Hybrid Estimation of CMB Polarization Power Spectra

0.92 MB·English

by  G. Efstathiou

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev1.4)
Hybrid Estimation of CMB Polarization Power Spectra
G. Efstathiou
Institute of Astronomy, Madingley Road, Cambridge, CB3 OHA.
5February2008
6 ABSTRACT
0 This paper generalises the hybrid power spectrum estimator developed in Efstathiou
0
(2004a)totheestimationofpolarizationpowerspectraofthecosmicmicrowaveback-
2
groundradiation.Thehybridpowerspectrumestimatorisunbiasedandweshowthat

n it is close to optimal at all multipoles, provided the pixel noise satisfies certain rea-
a
sonable constraints. Furthermore, the hybrid estimator is computationally fast and
J
can easily be incorporated in a Monte-Carlo chain for Planck-sized data sets. Sim-
5
ple formulae are given for the covariance matrices, including instrumental noise, and

1 thesearetestedextensivelyagainstnumericalsimulations.Wecomparethebehaviour
v of simple pseudo-Cℓ estimates with maximum likelihood estimates at low multipoles.
7
Forrealisticskycuts,maximumlikelihoodestimatesreduceverysignificantlythemix-
0 ingofE andB modes.Toachievelimitsonthescalar-tensorratioofr ≪0.1fromsky
1
1 maps with realistic sky cuts, maximum likelihood methods, or pseudo-Cℓ estimators
0 based on unambiguous E and B modes, will be essential.
6
0 Key words: Methods: data analysis,statistical; Cosmology:cosmic microwaveback-
/ ground, large-scale structure of Universe
h
p
-
o
r 1 INTRODUCTION
t
s
a In an earlier paper (Efstathiou 2004a, hereafter E04) a hybrid power spectrum estimator was developed for temperature
:
v anisotropiesofthecosmicmicrowavebackground(CMB)anisotropies.Thisestimatorcombinedquadraticmaximumlikelihood
i (QML) estimates at low multipoles with a set of pseudo-C (PCL) estimates at higher multipoles to produce a near optimal
X ℓ
power spectrum estimate over theentire multipole range for realistic sky coverages, scanning patterns and instrument noise.
r
a We used analytic arguments and large numbers of numerical simulations to demonstrate that the method was near optimal
and that the covariance matrix overthe full range of multipoles could be estimated simply and accurately.
A full maximum-likelihood estimator would require the inversion and multiplication of N N matrices, where N is
d d d
×
thesize of the data vector. For WMAP (Bennett et al. 2003a) or Planck (Bersanelli et al. 1996; The Planck Consortia 2005)
sized data sets, with N > 106 107, a brute force application of (N3) methods is impossible computationally. This has
d ∼ − O d
been portrayed as a major computational challenge in CMB data analysis (e.g. Bond et al. 1999; Borrill 1999) and various
approximate,but computationally demanding,techniquesfor solving thisproblem havebeen proposed (e.g. Oh,Spergeland
Hinshaw 1999, Dor´e, Knox and Peel (2001), Pen 20003). One of the major motivations for E04 was to show that an (N3)
O d
computationisunnecessaryandthatanear-optimalestimator(essentiallyindistinguishablefromanexact (N3)maximum-
O d
likelihood solution), with a calculable covariance matrix, could be estimated simply by combining PCL and QML estimates.
Furthermore, the analysis of realistic experiments must deal with uncertainties such as correlated instrument noise, beam
calibrations, foreground separation, point sources etc. It is simply not worth performing an exact (N3) power spectrum
O d
analysis if the assumptions under which it is optimal are violated by ‘real world’ complexities, and if these complexities
cannot be folded into the error estimates. With a fast hybrid estimator it is feasible to assess such ‘real-world’ complexities
usingahybridestimator within aMonte-Carlo chainand toquantifyanycorrrections tothecovariance matrix,orlikelihood
function.The ability to analyse such‘real-world’ effectsis likely tobe muchmore important than anyhypothetical marginal
improvement that might be gained by applying a full (N3) method. We refer the reader to E04 for a discussion of the
O d
rationale for a hybrid estimator. The arguments will not be repeated here. Instead, we will focus on generalising E04 to the
estimation of power spectra of theCMB polarization anisotropies.
Asiswell-known,theThomsonscatteringofananisotropicphotondistributionleadstoasmallnetlinearpolarization of
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2 G. Efstathiou
the CMB anisotropies (for an introductory review and references to earlier work see Hu and White 1997). This polarization
signal can be decomposed into scalar E-modes and pseudo-scalar B-modes. The separation of a polarization pattern into E
and B modes is of particular interest since scalar primordial perturbationsgenerate only E mode while tensor perturbations
generate E and B modes of roughly comparable amplitudes (Zaldarriaga and Seljak 1997; Kamionkowski, Kosowsky and
Stebbins 1997). The detection of an intrinsic B-mode signal in the CMB would provide incontrovertible evidence that the
Universeexperienced an inflationary phase and would fix theenergy scale of inflation (see e.g. Lyth 1984).
⋆
AnE-modepolarization signalwas firstdiscovered byDASI (Kovacet al. 2002, Leitchet al. 2004). Exquisitemeasure-
mentsofthetemperature-E-mode(TE) crosspowerspectrum havebeenreportedbytheWMAP† team (Kogut et al. 2003).
MeasurementsoftheE-modepowerspectrumhavebeenreportedbytheCBI‡ experiment(Readheadetal.2004)andbythe
2003flightofBoomerang(Montroyetal.2005).PrimordialB-modeanisotropieshavenotyetbeendetectedintheCMB.The
detection of E-mode and possibly B-mode anistropies is one of the main science goals of the Planck mission and of ground
based experimentssuch as Clover (Taylor et al. 2004).
Associatedwiththeseexperimentaldevelopments,therehavebeenmanyinvestigationsoftechniquesforanalysingE and
B modes from maps of the CMB sky.These analyses can be grouped, approximately,into thefollowing catagories:
(i) PCL estimators and correlation functions: Fast methods of estimating E and B-mode power spectra using correlation
functions are described by Chon et al. (2004). Statistically equivalent PCL estimators are described by Kogut et al. (2003),
Hansen andGorski (2003), Challinor and Chon (2005, hereafter CC05) and Brown et al. (2005). Inparticular, CC05 present
analyticapproximationstothecovariancematricesofPCLestimatesforthecaseofnoise-freedata,whileBrownet al.(2005)
developa Monte-Carlo method for calibrating covariance estimates from incomplete mapsof thesky.PCL estimators can be
evaluated using fast spherical transforms and hencescale as (N3/2).
O d
(ii) Maximum Likelihood Methods: Thegeneralization oftheiterativemaximumlikelihood powerspectrumestimation meth-
odsofBondetal.(1998)totheanalysisofpolarisationisstraightforwardandwillnotbediscussedfurtherhere.Tegmarkand
deOliveira-Costa(2001,hereafterTdO01)defineaquadraticestimatorwhichisbasedonassumedformsforthetemperature
and polarization power spectra, generalising earlier work of Tegmark (1997). This method,which we will refer toas QML, is
equivalent to a maximum likelihood solution if the guesses for the power spectra are close to their true values. As explained
above, thesemethods involvematrix inversions and multiplications which scale as (N3).
O d
(iii) Harmonic E and B Mode Decomposition: The Stokes’ parameters Q and U describing linear polarization define a rank
two symmetric trace-free polarization tensor on sphere. Over the complete sky, the polarization tensor can be decomposed
uniquely into E and B modes. However, since the E and B decomposition is non-local, it is non-unique in the presence of
boundaries. In any realistic situation, a sky cut must beimposed toexclude contamination of the CMB signal by high levels
of Galactic emission at low Galactic latitudes. Various authors have discussed ways of detecting pure E and B modes from
Q and U maps on an incomplete sky (Lewis et al. 2002; Bunn et al. 2003; Bunn 2003; Lewis 2003). A key motivation for
these analyses has been for diagnostic purposes (e.g. checking for a frequency dependent Galactic B-mode signal). However,
as thisarticle was nearingcompletion an interesting paper appeared by Smith(2005) describing howto construct pseudo-C
ℓ
estimators from unambiguous E and B modes on a cut sky. This type of analysis can effectively eliminate the severe E and
B mode mixing that afflicts simple pseudo-C estimators at low multipoles (see Section 3.4). (The method is not completely
ℓ
straightforward,norunique,becauseitrequiresa‘pre-estimation’steptodefineunambiguousmodes.Wewillthroughoutthis
paperuse theabbreviation PCL to refer to thesimple pseudo-C estimators as defined in Section 2.)
ℓ
As we will show in this paper, these three analysis techniques are closely related. To illustrate how these methods are
interelated it is useful to consider separately thecases of noise-free and noisy data:
Comparison of PCL and QML estimators for noise-free data: On a complete sky with noise-free data, PCL and QML
•
estimatesareidentical.However,askycutwillcoupleCMBmodesoverarangeofmultipoles∆L,andthiswillleadtomixing
ofE andB pseudo-multipolesdefinedovertheincompletesky(seeSection2).AlthoughPCLpowerspectrumestimatorscan
bedefinedwhichgiveunbiasedestimatesofthetrueE andB modepowerspectrainthe mean,themixingofE andB modes
is reflected in large variances and couplings between the estimated power spectra at multipoles ℓ < ∆L (Section 2). A QML
∼
estimator applied to noise-free data unscrambles the E- and B-mode multipoles, in much the same way as the direct modal
decompositionsdescribedbyLewiset al. (2002),Lewis(2003) andSmith(2005).TheQMLestimatorreturnsalmost optimal
power spectrum estimates with smaller variances than a PCL estimator (Section 3). In particular, for realistic sky cuts an
estimatorthatminimisesE andB modemixing,suchastheQMLestimator,isessentialifonewantstoprobelowamplitude
B-modesignals(tensor-scalarratiosr 0.1).Athighmultipoles,ℓ ∆L,E andB modemixingbecomesunimportantand,
≪ ≫
in the case of noise free data, QML and PCL estimators become statistically equivalent (Section 3).
⋆
DegreeAngularScaleInterferometer
† WilkinsonMicrowaveAnisotropyProbe
‡ CosmicBackgroundImager
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Polarization Power Spectrum Estimation 3
Comparison of PCL and QML estimators for noisy data: If the following conditions are satisfied: (a) the Q and U maps
•
have identical noise properties; (b) the noise is uncorrelated with variance per pixel σ2; (c) the estimates of E and B mode
i
powerspectraareformultipoleshigherthansomecharactersticmultipoleL ,wherenoisedominates;thenonecanshowthat
N
aninverse-varianceweightedPCLestimatorisstatistically equivalenttoaQMLestimator(Section4.2).Fortheintermediate
multipoles,L < l < ∆L,theoptimalweightingforPCLestimatesisintermediatebetweenequalweightperpixelandinverse
N
∼ ∼
variance weighting. By combining a set of PCL estimates with different weights, it is possible to define a fast polarization
estimator(inanalogywiththetemperatureestimatordiscussedbyE04)thatisstatisticallyindistinguishablefromamaximum
likelihood estimatoroverawiderangeofmultipoles. Dealingwith stronglycorrelated noiseismoreproblematic.Fortunately,
forPlanck-typescanningstrategies,thenoisepattern(inQ,U andT)shouldbeaccuratelywhiteathighmultipoles(Efstathiou
2005).Adetailedcorrelatednoisemodelshouldthereforeonlyberequiredatlowmultipoles.Itisstraightforwardtoincludea
modelforcorrelated noisein aQMLestimate at lowmultipoles, providedthepixelnoisecovariancematrix can beestimated
for low resolution maps (Section 4.2).
Thelayoutofthispaperissimilar tothatforthetemperatureanalysis presentedinE04.PCL polarization estimatesare
discussed in Section 2 together with analytic approximations for covariances matrices in the absence of instrumental noise.
The relationship of this work to the results presented in CC05 is discussed. Section 3 discusses QML polarization estimators
for noise free data. A large set of numerical simulations is used to illustrate the effects of E and B mode mixing on an
incomplete sky for both PCL and QML estimators. Section 4 discusses PCL and QML estimators including instrumental
noise and Section 5 discusses a hybrid polarization power spectrum estimator. To keep the discussion simple, most of the
analytic and numerical results presented in this paper refer to E and B mode power spectrum estimation. Our discussion
of the temperature-E mode cross power spectrum (denoted CX in this paper) is, intentionally, less complete since no new
concepts are required for its analysis. Ourconclusions are summarized in Section 6.
2 ESTIMATION USING PSEUDO-C
ℓ
We begin by relating the Q and U Stokes parameters (defined in direction nˆ with respect to a spherical coordinate system
ˆe ,ˆe ) and thespin two harmonics:
θ φ
Q(nˆ) iU(nˆ)= a Y (nˆ). (1)
±2ℓm±2 ℓm
±
ℓm
X
Note that the sign convention for polarization in this paper follows that of Zaldarriaga and Seljak (1997), which differs from
the IAU convention (see Hamaker and Bregman 1996). The coefficients a are related to the E and B mode multipole
±2ℓm
coefficients aE and aB by
ℓm ℓm
a = (aE iaB ). (2)
±2ℓm − ℓm± ℓm
If we have an incomplete sky, we can compute ‘pseudo-multipole’ coefficients a˜T , a˜E and a˜B by computing the following
ℓm ℓm ℓm
sums,
a˜T = ∆T w Ω Y∗ (θ ), (3a)
ℓm i i i ℓm i
i
X
1 1
a˜E = (Q+iU) w Ω Y∗ +(Q iU) w Ω Y∗ = (Q R+∗+iU R−∗)w Ω , (3b)
ℓm −2 i i i 2 ℓm − i i i −2 ℓm −2 i ℓm i ℓm i i
i i
X X
i i
a˜B = (Q+iU) w Ω Y∗ (Q iU) w Ω Y∗ = (Q R−∗+iU R+∗)w Ω , (3c)
ℓm 2 i i i 2 ℓm− − i i i −2 ℓm 2 i ℓm i ℓm i i
i i
X X
where in equations (3b) and (3c)
R+ = Y + Y , R− = Y Y , (4)
ℓm 2 ℓm −2 ℓm ℓm 2 ℓm− −2 ℓm
Ω is the area of pixeli and w is an arbitrary weight function with spherical transform
i i
w˜ = w Ω Y∗ (θ ). (5)
ℓm i i ℓm i
From thXesepseudo-multipole coefficients, we can form the following PCL power spectrum estimates
1 1 1 1
C˜T = a˜T a˜T∗, C˜X = a˜T a˜E∗, C˜E = a˜E a˜E∗, C˜B = a˜B a˜B∗. (6)
ℓ (2ℓ+1) ℓm ℓm ℓ (2ℓ+1) ℓm ℓm ℓ (2ℓ+1) ℓm ℓm ℓ (2ℓ+1) ℓm ℓm
m m m m
X X X X
The expectation valuesof thesePCL estimates are related to thetrue valuesCT, CX, CE, CB by
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4 G. Efstathiou
Figure 1. An example of the input maps used in the simulations described in Section 2. The pixel size is θc = 1◦ and the CMB sky
hasbeensmoothedwithaGaussianbeamofFWHMθs=2◦.TheconcordanceΛCDMmodelwiththeparametersgiveninthetexthas
been assumed anda tensor component with r=0.2has been added to give annon-zero B-modecontribution. Figures (a), (c) and (d)
showtheT,Q,andU maps,whileFigure(b)showstheWMAPKp0mask.
C˜T MT 0 0 0 CT
h i
C˜X 0 MX 0 0 CX
 h i =  , (7)
C˜E 0 0 MEE MEB CE
h i
 C˜B   0 0 MBE MBB  CB 
 h i    
    
where thematrices MT, MX, etc are given by (Kogut et al. 2003)
2
(2ℓ +1) ℓ ℓ ℓ
MT = 2 (2ℓ +1)W˜ 1 2 3 , (8a)
ℓ1ℓ2 4π 3 ℓ3 0 0 0
Xℓ3 (cid:18) (cid:19)
(2ℓ +1) ℓ ℓ ℓ ℓ ℓ ℓ
MX = 2 (2ℓ +1)W˜ (1+( 1)L) 1 2 3 1 2 3 , L=ℓ +ℓ +ℓ , (8b)
ℓ1ℓ2 8π 3 ℓ3 − 0 0 0 2 2 0 1 2 3
Xℓ3 (cid:18) (cid:19)(cid:18) − (cid:19)
2
(2ℓ +1) ℓ ℓ ℓ
MEE =MBB = 2 (2ℓ +1)W˜ (1+( 1)L)2 1 2 3 , (8c)
ℓ1ℓ2 ℓ1ℓ2 16π 3 ℓ3 − 2 2 0
Xℓ3 (cid:18) − (cid:19)
2
(2ℓ +1) ℓ ℓ ℓ
MEB =MBE = 2 (2ℓ +1)W˜ (1 ( 1)L)2 1 2 3 , (8d)
ℓ1ℓ2 ℓ1ℓ2 16π 3 ℓ3 − − 2 2 0
Xℓ3 (cid:18) − (cid:19)
and W˜ is the power spectrum of theweight function
ℓ
1
W˜ = w˜ 2. (9)
ℓ (2ℓ+1) | ℓm|
Xm
(cf equation (5)). In the absence of parity violating physics in the early universe, the primordial CTB and CEB spectra
shouldbeidenticallyzero.Evenifthereisnocompellingmotivationfromfundamentalphysics,theremaybeotherreasonsfor
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Polarization Power Spectrum Estimation 5
Figure 2.PCLpolarizationpower spectra.Thebluepoints showtheconvolved powerspectra,C˜ℓ,averaged over105 simulationswith
thesameparametersasthoseinFigure1andincludingtheKp0mask.ThegreenpointsshowthedeconvolvedspectraCˆℓ.Thesolidlines
showtheformsofC˜ℓ andCˆℓ computedfromthetheoreticalinputspectra.ThedashedlinesinFigures(a)–(c) showthecontributionto
Cˆℓ fromthetensorcomponent.
wantingtoestimatethesespectra, e.g.for consistency checksandtotest forsystematic errors. However,tokeeptheanalysis
as simple as possible, they will benot beincluded in this paper.
If the sky cut is small, then the matrix M appearing in equation (7) will be non-singular and hence invertible. If this is
thecase, then unbiased estimates of thetruepower spectra can be formed from thePCL estimates by evaluating §
Cˆℓ =Mℓ−ℓ′1C˜ℓ′. (10)
Asanexample,Figure1showsnoise-freeT,QandU mapsfromasingleGaussianrealisation oftheconcordanceΛCDM
modelfavouredbyWMAP(Spergelet al. 2003). (Theexact parametersadoptedinthispaperarethoseoftheΛCDMmodel
defined in Section 2 of Efstathiou (2003)). The maps have been created using software written by the author that uses an
‘igloo’ pixelisation scheme. Themapsshown in Figure 1haveapixelsize of 1◦ anda symmetrical Gaussian beam smoothing
of FWMH of θ = 2◦. In these simulations, we have assumed a tensor-scalar ratio of r = 0.2, where r is defined in terms of
s
therelative amplitudes of theensemble averages of thetemperature power spectra at ℓ=10,
CTtensor
r= ℓ=10 . (11)
CTscalar
ℓ=10
AsinE04, unlessotherwisestated,beam functionswillnot bewritten explicitlyinequationsandsoC will sometimes mean
ℓ
C b2, where b is theGaussian beam function
ℓ ℓ ℓ
1
b =exp ℓ(ℓ+1)(0.425θ )2 . (12)
ℓ −2 s
(cid:16) (cid:17)
Figure 1 also shows one of the Kpn family of WMAP masks (see Bennett et al., 2003b, for a discussion of the WMAP
masks). The Kp0 mask shown in Figure 1 removes about 21 per cent of the sky at low Galactic latitudes and is a relatively
conservativeGalacticmask.(Forreference,theWMAPTEanalysisreportedbyKogutetal.(2003)usedthelessconservative
Kp2mask which removes around 13 per cent of thesky).
Figure 2 shows the averages of PCL estimates C˜ and Cˆ computed from 105 noise-free simulations with the same
ℓ ℓ
cosmological parameters usedtoconstructFigure1andwiththeKp0skymask applied.FortheKp0mask,thematrix M in
§ Thenotation herefollowsthenotation introduced inE04. Theexpectation values ofthe hatted PCLestimates (Cˆℓ)(ifthey arewell
defined) are equal to the true power spectra. The expectation values of the tilde estimates (C˜ℓ) are given by convolutions of the true
powerspectra(equation 7).
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6 G. Efstathiou
Figure 3.Thediagonalcomponents ofthePCLpowerspectrum covariancematrixestimated fromnumericalsimulationscomparedto
the theoretical dispersions given in equations (15b) – (15d). The diagrams to the left show results for the unapodised Kp0 mask and
thosetotherightshowresultsfortheapodisedKp0maskillustratedinFigure4.
equation (10) is non-singular and can be inverted. Figure 2 shows that the PCL estimates Cˆ provide unbiased estimates of
ℓ
thetruepower spectra.
For the temperature polarization power spectrum, accurate expressions for the covariance matrices of PCL power spec-
trum at high multipoles (ℓ ∆L) can be derived quite easily (see E04). The analogous problem for polarization esti-
≫
mates is much more difficult (see CC05) because the covariances depend on the matrix products ±I(ℓm)(LM) ±I(LM)(ℓ′m′),
±I(ℓm)(LM) ∓I(LM)(ℓ′m′), where
1
±I(ℓm)(ℓ′m′) = 2dnˆw(nˆ)(2Yℓ∗m(nˆ)2Yℓ′m′(nˆ)± −2Yℓ∗m(nˆ)−2Yℓ′m′(nˆ)). (13)
Z
CC05 show that these products can be expressed as integrals ±I(ℓm)(ℓ′m′) as in equation (13), but with the window finction
w replaced byw2 andintegralsinvolvingthegradientsofw.ForPCL estimatesfrom noisefreedata,thecovariancematrices
canthereforebeapproximatedbytermswhichdependon thecouplingmatrices inequations(8a-8d),butwith W˜ replaced
ℓ
bythe power spectrum of thesquare of the window function,
w˜2 = w2Ω Y∗ (θ ), (14)
ℓm i i ℓm i
X
together with complicated terms that depend on gradients of the window functions wi.¶ Ignoring the gradient terms, the
covariances can be approximated by
h∆C˜ℓT∆C˜ℓT′i≈ (22Cℓ′ℓT+C1ℓT′)MℓTℓ′, (15a)
¶ Therearenosuchgradienttermsforthetemperaturecovariancematrixh∆C˜ℓT∆C˜ℓT′i,seeE04foradetailedanalysis.
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Polarization Power Spectrum Estimation 7
Figure 4.ApodisedKp0maskatapixelresolutionofθc=1◦ createdusingtheiterativealgorithmdescribedinthetext.
h∆C˜ℓX∆C˜ℓX′i≈ (CℓTC(2ℓT′ℓC′+ℓEC1)ℓE′)1/2MℓXℓ′ + (C2ℓℓX′+CℓX1′)MℓTℓ′, (15b)
h∆C˜ℓE∆C˜ℓE′i≈ (22Cℓ′ℓE+C1ℓE′)MℓEℓ′E, (15c)
h∆C˜ℓB∆C˜ℓB′i≈ (22Cℓ′ℓB+C1ℓB′)MℓBℓ′B, (15d)
h∆C˜ℓE∆C˜ℓB′i≈ [(CℓECℓE′)12/(22ℓ+′+(C1ℓB)CℓB′)1/2]2MℓEℓ′B. (15e)
The covariance matrix of the PCL estimates Cˆ is given by
ℓ
h∆Cˆℓ∆Cˆℓ′i=M−1h∆C˜ℓ∆C˜ℓ′i(M−1)T, (16)
where M is the matrix appearing in equation (7) (i.e. computed from W˜ , rather than from the power spectrum of w2).
ℓ i
Neglect of the gradient terms will be referred to somewhat loosely as the ‘scalar approximation’, since the expressions (15a-
15e) depend on only the spin-0 multipoles of the square of the window function (14). It is interesting to ask under what
circumstances thescalar approximation provides accurate estimates of the covariances. For relatively small sky cuts, such as
theKp0mask,theseexpressionsprovidequiteaccurateestimatesfortheX andE powerspectraatmultipolesℓ > ∆L,where
∼
∆Lis themode-coupling scale introduced bytheskycut. Theleft-hand panels in Figure 3show thediagonal componentsof
thecovariancematricesfortheC˜X,C˜E andC˜B powerspectraestimatedfromequations(15b)-(15d)comparedtoestimates
ℓ ℓ ℓ
from the105 simulations used to generate Figure 2. For the Kp0 mask, the characterstic coupling scale is ∆L 20, and one
∼
can see from Figure 3 that the simple analytic expressions provide quite accurate estimates of the errors of the X and E
power spectra at multipoles ℓ > 50, but fail by a factor of 2 or more for the B component. The failure of simple analytic
∼ ∼
expression for the B-component is related to mixing of the E and B modes. For the E power spectrum, mixing of E and B
modesisalwaysunimportantathighmultipoles,ℓ ∆L(sincetheE-modeamplitudeisfixedbythedominantscalarmode).
≫
WethereforeexpectthescalarapproximationtoprovideaccurateestimatesofthecovariancesfortheX andE powerspectra
at high multipoles. However, if the intrinsic amplitude of the B mode is low, mixing of E and B modes can dominate the
estimatesofC˜ℓB andthecovariancesh∆C˜ℓB∆C˜ℓB′iandh∆C˜ℓE∆C˜ℓB′iathighmultipoles.Forourchosentensor-scalaramplitude,
r=0.2, mode-mixing dominates the C˜B amplitude at high multipoles, which is why thescalar approximation fails so badly.
ℓ
Intuitively, one would expect that it would be possible to significantly reduce the effects of E and B mode mixing at
highmultipolesbyapodisingtheskycut.Apodisation down-weightsdataclosetotheedgewherethenon-localnatureofthe
E and B mode decomposition causes ambiguity between modes (cf CC05). Thus we would expect the scalar approximation
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8 G. Efstathiou
Figure5.CovariancematricesfortheB-modepolarizationspectra,illustratingtheeffectsofapodizationonthePCLestimator.Figures
5(a)and5(c)showthecovariancematricescomputedfrom105simulationswiththesameparametersasthoseinFigure1.Thesimulations
usedinFigure5(a)usetheunapodisedKp0maskwhilethoseusedinFigure5(c)usetheapodisedKp0maskshowninFigure4.Figures
5(b)and5(d)showthecorrespondinganalyticcovariancematricesusingtheapproximationofequation(15d)
to be much more accurate if we apodise the Kp0 mask. Of course, in this context apodisation will lead to some information
loss, but in practice the purpose of applying a mask is to reduce the effects of systematic errors caused by inaccuracies in
modelling Galactic emission. Hence, the data close to the edge of a mask will usually be less reliable than data far removed
from the mask. The slight loss of information caused by apodising is therefore likely to be more than compensated by the
reduction in E and B mode mixing at high multipoles.
Figure 4 shows an example of an apodisation algorithm applied to the Kp0 mask. The apodisation algorithm works
iteratively as follows. First, the Kp0 mask in this pixelisation is specified by w = 0 or 1 according to whether pixel i lies
i
within or outside themask. Then:
(i) compute the spherical transform of the set of weights, w˜ , (equation 5) and multiply these coefficients by a smoothing
ℓm
factor of b (equation 12) with Gaussian FWHM of θ =10◦;
ℓ s
(ii) construct a smoothed weight function w˜ byperforming an inversespherical transform;
i
(iii) set theweights w˜ that lie within the original Kp0 mask equal to zero;
i
(cid:13)c 0000RAS,MNRAS000,000–000
Polarization Power Spectrum Estimation 9
(iv) go back to step (i) and transform the smoothed weights to make a new set of coefficients w˜ and continue steps (i) –
ℓm
(iv) for a specified set of iterations.
TheexampleshowninFigure4showstheapodised maskafter4iterations. Theeffectivesmoothinglengththereforehas
aGaussianFWHMofθ =20◦ andonecanseethatthe‘islands’interiortothemaskedregionevidentinFigure1bhavebeen
s
smoothed tolowamplitudes. Theright-handpanelsof Figure 3showthediagonal componentsof thecovariancematrices for
the X, E and B mode PCL power spectra of the apodised maps compared to the predictions of the scalar approximation.
TheX andE dispersions nowagree almost perfectly withthescalar approximationfor ℓ > 30.FortheB mode,thevariance
caused by E and B mode mixing at ℓ > ∆L is much less than for the unapodised case s∼hown in Figure 3. For the diagonal
∼
components of the B mode power spectrum, the scalar approximation is accurate to a few percent or so. Figure 5 shows
the structure of the B mode covariance matrices for the unapodised and the apodised Kp0 masks. In the apodised case, the
covariancematrix isdiagonally dominantatℓ > 20andthediagonalcomponentsareaccuratelydescribed byequation(15d).
∼
The scalar approximation is therefore a very good approximation for all three power spectra X, E and B. Of course,
even with apodisation, thescalar approximation for the B modespectrum will fail at some critical valueof thetensor-scalar
ratio r, hence one cannot simply rescale equation (15d) by the appropriate amplitude of CB for arbitrarily low values of r.
ℓ
For low values of r, one can explicitly include the gradient terms which depend on the amplitude of CℓECℓE′ (equation (80)
of CC05) to get an accurate model of the covariance matrix at high multipoles. However, for most forseable experiments,
including Planck, instrument noise is likely to dominate the B-mode covariance matrix at high multipoles. In Section 4 we
will showthat in thissituation thescalar approximation is extremelyaccurate independentof theintrinsicamplitude of CB.
ℓ
3 QUADRATIC MAXIMUM LIKELIHOOD
3.1 Preliminaries
A QML estimator for temperatureCMB power spectra is discussed by Tegmark (1997). The generalization of this estimator
totheestimation ofCMBpolarization isstraightforward andisdiscussedindetailbyTdO01.Theestimatorwillbereviewed
here briefly. We work in pixel space and define an input data vector x consisting of the temperature differences and Stokes
parameters Q and U (definedwith respect to a fixed coordinate system, as in Figure 1) specified at each pixel,
x (∆T,Q,U). (17)
≡
The optimal QML power spectrum estimate is (TdO01)
yr =x x Erℓ, r (T,X,E,B), (18)
ℓ i j ij ≡
where thematrices Erℓ are given by
1 ∂C
Erℓ= C−1 C−1, (19)
2 ∂Cr
ℓ
and C is thecovariance matrix of thedata vector x,
CTT CTQ CTU
Cij = xixj = CQT CQQ CQU . (20)
h i  
CUT CUQ CUU
 
ThematricesErℓ definedinequation(19)giveformallyminimumvariancepowerspectrumestimatesifthecovariancematrix
C issetequaltothetruecovariancematrix.However,asnotedbyTdO01,equation(18)mixes∆T componentsofthedata
ij
vector x with Q and U components in providing estimates of the E and B mode power spectra. This is undesirable because
forrealistic noisy data,systematic errorsinthe∆T measurementscouldcontaminateestimates of themuchlower amplitude
E and B mode power spectra. It is therefore safer to separate the ∆T measurements from the Q and U measurements by
‘reshaping’ thematrix (20):
CTT 0 0
Cˇ = 0 CQQ CQU (21)
ij
 
0 CUQ CUU
 
and using thematrices
1 ∂C
Eˇrℓ= Cˇ−1 Cˇ−1. (22)
2 ∂Cr
ℓ
in place of the matrices Erℓ in equation (18). The estimates yr will give unbiased estimates of the true power spectra Cs,
ℓ ℓ
s (T,X,E,B),
≡
(cid:13)c 0000RAS,MNRAS000,000–000
10 G. Efstathiou
hyℓri=Fˇℓsℓr′Cℓs′, (23a)
where
1 ∂C ∂C
Fˇℓsℓr′ = 2Tr ∂Cs Cˇ−1∂CrCˇ−1 . (23b)
(cid:20) ℓ′ ℓ (cid:21)
By reshaping thecovariance matrix, theestimate (18) will nolonger beminimum variance. However,it isstraightforward to
show that for a Kp0-typemask, the increase in variance caused byusing (21) instead of (20) is almost imperceptible.
Following E04 it is useful todefinere-scaled power spectra
C˜ℓr =yℓr/ Fˇℓrℓr′, (24)
Xℓ′
which have similar shapes to the true power spectra Cr. Furthermore, if the matrix Fˇ is invertible, one can define unbiased
ℓ
estimates of thetrue power spectra via
Cˆr =Fˇ−1y. (25)
ℓ
These estimators can be considered theQML analogues to thePCL estimators definedby equations (6) and (10).
The covariance matrices of theQML estimates is given by
hyℓryℓs′i−hyℓrihyℓs′i≡Fℓrℓs′ =2Tr CEˇrℓCEˇsℓ′ , (26)
h i
where Frs is theFisher matrix. From equation (25), thecovariance matrix of the deconvolved QML estimates is given by
h∆Cˆℓ∆Cˆℓ′i=Fˇ−1FFˇ−1. (27)
Noticethat ifwehad usedthetruepixelcovariance matrix (20),ratherthan the‘reshaped’ form (21), thecovariance matrix
(26) would takethe more familiar form
1 ∂C ∂C
Fℓsℓr′ = 2Tr(cid:20)∂Cℓs′C−1∂CℓrC−1(cid:21). (28)
Notice also from equations (18) and (22) that theestimator yr can be written as
ℓ
(2ℓ+1)
yr = C˜zr, (29)
ℓ 2Ω2 ℓ
where theC˜zr are the PCL power spectra as definedin equation (6) but computed from themaps
z =Cˇ−1x . (30)
j ji i
(Ω in equation (29) is the solid angle of a single map pixel, all assumed to be of identical area) Thus, if one can invert
the matrix Cˇ−1, the QML estimator can be computed rapidly by applying fast spherical transforms to the maps defined
ij
by equation (30). This is particularly useful if one wants to compute Monte-Carlo simulations of the QML estimator to
characterisethecovariancematrix.Forapplicationstomapswithlargenumbersofpixels,itmaywellbemoreefficienttouse
Monte-Carlo simulations to estimate errors, rather than brute force evaluation of (26), particularly if the covariance matrix
(27) is accurately band diagonal (which is true for Kp0-like sky masks, see Figure 7 below).
3.2 Covariance Matrix at High Multipoles
InthisSection,wewill assume thatCB =0andderiveanapproximation tothecovariance matrixof CˆE forhigh multipoles
ℓ ℓ
andasmallsky-cut.TheanalysispresentedhereissimilartotheanalysisoftheQMLtemperaturepowerspectrumcovariance
matrix given in Section 3.4 of E04.
If the B-mode is absent, all of the information on the polarization anisotropies is contained in the spin 2 field P
2 i
≡
(Q +iU ).Thespin 2field(Q iU )containsnoadditionalinformation. Wecanthereforeconstructaquadraticestimator
i i i i
− −
yE = P w P∗w E , (31)
ℓ 2 i i2 j j ij
where E is as defined in equation (19) with,
ij
C = P w P∗w = CEw w Y (i) Y∗ (j). (32)
ij h2 i i 2 j ji ℓ i j 2 ℓm 2 ℓm
Xℓm
wherew isan arbitrary pixelweight function.Iftheskycutissmall, itwill beagood approximation toset theinverseof C
i
to
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