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Polarisation of the CMB An introduction

Orsay, Sept. 12 th 2005. Polarisation of the CMB An introduction. S. Prunet (IAP). Theoretical perspective. Theoretical introduction Gravitational waves signature in B modes Additional information on cosmological parameters Nature of perturbations: Scalars versus tensors

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Polarisation of the CMB An introduction

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

  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 In the electron rest frame • Generated by Thomson scattering • Due to quadrupolar anisotropy in the electron rest frame • Linked to velocity field gradients at recombination/reionisation

  4. Theory: a brief reminder Calculation of Theoretical Cl (e.g. CMBFAST) 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 Scalar perturbations Differential form in Fourier space

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

  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 • 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) Linearized Einstein and Boltzmann equations

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

  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 Delicate cancellation in the temperature spectrum (most poorly determined combination in the Fisher matrix) Introducing isocurvature modes invalidates parameter estimation… Degeneracies are broken with polarisation measurements 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 3 years results:sharpened E detection First detection of E-mode polarisation by DASI (2002) 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 WMAP CBI HEMTs- differential CAPMAP Interferometer - HEMTs Interferometer - HEMTs DASI Sing. dish - HEMTs Sing. Dish - Bolometers Montroy et al. 2005 Piacentini et al. 2005 B03

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

  19. Polarisation: on-going BICEP QUAD

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

  21. Space-borne polarimeter Specific needs • 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

  22. Polarisation: the future challenge Courtesy EPIC consortium • Primordial GW background: no theoretical prior on amplitude… • One-field inflationary models: Tensor amplitude varies as Einf4 • Lensing-induced B-modes: dominant at least on small scales • Polarized foreground emissions are nearly unknown ...

  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 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 Kesden, Cooray, Kamionkowski (2002)

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

  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 • 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) Lewis, Challinor, Turok 2001

  27. E-B mixing: statistical separation • 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 • 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 OR 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)

  28. The case of interferometers Visibilities: sample the convolved UV space: Relationship between (Q,U) and (E,B) in UV (flat) space Idem for Q and U Stokes parameters RL and LR baselines give (Q§iU) 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 Use in conjonction with an ML estimator Newton-like iterative maximisation Least squares solution • Hobson and Maisinger 2002 • Myers et al. 2003 • Park et al. 2003 Fisher matrix For an NGP pointing matrix: Covariance derivatives for one visibility Resultant noise matrix

  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 forprior 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 Likelihood (gaussian noise) Gaussian prior WIENER FILTER Tegmark &Efstathiou 1997 Hobson et al. 1998 Bouchet & Gispert 1999

  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 ClE: • Dotted: best channel (no forgs) • Dashed: combined channels (no forgs) • Solid: WIENER (CMB+dust+sync.) Bouchet, Prunet, Sethi 1999

  33. The entropic prior: MEM with u, v, positive • Initially designed for positive, uncorrelated distributions • Extended to non-positive distributions • Include correlations via Cholesky decomposition 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 (kinetic) Dust (thermal) SZ (thermal) Synchrotron Free-free 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 Bilinear model: MATCH Parameters to adjust • Problem is non-linear • Needs good minimisation engine • Arbitrary parametrisation of E.M. behaviour • Direct fit of component power spectra: spectral matching (gaussian priors) Delabrouille, Cardoso, Patanchon 2003 Gaussian likelihood:

  38. Relaxing assumptions CMB + DUST + SZ + NOISE HFI channels (6 frequencies) Delabrouille, Cardoso, Patanchon 2003 EM and BFGS combined algorithms • Relaxing (E.M.) frequency – space factorizatione.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 separatione.g. Vielva et al. 2001 • Statistical isotropy of CMB/foregroundse.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

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