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CMB and cluster lensing Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/ Lewis & Challinor, Phys. Rept. 2006 : astro-ph/0601594 Lewis & King, PRD 2006 : astro-ph/0512104

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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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LensPix sky simulation code: http://cosmologist.info/lenspix Full calculation: deflection angle on sky given in terms of lensing potential Lensed temperature given by Lewis 2005, astro-ph/0502469

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Lensed temperature C l Analogous results for CMB polarization. Essentially exact to order of weak lensing – very well understood effect on power spectra. Non-linear P k 0.2% on TT, ~5% on BB Lewis, Challinor Phys. Rept. 2006 : astro-ph/0601594 andlinear in lensing potential power spectrum Full-sky fully non-perturbative generalization of method by Seljak 1996

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Lensing effect on CMB temperature power spectrum: smoothing of acoustic peaks; small scale power Full-sky calculation accurate to 0.1%: Fortran code CAMB (http://camb.info)http://camb.info

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Polarization lensing: C x and C E Important ~ 10% smoothing effect

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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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Current 95% indirect limits for LCDM given WMAP+2dF+HST Polarization power spectra Lewis, Challinor : astro-ph/0601594; Lewis Moriond 2006

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Non-Gaussianity 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 - Zero unless correlation Large scale signal from ISW-induced T- Ψ correlation Small scale signal from non-linear SZ – Ψ correlation … Zaldarriaga astro-ph/9910498, Goldberg&Spergel, etc…

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Trispectrum: Connected four-point 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 Zaldarriaga astro-ph/9910498; Hu astro-ph/0105117

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Confusion with primordial non-Gaussianity? 1-point function - SZ-lensing correlation can dominate on very small scales - On larger scales oscillatory primordial signal should be easily distinguishable with Planck if large enough Komatsu: astro-ph/0005036 - ISW-lensing correlation only significant on very large scales Bispectrum - lensing only moves points around, so distribution at a point Gaussian - But complicated by beam effects Kesden, Cooray, Kamionkowski: astro-ph/0208325

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Trispectrum (4-point) Basic inflation: - most signal in long thin quadrilaterals Lensing: - broader distribution, less signal in thin shapes Can only detect inflation signal from cosmic variance if f NL >~ 20 Komatsu: astro-ph/0602099Hu: astro-ph/0105117 No analysis of relative shape-dependence from e.g. curvaton?? Lensing probably not main problem for flat quadrilaterals if single-field non-Gaussianity

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Cluster CMB lensing e.g. to constrain cosmology via number counts 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 Need sensitive ~ arcminute resolution observations

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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 can compute likelihood of given lens (e.g. NFW parameters) essentially exactly Unlensed CMB unknown, but statistics well understood (background CMB Gaussian) :

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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 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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Is CMB lensing better than galaxy lensing? Assume background galaxy shapes random before lensing Measure ellipticity after lensing by cluster Lensing On average ellipticity measures reduced shear Shear is γ ab = Constrain cluster parameters from predicted shear Assume numerous systematics negligible…

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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 10 14 h -1 M sun, c=5 Galaxies (500 gal/arcmin 2 )

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Summary Weak lensing of the CMB very important for precision cosmology - changes power spectra - potential confusion with primordial gravitational waves for r <~ 10 -3 - Non-Gaussian signal, but well known and probably not main problem Cluster lensing of CMB - Temperature lensing difficult because of confusions - CMB polarisation lensing needs high sensitivity but potentially useful at high redshift - galaxy lensing expected to be much better for low redshift clusters - CMB lensing has quite different systematics to galaxy lensing

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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 Parameters can be improved using BB/lensing reconstruction; non-Gaussianity important in the future; c.f. Wayne Hus talk

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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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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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