Lensed CMB power spectra from all-sky correlation functions(5)

Weak lensing of the CMB changes the unlensed temperature anisotropy and polarization power spectra. Accounting for the lensing effect will be crucial to obtain accurate parameter constraints from sensitive CMB observations. Methods for computing the lensed

4

whereJn(x)isaBesselfunctionofordern.Notethatthetrace-freetermisanalyticatr=0duetothesmall-rbehaviourofJ2(lr).FollowingRef.[1],letusdenote α(x)·α(x′) byCgl(r)sothat

Cgl(r)=

1

2π

sothat

αi(x)αj(x′) =

1

dll3ClψJ2(lr),

(13)

ClΘeil·r eil·[α(x) α(x)] ,2(2π)

′

(15)

wherewehaveassumedthattheCMBandlensingpotentialareindependent(i.e.weareneglectingthelargescale

correlationthatarisesfromtheintegrated-Sachs-Wolfee ectandhasonlyatinye ectonthelensedCMB).SinceweareassumingαisaGaussian eld,l·[α(x) α(x′)]isaGaussianvariateandtheexpectationvalueinEq.(15)reducesto

′1

eil·[α(x) α(x)] =exp

l2[σ2(r)+cos2(φl φr)Cgl,2(r)],(16)2wherewehaveusedliljr ir j =l2cos2(φl φr)/2andde nedσ2(r)≡Cgl(0) Cgl(r).Here,e.g.φlistheanglebetweenlandthex-axis.Thecos2(φl φr)terminEq.(16)isdi culttohandleanalytically.Instead,weexpandtheexponentialandintegratetermbyterm.ExpandingtosecondorderinCgl,2,we nd

1122 (r)=l4Cgll4Cgl(17)ξ,2(r)J0(lr)+,2(r)J4(lr).1616Expandingtothisorderissu cienttogetthelensedpowerspectrumtosecondorderinClψ;higherordertermsin

Cgl,2onlycontributeattheO(10 4)levelonthescalesofinterest.Notethattheexp( l2σ2/2)termiseasilyhandledwithoutresortingtoaperturbativeexpansioninClψ.Sinceσ2issigni cantlylessthanCgl,2(asshowninFig.2),theperturbativeexpansioninCgl,2convergesmuchfasterthanoneinσ2.Equation(17)extendstheresultofRef.[1]tosecondorderinCgl,2.

1.

Polarization

Thepolarizationcalculationisalsostraightforwardinthe at-skylimit[14].Weusethespin 2polarizationP≡Q+iU,whereQandUaretheStokes’parametersmeasuredwithrespecttothe xedbasiscomposedofthexand yaxes.ExpandingP(x)intermsoftheFouriertransformsofitselectric(E)andmagnetic(B)parts,wehave

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新人文社科Lensed CMB power spectra from all-sky correlation functions(5)全文阅读和word下载服务。