Polarisation of the CMB An introduction

Polarisation of the CMBAn introduction

S. Prunet (IAP)

Orsay, Sept. 12th 2005

Theoretical perspective

• Theoretical introduction

• Gravitational waves signature in B modes

• Additional information on cosmological parameters

• Nature of perturbations:– Scalars versus tensors– Adiabatic versus isocurvature

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

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 Cl

(e.g. CMBFAST)

Theory: a brief reminder

Scalar perturbations

Rotationally invariant polarisation variablesSummation on Fourier modes

Angle needed to rotate Stokes Parameters from k-dependentBasis to fixed frame on the sky

(Q+iU) is a spin-2 quantity

Decomposition on spin-2 harmonics

Rotationally invariant quantities

E and B modes of polarisation

Scalar quantity

Pseudo-scalar quantity

Scalar perturbations cannot produce B modesB modes are model-independent tracers of tensor perturbations

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)

Polarisation power spectra

Wayne Hu

Effect of reionisation on polarisationEffect of reionisation: rescattering of CMB photons

http://background.uchicago.edu/~whu/

Amplitude – optical depth to LSS degeneracy: polarization does help !

Models with constant C2eà2üC are hard to separate with T only

Zaldarriaga, Spergel, Seljak 97

Impact on optical depth measurement

Impact on other cosmological parametersAdding polarisation: improved parameters measurements as a function of multipole

Case of the Planck mission

Zaldarriaga, Spergel, Seljak 97

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

Measuring polarisation improves a lot !

Bucher, Moodley, Turok 2001

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

Polarisation: first measurement

First detection ofE-mode polarisationby DASI (2002)

3 years results:sharpened E detection

Kovac et al. 2002Leitch et al. 2005

PolarisationUpper limits and first measurement by DASI

Polarisation spectra: present observational status

EE power spectrumTE cross-spectrum

B03

DASICBI

CAPMAP

WMAP

Interferometer - HEMTs

HEMTs- differential

Interferometer - HEMTs

Sing. dish - HEMTs

Sing. Dish - Bolometers

Montroy et al. 2005

Piacentini et al. 2005

Cosmological consistency

WMAP TT+TE CBI+B03+DASI EE+TE WMAP TT+TE + CBI+B03+DASI TT+TE+EE

Sievers et al. 2005

Polarisation: on-going

QUAD

BICEP

Polarisation: Planck (>= 2007)

Nice propaganda ! But does not include systematics …

http://background.uchicago.edu/~whu/

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

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 ...

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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

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

Lensing “cleaning”: improvement ?

Hirata & Seljak 2003

• Iterative ML method• Gains in the low-noise limitby reducing the CV of the residual

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)

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

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)

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

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

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 1997Hobson et al. 1998Bouchet & Gispert 1999

Likelihood (gaussian noise)

Gaussian prior

WIENER FILTER

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

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

Input processes

CMB

SZ (thermal)

SZ (kinetic)

Dust (thermal)

Free-freeSynchrotron

Hobson et al. 1998

•Planck specs•Assumed known spectra•Assumed known mixing matrix•MEM full ICF

10x10 deg

Hobson et al. 1998

Wiener full ICF

Hobson et al. 1998

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

Relaxing assumptionsCMB + DUST + SZ + NOISEHFI 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 !

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