1. 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

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

3. 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)

4. 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

5. 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

6. 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

7. 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

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

9. 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

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

11. 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…

12. 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

13. 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

14. 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

15. 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

16. 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) :

17. 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

18. 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

19. 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…

20. 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 )

21. 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

22. 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

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

24. 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