthapproximation
≪
such as δN formalism. Here, in a simple example, we show that vertex integrations can yield the apparent secular growth
throughthenon-linearcontributionsfromthemodesataroundtheHubblescale. EvenifthevertexisconfinedintheregionO,
theintegrationregionofeachvertexisstillinfiniteinthe timedirectionas dtd3xa3( ) d(lna)/H4( ), whichmay
··· ≃ ···
causetheseculargrowth. Roughlyspeaking,theintegrand( )willbewrittenintermsofthedimensionlesstimedependent
··· R R
slowrollparametersandthewavenumberofthefieldsinthisvertexk /aH normalizedbytheHubblescale. Ifwefocusonthe
m
non-linearinteractioncomposedofthemodeswithk /aH oforderunity,theintegrand( )areexpressedonlyintermsofthe
m
···
parameterswhicharesupposedtochangeveryslowlyintimeandthenthecontributionfromtheinteractionvertexseemstoyield
thelogarithmicgrowth.Thisisanotheroriginoftheseculargrowth,whichshouldbedistinguishedfromtheoneinheritedfrom
theIRbehaviorof (ζ )2 . Ofcoursetheaboveargumentistoonative,butitshowsthattheabsenceoftheseculargrowthfrom
I
h i
thevertexintegrationisrathersubtle,requiringmorecarefultreatmentaboutthemodesaroundtheHubblescale.Becauseofthis
subtlety,introducingtheUVcutoffatthelengthscalelongerorequaltotheHubblelengthscalebyhandmakesthediscussion
incomplete. Infact,ifitwereallowedtosimplyneglecttheshortwavelengthmodes,thediscussioninRef.[28]withtheinitial
timet sentto wouldhavegivenaroughproofoftheabsenceofIRdivergencewithoutanylimitationtothequantumstate
i
−∞
bysendingthe initialtime tothe pastinfinity, whichcontradictsourcurrentclaim thatthe quantumstate is restricted in order
to avoidIR divergence. Recently, the absenceof the seculargrowthwas claimedrelyingonthe conservationof the curvature
perturbationinRefs.[37,38],buttheaspectsmentionedabovewerenotdiscussed. Inaddition,eveniftheconservationofζk
4
inthelimitk/aH 1isproved,thelogarithmicenhancementintheform (k/aH)2ln(k/a H )maygiverise,wherea and
i i i
≪
H arethescalefactorandtheHubbleparameterattheinitialtime. Thefactorln(k/a H )canbecomelargetoovercomethe
i i i
suppressionby(k/aH)whenwesendtheinitialtimetothepastinfinity.
B. Summaryofupcomingresults
Shortsummaryoftheresults. Inthissubsection,we summarizewhatwewillshowinthispaper. Takingaccountofthe
currentstatusofIRissuesmentionedabove,wewillestablishthefollowingthreestatementsinthispaper:
1. ThereisanalternativeequivalentHamiltonianthatdescribesthequantumdynamicsofourinterestandwhoseinteraction
partissolelycomposedoftheIRirrelevantoperators(,whichmeanthefieldoperatorsassociatedwiththeoperationsthat
manifestlysuppresstheIRcontributionsuchas∂ /aH and∂ /H).
i t
2. TheEuclideanvacuumstate, whichisspecifiedbytheregularitywhenthetimecoordinatesinthen-pointfunctionsare
analytically continuedto the imaginary in the complexplane, is physically the same both in the alternative description
mentionedinitem1,andintheoriginaldescription.
3. Then-pointfunctionsintheEuclideanvacuumstaterespectthespatialtranslationinvarianceandareregularintheIR.The
seculargrowthisabsent,evenifweincludetheverticeswithnon-IRmodes,aslongasveryhighorderofloopcorrections
arenotconcerned.
Belowweaddalittlemoredetailedexplanationsabouttheabovethreeitems.
Gauge issue. In this paper, the quantization and fixing the initial quantum state as a starting point of our discussion is
performed in the original system which describes the whole universe, where the residual gauge degrees of freedom are left
unfixed. Then,followingRefs.[25–27,30,41],weintroduceafieldoperatorwhichpreservestheinvarianceunderanyspatial
coordinatestransformations,includingresidualgaugetransformations.Werefertosuchanoperatorasagenuinegaugeinvariant
operator.Asarepresentative,weconsideragenuinegaugeinvariantcurvatureperturbation,gR.Aslongastheexpectationvalues
ofsuchgenuinegaugeinvariantoperatorsareconcerned,wecanperformtheresidualgaugetransformationwithoutaffectingthe
resultsofcomputations. Wewillshowthat,usingthisresidualgaugetransformation,theboundaryconditionsofthenon-local
operator∂−2intheactioncanbemodifiedtoberegularintheIR.
Requirement of the gauge invariance in quantum state. To calculate the n-point functions which preserve the invari-
anceundertheresidualgaugetransformations,theinitialstateshouldbealsospecified inagenuinelygaugeinvariantmanner.
However,whenweperformthequantizationconsideringthewholeuniverse,preservingtheresidualgaugeinvariancebecomes
obscure, because these residual gaugedegreesof freedomare notpresentas longas we dealwith the whole universe. In our
previous paper [27], we discovered a correspondence between the IR regularity and the invariance under the residual gauge
transformations,whichwillprovideanimportantcluetotheguidingprincipleinchoosingthegenuinelygaugeinvariantinitial
state. Todiscussthispoint,asidefromtheoriginalcanonicalvariablesζ(x)anditsconjugatemomentumπ(x),whoseevolution
is governed by the action (1.2), we introduced another set of the canonical variables corresponding to the description in the
coordinatesshiftedbyaconstantdilatationtransformation:
ζ˜(x):=ζ(t, e−sx), π˜(x), (1.7)
wheresisatimeindependentc-numberandπ˜(x)istheconjugatemomentumofζ˜(x). InRef.[27],weshowedthatrequesting
theequivalencebetweenthetwoquantumsystemsdescribedby ζ,π and ζ˜,π˜ guaranteestheIRregularityofloopcorrec-
{ } { }
tions.Here,theequivalenceoftwoquantumsystemsmeansthatthesameiterationscheme(orformallythesameinitialcondition
oftheinteractingsystem)givesphysicallythesamequantumstateinbothsystemsrelatedtoeachotherbythedilatationtransfor-
mation. Namely,alltheexpectationvaluesevaluatedinbothsystemsareequivalentifwetakeintoaccounthowtheytransform
underdilatation transformation. Requesting this equivalencewill be thoughtof as the invarianceof the initial state underthe
dilatationtransformation. InRef.[27],weemployedtheiterationschemeinwhichtheinteractionisturnedonatafinitepast.
Then, it turnedoutthatthe IR regularity/gaugeinvarianceconditioncannotbe consistentlyimposed. In the presentpaperwe
willsettheinitialquantumstateattheinfinitepast. Wewillshowthattheabovetransformationcanbeextendedtoallowatime
dependenceof the parameter s. As we described in the previoussection, this extensionplays a crucialrole in discussing the
absenceoftheseculargrowth.
The Euclidean vacuum. The second and third items are related with each other, once we establish the correspondence
betweenthegaugeinvarianceandtheIRregularity.Wewillshowthatthetwoquantumsystemsdescribedby ζ,π and ζ˜,π˜
{ } { }
areequivalentifwechoosetheEuclideanvacuum,whichisdefinedbyrequestingtheregularityofthen-pointfunctionsatthe
distantpastwiththetimepathrotatedtowardthecomplexplane. Tobemorespecific,astheseconditem,wewillshowthatthe
n-pointfunctionsforζ(x) calculatedby the canonicalvariables ζ,π with the boundaryconditionof the Euclideanvacuum
{ }
5
agreeswiththen-pointfunctionsforζ˜(t,es(t)x)calculatedbythecanonicalvariables ζ˜,π˜ underformallythesameboundary
{ }
condition,i.e.,
ζ(t, x )ζ(t, x ) ζ(t, x ) = ζ˜(t, es(t)x )ζ˜(t, es(t)x ) ζ˜(t, es(t)x ) . (1.8)
h 1 2 ··· n i{ζ,π} h 1 2 ··· n i{ζ˜,π˜}
Combinedwith the previouslymentionedtechniqueto deal with the gaugeissue, we will show that when we choose the Eu-
clideanvacuum,theHamiltoniandensityfor ζ˜,π˜ ,canbeexpressedonlyintermsoftheIRirrelevantoperators.
{ }
The IR regularityandthe absence of the secular growth. As forthe third item, we evaluatethe n-pointfunctionof the
genuinely gauge invariantoperator. Performingthe quantizationin the canonical system of ζ˜,π˜ , we will show that the IR
{ }
contributions do not diverge and that the secular growth is suppressed. We carefully investigate the contributions from the
modeswhicharecomparabletoorlessthantheHubblescale, i.e.,k & aH,withoutemployingtheasymptoticexpansionwith
respectto k/aH. As is stressed at the end ofthe precedingsubsection, thispointis one of the necessaryingredientsto show
theabsenceoftheseculargrowth. OnemaynaivelyexpectthattheUVmodeswithk/aH & 1willnoteffectivelycontribute
tothevertexintegrationbecauseoftheoscillatorybehaviour.Amorecarefulconsiderationtellsusthatthisnaiveexpectationis
notnecessarilycorrect.Ingeneral,vertexintegrationsbecomeamixtureofthepositiveandnegativefrequencymodefunctions,
whichyieldsthephaseintheUVlimiteiη(k1−k2+k3−···)whereηrepresentstheconformaltimewhichrunsfrom to0.Then,
−∞
thephasedoesnotnecessarilyexhibittherapidoscillationevenforthemodeswithk /aH k η &1,wherem=1, 2, ,
m m
≃− ···
whichcanbeacauseofseculargrowth.Intriguingly,choosingtheEuclideanvacuumplaysacrucialrolenotonlyintheIRlimit
butalso in theUV limit. Onecanshowthatthereisnomixingbetweenthe positiveandthe negativefrequencymodes, ifwe
choosetheEuclideanvacuum.Therefore,seculargrowthisevadedinthiscase.
Theoutlineofthepaper. Theoutlineofthispaperisasfollows. InSec.II,wewillbrieflyreviewthewaytoconstructthe
genuinelygaugeinvariantoperatorgR,followingRefs.[25,26]. Then,wewillintroducethecanonicalvariables ζ˜,π˜ andwill
{ }
derivetheHamiltonianforthesevariables.InSec.III,wewilldiscusstheitems1and2thatwementionedabove.InSec.IIIA,
we will describethe boundaryconditionsof the Euclideanvacuumand will proveEq. (1.8), which impliesthatthe boundary
conditionsoftheEuclideanvacuumselectthesamegroundstatebothin ζ,π and ζ˜,π˜ . InSec.IIIBandSec.IIIC,wewill
{ } { }
formulatethecanonicalquantizationintermsof ζ˜,π˜ andwillshowthattheinteractingverticesforthesecanonicalvariables
{ }
consist only of the IR irrelevant operators. Particularly in Sec. IIIC, we will show that using the residual gauge degrees of
freedom, the non-local operator ∂−2 can be made IR regular. In Sec. IV, we will discuss the item 3. In Sec. IVA, we will
showthattheboundaryconditionoftheEuclideanvacuumleadstotheso-callediǫprescriptioninaperturbativeexpansion. In
Sec. IVB, we will calculate the Wightmanpropagator,by which the n-pointfunctionsare expanded. Then, in Sec. IVC, we
explicitlyevaluaten-pointfunctionstoinvestigatetheIRregularityandtheseculargrowth.InSec.V,asconcludingremarks,we
discussanotherpossibilityoftheinitialstatewhichsatisfiestheIRregularity/gaugeinvarianceconditions.Wewillalsomention
therelatedpaperstoclarifywhatisnewinthispaper.
The advantage of the in-in formalism. In our previous publications [27–29], in calculating n-point functions, we used
the retarded Green function to solve the non-linearHeisenberg equation. This is because we thoughtthat using the retarded
Green function, whose Fourier mode is regular in the IR limit, makes the proof of the IR regularity transparent. However,
the perturbativeexpansionusing the retarded Green functionis not suitable forthe present purpose, because the positive and
negativefrequencymodesaremixedinthevertexintegrationsoncetheretardedGreenfunctionisused.Therefore,theboundary
conditionsoftheEuclideanvacuumdoesnotguaranteetheconvergenceofthetimeintegrationsforallthevertices.Bycontrast,
when we calculate the n-point functions in the in-in formalism, all vertex integrals can be made manifestly convergent by
adopting the boundary conditions of the Euclidean vacuum (see Sec. IIIA). Since the n-point functions obtained from the
solution written in terms of the retarded Green function agree with those obtained in the in-in formalism, the vertices which
donotconvergeshouldvanishinthefinalresultofthe n-pointfunctions. However,thecancellationisobscuredin anexplicit
perturbativeexpansion. Therefore,inthispaper,wecalculatethen-pointfunctiontotallybasedonthein-informalism,without
usingtheretardedGreenfunction.
II. CONSTRUCTINGTHEGAUGEINVARIANTQUANTITY
Inthispaper,asanexplicitmodelofinflation,weconsiderastandardsinglefieldinflationmodelwhoseactiontakestheform
M2
S = pl √ g[R gµνφ φ 2V(φ)]d4x, (2.1)
,µ ,ν
2 − − −
Z
whereM isthePlanckmassandwesetφtoadimensionlessscalarfield,dividingitbyM . However,aslongasweconsidera
pl pl
scalarfieldwiththesecond-orderkineticterm,anextensionproceedsinastraightforwardway.InSec.IIA,wewillconstructthe
genuinegaugeinvariantoperatorcorrespondingtothespatialcurvature ofaφ-constantsurface. InSec.IIB,wewillintroduce
thecanonicalsystem ζ˜,π˜ whoseHamiltoniandensityiscomposedonlyoftheIRirrelevantoperators.
{ }
6
A. Gauge invariantoperatorandquantization
Wefixthetimeslicingbyadoptingtheuniformfieldgaugeδφ=0. UndertheADMmetricdecomposition,whichisgivenby
ds2 = N2dt2+h (dxi+Nidt)(dxj +Njdt), (2.2)
ij
−
wetakethespatialmetrich as
ij
h =e2(ρ+ζ) eδγ , (2.3)
ij ij
wherea:=eρisthescalefactor,ζ istheso-calledcurvaturepertur(cid:2)batio(cid:3)nandδγ isatracelesstensor:
ij
δγi =0. (2.4)
i
Asspatialgaugeconditionsweimposethetransverseconditionsonδγ :
ij
∂ δγi =0. (2.5)
i j
Since thetime slicing isfixedbythe gaugeconditionδφ = 0, thereare remainingresidual gaugedegreesoffreedomonlyin
choosingthespatialcoordinates.Inthispaper,weneglectthevectorandtensorperturbations.Thetensorperturbation,whichis
massless,canalsocontributetotheIRdivergenceofloopcorrections.Wewilladdressthisissueinourfuturepublication.
FollowingRefs.[25,26],weconstructagenuinegaugeinvariantoperator,whichpreservesthe gaugeinvarianceinthelocal
observable universe. For the construction, we note that the scalar curvature sR, which transforms as a scalar quantity under
spatialcoordinatetransformations,becomesgenuinelygaugeinvariant,ifweevaluateitinthegeodesicnormalcoordinateson
eachtimeslice. Thegeodesicnormalcoordinatesareintroducedbysolvingthespatialthree-dimensionalgeodesicequation:
d2xi dxj dxk
gl +sΓi gl gl =0, (2.6)
dλ2 jk dλ dλ
where sΓi is the Christoffel symbol with respect to the three dimensional spatial metric on a constant time hypersurface
jk
and λ is the affine parameter. Here we put the index gl on the global coordinates, to reserve the simple notation x for the
geodesic normal coordinates, which will be mainly used in this paper. We consider the three-dimensional geodesics whose
affineparameterrangesfromλ=0to1withtheinitial“velocity”givenby
dxi (x,λ)
gl =e−ζ(λ=0)xi. (2.7)
dλ
(cid:12)λ=0
(cid:12)
A point xi in the geodesic normalcoordinates is identifi(cid:12)ed with the end point of the geodesic, xi (x,λ = 1) in the original
(cid:12) gl
coordinates. Using the geodesic normal coordinates xi, we perturbatively expand xi as xi = xi +δxi(x). Then, we can
gl gl
constructagenuinelygaugeinvariantvariableas
gR(t, x):=sR(t, xi (x))=sR(t, xi+δxi(x))), (2.8)
gl
wheretdenotesthecosmologicaltime.
B. Dilatationsymmetryintheglobaluniverse
Thefocusofthissubsectionisonthedilatationtransformation,shiftingtotherescaledspatialcoordinates:
x˜i :=es(t)xi. (2.9)
Solving the Hamiltonian and momentumconstraintequations, we can derive the action that is expressed only in terms of the
curvatureperturbationζ(x),whichisschematicallywrittenas
S = dtd3xL[∂ ζ(x),ζ(x)], (2.10)
t
Z
Usingthecurvatureperturbationζ andtheconjugatemomentumdefinedbyπ := δL/δ(∂ ζ),theHamiltoniandensityisgiven
t
bytheLegendretransformas
H[ζ(x),π(x)] :=π(x)∂ ζ(x) L[∂ ζ(x), ζ(x)]. (2.11)
t t
−
7
Whatisimportanthereisonlythefactthatthecurvatureperturbationζ appearintheactioneitherwithdifferentiationorin
theformofthecombinationofthephysicaldistanceeρ+ζdx[27]. Inthenewcoordinates(2.9),thephysicaldistanceiswritten
aseρ+ζ˜(t,x˜)−s(t)dx˜,withthedefinitionofanewvariable
ζ˜(t, x˜):=ζ(t, x). (2.12)
Thus,ifthefieldζ(x)isreplacedwithζ˜(t, x˜) s(t)underthechangeofthecoordinatesfromxtox˜,theactionbasicallyremains
−
invariant. Toexpress∂ ζ(x)intermsofthenewvariableζ˜,wedenotethepartialdifferentiationwiththespatialcoordinatesx
t
fixedas (∂tζ˜(t, x˜))x. Thesubscriptassociated with the parenthesesspecifies the spatialcoordinatesthat we fix in taking the
partialdifferentiation.Then,wehave
(∂tζ˜(t, x˜))x =∂tζ(x). (2.13)
Forbrevity,whenthefixedspatialcoordinatesareidenticaltotheonesintheargumentofthevariable,wesimplyuse∂ . Then,
t
wecanestablishanidentity
dtd3xL[∂tζ(x),ζ(x)] = dtd3x˜L[(∂tζ˜(t,x˜))x,ζ˜(t,x˜) s(t)]. (2.14)
−
Z Z
Recalling the relation between x and x˜ (2.9), this equality also means the equality at the level of Lagrangian density,
e−3s(t)L[∂tζ(x),ζ(x)] =L[(∂tζ˜(t,x˜))x,ζ˜(t,x˜) s(t)].
−
Weintroducethecanonicalconjugatemomentumcorrespondingtoζ˜(t,x˜)inthestandardwayas
π˜(t, x):= ∂L[(∂tζ˜(t, x˜))x,ζ˜(t, x˜)−s(t)]. (2.15)
∂(∂ ζ˜(t, x˜))
t
Noticingtherelation
∂tζ˜(t,x˜)=(∂tζ˜(t,x˜))x s˙(t)x˜ ∂x˜ζ˜(t,x˜), (2.16)
− ·
wehave
π˜(t, x˜)= ∂L[(∂tζ˜(t, x˜))x,ζ˜(t, x˜)−s(t)] =e−3s(t)∂L[∂tζ(x),ζ(x)] =e−3s(t)π(x). (2.17)
∂((∂tζ˜(t, x˜))x) ∂(∂tζ(x))
Asisexpected,usingthecommutationrelationsforζ andπtogetherwithEqs.(2.12)and(2.17),wecanverify
ζ˜(t, x˜), π˜(t, y˜) =e−3s(t)iδ(3)((x y))=iδ(3)(x˜ y˜), (2.18)
− −
h i
aswellas
ζ˜(t, x˜), ζ˜(t, y˜) =[π˜(t, x˜), π˜(t, y˜)]=0. (2.19)
h i
TheHamiltoniandensityforζ˜(x˜)andπ˜(x˜)isobtainedinthestandardwayas
H˜ ζ˜(t,x˜), π˜(t,x˜) :=π˜(t,x˜)∂tζ˜(t, x˜) L[(∂tζ˜(t, x˜))x,ζ˜(t, x˜) s(t)]
− −
h i=π˜(t,x˜)(∂tζ˜(t,x˜))x L[(∂tζ˜(t, x˜))x,ζ˜(t, x˜) s(t)] s˙(t)π˜(t,x˜)x˜ ∂x˜ζ˜(t, x˜)
− − − ·
=H[ζ˜(t, x˜) s(t),π˜(t, x˜)] s˙(t)π˜(t,x˜)x˜ ∂x˜ζ˜(t, x˜), (2.20)
− − ·
whereintheequalityonthesecondlineweusedEq.(2.16). ThelastequalityisexactlythesameLegendretransformationasin
theoriginalsystemandthereforewecanusethesamefunctionalformoftheHamiltoniandensityH.
Assumingthats(t)isassmallasζ˜(x)andπ˜(x),wedecomposetheHamiltoniandensitiesHandH˜ intothenon-interacting
parts,whichincludeonlythequadraticterms,andtheinteractingpartsas
H[ζ(x), π(x)]=H [ζ(x), π(x)]+H [ζ(x), π(x)], (2.21)
0 I
and
H˜ ζ˜(x), π˜(x) =H ζ˜(x), π˜(x) +H˜ ζ˜(x), π˜(x) . (2.22)
0 I
h i h i h i
8
Intheaboveweusedthecoordinatesxinsteadofx˜ forthe ζ˜,π˜ system,butitwillnotcauseanyconfusionaftertherelations
{ }
betweenthe ζ,π and ζ˜,π˜ systemshavebeenestablished. Here,wereplacedH [ζ˜(x) s(t), π˜(x)]withH [ζ˜(x), π˜(x)],
0 0
sinceζ(x)alw{ays}appea{rswit}hthespatialderivativeinH [ζ(x),π(x)]. Remarkably,thenon−-interactingpartoftheHamiltonian
0
density doesnot changeat all underthe dilatationtransformation. Using Eq. (2.20), we find that the interactionHamiltonian
H˜ [ζ˜, π˜]isgivenby
I
H˜I ζ˜(x), π˜(x) :=HI ζ˜(x) s(t), π˜(x) s˙(t)π˜(x)x ∂xζ˜(x). (2.23)
− − ·
h i h i
Inthisway,wecanwritedownH˜ onlyintermsofζ˜(x) s(t),ζ˜withdifferentiation,π˜ ands˙(t). InRef.[27],weintroduced
I
−
thetwosetsofthecanonicalconjugatevariableswhichareconnectedbythedilatationtransformationwithaconstantparameter
s. When we take the limit where s(t) is constant, the Hamiltonian density H˜(x) takes the same functional form as H(x)
exceptfortheconstantshiftofζ˜(x)by s. Itisbecause,withoutmodifyingthegaugecondition,wecanperformthedilatation
−
transformationwith the constant parameter s also in the whole universe. Then the action which preservesthe diffeomorphic
invariancebecomesinvariantunderthechangefrom ζ(x)toζ(t,e−sx) s. HerewehaveextendedtheargumentinRef.[27]
−
toallowstodependontime. AswementionedinSec.I,thisextensionplaysthecrucialroleinourdiscussionaboutthesecular
growth. Inthenextsection,wewillshowthatalltheinteractionverticesinthecanonicalsystem ζ˜,π˜ arecomposedonlyof
{ }
theIRirrelevantoperator.
III. INTERACTIONHAMILTONIANWITHTHEIRIRRELEVANTOPERATORS
In this section, we describe the first two of the three items we raised in Sec. I. In the preceding section, we derived the
Hamiltonianforthecanonicalvariablesζ˜(x)andπ˜(x). Since ζ,π and ζ˜,π˜ areconnectedbythecanonicaltransformation,
{ } { }
ifwechoosethesameinitialstateinbothofthetwocanonicalsystems,then-pointfunctionsforthesameoperator,forinstance
gR, calculated in these canonical systems should agree with each other. However, even if we adopt operationally the same
scheme to select the initial state in these two systems, it does not guarantee that the selected initial states are the same. In
Sec.IIIA,afterwedescribethedefinitionoftheEuclideanvacuum,wewillshowthattheconditionoftheEuclideanvacuum
operationallyselectsthesamequantumstateirrespectiveofthechoiceofthecanonicalvariables. Thisensurestheequivalence
ofthesetwocanonicalsystemsincludingthechoiceoftheinitialquantumstate,whichwementionedintheitem2. InSec.IIIB,
wewillperformthequantizationusingthecanonicalvariables ζ˜,π˜ . AswewillshowinSec.IIIC,byvirtueoftheequivalence
{ }
betweenthetwocanonicalsystems,theinteractionverticesfor ζ˜,π˜ canbeexpressedintermsofoperatorproductscomposed
{ }
onlyoftheIRirrelevantoperators.
A. Euclideanvacuumanditsuniqueness
In the case with a massive scalar field in de Sitter spacetime, the boundaryconditionspecified by rotatingthe time path in
thecomplexplanecanbeunderstoodasrequestingtheregularityofcorrelationfunctionsontheEuclideanspherewhichcanbe
obtainedbytheanalyticcontinuationfromtheonesondeSitterspacetime. ThevacuumstatethusdefinediscalledEuclidean
vacuumstate. Because of the similarity, here we also referto the state which is specified by a similar boundarycondition as
theEuclideanvacuum. Tobemoreprecise,wedefinetheEuclideanvacuumasfollows. Inthein-informalism,theinsertionof
interactionverticesisorderedalongtheclosedtimepath. Byrotatingthetimepathtowardtheimaginaryplane,theforwardtime
evolutionbeginsatη(t ) = (1 iǫ)andendsatthefinaltimet andthebackwardtimeevolutionbeginsatt andendsat
i f f
−∞ −
η(t ) = (1+iǫ). Herewesetǫtoasmallpositivenumber. Sincerotatingthetimepathcanbebetterunderstoodbyusing
i
−∞
theconformaltimeη,weintroducedtheconformaltimeηas
t dt′ ρ(t) dρ′
η(t):= = . (3.1)
eρ(t′) eρ′ρ˙(ρ′)
Z Z
We define the Euclideanvacuum, requestingthe regularityof the n-pointfunctionswith an arbitrarynaturalnumber n in the
limitofη(t ) (1 iǫ),i.e.,
i
→−∞ ±
F (x , x ):= T ζ(x ) ζ(x ) < as η(t ) (1 iǫ), (3.2)
n 1 n c 1 n a
··· h ··· i ∞ →−∞ ±
wherea = 1, nandT denotesthetimeorderingalongtheclosedtimepath. We first showthatthen-pointfunctionsofζ
c
···
areuniquelyfixedbyrequestingthecondition(3.2). Inthispaper,forsimplicity,weassumethateρρ˙(ρ)israpidlyincreasingin
timesothat
η(t) =O 1/eρ(t)ρ˙(t) . (3.3)
| |
(cid:16) (cid:17)
9
Next, we show that the boundary condition of the Euclidean vacuum uniquely determines the n-point functions
F (x , x ). WeschematicallydescribetheHeisenbergequationforζ(x)as
n 1 n
···
Lζ =S [ζ], (3.4)
NL
whereListhesecond-orderdifferentialoperator:
∂2
L:=∂2+(3 ε +ε )∂ . (3.5)
ρ − 1 2 ρ− e2ρρ˙2
Fornotationalconvenience,weintroducedthehorizonflowfunctions,
1 d 1 d
ε := ρ˙, ε := ε , (3.6)
1 n n−1
−ρ˙dρ ε dρ
n−1
with n 2, but we do not assume that these functionsare small to keep the backgroundevolution unconstrainedexceptfor
≥
requestingEq.(3.3), which is valid, forinstance, when ε are constantin time. Using the Heisenbergequation(3.4), we can
n
obtaintheevolutionequationofthepath-orderedn-pointfunctionsF (x , x )as
n 1 n
···
L F (x , , x )=V(a)[ F ], (3.7)
xa n 1 ··· n NL { m}m>n
whereL isthederivativeoperatorLgiveninEq.(3.5)withthecoordinatesxreplacedwithx . Sincetheequationofmotion
xa a
forζ(x)isnon-linear,theequation(3.7)includesthesourceterm(therighthandside)composedofm-pointfunctionsofζ(x)
withm > n. We canverifytheuniquenessofthen-pointfunctionsforζ(x) byshowingthatsolutionofEq.(3.7)isuniquely
fixedbytheboundarycondition(3.2). Toshowthisuniqueness,weformallysolvetheequation(3.7)as
F (x , , x )=f (x , , x )+L−1V(a)[ F ], (3.8)
n 1 ··· n n 1 ··· n xa NL { m}m>n
wheref (x , , x )isahomogeneoussolution,whileweassumethatthespecificsolutionL−1V(a)[ F ]satisfiesthe
n 1 ··· n xa NL { m}m>n
regularityconditioninthelimitsη(t ) (1 iǫ).Nowthequestioniswhethertheboundarycondition(3.2)allowsustoadd
a
anyhomogeneoussolutions.IntheFou→rie−rs∞pace±,fncanbeexpandedbye−ikη(ta)oreikη(ta)inthelimitsη(ta) (1 iǫ).
Theregularityatη(ta) (1+iǫ)acceptse−ikη(ta) only,whiletheregularityatη(ta) (1 iǫ)ac→cep−ts∞theo±ther.
→ −∞ → −∞ −
Thus the regularity condition in the two limits does not allow to add any homogeneoussolutions f , which implies that the
n
n-pointfunctionsF (x , , x )areuniquelyfixedbytheboundaryconditionoftheEuclideanvacuum.
n 1 n
···
Next, weshowthatthisuniquenessisensuredindependentofwhetherweusethecanonicalvariables ζ,π or ζ˜,π˜ . We
{ } { }
employtheboundaryconditionoftheEuclideanvacuumforthecanonicalvariableζ˜aswell,requesting
T ζ˜(x ) ζ˜(x ) < as η(t ) (1 iǫ). (3.9)
h c 1 ··· n i{ζ˜,π˜} ∞ a →−∞ ±
Then,wecanshowthatthepath-orderedn-pointfunctions
F˜n(x1, ···xn):=hTcζ˜(t1, es(t1)x1)···ζ˜(tn, es(tn)xn)i{ζ˜,π˜}, (3.10)
agreewiththen-pointfunctionsF (x , x )= T ζ(x ) ζ(x ) fixedbytheboundarycondition(3.2),i.e.,
n 1 n c 1 n {ζ,π}
··· h ··· i
F˜ (x , x )=F (x , x ). (3.11)
n 1 n n 1 n
··· ···
Hereputtingthesuffixes ζ,π or ζ˜,π˜ ,wedenotethecanonicalvariablesusedinimposingtheboundaryconditionexplicitly.
{ } { }
WeagainschematicallydescribetheHeisenbergequationforζ˜as
Lζ˜=S˜ [ζ˜]. (3.12)
NL
Sinceζ(x)andζ˜(x)areconnectedbythecanonicaltransformation,theequationofmotionobtainedbyoperatingLon
ζ(x)=ζ˜(t, es(t)x)=ζ˜(x)+s(t)x ∂xζ˜(x)+ , (3.13)
· ···
can be recast into Eq. (3.4) by using Eq. (3.12). A similar argument follows for the equations of motion for the correlation
functionsF andF˜ . Usingtheequationofmotionforthen-pointfunctionsofζ˜(x),whichcanbederivedfromEq.(3.12),we
n n
canconfirmthatanoperationofL on
xa
F˜n(x1, ···xn)=hTcζ˜(x1)···ζ˜(xn)i{ζ˜,π˜}+s(t1)hTcx1·∂x1ζ˜(t1, x1)···ζ˜(tn, xn)i{ζ˜,π˜}+··· (3.14)
10
leadsto
L F˜ (x , , x )=V(a)[ F˜ ]. (3.15)
xa n 1 ··· n NL { m}m>n
Thisequationtakesthesameformastheequationofmotion(3.7). We alsonotethattheboundaryconditionoftheEuclidean
vacuum(3.9)implies
F˜ (x , , x , , x )< as η(t ) (1 iǫ). (3.16)
n 1 a n a
··· ··· ∞ − →∞ ±
Theequivalence(3.11)isnowtransparent,becausetheequationsofmotion (3.7)and(3.15),andtheboundaryconditions(3.2)
and(3.16)arethesame,andthelatterspecifythesolutionsoftheformeruniquely. Thisequivalenceisadistinctivepropertyof
theEuclideanvacuum1. Herewetooktheboundaryconditionsforn-pointfunctionsasthedefinitionoftheEuclideanvacuum
state, assumingtheexistenceofsuchaquantumstate. InSec.IVA,weexplainsuchaEuclideanvacuum,ifexists, shouldbe
theonegivenbytheordinaryiǫprescription.
B. Rewritingthen-pointfunctions
Inthissubsection,werearrangetheexpressionforthe n-pointfunctionsofthegenuinelygaugeinvariantvariablegRintoa
moresuitableformtoexaminetheregularityoftheIRcontributions.First,solvingthethreedimensionalgeodesicequations,we
obtaintherelationbetweentheglobalcoordinatesxi andthegeodesicnormalcoordinatesxi as
gl
xi =e−ζ(t,e−ζx)xi+ , (3.17)
gl ···
wheretheellipsismeansthetermswhichvanishwhenζ(x)isspatiallyhomogeneous,i.e.,thetermssuppressedintheIRlimit.
NotethatchangingthespatialcoordinatesintothegeodesicnormalcoordinatesalsomodifiestheUVcontributions.Tsamisand
Woodard[45] showedthatusingthegeodesicnormalcoordinatescanintroduceanadditionaloriginofUV divergence,which
maynotbeabletoberenormalizedbylocalcounterterms[46]. Itshouldbeclarifiedwhetherthisissueisaseriousproblemor
not,butwedeferittoafuturestudy.Instead,tokeeptheUVcontributionsundercontrol,wereplaceζ(x)inEq.(3.17)withthe
smearedcurvatureperturbationgζ¯(t),i.e.,
xi =e−gζ¯(t)xi, (3.18)
gl
with
d3xW (x)ζ(t,e−gζ¯x)
gζ¯(t):= Lt , (3.19)
d3xW (x)
R Lt
where W (x) is a windowfunctionwhich is non-vanishinRgonlyin the localregion Σ O. We approximatethe averaging
scaleateLacthtimetbytheHubblescale,i.e.,L 1/ eρ(t)ρ˙(t) . Althoughgζ¯appearsotn∩theright-handsideofEq.(3.19), gζ¯
t
isdefinediterativelyateachorderoftheperturbat≃ion.W{ecalcula}tethen-pointfunctionsofR gζ(t,x),insteadofgR,with
x
gζ(t, x):=ζ(t, e−gζ¯(t)x). (3.20)
Here,R denotestheIRsuppressingoperatorsuchas
x
∂ , ∂x , 1 d3xWLt(x) , , (3.21)
ρ eρ(t)ρ˙(t) − d3yW (y) ···
(cid:18) R Lt (cid:19)
where x is the spacetime coordinatesof the field on which thesRe operatorsact. Although R gζ(t, x) is not genuinely gauge
x
invariant,itisstillinvariantunderthedilatationtransformation,whichisassociatedwiththedominantIRcontributions.Infact,
sincethesmearedcurvatureperturbationgζ¯(t)transformsintogζ¯(t) f underthedilatationtransformation: x e−fxwitha
constantf,R gζ(x)iskeptinvariantunderthistransformation.Byco−ntrast,theconstantpartofgζ(x)canbemo→difiedunderthe
x
dilatationtransformationasgζ(x) gζ(x) f. SincethegenuinegaugeinvariantvariablegR(x)shouldnotbeaffectedbythe
dilatationtransformation,whichis→apartof−theresidualgaugetransformations,gζ(x)appearsonlyintheformofR gζ(x)when
x
1The uniqueness ofthe Euclidean vacuum becomes intuitively clear when the Hamiltonian is time independent and the lowest energy eigenstate is non-
degenerate,becausetheiǫprescriptionselectstheuniquegroundstateofthesystem.