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**Institute of Astronomy, Cambridge**

Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge

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Talk based on recent review – this is recommended reading and fills in missing details and references : Physics Reports review: astro-ph/

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Recent papers of interest (since review): - Cosmological Information from Lensed CMB Power Spectra; Smith et al. astro-ph/ For introductory material on unlensed CMB, esp. polarization see Wayne Hu’s pages at Anthony Challinor’s CMB introduction: astro-ph/

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**Outline Review of unlensed CMB Lensing order of magnitudes**

Lensed power spectrum CMB polarization Non-Gaussianity Cluster lensing Moving lenses Reconstructing the potential Cosmological parameters

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**Evolution of the universe**

Opaque Transparent Hu & White, Sci. Am., (2004)

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**Observation as a function of frequency Black body spectrum observed by COBE**

Residuals Mather et al 1994 - close to thermal equilibrium: temperature today of 2.726K ( ~ 3000K at z ~ 1000 because ν ~ (1+z))

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**Observations: the microwave sky today**

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

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**Can we predict the primordial perturbations?**

Maybe.. Inflation make >1030 times bigger Quantum Mechanics “waves in a box” calculation vacuum state, etc… After inflation Huge size, amplitude ~ 10-5

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**Perturbation evolution**

photon/baryon plasma + dark matter, neutrinos Characteristic scales: sound wave travel distance; diffusion damping length

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**Observed ΔT as function of angle on the sky**

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**CMB power spectrum Cl Use spherical harmonics Ylm Observe:**

Theory prediction - variance (average over all possible sky realizations) - statistical isotropy implies independent of m

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**CMB temperature power spectrum Primordial perturbations + later physics**

diffusion damping acoustic oscillations primordial power spectrum finite thickness Hu & White, Sci. Am., (2004)

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**linearized GR + Boltzmann equations**

Calculation of theoretical perturbation evolution Perturbations O(10-5) Simple linearized equations are very accurate (except small scales) Fourier modes evolve independently: simple to calculate accurately Physics Ingredients Thomson scattering (non-relativistic electron-photon scattering) tightly coupled before recombination: ‘tight-coupling’ approximation (baryons follow electrons because of very strong e-m coupling) Background recombination physics Linearized General Relativity Boltzmann equation (how angular distribution function evolves with scattering) linearized GR + Boltzmann equations Initial conditions + cosmological parameters Cl

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Sources of CMB anisotropy Sachs Wolfe: Potential wells at last scattering cause redshifting as photons climb out Photon density perturbations: Over-densities of photons look hotter Doppler: Velocity of photon/baryons at last scattering gives Doppler shift Integrated Sachs Wolfe: Evolution of potential along photon line of sight: net red- or blue-shift as photon climbs in an out of varying potential wells Others: Photon quadupole/polarization at last scattering, second-order effects, etc.

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**Temperature anisotropy data: WMAP 3-year + smaller scales**

BOOMERANG Hinshaw et al + many more coming up e.g. Planck (2008)

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**What can we learn from the CMB?**

Initial conditions What types of perturbations, power spectra, distribution function (Gaussian?); => learn about inflation or alternatives. (distribution of ΔT; power as function of scale; polarization and correlation) What and how much stuff Matter densities (Ωb, Ωcdm);; neutrino mass (details of peak shapes, amount of small scale damping) Geometry and topology global curvature ΩK of universe; topology (angular size of perturbations; repeated patterns in the sky) Evolution Expansion rate as function of time; reionization - Hubble constant H0 ; dark energy evolution w = pressure/density (angular size of perturbations; l < 50 large scale power; polarization) Astrophysics S-Z effect (clusters), foregrounds, etc.

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CMB summary Time from big bang to last scattering (~300Mpc comoving; ~ years) – determines physical size of largest overdensity (or underdensity) Distance of last scattering from us (~14Gpc comoving; 14 Gyr) - determines angular size seen by us Damping scale (angular size ~ arcminutes) - determines smallest fluctuations (smooth on small scales) Thickness of last scattering (~Hubble time, 100Mpc) - determines line of sight averaging - determines amount of polarization (see later) Other parameters - determine amplitude and scale dependence of perturbations

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**Lensing of the CMB Last scattering surface Inhomogeneous universe**

- photons deflected Observer

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**Not to scale! All distances are comoving largest overdensity**

~200/14000 ~ degree Neutral gas - transparent Ionized plasma - opaque Recombination ~200Mpc Mpc ~100Mpc Good approximation: CMB is single source plane at ~ Mpc

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**1st order effects Zeroth-order CMB**

CMB uniform blackbody at ~2.7 K (+dipole due to local motion) 1st order effects Linear perturbations at last scattering, zeroth-order light propagation; zeroth-order last scattering, first order redshifting during propagation (ISW) - usual unlensed CMB anisotropy calculation First order time delay, uniform CMB - last scattering displaced, but temperature at recombination the same - no observable effect

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1st order effects contd. First order CMB lensing: zeroth-order last scattering (uniform CMB ~ 2.7K), first order transverse displacement in light propagation A B Number of photons before lensing Number of photons after lensing A2 ---- B2 Solid angle before lensing Solid angle after lensing = = Conservation of surface brightness: number of photons per solid angle unchanged uniform CMB lenses to uniform CMB – so no observable effect

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2nd order effects Second order perturbations at last scattering, zeroth order light propagation -tiny ~(10-5)2 corrections to linear unlensed CMB result First order last scattering (~10-5 anisotropies), first order transverse light displacement - this is what we call CMB lensing First order last scattering (~10-5 anisotropies), first order time delay - delay ~1MPc, small compared to thickness of last scattering - coherent over large scales: very small observable effect Others e.g. Rees Sciama: second (+ higher) order reshifting SZ: second (+higher) order scattering, etc…. Hu, Cooray: astro-ph/

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**CMB lensing order of magnitudes**

(set c=1) Ψ β Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ (β << 1) 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 ~ Mpc pass through ~50 lumps total deflection ~ 501/2 x 10-4 ~ 2 arcminutes assume uncorrelated (neglects angular factors, correlation, etc.)

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So why does it matter? 2arcmin: ell ~ 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 NOT because of growth of matter density perturbations!

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**Comparison with galaxy lensing**

Single source plane at known distance (given cosmological parameters) Statistics of sources on source plane well understood - can calculate power spectrum; Gaussian linear perturbations - magnification and shear information equally useful - usually discuss in terms of deflection angle; - magnification analysis of galaxies much more difficult Hot and cold spots are large, smooth on small scales - ‘strong’ and ‘weak’ lensing can be treated the same way: infinite magnification of smooth surface is still a smooth surface Source plane very distant, large linear lenses - lensing by under- and over-densities; Full sky observations - may need to account for spherical geometry for accurate results

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**Lensing Potential Lensed temperature depends on deflection angle**

Newtonian potential co-moving distance to last scattering Lensing Potential Deflection angle on sky given in terms of angular gradient of lensing potential c.f. introductory lectures

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**Power spectrum of the lensing potential**

Expand Newtonian potential in 3D harmonics with power spectrum Angular correlation function of lensing potential:

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**jl are spherical Bessel functions**

Use Orthogonality of spherical harmonics (integral over k) then gives Then take spherical transform using Gives final general result

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**Deflection angle power spectrum**

On small scales (Limber approx) Deflection angle power ~ Non-linear Linear Deflections O(10-3), but coherent on degree scales important! Computed with CAMB:

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**Lensing potential and deflection angles**

LensPix sky simulation code:

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**Note: can only observe lensed field**

Any bulk deflection is unobservable – degenerate with corresponding change in unlensed CMB: e.g. rotation of full sky translation in flat sky approximation Observations sensitive to differences of deflection angles

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**Correlation with the CMB temperature**

very small except on largest scales

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**Calculating the lensed CMB power spectrum**

Approximations and assumptions: - Lensing potential uncorrelated to temperature - Gaussian lensing potential and temperature - Statistical isotropy Simplifying optional approximations - flat sky - series expansion to lowest relevant order

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**Statistical isotropy:**

Unlensed temperature field in flay sky approximation Fourier transforms: Statistical isotropy: where So Similarly for the lensing potential (also assumed Gaussian and statistically isotropic)

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**Lensed field: series expansion approximation**

(BEWARE: this is not a very good approximation! See later) Using Fourier transforms, write gradients as Then lensed harmonics then given by

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**Lensed field still statistically isotropic:**

with Alternatively written as where (RMS deflection ~ 2.7 arcmin) Second term is a convolution with the deflection angle power spectrum - smoothes out acoustic peaks - transfers power from large scales into the damping tail

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**Lensing effect on CMB temperature power spectrum**

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**Small scale, large l limit: - unlensed CMB has very little power due to silk damping:**

- Proportional to the deflection angle power spectrum and the (scale independent) power in the gradient of the temperature

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**Accurate calculation - lensed correlation function**

Do not perform series expansion Lensed correlation function: Assume uncorrelated To calculate expectation value use

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where Have defined: small correction from transverse differences - variance of the difference of deflection angles

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**So lensed correlation function is**

Expand exponential using Integrate over angles gives final result: Note exponential: non-perturbative in lensing potential

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**Power spectrum and correlation function related by**

used Bessel functions defined by Can be generalized to fully spherical calculation: see review, astro-ph/ However flat sky accurate to <~ 1% on the lensed power spectrum

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**Series expansion in deflection angle?**

Only a good approximation when: - deflection angle much smaller than wavelength of temperature perturbation - OR, very small scales where temperature is close to a gradient CMB lensing is a very specific physical second order effect; not accurately contained in 2nd order expansion – differs by significant 3rd and higher order terms Series expansion only good on large and very small scales

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**Other specific non-linear effects**

Thermal Sunyaev-Zeldovich Inverse Compton scattering from hot gas: frequency dependent signal Kinetic Sunyaev-Zeldovich (kSZ) Doppler from bulk motion of clusters; patchy reionization; (almost) frequency independent signal Ostriker-Vishniac (OV) same as kSZ but for early linear bulk motion Rees-Sciama Integrated Sachs-Wolfe from evolving non-linear potentials: frequency independent

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Summary so far Deflection angles of ~ 3 arcminutes, but correlated on degree scales Lensing convolves TT with deflection angle power spectrum - Acoustic peaks slightly blurred - Power transferred to small scales large scales small scales

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**Lensing important at 500<l<3000 Dominated by SZ on small scales**

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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/

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**Observed Stokes’ Parameters**

- - Q U Q → -Q, U → -U under 90 degree rotation Q → U, U → -Q under 45 degree rotation Measure E field perpendicular to observation direction n Intensity matrix defined as Linear polarization + Intensity + circular polarization CMB only linearly polarized. In some fixed basis

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**Alternative complex representation**

e.g. Define complex vectors And complex polarization Under a rotation of the basis vectors - spin 2 field all just like the shear in galaxy lensing

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**E and B polarization “gradient” modes E polarization**

“curl” modes B polarization e.g.

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**E and B harmonics Harmonics are orthogonal over the full sky:**

Expand scalar PE and PB in scalar harmonics Expand P in spin-2 harmonics Harmonics are orthogonal over the full sky: E/B decomposition is exact and lossless on the full sky Zaldarriaga, Seljak: astro-ph/ Kamionkowski, Kosowsky, Stebbins: astro-ph/

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**On the flat sky spin-2 harmonics are**

Inverse relations: Factors of rotate polarization to physical frame defined by wavenumber l

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l Polarization Qxy=-1, Uxy=0 Pxy = -1 in bases wrt (rotated by –φ) Ql = 0, Ul = 1 Pl = i Pl = Pxy e-2iφ -φ y x

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**CMB Polarization Signals**

E polarization from scalar, vector and tensor modes B polarization only from vector and tensor modes (curl grad = 0) + non-linear scalars Average over possible realizations (statistically isotropic): Expected signal from scalar modes

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**Primordial Gravitational Waves (tensor modes)**

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 - cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc) Look at CMB polarization: ‘B-mode’ smoking gun

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**Lensing of polarization**

Polarization not rotated w.r.t. parallel transport (vacuum is not birefringent) Q and U Stokes parameters simply re-mapped by the lensing deflection field e.g. Last scattering Observed ~ ellipticities of infinitesimal small galaxies

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**Lensed spectrum: lowest order calculation**

Similar to temperature derivation, but now complex spin-2 quantities: Unlensed B is expected to be very small. Simplify by setting to zero. Expand in harmonics Calculate power spectrum. Result is

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Effect on EE and TE similar to temperature: convolution smoothing + transfer of power to small scales

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**Polarization lensing B mode power spectra**

BB generated by lensing even if unlensed B=0 On small scales, ClE lensed BB given by ClB Nearly white spectrum on large scales (power spectrum independent of l) l4Clφ l4Clφ l2ClE Can also do more accurate calculation using polarization correlation functions

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**Polarization power spectra**

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

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**Analogues of CMB lensing**

Lensing of temperature power spectrum: - lensed effect on galaxy number density/21cm power spectrum - smoothing of baryon oscillations (but much smaller effect ~ 10-3, low z) Q/U polarization: - e1/e2 ellipticity of a point source Q/U not changed by gravitational shear along path CMB polarization at last scattering - galaxy shape distribution in source plane - usually assume shapes uncorrelated ~ CE=CB=const - Intrinsic galaxy alignments can give something else Lensing of CMB polarization - white lenses to white CE → CE(1+4<κ2>), CB → CB(1+4<κ2>) - c.f. shape noise per arcminute: number density of galaxies depends locally on magnification - c.f. effect of magnification on intrinsic alignment power spectrum

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**Non-Gaussianity (back to CMB temperature)**

Unlensed CMB expected to be close to Gaussian With lensing: … For a FIXED lensing field, lensed field also Gaussian For VARYING lensing field, lensed field is non-Gaussian Three point function: Bispectrum < T T T > - Zero unless correlation <T Ψ> Large scale signal from ISW-induced T- Ψ correlation Small scale signal from non-linear SZ – Ψ correlation

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**Trispectrum: Connected four-point < T T T T>c**

Depends on deflection angle and temperature power spectra ‘Easily’ measurable for accurate ell > 1000 observations Other signatures correlated hot-spot ellipticities Higher n-point functions Polarization non-Gaussianity

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**Bigger than primordial non-Gaussianity?**

1-point function lensing only moves points around, so distribution at a point Gaussian But complicated by beam effects Bispectrum - ISW-lensing correlation only significant on very large scales - SZ-lensing correlation can dominate on very small scales - On larger scales oscillatory primordial signal should be easily distinguishable with Planck Komatsu: astro-ph/

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Trispectrum (4-point) Basic inflation: - most signal in long thin quadrilaterals Lensing: - broader distribution, less signal in thin shapes Komatsu: astro-ph/ Hu: astro-ph/ Can only detect inflation signal from cosmic variance if fNL >~ 20 Lensing probably not main problem for flat quadrilaterals if single-field non-Gaussianity No analysis of relative shape-dependence from e.g. curvaton?? Also non-Gaussianity in polarization…

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**Large scale lensing reconstruction**

As with galaxy lensing, ellipticities of hot and cold spots can be used to constrain the lensing potential But diffuse, so need general method Think about fixed lensing potential: lensed CMB is then Gaussian (T is Gaussian) but not isotropic - use off-diagonal correlation to constrain lensing potential

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**Can show that Define quadratic estimator**

Maximise signal to noise, write in real space: For more details see astro-ph/ or review

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**Method is potentially useful but not optimal**

Limited by cosmic variance on T, other secondaries, higher order terms Requires high resolution: effectively need lots of hot and cold spots behind each potential Reconstruction with polarization is much better: no cosmic variance in unlensed B Polarization reconstruction can in principle be used to de-lens the CMB - required to probe tensor amplitudes r <~ requires very high sensitivity and high resolution - in principle can do things almost exactly: a lot of information in lensed B at high l Maximum likelihood techniques much better than quadratic estimators for polarization (Hirata&Seljak papers)

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Input Quadratic (filtered) Approx max likelihood astro-ph/

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**Lensing potential power spectrum**

Hu: astro-ph/

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**Cluster CMB lensing e.g. to constrain cosmology via number counts**

Lewis & King, astro-ph/ 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 Need sensitive ~ arcminute resolution observations

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**RMS gradient ~ 13 μK / arcmin deflection from cluster ~ 1 arcmin**

Lensing signal ~ 10 μK BUT: depends on CMB gradient behind a given cluster Unlensed Lensed Difference Unlensed CMB unknown, but statistics well understood (background CMB Gaussian) : can compute likelihood of given lens (e.g. NFW parameters) essentially exactly

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**Add polarization observations?**

Difference after cluster lensing Unlensed T+Q+U Less sample variance – but signal ~10x smaller: need 10x lower noise Note: E and B equally useful on these scales; gradient could be either

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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) - Re-scattered thermal SZ (freq) - Kinetic SZ (higher order) - Other lenses Generally much cleaner

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**CMB polarization only (0.07 μK arcmin noise) **

Optimistic Futuristic CMB polarization lensing vs galaxy lensing Less massive case: M = 2 x 1014 h-1 Msun, c=5 CMB polarization only (0.07 μK arcmin noise) Galaxies (500 gal/arcmin2)

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**Moving Lenses and Dipole lensing**

Homogeneous CMB Rest frame of CMB: ‘Rees-Sciama’ (non-linear ISW) v Blueshifted hotter Redshifted colder Rest frame of lens: Dipole gradient in CMB T = T0(1+v cos θ) ‘dipole lensing’ deflected from hotter Deflected from colder

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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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**Cosmological parameters**

Essential to model lensing; but little effect on basic parameter constraints Planck (2007+) parameter constraint simulation (neglect non-Gaussianity of lensed field; BB noise dominated so no effect on parameters) Important effect, but using lensed CMB power spectrum gets ‘right’ answer Lewis 2005

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**Extra information in lensing**

Unlensed CMB has many degeneracies: e.g. distance and curvature flat closed θ θ Lensing introduces additional information: growth and scale of lensing deflection power break degeneracies - e.g. improve constraints on curvature, dark energy, neutrino mass

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Lensed CMB power spectra contain essentially two new numbers: - one from T and E, depends on lensing potential at l<300 - one from lensed BB, wider range of l astro-ph/ More information can be obtained from non-Gaussian signature: lensing reconstruction - may be able to probe neutrino masses ~ 0.04eV (must be there!)

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Summary Weak lensing of the CMB very important for precision cosmology - changes power spectra at several percent - potential confusion with primordial gravitational waves for r <~ Non-Gaussian signal - Generally well understood, modelled accurately in linear theory with small non-linear corrections Potential uses - Break parameter degeneracies, improve parameter constraints - Constrain cluster masses at high redshift - Reconstruction of potential to z~7

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