Lensing of the CMB

1
Lensing of the CMB

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

2
Talk based on recent review this is recommended
reading and fills in missing details and
references
Physics Reports review astro-ph/0601594
3

• Recent papers of interest (since review)-
  Cosmological Information from Lensed CMB Power
  Spectra Smith et al. astro-ph/0607315
• For introductory material on unlensed CMB, esp.
  polarization see Wayne Hus pages
  athttp//background.uchicago.edu/whu/Anthony
  Challinors CMB introduction astro-ph/0403344

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Outline

• Review of unlensed CMB
• Lensing order of magnitudes
• Lensed power spectrum
• CMB polarization
• Non-Gaussianity
• Cluster lensing
• Moving lenses
• Reconstructing the potential
• Cosmological parameters

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Evolution of the universe
Opaque
Transparent
Hu White, Sci. Am., 290 44 (2004)
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Observation as a function of frequencyBlack body
spectrum observed by COBE
Residuals Mather et al 1994
- close to thermal equilibrium temperature
today of 2.726K ( 3000K at z 1000 because ?
(1z))
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(almost) uniform 2.726K blackbody
Dipole (local motion)
O(10-5) perturbations (galaxy)
Observations the microwave sky today
Source NASA/WMAP Science Team
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Can we predict the primordial perturbations?

• Maybe..

Inflation make gt1030 times bigger
Quantum Mechanicswaves in a box
calculationvacuum state, etc
After inflation Huge size, amplitude 10-5
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Perturbation evolution
photon/baryon plasma dark matter, neutrinos
Characteristic scales sound wave travel
distance diffusion damping length
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Observed ?T as function of angle on the sky
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CMB power spectrum Cl

• Use spherical harmonics Ylm

Observe
Theory prediction
- variance (average over all possible sky
realizations)- statistical isotropy implies
independent of m
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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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Perturbations O(10-5)
Calculation of theoretical perturbation evolution
Simple linearized equations are very accurate
(except small scales)Fourier modes evolve
independently simple to calculate accurately
Physics Ingredients

• Thomson scattering (non-relativistic
  electron-photon scattering) - tightly coupled
  before recombination tight-coupling
  approximation (baryons follow electrons because
  of very strong e-m coupling)
• Background recombination physics
• Linearized General Relativity
• Boltzmann equation (how angular distribution
  function evolves with scattering)

linearized GR Boltzmann equations
Initial conditions cosmological parameters
Cl
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Sources of CMB anisotropy Sachs Wolfe
Potential wells at last scattering cause
redshifting as photons climb out Photon density
perturbations Over-densities of photons look
hotter Doppler Velocity of photon/baryons
at last scattering gives Doppler
shiftIntegrated Sachs Wolfe Evolution of
potential along photon line of sight net red-
or blue-shift as photon climbs in an out of
varying potential wellsOthers Photon
quadupole/polarization at last scattering,
second-order effects, etc.
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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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What can we learn from the CMB?

• Initial conditionsWhat types of perturbations,
  power spectra, distribution function (Gaussian?)
  gt learn about inflation or alternatives.(distrib
  ution of ?T power as function of scale
  polarization and correlation)
• What and how much stuffMatter densities (Ob,
  Ocdm) neutrino mass(details of peak shapes,
  amount of small scale damping)
• Geometry and topologyglobal curvature OK of
  universe topology(angular size of
  perturbations repeated patterns in the sky)
• EvolutionExpansion rate as function of time
  reionization- Hubble constant H0 dark energy
  evolution w pressure/density(angular size of
  perturbations l lt 50 large scale power
  polarization)
• AstrophysicsS-Z effect (clusters), foregrounds,
  etc.

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CMB summary

• Time from big bang to last scattering (300Mpc
  comoving 300 000 years) determines physical
  size of largest overdensity (or underdensity)
• Distance of last scattering from us (14Gpc
  comoving 14 Gyr)- determines angular size seen
  by us
• Damping scale (angular size arcminutes) -
  determines smallest fluctuations (smooth on small
  scales)
• Thickness of last scattering (Hubble time,
  100Mpc)- determines line of sight averaging-
  determines amount of polarization (see later)
• Other parameters - determine amplitude and scale
  dependence of perturbations

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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 Mpc
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Zeroth-order CMB

• CMB uniform blackbody at 2.7 K(dipole due to
  local motion)

1st order effects

• Linear perturbations at last scattering,
  zeroth-order light propagation zeroth-order
  last scattering, first order redshifting during
  propagation (ISW)- usual unlensed CMB anisotropy
  calculation
• First order time delay, uniform CMB - last
  scattering displaced, but temperature at
  recombination the same - no observable effect

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1st order effects contd.

• First order CMB lensing zeroth-order last
  scattering (uniform CMB 2.7K), first order
  transverse displacement in light propagation

A
B
Number of photons before lensing-----------------
----------------------------Number of photons
after lensing
A2 ---- B2
Solid angle before lensing -----------------------
------------ Solid angle after lensing


Conservation of surface brightness number of
photons per solid angle unchanged
uniform CMB lenses to uniform CMB so no
observable effect
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2nd order effects

• Second order perturbations at last scattering,
  zeroth order light propagation-tiny (10-5)2
  corrections to linear unlensed CMB result
• First order last scattering (10-5 anisotropies),
  first order transverse light displacement- this
  is what we call CMB lensing
• First order last scattering (10-5 anisotropies),
  first order time delay- delay 1MPc, small
  compared to thickness of last scattering-
  coherent over large scales very small observable
  effect
• Others e.g. Rees Sciama second ( higher) order
  reshiftingSZ second (higher) order scattering,
  etc.

Hu, Cooray astro-ph/0008001
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CMB lensing order of magnitudes
(set c1)
?
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

NOT because of growth of matter density
perturbations!
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Comparison with galaxy lensing

• Single source plane at known distance (given
  cosmological parameters)
• Statistics of sources on source plane well
  understood- can calculate power spectrum
  Gaussian linear perturbations- magnification and
  shear information equally useful - usually
  discuss in terms of deflection angle -
  magnification analysis of galaxies much more
  difficult
• Hot and cold spots are large, smooth on small
  scales- strong and weak lensing can be
  treated the same way infinite magnification of
  smooth surface is still a smooth surface
• Source plane very distant, large linear lenses-
  lensing by under- and over-densities
• Full sky observations- may need to account for
  spherical geometry for accurate results

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Lensed temperature depends on deflection angle
Newtonian potential
co-moving distance to last scattering
Lensing Potential
Deflection angle on sky given in terms of angular
gradient of lensing potential
c.f. introductory lectures
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Power spectrum of the lensing potential

• Expand Newtonian potential in 3D harmonics

with power spectrum
Angular correlation function of lensing potential
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jl are spherical Bessel functions
Use
Orthogonality of spherical harmonics (integral
over k) then gives
Then take spherical transform using
Gives final general result
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Deflection angle power spectrum
On small scales (Limber approx)
Deflection angle power
Non-linear
Linear
Deflections O(10-3), but coherent on degree
scales ? important!
Computed with CAMB http//camb.info
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Lensing potential and deflection angles
LensPix sky simulation code http//cosmologist.in
fo/lenspix
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• Note can only observe lensed field
• Any bulk deflection is unobservable degenerate
  with corresponding change in unlensed CMB e.g.
  rotation of full sky translation in flat sky
  approximation
• Observations sensitive to differences of
  deflection angles

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Correlation with the CMB temperature
very small except on largest scales
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Calculating the lensed CMB power spectrum

• Approximations and assumptions - Lensing
  potential uncorrelated to temperature - Gaussian
  lensing potential and temperature - Statistical
  isotropy
• Simplifying optional approximations - flat sky
  - series expansion to lowest relevant order

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Unlensed temperature field in flay sky
approximation

• Fourier transforms

• Statistical isotropy

where
So
Similarly for the lensing potential (also assumed
Gaussian and statistically isotropic)
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Lensed field series expansion approximation
(BEWARE this is not a very good approximation!
See later)
Using Fourier transforms, write gradients as
Then lensed harmonics then given by
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Lensed field still statistically isotropic
with
Alternatively written as
where
(RMS deflection 2.7 arcmin)
Second term is a convolution with the deflection
angle power spectrum
- smoothes out acoustic peaks- transfers power
from large scales into the damping tail
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Lensing effect on CMB temperature power spectrum
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Small scale, large l limit - unlensed CMB has
very little power due to silk damping
- Proportional to the deflection angle power
spectrum and the (scale independent) power in the
gradient of the temperature
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Accurate calculation -lensed correlation function

• Do not perform series expansion

Lensed correlation function
Assume uncorrelated
To calculate expectation value use
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where
Have defined
small correction from transverse differences
- variance of the difference of deflection angles
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So lensed correlation function is
Expand exponential using
Integrate over angles gives final result
Note exponential non-perturbative in lensing
potential
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Power spectrum and correlation function related by
used Bessel functions defined by
Can be generalized to fully spherical
calculation see review, astro-ph/0601594 However
flat sky accurate to lt 1 on the lensed power
spectrum
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Series expansion in deflection angle?
Only a good approximation when - deflection
angle much smaller than wavelength of temperature
perturbation - OR, very small scales where
temperature is close to a gradient
CMB lensing is a very specific physical second
order effect not accurately contained in 2nd
order expansion differs by significant 3rd and
higher order terms
Series expansion only good on large and very
small scales
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Other specific non-linear effects

• Thermal Sunyaev-ZeldovichInverse Compton
  scattering from hot gas frequency dependent
  signal
• Kinetic Sunyaev-Zeldovich (kSZ)Doppler from bulk
  motion of clusters patchy reionization(almost)
  frequency independent signal
• Ostriker-Vishniac (OV)same as kSZ but for early
  linear bulk motion
• Rees-SciamaIntegrated Sachs-Wolfe from evolving
  non-linear potentials frequency independent

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Summary so far

• Deflection angles of 3 arcminutes, but
  correlated on degree scales
• Lensing convolves TT with deflection angle power
  spectrum - Acoustic peaks slightly blurred -
  Power transferred to small scales

large scales
small scales
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Lensing important at 500ltllt3000Dominated by SZ
on small scales
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Thomson Scattering Polarization
W Hu
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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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Observed Stokes Parameters
-
-
Q
U
Q ? -Q, U ? -U under 90 degree rotation
Q ? U, U ? -Q under 45 degree rotation
Measure E field perpendicular to observation
direction nIntensity matrix defined as
Linear polarization Intensity circular
polarization
CMB only linearly polarized. In some fixed basis
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Alternative complex representation
e.g.
Define complex vectors
And complex polarization
Under a rotation of the basis vectors
- spin 2 field
all just like the shear in galaxy lensing
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E and B polarization
gradient modesE polarization
curl modes B polarization
e.g.
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E and B harmonics

• Expand scalar PE and PB in scalar harmonics
• Expand P in spin-2 harmonics

Harmonics are orthogonal over the full sky
E/B decomposition is exact and lossless on the
full sky
Zaldarriaga, Seljak astro-ph/9609170 Kamionkowski
, Kosowsky, Stebbins astro-ph/9611125
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On the flat sky spin-2 harmonics are
Inverse relations
Factors of
rotate polarization to physical frame defined by
wavenumber l
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l
Polarization Qxy-1, Uxy0 Pxy -1 in bases
wrt (rotated by f) Ql 0, Ul 1Pl i Pl
Pxy e-2if
-f
y
x
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CMB Polarization Signals

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

Average over possible realizations (statistically
isotropic)
Expected signal from scalar modes
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Primordial Gravitational Waves(tensor modes)

• Well motivated by some inflationary models-
  Amplitude measures inflaton potential at horizon
  crossing- distinguish models of inflation
• Observation would rule out other models -
  ekpyrotic scenario predicts exponentially small
  amplitude - small also in many models of
  inflation, esp. two field e.g. curvaton
• Weakly constrained from CMB temperature
  anisotropy

- cosmic variance limited to 10 - degenerate
with other parameters (tilt, reionization, etc)
Look at CMB polarization B-mode smoking gun
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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
ellipticities of infinitesimal small galaxies
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Lensed spectrum lowest order calculation
Similar to temperature derivation, but now
complex spin-2 quantities
Unlensed B is expected to be very small. Simplify
by setting to zero.Expand in harmonics
Calculate power spectrum. Result is
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• Effect on EE and TE similar to temperature
  convolution smoothing transfer of power to
  small scales

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Polarization lensing B mode power spectra

• BB generated by lensing even if unlensed B0

On small scales,
ClE
lensed BB given by
ClB
Nearly white spectrum on large scales(power
spectrum independent of l)
l4Clf
l4Clf l2ClE
Can also do more accurate calculationusing
polarization correlation functions
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Polarization power spectra
Current 95 indirect limits for LCDM given
WMAP2dFHST
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Analogues of CMB lensing

• Lensing of temperature power spectrum - lensed
  effect on galaxy number density/21cm power
  spectrum - smoothing of baryon oscillations (but
  much smaller effect 10-3, low z)
• Q/U polarization - e1/e2 ellipticity of a point
  sourceQ/U not changed by gravitational shear
  along path
• CMB polarization at last scattering - galaxy
  shape distribution in source plane - usually
  assume shapes uncorrelated CECBconst -
  Intrinsic galaxy alignments can give something
  else
• Lensing of CMB polarization - white lenses to
  white CE ? CE(14lt?2gt), CB ? CB(14lt?2gt) -
  c.f. shape noise per arcminute number
  density of galaxies depends locally on
  magnification - c.f. effect of magnification on
  intrinsic alignment power spectrum

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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
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Large scale lensing reconstruction

• As with galaxy lensing, ellipticities of hot and
  cold spots can be used to constrain the lensing
  potential
• But diffuse, so 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

Define quadratic estimator
Maximise signal to noise, write in real space
For more details see astro-ph/0105424 or review
69

• Method is potentially useful but not optimal
• Limited by cosmic variance on T, other
  secondaries, higher order terms
• 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- in principle
  can do things almost exactly a lot of
  information in lensed B at high l
• Maximum likelihood techniques much better than
  quadratic estimators for polarization
  (HirataSeljak papers)

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Quadratic (filtered)
Approx max likelihood
Input
astro-ph/0306354
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Lensing potential power spectrum
Hu astro-ph/0108090
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Cluster CMB lensinge.g. to constrain cosmology
via number counts
Lewis King, astro-ph/0512104
Following Seljak, Zaldarriaga, Dodelson, Vale,
Holder, 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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Optimistic Futuristic CMB polarization lensing vs
galaxy lensingLess massive case M 2 x 1014
h-1 Msun, c5
CMB polarization only (0.07 µK arcmin noise)
Galaxies (500 gal/arcmin2)
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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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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
80
Extra information in lensing
Unlensed CMB has many degeneracies e.g. distance
and curvature
flat
closed
?
?
Lensing introduces additional information growth
and scale of lensing deflection power break
degeneracies- e.g. improve constraints on
curvature, dark energy, neutrino mass
81

• 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
• More information can be obtained from
  non-Gaussian signature lensing reconstruction-
  may be able to probe neutrino masses 0.04eV
  (must be there!)

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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-
  Non-Gaussian signal- Generally well understood,
  modelled accurately in linear theorywith small
  non-linear corrections
• Potential uses- Break parameter degeneracies,
  improve parameter constraints- Constrain
  cluster masses at high redshift- Reconstruction
  of potential to z7