Frontiers of Inationary Cosmology
Robert H. Brandenberger
PhysisDepartment, BrownUniversity,
Providene,R.I.,USA
Reeivedon2February,2001
Afterabrief reviewthetheoryof osmologial perturbations,Ihighlight somereent progress in
theareaofreheatingininationaryosmology,fousinginpartiularonparametriampliationof
super-Hubbleosmologialutuations,andontheroleofnoiseintheresonanedynamis(yielding
a newproof of Andersonloalization). I thendisuss several important oneptual problems for
theurrentrealizationsofinationbasedonfundamentalsalarmatterelds,andreviewsomenew
approahesatsolvingtheseproblems.
I Introdution
Inationary osmology [1℄ has beome oneof the
or-nerstonesof modern osmology. Ination wastherst
theorywithinwhihitwaspossibletomakepreditions
aboutthestrutureoftheUniverseonlargesalesbased
onausalphysis. Thedevelopmentoftheinationary
Universesenariohasopenedupanewand extremely
promising avenue for onneting fundamental physis
withexperiment.
Afterabriefintrodutiontoinationaryosmology
(Setion II) and an overview of the theory of
osmo-logialperturbationsapplied to ination(Setion III),
I fous on reent improvements in our understanding
ofthetheoryofinationaryreheating(SetionIV),
fo-usinginpartiularonreentstudiesoftheparametri
ampliationofgravitationalutuationsandonthe
ef-fetsofnoiseontheresonanedynamis (whih yields
anewproofofAndersonloalization inonespatial
di-mension).
However, in spite of the remarkable suess of the
inationary Universe paradigm, there are several
se-riousoneptualproblemsforurrentmodelsin whih
inationisgeneratedbythepotentialenergyofasalar
matter eld. These problems are disussed in Setion
V.
SetionVIisasummaryofsomenewapproahesto
solvingtheabove-mentionedproblems. An attemptto
obtain ination from ondensatesis disussed, a
non-singular Universe onstrution making use of higher
derivativetermsinthegravitationalationisexplained,
and a framework for alulating the bak-reation of
osmologialperturbationsissummarized.
This short review fouses ona seleted numberof
topis at the forefront of inationary osmology. For
omprehensive reviews of ination, the reader is
re-tionarymodelbuildingintheontextof
supersymmet-rimodelsanbefoundin [6℄. Anextended andmore
pedagogialversionofthesenotesis[7℄. Notethat
Se-tions II, V and VI of this artile are idential to the
orrespondingsetionsin [8℄.
II Basis of Inationary
Cos-mology
MosturrentmodelsofinationarebasedonEinstein's
theoryofGeneralRelativitywithamattersouregiven
byasalareld'. Basedontheosmologialpriniple,
themetriofspae-timeonlargedistanesalesanbe
writteninFriedmann-Robertson-Walker(FRW)form:
ds 2
=dt 2
a(t) 2
dr 2
1 kr 2
+r 2
(d# 2
+sin 2
#d' 2
)
;
(1)
where the onstant k determines the topology of the
spatial setions. In the following, we shall set k =0,
i.e. onsideraspatially atUniverse. Inthisase, we
anwithoutlossofgeneralitytakethesalefatora(t)
tobeequalto 1atthe presenttime t
0
, i.e. a(t
0 )=1.
Theoordinates r;#and' areomovingspherial
o-ordinates.
ForahomogeneousandisotropiUniverseand
set-ting the osmologial onstant to zero, the Einstein
equationsredue totheFRWequations
_ a
a
2
= 8G
3
(2)
a
a =
4G
3
(+3p); (3)
den-toyieldtheontinuityequation(withHubbleonstant
H =a=a)_
_
= 3H(+p): (4)
Theequationofstateofmatterisdesribedbya
num-berwdened by
p=w: (5)
Theideaof ination[1℄ isverysimple. Weassume
there is a time interval beginning at t
i
and ending at
t
R
(the\reheatingtime")duringwhihtheUniverseis
exponentiallyexpanding,i.e.,
a(t)e Ht
; t[t
i ;t
R
℄ (6)
withonstantHubbleexpansionparameterH. Suh a
period is alled\deSitter" or \inationary."The
su-essofBigBangnuleosynthesissetsanupperlimitto
thetimet
R
ofreheating,t
R t
NS ,t
NS
beingthetime
ofnuleosynthesis.
Duringtheinationary phase, thenumber density
of any partiles initially present at t = t
i
deays
ex-ponentially. At t = t
R
, allof the energy whih is
re-sponsibleforinationisreleased(seelater)asthermal
energy. This is a non-adiabati proess during whih
theentropyinreasesbyalargefator.
A period of ination an solve the homogeneity
problem of standardosmology,the reasonbeingthat
during ination the physial size of the forward light
oneexponentiallyexpandsandthusaneasilybeome
larger than thephysial size of thepast light one at
t
re
,thetimeoflastsattering,thusexplainingthenear
isotropyof theosmi mirowavebakground(CMB).
Inationalsosolvestheatness problem[9,1℄.
Most importantly, ination provides a mehanism
whihinaausalwaygeneratestheprimordial
pertur-bationsrequired for galaxies, lusters and even larger
objets. In inationary Universe models, the Hubble
radius(\apparent"horizon)andthe(\atual")horizon
(the forwardlight one)do notoinideat late times.
Provided that the duration of ination is suÆiently
long,then all saleswithin ourpresentapparent
hori-zonwereinsidethehorizonsinet
i
. Thus,itisin
prin-iple possible to have a ausal generation mehanism
forperturbations[10,11,12,13℄.
Aswill bedisussedinSetion III,thedensity
per-turbationsprodued during ination aredue to
quan-tum utuationsin thematterand gravitationalelds
[11,12℄. Theamplitudeoftheseinhomogeneities
orre-spondstoatemperatureT
H
H,theHawking
temper-ature ofthe deSitterphase. This leadsoneto expet
that atalltimes tduring ination,perturbationswith
a xed physial wavelength H 1
will be produed.
Subsequently,thelengthofthewavesisstrethedwith
theexpansionofspae, andsoonbeomesmuh larger
than the Hubble radius `
H
(t) = H 1
(t). The phases
of the inhomogeneities are random. Thus, the
ina-tionaryUniversesenariopreditsperturbationsonall
beginningofinationtotheorrespondingquantityat
thetime of reheating. Inpartiular, provided that
in-ationlastssuÆientlylong,perturbationsonsalesof
galaxiesandbeyondwill begenerated. Note, however,
thatitisverydangeroustointerpretdeSitterHawking
radiationasthermalradiation. Infat,theequationof
stateofthis\radiation"isnotthermal[14℄.
In most urrent models of ination, the
exponen-tialexpansionisdrivenbythepotentialenergydensity
V(')ofafundamental salarmattereld 'with
stan-dardation
S
m =
Z
d 4
x p
gL
m
(7)
L
m
(') = 1
2 D
'D
' V('); (8)
whereD
denotestheovariantderivative,andgisthe
determinantofthemetritensor. Theresulting
energy-momentum tensor yields the followingexpressions for
theenergydensityandthepressurep:
(') = 1
2 _ ' 2
+ 1
2 a
2
(r') 2
+V(') (9)
p(') = 1
2 _ ' 2
1
6 a
2
(r') 2
V('): (10)
It thus follows that if the salar eld is homogeneous
and stati,but the potentialenergy positive,then the
equation ofstatep= neessaryforexponential
in-ationresults(see(4)).
Mostoftheurrentrealizationsof potential-driven
inationarebasedonsatisfyingtheonditions
_ ' 2
;a 2
(r') 2
V('); (11)
viatheideaofslowrolling[15,16℄. Considerthe
equa-tionofmotionofthesalareld':
'+3H'_ a 2
5 2
'= V 0
('): (12)
Ifthesalareldstartsoutalmosthomogeneousandat
rest, iftheHubble dampingterm(the seondterm on
the l.h.s. of (12) islarge, and ifthe potential isquite
at (so that the term on the r.h.s. of (12) is small),
then'_ 2
mayremainsmallomparedtoV('), inwhih
ase exponential inationwill result. Note that if the
spatial gradienttermsareinitially negligible,theywill
remainnegligiblesinetheyredshift.
Chaotiination [17℄ is aprototypialinationary
senario. Considerasalareld'whihisveryweakly
oupled toitself and other elds. Inthis ase,'need
notbeinthermalequilibrium atthePlanktime,and
mostof thephasespaefor 'will orrespondto large
values of j'j (typially j'j m
pl
). Consider now a
region in spae where at the initial time '(x) is very
large, and approximately homogeneousand stati. In
imme-andindue anequationofstatep' whihleadsto
ination. DuetothelargeHubbledampingterminthe
salareldequationofmotion,'(x)will onlyrollvery
slowlytowards'=0 (we aremaking the assumption
that V(') hasaglobalminimumat anite valueof'
whih an then be hosen to be ' = 0). The kineti
energy ontribution to and pwill remain small, the
spatialgradientontributionwillbeexponentially
sup-pressedduetotheexpansionoftheUniverse,andthus
ination persists. Note that the preise form of V(')
isirrelevanttothemehanism.
It is diÆultto realizehaoti ination in
onven-tional supergravity models sine gravitational
orre-tions to the potential of salar elds typially render
thepotentialsteepforvaluesofj'joftheorderofm
pl
and larger. This prevents the slow rolling ondition
(11) from beingrealizable. Even if this onditionan
besatised,thereareonstraintsfromtheamplitudeof
produeddensityutuationswhiharemuhharderto
satisfy(seeSetionV).
Hybrid ination [18℄ is a solution to the
above-mentioned problem of haoti ination. Hybrid
ina-tion requires at least two salar elds to play an
im-portant role in the dynamis of the Universe. As a
toymodel,onsider thepotentialofatheorywithtwo
salarelds'and :
V('; ) = 1
4 (M
2 2
) 2
+ 1
2 m
2
' 2
+ 1
2
0
2
' 2
: (13)
For valuesofj'jlargerthan'
'
=
0
M 2
1=2
(14)
theminimumof is =0,whereasforsmallervalues
of 'thesymmetry ! is brokenand theground
state valueof j j tends to M. Theidea of hybrid
in-ation isthat ' isslowlyrollinglikethe inatoneld
in haotiination,but thattheenergy density ofthe
Universeisdominatedby . Inationterminates one
j'j dropsbelowtheritialvalue'
, atwhihpoint
startstomove.
Notethatinhybridination'
anbemuhsmaller
than m
pl
and hene ination without super-Plank
sale values of the elds is possible. It is possible to
implementhybridinationintheontextof
supergrav-ity(seee.g. [19℄).
Atthe present time thereare many realizationsof
potential-drivenination,butthereisnoanonial
the-ory. Alotofattentionisbeingdevotedtoimplementing
ination in the ontext of unied theories, the prime
andidatebeingsuperstringtheoryorM-theory. String
theory orM-theorylivein 10or11spae-time
dimen-sions,respetively. Whenompatiedto4spae-time
dimensions,thereexistmanymodulields,salarelds
whih desribe at diretions in the ompliated
va-devoted to attempts at implementing ination using
modulields(seee.g. [20℄andreferenestherein).
Reently,ithasbeensuggestedthatourspae-time
isabranein ahigher-dimensional spae-time(see[21℄
forthebasionstrution). Waysofobtainingination
onthebranearealsounderativeinvestigation(seee.g.
[22℄).
It should also not be forgotten that ination an
arisefromthepurelygravitationalsetorofthetheory,
as in the original model of Starobinsky [23℄ (see also
SetionVI), orthatitmayarisefrom kinetitermsin
aneetiveationasinpre-big-bangosmology[24℄or
ink-ination[25℄.
Theorieswith(almost)exponentialination
gener-ially predit an (almost) sale-invariant spetrum of
densityutuations,aswasrstrealizedin [10,11,12,
13℄andthenstudiedmorequantitativelyin[26,27,28℄.
ViatheSahs-Wolfeeet[29℄,thesedensity
perturba-tionsindue CMBanisotropieswithaspetrumwhih
isalsosale-invariantonlargeangularsales.
Theheuristipitureisasfollows. Iftheinationary
periodwhihlasts fromt
i tot
R
isalmostexponential,
thenthe physialeets whih areindependentof the
smalldeviationsfromexponentialexpansion(an
exam-pleofsomethingwhihdoesdependonthesedeviations
iseetsonnetedwiththeremnantradiationdensity
during ination) are time-translation-invariant. This
implies,for example, that quantum utuations at all
times have the same strength when measured on the
samephysiallengthsale.
Iftheinhomogeneitiesaresmall,theyandesribed
by lineartheory, whih implies that allFouriermodes
kevolveindependently. Theexponentialexpansion
in-ates the wavelength of any perturbation. Thus, the
wavelengthof perturbationsgenerated earlyin the
in-ationaryphaseonlengthsalessmallerthanthe
Hub-bleradiussoonbeomesequalto the(\exits")Hubble
radius (this happens at the time t
i
(k)) and ontinues
to inrease exponentially. After ination, the Hubble
radius inreasesas t while the physial wavelength of
autuation inreases only asa(t). Thus, eventually
the wavelength will ross the Hubble radius again (it
will \enter" the Hubble radius) at time t
f
(k). Thus,
it is possible for ination to generate utuations on
osmologialsalesbyausalphysis.
Anyphysial proesswhihobeysthesymmetryof
the inationary phase and whih generates
perturba-tionswill generateutuations ofequalstrength when
measuredwhentheyrosstheHubbleradius(see,
how-ever,Setion V.2):
ÆM
M (k;t
i
(k)) = onst (15)
(independent of k). Here,ÆM(k;t) denotes the r.m.s.
massutuationonalengthsalek 1
attimet.
sales(see, however,the omments at the end of
Se-tionIV.1). Therefore,themagnitudeof ÆM
M
anhange
onlybyafatorindependentofk,andheneitfollows
that
ÆM
M (k;t
f
(k)) = onst; (16)
whih is the denition of a sale-invariant spetrum
[30℄.
III Theory of Cosmologial
Per-turbations
On sales larger than the Hubble radius the
Newto-niantheoryof osmologialperturbationsis
inapplia-ble, and a general relativisti analysis is needed. On
these sales, matter is essentially frozen in omoving
oordinates. However,spae-timeutuationsanstill
inreaseinamplitude.
In priniple, it is straightforward to work out the
generalrelativistitheoryoflinearutuations[31℄. We
linearizetheEinsteinequations
G
=8GT
(17)
(where G
is the Einstein tensor assoiated with
the spae-time metri g
, and T
is the
energy-momentumtensorofmatter)aboutanexpandingFRW
bakground(g (0)
;'
(0)
):
g
(x ;t) = g (0)
(t)+h
(x;t) (18)
'(x ;t) = ' (0)
(t)+Æ'(x ;t) (19)
andpikoutthetermslinearinh
andÆ'toobtain
ÆG
= 8GÆT
: (20)
Intheabove,h
istheperturbationinthemetriand
Æ'istheutuationofthemattereld'.
In pratie, there are many ompliations whih
make this analysis highly nontrivial. The rst
prob-lem is \gaugeinvariane" [32℄. Imagine starting with
a homogeneousFRW osmology and introduing new
oordinates whih mix x and t. In terms of the new
oordinates,themetrinowlooksinhomogeneous. The
inhomogeneouspieeofthemetri,however,mustbea
pureoordinate(or"gauge")artifat. Thus,when
ana-lyzingrelativistiperturbations,aremustbetakento
fatorouteets duetooordinatetransformations.
Therearevariousmethodsofdealingwithgauge
ar-tifats. Thesimplest andmostphysialapproahisto
fousongaugeinvariantvariables,i.e.,ombinationsof
themetriand matter perturbations whih are
invari-antunder linearoordinatetransformations.
Thegaugeinvarianttheory ofosmologial
pertur-bationsisinpriniplestraightforward,although
tehni-ally rathertedious. InthefollowingI willsummarize
themainstepsandreferthereaderto[33℄forthedetails
and further referenes (see also [34℄ for apedagogial
introdutionand[35,36,37,38,39,40,41,42℄forother
approahes).
We onsider perturbations about a spatially at
Friedmann-Robertson-Walkermetri
ds 2
=a 2
()(d 2
dx 2
) (21)
where isonformaltime(relatedtoosmitimet by
a()d=dt). Atthelinearlevel,metriperturbations
an be deomposed into salar modes, vetor modes
andtensormodes(gravitationalwaves). Inthe
follow-ing,wewillfousonthesalarmodessinetheyarethe
onlyoneswhihoupletoenergydensityandpressure.
Asalarmetriperturbation(see[43℄forapreise
def-inition) an be written in termsof four free funtions
ofspaeandtime:
Æg
=a
2
()
2 B
;i
B
;i 2( Æ
ij +E
;ij )
: (22)
The next step is to onsider innitesimal
oordi-nate transformations whih preservethe salarnature
of Æg
, and to alulate the induedtransformations
of; ;B andE. Thenwendinvariantombinations
tolinearorder.(Notethatthereareingeneralno
om-binationswhih areinvariantto allorders[44℄.) After
somealgebra,itfollowsthat
= +a
1
[(B E 0
)a℄ 0
(23)
=
a 0
a
(B E 0
) (24)
aretwoinvariantombinations(aprimedenotes
dier-entiationwithrespetto).
Perhapsthe simplest way [33℄ to derive the
equa-tionsofmotionforgaugeinvariantvariablesisto
on-sider the linearized Einstein equations (20) and to
write them out in the longitudinal gauge dened by
B = E = 0, in whih = and = , to diretly
obtaingaugeinvariantequations.
Forseveraltypesofmatter, inpartiular forsalar
eldmatter,ÆT i
j Æ
i
j
whihimplies= . Hene,the
salar-typeosmologialperturbationsan inthis ase
be desribed by a single gauge invariant variable. In
theaseofasinglesalarmattereld',theperturbed
Einstein equationsanbeombinedtoyield
+ H 2
'
_ '
_
+ k
2
a 2
+2 _
H 2H
'
_ '
= 0: (25)
Forutuationswith saleslargerthantheHubble
ra-dius,thisequationofmotionanbewrittenintheform
ofanapproximate onservationlaw[28,45,39,46,47,
48℄
_ ' 2
_
= 0 (26)
where
= 2H
1
_
+
During the period of slow-rolling of the salar eld
(andalsoforsingleperfetuids),(26)beomessimply
_
=0.
If the equationof state of matter is onstant, i.e.,
w=onst, then _
=0impliesthat therelativisti
po-tential is time-independent on sales larger than the
Hubble radius, i.e. (t)=onst. Duringatransition
from an initial phase with w = w
i
to a phase with
w=w
f
,hanges. Inmanyases,agood
approxima-tiontothedynamisgivenby(26)is
1+w (t
i ) =
1+w (t
f
); (28)
To make ontat with late time matter
perturba-tions andwith theNewtonian intuition, itis usefulto
note that, asaonsequeneof theEinstein onstraint
equations,atthetimet
H
(k)whenamodekrossesthe
Hubbleradius,isameasureofthefrationaldensity
utuations:
(k;t
H (k))
Æ
(k;t
H
(k)): (29)
Theprimordialperturbationsinaninationary
os-mologyaregeneratedbyquantumutuations(seealso
[49, 50℄). Sine the sale of the utuations of
inter-est todaywaslargerthantheHubbleradiusforalong
time, it is ruial to onsider notjust matter
utua-tions, but alsothegravitationalutuationsdesribed
at a lassiallevelin the previous paragraphs. Thus,
the generation and evolution of osmologial
utua-tions in inationary osmology beomes a problem of
quantumgravity. However,duetothefatthatgravity
is an attrative fore, we know that the amplitudeof
the utuations had to have been extremely small in
theveryearlyUniverse. Hene,aperturbativeanalysis
willbewelljustied. Whatfollowsisaverybrief
sum-maryoftheuniedanalysisofthequantumgeneration
and evolution of perturbations in an inationary
Uni-verse(foradetailedreviewsee[33℄). Thebasipointis
that atthelinearizedlevel,thesystemof gravitational
andmatterperturbationsanbequantizedina
onsis-tent way. Theuse of gauge invariant variables makes
theanalysisbothphysiallylearandomputationally
simple. Dueto theEinstein onstraintequationwhih
ouples metri and matter utuations, there is only
onesalareld degreeof freedomto bequantized(see
[51℄ and[52℄fortheoriginalanalysis).
Therststepofthisanalysisistoexpandthe
grav-itational and matter ationsto quadrati order in the
utuationvariablesaboutalassialhomogeneousand
isotropibakgroundosmology. Fousingonthesalar
metrisetor,itturns outthatoneanexpressthe
re-sultingationforthequantumutuationsintermsof
a single gauge invariant variable whih is a
ombina-tion ofmetri andmatter perturbations, and that the
resultingationreduestotheationofasinglegauge
[52,51℄(the timedependenereetstheexpansionof
thebakgroundspae-time)Weanthususestandard
methods toquantize this theory. If weemploy
anon-ialquantization, thenthe modefuntions of theeld
operatorobeythesameequationsaswederivedinthe
gauge-invariantanalysis oflassialrelativisti
pertur-bations.
Thetimedependeneofthemassleadstoequations
whih havegrowingmodeswhih orrespondto
parti-le prodution or equivalently to the generation and
ampliationof utuations. Sine ination
exponen-tially dilutes the density of pre-existing matter, it is
reasonableto assume that the perturbations start o
(e.g. atthebeginningofination)in thevauumstate
(dened asastatewith nopartiles with respetto a
loal omoving observer). The state dened this way
will not be the vauum state from the point of view
of anobserverat alater time. TheBogoliubovmode
mixingtehniqueanbe usedto alulatethenumber
density of partiles at alater time. In partiular,
ex-petation values of eld operators suh as the power
spetrumanbeomputed.
Ifthebakgroundsalareldisrollingslowly,then
theresultingmassutuationsaregivenby
ÆM
M (k;t
f (k))
3H 2
j'_
0 (t
i (k))j
_ ' 2
0 (t
i (k))
= 3H
2
j'_
0 (t
i (k))j
(30)
Thisresultannowbeevaluatedforspei modelsof
inationto nd theonditionson thepartile physis
parameterswhih giveavalue
ÆM
M (k;t
f
(k))10 5
(31)
whihisrequiredifquantumutuationsfromination
aretoprovidetheseedsforgalaxyformationandagree
withtheCMBanisotropydata.
Forhaotiinationwithapotential
V(')= 1
2 m
2
' 2
; (32)
we an solve the slow rolling equations for the
ina-tonandobtaintherequirementm10 13
GeVtoagree
with(31). Similarly, fora quarti potential with
ou-plingonstant,theondition10 12
isrequiredin
ordernotto onit with observations. Thus, in both
examplesoneneedsaverysmallparameterinthe
par-tilephysismodel. It hasbeenshownquitegenerally
[53℄thatsmallparametersarerequiredifinationisto
solvetheutuationproblem.
Tosummarize, the main results of the analysis of
density utuationsin inationary osmology are: (1)
QuantumvauumutuationsinthedeSitterphaseof
aninationaryUniversearethesoureofperturbations.
(2)As aonsequeneofthehangein thebakground
ra-Infat,unlessthepartilephysismodelontainsvery
smallouplingonstants,thepreditedutuationsare
in exess of those allowed by the bounds on osmi
mirowaveanisotropies. (3) The quantum generation
and lassial evolution of utuations an be treated
in aunied manner. The formalism is no more
om-pliated that thestudy of afree salareld in atime
dependentbakground. (4)InationaryUniverse
mod-elsgeneriallyprodueanapproximatelysaleinvariant
Harrison-Zel'dovihspetrum(16).
IV Parametri Resonane and
Reheating
Reheatingisanimportantstageininationary
osmol-ogy. It determines the stateof the Universe after
in-ation and has onsequenes for baryogenesis, defet
formationandotheraspetsofosmology.
Afterslowrolling,theinatoneld beginsto
osil-late uniformly in spae aboutthe true vauum state.
Quantum mehanially,thisorrespondstoaoherent
stateofk=0inatonpartiles. Duetointerationsof
theinatonwithitselfandwithotherelds,the
oher-entstatewilldeayintoquantaofelementarypartiles.
This orrespondsto post-inationary partile
produ-tion.
Reheatingisusuallystudiedusingsimplesalareld
toymodels. Theonewewilladopthereonsistsoftwo
realsalarelds,theinaton'interatingwitha
mass-less salar eld representing ordinary matter. The
Lagrangianis
L = 1
2
'
' 1
2 m
2
' 2
+ 1
2
1
2 g
2
' 2
2
; (33)
withm10 13
GeV(seeSetionIIIforajustiationof
this hoie), and g 2
denotingthe interation oupling
onstant. Thebaremassandself interationsof are
negleted.
Intheelementary theory of reheating (seee.g. [54℄
and[55℄),thedeayoftheinatonwasalulatedusing
rstorderperturbationtheory. Thedeayrate
B of'
typially turns outto bemuh smallerthan the
Hub-bleexpansionrateat theend ofination(see[7℄ fora
workedexample). Thedeayleadstoadereaseinthe
amplitudeof 'whih anbe approximatedbyadding
an extra dampingterm to the equation of motion for
':
'+3H'_+
B _ ' = V
0
('): (34)
From the above equation it follows that as long as
H >
B
, partileprodution isnegligible. Duringthe
phase of oherent osillation of ', the energy density
andheneHaredereasing. Thus,eventuallyH =
B ,
and at that pointreheating ours (the remaining
en-ergydensityin'isveryquiklytransferredto
parti-les). However,whenthisours,thematter
tempera-tureis muh smallerthantheenergysale ofination
(reheatingisaslowproess). ThiswouldimplynoGUT
baryogenesisand noGUT-saledefetprodution. As
we shall see, these onlusions hange radially if we
adoptanimprovedanalysis ofreheating.
As was rst realized in [56℄, the above analysis
misses anessentialpoint. Tosee this,wefouson the
equation of motion for the matter eld . The
equa-tions for the Fourier modes
k
in the presene of a
oherentinatoneld osillatingwithamplitudeA,
'(t) = Aos(mt); (35)
is
k +3H_
k +(k
2
p +m
2
+
1
2 g
2
A 2
os(2mt))
k
= 0; (36)
where k
p
= k=a is the time-dependent physial
wavenumber, and m 2
=
1
2 A
2
(for other toy models a
similar equation is obtained, but with a dierent
re-lationship betweenthe massand theoeÆient of the
osillatingterm).
Letusforthemomentneglettheexpansionofthe
Universe. In this ase,the fritionterm in (36)drops
out,k
p
istime-independent,andEquation(36)beomes
aharmoniosillatorequationwithaperiodially
vary-ingmass. Inthe mathematisliterature,thisequation
is alled the Mathieu equation. It is wellknown that
there is an instability. In physis, the eet is known
asparametri resonane(seee.g. [57℄). At
frequen-ies !
n
orresponding to half integermultiples of the
frequeny! ofthevariationofthemass,i.e.
! 2
k =k
2
p +m
2
= (
n
2 !)
2
n=1;2;:::; (37)
there are instabilitybands with widths!
n
. For
val-ues of !
k
within the instabilityband, the value of
k
inreasesexponentially:
k e
t
with g
2
A 2
!
: (38)
Inmodelsof haoti inationA m
pl
. Hene,unless
g 2
isunnaturallysmall(atypialvalueisg 2
m=m
pl ),
itfollowsthatH. Theonstantisalledthe
Flo-quetexponent.
Sine the widths of the instability bands derease
as a power of the (small) oupling onstant g 2
with
inreasingn, forpratialpurposesonlythelowest
in-stabilityband isimportant. Itswidth is
!
k
gA: (39)
Note, in partiular, that there is no ultraviolet
diver-genein omputing thetotalenergy transferfrom the
'totheeld duetoparametriresonane[56℄.
Itis easyto inlude the eetsof theexpansionof
theUniverse (seee.g. [56,58, 59℄). Themaineet is
modeslowlyentersandleavestheresonanebands. As
aonsequene,anymodeliesintheresonanebandfor
onlyanitetime.
The rate of energy transfer is given by the phase
spae volume of thelowestinstability band multiplied
bytherateofgrowthofthemodefuntion
k
. Usingas
an initial onditionfor
k
the value
k
H given by
the magnitude of the expeted quantum utuations,
weobtain
_
(
!
2 )
2
!
k He
t
: (40)
Hene,theenergytransferwillproeedfastonthetime
sale of the expansion of the Universe. There will be
explosive partile prodution, and the energy density
inmatterattheendofreheatingwillbeapproximately
equaltotheenergydensityattheendofination.
Theaboveisasummaryofthemainphysisofthe
modern theory of reheating. The atual analysis an
berened in manyways(see e.g. [58,59, 60℄, and, in
the toymodel onsidered here, [61℄). First ofall, itis
easytotaketheexpansionoftheUniverseintoaount
expliitly (bymeans ofatransformation of variables),
to employ an exatsolution of the bakgroundmodel
andtoreduethemodeequationfor
k
toanequation
whihalsoadmitsexponentialinstabilities.
The next improvement onsists of treating the
eld quantum mehanially (keeping ' as a lassial
bakground eld). At this point, the tehniques of
quantum eld theory in a urved bakground an be
applied. There is no need to impose artiial
lassi-al initial onditions for
k
. Instead, wemay assume
that starts in its initial vauum state. The
Bogoli-ubovmodemixing tehniquean be used to ompute
thenumberof partilesatlatetimes.
Notethatthestateofafterparametriresonane
is notathermalstate. Thespetrumonsists ofhigh
peaks in distint wavebands. An important question
ishowthisstatethermalizes. Forsomereentprogress
onthisissuesee[62,63℄. Sinethestateafterexplosive
partile prodution is not athermal state, it is useful
tofollow[58℄andallthisproess\preheating"instead
of reheating.
Note that the details of theanalysis of preheating
are quite model-dependent. In fat [58, 60℄, in most
modelsonedoesnotgetthekindof\narrow-band"
res-onanedisussedhere,but\broad-band"resonane. In
this ase,theenergytransferisevenmoreeÆient.
Reently [64℄ it has been argued that parametri
resonanemayleadto resonantampliationof
super-Hubble-sale osmologialperturbations. Thepointis
that inthepreseneof anosillatinginatoneld, the
equation of motion (25) for the osmologial
pertur-bations ontainsaontribution to themass term(the
oeÆient of )whih is periodiallyvarying in time.
Hene, the equation takes on a similar form to the
Mathieu equationdisussed above(36). Insome
mod-with wavelength larger than the Hubble radius,
lead-ingtotheapparentampliationofsuper-Hubble-sale
modes. Suh aproess doesnot violateausality[48℄
sine it is driven by the inaton eld whih is
oher-entonsuper-Hubble salesat theend ofination asa
onsequeneof theausal dynamis of an inationary
Universe.
Theanalysis ofEquation(25)during theperiod of
reheatingis, however, ompliated by asingularityin
theoeÆientsofboth _
andattheturningpointsof
thesalarmattereld'. Thissingularitypersistswhen
usingthe`onservationlaw'form (26)oftheequation:
when '_ = 0, one annot immediately draw the
on-lusion that _ = 0. Note, also, that a large inrease
in thevalue of during reheatingis predited by the
usual theory of osmologialutuationswhih treats
thereheating period asa smoothhange in the
equa-tionofstatefrom that of nearlydeSitterto that ofa
radiation-dominatedUniverse(see(28)). Careful
anal-ysesforsimplesingle-eld[48,65℄modelsdemonstrated
thatthere isindeedno netgrowthofthephysial
am-plitude of gravitational utuations beyond what the
usualtheoryofosmologialperturbationspredits(see
also [66, 67℄ for earlierresults supporting this
onlu-sion). Thereisinreasingevidenethatthisonlusion
holdsin general formodels withpurely adiabati
per-turbations[68,69℄.
Intheaseofmultiplemattereldmodelsthereare
extratermsontherighthand side oftheequations of
motion(25)and(26)whihare notexponentially
sup-pressedonlengthsaleslargerthantheHubbleradius.
Thesetermsarerelatedtotheexisteneofisourvature
utuations.Asrstdemonstratedin[73℄,insuh
mod-els exponential inrease in the amplitude of during
reheatingisindeedpossible. However,inmanymodels
theperturbationmodeswhih anundergoparametri
ampliation during reheating are exponentially
sup-pressedduringination[70,71,72℄,andtheythushave
anegligibleeet onthenalamplitudeof. The
ri-terion for models (suh as the one proposed in [73℄)
to have exponential growthof the physial amplitude
of osmologial perturbations during ination is that
thereisanisourvature/entropymodewhihisnot
sup-pressedduringination[74℄. Theresultingexponential
ampliation of utuations renders these models
in-ompatiblewiththeobservationalonstraints,even
in-ludingbak-reationeets[75℄.
Sinetheexponentialpartileprodutionrate
dur-ingreheatingreliesonaparametriresonane
instabil-ity,it isreasonableto beonernedwhethertheeet
willsurviveinthepreseneofnoise. There arevarious
souresof noiseto be onerned about. Firstly, there
arethequantumorthermalutuationsintheinaton
itself,the utuationswhihin inationary osmology
arethesoureoftheobservedstrutureintheUniverse.
onsideredtheeetsofnoiseintheinatoneldonthe
dynamisof.
Weassumethatthedynamisoftheinatonis
de-sribed by periodi motion with superimposed noise
givenbyafuntion q(t;x),i.e.
'(t;x) = Aos(!t)+q(t;x): (41)
Forsimpliity, we havenegletedtheexpansionofthe
Universe. In the ase of spatially homogeneous noise
onsidered in [76℄, the dynamis of still deomposes
into independentFouriermodes whih obeythe
equa-tion
k +k
2
k +
m 2
+p(!t)+q(t)
k
=0; (42)
wherep(y)isafuntionwithperiod2. Thisequation
anbewrittenintheusualwayasahomogeneousrst
order22matrixdierentialequationwithaoeÆient
matrixontainingnoise.
Forderivingtheresultsmentionedbelow,itis
suÆ-ienttomakeertainstatistialassumptionsaboutq(t).
Weassume that thenoiseis drawn from somesample
spae(whihforhomogeneousnoiseanbetakento
be =C(<)), and that the noiseis ergodi, i.e. the
time average of the noise is equal to the expetation
valueof thenoiseoverthesamplespae. Inthisase,
itanbeshownthat thegeneralizedFloquetexponent
ofthesolutionsfor
k
is welldened. Toobtainmore
quantitative information about (q) it is neessaryto
makefurtherassumptionsaboutthenoise. Weassume
rstly that the noise q(t) is unorrelated in time on
sales larger than the time period T of the periodi
motiongivenbyp,andisidentiallydistributed forall
realizationsof thenoise. Seondly, weassumethat
re-stritingthenoiseq(t;)tothetimeinterval0t<T,
thenoisesampleswithinthesupportoftheprobability
measurellaneighborhood,in C(0;T),oftheorigin.
The main result whih was proved in [76℄ is that
the Floquet exponent (q) in the presene of noise is
stritlylargerthantheexponent(0)intheabseneof
noise
(q) > (0); (43)
whihdemonstratesthatthepreseneofnoiseleadsto
astritinreaseintherateofpartileprodution. The
proofwasbasedonanappliationofFurstenberg's
the-orem,atheoremonerningtheLyapunovexponentof
produtsofindependentidentiallydistributedrandom
matriesf
j
:j=1;:::;Ng.
Forinationaryreheating, theaboveresultimplies
thatnoiseintheinatoneliminatedthestabilitybands
of the system, and that all modes
k
grow
exponen-tially. Theresultwasextendedtoinhomogeneousnoise
in[77℄. Theanalysisismathematiallymuhmore
om-pliatedsinetheFouriermodesnolongerdeoupleand
theproblemisaprobleminthetheoryofpartial(rather
similar result to (43) an be derived, with the strit
inequalityreplaedby.
If we replae time derivatives by spatial
deriva-tives (denoted by a prime) in (42), we obtain the
time-independent Shrodinger equation for a
one-dimensional non-relativisti eletrongas in a periodi
potentialsubjetto aperiodinoise
~
E (x) = 00
+[V
p (x)+V
q
(x)℄ ; (44)
where V
p
is periodi, V
q
represents the random noise
ontributiontothepotential,and ~
Eisaonstant. Our
resultsimplythatin thepreseneofrandomnoise,the
periodi solutionsoftheequationwithperiodi
poten-tial (the Bloh waves) beome loalized, and that the
loalizationlengthdereasesasthenoiseamplitude
in-reases[78℄. Thus, weobtainanewproofofAnderson
loalization [79℄.
V Problemsof Inationary
Cos-mology
V.A FlutuationProblem
A generiproblem for allrealizations of
potential-driven ination studied up to now onerns the
am-plitudeofthedensityperturbationswhihareindued
byquantumutuationsduringtheperiodof
exponen-tial expansion [27, 28℄. From the amplitude of CMB
anisotropiesmeasuredbyCOBE,andfromthepresent
amplitudeofdensityinhomogeneitiesonsalesof
lus-ters of galaxies, it follows that the amplitude of the
massutuationsÆM=Monalengthsalegivenbythe
omoving wavenumberk at thetime t
f
(k) when that
salerossestheHubbleradiusintheFRWperiodisof
theorder10 5
.
However,as wasdisussed in detailin Setion III,
the present realizations of ination based on salar
quantum eld matter generially [53℄ predit a muh
largervalueoftheseutuations,unlessaparameterin
the salar eld potential takes on a very small value.
For example, as disussed at the end of Setion III,
in asingle eld haoti inationary model with
quar-tipotentialthemassutuationsgeneratedareofthe
order 10 2
1=2
. Thus, in order notto onit with
ob-servations,avalueofsmallerthan10 12
isrequired.
There have beenmany attempts to justify suh small
parameters based on spei partile physis models,
butnosingleonviningmodelhasemerged.
With the reentdisovery[64, 48℄ that long
wave-length gravitationalutuations may beamplied
ex-ponentiallyduring reheating,anewaspet ofthe
u-tuationproblemhasemerged. Allmodelsinwhihsuh
ampliationours(seee.g.[74℄foradisussionofthe
large,independentofthevalueoftheouplingonstant
[75℄.
V.B Trans-Plankian Problem
Inmanymodelsofination,inpartiularinhaoti
ination, theperiodofinationis solongthat
omov-ing sales of osmologialinterest todayorresponded
toaphysialwavelengthmuhsmallerthanthePlank
length at the beginning of ination. In extrapolating
the evolution of osmologial perturbations aording
tolineartheorytotheseveryearlytimes,weare
impli-itly making the assumptions that the theory remains
perturbative to arbitrarily high energies and that it
anbedesribedbylassialgeneralrelativity. Bothof
these assumptionsbreak down onsuper-Planksales.
Thus thequestionarisesasto whetherthepreditions
of the theory are robust against modiations of the
modelonsuper-Planksales.
Asimilarproblemours inblakholephysis[80℄.
The mixing between the modes falling towards the
blak hole from past innity and the Hawking
radia-tion modesemanating to future innity takesplae in
the extreme ultraviolet region, and ould be sensitive
to super-Plankphysis. However,in theaseofblak
holesithasbeenshownthatforsub-Plankwavelengths
atfutureinnity,thepreditionsdonothangeundera
lassofdrastimodiationsofthephysisdesribedby
hangesinthedispersionrelationofafreeeld[81,82℄.
Aswasreently[83,84℄disovered,theresultinthe
aseofinationaryosmologyisdierent: thespetrum
ofutuationsmaydependquitesensitivelyonthe
dis-persionrelationonsuper-Planksales. Ifwetakethe
initial state ofthe utuationsat thebeginningof
in-ation to be given by the state whih minimizes the
energy density, then for ertainof thedispersion
rela-tions onsidered in [82℄, the spetrum of utuations
hangesquiteradially. Theindexofthespetruman
hange(i.e. thespetrumisno longersale-invariant)
andosillationsin thespetrummaybeindued. Note
that forthe lass of dispersionrelations onsidered in
[81℄ thepreditionsarethestandardones.
Theaboveresultsmaybebadnewsforpeoplewho
would like to onsider salar-eld driven inationary
models asthe ultimate theory. However, the positive
interpretationoftheresultsisthatthespetrumof
u-tuations may provide awindowon super-Plank-sale
physis. The present observations an already be
in-terpreted in thesense [85℄ that thedispersionrelation
of theeetiveeld theorywhih emergesfrom string
theoryannotdiertoodrastiallyfromthedispersion
relationofafreesalareld.
V.C Singularity Problem
Salareld-drivenination doesnot eliminate
sin-gularities from osmology. Although the standard
downif matterhas anequationof statewith negative
pressure,asistheaseduringination, neverthelessit
anbeshownthat an initialsingularitypersists in
in-ationaryosmology[86℄. Thisimpliesthatthetheory
is inomplete. In partiular, the physial initial value
problemisnotdened.
V.DCosmologialConstant Problem
Sinetheosmologialonstantatsasaneetive
energydensity,itsvalueisboundedfrom abovebythe
presentenergydensityoftheUniverse. InPlankunits,
the onstraint on the eetive osmologial onstant
eff
is(seee.g. [87℄)
eff
m 4
pl 10
122
: (45)
This onstraintapplies both to the bare osmologial
onstantandtoanymatterontributionwhih atsas
aneetiveosmologialonstant.
Thetruevauumvalue(takenon,tobespei,at
'= 0) of the potential V(') ats asan eetive
os-mologialonstant. Itsvalueisnotonstrainedbyany
partilephysisrequirements(intheabseneofspeial
symmetries). Thustheremustbesomeasyetunknown
mehanismwhihanels(oratleastalmostompletely
anels)thegravitationaleets ofanyvauum
poten-tial energy of any salar eld. However, salar
eld-driveninationrelies onthealmostonstantpotential
energyV(')duringtheslow-rollingperiodating
grav-itationally. How an one be sure that the unknown
mehanism whih anels the gravitational eets of
V(0) does not also anel the gravitational eets of
V(') during the slow-rolling phase? This problem is
theAhilles heelof anysalareld-driven inationary
model.
Supersymmetry alone annot provide a resolution
of this problem. It is true that unbroken
supersym-metryforesV(')=0in thesupersymmetrivauum.
However, supersymmetry breaking will indue a
non-vanishingV(')in thetruevauum after
supersymme-trybreaking,andaosmologialonstantproblemofat
least60ordersofmagnituderemainsevenwiththe
low-est sale of supersymmetry breaking ompatible with
experiments.
VI New Avenues
In the light of the problems of potential-driven
ina-tiondisussed in theprevioussetions, it isof utmost
importane to study realizations of inationwhih do
notrequirefundamentalsalarelds,orompletelynew
avenuestowardsearlyUniverseosmologywhih,while
maintaining(someof)thesuessesofination,address
andresolvesomeofitsdiÆulties.
os-thepre-big-bangsenario[24℄,andthevaryingspeedof
lightpostulate[88,89℄. Thepre-big-bangsenariois a
modelinwhihtheUniversestartsinanemptyandat
dilaton-dominatedphasewhihleadstopole-law
ina-tion. A nie feature of this theory is that the
meha-nism of super-inationary expansionis ompletely
in-dependent of a potential and thus independentof the
osmologialonstantissue. The senario,however,is
onfronted with agraeful exit problem [90℄, and the
initialonditionsneedtobeveryspeial[91℄(see,
how-ever,thedisussionin [92℄).
It isalsoeasy torealizethat atheory inwhihthe
speedoflightismuhlargerintheearlyUniversethan
atthepresenttimeanleadtoasolutionofthehorizon
andatness problemsof standardosmologyandthus
an provide an alternative to ination for addressing
them. Forrealizationsofthissenariointheontextof
thebraneworldideasseee.g. [93,94,95℄.
String theory may lead to a natural resolution of
someof thepuzzles of inationaryosmology. This is
an area of ative researh. The reader is referred to
[20℄forareviewofreentstudiesofobtainingination
with moduli elds, and to [22℄ for attempts to obtain
ination withbranes. Below, three moreonventional
approahestoaddressingsomeoftheproblemsof
ina-tionwillbesummarized.
VI.A Inationfrom Condensates
Atthepresenttimethereisnodiretobservational
evidenefortheexisteneoffundamentalsalareldsin
nature(inspiteofthefatthatmostattrativeunied
theoriesofnaturerequiretheexisteneofsalareldsin
thelowenergyeetiveLagrangian). Salareldswere
initiallyintroduedtopartilephysistoyieldanorder
parameterforthesymmetrybreakingphasetransition.
Manyphasetransitionsexistin nature;however,inall
ases,theorderparameterisaondensate. Hene,itis
usefulto onsider thepossibility ofobtainingination
usingondensates,andinpartiulartoaskifthiswould
yieldadierentinationarysenario.
Theanalysisofatheorywithondensatesis
intrin-siallynon-perturbative. Theexpetation valueofthe
HamiltonianhHiofthetheoryontainstermswith
ar-bitrarily high powers of the expetation value h'i of
the ondensate. A reent study of the possibility of
obtaining ination in a theory with ondensates was
undertakenin[96℄(seealso[97℄forsomeearlierwork).
InsteadoftrunatingtheexpansionofhHiatsome
ar-bitraryorder,theassumptionwasmadethatthe
expan-sionofhHiinpowersofh'iisasymptotiand,
spei-ally,Borelsummable(ongeneralgroundsoneexpets
thattheexpansionwill beasymptoti-seee.g. [98℄)
hHi = 1
X
n=0 n!( 1)
n
a
n h'
n
i (46)
= Z
1
ds
f(s)
s(sm
pl +h'i)
e 1=s
: (47)
Therst linerepresentsthe originalseries, theseond
line the resummed series. The funtion f(s) is
deter-minedbytheoeÆientsa
n .
Theosmologialsenariois asfollows: the
expe-tation value h'i vanishes at times before the phase
transitionwhentheondensateforms. Afterwards,h'i
evolvesaording to the lassialequations of motion
withthepotentialgivenby(46)(assumingthatthe
ki-netitermassumesitsstandardform). Itaneasilybe
heked that the slow rolling onditions are satised.
However, the slow roll onditions remain satisedfor
all valuesof h'i, thus leading to agraeful exit
prob-lem-inationwill neverterminate.
However, orrelation funtions, in partiular h 2
i,
arein generalinfrareddivergentinthedeSitterphase
of an expanding Universe. Onemust therefore
intro-dueaphenomenologialutoparameter(t)intothe
vauum expetation value (VEV), and replae h'i by
h'i=. It is natural to take (t) H(t) (see e.g.
[99,100℄). Hene,thedynamialsystemonsistsoftwo
oupledfuntionsoftimeh'iand. Aarefulanalysis
showsthatagraefulexitfrominationourspreisely
ifhHitendstozerowhenh'itendsto largevalues.
Asisevident,thesenarioforinationinthis
om-posite eld model is very dierent from the standard
potential-driven inationary senario. It is
partiu-larly interesting that the graeful exit problem from
ination is linked to the osmologial onstant
prob-lem. Note that models of ination based on
ompos-itespresumablydonotsuerfromthetrans-Plankian
problem,thereasonbeingthattheeetiveeldtheory
whihdesribesthestronglyinteratingsystemis
time-translation-invariant during the de Sitter phase. The
physialpitureisthatmodeinterationsareessential,
andareresponsibleforgeneratingtheutuationsona
salekwhenthissaleleavestheHubbleradiusattime
t
i (k).
VI.BNonsingular Universe Constrution
Anotherpossibilityofobtaininginationisby
mak-ing useofamodiedgravitysetor. Morespeially,
we an add to the usual gravitational ation higher
derivative urvature terms. These terms beome
im-portant only at high urvatures. As realized a long
time ago [23℄, spei hoies of the higher
deriva-tive terms an lead to ination. It is well motivated
to onsider eetive gravitational ations with higher
derivativetermswhenstudyingthepropertiesof
spae-time at largeurvatures, sineanyeetive ationfor
lassialgravityobtainedfromstringtheory,quantum
gravity, or by integrating out matter elds, ontains
suhterms. Inourontext,theinterestingquestionis
whether oneanobtainaversionofination avoiding
someoftheproblemsdisussedintheprevioussetion,
speiallywhetheroneanobtainnonsingular
Most higher derivativegravitytheories have muh
worse singularity problems than Einstein's theory.
However, it is not unreasonable to expet that in the
fundamental theory of nature, be it string theory or
some other theory, the urvatureof spae-time is
lim-ited. In Refs. [101, 102℄ the hypothesis was made
that when the limiting urvature is reahed, the
ge-ometry must approah that of a maximally
symmet-ri spae-time, namely de Sitter spae. The question
now beomes whether it is possible to nd a lass of
higherderivativeeetiveationsforgravitywhihhave
thepropertythatat largeurvaturesthesolutions
ap-proah de Sitter spae. A nonsingular Universe
on-strutionwhihahievesthisgoalwasproposedinRefs.
[103,104℄. Itisbasedonaddingto theEinstein ation
apartiularombinationofquadratiinvariantsofthe
Riemanntensorhosensuhthattheinvariantvanishes
onlyindeSitterspae-times. Thisinvariantisoupled
totheEinsteinationviaaLagrangemultipliereldin
awaythattheLagrangemultiplieronstraintequation
fores the invariant to zero at high urvatures. Thus,
the metri beomes de Sitter and hene geodesially
omplete andexpliitlynonsingular.
Ifsuessful,theaboveonstrutionwillhavesome
veryappealingonsequenes. Consider,forexample,a
ollapsing spatially homogeneousUniverse. Aording
to Einstein'stheory,this Universewill ollapseina
-nitepropertimeto anal\bigrunh"singularity. In
the newtheory, however,the Universewillapproaha
de Sittermodelas theurvatureinreases. Ifthe
Uni-verseislosed,therewillbeadeSitterbounefollowed
by re-expansion. Similarly, spheriallysymmetri
va-uumsolutionsofthenewequationsofmotionwill
pre-sumablybenonsingular,i.e.,blakholeswouldhaveno
singularities in their enters. In two dimensions, this
onstrutionhasbeensuessfullyrealized[105℄.
ThenonsingularUniverseonstrutionof[103,104℄
anditsappliationstodilatonosmology[106,107℄are
reviewedinareentproeedingsartile[108℄. What
fol-lowsisjust averybriefsummaryofthepointsrelevant
to theproblemslistedin SetionV.
TheproedureforobtaininganonsingularUniverse
theory[103℄isbasedonaLagrangemultiplier
onstru-tion. Startingfrom theEinstein ation,onean
intro-due Lagrangemultiplierselds '
i
oupledto seleted
urvatureinvariantsI
i
,andwithpotentialsV
i ('
i )
ho-sen suh that at low urvature the theory redues to
Einstein'stheory,whereasathigh urvaturesthe
solu-tionsaremanifestlynonsingular. Theminimal
require-ments for anonsingular theory are that all urvature
invariants remain bounded and the spae-time
mani-fold isgeodesiallyomplete.
It is possible to ahieve this by a two-step
proe-dure. First, we hoose a urvature invariant I
1 (g ) (e.g. I 1
=R )anddemandthatitbeexpliitlybounded
by the '
1
onstraint equation. In a seond step, we
demand that as I (g )approahesits limiting value,
the metri g
approah the de Sitter metri g DS
, a
denite nonsingular metri with maximal symmetry.
Inthisase, allurvatureinvariantsare automatially
bounded(theyapproahtheirdeSittervalues),andthe
spae-timeanbeextendedtobegeodesiallyomplete.
Theseondstepanbeimplementedby[103℄hoosing
aurvatureinvariantI
2 (g
)withthepropertythat
I
2 (g
)=0 , g
=g
DS
; (48)
introduingaseondLagrangemultipliereld'
2 whih
ouples to I
2
, and hoosing a potential V
2 (' 2 ) whih foresI 2
tozeroatlargej'
2 j: S= Z d 4 x p
g[R+'
1 I 1 +V 1 (' 1 )+' 2 I 2 +V 2 (' 2 )℄; (49)
withasymptotionditions
V 1 (' 1 ) ' 1 asj' 1
j!1 (50)
V
2 ('
2
) onst asj'
2
j!1 (51)
V i (' i ) ' 2 i asj' i
j!0 i=1;2: (52)
TherstonstraintrendersR nite, theseond fores
I
2
to zero,andthethird isrequiredinorder toobtain
theorretlowurvaturelimit.
Theinvariant I 2 =(4R R R 2 +C 2 ) 1=2 ; (53)
singles out the de Sitter metri among all
homoge-neousandisotropi metris(in whihase addingC 2
,
the Weyl tensor square, is superuous), all
homoge-neousandanisotropimetris,andallradially
symmet-rimetris.
As a spei example onean onsider the ation
[103,104℄ S = Z d 4 x p g (54)
R+'
1 R (' 2 + 3 p 2 ' 1 )I 1=2 2 +V 1 (' 1 )+V
2 (' 2 ) with V 1 (' 1
) = 12H 2
0 '
2
1
1+'
1
1
ln(1+'
1 )
1+'
1 (55) V 2 (' 2
) = 2
p 3H 2 0 ' 2 2
1+' 2
2
: (56)
It anbe shown that allsolutions of the equations of
motion whih follow from this ation are nonsingular
[103, 104℄. Theyare either periodi aboutMinkowski
spae-time('
1 ;'
2
)=(0;0) orelse asymptotially
ap-proahdeSitterspae(j'
2
j!1).
Oneofthemostinterestingpropertiesofthistheory
isasymptotifreedom[104℄,i.e., theouplingbetween
matterandgravitygoesto zeroat high urvatures. It
eld) to thegravitationalation in the standardway.
Onendsthat intheasymptotideSitterregions,the
trajetoriesofthesolutionsprojetedontothe('
1 ;'
2 )
plane are unhanged by adding matter. This applies,
for example, in a phaseof de Sitter ontrationwhen
the matter energy density is inreasing exponentially
but does not aet the metri. The physial reason
for asymptoti freedom is obvious: in the asymptoti
regions of phase spae, the spae-time urvature
ap-proahesitsmaximalvalueandthusannotbehanged
evenbyadding anarbitrarilyhighmatter energy
den-sity. Hene,thereisthepossibilitythatthistheorywill
admitanaturalsuppressionmehanismfor
osmologi-alutuations. Ifthiswerethease,thenthesolution
ofthesingularityproblemwouldatthesametimehelp
resolvethe utuationproblem of potential-driven
in-ationaryosmology.
VI.C Bak-Reation of Cosmologial
Perturba-tions
Thelinear theory of osmologialperturbations in
inationaryosmologyiswellstudied. However,eets
beyondlinearorderhavereeivedverylittleattention.
Beyondlinearorder,perturbationsaneetthe
bak-groundin whih theypropagate, an eet well known
fromearlystudies[109℄ofgravitationalwaves. Aswill
besummarizedbelow,thebak-reationofosmologial
perturbations in an exponentiallyexpanding Universe
atslikeanegativeosmologialonstant,asrst
real-izedin theontext ofstudiesofgravitationalwavesin
deSitterspaein [110℄.
Gravitationalbak-reationofosmologial
pertur-bationsonernsitselfwiththeevolutionofspae-times
whih onsist of small utuations about a
symmet-riFriedmann-Robertson-Walkerspae-timewith
met-ri g (0)
. The goalis to study the evolution of spatial
averagesofobservablesintheperturbedspae-time. In
linear theory, suh averaged quantities evolve likethe
orrespondingvariablesin thebakgroundspae-time.
However, beyond linear theory perturbations havean
eet ontheaveragedquantities. Intheaseof
gravi-tationalwaves,thiseet iswell-known[109℄:
gravita-tionalwavesarryenergyandmomentumwhihaets
thebakgroundinwhihtheypropagate. Here,weshall
fousonsalarmetriperturbations.
Theideabehindtheanalysisofgravitational
bak-reation [111℄ is to expand the Einstein equations to
seond order in theperturbations, to assumethat the
rst order terms satisfy the equations of motion for
linearized osmologialperturbations[33℄(hene these
terms anel), to take the spatial average of the
re-mainingterms,andtoregardtheresultingequationsas
equations foranew homogeneousmetrig (0;br)
whih
inludestheeetoftheperturbationstoquadrati
or-der:
G
(g
(0;br)
) = 8G h
T (0)
+
i
(57)
where the eetive energy-momentum tensor
of
gravitational bak-reation ontains the terms
result-ing from spatial averaging of the seond order metri
andmatterperturbations:
=<T
(2)
1
8G G
(2)
>; (58)
wherepointedbraketsstandforspatialaveraging,and
thesupersriptsindiatetheorderin perturbations.
As formulated in (57) and (58), the bak-reation
problem is not independent of theoordinatizationof
spae-timeandheneisnotwelldened. Itispossible
totakeahomogeneousandisotropispae-time,hoose
dierent oordinates, and obtaina non-vanishing
.
This\gauge"problemisrelatedtothefatthatin the
abovepresription,thehypersurfaeoverwhihthe
av-erageistakendependsonthehoieofoordinates.
The key to resolving the gauge problem is to
re-alize that to seond order in perturbations, the
bak-ground variables hange. A gauge independent form
of the bak-reation equation (57) an hene be
de-rived [111℄ by dening bakground and perturbation
variables Q = Q (0)
+ÆQ whih do not hange
un-der linearoordinate transformations. Here,Q
repre-sentsolletivelybothmetriandmattervariables. The
gauge-invariantformofthebak-reationequationthen
looks formally idential to (57), exept that all
vari-ablesarereplaedbytheorrespondinggauge-invariant
ones. We will follow the notation of [33℄, and use as
gauge-invariantperturbationvariablestheBardeen
po-tentials [35℄ and whih in longitudinal gauge
o-inide withtheatualmetriperturbationsÆg
.
Cal-ulations hene simplify greatlyifwework diretlyin
longitudinal gauge. Thesealulationshavebeen
on-rmed[112℄byworkinginaompletelydierentgauge,
makinguseoftheovariantapproah.
In[113℄,theeetiveenergy-momentumtensor
of gravitational bak-reation was evaluated for long
wavelength utuations in aninationary Universe in
whih the matter responsible for ination is a salar
eld 'withthepotential
V(') = 1
2 m
2
' 2
: (59)
Sine in this model there is no anisotropi stress, the
perturbedmetriin longitudinal gaugeanbewritten
[33℄in termsof asinglegravitationalpotential
ds 2
=(1+2)dt 2
a(t) 2
(1 2)Æ
ij dx
i
dx j
; (60)
wherea(t)istheosmologialsalefator.
ItisnowstraightforwardtoomputeG (2)
andT
(2)
in terms of the bakground elds and themetri and
matter utuationsand Æ', Bytakingaveragesand
makinguseof(58),theeetiveenergy-momentum
Thegeneralexpressionsfortheeetiveenergy
den-sity (2)
= 0
0
andeetivepressurep (2)
= 1
3
i
i
involve
manyterms. However,theygreatlysimplifyifwe
on-sider perturbations with wavelength greater than the
Hubbleradius. Inthis ase,alltermsinvolvingspatial
gradientsarenegligible. Fromthetheoryoflinear
os-mologialperturbations(seee.g. [33℄)itfollowsthaton
saleslargerthantheHubbleradiusthetimederivative
of is also negligibleas longasthe equation of state
of thebakgrounddoesnothange. TheEinstein
on-straintequationsrelatethetwoperturbation variables
and Æ', enabling salar metri and matter
utua-tionstobedesribedintermsofasinglegauge-invariant
potential . During the slow-rollingperiod of the
in-ationaryUniverse, theonstraintequationtakesona
verysimpleformandimpliesthatandÆ'are
propor-tional. Theupshot of these onsiderationsis that
is proportionaltothetwopointfuntion < 2
>,with
a oeÆient tensor whih depends on the bakground
dynamis. Intheslow-rollingapproximationweobtain
[113℄
(2)
' 4V < 2
> (61)
and
p (2)
=
(2)
: (62)
Thisdemonstratesthattheeetiveenergy-momentum
tensor of long-wavelength osmologial perturbations
hasthesameformasanegativeosmologialonstant.
Note that during ination, the phase spae of
in-fraredmodesisgrowing. Hene,themagnitudeofj (2)
j
isalsoinreasing. Hene,thebak-reationmehanism
may lead to a dynamial relaxation mehanism for a
bare osmologial onstant driving ination [114℄. A
similar eet holds for pure gravityat two looporder
in thepreseneofabareosmologialonstant[110℄.
Theinterpretationof (2)
asaloaldensityhasbeen
ritiized, e.g. in [115℄. Instead of omputing
physi-al observables from a spatially averaged metri, one
should omputethe spatial average of physial
invari-antsorreted to quadratiorder in perturbation
the-ory. Work on this issue is in progress [116℄ (see also
[68℄).
VII Conlusions
Inationary osmology is an attrative senario. It
solvessomeproblems ofstandardosmologyandleads
tothepossibilityofaausaltheoryofstruture
forma-tion. The spei preditionsofaninationary model
ofstrutureformation,however,dependonthespei
realizationofination,whihmakestheideaofination
hardtoverifyorfalsify. Manymodelsofinationhave
beensuggested,but atthepresenttimenoneare
suÆ-ientlydistinguishedtoform a\standard"inationary
theory.
There is now a onsistent quantum theory of the
bationswhihdesribestheoriginofutuationsfrom
aninitialvauumstateoftheutuationmodesatthe
beginningofination,andwhihformsthebasisforthe
preisionalulationsofthepowerspetrumofdensity
utuations and of CMB anisotropieswhih allow
de-tailedomparisons with urrent and upoming
obser-vations.
As explained in Setion IV, a new theory of
in-ationary reheating (preheating) has been developed
based on parametri resonane. Preheating leads to
arapid energy transferbetween the inatoneld and
matterat the end of ination, withimportant
osmo-logialonsequenes. Reentdevelopmentsinthisarea
are the realization that long wavelength gravitational
utuations may beamplied exponentiallyin models
with anentropy perturbation mode whih is not
sup-pressed during ination, and the study of the eets
ofnoise in theinaton eld onthe resonaneproess,
leadingto theresultthat suhnoiseatually enhanes
theresonane(thisresult alsoleadsto anewproof of
Andersonloalization).
Note, however, that there are important
onep-tualproblems for salareld-driven inationary
mod-els. Foursuhproblemsdisussedin SetionVarethe
utuation problem, the trans-Plankian problem, the
singularityproblemandtheosmologial onstant
prob-lem,thelastofwhihistheAhillesheelofthese
ina-tionarymodels.
It maybethat aonviningrealizationof ination
willhavetowaitforanimprovementinour
understand-ingoffundamentalphysis. Somepromisingbut
inom-pleteavenueswhihaddresssomeoftheproblems
men-tionedaboveandwhihyieldinationaredisussedin
SetionVI.
Aknowledgements
I wish to thank Prof. J.A.S. Lima for the
invita-tiontopresentthisleture, andforhis wonderful
hos-pitalityduringmyvisitto Brazil. Ialsowish tothank
Prof. RihardMaKenzieforhospitalityatthe
Univer-sitede MontrealandProfs. JimCline andRobMyers
forhospitalityatMGillUniversityduringthetimethis
manusriptwasompleted. Theauthorissupportedin
partbytheUS Departmentof Energyunder Contrat
DE-FG02-91ER40688,TaskA.
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