cmb lensing and delensing - antony lewis' home pagecosmologist.info/notes/imperial2016.pdf ·...
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CMB Lensing and DelensingAntony Lewis
http://cosmologist.info/
work with the Planck Collaboration, Geraint Pratten, Julien Carron and Julien Peloton
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O(10-5) perturbations
(+galaxy)
Dipole (local motion)
(almost) uniform 2.726K blackbody
Observations:
the microwave
sky today
0th order (uniform 2.726K) + 1st order perturbations (anisotropies)
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CMB temperature
Perturbations: End of inflation Perturbations: Last scattering surface
gravity+
pressure+
diffusion
0th order uniform temperature + 1st order perturbations:
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14 000 Mpc
z~1000
z=0θ
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Gaussian ⇒ information in power spectrum
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Last scattering surface
Inhomogeneous universe
- photons deflected
Observer
CMB Lensing
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Deflection angle 𝛼, shear 𝛾𝑖 , convergence 𝜅 , and rotation 𝜔
𝜅
𝛾
Convergence
Shear
Rotation 𝜔 = 0 from scalar perturbations in linear perturbation theory
𝜔
Rotation
𝜔 = 0 ⇒ 𝛼 = 𝛻𝜓
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Deflections O(10-3), but coherent on degree scales important!
Deflection angle power spectrum
Linear
Non-linear
On small scales
(Limber approx. 𝑘𝜒 ∼ 𝑙)
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Credit: D. Hanson
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Credit: D. Hanson
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UnlensedMagnified Demagnified
Local effect of lensing on the power spectrum
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Credit: Duncan Hanson
Averaged over the sky, lensing smooths out the power spectrum
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Lensing effect on CMB temperature power spectrum
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Effect on TE and EE polarization spectra
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Current 95% indirect limits for LCDM given WMAP+2dF+HST (bit old!)
B-mode polarization power spectrum
Lensing an important contaminant of searches for primordial gravitational waves
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1. Can we reconstruct the lensing?
- gives a powerful cosmological probe
(z~2 peak; constraints on LCDM, massive neutrinos, etc.)
2. Can we then delens?
- unsmooth the power spectra, clean the lensing B modes
3. Can we model the lensing signal accurately?
Outline
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𝑪𝒍 𝑪𝒍 𝑪𝒍 𝑪𝒍
𝑪𝒍 𝑪𝒍 𝑪𝒍 𝑪𝒍
𝑪𝒍 𝑪𝒍 𝑪𝒍 𝑪𝒍
𝑪𝒍 𝑪𝒍 𝑪𝒍 𝑪𝒍
Lensing reconstruction (concept)
Measure spatial variations in magnification and shear
Use assumed unlensed spectrum, and unlensed statistical isotropy
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For a given (fixed) lensing field, 𝑇 ∼ 𝑃 𝑇 𝑋 :
function easy to calculate for 𝑋(𝐊) = 0 Can reconstruct the
modulation field 𝑋
Lensing reconstruction- Maths and algorithm sketch
Flat sky approximation: modes correlated for 𝐤2 ≠ 𝐤3
𝑋 here is lensing potential,
deflection angle, or 𝜅
𝐴 𝐿, 𝑙1, 𝑙2 ∼
Zaldarriaga, Hu, Hanson, and others.
Full sky analysis similar, summing modes with optimal weights gives
First-order series expansion in the lensing field:
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→ process input maps
→ estimate lensing potential
from anisotropic 2-point
Lens Reconstruction Pipeline
Filtering
Quadratic Estimator
Power Spectrum Estimation
Filtering
Quadratic Estimator
Data / Sims Data / Sims
Cross-correlation
ϕ Tracer
Filtering
Data / Sims
Filtering
Data / Sims
→ estimate lensing power
spectrum.
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Best measured modes of
MV estimator have S/N=1.
Planck noise power spectra for lensing estimators.
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Planck 2015 lensing reconstruction (𝐸𝛁𝚽)
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Simulated Planck lensing reconstruction
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True simulation input
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Lensing Power Spectrum Results
ACTPOL
Credit: Julien Carron
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Kids-450 2016 (1606.05338)
Galaxy lensing (cosmic shear)Planck CMB Lensing
Are we heading for a clear breakdown of 𝐿𝐶𝐷𝑀?.. or statistical fluctuation, better understanding of galaxy shear systematics…?
+ weak Planck 𝐴𝑠 prior
Planck TT+lowP+lensing
Cosmological parameters in LCDM
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Full likelihood model in hand, accounting for non-independence of the signals
In future joint analysis of CMB and lensing power spectra
should be able to detect down to ∑𝑚𝜈~0.06eV (with external data)
arXiv: 1303.5379
arXiv:1611.01446
Code available on request to JP
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Delensing
Expected Planck internal delensing efficiencies
1. Use external tracer of matter, e.g. CIB. (Larsen et al. 2016)
2. Use the lensing reconstruction
a. Use best-estimate lensing field
find 𝛽 such that
b. Approximate 𝛽 ≈ −𝛼 where 𝛼 = 𝛻𝜙
c. Remap points using estimated 𝛼 to get delensed map
Carron, Lewis, Challinor in prep.
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BUT: Internal delensing causes biases
𝜅 > 0 𝜅 < 0𝜅 = 0
After Delensing:
BUT: fluctuation in scale could also just be random cosmic variance
⇒ Delensing removes random fluctuations in peak location
⇒ Delensing artificially sharpens the peaks, even with no actual lensing!
⇒ Must subtract bias expected even if no lensing
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PRELIMINARY
Carron, Lewis, Challinor in prep.
Power difference after delensing 𝐶𝑙delensed − 𝐶𝑙
dat
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PRELIMINARY
∼ 25𝜎 detection of TT delensing, 20𝜎 of polarization delensing; consistent with expectations
Carron, Lewis, Challinor in prep.
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(VERY) PRELIMINARY
Planck delensing of B-mode polarization
Detection of reduction in B-mode lensing power at 3-5𝜎
(but noise high, so does not yet help with tensor 𝑟 constraint)
Carron, Lewis, Challinor in prep.
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Using EE clear physics targets may be (just) within reach of S4… also delens EE
arXiv:1609.08143
Future prospects
Delensing can decrease
errors by up to ~ 20%
arXiv:1609.08143
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Theory – how to predict the expected lensing signal?
1. Linear Theory (+ approximate non-linear matter power)
lenses large and high redshift ⇒ nearly linear; Born approximation
2. Non-linear effects:
- Important to quantify
- Can give interesting new bispectrum and rotation signals
Post-Born lensing + non-linear structure formationPratten & Lewis: arXiv:1605.05662
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Ray-deflection: first lens changes location of second lensing event
+
Linear approximation
Post-Born lensing
=
e.g. more net lensing
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+
Linear approximation
Post-Born lensing
=
Lens-Lens coupling: Beam size (and shape) affected by first lensing event
e.g. less net lensing
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+
+ 𝜅 + 𝛾
Lens-Lens with two non-aligned shears ⇒ rotation
=
𝜔
- Post-Born effects can change power spectra
- Rotation could introduce new source of confusing B-modes
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- Negligible change to convergence spectrum
- Non-zero rotation spectrum
Effect on lensing convergence and rotation power spectra
Pratten & Lewis: arXiv:1605.05662
Safe to neglect for near future (< 10−3)
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B-mode signal from rotation
~2.5% of B mode amplitude from rotation
Pratten & Lewis: arXiv:1605.05662
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How Gaussian is the lensing potential field?
Non-Gaussianity potentially important:
- Useful extra signal? (Namikawa 2016)
- Biases on lensing quadratic estimators (Boehm et al 2016)
- Corrections to the lensed CMB power spectra (Lewis & Pratten 16)
Large distance to CMB ⇒ many independent lenses
Expected to be quite small:
⇒ Gaussianization by central limit theorem
But how non-Gaussian, and what shape?...
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Flat sky approximation:
If you know 𝜅 𝑙1 , 𝜅 𝑙2 , sign of 𝑏𝑙1𝑙2𝑙3tells you which sign of 𝜅 𝑙3 is more likely
Bispectrum
⟨𝜅 𝑙1 𝜅 𝑙2 𝜅(𝑙3)⟩ =1
2𝜋𝛿 𝑙1 + 𝑙2 + 𝑙3 𝑏𝑙1𝑙2𝑙3
𝑙2
𝑙3
𝑙1 𝒍𝟏 + 𝒍𝟐 + 𝒍𝟑 = 𝟎
𝜅 𝑙1 𝜅 𝑙2 = 𝛿 𝑙1 + 𝑙2 𝐶𝑙𝜅
- power spectrum encodes all the information
- modes with different wavenumber are independent
Gaussian + statistical isotropy
Flat sky approximation: 𝜅 𝑥 =1
2𝜋∫ 𝑑2𝑙 𝜅(𝑙)𝑒𝑖𝑥⋅ 𝑙
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+
+
+
𝑙1 + 𝑙2 + 𝑙3 = 0, 𝑙1 = 𝑙2 = |𝑙3|Equilateral
=
b>0
b<0
𝑙1
𝑙2
𝑙3
𝜅(𝑙1) 𝜅(𝑙2)
−𝜅(𝑙3)
𝜅(𝑙3)
Lewis: The Real Shape of Non-Gaussianities, arXiv:1107.5431
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In 2D projection (e.g. lensing)
Positive equilateral bispectrumClusters
(+sight-aligned filaments)
Millennium simulation
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Near-equilateral to flattened/folded:
b<0b>0
𝑙2
𝑙3𝑙1
𝑙2
𝑙3
𝑙1
b<0b>0
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Positive flattened bispectrum
In 2D projection (e.g. lensing)
Filaments
(+sight-aligned sheets)
Millennium simulation
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LSS has positive bispectrum, hence 𝜅 bispectrum from LSS also positive.
What about post-Born?
Big negative lens-lens effect Zero lens-lens effect
Negative flattened bispectrum
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Unexpectedly small folded Gaussianity of the CMB lensing convergence!
Post-born LSS
Convergence Bispectrum
+ = Total
Pratten & Lewis: arXiv:1605.05662
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This cancellation is a fluke, LSS dominates at lower redshifts
Pratten & Lewis: arXiv:1605.05662
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Naïve S/N for post-Born and total bispectrum
Can measure 𝜔 from
its bispectrum with 𝜅?
Negligible
Small but may be
important for S4
Pratten & Lewis: arXiv:1605.05662
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Conclusions
Lensing the leading secondary effect on the CMB anisotropies
Well measured by Planck
- 2017 release coming up; many others over next 10 years
Test LCDM, constrain parameters, ∑𝑚𝜈, dark energy, etc.
Delensing works! Will be important for future tensor mode searches.
Post-Born impact on power spectra negligible for near future
But LSS and post-Born important for CMB convergence bispectrum
- “lucky” cancellation makes CMB lensing remarkably Gaussian
- but bispectrum still detectable with future data
- could also detect lensing rotation using 𝜅𝜅𝜔 bispectrum