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Figure F.11. Figure similar to g. 5.2, here we added a smaller ontributionof

a ℓ0 = 100µ

Ktothemultipoles,suhthat weanobservetheonsetoftheseparation proessofthevetors. Mollweideprojetion oftheskywithquadrupole(upperrow)

andotopole(lowerrow)multipolevetors[equation(5.5) ℄. Themeshonsistsofsteps

in

30

. Displayedaretenpairsofquadrupolevetors(smalldots)andtheirtenarea

vetors[equation(5.6)(bigdots)℄aswellastentriplesofotopolevetors(smalldots)

and theirarea vetors(bigdots);togethernessisindiatedbyolour. Thearbitrary

signofthevetorshasbeenusedtogaugethemall tothenorthernhemisphere. The

statistially isotropi and Gaussianase (leftolumn) isbrokenbythe imprintof a

moderateaxialeet

a ℓ0 = 100µ

K(rightolumn)whereuponmultipolevetorsstart

to movetothepoleandareavetorsstart tomovetotheequatorialplane. Thefull

separation an be observed when adding strong ontributions

a ℓ0 ∼ 1000µ

K, .f.

g. 5.2.

0

20 40

60

80

100 0 20

40 60

80 100 1.4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

N(S nn )/10 3 3 2.5 2 1.5

a 30 axial [ µ K]

a 20 axial [ µ K]

0

50

100

150

200 0 50

100 150

200 1

1.5 2 2.5 3 3.5 4

N(S nn )/10 3 3.5 3 2.5 2 1.5

a 30 axial [µK]

a 20 axial [ µ K]

Figure F.12. Testing theeetofasymmetriadditiveontributions

a axial ℓ0

onthe

intrinsiross alignmentofquadrupoleandotopole. Here,theross alignmenttest

is denedas

S nn ≡ P |ˆ n (2;,1,2) · n ˆ (3;,i,j) |

;notethe dierent pre-fatorasompared

to (5.8) . Like in g. F.10, we have plotted the arbitrarily saled (

×10 −3

) number

ofMonteCarlosthat are,for givenvaluesof

a axial ℓ0

,onsistentwithanexperimental value of

S nn ≃ 2.62

,thathas beenobtainedfrom WMAP(1yr)leaned maps. The

totalnumberofMonteCarlosisagain

10 5

. Theupperandlowergureshowthesame

test, onlywith adierentrange ofsimulations. Fromtheuppergure,weseethat

indeed,intrinsialignmentsareapparentlyuredbyaddingaxialontributionsupto

∼ 100µ

K.Inthelowergurewe seethat,wheninreasing

a axial ℓ0

further,this isonly

aloalmaximum.Thisisperfetlyonsistentwithourndingsg.5.3.

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