Cosmological perturbation theory in a matter-dominated universe: the gradient expansion.

Fa oltà diS ienze Matemati heFisi hee Naturali

CorsodiLaurea Spe ialisti ainS ienze Fisi hee Astrofisi he

A. A.2004-05

Tesi di Laurea Spe ialisti a

Cosmologi al

Perturbation Theory in a

Matter Dominated

Universe: the Gradient

Expansion

Candidato Relatore

Introdu tion 1

1 Des ribingour Universe 3

1.1 Thestandard osmologi almodel . . . 3

1.2 Ination . . . 16

1.3 FoundamentalideasofStru tureFormation . . . 20

2 Dust Cosmology: frame and formalism 29

2.1 Spa e-time splittings,gauge hoi esandgeneralhypotheses . . . 29

2.2 Chara terizationofthematter ontent . . . 32

2.3 Thesyn rhonousand omovingsystemof oordinates . . . 34

2.4 Einstein Equationsin ADMformalism . . . 40

3 Standard Perturbation Theory at Firstand Se ondOrder 45

3.1 IdeasoftheStandardPerturbationTheory . . . 45

3.2 Implementingtheperturbations . . . 47

3.3 Gauge hoi eandgaugedependen e inperturbationtheory . . . 50

3.4 Standardperturbationsat

1

st

and

2

nd

orderofEinstein-deSitter

universeinthesyn hronous- omovinggauge. . . 54

4 Gradient ExpansionTe hnique 59

4.1 Thestartingspatialmetri andba kground omparison . . . 60

4.2 Theexpansions heme . . . 62

4.3 Gradientexpansionte hniqueat

1

st

order . . . 64

4.4 Gradientexpansionte hniqueat

2

nd

order. . . 67

4.5 Che kof onstraints . . . 74

5 ComparingPerturbative Te hniques. Other Results. 77

5.1 Comparisonbetweenstandardtheoryandgradientexpansion . . 77

5.2 Weyltensoranditsmagneti part . . . 80

Con lusions 85

A De ompositionof spatial ve tors and tensors 87

B Syn hronous gauge: geometri alquantities 89

Theideaunderlyingthetheoryof spa etimeperturbationsisthesamethatwe

have in any perturbative formalism: we try to nd approximate solutions of

some eld equations (Einstein Equations), onsidering them as "small"

devi-ations from aknown exa tsolution (the ba kground: usually the

Friedmann-Robertson-Walker(FRW) metri ).

The ompli ationsinGeneralRelativity,asinanyotherspa etimetheory,arise

fromthefa t thatwehavetoperturbenotonlytheeldsin agivengeometry

-eldsdes ribingthematter ontentinliteralsenseors alareldsasthe

ina-tonfortheInationorthequintessen efortheDarkEnergy-,butthegeometry

itself,thatisthemetri .

Thene essity forthedevelopmentof su h aformalismresidesin thedi ulty

ofEinstein Equationsresolution, andin the fa t thatrelativelyfew physi ally

interestingexa tsolutionsoftheEinsteinEquationsareknown. Fromthepoint

ofviewofCosmology,theultimateaim ofperturbationtheoryistoprovidean

appropriatetoolforunderstandingthelarge-s ale lusteringofmatterin

galax-iesand lustersof galaxies,itspropertiesanditsorigin.

Inthisthesiswelimitourselvestothestudyofuniversesdominatedbyaperfe t

pressurelessuid, alleddustorsimplymatter,thatweassumetobeirrotational

aswell. Inthesyn hronousand omovinggauge,wepresentthe al ulationat

rst and se ond order of the perturbative fun tions of the so- alled gradient

expansionte hnique,and omparesu h ate hniquewiththestandard

pertur-bationapproa h: ourapproa hisanalyti alandtheanalysisfullyrelativisti .

Thestandardtheoryisbasedontheperturbationsofahomogenousandisotropi

FRWba kgroundmetri onsideringthe(small)u tuationsofthatmetri ,

de-viationsin ludingaprioriallthethreeperturbationmodes: s alar,ve torand

tensormodes. Inotherwords,weassumeFRW asagoodzerothorder

approx-imation for des ribing ouruniverse. Observations tell us that the universe is

farfrombeinghomogenousandisotropi atsmall s ales. Totakeintoa ount

oftheseinhomogeneities,theperturbativeexpansionisneeded,anditis

imple-mentedthroughspa eandtimefun tions,whoseformin termsoftheso- alled

pe uliar gravitationalpotentialis determined at dierentorders solving

itera-tivelyEinsteinEquations(thelinearorrstorderapproa histhemost ommon

butin thelastde adesome osmologistshavebegunstoppingatse ondorder).

Inthethesisthestartingpointisexa tlythestandardone: twophysi al

vari-ablesareintrodu ed,the"volumeexpansion"andthe"shear",andtheEinstein

Equationsarewrittenin theADMformalism. Theperturbationpro edure,on

theotherhand,is dierent. Westartwithaspatial metri ontainingthe

formally relatedto FRW byan exponentialspa e-dependentfa tor. Then we

onsiderasperturbationparameternotthemagnitudeofthedeviationfromthe

ba kground,but thespatial gradients ontent,sothat thezerothorder metri

(orthezerothorderofanyothereld)istheonenot ontainingspatial

deriva-tives.

Counting thegradients ontent at dierent orders means onsidering the

typ-i al s ale lengthsonwhi h the metri (and otherelds) varies spatially being

larger, in dierentapproximation, than the hara teristi times on whi h the

samequantities vary intime: theresultisanon-linearapproximationmethod

whi h allows us to study how osmologi al inhomogeneities grow from initial

perturbations, our"seed"(generatedbyinationaryu tuations).

Therefore, in this thesis, after des ribing irrotationaldust dynami s (Chapter

1), ommentingour gauge hoi e (Chapter 2) and summarizing basi ideasof

osmologi alperturbationstheory(Chapter3),weget

Ψ

and

χ

ij

uptothe se -ondorder(the orderwithfourspatialgradients)solvingrespe tivelyexpansion

and shear evolution equations. We he k energy and momentum onstraints

(Chapter 4), we arry on omparing our result with the standard ones by a

suitablepro edure,andnallyweshowtheformthat themagneti partofthe

Des ribing our Universe

Thisthesisdealswithdeparturesfromanidealhomogenousandisotropi FRW

(Friedmann-Robertson-Walker) osmologi almodel. Beforegoingintothe

te h-ni alitiesof the osmologi alperturbations, wewantin this hapter tooutline

thestateofthe artof thepresent osmology,pointingout theideasand

te h-inquesunderlyingthestandarddes riptionoftheuniverseindierent ontexts

andphasesofitshistory.

In parti ular, from a qualitative point of view, we present the osmologi al

modelthat is abletogivethebest t tothe ompleteset ofhigh-qualitydata

available at present, that is the standard "

Λ

CDM Hot Big Bang" model; we briey show the problems left unsolved by this standard model and the

rea-sonswhi h leadus to invokealternative s enariosfor theearly universe, su h

asInation. Finally, as matter today is lustered in galaxies and lusters of

galaxies,a ompletedes riptionoftheuniverseshould in lude ades riptionof

deviationsfromhomogeneity: wethenresorttoInationasthesimplest viable

me hanismforgenerating theobservedperturbations,andbrieyoverviewthe

possibleapproa hesusedatpresenttostudytheevolutionofsu hperturbations

andhen etheobservablelarge-s alemassdistribution.

Thetreatmentof thisChapter isnotmeantto be exhaustiveandpre ise asit

ouldbe[4℄,[3℄,[1℄,...: somesubje tsandtheoverallformalismaregoneonin

mu hmoredetailin following hapters.

1.1 The standard osmologi al model

GeneralRelativity, togetherwithsymmetryassumptionsof themetri and

as-sumptionsaboutthematter ontentoftheuniverse,isoneofthefoundamental

toolsfor thestudyof osmology: itindeed hasprodu edin the lastde ades a

quiteremarkablysu essfullpi tureofthehistoryofouruniverse.

WhileGeneralRelativityisin prin iple apableofdes ribingthe osmologyof

any given distribution of matter, it is extremely fortunate that our universe

appears to be homogenous and isotropi on the largest s ales. Together,

ho-mogeneity and isotropy allow us to extend the Coperni an Prin iple to the

Cosmologi alPrin iple, statingthat allspatial positions intheuniverseare

ussionfo usedonsomewell-denedandusefulproblems(homogenousmodels,

theirrelativemeritsandpossibletests). Nowadays,pre ise testshaveemerged

and the results do agree with the idea of the Cosmologi al Prin iple at least

as a zeroth order guidelines. If on s ales

&

tens of Mp wesee galaxies and galaxies lustersinone-dimensionalandbidimensionalstru tures(lamentsand

sheets)andva uumregionswithoutgalaxiesevenupto50-100Mp ,threesets

of observations -galaxy ounting, extragala ti radio sour es, CMB

tempera-turesmoothness- givesome eviden e that matter distribution and motionare

quitea uratelyisotropi ons ales

≫ 10

2

Mp and omparableto ourHubble

length,atleastwithinourvisiblepat h[9℄. Flu tuationsfromhomogeneityand

isotropy are thought to be of the order of

δρ

ρ

∼ 10

−5

[10℄, thus they an be negle tedat arstapproa htothesubje t.

FRW osmologi almodels

A purely kinemati onsequen eof requiringhomogeneityand isotropyof our

spatial se tions 1

is theFriedman-Robertson-Walker(FRW) metri ,whi h

en-ablesustodes ribetheoverallgeometryandevolutionoftheuniverseinterms

of two osmologi al parameters a ounting for the spatial urvature and the

overallexpansionor ontra tionoftheuniverse:

dS

F RW

2

= a

2

(τ ) [−dτ

2

+

dr

2

1 − κ r

2

+ r

2

_dθ

2

_{+ r}

2

_sin

2

_{θ dφ}

2

_].

(1.1)

τ

is the onformal time related to the osmi proper time

t

by the relation

dt = a(t)dτ

. Byres aling the radial oordinate, we an hoose the urvature onstant

κ

to take only dis rete values +1, -1 or 0 orresponding to losed, open, or atspatial geometries. These are lo al statements, whi h should be

expe tedfromalo altheorysu hasGeneralRelativity: theglobaltopologyof

thespatialse tionsmaybethat ofthe overingspa esbutitneednotbe.

A ombinationofhighredshiftsupernovaandLargeS aleStru ture(LSS)data

and measurements of the osmi mi rowave ba kground (CMB) anisotropies

strongly favors for a spatially at model, then we will almost always assume

su ha onstraint.

We next turn to osmologi al dynami s, in the form of dierential

equa-tionsgoverningtheevolutionofthes alefa tor

a(t)

;these omefromapplying EinsteinEquations(E.E.):

R

µν

−

1

2

Rg

µν

= 8πGT

µν

+ Λg

µν

(1.2) whereitis ommontoassumethatthematter ontentoftheuniverseisaperfe t

uid,forwhi h

T

µν

_{= (ρ + p)u}

µ

_u

ν

_{+ pg}

µν

_.

(1.3)

Thepressure

p

is ne essarilyisotropi ,for onsisten ywiththeFRW metri ;

ρ

isthe energydensityin therest frameof theuid, and

u

µ

is the4-velo ityin

1

omoving oordinate(seelaterSe tion 2.2).

The osmologi al onstant

Λ

term anbeinterpretedasparti lephysi s pro- essesyieldinganee tivestress-energytensorfortheva uumof

Λg

µν

/8πG

,and wehaveintrodu edit in E.E. be ausere entobservations (luminosity-redshift

ofSN

I

A andthe CMB anisotropiesmeasurements) suggest ana elerationof theuniverseexpansionandthustherequirementofanonstandarduid, alled

DarkEnergy. With

Λ

wemeanthesimplestformofDarkEnergy,thatisan en-ergy omponentindipedentoftime,spatiallyhomogenousandwithanequation

ofstate:

p

Λ

= −ρ

Λ

= −

Λ

8πG

.

(1.4)

Thus,forbrevity,fromnowonwewillnotexpli ititintheequationsbut treat

itasanyother(evenifparti ular)energy omponent.

Withthissimplieddes riptionformatter,equations(1.2) anberewritten

asfollows

H

2

≡

 ˙a

_a



2

=

8πG

3

X

i

ρ

i

−

κ

a

2

(1.5a)

¨

a

a

= −

4πG

3

X

i

(ρ

i

+ 3p

i

),

(1.5b)

where

H(t)

istheHubbleparameter,overdotsdenotederivativeswithrespe tto time

t

and theindex

i

labels alldierentpossibletypesof energy omponents in the universe. The rst equation is often alled Friedmann equation and

isa onstraintequation,these ondoneissometimereferredtoasa eleration

equationandisanevolutionequation. Athirdusefulequation-notindependent

oftheselasttwo-isthe ontinuityequation

T

µν

;µ

. Withourassumptionsitreads

˙ρ = −3H(ρ + p)

(1.6)

whi h implies that theexpansionof the universe (as spe ied by

H

) an lead to lo al hanges in theenergy density. Letus note that there is no notionof

onservationof "total energy",asenergy anbeinter hangedbetweenmatter

andthespa etimegeometry.

TheFRWequations anbesolvedquiteeasilysupposingthatonesingleenergy

omponent dominates. Within auid approximation, dening anequation of

stateparameter

w

whi hrelatesthepressure

p

totheenergydensity

ρ

by

p = wρ

, the ordinaryenergy ontributions of our universe su h as dust and radiation

are distinguished by, respe tively,

w = 0

and

w = 1/3

. On the ontrary, a osmologi al onstantis hara terizedby

w = −1

(equation(1.4)).

Equation(1.6)iseasilyintegratedtoyield

ρ ∝ a

−3(1+w)

.

(1.7)

ThenFriedmannequation(1.5a)with

κ = 0

and

w 6= −1

issolvedby

a(t) ∝ t

2/[3(1+w)]

.

(1.8) GeneralqualitativefeaturesofthefutureevolutionofFRWuniverse annowbe

zero(apart from

t = 0

): thus, if the universe is presentlyexpanding, it must ontinuetoexpandforever. Indeed,foranyenergy ontentwith

p ≥ 0

,

ρ

must de rease as

a

in reases at least as rapidly as

a

−3

, the value for dust. Thus,

ρa

2

_{→ 0}

as

a → ∞

. Hen e for

κ = 0

theexpansionvelo ity

˙a

asymptoti ally approa heszeroas

t → ∞

,whileif

κ = −1

wehave

˙a → 1

as

t → ∞

. Otherwise, if

κ = +1

, the universe annot expand foreverbut there is a riti al value

a

c

su h that

a ≤ a

c

: atanitetimeafter

t = 0

theuniversea hievesamaximum size

a

c

andthenbeginsto re ontra t.

Thepresen eofava uumenergyaltersthefate oftheuniverseandtheabove

simple on lusions:if

Λ < 0

,theuniversewilleventuallyre ollapseindependent ofthesignof

κ

. Forlargevaluesof

Λ

evena loseduniversewillexpandforever. Table 1.1 summarizes the behaviour of the most important sour esof energy

densityin osmologyinthe aseof aatuniverse.

TypeofEnergy

w

ρ(a)

a(t)

H(t)

Dust 0

a

−3

_t

2/3

2

3t

Radiation

1

3

a

−4

t

1/2

1

2t

Cosmologi alConstant -1

const

e

Ht

q

Λ

3

Table1.1: Thebehaviourofthes ale fa torandHubble onstant applietothe ase

ofaatuniverse;behavioursofenergydensityareperfe tlygeneral.

There are three foundamental features of FRW spa etimes whi h we are

goingtodis uss:

•

expansion(or ontra tion)

=⇒

gravitationalredshift (orblueshift);

•

existen eofaninitialsingularity,theBig Bang;

•

existen eofparti lehorizons.

ExpansionandRedshift TherststrikingresultofFRWmodelsisthat

universe annotbestati butmustbeexpandingor ontra ting. This on lusion

followsimmediatelyfromequation(1.5b)written inthesimpleform

¨

a = −

4πG

₃

(ρ + 3p)a.

(1.9)

(1.9)tells us that

¨

a < 0

if

ρ + 3p > 0

and

a > 0

¨

if

ρ + 3p < 0

: in any ase, the universe must always either beexpanding (

˙a > 0

)or ontra ting(

˙a < 0

) (withthepossibleex eptionofaninstantoftimewhenexpansion hangesoverto

ontra tion,asinthe ase

κ = +1

). Letus ommentthenatureofthisexpansion or ontra tion: thedistan e s alebetweenallisotropi observers hangeswith

time,butthereisnopreferred enterofexpansionor ontra tion.Indeed,ifthe

distan e(measuredonthehomogenoussli e)betweentwoisotropi observerat

time

t

is

r

, therateof hangeof

r

is

v ≡

dr

_dt

=

r

a

da

where

H(t)

isthewell-knownHubbleparameterand(1.10)isknownasHubble Law. Letus stillnote thatthe expansionspeed anbegreater thanthespeed

oflightwithoutanyharmfulthought.

Theexpansionoftheuniverseis onrmedina ordan ewithequation(1.10):

themostdire t observationaleviden e forthat omes from theredshift ofthe

spe tral lines of distant galaxies. The idea is that a lo al observerdete ting

lightfrom adistantemitterseesaredshiftin frequen yor,in otherwords,the

wavelength

λ

ofea hphotonin reasesinproportiontotheamountofexpansion, asanyotherphysi als aleisstret hedbyexpansion. Thesolutionofallredshift

problems(asillustratedinFigure1.1)inSpe ialandGeneralRelativityis

gov-ernedbythefollowingtwofa ts: rst,lighttravelsonnullgeodesi s;se ondly,

thefrequen yofalightsignalofwaveve tor

k

µ

measuredbyanobserverwith

4-velo ity

u

µ

is

ν = −k

µ

u

µ

. Thus we analwaysnd theobserved frequen y

by al ulating the null geodesi determined by the initial value of

k

µ

at the

emissionpointandthen al ulatingtherighthandsideoftheformerexpression

attheobservationpoint[1℄. Theredshift fa toristhengivenby

z ≡

λ

2

_λ

− λ

1

1

=

ν

1

ν

2

− 1 =

a(t

2

)

a(t

1

)

− 1.

(1.11)

Σ

Σ

2

1

P

1

P2

u

u

1

2

k

µ

µ

µ

Figure1.1: A spa etimediagramshowingthe emissionofalight signal atevent

P

1

anditsre eptionatevent

P

2

Itispossibletorelatetheredshifttotherelativevelo ityofthetwoobservers

in the ase of small s ales (i.e. less than osmologi al s ales) su h that the

expansion velo ity is non-relativisti . In this ase, for light emitted say by

nearbygalaxies,wehave

t

2

− t

1

≈ r

,where

r

isthepresentproperdistan eto thegalaxy;furthermore,

a(t

2

) ≈ a(t

1

) + (t

2

− t

1

) ˙a

. Thuswend

z

non rel

≈

˙a

whi histhelinearredshift-distan erelationshipdis overedbyHubble. The

red-shiftsofdistantgalaxieswilldeviatefrom thislinearlawdependingonexa tly

how

a(t)

varieswith

t

.

The redshift

z

is often used in pla e of the s ale fa tor: to be omplete,

z, t, a(t), ρ(t)

andthetemperatureTareallusedasvariablestorefertodierent phasesoftheuniversehistory(Tables1.1).

Big Bang singularity Both matter and radiation dominated at

uni-versespresentasingularityat

t = 0

inwhi h

a = 0

. Thus,undertheassumption ofhomogeneityand isotropy, GeneralRelativity makesthe strikingpredi tion

that at a time

t =

R

1

0

da

a H(a)

=

2

3(1+w)H

0

∼ H

−1

0

ago the universe was in a

singularstate: thedistan ebetweenall"pointsofspa e"waszero,thedensity

ofmatterand the urvature ofspa etimeinnite. Thissingularitystateof the

universeisreferredtoasBigBang,andthequantity

H

−1

0

,knownastheHubble

time, provides a useful estimate of the time s ale for whi h the universe has

beenaround. 2

Thenatureof thissingularityisthat resultingfromahomogenous ontra tion

ofspa edownto "zerosize". TheBig Bangdoesnotrepresentanexplosionof

matter on entratedatapreexistingpoint: itdoesnotmakesensetoaskabout

the state of the universe "before" the Big Bang be ause spa etime stru ture

itself is singularat

t = 0

; thus General Relativity leadsto the viewpointthat universebeganattheBigBang. Formanyyearsitwasgenerallybelievedthat

thepredi tionofasingularoriginwasdue merelyto theassumptionsof exa t

homogeneityandisotropy,thatiftheseassumptionswererelaxedonewouldget

anon-singular"boun e" at small

a

ratherthan asingularity. TheSingularity TheoremofGeneralRelativity[1℄showsthatsingularitiesaregeneri featuresof

osmologi alsolutions. Of ourse,at theextreme onditionsveryneartheBig

Bangoneexpe tsthatquantumee tswillbe omeimportant,andpredi tions

of lassi alGeneralRelativityareexpe tedtobreakdown.

Parti le horizons We shall demonstratenow the third ru ial point of

FRW spa etimes: FRW osmologi almodelspresuppose theexisten e of

non-trivial parti le horizons, where, by this expression, we mean in general the

boundaryoftheobservableregionatageneri time

t

,ortheboundarybetween theworldlines that anbeseenbyanobserverata ertain pointofspa etime

andthoseonethat annotbeseen(seeFigure(1.2)). InGeneralRelativitythe

questionabout how mu h of ouruniverse anbe observedat agivenpointis

due, and indeed, in spite of the fa t that the universe was vanishingly small

at early times, theexpansionpre luded ausal onta t from being established

throughouttheuniverse.

The photons travel on null paths hara terized by

dr =

dt

a(t)

= dτ

: the physi aldistan e thataphoton ouldhavetravelledsin ethaBanguntiltime

t

,thedistan eto theparti lehorizon,is

R

H

(t) = a(t)

Z

t

0

dt

′

a(t

′

₎

(1.13) 2

r

t

SINGULARITY

r=0

observer

particles already

seen

particles not

yet seen

Figure 1.2: The ausalstru ture ofFRWspa etime nearthe Big Bangsingularity:

parti lehorizonsarisewhenthepast light oneof anobserverterminatesatanite

time

t

or onformaltime

τ

.

An observer at a time

t

is able to re eive a signal from all other isotropi observersif and only if the integral of (1.13) diverges : in this ase the at

FRW metri is onformally related to Minkowski spa etime and there is no

parti le horizon. On theother hand, ifthe integral onverges, FRW model is

onformallyrelated only to a portion of Minkowski spa etime (the one above

a

t = const

surfa e) and parti le horizon doeso ur. It is notdi ult to see thattheintegral onvergesinallFRWmodelswithequationofstateparameter

w ∈ (0, 1)

:

R

H

(t) =

(

2t = H

−1

_{(t) ∝ a}

2

(radiation)

3t = 2H

−1

_{(t) ∝ a}

3/2

(dust)

.

(1.14) As

H(t)

−1

istheageoftheuniverse,

H(t)

−1

is alledtheHubbleRadius,asitis

thedistan ethatlight antravelinaHubbletime

H(t)

. Iftheparti lehorizon exists thenit would oin ide, upto numeri alfa tor, with theHubble radius:

forthisreason,inthe ontextofstandard osmology(when

ω > −1/3

)horizon andHubbleradiusareusedinter hangeably.

These on lusions are not true anymore in the ase of non standard matter,

that is

w /

∈ (0, 1)

: in the aseof a osmologi al onstant(forexample, during Inationor in thelater timeof universe history),parti le horizon andHubble

radiusarenotequalasthehorizondistan egrowsexponentiallyintimerelative

totheHubbleradius.

Aphysi allengths ale

λ

iswithinthehorizonif

λ < R

H

∼ H

−1

;intermsofthe

orresponding omovingwavenumber

k

,

λ = 2πa/k

, wewillhavethefollowing rule:

k

a

≪ H

−1

_=⇒

s ale

λ

outsidethehorizonandno ausality

k

a

≫ H

−1

=⇒

s ale

λ

within thehorizonand ausality

.

any omoving length s ale evolves in time with a power law

t

α

with

α < 1

(

κ = 0

), thusits rateofin rease isalwayssmallerthan therateof in reasein the Hubble horizon size, whi h is linear in time. Thus, for example, the size

ofa omovingregion orrespondingat presenttoasuper luster(say

∼ 30Mpc

at

t ≈ 10

9

_years

) was omparable to the horizon at epo h shortlybefore the

re ombination(

t ≈ 10

5

_years

)andwasmu hgreater thanthehorizonat some

earlierepo h.

These onsiderationsabouttheexisten eofparti lehorizonsandof ausally

dis onne tedregionsin FRW models leadto veryinterestingissues. Webegin

presentingoneofthem(knownasHorizonproblem),postponingabrief

dis us-sionsoftheshort omingsofthestandard osmologi almodelasdes ribeduntil

hereto anextparagraph.

Asmentionedearlier,wehavegoodreasonstobelievethatthepresentuniverse

is homogenous and isotropi to a very high degree of pre ision. Now, many

ordinarysystems,su h asgas onnedin a box, often are foundin extremely

homogenous and isotropi states: the usual explanation of that state is that

they havehad an opportunity to self-intera t and thermalize, exa tly asin a

box lled with gas initially in an inhomogenous state, these inhomogeneities

qui kly "washout" ona times ale of theorder of thetransit time a rossthe

box. However this type of explanation annot possibly apply to a universe

withparti lehorizons,sin edierentportions annotevensendsignalstoea h

other, farless intera t su ientlyto thermalize ea hother. Thus, in order to

explain the homogeneity and isotropy of the present universe, one must

pos-tulate that either

(a)

the universe wasbornin anextremely homogenousand isotropi state,or

(b)

theveryearlyuniversedieredsigni antlyfromtheFRW models so that no horizons were present; the inhomogeneities and anisotropy

then"dampedout"bysomeme hanismsandtheuniverseapproa hedtheFRW

models that t present observations. Unfortunately, if the rst point of view

may appearrather unnatural and aprofessionof faith, the se ond onesuers

notonlyfromtheabsen e ofaplausiblepi tureofevolutionfrom a haoti to

a FRW state, but for the fa t that gravity promotes inhomogeneity, not

ho-mogeneity. Later we will see how athird way is now a epted, theone of an

inationaryphase oftheveryearlyuniverse.

Brief outlineof universe evolution

The above onsiderations should be almost su ient to understand and

jus-tify the basi aspe ts of the evolution of our universe from the Big Bang to

the present in the standard pi ture. Two points should be still laried for

ompleteness:

•

thevariousparti lesinhabitingtheuniverse anbeusefully hara terized a ordingtothree riteria: in equilibrium vs. outofequilibrium

(de ou-pled),bosoni vs. fermioni ,andrelativisti (velo itiesnearto

c

)vs. non relativisti (dust);

•

mu hofthehistoryofthestandardBigBangmodel anbeeasilydes ribed byassumingthatoneofthe omponentsdominatesthetotalenergy

den-Asmentionedearlier,the osmologi alenergy onservation(equation(1.6))tells

usthatthede reaseofthes alefa tor

a

asonegoesba ktowardsthepasthas thesamelo alee t onthematterasifthematterwerepla edinaboxwhose

walls ontra tatthesamerate. Thus(inagreementwithTable1.1)the

ontri-bution of radiation omparedwith ordinarymatter in reases in thepast, and

theremustbeaperiodintheearlytimesofuniverseevolutioninwhi hthis

ra-diationshouldhavebeenthedominant ontributiontotheenergy. Thepresent

radiationenergy ontribution tothe universe energydensity isrepresentedby

theCMBenergydensity,wi hisabout1000timessmallerthanthepresentmass

density ontribution ofmatter. Onewould expe ttheradiation-lledmodelof

theuniversetobeagoodapproximationforthedynami softheuniversebefore

astage inwhi hthes alefa tor

a

wasmorethanfew1000timessmallerthan itspresentvalue, whilethe dust lled model should be agoodapproximation

afterwards. Inthe ontextofthisseparation,anotherimportantissueiswhether

theintera tionsofmatterorradiationpro eedonarapidenoughtimes alefor

thermalizationto o urlo ally (within the parti le horizon). A given spe ies

remainsinthermal equilibriumwiththesurroundingthermalplasmaaslongas

itsintera tionrateislargerthantheexpansionrateoftheuniverse. Aparti le

spe ies for whi h the intera tion rateshavefallen below theexpansion rateis

saidtohavefrozenout orde oupled. Asgoodruleofthumb,theexpansionrate

intheearlyuniverseis "slow", andparti lestendtobeinthermalequilibrium

(unlesstheyareveryweakly oupled);inour urrentuniverse,nospe iesarein

equilibriumwiththeba kgroundplasma(representedbytheCMBphotons).

The basi pi ture of theevolutionof ouruniverse anthen be told as

fol-lows: the universe began with a singularity state as a hot (

T → ∞)

, dense (

ρ → ∞

)soupofmatterandradiationinthermalequilibrium. Theenergy on-tentofearlyuniversewasdominatedbyradiation: attheseearlytimesthermal

equilibrium held and other spe i phenomena took pla e su h as primordial

nu leosynthesis. However,astheuniverseevolved,thermalequilibriumwasnot

maintainedandtheordinarymatter ontributionbegantodominatetheenergy

ontentof theuniverse (about

4 × 10

4

yearsafter theBang): thedynami s of

theuniversebe amethatofadustlledFRWmodel hara terizedbytheCMB

photonsba kground,matter-antimatterasymmetryand osmologi alstru ture

formation.

There is noroomin this thesisto llthe details ofthis s hemati and full

ofgapsevolutionaryhistory,andto dis ussforexamplethevery omplexrst

few minutes of universe life hara terized by symmetry breakings and phase

transitions,andother[4℄: moreinteresting,eveninrelationtothefollowing

de-velopments,istounderlinethegoodpredi tionsoftheHotBigBang modeland

tounderstandhowitfa esre entobservationsandsometheorethi alquestions.

Parametrizing theuniverse: short omings ofthe standard model

Earlierweintrodu ed globalparameters su h asexpansionfa tor

a(t)

, spatial urvature

κ

andHubbleparameter

H(t)

, thelatterdenedby

H(t) +

˙a

a

=

a

′

a

2

or

H(τ) +

a

′

a

(1.15)

wherethedotdenotesdierentiationwithrespe tto

t

andtheprime dierentia-tionwithrespe tto

τ

. Inaddition,itisusefultodeneseveralothermeasurable osmologi alparameters.

The Friedmann equation (1.5a) suggests to dene a riti al density

ρ

c

and a osmologi aldensityparameter

Ω

tot

ρ

c

+

3H

2

8πG

and

Ω

tot

+

ρ

ρ

c

(1.16)

su hthat it anberewrittenasfollows

κ

a

2

= H

2

_(Ω

tot

− 1)

(1.17)

Fromequation(1.17),one andistinguishthedierent ases

ρ < ρ

c

↔ Ω

tot

< 1 ↔ κ = −1 ↔

open

ρ = ρ

c

↔ Ω

tot

= 1 ↔

κ = 0

↔

f lat

ρ > ρ

c

↔ Ω

tot

> 1 ↔ κ = +1 ↔ closed.

(1.18)

It is often ne essary to distinguish dierent ontributions to the density, and

therefore onvenient to dene present-day density parametersfor pressureless

matter

Ω

m

, relativisti parti les

Ω

r

, and for the va uum

Ω

v

. This last one is equal to

Ω

Λ

= Λ/3H

2

in models with osmologi al onstant, i.e. onstant

va uumenergydensity. ThentheFriedmannequationbe omes

κ

a

2

0

= H

2

0

(Ω

m

+ Ω

r

+ Ω

v

− 1)

(1.19) wherethesubs ript0indi ates present-dayvalues.

Onewaytoquantifythede eleration(ora eleration)oftheuniverseexpansion

ofequation(1.5b)isthede eleration parameter

q

0

denedas

q

0

+

−

 a¨a

˙a

2



0

=

1

2

Ω

m

+ Ω

r

+

1 + 3w

2

Ω

v

.

(1.20)

Theexpansiona eleratesif

q

0

< 0

andthisequationshowsthat

w < −1/3

for theva uummayleadtoana eleratingexpansion.

Itisusual toexpress theHubbleparameterand hen ealltheprevious

param-etersintermsofthes aledHubbleparameter

h

forwhi h

H ≡ 100h km s

−1

M pc

−1

.

(1.21) Theterm" osmologi alparameters"isin reasingitss opebe auseoftherapid

advan es in observational osmology of the last ten years whi h are leading

to theestablishmentof therst high pre ision osmologi almodel. The most

a urate model of the universe requires onsideration of a wide range of

dif-ferenttypesofobservations,with omplementaryprobesproviding onsisten y

he ks,liftingparameterdegenera ies,andenablingthestrongest onstraintsto

bepla ed. Hen e,nowadays,theterm" osmologi alparameters"notonlyrefers

totheoriginalusageofsimplenumbersastheaboveonesdes ribingtheglobal

dynami sandpropertiesoftheuniverse,butalsoin ludes theparametrization

of somefun tions des ribing the nature of perturbations in theuniverse, and

physi al parameters of the state of the universe. Typi al omparison of

os-mologi al models with observational data now feature about ten parameters,

Parameter Symbol Value

HubbleParameter

h

0.73 ± 0.03

Totalmatterdensity

Ω

m

Ω

m

h

2

_{= 0.134 ± 0.006}

BaryonDensity

Ω

b

Ω

b

h

2

_{= 0.023 ± 0.001}

Cosmologi alConstant

Ω

Λ

Ω

v

= 0.72 ± 0.05

RadiationDensity

Ω

r

Ω

r

h

2

_{= 2.47 × 10}

−5

Densityperturbationamplitude

∆

2

R

(k

∗

)

seelaterP(k) Densityperturbationspe tralindex

n

n = 0.97 ± 0.03

Tensor tos alarratio

r

r < 0.53 (95%conf )

Ionizationopti allenght

τ

τ = 0.15 ± 0.07

Table1.2: Thebasi setof osmologi alparameters:un ertainitiesareone-sigma/

68%

onden eunlessotherwisestated.

We have by now most of the ingredients needed to understand the rst

half of the shown parameters; the se ond one will be in part justied in the

ontinuation,while the ionization opti al depth will notbe ommented at all

inthis thesis. Thespatial urvaturedoesnotappear inthe listbe auseit an

be determinedfrom theother parametersusing(1.17) or(1.19), andthetotal

presentmatterdensityisindi atedasusualasasumofbaryoni matteranddark

matter densities, namely

Ω

m

= Ω

dm

+ Ω

b

. With appropriate arguments, the parameterset listedabove anberedu edtosevenparametersasthesmallest

set that an usefully be ompared to the present osmologi al data set. Of

oursethisisnottheuniquepossible hoi e: one ouldinsteaduseparameters

derivedfromthosebasi onessu hastheageoftheuniverse,thepresenthorizon

distan e, thepresentCMB and neutrinoba kgroundtemperatures, theepo h

ofmatter-radiationequality,theepo hoftransitiontoana eleratinguniverse,

thebaryonto photonratio,... Furthermore,dierenttypesofobservationsare

sensitivetodierentsubsetsof thefull osmologi alparameterset.

Having in mind the aboveparametrization and Table1.2 as mirrorof the

disposableobservationaldata, we anpro eed in evaluating thestandard

os-mologi almodel. Amongthemostnotablea hievementsofHotBigBangFRW

standardmodelare

•

thepredi tionof osmologi alexpansion;

•

thepredi tionandexplanationofthepresen eofareli ba kground radi-ationwithtemperatureoforderoffewK,theCMB;

•

theexplanationsofthe osmi abundan eoflightelements;

•

thepossibilitytoinsertinthispi turethestru tureformationphenomenon. Onthe ontrary,themostsevereproblemsthatithastofa e anbesummarized

inthefollowinginterestingissues.

•

Horizonproblem.

Under the term"horizon problem"awide rangeof fa ts is in luded,all

related to the existen e of parti le horizons in FRW models. We have

A ordingtothestandardmodel,photonsandtheother omponentssu h

asele tronsandbaryonsde oupledat atemperatureof0.3 eV.Re alling

the pre eding dis ussions, this happenedwhen the rateof intera tion of

photonswith,say,ele tronsandprotonsbe ameoftheorderoftheHubble

size(thatis,ofthehorizonsize),andtheexpansionmadenotpossiblethe

reverserea tionof

p+e

+

_{→ H +γ}

. Thetemperatureof0.3eV orresponds

totheso- alledsurfa eoflast-s attering,posedat aredshift

z

LS

≈ 1100

, after the matter-radiation equivalen e and hen e in matter era. From

the epo h of last-s attering onwards, photons free-stream and now are

measurable in the well known CMB, whose spe trumis onsistent with

that ofabla k-bodyatatemperatureof

2.726 ± 0.01

K.Thenletuslook attwophotonsfromdierentpartsofthesky: thelengh orrespondingto

ourpresentHubbleradiusatthetimeoflast-s atteringwas(remembering

that

T ∝ a

−1

)

λ

H

0

(t

LS

) = R

H

(t

0

)

 a(t

LS

)

a(t

0

)



= R

H

(t

0

)



_T

0

T

LS



Duringthematterdomination

H

2

_{∝ a}

−3

_{∝ T}

3

,andatlast-s attering

H

LS

−1

= R

H

(t

0

)



_T

0

T

LS



3/2

≪ R

H

(t

0

)

Being

T

0

∼ 2.7K ∼ 10

−4

eV

≪ T

LS

, the length orresponding to our present Hubble radius was mu h mu h larger that the horizon at that

time. Be auseCMBexperimentslikeCOBEandWMAPtellsusthatour

twophotonshavenearlythesametemperaturetoapre isionof

10

−5

,we

are for ed to saythat those two photons were verysimilar even if they

ouldnottalkto ea hother, andthat theuniverseatlast-s atteringwas

homogenous andisotropi in aphysi al regionaboutsomeorder greater

thanthe ausally onne tedone!

NotonlythehomogeneityoftheCMBisabletotellusimportantthings,

but nowadays the measured temperature u tuations ( onsequen es of

densityinhomogeneities)areamineofinformationtoo,andanother

strik-ingfeatureoftheCMBisthatphotonsatthelast-s atteringsurfa ewhi h

were ausallydis onne ted havethe samesmall anisotropies([10℄). The

standardmodel annotsayanythingwithreferen etothis.

•

Flatnessproblemandthepe uliarityofinitial onditions. TheFriedmannequationtellsus that

(Ω

tot

− 1) = κ/ H

2

a

2

therefore(weimpli itly onsiderfrom nowon

Ω ≡ Ω

tot

)

(Ω − 1) → 0

for

t → 0

inboth asesofradiationand matterdomination: in otherwords, given

(Ω(t) − 1)

at agiventime

t

,

Ω

hasto depart from 1bothin open and losed ases. Present observations tell us that

(Ω

0

− 1)

is of order unity(i.e.

∈ (0, ∼ 1)

). Letus al ulatethesamevalueatsomeearlytime ofuniverse, sayat Plan ktime(at

t ≈ 10

−43

sorT

∼ 10

19

GeV):

|Ω − 1|

T =T

P l

|Ω − 1|

T =T

0

≈

 a

2

_(t

P l

)

a

2

_(t

0

)



≈

 T

2

0

T

2

P l



≈ O(10

−64

)

Averyproblemati questionarises,be ausehow anitbepossiblethat

Ω

had been sonear the riti alvalueableto lead tothe universe observed

today? Evensmalldeviationsof

Ω

from1atearlytimewouldhaveledto the ollapseorthe oolingof the universe in few

10

−43

s, respe tively in

the aseof

κ = +1

or

κ = −1

. Inorder togetthe orre tvalue

(Ω

0

− 1)

atpresent,thevalue

(Ω − 1)

atearlytimeshadtobened-tunedtovalues amazingly losetozero,butwithoutbeingexa tlyzero. Thisisthereason

whytheatness problemisalsodubbedthe"ne-tuningproblem".

•

Existen eofDarkMatter.

Wehavearemarkable onvergen eonthevalueofthedensityparameter

in matter(

w = 0

):

Ω

m

= 0.28 ± 0.05

. We all baryoni matter orsimply ordinarymatteranythingmadeofatomsand their onstituents,andthis

wouldin ludeallofstars,planets,gasanddustintheuniverse. Ordinary

baryoni matter, it turns out,is notenoughto a ountforthe observed

matterdensity:

Ω

b

∼ 0.043 ± 0.002 ≪ Ω

m

This determination omes from avariety of methods: dire t evaluation

of baryons, onsisten y with the CMB powerspe trum, and agreement

withthepredi tionsof primordialnu leosynthetis,whi hpla es the

on-straint

Ω

b

≤ 0.12

. Most of the matter density must therefore be in the formofnon-baryoni matter,ordark matter. Candidatesfordarkmatter

in lude the lightest supersymmetri parti le, the axion, but in the past

essentiallyeveryknownparti leoftheStandardModelofparti lephysi s

and predi ted parti les of Supersymmetry theories have been ruled out

asa andidate for it. The things weknoware that it hasno signi ant

intera tions withother matter, so astohavees apeddete tion thus far,

andthatitsparti leshavenegligiblevelo ity,i.e. theyare" old".

•

Eviden eofa eleratedexpansion.

Astonishignly, in re ent years, it appears that an ee t of a elerating

expansion(

q

0

< 0

)hasbeenobservedin theSupernovaHubblediagram: the ommonpositioninthelastyearsistoinvoketheexisten eofanother

energy omponent(dierentfrommatterandradiation),and omparison

with the predi tion of FRW models leads of ourse to favor a

va uum-dominateduniverse. Inthispi ture, urrentdataindi atethattheva uum

energyisindeedthelargest ontributortothe osmologi aldensitybudget,

with

Ω

v

= 0.72 ± 0.05

,[11℄. Thenatureofthisdominanttermispresently un ertain,but mu h eortis being invested in dynami almodels, under

the at h-allheadingofquintessen e,orDarkEnergy.

•

Theproblemofperturbationsunknownorigin.

Therstissuesarisefrom a ombinationofobservationalfa ts andtheoreti al

prin iples,andtogetherwiththelastonetheyndthebestmodelsolutioninthe

Inationary paradigm. TheDarkMatter andthe DarkEnergy problemsfor e

us to take into a ount an ampler osmologi al model referred to by various

names,in luding"

Λ

CDMHotBigBang"model,the on ordan e osmology,or thestandard osmologi almodel. But thesense of a omplishmentat having

we do not understand very well any of them. For instan e, there are many

proposalsforthenatureofDarkMatter,butno onsensusastowhi his orre t.

Even thebaryondensity, nowmeasuredtoana ura y ofafewper ent, la ks

anunderlyingtheoryabletopredi titevenwithinordersofmagnitude. Finally

thenatureoftheDarkEnergyremainsamystery,evenifveryre entworkshave

suggestedviableme hanismsabletoexplainthea elerationwithoutinvoking

anextraenergy omponent[37℄.

1.2 Ination

Thehorizonproblem isarelevantproblem ofthestandard osmologybe ause

at itsheart there is simply ausality. Fromthe onsiderationsmadesofar, it

appears that solvingtheshort omings ofthe standardmodel requires atleast

an importantmodi ation to how the information an propagatein theearly

universe, and hen e that the universe has to go through a primordial period

during whi h the physi al s ale

λ

evolvesfaster than the horizon s ale

H

−1

.

Cosmologi alInation issu hame hanism.

Thefoundamental ideaof Ination is that the universe undergoesaperiod of

a eleratedexpansion,denedasaperiodwhen

¨

a > 0

,atearlytimes. Theee t ofthisa elerationisto qui kly expandasmall regionofspa eto ahuge size,

redu ing the spatial urvature in the pro ess, making the universe extremely

loseto at. In addition, thehorizon size isgreatlyin reased, so that distant

pointsontheCMBa tuallyarein ausal onta t.

Aninationarystage isdenedasaperiodoftheuniverseduringwhi hthe

lattera elerates. Frompreviousse tionswehavelearnedthat

¨

a > 0 ⇐⇒ (ρ + 3p) < 0

(1.22)

andthatsu ha onditionisnotsatisedneitherduringaradiation-dominated

phasenorinamatter-dominatedphase. Evenifitissu ientthat

p < −ρ/3

,in ordertostudythepropertiesoftheperiodofination,weassumetheextreme

ondition

p = −ρ

whi h onsiderably simplies theanalysis and that wehave alreadymet in termsof a osmologi al onstant. Were all brieythat in the

aseofsu hanenergy omponent

ρ ∝ const

(1.23)

H

I

∝ const

(1.24)

a(t) = a

i

e

H

I

(t−t

i

)

∝ e

H

I

t

(1.25)

R

I

H

(t) ∝ H

I

−1

e

H

I

t

(1.26)

where the subs ript (or supers ript) I indi ates that we refer to an ination

quantity and

t

i

denotes the time at whi h ination starts. Contrary to what happens in FRW dust or radiation lled universes, a omoving length s ale

in reasesfasterthantheparti lehorizonandmu hfasterthantheHubblesize.

Bytheway,Ination isaphaseof thehistoryof theuniverseo urring before

theeraofnu leosynthetis(

t ≈ 1s

,T

≈

1MeV)duringwhi hthelightelements abundan eswere formed: this is be ause nu leosynthetisis the earliest epo h

thepredi tionsoftheHotBigBangmodel. However,thethermalhystoryofthe

universebeforethatstageisalmostunknownandmanymodelsofInationare

settobearoundthePlanktime(

t

P l

≈ 10

−43

_s

). Itis ommon,evenin reponse

toother tasks, to think ofaperiodof reheating atthe endof Ination during

whi hthermalequilibriumisestablishedandradiationerabegins.

Itisuseful tohaveageneralexpressionto des ribehowmu hInation o urs

on eithasbegun. Thisistypi allyquantiedbythenumberofe-folds,dened

by

N (t) + ln

 a(t

f

)

a(t)



and

N

tot

= ln

 a(t

f

)

a(t

i

)



(1.27)

Resolutionof thehorizon problem ThankstoInationany omoving

lengths aleobservableatpresenthasbeen ausally onne tedatsome

primor-dialstageoftheevolutionoftheuniverse,removingthehorizonproblem. This

anbeeasily seenwiththehelp of Figure1.3. Letus onsider lengths ales

λ

whi harewithinthehorizontoday(

λ < H

−1

_(t

0

) ≡ H

0

−1

)butwereoutsidethe horizonforsomepreviousperiod (

λ > H

−1

_(t

past

)

)duringthematteror radia-tionera. Ifthereisaperiod(ination)duringwhi hphysi allengths alesgrow

faster than

H

−1

, su h today observable s ales had a han e to be within the

horizonin that early period again(

λ < H

−1

I

): in fa t, during theinationary

epo htheHubbleradiusis onstantandthe onditionsatised.

log a

legth scales

λ

H

H

H

−1

const

−1

−1

a

a

a

2

3/2

I

(MD)

(RD)

end of

Inflation

H

−1

λ =

problem and let the present day largestobservables ale re-enter the horizon

during Ination. The largestobservable s aleis of ourse the presentHubble

radius

H

0

and we want it to be redu ed during Ination to a value

λ

H

0

(t

i

)

smallerthanthevalueoftheHubblesize

H

−1

I

duringInation. Thisgives

λ

H

0

(t

i

) = H

−1

0

 a(t

f

)

a(t

0

)



_{ a(t}

i

)

a(t

f

)



= H

₀

−1

 T

0

T

f

)



e

−N

tot

_.

_H

−1

I

(wherewehavenegle tedforsimpli itytheshortperiodofmatter-domination).

Thenthe onditionforsolvingthehorizonproblemis

N

tot

&

ln(

T

0

H

0

) − ln(

T

f

H

I

) ≈ 67 + ln(

T

f

H

I

).

(1.28)

Morepre ise valutationsgive

N

tot

&

60

.

Ination and atness problem Ination solves elegantly the atness

problem,thankstothefa tthattheHubbles aleis onstantand

Ω − 1 =

_a

2

k

_H

2

I

∝ 1/a

2

.

We haveseen that to reprodu e a valueof

(Ω

0

− 1)

of order unity today the initialvalueof

(Ω−1)

atPlan ktimemustbe

|Ω−1| ∼ 10

−60

. Sin eweidentify

thebeginningoftheradiationerawiththeendofInation, andthetime s ale

ofInationisPlan ktime,werequire

|Ω − 1|

t=t

f

∼ 10

−60

. DuringInation

|Ω − 1|

t=t

f

|Ω − 1|

t=t

i

=

 a

i

a

f



2

= e

−2N

tot

Taking

|Ω − 1|

t=t

i

oforderunity,itisenoughtorequirethat

N

tot

≈ 60

tosolve theatnessproblem. Fromthepointofviewofthene-tuning,Inationavoids

thehindran e ofanenormousne-tuning, be ause thedensityparameter

Ω

is drivento1withexponentialpre ision.LetusnotethatiftheperiodofInation

lastslongerthan60e-foldingthepresent-dayvalueof

Ω

0

willbeequaltounity withagreatpre ision. Thuswe ouldsaythatageneri predi tionofInation

is

Ω

0

= 1

,and urrentdataonCMBanisotropies onrmthispredi tion. Inationas driven by a slowly-rolling s alareld

Knowing the various advantages of having a period of a elarated expansion

phase, the next task onsists in nding a model that satises the onditions

mentionedabove. TherearemanymodelsofInation. Todaymostofthemare

basedonanews alareld,theinaton

φ

.

We onsidermodellingmatterintheearlyuniversebytheinaton,areals alar

eldwhi hmoveswithapotential

V (φ)

. ItsLagrangianthenreads

L =

1

2

∂

µ

φ∂

ν

φ + V (φ)

(1.29) andthestress-energytensoris

T

µν

= φ

,µ

φ

,ν

− g

µν

 1

2

φ

,µ

φ

,ν

+ V (φ)



The orrespondingenergydensity

ρ

φ

andpressure

p

φ

are

T

00

= ρ

φ

=

˙

φ

2

2

+ V (φ) +

(∇φ)

2

2a

2

(1.31a)

T

ii

= p

φ

=

˙

φ

2

2

− V (φ) −

(∇φ)

2

6a

2

(1.31b)

where itis evident that ifthe gradientterm were dominant, wewould obtain

p

φ

= −

ρ

₃

φ

, notenoughtodriveInation.

Inthe ase ofan homogenouseld

φ(t, ~x) = φ(t)

, theinaton behaveswith a perfe tuid andexpression(1.31) be ome

T

00

= ρ

φ

=

˙

φ

2

2

+ V (φ)

(1.32a)

T

ii

= p

φ

=

˙

φ

2

2

− V (φ)

(1.32b)

Theequationofmotionforthehomogenousinatonis

φ =

dV

dφ

i.e.

φ + 3

¨

˙a

a

φ +

˙

dV

dφ

= 0

(1.33)

whi h an be thought of as the usual Klein-Gordon equation of motion for

a s alar eld in Minkowski spa e, but with a fri tion term

3H ˙

φ

due to the expansionof theuniverse. TheFriedmannequationwith su h as alareld as

thesole energysour eis

H

2

=

8πG

3

 1

2

φ

˙

2

_{+ V (φ)}



(1.34)

Letus nowquantify under whi h ir umstan es as alareldmaygiveriseto

aperiod of Ination. First of all, let us note that requiring

V (φ) ≫ ˙φ

2

im-pliesfromexpressions (1.32)that thepotentialenergyofthes alareldisthe

dominant ontributiontoboththeenergydensity andthepressure,and hen e

p

φ

≃ −ρ

φ

: from this simple al ulation, we realize that a s alar eld whose energydominatestheuniverseand whosepotentialenergydominatesoverthe

kineti term an mimi a osmologi al onstantdominateduniverse,and then

givesInation. Inationisdrivenbytheva uumenergyoftheinatoneld.

If

φ

˙

2

_{≪ V (φ)}

,thes alareldisslowlyrollingdownitspotentialandthisisthe

reasonwhysu haperiodis alled slow-roll. Theso- alledslow-roll

approxima-tion onsistsin two onditions:

•

negle tingthekineti termof

φ

ompared tothepotentialenergy;

•

assumingaatpotentialsothat

φ

¨

isnegligibleaswellin(1.33).

In this approximation, the Friedmann equation (1.34) and the eld equation

(1.33)arewritten

H

2

≃

8πG

₃

V (φ)

(1.35)

3H ˙

φ ≃ −V

′

(φ)

(1.36)

where in this ontext

V

′

_{(φ) =}

dV

dφ

. That is, thefri tion due to the expansion

• ˙φ

2

_{≪ V (φ) =⇒}

(V

′

₎

2

V

≪ H

2

;

• ¨

φ ≪ 3H ˙φ =⇒ V

′′

_{≪ H}

2

.

Ifwedenethefollowingslow-roll parameters

ǫ ≡ −

_H

H

˙

2

= 4πG

˙

φ

2

H

2

=

1

16πG

 V

′

V



2

(1.37a)

η ≡

_8πG

1

 V

′′

V



(1.37b)

theslow-roll onditionsholdif

|ǫ| ≪ 1

and

|η| ≪ 1

.

It is noweasy to see in anothersense how the slow-roll approximationyields

ination. Letusre allthat Inationisdenedby

¨

a > 0

,orinotherterms

¨

a

a

2

= ˙

H + H

2

_{> 0}

˙

H > 0

annotbeforas alarpotential(as

p

annotbe

< −ρ

): thea eleration ondition an betranslatedto

−

H

˙

H

2

= ǫ < 1

As soon asthis onditionfails, Ination ends: in general, slow-rollinationis

attainedif

ǫ ≪ 1

and

|η| ≪ 1

, where thelatter onditionhelps to ensurethat inationwill ontinueforasu ientperiod.

Withinthis approximation,thetotal numberofe-folds betweenthebeginning

andtheendofInationis

N

tot

≡ ln

 a(t

f

)

a(t

i

)



=

Z

t

f

t

i

Hdt ≃ −8πG

Z

φ

f

φ

i

V

V

′

dφ.

(1.38)

Con luding,Ination is osmologi allyattra tivebut seriousproblems areleft

unsolvedwithit: ontheonehand,we annotsayiftheuniversein itsearliest

stagessatisedthe onditionsforInationtolightup(i.e. forinatontoundergo

slowrollover);ontheotherhand,therearenoexperimentaleviden esevenfor

theexisten eofaneutralspinzerobosonfarlessfortheexisten eoftheinaton

inparti ular.

1.3 Foundamental ideas of Stru ture Formation

Asalreadymentioned,theCosmologi alPrin ipleandhen etheinhomogeneity

oftheuniverse haveplayeda uriousrole in thehistoryof modern osmology:

if the overall properties of the universe are very lose to being homogenous

andhen emu hof universedynami sasawhole anbesaidthankstothe

as-sumptionof homogeneityandisotropyonthelargests ales,ontheotherhand

teles opes reveal a wealth of details on s ales varying from single galaxies to

largestru turesofsize farex eeding

10

2

Mp . Understandingthe existen eof

thesestru tureisoneoftheprin ipaltaskofmodern osmology,andthisstudy

de-The interest in the large-s alemass distribution tra es ba k to the Thirthies

withLemaitre,whopointedoutthat iftheevolvinghomogenousandisotropi

worldmodelisareasonablerstappoximation(wenowsayzerothorder

approx-imation),thenthenextstepistoa ountforthedeparturesfromhomogeneities

in theobservedstru tures. As theCosmologi al Prin iple annotbeexpe ted

from general arguments and physi al prin iples, nor the existen e of galaxies

anbe dedu ed from generalprin iples be ausewe donot knowhowto

spe -ify initial onditions: we have been left with Lemaitre' s program onsisting

intryingtondthe hara terofdensity u tuationsin theearlyuniverseand

modelling the physi al pro esses that haveoperated subsequently to develop

su hu tuationsintotheirregularitiesweobservetoday.

Mu h work hasbeen donein the last de ades and now we anfollow agreat

part of the evolution of initial perturbations to present stru tures thanks to

alonglist of osmologi als hemesand methods. But before goinginto some

more detailed des ription of the idea of stru ture formation we want still to

stressonthenatureoftheCosmologi alPrin iple. Ifitwerereallyaprin iple,

asinitiallysuggestedbyMilne,theCosmologi alPrin ipleshouldbe ompared

to alaw ofnature: on the ontrary, now it is ommon sense to intend it asa

philosophi alassumptionwhi hallowsusto ir umeventourinabilitytoobtain

informationabouttheuniverseoutsideourpastlight- onebyassumingthat a

symmetryprin ipleexistseverywhere. ByassumingtheCosmologi alPrin iple,

we assumethat we are ableto determine onditionsmany Hubble radii away

fromusbyusingobservationaldatawithin ourpastlight- one,whoseregionof

inuen eis,bydenition,limitedtooneHubbleradius. Itisexa tlythispoint

that should lead us to treat the Cosmologi alPrin iple asa subtle approa h.

Moreover,homogeneity ouldonlyapplyontheaverageovermanygalaxies:we

shouldthenkeepinmindthatwhenwerefertohomogeneityandisotropyofthe

universeweta itly assumethat spatial smoothingoversomesuitablylarge

l-terings alehasbeenappliedexa tlywiththepurposeoflettingthene-grained

detailstobeignored.

Agreatdealofstru tureformationtheoryisbasedonthestudyofjust one

s alareld,namelythedensityperturbationeld denedas

δ(t, ~x) ≡

ρ(t, ~x) − ρ

_ρ

b

(t)

b

(t)

(1.39)

where

ρ

b

represents the unperturbed mean value of the ba kground universe density,intheFRW model. Inspe i ases,thiseld isrelatedto the

Newto-nian pe uliar gravitational potential

ϕ(~x)

throughthe Poisson equationwhi h inanexpandinguniversereads

∇

2

ϕ(t, ~x) = 4πG a

2

(t)ρ

b

(t) δ(t, ~x).

(1.40) Thereare manydierentnotationsused to des ribethedensity perturbations

andtheirevolution,bothintermsofthequantitiesusedtodes ribethe

pertur-bationsasmetri deviations andof thedenition ofan appropriatestatisti al

treatment. Theformerapproa hwill be leareronly in thefollowing hapters

anditistheheartofthethesis;fornow,wewanttogiveasket hofthelatter.

A riti alfeatureofthequantity

δ

isthatitinhabitsauniversethatisisotropi and homogenous in its large-s ale properties: this suggest a statisti al

shouldalsobestatisti allyhomogenous. Inother words,

δ

ree tsastationary randompro ess: everyspatialposition

~x

i

isasso iatedtoasto hasti variable

δ(~x

i

)

,with

i = 1, 2, ...N

and

N → ∞

,andalltheprobabilitydensitiesonanite numberof points

P

~

x

1

,~

x

N

,...,~

x

N

(δ

1

, δ

2

, ...δ

N

)

areinvariantundertranslations, ro-tationsandree tionof thepointsset

~x

1

, ~x

N

, ..., ~x

N

. Theuniverseweobserve is the statisti al realization of

δ(~x)

thought as a sto hasti eld, and in this languagetheunperturbeddensityofFRW ba kgrounduniverse orrespondsto

theaverageoverthestatisti alensemble,

ρ

b

≡ hρ(~x)i

.

Cosmologi aldensityeldsareanexampleofergodi pro ess,inwhi hthe

aver-ageoveralargevolumetendstothesameanswerastheaverageoverastatisti al

ensemble.

Itisusualtodes ribe

δ

asaFouriersuperposition:

δ(~x) =

X

δ(~k) e

ˆ

−i~

k~

x

(1.41)

The ross-termsvanishwhenwe omputethevarian eintheeld,whi hisjust

asumovermodesofthepowerspe trum

hδ

2

i =

X

|ˆδ(~k)|

2

≡

X

P (k)

(1.42)

wherethestatisti alisotropi natureoftheu tuationsallowsustowrite

P (k)

ratherthan

P (~k)

. Anotherquantitywhi hdes ribesthestatisti alpropertiesof

δ

istheauto orrelationfun tion,whi hisrelatedtothepowerspe trumthrough Fouriertransformationandhen egivesthesamedes riptionofthedensityeld:

forthisreason,weskipforbrevitytheintrodu tionofthisfurther on ept.

Thephysi al meaning ofthepowerspe trum isthe following:

P (k) ∝ |ˆδ(~k)|

2

,

thelatterbeingtheamplitudeofplanewaveswithwavelength

λ = 2π/k

;then thevalueof thespe trumat every

k

tellsus howmu hthe ontribution of

k

-s aleu tuationsisimportantin theFouriersumin orderto form thegeneri

perturbation

δ(~x)

in ongurationsspa e. Inotherwords,

P (k)

isameasureof thepoweroftheu tuationsofwavenumber

k

.

A sto hasti eld is said to be Gaussian if the phases of the Fourier modes

des ribingu tuationsatdierents ales

λ

are un orrelated,that isif the am-plitudesofwavesofdierentwavenumbersarerandomlydrawnfromaRayleigh

distributionof width given bythe powerspe trum. Thedensity perturbation

eld isGaussian(see later): this meansthatif we oulddoaverybignumber

ofstatisti alrealizations oftheuniverse, inanypoint

~x

thedistributionof the observedvalue of

δ(~x)

in all those universes would be aGaussian entered in zero. Inmomentumspa e,be ausetheFouriertransformationofaGaussianis

stillaGaussian,thesamedes riptionapplies.

AGaussiandistributionisunivo allydes ribedbyitsaverageanditsvarian e:

thus, inour ase,whatweneedfordes ribingthedensityu tuationeld

δ(~x)

isjustitspowerspe trum.

Assumingfor

P (k)

asimplefun tional formallowsus doingsimpleanduseful onsiderations. Themost onvenientpowerspe traaretheso- alledpower-law

powerspe tra

P (k) ∝ k

n−1

(1.43)

s ale,andhen etheyare hara terizedbynoparti ularphysi als ale. Among

the others, a ase of parti ular interest is the Harrison-Zel'dovi h spe trum,

whi h orrespondsto apowerspe trumwith

n = 1

. Inationand osmologi alperturbations

Inorderforstru tureformationtoo ur,theremusthavebeensmallpreexisting

u tuationsonphysi al lengths ales when they rossed theHubble radius in

theradiation-dominatedormatter-dominated eras. Inthe standardBigBang

modelthesesmallperturbationshavetobeputbyhand,be auseitisimpossible

toprodu eu tuationsonanylengths alewhileitis largerthanthehorizon.

Sin ethegoalof osmologyistounderstandtheuniverseonthebasisofphysi al

laws, this appeal to initial onditionis unsatisfa tory. The hallenge is

there-foreto givean explanation to thesmall "seed"perturbationswhi h allowthe

gravitationalgrowthofthematterperturbations.

The simplest me hanism for generating the observed perturbations is the

in-ationary osmology, as mentioned in previous se tions. Although originally

introdu edasapossiblesolutionsofalreadyseenproblemssu hasthehorizon

and atness problems, asanunexpe ted bonus, Ination hasthe useful

prop-ertytogeneratespe traofbothdensityperturbationsandgravitationalwaves,

throughtheampli ationofquantumu tuations: these perturbationsextend

from extremely short s ales to s ales onsiderably in ex ess of the size of the

observableuniverse.

In the simplest inationary model introdu ed earlier, Ination is driven by a

slowly-rollings alareld,theinaton: thislatter anbesplitin

φ(t, ~x) = φ

0

(t) + δφ(t, ~x),

(1.44) where

φ

0

isthe lassi al(innitewavelength)eld,thatistheexpe tationvalue oftheinatoneldontheinitialisotropi andhomogenousstate,whose

stress-energytensorandequationofmotionhavebeenalreadyexpressedin(1.32)and

(1.33);

δφ(t, ~x)

representsthequantumu tuationsaround

φ

0

. Thisseparation is justied by the fa t that quantum u tuations are mu h smaller than the

lassi alvalueandthereforenegligibilewhenlooking atthe lassi alevolution,

asdoneinpreviouspages. Nevertheless,exa tlythosequantumu tuationsare

responsibleforthe reationofinitialperturbationswhoseevolution annowbe

seenin thelarge-s alestru tureoftheuniverse.

Itisnotpossibleto des ribethegenerationofperturbationsofas alareld in

this ontext: thema hineryneededfotsu hataskisalmostthesameformalism

developedthroughoutthethesis,atleastalineartheoryof osmologi al

pertur-bationswouldbeneeded. Anyway, we angiveaheuristi explanationof why

we expe t that during Ination su h u tuations are indeed present and how

theseinatonu tuationswillindu e inturnpertubationsofthemetri [10℄.

Ifwetakeequation(1.33)addingthenon-homogenousterm

−∇

2

_φ/a

2

,andsplit

theinatoneldasin(1.44),thequantumperturbation

δφ

satisestheequation ofmotion

δ ¨

φ + 3Hδ ˙

φ −

∇

2

_δφ

a

2

+ V

′′

_{δφ = 0.}

(1.45)

Dierentiating (1.33) with respe t to time

t

and taking

H

onstant (we are duringinationaryphase!) wend