1. Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/ Review ref: Lewis, Challinor, Phys. Rep: astro-ph/0601594

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

3. 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

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

5. Temperature anisotropy data: WMAP 3-year + smaller scales BOOMERANG Hinshaw et al + many more coming up e.g. Planck (2008)

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

7. 14 000 Mpc Not to scale! All distances are comoving ~100Mpc ~200/14000 ~ degree largest overdensity Neutral gas - transparent Ionized plasma - opaque Good approximation: CMB is single source plane at ~14 000 Mpc Angular diameter distance well measured by angle of acoustic peaks Recombination ~200Mpc

8. 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)

9. So why does it matter? 2arcmin: ell ~ 3000 - 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

10. In detail, lensed temperature depends on deflection angle: Lensing Potential Deflection angle on sky given in terms of lensing potential

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

12. Simulated full sky lensing potential and (magnified) deflection angle fields Easily simulated assuming Gaussian fields - just re-map points using Gaussian realisations of CMB and potential

13. Lensed temperature C l Essentially exact to order of weak lensing by Gaussian field – 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 Full-sky fully non-perturbative generalization of method by Seljak 1996 - convolution of unlensed C l - W is non-linear in lensing potential power

14. Lensing effect on CMB temperature power spectrum CAMBs 0.1% calculation; http://camb.info : Challinor & Lewis 2005, astro-ph/0502425http://camb.info

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

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

17. 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 or spin-2 field - just like shear

18. E and B polarization gradient modes E polarization curl modes B polarization e.g. e.g. cold spot

19. 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 B modes only expected from gravitational waves and CMB lensing

20. 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 Last scattering Observed e.g.

21. Polarization lensing: C x and C E

22. Polarization lensing: C B Nearly white BB spectrum on large scales

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

24. 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 … Specific form of non-Gaussianity - e.g. 1 point still Gaussian, very small 3-point function - should be able to distinguish from primordial non-Gaussianity Modifies covariance of lensed C l (esp. BB) Can be used to learn about lensing potential – reconstruction methods…

25. Likelihoods Small number of lensing modes: BB C l correlated between l. (Smith, Challinor, Rocha 2006) Correction to temperature likelihood is small; on full sky usual result is quite good Correct BB (and others) using covariance from simulations. Good approx is Smith, Challinor, Rocha 2006 ASIDE: Also works for cut sky – can use for convergence power spectrum For multiple redshift bins can generalise for correlated fields: X= (k 11,k 22,k 12,…) for details see Hammimeche & Lewis (in prep).

26. Large scale lensing reconstruction As with galaxy lensing, ellipticities of hot and cold spots could be used to constrain the lensing potential But diffuse, know source statistics, can use magnification - 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

27. Can show that optimal quadratic estimator is - simple function of filtered fields For more details see Hu astro-ph/0105424 or review; c.f. Metcalf & White 2007 Analogous results for CMB polarization

28. e.g. estimate lensing potential power spectrum - more information on cosmological parameters Hu: astro-ph/0108090 (ideal is limit using non-optimal quadratic estimator)

29. e.g. reconstruct lensing potential field should correlate with other matter tracers Constrain large-scale matter distribution to redshift z ~ 6 De-lens the CMB (remove B-mode lensing contamination to see primordial B modes)

30. First claimed detection in cross-correlation (see talk by Olivier Doré) (http://cosmocoffee.info discussion)http://cosmocoffee.info

31. Limited by cosmic variance on T, other secondaries, higher order terms Quadratic method useful but not optimal -especially for polarization (Hirata&Seljak papers) 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 <~ 10 -4 - requires very high sensitivity and high resolution Reconstruction complications

32. astro-ph/0306354 Input Quadratic (filtered)Approx max likelihood

33. 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/0607315 Can break degeneracies in linear CMB: improve constraints on dark energy, curvature, etc. May be able to probe neutrino masses ~ 0.04eV (must be there! see astro-ph/0603494) Other information in CMB lensing (>> arcminute)

34. Cluster CMB lensing e.g. to constrain cosmology via number counts GALAXY CLUSTER Last scattering surface What we see Seljak, Zaldarriaga, Dodelson, Vale, Holder, Lewis, King, Hu. Maturi,. etc. CMB very smooth on small scales: approximately a gradient 0.1 degrees Need sensitive ~ arcminute resolution observations

35. 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) :

36. 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

37. 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

38. CMB polarization only (0.07 μK arcmin noise) Optimistic Futuristic CMB polarization lensing vs galaxy lensing e.g. M = 2 x 10 14 h -1 M sun, c=5 Galaxies (500 gal/arcmin 2 ) Lewis & King 2006 Fitting profiles. e.g. to measure mass and concentration Can stack for constraints from multiple clusters

39. General cluster mass reconstruction Can use quadratic reconstruction methods similar to those on large scales Potential problems with bias due to large central magnifications - use full likelihood function (e.g. Hirata et al, though prior less clear) - various ad hoc methods also work (Maturi, Hu..) Not competitive with galaxy lensing except possibly for high redshift But systematics very different; may be useful cross- check

40. CMB/Galaxy lensing comparison CMB Lensing - single source plane, lenses 0.5<~z<~7 - accurate source plane distance - statistics of source plane well understood - systematics: pointing/beam uncertainty, SZ, foregrounds,… - Small corrections from non-linear P k - Smoothes temperature power spectrum - B modes generated by lensing of E Galaxy lensing - many source planes, lenses <~1.5 - often only photo-z redshifts - make no assumption about source distribution - systematics: PSF modelling, source selection, noise bias, …. - Non-linear P k crucial -magnification effect on source number counts (e.g. smoothes baryon oscillations; c.f. original Vallinotto talk) - Mixing of intrinsic alignment source plane E and B fields by lensing

41. Lensing of 21cm Very similar to CMB lensing, but 21cm power spectrum much more small scale power and many source planes/3D information Lensed angular power spectrum result simple generalization from lensed CMB temperature (Lewis & Challinor 2007 c.f. Mandel & Zaldarriaga 2006) Can reconstruct potential from lensed 21cm – lots of information in 3D (Hilbert, Metcalf, White, Zaldarriaga, Zahn, Cooray... see Metcalf poster) C l (z=50,z=52) C l (z=50,z=50)

42. 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 <~ 10 -3 - introduces non-Gaussian signal - well understood in theory – accurately modelled with linear theory + small non-linear corrections Potential uses - Break parameter degeneracies, improve parameter constraints - Constrain cluster masses to high redshift - Reconstruction of potential at 0.5 <~ z <~ 7

44. Correlation with the CMB temperature very small except on largest scales

45. 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 Cosmological parameters Essential to model lensing; but little effect on basic parameter constraints

46. 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

47. 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

48. 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 - Zero unless correlation Large scale signal from ISW-induced T- Ψ correlation Small scale signal from non-linear SZ – Ψ correlation …

49. 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

50. Bigger than 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 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

51. 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 Also non-Gaussianity in polarization…