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Jan 02, 2017

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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

    TT

    n( ) = almT Ylmlm 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 =

    4TYm

    aEm =

    4(Q,U)DEYm

    aBm =

    4(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(N3pix)

    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

    aXma

    Y m

    =

    MBX B

    Y FaXma

    Y m

    +

    nXmn

    Y 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

  • CEECBB

    =

    M+ M

    M M+

    CEECBB

    aEmaBm

    =

    W+ iWiW W+

    aEmaBm

    M. Tristram Rencontres de Moriond 2010B-mode power spectrum estimator

    E/B mixing

    CMB measurements are not full skycut 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 =

    4(Q,U)DB(WYm)

    =

    W DB(Q,U) Ym +

    C

    (Q,U) (WYm) +

    C

    (Q,U)WYm

    aBm =

    4M (Q,U)DBYm

    =

    DB(Q,U) Ym +

    C

    (Q,U)Ym +

    C

    (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) + 1,laB ,lm(1) + 0,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

    http://www.apc.univ-paris7.fr/~radek/s2hat.htmlhttp://www.apc.univ-paris7.fr/~radek/s2hat.html

  • 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

    CEECBB

    =

    M+ M

    M M+

    CEECBB

  • 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

    leakageCMB 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

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