1. Polarisation of the CMB An introduction S. Prunet (IAP) Orsay, Sept. 12 th 2005

2. Theoretical perspective Theoretical introduction Gravitational waves signature in B modes Additional information on cosmological parameters Nature of perturbations: –Scalars versus tensors –Adiabatic versus isocurvature

3. Polarisation: Physical origin Generated by Thomson scattering Due to quadrupolar anisotropy in the electron rest frame Linked to velocity field gradients at recombination/reionisation In the electron rest frame

4. Boltzmann transport equation describes the evolution of the photon distribution function Collisional part describes the scattering of the photons with electrons Gravitational part describes the motion of the photons in the perturbed background Differential form in Fourier space Calculation of Theoretical C l (e.g. CMBFAST) Theory: a brief reminder Scalar perturbations

5. Rotationally invariant polarisation variables Summation on Fourier modes Angle needed to rotate Stokes Parameters from k-dependent Basis to fixed frame on the sky (Q+iU) is a spin-2 quantity Decomposition on spin-2 harmonics Rotationally invariant quantities

6. E and B modes of polarisation Scalar quantity Pseudo-scalar quantity Scalar perturbations cannot produce B modes B modes are model-independent tracers of tensor perturbations

7. Computation of the spectrum Linearized Einstein and Boltzmann equations System of coupled ODEs in multipole space (e.g. Bond & Efstathiou) Closure relation at Transport, gravity and collision phenomena intricately mixed Slow codes (up to ½ hour per spectrum) “Line of sight” integration (e.g. Seljak & Zaldarriaga) Closure relation of ODEs at low Disentangle transport / projection effects from anisotropy sources Fast codes (a few seconds per spectrum) Approximate calculations (e.g. DASH) Very fast Useful for parameter estimation with Markov chains (IS)

8. Polarisation power spectra Wayne Hu

9. Effect of reionisation on polarisation Effect of reionisation: rescattering of CMB photons http://background.uchicago.edu/~whu/

10. Amplitude – optical depth to LSS degeneracy: polarization does help ! Models with constantare hard to separate with T only Zaldarriaga, Spergel, Seljak 97 Impact on optical depth measurement

11. Impact on other cosmological parameters Adding polarisation: improved parameters measurements as a function of multipole Case of the Planck mission Zaldarriaga, Spergel, Seljak 97

12. Isocurvature perturbations Introducing isocurvature modes invalidates parameter estimation… Degeneracies are broken with polarisation measurements Delicate cancellation in the temperature spectrum (most poorly determined combination in the Fisher matrix) Bucher, Moodley, Turok 2001

13. Measuring polarisation improves a lot ! Bucher, Moodley, Turok 2001

14. Observing the polarisation Signal is tiny !  the field is (very) young ! Multiple methods: –HEMTs vs bolometers –Interferometers versus scanned single dish –In common: huge increase of #detectors Data processing, detectors, are still in R&D phase

15. Polarisation: first measurement First detection of E-mode polarisation by DASI (2002) 3 years results: sharpened E detection Kovac et al. 2002 Leitch et al. 2005

16. Polarisation Upper limits and first measurement by DASI

17. Polarisation spectra: present observational status EE power spectrum TE cross-spectrum B03 DASI CBI CAPMAP WMAP Interferometer - HEMTs HEMTs- differential Interferometer - HEMTs Sing. dish - HEMTs Sing. Dish - Bolometers Montroy et al. 2005 Piacentini et al. 2005

18. Cosmological consistency WMAP TT+TECBI+B03+DASI EE+TEWMAP TT+TE + CBI+B03+DASI TT+TE+EE Sievers et al. 2005

19. Polarisation: on-going QUAD BICEP

20. Polarisation: Planck (>= 2007) Nice propaganda ! But does not include systematics … http://background.uchicago.edu/~whu/

21. Space-borne polarimeter Specific design to control instrumental systematics –Thermal stability (tiny signals !) –Instrumental polarisation control –Optimized scanning strategy Detectors are ~background limited –Need a lot of them !! –Detector arrays, no horns, big focal planes Specific needs

22. Polarisation: the future challenge Primordial GW background: no theoretical prior on amplitude… One-field inflationary models: Tensor amplitude varies as E inf 4 Lensing-induced B-modes: dominant at least on small scales Polarized foreground emissions are nearly unknown... Courtesy EPIC consortium

23. Polarisation from space: requirements Large scales: space required Stable environment: space … Detectors are background limited need lots of them ! detector arrays large telemetry … Stringent systematics control Courtesy EPIC consortium

24. Lensing-induced B-mode cleaning Kesden, Cooray, Kamionkowski (2002) Substract lensing-induced BB by reconstruction of deflection angle using 4-point minimum variance estimators (Hu & Okamoto 2002) Exponantial cut-off of CMB anisotropies at small scales limits lensing reconstruction

25. Lensing “cleaning”: improvement ? Hirata & Seljak 2003 Iterative ML method Gains in the low-noise limit by reducing the CV of the residual

26. Cut-sky effects: E-B mixing Mixing occurs from line integrals on the border Define STF windows that project out E contribution This can be achieved by SVD of coupling matrix For each m, 2 modes are lost Lewis, Challinor, Turok 2001 Separation is done at the map level Block-diagonal structure of coupling allows to gain CPU time for azimuthally symmetric patches Pixel effects can be important if no quadrature sampling … (e.g. Bunn et al. 2002)

27. E-B mixing: statistical separation Use integrals of the Stokes correlations functions over observed angular range to construct pure E and B statistics Originally derived for lensing (Crittenden et al. 2002) Generalized to the sphere (Chon et al. 2004) and coupled to fast, edge-corrected estimation of correlation functions Fast decoupled, edge-corrected estimators of polarized spectra available E-B separation only in the mean ! E-mode cosmic variance leaks into B-mode variance Only valid for sufficiently large surveys (Challinor & Chon 2005) OR Use the coupling kernels of polarised pseudo-Cls (Hansen & Gorski 2003) Generalise MASTER (or FASTER) method Regularised (binned) inversion of coupling kernel This was used in the B03 data processing

28. The case of interferometers Visibilities: sample the convolved UV space: Idem for Q and U Stokes parameters RL and LR baselines give (Q§iU) Relationship between (Q,U) and (E,B) in UV (flat) space Visibilities correlation matrix UV coverage of a single pointing of CBI (10 freq. bands) ( Pearson et al. 2003)

29. Pixelisation in UV/pixel space Redundant measurements in UV-space Possibility to compress the data ~w/o loss Hobson and Maisinger 2002 Myers et al. 2003 Park et al. 2003 Least squares solution For an NGP pointing matrix: Resultant noise matrix Use in conjonction with an ML estimator Newton-like iterative maximisation Fisher matrix Covariance derivatives for one visibility

30. Foregrounds: component separation Unlike temperature, foregrounds are a big problem Bayesian methodology –Maximum A Posteriori maximization –Different priors give different methods Relaxing assumptions on foreground properties –Blind & semi-blind methods (spectral matching, ICA) –Spatially varying spectral indices Too many parameters  Not good ! –Astrophysics needed !! –Statistical (proper) characterization on templates for prior building … Tucci et al. 2005 Ponthieu et al. 2005 Giardino et al. 2002 Tucci et al. 2000 Prunet et al. 1998

31. Bayesian formulation Assume a linear problem: Add in some important assumptions: Azimuthally symmetric beams Spatial-(E.M.) frequency decoupling Statistical isotropy Known mixing matrix Tegmark &Efstathiou 1997 Hobson et al. 1998 Bouchet & Gispert 1999 Likelihood (gaussian noise) Gaussian prior WIENER FILTER

32. Wiener filter Optimal linear filter (min residual power) Analytical predictions for errors on the maps on the power spectra Easily extended to polarized data Everything is assumed gaussian… Fractional error on C l E : Dotted: best channel (no forgs) Dashed: combined channels (no forgs) Solid: WIENER (CMB+dust+sync.) Bouchet, Prunet, Sethi 1999

33. The entropic prior: MEM Initially designed for positive, uncorrelated distributions Extended to non-positive distributions Include correlations via Cholesky decomposition with u, v, positive Cross-entropy of the distributions: Non-linear maximization problem Approximate error predictions (Fisher approach) Recover Wiener in small-fluctuations limit Hobson et al. 1998

34. Input processes CMB SZ (thermal) SZ (kinetic) Dust (thermal) Free-free Synchrotron Hobson et al. 1998

35. Planck specs Assumed known spectra Assumed known mixing matrix MEM full ICF 10x10 deg Hobson et al. 1998

36. Wiener full ICF Hobson et al. 1998

37. Relaxing assumptions … Reality is more complex !!! Keep (E.M) frequency – space factorization (no varying spectral indices) Unknown Mixing matrix BLIND METHODS Problem is non-linear Needs good minimisation engine Arbitrary parametrisation of E.M. behaviour Direct fit of component power spectra: spectral matching (gaussian priors) Bilinear model: MATCH Delabrouille, Cardoso, Patanchon 2003 Gaussian likelihood: Parameters to adjust

38. Relaxing assumptions CMB + DUST + SZ + NOISE HFI channels (6 frequencies) EM and BFGS combined algorithmsDelabrouille, Cardoso, Patanchon 2003 Relaxing (E.M.) frequency – space factorization e.g. Tegmark et al. 2000, Bennett et al. 2003, Eriksen et al. 2005 High S/N limit: maximize non-gaussianity (ICA) e.g. Maino et al. 2003, Stivoli et al. 2005 Combined MEM/Wavelets: emissions and PS separation e.g. Vielva et al. 2001 Statistical isotropy of CMB/foregrounds e.g. Hajian & Souradeep 2003, de Oliveira-Costa et al. 2004, Prunet et al. 2005 Etc !

39. Conclusions CMB physics OK Data processing: OK, but still in development phase Component separation: pfff… –Foregrounds are ugly (yes,yes) –Break about all assumptions you can do –Need to assess robustness of techniques –ASTROPHYSICAL INSIGHT NEEDED