On the influence of local inhomogeneities on cosmological observables : from galaxies to the microwave background

On the Inuen e of

Lo al Inhomogeneities on

Cosmologi al Observables

From Galaxies to the Mi rowave Ba kground

On the Inuen e of

Lo al Inhomogeneities on

Cosmologi al Observables

From Galaxies to the Mi rowave Ba kground

Aleksandar Raki¢

Department of Physi s

BielefeldUniversity

From Galaxiesto the Mi rowave Ba kground

Dissertation zur Erlangung des Grades eines Doktors der

Naturwissens haften (Doktor rerum naturalium)

amFa hberei h Physik der Universität Bielefeld

vorgelegt von: Aleksandar Raki¢

geboren am28 Mai 1979in Menden

Guta hterundPrï¾

1

2

fer- Referees: Prof. Dr. Dominik J.S hwarz

Prof. Dr. Dietri hBödeker

Prof. Dr. ReinhartKögerler

Prof. Dr. Andreas Hütten

Abstra t. Despite thegood onsisten y ofthe osmologi alstandard modelwiththebulk ofpresent

obser-vations,anumberofunanti ipated featureshavere entlybeendete ted withinlarge-angledata oftheCosmi

Mi rowaveBa kground. Amongthesefeatures arethe anomalousalignmentsofthe quadrupole and o topole

withea hother,theirunexpe tedalignmentswith ertainastrophysi aldire tions(e.g.equinox,e lipti )aswell

asthestubbornla kofangularauto orrelationons ales

> 60

◦

.Wepursuetheideathatpro essesofnon-linear

stru tureformation ould ontributetothe large-s aleanomaliesviaalo al Rees-S iamaee t. Wendthat

existingstru turesareabletoprodu eCMB ontributionsupto10

−5

.Foranaxiallysymmetri setupweshow

thatthisee tdoesindu ealignments,albeit notof thesameformasextra ted fromthe data, andthatyet

aSolarsystemee tseemspreferredbythe data. Moreover,weaddresstherelationshipbetweentheintrinsi

alignmentofquadrupoleando topoleontheonehandandtheanomalousangulartwo-point orrelationfun tion

ontheotherhand. Wedemonstratetheabsen eofany orrelationsbetweenthemandareabletoex ludethe

joint aseathigh onden ewithrespe ttore entdata. Thisresultenablesustoputstringent onstraintson

anyrelevantmodelthatexhibitsanexpli itaxialsymmetry.

Keywords. gala ti dynami s,darkmatter, osmi mi rowaveba kground,large-s alestru tureofuniverse,

darkenergy,generalrelativity, osmology

Abriss. Trotzder gutenÜbereinstimmungdes aktuellen kosmologis henStandardmodellsmit dem Groÿteil

der vorhandenen Daten, wurdenkürzli h unerwartete Eigens haften der kosmis hen

Mikrowellenhintergrund-strahlungbezügli hdergöÿtengemessenenWinkelskalenbekannt. Diesebeinhalten: dieanomale

Ri htungskor-relationzwis henQuadrupolundOktupolselbst,ihreunverstandeneAusri htungbezügli hbestimmter

astro-physikalis her Ri htungen (z.B.Equinox, Ekliptik) alsau heine Temperatur-Zweipunktskorrelationsfunktion,

dieaufWinkelskalen

> 60

◦

unerwarteterweisevers hwindet.Wiruntersu hendieMögli hkeit,dassProzesse,die derni htlinearen Strukturbildungangehören,zudenAnomalienbeitragenkönnen,undzwardur hdenlokalen

Rees-S iamaEekt.Wirnden,dassderRees-S iamaEektdur htatsä hli hvorhandene,sehrmassive

Struk-turen, dieGröÿenordnung 10

−5

inden Temperaturanisotropien errei henkann. Wir könnenzeigen, dass,im

Rahmeneineraxial-symmetris henGeometrie,inder Tat bestimmte Ri htungskorrelationen dur hdenEekt

induziertwerden,diesejedo hni htvon derglei henFormwiedieindenDaten gefundenensind. Glei hwohl

wirdeineKorrelationmitdenRi htungenunseresSonnensystemsvondenDatenbevorzugt. Auÿerdem

unter-su henwirinwiefernzwis hender intrinsis henAusri htung von Quadrupol undOktupolzueinanderund der

anomalenZweipunktskorrelationsfunktioneineAbhängigkeitbestehenkönnte.Wirdemonstrieren,dasskeinerlei

Abhängigkeitzwis hendiesenAnomalienbestehtundwirkönnendaskombinierteSzenariomithoherSignikanz

auss hlieÿen. Dadur h sindwirinderLage,s harfeEins hränkungenanzugeben, diefürallerelevanten

axial-symmetris henModellebindendseinmüssen.

S hlagwörter. Galaxiendynamik,dunkleMaterie,kosmis heMikrowellenhintergrundstrahlung,groÿräumige

FromGalaxiesto theMi rowave Ba kground

Thisthesisisbaseduponthefollowingpubli ations:

◮

Mi rowaveSkyandtheLo alRees-S iamaEe t

AleksandarRaki¢,SyksyRäsänenandDominikJ.S hwarz;Mon. Not. Roy. Astron. So . Lett. 369:

L27L31,2006;astro-ph/0601445

◮

CorrelatingAnomaliesoftheMi rowaveSky: TheGood,theEvilandtheAxis

AleksandarRaki¢andDominikJ.S hwarz;Phys.Rev. D75: 103002,2007;astro-ph/0703266

◮

CanExtragala ti ForegroundsExplaintheLarge-AngleCMBAnomalies?

AleksandarRaki¢, Syksy Räsänenand DominikJ. S hwarz; astro-ph/0609188; to appear inthe

pro- eedingsofthe11thMar elGrossmannMeetingongeneralrelativity

Publi ationsinpreparation:

◮

GeneralRelativisti Gala ti Dynami sandtheNewtonianLimitofLewis-PapapetrouSpa e-Times AleksandarRaki¢andDominikJ.S hwarz

◮

Ba krea tionEe tsontheObserver'sPastLightCone ThomasBu hert,AleksandarRaki¢andDominikJ.S hwarz

Thework ontainedinthisthesisispartoftheresear hdonewithintheInternationalResear hTraining

Group (GRK 881) entitled as Quantum Fields and Strongly Intera ting Matter: From Va uum to

ExtremeDensityandTemperatureConditions. Thisgraduates hoolisajointproje toftheUniversity

ofBielefeldand theUniversitéParis-SudXI (Paris VI,Paris VII,Sa lay);itis fundedby thegerman

resear hfoundation(DFG)andsowastheauthor.

GRK881

PhDthesisintheoreti alphysi s

Author: AleksandarRaki¢

E-mailaddress: araki web.de

Typefa e: ComputerModernRoman8pt,9pt,10pt ,11pt,12pt

Distribution: L A T E X2

ε

using

AMS

L A T E Xandhyperref

Notation 1

Prefa e 3

Part I. Exa t Solutions asToy Models 11

Chapter1. TheCosmologi alProblemofDarkEnergy 13

1.1. Fa etsoftheProblem 14

1.2. DarkEnergy andtheStandardCosmologi alModel 15

1.3. AnInhomogeneousAlternative? 27

Chapter2. TheCosmologi alProblemofDarkMatter 45

2.1. Dire tEviden eandLensing 45

2.2. Classi alEviden efromDynami s 51

2.3. ModellingGalaxieswithGeneralRelativity 55

Part II. Axisymmetri Ee ts in the CMB 75

Chapter3. OntheCosmi Mi rowaveBa kground 77

3.1. OverviewofSour esofCMBAnisotropy 77

3.2. Re ombination 80

3.3. Observablesof theCMB 85

Chapter4. Extrinsi AlignmentsintheCMB 95

4.1. TheAlignmentAnomalies 96

4.2. Lo alRees-S iamaEe t 97

4.3. AngularPowerAnalysis 101

4.4. Extrinsi AlignmentAnalysis 103

4.5. Con lusion 106

Chapter5. Intrinsi Alignmentsin theCMB 109

5.1. Introdu tion 110

5.2. Choi eofStatisti 112

5.3. StandardModelPredi tions 113

5.4. In lusionofaPreferredAxis 117

5.5. Con lusion 119

SummaryandOutlook 121

A knowledgements 123

Part III. Appendi es 125

AppendixA. Criti alValuesof

Ω

m

and

Ω

Λ

intheFRWModel 127

B.1. GeneralSpheri allySymmetri Spa etimewithZeroVorti ity 131

B.2. EinsteinEquationsoftheLemaître-Tolman-BondiModel 132

AppendixC. RotatingPost-NewtonianMetri s 135

C.1. FullDierentialRotation 135

C.2. SpatialCurvatureTerms 135

AppendixD. Aspe tsofStru tureFormation 137

D.1. GravitationalInstabilitiesandPe uliarVelo ities 137

D.2. Statisti alPropertiesoftheDensityField 138

D.3. SilkDampingandHierar hy 139

AppendixE. ThermalHistoryin aNutshell 143

E.1. NeutrinoDe oupling 143

E.2. Ele tron-PositronAnnihilation 144

E.3. Nu leosynthesis 145

AppendixF. AdditionalPlotsandResults 147

Throughoutthisworkwewill usethefollowingmetri signature,

(−, +, +, +) .

Bysmalllatinindi es,runningfrom

1

to

3

,wedenotespatial omponentsoftensors,e.g.

K

ij

. Usingsmallgreekindi es,runningfrom

0

to

3

,wedenote four-dimensional omponentsof ten-sors,e.g.

K

µν

. Wemakeuseof theEinstein summation onvention.

Partialderivativesareindi atedbya omma,

K

µν,λ

≡

∂

∂x

λ

K

µν

and ovariantderivativesbyasemi olon

K

µν;λ

≡

∂

∂x

λ

K

µν

− Γ

ρ

λµ

K

ρν

− Γ

ρ

λν

K

ρµ

.

Thesign onventionswhi h weuseforthe osmologi al onstant, forthedenitionof the

Rie-mann urvature tensor as well as for the other relevant quantities in the Einstein equations

are given in app. B. The spatial Ri i s alar is written aligraphi ally throughout the text,

R

_≡

(3)

_R

i

i

.

Ve torsandve toreldsare writtenin boldfa e,e.g.

ξ

,

L

σ

. Normalve torsaredenoted bya hat,e.g.

x

ˆ

.

Wedenotethesymmetrisationandantisymmetrisationoftensorsby

K

{µν}

≡

1

2

(K

µν

+ K

νµ

) ,

K

[µν]

≡

1

2

(K

µν

− K

νµ

) .

In hap. 2wewill dealwith axisymmetri systems, andthereforethe operators

∆

(3)

and

∆

(2)

denotethethree-dimensionalandtwo-dimensionalLapla eoperatorsin ylindri al oordinates.

Theuseof artesian oordinatesisexpli itlyindi ated,e.g.

∆

(3)

cart

.

Themostfundamental osmologi alobservationone anthinkofisthedarknessofournight

sky. Atrstglan e,thismightappeartrivial,buttheappropriatequestionis,howisitpossible

thatourskyisdarkatnight? Theproperanswertoithas ru ialimpli ationsfor osmology. In

theearlydaysof astronomy,the ommon osmologi alparadigmstatedthat theUniverse was

eternal,innite andof Eu lidean geometry. Followingthis paradigm, in 1826Heinri hOlbers

al ulatedthetotalradiationenergydensityofstarsthatwouldbepresentin su h aUniverse.

Thestarsweretaken aspointsour eswith onstantluminosity andtheir numberdensity was

also onstant. Theresultofthe al ulationisastonishingly absurd: therewouldbeaninnite

radiationdensity omingfromstarlight. Interpretedwithinastati ,inniteandEu lideanworld

model, the ommon fa t that ournightskyis darkbe omes suddenlyamystery. This la kof

opti alba kgroundlightis usuallyreferredto asOlbers'paradox,but itshould bementioned

thattheproblemwasdis ussedalreadymu hearlier,forinstan ebydeCheseauxin1744.

Withinthemodernstandardmodelof osmology,a ommonwayofresolvingOlbers'

para-doxliesinassumingaBigBangandtakingthe osmologi alexpansionofspa etimeintoa ount.

InaUniversethathasexistedforanniteamountoftime,theextensionoftheobservablepart

of the Universe the horizon is also nite, and therefore only a limited number of stars is

potentially observable. In this formulation of Olbers' paradox we assumed a distribution of

pointsour es. We ouldgoonestepfurther and onsidertheextended surfa esoftheemitting

stars. Then it turns out that every line of sight towardus must start at some nite surfa e

andwithintheoldworldviewwewouldinevitablybeledtoaskythatis,duetoproje ted

overlap,fully overedbytheluminoussurfa esofthestars. Thebrightnesstemperatureofstars

isindependentofdistan e intheEu lideanpi ture,andsothisformulationofOlbers'paradox

statesthatthewholeskyshouldbeashotasthesurfa eofatypi alstar. Nowtheresolutionof

Olbers'paradoxwithin modern osmologybe omessomewhatdierent. AssumingaBig Bang

and ontinuous osmi expansion, one an extrapolate that there indeed must have existed a

ommonhotemissionsurfa e,namelythesurfa eoflasts atteringatwhi htheUniversebe ame

transparentfor photons. This instant marksthe birth of the Cosmi Mi rowave Ba kground

(CMB)radiation. Now,sin elasts attering o urred alongtimeagowhenthetemperature

of theUniverse was around

3000

K and the Universe has expanded eversin e, one annd thattheCMBphotonshaveundergonearedshiftingbyafa torofroughly

1100

uptoday. This resultsin apresent-day ba kgroundtemperature of

2.73

K. In this sense, the existen e of the CMBrepresentstheresolutionofOlbers'paradox: we annotobservea

3000

Khotsky,be ause the osmi expansionhas ooleddowntheprimordialradiation.

Today,measurementsofthetinyanisotropiesinthemi rowaveba kgroundradiationprovide

a osmologi alprobeofutmostrelevan e. With satellitemeasurementsoftheCMB likethe

Wilkinson Mi rowave Anisotropy Probe (WMAP)  a onsiderable pre ision in osmologi al

datahasbeenrea hed.

Due to itsverygood a ordan ewith CMBmeasurements,aswell aswith otherdatasets

from the observation of the large-s ale stru ture at lower redshifts, a osmologi al standard

model has emerged, the inationary

Λ

Cold Dark Matter model. Among the energy density ingredientsofthatmodelarethe ontributionsofDarkEnergy (

76%

),DarkMatter(

20%

)and baryoni matter(

4%

). Althoughtheyrepresentdominant ontributions,thestandardmodelis notexplanatorywithrespe ttothenature andoriginofthedark omponentsoftheUniverse.

Althoughalotofeortisinvested,andalthoughnumerousattemptstoatta ktheproblem an

be found, there exists nosettled explanation for thedark omponentsof the standardmodel;

theyremainpoorelyunderstooduptoday. Moreover,the urrent osmologi alstandardmodel

isbaseduponarelativelysimple,homogeneousandisotropi solutionoftheunderlyinggeneral

relativisti eld equations, the Friedmann-Robertson-Walker spa etime. Within this model,

bothCMBandotherdatarequiretheUniversetobespatiallyat.

In hap. 1 we review the phenomenology of the urrent standard model of osmology as

wellasitstheoreti alframework. Wefo usonthe osmologi alproblemofDarkEnergyandwe

explainitsbasi experimentaleviden e. Thevalidityofthe rudestandardmodelassumptions

ofhomogeneityandisotropyonlarges ales anbequestioned. Itissubje tto urrentdebatein

howfarinhomogeneousmodels anttheavailabledatathatindi atesana eleratedexpansion

of the Universe. The ru ial dieren e is that inhomogeneous models are potentially able to

a hievethiswithoutDarkEnergy. Inparti ularweanalysethespheri allysymmetri

Lemaître-Tolman-Bondimodelanddis usshowitmay hangetheinterpretationofsupernovaandCMB

data. InordertousetheinhomogeneousmodelfortheCMBanalysisin thelater hapters,we

nallypresentanalyti al ulationsoftheintegratedSa hs-Wolfeee tin thatmodel.

Chap. 2dealswith the osmologi alproblemofDarkMatter. Wereview presenteviden e

forDarkMatter andfo usespe ially ontheatgala ti rotation urves. Weomit dis ussions

ofparti le andidatesfor DarkMatter andfo uson anunusual approa h, namely thegeneral

relativisti modelling ofgalaxies. Regardingrotation urves,the omparison from whi h Dark

Matterfollowsinthestandardpi ture,isalwaysa omparisonbetweenNewtonianphysi sand

thedata. It anbequestioned whether generalrelativisti termsreally anbefully negle ted.

Infa t,re entlyageneralrelativisti model ofagalaxyhasbeenpresented(the

Coopersto k-Tieu model) in whi h it is laimed that Dark Matter is made superuous. Partly, hap. 2

is very te hni al; we arry out various analyti al analyses in order to better understand the

Coopersto k-Tieumodelandespe iallyitsNewtonianlimit.

A ru ial omponent of the standard model is the inationary s enario. Ination

pre-di ts anearly epo h of dramati global expansionof spa etime and so providesthe seeds for

theformationof large-s alestru ture throughafreeze-outof primordialquantum u tuations

on ma ros opi s ales. As a onsequen e, the simplest inationary theories, predi t a nearly

s ale-invariant power spe trum of statisti ally isotropi , adiabati and gaussianly distributed

primordialu tuations.

Despite theremarkablea hievementsofthestandardmodel,there arealso someproblems

withit. WhenanalysingWMAPdatafromthelargestangularseparations ales,several

anom-aliesarefound,whi h arein oni twiththepredi tionofstatisti alisotropyoftheCMB.

Afterreviewingthebasi physi alme hanismsthat ontributetotheCMB, anddis ussing

theunderlyingtheoreti alframeworkin hap.3,weapproa htheproblemofthelarge-s aleCMB

anomaliesin hap.4and hap.5. In hap.4ouransatzisalo alRees-S iamaee t the

non-linearanalogueoftheintegratedSa hs-Wolfeee t. Westatethatthelo al Rees-S iamaee t

ofvast,yetnon-virialisedstru turesindu essigni ant ontributionstothelarge-s aleCMB.We

omputeitsinuen eonthephaseanomalieswiththehelpofastatisti alanalysisandndthat

anRees-S iamaee t modelled byasimplyspheri aloverdensity anbeex ludedat high

onden e. In ontrastto hap.4, hap.5 opesonlywithintrinsi alignmentsamongthelowest

CMBmultipoles. Therearetwo lassesofanomalies,phase(dire tional)anomaliesandangular

poweranomalies. Weaskto what extentanomalies ofthetwo lassesare orrelatedwith ea h

other, be ause this isof importan e for modelbuilding. Weperforman exhaustivestatisti al

analysis anddemonstrate theabsen e of su h orrelationswith high signi an e. Further, we

nd stringent onstraints on any models, trying to explain the anomalies, that exhibit axial

sondern knapp überdemBoden.

Es s heint mehrbestimmtstolpernzu

ma hen, als begangenzu werden.

FranzKafka(18831924)

AphorismenBetra htungenüberSünde,Leid,

if anytherebewhi h isintelligible tous,of

thevasta umulations ofmatter whi h

appear, onourpresentinterpretationsof

spa e andtime,tohave been reatedonlyin

orderthattheymay destroythemselves?

Whatisthe relationoflife tothatUniverse

of whi h,if weareright, it ano upyonly

sosmall a orner? Whatif anyisour

relationtothe remotenebulae, for surely

theremustbesomemore dire t onta tthan

thatlight an travel betweenthemandusin a

hundredmillion years? Do their olossal

in omprehending masses omenearer to

representing themain ultimaterealityof the

Universe,ordo we? Arewemerelypartof

thesamepi tureasthey, orisitpossiblethat

wearepartof the artist? Aretheyper han e

onlyadream, whilewearebrain ellsin the

mindof thedreamer? Orisour importan e

measuredsolely by thefra tionsof spa e and

timeweo upyspa e innitelyless thana

spe kofdustin alarge ity,andtimeless

thanoneti kof a lo kwhi h hasenduredfor

agesand will ti kon for agesyet to ome?

SirJamesJeans(18771946)

The Cosmologi al Problem of Dark Energy

WhydoesDarkEnergyseemtodominatetheenergybudgetofthe osmos? Whatdoesthis

major ontributor onsist of at all? Why isthe absolute valueof theDark Energy density so

tiny as ompared tothe expe tation fromquantum theory? Undoubtedly,the hallengeposed

byDarkEnergyisthe mostfar-rea hingofthegrandopen questionsin modern osmology. It

is tightly relatedto the questionof howfar there is ru ial physi s missing in the underlying

theoriesat themoment;anexamplethereofwould beauniedtheoryofgravityandquantum

elds. There is ageneri relationto the veryfundamental questionof howthe absolute

zero-pointenergiesofquantagravitate. ThenotionofDarkEnergygoeshandinhandwithEinstein's

osmologi al onstant

Λ

. Ontheotherhand,alsodynami als alareldsthatwould ontribute to

Λ

in atime-dependentwayare onsidered,likeforinstan equintessen eormodulields.

Figure 1.1. Theinuen eofDarkEnergyrea hesfromthesmallesttothelargest

stru turesintheUniverse. Left: mi ros opi imageofatinyball(

d ≃ 10

−1

mm)that

ismountedatasmalldistan euponasmoothplateinordertomeasuretheo urring

(ele tromagneti )Casimiree t. TheminuteCasimirfor epullstheballtowardthe

plate be ause the numberof va uumu tuation modes inthe smallspa e between

ball and plateis limited,whereasthe wavelengths ofva uumu tuationso urring

inthe`freespa e'ontheoppositesideoftheplate antakearbitraryvalues. Va uum

u tuationssimilartothosefromtheCasimiree tareasso iatedwithDarkEnergy

butinthis aseare generatedbyspa e itself. ThenowadaysdominantDarkEnergy

a ts as a repulsive for e on the largest s ales, eventually ausing the Universe to

expandforever. Right: animageofthe lusterofgalaxiesnamedSDSSJ

1004 + 4112

after itsdete tion withinthe SloanDigitalSkySurvey. The lusteris aroundseven

billion light years away (

z = 0.68

), lo ated inthe onstellation of Leo Minor, and

represents a beautiful sampleof Large-S ale Stru ture. Also, due to gravitational

lensing o thehugelensing mass ofthe luster,ar imagesof moredistantgalaxies

in the ba kground an be seen in the image. A ording to observations of distant

supernovae(

z & 0.2

) there essionofgalaxiesis urrentlyspeedingupas duetothe

1.1. Fa ets of the Problem

Thefamousmismat hof

∼ 120

ordersofmagnitudethat resultsfromtryingtoestimate

Λ

fromquantumeldtheoryillustrateswelltheamountofourignoran eregardingthefundamental

physi sthat may be involved. Likewise the Dark Energy whi h is so poorly understood does

in fa t onstituteawhole

∼ 70%

of theenergydensity ontentof theUniverse, whi h readily indi ates the weight of the problem. Still, it is always adequate to arefully re onsider all

assumptions that are madein order to get aphysi al result, espe ially if it is su h a weighty

one. In fa t, theabovesituation resultsfrom a omparison ofa largevarietyof astronomi al

testswiththe osmologi alstandardmodel. Additionally,the omparisonof

Λ

withtheabsolute zero-pointenergy takespla e within quantum eld theory whi h is at the basis of the a tual

standardmodelofelementaryparti lephysi s. Wewanttoemphasisethattheempiri albasisof

the osmologi alstandardmodelisfarlesssubstantialthanthatofthestandardmodelofparti le

physi s. Oneof themain dieren esisof oursetheinherentimpossibilityto doastronomi al

measurements in su h a repeatable and ontrolled way as it is done in a laboratory. That

is,mostly astronomersare leverspe tators,waiting for therightmomentof observation, but

allwaysbeingin apableoftou hingorturningthesour einordertorepeattheirmeasurement.

Aswewillseebelow,oneofthemostweightyeviden efor

Λ

omesfrom su hanastronomi al measurement,namelytheobservationofdistantsupernovae.

Withinthestandard osmologi almodeltheenergy-matter ontentoftheUniverseis

har-a terisedbyfourdimensionlessdensityparameterswiththefollowingnormalisation:

(1.1)

Ω

m

+ Ω

r

+ Ω

Λ

+ Ω

k

= 1 .

Here,

Ω

m

isthedensityofmatterinvolvingallkindsofmatterpresentwhetherdarkorluminous, baryoni ornon-baryoni ;

Ω

r

∼ 10

−4

standsfortheenergypresentin the osmi mi rowaveas

well as in the primordial low-mass neutrino ba kground radiation;

Ω

k

stands for the energy-matter ontributionasso iatedwiththe urvatureofspa eduetoGeneralRelativityandnally

Ω

Λ

isthe ontributionof DarkEnergy. From measurementsof e.g.the CMBit isknownthat thethree-geometryofspa eisattoahighdegreeofa ura ysu hthat

Ω

k

anbesettozero. Also negle tingthe minor ontribution from

Ω

r

, a ouple of dierent lasses of astronomi al observationssuggesttheso alled osmi on ordan e:

(1.2)

Ω

b

≃ 0.04 ,

Ω

DM

≃ 0.20 ,

Ω

Λ

≃ 0.76 ,

where, a ordingto usual notation,wesplitthe matterdensityparameter

Ω

m

into abaryoni ontribution and a ontribution from DarkMatter. The issue of Dark Matter is dis ussed in

moredetailin hapter2. Butwhatevertheparti ular ompositionofthenumeri alvaluesofthe

dierentenergy-matter omponents,asinferredin theframeworkof the osmologi alstandard

model may try to tell us, one result is parti ularly striking: only

4%

of the whole is due to well-understood physi s, i.e.to baryons. Anothersurprising feature of DarkEnergy is known

asthe oin iden eproblem. Itrefersto thefa tthat the ontributionof thetime-independent

Λ

parameter,if wewould measureit together with theother osmologi aldensity parameters in the past when the universe had only around one tenth of its present size, would be only

Ω

Λ

≃ 0.003

. That is, the inuen e of

Λ

, ausing theexpansionof theUniverseto a elerate, appears to be ome signi ant at just around at the present time. It is un laried in howfar

these` oin iden es'areree tingsomedeepphysi al ontiguity. However,itis on eivablethat

the osmologi al onstantmightbe arunningand would approa h somenaturalvalueat late

times[PR03℄.

We onsider thepossibilityof

Λ

itselfbeingasuperpositionofdierentphysi alee ts: (1.3)

Ω

Λ

= Ω

Λ,Einstein

+ Ω

Λ,QF

+ Ω

Λ,unknown

.

The term

Ω

Λ,Einstein

is nothing else than the original osmologi al onstantas introdu ed by Einstein in order to maintain stati osmologi al solutions of his eld equations;

Ω

Λ,QF

is a ontributionfromvirtualparti le-antiparti leu tuations in thequantum va uum;

Ω

Λ,unknown

thatquantumu tuations

Ω

Λ,QF

reallydoexistisimpressivelydemonstratedbymeasurements ofthe(ele tromagneti )Casimiree t,seeg.1.1. TheCasimiree t anbemeasuredbetween

mi ros opi obje ts,forexamplesmall ondu tingplates,thatarepositionedatatinydistan e

toea hother. Whereasthequantumu tuations oftheva uum,aspredi tedwithinquantum

eld theory, an populate arbitrary modes in empty spa e, the number of possible modes in

betweenthemi ros opi obje tsislimitedandsotheenergyofthesystemissuppressed. This

resultsinanattra tivefor ethatisofmeasurablestrengthfore.g.theele tromagneti eldand

ispurelyduetosubtlequantumee ts.

The problem one naturally en ounters with the ontribution of

Λ

may be demonstrated byusing the CMB asan example[PR03℄. TheCMB has amonopole temperature of

≃ 2.7

K and energy density

Ω

CMB

∼ 10

−5

rea hing its maximum at the Wien peak

λ ∼ 2

mm. Here the photon o upation number is

∼ 1/15

. Given a ertain frequen y, the zero-point energy amountstohalftheenergyofthephoton. Thereforethezero-pointenergyoftheele tromagneti

eld at theWien peak translatesinto a ontribution of

δΩ

Λ,CMB

∼ 10

−4

to the DarkEnergy

densityparameter. Asitwillbe ome learfromequation(1.32)thesumoverwavelengthss ales

a ordingto

λ

−4

andthus wewouldhave

δΩ

Λ,CMB

∼ 10

10

at visiblewavelengths! Thisnaive

extrapolationalreadyyields su h anabsurdgure. However,aswasalreadymentionedabove,

itmaybehypothesised[PR03℄ thattheDarkEnergydensityasso iatedwith

Λ

is runningand hasrea hednowadaysbe auseDarkEnergyhadalmost

13.4

billionyearstimeforrunningby now losetoavaluethat wouldbesomewhatnatural,namelyzero.

1.2. Dark Energy and the Standard Cosmologi alModel

Beforewearegoingtodis ussratherdire teviden eforare enta elerationofthe osmi

expansion,wewill on iselyreviewthe urrentstandardmodelof osmology. This omprisesthe

underlying symmetriesof the Friedmann-Robertson-Walkerspa etimeas well asthe resulting

generalrelativisti dynami softhemodel. Alsothebasi on eptsandthe onsequen esofthe

standardinationarys enarioarereviewed.

In osmology there exist several denitions of what may be attributed as an observable

distan etoanastronomi alobje t. Thenon-trivialpointisthatthevariousdistan emeasures

giveapproximatelythesameresultonlyfornearbyobje tsandmoreoverthattheirmeasurement

fordistantobje tsissensitivetotheparti ulardynami softheunderlyingtheory. Thereexists

re enteviden e that supportsthepresen eof DarkEnergyprovidedbytheanalysis ofdistant

supernovae. Under theassumptionthatsupernovaeoftypeIaforma lass ofstandard andles

their measured brightness an be used to dire tly test the distan e-redshift relation within

dierentdynami alrealisationsofthestandardmodel.

1.2.1. TheStandardModelina Nutshell. Avery ru ialstatementthatismaderight

fromthebeginningisthattheUniverseappearsisotropi tousinaglobalsensewhenobserved

fromearth. Se ond, followingtheCoperni anstandpoint itis assumedthat anobservationof

the Universe made from any other galaxy should also look isotropi for the observers there.

On e wea ept this, the Universe must also be homogeneous be auseof its isotropy around

anypoint. Of ourse,observationsofournearneighbourhooddoneitherlookhomogeneousnor

isotropi at rstglan e. Inthestandardmodelitis assumedthat there is atransition from a

lumpy toan approximatelysmooth pi tureat as aleof roughly

100

Mp . This implies, that whenwepla eballsofradius

100

Mp in theUniverseatrandomlo ationsandwemeasure the mass prole within an ensemble of balls then the root mean square u tuation of the values

takenat

100

Mp isroughlyequalto themeanvalue,su hthat we anregardtheu tuations at large s ales as perturbations on top of the homogeneous model. On the other hand, the

smallerthe s ale,themorenon-lineararethedepartures ofu tuationsfrom homogeneity. In

thefollowingwereviewtheni eoverviewpaperbyPeeblesandRatraonDarkEnergy andthe

standardmodel[PR03℄.

distan e

D

P

betweentwowell-separatedgalaxies asafun tion of osmi time

t

is

(1.4)

D

P

(t) ∝ a(t) ,

where

a

isthes alefa tor. But

a

isdenedsu hthatitisindependentofthe hoi eofgalaxies wemakeforthe omparison. Thustheexpansion(1.4)preserveshomogeneityandisotropy. The

derivativeof (1.4)givesustheproperspeed

(1.5)

v

P

(t) =

dD

P

dt

= H(t)D

P

,

H(t) ≡

˙a(t)

a(t)

,

introdu ingtheHubbleparameter

H

anddenotingderivativeswithrespe tto osmi timewith adot. ThevalueoftheHubbleparameterasmeasuredtodayisa entralparameterandsowe

givehereits urrentmeasure(2007)a ordingto [Y

+

06℄

(1.6)

H

0

= 100 h km s

−1

_Mpc

−1

_{= h (9.78 Gyr)}

−1

_{with h = 0.73}

+0.04

−0.03

.

The a tual expansion of the Universe was rst observed in 1929 and it is referred to as the

Hubbleexpansionduetoitsdis overer[Hub29℄.

A law similar to (1.4) also holds for the wavelengths of light signals that are ex hanged

betweentwogalaxies.The hangeinwavelengththatasignalagivenfeatureinthespe trum

undergoesthat hasbeenemittedfrom adistantsour eamountsto

(1.7)

λ

ob

λ

em

=

a(t

ob

)

a(t

em

)

≡ 1 + z ,

and

z

is alled the osmologi alredshift. Theredshift providesthemost onvenient hara ter-isti to label observations of theUniverse that rea h into the veryfar past. Forexample, the

de ouplingofmatterandradiationin theyoungUniversewhi histheoriginoftheCMB

radi-ation, o urred at around

z = 1088

. TheUniverse isionised today;from CMBmeasurements oneinfers thatreionisationtookpla e atredshiftsof around

z ≃ 10

. Thegalaxy lusterSDSS J

1004 + 4112

showning.1.1isobservedat aredshiftofaround

z ≃ 0.68

. Howingeneralthe redshiftistranslatedintodistan es,orvi eversa,isgeneri allydependingontheparametersof

theunderlyinggeneralrelativisti model. However,givenasmallredshift

z < 1

,equation(1.7) be omesHubble'slaw,whi hthenreadstolowestorder:

cz = HD

C

.

Theresultssofarhavebeenobtainedbyusinghomogeneityandisotropyonly,andrepresent

the low-redshift limit of the standard model. However, for extrapolation to higher redshifts

z > 1

,thegeneralrelativisti formulationofthetheoryistobeused. The ru ialassumptionsof homogeneityandisotropyareree tedbythewell-knownFriedmann-Robertson-Walker(FRW)

spa etime (1.8)

ds

2

= −dt

2

+ a

2

(t)



1

1 − kr

2

dr

2

_{+ r}

2

_dθ

2

_{+ sin}

2

_θdϕ

2



.

Throughremapping of the radial oordinate one usually normalisesthe spatial urvature

pa-rameter

k

su hthatittakesthevalues

k = 1, 0, −1

,whi hstandfora losed,atoropenspatial geometryofthemodel. Themetri anberewrittenas

(1.9)

ds

2

= −dt

2

+ a

2

(t)



dχ

2

+ S

k

2

(χ) dθ

2

+ sin

2

θdϕ

2



,

byintrodu ingthefun tion

S

k

(χ)

with

(1.10)

S

k

(χ) =







sinχ

for k = 1

χ

for k = 0

sinhχ

for k = −1

.

EmployingtheFriedmann-Robertson-Walkermetri andtheassumptionthatonlarges alesthe

galaxiesbehavelikethe onstituentsofaperfe tuid, one ansolvetheeldequations

(1.11)

G

µν

≡ R

µν

−

1

and,denoting osmi timederivativeswithadot,obtaintheresult: (1.12)

¨

a

a

= −

4

3

πG (ρ + 3p) +

Λ

3

.

The ovariant onservationofenergyandmomentum

T

µν

;µ

= 0

impliesthenadditionally

(1.13)

˙ρ = −3H (ρ + p) .

Integratingtheequations(1.12) and(1.13)yieldstheimportantFriedmannequation

(1.14)

H

2

₌

8

3

πGρ −

k

a

2

+

Λ

3

,

andtheintegration onstant

k

isrelatedtothepresentvalueofthespatial urvaturevia

(1.15)

Ω

k

= −

k

H

2

0

a

2

0

.

If

Λ

is onstant,ausefulwayofwritingtheFriedmannequationis

(1.16)

H

2

_{(z) = H}

2

0



Ω

m

(1 + z)

3

+ Ω

r

(1 + z)

4

+ Ω

Λ

+ Ω

k

(1 + z)

2



,

andsimilarlyonerewritestheequation(1.12)

(1.17)

¨

a

a

= −H

2

0



Ω

m

(1 + z)

3

2

+ Ω

r

(1 + z)

4

− Ω

Λ



,

wherebytheremainingdensityparametersofthestandardmodel

Ω

i

aregivenby

(1.18)

Ω

m,r

=

ρ

m,r

ρ

crit

,

ρ

crit

≡

3H

2

0

8πG

,

Ω

Λ

=

Λ

H

2

0

.

Theuseof (1.16) liesin thefa t thatone animmediatelyread otheredshift dependen eof

therespe tive omponentsoftheFriedmannmodel. Therein,

Ω

m

standsforallnon-relativisti matter whose pressure we negle t (

p

m

≪ ρ

m

). We see that the mass density is diluted by the expansion of the Universe as

ρ

m

∝ a

−3

_{∝ (1 + z)}

3

. Further,

Ω

r

stands for radiation (e.g.theCMB)as wellasrelativisti matterwithequationofstate

a

w = 1/3

,andbehaveslike

ρ

r

∝ a

−4

∝ (1 + z)

4

under expansion. By onstru tion,

Λ

is onstant for the moment, and furtherthedensity orrespondingtospatial urvature(1.15)isdilutedas

ρ

k

∝ a

−2

_{∝ (1 + z)}

2

.

eq. ofstate density s aling Hubble

w

ρ ∝ a

−3(1+w)

_{a(t) ∝ t}

3(1+w)

2

_{H(t) =}

2

3(1+w)

1

t

radiation,

w =

1

3

ρa

−4

a(t) ∝ t

1/2

H(t) =

1

2t

matter,

w = 0

ρa

−3

_{a(t) ∝ t}

2/3

_{H(t) =}

2

3t

Table1.1. StandardsolutionstotheFriedmannequationforaradiationdominated

andamatterdominatedUniverse. TheFRWexpressionsfordensity,s alefa torand

Hubbleparameterassuminga ontributionwithequationofstate

w

aregiveninthe

rstline. RegardingaDarkEnergy ontributionwith

w = −1

thedensityis onstant andintegrationoftheFriedmannequationyieldstheexponentialbehaviour(1.25) .

Next, we want to onsider the properties of

Λ

in further detail. As inspired by spe ial relativity, we an make the assumption that every inertialobservershould measure the same

va uum. An inertialobserverisan observerwholiveslo ally in aMinkowskianframe,that is

hismetri is hara terisedby

η

µν

= diag(−1, 1)

. Now,theform ofthe metri isleft invariant byLorentztransformationtosomeotherinertialobserver'sframe. Be auseweassumedthatall

inertialobserversshouldsee thesameva uum,theenergy-momentumtensoris

(1.19)

T

Λ

µν

= ρ

Λ

g

µν

,

a

witha onstantva uumenergydensity

ρ

Λ

. Thustheeldequations anbewrittenintheform (1.20)

G

µν

= 8πG (T

µν

+ ρ

Λ

g

µν

) ,

whi h ree tsEinstein's originalidea b

ofmodifyingtheenergy-matter ontentoftheUniverse

by adding a onstant

Λ

. We see that Dark Energy behaveslike an ideal uid with negative pressurea ordingtotheequationofstate

(1.21)

p

Λ

= −ρ

Λ

.

At the time Einstein thought about this modi ation, the Hubble re ession of nebulae was

notyet established; quitethe ontrary,astati osmoswasthestate oftheart, whi hwasan

extrapolation of the nding that nearby stars moved at low velo ities. In order to obtain a

stati solutionwith

¨

a = 0

Einstein introdu ed an

Ω

Λ

in modern languagetoneutralise the (positive) ontributions ofthe otheringredientsof matterand radiation, .f. (1.17). However,

thebalan e

¨

a = 0

is not astable onebe ausealready smallperturbations to either the mean massdensityor thedistribution ofmass will ause theUniverseto ontra torexpand. Note

that,ifthedensity

ρ

Λ

isnot onstantintimewhi histhe aseinmanymodernDarkEnergy s enariosalsotheDarkEnergymomentumtensorwouldhaveaform thatdiersfrom(1.19),

su hthat intheendthe hara teristi sof theva uum dodependontheobserver'svelo ity.

Inthe ontext ofgravitationaluid dynami soneusually distinguishesbetweenthea tive

andpassivegravitationalmassdensity. Thea tivemass density (

ρ + 3p

)standsfor the gravi-tationaleldthat is generatedbythe uid,the passivegravitationalmassdensity(

ρ + p

)is a measureof howtheuid streamingvelo ityisae ted byagravitationalsour e. Thus, in the

DarkEnergy model hara terisedby (1.19)and(1.21), thea tivegravitationalmassdensityis

negative(assumingapositive

ρ

Λ

)andifthisdark omponentdominatestheenergy-momentum tensorthen

¨

a

will bepositive. This ree tsthe fa t that theexpansionof theUniverse a el-erates. Thus one an summarise the ee t of

Λ

in physi al terms asfollows: the a elerated expansionisnottheresultofsomenewfor e,ratheritisduetothenegativea tivegravitational

massdensity that we anasso iatewith the Dark Energy. Then, onsidering non-relativisti

movement, therelativea eleration

g

of freefalling testbodiesis modied byahomogeneous a tivemassdensityduetothepresen eof

Λ

to

(1.22)

d

2

_r

dt

2

= g + H

2

0

Ω

Λ

r

.

We analreadyguessthatthemagnitudeofthis ee tisprobablysmall. We anestimatethe

sizeoftheratioofa elerations

g

Λ

/g

. LetusassumethattheSolarSystemmovesina ir ular orbitaroundthe entre oftheMilky Waywitha ir ular speedof

v ≃ 220

km/sat aradiusof

b

Tobeexa t,thisisnotstri tlytrue. Thoughmathemati allythesame,Einstein[Ein17℄addedthe new

termtothelefthandsideoftheeldequations,thatistothe`geometri side':

G

µν

−

Λg

µν

= 8πGT

µν

. Note thatEinsteinfurther motivatedthismodi ationbyananalogy to NewtonGravity. Interestingly,inNewton

Gravityoneen ountersaseriousproblemwithaworldmodelthatishomogeneousandinnite. Itwasalready

seenbyNewtonhimselfthatthegravitationalpotentialenergyofsu hasystemdiverges: thevolumeofashell

atdistan e

r

to

r + δr

fromanobserveris

δV = 4πr

2

_δr

andwiththeassumptionofhomogeneousmassdensity

ρ

,the masswithin

δV

amountsto

δM = 4πρr

2

_δr

. Thusthegravitationalpotentialenergy a ordingtothis

massbe omes

δU = GδM/r = 4πGρrδr

. Integrating

δU

weseethat

U

diverges like

r

2

when

r

be omesvery large[Pee93 ℄. Einsteinandafterhimothers, .f.[PR03℄,suggesteda ureforthissituationbyamodi ationof

thePoissonequationa ordingto

∆

(3)

_{φ − λφ = 4πGρ}

,whi hgivesthepotentialofapointmassaYukawaform

φ ∝ e

−

√

λr

(thesesolutionsarealso alledSeeliger-Neumannsolutions). Now, themodiedPoisson equation

allowsfor ahomogeneous stati solution

φ = −4πGρ/λ

. Butthe analogy shouldnot betakentoo seriously: notethatthe modiedPoisson equationdoesnot omeout asaNewtonian limitfromthe generalrelativisti

equationwith osmologi al onstant. Thatis,

Λ

doesnota tlikealong-range utoingravitation,itisrather arepulsiveformofenergythatisinoppositiontothemeangravitationalattra tionofmatter.

Also,theinstabilityofthestati Einsteinsolution anbeseenfromequation(1.22) . Amassdistribution

r ≃ 8

kp . Theratioof

g

Λ

tothetotalgravitationala eleration

g = v

2

_/r

isthenestimatedby (1.23)

g

Λ

g

=

H

2

0

Ω

Λ

r

2

v

2

∼ 10

−5

_.

Thisisalreadyasmallnumberbutitbe omesmu hsmallerwhentheradiusisredu ed. Sin ethe

Sunisalreadylo atedattheveryoutskirtsoftheluminousdis oftheMilkyWay,thepossibility

of dete tingthis ee t by measuringdeviations from theordinary internal dynami s in other

galaxiesisnotverypromising. Thea ura yofpre isiontestsofgravitationonthelevelofour

SolarSystemismu hbetter. Butontheses alestheratio(1.23)isoftheorder

g

Λ

/g ∼ 10

−22

.

Nextwewantto onsidera ompli ation,namelyaworking modelforadynami al

ρ

Λ

. Theaforementionedme hanismof oupling

Λ

toanegativea tivegravitationalmass den-sity is loselyrelated to the on ept of osmologi alination. There exists aproblem that is

en ounteredifweassumethattheUniversewasevolvingduetoaFRWsolutionwithinitsentire

history. Letusre alltheexpressionfortheparti lehorizon

(1.24)

x =

Z

_dt

a(t)

,

whereweassumedspatialatness. Itisameasureoftheintegrated oordinatedispla ementas

alightraymovestheproperdistan e

dl = a(t)dx

duringthetime

dt

. Nowthepointisthatfor vanishing

Ω

Λ

theintegral(1.24)does onvergeinthepast(

ax

istheproperradiusoftheparti le horizon),thatisourviewshould fallonseveral ausallydis onne tedpartsoftheUniverse. In

orderto makethe Universe homogeneous,signalsmust travel betweenthe regionsthat are in

onta twithat mostthespeedoflight. Thus, noregions thatare morethan

2ax

apart ould haveeverbeenin ausal onta t. Letustryanestimate: assumingthat thetemperatureofthe

youngUniversewas

T ≃ 10

14

GeVatsomeinitialtime

t

init

,we anthenimaginea orresponding ausally onne ted ballwith radius

2ax

that hasexpanded and todayshould form the border ofthe urrentlyobservableUniverse. Inoursimple estimate,the temperatureof theUniverse

hasevolvedfrom that initial epo h at

T ≃ 10

14

GeV to

T

0

≃ 2.7

K

≃ 2.4 × 10

4

eVtoday, thus

givingafa torofexpansionoftheUniverseof

T /T

0

≃ 4 × 10

26

. Moreover,atthetemperature

T ≃ 10

14

GeV,thehorizonsizehasbeen

2ax ≃ 6×10

−25

matatimeof

t

init

≃ 10

−35

s. Therefore

theprimordial ausal ballwould haveexpandedto a size of

2.4

m todaywhi h is rathersmall forthe urrentsize of theUniverse. Andhow anthen galaxies asobservedtodayin dierent

dire tionsontheskylook sosimilar d

toea hother? Theanswerisprovidedbythestatement

that theexpansion historyof theUniverse wasnotFRW-likefor a ertaintime period in the

youngUniverse. InsteadoneassumesaDeSittersolutionwith

Λ > 0

and

T

µν

= 0

andthes ale fa torbehaviour

(1.25)

a(t) ∝ e

H

Λ

t

_,

with

H

Λ

being onstant. That is, in the DeSittermodel, the Universe undergoes aphase of exponentialblowup and

Λ

be omesessential.

IntheinationaryviewtheearlyuniverseisdominatedbyalargeDarkEnergydensity

ρ

Λ

. ThentheDarkEnergy anbemodelledwiththehelp ofanapproximatelyhomogeneouss alar

eld

Φ

inanalogyto modelsknownfrom quantum eldtheory. Thea tiontakestheform

(1.26)

S =

Z √

−g

 1

2

g

µν

_∂

µ

Φ∂

ν

Φ − V (Φ)



d

4

_{x ,}

d

One angiveanotherveryinstru tiveillustrationofthehorizonproblemregardingtheCMB.Usingthe

on eptoftheangulardiameterdistan e(1.38)(whi hisameasurablequantity)one an omputethatuptothe

timeoflasts atteringoftheCMBphotons,regionsthat ouldhavehad ausal onta ttoea hother,todayhave

thesizeofapproximatelyonedegreeonthesky.ThatmeansanimageoftheCMBshould ontainmanypat hes

where

g

isthedeterminantofthemetri

g = det(g

µν

)

andweused

~

= 1

. Thefun tion

V (Φ)

isthepotentialenergydensityandwithvanishingspatial urvaturewegettheeldequation

(1.27)

Φ + 3

¨

˙a

a

Φ +

˙

dV (Φ)

dΦ

= 0 .

We andenetherestframeofanobserverwhoismovingsu hthattheUniverselooksisotropi ;

thentheenergy-momentum tensorofthehomogeneouseld

Φ

isdiagonalwith

(1.28)

ρ

Φ

=

1

2

Φ

˙

2

_{+ V (Φ) and p}

Φ

=

1

2

˙Φ

2

_{− V (Φ) .}

From these equationsit is lear that ifthe s alareld varies slowly with time

˙Φ

2

_{≪ V}

, then

theequationofstateofthe osmologi al onstant anbere overed:

p

Φ

≃ −ρ

Φ

.

Normally it is assumed in inationary theory that the exponential phase (1.25) lasts so

longthat all regions in theobservableUniversehaverea hed ausal onta twith ea h other.

Eventually

Φ

anstarttovaryrapidlythusprodu ingentropyfortheUniverse. Thisisbe ause after a rollover phase the eld falls into the potential well of the real va uum and starts to

os illateduetoitskineti energy. Thelargeinitialva uumenergyistransformedinto oherent

os illations of the eld

Φ

and these u tuations are damped  besides the Hubble fri tion

3H ˙Φ

by parti leprodu tionorthe intera tionof

Φ

withother elds, whi h is equivalentto athermalisation oftheeld energyandentropyprodu tion. Throughthisso alled reheating,

e.g.baryons anbeprodu edandintheend

ρ

Φ

remainssmallorzero. However,itis on eivable that

ρ

Φ

ould havea veryslow late-time behaviour,possiblyslowerthan theevolution of the matterdensity. Then

ρ

Φ

willbedominantagain,after a ertaintimeandthis ouldprovidean answertothe oin iden eproblem. A on reteansatzthat leadsto su halatetime evolution

of

ρ

Φ

is

V

κ

= κ/Φ

α

with a onstant

κ

that has the dimension of mass

α+4

[PR03℄. We an

onstraintheformofthes alefa torbyassumingthataftertheinationaryphasetheUniverse

is dominated by matter or by radiation whi h leads to a power law expansion behaviour of

a ∝ t

n

, .f. tab.1.1. With this form of thes ale fa torwe ansolvetheeld equation(1.27)

and obtain

Φ ∝ t

2/(2+α)

. The mass density asso iated with the s alar eld

Φ

behaves like

ρ

φ

/ρ ∝ t

4/(2+α)

withrespe ttothematterorradiationdensity. Thuswe anre overEinstein's osmologi al onstant

Λ

fromthismodelinthelimitof

α → 0

whi h orrespondstoa onstant

ρ

Φ

. In the ase

α > 0

the eld

Φ

angrowverylarge and due to

V

κ

= κ/Φ

α

the a ording

density will go to zero,

ρ

Φ

→ 0

, whi h implies that the Universe approa hesa Minkowskian state. Su h a powerlaw model with

α > 0

has two important hara teristi s [PR03℄. First, theenergydensityof matterandradiationde reasesmorerapidlythanthat ofthes alareld

solution. This implies that it is possible to havea

ρ

Φ

that is small right after ination (but stillat highredshift) andthusdoesnotinterferewiththestandardprodu tion s enarioof the

light elements. However, after some time

ρ

Φ

an dominate again, mimi king a osmologi al onstant. Se ond,ithasbeenshownbyRatraandPeeblesthatthe lassofsolutions

α > 0

has theattra tor hara teristi ,thatisavastrangeofinitial onditionseventuallyendupwiththis

solution.

Theinationarys enarioexplainsthelarge-s alehomogeneityoftheUniversetodayby

pos-tulatingaDeSitter-likephaseofexponentialgrowthoftheUniverseatveryearlytimes.

More-overitprovidestheinitial onditionsfor stru tureformationby thevastfreezingof zero-point

quantumeldu tuationsto osmologi als ales. Thustheseedsfortheobservedstru tureson

osmologi als alestodayhaveoriginatedfromquantumu tuationsoftheearlyUniverse. The

powerspe trumof the lassi aldensity u tuations that havebeenfrozenout from quantum

u tuationsis

(1.29)

P (k) = h|δ(k, t)|

2

i = Ak

n

T

2

(k) ,

where

δ(k, t)

is the Fourier transform of the density ontrast,

δ(x, t) = ρ(x, t)/ ¯

ρ(t) − 1

at wavenumber

k

,withthe massdensity

ρ

and itsmean

ρ

¯

.

A

is a onstantthat omesoutfrom the on reteformofthepotential

V

one hooseswithinagiveninationarymodel. Thetransfer fun tion

T (k)

governshowthedensity ontrast

δ(k, t)

evolvesunder theinuen e ofradiation

pressureand thedynami s ofmatter atredshifts

z . 10

4

. Now,foraninationary expansion

followinganapproximateDeSittersolution(1.25),thespe tralindex

n

willbe losetounity e

. A

spe trumwithexa tly

n = 1

is alledHarrison-Zel'dovi hpowerspe trum. Thestrikingfeature ofsu haspe trumisthatitwouldhaveequalpower(amplitude)inallitsmodesatthetimeit

entersHubblehorizonandisthisalsonameds aleinvariant. Anti ipatingresultsforthe

Sa hs-Wolfe ee t from se . 1.3.3we an understand the notionof s ale invarian e alternatively by

thefollowingresult[Lon98℄fortheangulars aledependen e ofCMBtemperatureu tuations

originatingfrom aninitialpowerspe trumproportionalto

k

n

, (1.30)

∆T

T

≃

∆φ

c

2

∝ θ

(1−n)/2

_,

with

∆T /T

beings ale-freeintheHarrison-Zel'dovi h ase

n = 1

. Notethatmore ompli ated s alareldpotentials anbeimagined(e.g.exponentialformpotentials)underwhi hthe

spe -tralindexistilted

n 6= 1

and anbeusedasanadditionalfreeparameterofthemodel. However, re ent CMB measurementsindi ate that

n = 1

is very lose to the best t

f

. The initial

on-ditionsforthemassdistributionin theseinationary modelsareprovidedbyasinglefun tion

δ(x, t)

,whi hisarealisationofaspatiallyrandomGaussianpro esssin ethema ros opi per-turbationsarefrozenoutfromalmostfreeandpurequantumu tuations. Thisisalsoreferred

toasadiabati itybe ausesu hu tuations anbeunderstoodastheresultofpurelyadiabati

ompressions and de ompressions of regions of an homogeneous (post-inationary) Universe.

A onsequen eof thefa t that thesimplest inationary models obeythe above onditionsis

thattheinitial onditionasdes ribedbyasinglefun tionofposition

δ(x, t)

isstatisti allyfully hara terisedby itspowerspe trum(1.29). More ompli ated models ofination forinstan e

produ eu tuationsthatarenotexa tlyGaussianorhavepowerspe trathat annotbebrought

intoapowerlawform.

Before we ome tothe osmologi altests ofthestandardmodel letusreturn tothe

prob-lem of the smallness of the va uum energy density. The zero-point energy of quantum elds

ontributestotheDarkEnergydensity. Arelativisti eld anbeunderstoodasa olle tionof

quantumme hani alharmoni os illatorswithallpossiblefrequen ies

ω

. Thezero-pointenergy willbenon-vanishingand amounts,bysuperposition offrequen ies,to

E

0

=

P

i

ω

i

/2

,where

i

labelsos illatorsand

~

= 1

. We anthink ofthesystemaslo kedinaboxof length

L

andwe then onsiderthelimit

L → ∞

under appropriateperiodi boundary onditions. Wethenhave

(1.31)

E

0

=

L

3

2

Z

_ω

k

(2π)

3

d

3

_{k ,}

withthewavenumber

k = 2π/λ

. Weare onsidering amassivebosoni eld

Φ

˜

. Byemploying thedispersionrelation

ω

2

k

= k

2

+ m

2

andintrodu inga utofrequen y

k

max

≫ m

in orderto makephysi alsense

g ,wearriveat[KKZ97℄ (1.32)

ρ

˜

Φ

= lim

_L→∞

E

0

L

3

=

Z

k

max

0

4πk

2

(2π)

3

√

k

2

_{+ m}

2

2

dk =

k

max

4

16π

2

.

e

Letusaddasmallnoteontheapproximationof

n = 1

ininationarymodels. Ingeneral,itdependson theparti ularunderlyings alarelddynami softhemodelinhowfars aleinvarian eisrealised. Inslowroll

inationtheeldisinitiallyrollingdowntheinationarypotentialslowlyanditsmovementissizeablydamped

bythe Hubble fri tionterm

3H ˙

Φ

. Imagine a limitwhere the dampingisextremely intense and the rollover be omesinnitelyslow,then thiswould orrespond toexa ts aleinvarian e

n = 1

. Consequently,agenuine inationarypredi tionis

n = 1 ± ε

withsomesmall

ε

.The(small)deviationsofaparti ularmodelofination formexa t s ale invarian e quantify how slow the eld a tually has rolledand how stronglyit was damped

meanwhile,seealso[DS02℄.

f

A tually,fromWMAP(3yr)data aloneavalue of

n = 0.958 ± 0.016

isobtained[S

+

07 ℄. Nevertheless,a

runningspe tralindex,thatisan

n

thatvariesabitwiththewavenumber

k

oftheperturbationmodes,isslightly preferredbytheWMAP(3yr)data.

g

Notethat,asweintrodu ea utowavenumber

k

max

,weatthesametimehavetospe ifyinwhatframe the utoisdened,thusinvokingapreferredframe.ThisviolationofLorentzinvarian eposesaproblemofthe

If we assume General Relativity to be valid up to, say the Plan k s ale and set

L

Planck

=

(8πG)

−1/2

_{= k}

max

weobtainavaluefortheva uumenergydensityof

(1.33)

ρ

˜

Φ

∼ 10

92

gcm

−3

,

whi h is121ordersofmagnitudeo theobservedvalueof

∼ 10

−30

. Redu ingthe uto s ale

to the ele troweak s aleof

∼ 200

GeVstill produ esa dis repan y of 54orders of magnitude; insertingtheQCDs ale

Λ

QCD

as utoresultsinamismat hof42ordersofmagnitude. These dis repan ies ould indi ate a massivein ompleteness of the urrent underlying physi s; it is

thinkablethattheremightbesome onnetionbetweenthedierent omponentsin(1.3) oming

from yet undis overedphysi s that auses the almost omplete an ellation of the seemingly

un orrelatedtermsin(1.3), .f.[KKZ97℄.

1.2.2. Distan eMeasuresand Dark EnergyEviden e. Inordertodes ribethe

ur-rentphenomenologyofthestandardmodelwerstshouldre allthe ommondistan emeasures

in osmology. We have already introdu ed the proper distan e

D

P

through (1.4). Another naturaldistan eisthat asso iatedwiththe urrentHubblevolume,theHubbledistan e

(1.34)

D

H

≡

c

H

0

.

Assuming ontinuous FRW evolution, an obje t that would be seen at a distan e of roughly

theHubbledistan e isseenasit wasaroundaHubbletime in thepast. TheHubbledistan e

representsameasure oftheobservableUniverse, .f. g.1.2.

The denition oftheHubble parameterasafun tionof redshift (1.16) willbeveryuseful

in the following. The onstant of proportionality of the properdistan e s aling (1.4) an be

expressedbythe omovingdistan e. The omovingdistan ealongthelineofsightisdenedby

(1.35)

D

C

≡ D

H

H

0

Z

z

0

dz

′

H(z

′

₎

.

The omovingdistan ebetweentwopointsthatwere loseinredshiftinthepastisthedistan e

wewould measuretodaybetweenthe pointsif theywere glued to theexpanding ba kground,

.f.[Hog00℄. Seeg.1.2foranillustrationofproperand omovingdistan es andtheirrelation

toimportant osmologi als alesliketheparti lehorizonandtheHubbledistan e.

Going further, one an dene a omoving distan e in a lateralsense. If we measure two

obje ts atthe sameredshiftthat are separatedbyan angle

θ

onthesky then their omoving distan eis

D

TC

θ

withtransverse omovingdistan edenotedby

D

TC

anddenedby

(1.36)

D

TC

≡











D

H

Ω

−1/2

k

sinh(Ω

1/2

k

D

C

/D

H

)

for Ω

k

> 0

D

C

for Ω

k

= 0

D

H

Ω

−1/2

_k

sin(Ω

1/2

_k

D

C

/D

H

)

for Ω

k

< 0

.

Ifthe osmologi al onstantvanishesthereexistsa losedsolution

(1.37)

D

TC

= 2D

H

2 − (1 − z)Ω

m

− (2 − Ω

m

)(1 + zΩ

m

)

1/2

(1 + z)Ω

2

m

for Ω

Λ

= 0 .

It anbeshownthatthereisa orresponden ebetweentransverse omovingdistan eandtheso

alledpropermotiondistan e. Thepropermotiondistan e isdened astheratioof transverse

velo itytopropermotionofanobje tandismeasuredin radianspertime, .f. [Wei72℄.

Theratioofthelateralphysi alsizeofanobje ttoitsangularsizeisanexpli itobservable

alledtheangulardiameterdistan e. Itisveryusefulfor osmologi almeasurements. Espe ially

when onsidering theCMB whi h anbe mapped onto a sphereat

z = 1088

, it is ru ial to onvert angular separations measured by an instrument to proper separations in the sour e

plane. Theangulardiameterdistan eis givenby

(1.38)

D

A

≡

D

TC

Figure1.2. Spa etimediagramsof osmologi altimeversusproperdistan e(upper

gure;

D

P

inour notation) and versus omoving distan e (lowergure;

D

C

in our notation) withinadu ialFRW modelwith

(Ω

m

, Ω

Λ

) = (0.3, 0.7)

and

H

0

= 70

km s

−1

Mp

−1

.Thereinthedottedlines,thatarelabelledbyvaluesofredshift,represent

theworldlinesof omovingobje ts. Thepastlight one(belongingtotheobserverwith

entral worldline atzero distan e)enfoldsall eventsthat we are urrently(

t =

now) observing. Further, there are three kinds of horizons in the gures: the parti le

horizon namesthe distan ethatlight anprin ipallyhavetravelledfrom

t = 0

until some given

t

, .f. (1.24) , and the redshift of obje ts at parti le horizon be omes innite; theeventhorizon representsthedistan ethat light anhave travelledfrom

a giventime

t

until

t = ∞

; the Hubblesphereenfolds the setof spa etime events beyondwhi h omovingobje tsarere edingfasterthanlighttheHubblesphereis

notreallyahorizonbe ause

z 6= ∞

forobje tsatHubbledistan eandmoreoveritis possibletoseebeyondit in osmologi almodelswith

q < −1

. As anbeseen from

the slope of thelight one,the speedof photonsrelative to the observer

v

rec

− c

is not onstant. Photonsfromthe regionof superluminalre ession(hat hed) anonly

rea huswhen omingtotheregionofsubluminalre ession(noshading). As anbe

seeninthegure,initiallyobje tsbeyondtheHubblespherehavebeenre edingfrom

usnotethebulge ofthelight oneat

t . 5

Gyr. Notethatthelight onedoesnot

hit the line

t = 0

asymptoti ally; ratherit rea hes a nitedistan e of

∼ 46

Glyr at

t = 0

whi hisexa tlythe urrentdistan etotheparti lehorizon. Thus,thelightof

anyobje tsthatare urrentlyobservabletous,whoselighthaspropagatedtowardus

sin e

t = 0

,hasbeenemittedfrom omovingpositions around

46

Glyr(

14

Gp )away

fromus. Notethatthe aspe tratioofthegures

∼ 3/1

ree tstheratio ofthesize ofobservableUniversetoitsage

∼ 46/14

. Thepi turesaretakenfrom[DL03 ℄.

In ontrasttoseveralotherdistan e measures,the angulardiameterdistan e doesnotdiverge

for

z → ∞

,infa titisnotamonotoni fun tion of

z

;itrea hesamaximumataround

z ∼ 1

. Athighredshiftsone ansay,asaruleofthumb, thattheangulardiameterdistan erelatesan

angularseparationofonear se ondtoasize of

∼ 5

kp [Hog00℄.

The luminosity distan e measures the ratio of total bolometri (i.e. integrated over all

frequen y bands) luminosity

L

to the apparent luminosity

L

A

. The apparent luminosity or bolometri ux

L

A

is the power re eived per unit mirror area. The apparent luminosity of a non-movingsour eatsomedistan e

l

inEu lideanspa ewouldbe

L/(4πl)

. Thereforeitmakes sensetogeneralisethis anddenetheluminositydistan eas[Wei72℄

(1.39)

D

L

≡



L

4πL

A



1/2

.

However,inastronomywhatisreallymeasuredistheapparentmagnitude

m

. Afterttingfor the alibrationfa tor

M

(absolutemagnitude)oneusuallyusesthedieren eofthesemagnitudes foranalysis: thedistan e modulus

m − M

. Thedistan e modulusis relatedtotheluminosity distan e through

m − M = 5 log(D

L

/1 Mpc) + 25

, with thenumber

25

omingfrom the fa t thatthedistan emodulusisdenedtovanishat

10

p . Notethatduetoafundamentalresult there ipro itytheorem, .f[EvE98℄theangulardiameterdistan eandtheluminositydistan e

anberelateddire tlyby

(1.40)

D

L

= (1 + z)

2

_D

A

= (1 + z)D

TC

.

Based on the on ept of the luminosity distan e, in 1998 the rst dire t eviden e for an

apparenta elerated expansionof the Universe waspublished [R

+

98, P

+

99℄. This was made

possible by measurements of the redshift and the (luminosity) distan e of supernovae. The

appearan e ofthis kind ofeviden e wasdubbed a osmologi alrevolution,for itprovided the

rstdire teviden ethattheUniversemayre entlyhavebe omedominatedbysomemysterious

form of energy. After this dis overy, measurements of the CMB and statisti al analyses of

galaxy-redshift surveys have onrmed the supernova ndings, albeit in a moreindire t way.

However,thesupernovameasurementsremainupto todaythemostdire t meansofprobinga

presentlarge-s alea elerationof the Universe. What one ne essarilyneedsin orderto make

reliable measurements with the help of the luminosity distan e (1.39) is a standard andle.

A standard andle would be  in a mu h simplied sense  somethinglike a onstant

100

W light bulb. That means, if we an rely onthe fa t that the light bulb is standardised, i.e. it

allways will emit a power of

100

W, then we an inferthe distan e to the bulb by measuring its apparent luminosity. Now, in osmology it appeared at rst not promising to think of

supernovaeasstandard andlesbe ause theirobservation yieldsa veryheterogeneous lassof

light urves. Originally, the lassi ation s heme for supernovae wassu h that the typeSNI

was hara terised by thela k ofhydrogen features in thesupernova spe trum. From 1980 on

theastronomersdividedthetypeI supernovaeinto twosub lasses: Iaand Ib. Thedistin tion

wasmade due to the presen e or absen e of a ertain sili on absorption feature at

6150

Å. In the light of this re lassi ation a remarkable uniformity in the light urvesof supernovae Ia

suddenlybe ameapparent.

But,areSNIareallystandard andlesinastri tsense? Onespe ulatesthatSNIaoriginate

from exploding whitedwarfs. But why should the white dwarfs explode and why should this

thenhappenatauniformthreshhold? Normally,whitedwarfsareprodu edasremnantsof

Sun-like starsthat haveused uptheirnu lear fuelforfusion. Theonlything that savesthe dwarf

fromfurther ollapseistheee tivepressureupheldbyele trondegenera y. Now,ifithappens

thatthewhitedwarfisprovidedwithsomesteadystreamofmattera retingontoitssurfa e,it

woulda umulatemassuntil a ommonphysi althresholdwhi h isnear theChandrasekhar

mass of

≃ 1.4M

⊙

 and then suddenly erupt within a massive thermonu lear explosion. If thiss enarioistruethenessentiallyalwaysthesamephysi al pro esstriggersSNIaexplosions,

whi h then would ba k the assumption of regarding SNIa as standard andles. Still, taking

an a urate look, the un orre ted light urves of SNIa do show some oset. Their maximal

luminosities exhibit aslightbut obviousdispersion of roughly

0.4

magnitudesas measuredin the blue band [S h06℄. One nds a strong orrelation between intrinsi brightness and the

shapeoftherespe tivelight urves: thesupernovaethathaveahighermaximalbrightnessalso

de reaseslower(asmeasuredfromtheirmaximum)thanthosewithsmallermaximalbrightness.

Moreoveritturnedoutthatsupernovaethatwerefainteralsoappearedredderorwereobserved

in highly in lined host galaxies. This ee t an be attributed to an extin tion in the host

galaxy additional to the extin tion in the Milky Way. Altogether it is possible to quantify

these systemati s with aphenomenologi alre alibration that takes areof boththe maximal

brightness-duration orrelationand theextin tion. The fundamental alibrationis gauged to

asample ofsupernovaethat werelo atedin hostgalaxiesto whi h thedistan es areverywell