ray survey: dipole velocity profile

Lepus

X- ray survey: dipole velocity profile

Figure 4.3. Left: A projetion of the intermediate-sale struture inour

neigh-bourhoodto thesupergalati

x − y

^plane. ^The^ontours^indiate^density^aording

(1, 3, 0.5) × 10 ⁻ ³

^lusters ^Mp

⁻ ²

respetively. The Shapley onentration repre-sents the most massive strutureinthe shown distane range. Onean learly see

thezoneofavoidane,fromwhihoptialdataannotbetaken. Right: dipoleprole

as derived fromreent X-ray galaxy surveys. TheShapley superluster dominantly

ontributesto thedipolebetween

∼ 100

^Mp^and

∼ 200

^Mp. ^The^pitures^are ^taken

from[TSVZ92 ℄and[KME04 ℄.

onditionandhave

v ² _c ≃ φ

^, ^whih^leads^us^to

(4.2)

∆T (θ, ϕ) T

RS

∼ φ ^3/2 ∼ δM

d 3/2

.

Wearegoingtomodelthenon-linearstruturebyaspheriallysymmetriLTBmodelembedded

in a at (

Ω = 1

⁾ Friedmann-Robertson-WalkerUniverse. Substituting theexpression for the massexesswithinthismodel[Pan92℄weobtainthePaneksaling

(4.3)

∆T (θ, ϕ) T

RS

∼ δρ

ρ 3/2

d t

3 .

Werepeat,

t

^is^the^osmi^time^at^whih^the^CMB^photons^rossed^the^struture,

d

^is^its^physial

sizeand

δρ/ρ

^its^density^ontrast. ^Inserting^theharateristisoftheShapleysuperluster,we seethatindeed aCMBanisotropyof

10 ⁻⁵

^due^to ^a^loal^Rees-Siama^eet^isreasonable.

Foralargeangularsaleof thesoureloaland nearbystrutures this indues

ontri-butionsto thelow-

ℓ

multipoles, espeiallythedipole,quadrupole and otopole. This, in turn, ould inlude a non-Doppler ontribution to thedipole. This would imply ahange of afew

perent in the inferred dipole veloity, whih might also explain someof theCMB anomalies

[FGM

+

06℄. The Shapleyonentrationis anon-linear struture,and the amplitudeof the

in-duedanisotropiesannot be reliablyalulatedin linearperturbation theory. Aordingto a

omparisonoflinearandexatalulationsforGreatAttrator-likeobjetswiththeLTBmodel

in [FSA94℄, linear theoryis reliableat distanes omparableto the Hubblesale, but failsfor

strutureswithin1000

h ⁻¹

^Mp^or^so.

The advantage of the spherial symmetry of the LTB model is that it allows exat

al-ulations fornon-linear objets; thedrawbakis that theobserved non-linearobjetssuh as

theGreatAttratorandtheShapleyonentrationdonotappearto bespheriallysymmetri.

Figure4.4. AMollweidemapshowingtheforegroundswhihtheWMAP

ollabora-tiontakesintoaountformapleaning.Theonlyextendedforegroundisrepresented

by the galatiregion. Theregions shownin pinkand beige indiatethe so alled

Kp0andKp2diuseemissionmasksusedbytheWMAPollaborationtoobtain

os-mologialmaps. Fordetails ofthemap-makingproess see[J

+

07a ℄ and[H

+

07℄. All

oftheremainingforegroundsthathavebeentakenintoaountarepointsoures. In

this workwe areonsideringthe(Rees-Siama)eetofextendedloalforegrounds,

seeg.4.8. Thepitureistakenfrom[WMAa ℄.

theShapleyonentrationdoesseem toberoughlyspherial[PQC

+

06℄. Also,if thepreferred

diretion indiated by the low-

ℓ

^anomalies ^is ^due ^to ^loal ^strutures, ^this ^implies ^that ^there

indeedisadegreeofsymmetryintheloalmassdistribution.

In addition, there is aseond motivation for studying aspheriallysymmetri

inhomoge-neous model, namely dark energy. If interpreted in the framework of isotropi and

homoge-neousosmology,observationsofSNIaimplythattheexpansionoftheUniverseisaelerating,

.f. se. 1.2.2. However, in an inhomogeneous spaetime the observations are not neessarily

inonsistentwith deeleration, see se. 1.3. Inpartiular, in theLTB model theparameter

q 0

denedwiththeluminositydistane isnolongeradiret measureofaeleration[HMM97℄. It

hasbeensuggestedbyseveralgroupsthataspheriallysymmetriinhomogeneityouldbeused

to explain the SNIa data, see se. 1.3, though it is not lear whether suh amodel ould be

onsistentwithwhatisknownaboutstruturesintheloalUniverse[Bol05℄ortheobservation

ofbaryonosillationsinthematterpowerspetrum. HerewewillonernonlytheCMB.

ThepitureoftheloalUniversethatweadoptisaspheriallysymmetridensity

distribu-tion,withtheloalgroupfallingtowardstheoreoftheoverdensityat theentre,.f. g.4.2.

Thelinebetweenourloationandtheentredenesapreferreddiretion

z ˆ

^,^whihⁱⁿ^the^present

aseorrespondstothediretionofthedipole aftersubtratingourmotionwithrespettothe

loalgroupandtheloalgroup'sinfalltowardsthenearbyVirgolusterassumingthe

primor-dialomponentofthedipoletobenegligible. Thediretions ontheskythatareimportantfor

ouranalysis aregivenin tab.4.2. Thissetupexhibits rotationalsymmetrywithrespetto the

axis

z ˆ

^negleting^transverse^omponents^of^our^motion. Consequently,onlyzonal harmonis (

m = 0

ⁱⁿ ^the

z ˆ

^-frame)^are^generated. ^We^have^already^antiipated ^this^result,^it ^is^onsistent

withourpreditionthatameoutfromtheanalytialtreatmentoftheRees-Siamaeetusing

anLTBmodel inse. 1.3.3. Notethat anyothereet withaxialsymmetrywouldalsoindue

anisotropyonlyin thezonalharmonis.

ThedensityeldhastwoeetsontheCMBseenbyano-entreobserver.First,photons

omingfrom dierentdiretions traveldierent routes throughtheloal overdensity,and this

reatesanisotropyevenwithaperfetlyhomogeneousdistributionofphotons. Inastationary

Figure 4.5. Left: thegalatioordinatesystem. Thegalatioordinatesystem

is denedasbeing parallelwiththeplaneoftheMilkyWayandentredonthesun.

Sotheequatoringalatioordinates(redirle,

0 ^◦

^galati^latitude)^liesⁱⁿ^the^plane

of ourgalaxy. Thegalatilatitude

b

^is ^the^angle ^above^or ^below^this ^plane^(yellow

angle)andthegalatilongitude

l

^(green^angle)^is^measured^from

0 ^◦

^to

360 ^◦

^,^ounter

lokwisewithrespettothenorthgalatipole.

0 ^◦

^of^galati^longitude^isarbitrarily dened asthe diretionpointing tothe galatientre(Sagittarius). Sometimes,in

astronomytheequatorialoordinatesystemisused. Right: therelationofthegalati

oordinatesystemtotheequatorialoordinatesystem. Thelatterisdenedthrough

the planeofthe Earth'sequator. Importantreferene diretionsontheskythat we

usehereare,ingalatioordinates: thenortheliptipole

(l, b) ≃ (96.4 ^◦ , 29.8 ^◦ )

^,^the

equinox

(l, b) ≃ (276.3 ^◦ , 60.2 ^◦ )

^and^the^north^galati^pole

(l, b) = (0 ^◦ , 90 ^◦ )

^. ^Pitures

are takenfrom[Ast℄and[Org ℄.

CMB. Seond, the environmentwill aet the evolution of the intrinsi anisotropies as the

homogeneousbakgroundspaedoes,byhangingtheangulardiameterdistane. Theomplete

alulation taking into aount both of these eets would be to study the evolution of the

CMBanisotropiesastheytravelarossthedensityeldusing perturbationtheoryontheLTB

bakground. Asinearliertreatments,weneglettheseondeetandsimplyaddtheanisotropy

generatedbytheLTBmodelontopoftheintrinsiontribution. Itispossiblethatthistreatment

missessomeeetsofproessingtheanisotropiesalreadypresent. Inpartiular,simplylinearly

adding a new soure of anisotropy will in general add multipole power, not redue it, while

aproperanalysis of the proessing of the intrinsi anisotropies ould leadto amultipliative

modiationoftheamplitudes ofthelowmultipoles, asmentionedin[GHHC05℄.

It hasbeen suggestedthat spheriallysymmetri inhomogeneities of the order of horizon

sizeorlargerwouldontributetothelowCMBmultipoles [DZS78,RT81,PP90, LP96℄; itwas

laimedin[Mof05℄ thatthis ouldexplaintheobservedpreferredaxis. Leavingasidethe issue

thatassumingspherialsymmetryfortheentireUniverseseemsquestionable,theobservational

signatureonthelowmultipoles isidentialto thatfrom theLTB modelused todesribeloal

strutures,possiblyapartfromtheamplitude.

4.3. Angular PowerAnalysis

Firstweaddressthequestionhowtheosmimirowaveskyis aetedbythe loal

Rees-Siama eet. We are going to study how maps of the CMB are aeted by the anisotropy

induedbyadditionalaxisymmetri ontributions

a ^axial _ℓ0

^on^the^largest^angular ^sales^by^using

0 1 2 3 4 5 6

0 0.05 0.1 0.15 0.2 0.25 0.3

Likelihood

C ₂ in [0.1 mK] ²

WMAP(1yr) WMAP(3yr) a ₂₀ ^axial = 0 µK [WMAP(1yr) best-fit ΛCDM]

20 µK 40 µK 70 µK

0 2 4 6 8 10 12 14

0 0.05 0.1 0.15 0.2 0.25 0.3

Likelihood

C ₃ in [0.1 mK] ²

WMAP(1yr) WMAP(3yr) a 30 axial = 0 µ K [WMAP(1yr) best-fit Λ CDM]

20 µ K 40 µ K 70 µK

Figure 4.6. Likelihood of quadrupole and otopole power for inreased axial

ontributions. Vertial lines denote experimental data: WMAP(1yr) ut-sky and

WMAP(3yr)maximumlikelihoodestimate. Consideringthe quadrupoleaddingany

multipolepower was exluded at

> 99%

^C.L. ^with ^respet ^to ^WMAP(1yr) ^but ^it

is possible to add up to

60µ

^K ^within ^the ^same ^exlusion ^level ^with ^respet ^to ^the

WMAP(3yr)value. Adding

80µ

^K⁽

100µ

^K)^to^the^quadrupole^leads^to^an^exlusion^of

99.7%

^C.L.⁽

99.9%

^C.L.). ^The^otopole^is^more^resistant^against^axialontaminations as itispossibletoadd awhole

100µ

^K^before^reahing^the^same^exlusion^level^with

respettotheupdatedWMAPdata.

Wesawin se. 3.3.2that theangularpowerspetrumin termsof theoeients

a ℓm

^an

beexpressedas

(4.4)

C ℓ = 1

2ℓ + 1 X ℓ

m=−ℓ

| a ℓm | ² .

Aspreditedbythestandardpereptionofinationaryosmology,theprimordialperturbations

arebelievedtofollowagaussianstatisti. Deviationsfromthiswouldbehardtoreonilewith

thestandardinationary paradigm. Therefore,theomplexoeients

a ℓm = a ^Re _ℓm + i a ^Im _ℓm

^are

expetedtobegaussianlydistributed withzeromeanand varianegivenbytheangularpower

C ℓ

^,^aording^to

(4.5)

f (a ℓ0 ) = 1

√ 2πC ℓ

exp

− (a ^Re _ℓ0 ) ² 2C ℓ

and f (a ^Re,Im _ℓm ) = 1

√ πC ℓ

exp − (a ^Re,Im _ℓm ) ² C ℓ

! .

Therefore,in thestandardmodel, theoeients

a ℓm

^are ^fullyharaterised by theirangular power,forwhihweusethevaluesfromthebestt

Λ

^CDMtemperaturespetrumtotheWMAP data. Inouraxisymmetrimodel,weparameterisetheeetofaloalstruturebyaddingaxial

ontributions

a ^axial _ℓ0

^to ^the^quadrupole^and^otopole. ^It^is ^obvious^that^the^additive^mehanism

annotmakethepowerdeitanomaly disappear. Forthestatistial analysiswegenerate

10 ⁵

MonteCarlorealisationsof thequadrupoleand theotopole. Inthe followingwedesribethe

resultsofourMonte Carloanalysisfortheangularpower(4.4)withrespettoone-yearaswell

asthree-yearWMAPdata.

4.3.1. WMAP(1yr) Angular Power. Consideringone-yeardata,thevaluesof

C 2

^and

C 3

^determined^from ^the^WMAP ^ut-sky^[H

+

03℄, thesoalled TOHmap [TdOCH03℄, the

La-grange ILC map [EBGL04℄ and the ILC map [B

+

C 3

^as^ompared

0 0.5 1 1.5 2 2.5 3 3.5

0 0.2 0.4 0.6 0.8 1

Likelihood

S _nNEP

ILC(3yr)

ILC(1yr)

axis in direction of WMAP dipole a l0 axial = 40 µK 70 µK 0 µK

0 0.5 1 1.5 2 2.5 3 3.5

0 0.2 0.4 0.6 0.8 1

S _nEQX

ILC(3yr) ILC(1yr) axis in direction of WMAP dipole a _l0 ^axial = 0 µK 40 µK 70 µK

0 0.5 1 1.5 2 2.5 3 3.5

0 0.2 0.4 0.6 0.8 1

S _nNGP

ILC(3yr) ILC(1yr) axis in direction of WMAP dipole

a l0 axial = 40 µK 70 µK 0 µK

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1

Likelihood

S _nNEP

ILC(3yr)

ILC(1yr)

axis in direction of north ecliptic pole a l0 axial = 40 µK 70 µK 0 µK

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1

S _nEQX

ILC(3yr) ILC(1yr) axis in direction of north ecliptic pole

a l0 axial = 40 µK 70 µK 0 µK

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1

S _nNGP

ILC(3yr) ILC(1yr) axis in direction of north ecliptic pole

a l0 axial = 40 µK 70 µK 0 µK

Figure 4.7. WMAPone- and three-year ILCmapsomparedtothe likelihoodof

analignment(4.7)ofquadrupoleandotopolenormalswithastrophysialdiretions

[northeliptipole(NEP),equinox(EQX)andnorthgalatipole(NGP)inolumns℄,

for twoorthogonal realisations ofthe preferred diretion

ˆ z

^(WMAP ^dipole, ^NEPⁱⁿ

rows). The bold histograms represent statistially isotropi and gaussian skies as

preditedbythe

Λ

^CDM^model. ^Inreasing^the^axialontributionmakestheanomalies worsefor

ˆ z

^being^aligned^with^the^WMAP^dipole,^but^with^the^exlusions^being ^less

signiantfor theILC(3yr)thanfor theILC(1yr). AtthesametimeaSolarsystem

eet is preferred by the data. Thenumberof MonteCarlo realisations pertest is

always

10 ⁵

with the duial CMB sky. For

a ^axial _ℓ0 = 70 µ

^K, ^the ^number ^of ^onsistent ^Monte ^Carlo ^hits

for

C 2 (C 3 )

^is ^redued ^by ^a ^fator ^of

∼ 5(15)

^ompared ^with ^the ^standard ^CMB ^sky. ^Note

thatadding anypowerto thetheoretiallyexpeted quadrupole isexluded atthe

> 99%

^C.L.

level from the ut-sky analysis, but for the otopole the same exlusion level is not reahed

until

a ^axial ₃₀ = 80µ

^K.^F^urther,^adding

50µ

^K⁽

100µ

^K)^to ^the^quadrupole^leads^to ^an^exlusion^of

99.6%

^C.L.⁽

99.9%

^C.L.). ^In^g.^4.6 ^we^show^a^omparison^of^one-^and^three-year^data.

4.3.2. WMAP(3yr) Angular Power. In g. 4.6 we show howthe histograms for the

quadrupoleandotopolepoweromparewiththemeasuredvaluesfromWMAP(1yr,3yr).

Con-sideringtheWMAP(1yr)ut-sky,addinganypowertothequadrupolewasalreadyexludedat

> 99%

^C.L. ^whereas^the ^WMAP(3yr) ^data ^allows^for ^adding^up ^to

a ^axial ₂₀ = 60µ

^Kⁱⁿ ^order ^to

reahthesameexlusionlevel. Theotopoleisquiterobustagainstaxialontaminations asit

liesbetteronthet: inordertoreahthesameexlusionlevelof

> 99%

^C.L.^it^is^neessary^to

add

a ^axial ₃₀ = 80µ

^K^with^respet^to^the^WMAP(1yr)^ut-sky^and^a^whole

a ^axial ₃₀ = 100µ

^K^with

re-spettotheWMAP(3yr)value. Addingamoderateaxialontributionof

a ^axial _ℓ0 = 40µ

^K^leads^to

anapproximatebisetionof thenumberofonsistentMonteCarlohits regardingWMAP(1yr)

data(exluded at

99.5%

^C.L ^for

C 2

^and

91.5%

^C.L ^for

C 3

^), ^where ^for ^the^updated ^ut-sky^a

ontributionof

a ^axial _ℓ0 = 40µ

^K^an^be^exluded^at

> 98%

^C.L.^for

C 2

^and^only^at

∼ 71%

^C.L^for

theotopole.

4.4. Extrinsi AlignmentAnalysis

Nowweaskwhat kindwhat kindofdiretionalpatterns theontribution

a ^axial _ℓ0

^indues ^on

theCMBsky. Inthemultipolevetorrepresentation[CHS04℄anyrealmultipole

T ℓ

^on^a^sphere

anbeexpressedwith

ℓ

^unit ^vetors

v ˆ ^(ℓ,i)

^and^one^salar

A ^(ℓ)

^as

(4.6)

T ℓ (θ, ϕ) =

X ℓ

m=−ℓ

a ℓm Y ℓm (θ, ϕ) ≃ A ^(ℓ) Y ℓ

i=1

ˆ

v ^(ℓ,i) · ˆ e(θ, ϕ) ,

where

e(θ, ϕ) = (sin ˆ θ cos ϕ, sin θ sin ϕ, cos θ)

^is^a^radial^unit^vetor.^Note^that^the^right^hand^side

ofequation(4.6)ontainsontributionswith`angularmomentum'

ℓ − 2

ℓ − 4

^,.^.^.^The^uniqueness

ofthemultipolevetorsisensuredbyremovingthese termsbytakingtheappropriatetraeless

symmetriombination;for detailssee [CHS04℄. Beausethesigns ofallthemultipole vetors

anbeabsorbedintothequantity

A ^(ℓ)

^, ^their^signs^are^unphysial^and ^so^one^is^free^to ^hoose

the hemisphere of eah vetor. Also note that the multipole vetors are independent of the

angularpower. Withthedeomposition(4.6)weahievedauniquefatorisationofamultipole

intoasalarpart

A ^(ℓ)

^,^whih^measures^its^total^power,^and

ℓ

^unit^vetors

v ˆ ^(ℓ,i)

^that^ontain^all

thephaseinformation.

Now it is neessary to dene a suitable statisti to ope with the information from the

multipolevetors. Introduingthe

ℓ(ℓ − 1)/2

^oriented^areas

n ^(ℓ;i,j) ≡ ˆ v ^(ℓ,i) × v ˆ ^(ℓ,j) / | v ˆ ^(ℓ,i) × ˆ v ^(ℓ,j) |

wearereadytodeneastatistiinordertoprobealignmentofthenormals

n ^(ℓ;i,j)

^with^a^given

physialdiretion

x ˆ

^[SSHC04℄,

(4.7)

S nx ≡ 1

4 X

ℓ=2,3

X

i<j

n ^(ℓ;i,j) · x ˆ .

This statistiis a sumoverall dot produtsfor a given

x ˆ

^, ^so ^it ^does ^not^imply ^any^ordering

betweenthetermsandisauniqueandompatquantity. Foromputingthemultipole vetors

we use themethod introdued by [CHS04℄. For mathematial details of the multipole vetor

formalismwereferto e.g.[Fis07℄.

AstheontributionofthestruturedesribedbytheLTBmodel,weaddtothequadrupole

andtheotopoleaomponent,denotedby

a ^axial _ℓ0

^,^whih^is^a^pure

m = 0

^mode^with^respet^to

agivenphysialdiretion

z ˆ

^. ^For^the^diretion

x ˆ

^wee^want^to ^insert^the^relevantastrophysial diretionswhih giveriseto alignment,likethediretionoftheelipti plane,theequinoxet.

Butthereisaath. Onewerotatethe

z ˆ

^axis^of^our^initial^oordinate^system^into^the^diretion

ofthepreferredaxisofourmodel,thediretionsonthesky,likenortheliptipoleet.,haveto

berealulatedinthatframe. ThisanbedoneintermsofWignerrotationmatries[CHSS06℄.

Writtenasvetors,theoeients

a ^′ _ℓ

^transform^under^rotations^as

a ^′ _ℓ = D ^† a ℓ

^,^where^the^vetor

notationmeansthat

a ℓ

^is^a^vetor^of^the

ℓ

^-th^multipole ^oeient^with

(2ℓ + 1)

^entries ^and

D

denotingtherotation. TherotationsanbeparameterisedintermsofthesoalledEulerangles

α, β, γ

^and^are^givenⁱⁿ ^matrix^form ^by^[CHSS06℄

D _m ^(ℓ) ′ m (α, β, γ) = e ^im ^′ ^γ d ^(ℓ) _m ′ e ^imα with d ^(ℓ) _m ′ m = X

k

( − 1) ^ℓ−m ^′ ^−k [(ℓ + m ^′ )!(ℓ − m ^′ )!(ℓ + m ^′ )!(ℓ − m ^′ )!] ^1/2 k!(l − m ^′ − k)!(l − m − k)!(m + m ^′ + k)!

×

cos β 2

2k+m ^′ +m sin β

2 2ℓ−2k−m ^′ −m

.

(4.8)

WehavearriedouttherotationswiththehelpofaMATHEMATICAroutine. Next,letusreviewour

resultsoftheMonteCarloanalysisforthealignmentstatisti(4.7)withrespettoastrophysial

diretions.

4.4.1. WMAP(1yr) Alignment. Welook for alignmentwith three dierentdiretions

ˆ

x

^: ^the^north^elipti ^pole,^the^equinox^and^the^north^galati^pole. ^The^rst^two^are^preferred

diretionsin theSolar systemandthe lastdenes theplaneof thedominantforeground. The

observed

S

^-values^from^the^dierent^CMB^maps^are^givenⁱⁿ^tab.^4.3. ^The^results^of^the

orre-dipole and equinox lieveryloseto eah other, so analignmenttest with the dipole will give

resultsverysimilartotheonewiththeequinox.

In g.F.2 thepreferredaxis

z ˆ

^is^hosen^to^be^the^measured ^WMAP(1yr)^dipole ^[B

⁺

^03b℄.

Weperformalignmenttests(4.7)withrespettothethreetestdiretions

x ˆ

^. ^F^or^all^three^tests

theanomaly gets learly worse, that is the axial mehanismdrives thehistograms awayfrom

thedata. Next,instead of using themotion of theloal group with respet to theCMB rest

frame [KLS

+

93℄ as the test diretion, we takethe veloityof theloal groupwhen orreted

forVirgoentrimotion[PK98℄,sinethisdiersmorefromtheWMAPdipole. Theresultsare

shown in g. F.3. The situation forthe alignment with theequinoxis again worse, but there

is not muh eet on the elipti alignment. For the alignment with the galati plane, the

axialontributionmakesanapparentgalatiorrelationmoreprobable,i.e.thereis aertain

probability of overestimating the galati foreground. For both test diretions by now, the

alignment with the equinox gets worse. Forexample, in the diretion of the Virgo-orreted

loal groupmotionanexlusion of

∼ 99.9%

^C.L. ^for

a ^axial _l0 = 50µ

^K ^an^be ^given^with ^respet

to all three leaned maps. Note that adding any multipole powerin this test analready be

exludedat the

≥ 99.4%

^C.L.

Asaomplementarytestweshowthealignmentlikelihoodwithregardtoanorthogonaltest

diretion,namelythenortheliptipole,ing.F.4. Anelipti extraontributionintheCMB

⁵⁴⁵

µ

²

S nNEP

^- ^0.194 ^0.193 ^0.210 ^0.252

-S nEQX

^- ^0.886 ^0.866 ^0.870 ^0.846

-S nNGP

^- ^0.803 ^0.803 ^0.810 ^0.794

-4.4.2. WMAP(3yr) Alignment. Similarly,wetestforalignmentwiththethreegeneri

diretions

x ˆ

^: ^north^elipti^pole,^equinox^and^north^galati^pole. ^The^results^of^the^orrelation

analysis are shown in g. 4.7: in the rst row the preferred diretion

z ˆ

^oinides ^with ^the

diretion of loal motion, the dipole. Here the anomaly beomes worse when inreasing the

amplitudeoftheaxialontribution. Butfor

x ˆ = NEP

^the^exlusion^beomes^somewhat^milder

goingfromone-yeartothree-yeardata;e.g.

a ^axial _ℓ0 = 40µ

^K^leads^to^an^exlusion^of

99.2%

^C.L.^for

ILC(1yr)butonly

98.2%

^C.L.^for^the^updated^ILC^map. ^Finding^an^alignment^with^the^equinox

thoughisstronglyexludedat

> 99.2%

^C.L.,^even^with^an^vanishing^axialontributionforboth one-and three-yeardata. Forinstane, for

x ˆ = EQX

^adding ^aontribution of

a ^axial _ℓ0 = 20µ

(

a ^axial _ℓ0 = 70µ

^K) ^leads ^to ^an ^exlusion ^level ^of

99.4%

^C.L. ⁽

99.9%

^C.L.) ^with ^respet ^to

three-year data. Similarlyto above,aSolarsystemeet is preferredbythe data. Forexample, an

alignmentwiththeeliptiitself(

x ˆ = NEP

⁾^may^only^be^exluded^at^the^level^of

92.3%

^C.L.^after

addingan axialontribution of

a ^axial _ℓ0 = 40µ

^K.^For^the^same^axial ontribution, thealignment withtheequinoxbeomeslessanomalousas

99.2%C.L. → 98.2%

^C.L.

Im Dokument On the influence of local inhomogeneities on cosmological observables : from galaxies to the microwave background (Seite 109-119)

Lepus

X- ray survey: dipole velocity profile

x − y

(1, 3, 0.5) × 10 − 3

− 2

∼ 100

∼ 200

v 2 c ≃ φ

∆T (θ, ϕ) T

RS

∼ φ 3/2 ∼ δM

d 3/2

.

Ω = 1

∆T (θ, ϕ) T

RS

∼ δρ

ρ 3/2

d t

3

.

t

d

δρ/ρ

10 −5

ℓ

+

h −1

+

+

+

ℓ

q 0

z ˆ

z ˆ

m = 0

z ˆ

0 ◦

b

l

0 ◦

360 ◦

0 ◦

(l, b) ≃ (96.4 ◦ , 29.8 ◦ )

(l, b) ≃ (276.3 ◦ , 60.2 ◦ )

(l, b) = (0 ◦ , 90 ◦ )

a axial ℓ0

0 1 2 3 4 5 6

0 0.05 0.1 0.15 0.2 0.25 0.3

Likelihood

C 2 in [0.1 mK] 2

WMAP(1yr) WMAP(3yr) a 20 axial = 0 µK [WMAP(1yr) best-fit ΛCDM]

20 µK 40 µK 70 µK

0 2 4 6 8 10 12 14

0 0.05 0.1 0.15 0.2 0.25 0.3

Likelihood

C 3 in [0.1 mK] 2

WMAP(1yr) WMAP(3yr) a 30 axial = 0 µ K [WMAP(1yr) best-fit Λ CDM]

20 µ K 40 µ K 70 µK

> 99%

60µ

80µ

100µ

99.7%

99.9%

100µ

a ℓm

C ℓ = 1

2ℓ + 1 X ℓ

m=−ℓ

| a ℓm | 2 .

a ℓm = a Re ℓm + i a Im ℓm

C ℓ

f (a ℓ0 ) = 1

√ 2πC ℓ

exp

− (a Re ℓ0 ) 2 2C ℓ

and f (a Re,Im ℓm ) = 1

√ πC ℓ

(1, 3, 0.5) × 10 ⁻ ³

⁻ ²

v ² _c ≃ φ

∼ φ ^3/2 ∼ δM

10 ⁻⁵

h ⁻¹

0 ^◦

0 ^◦

360 ^◦

0 ^◦

(l, b) ≃ (96.4 ^◦ , 29.8 ^◦ )

(l, b) ≃ (276.3 ^◦ , 60.2 ^◦ )

(l, b) = (0 ^◦ , 90 ^◦ )

a ^axial _ℓ0

C ₂ in [0.1 mK] ²

WMAP(1yr) WMAP(3yr) a ₂₀ ^axial = 0 µK [WMAP(1yr) best-fit ΛCDM]

C ₃ in [0.1 mK] ²

| a ℓm | ² .

a ℓm = a ^Re _ℓm + i a ^Im _ℓm

− (a ^Re _ℓ0 ) ² 2C ℓ

and f (a ^Re,Im _ℓm ) = 1

exp − (a ^Re,Im _ℓm ) ² C ℓ

a ^axial _ℓ0

10 ⁵

a ^axial _ℓ0

a ^axial _ℓ0 = 40µ

S _nNEP

S _nEQX

ILC(3yr) ILC(1yr) axis in direction of WMAP dipole a _l0 ^axial = 0 µK 40 µK 70 µK

S _nNGP

S _nNEP

S _nEQX

S _nNGP

10 ⁵

a ^axial _ℓ0 = 70 µ

a ^axial ₃₀ = 80µ

a ^axial ₂₀ = 60µ