what can be learned from the lensed b-mode cmb...
TRANSCRIPT
What can be learned from the lensed B-mode CMB
polarization power spectrum?
Anthony ChallinorCavendish Laboratory, University of Cambridge
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
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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):
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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
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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
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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
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REMOVING BIAS
• Gaussian likelihood and WMAP1 give significant bias in ‘dark’ parameters (where); new likelihood removes bias
– Confirmed with 150 simulations on flat patches
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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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