Removing beam asymmetry bias in precision CMB temperature and polarisation experiments

3.3 MB·

by  Christopher G. R. Wallis

Mon.Not.R.Astron.Soc.000,1–18(2014) Printed17March2015 (MNLATEXstylefilev2.2)
Removing beam asymmetry bias in precision CMB temperature and
polarisation experiments
5
1 1⋆ 1 1
Christopher G. R. Wallis , Michael L. Brown , Richard A. Battye ,
0
1 2
2 Giampaolo Pisano and Luca Lamagna

r
a 1JodrellBankCentreforAstrophysics,SchoolofPhysicsandAstronomy,TheUniversityofManchester,ManchesterM139PL
M 2DipartimentodiFisica,SapienzaUniversita`diRoma,PiazzaleAldoMoro5,00185Roma,Italy

6
1 Accepted2014XXXXXXX.Received2014XXXXXXX;inoriginalform2014XXXXXXX

]
O
ABSTRACT
C Asymmetric beamscan create significantbias in estimates of the power spectra from CMB
. experiments.Withthetemperaturepowerspectrummanyordersofmagnitudestrongerthan
h
theB-modepowerspectrumanysystematicerrorthatcouplesthetwomustbecarefullycon-
p
trolled and/or removed. Here, we derive unbiased estimators for the CMB temperature and
-
o polarisationpowerspectratakingintoaccountgeneralbeamsandgeneralscanstrategies.A
r simpleconsequenceofasymmetricbeamsisthat,evenwithanidealscanstrategywhereevery
t
s skypixelisseenateveryorientation,therewillberesidualcouplingfromtemperaturepower
a toB-modepoweriftheorientationofthebeamasymmetryisnotalignedwiththeorientation
[
of the co-polarisation.We test our correctionalgorithm on simulations of two temperature-

2 onlyexperimentsanddemonstratethatitisunbiased.Thesimulatedexperimentsuserealistic
v scan strategies, noise levels and highly asymmetric beams. We also develop a map-making
5 algorithmthatiscapableofremovingbeamasymmetrybiasatthemaplevel.Wedemonstrate
7 itsimplementationusingsimulationsandshowthatitiscapableofaccuratelycorrectingboth
0 temperature and polarisation maps for all of the effects of beam asymmetry including the
2
effectsoftemperaturetopolarisationleakage.
.
1
Key words: methods: data analysis - methods: statistical - cosmology:cosmic microwave
0
4 back-ground-cosmology:large-scalestructureofUniverse
1
:
v
i
X
1 INTRODUCTION fainter B-mode signal presents a much greater challenge for ex-
r perimentalists,thoughthefirsttentativedetectionsofB-modepo-
a The cosmic microwave background (CMB) has proved to be an
larization on small scales are now being made through cross-
incrediblyusefultoolfortestingcosmological models. TheCMB
correlations (Hansonetal. 2013; PolarbearCollaborationetal.
two-point correlation functions, or alternatively in Fourier space
2013).
thepowerspectra,ofthetemperatureandpolarisationfluctuations
TheE-modepowerspectrumisapproximatelytwoordersof
areofparticular interest.Therehasbeen awealthofexperiments
magnitudefainterthanthetemperaturepower spectrumwhilethe
whichhavebeensuccessfulatcharacterisingtheCMBpowerspec-
traandtheseobservationsareconsistentwiththestandardΛCDM B-modepowerspectrumisexpectedtobeatleast2ordersofmag-
nitudefainterstill(Challinor2013).Thismeansthatinanyexperi-
cosmological model (see e.g. PlanckCollaborationetal. 2013c).
mentaimedatdetectingB-modes,systematiceffectsthatcouldpo-
The temperature power spectrum is now extremely well charac-
terised(PlanckCollaborationetal.2013a)andongoingandfuture tentiallycouplethetemperatureorE-modesignaltotheB-mode
power spectrummust bestrictlycontrolled. Onesourceofpoten-
experimentswillfocusoncharacterisingthepolarizationmoreac-
tialerrorisanasymmetricopticalresponsefunction(i.e.theexper-
curately.
imentalbeam).DuringthedataanalysisforCMBexperiments,one
The polarisation of the CMB, being a spin-2 field, can be
oftenassumesthatthebeamisaxisymmetricsothatitseffectisto
decomposed intocurl-free(E-mode) andgradient-free (B-mode)
simply(andisotropically)smooththesky.Thisassumptioncanre-
components. E-modes have been successfully detected and char-
sultinabiasinsubsequentpowerspectrumestimatesif,asisoften
acterised, and have helped to constrain cosmology (Kovacetal.
thecaseinreality,thebeamisnotperfectlyaxisymmetric.
2002; Readheadetal. 2004; Montroyetal. 2006; Brownetal.
2009; Chiangetal. 2010; QUIETCollaborationetal. 2012). The AtpresentthishasnotbeencrucialinextractingCMBpower
spectra,butwithincreasedsensitivityofinstruments,thisbiaswill
beimportantforfutureexperiments. Currentlytherearethreeap-
⋆ E-mail:[email protected] proachesintheliteraturethatattempttodealwiththiseffect.The
2 ChristopherG.R. Walliset al.
first is simply to quantify the systematic error on the cosmologi- modesproposedforfutureCMBsatelliteexperimentssuchasthose
calparameterscausedbytheasymmetryandbesatisfiedthatthey describedinBocketal.(2009).
are below the statistical uncertainty. This can be done by simu- Removingtheasymmetryatthemaplevelhastwomainben-
latingan experiment’s full beam response (Mitraetal. 2011) and efits.Firstly,incontrasttothecaseofthepseudo-C approach,the
ℓ
propagatingtheerror(PlanckCollaborationetal.2013b).Thisap- polarisationpowerspectrumestimatorerrorbarsarenotaffectedby
proach is effective as the error in the maps can be estimated. In thesamplevarianceofthetemperaturepowerspectrum.Secondly,
their analysis the Planck team accounted for the effects of beam theresultingbias-freetemperatureandpolarisationmapscanalso
asymmetry by finding the most suitable axisymmetric effective be used for science other than power spectrum estimation. Fore-
beam transfer function to deconvolve the recovered power spec- groundscanberemovedafterthemap-makinghasbeenperformed
trum. Any further asymmetry bias was shown to have little to meaningthatcurrentcomponent separationtechniquescanbeap-
no effect on the science (PlanckCollaborationetal. 2013a). The plied.
second method investigated attempts a full deconvolution of the The paper is organized as follows. We begin in Section 2
TimeOrderedData(TOD)fromaCMBexperimenttoremovethe where we present some basic definitions and develop the mathe-
effect completely (Wandelt&Go´rski 2001; Challinoretal. 2000; maticalformalismonwhichouralgorithmsarebased.InSection3,
Keiha¨nen&Reinecke2012).Theresultsofthismethodareencour- wepresent thepseudo-C based approach tocorrecting forbeam
ℓ
agingbutitisnotabletodealwithcertainunavoidablereal-world asymmetry.Section4discussesthepotentialimpactofbeamasym-
complications. In particular when noise is added to the TOD the metries on CMB polarization experiments, if they are left uncor-
deconvolutionnolongerworksforhighmultipoles.Inaddition,the rected.InSection5,wepresentatechniquetocorrectfortheeffects
deconvolutiononlyworksiftheexperimentobservestheentiresky. ofbeamasymmetryinthemapdomain.Section6discussessome
Thisisnotasignificantproblemforsatellite-basedexperimentsbut detailsofthedecompositionofthebeamwhichisrequiredforboth
for ground- or balloon-based experiments this will obviously not ofourapproaches.Wedemonstrateourtechniquesonsimulations
bethecase. inSections7&8andourconclusionsarepresentedinSection9.
A third approach is to calculate, and subsequently correct
for, the asymmetry bias on the measured pseudo-C in an ex-
ℓ
periment (Ramamonjisoaetal. 2013; Souradeepetal. 2006). As 2 BASICDEFINITIONSANDPRELIMINARIES
presentlyformulatedthisisunabletodeal withacut skywithout
Ourobjectiveistoconstructestimatorsforthetemperatureandpo-
apodisingtheazimuthaldependence ofthemask.Thisinturnre-
larisationfluctuationpowerspectragivenaTOD.Weassumethat
sultsinareductionofthecosmologicalinformationcontentofthe
anynon-astrophysical signalsintheTODhavebeenflagged and,
TODwhichissomethingthatwewouldliketoavoid.Inaddition,
forthepseudo-C approach, thatforegroundshavebeenremoved
ℓ
theseauthorsassumethateachskypixelisseeninasingleorien-
and/or masked. We consider an asymmetric beam and a general
tationonly,whereas,ingeneral,experimentswillobserveeachsky
scanstrategy.Webeginbydefiningsomerelevantquantities.
pixelinanumberofdifferentorientations.
The CMB temperature and polarisation fluctuations,
In this paper, we present two methods to remove the effect ∆T(θ,φ),Q(θ,φ) and U(θ,φ), can be decomposed into spin-
of asymmetry bias. The firstisan algorithm to recover theCMB weightedsphericalharmonics
powerspectrausingthepseudo-C approach.Wemakenoassump-
ℓ
tions about the beam or the scan strategy indeveloping unbiased a0ℓm = dΩ0Yℓ∗m(Ω)∆T(Ω) and (1)
estimators for the underlying CMB temperature and polarisation Z
power spectra. We also show that noise can be easily accommo- a = dΩ Y∗ (Ω)[Q(Ω)∓iU(Ω)], (2)
±2ℓm ±2 ℓm
dated.Thepseudo-Cℓ estimatorthatweproposeisbasedonacal- Z
culation similar to one presented in Hanson,Lewis,&Challinor wheresYℓm arethespinweightedspherical harmonics.Thetem-
(2010). Here, we extend the analysis to polarisation and demon- perature,E-modeandB-modeharmoniccoefficientsarerelatedto
strate its implementation on simulated temperature-only experi- theseby
ments.Sincetheestimatorworksdirectlyonthetimeordereddata
aT = a , (3)
(TOD),it is sub-optimal for polarization experiments that do not ℓm 0ℓm
1
directlymeasurebothQandU Stokesparametersinthetimeline aE = − (a +a ), (4)
ℓm 2 2ℓm −2ℓm
(e.g.viadetectordifferencing).Forsuchexperiments,theestima-
1
torwillperformthedecompositionintoQandU atthelevelofthe aB = − (a −a ). (5)
ℓm 2i 2ℓm −2ℓm
powerspectrumwhichwillcontributetothestatisticalerror.How-
ever,thisestimatoriswellsuitedtoanexperimentsuchasPlanck Weareinterestedinobtainingunbiasedestimatesforthepowerand
whichhas both instrument-Q andinstrument-U detector pairson crossspectraoftheCMBdefinedas
itsfocalplane.
ℓ
1
Inadditiontothepseudo-Cℓapproach,wepresentanewmap- CℓXY = 2ℓ+1 aXℓmaYℓm∗, (6)
makingalgorithmthatiscapableofmakingtemperatureandpolar- mX=−ℓ
isation maps cleaned of asymmetry bias. The map-making algo- whereX,Y={T,E,B}.
rithmproducesmapscontainingtheskysignalsmoothedwithjust Theresponse of atelescope tothe Stokesparameters on the
theaxisymmetriccomponentsofthebeamandnoise.Notethatthis sky(T,Q,U)canbedescribedbysomeresponse(beam)functions
map-makingschemerequiresasuitablescanstrategy—ingeneral, (T˜,Q˜,U˜).Thetotaldetectedpoweris
the more complex the beam asymmetry is, the more redundancy
(in termsof polarization angle covereage) isrequired in the scan W ∝ (TT˜+QQ˜+UU˜)dΩ. (7)
strategy. For the asymmetric beams that we have investigated in Z
thispaper,thescanstrategyrequirementsarefullymetbyscanning For details of how polarised detectors respond to the CMB see
Removing beamasymmetrybiasinCMB experiments 3
Challinoretal.(2000).OneelementoftheTODisthenthispower TheWignerD-matricesprovideacompleteorthogonal basis
W integratedoverthetimeintervalbetweentwosamples.Wede- forthisspace,soweusethemtodecomposeboththeTODandthe
finethespinweightedsphericalharmonictransformsofthebeam hitcubeas
tobe
Tℓ∗ = d3ωDℓ (ω)t˜(ω)n(ω) and (13)
mk mk
b0ℓk = dΩ0Yℓ∗k(Ω)T˜(Ω) and (8) Z
2ℓ+1
Z wℓ = d3ωDℓ∗ (ω)W(ω)n(ω). (14)
b±2ℓk = dΩ±2Yℓ∗k(Ω)[Q˜(Ω)∓iU˜(Ω)], (9) mk 8π2 Z mk
Z Here, we have introduced the weighting function n(ω), which
whenthebeamispointinginthez-directioninafiducialorientation rangesfrom0to1.Weusethisfunctiontodown-weightnoisypix-
suchthattheco-polarisationisalignedwiththey-direction.Note els,applyaforegroundmaskand/orapodisethehitcubesothatis
thatthisformalismcandescribeallaspectsofadetector’sresponse canbedescribedwellwithinourexpansion.1Weusetheprefactor
function.Forexample,wecanincludeboththeasymmetryofthe inequation(14)tocorrectlynormalisethecoefficientssothatwe
beam and any cross-polarisation response. This will allow us to canwrite
remove any bias that these beam imperfections would impart on
W(ω)n(ω)= wℓ Dℓ (ω). (15)
theestimatedpowerspectra. mk mk
We truncate our expansion of both the beam and the sky at ℓXmk
somemaximummultipole,ℓ .Wealsocaptheexpansionofthe Forthisreconstruction of thehit cube to beexact, one would re-
max
beaminkatsomemaximumvalue,kmax.Thisisareasonableap- quirektorangefrom−ℓtoℓ,notaswehaveherefrom−kmaxto
proximationtomakeasbeamresponsefunctionsaretypicallyclose kmax.Thisisnotaproblemforourproposedtechniquessince,as
toaxisymmetric.InSection6wewillexaminethisassumptionfor weshallseelater,weonlyneedtocapturethefeaturesintheψdi-
somespecificcases.Thebeamisthenrotatedaroundtheskyina rectionwithFouriermodesuptosomekmax,thevalueofwhichis
scan to measure the CMB. We describe this rotation using Euler determinedbythecomplexityofthebeamasymmetry.Thisisanal-
anglesω=[φ,θ,ψ].Thisisreallythreeactiverotations.Theyare ogoustotheaxisymmetriccase,wheretorecoverthetemperature
active as the beam moves with respect to the coordinate system. fluctuationpowerspectrumtheanalysisusesthehitmap,whichis
Thefollowingseriesofstepsdescribehowtorotatethebeamfrom justthek=0Fouriermodeofthehitcube.Sowhilewecannotfully
thefiducialorientationtotheorientationdescribedbyω,allrota- recoverthecompletehitcubewedorecoveritsimportantfeatures.
tionsbeingperfomedanticlockwisewhenlookingdowntheaxisby
whichtheyaredefined.
3 BEAMASYMMETRYCORRECTIONWITHINTHE
(i) Thebeamisrotatedaroundthezaxisbyψ.
PSEUDO-C FRAMEWORK
(ii) Thebeamisrotatedbyθaroundtheyaxis. ℓ
(iii) Thebeamisrotatedaroundthezaxisagainbyφ. In this section, we develop a technique to correct for the effects
The Wigner D-matrix, Dℓ (ω), performs the required ro- of beam asymmetry within the framework of pseudo-Cℓ power
mk spectrum estimators (Hivonetal. 2002; Brown,Castro,&Taylor
tations on the spherical harmonic decomposition of a function.
2005). As described earlier, thisapproach to correcting for beam
ThereforewecanwriteoneelementoftheTOD(tj)as
asymmetries is well suited to experiments that can measure the
tj = Dmℓ∗k(ωj)b∗sℓkasℓm. (10) Q and U Stokes parameters simultaneously in the timestream
(e.g. differencing experiments). In Appendix A we show that for
sℓmk
X
such an experiment the pseudo-C presented here is similar to
For simplicity, in this paper, we only write the index which is ℓ
thatofthestandardpseudo-C approach(Brown,Castro,&Taylor
being summed over and not the ranges. For the rest of this pa- ℓ
2005).
per one should assume ℓ ranges from 0 to ℓ , m from −ℓ to
max
ℓands=0,±2. Theindexk rangesfrom−k tok ,unless
max max
ℓ<k ,inwhichcasetherangeisthesameasform.
max 3.1 Definitionofthepseudo-C spectra
ℓ
Beforewecangofurtherwemustdefinesomemoremathe-
maticalconstructs.Thefirstisthe“hitcube”,W(ω)≡ δ(ω− We wish to define a two-point statistic that contains the relevant
j
ωj),whichdescribeswhichskypositionstheexperiment hasob- information from the TOD and which can also be related to the
served, and in which orientations. W(ω) isdefined onPthe space powerspectradefinedinequation(6)viaacouplingmatrix.Asthe
running from0toπ inθ andfrom0to2π inbothφandψ.The quantityTmℓk containsallof theinformationpresent intheTOD,
infinitesimalvolumeelementofthisspaced3ω=sinθdθdφdψ. itmustthereforecontainalloftheinformationrequiredtorecover
Theconvolutionoftheskywiththebeamt(ω)isacontinuous theCMBspectra.Wedefinethepseudo-Cℓspectraas
functionwhichweonlyhavelimitedknowledgeof.Theknowledge
wcuebhe.avFeorismdailclytawteedhbayvtehescanstrategy,orinthisformalismthehit C˜ℓkk′ ≡ Tmℓ∗kTmℓk′, (16)
m
X
t(ω) = Dℓ∗ (ω)b∗ a , (11)
mk sℓk sℓm
sXℓmk 1 Specificallyweconsideraweightingfunctionthatopenlydependsonθ
t˜(ω) = W(ω)t(ω). (12) andφ.Ourchoiceofnormalisedweightingfunctionforagivenskypixel
t˜(ω)isafunctionwhichcontainsalloftheastrophysicalinforma- isn(p)=m(p)/Nhits(p),wherem(p)isthemaskappliedtopixelpand
Nhits(p)isthenumberofhits thatpixel preceives inthescanstrategy.
tionpresentintheexperiment.ItissimplyarewritingoftheTOD, WeightingaTODelementwithafactorof1/Nhits(p)isequivalenttothe
whereeachelementisdecribedbyaδ-functionattherelevantori- weighttheelementreceiveswhenmakingabinnedmapandshouldnotbe
entationω. consideredtobedownweightinghighsignaltonoiseregionsofthemap.
4 ChristopherG.R. Walliset al.
whichcan becomputed directlyfromtheTOD.Theappropriate- the coupling operator that describes the contribution of each sky
ness of this choice becomes clear when we write the Wigner D- spectrumCs1s2 toeachoftheC˜kk′,i.e.wecanwrite
ℓ ℓ
matricesintermsofthespinweightedspherical harmonics, from
equation(3.10)inGoldbergetal.(1967): hC˜k1k1′i = Ok1k1′s1s2Cs2s3, (23)
ℓ1 ℓ1ℓ2 ℓ2
4π ℓ2Xs2s3
Dmℓ k(φ,θ,ψ)= 2ℓ+1eikψ−kYℓm(θ,φ). (17) where,
r
WeseethatTmℓ0willbesimilartothea0ℓmcoefficientsofabinned Oℓk11ℓk21′s1s2 = b∗s2ℓ2k2bs3ℓ2k3Mkℓ11kℓ21′k2k3. (24)
mapmadefromtheTODwhileTmℓ,±2willbesimilartothea±2ℓm kX2k3
coefficients.ThisisduetothefactthattheWignerD-matricesare Certainsymmetries can be used to reduce the number of M ma-
decomposingthe3Dspace,ω,withbasisfunctionsovertheθand tricesrequiredtoevaluateequation(24).Thesesymmetries,which
φdimensionswhicharethespinweightedsphericalharmonics. arederivedinAppendixC,are
Mkℓ11kℓ21′k2k3 = Mkℓ1′1kℓ21∗k3k2, (25)
3H.e2rewCealaciumlattoinfigntdhaencoanuapllyitnicgeoxpperreastsoiornfortheC˜kk′ definedin Mkℓ11kℓ21′k2k3 = (−1)k2−k1+k3−k1′M−ℓ1kℓ12−∗k1′−k2−k3. (26)
ℓ
equation(16)intermsofthetrueskyspectraandacouplingmatrix
thatdependsonlyonthescanstrategyandthebeam.Webeginby
3.3 RecoveringthetrueCMBspectra
re-writingthedecompositionoftheTODusingequation(10)and
thedefinitionofthewindowfunction.Wedothistoreplacethesum We are now in a position to obtain unbiased estimators for the
overjwithanintegral, true CMB power spectra CℓXY defined in equation (6). We start
bydefiningalargevectormadeupofthepseudo-C spectra,C˜kk′,
Tmℓ11∗k1 = Dmℓ11k1(ωj)tjn(ωj) and another comprised of the truefull-sky spin spℓectra, Cℓss′ℓde-
Xj finedinequation(21):
Using th=ejdse2fiℓX2nmiti2oDkn2mℓ1o1fk1th(eωhj)itDcmℓu2b2∗ek2W(ω(jω)b)∗s=2ℓ2k2jaδs(2ωℓ2m−2ωn(jω)jw)e. CC˜ii == ((CC˜ℓℓ0000,,CC˜ℓℓ0022,,CC˜ℓℓ00−−22,,CC˜ℓℓ2222,,CC˜ℓℓ22−−22,,CC˜ℓℓ−−22−−22))TT. ((2287))
deducethat Eachofthesearevectorsoflength6(ℓ +1).Usingthesedefi-
max
P
nitionswecanwriteouroverallcouplingmatrixequationas
Tℓ1∗ = b∗ a Kℓ1ℓ2 , (18)
m1k1 s2ℓX2m2k2s2ℓ2k2 s2ℓ2m2 m1k1m2k2 C˜i1 = Oi1i2Ci2. (29)
whereinthelastlinewehaveintroducedthecouplingkernel, Xi2
Kℓ1ℓ2 ≡ d3ωDℓ1 (ω)Dℓ2∗ (ω)W(ω)n(ω). (19) WewritetheoverallcouplingoperatorOi1i2 explicitlyintermsof
m1k1m2k2 m1k1 m2k2 the individual O-matrices of equation (24) in Appendix D. Once
We are now in aRposition to calculate the coupling operator. We this operator has been calculated, equation (29) can be inverted
startfromthedefinitionofC˜ℓk1k1′, to recover the true spin spectra Cℓss′, properly deconvolved for
both the mask and the asymmetric beam. A further simple trans-
C˜ℓk11k1′ ≡ Xm1 Tmℓ11∗k1Tmℓ11k1′ fthoermfiantaiolne,stwimhiacthesisofextphleicsiitxlypwosrsititbelnedCoMwBnipnoAwperpesnpdeicxtrEa,,CyiℓXelYds.
= b∗ a Kℓ1ℓ2 Note that, as with normal pseudo-Cℓ estimators, in the presence
m1 s2ℓ2k2 s2ℓ2m2 m1k1m2k2 of a severe sky cut, the matrix, Oi1i2 will be singular and must
s2ℓX2m2k2 thereforebebinnedbeforeitcanbeinverted.Thisisstandardprac-
s3ℓ3m3k3
ticewithpseudo-C powerspectrumestimators(Hivonetal.2002;
× bs3ℓ3k3a∗s3ℓ3m3Kmℓ11ℓk31′∗m3k3. (20) Brown,Castro,&Tℓaylor2005).
IfwenowassumethattheCMBtemperatureandpolarizationfluc-
tuationsareGaussiandistributedwithisotropicvariance, thenwe
canwritehasℓma∗s′ℓ′m′i=Cℓss′δℓℓ′δmm′ wherewehavedefined 3.4 Includingnoise
Cℓss′ = 2ℓ1+1 asℓma∗s′ℓm. (21) Amnuystuaslesfoulbealgabolreithtomdfeoarlrwemithoviinnsgtruthmeeenftfaelctnsooisfeb.eSaimncaesoyumrmalegtroy-
m
X
rithmworkswithintheframeworkofthestandardpseudo-C tech-
Usingthisresult,equation (20)simplifiesto ℓ
nique,wecanuseexactlythesameapproachtoremovethenoise
hC˜ℓk11k1′i = m1 b∗s2ℓ2k2Kmℓ11ℓk21′m2k2bs3ℓ2k3Kmℓ11ℓk21∗′m2k3Cℓs22s3 bTiOasDaesleismaednotpintecdluidnintghenositsaendcaarndbaenwalryitstiesn(aHsivonetal. 2002). A
s2ℓX2m2k2
s3k3 tj = Dmℓ∗k(ωj)b∗slkaslm+nj. (30)
= b∗s2ℓ2k2bs3ℓ2k3Mkℓ11kℓ21′k2k3Cℓs22s3, (22) sXlmk
ss2X3ℓ2kk32 Ifweassumethatthenoiseisnotcorrelatedwiththepointingdi-
rectionofthetelescopethen
where,inthesecondstep,wehaveusedtheproductoftwocoupling
kernels,Mkℓ11kℓ21′k2k3,derivedinAppendixB.Wecannowidentify (C˜ℓkk′)SN =(C˜ℓkk′)S+Nℓkk′, (31)
Removing beamasymmetrybiasinCMB experiments 5
wheretheSNandSsuperscriptsdenotethesignal-plus-noiseand where we have used the orthogonality of the Wigner D-matrices
thesignal-only pseudo-C spectrarespectively. Anunbiased esti- (Goldbergetal. 1967). Equation (37) shows that the asymmetry
ℓ
mate of the noise power spectra, Nkk′, can be obtained by per- willcoupletemperautretopolarizationeveninthecaseofanideal
ℓ
formingasetofsimulationscontainingonlyinstrumentnoiseand scanstrategy.Thiswastobeexpected,sinceboththeCMBpolari-
calculatingC˜kk′ foreachasbeforeusingequation(16).Thenoise sationfieldandtheconvolutionofthetemperaturefluctuationswith
ℓ
biasisthenestimatedastheaverageoverthesetofnoiserealisa- thek = ±2termofthebeamarespin-±2quantities.Fromequa-
tions,Nkk′ = hCkk′i.Thefinalestimatorforthefull-sky,noise- tions(33)and(34),theeffectonthemeasuredtheE-andB-mode
ℓ ℓ
polarisationis
debiased and asymmetry-cleaned spin power spectra can then be
writtenas 4π2
∆Tℓ = − (b +b )a∗ and (38)
Ci1 = O−i11i2(C˜i2 −hNi2i), (32) mE 2ℓ+1 0ℓ2 0ℓ−2 0ℓm
wherehXNi2i2iisavectoroflength6(ℓmax+1)comprisedofallof ∆TmℓB = −i(24ℓπ+21)(b0ℓ2−b0ℓ−2)a∗0ℓm. (39)
theindividualnoisebiasspectra,constructedinananalgousfashion For the coupling from temperature to B-mode polarisation to be
toequations(27)and(28).Asbefore,thesixCMBpowerspectra non-zerointhiscase,thenb mustbecomplex.Thiswillonly
0ℓ±2
arethenrecoveredtriviallyusingtherelationinAppendixE. bethecaseifthebeamasymmetryisorientatedatanangletothe
polarizationsensitivitydirectiondefinedbytheco-polarresponse,
whichwillnotbethecaseingeneral.
This result was first found by ?, where they examined sta-
4 IMPACTOFBEAMASYMMETRIESONCMB tistically varying systematic errors. A statistically varying dif-
POLARIZATIONEXPERIMENTS ferential beam ellipticity between a detector pair would couple
temperature to polarisation just as our constant ellipticity has.
Withtheanaysisoftheprevioustwosectionsinplace,wecannow
O’Dea,Challinor,&Johnson(2007)alsostudiedthesystematicer-
examinetheeffectthatbeamasymmetrieswillhaveontheE-and
rors induced when an elliptical Gaussian beam is used in a B-
B-modepolarizationpowerspectra.Webeginbynotingagainthat
mode experiment. They considered a specific type of ellipticity:
the Tℓ of equation (13) are closely related to the spin-2 har-
m±2 onewherethebeamiseitherperturbed alongthedirectionofthe
moniccoefficientsa ofQandU mapsconstructedfromthe
±2ℓm polarisation, or perpendicular to it. This type of perturbation has
sameTOD.Inanalogywithequations(4)and(5),wecantherefore
theuniquepropertyofhavingadecompositionwhereb isreal,
definethefollowingE-mode-likeandB-mode-likelinearcombi- 0ℓ±2
andthereforesuchanasymmetrycannotcoupletemperaturefluc-
nationsandtwo-pointcorrelationsoftheTℓ :
m±2 tuationstoB-modefluctuations.AsO’Dea,Challinor,&Johnson
Tℓ ≡ −1 Tℓ +Tℓ (33) (2007)shows,ifthebeamisperturbedinanyotherwaythanthis
mE 2 m2 m−2 special case, then temperature fluctuations will be coupled to B-
Tℓ ≡ −1(cid:16) Tℓ −Tℓ (cid:17) and (34) modefluctuationseven inthecaseof anidealscan strategy.This
mB 2i m2 m−2 resultisinagreementwiththefindingsofShimonetal.(2008)who
=⇒ C˜EE = 1 C˜(cid:16)22+C˜−2−2+(cid:17)C˜−22+C˜2−2 (35) alsoconsideredthecouplingbetweentemperatureandB-modepo-
ℓ 4 ℓ ℓ ℓ ℓ larisationduetobeamasymmetryeffects.
1(cid:16) (cid:17)
=⇒ C˜BB = − C˜22+C˜−2−2−C˜−22−C˜2−2 (36)
ℓ 4 ℓ ℓ ℓ ℓ
(cid:16) (cid:17)
In Appendix A, we show that the quantities defined in equa-
tions(33) and(34) aresimilartothe standard pseudo-C E- and
ℓ
B-modes defined in Brown,Castro,&Taylor (2005). These rela-
tionscanthenbeusedtoinvestigatetheimpactofbeamasymme-
5 BEAMASYMMETRYCORRECTIONDURING
triesand/oranon-zerocross-polarbeamresponsefunction.
MAP-MAKING
Of particular concern is the potential coupling between the
temperature power spectrum and the B-mode polarization power In some cases, the approach to correcting for beam asymmetry
spectrum. As the temperature power spectrum is known to be at presented in Section 3 will be sub-optimal. For example, in the
least four orders of magnitude stronger than the B-mode power, casewhereonecorrectsforsignificanttemperature-to-polarization
anycouplingbetweenthetwocouldbecatastrophicifnotproperly leakage, there will be a contribution to the error-bars on the re-
accountedfor.Hereweshowhowthemostprominentasymmetric constructedpolarizationpowerspectraduetothesamplevariance
modesofthebeamcanpotentiallycreatesuchacoupling. associatedwiththeleakedtemperaturesignal.Inaddition,forpo-
As described in Section 6 (see Figs. 1 and 2), the k = ±2 larizationexperimentsthatdonotmeasureQandUsimultaneously
modeisbyfarthemostprominentasymmetrictermfortherepre- inthetime-stream,theB-modepowerspectrumerrorswillbeaf-
sentativebeamsthatweconsiderlaterinthispaper.Toexaminethe fectedbythesamplevarianceassociatedwiththemuchlargerE-
impactofthek = ±2asymmetry, weconsider asituationwhere modepolarisation.Atechniquethatcorrectsforbeamasymmetry
all possible orientations are observed, i.e. where W(ω)n(ω)=1, atthemaplevelwillbeimmunetotheseissuesandisthereforean
forthenormalisationdescribedinsection2.Inthiscase,thecon- attractiveprospect.Here,wedevelopsuchamethodtocorrectfor
tributiontoTℓ∗ fromthetemperaturefluctuationsis beamasymmetryeffectsduringthemap-makingstep.
m±2
Webegin by recallingthat the convolution of the sky signal
Tmℓ∗±2 = b∗0ℓ′k′a0ℓ′m′ d3ωDmℓ ±2(ω)Dmℓ′∗′k′(ω) withageneralbeamisgivenbyequation(11).Inanexperimenta
ℓ′Xm′k′ Z telescopewillscanthesky,givingusasetofmeasurementsofthis
8π2 functiont(ω).Foreachpixelontheskywethereforehaveasetof
= 2ℓ+1b∗0ℓ±2a0ℓm, (37) measurementsatvariousorientations.
6 ChristopherG.R. Walliset al.
5.1 Extractingthetemperatureandpolarisationofapixel cancalculatethetemperatureandpolarisationofthepixelfromthe
inverseFouriertransformoftheestimatedS˜ as
Thisdiscussionisconcerned withestimatingthetemperatureand k
polarisationsignalatthepositionofasingleskypixel.Thedetected T = S˜ , (43)
0
signalatthepositionofapixelSwilldependontheinstrumentori- Q = 2ℜ(S˜ ), (44)
2
entationatthetimeofobservationψduetothepolarisationofthe
U = 2ℑ(S˜ ). (45)
skyandthebeamasymmetry.Ifthebeamwasaxisymmtricandhad 2
no polarised response then S(ψ) would be constant and equal to Performingthisprocedureforallobservedpixels,wewillthenhave
thebeam-smoothedCMBtemperatureatthepixellocation.2This estimatesoftheT,QandUmapswhicharefreeofsystematicsthat
can be seen from equation (11) by setting bsℓk=b0ℓ0δk0δs0 and haveadifferentspintoourdesiredquantity.Notewewillhavenot
noticingthattheψdependenceoft(ω)comesfromtheWignerD removedsystematicsthathavethesamespin.Specificallythecou-
matrices.ForapolariseddetectorS(ψ)wouldthencontaink=±2 plingfromtemperaturetopolarisationduetotheasymmetryinthe
Fouriermodes,duetothespin-2natureofpolarisation.Thisprop- beamdiscussedinSection4willstillcontaminateourestimatesof
ertyisexploitedbymap-makingalgorithmstofindthetemperature QandU.Wewillconsiderthisprobleminthefollowingsubsec-
andpolarisationofapixel.Inthesealgorithmstheψdependenceof tion.AsweshowinthesimulationtestsinSection8therewillbea
thedetectedsignalisassumedtobeduetothepolarisationsignal. noisepenaltyassociatedwiththiseffectivere-weightingofthedata.
Here, we relax this assumption and develop a map-making algo- Forthisreasonk shouldbeaslargeasitneedstobetocapture
max
rithmthatprovidesestimatesofthetemperatureandpolarisationof alltheasymmetryinthebeambutnolargerasthiswillincreasethe
apixelthatarefreeofsystematicsofdifferentspins. noiseinthemap.
We represent the orientations at which a pixel is seen in an Thismethod issimilar tothe re-weighting of the data tore-
experimentbydefiningawindowfunctionh(ψ)≡ 1 δ(ψ− movesystematicsofdifferentspinpresentedinBocketal.(2009).
nhits j
ψj),wherenhits isthenumber of hitsonthepixel.h(Pψ)willbe NotehoweverthatinBocketal.(2009)studysuggestsdescarding
differentforeachpixelandwillbedependantonthescanstrategy. (ordownweighting)alldatathatdoesnothavea”counterpart”that
ThedetectedsignalSd(ψ)istherefore couldbeusedtoaveragesystematicstozero.Forexample,aspin-
1systematiccanberemovedifeveryTODelementhasacounter
Sd(ψ)=h(ψ)S(ψ). (40)
part where the pixel was hit with ψ orientation π away from the
InFourierspace3thismultiplicationtakestheformofaconvolution first.Conversely, themethodweproposeinthispapercanuseall
ofthedatatocharacterisethespin-1systematicandremoveit.
∞
S˜kd = Hkk′S˜k′, (41)
k′X=−∞
5.2 Removingtheleakagefromtemperaturetopolarisation
wherewehavedefinedHkk′≡h˜k−k′.Therefore, ifwecaninvert
thematrixHkk′ thenwecanrecoverthetrueS˜k.Recoveringthe WeshowedinSection4thattheasymmetryofthebeamwillleak
spin-0 and spin-2 features of S(ψ) isthe main goal of this work temperaturefluctuationstopolarisationregardlessofthescanstrat-
sincethesearetheCMBtemperatureandpolarisationofthepixel. egy.Thisisduetothespin-2dependenceofthetemperatureofthe
Therefore,wewouldliketoobtainestimatesforS˜ andS˜ . CMBconvolvedwiththek=±2modeofthebeamresponsefunc-
0 ±2
Inverting the matrix Hkk′ as it is written in equation (41) tion. The polarisation maps made using the map-making method
wouldbeimpossible:firstlyitisinfinitelylarge,andsecondlyfor describedabovewillstillcontainthisspin-2leakagefromtemper-
any realistic h(ψ) the matrix will be singular.4 However, if we aturetopolarisation.However,sinceweknowthatthisleakageis
maketheassumptionthatS˜ cutsoffatasmallvalueofk andif fromthetemperaturefluctuationscouplingtothek=±2asymme-
k max
theψanglecoverageofthatpixelissufficentsuchthatthisreduced tryinthebeam,andsincewealsonowhaveanunbiasedestimate
matrixisinvertiblethenwecanusetheapproximation, ofthetemperaturemap,wecanthereforecalculateandremovethis
leakage.
kmax Westartfromtheestimatedtemperaturemap.Ifwehaveob-
S˜k = Hk−k1′S˜kd′. (42) servedonlypartoftheskyitisnecessarytoapodisethemapsuch
k′=X−kmax thattherearenofeaturesduetothemaskthataresmallerthanthe
We can choose k by measuring the azimuthal dependence of beamscale.Wethentakethesphericalharmonictransformofthis
max
thebeamandensuringthatweincludeenoughk-modestocapture map,
alloftheFouriermodesinS(ψ).Asinthecaseofthepseudo-C
ℓ
approach (Section 3), the kmax should chosen so that the asym- a˜0ℓm = dΩ0Yℓ∗m(Ω)T˜(Ω). (46)
metryofthebeamisfullycaptured.Wereturntothisissueinthe Z
followingsectionwhereweexaminetheharmonicdecomposition Wecannowcalculatethepolarizationleakageusingequations(38)
ofsomerepresentativeasymmetricbeams. and (39). Onecan show that thespherical harmonic transformof
AdoptingtheHEALPix5definitionoftheStokesparameters,we thetemperatureleakageisgivenby
b
∆aE = 2ℜ 0ℓ2 a˜ , (47)
ℓm b 0ℓm
2 Thedectectedsignalforapixeliisrelatedtoourpreviousnotationby (cid:18) 0ℓ0(cid:19)
Si(ψ)=t(ψ,θ=θi,φ=φi) ∆aB = 2iℑ b0ℓ2 a˜ . (48)
3 WedefinetheFouriertransformoff(ψ)andtheinversetransformtobe ℓm b 0ℓm
4f˜kN=ote21πthaRt02“πredalψiseti−c”ikiψnfth(iψsc)oanntdexft(eψxp)li=citPlye∞kx=c−lu∞deseitkhψecf˜aks.eofanideal Upontransforming(cid:18)ba0cℓk0(cid:19)torealspace,thesetermscanthenbere-
scanstrategyforwhichh(ψ)=1. moved from the polarisation maps. Note that the regions within
5 Seehttp://healpix.sourceforge.net a beam-scale of the locations where the temperature map was
Removing beamasymmetrybiasinCMB experiments 7
-40 -30 -20 -10 0 -40 -30 -20 -10 0 -40 -30 -20 -10 0
B(x,y)/dB R(x,y)/dB |B(x,y)-R(x,y)|/dB
222 222 222
111 111 111
s s s
e e e
e e e
gr 000 gr 000 gr 000
e e e
d d d
y/ y/ y/
---111 ---111 ---111
---222 ---222 ---222
---222 ---111 000 111 222 ---222 ---111 000 111 222 ---222 ---111 000 111 222
x/degrees x/degrees x/degrees
Figure1.Leftpanel:Thesimulatedbeamofa17modedhornplannedtobeonboardtheLSPEballoonexperiment.Centrepanel:thereconstructionofthe
beamretainingonlymodeswithk=0,±2.Rightpanel:Theabsoluteerrorbetweenthesimulatedbeamandthereconstruction.
aposidedshouldbedisregardedastheleakagewillnothavebeen resultsinacomplicated and potentiallyasymmetricoverall beam
removedintheseregions. shape.Notethatthenumericalsimulationusedhereisofthehorn
onlyandasimplescalingoftheoverallbeamsizehasbeenapplied
toroughlyapproximatetheeffectofthetelescopelens.
6 EVALUATINGTHEPARAMETERk FORA We note that the LSPE experiment will also include a half-
max
TEMPERATUREONLYEXPERIMENT waveplate(HWP)infrontoftheopticswhichwillbeusedtoin-
creasethepolarizationanglecoverageoftheexperiment.However,
Both the pseudo-Cℓ technique and the map-making approach for forthepurposesofthisdemonstration,weconsideratemperature-
removingbeamasymmetriesrequireustoimposeacaponthehar- onlyexperimentandwethereforeignoretheeffectoftheHWP.
monic expansion of the beam, kmax. We now look at a realistic Upon performing the spin-0 decomposition of the beam,
set up for a CMB experiment in order to ascertain how large the
(equation8),wefindthatonlyafewazimuthalmodesarerequired
requiredkmaxislikelytobeforrealexperiments.Wemakenoas- toaccuratelymodelwhatmightbeconsideredaratherasymmetric
sumptiononthescanstrategy.However,beamshapesaregenerally
beam.Fig.1showstheoriginalsimulatedbeamalongsidearepre-
designed to be as axisymmetric as possible. We expect therefore
sentationofitretainingonlythetwomostsignificantmodes(k=0
that the beam expansion can be truncated after only a few terms
andk=±2)ofthedecomposition.Themajorfeaturesofthebeam
with minimal loss of accuracy. We note that the pseudo-Cℓ cor- asymmetry are clearly well captured using this heavily truncated
rectiontechnique caninprinciplebeappliedforanykmax thatis expansion.Wethereforechoosetosetk =2.Thebeamsshown
max
deemed necessaryinorder toachieve therequiredaccuracy. This
hereareinvariantwhenrotatedbyπ,thereforetheoddazimuthal
will come with the obvious computational cost of increasing the
terms will be zero. Explicitly the truncated decomposition of the
numberofsummationsrequiredtoevaluateequation(24).Incon-
beamcanbewrittenas(nowwritingb =b etc.forclarity)
0lk lk
trast,forthemap-makingapproach,therewillbesomemaximum
valueofkmax forwhichthematrixHkk′ ofequation(42)willbe blk =bl0δk0+bl+2δk,+2+bl−2δk,−2. (49)
invertible.Wealsonotethatincreasingk willincreasethesta-
max InFig.1wealsoshowtheresidualsbetweenthefullbeamandthis
tisticalerror on therecovered map. It should thereforebe chosen
representationwhichisseentobeagoodapproximation. Wecan
to be only as largeas isrequired by the asymmetry in the beam.
immediatelyseefromequation(22)thatthiswillgreatlysimplify
Inpractice,whetherthemap-makingapproachwillbeappropriate
thepseudo-C couplingmatrixcalculationbylimitingthesumover
ℓ
for any given experiment will depend on both the complexity of
k andk toonlyincludethek ,k =0,±2modes.
2 3 2 3
thebeamasymmetryandonthepolarizationanglecoverageofthe
WealsoexamineageneralellipticalGaussianbeamdescribed
experiment.
by
6.1 Beamdecompositionsforsomerepresentativecases B(θ,φ)= 2π1qσ2e−2θσ22(cos2φ+q−1sin2φ), (50)
As a demonstration of how a realistic and asymmetric beam where q is a parameter that defines the asymmetry of the beam
can be represented using just a few k-modes, we consider a and the full with at half maximum (FWHM) θ = 2.35σ.
FWHM
numerical simulation of an instrumental beam corresponding to If q=1 then the beam is axisymmetric. To test the expansion we
the multi-moded 145 GHz feed horns planned to be flown on- choose θ =7 arcmin and q=1.5, which represents a highly
FWHM
boardtheballoon-borneLargeScalePolarisationExplorer(LSPE, ellipticalbeam,significantlymoreellipticalthanonemightexpect
TheLSPEcollaborationetal. 2012). By coupling 17 wave-guide inarealCMBexperiment.Thissimulatedbeamthereforeprovides
modes,thesensitivityofasingleLSPEhorn+detectormoduleis a stringent test of our correction algorithms. We plot the spin-0
greatlyincreased.However,thelargenumberofpropagatedmodes decompositionofthisbeaminFig.2.Itisencouragingtoseethat
8 ChristopherG.R. Walliset al.
Howeverinthecasewherethepowerinthemaskextendsto
highℓthesecrossspectratermscanbecomesignificant.Thisresults
inasignificantcontributiontotheoperatorbyatermoftheform
b∗ b Mℓ1ℓ2 ,wherek 6=k .Thisadditionalcontribution
0ℓ2k2 0ℓ2k3 00k2k3 2 3
couldpotentiallybelargerthanthek =k termif|b |≫|b |.
2 3 0ℓk2 0ℓk3
|bLk|2
The
7 TESTINGTHEPSEUDO-C ONSIMULATIONS
ℓ
Inthissection,wedemonstratetheimplementationofthepseudo-
C techniquedeveloped inSection3onnumerical simulationsof
ℓ
tworepresentativetypesofCMBexperiment–agenericballoon-
borneexperiment andafuturesatellitemission.Notethatwetest
theperformanceofthealgorithminthetemperature-onlycase.The
Figure2.Theamplitudesofthek=0,2,4,6,8,10termsofthebeamex- implementationofourtechniqueforpolarizationexperimentswill
pansion,|bℓk|2,asafunctionofmultipole,ℓforanellipticalGaussianbeam bepresentedinafuturepaper.Foratemperature-onlyanalysis,the
asdefinedinequation(50)withσ=3arcmincorespondingtoaθFWHM=7 couplingisreducedto
arcminandellipticityparameter,q=1.5.
C˜00 = O0000C00 with, (53)
ℓ1 ℓ1ℓ2 ℓ2
Xℓ2
onceagain,wecandescribethebeamasymmetryusingarelatively
O0000 = b∗ b Mℓ1ℓ2 (54)
smallnumberofterms. ℓ1ℓ2 0ℓ2k2 0ℓ2k3 00k2k3
kX2k3
Themainstagesofthesimulationpipelineareasfollows:
6.2 Usingthenoisepowertosetk inatemperatureonly
max (i) WeusetheHEALPixpackagetocreatesimulationsoftheCMB
experiment
temperaturefieldbasedonthetheoreticalCMBpowerspectrumfor
Fromtheabovedemonstration,itappearsthatonlyasmallnumber theconcordanceΛCDMcosmologicalmodel.
oftermsintheharmonicexpansionofthebeamneedtoberetained (ii) WethencreateaTODbyconvolvingtheCMBwiththetele-
inordertocapturetheglobalfeaturesofthebeamresponseinclud- scopebeamintheappropriateorientationgiventhescanstrategy.
ingtheeffectsofasymmetry. (iii) WethencalculatetheC˜ℓ00usingequation(16).
However,tobetrulyconfidentthatthetruncatedexpansionis (iv) Thecouplingoperatoriscalculatedusingtheexpressionin
anaccurateenoughrepresentationofthebeam,onewouldideally equation(54).
choosethevalueofkmaxsuchthattheanyresidualerrorfrommis- (v) WiththeoperatorOℓ010ℓ020 calculated,wetheninvertittore-
representingthebeamissmaller(tosometolerancelevel)thanthe cover an estimate of the full-sky and beam-deconvolved power
statisticalerror inanexperiment. Foraparticularasymmetricex- spectrum.
pansiontermtheleadingordercontributiontoC˜00is∼|˜b |2C00,
ℓ 0ℓk ℓ Notethatweusetwodifferentconvolutioncodesforthetwo
where˜b0ℓk=b0ℓk 2ℓ4+π1.Thereforewecansaythattheasymmet- classes of experiment that we simulate. For the balloon-like test
rictermshouldbeqincludedif we calculate each element of the TOD using equation (10). The
HEALPixROTATEALMSroutineoperatesonthebeammultipoleco-
hpkipix|˜b0ℓk|2Cℓ00 >fhNℓ00iσ, (51) efficients. Thisfunction uses the Wigner D-matricesto rotatethe
wheref isatoleranceleveltowhichwerequireoursystematicer- beamcoefficientsbyasetofEuleranglesgivenoninput.TheTOD
rorstobebelow thestatisticalerror.Areasonable choicemaybe elementcanthenbecalculatedusing,
f=0.1,toensurethatthesystematicerroris10%ofthenoisesta-
tisticalerror.hNℓ00iσ isthestandarddeviationofthenoisepower, tj = bjlm∗alm, (55)
lm
seeequation(31),whichisthedominatingsourceofstatisticalerror X
athighℓ.Thetermhp i isthekthmodeoftheψanglecoverage wherebj∗ isthemultipoleexpansionofthebeamwhenithasbeen
k pix lm
qualityofthescanstrategydefinedas rotatedthroughtheEuleranglesωj.InthiswaythewholeTODcan
becreated.Whilethisisaccurateandsimpleitwouldbeimpossible
2
hp i = 1 eikψi , (52) tousethisapproachinthesatellite-likeexperimentasthenumber
k pix *n2hits (cid:12)(cid:12)(cid:12)Xi (cid:12)(cid:12)(cid:12) +pix toifonTaOllDyienlfeemaseinbtlse.aTnodctrheeatreetqhueirseidmuℓmlaatexdwToOuDldimnathkeesiatteclolmitep-ulitkae-
wherehipix denote(cid:12)(cid:12)sanaverag(cid:12)(cid:12)eoverallpixelsonthesky,nhits is case,weuseamethodsimilartotheFEBeCoPapproach(Mitraetal.
thenumberofhitsapixelreceivesandthesumovertheindexiis 2011)exceptthatdonotcalculatetheeffectivebeamsbutratherwe
over allobservations of aparticularpixel.Thelower thevalueof producetheindividual TODelements. Theconvolution issimpli-
hpkipix,thebetterthescanstrategyisatremovingthebiascreated fiedbydescribingthebeaminpixelspaceandweonlycalculatethe
bythebeamasymmetryduetothekthazimuthalmode. valueofthebeamonasetofneighbouring pixelscentredaround
Theruleofthumbinequation(51)becomesproblematicwhen thepointingcentreof thebeam. Weusepixelswithin5σq of the
theskycoverageissmallandinthepresenceofahighlyasymmet- centreofthebeam(seeequation50)todescribethebeamoverthis
ricbeamduetocouplingbetweenasymmetrictermsofdifferentk. localisedregionofsky.
This coupling is suppressed in the all-sky case because the cross A second technical challenge was to obtain the Wigner D-
spectraWkℓ2k3 aresmallfork26=k3,whichinturnmeansthecou- matrix transforms of the TOD and the window function. We
plingmatrixMℓ1ℓ2 issmall. achieved this by implementing equations (13) and (14) using the
00k2k3
Removing beamasymmetrybiasinCMB experiments 9
expression for the Wigner D-matrix in terms of spin weighted
spherical harmonics (equation 17). First, the integral over ψ was
performed using a simple Newton-Cotes numerical integration
method.TheHEALPixroutineMAP2ALMSPINwasthenusedtoper-
formtheintegralsoverθandφ.
7.1 Balloon-likeexperiment
ForourfirsttestweusethesimulatedLSPEbeamshowninFig.1
withamodification.Wegreatlyamplifythesizeoftheasymmetry
in order to produce a more stringent test of the asymmetry cor-
rection algorithm. The spherical harmonic decomposition of this
exaggerated beam is related to the original LSPE decomposition
by Figure4.Reconstructionofthepowerspectrumfortheballoon-likesimu-
bexagg =bLSPEδ +bLSPEδ ×100. (56) lationusingthepseudo-Cℓ method.Theinputtheoreticalpowerspectrum
0ℓ,k 0ℓ,0 k0 0ℓ,±2 k±2 isplotted(black)alongwiththerecovered powerspectrum whentheex-
WeusetheplannedscanstrategyforLSPEbutwestressthat agerratedbeamisused(red),averagedover1000realisations.Wealsoplot
therecoveredpowerspectrumonewouldrecoveriftheaxisymmetricMAS-
wehavenotincludedtheeffectsofthehalf-waveplate(HWP)that
TER(Hivonetal.2002)analysiswasused(blue)andifthenoisewasig-
LSPEwilluse. TheLSPEexperiment willincorporateaHWPin
nored(yellow).Theverticalerrorbarsshowthestandarddeviationofthe
frontoftheopticswhichwillbeusedtoincreasethepolarization
realisationsfromthemean.
angle coverage of the experiment. However, for the purposes of
this demonstration where we consider a temperature-only exper-
iment, we have ignored the (advantageous) effects of the HWP.
σ=3 arcmin corresponding to a FWHMof 7 arcmin and q=1.5.
LSPEwillbelaunchedatalatitudeof780Northandwillfollowa
Thescanstrategyusedissimilartothatproposed for theExperi-
jet stream around the globe staying at roughly the same latitude.
mentalProbeofInflationaryCosmology(EPIC,Bocketal.2009).
The payload will spin at 3 rpm and the telescope will perform
In Fig. 5 we show the hit map, the p quality (equation 52) and
constant allevation scans with regular (§daily) steps in elevation 2
theweightingfunctionthatwehaveusedtomaskthegalaxyand
(TheLSPEcollaborationetal.2012).
extragalactic sources. We use the same mask as was used by the
We created a set of TOD following the above procedure in-
Planck collaboration (PlanckCollaborationetal. 2013b) for their
cludingwhitenoise.TheLSPEscanstrategycovers∼25%ofthe
powerspectrumanalysis.Theweightingfunctionalsoapodisesthe
skyandsoprovidesatestofthealgorithm’sabilitytoaccountfora
hitmap.NotethattheEPICscanstrategywasdesignedtoimprove
cutskyinadditiontotheeffectsofbeamasymmetry.Fig.3shows
the2nd ψ coveragequalitydefinedinSection6.2.Thischoiceof
anexampleofasimplebinnedmapcreatedfromoneofoursimu-
scanstrategywillthereforebeeffectiveinmitigatingbeamasym-
lations.Thisfigurealsoshowsthehitmapofthescanstrategy,the
metriesevenintheabsenceofadedicatedcorrectionalgorithm.We
2ndψcoveragequalityp definedinSection6.2andtheweighting
2 thereforeexpectthebias,intheabsenceofadedicatedcorrection,
functionthatweusetoapodisethehitmap.Weincludean(unre-
tobemuch smallerthan seenintheprevious section (although it
alisticallyhigh)noiselevelof1.6mK-arcmininoursimulationsin
will still be non-zero). The mean recovered power spectra using
order to test the removal of noise bias from the recovered power
k =0,2areshowninthetoppanel ofFig.6.Thelowerpanel
spectrum. max
ofthisfigureshowsthefractionalresidualbiasinthepowerspec-
The C˜00 for 1000 realisations were computed according to
ℓ trumrecoveryanddemonstratesanunbiasedresulttohigh-ℓwhen
equation(53)andthecouplingoperatorwascalculatedusingequa-
k =2isused.Thesesimulationsincludedwhitenoisesuchthat
tion(54).Inordertoinvertthecouplingmatrixweneededtobinthe max
the error on the map is equivalent to 8.5µK-arcmin. We can see
C˜00andthecouplingoperator,asinHivonetal.(2002).Thisbin-
ℓ thatevenwithascanstrategydesignedtooptimisecrossingangles
ningensuresthatthematrixisinvertibleandthattheresultingdata
andtherebyreducetheasymmetrybias,thereremainsabiasatthe
pointsareuncorrelated.Hereweuseabinsizeof∆ℓ=20.Acorrec-
levelof∼1−2σwhentheaxisymmetricapproximationismade.
tionfornoisebiaswasalsoimplementedfollowingtheprocedure
The importance of this test is two-fold. It demonstrates that
outlinedinSection3.4.Fig.4showsthemeanpowerspectrumre-
the algorithm can deal with a more conventional elliptical Gaus-
coveredfromthesimulationsincomparisontotheinputspectrum.
sian beam and also that it can be extended to high-ℓ. The cou-
In this figure, we also plot the mean power spectrum recovered
plingoperatorwascalculatedin9hrsonone2.26GHzprocessor
whenthebeamwasassumedtobeaxisymmetric,byincludingonly
toℓ = 4000. Wecalculated the coupling operator to amuch
the b term on the pseudo-C . This figureclearly demonstrates max
0ℓ0 ℓ higherℓthanthemaximummultipoleofinteresttoensurethatwe
thatouralgorithmcansuccessfullyrecoverthepowerspectrumin
did not miss any effects from the aliasing of power from higher
asituationwheretheaxisymmetricapproximationclearlyfails.The
multipoles.
power spectrumobtainedinthecasewherethenoiseintheTOD
was ignored is also shown demonstrating the ability of our algo-
rithmtosuccessfullycorrectfornoisebias.
8 TESTINGTHEMAP-MAKINGALGORITHMON
SIMULATIONS
7.2 Satellite-likeexperiment
Wetest the map-making algorithm described inSection 5 in two
Thesecondtestthatweperformisusingasatellite-likescanstrat- ways. First, we use the algorithm on multiple realisations of the
egy.Forthistest,weuseabeamdescribedbeequation(50)with samesimulatedexperimentasdescribedinSection7.2toshowthat
10 ChristopherG. R. Walliset al.
Figure3. Topleft: Anexample binned mapmadeinthe balloon-like test. Theexaggerated beam was usedwith aspherical harmonic decomposition as
describedinequation(56).Thescanstrategyimplementedistypicalofaballoonbasedexperiment.Topright:Thehitmapfromonedayofobservationsfor
theballoon-likeexperiment.Bottomleft:Thep2quality,definedinequation(52),ofthescanstrategy.WenoteagainthataHWPwasnotincludedwhichis
whythepolarizationanglecoverageispoorforthissimulation.Bottomright:Theweightingfunctionappliedtoapodisethehitmap.
thealgorithmcancorrectlyremovetheasymmetrybiasonthetem- 8.1 Removingtheasymmetrybiasonthetemperaturepower
peraturepowerspectrum.Thesecondtestisperformedonasingle spectrum
noise-free full sky simulation of the satelliteexperiment. We use
this latter simulation to test if the algorithm can successfully re- AsinSection7.2,wecreate48TODrealisationsusingthesatellite-
movetheasymmetrybiasdirectlyfromthetemperatureandpolar- like scan strategy, and using an elliptical Gaussian beam with
isationmaps.WedonotincludeaCMBpolarisationsignalinour θFWHM=7 arcmin and q=1.5 (see equation 50). As before, the
TODand so asuccessful asymmetry correction algorithm should TODincludedwhitenoiseequivalenttoamapnoiselevelof8.5µK
recoverazeropolarisationsignal. arcmin.
From the simulated TOD, we then compute both a simple
binnedmapandamapconstructedusingthealgorithmpresented
in Section 5 with k =2. We then apply the same galactic and
max
point source mask to these maps as used in Section 7.2. A stan-