DMR UBC

LBL-Italy Princeton Cyanogen

COBE satellite COBE satellite sounding rocket

White Mt. & South Pole ground & balloon optical

2.726 K blackbody

I ν (W m − 2 sr − 1 Hz − 1 )

Figure 3.2. SpetraldistributionoftheCMBplottedasintensityvs. wavelength.

Thedatapointslieperfetly onablak bodyspetrum peakingataround

160

^GHz.

Besides data from the COBE mission [MCC

+

94℄, there is also data shownfrom a

roketexperiment(UBC Roket)[GHW90 ℄,ground basedexperiments(LBL/Italy)

[SBL91 ℄, (Prineton) [SJWW95 ℄as well as spetrosopyof therotational exitation

ofyanogen[RM95 ℄. Pitureistakenfrom[Smo97℄.

Lyman-

α

^{radiation by osmi expansion is also taken into aount. However, aslong as we}

lookatrelevantredshiftsof

10

^-

1000

^{,theseeets shallnotsigniantlyaet thebasiresults}

ofoursimplied onsideration. Note that therateequation (3.5)obeysasimplesaling. The

righthand side involves

Γ up

^and

R

^{, bothare funtions ofonly} temperature(redshift). Hene, parameterdependene is only arried by

n ² _p

^{whih sales as}

(Ω b h ² ) ²

^{on the righthand side,}

and aordingly the saling is proportional to

Ω b h ²

^{on the right hand side. It is onvenient}

toexpressthings intermsofredshift,andso weanusethefollowingtransformationvalidfor

matterdominationandat largeredshifts:

(3.6)

dt

dz ≃ − 3.09 × 10 ¹⁷ (Ω m h ² ) ^−1/2 z ^−5/2 s ,

Combiningthiswiththeaforementioned,weobtainasalinglawforthefrationalionisation:

(3.7)

x(z) ∝ (Ω m h ² ) ^1/2

Ω b h ² .

Notethatthissalingisompletelydierentastheoneobtainedfrom Sahatheory.

Inordertosolvetherateequationweonsiderlatetimes;thatiswerestritto timeswhen

theUniversehasooledsofarthat weannegletexitedtransitionofthe

2

^{S states. Therate}

equationthenbeomes

(3.8)

dln x

dln z ≃ 60xz Ω b h ² (Ω m h ² ) ^1/2 .

Reallthatforthisequationwehavenegletedtheosmiexpansionandsotheequationisnot

validanymorewhenthelefthand sidebeomeslessthanunity.

Now,oneaninludeallrelevanteetsandsolvefortheionisationfrationintheredshift

intervalinterestingforreombination,

800 . z . 1200

^{. Itis foundthat theionisationfration}

maybewellapproximatedbytheriterion[JW85℄

(3.9)

x(z) ≃ 2.4 × 10 ⁻³ (Ω m h ² ) ^1/2 Ω b h ²

z 1000

12.75

.

From(3.9)weanlearnthattheionisationfrationhasaverystrongredshiftdependene;thatis,

theredshifthangesoverarathersmallintervalwhiletheionisationfrationhangesdrastially:

from

x = 1

^(ompleteionisation)downto

x ∼ 10 ⁻⁴

^{(nearly omplete}reombination). Thefat that theionisation doesnotgo to exatlyzero reetsthe inuene of osmi expansionthat

wenegletedbefore. Atsmallvaluesofthe ionisationfrationtherateof reombinationdrops

belowtherateofexpansionoftheUniverse: thenithappensthatsomeionsdonothaveenough

timetondthemselvesapartnereletrontoreombinewithbeforethedensityoftheUniverse

beomes too muh diluted. Plugging (3.9) into the formula for the optial depth as due to

Thomsonsattering,oneobtainstheimportantresult

(3.10)

τ(z) ≡

Z

n e xσ T dl ≃ 0.37 z 1000

14.25

,

where we integrated overtheproperdistane

l

^{along the line of sight. The remarkablepoint}

isthat in theexpressionfortheoptial depth,the osmologialparameterdependene anels

out. Thereason for that is the saling of thefrational ionisation (3.9) that ame out of the

rate equation. Again,

τ

^{is very sensitive to hanges in}

z

^{and so the last sattering shell is a}

rathersharptransition. Thedistributionfuntion

e ^−τ dτ /dz

^{forthelast satteringredshiftan}

beexpressedbyaGaussianwith mean

z ≃ 1088

^{andastandarddeviation}

z ≃ 60

^{. Thisis the}

reasonwhyweobserveaveryuniformprimordialradiationfromanalmostsynhronousemission

surfae(`snapshot') in theearly Universe: the last sattering surfae. The redshifting during

thebillionsofyearsthephotonshavetravelledsinethenhasbroughttheCMBradiationinto

themirowaveband,whereitwasrstobservedbyPenziasandWilson in1965[PW65℄.

The spetrum ofthe CMBradiation is aPlankspetrum. In fat,itsspetrumwasrst

auratelymeasuredbytheFarInfraredAbsoluteSpetrophotometer(FIRAS)mountedonthe

CosmiBakgroundExplorerCOBE satellite[MCC

+

94℄,and isthebest blakbodyspetrum

everobtainedfromarealmeasurement,seeg.3.2. LetusshortlyderivehowaninitialPlank

spetrumfortheprimordialradiationkeepsitsformduringtheevolutionoftheUniverse.

Con-sideraPlankspetrumofphotonsataninitialtemperature

T 0

^attime

t 0

^{,thenthefuntion}

(3.11)

B ν (T 0 ) = 2hν ³

c ²

1 e ^{hν/(k B T 0 )} − 1

measurestheblakbodysurfaebrightness;here

h

^{isofoursethePlankonstant,notto be}

onfusedwiththenormalisedHubbleparameter. Thesurfaebrightnessistheluminositythat

goesthrough aunit areaduring aunit time interval, perunit solid angle and unit frequeny

interval. Then the numberdensity of photons in a frequenyrange between

ν

^and

ν + dν

^is

givenby

(3.12)

dN ν

dν = 4π hc

B ν

ν = 8πν ² c ³

1 e ^{hν/(k B T 0 )} − 1 .

Nowletusonsideraninstant

t 1 > t 0

^{,inwhihtheUniversewouldhaveexpandedbythefator}

a(t 1 )/a(t 0 )

^{andanobserverseestheinitialphotonredshifted bythefator}

1 + z = a(t 1 )/a(t 0 )

^.

Aordingly, aninitialfrequenyinterval

dν

^{is beingredshifted to}

dν ^′ = dν/(1 + z)

^{. Sinewe}

arewithin matterdomination,thenumberdensityofphotonsisdilutedwith

a ⁻³

^(.f.tab.1.1)

andso

dN _ν ^′ ′ = dN ν /(1 + z) ³

^{. Therefore,thenumberdensityofphotonsin thefrequenyrange}

between

ν ^′

^and

ν ^′ + dν ^′

^beomes

(3.13)

dN _ν ^′ ′

dν ^′ = dN ν /(1 + z) ³ dν/(1 + z) = 8π

c ³ 1 (1 + z) ²

(1 + z) ² ν ^′2

e ^{hν ′ (1+z)/(k B T 0 )} − 1 = 8πν ^′2 c ³

1

e ^{hν ′ /(k B T 1 )} − 1 ,

and so the form of the Plank distribution is left invariant under global expansion; only the

temperature

T 0

^{is replaed by the redshifted} temperature

T 1 = T 0 (1 + z)

^{. Thus, sine we}

observethespetrumoftheCMBtobetheoneofablakbodytoday,weanextrapolatethat

ithashadthisform uptodistortionsduetoadditionalphysiseversinelast sattering.

Note that, althoughthere isaverysmall osetbetweentheinstantof reombinationand

theeventualeetive deouplingof the primordial photons,weare using

z rec

^{throughout this}

worktodenote theinstantoflastsattering.

3.3. Observables of the CMB

In theourse ofse. 3.1wegotto knowthebasimehanismsthat areresponsiblefor the

CMBanisotropy. Thenextquestionis,howthemainphysialeetstranslateintoquantiable

observables. In se. 1.3.3, we have antiipated a basi part of the answer: the (integrated)

Sahs-Wolfeeet. TheSahs-Wolfeformulaparameterisestheinueneofthemostimportant

primaryandseondarysouresoftheCMBtemperatureanisotropy

∆T /T

^{,whih isaphysial}

observableaessible through dierential measurements. What remains to be done is to nd

astatistialframeworkof thetemperatureanisotropiesthat isonvenientandsuitablefor the

omparisonof theoryand experiment. Inorder to dothis aurately,onemust opewith the

fatthat theapproximationofthematter-photonmedium asaperfetuidbreaksdownafter

reombination. An adequatetreatmenttheninvolvesthesolutionoftheorrespondingkineti

equation,thefullBoltzmannequationforthephotondistributionfuntion. Seljakand

Zaldar-riaga[SZ96℄havedevelopedapublilyavailableFORTRANode,alledCMBFAST[CMB℄,thatan

beusedforstate-of-the-artomputation. Herewerestritourselvestoabasiunderstandingof

theCMB powerspetrumanditsuseforphenomenology. However,seeforinstane [HS95℄for

anexhaustivedisussion.

3.3.1. Fourier Analysis of the Temperature Power Spetrum. Howanwerelate

thethree-dimensionaldensityperturbationsfrom inationtothetwo-dimensionaltemperature

eldthatweobserveintheCMB?Thedensityperturbationsseealsoapp.Dareharaterised

bytheirpowerspetrum

P(k)

^{fromequation(1.29). Sometimesthepowerspetrumisexpressed}

as[Pea99℄

(3.14)

∆ ² (k) ≡ V

(2π) ³ 4πk ³ P (k) ,

fora given volume

V

^{. The quantity}

∆ ² (k)

^isdimensionless andhas the interpretation of the varianeofperturbationsperintervalof

ln k

^{;that is,}

∆ ² (k) = h δ ² i ,ln k ∝ k ³ P (k)

^{. Forinstane}

ifwehad

∆ ² (k) = 1

^{thiswouldmeanthat,perlogarithmi}

k

^{interval,therearedensity}

pertur-bationsof order unity. Here,weonsider a simplied Fourieranalysis following[Pea99℄. The

simpliationisprovidedbytheassumptionofloalthermodynamiequilibriumoftheprimeval

photonsas well as the assumption of spatial atness this will be agood approximationfor

intermediatesales.

Givenanobservedintensity

I ν

^{,thebrightness}temperatureisthetemperatureablakbody would need to havein order to radiate that intensity. Thereforeonean invert the

Rayleigh-Jeanslawtodenethebrightnesstemperatureas

(3.15)

T B ≡ I ν c ²

2k B ν ² .

Now,wean think ofthemeasuredCMBasatwo-dimensionalrandomeld ofanisotropiesin

thebrightnesstemperature. Considerapathof thetwo-dimensionalCMBskyofside

L

^{, but}

beingsmall enoughto be at. Itis usefulto introduetheFouriertransformofthe frational

temperaturedierenes,

(3.16)

∆T

T (X) = L ² (2π) ²

Z

T K e ^−iK·X d ² K and T K (K) = 1 L ²

Z ∆T

T (X )e ^iK·X d ² X .

Here,by

K

^and

X

^wedenotetwo-dimensionalvetorsofpositionandwavenumberrespetively, andmoreoverthe temperatureanisotropy

∆T /T

^{isaentralquantity ofCMB analysis,being}

denedas

∆T /T ≡ (T (θ, φ) − T 0 )/T 0

^{withthemonopolebakground}temperature

T 0

^.

In analogyto thetreatmentof the three-dimensionaldensity perturbations, we anwrite

downadimensionlesspowerspetrumofthetemperatureutuationsintwodimensions

(3.17)

T _2D ² ≡ L ²

(2π) ² 2πK ² | T K | ² .

Similarto(3.14),butnowin two-dimensions,thisisameasureofthevarianeinthefrational

temperature dierenesof the CMB, oming from modes of unit length in

ln K

^{. In fat, the}

Fouriertransformofthetemperaturepowerspetrumyieldsthetwo-pointorrelationfuntion

(3.18)

C(θ) 2D ≡

Z T ²

2D (K) J 0 (Kθ) K dK ,

whihistheobservablewewerelookingfor.

J 0

^{denotestheBesselfuntion;itenterstheformula}

viatheangularpartoftheFourierintegration.

Weanreonstrutthetwo-dimensionaltemperatureutuationeldfromtheatual

three-dimensionalonebyintegratingovertheoptialdepthatlastsatteringandoverthewavenumber,

(3.19)

∆T T = V

(2π) ³ Z Z

T _k ^3D e ^−ik·r d ³ k e ^−τ dτ .

Theoptialdepth expressionanbeapproximatedbyaGaussianwith

(3.20)

e ^−τ dτ ∝ e ^{−(r−r rec )/(2σ 2 r )} dr ,

and

r

^{being theomovingradius. This meansthat theentral distane to thelast sattering}

shellisgivenby

r rec

^{, whih inturn anbe}approximatedbytheHubble radiusbeauseof the highredshiftofthelast satteringshell. Above,wealreadyused anestimateforthethikness

ofthe last sattering shellof

z ≃ 70

^{. Infat one anshow [Pea99℄that the thiknessanbe}

expressedas

(3.21)

σ r ≃ 7Mpc

(Ωh ² ) ^1/2 .

Applyingananalogousdenitiontothespatialtemperaturepowerspetrumasin the

two-dimensionalase,weanwrite

(3.22)

T _3D ² ≡ V

(2π) ³ K ² | T K | ² .

By equating the respetive two-dimensional and three-dimensional two-point funtions, one

obtainsthenalprojetionformula[Pea99℄

(3.23)

T ²

2D = K ² Z ∞

0

T ²

3D [(K ² + w ² ) ^1/2 ] e ^{−w 2 σ r 2} dw (K ² + w ² ) ^3/2 .

Thisprojetionformula nallyrepresentstherelationbetweenthetwo-dimensionaland

three-dimensionaltemperaturepowerspetra. Thetwo-dimensionalpowerspetrumreeives

ontri-butions from all the three-dimensional modes with wavenumbers smaller than

K

^{, the other}

modesareintegratedout. Therefore,what theprojetioneetivelydoesissmearing. Through

smearingonegetsthe two-dimensional temperaturespetrumfrom thethree-dimensionalone.

Anyfeature presentat aertainsale in thespatial eld anbefound at theverysamesale

in the projeted spetrum. Also note that, as long as

T ²

3D

^{is not a very strongly inreasing}

funtion,thedampingtermwillausetheintegraltobedominatedbytheontributionaround

w = 0

^{. Ifthisis notthease,thenitethikness}

σ r

^{beomesrelevant.}

Inse.3.1wedisussedvarioussouresofCMBanisotropy. Nowweneedsomequantitative

expressionsfortheanisotropyontributions. Weonsideronlysomeofthemin ordertoobtain

arstpitureofthestandardinterpretationwithinsynhronousandomovinggauge.

•

^{Sahs-Wolfe soure} Perturbations in the primordial density eld ause anisotropy via: (a)additionalredshiftingof thephotonsthat arelimbingoutofpotentialwells,

Figure 3.3. A Mollweide map of the intrinsi CMB temperature anisotropies

[

O (10 ⁻⁵ )

^{℄ as derived from three years of WMAP mission data. Here, red olours}

indiatewarmerspotsandblueoloursindiateolderregions. Asuperposition

teh-nique the Internal LinearCombination(ILC) has beenapplied to theraw data

inordertosubtrat astrophysialforegrounds. TheILCmethodintrodues free

o-eientsthatarettedinordertondamaximallyleanmap,withtheonstraints

that thevarianeof theresulting mapisminimisedand, atthe sametime,the

am-plitude of thesignal is preserved. Other (more obvious)leaninghas to bedone in

addition: removingthedipoleontributiong.3.4andthelargeMilkyWay

ontam-ination,.f.g. 4.4aslieof

∼ 30 ^◦

^{isutawaytobothsides oftheequatorandis}

to bereonstrutedproperly. Providedthe leaningtehniquesworkattherequired

auray, the residualtiny anisotropies are of osmologial origin; theyrepresent a

snapshotoftheprimevalquantumutuationsfrozenoutintheearlyUniverse. The

pitureistakenfrom[WMAa℄.

relativistiperturbation alulationrevealsthat thenetresultis exatlyonethird of

theNewtonianexpression, thatis

(3.24)

∆T T

SW

= ∆φ 3c ² .

The fator of

1/3

^is non-intuitive; it an be shown [HPLN02℄ that it is a peuliar preditionfrom GR,and annotbeobtainedfrom anykindof Newtonian reasoning.

Moreover, the fator is unique also onerning the physial setting (standard model

plusadiabatiperturbations). Inpartiular,takinganisourvaturesetting,the

result-ingSahs Wolfeontribution is

∆T /T = 2∆φ

^{. The}orresponding Fourier-expanded expressionis

(3.25)

T _k ^SW = − Ω(1 + z rec ) 2

H 0

c 2

δ k (z rec ) k ² .

•

^{Doppler veloity The eletrons, o whih the photons last satter, are subjet to}

induedpeuliarveloity,whihresultsinanadditionalfrequeny-shift. Theresulting

anisotropyisgivenby

(3.26)

∆T T

DV

= δv · ˆ r c ,

andtheaordingresultin Fourierspaeis

(3.27)

T _k ^DV = − i[Ω(1 + z rec )] ^1/2 H 0

c

δ k (z rec )

k ˆ k · ˆ r .

Figure 3.4. TheunorreteddipoleasmeasuredwithCOBEshowninMollweide

projetion. Thisdistributionisinterpreted as being theresultof theDoppler eet

induedbyourloalmotion againsttheCMBrestframe. Ourveloityvetoristhe

end resultof a superpositionof various peuliarveloities up to the last sattering

surfae;itsmagnitudeis

≃ 370

^{km/s. The}temperatureexessarisingfromthedipole is

≃ 3.3

^{mKandanbeusedfor}alibration. Thepitureistakenfrom[WMAb℄.

•

^{AdiabatisoureBeauseoftheirtightouplingbefore}reombination,anyadiabati perturbations in the matter-radiation density are likewise imprinted on the photons

too,resultingin additionalanisotropy. Therespetiveformulaeread

(3.28)

∆T T

AS

= − δz 1 + z = δρ

ρ and T _k ^AS = δ k (z rec ) 3 .

•

Isourvature soure As opposed to the adiabati perturbations, the isourvature perturbations allow the entropy to vary. In the adiabati senario all the dierent

energy speies undergoa ommon density perturbation. Isourvature perturbations

aredenedasaninitialondition,whihstatesthattheredonotexistanydeviationsin

totalenergydensityfromthebakgroundattheinitialtime. Thereforetheurvature

is spatially onstant and so the name beomes lear. A formal means to dene an

isourvature setting is given by

Φ ˜ → 0

^while

t → 0

^{[MFB92℄. Here}

Φ ˜

^{is the gauge}

invariantversionof the metriperturbation in (1.92). The gaugeinvariant Bardeen

potentials

Φ ˜

^and

Ψ ˜

^{areonstrutedfrom (1.92)asfollows}

(3.29)

Φ ˜ ≡ Φ + 1

a [(B − E ,η )a] ,η , Ψ ˜ ≡ Ψ − a ,η

a (B − E ,η ) ,

where

η

^{denotesonformaltimeasusual. Anexampleofan}isourvaturesettingwould betoinitiallydistributedierentspeieslikebaryonsandphotonsinhomogeneously

but adjust the total energy density in a homogeneous way. As it is pointed out in

[MFB92℄,isourvaturemodesarepreditedbysomeaxionmodels,modelswith

topo-logialdefets(e.g.osmistrings)orsomeexotiinationarymodels. Experimentally,

isourvaturemodesannotbeexludedfully,butstringentboundsonsuhadmixtures

anbegiven,espeiallyonerningtheross-orrelationofCMBandlarge-sale

stru-ture, as wellasfrom theCMB alone, asis shownin [KS07℄or[Tro07℄. However,we

Inludingthesesouresofanisotropy,thethree-dimensionaltemperaturepowerspetrumis

givenby[Pea99℄

(3.30)

T ²

3D = h

(f AS + f SW ) ² (k) + f _DV ² (k)(ˆ k · r) ˆ ² i

∆ ² _k (z rec ) ,

withthedimensionless fators

f

parameterisingthedierentsouresas

(3.31)

f SW ≡ − 2

(kD ^rec _H ) ² , f DV ≡ 2

kD ^rec _H , f AS ≡ 1 3 .

Here

D _H ^rec

^{denotestheHubblehorizonsaleat lastsattering}

(3.32)

D ^rec _H ≡ 2c

Ω ^1/2 m H 0

(1 + z rec ) ^−1/2 ≃ 184(Ωh ² ) ^−1/2 Mpc .

Equation(3.30)providesthenalanswertothequestionofthissubsetion. Itrelatesthe

three-dimensionaltemperaturepowerspetrumtothethree-dimensionalmatterpowerspetrum. The

two-dimensional temperature power spetrum is onneted to the three-dimensional one via

theprojetion(3.23). Theanalysis isdone in Fourierspae. Thethree basisoures ofCMB

anisotropyweonsideredherebeomesigniantondierentsales. SinetheomovingHubble

saleamountsroughlyto

∼ 300

^{Mpatlastsattering,weanlearnfrom(3.30)thatthe}

Sahs-Wolfe term is vital at wavelengths largerthan

∼ 300

^{Mp. Going to smaller sales, rst the}

Dopplerterm beomes dominant, andeventually theadiabatiutuations take overat small

sales.

3.3.2. The CMB Angular Power Spetrum. The preeding formalism relies on the

assumptionofatness;bothatnessofthethree-spaeof theUniverseand atnessofthe

on-sideredpathesoftheCMB.Forseveralreasons,thesimpliedtreatmentbreaksdown,asbeing

toonaïve,bothon thesmallestand thelargestCMB sales. Here, wewant toshortlyreview

themodernstandardtoolkitforanadequatestatistialomparisonofCMBmeasurementswith

theory,following[Lon98℄and[CHSS07℄.

The information we reeive in form of CMB photons from the epoh of deoupling, is a

temperatureelddistributed ontheinner surfaeofourlastsatteringsphere. Fromquantum

mehanis, it is known that the appropriate mahinery for expanding physial funtions that

liveon a sphereis provided by the analysis of spherial harmonis. The spherial harmonis

providetheorretbasisin whihweanattempt to expandtemperatureanisotropyreorded

overthewhole CMBsky. Weanwrite

(3.33)

∆T

T (θ, φ) = X ∞

ℓ=0 m=ℓ X

m=−ℓ

a ℓm Y ℓm (θ, φ) ,

withexpansionoeients

a ℓm

^{,ontainingallthephysis,andthespherialharmonis}

Y ℓm (θ, φ)

^.

Forthelatter,wenotethefollowingnormalisationinvolvingtheassoiatedLegendrepolynomials

(3.34)

Y ℓm (θ, φ) =

2ℓ + 1 4π

(ℓ − | m | )!

(ℓ + | m | )!

1/2

P ℓm (cosθ) e ^imφ ×

( − 1) ^m for m ≥ 0 1 for m < 0 .

The (assoiated) Legendre polynomials an be found tabulated, for instane in [AS72℄.

A-ordingto this normalisation,thespherialharmonisare aset oforthonormalbasiselements

with

(3.35)

Z

Y _ℓm ^∗ Y ℓ ^′ m ^′ dΩ = δ ℓℓ ^′ δ mm ^′ ,

where the

δ ℓm

^{is just the Kronekerdelta and}

dΩ

^{stands for the full element of solid angle.}

Heneitispossibletoreonstrut theoeients

a ℓm

^byinversion,

(3.36)

a ℓm =

Z ∆T

T (θ, φ)Y _ℓm ^∗ dΩ .

Itisveryusefultounderstandhowthemultipolepowerinaspherialharmoniofmultipole

ℓ

^{relatesto theaordingportion of angular powerat asale}

θ

^{. Longair [Lon98℄ arguesthat}

therootsof

Re(Y ℓm )

^and

Im(Y ℓm )

^{providealattiestrutureon theskythat divides theeld}

into approximatelyretangular pathes. Whenlooking at that sky from low latitude(

θ

^{), the}

minimalsidesofthepathesarewellapproximatedby

π/ℓ

^{. Ontheotherhand,whendeparting}

from lowlatitude moving to thepoles the rootsof the azimuthal parts

sinmφ

^and

cosmφ

lustermoreandmorelosetoeahother. ButthisisompensatedbytheassoiatedLegendre

Polynomials,sinetheyapproahzerointhese regions. Together,thisleadstotheremarkable

fatthatto everyspherialharmoniauniqueangularresolutionanbeattributed

(3.37)

θ ≃ π

ℓ .

NowweometotheissueofGaussianity. Wedisussedabove,thatthestandardinationary

model predits utuations that are among other requirements purely Gaussian. This is

beause,intheinationaryview,theinitialperturbationsinthedensityoftheearlyUniverseare

providedbypurequantumutuationswhiharefrozenout. WithGaussianity,itismeantthat

thephasesofthewavesthatonstitutetheharmonideomposition(3.33),arepurelyrandom.

The assumption of Gaussianity leads to a ouple of appealing simpliations. Nevertheless,

therearemodelsthatpreditnon-Gaussianfeatures intheCMB.Suhareforinstanemodels

withtopologial defets likeosmi stringsorosmi textures aswell asompliatedination

models.

AssumingGaussianityoftheCMButuationsimpliesthat utuationsaresuperimposed

fromwaveswithrandomphases. Thereforeeahoftheexpansionoeientsin(3.33)provides

anestimateofthe amplitudeontainedintheonsidered utuationmode. Beause thereare

(2ℓ + 1)

^oeients

a ℓm

^permultipole

ℓ

^{, oneobtainsanensembleofamplitudeestimatesover}

whih weansimply average,if we further assumethe statistial isotropyof thetemperature

anisotropy eld. Statistial isotropy implies that the power spetrum is irular symmetri

aroundanypointontheskyandonsequentlyweanonstrutawell-denedestimatorforthe

powerofamultipolebytakingthemeanof

a ℓm a ^∗ _ℓm

^{andperforminganall-skyaverage,}

(3.38)

C ℓ = 1

2ℓ + 1 X

m

a ℓm a ^∗ _ℓm .

Thebulkof urrentCMB analysesis well onsistentwith Gaussian temperature anisotropies;

thequantities that arefound suitablefor probingnon-Gaussianity, aspredited bysome

non-standardmodels,arethebispetrum(three-pointorrelationfuntionofthe

a ℓm

^),trispetrum, analysesoftheMinkowskifuntionalsaswellasothermahinery,seee.g.[S

+

07,C

+

06b℄assome

representativestudies. Fromthe side of model-building, non-Gaussianfeatures appear rather

naturallyin thepreditionsof moreinvolvedmodels,likemulti-eldination. Itis speulated

that non-Gaussianity may be detetable with future experiments that reah higher auray.

Theaordingtheoretialtoolsforanalysisdoexistalready,seee.g.[FS07℄. However,itshould

benotedthattherearestudiesthatlaimtohavedeteteddeparturefromGaussianity[BTV07℄.

Moreover,wenotethatonlyinaseofstatistialisotropyofthemirowaveskyweanwrite

theensembleaverageovertheprodutofspherialharmonioeientsas[CHSS07℄

(3.39)

h a ^∗ ℓm a ℓ ^′ m ^′ i = C ℓ δ ℓℓ ^′ δ mm ^′ .

As for the point with statistial isotropy, the whole next two hapters of this thesis will be

onernedwiththeanalysisofexistingevidenethesoalledlow-

ℓ

^{CMBanomaliesindiating}

violationof statistialisotropyonthelargestangularsalesin theCMB.

Letusproeedfurtherwiththestandardstatistialframeworkoftemperatureanisotropies.

Theapproahwepursuedabovewastorstdene theangularpowerspetrumofutuations

(3.38), whih represents, in ase of Gaussianity and statistial isotropy, aomplete statistial

Im Dokument On the influence of local inhomogeneities on cosmological observables : from galaxies to the microwave background (Seite 93-107)