Lensing of the CMBAntony Lewis

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

Review ref: Lewis, Challinor , Phys. Rep: astro-ph/0601594

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

Evolution of the universe

Opaque

Transparent

Perturbation evolution – what we actually observeCMB monopole source till 380 000 yrs (last scattering), linear in conformal time

scale invariant primordial adiabatic scalar spectrum

photon/baryon plasma + dark matter, neutrinos

Characteristic scales: sound wave travel distance; diffusion damping length

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

CMB temperature power spectrumPrimordial perturbations + later physics

diffusiondampingacoustic oscillations

primordial powerspectrum

finite thickness

Temperature anisotropy data: WMAP 3-year + smaller scales

BOOMERANG

Hinshaw et al

+ many more coming up

e.g. Planck (2008)

Weak 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 MpcAngular diameter distance well measured by angle of acoustic peaks

Recom

bination

~200Mpc

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)

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

In detail, lensed temperature depends on deflection angle:

Lensing PotentialDeflection angle on sky given in terms of lensing potential

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

Deflection angle power spectrum

Computed with CAMB: http://camb.info

Linear

Non-linear

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

Lensed temperature Cl

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

- convolution of unlensed Cl

- W is non-linear in lensing potential power

Lensing effect on CMB temperature power spectrum

CAMB’s 0.1% calculation; http://camb.info : Challinor & Lewis 2005, astro-ph/0502425

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

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

Hu astro-ph/9706147

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

E and B polarization

“gradient” modesE polarization

“curl” modes B polarization

e.g.

e.g. cold spot

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

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.

Polarization lensing: Cx and CE

Polarization lensing: CB

Nearly white BB spectrum on large scales

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

Polarization power spectra

Lewis, Challinor : astro-ph/0601594

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…

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 spectrum

For multiple redshift bins can generalise for correlated fields:

X= (k11,k22,k12,…)

for details see Hammimeche & Lewis (in prep).

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

• Can show that ‘optimal’ quadratic estimator is

- simple function of filtered fields

For more details see Hu astro-ph/0105424 or review; c.f. Metcalf & White 2007

Analogous results for CMB polarization

e.g. estimate lensing potential power spectrum- more information on cosmological parameters

Hu: astro-ph/0108090 (‘ideal’ is limit using non-optimal quadratic estimator)

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)

First claimed detection in cross-correlation (see talk by Olivier Doré)

(http://cosmocoffee.info discussion)

• Limited by cosmic variance on T, other secondaries, higher order terms

• Quadratic method useful but not optimal-especially for polarization (Hirata&Seljak 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 <~ 10-4

- requires very high sensitivity and high resolution

Reconstruction complications

astro-ph/0306354

Input Quadratic (filtered) Approx max likelihood

• 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/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)

Other information in CMB lensing (>> arcminute)

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

GALAXYCLUSTER

Last scattering surface What we see

Seljak, Zaldarriaga, Dodelson, Vale, Holder, Lewis, King, Hu. Maturi,. etc.

CMB very smooth on small scales: approximately a gradient

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 lensinge.g. M = 2 x 1014 h-1 Msun, c=5

Galaxies (500 gal/arcmin2)

Lewis & King 2006

Fitting profiles. e.g. to measure mass and concentration

Can stackfor constraintsfrom multipleclusters

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

CMB/Galaxy lensing comparisonCMB Lensing

- single source plane, lenses 0.5<~z<~7

- 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 <~1.5

- 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

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)

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

Cl(z=50,z=52)Cl(z=50,z=50)

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

- 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 <~ z <~ 7

Correlation with the CMB temperature

very small except on largest scales

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

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

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…