Weak Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge

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

Hu & White, Sci. Am., (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… Inflation make >10 30 times bigger After inflation Huge size, amplitude ~ New physicsKnown physics

Perturbation evolution – what we actually observe CMB monopole source till 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., (2004) CMB temperature power spectrum Primordial perturbations + later physics diffusion damping acoustic oscillations primordial power spectrum 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 β ~ Characteristic size from peak of matter power spectrum ~ 300Mpc Comoving distance to last scattering surface ~ MPc pass through ~50 lumps assume uncorrelated total deflection ~ 50 1/2 x ~ 2 arcminutes (neglects angular factors, correlation, etc.) (β << 1)

So why does it matter? 2arcmin: ell ~ 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

Deflections O(10 -3 ), but coherent on degree scales important! Deflection angle power spectrum Computed with CAMB:

LensPix sky simulation code: Lewis

Lensing effect on CMB temperature power spectrum Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/

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/

Thomson Scattering Polarization W Hu

CMB Polarization Generated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution Hu astro-ph/

Polarization: Stokes Parameters - - QU 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 modes E 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: 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/ , Okamoto, Hu: astro-ph/ Lewis, Challinor : astro-ph/

Polarization lensing: C x and C E Lewis, Challinor : astro-ph/

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/

Cluster CMB lensing GALAXY CLUSTER 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/ degrees

Toy model: spherically symmetric NFW cluster M 200 ~ 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 2

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

Constraining cluster parameters 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 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 arcmin0.07 μK arcmin TemperaturePolarisation 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 γ ab = Constrain cluster parameters from predicted shear

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

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

Rest frame of CMB: Redshifted colder Blueshifted hotter Moving Lenses and Dipole lensing Homogeneous CMB Rest frame of lens:Dipole gradient in CMB Deflected from colderdeflected from hotter v T = T 0 (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/

arXiv paper filtering, discussion and comments Currently 420 registered readers

Calculate C l by series expansion in deflection angle? Series expansion only good on large and very small scales Accurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005 No

arXivJournal.org

Is this right? Lieu, Mittaz, ApJ L paper: astro-ph/ Claims shift in CMB peaks inconsistent with observation - ignores effect of matter. c.f. Kibble, Lieu: astro-ph/ Lieu, Mittaz, ApJ paper:astro-ph/ 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 < : expect rigorous linear argument to be very accurate (esp. with non-linear corrections)