Lensing of the CMBAntony Lewis

Institute of Astronomy, Cambridgehttp://cosmologist.info/

Physics Reports review: astro-ph/0601594

Talk based on recent review – this is recommended reading and fills in missing details and references :

• 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 Hu’s pages athttp://background.uchicago.edu/~whu/

Anthony Challinor’s CMB introduction: astro-ph/0403344

• Review of unlensed CMB• Lensing order of magnitudes• Lensed power spectrum

• CMB polarization• Non-Gaussianity• Cluster lensing• Moving lenses• Reconstructing the potential• Cosmological parameters

Outline

Hu & White, Sci. Am., 290 44 (2004)

Evolution of the universe

Opaque

Transparent

Observation as a function of frequencyBlack body spectrum observed by COBE

- close to thermal equilibrium: temperature today of 2.726K ( ~ 3000K at z ~ 1000 because ν ~ (1+z))

Residuals Mather et al 1994

Source: NASA/WMAP Science Team

O(10-5) perturbations (+galaxy)

Dipole (local motion)

(almost) uniform 2.726K blackbody

Observations:the microwave sky today

Can we predict the primordial perturbations?

• Maybe..

Quantum Mechanics“waves in a box” calculation

vacuum state, etc…

Inflationmake >1030 times bigger

After inflationHuge size, amplitude ~ 10-5

Perturbation evolution

photon/baryon plasma + dark matter, neutrinos

Characteristic scales: sound wave travel distance; diffusion damping length

Observed ΔT as function of angle on the sky

CMB power spectrum Cl

• Use spherical harmonics Ylm

2|| lml aCTheory prediction

- variance (average over all possible sky realizations)- statistical isotropy implies independent of m

*lmlm YTda

Observe:

Hu & White, Sci. Am., 290 44 (2004)

CMB temperature power spectrumPrimordial perturbations + later physics

diffusiondampingacoustic oscillations

primordial powerspectrum

finite thickness

Perturbations O(10-5)

Simple linearized equations are very accurate (except small scales)

Fourier modes evolve independently: simple to calculate accurately

Calculation of theoretical perturbation evolution

•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)

Physics Ingredients

ClInitial conditions + cosmological parameters

linearized GR + Boltzmann equations

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.

Temperature anisotropy data: WMAP 3-year + smaller scales

BOOMERANG

Hinshaw et al

+ many more coming up

e.g. Planck (2008)

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 stuffMatter densities (Ωb, Ωcdm);; neutrino mass(details of peak shapes, amount of small scale damping)

• Geometry and topologyglobal curvature ΩK 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 < 50 large scale power; polarization)

• AstrophysicsS-Z effect (clusters), foregrounds, etc.

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

Lensing of the CMB

Last scattering surface

Inhomogeneous universe - photons deflected

Observer

14 000 Mpc

Not to scale!All distances are comoving

~100Mpc

~200/14000 ~ degree

largest overdensity

Neutral gas - transparent

Ionized plasma - opaque

Good approximation: CMB is single source plane at ~14 000 Mpc

Recom

bination

~200Mpc

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

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

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

CMB lensing order of magnitudes

β

Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ

Ψ

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 ~ 14000 Mpc

pass through ~50 lumps

assume uncorrelated

total deflection ~ 501/2 x 10-4

~ 2 arcminutes

(neglects angular factors, correlation, etc.)

(β << 1)

(set c=1)

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!

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

Lensed temperature depends on deflection angle

Lensing PotentialDeflection angle on sky given in terms of angular gradient of lensing potential

co-moving distance to last scattering

c.f. introductory lectures

Newtonian potential

Power spectrum of the lensing potential

• Expand Newtonian potential in 3D harmonics

with power spectrum

Angular correlation function of lensing potential:

Use jl are spherical Bessel functions

Orthogonality of spherical harmonics (integral over k) then gives

Then take spherical transform using

Gives final general result

Deflections O(10-3), but coherent on degree scales important!

Deflection angle power spectrum

Computed with CAMB: http://camb.info

Linear

Non-linear

On small scales (Limber approx)

Deflection angle power ~

Lensing potential and deflection anglesLensPix sky simulation code: http://cosmologist.info/lenspix

• 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

Correlation with the CMB temperature

very small except on largest scales

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

• Fourier transforms:

• Statistical isotropy:

Unlensed temperature field in flay sky approximation

So where

Similarly for the lensing potential (also assumed Gaussian and statistically isotropic)

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

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

Lensing effect on CMB temperature power spectrum

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

Accurate calculation -lensed correlation function

• Do not perform series expansion

Assume uncorrelated

Lensed correlation function:

To calculate expectation value use

where

Have defined:

- variance of the difference of deflection angles

small correction from transverse differences

So lensed correlation function is

Expand exponential using

Integrate over angles gives final result:

Note exponential: non-perturbative in lensing potential

Power spectrum and correlation function related by

used Bessel functions defined by

Can be generalized to fully spherical calculation: see review, astro-ph/0601594However flat sky accurate to <~ 1% on the lensed power spectrum

Series expansion in deflection angle?

Series expansion only good on large and very small scales

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

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

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

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

Thomson Scattering Polarization

W Hu

CMB PolarizationGenerated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution

Hu astro-ph/9706147

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

Alternative complex representation

Define complex vectors

And complex polarization

e.g.

Under a rotation of the basis vectors

- spin 2 field

all just like the shear in galaxy lensing

E and B polarization

“gradient” modesE polarization

“curl” modes B polarization

e.g.

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/9609170Kamionkowski, Kosowsky, Stebbins: astro-ph/9611125

On the flat sky spin-2 harmonics are

Inverse relations:

Factors of rotate polarization to physical frame defined by wavenumber l

l

x

y

Polarization Qxy=-1, Uxy=0Pxy = -1

in bases wrt (rotated by –φ)Ql = 0, Ul = 1Pl = i

Pl = Pxy e-2iφ

-φ

CMB Polarization Signals

Average over possible realizations (statistically isotropic):

• E polarization from scalar, vector and tensor modes

• B polarization only from vector and tensor modes (curl grad = 0) + non-linear scalars

Expected signal from scalar modes

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

Look at CMB polarization: ‘B-mode’ smoking gun

- cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc)

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

Last scattering Observed

e.g.

~ ellipticities of infinitesimal small galaxies

Lensed spectrum: lowest order calculationSimilar 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

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

Polarization lensing B mode power spectra

Nearly white spectrum on large scales(power spectrum independent of l)

l4Clφ

l4Clφ l2Cl

E

ClB

ClE

On small scales,

lensed BB given by

BB generated by lensing even if unlensed B=0

Can also do more accurate calculationusing polarization correlation functions

Current 95% indirect limits for LCDM given WMAP+2dF+HST

Polarization power spectra

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

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

…

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

Bigger than primordial non-Gaussianity?

• 1-point function

- 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

- ISW-lensing correlation only significant on very large scales

• Bispectrum

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

• Trispectrum (4-point)

Basic inflation:- most signalin long thin quadrilaterals

Lensing:- broader distribution, lesssignal in thin shapes

Can only detect inflation signal from cosmic variance if fNL >~ 20

Komatsu: astro-ph/0602099 Hu: astro-ph/0105117

No analysis of relative shape-dependence from e.g. curvaton??

Lensing probably not main problem for flat quadrilaterals if single-field non-Gaussianity

Also non-Gaussianity in polarization…

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

• Can show that

Define quadratic estimator

Maximise signal to noise, write in real space:

For more details see astro-ph/0105424 or review

• 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 <~ 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 (Hirata&Seljak papers)

astro-ph/0306354

Input Quadratic (filtered) Approx max likelihood

Lensing potential power spectrum

Hu: astro-ph/0108090

Cluster CMB lensinge.g. to constrain cosmology via number counts

GALAXYCLUSTER

Last scattering surface What we see

Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, etc.

CMB very smooth on small scales: approximately a gradient

Lewis & King, astro-ph/0512104

0.1 degrees

Need sensitive ~ arcminute resolution observations

Unlensed Lensed Difference

RMS gradient ~ 13 μK / arcmindeflection from cluster ~ 1 arcmin Lensing signal ~ 10 μK

BUT: depends on CMB gradient behind a given cluster

can compute likelihood of given lens (e.g. NFW parameters) essentially exactly

Unlensed CMB unknown, but statistics well understood (background CMB Gaussian) :

Unlensed T+Q+U Difference after cluster lensing

Add polarization observations?

Less sample variance – but signal ~10x smaller: need 10x lower noise

Note: E and B equally useful on these scales; gradient could be either

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

CMB polarization only (0.07 μK arcmin noise)

Optimistic Futuristic CMB polarization lensing vs galaxy lensingLess massive case: M = 2 x 1014 h-1 Msun, c=5

Galaxies (500 gal/arcmin2)

Rest frame of CMB:

Redshiftedcolder

Blueshiftedhotter

Moving Lenses and Dipole lensing

Homogeneous CMB

Rest frame of lens: Dipole gradient in CMB

Deflected from colderdeflected from hotter

v

T = T0(1+v cos θ)

‘Rees-Sciama’(non-linear ISW)

‘dipole lensing’

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

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

Cosmological parametersEssential to model lensing; but little effect on basic parameter constraints

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

θ

θ

• 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 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!)

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 <~ 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 z~7