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Weak Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/

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

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

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

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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 ~ 10 -5 New physicsKnown physics

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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

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

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Weak lensing of the CMB Last scattering surface Inhomogeneous universe - photons deflected Observer

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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 ~ 50 1/2 x 10 -4 ~ 2 arcminutes (neglects angular factors, correlation, etc.) (β << 1)

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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

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Full calculation: Lensed temperature depends on deflection angle: Lensing Potential Deflection angle on sky given in terms of lensing potential

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

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LensPix sky simulation code: http://cosmologist.info/lenspix Lewis 2005 http://cosmologist.info/lenspix

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Lensing effect on CMB temperature power spectrum Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/0502425

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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

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Thomson Scattering Polarization W Hu

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CMB Polarization Generated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution Hu astro-ph/9706147

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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

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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

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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

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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

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Polarization lensing: C x and C E Lewis, Challinor : astro-ph/0601594

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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

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

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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/0512104 0.1 degrees

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Toy model: spherically symmetric NFW cluster 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 2

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

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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

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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

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0.5 μK arcmin 0.7 μK arcmin0.07 μK arcmin TemperaturePolarisation Q and U Noise: less dispersion in error

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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

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Galaxy lensing comparison Massive case: M = 10 15 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)

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

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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

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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

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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

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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

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Convergence dipole expected ~ 5 x 10 -4

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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

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Physics Reports review: astro-ph/0601594

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http://CosmoCoffee.info arXiv paper filtering, discussion and comments Currently 420 registered readers

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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

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arXivJournal.org

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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)

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