Weak Lensing of the CMBAntony Lewis

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

• From the beginning• Lensing order of magnitudes• Lensed power spectrum• Effect on CMB polarization• Cluster masses from CMB lensing

Outline

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

Evolution of the universe

Opaque

Transparent

Source: NASA/WMAP Science Team

O(10-5) perturbations (+galaxy)

Dipole (local motion)

(almost) uniform 2.726K blackbody

Observations:the microwave sky today

Where do perturbations come from?

Quantum Mechanics“waves in a box” calculation

vacuum state, etc…

Inflationmake >1030 times bigger

After inflationHuge size, amplitude ~ 10-5

New physics Known physics

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

Weak lensing of the CMB

Last scattering surface

Inhomogeneous universe - photons deflected

Observer

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

Full calculation: 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

LensPix sky simulation code:http://cosmologist.info/lenspixLewis 2005

Lensing effect on CMB temperature power spectrum

Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/0502425

Planck (2007+) parameter constraint simulation (neglect non-Gaussianity of lensed field)

Important effect, but using lensed CMB power spectrum gets ‘right’ answer

Lewis 2005, astro-ph/0502469

Thomson Scattering Polarization

W Hu

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 tensor

E and B polarization

“gradient” modesE polarization

“curl” modes B polarization

e.g.

B modes only expected from gravitational waves and CMB lensing

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

Polarization lensing: CB

Nearly white BB spectrum on large scales

Lensing effect can be largely subtracted if only scalar modes + lensing present, but approximate and complicated (especially posterior statistics).Hirata, Seljak : astro-ph/0306354, Okamoto, Hu: astro-ph/0301031

Lewis, Challinor : astro-ph/0601594

Polarization lensing: Cx and CE

Lewis, Challinor : astro-ph/0601594

Primordial Gravitational Waves

• 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 - significant power only at l<100, cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc)

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

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

Polarization power spectra

Lewis, Challinor : astro-ph/0601594

Cluster CMB lensing

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

Toy model: spherically symmetric NFW cluster

)()(

vrcrr

Ar

M200 ~ 1015 h-1 Msun

c ~ 5, z ~ 1 (rv ~ 1.6Mpc)Deflection ~ 0.7 arcmin

(approximate lens as thin, constrain projected density profile)

assume we know where centre is

2

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

Constraining cluster parameters

Calculate P(c,M200 | observation)

Simulated realisations with noise 0.5 μK arcmin, 0.5 arcmin pixelsSomewhat futuristic: 160x lower noise 14x higher resolution than Planck; few times better than ACT

CMB approximately Gaussian – know likelihood function

Unlensed T+Q+U Difference after cluster lensing

Add polarization observations?

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

Plus side: SZ (etc) fractional confusion limit probably about the same as temperature

0.5 μK arcmin 0.7 μK arcmin 0.07 μK arcmin

Temperature Polarisation Q and U

Noise:

less dispersion in error

Is it better than galaxy lensing?

• Assume galaxy shapes random before lensing• Measure ellipticity after lensing

Lensing

• On average ellipticity measures reduced shear

• Shear is γab = ∂<a αb>

• Constrain cluster parameters from predicted shear

Galaxy lensing comparisonMassive case: M = 1015 h-1 Msun, c=5

CMB temperature only (0.5 μK arcmin noise) Galaxies (100 gal/arcmin2)

(from expected log likelihoods)

Ground (30/arcmin)

CMB temperature only (0.07 μK arcmin noise)

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

Galaxies (500 gal/arcmin2)

CMB 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)- Kinetic SZ (higher order)- Other lenses

Generally much cleaner

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

Observable Dipoles• Change of velocity:

- Doppler change to total CMB dipole- aberration of observed angles (c.f. dipole convergence)

• Can observe: actual CMB dipole: (non-linear) local motion + primordial contribution

• Can observe: Dipole aberration (dipole convergence + kinetic aberration)

• So: Lensing potential dipole ‘easily’ observable to O(10-5)

- Can find zero-aberration frame to O(10-5) by using zero total CMB-dipole frame

- change of frame corresponds to adding some local kinematic angular aberration to convergence dipole

- zero kinematic aberration and zero kinematic CMB dipole frame = Newtonian gauge

Convergence dipole expected ~ 5 x 10-4

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

- changes power spectra

- potential confusion with primordial gravitational waves

• Cluster lensing of CMB

- gravitational lensing so direct probe of mass (not just baryons)

- mass constraints independent of galaxy lensing constraints; source redshift known very accurately, should win for high redshifts

- galaxy lensing expected to be much better for low redshift clusters

- polarisation lensing needs high sensitivity but cleaner and less sample variance than temperature

Physics Reports review: astro-ph/0601594

http://CosmoCoffee.infoarXiv paper filtering, discussion and comments

Currently 420 registered readers

Calculate Cl by series expansion in deflection angle?

Series expansion only good on large and very small scalesAccurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005No

arXivJournal.org

Is this right?• Lieu, Mittaz, ApJ L paper: astro-ph/0409048

- Claims shift in CMB peaks inconsistent with observation - ignores effect of matter. c.f. Kibble, Lieu: astro-ph/0412275

• Lieu, Mittaz, ApJ paper:astro-ph/0412276Claims large dispersion in magnifications, hence peaks washed out

- Many lines of sight do get significant magnification - BUT CMB is very smooth, small scale magnification unobservable - BUT deflection angles very small - What matter is magnifications on CMB acoustic scales i.e. deflections from large scale coherent perturbations. This is small. - i.e. also wrong

• Large scale potentials < 10-3 : expect rigorous linear argument to be very accurate (esp. with non-linear corrections)