Table Of ContentGlueball Inflation and Gauge/Gravity Duality
LiliaAnguelova
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Abstract We summarize our work on building glueball inflation models with the
1
1 methodsofthegauge/gravityduality.Wereviewtherelevantfive-dimensionalcon-
sistent truncation of type IIB supergravity.We consider solutions of this effective
] theory,whosemetrichastheformofadS foliationoveraradialdirection.Byturn-
h 4
t ing on small (in an appropriate sense) time-dependentdeformationsaround these
-
solutions, one can build models of glueball inflation. We discuss a particular de-
p
e formedsolution,describinganultra-slowrollinflationaryregime.
h
[
1
1 Introduction
v
9
4
Compositeinflationmodels[1,2]provideapossibleresolutiontothewell-knownh -
4
2 problem[3,4]ofinflationarymodel-building.However,theyarequitechallenging
0 to studywith standardQFT methods,sincethey involvea strongly-coupledgauge
. sector. This has motivated interest in developing descriptions of such models via
1
0 a string-theoretic tool aimed precisely at studying the nonperturbative regime of
6 gaugetheories,namelythegauge/gravityduality.Gravitationalduals,inwhichthe
1 inflaton arises from the position of a D3-brane probe have been considered in [5,
:
v 6, 7, 8, 9]. Instead, in [10, 11, 12] we studied models, whose inflaton arises from
i thebackgroundfieldsofthegravitationalsolutionandisthusaglueballinthedual
X
gaugetheory.
r
a Thebackgroundsofinterestforussolvetheequationsofmotionofthe5dconsis-
tenttruncationoftypeIIBsupergravityestablishedin[13].Thelatterencompassesa
widevarietyofprominentgravityduals,like[14,15,16,17,18],asspecialsolutions
and thus provides a unifying framework for gauge/gravity duality investigations.
Thework[10]obtainednewnon-supersymmetricclassesofsolutionsofthistheory,
LiliaAnguelova
InstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,Sofia1784,
Bulgaria,e-mail:[email protected]
1
2 LiliaAnguelova
whosemetricisoftheformofadS fibrationoverthefifthdirection.Theseback-
4
grounds provide a useful playground for studying certain strongly-coupledgauge
theoriesindeSitterspace.Tohaveaninflationarymodel,however,oneneedsatime-
dependentHubbleparameter.Therefore,in[12]weinvestigatedtime-dependentde-
formationsaroundasolutionof[10],inordertosearchforgravitydualsofglueball
inflation.
Itisworthpointingoutthatthemaincosmologicalobservablesofaninflationary
model(likethescalarspectralindexn andthetensor-to-scalarratior)areentirely
s
determined by the Hubble parameter and inflaton field as functions of time [19].
Hence,onceonehasadeformedbackgroundintheaboveset-up,onecanimmedi-
ateycomputethedesiredquantities.Thisisthesense,inwhichthetime-dependent
deformationsof the previous paragraph give models of cosmological inflation. In
that vein, in [12] we calculated the slow roll parameters for a solution we found
thereandthusestablishedthatitgivesagravitydualofultra-slowrollglueballin-
flation.Theultra-slowrollregimemayplayanimportantroleinunderstandingthe
observedlowl anomalyinthepowerspectrumoftheCMB.Henceitdeservesfur-
therstudy.Wealsodiscusshereperspectivesforbuildinggravitydualsofstandard
slowrollinflationarymodels.
2 Effective 5d theory
Inthissectionwesummarizenecessarymaterialaboutthe5dconsistenttruncation
oftypeIIBsupergravityrelevantforourconsiderations.Wealsorecallaparticular
solution of this theory,whose time-dependentdeformationswe will investigatein
thenextsubsection.
2.1 Actionand field equations
Letusbrieflyreviewthebasiccharacteristicsofthefive-dimensionalconsistenttrun-
cationof[13].UsingaparticularansatzforthebosonicfieldsoftypeIIBsupergrav-
ity in terms of certain 5d fields and integrating out five compact dimensions, one
reducestheten-dimensionalIIBactiontothefollowingfive-dimensionalone:
R 1
S= d5x detg + G (F )¶ F i¶ IF j+V(F ) . (1)
ij I
− −4 2
Z (cid:20) (cid:21)
p
Here F i isasetof5dscalarfields,thatarisefromthecomponentsofthe10dones
includ{ing}metricwarpfactors,V(F ) isa rathercomplicatedpotential,G (F ) isa
ij
diagonalsigma-modelmetricand,finally,RistheRicciscalarofthe5dspacetime
metric g . The full expressionsforV(F ) and G (F ) can be found in [13]; for a
IJ ij
moreconcisesummary,seealso[10].Thefieldequationsthattheaction(1)implies
GlueballInflationandGauge/GravityDuality 3
are:
(cid:209) 2F i+Gi gIJ(¶ F j)(¶ F k) Vi = 0 ,
jk I J
−
4
R +2G (¶ F i)(¶ F j)+ g V = 0 , (2)
IJ ij I J IJ
− 3
whereVi =GijV , V = ¶V and Gi are the Christoffel symbols of the sigma-
j i ¶F i jk
modelmetricG .
ij
2.2 AsolutionwithdS slicing
4
We willbe interestedin time-dependentdeformationsarounda particularsolution
ofthesystem(2)foundin[10].Soletusfirstrecallitsform.Inthenotationof[10]
and working within the same subtruncation as there (i.e., with zero NS flux), we
havesixscalarsinthe5deffectivetheory:
F i(xI) = p(xI),x(xI),g(xI),f (xI),a(xI),b(xI) . (3)
{ } { }
Thework[10]foundthreefamiliesofsolutionsof(2)witha5dmetricoftheform
3
ds2=e2A(z) dt2+s(t)2 (cid:229) (dxm)2 +dz2, (4)
5 "− m=1 #
where s(t)=eHt with H =const. In all of them, three of the scalars F i vanish
identically,namely:
g(xI)=0 , a(xI)=0 , b(xI)=0 . (5)
Twoofthosesolutionsarenumericalandoneisanalytical.Forconvenience,wewill
studydeformationsaroundthelatter.Denotingitsmetricfunctionsandscalarfields
bythesubscript0,wehave[10]:
1 7
A (z)= ln(z+C)+ ln H2 ,
0 2 3 0
(cid:18) (cid:19)
1 1 7N2
p (z)= ln(z+C) ln ,
0
−7 −14 9
(cid:18) (cid:19)
x (z)= 6p (z) , f =0 , (6)
0 0 0
−
whereCandN areconstants.
Letusmentioninpassingthattheformof(6)isconsistentwithALD(asymptot-
icallylineardilaton)behavioratlargez.Thisisnotobviousatfirstsightduetothe
useofadifferentcoordinatesystem(instringframe)comparedtotheconventional
one (in Einstein frame), in which the holographic renormalization of ALD back-
4 LiliaAnguelova
grounds was developed [20]. This issue was discussed in more detail in [10, 11],
where it was also pointed out that the same kind of asymptotics characterizesthe
walkingsolutionsof[16]aswell.
3 Deforming the dS solution
4
Nowwearereadytoturntotheinvestigationofsolutionsofthesystem(2),which
aredeformationsaroundthezerothorderbackground(6).Sinceouraimistostudy
glueballinflation, we wouldlike to find solutions, whose 5d metric is of the form
(4)butwithH˙ =0.RecallthattheHubbleparameterisdefinedas
6
s˙
H = , (7)
s
¶
whereforconveniencewehavedenoted˙ . Now,oneoftheslowrollconditions
≡ ¶ t
widelyusedininflationarymodelbuilding1isthefollowing[19]:
H˙
<< 1 . (8)
−H2
Inviewofthat,wewilllookforsolutionswithtime-dependentH byconsidering
small,inthesenseof(8),deformationsaroundanH =constsolution.
Forthatpurpose,letusintroduceasmallparameterg ,satisfying
g <<1, (9)
andsearchforsolutionsthatareexpansionsinpowersofthisparameter.Todothis,
wemakethefollowingansatzforthenonvanishing5dfields:
p(t,z)= p (z) , x(t,z)=x (z) ,
0 0
f (t,z) = gf (t,z)+g 3f (t,z)+O(g 5) ,
(1) (3)
A(t,z)= A (z)+g 2A (t,z)+O(g 4) ,
0 (2)
H(t,z)= H t+g 2H (t,z)+O(g 4) , (10)
0 (2)
whereH(t,z)isawarpfactordefinedvia
3
ds2=e2A(t,z) dt2+e2H(t,z) (cid:229) (dxm)2 +dz2 . (11)
5 "− m=1 #
1Oneshouldkeepinmind,though,thattherearemoreexoticinflationaryregimes,inwhichone
ormoreoftheslowrollconditionscanbeviolated;see[21,22,23],forinstance.
GlueballInflationandGauge/GravityDuality 5
Inotherwords,wekeepthescalarsp(xI)andx(xI)thesameasin(6),whileallowing
small deviations around that zeroth order solution in the scalar f and the metric
functionsAandH.
It is worth commentinga bit more on the formof the deformationansatz (10).
First of all, in orderto obtainsolutionswith H˙ =0, we need to turn on time de-
pendence in at least one scalar. It is convenient6to take this scalar to be f since,
unlike p andx,itvanishesatzerothorderand,furthermore,itisaflatdirectionof
thepotential;see[12].Therefore,f willplaytheroleoftheinflatoninourset-up.
Alsonotethat,althoughwewouldliketohavet-dependentHonly,wehaveallowed
z-dependencetoo,formoregenerality.And,finally,thedifferentpowersofg inthe
expansion of f , compared to the expansions of the warp factors, will be of great
significanceforfindingananalyticalsolution,aswillbecomeclearbelow.
3.1 Equations ofmotion
Letusnowsubstitutetheansatz(10)inthesystem(2)andstudytheresultorderby
ordering .Clearly,sinceweareexpandingaroundazerothordersolution,thereis
nocontributionatorderg 0.
Atorderg ,wehavethefollowingfieldequation[12]:
f¨(1)+3H0f˙(1)=e2A0 f (′1′)+4A′0f (′1) , (12)
(cid:16) (cid:17)
¶
where .Tofindasolution,letusmaketheansatz
′≡ ¶ z
f =F (t)F (z) (13)
(1) 1 2
andsolvetheeigenproblems
F¨1+3H0F˙1=lF 1 and e2A0 F 2′′+4A′0F 2′ =lF 2 (14)
withl beingsomeconstant.Oneeasilyobtainsth(cid:0)at[12]: (cid:1)
F (t)=C ek+t+C ek t , where k = 3H0 9H02+4l (15)
1 1 2 − ± − 2 ±q 2
andC areintegrationconstants,while
1,2
3 3 4l
F (z)=C (z+C)a ++C (z+C)a with a = 1+ (16)
2 3 4 − ± −2±2s 21H02
andC beingintegrationconstants.
3,4
Notethatifl =0,thenoneisfreetoaddanarbitraryconstanttothef solution,
(1)
determinedby(14).Thiswillbeimportantinthefollowing.
6 LiliaAnguelova
Atorderg 2,wefindacoupledsystemforthewarpfactordeformationsA and
(2)
H ,namely[12]:
(2)
7 56
E1: H2 (z+C)2A + (z+C)A +7(z+C)H +6A
− 0 3 ′(′2) 3 ′(2) (′2) (2)
(cid:18) (cid:19)
1
+H 3A˙ +6H˙ +3A¨ +3H¨ + f˙2 =0 ,
0 (2) (2) (2) (2) 2 (1)
(cid:0)7 (cid:1) 56 49
E2: H02 3(z+C)2 A′(′2)+H(′2′) + 3 (z+C)A′(2)+ 3 (z+C)H(′2)
(cid:18) h i
+6A H 5A˙ +6H˙ A¨ H¨ =0 ,
(2) − 0 (2) (2) − (2)− (2)
(cid:19)
(cid:0) 2 (cid:1) 1
E3: 4A +3H + 4A +3H + f 2 =0 ,
′(′2) (′2′) z+C ′(2) (′2) 2 (′1)
(cid:16) 1 (cid:17)
E4: 3A˙′(2)+3H˙(′2)+3H0H(′2)+2f˙(1)f (′1)=0 . (17)
Tosolvethisratherinvolvedsystem,letustakeforconveniencethef solutionto
(1)
be:
f (1)=Cf +C˜ekt(z+C)a with Cf ,C˜=const , (18)
where k is any of k and a is any of a . Note that the addition of the arbitrary
± ±
constantCf in(18)makesnodifferenceforthesolutionsof(17),sincethefunction
f (1)entersthoseequationsonlythroughitsderivatives.However,thepresenceofCf
willturnouttobeusefullater.Plus,itwillbecomeclearshortlythatitisconsistent
with(14).
Now,theformofE3in(17),togetherwith(18),suggestslookingforasolution
withthefollowingansatz:
A (t,z)=e2ktAˆ(z) and H (t,z)=Cˆ +e2ktHˆ(z), (19)
(2) (2) H
whereCˆ =const. Again, we have included an arbitrary constantCˆ , since H
H H (2)
entersthesystem(17)onlyviaitsderivatives.Substituting(19)and(18)into(17),
onecan see thatthe time-dependencefactorsout. Thus,oneis leftwith a coupled
system of ODEs for the functionsAˆ(z) and Hˆ(z). A detailed investigationin [12]
showedthatthissystemhasasolutiononlyfor2
a =0 , (20)
in which case both Aˆ =const and Hˆ =const. Substitutinga =0 in (16), we find
thatl =0 as well. This in turn implies thatwe are free to add the constantCf in
(18),ascommentedbelow(16).Finally,from(15)wenowhave:
k= 3H , (21)
0
−
2 To prove this, one also needs to use the fact that (15) and (16) imply the following relation
betweenkanda : k=−32H0±√84a 2+6252a +81H0 .
GlueballInflationandGauge/GravityDuality 7
wherewehavetakenthevalueofk inordertohavetime-dependenceintheinflaton
fieldf . −
3.2 Ultra-slowrollinflation
Thesolutionwedescribedabovegivesadualdescriptionofanultra-slowrollglue-
ballinflationmodel.Toseethis,letuscomputetheinflationaryslow-rollparameters.
TheyaredefinedintermsoftheinflatonfieldandHubbleparameteras[19]:
H˙ f¨
e = and h = . (22)
−H2 −Hf˙
FromtheresultsofSubsection3.1,wehavethatf andH aregivenby:
f = Cf +C˜e−3H0t g +O(g 3),
H =(cid:16)H0 CH e−6H0(cid:17)tg 2+O(g 4), (23)
−
whereCH issomeconstant;formoredetails,see[12].3
Substituting (23) in (22), we find that the slow roll parameters behave as e =
O(g 2)andh =3+O(g 2);see[12]formoredetailedexpressions.Inotherwords,
atleadingorderwehave:
e <<1 and h =3 . (24)
These are preciselythe valuesof e andh for the ultra-slowregime,consideredin
[24,25]. Infact,ourresultfortheinflatonin(23)also agreescompletelywiththe
expressionin[25].
Itisworthpointingoutasimilaritybetweenourmodelandtheconstant-rate-of-
rollsolutionsof[23].Forthatpurpose,letusintroducethefollowingseriesofslow
rollparameters:
H˙ e˙
e = and e = n , (25)
1 −H2 n+1 Hen
whereobviouslye e .Onecaneasilycomputethat,atlarget,oursolutiongives
1
≡
[12]:
e 0 and e 6 . (26)
2n+1 2n
→ →−
3 Notethat,sincethecorrectionA tothewarpfactorA(t,z)in(11)alsodependsont,ascan
(2)
beseenfrom(19),oneshould,inprinciple,firstperform acoordinatetransformationt t that
absorbs that dependence, before computing the physical Hubble parameter H(t ) and→inflaton
fieldf (t ).However, inthepresentcase,thisleadstoexactlythesameexpressions as(23)with
t substitutedbyt ,withtheonlydifferencebeingthenumericalvalueoftheconstantCH.Sowe
willnotdiscussthedetailsofthattransformationhere.
8 LiliaAnguelova
Thisisconsistentwiththeasymptoticsin[23].Itwouldbeinterestingtoinvestigate
whetherthereisadeeperunderlyingreasonforthat.
Inconclusion,letusmakeafewcommentsregardingotherinflationarymodels
in our framework. Although an ultra-slow roll inflationary regime may be desir-
able to account for the low l anomaly in the CMB power spectrum, it is rather
short-lived.Soithastobesucceededbyregularslowroll,inordertohaveenough
expansionandthusgiveacompleteinflationarymodel.Toobtainsuchsolutionsin
our gauge/gravity duality set-up, one may need to study deformations around the
numericalsolutionsof [10], instead of the analyticalone (6). It could also be that
dualsofregularslowrollcanbefoundbymodifyingtheinitialansatzforthedefor-
mationsaroundtheanalyticalsolution.Finally,itwouldbeinterestingtoinvestigate
whatkindofmodelscanbeobtainedbygoingtothenextordering intheexpan-
sions(10),whiletakingf ,A andH tovanish.Thisseemstoopenmuchwider
(1) (2) (2)
possibilitiesforinflationarymodelbuilding,astheequationsofmotionforA and
(4)
H wouldbe independentoff . Thus, manyof the restrictionswe encountered
(4) (3)
here(andasaresultofwhichweendedupwithultra-slowroll)wouldnotoccur.
Acknowledgements IhavereceivedpartialsupportfromtheEuropeanCOSTActionMP-1210
andtheBulgarianNSFgrantDFNIT02/6.
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