transparencies (pdf)

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


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 )


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


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


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


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



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



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


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


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




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


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


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


variance

Tristram et al. 2005

Chon et al. 2004

pure algorithmXpure Grain et al. 2009

fisher analysisfisher

figure by H. Nishino (KEK)


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

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transparencies (pdf)