Blind Component Separation for Polarized Obseravations of the CMB
Jonathan Aumont,Juan-Francisco Macias-Perez24-03-2006
Rencontres de Moriond 2006La Thuile, Italy
Jonathan Aumont, LPSC Grenoble Moriond 2006
Overview
• Model of the microwave sky• Spectral matching algorithm extended to polarization• Planck simulations• Performances of the algorithm• Results• Conclusions
Jonathan Aumont, LPSC Grenoble Moriond 2006
Model of the microwave sky
• Data in the spherical harmonics space for X = { T,E,B }:
• Density matrices:
• Then data read:
• In real space: { I,Q,U } { T,E,B } in Fourier space
Jonathan Aumont, LPSC Grenoble Moriond 2006
Spectral matching
• Expectation-Maximization (EM) algorithm [Dempster et al. JRSS 1977]:
Set of parameters: iRS l ), RN ( l ), A }
• Iterations:
• E-step: expectation of the likelihood for i (gaussian prior)• M-step: maximization of the likelihood to compute i+1
[Delabrouille, Cardoso & Patanchon, 2003, MNRAS]
• In this work, 10000 EM iterations are generally performed
Jonathan Aumont, LPSC Grenoble Moriond 2006
I, Q and U sky maps simulations
• White noise maps normalized to the instrumental noise level for each frequency
• Thermal dust emission: • Power-law model• Normalized with respect to Archeops 353 GHz data [Ponthieu et al. A&A 2005]
• Galactic synchrotron emission:• Template maps [Giardino et al. A&A 2002]:• Isotropic spectral index ( -2.7 )
• CMB• Spectra generated with CAMB [Lewis et al. ApJ 2000] for concordance model according to WMAP1 [Bennett et al. ApJS 2003], r = 0.7 and gravitational lensing I Q
I Q
I Q
Jonathan Aumont, LPSC Grenoble Moriond 2006
Priors and Planck Simulations
• Blind analysis: • { RS , RN , A } • no priors
• Blind with A(T) = A(E) = A(B): • = { RS , RN , A } • we suppose that emission laws are the same in temperature and polarization
• Semi-Blind analysis: • = { RS , RN , A(dust,sync) }• we suppose that the CMB electromagnetic spectrum is known and we fix it
• Planck simulations:• LFI and HFI polarized channels: [30,40,70,100,143,217,353] GHz• 14 months nominal mission• complete sky coverage• infinite resolution • no systematics
Jonathan Aumont, LPSC Grenoble Moriond 2006
Blind separation (CMB + Foregrounds + Noise)
• CMB + Synchrotron + Dust + Noise• nside = 128
TT EE BB
TE TB EB
• Separation is efficient for TT, EE, TE, TB and EB• No detection of BB modes• Small bias in TT for l < 30
CMB
Jonathan Aumont, LPSC Grenoble Moriond 2006
TT EE BB
TE TB EB
• Separation is efficient for TT, EE, BB, TE, TB, and EB
DustBlind separation (CMB + Foregrounds + Noise) (2)
Jonathan Aumont, LPSC Grenoble Moriond 2006
TT EE BB
TE TB EB
• Separation is efficient for TT, EE, BB, TE, TB, and EB• Small bias in TT for l < 50
SynchrotronBlind separation (CMB + Foregrounds + Noise) (3)
Jonathan Aumont, LPSC Grenoble Moriond 2006
Blin
dB
lind,
ass
umin
g T
= E
= B
Dust Synchrotron CMB
Mixing matrix reconstruction (arbitrary units)
(GHz) (GHz) (GHz)
(GHz) (GHz) (GHz)
Jonathan Aumont, LPSC Grenoble Moriond 2006
Assuming A(T) = A(E) = A(B) (CMB + Foregrounds + Noise)
TT EE BB
TE TB EB
CMB
• Detection of BB modes for l < 50• No bias at low l in TT
Jonathan Aumont, LPSC Grenoble Moriond 2006
Semi-blind exploration of small angular scales (CMB + Fgds + Noise)
TT EE BB
TE TB EB
• Reconstruction of TT, TE, TB, EB up to l ~ 1500• Reconstruction of EE up to l ~ 1200• Reconstruction of BB up to l ~ 50
CMB• nside = 512
Jonathan Aumont, LPSC Grenoble Moriond 2006
Error bars of the reconstruction
CMB only / A fixedCMB + fgds / A(CMB) fixedCMB + fgds / Blind
• Presence of foregrounds increases the error bars by at least a factor of 2
TT EE BB
TE TB EB
Jonathan Aumont, LPSC Grenoble Moriond 2006
Conclusions
• Spectral matching algorithm extended to polarization to jointly deal with TT, EE, BB modes and also with cross power spectra TE, TB and EB We are able to separate blindly A, RN and RS, except for the CMB BB modes
• When we suppose A(T) = A(E) = A(B) we are able to recover CMB BB modes for l < 50 at 5
• Effect of the presence of foregrounds increases the error bars of the reconstruction. Decreases by addition of priors
• Improvements: - beam smoothing - filtering smoothing - incomplete sky coverage effect - components with anisotropic spectral index
[Aumont & Macias-Perez, 2006, submitted to MNRAS, astro-ph/0603044]
Jonathan Aumont, LPSC Grenoble Moriond 2006
Model of the microwave sky (2)
• Density matrix expressions:
• Example: 2 frequencies, 2 components data
Jonathan Aumont, LPSC Grenoble Moriond 2006
Formalism (2)
• Density matrices:
• Then data reads:
Likelihood maximization
• Bayes Theorem:
• Wiener solution:
Jonathan Aumont, LPSC Grenoble Moriond 2006
Sky maps simulations
• Thermal dust emission: • Dust power-law model [Prunet et al. 1998] :
• Normalized with respect to Archeops 353 GHz data [Ponthieu, …, Aumont et al. 2005]
• Galactic synchrotron emission:• Template maps for I, Q and U [Giardino et al. 2002]:• Isotropic spectral index ( -2.7 )
• CMB• Spectra generated with CAMB [Lewis et al. 2000] for concordance model with WMAP [Bennet et al. 2003] with gravitational lensing
• White noise maps for each frequency
Jonathan Aumont, LPSC Grenoble Moriond 2006
CMB power spectra
• CMB:Spectra generated with CAMB [Lewis et al. 2000] for: •
mb• • Gravitationnal lensing• r [10-4, 0.7]
Jonathan Aumont, LPSC Grenoble Moriond 2006
The mixing matrix, A
• Noise levels are relative noise levels with respect to the 143 GHz channel• At this frequency, noise levels are 6,3 KCMB (T) and 12,3 KCMB (E,B)per square pixels of side 7 arcmin and for a 14-months Planck survey
Jonathan Aumont, LPSC Grenoble Moriond 2006
Blind reconstruction of the noise at 100 GHz
• Efficient reconstruction of the noise power spectra for T, E and B for nside = 512
Jonathan Aumont, LPSC Grenoble Moriond 2006
TT EE BB
TE TB EB
• No bias at low l in TT
Synchrotron
Assuming A(T) = A(E) = A(B) (CMB + Foregrounds + Noise)
Jonathan Aumont, LPSC Grenoble Moriond 2006
Semi-blind exploration of small angular scales (CMB + Fgds + Noise)
TT EE BB
TE TB EB
• Reconstruction of TT, EE, BB, TE, TB, EB up to l ~ 1500
Dust• nside = 512
Jonathan Aumont, LPSC Grenoble Moriond 2006
Semi-blind exploration of small angular scales (CMB + Fgds + Noise)
TT EE BB
TE TB EB
Synchrotron• nside = 512
• Reconstruction of TT, EE, BB, TE, TB, EB up to l ~ 1500
Jonathan Aumont, LPSC Grenoble Moriond 2006
Reconstruction of the CMB BB modes (CMB + Fgds + Noise)
A(CMB) fixedA fixed
Jonathan Aumont, LPSC Grenoble Moriond 2006
Reconstruction of the CMB BB modes with SAMPAN
r = 10-3 r = 10-2 r = 10-1
• Satellite prototype experiment with polarized bolometers at 100, 143, 217, 353 GHz• Sensitivity 10 times better than Planck• Simulations with CMB + Dust