Random Axial + Random

Figure 5.2. Mollweideprojetionoftheskywithquadrupole(upperrow)and

o-topole(lowerrow) multipolevetors [equation(5.5) ℄. Themeshonsists of stepsin

30 ^◦

^.^Displayed^are^ten^pairs^of^quadrupole^vetors^(small^dots)^and^their^ten^area

ve-tors [equation(5.6) (bigdots)℄as wellasten triplesofotopolevetors (smalldots)

and theirarea vetors(bigdots);togethernessisindiatedbyolour. Thearbitrary

signofthevetorshasbeenusedtogaugethemall tothenorthernhemisphere. The

statistially isotropi and Gaussianase (leftolumn) isbrokenbythe imprintof a

strongaxialeet

a ℓ0 = 1000µ

^K^(right^olumn)^whereupon^multipole^vetors^move^to

thepoleandareavetorsmovetotheequatorialplane. Theonsetoftheshown

sep-arationofmultipolevetorsandrossprodutsanalreadybeobservedatmoderate

axialontributionsof

a ℓ0 ∼ 100µ

^K,^.f.^g.^F.11.

[RLLA07℄ross-orrelationanalysisofCMBdataandgalaxysurveydatashowsnoevidenefor

an`Axis ofEvil' in theobservedlarge-salestruture. Inontrast, reentlyanopposite laim

hasbeenput forward[Lon07℄,whereitwaslaimedthatananalysisofSDSSdatagivesriseto

apreferredaxisintheUniverse.

Motivatedby these observedCMBanomalies, several mehanismsbasedon some

axisym-metri eet have been proposed, although the operational denition of the `Axis of Evil'

[LM05, LM07℄ does not neessarily imply the existene of suh a strong symmetry. Among

thevarious eetsthathavebeensuggestedtopossiblyintrodueapreferredaxisinto

osmol-ogyare: a spontaneous breaking of statistial isotropy [GHHC05℄, parityviolation in general

relativity[Ale06℄, anisotropiperturbations ofdarkenergy[KM06,BM06℄, residuallarge-sale

anisotropiesafter ination[CCT06,GCP06℄, oraprimordial preferreddiretion [ACW07℄. At

thesametime, ithas been studied [RRS06b, IS06℄ howthe loal Rees-Siamaeet ofan

ex-tendedforeground,non-linearindensityontrast,aetsthelowmultipolemomentsoftheCMB

viaitstime-varyinggravitationalpotential,seetheprevioushapter. Inasenariowithasingle

overdensitytheoeientsofthespherialharmonideomposition,the

a ℓm

^,^beome^modied

byonlyzonalharmonis,i.e.

m = 0

^modes. ^This^is ^equivalent^to^an ^axial^eet ^along^the^line

onnetingourposition withtheentreofthesoure.

Infat,theobservedpatternin theCMBforquadrupoleandotopoleisanearly pure

a ℓℓ

moderespetively; asseeninaframe where the

z

^-axis^equals^the ^normal^of^the ^plane^dened

bythe twoquadrupole multipole vetors[CHSS06℄. In [CHSS07℄ it hasalready beenargued,

thatforegroundmehanismsoriginatingfromarelativelysmallpath oftheskywouldmainly

theprimordial utuationswould havediulties explainingthelowmultipole powerat large

saleswithoutahaneanellation.

It is important to study how theinlusion of apreferred axisompareswith the intrinsi

multipoleanomalies atlargestsales. Ouranalysis isrestritedto axisymmetrieets ontop

oftheprimordialutuationsfromstandardination,thusseondaryorsystematieets. We

aregoing to quantify howpoorlyan axisymmetri eet at lowmultipoles of whateverorigin

mathesthethreeyear-dataofWMAP.Further,wewilldemonstratethatthereisnoorrelation

betweenthetwotypesofintrinsilow-

ℓ

^anomalies: ^the^two-point^orrelation^deit^and^intrinsi

alignment;andthatthereremainsnoneevenwhenapreferredaxisisintroduedtotheproblem.

5.2. Choieof Statisti

A ommon observable is the multipole power. Aording to the standard pereption of

inationary osmology, the CMB utuations are believed to follow a Gaussian statisti and

tobedistributed in astatistially isotropiway. Thenotionofstatistial isotropymeansthat

theexpetationvalueofpairsofoeients

h a ^∗ _ℓ ′ m ^′ a ℓm i

^isproportionalto

δ ℓ ^′ ℓ δ m ^′ m

^,^.f. ^(3.39).

Theproportionalityonstantmeasuringthe expetation valueofthe multipole onthefull sky

isommonlyestimated by

C ℓ

^,^.f. ^se.^3.3.2. ^The^angular^power^an^also^be^written^as

(5.1)

C ℓ ≡ 1

2ℓ + 1 X ℓ

m=−ℓ

| a ℓm | ² = 1 2ℓ + 1

Z

dΩ T _ℓ ² (θ, ϕ) ,

with

T ℓ

^being^the

ℓ

^-th^multipole^of^the^CMBtemperatureanisotropy. Itanbeexpandedwith thehelp ofspherialharmonisas:

T ℓ = P

m a ℓm Y ℓm

^. ^Note^that, ^sine ^we^onsider ^multipole

momentsthat arereal,the

a ℓm

^must^full^the^additional^ondition:

a ^∗ _ℓm = ( − 1) ^m a ℓ−m

^. ^Using

theestimator(5.1))theangulartwo-pointorrelationfuntion isgivenby

(5.2)

C(θ) = 1

4π X ∞

ℓ=0

(2ℓ + 1)C ℓ P ℓ (cos θ) ,

wherethe

P ℓ

^are^the^LegendrePolynomialsof

ℓ

^-th^order.

Besides of the multipole power itself, it is useful to introdue an all-sky quantity that

embraesallsales. Asinspiredbythe

S 1/2

^statisti,^presentedⁱⁿ^[S

+

03℄formeasuringthelak

ofangularpoweratsaleslargerthan

60 ^◦

^,^we^use^here^an^analogous^all-sky^statisti^[CHSS07℄

(5.3)

S full ≡

Z 1

−1

C ² (θ) d(cosθ) .

It is a measure of the total power squared on the full-sky. In ontrast to the

S 1/2

^statisti

+

03℄, the

S full

^statisti^does ^not^ontain ^any ^a^priori ^knowledge ^on^the ^variation ^of ^the^two

pointangular orrelation (5.2)for angles

> 60 ^◦

^. ^Here ^we^are ^onsidering ^espeially ^the ^large

angularsalesbut wearenotinterestedinthemonopoleand dipoleandthusarriveat

(5.4)

S _full ^trunc = 1

8π ² 5C ₂ ² + 7C ₃ ² .

Ofourse,allmultipoleshavetobeonsideredforthefull-skystatisti(5.3)butweanusethe

trunated part(5.4), beause herethe anomalies are mostpronouned and wewant to hek

for the interplay of this part of the full-sky powerstatisti with the other (phase) anomalies

withinquadrupoleandotopole. Thispartisthensimplytobeaddedtotherestofthesumof

(squared)multipolepowerin(5.3),reoveringtheexpressionforthefull-sky.

Next weturn to thestatistisinvolvingthephaserelationshipsof multipoles. Weuse the

oneptofMaxwell'smultipolevetors[Max79℄inordertoprobestatistial isotropy,sinethis

representationproved to be useful foranalyses of geometri alignments and speial diretions

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Likelihood ILC(3yr) DQ-corrected

100 µK 0 µK 200 µK 700 µK 1000 µK 2000 µK a _ℓ0 =

S nn

0 2 4 6 8 10 12 14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Likelihood ILC(3yr) DQ-corrected

100 µK 0 µK 200 µK 700 µ K 1000 µ K 2000 µK a _ℓ0 =

S ww

Figure5.3. EvolutionoftheMonteCarlolikelihoodofthealignmentstatistis

S nn

(5.7)and

S w w

^{(5.8) .} ^The^eet^of^an^axisⁱⁿ^the^CMB^is^modeled^via^inreasing

addi-tionalzonalharmoniswithoeients

a ℓ0

^. ^At

a ℓ0 = 1000µ

^K^the^multipoles^beome

purely zonalingoodapproximation. RegardingWMAP'sILC(3yr) map

S nn

^is

un-expetedat

98.3%

^C.L.^and

S w w

^is^odd^at

99.5%

^C.L.^with^respet^to^thestatistially isotropi and Gaussiansky (bold histograms). Thebestimprovementisreahedfor

bothstatistisatroughly

a ℓ0 = 100µ

^K.

oeients

a ℓm

^ontaining ^the ^physis. Alternatively, with the use of the multipole vetors formalismweanexpandanyrealtemperaturemultipolefuntion onasphereinto

(5.5)

T ℓ (θ, ϕ) = X ℓ

m=−ℓ

a ℓm Y ℓm (θ, ϕ) = A ^(ℓ)

" _ℓ Y

i=1

v ˆ ^(ℓ,i) · ˆ e(θ, ϕ)

− L _ℓ (θ, ϕ)

# ,

and

ˆ e

^is^a^radial^unit^vetor,^just^likeⁱⁿ^(4.6). ^The^`angular^momentum'^residuals^are^subtrated

withthehelpoftheterm

L _ℓ (θ, ϕ)

^. ^We^hoose^the^sign^of^the^multipole^vetors^so^that^they^all

pointtothenorthernhemisphere.

Inordertodisloseorrelationsamongthemultipolevetorswerstonsiderforeah

ℓ

^the

ℓ(ℓ − 1)/2

independentorientedareasbuiltfromtherossproduts

(5.6)

w ^(ℓ;i,j) ≡ ± v ˆ ^(ℓ,i) × v ˆ ^(ℓ,j) ,

whereofwewill also usethe normalisedvetors

n ^(ℓ;i,j) ≡ w ^(ℓ;i,j) / | w ^(ℓ;i,j) |

^. ^Now,ⁱⁿ ^[SSHC04℄

andsubsequentworks,thedotprodutsoftheareavetorshaveproventobeahandyexpression

inordertoquantifyalignmentsofthemultipolevetorsamongeahotherandalsowithexternal

diretions(whih wedo notonsider here). The following measure,asstated in [Wee04℄, and

usedin[SSHC04,CHSS06,CHSS07℄servesasanaturalhoieofastatistiinordertoquantify

theintrinsialignmentofquadrupole andotopoleorientedareas:

(5.7)

S ww ≡ 1

3 X

i<j

w ^(2;1,2) · w ^(3;i,j) .

Note that weonsider onlythe verylargestsales, i.e. we use thestatisti only for

ℓ = 2, 3

Analogously,astatistiinvolvingthenormalisedareavetorsisgivenby:

(5.8)

S nn ≡ 1

3 X

i<j

n ^(2;1,2) · n ^(3;i,j) .

5.3. Standard Model Preditions

Standardinationary

Λ

^CDM^osmology^requires^the^CMBanisotropiestobeGaussianand statistiallyisotropi. Forthesubsequentanalysis wehaveprodued Monte Carlorealisations

of the harmoni oeients

a ℓm

^following ^the ^underlying

Λ

^CDM ^theory. ^F^rom ^[CHS04℄ ^an

0 1 2 3 4 5 6 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Likelihood ILC(3yr) DQ-corrected

+100 µK

± 10 µ K

± 100 µ K

± 1000 µ K

S nn a 20 =

a 30 =

0 1 2 3 4 5 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Likelihood ILC(3yr) DQ-corrected

+100 µK

± 10 µK

± 100µK

± 1000µK

S ww a 20 = a ₃₀ =

Figure 5.4. The sign of additional axial ontributions

a ℓ0

^has ^no^physial ^eet

onthestatistis

S nn

^and

S w w

^. ^For^the ^quadrupole^this^follows ^from^the^symmetry

of the LegendrePolynomial

P 2

^[see ^equation ^(5.13)℄. ^The ^quadrupoleontribution is kept xed at

a 20 = 100µ

^K^while ^the ^axialontribution to the otopole is varied bothinmagnitudeandinsign. Respetive pairsof

±a 30

^histograms^lie^virtually ^on

eah otherandtheir statistis are thusindistinguishable. Thereferene histograms

following fromthe axially unmodied

Λ

^CDM ^model^(bold ^histograms ⁱⁿ^g. ^5.3 ^lie

nearlyontopofthedisplayed

a 20 = 100µ

^K^and

a 30 = ±10µ

^K^ases,^and^are^thus^not

shown.

Mollweide maps of a sample of random Gaussian and statistially isotropi quadrupole and

otopolevetorsas wellastheirnormalsaregivenin g.5.2(leftolumn).

Conerning thequestionoforrelationsbetweenthemultipole powerandthealignmentof

multipolevetors,itappearsnaturalto expetthat there isnone. That isbeauseweinvoked

Gaussianrandomandstatistiallyisotropiskies,leadingtomultipolevetors(5.5)independent

ofthemultipole power(5.1). Thisassumptionneedstobetestedand quantied.

Nevertheless,asmallorrelationouldbeexpetedfrom thefollowingreason: Considering

onlymultipolesuptosomelimitingpower,theresultingprobabilitydensitydistributionforthe

a ℓm

^must ^be non-Gaussian. In fat, this restrition leads to a negative kurtosis for the

a ℓm

distribution(the skewnessvanishes). Havingthatinmind,itappearssuddenlyunlearwhether

thenaiveexpetationofvanishingorrelationofpowerwithintrinsialignmentwillhold. Below

wesubstantiatetheabseneoforrelationsbymeansofaMonte Carloanalysis.

Let us rst look at thealignment anomalies. In g. 5.3 the likelihood of the quadrupole

and otopole alignment statistis

S ww

^and

S nn

^is ^shown. ^The ^preditions ^of ^the ^standard

inationary

Λ

^CDM ^model ^are ^shown ^as ^the ^bold ^histograms respetively (

=

^vanishing ^axial

ontamination). Aordingtothethree-yearILCmapfromWMAP[WMAa ℄wegetthe

follow-ingmeasuredvaluesforthealignmentstatistis:

S _nn ^ILC(3yr) = 0.8682 and S _ww ^ILC(3yr) = 0.7604 ,

when[CHSS07℄ orreted for theDoppler-quadrupole. The totalnumber of Monte Carloswe

produedpersampleis

N = 10 ⁵

^. ^We^infer^that ^the^unmodied^inationary

Λ

^CDM^predition

isunexpetedat

98.3%

^C.L.^with^the

S nn

^statisti^and^unexpeted^at

99.5%

^C.L.^a^with^respet

tothe

S ww

^statisti.

Next weonsidertheross-orrelationbetweentheintrinsiphaseanomalies andthe

mul-tipole power(5.1) within thelow-

ℓ

^. ^For^this^we^hose ^those

a ℓm

^that ^allow^for ^say^the ^lowest

possible

5%

ⁱⁿ^the^left^tail^of^thedistributionsfor

C 2

^and

C 3

^that^follow^from^statistial^isotropy,

Gaussianityandthe

Λ

^CDM^best-t^to^the^WMAP^data. ^Then^we^ompute^the^expression

S ww

fortheseleted

a ℓm

^and^ompare^it^to^the^aording^ILC(3yr)^value. ^As^expeted,^no^orrelation

Thevaluequotedabovewas[CHSS07℄

99.6%

^C.L.^The^small^dierene^is^due^to^theinorporationofthe

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 WMAP(3yr)

S w w

S _full ^trunc [0.1mK] ⁴

Figure5.5. Contourofthesatterofintrinsialignment(5.7)versusfull-skypower

squared(5.4) . The shapeanbe understoodfrom thefolding ofthe tworespetive

distributions. The total numberof Monte Carlopoints is

N = 10 ⁵

^. ^The ^measured

data point from WMAP three-year data is inluded. The maximum of likelihood

requires

S w w

^far^smaller^than^obtained^from^ILC(3yr). ^Consisteny^with^the^data^an

beexludedat

99.95%

^C.L.^Contours^orrespond ^to^lines^of

1/2 ⁿ

^times^the ^maximal

likelihood,with

n = 1, . . . , 5

isfound,thatisneithertheshapenortheexpetationvalueofthealignmentstatistiisshifted.

Wendthesamealsofortheombinationofthe lowest allowed

5%

ⁱⁿ

C 2

^and^the^highest

5%

fromtherighttailofthedistributionof

C 3

^and^the^remaining^two^possibleombinationsthereof.

Aswedo notndanyorrelations, we anonludethat the

S ww

^and

S nn

^statistis^are ^not

sensitivetothenon-Gaussianityinduedbytherestritiontolowmultipole power.

Moreover,weprobetheoppositediretionbytaggingthose

a ℓm

^that^lieⁱⁿ^the^allowed^right

tailof the

S ww

distributionwithrespetto

S ww ^ILC(3yr)

^. ^Thedistribution ofthemultipole power for

C 2

^and

C 3

^made^of^these

a ℓm

^remains^unhanged. ^The^latter^nding^onrms^that^multipole

powerandtheshapeofmultipoles(phases)areunorrelated.

UsingEquation(5.4),the[WMAa ℄MaximumLikelihoodEstimate(MLE)fromtheWMAP

ILC(3yr)map for the angular power spetrum yields

S _full ^trunc,MLE = 29431µK ⁴

^. ^Compared ^to

thevalueof

136670µK ⁴

^from^the

Λ

^CDM^best-t ^to^WMAP(3yr) ^data,^this ^is^not^signiantly

unexpeted, withanexlusionlevelofonly

92.1%

^C.L.

Nowwewanttohekfororrelationsbetweentheall-skymultipolepowerandthemultipole

alignment. Asforreasonsexplainedinthenextsetionwepreferthe

S ww

^statisti^to

S nn

ⁱⁿ^the

followingorrelationanalysis. InFigure5.5thesatterplotof

S ww

^against

S _full ^trunc

^is^shown. ^The

formoftheontouranbeunderstoodasjustthefoldingofthe

χ ²

^-like^form^of^thedistribution for

S _full ^trunc

^with^thegaussian-likeformofthe

S ww

distribution. AtrstglaneweseefromFigure 5.5that theMLEfromWMAP(3yr)

S _full ^trunc,MLE = 29431µ

⁴

^requires^the^alignment^statisti^to

beofmiddlevalues(around

0.4

^),^whih^isinonsistentwiththerespetivemeasuredanomalous valuefrom ILC(3yr). Moreoverthe lak of any linear behaviour in the ontour suggeststhat

thereisnoorrelationbetweenthetwostatistis.

Giventhatnoorrelationispresentbetween

S ww

^and

S _full ^trunc

^,^we^would^expet^that^the^joint

probabilitythatbothpowerandalignmentarein aordanewithdatafatorisesaordingto:

(5.9)

p S _full ^trunc ≤ data ∧ S ww ≥ data

= p 1 S _full ^trunc ≤ data

p 2 (S ww ≥ data) .

But in realitywe anonly aess nitestatistial samples ofthese quantities andthe

fa-torisationwill notbe exat. However, we want to will hek the validity of (5.9) within our

statistialensemble. Whenusing thefull sample with

N = 10 ⁵

respetively weobtainajoint likelihood of

p ≃ 0.05%

^. ^The ^error

∆

^of ^the fatorisation, whih we dene asthe dierene betweenthelefthandsidein(5.9)andtherighthandside,isoftheorder

O (10 ⁻⁵ )

^,^that^is^of^the

orderoftheMonteCarlonoise. Inorderto traktheevolutionoftheerror

∆

^we^also^ompute

thejointlikelihood(5.9)forsmallersubsamples;seetab.5.3. Reduing

N

^to

N = 10 ⁴

^we^obtain

anevensmallerjointlikelihood of

p = 0.02%

^but ^with^an^error^that^is ^of^the^same^magnitude.

With

N = 10 ³

^we^do^not^have^a^single^hit ^for ^the^joint ^Monte ^Carlos^leading^to

p = 0%

^with

thesameerror asin the

N = 10 ⁴

^ase ^of

∆ = 0.02%

^. ^Note^that ^just ^one ^Monte ^Carlo ^hit ⁱⁿ

favourofthejointasewouldraisetheerrorhereto

∆ = 0.08%

^. ^In^the^end,^the^onvergene^of

thejointlikelihoodappearstobeveryslowwithrespettothesamplesize

N

Furthermore weareinterestedin thestabilityoftheresultsfor

∆

^with ^respet^to ^hanges

inthemeasureddata. Forthiswehoose theWMAP(1yr)values:

(5.10)

S trunc,pseudo

C ℓ

full = 10154µK ⁴ and S _ww ^ILC(1yr) = 0.7731 .

We use a sample of the full size

N = 10 ⁵

^and ^obtain ^a ^joint ^likelihood ^with ^respet ^to ^the

one-yeardataof

p = 0.001%

^with^an^error

∆ = 0.002%

^. ^That ^is,^with^respet^to ^one-year^data

both the joint likelihood and its error are of the order of the Monte Carlo noise. From the

WMAP(1yr) data aloneweould exludethejointase(5.9) rather onservativelyat

99.99%

C.L.Thisappearstobeastrongerexlusionthantheonefromthree-yeardata. Butwedonot

bothermuhaboutthedierenebeauseofthedierentestimatorsthathavebeenusedbythe

WMAPteamfortheangularpowerspetrum(pseudo-

C ℓ

^vs. ^MLE)^[WMAa℄.

samplesize

N

^joint

p

^error

∆

100000

0.048% 0.008%

100000

0.001% 0.002%

10000

0.02% 0.02%

1000

0% 0.02%

Table5.1. Jointlikelihoods(5.9)for

S _full ^trunc

^and

S w w

^beingⁱⁿ^aordane^with^data

simultaneously. Theexperimental values referto WMAP's ILC(3yr) map [WMAa℄

exept forthe seondrow. Theerror

∆

^of^the fatorisationinequation(5.9) isthe dierenebetweenlefthandsideandrighthandsideinthatequation.

Wequoteherethemostonservativeresult,namelythefullsamplejointlikelihoodasefor

S ww

^and

S _full ^trunc

^with^respet ^to ^the^WMAP(3yr) ^data. ^Therefore^we^an^exlude^that ^ase^at

> 99.95%

^C.L. ^with ^an^errorⁱⁿ ^the^third ^digit ^after ^the^omma^lying^within ^the ^Monte ^Carlo

erroroftheusedsample(

N = 10 ⁵

^).

Finally weattempt to analyse theorrelationof theall-skypowerstatisti

S _full ^trunc

^and ^the

intrinsimultipolealignment

S ww

^byquantitativemeans. Itiswellknownfrom statistis,that whenhekinganitetwo-dimensionalsamplefororrelations,theempiriovariane

(5.11)

cov[ S _full ^trunc , S ww ] ≡ 1 N − 1

X N

i=1

S ^trunc _full,i − S ¯ _full ^trunc

S ww, i − S ¯ ww

isa ruial quantity. Thebar standsfor the meanof avariable. As theovarianeis a sale

dependentmeasure, i.e.depending onthe magnitudesof thesample values

S ww, i

^and

S ww, i

the dimensionless Bravais-Pearsonoeient orempirial orrelation oeient is the better

expressiontouse:

(5.12)

ρ S ^trunc _full , S ww ≡ cov[ S ^trunc _full , S ww ]

p cov[ S _full ^trunc , S _full ^trunc ] cov[ S ww , S ww ] .

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 WMAP(3yr) 0.8

S w w

S ^trunc _full [0.1mK] ⁴

Figure5.6. Satterontourofpairsof

S w w

^and

S _full ^trunc

^after^an^axial^modiation^of

a ℓ0 = 70µ

^K^has^been^applied;^this^is^theontributioninvolvingmaximalimprovement in

S w w

^(see^g.^5.3). ^The^total^number^of^Monte^Carlo^pairs^is

N = 10 ⁵

^. ^Note^that

the horizontal axis now runs from zero to

1.4 × 10 ⁻⁶ mK ⁴

^, ^whereas ⁱⁿ ^g. ^5.5 ^the

maximal displayed value is

4 × 10 ⁻ ⁷ mK ⁴

^. ^The^inlusion ^of ^a ^preferred ^axis ^leaves

all-skymultipolepowerandintrinsialignmenttotallyunorrelatedandinonsistent

withtheWMAP(3yr)data. Contourlinesaredenedasing.5.5.

Finally,employingtheWMAP(3yr)dataweobtainanempirialorrelationoeientof

ρ S _full ^trunc ,S ww = − 0.0027 ,

withrespettothefullsample

N = 10 ⁵

^,^whih^indeed^indiates ^only^marginalorrelation.

5.4. Inlusionof a Preferred Axis

Now we ask what happens when introduing axial ontributions on top of a statistially

isotropiandgaussianmirowavesky. Thepresene ofapreferreddiretion withaxisymmetry

in theCMB will exlusively exitethe zonal modes in asethe axisis ollinearto the

z

^-axis.

Herewedonotbotheraboutexternaldiretionssinetheinternalalignmentsareindependent

ofthese. Thereforesuh anaxiswill manifest itself throughadditionalontributions

a ℓ0

^. ^We

areonsidering thequadrupoleand theotopoleandthequestionarises,in howfarthesignof

theaxialontributions

± a ℓ0

^plays^a^role. ^The^oeients

a ℓm

^an^bereonstrutedfrom

(5.13)

a ℓm =

Z ∆T

T (θ, ϕ) Y _ℓm ^∗ dΩ .

Obviously, within the quadrupole the sign of

± a 20

^is ^irrelevant ^beause ^of ^the ^symmetry ^of

theLegendre Polynomial

P 2

^with ^respet ^to

θ = 90 ^◦

^. ^The ^Legendre ^Polynomial

P 3

^however

isantisymmetriwith respet to

θ = 90 ^◦

^. ^Therefore ^the^relevane^of ^the ^sign^of ^the ^otopole

ontributions

a 30

^has^to ^be^laried. Consequentlywehavehosen axed valuefor theaxial quadrupoleontribution

a 20

^and^have ^then^varied ^the^aording^otopole ontributionin sign andin magnitude. Theresultsaredisplayeding.5.4. Apparentlythe

S nn

^and

S ww

^statistis

thatareimportanthere,donotdistinguishbetweenthesignoftheappliedaxialeet. Therefore

weneednottobotheraboutthesignsofthe

a ℓ0

^and^let^them^heneforth^be^positive.

In Figure5.3 theevolutionof the

S ww

^and

S nn

^statistis^with ^respet^to ^inreasing^axial

Letusrstlookattheevolutionofthe

S nn

^statisti. ^This^expression^measures^the^average

| cos |

^of ^the ^angles ^between ^the ^quadrupole ^oriented ^area ^and ^the ^otopole ^areas. ^The ^pure

Monte Carlo peaks at

0.5

^reeting ^the ^fat ^that ^the ^average ^distane ^of ^four isotropially distributedvetorsonahalf-sphere fromeahother is

60 ^◦

ⁱⁿ^the^ase^of^statistial^isotropy. ^It

isahalf-spherebeausethesignsofthemultipole vetorsarearbitraryandso wehoosethem

alltopointtothenorthernhemisphere. Wheninreasingtheontributionoftheaxialeetthe

multipolesbeomeinreasinglyzonalandarriveatbeingpurelyzonalinagoodapproximation

at values of

a ℓ0 = 1000µ

^K. ^On ^the ^level ^of ^the ^multipole ^vetors ^this ^means ^that ^their ^ross

produtsallmovetotheequatorialplane(seeg.5.2). Thatisthereasonwhythehistogramin

g.5.3(left)movestotherightwhenweinreasetheaxialeet,beausenowisotropyisbroken

fromthehalf-spheretothehalf-irlemakingthe

S nn

^histogram^peak^sharper^at^higher^values.

ThemeasuredvaluefromtheILC(3yr)mapof

S nn ^ILC(3yr) = 0.868

^is^anomalous^at

98.3%

^C.L.^with

respet to the pure Monte Carlo (bold histogram in g. 5.3 whih standsfor the statistially

isotropiandgaussianmodel. Byaddingaxialontributionthemaximalimprovementisreahed

a ℓ0 = 100µ

^K^where ^the^ILC(3yr)^beomes^unexpeted ^at

96.7%

^C.L. ^Further^enhanement

oftheaxialeet makesthe

S nn

^statisti^more^and^more^narrow^around^an^expetation ^value

< 0.7

^. ^This^makes^it^impossible^to^remove^the^anomalyⁱⁿ^the

S nn

ross-alignmentwithrespet to the ILC(3yr) experimental value only by inreasing the axial ontribution to high enough

values.

On the other hand the

S ww

^statistiadditionally measures themodulus of the sin of the anglesbetweenthemultipolevetorsthemselves. Asanbeseenfromg.5.2multipolevetors

are all moving toward the north pole lustering more and more as the axial ontribution is

enhaned. The

S ww

^statisti ^measures ^the^average ^of ^the^modulus^of ^the ^produts ^of^the ^sin

of angles between quadrupole vetors, otopole vetors and the os of the angle between the

areavetors. Therefore ontopof theinformation alreadyontainedin

S nn

^the

S ww

^statisti

isable to goto zerofor highestzonal ontamination asthe losenessof themultipole vetors

in that asedampenstheprodut ofsines andosines quadratiallyto arbitrarysmall values.

Thus wendthat

S ww

^is ^the^more^onvenient^statisti^for^further ^analyses,^as^it ^does^ontain

moreinformationthan the

S nn

^statisti^andadditionallyshowsasimpleandlearasymptoti behaviour. In the ase of this statisti the anomaly is signiant at

99.5%

^C.L. ^with ^respet

S ww ^ILC(3yr) = 0.7604

^. ^Similarly^to ^before ^the^maximal improvement is reahed with anaxial ontributionof

a ℓ0 = 70µ

^K,^whih ^degrades^the^anomalyⁱⁿ

S ww

^to

99.2%

^C.L.

Nowwereturn totheorrelation analysisof thealignmentwiththepure multipole power

C ℓ

^. ^When^introduing^an^axial^eet,^say

a ℓ0 = 100µ

^K,^we^improve^the^t^to^the

S ww

^statisti,

butinterestinglythemultipolepoweranomalybeomesmuhmorepronouned. Thisbehaviour

isexpeted[RRS06b,RRS06a℄forthe

C ℓ

-distribution(beingamodied

χ ²

-distribution)when theaxial ontribution is enhaned, but it is unexpeted that exatly the samehappens for a

multipolepowerdistribution`thatknowsoftheintrinsialignmentofquadrupoleandotopole'.

Thisindiatesthatthereisnoorrelationatallbetweenmultipolepowerandthephasealignment

evenwhentheyaretuned toeahother.

Proeedingwiththeanalysisoforrelationsbetweenalignmentandthefull-skypower

statis-ti,again wetryto provokeorrelationwith the help of axial symmetryin theCMB. In fat

weapply anaxial eet oftheideal magnitude(

a ℓ0 = 70µ

^K) ⁱⁿ ^order ^ahieve^larger^valuesⁱⁿ

S ww

^. ^The ^negative^result^is ^shown ⁱⁿ ^g.^5.6: ^as

S _full ^trunc

^is ^a^linear^ombination^of ^squared

C ℓ

distributions it is asharplypeaked

χ ²

^-likedistribution beingverysensitive to axial ontribu-tions. Thereforetheontouring.5.7 isfairlyshiftedto theright(tohigher valuesin

S _full ^trunc

⁾

andbroadenedwithrespet tothe axiallyunmodiedase,obviatinganyorrelationwith the

intrinsialignment. Theshapeoftheoverallontourisroughlyleftinvariantbythesaleshift

S _full ^trunc

Theg.5.7illustratesthepurezonalase. Hereawhole

a ℓ0 = 1000µ

^K^has^been^indued^into

themultipolevetors. Again,duetothesensitivityof

S ^trunc _full

^to^axialontaminationthispushes theallowedregioninthesatterplottoveryhighvaluesinfull-skypowersquared,degenerating

Im Dokument On the influence of local inhomogeneities on cosmological observables : from galaxies to the microwave background (Seite 121-166)

30 ◦

a ℓ0 = 1000µ

a ℓ0 ∼ 100µ

a ℓm

m = 0

a ℓℓ

z

ℓ

h a ∗ ℓ ′ m ′ a ℓm i

δ ℓ ′ ℓ δ m ′ m

C ℓ

C ℓ ≡ 1

2ℓ + 1 X ℓ

m=−ℓ

| a ℓm | 2 = 1 2ℓ + 1

Z

dΩ T ℓ 2 (θ, ϕ) ,

T ℓ

ℓ

T ℓ = P

m a ℓm Y ℓm

a ℓm

a ∗ ℓm = ( − 1) m a ℓ−m

C(θ) = 1

4π X ∞

ℓ=0

(2ℓ + 1)C ℓ P ℓ (cos θ) ,

P ℓ

ℓ

S 1/2

+

60 ◦

S full ≡

Z 1

−1

C 2 (θ) d(cosθ) .

S 1/2

+

S full

> 60 ◦

S full trunc = 1

8π 2 5C 2 2 + 7C 3 2 .

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Likelihood ILC(3yr) DQ-corrected

100 µK 0 µK 200 µK 700 µK 1000 µK 2000 µK a ℓ0 =

S nn

0 2 4 6 8 10 12 14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Likelihood ILC(3yr) DQ-corrected

100 µK 0 µK 200 µK 700 µ K 1000 µ K 2000 µK a ℓ0 =

S ww

S nn

S w w

a ℓ0

a ℓ0 = 1000µ

S nn

98.3%

S w w

99.5%

a ℓ0 = 100µ

a ℓm

T ℓ (θ, ϕ) = X ℓ

m=−ℓ

a ℓm Y ℓm (θ, ϕ) = A (ℓ)

" ℓ Y

i=1

v ˆ (ℓ,i) · ˆ e(θ, ϕ)

− L ℓ (θ, ϕ)

# ,

ˆ e

L ℓ (θ, ϕ)

ℓ

ℓ(ℓ − 1)/2

w (ℓ;i,j) ≡ ± v ˆ (ℓ,i) × v ˆ (ℓ,j) ,

n (ℓ;i,j) ≡ w (ℓ;i,j) / | w (ℓ;i,j) |

S ww ≡ 1

3 X

i<j

30 ^◦

h a ^∗ _ℓ ′ m ^′ a ℓm i

δ ℓ ^′ ℓ δ m ^′ m

| a ℓm | ² = 1 2ℓ + 1

dΩ T _ℓ ² (θ, ϕ) ,

a ^∗ _ℓm = ( − 1) ^m a ℓ−m

60 ^◦

C ² (θ) d(cosθ) .

> 60 ^◦

S _full ^trunc = 1

8π ² 5C ₂ ² + 7C ₃ ² .

100 µK 0 µK 200 µK 700 µK 1000 µK 2000 µK a _ℓ0 =

100 µK 0 µK 200 µK 700 µ K 1000 µ K 2000 µK a _ℓ0 =

a ℓm Y ℓm (θ, ϕ) = A ^(ℓ)

" _ℓ Y

v ˆ ^(ℓ,i) · ˆ e(θ, ϕ)

− L _ℓ (θ, ϕ)

L _ℓ (θ, ϕ)

w ^(ℓ;i,j) ≡ ± v ˆ ^(ℓ,i) × v ˆ ^(ℓ,j) ,

n ^(ℓ;i,j) ≡ w ^(ℓ;i,j) / | w ^(ℓ;i,j) |

w ^(2;1,2) · w ^(3;i,j) .

n ^(2;1,2) · n ^(3;i,j) .

S ww a 20 = a ₃₀ =

S _nn ^ILC(3yr) = 0.8682 and S _ww ^ILC(3yr) = 0.7604 ,

N = 10 ⁵

S _full ^trunc [0.1mK] ⁴

N = 10 ⁵

1/2 ⁿ