What can be learned from the lensed B-mode CMB

polarization power spectrum?

Anthony ChallinorCavendish Laboratory, University of Cambridge

a.d.challinor@mrao.cam.ac.uk

Melchiorri memorial conference, 14 April 2006(astro-ph/0502425; astro-ph/0511703; astro-ph/0601594)

WEAK GRAVITATIONAL LENSING OF THE CMB

• Weak lensing by density perturbations ⇒ deflection is gradient, α = ∇ψ, in Bornapproximation:

ψ(n) = −2∫ χ∗0

dχΨ(χn; η0 − χ)χ∗ − χ

χχ∗

• R.m.s. deflection 2.4 arcmin; coherent over several degrees

1

WEAK LENSING AND THE CMB POWER SPECTRA

• Currently not detected – inclusion changes WMAP3 fits very little (Lewis 2006) butimportant bias for e.g. Planck if neglected

• Non-zero four-point function ⇒ important for future CBl errors and correlations

• Background ‘noise’ for large-angle gravity-wave searches:√CBl ∼ 1.3nK

dominates primordial for V 1/4 < 1.2× 1016 GeV

– Limits r > 10−4 if no cleaning

2

THEORETICAL SYSTEMATICS: GRADIENT APPROXIMATION AND SKYCURVATURE∗

• Gradient approx. (Hu 2000), Θ = Θ + α ·∇Θ, inaccurate at percent level forl > 1000

• Flat-sky approx. (Seljak 1996) inaccurate due to large-scale coherence of α

∗Challinor & Lewis 2005

3

THEORETICAL SYSTEMATICS: NON-LINEAR MATTER POWER∗

• Small-scale non-linear corrections (e.g. Smith et al. 2003) to matter powerspectrum important at > 5% on all scales for CBl

2 10 40 100 200 400 700 1000 1500 2000 3000−0.01

0

0.01

0.02

2 10 40 100 200 400 700 1000 1500 2000 3000−0.01

0

0.01

2 10 40 100 200 400 700 1000 1500 2000 3000−0.01

0

0.01

2 10 40 100 200 400 700 1000 1500 2000 30000

0.1

0.2

0.3

0.4

PSfrag replacements

∆C

Θ l/C

Θ l∆

CX l

/(C

E lC

Θ l)1

/2

∆C

E l/C

E l∆

CB l

/C

B l

l

101 102 103

10−8

10−7

10−6

PSfrag replacements

l4Cψ l

l

∗Challinor & Lewis 2005

4

APPLICATIONS OF CMB LENSING

• Lensing potential sensitive to parame-ters not constrained by primary CMB(e.g. ∆mν = 0.04eV and ∆w = 0.2

; Kaplinghat et al. 2003) but requires re-construction of deflections using CMBnon-Gaussianity (e.g. Hu 2001) for fullexploitation

• Deflection and T -T and E-B reconstruction on 10◦ × 10◦ patch (1 µK-arcminnoise on T , 4-arcmin beam; Hu & Okamoto 2001):

5

IS THE OBSERVED POWER SPECTRUM ENOUGH?

• Measured power spectrum Cl not a sufficient statistic for non-Gaussian fields butstill a useful (lossy) compression for efficient, robust analysis

– CBl most promising but parameter degeneracy ⇒ can’t separate e.g. w andsub-eV mν

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0 200 400 600 800 1000

l(l+1

)(dC

lBB/d

x) /(

2π)/µ

K2

l

6

NON-GAUSSIAN COVARIANCE

• Lens-induced B ∼ Eψ non-Gaussian ⇒ trispectrum contribution to covariance ofmeasured B-mode power ∼ (CEl C

ψl )2 (Smith et al. 2004)

– Increases errors on dark parameters from lensed CBl c.f. if lens-induced Bwere Gaussian or could implement lensing reconstruction

7

SIMULATION METHODOLOGY

• How do we account for this non-Gaussianity in power-spectrum analysis?

• Full-sky simulations with Antony Lewis’s LensPix code(http://cosmologist.info/lenspix/)

– Re-maps polarization on sphere with parallel-transport prescription of Challinor& Chon (2002)

8

LIKELIHOOD APPROXIMATIONS

• Want approximation to sampling distribution P (Cl|θ) depending only on mean〈Cl〉 = Cl(θ) and covariance Sll′(θ)

– Look for transformation xl(Cl) in which likelihood approx. Gaussian:

lnP (Cl|θ) ≈ lnA− 12∑ll′M

−1ll′ (xl − µl)(xl′ − µl′)

– Demand third derivative of (complicated) likelihood vanish everywhere w.r.t. xlwhen derivs approximated by sampling distribution for full-sky, Guassian data:

lnP (Cl|Cl) = −∑l

(2l+ 1

2

ClCl

+2l+ 1

2lnCl −

2l − 1

2ln Cl

)

– End up with differential equation 2(dCl/dxl)2 = 3Cld

2Cl/dx2l ⇒ xl ∝ C

1/3l –

likelihood approximately Guassian in C1/3l !

– Correct mean and covariance dictate (with A−1 ∝√

detMll′Πlµ2l )

µl =(2l − 1

2l+ 1Cl

)1/3, M−1

ll′ = 3C2/3l

(2l − 1

2l+ 1

)1/6S−1ll′ 3C

2/3l′

(2l′ − 1

2l′ + 1

)1/6

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LIKELIHOOD COMPARISONS

• For full-sky, Gaussian data compare exact, log-normal approx. (Bond et al. 2000),WMAP1 approx. (Verde et al. 2003; combination of Gaussian and log-normal) andnew pCl approx.

• More critical at higher l for lensing application since fewer independent modes

10

REMOVING BIAS

• Gaussian likelihood and WMAP1 give significant bias in ‘dark’ parameters (where); new likelihood removes bias

– Confirmed with 150 simulations on flat patches

11

SUMMARY

• Weak lensing of CMB contains information on ‘dark parameters’ (e.g. sub-eVneutrino masses and dark energy model) not accessible with primary CMB

• Predictions for lensed spectra now limited by accuracy of non-linear matter powerspectrum modelling

• Lensing reconsruction required to avoid loss and break degeneracies in lensedB-mode power spectrum

– Compressing to power spectrum still useful for efficient, robust initial analysis ofnear-future data

– Non-Gaussianity (⇒ fewer independent modes) further degrades errors andrequires careful treatment of likelihood to avoid parameter bias

– New likelihood more generally applicable (e.g. for spectra of Gaussian fields onlarge scales)

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