Lensing of the CMB

Antony Lewis

http://cosmologist.info/

Last scattering surface

Inhomogeneous universe

- photons deflected

Observer

CMB Lensing

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 Potential

Deflection 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

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

Planck forecasts for BB

tensors

r=0.05

lensing

Unlensed Magnified Demagnified

Beyond the power spectrum

Bispectrum in ultra-squeezed limit

Lensing potential-temperature correlation Slope of the lensed temperature

power spectrum

Reduced bispectrum

Creminelli & Zaldarriagaa 2004; c.f. Maldacena 2003, Creminelli & Zaldarriaga 2004 for primordial bispectrum

+ similar argument for shear

Why is there a correlation between large-scale

lenses and the temperature?

Overdensity: magnification correlated with positive Integrated Sachs-Wolfe (net blueshift)

Underdensity: demagnification correlated with negative Integrated Sachs-Wolfe (net redshift)

(small-scales: also SZ , Rees-Sciama..)

T T

E E

- Lensing bispectrum depends on power difference: has phase shift

compared to any adiabatic primordial bispectrum (and different scale

dependence)

BUT:

Lensing

CMB temperature lensing

Lensing bispectrum also squeezed triangles but quite distinctive

See Lewis, Challinor & Hanson 2010 (in prep) for details

Lensing field is FIXED:

Anisotropic Gaussian temperature distribution

- Different parts of the sky magnified and demagnified

- Re-construct the actual lensing field

Lensing field is RANDOM:

Non-Gaussian statistically isotropic temperature distribution

- Significant connected 4-point function

- Excess variance to anisotropic-looking realizations

- Lensed temperature power spectrum

We see only one sky - both interpretations can be useful

Trispectrum – or anisotropy?

Anisotropy estimators

Maximum likelihood:

First iteration solution: Quadratic Maximum Likelihood (QML)

Hanson & Lewis, 0908.0963

Hanson, Challinor & Lewis, 0911.0612

Reconstruct statistical anisotropy

in WMAP data?

E.g. primordial power spectrum

anisotropy anomaly?

Look for direction-dependence in primordial power spectrum:

Simple case:

e.g.

Ackerman et.al. astro-ph/0701357

Gumrukcuoglu et al 0707.4179

• Reconstruct g(k).

QML estimator:

Quadrupole primordial

power asymmetry??

Hanson & Lewis, 0908.0963

Very significant evidence for

~ 10% quadrupole angular dependence!

Variance from

simulations

a(k)=1

Dashed: KQ75

Solid: KQ85

WMAP team agree!: Bennet et al. 1001.4758

Direction close to ecliptic!

Fully explained by analytic model of beam asymmetries

can be explained as correlated noise

No detection..

Consistent with Pullen et al 2010

constraint from large-scale structure 1003.0673

Hanson, Lewis & Challinor 1003.0198

Back to CMB lensing..

Use Quadratic Maximum Likelihood estimators to

reconstruct the lensing potential:

.. then estimate lensing potential power spectrum

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

Observed CMB temperature power spectrum

Primordial perturbations + known physics with unknown parameters

Observations Constrain theory of early universe

+ evolution parameters and geometry

Nolta et al.

e.g. Geometry: curvature

flat closed

θ

θ

We see:

or is it just closer??

flat

We see:

θ

Degeneracies between parameters

flat

θ

Dunkley et al. 2009

WMAP 5

Need other information to break

remaining degeneracies

Smith et al. 0811.3916

But lensing provides information over a range of different redshifts:

Already will help with Planck

Perotto et al. 2009

(simulation)

Neutrino mass fraction

with and without

lensing (Planck only)

Perotto et al. 2006

Probe 0.5<~ z <~ 6: depends on geometry and matter power spectrum

Hanson et al. 0911.0612

Fisher limit

with no lensing

Hirata et al astro-ph/0306354

Input Quadratic (filtered) Approx max likelihood

CMB polarization only (0.07 μK arcmin noise)

Optimistic Futuristic CMB polarization lensing vs galaxy lensing

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

Galaxies (500 gal/arcmin2)

Lewis &

King 2006

CMB cluster lensing. e.g. to measure mass and concentration

Can stack

for constraints

from multiple

clusters

14 May 2009

Planck and the future, 2009+

High sensitivity and resolution

CMB temperature and polarization

ACTpol 1006.5049

From 2013

Summary • Weak lensing of the CMB very important for precision cosmology

- changes power spectra at several per cent - potential confusion with primordial gravitational wave B-modes for r <~ 10-3 - introduces non-Gaussian signal; important potential confusion for Planck - well understood in theory – accurately modelled with linear theory + small non-linear corrections

• Potential uses - Break parameter degeneracies, improve parameter constraints : dark energy, neutrino mass, curvature - Reconstruction of potential at 0.5 <~ z <~ 7 - cluster lensing at high redshift