Lensed CMB power spectra from all-sky correlation functions(12)

1– ′ 1andn 2.thenn′1basisatn1)tothegeodesicbasisconnectingn

2alittlemoreworkisrequired.Letχ′denotetheanglebetweenthegeodesicsForthelensedpolarizationatn

2ton ′ 1(alongthez-axis)ton ′ 2onthegeodesicbasisconnectingn2,andn2(seeFig.3).Thelensedpolarizationatn 1andn 2isthenadaptedton

2iχ′ 2iψ2 ( .Pn2)=P( n′)ee2

(47)

′Wecanwriten2asthedirectionobtainedbyrotatingadirectionwithpolarangles(α2,ψ2)byanangleβaboutthe

′y-axis,i.e.n2=D(0,β,0)(α2,ψ2).WritingPas(Q iU),andusingEq.(42),wehave

P( n)=(Elm+iBlm) +2Ylm( n).(48)

lm

Usingtherotationpropertiesofthespin-sharmonics(seeAppendixA),wethen nd

2iχ′ 2iψ2 2iκl eP( n2)=ee(Elm+iBlm) Dmm′(0,β,0)2Ylm′(α2,ψ2).

lmm′

(49)

′Theangleκistherotationaboutn2thatisrequiredtobringthepolarbasisthereontothatobtainedbyrotating (0,β,0).Sincethelatterisalignedwiththegeodesicbasisadaptedton 2andn ′thepolarbasisat(α2,ψ2)withD2,

′ 2simpli estowehaveκ=χandthelensedpolarizationatn

( Pn2)=e 2iψ2(Elm+iBlm) dl(50)mm′(β)2Ylm′(α2,ψ2).

lmm′

Wecannowquicklyproceedtothefollowingexpressionsforthelensedpolarizationcorrelationfunctions:

2iψ1 2iψ2 ξ+(β)=(ClE+ClB)dl ,2Ylm(α1,ψ1)2Ylm′(α2,ψ2)emm′(β) e

lmm′

(51)(52)(53)

(β)=ξ

lmm′

X(β)=ξ

lmm′

2iψ1 2iψ2

(ClE ClB)dl , 2Ylm(α1,ψ1)2Ylm′(α2,ψ2)emm′(β) e 2iψ2ClXdl ,mm′(β) Ylm(α1,ψ1)2Ylm′(α2,ψ2)e

wheretheexpectationvaluesareoverlensingrealizations.Here,ClEandClBarethepowerspectra |Elm|2 and

|Blm|2 respectively.Thecross-correlationpowerspectrumisClX≡ ΘlmElm .

WeevaluatetheexpectationvaluesinEqs.(51–53)followingtheearliercalculationforthetemperature,i.e.ex-pandingPr(α1,α2,ψ1,ψ2)tosecondorderinCgl,2beforeintegrating.Asforthetemperature,Cgltermscontribute

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