Weak Lensing of the CMB

Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/

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

Evolution of the universe Opaque Transparent Hu & White, Sci. Am., 290 44 (2004)

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

Where do perturbations come from?

New physics Known physics

Inflation

make >10 30 times bigger

Quantum Mechanics

“waves in a box” calculation vacuum state, etc…

After inflation

Huge size, amplitude ~ 10 -5

Perturbation evolution – what we actually observe CMB 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

CMB temperature power spectrum Primordial perturbations + later physics acoustic oscillations diffusion damping Hu & White, Sci. Am., 290 44 (2004) finite thickness primordial power spectrum

Weak lensing of the CMB

Last scattering surface Inhomogeneous universe - photons deflected Observer

Lensing order of magnitudes

Ψ β Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ Potentials linear and approx Gaussian: Ψ ~ 2 x 10 -5 β ~ 10 -4 ( β << 1) Characteristic size from peak of matter power spectrum ~ 300Mpc Comoving distance to last scattering surface ~ 14000 MPc pass through ~50 lumps total deflection ~ 50 1/2 x 10 -4 assume uncorrelated ~ 2 arcminutes (neglects angular factors, correlation, etc.)

So why does it matter?

• 2arcmin: ell ~ 3000 - o n 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 Potential

Deflection angle on sky given in terms of lensing potential

Deflection angle power spectrum Deflections O(10 -3 ), but coherent on degree scales  important!

Computed with CAMB: http://camb.info

LensPix sky simulation code: http://cosmologist.info/lenspix Lewis 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 Polarization

Generated 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

e.g.

“gradient” modes E polarization “curl” modes B polarization e.g. cold spot

B modes only expected from gravitational waves and CMB lensing

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: C

B 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: C

x

and C

E 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

Polarization power spectra Current 95% indirect limits for LCDM given WMAP+2dF+HST Lewis, Challinor : astro-ph/0601594

Cluster CMB lensing

Lewis & King, astro-ph/0512104 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

Toy model: spherically symmetric NFW cluster

 (

r

) 

r

(

cr A



r v

) 2 M 200 ~ 10 15 h -1 M sun c ~ 5, z ~ 1 (r v ~ 1.6Mpc) Deflection ~ 0.7 arcmin (approximate lens as thin, constrain projected density profile) assume we know where centre is

RMS gradient ~ 13 μK / arcmin deflection from cluster ~ 1 arcmin Lensing signal ~ 10 μK Unlensed BUT: depends on CMB gradient behind a given cluster Lensed Difference

Constraining cluster parameters

CMB approximately Gaussian – know likelihood function Calculate P(c,M 200 | observation) Simulated realisations with noise 0.5 μK arcmin, 0.5 arcmin pixels Somewhat futuristic: 160x lower noise 14x higher resolution than Planck; few times better than ACT

Add polarization observations?

Unlensed T+Q+U Difference after cluster lensing Less sample variance – but signal ~10x smaller: need 10x lower noise Plus side: SZ (etc) fractional confusion limit probably about the same as temperature

Temperature Noise: 0.5 μK arcmin Polarisation Q and U 0.7 μK arcmin 0.07 μK arcmin 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 γ ab = ∂ • Constrain cluster parameters from predicted shear

Galaxy lensing comparison

Massive case: M = 10 15 h -1 M sun, c=5 (from expected log likelihoods) Ground (30/arcmin) CMB temperature only (0.5 μK arcmin noise) Galaxies (100 gal/arcmin 2 )

Optimistic Futuristic CMB polarization vs galaxy lensing Less massive case: M = 2 x 10 14 h -1 M sun , c=5 CMB temperature only (0.07 μK arcmin noise) Galaxies (500 gal/arcmin 2 )

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

Moving Lenses and Dipole lensing

Rest frame of CMB:

Homogeneous CMB `Rees Sciama’ (non-linear ISW)

v Rest frame of lens:

Blueshifted hotter Redshifted colder Dipole gradient in CMB ‘dipole lensing’ deflected from hotter Deflected from colder T = T 0 (1+v cos θ)

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.info

arXiv paper filtering, discussion and comments Currently 420 registered readers

Calculate C l by series expansion in deflection angle?

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

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/0412276 Claims 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)