Weak Lensing of the CMB

1. Weak Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/

2. Outline • From the beginning • Lensing order of magnitudes • Lensed power spectrum • Effect on CMB polarization • Cluster masses from CMB lensing

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

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

5. Where do perturbations come from? New physics Known physics Inflation make >1030 times bigger Quantum Mechanics“waves in a box” calculationvacuum state, etc… After inflation Huge size, amplitude ~ 10-5

6. 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

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

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

9. 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.)

10. 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

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

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

13. LensPix sky simulation code:http://cosmologist.info/lenspixLewis 2005

14. Lensing effect on CMB temperature power spectrum Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/0502425

15. 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

16. Thomson Scattering Polarization W Hu

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

18. 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 tensor

19. E and B polarization “gradient” modesE polarization “curl” modes B polarization e.g. e.g. cold spot B modes only expected from gravitational waves and CMB lensing

20. 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

21. Polarization lensing: CB 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

22. Polarization lensing: Cx and CE Lewis, Challinor : astro-ph/0601594

23. Primordial Gravitational Waves • 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 - significant power only at l<100, cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc) Look at CMB polarization: ‘B-mode’ smoking gun

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

25. Cluster CMB lensing Lewis & King, astro-ph/0512104 Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, etc. CMB very smooth on small scales: approximately a gradient What we see Last scattering surface GALAXYCLUSTER 0.1 degrees

26. Toy model: spherically symmetric NFW cluster 2 M200 ~ 1015 h-1 Msun Deflection ~ 0.7 arcmin c ~ 5, z ~ 1 (rv ~ 1.6Mpc) (approximate lens as thin, constrain projected density profile) assume we know where centre is

27. 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

28. Constraining cluster parameters CMB approximately Gaussian – know likelihood function Calculate P(c,M200 | observation) Simulated realisations with noise 0.5 μK arcmin, 0.5 arcmin pixels Somewhat futuristic: 160x lower noise 14x higher resolution than Planck; few times better than ACT

29. Add polarization observations? Difference after cluster lensing Unlensed T+Q+U Less sample variance – but signal ~10x smaller: need 10x lower noise Plus side: SZ (etc) fractional confusion limit probably about the same as temperature

30. Temperature Polarisation Q and U Noise: 0.7 μK arcmin 0.07 μK arcmin 0.5 μK arcmin less dispersion in error

31. Is it better than galaxy lensing? • Assume galaxy shapes random before lensing • Measure ellipticity after lensing Lensing • On average ellipticity measures reduced shear • Shear is γab = ∂<a αb> • Constrain cluster parameters from predicted shear

32. Galaxy lensing comparison Massive case: M = 1015 h-1 Msun, c=5 (from expected log likelihoods) Ground (30/arcmin) CMB temperature only (0.5 μK arcmin noise) Galaxies (100 gal/arcmin2)

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

34. CMB 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)- Kinetic SZ (higher order)- Other lensesGenerally much cleaner

35. 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

36. 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

37. Observable Dipoles • Change of velocity: - Doppler change to total CMB dipole - aberration of observed angles (c.f. dipole convergence) • Can observe: actual CMB dipole: (non-linear) local motion + primordial contribution • Can observe: Dipole aberration (dipole convergence + kinetic aberration) • So: Lensing potential dipole ‘easily’ observable to O(10-5)- Can find zero-aberration frame to O(10-5) by using zero total CMB-dipole frame - change of frame corresponds to adding some local kinematic angular aberration to convergence dipole - zero kinematic aberration and zero kinematic CMB dipole frame = Newtonian gauge

38. Convergence dipole expected ~ 5 x 10-4

39. Summary • Weak lensing of the CMB very important for precision cosmology- changes power spectra- potential confusion with primordial gravitational waves • Cluster lensing of CMB- gravitational lensing so direct probe of mass (not just baryons)- mass constraints independent of galaxy lensing constraints; source redshift known very accurately, should win for high redshifts- galaxy lensing expected to be much better for low redshift clusters- polarisation lensing needs high sensitivity but cleaner and less sample variance than temperature

40. Physics Reports review: astro-ph/0601594

41. http://CosmoCoffee.info arXiv paper filtering, discussion and comments Currently 420 registered readers

42. Calculate Cl by series expansion in deflection angle? Series expansion only good on large and very small scalesAccurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005 No

43. arXivJournal.org

44. Is this right? • Lieu, Mittaz, ApJ L paper: astro-ph/0409048 - Claims shift in CMB peaks inconsistent with observation - ignores effect of matter. c.f. Kibble, Lieu: astro-ph/0412275 • Lieu, Mittaz, ApJ paper:astro-ph/0412276Claims large dispersion in magnifications, hence peaks washed out - Many lines of sight do get significant magnification - BUT CMB is very smooth, small scale magnification unobservable - BUT deflection angles very small - What matter is magnifications on CMB acoustic scales i.e. deflections from large scale coherent perturbations. This is small. - i.e. also wrong • Large scale potentials < 10-3 : expect rigorous linear argument to be very accurate (esp. with non-linear corrections)