Lensing of the CMB

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Lensing of the CMB

• Antony Lewis
• Institute of Astronomy, Cambridge
• http//cosmologist.info/

Review ref Lewis, Challinor , Phys. Rep
astro-ph/0601594
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Evolution of the universe
Opaque
Transparent
Hu White, Sci. Am., 290 44 (2004)
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Perturbation evolution what we actually
observeCMB monopole source till 380 000 yrs
(last scattering), linear in conformal timescale
invariant primordial adiabatic scalar spectrum
photon/baryon plasma dark matter, neutrinos
Characteristic scales sound wave travel
distance diffusion damping length
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CMB temperature power spectrumPrimordial
perturbations later physics
diffusion damping
acoustic oscillations
primordial powerspectrum
finite thickness
Hu White, Sci. Am., 290 44 (2004)
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Temperature anisotropy data WMAP 3-year
smaller scales
BOOMERANG
Hinshaw et al
many more coming up
e.g. Planck (2008)
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Weak lensing of the CMB
Last scattering surface
Inhomogeneous universe - photons deflected
Observer
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Not to scale!All distances are comoving
largest overdensity
200/14000 degree
Ionized plasma - opaque
Neutral gas - transparent
Recombination
200Mpc
14 000 Mpc
100Mpc
Good approximation CMB is single source plane at
14 000 MpcAngular diameter distance well
measured by angle of acoustic peaks
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Lensing order of magnitudes
?
ß
Newtonian argument ß 2 ? General
Relativity ß 4 ?
(ß ltlt 1)
Potentials linear and approx Gaussian ? 2 x
10-5
ß 10-4
Characteristic size from peak of matter power
spectrum 300Mpc
Comoving distance to last scattering surface
14000 MPc
total deflection 501/2 x 10-4
pass through 50 lumps
2 arcminutes
assume uncorrelated
(neglects angular factors, correlation, etc.)
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So why does it matter?

• 2arcmin ell 3000- On small scales CMB is
  very smooth so lensing dominates the linear
  signal
• Deflection angles coherent over 300/(14000/2)
  2 - comparable to CMB scales- expect
  2arcmin/60arcmin 3 effect on main CMB acoustic
  peaks

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In detail, lensed temperature depends on
deflection angle
Lensing Potential
Deflection angle on sky given in terms of lensing
potential
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Deflection angle power spectrum
Non-linear
Linear
Deflections O(10-3), but coherent on degree
scales ? important!
Computed with CAMB http//camb.info
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Simulated full sky lensing potential and
(magnified) deflection angle fields
Easily simulated assuming Gaussian fields - just
re-map points using Gaussian realisations of CMB
and potential
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Lensed temperature Cl
- convolution of unlensed Cl- W is non-linear in
lensing potential power
Essentially exact to order of weak lensing by
Gaussian field very well understood effect on
power spectra.Non-linear Pk 0.2 on TT, 5 on BB
Lewis, Challinor Phys. Rept. 2006
astro-ph/0601594
Full-sky fully non-perturbative generalization of
method by Seljak 1996
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Lensing effect on CMB temperature power spectrum
CAMBs 0.1 calculation http//camb.info
Challinor Lewis 2005, astro-ph/0502425
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Lensing important at 500ltllt3000Dominated by SZ
on small scales
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CMB Polarization
Generated during last scattering (and
reionization) by Thomson scattering of
anisotropic photon distribution
Hu astro-ph/9706147
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Polarization Stokes Parameters
-
-
Q
U
Q ? -Q, U ? -U under 90 degree rotation
Q ? U, U ? -Q under 45 degree rotation
Rank 2 trace free symmetric tensoror spin-2
field- just like shear
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E and B polarization
gradient modesE polarization
curl modes B polarization
e.g.
e.g. cold spot
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Why polarization?

• E polarization from scalar, vector and tensor
  modes (constrain parameters, break
  degeneracies)
• B polarization only from vector and tensor modes
  (curl grad 0) non-linear scalars

B modes only expected from gravitational waves
and CMB lensing
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Lensing of polarization

• Polarization not rotated w.r.t. parallel
  transport (vacuum is not birefringent)
• Q and U Stokes parameters simply re-mapped by the
  lensing deflection field

e.g.
Observed
Last scattering
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Polarization lensing Cx and CE
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Polarization lensing CB
Nearly white BB spectrum on large scales
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Polarization power spectra
Current 95 indirect limits for LCDM given
WMAP2dFHST
Lewis, Challinor astro-ph/0601594
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Non-Gaussianity

• Unlensed CMB expected to be close to Gaussian
• With lensing

• For a FIXED lensing field, lensed field also
  Gaussian
• For VARYING lensing field, lensed field is
  non-Gaussian

• Specific form of non-Gaussianity - e.g. 1 point
  still Gaussian, very small 3-point function -
  should be able to distinguish from primordial
  non-Gaussianity
• Modifies covariance of lensed Cl (esp. BB)
• Can be used to learn about lensing potential
  reconstruction methods

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Likelihoods

• Small number of lensing modes BB Cl correlated
  between l. (Smith, Challinor, Rocha 2006)
• Correction to temperature likelihood is small on
  full sky usual result is quite good

Correct BB (and others) using covariance from
simulations. Good approx is
Smith, Challinor, Rocha 2006
ASIDE Also works for cut sky can use for
convergence power spectrumFor multiple redshift
bins can generalise for correlated fields
X (k11,k22,k12,)
for details see Hammimeche Lewis (in prep).
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Large scale lensing reconstruction

• As with galaxy lensing, ellipticities of hot and
  cold spots could be used to constrain the lensing
  potential
• But diffuse, know source statistics, can use
  magnification- need general method
• Think about fixed lensing potential lensed CMB
  is then Gaussian (T is Gaussian) but not
  isotropic- use off-diagonal correlation to
  constrain lensing potential

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• Can show that optimal quadratic estimator is

- simple function of filtered fields
Analogous results for CMB polarization
For more details see Hu astro-ph/0105424 or
review c.f. Metcalf White 2007
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e.g. estimate lensing potential power spectrum-
more information on cosmological parameters
(ideal is limit using non-optimal quadratic
estimator)
Hu astro-ph/0108090
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e.g. reconstruct lensing potential field

• should correlate with other matter tracers
• Constrain large-scale matter distribution to
  redshift z 6
• De-lens the CMB (remove B-mode lensing
  contamination to see primordial B modes)

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First claimed detection in cross-correlation (see
talk by Olivier Doré)
(http//cosmocoffee.info discussion)
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Reconstruction complications

• Limited by cosmic variance on T, other
  secondaries, higher order terms
• Quadratic method useful but not
  optimal-especially for polarization
  (HirataSeljak papers)
• Requires high resolution effectively need lots
  of hot and cold spots behind each potential
• Reconstruction with polarization is much better
  no cosmic variance in unlensed B
• Polarization reconstruction can in principle be
  used to de-lens the CMB - required to probe
  tensor amplitudes r lt 10-4- requires very high
  sensitivity and high resolution

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Quadratic (filtered)
Approx max likelihood
Input
astro-ph/0306354
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Other information in CMB lensing (gtgt arcminute)

• Lensed CMB power spectra contain essentially two
  new numbers - one from T and E, depends on
  lensing potential at llt300 - one from lensed
  BB, wider range of lastro-ph/0607315
• Can break degeneracies in linear CMB improve
  constraints on dark energy, curvature, etc.
• May be able to probe neutrino masses 0.04eV
  (must be there! see astro-ph/0603494)

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Cluster CMB lensinge.g. to constrain cosmology
via number counts
Seljak, Zaldarriaga, Dodelson, Vale, Holder,
Lewis, King, Hu. Maturi,. etc.
CMB very smooth on small scales approximately a
gradient
What we see
Last scattering surface
GALAXYCLUSTER
0.1 degrees
Need sensitive arcminute resolution observations
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RMS gradient 13 µK / arcmindeflection from
cluster 1 arcmin
Lensing signal 10 µK
BUT depends on CMB gradient behind a given
cluster
Unlensed
Lensed
Difference
Unlensed CMB unknown, but statistics well
understood (background CMB Gaussian)
can compute likelihood of given lens (e.g. NFW
parameters) essentially exactly
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Add polarization observations?
Difference after cluster lensing
Unlensed TQU
Less sample variance but signal 10x smaller
need 10x lower noise
Note E and B equally useful on these scales
gradient could be either
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Complications

• Temperature - Thermal SZ, dust, etc. (frequency
  subtractable) - Kinetic SZ (big problem?) -
  Moving lens effect (velocity Rees-Sciama,
  dipole-like) - Background Doppler signals -
  Other lenses

• Polarization - Quadrupole scattering (lt
  0.1µK)- Re-scattered thermal SZ (freq)- Kinetic
  SZ (higher order)- Other lensesGenerally much
  cleaner

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Fitting profiles. e.g. to measure mass and
concentration
Optimistic Futuristic CMB polarization lensing vs
galaxy lensinge.g. M 2 x 1014 h-1 Msun, c5
Can stackfor constraintsfrom multipleclusters
Lewis King 2006
CMB polarization only (0.07 µK arcmin noise)
Galaxies (500 gal/arcmin2)
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General cluster mass reconstruction

• Can use quadratic reconstruction methods similar
  to those on large scales
• Potential problems with bias due to large central
  magnifications- use full likelihood function
  (e.g. Hirata et al, though prior less clear)-
  various ad hoc methods also work (Maturi, Hu..)
• Not competitive with galaxy lensing except
  possibly for high redshift
• But systematics very different may be useful
  cross-check

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CMB/Galaxy lensing comparison

• CMB Lensing
• single source plane, lenses 0.5ltzlt7
• accurate source plane distance
• statistics of source plane well understood
• systematics pointing/beam uncertainty, SZ,
  foregrounds,- Small corrections from
  non-linear Pk- Smoothes temperature power
  spectrum- B modes generated by lensing of E

• Galaxy lensing
• many source planes, lenses lt1.5
• often only photo-z redshifts
• make no assumption about sourcedistribution-
  systematics PSF modelling, source selection,
  noise bias, .- Non-linear Pk
  crucial-magnification effect on source
  numbercounts (e.g. smoothes baryon
  oscillations c.f. original Vallinotto talk)
• - Mixing of intrinsic alignment source plane E
  and B fields by lensing

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Lensing of 21cm

• Very similar to CMB lensing, but 21cm power
  spectrum much more small scale power and many
  source planes/3D information
• Lensed angular power spectrum result simple
  generalization from lensed CMB temperature(Lewis
  Challinor 2007c.f. Mandel Zaldarriaga 2006)

Cl(z50,z52)
Cl(z50,z50)

• Can reconstruct potential from lensed 21cm
  lots of information in 3D(Hilbert, Metcalf,
  White, Zaldarriaga, Zahn, Cooray... see Metcalf
  poster)

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Summary

• Weak lensing of the CMB very important for
  precision cosmology- changes power spectra at
  several percent- potential confusion with
  primordial gravitational waves for r lt 10-3-
  introduces non-Gaussian signal- well understood
  in theory accurately modelled with linear
  theory small non-linear corrections
• Potential uses- Break parameter degeneracies,
  improve parameter constraints- Constrain
  cluster masses to high redshift- Reconstruction
  of potential at 0.5 lt z lt 7

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Correlation with the CMB temperature
very small except on largest scales
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Cosmological parameters
Essential to model lensing but little effect on
basic parameter constraints
Planck (2007) parameter constraint simulation
(neglect non-Gaussianity of lensed field BB
noise dominated so no effect on parameters)
Important effect, but using lensed CMB power
spectrum gets right answer
Lewis 2005
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Moving Lenses and Dipole lensing
Homogeneous CMB
Rest frame of CMB
Rees-Sciama(non-linear ISW)
v
Redshiftedcolder
Blueshiftedhotter
Rest frame of lens
Dipole gradient in CMB
T T0(1v cos ?)
dipole lensing
Deflected from colder
deflected from hotter
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Moving lenses and dipole lensing are equivalent

• Dipole pattern over cluster aligned with
  transverse cluster velocity source of confusion
  for anisotropy lensing signal
• NOT equivalent to lensing of the dipole observed
  by us, -only dipole seen by cluster is lensed
  (EXCEPT for primordial dipole which is physically
  distinct from frame-dependent kinematic dipole)

Note

• Small local effect on CMB from motion of local
  structure w.r.t. CMB(Vale 2005, Cooray 2005)
• Line of sight velocity gives (v/c) correction to
  deflection angles from change of framegenerally
  totally negligible

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Non-Gaussianity(back to CMB temperature)

• Unlensed CMB expected to be close to Gaussian
• With lensing

• For a FIXED lensing field, lensed field also
  Gaussian
• For VARYING lensing field, lensed field is
  non-Gaussian

Three point function Bispectrum lt T T T gt
- Zero unless correlation ltT ?gt

• Large scale signal from ISW-induced T- ?
  correlation
• Small scale signal from non-linear SZ ?
  correlation

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• Trispectrum Connected four-point lt T T T Tgtc

• Depends on deflection angle and temperature
  power spectra
• Easily measurable for accurate ell gt 1000
  observations

Other signatures

• correlated hot-spot ellipticities
• Higher n-point functions
• Polarization non-Gaussianity

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Bigger than primordial non-Gaussianity?

• 1-point function

• lensing only moves points around, so
  distribution at a point Gaussian
• But complicated by beam effects

• Bispectrum

- ISW-lensing correlation only significant on
very large scales
- SZ-lensing correlation can dominate on very
small scales
- On larger scales oscillatory primordial signal
should be easily distinguishable with Planck
Komatsu astro-ph/0005036
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• Trispectrum (4-point)

Basic inflation- most signalin long thin
quadrilaterals
Lensing- broader distribution, lesssignal in
thin shapes
Komatsu astro-ph/0602099
Hu astro-ph/0105117
Can only detect inflation signal from cosmic
variance if fNL gt 20
Lensing probably not main problem for flat
quadrilaterals if single-field non-Gaussianity
No analysis of relative shape-dependence from
e.g. curvaton??
Also non-Gaussianity in polarization