Lensed CMB power spectra from all-sky correlation functions(17)

140.02

ClΘ/ClΘ

0.01 0 0.01

2 10

40

100

200

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1000

1500

2000

3000

ClX/(ClE ClΘ )1/2

0.01

0

0.01

2 10

40

100

200

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1000

1500

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0.01

ClE/ClE

0

0.01

2 10

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1000

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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

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