Primordial non-Gaussianities from inflation

0.23 MB·English

by  Hael Collins

Primordialnon-Gaussianitiesfrom inflation
HaelCollins1
TheNielsBohrInternational Academy&TheDiscovery Center
TheNielsBohrInstitute, Copenhagen, Denmark
(Dated:November29,2010)
Abstract
1 ThesenotespresentadetailedintroductiontoMaldacena’scalculation[1]of
1 thecubictermsintheinflationaryaction. Theseinteractionsareimportantsince
0 theyproducethemostreadilyobservableevidenceforanon-Gaussiancomponent
2
in the pattern of primordial fluctuations produced by inflation. In the simplest

n classofinflationarytheories,thosewithonlyasinglescalarfieldparticipatingin
a theinflationaryera, thesenon-Gaussianitiesarepredictedtobeextremelysmall,
J
aswillbereviewedhere.
6

]
O
Anycosmological modelthatseeksadeeperoramorecompleteexplanation ofthe
C
universe must be able to account for the origin of the great variety of structures that
.
h appear in it today. Most of this structural complexity in the universe appears to have
p
arisenthroughprocessesthatarelargelyunderstoodandfrommaterialthatwasoncein
-
o amuchsimplerstatethanitistoday. Starsandgalaxies, forexample, whichnowstand
r out so clearly from much emptier surroundings, could have grown through the slow
t
s gravitationalcollapseofgaseswhichwereinitiallydistributedfarmoreuniformly. This
a
basic picture suggests that the universe in its primeval stages was once extremely ho-
[
mogeneous,apredictionthatmatcheswellwiththeobservationsmadesofar. However,

1
this mechanism also requires that, even at the very earliest times, some spatial varia-
v
tion must always have existed. In aperfectly homogeneous and isotropic universe, the
8
0 process ofcollapse and growth, andthe development ofevermorecomplex structures,
3 wouldneverhavebegun.
1 The theory of inflation provides one possible mechanism for generating the initial
1. spatialvariations intheuniverse. Intheinflationary picture,space-timeitselffluctuates
0 quantum mechanically about a background that is expanding at an accelerating rate.
1 Thisextremeexpansionspreadsthefluctuations, whichbeginwithatinyspatialextent,
1
throughout avastregion oftheuniverse, wheretheyeventually become smallclassical
:
v fluctuationsinthespace-timecurvature—orequivalently, smallspatialvariationsinthe
i strengthofgravity. Sinceeverythingintheuniversefeelstheinfluenceofgravity,these
X
fluctuations in the gravitational field are transferred to the matter and radiation fields,
r
a creating slightly overdense and underdense regions. The resulting matter fluctuations
then become the ‘initial conditions’ that start the process of collapse which forms the
starsandgalaxies oflaterepochs.
Totestwhetherthispictureiscorrect,itisnecessarytodescribeveryaccuratelythe
properties ofthe pattern generated by inflation for the original, primordial fluctuations
inspace-timewhichcanthenbecomparedwithwhatisinferredfromobservations. Be-
fore entering into a detailed calculation of these primordial perturbations, it is instruc-
tivetoexplain firstfrom amoregeneral perspective howthe fluctuations ofaquantum
1Electronicaddress:[email protected]
1
field lead to a classical pattern of perturbations. Since a quantum field is never mea-
sureddirectly, itisnecessaryalsotodeterminehowandinwhatformitsinformationis
available.
Tobegin,consider aquantum field,whichwillbewrittenasz (t,~x). Laterthisfield
will be connected with the fluctuations of the background space-time, but for now its
exact physical meaning will be left a little vague. Although z (t,~x) is a quantum field,
its influence is inferred by how itaffects classical things, so the information contained
inz (t,~x)needstobeconvertedintoaclassicalfunction,orrather,intoasetofclassical
functions. Onewaytodosoisbytakingtheexpectation valuesofproducts ofthefield
z (t,~x)atdifferent places. Ifthefield isinaparticular, time-dependent, quantum state,
0(t) ,thenthefunctions,
| i
0(t)z (t,~x )z (t,~x ) z (t,~x )0(t) , (1)
1 2 n
h | ··· | i
tell the extent to which fluctuations in different places are correlated with each other
in that state, and they are accordingly called n-point correlation functions. Usually,
most of the emphasis on deriving the inflationary prediction for the initial fluctuations
focusesonthefield,butitisequally importanttounderstand thestatetoo.2 Becauseof
thedramaticexpansionduringinflation,thedetailedassumptionsmadeabouttheprop-
erties of this state—even at seemingly infinitesimal distances—can have an important
influenceonthepredictions forthepatternofthefluctuations.
Each n-point function potentially contains unique information not found in any of
theothers. Ifthefluctuationsintheveryearlystagesoftheuniversearesmallcompared
withthevalueofthebackground,thenitshouldbepossibletoevaluatethesecorrelation
functionsperturbatively. Thisfactimpliesfurtherthatthehigherordern-pointfunctions
will be progressively more suppressed by this inherent smallness of z (t,~x). So given
a limited experimental accuracy, only the lowest few correlators will be observable in
practice. If the spatially independent background has been chosen correctly, then the
one-point function vanishes,
0(t)z (t,~x)0(t) =0, (2)
h | | i
meaning that thefirstobservable measure oftheprimordial fluctuations isprovided by
thetwo-pointfunction,
0(t)z (t,~x)z (t,~y)0(t) , (3)
h | | i
followednextbythethree-point function,
0(t)z (t,~x)z (t,~y)z (t,~z)0(t) , (4)
h | | i
andsoon.
To calculate these correlation functions requires knowing something about the dy-
namics governing the field z (t,~x). Suppose that these are determined by an action S
withanassociated Lagrangedensity L[z ],
S= d4xL[z (x)]. (5)
Z
Sincez issmall,theLagrangian canbeexpandedasaseriesinpowersofthefield,
S= d4x L +L(2)[z (x)]+L(3)[z (x)]+L(4)[z (x)]+ , (6)
0
Z ···
(cid:8) (cid:9)
2Sometimesrathercontrivedpotentials,withdramaticsuddenchangesarestudied,althoughthestate
isassumedtobeotherwisequitetame.Sometimestooonlyafiniteamountofinflationisassumedtohave
occurred,andwhatcamebeforethiseramighthaveleftsomeimprintinthestateaswell.
2
whereL(2) isquadratic inthefield,L(3) iscubic, etc. Thezerothpiece fixesthetime
evolution of the background; and since this background isalways implicitly asolution
to the equations of motion, the L(1) term vanishes. Associating a vertex with each of
theseterms,withonelegforeachfactorofthefield,
L(2)∼ ............................................................................................................................................................................................................................................................................................................................................................................., L(3)∼ ............................................................................................................................................................................................................................................................................................................................................................................, L(4)∼ ....................................................................................................................................................................................................................................................................................................................................................................................................................... , etc., (7)
thefirstfewcontributions tothetwo-pointfunction, inaperturbativeexpansion, canbe
diagrammatically represented by
h0(t)|z (x)z (y)|0(t)i = x..........................................................................................................................................................................................................................................................................................................................................................................y... + x..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................y... + ··· (8)
whilethethree-point function is
z
h0(t)|z (x)z (y)z (z)|0(t)i = ............................................................................................................................................................................................................................................................................................................................................................................ + ··· (9)
x y
plus further higher order corrections which have not been written explicitly. In these
expressions, the locations of the individual fields have been abbreviated as x=(t,~x),
y=(t,~y)andz=(t,~z). Beyondcubicorder,theleadingcontributions tothecorrelation
functionsaremadeupinpartbypowersofthelowerordercorrelators. Atquarticorder,
for instance, the leading behavior receives contributions both from the three possible
pairingsoftwo-pointfunctionsaswellasfromtheconnectedgraphassociatedwiththe
L(4) vertex,
x x x x x x
xx13......................................................................................................................................................................................................................................................................................................................................................................................................................................................xx24 + ...........................................................................................................................................................1................................................................ ............................................................................................................................................................2............................................................... + 1.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 2 + 1................................................................................................................................................................................................................................................................................................................................................................................................................................... 2 + ···
x3 x4 x3 x4 x3 x4
(10)
whichisequivalently writtenas
0(t)z (x )z (x )z (x )z (x )0(t) = 0z (x )z (x )0 0z (x )z (x )0
1 2 3 4 1 2 3 4
h | | i h | | ih | | i
+ 0z (x )z (x )0 0z (x )z (x )0
1 3 2 4
h | | ih | | i
+ 0z (x )z (x )0 0z (x )z (x )0
1 4 2 3
h | | ih | | i
+ 0(t)z (x )z (x )z (x )z (x )0(t)
h | 1 2 3 4 | i connected
+ . (cid:12) (11)
··· (cid:12)
Ascenariothathasnohigherorderinteractions beyondthequadratic ones,
L[z (x)]=L +L(2)[z (x)], (exactly) (12)
0
is said to be a Gaussian theory. In such theories, all the odd-point functions vanish
(sincediagrammaticallyitisquiteobviousthatitisimpossibletodivideanoddnumber
3
ofpointsintopairsconnectedbyL(2)withoutalwaysleavingonepointunpaired—and
thisone-pointfunctionvanishes),whilealltheeven-pointfunctionscanbedecomposed
entirely into products of the two-point function. In the Gaussian case, the four-point
function would then be equal to just the first three terms shown above, without the
connectedpart. Thereisnoreasontosupposethattheprimordialpatternoffluctuations
in the actual universe forms a perfectly Gaussian pattern. But inflationary models do
typically have cubic (and higher) interactions which are suppressed, so they usually
predictapatternthatisstilllargelyGaussianinitscharacter. Onetestoftheinflationary
picture then is to see whether the actual primordial fluctuations have a non-Gaussian
component that islarge orsmall. Todoso, itisimportant toknow the explicit form of
thethree-point function expected byinflation.
These notes will derive the standard inflationary source for the most readily ob-
served signal of a non-Gaussian component within the pattern of primordial fluctua-
tions, that generated by the cubic terms in the action, L(3). This set of terms will
moreover becalculated here in thesimplest possible setting, where there is only a sin-
glescalarfieldthatisparticipating intheinflation.
This derivation will be developed in several stages, each of a successively greater
complexity. The first will analyze how the background evolves during inflation, and
itwillalsointroduce afewdimensionless parameters associated withthis background,
before proceeding to a calculation of the quadratic action for the fluctuations and the
resultingbehaviorforthetwo-pointfunction. Thiscalculationofthetwo-pointfunction
willalsodefineasetofcoordinatesandageneralmethodthatwillbeusedinthesenotes
todescribe thequantum fluctuations about thebackground. Onlyonce themethod that
is used to describe the fluctuations has been illustrated for the quadratic terms will the
fullcalculation ofthecubictermsfinallybegin.
Whilederiving thisstandard prediction, alongthewayitwillbenecessary tomake
variousassumptionsaboutthepropertiesoftheuniverseduringinflationandaboutwhat
constitutes a‘well behaved’ theoretical description ofa quantum fieldtheory in anex-
panding background. As long as these assumptions are part of a sound and consistent
framework, there is nothing inherently wrong about adopting them—but they are as-
sumptions all the same, and they might not necessarily hold in the actual universe. It
is therefore always a good practice to be a little wary about what ideas are believed to
hold during an era so far removed from what can be tested directly; at the very least,
anysuchassumptions oughttobestatedopenlyandclearlywhenevertheyaremade.
I. THEBACKGROUND
To begin, consider a simple inflationary model with a scalar field f which is re-
sponsible for the accelerated expansion. If this field has a potential energy described
byV(f ), then the total action, including a part both for the dynamics of f and for the
dynamicsofthespace-time itself,is
S=Z d4x√−gn12Mp2lR+21gmn ¶ m f¶ n f −V(f )o, (13)
where M =(8p G) 1/2 is the reduced Planck mass, G being Newton’s constant. R is
pl −
thescalarcurvatureassociatedwiththespace-time,whosegeometryisdescribedbythe
metricgmn . Allofthefieldscanbeplacedonthesamefootingbychoosingunitswhere
M =1; since M only accompanies the gravitational field and not f , it is usually a
pl pl
simplemattertorestoreitwheneveritmightbeneeded.
4
Varyingtheaction withrespect toasmallchange inthemetric, gmn gmn +d gmn
→
producesEinstein’sequation, whichrelateshowthepresenceofmatterandenergy—in
this case, that produced by the scalar field f —affect the geometry of the space-time
wheretheyexist,
Rmn 1gmn R=Tmn . (14)
−2
Tmn istheenergy-momentum tensorofthefieldf ,
Tmn =¶ m f¶ n f 1gmn gls ¶ l f¶ s f +gmn V. (15)
−2
Varying with respect to the scalar field f f +df yields the equation of motion for
→
thescalarfield,
d V
(cid:209) 2f + =0. (16)
df
Before attempting to explain the structure of the quantum fluctuations produced
by the inflationary era, it is good to begin by following how the classical background
evolves. Empirically, at very large scales or at very early times—at least at those that
are visible—the universe appears to be quite uniform spatially. Inflation assumes this
spatial uniformity holds over alarge enough patch ofthe universe, from the beginning
of inflation to its end, that the classical part of the metric can be treated as though it
depended only on the time coordinate. Such a metric can be put into the following
standard form,3
ds2=dt2 e2r (t)d dxidxj. (17)
ij
−
Therateatwhichr (t)ischanging corresponds totheHubblescale,
dr
H(t)=r˙(t) , (18)
≡ dt
which is the characteristic energy scale associated with the gravitational evolution.
Evaluatingthecomponents oftheEinsteinequation forthismetricyields
1
3r˙2 = f˙2+V
2
1
2r¨ 3r˙2 = f˙2 V, (19)
− − 2 −
whiletheequationforthefielditselfis
d V
f¨+3r˙f˙+ =0. (20)
df
Thus far, other than restricting to the case of a single scalar field and a particular
assumption about the approximate homogeneity of themetric, the setting has notbeen
otherwise constrained. Mostespecially, thepotential energy ofthefieldV(f )hasbeen
left unspecified. Most inflationary models, if they are to produce a sufficient amount
of expansion, must be in a ‘slowly rolling’ phase where the value of the field is not
changing too rapidly—that is, f˙ should be ‘small’. Further, to maintain this condition
3Thischoice of themetricimplicitlyassumes that space-timeisspatiallyflat. Empiricallytheearly
universedoesindeedappeartobespatiallyflat—orisatleastverynearlyso—sothroughoutthesenotes
suchabackgroundwillbeassumed.
5
overasufficientlylongperiodtoproduceanecessaryamountofinflation,thefield’sac-
celeration(f¨)mustalsobetiny. Thesetwoconditionscanbeputalittlemoreprecisely
byrequiring thefollowingdimensionless parameters tobesmall,
d 1 r¨ 1 f¨ f¨
e = = , d = = . (21)
dt H −r˙2 H f˙ r˙f˙
In the limit where both e 1 and d 1, the derivatives of these parameters are still
smaller, e˙,d˙ He ,Hd , s≪o e and d w≪ill usually be treated as though they were con-
stants. Byapp≪lying the background equations of motion tosolve forr¨ = 1f˙2, e can
−2
alsobeexpressed as
1f˙2
e = . (22)
2r˙2
Thisformfore makeswhatismeantbya‘small’valueforf˙ moreprecise: thevalueof
the field should change slowly when compared with the rate at which the background
itselfischanging,r˙ =H. RestoringM foramoment,thevaluefore ismoregenuinely
pl
1 1 f˙2
e = , (23)
2M2 r˙2
pl
whichshowsthatitisindeeddimensionless.
Whene =0exactly,thenf˙ =0;inthiscase,bothV(f )andr˙ =H areconstantsas
well. Aspace-timewithaconstant,positive(V >0)vacuumenergydensityiscalledde
Sitter space. Therefore, e also characterizes by how much an inflationary space-time
departsfromapurelydeSitterbackground.
II.THE QUADRATICACTION
Oneveryappealing property ofinflationisthatsometinyamountofspatial depen-
dence isinevitable. These tiny, primordial variations inthe space-time, which mustbe
present in inflation, provide the initial inhomogeneities that are needed to explain the
beginnings of the structures that are observed today. Their origin in the theory lies in
thequantumbehaviorofboththefieldandthespace-time. Therefore,inadditiontothe
classical quantities f (t) and gmn associated with the background, consider a quantum
correction toeachaswell,
gmn (t) gmn (t)+d gmn (t,~x)
→
f (t) f (t)+df (t,~x). (24)
→
Since d gmn and df are quantum mechanical, they are always fluctuating, and they
therebyintroducesomespatialdependenceintowhatwouldbeanotherwisefeatureless
background. The extreme expansion during inflation in turn stretches these fluctua-
tions,whichwouldotherwiseremainofatinyspatialextent,tovastsizes. Thisprocess
happens sorapidly thatthefluctuations aresoonfrozen intothespace-time andremain
beyond any further causal influence while the inflationary stage lasts. It is this basic
mechanismthatfillstheuniversewithapatternoftinyprimordialperturbationsaccord-
ing to inflation. This section of the notes will derive the leading, quadratic component
ofthispatternasapreliminarysteptoanalyzingtheleadingsourceofthenon-Gaussian
component.
6
One difficulty in describing these fluctuations is that general relativity contains a
fair amount of redundancy. Although a coordinate system must be chosen to be able
to compare a prediction with what is observed, nothing that depends in detail on this
choice corresponds to a genuine physical effect. These notes will only be considering
thescalarfluctuations, whichareultimatelymeanttoberesponsibleforthedensityand
the temperature fluctuations in the matter of the early universe. A scalar fluctuation is
onethattransformsasascalarquantityundertheunbrokenspatialrotationsandtransla-
tionsratherthanasascalarunderageneralfour-dimensionalcoordinatetransformation.
To count the number of distinct scalar functions that are possible in a general pertur-
bation to the classical background, d gmn (t,~x), first divide the fluctuation in the metric
intoblocks,
d g (t,~x) d g (t,~x)
d gmn (t,~x)=(cid:18)d g0i00(t,~x) d g0iji(t,~x)(cid:19). (25)
d g itself provides one scalar field. Another is generated by making a three-vector
00
from a scalar field by taking its spatial derivative; thus d g ¶ B also contains one
0i i
scalar field. d g contains twoscalar fields—specifically, itstra∼ce andonemoregener-
ij
ated by taking two derivatives of a scalar function, ¶ ¶ x . Including a final fluctuation
i j
fortheactualscalarfielddf ,alltoldtherearefiveseparate scalarfields.
Most of these fields have no physical meaning. Two of them, for example, are
related tohowthecoordinates werechosen. Dividing ageneral coordinate transforma-
tion, xm xm +d xm , into its temporal and spatial pieces, there is one scalar function
in d x0 an→d it is again possible to form a spatial vector by taking a spatial derivative of
a second, d xi d ij¶ f. Additionally, it will soon become apparent that two more are
j
∼
non-propagating degrees of freedom fixed by two constraints. Thus, the five potential
scalar fields are reduced by four, leaving but a single physical field. This section will
show how this reduction proceeds before isolating this remaining physical scalar field
andanalyzing howitsdynamicsleadtoaprediction forthetwo-pointfunction.
To analyze the fluctuations about the simple, spatially invariant inflationary back-
ground, writethemetricinthefollowinggeneralform,
ds2 = N2dt2 h Nidt+dxi Njdt+dxj
ij
−
= (N2 NNi)(cid:0)dt2 2N dt(cid:1)d(cid:0)xi h dxid(cid:1)xj. (26)
i i ij
− − −
This metric was originally introduced by Arnowitt, Deser and Misner [2] to analyze
gravity from a Hamiltonian perspective. In this framework, N(t,~x) is called the lapse
function whileNi(t,~x)istheshift vector. NotethatN isdefinedtobeN h Nj. The
i i ij
≡
components oftheinverse metricare
1 1 1
g00= , g0i = Ni, gij = N2hij NiNj . (27)
N2 −N2 −N2 −
(cid:2) (cid:3)
The spatial components, h , can be used to define a metric for the three-dimensional
ij
spatial hypersurfaces ofthefull space-time. Todistinguish the curvature andcovariant
derivatives calculated using this metric, h , from those evaluated with the full space-
ij
timemetric,gmn ,theformerwillbewrittenwithacaret—as,forexample,(cid:209)ˆi orRˆ. The
fullactionofthetheory, rewrittenintermsofthismetric,thenbecomes
1 1 1
S= d4x√h NRˆ+ (E Eij E2)+ (f˙ Ni¶ f )2 Nhij¶ f¶ f 2NV(f ) .
ij i i j
2Z N − N − − −
n o
(28)
7
Inthisexpression, anewspatial tensorE wasintroduced, whichisdefinedby
ij
E = 1 h˙ (cid:209)ˆ N (cid:209)ˆ N
ij 2 ij− i j− j i
= 1(cid:2)h˙ h ¶ Nk h (cid:3)¶ Nk Nk¶ h , (29)
2 ij− ik j − jk i − k ij
(cid:2) (cid:3)
and which is closely related to the extrinsic curvature associated with how the spatial
surfaces are embedded in the full space-time. Once again, the spatial indices are im-
plicitlycontracted usingthemetrich ,sothat
ij
E Eij E2= hikhjl hijhkl E E . (30)
ij ij kl
− −
(cid:2) (cid:3)
One advantage of this approach is that the fields N and Ni are both Lagrange mul-
tipliers, with no underlying dynamics, so their equations of motion produce two con-
straintsthatreducethenumberofindependentscalardegreesoffreedombytwo,aswas
mentioned earlier. VaryingwithrespecttoN N+d N yields
→
Rˆ N 2(E Eij E2) N 2(f˙ Ni¶ f )2 hij¶ f¶ f 2V =0, (31)
− ij − i i j
− − − − − −
whileNi Ni+d Ni gives
→
(cid:209)ˆ N 1(E j d jE) =N 1(f˙ Nj¶ f )¶ f . (32)
j − i − i − − j i
(cid:2) (cid:3)
Since the behavior of the gravitational part of the action is so complicated in gen-
eral, and since the actual quantum fluctuations of the field and the metric are small
when compared with the classical background values, the fluctuations can be studied
byexpandingtheactiononlytothenecessaryorderinthefluctuationsfortheparticular
quantitythatisbeinganalyzed. Atfirst,whencalculatingthetwo-pointfunction,itwill
beonlynecessarytokeepthosetermsintheactionthatarequadraticinthefluctuations;
butlater,whenevaluatingthethree-pointfunction,theleadingsignalofanon-Gaussian
pattern, thecubictermswillbekeptaswell.
Ageneralparameterization ofthescalarfluctuations inthemetricisprovided by
N =1+2F (t,~x), Ni=d ij¶ B(t,~x), h =e2r (t) (1+2z (t,~x))d +¶ ¶ x ,
j ij ij i j
(cid:2) (3(cid:3)3)
andinthefield’sfluctuations by
f =f (t)+df (t,~x). (34)
0
This particular parameterization is not left unchanged under a small change of the co-
ordinates; butsince the calculation herewilluse thefreedom tochoose aparticular set
ofcoordinates thatsimplifiescertainpartsoftheanalysis,itisnotnecessarytowritean
explicitlycoordinate-invariant formforthefluctuationsfromthestart. Onefinalraising
and lowering convention that will be followed is that the indices of derivatives acting
on any one of the scalar fluctuations—z , F , B, or x —will always be implicitly raised
orcontractedwithaKroneckerd ij,aswasdoneaboveinthedefinitionofNiasd ij¶ B.
j
By redefining the time coordinate and shifting the spatial coordinate through the
derivativeofascalarfunction, xi xi+d ij¶ f(t,~x),twoofthefivescalarfunctions in
j
→
thegeneral parameterization can beremoved. Thisfreedom willbeused tochoose the
coordinatessothatthefluctuationsofthescalarfieldvanish,df =0,andsothatx =0,
leaving
N =1+2F (t,~x), Ni=d ij¶ B(t,~x), h =e2r (t)+2z (t,~x)d , f =f (t).
j ij ij
(35)
8
Noticethatz hasbeenslightlyredefinedsothatitnowappearsintheexponent,though
to first order the coordinates are the same as the initial definition above; however, this
form will be much more useful when analyzing the three-point function later. This
choice for the coordinates simplifies the constraint equations quite a bit, since f will
thenhavenospatialderivatives,
Rˆ N 2(E Eij E2) N 2f˙2 2V =0
− ij −
− − − −
(cid:209)ˆ N 1(E j d jE) =0. (36)
j − i − i
(cid:2) (cid:3)
Notice that the potential energy V(f ) has still been left quite general; but it can be
removedentirelyfromthefirstconstraint equation byapplying thett component ofthe
Einsteinequation,
Rˆ N 2 E Eij E2 6r˙2+ 1 N 2 f˙2=0. (37)
− ij −
− − − −
(cid:0) (cid:1) (cid:0) (cid:1)
Now solve these constraints tofirst order inthe coordinates that were chosen. The
scalarcurvatureRˆ associated withthespatialmetrich is
ij
Rˆ=e 2r 2z 4¶ ¶ kz 2¶ z¶ kz = 4e 2r ¶ ¶ kz +O(z 2), (38)
− − k k − k
− − −
(cid:2) (cid:3)
whileE is
ij
E =e2r r˙(1+2z )d +zd˙ ¶ ¶ B + (39)
ij ij ij i j
− ···
(cid:2) (cid:3)
sothat
E Eij E2= 6r˙2 12r˙z˙+4r¶˙ ¶ B+ . (40)
ij i j
− − − ···
TheN constraint equation thenbecomes—again, tofirstorderinthefluctuations—
3r˙ 2r˙F z˙ ¶ ¶ k r˙B+e 2r z +f˙2F =0. (41)
k −
− − −
(cid:2) (cid:3) (cid:2) (cid:3)
Similarlyexpanding theNi constraint tofirstorderyields,
2¶ 2r˙F z˙ =0. (42)
i
−
(cid:2) (cid:3)
Thisequation removesoneofthescalardegreesoffreedombyfixingF ,
1z˙
F = . (43)
2r˙
Although this constraint would have seemingly allowed the addition of an arbitrary
constant aswell,itwaschosen tobezerosothattheoriginal background metricwould
be restored when the fluctuations are removed. When this result is inserted into the N
constraint, itsimilarlycompletely fixesanotherofthescalarfields,B,
e 2r 1f˙2
B= − z +c with ¶ ¶ kc = z˙. (44)
− r˙ k 2r˙2
Having solved the constraint equations and thereby eliminated two of the scalar
fields, F and B, by expressing them interms ofthe one remaining fieldz , it is timeto
determinethequadratic actionforthisremainingscalarfield. First,againsubstitute the
9
background equation,V =3r˙2 1f˙2,intotheaction, stillusingthecoordinate system
wherethefluctuationofthefiel−ddf2 =0vanishes, toobtain
1
S= d4x√h NRˆ+N 1(E Eij E2) 6Nr˙2+(N+N 1)f˙2 . (45)
− ij −
2Z n − − o
To calculate the two-point function of z , expand this integrand to second order in the
smallfluctuations. Atthisorder,
Rˆ=e 2r 4¶ ¶ kz +8z¶ ¶ kz 2¶ z¶ kz + (46)
− k k k
− − ···
(cid:8) (cid:9)
and
E Eij E2 = 6r˙2 12r˙z˙+4r¶˙ ¶ kB 6z˙2+6r¶˙ z¶ kB+4z¶˙ ¶ kB
ij k k k
− − − −
+(¶ ¶ B)(¶ i¶ jB) (¶ ¶ kB)2+ (47)
i j k
− ···
and
1 z˙2
N+ =2+ + . (48)
N r˙2 ···
Thefirsttermintheactionisresponsible forthespatialpartofthekinetictermforz ,
r¨ 1
√hNRˆ = 2er ¶ z¶ kz ¶ 2er z¶ ¶ kz
r˙2 k − 0(cid:26) r˙ k (cid:27)
r¨ z˙ 1
2er ¶ 2+z + z + ¶ kz z¶ kz˙ + ; (49)
− k(cid:26)(cid:18) r˙2 r˙(cid:19) −r˙ (cid:27) ···
usingthebackground equations toreplacer¨ = 1f˙2,onlyonetermremains
−2
f˙2
√hNRˆ= er ¶ z¶ kz + , (50)
− r˙2 k ···
up to derivative terms which have no dynamical effect. Taken together, the rest of the
termstosecondorderare
1 1
√h E Eij E2 6Nr˙2+ N+ f˙2
ij
(cid:26)N − − (cid:18) N(cid:19) (cid:27)
(cid:0) (cid:1)
f˙2
= e3r z˙2+e3r 2r¨+f˙2 2+6z +9z 2
r˙2
(cid:2) (cid:3)(cid:2) (cid:3)
2¶ e3r r˙ 2+6z +9z 2
0
−
+e3r ¶(cid:8) 4r˙(cid:2)+12rz˙ ¶ ¶(cid:3)(cid:9)jB ¶ kB+(¶ B)(¶ j¶ kB) + . (51)
k j j
− ···
(cid:8)(cid:0) (cid:1) (cid:9)
Usingthebackground equation f˙2= 2r¨ onceagainleaves justoneterm thatisnota
−
totalderivative,
1 1 f˙2
√h E Eij E2 6Nr˙2+ N+ f˙2 =e3r z˙2+ . (52)
(cid:26)N ij − − (cid:18) N(cid:19) (cid:27) r˙2 ···
(cid:0) (cid:1)
Sowhatremains,afterapplyingthebackground equationsandignoringtotalderivative
terms,isaremarkably simpleexpression forthefluctuations,
f˙2
S= dt d3~xe3r 1z˙2 1e 2r ¶ z¶ kz + . (53)
Z r˙2Z n2 −2 − k ···o
10