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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

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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 +

(Q,U)× ∂(WYm) +

∂(Q,U)×WYm

aBm =

4πM · (Q,U)×DBYm

=

ΩDB(Q,U)× Ym +

(Q,U)∂Ym +

∂(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