Strong restriction on inflationary vacua from the local gauge invariance I: Local gauge invariance and infrared regularity

0.18 MB·English

by  Takahiro Tanaka

Strong restrictionon inflationary vacua from the local“gauge”invariance I
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:September11,2012)
Theprimordialperturbationiswidelyacceptedtobegeneratedthroughthevacuumfluctuationofthescalar
fieldwhichdrivesinflation. Itis,however,notcompletelyclearwhatisthenaturalvacuumintheinflationary
universe particularly in the presence of non-linear interactions. In this seriesof papers, wewill address this
issue, focusing on the condition required for the removal of the divergence from the infraredcontribution to
loopdiagrams. TheIRregularityisdeeplyrelatedtopreservingthecoordinateindependenceindescribingthe
localobservableuniverse. Inourpreviousworks,theIRregularityconditionwasdiscussedusingtheslowroll
2
expansion,whichrestrictsthebackgroundevolutionoftheinflationaryuniverse. Wewillshowmoregenerally
1
thattheabsenceofinfrareddivergenceleadstonon-trivialconstraintsontheallowedquantumstates.
0
2

p
e
S I. INTRODUCTION

0
1 Itiswidelyacceptedthatprimordialcurvatureperturbationsoriginatefromthevacuumfluctuationoftheinflatonfieldinthe
inflationaryuniverse.Differentchoicesofquantumstatesleadtodifferentstatisticsoftheprimordialfluctuation.Amongvarious

]
optionsofvacuumstates,weusuallychoosetheadiabaticvacuum.Afreescalarfieldcanbeunderstoodasasetofindependent
h
t harmonicoscillators. Takingtheadiabaticvacuumlooksreasonable,becauseeachoscillatortendstohaveafixedfrequencyas
-
p thetimescalebecomesshorterthanthatofthecosmicexpansion.However,oncewestarttotakeintoaccounttheself-interaction
e ofthe field, thereasoningofchoosingthe adiabaticvacuumbecomesobscure. Thereforeitis aninterestingquestionwhether
h somephysicalrequirementprohibitstakinganarbitraryquantumstateornot. Thisisthequestionweaddressinthispaper.
[ Thekeyideaistofocusonfluctuationsthatwecanmeasureinobservations. Sincetheregionwecanobserveislimitedtoa

1 portionofthewholeuniverse,weneedtotakeintoaccountthelocalpropertyoftheobservablefluctuations. Incasewecanget
v accesstoinformationonlyonaportionofthewholeuniverse,thereappearadditionaldegreesoffreedomtochooseboundary
4 conditionsinfixingthecoordinates.Asshowninourrecentpublications[1–6]andwillbebrieflydescribedlater,thesedegrees
1 of freedomin the boundaryconditionscan be thoughtof as residualdegreesof freedomin fixing the coordinatesofthe local
9
observableuniverse.Theobservablefluctuationsshouldbeinsensitivetothecoordinatechoiceinthelocalobservableuniverse.
1
We showed that removing this ambiguity significantly affects the infrared (IR) behaviour of primordial fluctuations. In the
.
9 conventionalgauge-invariantperturbation,inwhichthisambiguityisnottakencareof,itiswidelyknownthatloopcorrections
0 of massless fields such as the inflaton diverge because of the IR contributions [7–28]. One may expect that performing the
2
quantizationafterweremovetheseresidualcoordinatedegreesoffreedominthelocaluniversecancurethesingularbehaviour
1
oftheIRcontributions.However,theseresidualcoordinatedegreesoffreedomarenotpresentifwedealwiththewholeuniverse
:
v includingthespatialinfinity,sincewhatwecalltheresidualcoordinatetransformationdivergesatspatialinfinity. Inthissense,
i
X if we think of the whole universe, they are not gauge degrees of freedom which preserve the action to be invariant. They
can be removedonly when our concern is limited to our local observableregion of the universe. However, if we consider to
r
a quantizethesystemexcludingtheregionoutsideofourobservableuniversefromthebeginning,theselectedquantumstate,in
general,dependsonthechoiceoftheboundaryconditions. InRef.[1],weassumedthattheboundaryconditionsareimposed
ata finitedistanceto removethedegreesoffreedominchoosingcoordinates,butthequantization(orequivalentlysetting the
initial quantumsate) was performedconsideringthe wholespatial section of the universe. In thatanalysis we concludedthat
there is no IR divergencesirrespectiveof the initial quantumstate. However, there were severaloverlookedpoints. After we
settheinitialquantumstate atafiniteinitialtime, weneedtotranslateitintothelanguagewrittensolelyintermsofthelocal
quantities. Namely, the Heisenbergoperatorscorrespondingto the perturbationvariablesshouldbe transformedinto the ones
thatsatisfygivenboundaryconditionsusinganappropriateresidualspatialcoordinatetransformation.Sincethenon-linearpart
ofthistransformationisnotguaranteedtobeIRregular,itcanproduceIRdivergences.
Ifwesendtheinitialtimetothepastinfinity,onemaythinkthatthisproblemmaydisappearbecauseourobservableregion
in comovingcoordinatesbecomesinfinitely largein this limit. However, this argumentis notso obvious. If we think of, for
instance,theconformaldiagramofdeSitterspace,thecausalpastofanobservernevercoversthewholeregionofatimeconstant
∗Electronicaddress:tanaka˙at˙yukawa.kyoto-u.ac.jp
†Electronicaddress:yurakawa˙at˙ffn.ub.es
2
sliceinflatchart.Furthermore,inthiscasewealsoneedtocareabouttheseculargrowthoffluctuation.InRef.[1],wediscussed
the absence of the secular growth, focusing only on modes beyondthe horizonlength scale. To complete the discussion, we
alsoneedtoincludethevertexeswhichcontainmodesfarbelowthehorizonlengthscale,becausethesevertexesmayyieldthe
seculargrowththroughcorrelationsbetweensubhorizonmodesandthesuperhorizonmodes.
Later in Refs. [3, 4], leavingthe residualcoordinatedegreesof freedomunfixed,we calculatedquantitiesthatare invariant
undertheresidualcoordinatetransformationinthelocaluniverseunderthesocalledslow-rollapproximation.Then,wederived
theconditionsontheinitialstatesthatguaranteetheIRregularity. Theseconditionsaretheonesthatguaranteetheabsenceof
additionalIRdivergencesoriginatingfromthecoordinatetransformationmentionedabove. However,thephysicalmeaningof
theseconditionsisnottransparentfromtheirderivationinRefs.[3,4].
Inthispaper,asinthecaseoftheconventionalperturbationtheoryandalsoasinRefs.[3,4],wewillperformthequantization
overthewholeuniverse. Then,calculatingobservablequantities,whichareinvariantundertheresidualcoordinatetransforma-
tion in the localuniverse, we will derivethe necessaryconditionsforthe initialquantumstate to be freefromIRdivergences
moregenerally,removingthelimitationtotheslow-rollapproximation.WewillalsoshowthattheIRregularityconditionscan
bethoughtofastheconditionsthatrequesttheequivalencebetweentwosystemsrelatedbymeansofasortofscaletransforma-
tionofspatialcoordinates. Inthispaper,wewillconsideracaseinwhichtheinteractionisturnedonatafiniteinitialtime. In
thenextpaper,wewilldiscussthecasewhentheinitialtimeissenttothepastinfinity. Thispaperisorganizedasfollows. In
Sec.II,wefirstshowasymmetryofthesystem,whichplaysanimportantroleinourdiscussion.Then,followingRef.[3,4],we
constructanoperatorwhichremainsinvariantundertheresidualcoordinatetransformation.InSec.III,weaddresstheconditions
ontheinitialstates. InSec.IV,wewillsummarizetheresultsofthispaperandwilloutlinefutureissues.
II. PREPARATION
Inthissection,wewillintroducebasicingredientstocalculateobservablefluctuations. Inthispaper,weconsiderastandard
singlefieldinflationmodelwhoseactiontakestheform
M2
S = pl √ g[R gµνφ φ 2V(φ)]d4x, (2.1)
,µ ,ν
2 Z − − −
where M is the Planck mass and φ is a dimensionalscalar field divided by M . In Sec. IIA, we show a symmetryof this
pl pl
system,whichplaysakeyroleinourargument.InSec.IIB,afterweintroduceavariablewhichpreservesinvarianceregarding
thecoordinatechoiceinthelocaluniverse,weprovideawayofquantizingthesystem.
A. Symmetryofthesystem
Tofixthetimeslicing,weadopttheuniformfieldgaugeδφ=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
(cid:2) (cid:3)
whereeρdenotesthebackgroundscalefactor,ζ istheso-calledcurvatureperturbationandδγ istraceless:
ij
δγi =0, (2.4)
i
wherethespatialindexwasraisedbyδij. Asspatialgaugeconditionsweimposethetransverseconditionsonδγ :
ij
∂ δγi =0. (2.5)
i j
In this paper, we neglect the vector and tensor perturbations. The tensor perturbation, which is a massless field, can also
contributetotheIRdivergenceofloopcorrections.Wewilladdressthisissueinourfuturepublication.
SolvingtheHamiltonianandmomentumconstraints,wecanderivetheactionwhichisexpressedonlyintermsofthecurvature
perturbation[29]. Sincethespatialmetricisgivenintheforme2ρe2ζdx2,wenaivelyexpectthattheexplicitdependenceonthe
spatialcoordinatesappearintheactionforζ onlyintheformofthephysicaldistanceeρeζdx. Wefirstexaminethisproperty.
UsingtheLagrangiandensityinthephysicalcoordinatesL ,weexpresstheactionas
phys
S = dtdxL(x)= dtdxe3(ρ+ζ)L (x). (2.6)
phys
Z Z
3
We can confirm that the Lagrangian density L is composed of terms which are in a covariant form regarding a spatial
phys
coordinatetransformationsuchashij∂ N andhijN ∂ ζ. Using
i j i j
dxˇi =eρ+ζdxi, ∂/∂xˇi =e−(ρ+ζ)∂/∂xi, Nˇ =e−(ρ+ζ)N , (2.7)
i i
wecanabsorbthecurvatureperturbationζ withoutdifferentiationwhichappearsfromthespatialmetric,forinstance,as
∂ ∂
hij∂ N =δije−2(ρ+ζ)∂ N =δij Nˇ +Nˇ ζ . (2.8)
i j i j (cid:18)∂xˇi j j∂xˇi (cid:19)
Afterastraightforwardrepetition,theLagrangiandensitycanberecastinto
S = dtdxe3(ρ+ζ)L [Dζ,N, Nˇ ]. (2.9)
phys i
Z
Here, to stress the factthat all ζs which remainunabsorbedare associated with differentiation∂ or ∂/∂xˇi, we introducedD
t
whichdenotesdifferentiationsingeneral.Usingthisexpressionoftheaction,theHamiltonianconstraintisgivenby
δL δL
C[Dζ,N, Nˇ ]:= phys =e−3(ρ+ζ) =0, (2.10)
i
δN δN
andthemomentumconstraintsaregivenby
δL δL
C [Dζ,N, Nˇ ]:= phys =e−2(ρ+ζ) =0. (2.11)
i i δNˇi δNi
Perturbing these constraints, we can show that the Lagrange multipliers N and Nˇ are given by solutions of the elliptic type
i
equations:
2 2
∂ ∂
N =f[Dζ(x)], Nˇ =f [Dζ(x)]. (2.12)
(cid:18)∂xˇ(cid:19) (cid:18)∂xˇ(cid:19) i i
Toaddressthesystemofthewholeuniverse,theintegrationintheactionisassumedtobetakenoverthewholeuniverse.Then,
byassumingtheregularityatthespatialinfinity,thesePoissonequationscanbeuniquelysolvedas
N =N[Dζ(x)], Nˇ =Nˇ [Dζ(x)]. (2.13)
i i
SubstitutingtheseexpressionsofN andNˇ intotheaction(2.9),wecanshowthattheactiontakestheform:
i
S = dtd3xˇL [D ζ(x)]. (2.14)
phys dxˇ
Z
Here,weputthesubscriptdxˇ onDtospecifythespatialcoordinatesusedinD. Theequation(2.14)explicitlyshowsthatthe
actionforζ onlydependsonthephysicalspatialdistance.
Nowwearereadytoshowthedilatationsymmetryofthesystem,whichplaysacrucialroleindiscussingtheIRregularity.
(Seealsothediscussionsin[30–36].) Changingthecoordinatesintheintegralfromxtoe−sxwithaconstantparameters,we
canrewritetheaction(2.14)as
S = dtd3xe3(ρ+ζ)Lphys[Deρ+ζdxζ(x)]= dtd3xe3(ρ+ζ−s)Lphys[Deρ+ζ−sdxζ(t, e−sx)]. (2.15)
Z Z
Thisimpliesthattheactionforζ possessesthedilatationsymmetry
S = dtd3xL[ζ(x)]= dtd3xL ζ(t, e−sx) s . (2.16)
Z Z −
(cid:2) (cid:3)
As long as we consider a theory which preserves the three-dimensionaldiffeomorphisminvariance, the system preserves the
dilatationsymmetry. Itisbecausetheabove-mentioneddilatationsymmetrycanbethoughtofasoneofthespatialcoordinate
transformations.
Tomakeuseofthedilatationsymmetry,weintroduceanothersetofcanonicalvariablesthanζ(x)anditsconjugatemomentum
π(x). Wecanshowthat
ζ˜(x):=ζ(t, e−sx), π˜(x):=e−3sπ(t, e−sx) (2.17)
4
satisfythecanonicalcommutationrelationsaswellasζ(x) andπ(x). Actually,usingthecommutationrelationsforζ(x) and
π(x),wecanverify
ζ˜(t, x), π˜(t, y) = ζ(t, e−sx), e−3sπ(t, e−sy) =e−3siδ(3)(e−s(x y))=iδ(3)(x y), (2.18)
h i (cid:2) (cid:3) − −
andalso
ζ˜(t, x), ζ˜(t, y) =[π˜(t, x), π˜(t, y)]=0. (2.19)
h i
UsingEq.(2.16),wecanshowthattheHamiltoniandensitiesexpressedintermsofthesetwosetsofthecanonicalvariables
arerelatedwitheachotheras
d3xH[ζ(x), π(x)]= d3x π(x)ζ˙(x) L[ζ(x)]
Z Z n − o
= d3x e−3sπ(t, e−sx)ζ˙(t, e−sx) L[ζ(t, e−sx) s]
Z n − − o
= d3xH[ζ˜(x) s, π˜(x)]=: d3xH˜[ζ˜(x), π˜(x)], (2.20)
Z − Z
whereonthesecondequality,weagainchangedthecoordinatesintheintegralasx e−sx. NotethattheHamiltoniandensity
forζ˜andπ˜isgivenbythesamefunctionalastheoneforζ andπwithζ˜shiftedbyζ˜→ s.
Wedefinethenon-interactingpartoftheHamiltoniandensityforζ˜andπ˜bythequa−draticpartofHinperturbation,assuming
thatsisassmallasζ˜andπ˜,asfollows,
H˜ [ζ˜(x), π˜(x)]=H [ζ˜(x) s, π˜(x)], (2.21)
0 0
−
where H is the free part of the Hamiltonian density for ζ and π. Using Eq. (2.20) with Eq. (2.21), we can show that the
0
interactionHamiltoniandensitieswhicharedefinedbyH :=H H foreachsetofthecanonicalvariablesarerelatedas
I 0
−
H˜ [ζ˜(x), π˜(x)]=H [ζ˜(x) s, π˜(x)]. (2.22)
I I
−
NotethattheHamiltoniandensitiesH andH˜ alsotakethesamefunctionalformexceptfortheconstantshiftofζ˜.
I I
B. Residualcoordinatetransformationsandquantization
Inthissubsection,weconsiderawaytocalculateobservablefluctuations.First,webeginwiththeclassicaltheory.Toobtain
theactionforζ byeliminatingtheLagrangemultipliersN andN ,weneedtosolvetheHamiltonianandmomentumconstraint
i
equations.AsisschematicallyexpressedinEqs.(2.12),theseconstraintequationsaregivenbytheelliptic-typeequations.Here
wenotethattheregionwherewecanobservationallyaccessisrestrictedtothecausallyconnectedregionwhosespatialvolume
is boundedat a finite past. As far as we are concerned only with the observable region, boundaryconditionsfor Eqs. (2.12)
cannot be restricted from the regularity of the spatial infinity, which is far outside of the observable region. The degrees of
freedomfor solutionsof N and N can be understoodas degreesof freedomin choosing coordinates. Since the time slicing
i
is fixed by the gauge conditionδφ = 0, there are remaining degreesof freedomonly in choosing the spatial coordinates. In
thefollowing,werefertosuchdegreesoffreedomwhichcannotbeuniquelyspecifiedwithouttheknowledgefromtheoutside
of the observableregion as the residual “gauge” degreesof freedom in the local universe. We put “ ”, because changing the
boundaryconditionsforN andN inthelocalregionmodifiestheactionforζ obtainedaftersolvingconstraintequations,while
i
itkeepstheactionexpressedbyN,N ,andζ invariant. Inthissense,thechangeoftheboundaryconditionisdistinctfromthe
i
usualgaugetransformation,whichkeepstheactioninvariant.
Theobservablefluctuationsshouldbefreefromsuchresidualgaugedegreesoffreedom.FollowingRefs.[3,4],weconstruct
agenuine“gauge”invariantoperator,whichpreservesthe“gauge”invarianceinthelocalobservableuniverse.Fortheconstruc-
tion,wenotethatthescalarcurvaturesR,whichtransformsasascalarquantityforspatialcoordinatetransformations,become
genuinely“gauge”invariant,ifweevaluateitinthegeodesicnormalcoordinatesspanoneachtimeslice. Thegeodesicnormal
coordinatesareintroducedbysolvingthespatialthree-dimensionalgeodesicequation:
d2xi dxj dxk
+sΓi =0, (2.23)
dλ2 jk dλ dλ
5
wheresΓi istheChristoffelsymbolwithrespecttothethreedimensionalspatialmetriconaconstanttimehypersurfaceand
jk
λistheaffineparameter. Weconsiderthethree-dimensionalgeodesicswhoseaffineparameterrangesfromλ=0to1withthe
initial“velocity”givenby
dxi(X,λ)
=e−ζ(λ=0) e−δγ(λ=0)/2 i Xj. (2.24)
dλ (cid:12) j
(cid:12)λ=0 h i
(cid:12)
(cid:12)
A pointXi in the geodesic normalcoordinatesis identified with the end pointof the geodesic, xi(X,λ = 1) in the original
coordinates. Using the geodesic normalcoordinatesXi, we perturbativelyexpandxi as xi = Xi +δxi(X). Then, we can
constructagenuinely“gauge”invariantvariableas
gR(X):=sR(t, xi(X))=sR(t, Xi+δxi(X))), (2.25)
wheretdenotesthecosmologicaltimeandweintroducedtheabbreviatednotationX := t, X . Aslongasthedeviationfrom
{ }
theFRWuniverseiskeptperturbativelysmall,wecanfoliatetheuniversebythegeodesicnormalcoordinates.
Next,wequantizethesystemtocalculatequantumcorrelationfunctionswhichbecomeobservable.Astraightforwardandfre-
quentlyusedwaytopreservethegaugeinvarianceistoeliminategaugedegreesoffreedombyfixingthecoordinatescompletely.
Inthelocalobservableuniverse,thecompletegaugefixingrequiresustofixtheboundaryconditionsinsolvingtheconstraint
equations. However,toquantizethelocallyrestrictedsystem,weneedtoabandonseveralpropertieswhichareavailableinthe
quantizationofthewholeuniverse.Oneisthatthequantizationrestrictedtoalocalsystemcannotbecompatiblewiththeglobal
translation symmetry at least in a manifest way. Another but related aspect is that basis functionsfor a mode decomposition
becomerathercomplicated(evenifitexists) thantheFouriermodes. To preservethe globaltranslationsymmetrymanifestly,
wetakeanotherwayofquantizationthanthequantizationinthecompletelyfixedgauge.
Inthispaper,weperformquantizationinthewholeuniversewiththeinfinitespatialvolume.Theglobaltranslationsymmetry
inthespatialdirectionsisthenmanifestlyguaranteedinthesensethatashiftofthespatialcoordinatesxtox+ajustchanges
theoverallphasefactorintheFouriermodebyeika. Basedonthisidea,weconsiderandcalculateobservablequantities. Since
the gauge invariant variable gR does not include the conjugate momentum of ζ, we can consider products of gR at an equal
timewithouttheproblemofoperatorordering. Then-productofgR, i.e.,gR(x ) gR(x ),preservesthe“gauge”invariance
1 n
in the localuniverse. To calculate the n-pointfunctionsof gR, we need to specif·y·t·he quantumstate as well. One maythink
thatthequantumstateshouldbealsoinvariantundertheresidual“gauge”transformations.However,wecannotdirectlydiscuss
thisinvarianceasaconditionforallowedquantumstatesinthisapproach,becausetheresidual“gauge”degreesoffreedomare
absentwhenwequantizefieldsinthewholeuniverse.
HerewenotethateventhoughtheoperatorgRisnotaffectedbytheresidual“gauge”degreesoffreedom,thisdoesnotimply
thatthen-pointfunctionsofgRareuncorrelatedtothefieldsinthecausallydisconnectedregion. Toexplainthisaspectmore
clearly,letusconsiderthen-pointfunctionofgRatt = t whosevertexesarelocatedwithintheobservableregionO . Fora
f f
later use, we refer to the spacetime regionwhich is causally connectedto spacetime pointsin O as the observableregionO.
f
After expandingof the operatorgR in terms of ζ , the interaction picture field of ζ, even if interaction vertexeswhich affect
I
gR(t , x)areconfinedwithintheobservableregionO,then-pointfunctionsofgRhavecorrelationtotheoutsideofOthrough
f
theWightmanfunctionofζ ,whichcanbeexpressedintheFourierspaceas
I
d3k
G+(x , x )= ζ (t, x )ζ (t, x ) = eik·(x1−x2) v (t)2, (2.26)
1 2 h I 1 I 2 i Z (2π)3 | k |
wherev is themodefunctionfor ζ . Since allthe vertexesare confinedwithin O, the spatialdistance x x is bounded
k I 1 2
fromabove. However,theIRmodeswithk x x −1 arenotsuppressedandlettheWightmanfun|ctio−nG+| divergefor
1 2
≤ | − |
scale-invariantorred-tiltedspectrums.
This long-rangecorrelationbecomesthe origin of the IR divergenceof loop contributions. Since the observablequantities
should take a finite value, we request the IR regularity of observable fluctuations, which is achieved only when the quantum
stateisselectedsothatthelong-rangecorrelationisproperlyisolatedfromtheobservablequantities. Inthefollowingsection,
wewillshowthatrequestingtheabsenceoftheIRdivergenceinfactconstrainsthequantumstateoftheinflationaryuniverse.
III. IRREGULARITYCONDITIONANDTHERESIDUAL“GAUGE”INVARIANCE
Asimplewaytoaddresstheevolutionofanon-linearsystemistosolvetheHeisenbergequationperturbativelybyassuming
thattheinteractionisturnedonattheinitialtime. Inthissection,takingthissetup,wecalculatethecorrelationfunctionofgR
uptoone-looporderandinvestigatetheIRbehaviour.
6
A. Specifyingtheiterationscheme
Fornotationalconvenience,weintroducethehorizonflowfunctionsas
ρ˙ 1 d
ε := i, ε := ε , (3.1)
0 n n−1
ρ˙ ε dρ
n−1
with n 1, but without assuming these functions are small we leave the background inflation model unconstrained. The
≥
assumptionthattheinteractionisturnedonattheinitialtimet requests
i
ζ(t , x)=ζ (t , x), π(t , x)=π (t , x) (3.2)
i I i i I i
whereπ istheconjugatemomentumdefinedfromthenon-interactingaction:
I
e−2ρ
S =M2 dtd3xe3ρρ˙2ε (∂ ζ)2 (∂ζ)2 , (3.3)
0 plZ 1(cid:20) ρ − ρ˙2 (cid:21)
as π := 2M2e3ρε ζ˙ . Employingthe initialconditions(3.2), we can relate the Heisenbergpicturefield ζ to the interaction
I pl 1 I
picturefieldζ as
I
ζ(t,x)=U†(t, t )ζ (t,x)U (t, t ), (3.4)
I i I I i
whereU istheunitaryoperatorgivenby
I
t1
U (t , t )=T exp i dtH (t) . (3.5)
I 1 2 I
(cid:20)− Z (cid:21)
t2
Since the Heisenberg fields are related to the interaction picture fields by the unitary operator, the canonical commutation
relationsforthecanonicalvariablesζ andπguaranteethoseforζ andπ as
I I
[ζ (t, x), π (t, y)]=iδ(3)(x y), [ζ (t, x), ζ (t, y)]=[π (t, x), π (t, y)]=0. (3.6)
I I I I I I
−
Wenotethatthen-pointfunctionsforζ giveninEq.(3.4)agreewiththosegivenbythesolutionofζ writtenintermsofthe
retardedGreenfunction.Weexpresstheequationofmotionforζ schematicallyas
Lζ =S , (3.7)
NL
whereLdenotesthedifferentialoperator
∂2
L:=∂2+(3 ε +ε )∂ , (3.8)
ρ − 1 2 ρ− e2ρρ˙2
andtheleft-handsideofEq.(3.7)isthesameequationofmotionasisderivedfromthenon-interactingpartoftheaction(3.3).
ByusingtheretardedGreenfunction:
G (x, x′)=iθ(t t′)[ζ (x), ζ (x′)] , (3.9)
R I I
−
whichsatisfies
1 1
LG (x,x′)= δ(4)(x x′), (3.10)
R −2M2 ε e3ρρ˙2 −
pl 1
withδ(4)(x x′):=δ(t t′)δ(3)(x x′),thesolutionoftheequationofmotionwiththeinitialcondition(3.2)isgivenby
− − −
ζ(x)=ζ (x)+L−1S (x) (3.11)
I R NL
wherethenon-lineartermisgivenby
t
L−1S (t, x):= 2M2 dt′ d3x′ε (t′)e3ρ(t′)ρ˙(t′)2G (x,x′)S (x′). (3.12)
R NL − plZ Z 1 R NL
ti
UsingEq.(3.11),wecansolvetheequationofmotioniteratively.
7
Weexpandtheinteractionpicturefieldζ ,whichsatisfies
I
Lζ (x)=0, (3.13)
I
asfollows
d3k
ζ (x)= a v eik·x+h.c. . (3.14)
I Z (2π)3/2 k k
(cid:0) (cid:1)
Themodefunctionv satisfiesL v =0where
k k k
e−2ρ
L :=∂2+(3 ε +ε )∂ + k2. (3.15)
k ρ − 1 2 ρ ρ˙2
Themodefunctionisnormalizedas
v eik·x, v eip·x =(2π)3δ(3)(k p), (3.16)
k p
−
(cid:0) (cid:1)
wheretheKlein-Gordoninnerproductisdefinedby
(f ,f ):= 2i d3xe3ρε f ∂ f∗ (∂ f )f∗ . (3.17)
1 2 − Z 1{ 1 t 2 − t 1 2}
Withthisnormalization,weobtainthecommutationrelationsforthecreationandannihilationoperatorsas
a , a† =δ(3)(k p), [a , a ]=0. (3.18)
k p − k p
(cid:2) (cid:3)
InsertingEq.(3.14)intoEq.(3.9),wecanrewritetheretardedGreenfunctionas
G (x, x′)=iθ(t t′) d3k eik·(x−x′)R (t, t′), (3.19)
R − Z (2π)3 k
whereR (t,t′)isgivenby
k
R (t,t′):=v (t)v∗(t′) v∗(t)v (t′). (3.20)
k k k − k k
Wecalculatethen-pointfunctionofgR,settingtheinitialstatetothevacuumdefinedbya 0 =0.
k
| i
Next,weexplicitlycalculatenon-linearcorrections. SinceweareinterestedintheIRdivergence,wepickuponlytheterms
whichcancontributetotheIRdivergences,employingtheiterationschemeofL−1. Fortheconvenience,wehereintroducethe
R
IR
symbol“ ”asin[4]todenoteanequalityundertheneglectofthetermsunrelatedtotheIRdivergences. TheIRdivergences
meanthe≈appearanceofthefactor ζ2 . Forthescaleinvariantspectrum,thisvariancedivergeslogarithmically.Onceatemporal
h Ii
orspatialdifferentiationactsononeoftwoζ sin ζ2 ,thevariancenolongerdiverges.Inthissensewekeeponlytermswhich
can yield ζ2 . When we write down the HIeisenhbeIrig operator ζ in terms of the interaction picture field ζ , at the one loop
h Ii I
level,thisisequivalenttokeeponlythetermswithouttemporaland/orspatialdifferentiationsandthetermscontainingonlyone
interactionpicturefieldwithdifferentiation.
Inthispaper,we referto aterm whichdoes(not)contributetothe IRdivergencessimplyasanIR(ir)relevantterm. In the
following,wewilluse
L−1ζ Dζ IRζ L−1Dζ , (3.21)
R I I ≈ I R I
whereDisaderivativeoperatorwhichsuppressestheIRmodesofζ suchas∂ and∂i/ρ˙eρ. Equation(3.21)canbeshownas
I ρ
follows.TheFouriertransformationofL−1ζ Dζ isproportionalto
R I I
t
d3p dt′ε (t′)e3ρ(t′)ρ˙2(t′)R (t, t′)ζ (t′)(Dζ ) (t′),
Z Z 1 k Ip I k−p
ti
where ζ and (Dζ ) denote the Fourier modes of ζ and Dζ . Since (Dζ ) (t′) (Dζ ) (t′) is suppressed and ζ
Ik I k I I I k−p − I k I,p
becomestimeindependentinthelimitp 0,theIRrelevantpieceoftheintegrandofthemomentumintegralcanbereplaced
→
by
t
ζ dt′ε (t′)e3ρ(t′)ρ˙2(t′)R (t, t′)(Dζ ) (t′).
IpZ 1 k I k
ti
8
ToobtainEq.(3.21),wewillalsouse
L−1f(x)IR0 for f(x)IR0. (3.22)
R ≈ ≈
KeepingonlytheIRrelevantterms,theADMactionissimplygivenby
S IRM2 dtd3xe3(ρ+ζ)ρ˙2ε (∂ ζ)2 e−2(ρ+ζ)(∂ζ)2 , (3.23)
≈ plZ 1(cid:20) ρ − ρ˙2 (cid:21)
whereweusedthelapsefunctionN andtheshiftvectorN ,givenbysolvingtheconstraintequationsasfollows,
i
IR
N 1+∂ ζ, (3.24)
ρ
≈
1∂ Ni IR e−2ρ∂2ζ+ε ∂ ζ. (3.25)
ρ˙ i ≈ − ρ˙2 1 ρ
Theaction(3.23)canbeeasilyderivedfromtheactionforthenon-interactingtheory.Extendingtheaction(3.3)toanun-linear
expressiontorecoverthedilatationsymmetry,weobtain
e−2(ρ+ζ)
S =M2 dtd3xe3(ρ+ζ)ρ˙2ε (∂ ζ)2 (∂ζ)2+ , (3.26)
plZ 1(cid:20) ρ − ρ˙2 ···(cid:21)
wheretheabbreviatedtermsdonotappearinthenon-interactingaction,whichmeansthattheyshouldhavemorethantwofields
withdifferentiation.SuchtermsareIRirrelevantuptotheone-looporder.Thus,weobtainEq.(3.23).
Takingthevariationoftheactionwithrespecttoζ,weobtaintheevolutionequationas
1 ∂ e3ρρ˙ε ∂ ζ e−2(ρ+ζ)∂2ζ IR0.
e3ρρ˙ε ρ 1 ρ − ρ˙2 ≈
1 (cid:0) (cid:1)
Byexpandingζ asζ =ζ +ζ +ζ ,theequationofmotionisrecastinto
I 2 3
···
Lζ =0, (3.27)
I
Lζ IR 2ζ ∆ζ , (3.28)
2 I I
≈ −
Lζ IR 2 ζ ∆ζ +ζ ∆ζ ζ2∆ζ , (3.29)
3 ≈ − 2 I I 2− I I
(cid:0) (cid:1)
wherewedefined
e−ρ
:= ∂ , ∆:=δij . (3.30)
i i i j
∇ ρ˙ ∇ ∇
SolvingEqs.(3.28)and(3.29),weobtain
ζ IR 2ζ L−1∆ζ , (3.31)
2 ≈ − I R I
ζ IRζ2L−1 4∆L−1∆+2∆ ζ , (3.32)
3 ≈ I R R I
(cid:0) (cid:1)
wherewenotedthepropertiesofL−1 giveninEqs.(3.21)and(3.22).
R
B. Calculatingthe“gauge”invariantoperator
Solvingthethreedimensionalgeodesicequations,weobtaintherelationbetweentheglobalcoordinatesxi andthegeodesic
normalcoordinatesXias
xi IRe−ζ(x)Xi. (3.33)
≈
Usingthegeodesicnormalcoordinates,thegauge-invariantcurvaturegRisexpressedas
gR(t,X)IRsR(t, e−ζ(t,e−ζX)X). (3.34)
≈
9
Here, we introduce the spatial average of ζ at the local observable region whose spatial scale is L in the geodesic normal
t
coordinatesas
d3XW (X)ζ(t,e−ζX)
gζ¯(t):= Lt , (3.35)
R d3XW (X)
Lt
R
whereW (X)isthewindowfunctionwhichspecifiesourobservableregion.Weapproximatethisaveragingscaleateachtime
tbythehLotrizonscale,i.e.,L eρ(t)ρ˙(t) −1. Usinggζ¯(t),wedecomposethespatialcoordinatesas
t
≃{ }
gR(t, X)IRsR(t, e−{ζ(t,e−ζX)−gζ¯(t)}e−gζ¯(t)X). (3.36)
≈
UsingthewindowfunctionW (X),wedefinealocalaverageofanoperatorO(t, X)as
Lt
d3XW (X)O(t,X)
O¯(t):= Lt . (3.37)
R d3XW (X)
Lt
Wenotethatthefactore−{ζ(t,e−ζX)−gζ¯(t)},whichcanbeexRpandedinperturbationas
e−{ζ(t,e−ζX)−gζ¯(t)}
1
=1 (ζ ζ¯ )+(ζ ζ¯) (ζ X ∂ ζ ζ X ∂ ζ + (ζ ζ¯ )2+ , (3.38)
I I 2 2 I X I I X I I I
−{ − − − · − · } 2 − ···
yields only IR regular contributions up to the one-loop order such as (ζ ζ¯ )ζ and ζ X ∂ ζ . The latter one also
I I I I X I
becomesfinitebecauseofthederivativesuppressionandthefinitenessofh X−1. Fortihereashonme·ntionediabove,neglectingthe
factore−{ζ(t,e−ζX)−gζ¯(t)},werewritethegaugeinvariantoperatorgRas | |
gR(t, X)IRsR(t, e−gζ¯(t)X). (3.39)
≈
SincethespatialcurvaturesRisgivenby
sR(x)IR 4e−2ρe−2ζ∂2ζ,
≈ −
usingthecurvatureperturbationinthegeodesicnormalcoordinates:
gζ(t, X):=ζ(t, e−gζ¯(t)X), (3.40)
wecandescribethegauge-invariantspatialcurvaturegRas
gR(t, X)IR 4e−2ρe−2(ζ−gζ¯)∂2 ζ(t,e−gζ¯(t)X)
≈ − X
IR 4e−2ρ∂2 gζ(t, X) (3.41)
≈ − X
atleastuptothethirdorderinperturbation.Atthesecondequality,weagainusedthefactthattheexponentialfactore−{ζ−gζ¯(t)}
do notgiveIR relevantcontributions. As is pointedoutby Tsamisand Woodardin Ref. [37] and also in Ref. [38], using the
geodesicnormalcoordinatescanintroduceanadditionaloriginofUVdivergence. Wesupposethatreplacingζ inthegeodesic
normalcoordinateswithgζ¯,whichissmoothedbythewindowfunction,canmoderatethesingularbehaviour.
Insertingthesolutionofζ giveninEqs.(3.31)and(3.32)intoEq.(3.40),weobtain
gζ IR ζ¯ 2L−1∆+X ∂ ζ , (3.42)
2 ≈ − I R · X I
gζ IR 1ζ¯2(cid:0)2L−1∆+X ∂ (cid:1)2ζ +ζ¯2(1 L−1L)X ∂ L−1∆ζ . (3.43)
3 ≈ 2 I R · X I I − R · X R I
(cid:0) (cid:1)
Inderivingtheexpressionofgζ ,weused
3
[L, x ∂ ]= 2∆. (3.44)
x
· −
We can verifythat the term with (1 L−1L) in Eq. (3.43) vanishes. Since this term is eliminatedby operatingL, it always
− R
vanishesifitsvalueanditsfirsttimederivativearebothzeroattheinitialtime,whichisautomaticallysatisfiedbythedefinition
oftheretardedintegral.
1OnemaythinkthatinteractionvertexeslocatedoutsidethepastlightconecanappearfromtheinverseLaplacian∂−2,whichareincludedinthelapsefunction
andshiftvectors. Inthatcase,|X|becomesunbounded. However, tuningtheboundaryconditionoftheconstraintequations, whicharetheelliptictype
equations,wecanmanifestlyshutofftheinfluencesfromoutsidetheobservableregionO. Sincethischangecanbethoughofasthechangeoftheresidual
coordinatedegreesoffreedom,itdoesnotinfluenceonthe“gauge”invariantoperatorgR.
10
C. ConditionsfortheabsenceofIRdivergence
WenextdiscusstheconditionsthattheIRdivergencedoesnotariseintheexpectationvaluesofthegaugeinvariantvariable
gR. Hereafter, since we calculate the fluctuation only in the geodesic normal coordinates, we use x to express the geodesic
normalcoordinates.Then,usingtheaboveexpression,twopointfunctionofgRuptotheonelooporderisobtainedas
gR(x )gR(x ) IR8e−4ρ ζ¯2 ∂2 ∂2 2 2L−1∆ +x ∂ ζ (x ) 2L−1∆ +x ∂ ζ (x )
h 1 2 i ≈ h Ii (1) (2)D (cid:0) R (1) 1· x1(cid:1) I 1 (cid:0) R (2) 2· x2(cid:1) I 2
+ 2L−1∆ +x ∂ 2ζ (x )ζ (x )+ζ (x ) 2L−1∆ +x ∂ 2ζ (x ) , (3.45)
(cid:0) R (1) 1· x1(cid:1) I 1 I 2 I 1 (cid:0) R (2) 2· x2(cid:1) I 2 E
whereweusedanabbreviatednotation∂2 := ∂2 and∆ := (eρρ˙)−2∂2 fora = 1,2. Aswementionedattheendofthe
(a) xa (a) xa
precedingsection,thecorrelationfunctionofgRcancontainIRdivergences.OnemaythinkthattheIRregularityisguaranteed
if(2L−1∆+x∂ )ζ =0isimposed.However,thisconditionisinconflictwiththeuseoftheordinaryretardedintegralinthe
R · x I
iterationprocess. Infact,onecancalculate(2L−1∆+x∂ )ζ as
R · x I
d3k k2
2L−1∆+x∂ ζ = eik·xa 2L−1 +ix k v +h.c. , (3.46)
R · x I Z (2π)3/2 (cid:26) k(cid:18)− R,ke2ρρ˙2 · (cid:19) k (cid:27)
(cid:0) (cid:1)
whereL−1 istheFouriermodeofL−1. Therequirementthatthisexpressionshouldidenticallyvanishleadsto
R,k R
k2
2L−1 +ix k v =0. (3.47)
(cid:18)− R,ke2ρρ˙2 · (cid:19) k
Thefirsttermisindependentofx,whilethesecondonemanifestlydependsonx. Thisshowsthatthecondition(2L−1∆+x
R ·
∂ )ζ =0isincompatiblewiththeuseoftheretardedintegral.
x I
Inplaceofthisnaivecondition,analternativepossibilityonecanthinkofistoimpose
d3k
2L−1∆+x∂ ζ = a Deik·xv +h.c. (3.48)
R · x I Z (2π)3/2 k k
(cid:0) (cid:1) (cid:0) (cid:1)
with
D :=k−3/2e−iφ(k)k ∂ k3/2eiφ(k), (3.49)
k
·
whereφ(k)isanarbitraryrealfunction.Thisconditioncontinuestoholdonceitanditstimederivativearesatisfiedonacertain
timeslicebecausebothsidesofthisequationvanishunderanoperationofthesecondorderdifferentialoperatorL. Thecondition
canberewrittenasaconditiononmodefunctions
k2
2L−1 v =Dv , (3.50)
− R,ke2ρρ˙2 k k
which is not contradictory as in the case of Eq. (3.47). With these conditions, the expectation value in Eq. (3.45) can be
summarizedintotalderivativeformas
gR(x )gR(x ) IR8e−4ρ ζ¯2 d(lnk)dΩk∂2 k7 v 2eik·(x1−x2) ,
h 1 2 i ≈ IiZ (2π)3 lnkn | k| o
(cid:10)
whereweusedk3/2D =e−iφ(k)k ∂ k3/2eiφ(k) and dΩ denotestheintegrationovertheangulardirectionsofk. Sincethis
k k
·
integraloftotalderivativevanishes,theIRdivergenceRdisappears.
Although requesting the condition (3.50) can make the IR divergence vanish, a bit more careful thought rules out this
possibility. Since the left hand side of (3.50) and its time derivative vanish at the initial time, the condition (3.50) requests
Dv (t )=Dv˙ (t )=0. Operatingthedifferentiationk ∂ onthenormalizationconditionofthemodefunctions,
k i k i k
·
2ie3ρε (v v˙∗ v˙ v∗)=1 (3.51)
− 1 k k− k k
leadstoacontradiction.Therighthandsidetriviallyvanishesaftertheoperationofk ∂ ,whilethelefthandsidegives3.
k
·
EventhoughtheIRregularitycondition(3.50)cannotbecompatiblewiththeinitialconditions(3.2),itisstillinstructiveto
give an alternative interpretation of the condition (3.50). We adopted the initial condition (3.2) for ζ/π, which identifies the
Heisenbergfieldswiththecorrespondinginteractionpicturefieldsattheinitialtimeandselectedthevacuumstateforthefree
fieldattheinitialtime.Wedenoteasetofoperationswhichspecifythequantumstateasmentionedabovebyaniterationscheme.