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B-mode CMB spectrum estimation
pure pseudo cross-spectrum approach
J. Grain, M. Tristram, R. Stompor
Grain, Tristram, Stompor 2009 PRD 79 123515
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
B-mode physics
• large scale gaussian B-modes- signature of primordial gravitational waves- Energy scale of inflation
•small scale non gaussian B-modes- leakage from E to B due to CMB lensing- GR tests
• systematics- foreground residuals- instrumental effects- ...
Hu et al. 2003
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
CMB anisotropies in polarization
ΔTT
n( ) = almT Ylm
lm∑ n( )
Q ± iU( ) n( ) = ±2alm ±2Ylmlm∑ n( )
Cl
XY =1
2l +1almX alm
Y *
m=−l
l
∑
ClTT ,Cl
EE ,ClBB ,Cl
TE ,ClTB ,Cl
EB
X,Y( )∈ T ,E,B
U
Q
I
• Spherical harmonic coefficients
• Angular power spectra
spin 0
spin 2
contains all needed information if the field is gaussian
aTm =
4πT×Ym
aEm =
4π(Q,U)×DEYm
aBm =
4π(Q,U)×DBYm
- direct estimation of pseudo-spectra from data- debiasing from beam effect ( ) and cut-sky ( )
- very fast (requiring time)- frequently near-optimal in practice (in temperature !)
- from the pixel-pixel correlation matrix M
- computationally expensive, requiring time and memory- not adapted to large survey and high resolution maps
O(N2pix)O(N3
pix)
O(N3/2pix )
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
power spectra estimators
•maximum likelihood
•quadratic estimator (pseudo-Cl)
Bond et al. 1998, Tegmark 1997, Borrill 1999
Peebles & Hauser 1974, Wandelt et al. 2001, Hivon et al. 2002, Tristram et al. 2005
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
cross-spectra pseudo-Cl
instrumental effects decorrelated noise between detectors
aX
maY ∗m
=
MBX BY F
aX
maY ∗m
+
nX
mnY ∗m
•Compute cross-correlation between two independent maps
•After pre-processing of data, noise could be consider as not correlated from one map to another
no noise biasno noise Monte-Carlo needed
Kogut et al. 2003, Hinshaw et al. 2003, Tristram et al. 2005
CEE
CBB
=
M+
M−
M− M+
CEE
CBB
aE
maB
m
=
W+ iW−−iW− W+
aE
maB
m
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
E/B mixing
• CMB measurements are not full sky
cut sky mix E and B
- can achieve E-B separation on ensemble average (mixing matrices can be computed analytically)
- BUT for any realization : B variance feels leakage
• pseudo-Cl correct for mixing
Lewis et al. 2002, Bunn et al. 2003small-scale experiment simulation
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
why we need to separate ?
B-mode
1 10 100multipole
10-5
10-4
10-3
10-2
10-1
l(l+1
)/2pi
Cl
r=0.05
ModelFisher estimate
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
why we need to separate ?
B-mode
1 10 100multipole
10-5
10-4
10-3
10-2
10-1
l(l+1
)/2pi
Cl
r=0.05
ModelFisher estimateModelFisher estimatepseudo-Cl estimator
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
why we need to separate ?
B-mode
1 10 100multipole
10-5
10-4
10-3
10-2
10-1
l(l+1
)/2pi
Cl
r=0.01
r=0.05
r=0.10
ModelFisher estimateModelFisher estimatepseudo-Cl estimator
aBm =
4π(Q,U)×DB(WYm)
=
ΩW · DB(Q,U)× Ym +
CΩ
(Q,U)× ∂(WYm) +
CΩ
∂(Q,U)×WYm
aBm =
4πM · (Q,U)×DBYm
=
ΩDB(Q,U)× Ym +
CΩ
(Q,U)∂Ym +
CΩ
∂(Q,U)Ym
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
pure estimator
• change the base for harmonic decomposition to keep only B modes
Bunn et al. 2003Smith 2005
Smith & Zaldarriaga 2007
ambiguous modes (contains E and B)
choose W to zero contour integrals remove the leakage reduce the variance
• standard approach
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
Xpure implementation
• implementation of pure algorithm in two steps1. window function computation2. pure pseudo cross-spectrum estimator
• using Scalable Spherical HArmonic Transform package (S2HAT)- fully parallel (in CPU time and memory)- very fast : less than 30min for 1000 simulations on 1024 procs
aB ,lmpure = aB, lm
(2) + λ1,laB ,lm(1) + λ0,laB ,lm
(0)
W(r n )
W1(r n ) = ∇W (r n )
W2(r n ) = ∇2W (r n )
http://www.apc.univ-paris7.fr/~radek/s2hat.html
Grain, Tristram, Stompor PRD 2009
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
Analytic weighting
Optimal weighting
apodization
Smith & Zaldarriaga 2007 Grain, Tristram, Stompor PRD 2009
• need for an appropriate apodization for spin 0,1,2 numerical derivatives (pixelization and edge issues)
• we propose and compare several type of apodization with any shape of the sky patch
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
mixing
M+ M−
standard
pure
• drop the ambiguous modes remove the leakage reduce the variance
CEE
CBB
=
M+
M−
M− M+
CEE
CBB
grain
10 100 1000multipole
10-8
10-6
10-4
10-2
100
102
l(l+
1)/2
pi C
l
B
E
model (r=0.05)
pure
standard
8 deg
5 deg
3 deg
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
leakage•CMB simulations without B mode for a small-scale B-mode dedicated experiment
- sky coverage 1%- illustrate the leakage from E to B in std and pure
E to B leakage(standard estimator)
B mode signal (r=0.05)
E to B leakage(pure estimator)
standard pseudo-ClXpolSpicePol
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
variance
Tristram et al. 2005
Chon et al. 2004
pure algorithmXpure Grain et al. 2009
fisher analysisfisher
figure by H. Nishino (KEK)
M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator
conclusions
•B-mode studies require specific power spectrum algorithm
•pure pseudo cross-spectra methods are well adapted- fast : (Npix)3/2 au lieu de (Npix)3
- allow for large MonteCarlo- pure estimator : accurate enough for B-mode detection
•Xpure is used for B-mode dedicated experiments- EBEx : balloon-borne- PolarBear : ground based
Oxley et al. 2004
Grain, Tristram, Stompor PRD 2009