1. Institute of Astronomy, Cambridge
Lensing of the CMB
Antony Lewis
Institute of Astronomy, Cambridge

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

3. Recent papers of interest (since review): - Cosmological Information from Lensed CMB Power Spectra; Smith et al. astro-ph/
For introductory material on unlensed CMB, esp. polarization see Wayne Hu’s pages at Anthony Challinor’s CMB introduction: astro-ph/

4. 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

5. Evolution of the universe
Opaque
Transparent
Hu & White, Sci. Am., (2004)

6. Observation as a function of frequency Black body spectrum observed by COBE
Residuals Mather et al 1994
- close to thermal equilibrium: temperature today of 2.726K ( ~ 3000K at z ~ 1000 because ν ~ (1+z))

7. Observations: the microwave sky today
(almost) uniform 2.726K blackbody
Dipole (local motion)
O(10-5) perturbations (+galaxy)
Observations:
the microwave
sky today
Source: NASA/WMAP Science Team

8. Can we predict the primordial perturbations?
Maybe..
Inflation
make >1030 times bigger
Quantum Mechanics “waves in a box” calculation vacuum state, etc…
After inflation
Huge size, amplitude ~ 10-5

9. Perturbation evolution
photon/baryon plasma + dark matter, neutrinos
Characteristic scales: sound wave travel distance; diffusion damping length

10. Observed ΔT as function of angle on the sky

11. CMB power spectrum Cl Use spherical harmonics Ylm Observe:
Theory prediction
- variance (average over all possible sky realizations) - statistical isotropy implies independent of m

12. CMB temperature power spectrum Primordial perturbations + later physics
diffusion
damping
acoustic oscillations
primordial power spectrum
finite thickness
Hu & White, Sci. Am., (2004)

13. linearized GR + Boltzmann equations
Calculation of theoretical perturbation evolution
Perturbations O(10-5)
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

14. 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 shift Integrated Sachs Wolfe: Evolution of potential along photon line of sight: net red- or blue-shift as photon climbs in an out of varying potential wells Others:
Photon quadupole/polarization at last scattering, second-order effects, etc.

15. Temperature anisotropy data: WMAP 3-year + smaller scales
BOOMERANG
Hinshaw et al
+ many more coming up
e.g. Planck (2008)

16. What can we learn from the CMB?
Initial conditions What types of perturbations, power spectra, distribution function (Gaussian?); => learn about inflation or alternatives. (distribution of ΔT; power as function of scale; polarization and correlation)
What and how much stuff Matter densities (Ωb, Ωcdm);; neutrino mass (details of peak shapes, amount of small scale damping)
Geometry and topology global curvature ΩK of universe; topology (angular size of perturbations; repeated patterns in the sky)
Evolution Expansion rate as function of time; reionization - Hubble constant H0 ; dark energy evolution w = pressure/density (angular size of perturbations; l < 50 large scale power; polarization)
Astrophysics S-Z effect (clusters), foregrounds, etc.

17. CMB summary
Time from big bang to last scattering (~300Mpc comoving; ~ 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

18. Lensing of the CMB Last scattering surface Inhomogeneous universe
- photons deflected
Observer

19. Not to scale! All distances are comoving largest overdensity
~200/14000 ~ degree
Neutral gas - transparent
Ionized plasma - opaque
Recombination
~200Mpc
Mpc
~100Mpc
Good approximation: CMB is single source plane at ~ Mpc

20. 1st order effects 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

21. 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

22. 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 reshifting SZ: second (+higher) order scattering, etc….
Hu, Cooray: astro-ph/

23. CMB lensing order of magnitudes
(set c=1) Ψ β
Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ
(β << 1)
Potentials linear and approx Gaussian: Ψ ~ 2 x 10-5
β ~ 10-4
Characteristic size from peak of matter power spectrum ~ 300Mpc
Comoving distance to last scattering surface ~ Mpc
pass through ~50 lumps
total deflection ~ 501/2 x 10-4
~ 2 arcminutes
assume uncorrelated
(neglects angular factors, correlation, etc.)

24. So why does it matter?
2arcmin: ell ~ 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!

25. 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

26. Lensing Potential 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

27. Power spectrum of the lensing potential
Expand Newtonian potential in 3D harmonics
with power spectrum
Angular correlation function of lensing potential:

28. jl are spherical Bessel functions
Use
Orthogonality of spherical harmonics (integral over k) then gives
Then take spherical transform using
Gives final general result

29. 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:

30. Lensing potential and deflection angles
LensPix sky simulation code:

31. 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

32. Correlation with the CMB temperature
very small except on largest scales

33. 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

34. Statistical isotropy:
Unlensed temperature field in flay sky approximation
Fourier transforms:
Statistical isotropy:
where So
Similarly for the lensing potential (also assumed Gaussian and statistically isotropic)

35. 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

36. 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

37. Lensing effect on CMB temperature power spectrum

38. 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

39. Accurate calculation - lensed correlation function
Do not perform series expansion
Lensed correlation function:
Assume uncorrelated
To calculate expectation value use

40. where
Have defined:
small correction from transverse differences
- variance of the difference of deflection angles

41. So lensed correlation function is
Expand exponential using
Integrate over angles gives final result:
Note exponential: non-perturbative in lensing potential

42. Power spectrum and correlation function related by
used Bessel functions defined by
Can be generalized to fully spherical calculation: see review, astro-ph/
However flat sky accurate to <~ 1% on the lensed power spectrum

43. 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

44. Other specific non-linear effects
Thermal Sunyaev-Zeldovich Inverse 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-Sciama Integrated Sachs-Wolfe from evolving non-linear potentials: frequency independent

45. 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

46. Lensing important at 500<l<3000 Dominated by SZ on small scales

47. Thomson Scattering Polarization
W Hu

48. CMB Polarization
Generated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution
Hu astro-ph/

49. 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 n Intensity matrix defined as
Linear polarization + Intensity + circular polarization
CMB only linearly polarized. In some fixed basis

50. 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

51. E and B polarization “gradient” modes E polarization
“curl” modes B polarization
e.g.

52. E and B harmonics Harmonics are orthogonal over the full sky:
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/
Kamionkowski, Kosowsky, Stebbins: astro-ph/

53. On the flat sky spin-2 harmonics are
Inverse relations:
Factors of
rotate polarization to physical frame defined by wavenumber l

54. l
Polarization Qxy=-1, Uxy=0
Pxy = -1
in bases wrt
(rotated by –φ)
Ql = 0, Ul = 1 Pl = i
Pl = Pxy e-2iφ -φ y x

55. 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

56. 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

57. 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.
Last scattering
Observed
~ ellipticities of infinitesimal small galaxies

58. 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

59. Effect on EE and TE similar to temperature: convolution smoothing + transfer of power to small scales

60. Polarization lensing B mode power spectra
BB generated by lensing even if unlensed B=0
On small scales,
ClE
lensed BB given by
ClB
Nearly white spectrum on large scales (power spectrum independent of l)
l4Clφ
l4Clφ l2ClE
Can also do more accurate calculation using polarization correlation functions

61. Polarization power spectra
Current 95% indirect limits for LCDM given WMAP+2dF+HST

62. 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 source Q/U not changed by gravitational shear along path
CMB polarization at last scattering - galaxy shape distribution in source plane - usually assume shapes uncorrelated ~ CE=CB=const - Intrinsic galaxy alignments can give something else
Lensing of CMB polarization - white lenses to white CE → CE(1+4<κ2>), CB → CB(1+4<κ2>) - c.f. shape noise per arcminute: number density of galaxies depends locally on magnification - c.f. effect of magnification on intrinsic alignment power spectrum

63. 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 < T T T >
- Zero unless correlation <T Ψ>
Large scale signal from ISW-induced T- Ψ correlation
Small scale signal from non-linear SZ – Ψ correlation

64. Trispectrum: Connected four-point < T T T T>c
Depends on deflection angle and temperature power spectra
‘Easily’ measurable for accurate ell > 1000 observations
Other signatures
correlated hot-spot ellipticities
Higher n-point functions
Polarization non-Gaussianity

65. 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/

66. Trispectrum (4-point)
Basic inflation: - most signal in long thin quadrilaterals
Lensing: - broader distribution, less signal in thin shapes
Komatsu: astro-ph/
Hu: astro-ph/
Can only detect inflation signal from cosmic variance if fNL >~ 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…

67. 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

68. Can show that Define quadratic estimator
Maximise signal to noise, write in real space:
For more details see astro-ph/ 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 <~ 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 (Hirata&Seljak papers)

70. Input
Quadratic (filtered)
Approx max likelihood
astro-ph/

71. Lensing potential power spectrum
Hu: astro-ph/

72. Cluster CMB lensing e.g. to constrain cosmology via number counts
Lewis & King, astro-ph/
Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, etc.
CMB very smooth on small scales: approximately a gradient
What we see
Last scattering surface
GALAXY CLUSTER
0.1 degrees
Need sensitive ~ arcminute resolution observations

73. RMS gradient ~ 13 μK / arcmin deflection 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

74. Add polarization observations?
Difference after cluster lensing
Unlensed T+Q+U
Less sample variance – but signal ~10x smaller: need 10x lower noise
Note: E and B equally useful on these scales; gradient could be either

75. 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 (< 0.1μK) - Re-scattered thermal SZ (freq) - Kinetic SZ (higher order) - Other lenses Generally much cleaner

76. CMB polarization only (0.07 μK arcmin noise)
Optimistic Futuristic CMB polarization lensing vs galaxy lensing Less massive case: M = 2 x 1014 h-1 Msun, c=5
CMB polarization only (0.07 μK arcmin noise)
Galaxies (500 gal/arcmin2)

77. Moving Lenses and Dipole lensing
Homogeneous CMB
Rest frame of CMB:
‘Rees-Sciama’ (non-linear ISW) v
Blueshifted hotter
Redshifted colder
Rest frame of lens:
Dipole gradient in CMB
T = T0(1+v cos θ)
‘dipole lensing’
deflected from hotter
Deflected from colder

78. 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 frame: generally totally negligible

79. 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 θ θ
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 l<300 - one from lensed BB, wider range of l astro-ph/
More information can be obtained from non-Gaussian signature: lensing reconstruction - may be able to probe neutrino masses ~ 0.04eV (must be there!)

82. 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 <~ Non-Gaussian signal - Generally well understood, modelled accurately in linear theory with small non-linear corrections
Potential uses - Break parameter degeneracies, improve parameter constraints - Constrain cluster masses at high redshift - Reconstruction of potential to z~7