Lensed CMB power spectra from all-sky correlation functions

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

LensedCMBpowerspectrafromall-skycorrelationfunctions

AnthonyChallinor1, andAntonyLewis2,

1

AstrophysicsGroup,CavendishLaboratory,MadingleyRoad,CambridgeCB30HE,U.K.

2

CITA,60St.GeorgeSt,TorontoM5S3H8,ON,Canada.

arXiv:astro-ph/0502425v2 14 May 2005

WeaklensingoftheCMBchangestheunlensedtemperatureanisotropyandpolarizationpowerspectra.Accountingforthelensinge ectwillbecrucialtoobtainaccurateparameterconstraintsfromsensitiveCMBobservations.Methodsforcomputingthelensedpowerspectrausingalow-orderperturbativeexpansionarenotgoodenoughforpercent-levelaccuracy.Non-perturbative at-skymethodsaremoreaccurate,butcurvaturee ectschangethespectraatthe0.3–1%level.Wedescribeanew,accurateandfast,full-skycorrelation-functionmethodforcomputingthelensinge ectonCMBpowerspectratobetterthan0.1%atl 2500(withintheapproximationthatthelensingpotentialislinearandGaussian).Wealsodiscussthee ectofnon-linearevolutionofthegravitationalpotentialonthelensedpowerspectra.Ourfastnumericalcodeispubliclyavailable.

I.INTRODUCTION

TheCMBtemperatureandpolarizationanisotropiesarebeingmeasuredwitheverincreasingprecision.Thestatisticsoftheanisotropiesalreadyprovidevaluablelimitsoncosmologicalparameters,aswellasconstraintsonearly-universephysics.Asweentertheeraofprecisionmeasurement,withsignal-dominatedobservationsouttosmallangularscales,non-lineare ectswillbecomeincreasinglyimportant.Oneofthemostsigni cantoftheseoverscalesofmostinterestforparameterestimationisweakgravitationallensingbylargescalestructure.Fortunatelyitcanbemodelledaccuratelyasasecond-ordere ect:thelineargravitationalpotentialalongthelineofsightlensesthelinearperturbationsatthelastscatteringsurface(seee.g.Refs.[1,2,3]andreferencestherein).Modellingoffullynon-linearevolutionisnotrequiredforthenearfutureonscalesofseveralarcminutes(correspondingtomultipolesl 2000)forthetemperatureandelectricpolarizationpowerspectra.Non-linearcorrectionscaneasilybeappliedtothelensingpotentialifandwhenrequired,providedthatitsnon-Gaussianitycanbeignored[1].

Inprinciple,theweak-lensingcontributiontotheobservedskycanprobablybesubtractedgivensu cientlyaccurateandcleanhigh-resolutionobservations.Earlyworkinthisarea[4,5,6,7]suggestedalimitontheaccuracyofthisreconstructionduetothestatisticalnatureofthe(unknown)unlensedCMB elds.Morerecently,ithasbeenarguedthatpolarizationremovesthislimitinmodelswherelensingistheonlysourceofB-modepolarizationonsmallscales[8].IfsubtractioncouldbedoneexactlywecouldrecovertheunlensedGaussiansky,andusethisforallfurtheranalysis.Howevercurrentmethodsforsubtractingthelensingcontributionareapproximate,andnoteasytoapplytorealisticsurveygeometries.Theresultofimperfectlensingsubtractionisaskywithcomplicated,non-Gaussianstatisticsofthesignal,andsigni cantlymorecomplicatednoisepropertiesthantheoriginal(lensed)observations.Forobservationsinthenearfuture,amuchsimplermethodtoaccountforthelensinge ectistoworkwiththelensedskyitself,modellingthelensinge ectbytheexpectedchangeinthepowerspectraandtheircovariances.Thee ectsoflensingnon-GaussianitiesonthecovarianceofthetemperatureandE-modepolarizationpowerspectraarelikelytobesmall,butthiswillnotbethecasefortheB-modespectrumoncethermal-noiselevelspermitimagingofthelens-inducedBmodes[9].Inthispaperwediscusshowtocomputethelensedpowerspectraaccurately.Thesimulationoflensedskiesandthee ectonparameterestimationisdiscussedinRef.[10].

Onscaleswherethenon-Gaussianityofthelensingpotentialcanbeignored,thecalculationofthelensedpowerspectraisstraightforwardinprinciple.However,achievinggoodaccuracyonbothlargeandsmallscalesforalltheCMBobservablesissurprisinglydi cult.ThelensingactionontheCMB eldsatscalesapproachingther.m.s.ofthelensingde ectionangle(～3arcmin)cannotbeaccuratelydescribedwitha rst-orderTaylorexpansion,asinthefull-skyharmonicmethodofRef.[3].ThereisnotmuchpowerintheunlensedCMBonsuchscales,buta rst-orderTaylorexpansionstillgiveslensedpowerspectrathatareinaccurateatthepercentlevelforl 1000.ThelensedCMBonscaleswellbelowthedi usionscaleisgeneratedbytheactionofsmall-scaleweaklensesonthe(relatively)large-scaleunlensedCMB,andaTaylorexpansionshouldbecomemoreaccurateagain[11].(However,non-lineare ectsarealsoimportantonsuchscales.)ThebreakdownoftheTaylorexpansioncanbeeasily xedbyusingthe at-skycorrelation-functionmethodsofRefs.[1,2],whichcanhandlethedominante ectofthelensingdisplacement

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

2

inanon-perturbativemanner.However,anewproblemthenarisesonscaleswherethe at-skyapproximationisnotvalid.AsnotedinRef.[3],thisisnotcon nedtolargescalesduetothemode-couplingnatureoflensing:degree-scalelensescontributesigni cantlytothelensedpoweroverawiderangeofobservedscales.Inthispaperwedevelopanewmethodforcomputingthelensedpowerspectrathatisaccurateonallscaleswherenon-Gaussianityduetonon-lineare ectsisnotimportant.Wedothisbycalculatingthelensedcorrelationfunctionsonthesphericalsky.Thisallowsustoincludeboththenon-perturbativee ectsofdisplacingsmall-scaleCMB uctuations,andthee ectsofskycurvature.

Thispaperisarrangedasfollows.WestartinSectionIIwithabriefintroductiontoCMBlensing,theninSectionIIIwereviewpreviousworkon at-skycorrelation-functionmethodsandpresentournewfull-skymethodandresults.InSectionIVwecompareournewresultswiththe at-skycorrelation-functionresultsofRefs.[1,2]andtheperturbativeharmonicresultofRef.[3],andexplainwhythelatterisnotaccurateenoughforprecisioncosmology.Thee ectofnon-linearevolutionofthedensity eldonthelensedpowerspectraisconsideredinSectionV.Weendwithconclusions,andincludesometechnicalresultsintheappendices.

II.

CMBLENSING

Gradientsinthegravitationalpotentialtransversetothelineofsighttothelastscatteringsurfacecausedeviations

actuallycomefrompointsonthelastscatteringsurfaceininthephotonpropagation,sothatpointsinadirectionn

′ n)andtheunlensedtemperaturebyΘ( .DenotethelensedCMBtemperaturebyΘ( adisplaceddirectionnn),so

′ n)=Θ( thelensed eldisgivenbyΘ( n).Thechangeindirectionontheskycanbedescribedbyadisplacement

′=n + ψ.Hereψisthelensingpotentialwhichencapsulatesthevector eldα( n)≡ ψ,sothat(symbolically)n

′isobtainedfromdeviationscausedbypotentialsalongthelineofsight.Morerigorously,onaunitspherethepointn

bymovingadistance| ψ|alongageodesicinthedirectionof ψ( nn),where isthecovariantderivativeonthesphere[12].Weassumethatthelensingisweak,sothatthepotentialsmaybeevaluatedalongtheunperturbedpath(i.e.weusetheBornapproximation).Lensingde ectionsareafewarcminutes,butarecoherentoverdegreescales,sothisisagoodapproximation.

Intermsofthezero-shearaccelerationpotentialΨ,thelensingpotentialina atuniversewithrecombinationatconformaldistanceχ isgivenbytheline-of-sightintegral

χ

χ χ

;η0 χ)dχΨ(χnψ( n)= 2

(2π)3/2

andpowerspectrum

Ψ(k;η)Ψ(k;η) =

′

′

Ψ(k;η)eik·x,

(2)

2π2

χ

k

dχ

χ

00

dχ′PΨ(k;η0 χ,η0 χ′)jl(kχ)jl(kχ′)

χ χ

χ χ′

.(4)

Inlineartheorywecande neatransferfunctionTΨ(k,η)sothatΨ(k;η)=TΨ(k,η)R(k)whereR(k)isthe

primordialcomovingcurvatureperturbation(orothervariableforisocurvaturemodes).Wethenhave

2

dkψ

Cl=16π(5)

χ χ

wheretheprimordialpowerspectrumisPR(k).Thiscanbecomputedeasilynumericallyusingcamb1[13],andatypicalspectrumisshowninFig.1.

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

liljCψleil·(x x′

)(2π)

2.canonlydependonδijandthetrace-freetensorr irj ,wherebytakingthetraceofthecorrelator,anditscontractionx)α) =1

j(x′2π

dll3CψlJ2(lr) r ir j xri3

(10)

Bysymmetry,thecorrelatorr≡ x′.Evaluating

thecoe cientsofthesetwotermswithrj,we nd

αi(,(11)

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

2

dl

P(x)=

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

5

FIG.2:Thefunctionsσ2(β)≡Cgl(0) Cgl(β)[solid]andCgl,2(β)[dashed]asafunctionofangularseparationβ(inradians)foratypicalconcordancemodel.Theresultsarecalculatedusingthefull-skyde nitionsofEqs.(35),andusethelinearpower

ψ

spectrumforCl.

whereπ φristheangletorotatethex-axisontothevectorjoiningxandx′,sothate.g.e2iφrP(x)isthepolarizationonthebasisadaptedtoxandx′.Thenthelensedcorrelationfunctionstosecond-orderinCgl,2are

12 +(r)=ξl4Cgl,2(r)J0(lr)16

12+l4Cgl(22),2(r)J4(lr),16

12 (r)=l4Cglξ,2(r)J4(lr)16

11+[J2(lr)+J6(lr)]+[J0(lr)+J8(lr)],(23)

22

112 X(r)=l4Cgl[J0(lr)+J4(lr)]ξ,2(r)J2(lr)+162

1+[J2(lr)+J6(lr)].(24)

2HereClEandClBaretheE-modeandB-modepowerspectra,andClXistheΘ-Ecross-correlation.ThisisthestraightforwardextensionoftheresultinRef.[14]tohigherorder;2seethatpaperforfurtherdetailsofthecalculation.

+(r)hasthesamestructureasforthetemperaturesincetheunlensedcorrelationfunctionsinvolveThelensedξ

thesameJ0(lr),andtherearenocomplicationsduetothedi erentlocalbasesde nedbythedisplacementranditsimageunderlensingr α′+αsincethephasefactorsfromtherotationscancel.Thisisnotthecaseforthelensed (r)andξ X(r).ξ

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

2.

Limberapproximation

AthighlthepowerspectrumPΨ(k)variesslowlycomparedtothesphericalBesselfunctionsinEq.(4),whichpickoutthescalek～l/χ.Using

π

k2dkjl(kχ)jl(kχ′)=

χ

l3

χdχPΨ(l/χ;η0 χ)

χ χ

χ

2

J0(kχr),(27)

inagreementwithRef.[1]ifwenotethathisPφ(k)=PΨ(k)/(4πk3)outsideradiation-domination.Ref.[1]alsode nesCgl,2(r)as(inournotation)

Cgl,2(r)≈4π

dk

dχPΨ(k;η0 χ)

χ χ

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

4π

l(l+1)Clψdl11(β)≡Cgl(β).

(35)

Asinthe at-skylimit,itisconvenienttode neσ2(β)≡Cgl(0)

Cgl(β).ThecovarianceoftheGaussianvariates

1α¯( n1), 1α¯( n1), 1α¯( n2)and 1α¯( n2)aredeterminedbyEq.(35).Transformingvariablestoα1,ψ1,α2andψ2

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

we ndtheirprobabilitydistributionfunction

Pr(α1,α2,ψ1,ψ2)=

4α1α2

22

2(α1cosψ1+α2cosψ2)/(σ+2Cgl Cgl,2)

2σ+2Cgl Cgl,2

e 1

(α1cosψ1 α2cosψ2)2/(σ2

+Cgl,2)

×2σ2+Cgl,2

1

×

e

σ2

α

4π

ClΘ

X2

8

000dl00+

4

4π

ClΘ=

2l+1

l

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

thoroughintroduction).Thepolarizationcorrelationfunctionscanbede nedintermsofthespin±2polarizationde nedinthephysicalbasisofthegeodesicconnectingthetwopositions.Asforthetemperature,weevaluatethe

1alongthez-axisandn 2inthex-zplaneatangleβtothez-axis.polarizationcorrelationfunctionsbytakingn

1andn 2sothatthelensedWiththisgeometry,thepolar-coordinatebasisisalreadythegeodesicbasisconnectingn

correlationfunctionsare

+(β)≡ P ( ( ξn1)Pn2) , (β)≡ P ( ( ξn1)Pn2) , X(β)≡ Θ( n1)P ( ξn2) .

(43)(44)(45)

Underalensingde ectionthepolarizationorientationispreservedrelativetothedirectionofthede ection(weare

neglectingthesmalle ectof eldrotation[8]),i.e.thepolarizationundergoesparalleltransport.Thegeometryofthede ectionsisshowninFig.3.

1andn 2)asWecaneasilyevaluatethelensedpolarizationontheconnectinggeodesicbasis(betweenn

( Pn1)=P(α1,ψ1)e 2iψ1.

(46)

Therotationangleψ1isthatneededtorotatethespin 2polarizationfrompolarcoordinates(coincidingwiththe

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

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

negligibly(seeAppendixCforthefullresult).We ndthefollowingresultsforthelensedpolarizationcorrelationfunctionstosecondorderinCgl,2:

+≈ξ

2l+1

lmm′

(ClE ClB)

X0222dl2 2+Cgl,2[X1212dl1 1+X1322dl

4π

3 3]

+

1

C2XlX

022X000dl02+Cgl,2

000

′

4π

Xl(l+(X112dl11+X132dl

1)

3 1)

+1

2

2

4

4π

(ClE+ClB)=

2l+1

l

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

11

ClΘ/ClΘ

0.1 0.05 0 0.05 2 10 40 100 200 400 700 1000 1500 2000

ClX/(ClE ClΘ )1/2

0.1 0.05 0 0.05 0.1 2 10 40 100 200 400 700 1000 1500 2000

0.2

ClE/ClE

0.1 0 0.1 2 10 40 100 200 400 700 1000 1500 2000

l(l+ 1)ClB/2πµK 2

0.2 0.15 0.1 0.05 0 2 10

40 100 200 400 700 1000 1500 2000

l

FIG. 4: Di erence between the lensed and unlensed temperature, cross-correlation and E-polarization power spectra (top three plots), and the lensed B power spectrum (bottom) for a ducial concordance model. The unlensed model has no tensor component (so no B-mode power), and the lensed B power spectrum shown is not highly accurate due to the neglect of non-linear evolution in the lensing potential. The magnitude of the lensing e ect depends on the uctuation amplitude in the model; here the model has curvature perturbation power As= 2.5× 10 9 on 0.05 Mpc 1 scales and spectral index ns= 0.99.

3.

Numerical implementation

The correlation function method is inherently very e cient, only requiring the evaluation of one dimensional sums and integrals. For an accurate calculation of ClB it is essential to compute the full range of the correlation function because it is sensitive to large and small scales. However, when ClB is not needed the lensing is only a small-scale e ect and we only need to integrate some of the angular range to computeξ(β) ξ(β) (and hence the lensing contribution l Cl ). We nd that usingβmax=π/16 is su cient for 0.1% accuracy to l= 2000, providing a signi cant factor of C 16 gain in speed. Truncating the correlation function does induce ringing on very small scales, so if accuracy is needed on much smaller scales the angular range can be increased. For all but very small scales, and the ClB spectrum, we can accurately evaluate the sums over l to compute the lensed correlation functions by sampling only every 10th l, yielding an additional signi cant time saving. Our code is publicly available as part of camb,3 with execution time being dominated by that required to compute the transfer functions for the CMB and the lensing potential. Once these have been computed, the time required to

3

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

bd336x280();rc="-15-png_6_0_0_0_0_0_0_918_1188-164-0-246-164.jpg" alt="Lensed CMB power spectra from all-sky correlation functions" />

12

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

resultisequivalenttoexpandingthecorrelationfunctionresultself-consistentlyto rstorderinClψ.AsdiscussedinSec.IIIB,thisisinaccuratebecausel2σ2intheisotropictermsisnotverysmallforlargel,somanytermsneedtoberetainedtogetaccurateresults.Itispossibletoextendtheharmonicresulttohigherorder[20],howeverthemulti-dimensionalintegralsrequiredscaleexponentiallybadlywithincreasingorder.Evenaself-consistentexpansiontosecondorderinClψisnotgoodenoughatl>2000,soatleastthirdorderwouldberequired.Furthermorewe

Bthemethodisalsosomewhatinaccurateonlargescales:becausetheB-modesignalcomesfromaseethatforCl

widerangeofl,andtheE-modepowerpeaksonsmallscales,thenon-perturbativee ectscanbesigni cantonallscales.Infact,thelarge-scalelensedE-modepoweralsoreceivesmostofitscontributionfromsmall-scalemodessincetheunlensedpolarizationpowerspectrumrisessteeplywithlonlargescales.However,lensingisstillonlyasmallfractionale ectforE-polarizationonlargescalesandsotheperturbativeexpansionisrelativelymoreaccurateforEthanB.

Thecorrelationfunctionmethodscaneasilyhandletheisotropictermnon-perturbatively.Theaccurate at-skyresultismuchmoreaccuratethanthelowest-orderharmonicfull-skyresult,withonly～0.3%curvaturecorrections

B.Althoughthisistothetemperature.5Thepolarizationerrorsareratherlarger,withpercent-leveldi erenceonCl

smallerthanthee ectofnon-linearitiesinthelensingpotential(seeSectionV),thelattercanbeaccuratelyaccountedforwithbettermodelling(e.g.Ref.[21])orsimulations.Whiletheaccurate at-skyresultisprobablysu cienttoPlancksensitivities,curvaturee ectsmustbetakenaccountfortrulyaccurateresultsapproachingthecosmic-variancelimit.Althoughthecurvatureisnegligibleonthescaleofthede ectionangles,putingourfull-skyaccurateresultisnotmuchharderorslowerthancomputingthe atresult,sowerecommendournewcalculationforfuturework.

NotethattheabsoluteprecisionofthelensedresultsislimitedbytheaccuracyofthecomputedlensingpotentialandtheunlensedCMBpowerspectra.Inparticular,uncertaintiesintheionizationhistorymaygenerateerrorssigni cantlyabovecosmicvarianceontheunlensedCl.WeusetherecfastcodeofRef.[22]thatmaywellbeinaccurateatabovethepercentlevel6[23,24].Howeveriftheionizationhistorycanbecomputedreliablytohighaccuracyournewlensingmethodcanthenbeusedtocomputethelensedpowerspectraaccurately.

V.

NON-LINEAREVOLUTION

ThemostimportantassumptionwehavemadesofaristhatthelensingpotentialislinearandGaussian.Onsmallscalesthiswillnotbequitecorrect.Althoughourmethoddoesnotallowustoaccountforthenon-Gaussianity,wecantakeintoaccountthee ectofnon-linearevolutiononthepowerspectrumofthelensingpotential[andhenceσ2(β)andCgl,2(β)[1]].Onscaleswherethenon-Gaussianityofthede ection eldissmallthisshouldbeagoodapproximation,assumingwehaveanaccuratewaytocomputethenon-linearpowerspectrumofthedensity eld.WeusethehalofitcodeofRef.[21]tocomputeanapproximatenon-linear,equal-timepowerspectrumgivenanaccuratenumericallinearpowerspectrumatagivenredshift.halofitisexpectedtobeaccurateatthefewpercentlevelforstandardΛCDMmodelswithpowerlawprimordialpowerspectra(butcannotbereliedonforothermodels,forexamplewithanevolvingdarkenergycomponent).WesimplyscalethepotentialtransferfunctionsTΨ(k,η)ofEq.(5)sothatthepowerspectrumofthepotentialΨhasthecorrectnon-linearformatthatredshift:

TΨ(k,η)→TΨ(k,η)

PΨ(k,η)

.

(67)

Sincenon-lineare ectsonClψareonlyimportantwheretheLimberapproximationholds,Eq.(67)shouldbeveryaccurate.

Thee ectofthenon-linearevolutiononthepowerspectrumofthelensingpotentialisshowninFig.1.Althoughthereisverylittlee ectonscaleswherethepowerpeaks(l～60),non-linearevolutionsigni cantlyincreasesthepoweronsmallscales.ThecorrespondingchangestothelensedCMBpowerspectraareshowninFig.6.Thetemperature

Θischangedby 0.2%forl～2000,buttherearepercentlevelchangesonsmallerscales.ThuspowerspectrumCl

inclusionofthenon-linearevolutionwillbeimportanttoobtainresultsaccurateatcosmic-variancelevels,butisnotlikelytobeimportantatl<2000forthenearfuture.Thee ectontheB-modepowerspectrumismoredramatic,

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

140.02

ClΘ/ClΘ

0.01 0 0.01

2 10

40

100

200

400

700

1000

1500

2000

3000

ClX/(ClE ClΘ )1/2

0.01

0

0.01

2 10

40

100

200

400

700

1000

1500

2000

3000

0.01

ClE/ClE

0

0.01

2 10

40

100

200

400

700

1000

1500

2000

3000

0.4

ClB/ClB

0.3 0.2 0.1 0 2 10 40 100 200 400 700 1000 1500 2000 3000

lFIG. 6: The fractional change in the lensed Cl due to non-linear corrections using halofit[21] for the same model as Fig. 4. The lensed Cl are computed using our new accurate method.

giving a> 6% increase in power on all scales. On scales beyond the peak in the B-mode power (l 1000) the extra non-linear power becomes more important, producing an order unity change in the B-mode spectrum on small scales. On these scales the assumption of Gaussianity is probably not very good, and the accuracy will also be limited by the precision of the non-linear power spectrum. For more accurate results, more general models, and on very small scales where the non-Gaussianity of the lensing potential becomes important, numerical simulations may be required (e.g. see Refs.[25, 26]). There are, of course, other non-linear e ects on the CMB with the same frequency spectrum as the primordial (and lensed) temperature anisotropies and polarization. The kinematic Sunyaev-Zel’dovich (SZ) e ect is the main such e ect for the temperature anisotropies, and current uncertainties in the reionization history and morphology make the spectrum ClΘ uncertain at the few percent level at l= 2000[27]. This is a little larger than the error in the rst-order harmonic lensing result, but this doesn’t mean that one should be content with the error in the latter. Precision cosmology from the damping tail will require accurate modelling of both lensing and the kinematic SZ e ect. Errors at the percent level in the lensing power on these scales would seriously limit our ability to constrain reionization scenarios with future arcminute-resolution observations. For the polarization spectra, the kinematic SZ e ect is much less signi cant[28].VI. CONCLUSIONS

We have presented a new, fast and accurate method for computing the lensed CMB power spectra using spherical correlation functions. Previous perturbative methods were found to be insu ciently inaccurate for precision cosmology,

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

andnon-perturbativeresultsinthe at-skyapproximationareinerroratabovethecosmic-variancelevel.Themethoddevelopedhereshouldenableaccuratecalculationofthelensinge ecttowithincosmic-variancelimitstol 2500undertheassumptionsoftheBornapproximationandGaussianityoftheprimordial elds.Non-linearcorrectionstothelensingpotentialhaveonlyasmalle ectonthelensedtemperaturepowerspectrum,butareimportantonallscalesforanaccuratecalculationofthelensedB-modepowerspectrum.

VII.

ACKNOWLEDGMENTS

WethankGayoungChonforherworktowardsimplementingthefull-skylowest-orderlensingresultofRef.[3]incamb,andALthanksMatiasZaldarriaga,MikeNolta,OliverZahn,PatriciaCastro,PatMcDonaldandBenWandeltfordiscussionandcommunication.ACacknowledgesaRoyalSocietyUniversityResearchFellowship.

APPENDIXA:ROTATINGSPIN-WEIGHTHARMONICS

n istherotationoperatorcorrespondingtoEuleranglesα,βandγ.This ,whereDConsiderevaluatingsYlmatD

1andevaluatingatn .Forspin-0harmonicsweisthesameasrigidlyrotatingthefunctionsYlm(asascalar)byD

knowthat

l n )=Dmn).Ylm(D′m( γ, β, α)Ylm′(

(A1)

Forspin-sharmonics,wenotethat

n)sYlm(

,sothatwhere(θ,φ)referton

l

Dmn)=( 1)m′m( γ, β, α)sYlm′(

′

=( 1)m

4π

l

D ms(φ,θ,0),

(A2)

=( 1)m=( 1)m

4π4π

llDm′m( γ, β, α)D m′s(φ,θ,0)llD mm′(α,β,γ)Dm′s(φ,θ,0)l′′D ms(φ,θ,κ)

4π

n )e isκ.=sYlm(D

(A3)

(α,β,γ)D (φ,θ,0)=D (φ′,θ′,κ),sothat(θ′,φ′)refertotheimageofn (α,β,γ),andκ underDHere,wehaveusedD

n tomapthepolarbasisvectorsthereontotheimageofthepolarbasisistheadditionalrotationrequiredaboutD

′ underD(α,β,γ).Denotingthepolarbasis(unit)vectorsatn byeθandeφ,andatn ′bye′atnθandeφ,wehave

′±iκ e′D(eθ±ieφ).θ±ieφ=e

(A4)

Thisensuresthatthe2l+1rank-stensor elds±Ylm( n)≡±sYlm( n)(eθ ieφ) ··· (eθ ieφ)transformirreducibly

l ±Ylm=underrotationsasDm′Dm′m±Ylm′.

APPENDIXB:EVALUATIONOFXimn

Theintegrals

Ximn≡

∞

2α

σ2

thatarerequiredforthenon-perturbativecalculationofthelensedpowerspectraonthesphericalskycaneasilybe

evaluatedasseriesinσ2.Fromthede nitionoftherotationmatrices,wehave

iαLy

|ln ,dlmn(α)= lm|e

i

e α

2

/σ2l

dmn(α)dα

(B1)

(B2)

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

zandL 2angularmomentumoperators,whereweadopttheCondon–Shortleyphasefortheeigenstates|lm oftheL

andwehaveset =1.Expandingtheexponentialasaseriesinα,wehave

Ximn=

∞ 1j=0

2

2

2

i

e (l+1/2)

2

σ2/4

fori=m n.(B6)

Inpractice,weobtainanexcellentapproximationtoXimn,validforalll,byadjustingthel-independentterminthe

exponentoftheasymptoticresult,andtheprefactor,sothatitsseriesexpansionagreeswithadirectevaluationofEq.(B3)toO(σ2).

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

APPENDIXC:FULLSECONDORDERRESULT

ThefullresultforthelensedcorrelationfunctionsaccuratetosecondorderinCglandCgl,2is8 ≈ξ

2l+1

l

+≈ξ

2l+1

l

22l′′′′l

X000(X000+2CglX000)Cgl,2dl+Cdgl11+Cgl,2X220d2 21 1

l(l+1)

22′l+CglX220dl22 4CglCgl,2X000X220d20

(C1)

2

22

CglX220dl00

2l′l′l

+Cgl,2X220X242d40 2CglCgl,2(X220X022d20+X242X022d42)+

1

4π

EB2′2′′′22′2

(Cl Cl)X022+2CglX022X022+Cgl(X022X022+2X022)+Cgl,2X022dl2 2

′′l

+Cgl,2X132(X132+2CglX132)dl3 3+Cgl,2X121(X121+2CglX121)d1 1

1′′

+2Cgl(X121X132+Cgl[X121X132+X132X121])dl+3 1

22l

Cgl,2X242d4 4

2

X≈ξ

2l+1

l

2

X220)

(C3)

2

+

2l2l

X242X220(Cgl,2d4 2+Cgld42).

(C4)

1

2

Asdiscussedinthemaintext,theCgltermsmaybeneglectedattheO(10 4)levelforrealisticlensingde ection

amplitudes.

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