Cosmic Microwave Background Anisotropies: Beyond Standard Parameters

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arXiv:astro-ph/0410115 v1 5 Oct 2004

Cosmi Mi rowave Ba kground Anisotropies:

Beyond Standard Parameters

Ph.D. Thesis

submitted atthe Departmentof Theoreti al Physi s of the

UNIVERSITY OF GENEVA toobtainthe degreeof

Do teur ès S ien es,mention Physique by

Roberto Trotta

Thesis N. 3534

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Supervisor: Prof.Ruth Durrer Members of the Jury: Prof.Joseph Silk

Prof.Thierry Courvoisier Dr.Pedro G. Ferreira

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nobody,noteven therain,has su hsmall hands E. E. Cummings

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Itisa pleasureto thankthe peoplewhohave ontributed totherealizationof thiswork and whohave a ompanied mealong theway. During these threeyears, Ihave enjoyed working with and learning from all the members of the Geneva Cosmology Group and of the group of Mi hele Maggiore. I have had the han e to ollaborate withseveral people, to whom I wouldliketoexpressmyappre iation: PedroP.Avelino,Ra helBean,Rebe aBowen,Ruth Durrer,SteenH.Hansen,CarlosJ.A.Martins,AlessandroMel hiorri,AlainRiazuelo,Graça Ro ha,Joseph Silkand Pedro T.P.Viana.

Ruth Durrerhas been a wonderful supervisor,never ounting thehours shespenton my questions, onstantly stimulating my interests while allowing me to pursue my resear h in all freedom. I thank her for her tea hing and for her example. The help of Alain Riazuelo was pre ious during my rst year and gave me a swift start into real resear h work. I am gratefultoAlessandro Mel hiorriforinvolvingmeinmany ollaborationswhi h onstitute a major partof thisthesis. Ienjoyedhaving various interesting andpromising dis ussionwith FilippoVernizzi, MartinKunzandCélineBoehm,whi hIamsureonedaywewillbeableto nalize. SamLea hisguiltytohave onvertedmetotheBayesians hoolduringmanyrestless and humorous dis ussions. I am indebted to Christophe Ringevaland Thierry Baerts higer for their help insolving myvarious omputer problems, and to Andreas Malaspinas for his Sisyphusworkoftroubleshootingour omputernetwork. Ishalljoinmythankstotheonesof the osmology ommunityto AnthonyLewisfordeveloping,supportingandmakingpubli ly available the amband osmom odes. Iwouldlike to expressmygratitude toProf. Silk, Prof.Courvoisierand Dr. Ferreira fora epting to be partof thejury.

TheEuropeanNetworkCMBNetandtheS hmidheinyFoundationhaveprovidedgenerous supportforsomeofthe ollaborationsIwasinvolved with,intheformoftravelgrants. Iam indebtedtoOxfordAstrophysi sandtoProf.Silkfortheirkindhospitalityinmanyo asions, andto Profs. Spergel and Kosowsky andto Prin etonUniversityfor nan ial supportofmy visit.

But there is not only s ien e, even to the life of a PhD student. My time in Geneva wouldnothavebeenthesamewithoutthefriendshipofTimonLin oln Boehm: Iwishhim all the best on his new path. I have had the pleasure of sharing many refreshing moments with Stefano Foa, Marj Tonini, Yasmin Friedmann, Simone Lelli, Anna Rissone, Martin Zimmermann, Davide Dutturi Lazzati. To my parents, myae tionate thoughts for their en ouragingpresen e. To Elisa,myan ée,mydeepestgratitudefor having been atmyside inall marvellousand alldi ult moments, andfor theones whi h arestill to ome.

This work was mu h improved both in form and ontents by the areful reading of Christophe Ringeval and Elisa Cunial (for the fren h part), Sam Lea h and Ruth Durrer for theEnglish part. I thank themfor their time and ompeten e. I alone bearthe respon-sibilityfor anymistake whi hmight still be present.

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Contents vii

List of Figures xi

List of Tables xiii

Overview and on lusions 1

Towards a osmologi al standard model . . . 1

Testingthe on ordan e modelwiththe CMB . . . 3

Outlookand on lusion . . . 6

I BASICS 9 1 Introdu tion 11 1.1 Notationand onventions . . . 11

1.2 Friedmann-Robertson-Walker osmology . . . 12

1.2.1 Einstein equations . . . 12 1.2.2 Boltzmann equation . . . 16 1.3 Cosmologi alobservations . . . 17 1.3.1 Big-Bang Nu leosynthesis . . . 19 1.3.2 Matter distribution . . . 19 1.3.3 Type Iasupernovæ . . . 21

2 Cosmologi al perturbation theory 23 2.1 Perturbation variables . . . 23

2.1.1 Metri perturbations . . . 23

2.1.2 Perturbations oftheenergy-momentum tensor. . . 24

2.1.3 Gauge transformations . . . 25

2.1.4 Gauge invarian e . . . 27

2.1.5 Multipleuids . . . 29

2.1.6 Entropy perturbations . . . 30

2.2 Perturbation equations . . . 31

2.2.1 Einstein equations . . . 31

2.2.2 Conservation equations . . . 33

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2.2.4 CollisionlessBoltzmann equation . . . 34

2.2.5 Thomsons attering . . . 38

2.2.5.1 Stokesparameters . . . 38

2.2.5.2 S attering rossse tion . . . 39

2.2.5.3 Temperature hierar hy. . . 40

2.2.5.4 Polarizationhierar hy . . . 42

2.2.5.5 E and Bpolarization. . . 43

II COSMIC MICROWAVE BACKGROUND 45 3 Fundamental equations 47 3.1 One perfe t uid . . . 47

3.2 Cold dark matterand radiation . . . 49

3.2.1 Adiabati andiso urvature modes . . . 49

3.2.2 A ousti os illations . . . 53

3.3 Neutrinos and initial onditions . . . 56

3.3.1 Evolutionequations for athree omponentsmodel . . . 56

3.3.2 Neutrinoentropymode . . . 60

3.3.3 Neutrinovelo itymode . . . 61

3.3.4 Thedivergent nature oftheanisotropi stress mode. . . 63

3.4 The role ofbaryons . . . 64

3.5 Damping. . . 65

3.6 Observablequantities . . . 67

3.6.1 Temperature u tuations . . . 67

3.6.2 Angular powerspe tra . . . 69

3.6.3 Matterpower spe trum . . . 73

4 Parameter dependen e 77 4.1 Standard parameters . . . 77 4.1.1 Larges ales . . . 78 4.1.2 A ousti region . . . 81 4.1.2.1 Peaklo ations . . . 81 4.1.2.2 Baryon signature . . . 84

4.1.2.3 EarlyISWee t . . . 85

4.1.3 Dampingtail . . . 85

4.1.3.1 Re ombination . . . 85

4.1.3.2 Reionization . . . 86

4.2 Normal parameters . . . 88

4.3 General initial onditions . . . 93

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III PARAMETER EXTRACTION 97

5 Statisti s and data analysis 99

5.1 Elementsof probabilityand statisti s. . . 99

5.1.1 Some on epts of probabilitytheory . . . 99

5.1.2 Theoriginof osmi varian e . . . 101

5.1.3 Theprin iple of MaximumLikelihood . . . 102

5.1.4 Orthodoxprobabilities Conden e intervals . . . 103

5.1.5 Statisti al inferen e Likelihoodintervals . . . 105

5.1.6 Gridding method . . . 110

5.1.7 Markov hainMonte Carlo . . . 112

5.2 Fishermatrix fore asts . . . 116

5.2.1 Experimental parameters . . . 116

5.2.2 Generalizations . . . 117

5.2.3 A ura yissues . . . 118

5.3 CMBobservations: abrief histori ala ount. . . 119

6 Beyond standard parameters 123 6.1 Extra relativisti parti les . . . 123

6.1.1 Motivation . . . 124

6.1.2 Ee tive numberofrelativisti spe ies . . . 124

6.1.3 CMBtheory anddegenera ies . . . 126

6.1.4 Pre-WMAP onstraintsfrom CMBand otherdata-sets . . . 128

6.1.5 Fisher matrixfore ast . . . 131

6.2 Theprimordial heliumfra tion . . . 136

6.2.1 Motivation . . . 136

6.2.2 Theimpa t ofhelium ontheCMB:ionization history revisited . . . . 137

6.2.3 Astrophysi al measurementsand BBN predi tions . . . 141

6.2.4 WMAPMonteCarloanalysis . . . 143

6.2.5 Potential offuture CMBobservations . . . 147

6.3 Timevariations of the ne-stru ture onstant . . . 152

6.3.1 Motivation . . . 152

6.3.2 Theobservationalstatus . . . 153

6.3.3 Ee tsof

α

ontheionization history . . . 155

6.3.4 Therole of reionization . . . 156

6.3.5 CMB onstraintson

α

fromWMAP alone . . . 158

6.3.6 Fisher matrixfore asts and degenera ies . . . 160

7 Testing the paradigm of adiabati ity 173 7.1 Introdu torysurvey . . . 173

7.2 Pre ision osmologyand general initial onditions . . . 175

7.2.1 Pre-WMAPdataanalysis . . . 175

7.2.2 Howimportant istheassumption of adiabati ity? . . . 178

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7.3.2 CMBand large s ale stru turedataanalysis . . . 181

7.3.3 Adiabati perturbations . . . 182

7.3.4 Mixedadiabati andiso urvature perturbations . . . 184

7.3.5 Doiso urvature perturbations mitigate the

Λ

problem? . . . 188

7.4 Pre ision osmologyindependent of initial onditions . . . 191

Publi ationlist 193

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1.1 Illustrationof the determination of(

Ω

m

, Ω

Λ

)

using supernovæ data.. . . 22 2.1 Geometryof the theThomsons attering pro ess. . . 39 3.1 CMBtransfer fun tionsfor adiabati and iso urvatureinitial onditions. . . . 72 4.1 Illustrationof the geometri al degenera y. . . 84 4.2 Individual ontributions to the adiabati temperature spe trum.. . . 85 4.3 Impa tof theshift parameteron the temperature andpolarizationspe tra. . 89 4.4 Impa tofa hangeintheepo hofequalityonthetemperatureandpolarization

spe tra. . . 90 4.5 Impa t of the energy density in the osmologi al onstant on the CMB

tem-perature and polarizationspe tra.. . . 90 4.6 Impa tof thebaryon densityon thetemperature andpolarizationspe tra. . . 91 4.7 Impa tofa degenerate ombination ofthenormalization andthereionization

opti aldepthon the temperatureand polarizationspe tra.. . . 92 4.8 Impa tofthes alarspe tralindex onthetemperatureand polarizationspe tra. 92 4.9 Temperature andpolarizationspe tra for general initial onditions(I). . . 94 4.10 Temperature andpolarizationspe tra for general initial onditions(II). . . . 95 5.1 Illustrationof the burn-in periodfor MonteCarloMarkov hains. . . 115 5.2 The smalls ale temperature spe trumobserved bytheCBI and ACBAR

ex-periments. . . 120 5.3 A ompilation of pre-WMAP CMB temperature anisotropy data ompared

withtheWMAPtemperature powerspe trum. . . 121 6.1 CMBdegenera ies in luding

ω

rel

. . . 127 6.2 Theshift parameter asafun tion of the ee tivenumber ofrelativisti spe ies.128 6.3 Two-dimensional likelihood plots for

ω

rel

and otherparameters. . . 129 6.4 Likelihood probability distribution fun tionfor theredshiftof equality. . . . 130 6.5 Derivativesof

C

ℓ

withrespe tto the9parameters oftheFishermatrixanalysis.135 6.6 Ionizationhistory for dierent values ofthehelium fra tion. . . 138 6.7 Temperature and polarizationpowerspe tra fordierent valuesof thehelium

massfra tion. . . 140 6.8 Comparison between urrent astrophysi al errors on the helium fra tion and

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6.9 Likelihooddistribution for theheliummassfra tionfrom CMBdataonly. . . 144 6.10 Joint likelihood intervalsinthe (

ω

b

, Y

p

)-planefromCMB dataalone. . . 145 6.11 Joint likelihood ontours in the (

Y

p

, z

re

)-plane and (

Y

p

, τ

re

)-plane from CMB dataalone. . . 147 6.12 S atter plotinthe

ω

b

− n

s

plane. . . 148 6.13 FMA fore astfor the expe ted errorsonthehelium fra tion.. . . 151 6.14 Degenera y between theshift parameter andthe ne-stru ture onstant. . . . 156 6.15 Ionizationfra tionandvisibilityfun tionfordierentvaluesofthene-stru ture

onstant at the epo h ofde oupling. . . 157 6.16 Impa tof variations ofthe ne-stru ture onstant and of thereionization

op-ti aldepthonthe CMBspe tra. . . 158 6.17 Ee tof variations ofthene-stru ture onstant on theCMBpowerspe tra. 159 6.18 Likelihood distribution fun tion for variations in the ne-stru ture onstant

fromCMB alone. . . 160 6.19 Likelihood ontour plotinthe

α

de

/α

0 − τ

re

plane. . . 161 6.20 Likelihood ontour plotinthe

α

de

/α

0 −

d

n

s

/

d

ln k

plane. . . 162 6.21 Fishermatrix fore asts for Plan kfor all ouples ofstandard parameters.. . . 167 6.22 Fishermatrixfore astsforanidealCMBexperimentforall ouplesofstandard

parameters. . . 168 6.23 Fisher matrix fore asts for Plan k in luding variations in the ne-stru ture

onstant.. . . 170 6.24 Fisher matrix fore asts for an ideal CMB experiment in luding variations in

thene-stru ture onstant. . . 171 6.25 Fore astsin the

α

de

/α

0 − τ

re

plane. . . 172 7.1 Best-tmodels for purelyadiabati andmixed initial onditions. . . 176 7.2 Likelihood ontours for purely adiabati and mixed iso urvature models and

iso urvature ontent ofthe bestt general iso urvaturemodels. . . 177 7.3 Bayesian andfrequentistlikelihood ontours inthe

(Ω

Λ

, h)

plane. . . 183 7.4 Bestt of CMB and 2dF data ompatible with

CDM osmologi al on ordan e model. . . 18 5.1 Chi-squaredieren eforone-andtwo-dimensional marginalizedlikelihoodplots.111 6.1

2σ

likelihood intervals on the ee tive energy density of relativisti parti les

frompre-WMAP data. . . 131 6.2 Experimental parameters usedinthe Fisher matrix analysis. . . 132 6.3 Fishermatrix fore astsfor theerrorson theenergy densityinrelativisti

par-ti les. . . 133 6.4 Fidu ial modelfor the Fisher matrix analysis. . . 149 6.5 Fishermatrixfore astsand omparisonwithpresent-dayresultsforthehelium

massfra tion. . . 150 6.6 Experimental parameters for theFisher matrixanalysis. . . 161 6.7 Fore asts for theWMAPfour yearmissionin luding reionization. . . 163 6.8 Fore asts for the WMAP four year mission in luding ne-stru ture onstant

variations andreionization. . . 164 6.9 Fishermatrix fore ast forthe Plan ksatellite andand ideal experiment. . . . 166 6.10 Fisher matrix fore ast for the Plan k satellite and and ideal experiment

in- luding variations of thene-stru ture onstant. . . 169 7.1 Likelihood (Bayesian)and onden e (frequentist)intervals for

Ω

Λ

alone. . . 190

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esanza uraaver d'al un riposo salimmosu, elprimo e iose ondo,

tanto h'i'vidide le ose belle

he porta 'l iel, perun pertugio tondo; e quindius immo ariveder lestelle.

Dante,La divina ommedia InfernoXXXIV, 133-139.

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Towards a osmologi al standard model

The study of osmi mi rowave ba kground anisotropies is one of the pillars of modern osmology. The osmi mi rowave ba kground (hereafterCMB) onsistsofphotonsleftover by the hot phase after the Big-Bang and is very homogeneous and isotropi . Its existen e waspredi tedbyGamov(1946),and a identally dis overedonly mu h laterbyPenzias and Wilson (Penzias & Wilson, 1965), but it was only in 1992 that theCOBE satellite (Smoot etal.,1992)dete tedthepresen eoftinytemperatureu tuations(1partin100'000),whi h arethought tohavebeengenerated byquantumu tuationsintheveryearlyuniverse. The observationalstudyofthesetemperatureu tuations,knownasanisotropies,hasbeenagreat te hnologi ala hievement. Overthelasttenyears,therehasbeenaspe ta ularadvan ement inthea ura yofmeasurements, usingground-based, balloon-bornand orbitalinstruments. The WMAP satellite (Bennett et al., 2003) has re ently measured the anisotropies with a pre ision whi h, on ertain s ales, is lose to a fundamental statisti al limit, alled osmi varian e.

The importan e of su h awealthof datafor theoreti al osmology annotbeoverstated. Inafewse ondsonadesktop omputer,itisnowadayspossibletoprodu ea uratenumeri al predi tionsof thestatisti al distribution ofthe anisotropies on theskyfor any osmologi al modelofinterest, i.e.oftheCMBangularpowerspe trum. Iftheprimordialu tuationsare Gaussian distributed,thenthe powerspe trumen odesall ofthestatisti alinformation: its omputationisbasedonlinearperturbationtheoryandtheunderlyingphysi s iswell under-stood. Thedetailedshapeof thepowerspe trum arries hara teristi signatures depending on thevalue of the late Universe osmologi al parameters and on the initial onditions for theperturbations. By late Universe osmologi al parameters we mean thequantities on-trolling theexpansion historyofthe Universe,i.e.itsmatterbudget, omplementedbysome des ription of the reionization history. In the former ategory, an in omplete list would in lude the Hubble parameter, the energy density in baryons, old dark matter and dark energy, the dark energy equation of state parameter (possiblyin luding a des riptionof its time evolution), the neutrino masses and the number of massless families plus the density parameters andee tiveequationofstate ofanyotherexoti formofmatterone mightwish to in lude;spe ifying howtheUniversewasreionized inthe ontext of stellarevolution the-ory might require three or four additional parameters, whi h however usually redu e to the opti aldepthtoreionizationorequivalentlytotheredshiftofreionization,asfarastheCMB is on erned. Spe ifying theinitial onditions requiresthe value of primordial parameters

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s ale dependen e.

The fa t thatCMB anisotropies are sensitive both to the late Universe osmologi al pa-rameters and to primordial parameters means that CMB observations only onstrain a (de-generate) ombination of both: until now, disentangling the former required rather strong assumptionsaboutthe natureofinitial onditions. Someguidan eisoeredbythe ination-ary paradigm: inits simplest in arnation,the de ayof theinaton eld produ esadiabati initial onditions, in whi h there is no u tuation in the relative number density of the spe ies, hen e no entropy perturbations (adiabati ). The presen eof entropy u tuations anex iteup tofour othernon-de aying modesfortheperturbations. Those are olle tively termed iso urvature, be ause in three ases the total matter density is unperturbed and hen e there is no urvature perturbation in the spatial se tions either. The observation of thersta ousti peakinthe CMBpowerspe trum(Page etal.,2003)at

ℓ = 220.1 ± 0.8

has substantially onrmed the predominan e of the adiabati mode. However, a subdominant iso urvature ontributiontotheprevalentadiabati mode annotbeex luded: afterall,there is no ompelling reason why the physi s of the early universe should boil down to only one degreeof freedom.

dm

+ Ω

b

+ Ω

Λ

∼ 1

implyaatUniverse. The ru ialpointisthatforthe CMBtheseresults onlyholdon ewe make therather strong assumptionof purelyadiabati initial onditions. Inthat ase,theprimordialparametersredu etothespe tralindexforthe u tuations,

n

s

∼ 1

,andan overalladiabati amplitude

A

AD

. Thesetwoquantities together withthe above ve late Universe parameters are what we all standard CMB parameters, be ause they buildthe basisof the on ordan emodel of present-day osmology

1 .

By ombiningCMBdatawithother osmologi al andastrophysi almeasurementssu h asgalaxydistributionstatisti s,supernovæ luminositydistan emeasurements,gravitational lensing statisti s, Lyman

α

absorption lines, lo al determination of the Hubble parameter, light elements abundan e we have rea hed an unpre edented pre ision indetermining the standard osmologi al parameters,whi h arenowknownwithan a ura yof afew per ent. Thisisevenmoreastonishingifwethinkthatonlytenyearsagoitwasonlypossibleformost parameters to estimate their order of magnitude. Most importantly, various independent

1

Wedo notdis ussthe possibility ofgravitational waves, whi hare indeedpredi ted by any inationary s enario;presentlytherearemerelyupperlimitstotheir ontribution,whi h ouldbesmallenoughtobe verydi ultto dete tintheCMB. Ourdis ussionhere andinthe followingfo usesonthe s alarse tor

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observations whi h probe very dierent epo hs of the osmi history and are based on totally dierent physi al pro esses seemto be onverging to thesame answer.

Wearenowinapositionwhere we anmoveon fromparameter ttingto modeltesting: inother words, inorder to establisha osmologi al standard model we need to assess the onsisten y and ompleteness ofour theoreti al framework. Inorder to besure thatwe an trust theerror-bars onthe standard parameters beyond thequoted statisti alerror,we have to onfrontourselveswiththe questionofpossiblesystemati errorsinthemeasurements on oneside,andofhiddenawsinourtheoreti alinterpretationofthedataontheother. Given theintrinsi di ultyofmany osmologi al observations,an assessment ofsystemati errors fora ertaindata-set an omefromthe ombination withother,independent measurements ofthesamequantity. Dis repan iesintheresultswillindi ateaawintheunderlyingtheory, or inthe data, or inboth. Thisisone of thereasonswhythe omparison of manydata-sets is soimportant, the other beingthat often the ombined datahave a superior onstraining powerduetothebreakingofdegeneratedire tionsinparameterspa e. Fromthepointofview ofmodel-building, itisnowbe oming possible torelaxsome assumptions whi h werebefore ne essaryinordertoextra tfromthedataanyinformationatall,andthereby he kwhether our results arerobustor else whetherthey riti allydepend onour prejudi es. Ifitis found that our on lusions depend strongly on the underlying model assumptions, then we need to riti ally review our theoreti al paradigm and open our mind to alternative expli ative models.

Testing the on ordan e model with the CMB

The CMB is an ex ellent testing ground to arry out this program: our theoreti al under-standing is based on General Relativity and linear perturbation theory, whi h su es to des ribe almost all of the relevant physi al pro esses. This makes us ondent that we un-derstand quite well CMB anisotropies,and we an exploit them to go beyond thestandard osmologi al parameters intwo dierentways: the rstpathleads dire tlyto theprimordial Universe, via the dependen e of the CMB on the nature of initial onditions; the se ond approa h makes use of the high quality of re ent CMB data to look for ee ts whi h were previouslyignored be ausethoughtto beirrelevant,butwhi harenowwithinthe onstrain-ing power ofthe observations. In both ases, themi rowave ba kground plays the role of a Universe-sized laboratory for the study of fundamental physi s whi h is often una essible to anyparti lephysi slaboratory. Thisworkpursues both thoseaspe ts,aswe detailinthe following.

In the rst part, we introdu e in Chapter 1 the homogeneous and isotropi F riedmann-Robertson-Walker universe,whi h is the ba kground on whi h perturbation theoryis built, and we briey present a few other observations whi h we later ompare and ombine with the CMB.We then give thederivation of all therelevant perturbation equations needed to des ribe the CMB in Chapter 2. Those are applied to the temperature u tuations in the osmi photons in the se ond part: in Chapter 3 we obtain under various approximations analyti al expressions for the growth of perturbations in an Universe ontaining photons, olddark matter, masslessneutrinos, baryonsand a osmologi al onstant;inChapter 4we

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asmeasuredbythetwo osmologi alprobeswedis uss, namely Big-Bang Nu leosynthesis (BBN) ombined with observations of the light elements abundan es, and CMB. This is be ause both of them are sensitive not only to the number ofweaklyintera tingneutrinos, butrathertothe totalenergy densityofrelativisti parti les whi hsetstheexpansionrateatearlytimes,andtherefore an onstraine.g.theexisten eof sterile neutrinos unobservable inZ-de ay experiments. Using pre-WMAPCMB data alone, we obtain fairly broad bounds on

N

eff

,

0.04 < N

eff

< 13.37

with

2σ

likelihood ontent, whi hareredu edbyin ludingpriorinformation omingfromsupernovæluminositydistan e measurements and large s ale stru ture observations. We show that

N

eff

, or equivalently

ω

rel

≡ Ω

rel

h

2

, the energy density parameter in relativisti parti les, is nearly degenerate withthe amountof energyinmatter,

ω

m

≡ Ω

m

h

2

,andthatitsin lusion inCMBparameter estimationalsoae tsthe onstraintsonotherparameterssu hasthe urvatureorthes alar spe tralindexofprimordialu tuations. However,eventhoughthisdegenera yhastheee t oflimitingthea ura yofparameterestimationfromtheWMAPsatellite,wendthatit an be brokenbymeasurements onsmaller s alessu h asthose provided bythe Plan ksatellite mission. We fore astthat Plan kwill be ableto onstrain

N

eff

within

0.24

(

1σ

).

Theprimordial

4

Hemassfra tion,

Y

p

,ispredi tedbyBBNalong withtheabundan esof theotherlight elements asa fun tion of two free parameters, namelythe baryon density

ω

b

andtherelativisti energydensity

ω

rel

. Ifwex

N

eff

= 3

andthereby

ω

rel

asmotivatedbythe parti le physi s standard model, then instandard BBN theabundan es of D,

3

He,

4

He and

7

Lidependonthebaryondensityalone: omparisonwiththeobservedvaluesinastrophysi al systemsindi atesaslightdis repan y,whi hhoweverpresently annot learlybeas ribedto systemati alerrorsor todeviationsfromthestandard BBNs enario. Weexplorein6.2the potentiality of using the CMB asa totally independent way of measuring

Y

p

via its impa t onthereionizationhistory,therebypossiblyallowingtodis riminatebetween thevarious hy-pothesis(Trotta&Hansen,2004). WendthatWMAPdatagiveonlyamarginaldete tion,

0.160 < Y

p

< 0.501

at 68% likelihood ontent. We estimate that the Plan k satellite will determinetheheliummassfra tionwithin

5%

(or

∆Y

p

∼ 0.01

),whi hhoweverwillonlyallow amarginaldis rimination betweendierentastrophysi almeasurements. Equallyimportant, we identify degenera ies between

Y

p

and other osmologi al parameters, most notably the

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baryon abundan e, the redshiftand opti aldepthof reionization andthe spe tralindex;we on lude that even though present-day CMB data a ura y does not require the in lusion of

Y

p

asa freeparameter, the un ertainty of the helium fra tion will have to be taken into a ount inorder to orre tly estimatetheerrorson thebaryon densityfrom Plan k.

0 < 1.02

with

95%

likelihood ontent, where

α

0

denotes the present value. We larify the issue of degenera ies between

α

and other standard parameters, and give exhaustive fore asts of the expe ted performan e of thefull four year WMAP data, of thePlan ksatellite andof anideal CMBexperiment. We emphasize therole ofpolarization measurements to lift at dire tions (i.e., degenera ies) in parameter spa e, and dis uss the role ofreionization inthedetermination of

α

de .

In Chapter 7 we relax the assumption of adiabati ity by allowing for the most general initial onditions(Bu heretal.,2000)andweinvestigatetwo omplementaryaspe ts: therst isthedegradationinthea ura yofthelateUniversestandardparametersasa onsequen e of the introdu tion of new degrees of freedom in the primordial Universe (Trotta et al., 2001); the se ondistherobustness ofthemeasurement of anon-zero osmologi al onstant,

Ω

Λ

6= 0

,when dierent statisti al approa hes (frequentistrather thenBayesian)areapplied to the data, or when general iso urvature modes arein luded intheanalysis (Trotta et al., 2003). We also expli itly test the paradigm of adiabati ity by using CMB observations to put onstraints ontheiso urvature ontribution.

Forthe rst point, the resultsin7.2 demonstrate thatthedeterminationof theHubble parameterandthebaryondensityfrompre-WMAPCMBdataisessentiallyimpossible with-out strong assumptions about the nature of initial onditions. Conversely, it be omes very di ult to put limitson thetype ofthe initial onditions without using external,non-CMB priors onthe late Universe parameters. Indeed, theCMB isperhaps themost ee tive way to dire tlyprobethe veryearlyUniverse,and thereby onstrainor falsifythemodels forthe generation of perturbations. It is therefore very important to extra t the most information aboutthe onditionsintheearlyUniverse. Addingpolarizationinformationgreatlyenhan es thepoweroftheCMBtosimultaneously onstrainthelateUniverseparameters andthe pri-mordial ones: we showin 7.4that thefullfour yearWMAPdatawill measure orthogonal ombinations of the late Universe parameters with an a ura y of theorder

10% − 30%

for most parameters even inthegeneral initial onditions ase. The Plan k missionwill have a betterpolarizationresolutionandwillbeabletodopre ision osmologyalmostindependently

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the osmologi al onstant problemby introdu ing iso urvature modes, our ndings in 7.3 indi atethat

Ω

Λ

6= 0

,asobtainedfroma ombination ofCMBandlarges alestru turedata, is indeed robust even in the presen e of iso urvature ontributions. The more onservative frequentiststatisti s as ompared to the usual Bayesian approa h ex ludes

Ω

Λ

= 0

only atthe

2σ

onden elevelforpre-WMAPCMBdata ombinedwiththe2dFGalaxyRedshift Survey,butthis onlyifwe admit arather lowvaluefor theHubble onstant,

h ∼ 0.5

,whi h would be in ontradi tion with the result of the Hubble Spa e Teles ope,

h = 0.72 ± 0.08

(Freedman etal.,2001).

Outlook and on lusion

TheCMBhasbe omeawellestablishedtoolforthestudyofourUniverse,andanunavoidable testing ground for any theoreti al model. The ever improving quality of the data permits on one side to look for new physi s in the early Universe, as shown in our study of time variations of

α

, on the presen e of extra relativisti parti les and on the existen e of non-adiabati modes; on the other hand, it also requires an upgrade of our modelling, so to properlytreat subtleee tssu hasthe un ertainty omingfrom ourunpre iseknowledgeof theprimordialHeliumfra tion,orfromourignoran e onthe orre tmodelforthegeneration of u tuations. For this reasons, it is important to look ahead, to the goals for the next generationofexperiments,andtotheirpotentialto onstrainorfalsifythetheoreti almodels. More than ever, the entral issue is be oming how to e iently and reliably extra t the most information from up oming high-quality data: there are about 2000 observable independent multipoles for ea h of the three angular power spe tra, namely temperature, E-polarization and temperature-polarization ross- orrelation, whi h however are highly re-dundant due to the smooth os illatory nature of the spe tra. The amount of information whi h an beextra ted ismu h less,and an be ondensed inmaybe adozenof well- hosen parameters. The best hoi e for those quantities is the one whi h takes into a ount the physi sandsele tsorthogonaldire tionsinparametersspa eonthebasisoffundamental de-genera ies. Thisideahasbeenaleitmotiv oftheworkspresentedhere,and thereisprobably stillspa e to apply itfurther,espe iallyin onne tion withtheprimordialparameters.

Despite this en ouragingpi ture, there arestill open hallenges for our understandingof theUniverse: thenatureofdarkenergyand darkmatter,thedetailsoftheinitial onditions and the epo h of reionization, for example. The CMB will provide key advan ements on all these issues over the next years. The polarization of the anisotropies has been dete ted bythe experiments DASI (Kova etal.,2002) and WMAPand willbe pre iselymappedby theforth oming experiments PolarBear, Bi ep, SPOrt, AMiBA and QUEST, opening up a newlineofresear handallowingtore onstru tthe osmologi al parameterswithstillhigher pre ision. This pro ess will ulminate with the European Spa e Agen y satellite Plan k (Plan k Website, 2004), whi h starting in 2007 will observe the temperature spe trum with theultimate possible pre ision and provide a urate mapping ofthepolarizationaswell. In view of this wealth of data, and in order to fully exploit its potential, it is of fundamental importan e thattheoreti alresear hon thesubje tadvan esa ordingly. Thereis aneedof more powerful and e ient omputational and statisti al te hniques whi h an handle the

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buildinghas to berened and inparti ular we need to further develop theinterdis iplinary link between models oming from high energy physi s, string theory,astrophysi s and their observationalsignatureonthe CMB.Thisapproa hwillstrengthentheroleoftheCMBasa universe-sizelaboratory for investigatingthemost elusivedomains offundamentalphysi s.

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enterprise, whethersmall or great, always all upon God. And we, too,who are go-ing to dis ourse of the nature of the uni-verse, how reated or how existing with-out reation, ifwebenotaltogetheroutof our wits, mustinvoketheaidofGods and Goddesses and pray that our words may bea eptableto themand onsistentwith themselves.

Plato Timaeus

Part I

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Introdu tion

1.1 Notation and onventions

We beginbyintrodu ing thenotation and onventions whi h areusedthroughout thiswork.

•

Themetri signature is

− + ++

.

•

The spa etime metri is denoted by

g

µν

, where thespa etime oordinate are

x

µ

_{, µ =}

0, 1, 2, 3

. Greek indexesalwaysrun from 0to 3.

•

The 3-spa e of onstant urvature has metri

γ

ij

. Latin indexes always run from1 to 3.

•

Whenwedis ussperturbations,theba kground,unperturbedquantitiesaredenotedby anoverline. Thereforefor instan e

ρ = ¯

ρ + δρ

,where

Thesymbol

Ω

X

X

≡ Ω

X

(η

0 )

, where

X

(η

0 )h

2

.

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1.2 Friedmann-Robertson-Walker osmology

In this se tion, we briey review the standard treatment of an homogeneous and isotropi universe. We present the ba kground Einstein and onservation equations for perfe t uids, alongwiththeunperturbed Boltzmann equation des ribing relativisti parti les.

1.2.1 Einstein equations

The osmi mi rowave ba kground is homogeneous and isotropi to better than one part in 100'000. This justies the assumption that the universe, on large enough s ale, an be treated as being homogeneous and isotropi . We then onsider a 4-dimensional manifold

M

endowed witha metri

g

µν

,so that onstant-time hypersurfa es are onstant- urvature, maximallysymmetri 3-spa es. The Friedmann-Robertson-Walker (FRW) metri reads

g

µν

x

i

d

x

j

_,

(1.1) withthe3-spa e metri of urvature

K = {0, +1, −1}

given by

γ

ij

φ

2 _{) .}

(1.2) Herethe s ale fa tor

a(t)

dependsonly on time,and

χ(r) =











x

i

d

x

j

_{) .}

(1.4) Following theassumptions of homogeneityand isotropy,the ba kground energy-momentum tensor,

T

µν

isboundto beof theperfe t uidform

T

µν

= (ρ + P )u

µ

u

ν

+ P g

µν

,

(1.5)

where

ρ, P

arefun tionsofthe onformal time

η

only,andrepresent theuidenergy density andpressure, respe tively. Theuid4-velo ityis thetimelike 4-ve tor

u

,with

u

µ

=

1 a

, 0, 0, 0

and

u

µ

u

µ

= −1 .

(1.6)

We supposethatthe equation ofstate of theuidisof theform

P = w(ρ)ρ ,

(1.7)

Λ

= −1

for a osmologi al onstant (va uum energy). Theenergy density

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of a osmologi al onstant is ontained in

T

µν

, and is of theform

ρ

Λ

= Λ/(8πG)

. Another relevant quantity istheadiabati soundspeed of theuid, dened as

c

2 _s

_{≡ ˙}

P / ˙ρ .

(1.8)

TheEinstein equations

G

µν

= 8πGT

µν

(1.9)

withthe FRWmetri (1.4) andthe energy-momentum tensor(1.5) yieldthetwo Friedmann equations. TherstFriedmannequationisarstorderdierentialequationforthe onformal Hubble parameter

H(η) ≡ ˙a/a

˙

H = −

4πG

₃

a

2 (ρ + 3P ) .

(1.10)

These ond one isa onstraint equation,

H

2 + K =

8πG

₃

a

2 ρ .

(1.11)

An evolution equation for the uid energy density follows from the 0 omponent of the energy-momentum onservation equation,

∇

µ

T

µν

_{= 0}

:

˙ρ + 3H(ρ + P ) = 0 ,

(1.12)

supplemented with the uid equation of state, Eq. (1.7) . If the universe ontains (or is dominated by) only one uid with

w =

onst , it follows from Eq. (1.12) that its energy densitybehavesas

ρ ∝ a

−3(1+w)

,

(1.13)

hen efrom Eq.(1.10) the s ale fa tor ofa at universe(

K = 0

) is

a =

2A

1 + 3w

η

2 1+3w

for

w 6= −1/3 .

(1.14) with

A

2 _{= 8πG/3ρa}

3(1+w)

₌

onst. In parti ular, in the radiation dominated universe (

w = 1/3

) we have

a ∝ η

,while inthematterdominated universe (

w ≈ 0

)

a ∝ η

2

.

In the standard osmologi al pi ture, the universe ontains non-relativisti , pressureless matter (baryons and old dark matter), photons, massless neutrinos and a va uum energy omponent. Inthis ase,the stress-energytensor isthesum oftheuid omponents

T

µν

=

X

α

T

_α

µν

.

(1.15)

The Friedmannequations (1.10,1.11) apply to thetotal energy densityand pressure, whi h are just the sum of the ontributions from ea h uid. The energy onservation equation, Eq. (1.12) ,still appliesto thetotal variables,whileingeneral for ea h omponent we have

∇

µ

T

α

µν

= Q

ν

α

,

(1.16)

where the4ve tor

Q

µν

α

des ribetheenergy-momentumtransferfromthe omponent

α

. The onservation oftotal energy requires

X

α

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Inthegeneral ase,the Friedmannequationshave tobesolved numeri ally. However, we aneasily writedownsolutions of simple ases. From Eq.(1.13) itfollows thatforradiation

ρ

r

∝ a

−4

while for matter

ρ

m

∝ a

−3

. Physi ally, the energy density of matter is diluted bythe growth ofthe physi al volume of the3-spa e, whilefor radiation an extra

a

−1

fa tor omes in from the redshifting of the parti les energy. Hen e, sin e

a

is growing, at early enough timetheuniverseisradiation dominated. The equality timeisdened asthetimeat whi hthetwo ontributions areequal,i.e.

ρ

r

= ρ

m

,afterwhi htheuniversebe omesmatter dominated. Therefore

a

eq

a

0 =

ρ

r

ρ

m

η0

≈ 3 · 10

−3

,

(1.18)

or intermsofthe redshift

z ≡ a

0 /a − 1

we have

z

eq

≈ 3000 .

(1.19) The subs ript

0

indi ates that the quantity is evaluated today. The numeri al estimate omes from the measurement of the present day radiation densityin the osmi mi rowave ba kground,whi h togetherwiththe assumption ofthree massless neutrino familiesyields

ρ

r

= 7.94 · 10

−34

T

CMB

2.737

K

4

g/ m

3 .

(1.20)

The matter ontent of the Universe is obtained from the ombination of CMB, large s ale stru ture and supernovæ type IAmeasurements. We shall seein 4.2 thatthe CMB itself isa goodprobe to determine theredshiftof equality.

Sin e for a osmologi al onstant

w

Λ

= −1

,

ρ

Λ

=

onst, its ontribution is negligible in theearlyuniverse,and indeedfor a redshift

z ≫

Ω

_Ω

m

Λ

3 − 1 ≈ 0.5 .

(1.21)

However, if

Λ 6= 0

,the lateuniverse will be dominated by theva uumenergy term. In that ase,

a(t) ∝ exp

(Λ/3)

1/2

_t

andthe expansion be omesexponential (inphysi al time). It is ustomary to introdu e the riti al energy density as the energy density for whi h theuniverse is at

ρ

rit

≡

3H

2 8πGa

2 .

(1.22)

We alsodene the Hubble parameter

H

0 ≡ H/a

0

and thefudge fa tor

h

H

0 ≡ 100 h km s

−1

Mpc

−1

.

(1.23)

The riti alenergy density todaythenevaluatesto

ρ

rit

(η

0 ) ≈ 1.88 · 10

−29

_h

2

g/ m

3 _.

(1.24) At all times, the density parameters

Ω

X

give the ontribution of the omponent

ρ

rit

=

Λ

8πGρ

rit

,

(1.27)

Ω

_K

_{(η) ≡}

−3K

8πGa

2 _ρ

rit

.

(1.28)

Bydenition thesumof thedensityparameters hasto beunity

Ω

r

(η) + Ω

m

3 .

(1.31)

The denition (1.28) expresses the energy density due to the urvature of the spatial se tions for

K = ±1

. Sin e

Ω

K

∝ H

−2

_{∝ η}

2

, the urvature is always negligible inthe early universe. Various osmologi al observations indi ate that today

Ω

K

≈ 0

. However, if the universe is not exa tly at, this would imply that at Plan k time

|Ω

K

| ≈ O(10

−60

₎

. The smallnessofthisnumberistheessen eoftheatnessproblem. Theinationaryme hanism indeed naturally provides a solution for this ne tuning problem: as the universe inates quasi-exponentially, its urvatureis driven to 0.

A key quantity is the angular diameter distan e

D

A

(z)

: onsider an obje t of physi al length

d

sitting at aredshift

z

1

( orresponding to onformal time

η

1

andradialdistan e

r

1

), whi hisobservedatourpresentposition(

z

0 = 0, r

0 = 0

)underanangle

θ

. Thentheangular diameter distan eis dened as

D

A

(η

1 ) ≡

d

θ

= a(η

1 )χ(η

0 − η

1 ) ,

(1.32)

where in the se ond equality we have used

d = λa(η

1 )

, with

λ

the omoving length of the obje t, and

θ = λ/χ(r

1 )

,noting that

r

1 = η

0 − η

1

sin e light travels on null geodesi s. We an now integrate Eq. (1.11) to nd

∆η ≡ η

0 − η

1 =

1 H

0 a

2

0 Z

a0

a1

d

a

Ω

r

+ Ω

m

a

0 + Ω

_K

a

2 a

2

0 + Ω

Λ

a

4 a

4

0 1/2

,

(1.33)

Thisequation ismore onveniently written inredshiftspa e

∆η =

1 H

0 a

0 Z

z1

0

d

z

[Ω

r

(1 + z)

4 + Ω

m

(1 + z)

3 + Ω

K

(1 + z)

2 + Ω

Λ

]

1/2

.

(1.34)

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it on the sky, we are in prin iple able to extra t the value of the osmologi al parameters using Eq. (1.34) . The CMB provides exa tly su h a standard rod on the sky: the a ousti os illations of thephoton uid justbefore re ombination have a hara teristi length s ale, whi hshows upasthe rst peakinthe angular powerspe trum, see 4.1.2. Theredshiftof re ombinationisalsoknownwithgooda ura y,hen etheCMBmeasureswithhighpre ision theangular diameterdistan e to thelast s attering surfa e. Thispie e ofinformation alone ishoweverinsu ient to re onstru t ompletely the matter-energy ontent ofthe Universe: thisproblem isknown asgeometri al degenera y, and itisexplained in 4.1.2.

1.2.2 Boltzmann equation

At early time, the energy density of the universe is dominated by the relativisti spe ies, and to leading order we an negle t in the ontribution of non-relativisti omponents to thetotal energy. As longas photons are inlo al thermodynami al equilibrium, the photon temperature

T

isrelatedto theenergy densityof radiationby

ρ

r

=

π

2

30 g

⋆

T

4 _,

(1.35) where

g

⋆

ountsthe total numberof relativisti degrees of freedom

g

⋆

≡

X

b

g

b

T

_b

4 T

4 +

X

f

g

f

T

_f

4 T

4

(1.36)

and

b

and

f

run over the bosoni and fermioni spe ies respe tively. The fa tors

T

b

and

T

f

take into a ount possible temperature dieren es between the photons and the other relativisti parti les. From Eq. (1.35) and

ρ

r

∝ a

−4

itfollows that whilethe photons arein thermodynami alequilibrium,

T ∝ 1/a

.

For

T > 4000

K

≈ 0.4

eV hydrogen nu lei areionized, andphotons are oupled tobaryons via non-relativisti Thomson s attering o free ele trons, see 2.2.5. As the temperature drops below

0.30

eV, orresponding to

z

de

≈ 1100

, almost all the hydrogen nu lei qui kly re ombine, themean free path of photonsbe omeslarger than theHubble length

1/H

: the universe be omestransparent. Thisevent is alledlast s attering or de oupling.

After re ombination, thephoton distributionfun tion

f (η, E) =

1 exp(E/T ) − 1

(1.37)

evolvesa ordingtothe ollisionless Boltzmannequation,whi h an be derivedbyrequiring thatthe total derivative of

f

withrespe tto theaneparameter

λ

vanishes

d

f

d

λ

= 0 .

(1.38)

Ingeneral

i

p

i

≡ |p| , n

i

n

j

γ

ij

= 1 .

(1.41)

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From Eq.(1.38) we have

∂f

∂η

+

∂f

∂x

i

n

i

₊

∂f

∂E

E +

˙

∂f

∂n

i

˙n

i

_{= 0 .}

(1.42) Be ause of isotropy,

∂f /∂n

i

_{= 0}

, while homogeneity implies

∂f /∂x

i

_{= 0}

. Using the 0 omponent of the geodesi sequation

d

p

α

d

λ

+ Γ

α

_µν

p

µ

p

ν

= 0 ,

(1.43)

whi h inthe FRW universe reads

˙

E + HE = 0

(1.44)

we obtain fromEq. (1.42) the ba kground Boltzmann equation

∂f

∂η

− HE

∂f

∂E

= 0 .

(1.45)

Thisequationissatisedbyany

f

oftheform

f = f (aE)

. We on ludethatafterde oupling the energy of the osmi photons is redshifted by the expansion as

E ∝ a

−1

. The bla k body distribution, Eq. (1.37) , retains its spe trum. The spe trumof the osmi mi rowave ba kgroundphotonshasbeenmeasuredverya uratelybytheFIRASspe trometeronboard theCOBE satellite (Fixsenet al.,1996), and wasfound to be ex eedingly lose to thermal. Deviations from a perfe t bla k body spe trum an be measured by the Comptonization parameter

y

, the hemi al potential

µ

and the parameter

Y

f f

des ribing ontamination by free-freeemission. The

95%

onden elimits onthose parameters are

|µ| < 9 · 10

−5

,

_{|y| < 1.2 · 10}

−5

,

_|Y

f f

| < 1.9 · 10

−5

.

(1.46)

After de oupling,

T

is no longer a temperature in the thermodynami al sense, rather a parameter inthe distributionfun tion, whi h drops as

T ∝ a

−1

.

1.3 Cosmologi al observations

It is only in omparatively re ent times that osmology has be ome a data driven s ien e, inwhi h theoreti alhypothesis an befalsied orvalidated againstobservationaldata. It is amazingthatonly15yearsagothetotalenergydensityoftheuniversewasknownwith order-of-magnitudea ura yonly. Nowadays,most osmologi alparametersare onstrainedwithin a few per ent. Thedis overy and a urate mapping of CMB u tuations has onstituted a majorpillarinthisevolutionandrepresentsafundamental ornerstoneofmodern osmology, see 5.3for an overview.

It is nevertheless of equal importan e that many other osmologi al probes have been developed in parallel, and this for at least two goodreasons. Firstly, all observation suers inoneformor inanotherfromthedegenera yproblem: only a ertain ombination of osmo-logi al parameters an bemeasured a urately. Sin e degenera y dire tionsaredierent for dierent observations, ombining two or more measurements leads to tighter onstrains on theparameters we areinterested in. The se ondreason isthat osmologi allyrelevant

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mea-Quantity Value Observations

Baryon density

ω

b

0.024

CMB,BBN,light elementsabundan e Cold dark matterdensity

ω

dm

0.116

CMB+LSS+SN, lusters

Λ

density

ω

Λ

0.378

CMB+LSS+SN+weak lensing

Hubble onstant

h

0.72

HST, SZ,strong lensing

Opti al depth

τ

re

0.17

CMB

Spe tralindex

n

s

1.00

CMB,LSS,Lyman-

α

, lusters

Baryons

Ω

b

7.95 · 10

−5

CMB Massless

ν

families

N

ν

3.04 CMB+LSS Curvature

Ω

K

0.00

CMB+LSS+SN+weak lensing Initial onditions purelyadiabati CMB

Table1.1: Parametersoftoday's

Λ

CDM osmologi al on ordan emodel,whi hisingood agreementwithmostofthe urrentobservationaleviden e omingfromCMB(Spergeletal., 2003),larges alestru tures(LSS)(Tegmarketal.,2004b),Big-BangNu leosynthesis(BBN) (Fields & Sarkar, 2004), supernovæ type Ia (SN) (Tonry et al., 2003), strong (Ko hanek & S he hter, 2004) and weak lensing (Contaldi et al., 2003), Lyman-

α

absorption systems (Seljaketal.,2003a) andgalaxy lusters (Bah all etal., 2003)observations.

whi htheexperimental onditions annotbemanipulated atwill. Veryoftentheinteresting physi s is hidden behind foreground emissions, poor statisti al sampling, faint signals and non-linearities. It is ommon to try and extra t osmologi al information by using obje ts whosephysi alpropertiesarepoorlyunderstood,andingeneral systemati sareverydi ult toassessin osmology. Hen ea osmologi almeasurementisusually onsideredasvalidonly if onrmed byone or more independent pie esofeviden e.

The so- alled

Λ

CDM on ordan e model is strongly supported by several independent observationaldata. It is generallya epted that our universe isvery lose to at (

Ω

K

≈ 0

); that it is dominated by dark energy (

Ω

Λ

≈ 0.7

), perhaps in form of va uum energy, or quintessen e or atra kings alar eld;thataround 25%isnon-intera ting old darkmatter, and that only the remaining 5% is onstituted of baryons. If the three neutrino familiesof theStandard Modelof parti le physi s are not massless (as thelarge mixing angle solution to the solar neutrino problem seems to suggest), than their massis bounded from above to be

m

ν

<

∼ O(1)

eV. Stru ture formation pro eeded by gravitational instability from quantum u tuationsstret hedtosuper-horizons alebyaperiodofsuperluminalexpansion(ination). Thesimplestinationarymodel, inwhi h ination isdrivenbyone singleslow-rolling s alar eld, su essfully predi ts the absen e of non-Gaussianity, the (predominantly) adiabati nature of the u tuations and the almost s ale invariant spe tral index (

n

s

∼ 1

) for the perturbations. The age of the universe, around

13

Gyrs, easily a ommodates the oldest observed obje ts. For deniteness, in Table 1.1 we give the parameters of what we believe is a urrently widely a epted on ordan e model, to whi h we will refer throughout this

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Apart fromCMBanisotropies,whi hwewill dis ussindepthintherestof thiswork,we brieypresentsome ofthe pie esofobservationaleviden ewhi h orroboratethe(presently) standard

Λ

CDM s enario.

1.3.1 Big-Bang Nu leosynthesis

Big-Bang Nu leosynthesis is based on the Standard Model of parti le physi s, and gives predi tionsfor theabundan eoflightelementsD,

3

He,

4

He and

7

Lisynthesizedintheearly Universe,whi hareingoodoverallagreement withtheobservedabundan es,seeOliveet al. (2000) for areviewand Fields &Sarkar (2004)for more re ent results.

Belowatemperature

T ∼ 1

MeVtheneutron-proton onversionratefallsbelowthe expan-sionrate, andthe neutronto protonratiofreezesout atthevalue

n/p = exp (−Q/T ) ≈ 1/6

, where

Q = 1.293

MeV is theneutron-proton massdieren e. Thelight elements produ tion starts slightly afterwards, at a temperature

T ∼ 0.1

MeV, whi h is well below thebinding energy of deuterium,

B

D

= 2.23

MeV be ause photo-disso iation prevents the formation of deuterium and other nu lei until then. By this time,

β

-de ay has further redu ed the neutron-to-proton ratio to

n/p ≈ 1/7

. The surviving neutrons end up almost ompletely in

4

He, while the abundan e of the other elements is sensitively dependent on the nu lear rea tions rates, whi h inturndepend on thebaryondensity,usually expressedwith respe t to thephotondensitybydeningthe parameter

η

10

as

η

10 ≡

n

b

n

γ

× 10

10 _{≈ 274 · ω}

b

(η

0 ) ,

(1.47)

where

η

0

isthe onformaltime today. Asimple ountingargument,seeEq.(6.16,page 136), yieldsthattheprimordial

4

He massfra tionisabout

25%

,whilethenumberdensitiesofthe other elements relative to hydrogen turn out to be of the order D/H

∼

3

He/H

∼ 10

−5

and

7

Li/H

∼ 10

−10

. The predi tions are very reliable and a urate, with a residual numeri al un ertaintywhi h dependsontheexperimentallydetermined rea tionrates; interestingly,it turnsout thatmostof thisun ertaintyisasso iatedwithour onlyapproximative knowledge oftheneutronlifetime(Cuo oetal.,2003). TheotherfreeparameterofBBNistheradiation densityintheearlyUniverse,whi hsetstheHubbleexpansionrateandthereforedetermines the freeze-out temperature for the weak rea tions and is usually parameterized with the equivalent number of(massless) neutrino families. We omment on thepossibilityofa non-standard numberof neutrino familiesand dis ussBBN-related issues in6.1.2.

In summary, agreement between the abundan e of the light elements as inferred from astrophysi al measurement and the orresponding predi tion of BBN is a powerful tool to verifytheStandardModelofparti lephysi s. In6.2.3wepresentindetailthedetermination oflightelements,dis ussthe slightdis repan iesbetween themandtheBBNpredi tionsand givesome possible interpretations. However, theoverallagreement issatisfa tory,and (fora standard number ofneutrino families)the light elementsabundan es an be explained by a baryon density ompatible withtheoneindependently inferredfromCMB,namely

η

10 ∼ 5.5

or

ω

b

∼ 0.02

.

1.3.2 Matter distribution

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ies and lusters observed today. From the determination of the statisti al distribution of matter one tries to re onstru t the properties of the primeval u tuations, and to validate thestru tureformation model.

In 3.6.3 we introdu e the linear matter power spe trum

P

m

(k)

, whi h represents the Fourier transformof the2-point orrelationfun tion for thematterdensity ontrast. Obser-vationsofthedistributionofgalaxiesouttoaredshift

z ∼ 0.1

probethegalaxy-galaxypower spe trum,

P

gg

;the SloanDigital SkySurvey, for example, urrently ontains approximately

2 × 10

5

galaxies (Tegmark et al., 2004a), and upon ompletion will a hieve

10

6

galaxies. The problem is then to relate

P

−1

(1.48) withthe biasparameter

b

whi h isjustan unknown onstant fa tor(seehowevere.g. Durrer et al., 2003afor a riti aldis ussion). In pra ti e, this pres ription amounts to introdu ing a free parameter whi h ontrols the amplitude of the matter power spe trum. There are methods whi h allow to determine the bias from the higher-order n-point fun tion of the distribution: forinstan e Verde etal. (2002)found

b = 1.04 ± 0.11

from thedataof the2dF GalaxyRedshiftsurvey(Colless etal.,2001), whi hplans to measure

2.5 × 10

5

galaxies. One analso onsiderthedistributionofgalaxy lustersasafun tionofredshift,whi hin prin iple one should be able to predi t byusing hydro-dynami al simulations. Comparison with the observed distribution would then allow to onstrain the osmologi al parameters. Thissimplesoundingprogramisinpra ti e ompli atedbytheneedofa uratelysimulating all the relevant physi s, and despite the great amount of omputational power nowadays available, re ent works in the eld still involve many approximations. As a result, luster datamainly onstraina ombinationofthematterpowerspe trumat lusterss alesandthe value of

Ω

m

,seee.g. Bah all etal.(2003).

Anotherwaytoprobethe massdistribution isoeredbytheLyman

α

forest,the absorp-tion lines in the spe tra of distant quasars produ ed by the neutral hydrogen in regions of overdense intergala ti gasalongthelineofsightataredshift

2 −4

(Croftetal.,2002). Sin e the overdensities probed at these redshifts are still lose to thelinear regime, one hopes to be ableto onne tthe observations to the matterpowerspe trumbymodellingnumeri ally therelevant physi s (Mandelbaumetal.,2003; Seljaketal.,2003a).

Weak gravitational lensingis very promising asa toolto onstrain osmologi al parame-ters,andinparti ular the matterdistribution. It usesthedistortionintheimagesofdistant galaxiesindu edbyinhomogeneitiesintheinterveningmatterdistribution(Kaiser&Squires, 1993),andre onstru tswithastatisti alanalysistheso- alled osmi shear (Wittmanetal., 2000;Bartelmann&S hneider,2001). Thete hniqueisnowrapidlybe omingmaturetohelp

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One ofthemost important aspe tsis thatallof theabove observations anbe ombined to a hieve superior onstraining power on the CDM model parameters, while testing the onsisten y of the theory itself, or the soundness of ea h data-set. A te hnique to merge galaxysurveys, lusterdistribution,weaklensingandLyman

α

datawiththeCMBto probe alargerportionofthematterpowerspe trumispresentedinTegmark&Zaldarriaga(2002). Thereispresentlyageneralagreementthatthematter ontentoftheUniverseislow,around

Ω

m

∼ 0.3

.

1.3.3 Type Ia supernovæ

Supernovæ (SN) are lassied a ording to their spe trum: the type Ia is hara terized by the absen e of hydrogen (the I), and by strong sili on features (the a). The standard pi ture is a progenitor binary system, with a white dwarf whi h a retes matter from its ompanion until it rea hes the Chandrasekhar limit, and the gravitational infall triggers a thermonu lear explosionwhi h we observe asa supernova. At the peakof its brightness, a SN an easily ex eed the luminosity of its host galaxy, making it a promising andidate to measuredistan es out to very high(

z ∼ 1 − 2

) redshifts.

Their most important property is the remarkable homogeneity in their spe tra, in the shape of their light- urve and in their peak absolute magnitude, whi h makes them nearly standard andles. In fa t, it was dis overed that intrinsi ally brighter SNIa de line more slowly than dim ones(Hamuy etal., 1996). By exploiting an empiri al orrelationbetween theshapeofthelight urveand theintrinsi luminosity,and orre tingfor extin tionee ts viameasurementsatdierentwavelengths,itisneverthelesspossibletoprodu ea alibrated andle, witha very narrowpeakmagnitude dispersion(Riess etal.,1996). For a review of the osmologi al appli ations, seee.g. Filippenko (2004).

The measured apparent magnitude

m

is related to the absolute magnitude

M

via the luminositydistan e

D

L

m = M + 5 log [H

0 D

L

(z, Ω

m

, Ω

Λ

)] + K

(1.49)

where the

K

- orre tion ompensates for the dieren e in wavelength of the emitted and re eived photons due to theexpansion, and theluminositydistan e of an obje t at redshift

z

isdened intermsof theintrinsi luminosity

L

and ofthemeasured ux

ℓ

as

D

L

(z) ≡

L

4πℓ

1/2

.

(1.50)

Theluminositydistan eisrelatedtotheangulardiameterdistan eby

D

L

(z) = (1+z)

2 _D

A

(z)

. Supernovæ essentially measure the angular diameter distan e over a redshift range of

z ∼

0.5 − 2

, mu h lower than range probed by the CMB. At su h low redshift, the radiation ontent isnegligible,and with