Table Of ContentImpactofInstrumental SystematicContaminationonthe Lensing MassReconstruction using the
CMB Polarization
Meng Su1, Amit P.S. Yadav1, and Matias Zaldarriaga1,2
∗
1Harvard-Smithsonian CenterforAstrophysics, 60Garden St., Cambridge, MA02138, USAand
2Jefferson Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA
Inthispaper, westudytheeffectsofinstrumentalsystematicsonthereconstructionofthedeflectionangle
power spectrum from weak lensing of Cosmic Microwave Background (CMB) temperature and polarization
observations. Weconsider7typesofeffectswhicharerelatedtoknowninstrumentalsystematics: calibration,
rotation, pointing, spin-flip, monopole leakage, dipoleleakageandquadrupole leakage. Theseeffectscanbe
characterizedby11distortionfields. Eachofthesesystematiceffectscanmimictheeffectiveprojectedmatter
9 power spectrumandhence contaminatethelensingreconstruction. Todemonstratetheeffect oftheseinstru-
0 mentalsystematicsonCMBlensingmeasurements,weconsidertwotypesofexperiments,onewithadetector
0
noiselevelforpolarizationof9.6µK-arcminandFWHMof8.0,typicalofupcominggroundandballoon-based
′
2
CMBexperiments,andaCMBPol-likeinstrumentwithadetectornoiselevelforpolarizationof2.0µK-arcmin
n andFWHMof4.0′,typicaloffuturespace-basedCMBexperiments. Foreachsystematicsfield,weconsider
a various choices of coherence scale α , starting from α = 10 to the maximum of α = 120. Among all
s s ′ s ′
J the11systematicparameters,rotationω,andmonopoleleakageγ &γ placethemoststringentrequirements,
a b
5 whilequadrupole leakageq, pointingerror pa and pb,andcalibrationaareamongtheleastdemanding. The
requirements from lensing extraction are about 1-2 orders of magnitude less stringent than the requirements
] tomeasure the primordial B-modes withinflationary energy scale of 1.0 1016 GeV. On the other hand the
O ×
requirementsforlensingreconstructionarecomparableorevenmorestringentforsomesystematicparameters
C thantherequirementstodetectprimordialB-modeswithinflationaryscaleE =3.0 1016GeV.
i ×
.
h PACSnumbers:
p
-
o
r
t
s
I. INTRODUCTION
a
[
ObservationsofthetemperatureanisotropiesoftheCosmicMicrowaveBackground(CMB) havebeena majortoolto con-
1
v strain cosmological parameters. The polarization data of the CMB can help us to extract additional information beyond the
5 temperatureinformation[1,2]. ThenextgenerationofCMBobservationswillfocusontheprecisemeasurementofpolarization
8 oftheCMB,especiallythesocalledB-modepolarization,whichisatleasttwoordersofmagnitudesmallerthanthetempera-
2 tureanisotropysignal. IncontrasttoE-modepolarization,whichcanbegeneratedbyscalarortensorperturbationsintheearly
0 universe,theprimordialB-modesaregeneratedonlybytensorperturbations[3,4]. Howeverevenintheabsenceofprimordial
.
1 B-modes,subsequentgravitationallensingbythelargescalestructureoftheUniverseconvertsE-modepolarizationtoB-mode
0 polarization[5,6,7,8]. Althoughtheamplitudeoftheprimordialgravitationalwavesignalisuncertainbymanyordersmagni-
9 tudeandmightnotbedetectablebythenextgenerationofpolarizationexperiments,thelensingB-modesignalisaguaranteed
0
predictionofthecurrentcosmologicalmodel. Inaddition,theB-modelensingsignalwillhelptobreakdegeneraciesbetween
:
v cosmologicalparameters[9,10,11,12,13,14].
i The weak gravitational lensing of CMB anisotropies provides a unique opportunity to map the matter distribution of the
X
universe. Thenon-GaussianhigherordercorrelationsintheCMBgeneratedbytheweaklensingcanbeusedtoreconstructthe
r
a massdistributionoftheinterveninglarge-scalestructure. Theprincipleiseasytounderstand. TheCMBphotonsareremapped
by gravitationallensing which introducescorrelationsbetween differentangularmoments. One way of extractingthe lensing
information is to use a quadratic combination of the CMB multipoles to define an estimator for the projected gravitational
potential[15,16,17,18,19,20,21,22].
AlthoughtheB-modepolarizationobservationsarecurrentlystillnoisedominated,thenextgenerationofCMBpolarization
instruments has the sensitivity to make first detections, at least of the lensing induced B-mode signal. However, there are
severalchallengesforCMBlensingdetection,mainlycomingfromastrophysicalforegroundsandinstrumentalsystematics. It
is important to estimate and control those spurious signals as well as possible when analyzing upcoming CMB data. These
challengeswillhavetobeovercomeinordertoprobethephysicsoftheearlyuniversethroughB-modepolarizationortoinfer
theprojectedlargescalematterdistributionfromE/Bpolarization.
∗Electronicaddress:[email protected]
2
Lensing studies can be considered as secondary science for an experimentdevoted to B-mode detection. Impact of instru-
mental systematics on the projected matter power spectrum is helpfulto both instrumentdesign and futuredata forecast. On
theotherhand,lensinginducedB-modepolarizationisacosmologicalcontaminantforthedetectionofprimordialB-modes,a
systemicstudyofinstrumentalsystematicsforlensingreconstructionmaywellberequiredofsuchanexperimenttodelensethe
observedCMBfields[23,24,25,26,27,28,29,30,31].
Intheliterature,instrumentalsystematicshavebeendiscussedextensively [32,33,34,35,36,37,38,39]. Themaingoalof
thispaperistoillustratestheeffectsofinstrumentalsystematicsandsystematicallystudytheimpactonthemassreconstruction
process for upcomingCMB experiments[24, 25, 26, 27, 28, 29, 30]. To calculate the effects of instrumentalsystematics on
theprojectedmatterpowerspectrum,wemakeuseofthequadraticestimatortoreconstructtheprojectedgravitationalpotential
[15],andclassifyandparameterizethesystematicsfollowing[32]. Wedividepolarizationcontaminationsintotwocategories:
those which are associated with a transfer between the polarization state of the incoming radiation (from detection system),
includingcalibrationandrotation, spin-flipcouplingandmonopoleleakageerrors, andthosewhich areassociated with CMB
anisotropyinducedbythefiniteresolutionorbeamofthetelescope. Forthepolarizationtransfersystematics,weonlyconcern
ourselves with polarization transfer in a single, perfectly known, direction on the sky. However, in reality, every experiment
necessarilyhasfiniteresolutionandthisthereforeisanadditionalclassofcontaminationassociatedwiththeresolutionorbeam
oftheexperiment. We referthereaderto[32]fora detaileddiscussionoftheparametrizationofthe systematicerrorsweuse
inthispaper. We calculatetheunlensedandlensedCMB powerspectrumusingCMBFAST [40]. Inthecalculation,wehave
assumed a flat ΛCDM cosmology with following cosmological parameter values: Ω = 0.045,Ω = 0.23,H = 70.5,n =
b c 0 s
0.96,n =0.0,τ=0.08.
t
Thispaperisorganizedasfollows: InSec.IIwereviewthebasiclensingformalism,defineournotation,andintroducethe
quadraticestimator of the lensing potentialreconstructionwhich we later use to explorethe effects of instrumentalsystemat-
ics. InSec.III,wefirstconsiderarelativelysimplecaseofthecalibrationsystematicsonthetemperatureestimatoroflensing
reconstruction. Thenwe considerEB estimatoras an example, studying7 typesof instrumentalsystematic effects onlensing
reconstruction,andcomparetothesystematiccontaminationoftheB-modepowerspectrumdetection.Weconsidertwoinstru-
mental configurations, one with noise sensitivity for polarization of 9.6 µK-arcmin and FWHM of 8.0 (Exp1from here on),
′
and another CMBPol like instrument with noise sensitivity for polarization of 2.0 µK-arcmin and FWHM of 4.0 (reference
′
experimentfrom here on). In Sec. IV and V, we describe our results and conclude with a discussion of the implications for
experimentsdedicatedtomeasureprimordialB-modesorthesecondarylensingsignal. Weleavediscussionsoftheotherthree
lensingpotentialestimators(EE,TE,TB)toAppendix.
II. LENSINGFORMALISM
GravitationallensingdeflectsthepathofCMBphotonsfromthelastscatteringsurfaceresultinginaremappingoftheCMB
temperature/polarizationpattern on the sky. In this section, we review the basic lensing calculation for both temperatureand
polarizationfieldsasthestartingpointofourdiscussion.WeformulateCMBlensingusingtheflat-skyapproximation[22]. The
flat-sky approachsimplifies the derivationby replacingsummationsoverWigner symbolsof sphericalharmonicmomentsby
integralsinvolvingmodecouplingangles[7]. MoredetailsofCMBlensingcanbefoundinthenicereviewpaper[41].
Atacertainpositionnˆ onthesky,theobservedCMBfieldX˜(nˆ)islensedfromanotherdirectionintheprimordialCMBsky
X(nˆ +d(nˆ))atz=1090.Theremappingprocesscanbedescribedas
T˜(nˆ) = T(nˆ +d(nˆ)), (1)
[Q˜ iU˜](nˆ) = [Q iU](nˆ +d(nˆ)),
± ±
whereT˜(nˆ)(T(nˆ))representsthelensed(unlensed)temperaturefluctuationfield, Q˜(nˆ)(Q(nˆ))andU˜(nˆ)(U(nˆ))arelensed(un-
lensed)polarizationStokesparameters,andd(nˆ)isthedeflectionanglewhichisrelatedtoφ(nˆ),thelensinggravitationalpoten-
tial,byd(nˆ)= φ(nˆ). Hereandthroughoutthispaper,weuseboldfacequantitiestoidentifyvectors,andX˜ (X)standsforlensed
∇
(unlensed)temperatureandpolarizationfields. Thelensingpotentialφ(nˆ)isgivenby
φ(nˆ) = 2 r0dr dA(r0−r) Φ(r,rnˆ), (2)
− Z0 dA(r)dA(r0)
whered isthecomovingdistancealongthelineofsight;r isthecomovingdistancetothesurfaceoflastscattering,andΦis
A 0
gravitationalpotential. ThelensingremappingprocessconservesthesurfacebrightnessdistributionoftheCMB,thusdoesnot
changetheone-pointstatistics.
The observed temperature and polarization fluctuations also include secondary effects, such as Sunyaev-Zel’dovich (SZ)
effect[42] and IntegratedSachs Wolfe (ISW) effect[43], which comefromthe first orderdensityor potentialfluctuationand
3
thusalsocorrelatewiththelensingdeflectionangle.WedenotethesephysicalcontaminationtolensingreconstructionbyXsec(nˆ).
WedenotethenoisecomponentbyXn(nˆ). ThetotalobservedCMBanisotropythereforeincludesthelensedprimarysignal,any
secondaryeffects, andnoise, i.e. Xt(nˆ) = X˜(nˆ)+Xsec(nˆ)+Xn(nˆ). Inthe nextsection, we will introduceanothercontribution
to Xt(nˆ), which comes from instrumentalsystematics Xsys(nˆ). We define the observedCMB field Xobs(nˆ) = Xt(nˆ)+Xsys(nˆ).
Here we write the secondary contributionas an independentcomponentfrom the lensed CMB. However, in reality it is hard
toseparateXsec(nˆ)fromXt(nˆ)becausesecondariesarealsolensedbygravitationalpotentialswithdeflectionanglesdepending
ontheirredshifts. Inthispaper,wesimplydropthecontributioneffectsXsec(nˆ),asthistopicisbeyondthefocusofthispaper.
We refer the readers to [22] for a treatment of the secondary anisotropy as a physical contamination to the lensing potential
reconstructionanalysis1.
It is convenientto work in Fourier space. If one considersa small enough patch of sky, spherical harmonic modes can be
replaced by Fourier modes. Generalization from the flat-sky to the full sky is straightforward. The Fourier transform of the
TaylorexpendedlensedCMBtemperatureandpolarizationfieldis
d2l
T˜(l) = dnˆT˜(nˆ)e ilnˆ =T(l) ′ T(l)L(l,l), (3)
− · ′ ′
Z −Z (2π)2
d2l
E˜ ±iB˜ (l) = Z dnˆ[Q˜(nˆ)±iU˜(nˆ)]e∓2iϕle−iˆl·nˆ =[E(l)±iB(l)]−Z (2π)′2[E(l′)±iB(l′)]Lp(l,l′),
h i
φ(l) = dnˆφ(nˆ)e ilnˆ, (4)
− ·
Z
where
1 d2l
L(l,l) φ(l l) (l l) l + ′′ φ(l ) φ(l l l )(l l) (l +l l) l +... , (5)
′ ′ ′ ′ ′′ ′ ′′ ′′ ′ ′′ ′ ′
≡ − − · 2Z (2π)2 × − − · − ·
(cid:2) (cid:3) (cid:2) (cid:3)
1 d2l
LP(l,l′) ≡ e±2i(ϕl′−ϕl)φ(l−l′) (l−l′)·l′ + 2Z (2π)′′2e±2i(ϕl′−ϕl)φ(l′′)×φ(l−l′−l′′)(l′′·l′) (l′′+l′−l)·l′ +... .
(cid:2) (cid:3) (cid:2) (cid:3)
We can immediately see that lensing induces remapping of CMB fields by lensing potential gradients, hence in Fourier
space lensing acts as a convolutionwhichcouplesdifferentharmonicmodes. Fouriermoments, powerspectrum, bispectrum,
trispectrumandsoonoftheCMBfieldsandthelensingpotentialcanbedefinedintheusualmanner:
Xi(l )X j(l ) (2π)2δ (l +l )Cij (l ),
D 1 ′ 2 E ≡ D 1 2 XiX′j 1
Xi(l )X j(l )X k(l ) (2π)2δ (l +l +l )Bijk (l ,l ,l ),
1 ′ 2 ′′ 3 c ≡ D 1 2 3 XX′X′′ 1 2 3
D E
Xi(l )X j(l )X k(l )X m(l ) (2π)2δ (l +l +l +l )Tijkm (l ,l ,l ,l ),
1 ′ 2 ′′ 3 ′′′ 4 c ≡ D 1 2 3 4 XX′X′′X′′′ 1 2 3 4
D E
... (6)
wheretheanglebracketsrepresentensembleaveragesoverrealizationsoftheprimordialCMBfields,thelarge-scalestructure
between observers and the last scattering surface, and the experimental noise. The connected part of the n-point function is
denotedbythesubscriptc. Thefields X, X , X , X areamong T(l), E(l), B(l),φ(l) . Thesuperscriptsi, j,k,mrepresentthe
′ ′′ ′′′
unlensedfieldX,thelensedfield X˜,theinstrumentalnoiseXn,the{ instrumentalsystem}aticsXsys,CMBsecondarycontribution
Xsec,thetotalsignalXt,ortheobservedfieldXobs. Wenotethatthebispectrumandhigherorderodd-correlationsvanishifone
ignoresthesecondaryeffects. ThisisbecauseoddmomentscontainsampleaveragesovertheoddprimordialCMBfieldswhich
weassumetobeGaussian.
Wemaketheassumptionthatfluctuationsinthelarge-scalestructurebetweentheobserverandthelastscatteringsurfaceare
Gaussianandhencecanbefullydescribedbyapowerspectrum. Weusethelensingpotentialpowerspectrumcalculatedfrom
CAMBFAST.TheinstrumentalnoiseXnisalsoassumedtobeGaussian. Wenotethatprimordialnon-Gaussianitycanpossibly
contributepercentleveluncertaintytoouranalysiswhichisconsideredin[45].
We are now in the position to calculate any order (cross) correlation functions of CMB fields and the lensing potential in
Fourierspace. WewillassumeuniformGaussiannoisewiththepowerspectrumCXXngivenby
l
ClXXn =w−X1el2σ2b. (7)
1 Asanote,thermalSZeffectcaninprincipalbeseparatedfromtheprimaryfluctuationsbyitsspectraldependence. ForthekineticSZeffect,itwasclaimed
thatbyusingaspeciallydesignedestimator[44],itispossibletoseparateitoutfromreallensingsignal. However,someimportantsecondarycontributions
suchastheISWeffectcannotbeseparatedeasilyandwillleadtoadditionalnoisecontributionsduetocorrelationswiththelensingpotentials[22].
4
XX 1W (l ,l ) 2W (l ,l )
′ XX′ 1 2 XX′ 1 2
TT (L l ) (L l )
· 1 · 2
TE cos2(ϕ ϕ )(L l ) (L l )
l1− l2 · 1 · 2
TB sin2(ϕ ϕ )(L l ) 0
l1 − l2 · 1
EE cos2(ϕ ϕ )(L l ) cos2(ϕ ϕ )(L l )
l1− l2 · 1 l1− l2 · 2
EB sin2(ϕ ϕ )(L l ) sin2(ϕ ϕ )(L l )
l1 − l2 · 1 l1− l2 · 2
BB cos2(ϕ ϕ )(L l ) cos2(ϕ ϕ )(L l )
l1− l2 · 1 l1− l2 · 2
TABLEI:WindowfunctionswhichappearinEq.(11)andEq.(12);hereL=l +l
1 2
w 1isthedetectornoisevariancepersteradianareafortemperature(X=T)orpolarization(X=EorB),andσ =θ /√8ln2
−X b fwhm
istheeffectivebeamwidthoftheinstrumentcalculatedfromitsfull-widthhalf-maximumresolutionθ . Wewillassumefully
fwhm
polarizeddetector,forwhich2w =w =w .
T E B
Quadratic combinations of CMB fields can be used as estimators of the lensing potential field and hence the intervening
projected mass between us and the last scattering surface. Furthermore, a CMB-field-squared map appropriately filtered in
Fourier space can serve as an optimal estimator for the deflection field. Optimal filters for quadratic estimators have been
designed[15]
A (L) d2l
d (L) XX′ 1 Xt(l )Xt(l )F (l ,l ), (8)
XX′ ≡ L Z (2π)2 1 ′ 2 XX′ 1 2
whereXandX canbeT,E,andB.ThenormalizationA ischosensuchthat d (L) =d(L) Lφ
′ XX XX CMB
′ h ′ i ≡
A (L) L2 d2l1 f (l ,l )F (l ,l ) −1 , (9)
XX′ ≡ "Z (2π)2 XX′ 1 2 XX′ 1 2 #
where
F (l ,l ) = ClX1′X′tClX2XtfXX′(l1,l2)−ClX1X′tClX2X′tfXX′(l2,l1), (10)
XX′ 1 2 CXXtCX′X′tCX′X′tCXXt (CXX′tCXX′t)2
l1 l2 l1 l2 − l1 l2
whereforXX =TT,EE,BB,andTE,
′
fXX′(l1,l2)=ClX1X′1WXX′(l1,l2)+ClX2X′2WXX′(l1,l2), (11)
andforX = T,E ,X = B,
′
{ }
fXX′(l1,l2)=ClX1E1WXX′(l1,l2)+ClX2′X′2WXX′(l1,l2). (12)
ThewindowfunctionsW aregiveninTable I. InFig. 1, we showthe inputpowerspectrumof thelensed CMB fields, the
XX
′
reconstructed deflection field, and the corresponding Gaussian noise as obtained by using the estimators defined in Eq. (8).
Notethattheaverage denotesanensembleaveragerestrictedonlytodifferentGaussianrealizationsoftheprimordial
CMB
h i
CMB and instrumentnoise but assuming a fixed realization of the large-scale structure. The unmarkedaverage, , means
h i
theaverageovertheprimordialCMBfieldand thelarge-scalestructurerealizationsasdefinedinEq.(6). Forthepurposesof
estimatingthelarge-scalestructureintherealobservableuniverse,itisessentialtoensurethattheestimatorsafterappropriate
averagingoverrealizationsaretrulyunbiasedforatypicalrealizationoftheprimordialGaussianCMBfield. Aswewillseein
thenextsectionthatinthepresenceofnon-zerosystematiccontamination,theestimatorsarebiased.
III. INSTRUMENTALSYSTEMATICSEFFECTONTHELENSINGPOTENTIALPOWERSPECTRUM
Inthissection,weusethequadraticestimatorstoderivetheeffectsofinstrumentalsystematicsonthereconstructedlensing
potentialpowerspectrum. Weshowthatinstrumentalsystematicscanintroducenon-GaussiancorrelationsofCMBfields. The
instrumental systematics-induced CMB trispectrum gives an extra contamination to the reconstructed deflection angle power
spectrum. In subsection IIIA, we consider the simple case of the TT estimator to explain how systematics contaminate the
lensingreconstructionprocess.InsubsectionIIIB,wetaketheEBestimatorasanexampletoshowhowinstrumentalsystematics
in CMB polarization measurementsaffect the lensing reconstruction. In order to compare the experimentalrequirementsfor
primordialB-modedetectionandlensingpotentialreconstruction,wealso calculatetheeffectsofinstrumentalsystematicson
B-modedetectionforagiveninflationaryenergyscale.
5
4
10 Cldd Cldd
K)1023 DetecTtorT noise for reference experiment 10-5 Exp1 NN((00)) ffoorr TTTB eessttiimmaattoorr RExepfeerreinmceen t NNN(((000))) fffooorrr TTETBE eeessstttiiimmmaaatttooorrr
l(l+1)C(/21l 1110000-011 DetecTtorE noise for Exp1 BB dd ()+ll1C/2 11l00--76 NN((00)) ffoorr EEBE eessttiimmaattoorr TETEBETB dd ()+ll1C/21l0-7 ETTETB N(0) for EB estimator
-2 EE -8
10 10
EB
-3 -8
10 10
10 100 1000 10 100 1000 10 100 1000
l l l
FIG.1: Leftpanel: CMBpowerspectrumfortheTT,TE,EEandlensedBBpowerspectrum. Thelower(upper)dashedblacklineshowsthe
temperature(polarization)noiseforExp1. Thelower(upper)dottedblacklineshowsthetemperature(polarization)noiseforCMBPol-like
referenceexperiment. Centerpanel: GaussiannoisefordifferentquadraticestimatorsfortheExp1. Rightpanel: Sameascentralpanelbut
forCMBPol-likereferenceexperiment. Notethat theEBestimatorhasthelowestGaussian noise, andmaythusbeconsidered asthebest
estimatoramongallthepossiblequadraticestimators.
A. Asimpleexampleoftemperaturesystematics
Weintroducethecalibrationparameter(gainfluctuationofreceivers)a(nˆ)fortemperaturemeasurement,definedas:
T˜obs(nˆ)=[1+a(nˆ)]T˜t(nˆ). (13)
Ifweassumethatthereisnocorrelationbetweenthelensingpotentialφandtheinstrumentalsystematica(nˆ),thepowerspec-
trumofthelensedCMBtemperaturewithsystematicscorrectiontermcanbefoundas(moredetailsaregiveninAppendixA):
d2l d2l d2l
C˜TT = 1 1 Cφφ(l l)2 CTT + 1 CTT Cφφ[(l l ) l ]2 + 1 Caa CTT. (14)
l " −Z (2π)2 l1 1· # l Z (2π)2 |l−l1| l1 − 1 · 1 Z (2π)2 |l−l1| l1
This result is given to linear order in the lensing-potential power spectrum Cφφ and the gain fluctuation systematics power
l
spectrumCaa. Thelasttermrepresentsthebiasintroducedbythecalibrationsystematics. Intheabsenceofsystematiceffects,
l
itiseasytoprovethatthedeflectionangleestimatoris d (L) = Lφ(L) d (L),asdesired. Butoncethecontribution
TT CMB TT
h i ≡
fromthecalibrationparametera(nˆ)isconsidered,onefinds:
A (L) d2l
d (L) =d (L)+ TT 1 F (l ,l )a(L)(CTT +CTT), (15)
h TT iCMB TT L Z (2π)2 TT 1 2 l1 l2
i.e. in the presenceof a(L)the estimator d (L) is a biased estimatorfor thedeflectionfield in Fourierspace. Consequently,
TT
the deflection angle power spectrum d (L)d (L) would be biased due to systematic contamination, and is given by (see
TT TT
h i
AppendixAfordetails)
6
A (L)A (L)
d (L) d (L) = TT TT ′
TT TT ′ CMB
h · i LSS SYS L L
DD E E ′
d2l1 d2l′1 (2π)2F (l ,l )F (l ,l )
×Z (2π)2 Z (2π)2 TT 1 2 TT 1′ 2′
Cφφf (l ,l )f (l ,l )δ (L+L)
( L TT 1 2 TT 1′ 2′ D ′
+(2π)2CTTtCTTt δ (l +l )δ (l +l )+δ (l +l )δ (l +l )
l1 l2 D 1′ 1 D 2′ 2 D 2′ 1 D 1′ 2
h i
+ Cφφ f (l ,l )f (l ,l )+Cφφ f (l ,l )f (l ,l ) δ (L+L)
h |l1+l1′| TT 1 1′ TT 2 2′ |l1+l2′| TT 1 2′ TT 2 1′ i D ′
+ Caaf (l ,l )f (l ,l )+Caa f (l ,l )f (l ,l )
h L aa 1 2 aa 1′ 2′ |l1+l1′| aa 1 1′ aa 2 2′
+Caa f (l ,l )f (l ,l ) δ (L+L)
|l1+l2′| aa 1 2′ aa 2 1′ i D ′ )
= (2π)2δ (L+L) Cdd+N(0) (L)+N(1) (L)+N(S) (L)+... , (16)
D ′ " L TT,TT TT,TT TT,TT #
where we define f (l ,l ) = CTT +CTT. In the last line, the first term in the square bracketis the deflectionangle power
aa 1 2 l1 l2
spectrumCdd. Thesecondterm is the so calledGaussian noise N(0) (L) whichgivesthe dominantnoise contributionto the
L TT,TT
variance of the deflection power spectrum. The third term N(1) (L) is the the leading order non-Gaussian noise which is
TT,TT
first order in Cφφ and gives correction to the dominant Gaussian noise N(0) (L). The forth term, N(S) (L), is the leading
L TT,TT TT,TT
orderinstrumentalsystematiccontributiontothevarianceandisfirstorderinCSS. TheGaussiannoiseN(0) (L)andthefirst
L TT,TT
ordernon-Gaussiannoise N(1) (L)havebeenpreviouslycalculatedin[15]and[21],respectively. Thesystematicnoiseterm
TT,TT
N(S) (L)isanewcontributiontolensingpowerspectrum. Inprinciple,oneshouldincludenoisetermswhicharehigherorder
TT,TT
inCφφ andCSS,howeversincebothofthemaresmall,wetruncateatthefirstorderandexpectthathigherordercontributions
L L
aremuchsmaller.
The quadraticestimator given in Eq. (8) is optimizedin the presence of N(0) (L), and assuming no contributionfrom the
TT,TT
firstordernon-GaussiannoiseN(1) (L)andinstrumentalsystematicnoiseN(S) (L). Hencetheestimatorisoptimal2 aslong
TT,TT TT,TT
asN(1) (L) N(0) (L),andN(S) (L) N(0) (L). Ithasbeenshownthatthenon-GaussiannoiseN(1) (L)isaboutone
TT,TT ≪ TT,TT TT,TT ≪ TT,TT TT,TT
orderof magnitudesmaller than the Gaussian noise contribution[21]. In IIIB, we calculate the systematic noise term NS(L)
contributiontothelensingpotentialreconstructionfortheEBestimator.
B. Generalanalysisonpolarizationsystematics
WeparametrizethefieldsofinstrumentalsystematicsforCMBpolarizationmeasurementsfollowing[32]. Thepolarization
contaminations fall into two categories, one associated with the detector system which distorts the polarization state of the
incomingpolarizedsignal(TypeIhereafter),andanotherassociatedwithdistortionoftheCMBsignalduetothebeamanisotropy
(TypeIIhereafter). Thisparametrizationcanbe generalizedto differentpolarimeters. The instrumentalresponsetoincoming
CMBradiationisusuallydescribedbytheJonestransfermatrix. BiasinducedinthematrixdeterminationwillmixtheStokes
parametersdeterminedfromit. Tofirstorder,theeffectofTypeIsystematicsontheStokesparameterscanbewrittenas[32]
δ[Q iU](nˆ)=[a i2ω](nˆ)[Q iU](nˆ)+[f if ](nˆ)[Q iU](nˆ)+[γ iγ ](nˆ)T(nˆ). (17)
1 2 1 2
± ± ± ± ∓ ±
a is a scalar field which describesthe miscalibrationof the polarizationmeasurements(recallthatin last subsection, we used
a to denote the miscalibration of temperature measurements), ω is also a scalar field that describes the rotation angle of the
instrument,(f if )arespin 4fieldsthatdescribethecouplingbetweentwospinstates(spin-flip),and(γ iγ )arespin 2
1 2 1 2
± ± ± ±
fieldsthatdescribemonopoleleakagefromthetemperaturetopolarization.
2IfthesystematiccontributionsarecomparabletotheGaussiannoisecontributionthenweneedtodesignnewoptimallensingreconstructionestimatorsto
takeintoaccounttheinstrumentalsystematicseffect.
7
SimilartotheTypeIsystematics,theeffectofTypeIIsystematicsontheStokesparameterscanbewrittenas[32]
δ[Q iU](nˆ;σ)=σp(nˆ) [Q iU](nˆ;σ)+σ[d id ](nˆ)[∂ i∂ ]T(nˆ;σ)+σ2q(nˆ)[∂ i∂ ]2T(nˆ;σ) (18)
1 2 1 2 1 2
± ·∇ ± ± ± ±
the systematic fields are smoothed over the average beam σ of the experiment. Therefore the type II systematic fields are
sensitivetotheimperfectionofthebeamonthescaleσ. (p ip )arespin 1fieldsthatdescribepointingerrors,(d id )are
1 2 1 2
± ± ±
alsospin 1fieldsthatdescribedipoleleakagefromtemperaturetopolarization,andqisascalarfieldthatdescribesquadrupole
±
leakage[32].
As a simple model, we willassume thatthe contaminationfields, as definedin (17) and (18), are statistically isotropicand
Gaussian(althoughsomeofthesystematicsfieldsneednotbeso),thustheirstatisticalpropertiescanbefullydescribedbytheir
powerspectra,
S(l)S(l) =(2π)2δ(l+l)CSS , (19)
′ ′ l
(cid:10) (cid:11)
whereS standsforanyofthe11systematicfields. Thesystematicfieldscanbemodeledwiththepowerspectraoftheform
CSS =C exp( l(l+1)α2), (20)
l 0 − S
i.e.whitenoiseabovecertaincoherencescaleα ,whichisakeyquantitytoaffectthelevelofcontaminationofeachsystematics
S
effects. ThenormalizationfactorC canbedeterminedby
0
d2l −1
C2 =A2 exp( l(l+1)α2) , (21)
0 S"Z (2π)2 − S #
whereA characterizesthermsofthecontaminationfieldS.
S
TheinstrumentalsystematicsinducedistortionsontheCMBfileds. ThecontaminationstotheBBandEEpowerspectradue
todifferentmeasurementsystematicstaketheform
d2l d2l
δClBB =XSS Z (2π)′2C|Sl−Sl′′|C|El′E| (σ)[WBS(l,l′)]2 +XSS Z (2π)′2C|Sl−Sl′′|C|Tl′T| (σ)[WBS(l,l′)]2, (22)
′ ′
d2l d2l
δClEE =XSS Z (2π)′2C|Sl−Sl′′|C|El′E| (σ)[WES(l,l′)]2 +XSS Z (2π)′2C|Sl−Sl′′|C|Tl′T| (σ)[WES(l,l′)]2. (23)
′ ′
The explicit forms of WS(l ,l ) and WS(l ,l ) are given in Table II, which are the window functionsof each systematic S
B 1 2 E 1 2
forB-modeandE-modeharmonics,respectively. The summationsofthe first termonthe RHS of Eq. (22) andEq. (23) run
overcalibrationa,rotationω,spinflip f and f ,andpointingerrorγ andγ . Thesummationsofthesecondtermrunoverthe
a b a b
restofthesystematicsparameterswhichdescribethetemperatureleakagegiveninTableII. CEE(σ)andCTT(σ)arethebeam
l l
smoothedtemperatureandE-modepolarizationpowerspectra.
CEE(σ)=CEEexp( l(l+1)σ), CTT(σ)=CTTexp( l(l+1)σ). (24)
l l − l l −
WeuseEq. (22)andEq. (23)tocalculatethesystematicrequirementsforB-modedetectioninordertocomparetherequire-
mentsforlensingreconstruction.WeshowtheresultsinTablesIIIandIV.
Now we move on to calculate the systematic contamination on the lensing power spectrum. The polarization fields are
essentiallyuncorrelatedwiththelensingpotential,soifwedonotconsidersecondaryeffects,then-pointfunctionswithnodd
arezero. Thenextnon-zeroorderisthe trispectrum. Thecalculationfortheconnectedpartofthetrispectrumof polarization
in the presence of instrumental systematics is similar to the connected temperature trispectrum presented in the last section.
Here we give the results for the trispectrum related to lensing reconstructionusing EB estimator, and refer the readersto the
AppendixCfortheexplicitcalculationforotherquadraticestimators. Atleadingorderwehave
8
TypeofS WS(l ,l ) WS(l ,l )
B 1 2 E 1 2
Calibrationa sin[2(ϕ φ )] cos[2(ϕ ϕ )]
l2− L l2− L
Rotationω 2cos[2(ϕ ϕ )] 2sin[2(ϕ ϕ )]
l2− L − l2− L
Pointingp σ(l ˆl ) zˆsin[2(ϕ ϕ )] σ(l ˆl )sin[2(ϕ ϕ)]
a 2× 1 · l2− L 2· 1 l2− l
Pointingp σ(l ˆl )sin[2(ϕ ϕ)] σ(l ˆl ) zˆsin[2(ϕ ϕ )]
b 2· 1 l2− l − 2× 1 · l2− L
Flip f sin[2(2ϕ ϕ ϕ )] cos[2(2ϕ ϕ ϕ )]
a l1 − l2− L l1− l2− L
Flip f cos[2(2ϕ ϕ ϕ )] sin[2(2ϕ ϕ ϕ )]
b l1 − l2− L − l1− l2− L
Monopoleγ sin[2(ϕ ϕ)] cos[2(φ ϕ)]
a l1− l l1 − l
Monopoleγ cos[2(ϕ ϕ)] sin[2(ϕ φ)
b l1− l − l1− l
Dipoled (l σ)cos[ϕ +ϕ 2ϕ] (l σ)sin[ϕ +φ 2ϕ]
a − 2 l1 l2− l 2 l1 l2 − l
Dipoled (l σ)sin[ϕ +φ 2ϕ] (l σ)cos[ϕ +ϕ 2ϕ]
b 2 l1 l2− l 2 l1 l2− l
Quadrupoleq (l σ)2sin[2(ϕ ϕ)] (l σ)2cos[2(ϕ ϕ)]
− 2 l2− l − 2 l2− l
TABLEII:Windowfunctionsforallthe11systematicparameters. Firstcolumnindicatesthetypeofsystematicparametersinconsideration.
SecondandthirdcolumnsshowwindowfunctionsforsystematicsinducedB-modeWS(l ,l ),andforE-modeWS(l ,l )respectively. These
B 1 2 E 1 2
windowfunctionsareneededtocalculatesystematiccontaminationonprimordialgravitationalwavedetectionorthedeflectionanglepower
spectrumreconstruction.Wenotethatl =l ˆl , l =L l ,andl =l ˆl .
1 1 1 2 − 1 2 22
E˜(l )obsB˜(l )obsE˜(l )obsB˜(l )obs = (2π)2δ (l +l +l +l )
1 2 ′1 ′2 c D 1 2 ′1 ′2 ×
D E
CEECEE Cφφ W (l , l )W (l , l )+Cφφ W (l , l )W (l , l )
( l1 l′1 " |l1+l2| B 2 −1 B ′2 −′1 |l1+l′2| B 2 −′1 B ′2 −1
P distortion P distortion
− −
+ CSS WS(l , l )WS(l , l ) + CSS WS(l , l )WS(l , l )
XS |l1+l2| B 2 −1 B ′2 −′1 XS |l1+l′2| B 2 −′1 B ′2 −1 i
T leakage
−
+CTECTE CSS WS(l , l )WS(l , l ) +
l1 l′1 h XS |l1+l2| B 2 −1 B ′2 −′1
T leakage
−
CSS WS(l , l )WS(l , l ) , (25)
XS |l1+l′2| B 2 −′1 B ′2 −1 #)
wherewe definedthelensingB-modewindowfunctionW (l,l) l (l l)sin2(ϕ ϕ )andWS(l , l )isthesystematics
B ′ ≡ ′ · − ′ l− l′ B 1 −′1
windowfunctionforanyofthe11systematicsparameters. TheformulaforeachofthesystematicwindowfunctionsWS(l,l)
B ′
canbefoundinTableII.
Differenttrispectrumcanbeconstructedfromcombinationsofthetemperatureandpolarizationfields. Wediscussothercases
intheAppendixCinordertocalculateEE,TEandTBestimatorsincludingcontributionsfromsystematicscontamination.The
formulasshownherearereadilygeneralizedtothefullsky. Foradiscussionofthesphericalgeneralizationofthepolarization
trispectra, see [47]. We have shown that E and B-modes are mixed not only by weak lensing, but also by instrumental sys-
tematics. Evenifthereisnolensinginducedcorrelation,certainkindsofsystematicscangivea non-zerocontributionsto the
trispectrum. Now we move on to construct quadratic lensing estimators from E/B polarization modes, and we quantitatively
showhowinstrumentalsystematicscontaminationaffectsthe lensingreconstructionprocess. Againwe take the EBestimator
as an example, and leave the discussions of other estimators to the Appendix C. The variance of the deflection angle power
spectrumincludingsystematiceffectscanbewrittenas
d (L) d (L) = AEB(L)AEB(L′) d2l1 d2l′1 F (l ,l )F (l ,l )
h EB · EB ′ iCMB LSS SYS L L Z (2π)2 Z (2π)2 EB 1 2 EB 1′ 2′
DD E E ′
E˜(l )obsB˜(l )obsE˜(l )obsB˜(l )obs
× 1 2 ′1 ′2
D E
= (2π)2δ (L+L) Cdd(L)+N(0) (L)+N(1) (L)+N(S) (L)+... , (26)
D ′ " EB,EB EB,EB EB,EB #
whereL = l +l , andCdd(L)isthedeflectionanglepowerspectrum. ThetermsN(0) (L), N(1) (L),and N(S) (L)arethe
1 2 EB,EB EB,EB EB,EB
Gaussian noise, first order non-Gaussiannoise, and the first order systematics noise. The Gaussian noise contributioncomes
9
Cldd s=20’ Cldd s=20’
10-6 a Exp1 NN((01)) ss==6300’’ a Reference NN((01)) ss==6300’’
d/2 -7 ss==1105’’ s=120’ experiment ss==1105’’ s=120’
d 10
Cl
1)
+ -8
(l 10
l
-9
10
-10
10
CNl(d0d) ss==2300’’ Cldd s=20’
-6 Exp1 N(1) s=60’ Reference N(0) s=30’
10 s=10’ s=120’ experiment N(1) s=60’
2 s=15’ s=10’ s=120’
dd/ s=15’
Cl -7
1) 10
+
(l
l
-8
10
-9
10
10 100 1000 10 100 1000
l l
FIG.2: Upperpanels: ContaminationfromthecalibrationsystematicsatothedeflectionanglepowerspectrumusingtheEBestimator. The
rmsfluctuation, A isassumedtobe10%. TheleftandrightpanelareforExp1andreferenceexperimentrespectively. Inboththepanels,
S
the solid black, dashed blue, and dot red curves show deflection angle power spectrumCdd(L), Gaussian noise N(0)(L), and the firstorder
non-GaussiannoiseN(1)(L). Theremainingcurvesshowtheabsolutevalueofthesystematicbiasforvariouschoicesofcoherencelengthα ,
s
startingfromα =10 toα =120.Lowerpanels:Sameastheupperpanelsbutforrotationsystematicsω.
s ′ s ′
fromthedisconnectedpartofthefour-pointfunction,whileboththefirst-ordernon-Gaussiannoise N(1) (L)andsystematics
EB,EB
noise N(S) (L) contributioncomes from the connected part. In derivingEq. (26) we have used filters F which are given
EB,EB EB
inEq. (10),andthetrispectrumwhichisgiveninEq. (25). Theellipsesstandsfortermsbeyondfirstorderinthesystematics
power spectra or the deflection angle power spectrum. We note that the Gaussian noise term also includessystematic effects
implicitlysinceinstrumentalsystematicsbiasthemeasuredpowerspectrumaswehaveshown.TheGaussianandnon-Gaussian
noisetermsin theabsenceof systematiccontributionhavebeenpreviouslyreportedin [15]and [21]. Thesystematicsnoise
termN(S) (L)isnew,forwhichtheexplicitformisgivenas
EB,EB
N(S) (L) = AEB(L)AEB(L′) d2l1 d2l′1 F (l ,l )F (l ,l )
EB,EB L L Z (2π)2 Z (2π)2 EB 1 2 EB 1′ 2′
′
P distortion P distortion
CEECEE − CSS WS(l , l )WS(l , l ) + − CSS WS(l , l )WS(l , l )
( l1 l′1 " XS |l1+l2| B 2 −1 B ′2 −′1 XS |l1+l′2| B 2 −′1 B ′2 −1 #
T leakage T leakage
− −
+CTECTE CSS WS(l , l )WS(l , l ) + CSS WS(l , l )WS(l , l ) . (27)
l1 l′1 " XS |l1+l2| B 2 −1 B ′2 −′1 XS |l1+l′2| B 2 −′1 B ′2 −1 #)
Eq. (27)isourmainresultforthesystematicscontaminationonlensingpowerreconstructionusingtheEBestimator. Wewill
usethisequationtonumericallycomputethesystematic-inducedbiasforthe11systematicparameters. Theresultsareshown
10
-6
10 Cldd s=20’ Cldd s=20’
fa Exp1 NN(s(0=1)) 1 0 ’ ss==63s=001’’ 20’ fa Rexepfeerriemnceen t NN((01)) ss==6300’’
s=15’ s=10’ s=120’
d/210-7 s=15’
d
Cl
)
1
+
(l -8
l10
-9
10-6
10 Cldd s=20’ Cldd s=20’
fb Exp1 NN(s(0=1)) 1 0 ’ ss==63s=001’’ 20’ fb Rexepfeerriemnceen t NN((01)) ss==6300’’
s=15’ s=10’ s=120’
-7
210 s=15’
d/
d
Cl
)
1
+
(l10-8
l
-9
10
10 100 1000 10 100 1000
l l
FIG.3:Upperpanels:Contaminationfromthespin-flipsystematics f tothedeflectionanglepowerspectrumusingtheEBestimator.Therms
a
fluctuation,A isassumedtobe10%.TheleftandrightpanelareforExp1andreferenceexperimentrespectively.Inboththepanels,thesolid
S
black,dashedblue,anddotredcurvesshowdeflectionanglepowerspectrumCdd(L),GaussiannoiseN(0)(L),andthefirstordernon-Gaussian
noiseN(1)(L). Theremainingcurvesshowtheabsolutevalueofthesystematicbiasforvariouschoicesofcoherencelengthα ,startingfrom
s
α =10 toα =120.Lowerpanels:Sameastheupperpanelsbutforspin-flipsystematics f .
s ′ s ′ b
inFigures(2)to(7)andarediscussedinnextsection3.
IV. RESULTS
Figures2to7andTablesIII&IVsummarizeourmainfindings.Wehavefocusedonthesystematics-inducedbiasfortheEB
estimator(resultsfortheotherestimatorsareprovidedintheAppendixC)becauseithasthehighestGaussiansignal-to-noise
ratioforreconstructingtheprojectedmatterpowerspectrum(SeeFig.1or[15]).Figures2-7showthecontaminationintroduced
bydifferentsystematiceffects(thetermN(S) (L)inEq. (27))inthedeflectionanglepowerspectrumreconstruction. Wehave
EB,EB
assumedthermsfluctuationofthesystematicsfieldstobe10%,andvariedcoherencelengthstartingfromminimumα = 10
s ′
tothemaximumα =120. Forcomparison,wealsoshowthelevelofGaussiannoiseandfirstordernon-Gaussiannoiseofthe
s ′
EB estimator, the terms N(0) (L) and N(1) (L) respectivelyin Eq. (27). Systematics-inducedbiasgenerallyincreaseswith
EB,EB EB,EB
3Asanote,foreachinstrumentalsystematicsparameterS,forboththepolarizationdistortionandtemperatureleakage,therearetwotermswhichcontribute
tothefinalresults. ThetermsproportionalCSS isgenerallymuchsmallerthanthetermsproportionaltoCSS butfornon-GaussiannoiseN(1)[21],the
termsproportionaltoCφφ areaboutanord|el1r+ol2f|magnitudelargerthanothertwotermsproportionaltoCφφ |l1+al′2n|dCφφ .
|l1+l2| |l1+l1′| |l1+l2′|
