Lensing of the CMB

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

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

3. Perturbation evolution – what we actually observeCMB monopole source till 380 000 yrs (last scattering), linear in conformal timescale invariant primordial adiabatic scalar spectrum photon/baryon plasma + dark matter, neutrinos Characteristic scales: sound wave travel distance; diffusion damping length

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

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

8. Lensing order of magnitudes 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 ~ 14000 MPc total deflection ~ 501/2 x 10-4 pass through ~50 lumps ~ 2 arcminutes assume uncorrelated (neglects angular factors, correlation, etc.)

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

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 Cl - convolution of unlensed Cl- W is non-linear in lensing potential power Essentially exact to order of weak lensing by Gaussian field – very well understood effect on power spectra.Non-linear Pk 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

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

15. Lensing important at 500<l<3000Dominated 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 - - Q U Q → -Q, U → -U under 90 degree rotation Q → U, U → -Q under 45 degree rotation Rank 2 trace free symmetric tensoror spin-2 field- just like shear

18. E and B polarization “gradient” modesE 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 e.g. Observed Last scattering

21. Polarization lensing: Cx and CE

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

23. Polarization power spectra Current 95% indirect limits for LCDM given WMAP+2dF+HST 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 Cl (esp. BB) • Can be used to learn about lensing potential – reconstruction methods…

25. Likelihoods • Small number of lensing modes: BB Cl 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 spectrumFor multiple redshift bins can generalise for correlated fields: X= (k11,k22,k12,…) 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 Analogous results for CMB polarization For more details see Hu astro-ph/0105424 or review; c.f. Metcalf & White 2007

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

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)

31. Reconstruction complications • 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

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

33. Other information in CMB lensing (>> arcminute) • 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 lastro-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)

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

35. RMS gradient ~ 13 μK / arcmindeflection 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

36. 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

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 lensesGenerally much cleaner

38. Fitting profiles. e.g. to measure mass and concentration Optimistic Futuristic CMB polarization lensing vs galaxy lensinge.g. M = 2 x 1014 h-1 Msun, c=5 Can stackfor constraintsfrom multipleclusters Lewis & King 2006 CMB polarization only (0.07 μK arcmin noise) Galaxies (500 gal/arcmin2)

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 Pk- 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 sourcedistribution- systematics: PSF modelling, source selection, noise bias, ….- Non-linear Pk crucial-magnification effect on source numbercounts (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 2007c.f. Mandel & Zaldarriaga 2006) Cl(z=50,z=52) Cl(z=50,z=50) • Can reconstruct potential from lensed 21cm – lots of information in 3D(Hilbert, Metcalf, White, Zaldarriaga, Zahn, Cooray... see Metcalf poster)

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. 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

46. Moving Lenses and Dipole lensing Homogeneous CMB Rest frame of CMB: ‘Rees-Sciama’(non-linear ISW) v Blueshiftedhotter Redshiftedcolder Rest frame of lens: Dipole gradient in CMB T = T0(1+v cos θ) ‘dipole lensing’ deflected from hotter Deflected from colder

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 < 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

49. 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

50. 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/0005036