Table Of ContentAstronomy&Astrophysicsmanuscriptno.paper_cross_corr_rnote_published_arxiv cESO2016
(cid:13)
August9,2016
Optimal cosmic microwave background map-making in the
presence of cross-correlated noise
(ResearchNote)
G.deGasperis1,2,A.Buzzelli2,3,P.Cabella1,2,P.deBernardis3,4,andN.Vittorio1,2
1 DipartimentodiFisica,UniversitàdiRoma“TorVergata”,viadellaRicercaScientifica1,I-00133,Roma,Italy
2 SezioneINFNRoma2,viadellaRicercaScientifica1,I-00133,Roma,Italy
6
3 DipartimentodiFisica,SapienzaUniversitàdiRoma,piazzaleAldoMoro5,I-00185,Roma,Italy
1
4 SezioneINFNRoma1,piazzaleAldoMoro5,I-00185,Roma,Italy
0
2
Received/Accepted
g
u
ABSTRACT
A
Aims.WepresentanextensionoftheROMAmap-making algorithmforthegenerationofoptimalcosmicmicrowavebackground
8
polarizationmaps.Thenewcodeallowsforapossiblecross-correlatednoisecomponentamongthedetectorsofaCMBexperiment.
ApromisingapplicationistheforthcomingLSPEballoon-borneexperiment,whichisdevotedtotheaccurateobservationofCMB
]
O polarizationatlargeangularscales.
Methods.Wegeneralized thenoisecovariance matrixintimedomaintoaccount foralltheoff-diagonal termsduetothedetector
C
cross-talk.Hence,weperformedpreliminaryforecastsoftheLSPE-SWIPEinstrument.
. Results.Wefoundthatconsideringthenoisecross-correlationamongthedetectorsresultsinamorerealisticestimateoftheangular
h
powerspectra.Inparticular,theextendedROMAalgorithmhasprovidedaconsiderablereductionofthespectraerrorbars.Weexpect
p
thatthisimprovementcouldbecrucialinconstrainingtheB-modepolarizationatthelargestscales.
-
o
Keywords. Cosmology:Cosmicmicrowavebackgroundpolarization–Methods:dataanalysis
r
t
s
a
[1. Introduction thatallowsustotakecommon-modenoiseintoaccountexplic-
itlyandproperlyisthatbyPatanchonetal.(2008),forintensity
2The temperature and polarization patterns of cosmic mi-
measurementsonly.GiventhepressinginterestonprimordialB-
vcrowave background (CMB) have been an invaluable
modedetection,aspecifictreatmentofnoisecross-correlationin
4source of cosmological information. While polarization
6 polarizationmeasurementsisnownecessary.
E-modes have been widely observed and analyzed (e.g.,
4 Amongthenextgenerationof CMB experiments,we focus
Planck: PlanckCollaborationetal. 2015b), B-modes are still
4 ontheLarge-ScalePolarizationExplorer1 (LSPE)balloonmis-
buried into foreground signal and experimental noise (e.g.,
0 sion, which is devoted to accurate observationsof CMB polar-
BICEP2/KeckandPlanckCollaborationsetal. 2015). Nowa-
. izationatlargeangularscales(seee.g.,deBernardisetal.2012).
1days, the hunting for B-modes represents one of the most
0 TheLSPEisexpectedtoimprovethelimitontheratiooftensor-
relevant and exciting research fields in cosmology. In fact,
6 to-scalarperturbationamplitudesdowntor .0.03.
a B-mode detection in CMB polarization would provide a
1 In this work we present an extension of the ROMA
definitiveconfirmationoftheexistenceofagravitationalwaves
: map-making code (Roma optimal map-making algorithm,
vprimordial background, as expected in the inflation paradigm
deGasperisetal. 2005) to produce optimal CMB polarization
i(Lyth&Riotto1999).
X mapsoutofcross-correlatedmulti-detectorCMB observations.
Becauseofthelowsignal-to-noiseratio,polarizationobser-
r Inparticular,wediscusshowaproperinclusionofthedetector
avationsrequirelargemultidetectorarrays.Theincreaseddetec- noisecross-correlationsresultsinmorerealisticestimatesofthe
tor array size and the integration of many pixels on the same
angular power spectra error bars. This benefit could be crucial
wafer rise the problem of cross talk among the detectors. This
for the detection of a primordial B-mode signal at low multi-
canbeduetoeitherfocalplanetemperaturevariationsand/orat-
poles.
mospheric fluctuations. Both these effects are expected to pro-
duce common-mode noise in the detectors. This noise cross-
correlation has to be properly taken into account in the map- 2. Formalism,algebraandnoisemodel
makingprocedure.
After the pioneering work of Wright (1996), map-making A CMB experimentobservesthe sky at a given resolution(i.e.
has been thoroughly studied in the literature (see e.g., with Np pixels in the sky) and collects Nd samples in a given
Cantalupoetal. 2010, and referencestherein). However,cross- temporalsequence,thetimeordereddata(TOD).Oncethedata
correlated noise among the detectors has not been discussed arecalibrated,anyartifactisflaggedoutandthepointingisre-
as much and is crudely neglected. The only detailed treatment constructed,the first step in the CMB data analysis is the esti-
Sendoffprintrequeststo:[email protected] 1 http://planck.roma1.infn.it/lspe
Articlenumber,page1of5
A&Aproofs:manuscriptno.paper_cross_corr_rnote_published_arxiv
mationofoptimalskymapsfromtheTOD.Intheliteraturetwo
main strategies are present: a maximum-likelihood(minimum-
variance)approach(see e.g.,deGasperisetal. 2005, andrefer-
encestherein)anddestripingtechniques(seee.g.,Tristrametal.
2011,andreferencestherein).
Wetakethecross-correlatednoisecomponentamongdetec-
tors into account in the following way. The observational data
fromonedetectorcanbemodeledas
D=AS+n. (1)
Here, D is the TOD, A is the generalizedpointing matrix, S =
(I,Q,U)isthemaptripletandnistheinstrumentalnoise,which
accountsfor anysystematic effects, cosmic ray hits, etc. When
dealingwithmultipledetectorobservations,theTODsfromin-
dividual detectors are simply concatenated end-to-end. Under
the assumption of a Gaussian and stationary noise, the gener-
alized least squared approach yields the following maximum-
likelihoodestimatorforthesignal
S=(cid:16)ATN−1A(cid:17)−1ATN−1D. (2)
e
The noise covariancematrix in the time domain, N nn ,
t t
≡ h ′i
isblockdiagonalonlyinthecaseofnocross-correlationamong
samplesofdifferentdetectors.ThepresentversionoftheROMA
algorithm takes into account all the off-diagonal terms, which
have usually beenneglectedin past works. The solutionof Eq.
2isaverycomputationallydemandingtaskbecauseofthelarge
size of the matrix N, kN kN , where k is the number of de-
p p Fig.1.Skyregionscannedby18sampledetectors,sparselylocatedin
×
tectors. Our algorithm adopts a Fourier-based, preconditioned the two focal planes of LSPE-SWIPE, in ecliptic coordinates (upper
conjugate-gradientiterativemethod. panel)andtheinversepixelconditionnumberR asanestimatorof
cond
The choice of the noise model is clearly a crucial issue. In polarization angle coverage per pixel (lower panel). In our definition,
general,thedetectornoisespectrumisinthesumofastationary R =1/2incaseofperfectanglecoverageuniformity.Bothmapsare
cond
Gaussianwhitenoiseathigherfrequenciesanda1/f component atHEALPix(seeGórskietal.2005)Nside=128.
at lower frequencies,which implies an unavoidablecorrelation
among different samples in the single detector time stream. A
furthercomplicationisthepresenceofthenoisecommon-mode 106
seenbyallthedetectors.Itscontributionhasbeenestimateddi-
rectlyfromdata(seee.g.,Masietal.2006).Thecross-correlated 104
noiseistypicalofground-andballoon-basedexperiment,asitis ]
1
sourced mainly by atmospheric fluctuations. However, it could z− 102
also affectspace missions, due to common-modedetectortem- H
2
peraturedrifts.Thisworkisaddressedtotheballoonregime,but K 100
µ
ourtreatmentiscompletelygeneral. [
/
r 10-2
e
w
3. ForecastsoftheLSPE-SWIPEexperiment o
P 10-4
In this work we focuson the forthcomingLSPE balloon-borne
experimentthatisdevotedtotheaccuratemeasurementofCMB 10-6
polarizationatlargeangularscales.
0.001 0.010 0.100 1.000 10.000
Weusethelevelofnoisecross-correlationamongtheCMB Frequency/[Hz]
polarimetersestimated for BOOMERanG (Masietal. 2006) as
abenchmarkforoursimulations.Sincethedeepsimilaritiesbe-
Fig.2.Powervsfrequencyfortemperature(indashedblackline)and
tween BOOMERanG and the LSPE-SWIPE experiment (long
polarizationintensity(insolidredline)inthecaseof halfwaveplate
duration stratospheric bolometric experiments subject to atmo-
(HWP) steps of 11.25 /min. In our simulations, we focused on two
◦
spheric common-modefluctuations),we foundit naturalto use noisekneefrequencies:0.02and0.1Hz.
the BOOMERanG noise properties for the LSPE-SWIPE pre-
liminaryforecasts.
55 deg. The azimuth scan speed will be set around 2 rpm, i.e.
12 deg/s. A large portion of the northern sky (around 25% of
3.1.Theinstrument
thecelestialsphere)isexpectedtobeobservedwithanangular
During its circumpolar Arctic flight, LSPE will scan the sky resolutionofabout1.5degreesFWHM. Thepayloadwillhost
by spinning around the local vertical, while keeping the tele- twoinstruments:theShortWavelengthInstrumentforthePolar-
scope elevation constant for long periods, in the range 30 to ization Explorer(SWIPE; deBernardisetal. 2012), which will
Articlenumber,page2of5
G.deGasperisetal.:OptimalCMBmap-makinginthepresenceofcross-correlatednoise(RN)
maptheskyinthreefrequencybandscenteredat140,220,and f =0.02Hz
k
240GHz; and the STRatospheric Italian Polarimeter (STRIP,
0
Bersanellietal. 2012), which will survey the same sky region
intwofrequencybandscenteredat43and90GHz.
In this work we focus on the SWIPE bolometric po- %) -10
larimeter instrument, which is characterized by two symmetric e (
nc -20
orthogonally-placedfocal planes hosting an overall number of e
er
1th1e0pdreetseecntcoersopfear fhraelqfuwenavcye.pTlahteep(eHcWuliPa)riptyoloafriSzaWtiIoPnEmwoidllulbae- al diff -30
torasfirstopticalelement,followedbya50cmaperturerefrac- n -40
o
tive telescope, a beam-splittingpolarizer,and finally the multi- cti
a
moded focal planes. The SWIPE detectors are multimode spi- Fr -50
derweb TES bolometers operating at 0.3K (see Gualtierietal.
2016). -60
In Fig. 1 we show the sky region as seen by a subset of
0 20 40 60 80 100
18 LSPE-SWIPE detectors, which are arranged in three triples
Multipoleℓ
sparsely located in each of the two focal planes, and the cor-
responding angle coverage estimator, for one observation day.
Fig.3. FractionaldifferenceoftheBBpowerspectraerrorbarsfrom
The maps are at HEALPix2 resolution N = 128 (27.5’ per
side LSPE-SWIPE simulations estimated considering and neglecting the
pixel,seeGórskietal.2005).InFig.2weshowthetemperature
bolometer noise cross-correlation of case 1 (in dashed red line) and
and polarization intensity power as a functionof frequencyfor
case2(insolidblackline)forakneefrequency f =0.02Hz.
k
onebolometeroftheSWIPEinstrument,assumingthenominal
HWPstepsof11.25 /min.
◦ f =0.1Hz
k
Becauseofthelargenumberofdetectorsweexpectthatthe
noisecross-correlationmayrepresentacriticalissue andthere- 0
fore,apropertreatmentisnecessary.However,wehighlightthat
thecross-correlationamongthedetectorsmustbeaccountedfor %) -10
inrelationtothepossiblefilteringofthedata;thebenefitofcon- e (
sideringthenoisecommonmodecruciallydependsonthemag- nc -20
e
nitudeofthelow-frequencycut.Forinstance,wetestedtheex- er
tendedROMA codeonthe BOOMERanGrealdata setandwe diff -30
found no remarkable benefits with respect to the past analysis al
n -40
that completely neglected the noise cross-correlation. The low o
impact of the new treatment is due to the heavy filter of low- acti
frequency data streams performed on the real data set, which Fr -50
hascrudelycutouttheinformationatlargeangularscaleswhere
-60
the cross-correlatednoise effectis morerelevant.However,us-
ing simulated unfiltered data we found that a proper treatment 0 20 40 60 80 100
ofthenoisecross-correlationresultsinconsiderablybettermaps Multipoleℓ
and angular power spectrum estimates (the spectrum standard
deviationshavebeenreducedupto20%;seeBuzzelli2015and Fig.4. FractionaldifferenceoftheBBpowerspectraerrorbarsfrom
Buzzellietal. 2016). In the following simulations, we first as- LSPE-SWIPE simulations estimated considering and neglecting the
sumethatnofilterwasappliedonthedata,hencewefaceaspe- bolometer noise cross-correlation of case 1 (in dashed red line) and
cificcasewherereasonablefrequencycutshavebeenperformed. case2(insolidblackline)forakneefrequency fk =0.1Hz.
3.2.Simulations 1. The cross-correlationis present in both the 1/f2 and white
noisepartofthespectrum.Intheformer,thecross-correlated
WegeneratesimulatedTODbasedontheSWIPEscanningstrat- componentissharedbyallthedetectors;inthelatterweas-
egy and polarimeter angles for one day of observations (Nd = sume 10% of cross-correlation among each triple and 1%
8.64 106 samples per detector) and telescope elevation of 45 among any other detector, with respect to the auto noise
×
deg. In particularwe choose an HWP stepping of 11.25◦/min, spectrum. This situation is the analog to that found in the
repeatedlyscanningthe range0◦ 78.75◦, andwe consider18 BOOMERanGanalysis;
−
detectorsasdescribedabove. 2. Thecross-correlationislimitedtothe1/f2 partofthenoise
We assume the cosmological parameters estimated by spectrumwithnowhitenoisecommonmode.
Planck (PlanckCollaborationetal. 2015a) with a tensor-to-
scalar ratio r = 0.09. The noise is simulated assuming the In the four cases, we produced 50 signal-only, noise-only, and
bolometer to be photon-noise limited with a white plateau of signalplusnoiseMonteCarlosimulatedmaps,bothtakinginto
15µK √sathigherfrequenciesanda1/fα noisewithα=2 accountornotthenoisecross-correlationamongdetectorsinthe
CMB
at lower frequencies, as expected for the multimoded 140GHz map-makingcode.Then,weappliedtheMASTERpowerspec-
channel.Wefocusontwokneefrequencies,0.02and0.1Hz,the trumestimator(Hivonetal.2002)tothemaps.
expectedbestandworstcase.Foreachkneefrequency,wecon- Itshouldbetakenintoaccountthat,atlargeangularscales,
sidertwodifferentcommon-modeconfigurations: the MASTER spectrumestimator methodmay notbe the most
convenient choice. As mentioned by Molinarietal. (2014),
2 http://healpix.sourceforge.net at low multipoles a quadratic maximum likelihood estimator
Articlenumber,page3of5
A&Aproofs:manuscriptno.paper_cross_corr_rnote_published_arxiv
(QML) is preferable. Nonetheless, the aim of this work is to
check the reliability of an improved map-making code, which 20
0.02 Hz
can also be successfully tested by using quicker but less accu- 0.05 Hz
0 0.1 Hz
rateangularpowerspectraestimates.
In Fig. 3 we show the fractional difference of BB power %)
spectraerrorbarsestimatedconsideringandneglectingthenoise e ( -20
c
n
cross-correlation in the algorithm, assuming the data set to be ere -40
cfoorntaamknineeatferdeqbuyenthceytfwko=n0o.i0s2eHcoz.nfiTghuerraetisounltsdefoscrrifbke=d a0b.1oHvez, al diff -60
are shown in Fig. 4. It is evident that neglecting the cross- n
o
correlation affects heavily the power spectra error bars. For cti -80
a knee frequency fk = 0.02Hz, the inclusion of the cross- Fra
correlation has the effect of reducing the spectra error bars up -100
to 40 50%. For f = 0.1Hz, we find an improvement up to
− k -120
50 60%.
− 0 20 40 60 80 100
Multipoleℓ
3.3.Comparisonwithfilteringtechniques
Fig. 5. Fractional difference between the BB power spectrum error
As mentionedabove,to reducethe low-frequencynoise contri-
bars estimated including the noise cross-correlation and applying a
bution, it is a common choice to high pass the data stream at low-frequency data filtering (neglecting the cross-correlated noise) at
somecutfrequency fc.However,thismethodisnotlossless, as fc=0.02Hzindottedline, fc=0.05Hzindashedlineand fc=0.1Hz
partofthesignalinformationisfilteredoutaswell. insolidline.TheBBspectracorrespond tothenoiseconfigurationof
While this option appears to be feasible when the low- case2with fk =0.1Hz.
frequency cutoff f is 0.02Hz since the cosmological signal at
c
lower frequencies is negligible, we found that (see Fig. 2) if
scaleswiththeprimaryaimtoconstraintheprimordialB-mode
f = 0.05 or 0.1Hz the high-pass filter will cut the polariza-
c polarization.
tionintensitypowerof 17and25%andtheI Stokespowerof
≃ WeappliedtheextendedROMAalgorithmtosimulateddata
31and48%,respectively.
≃ oftheLSPE-SWIPEinstrumentandfoundthattheinclusionof
To estimate the effectof data filteringon the powerspectra
cross-correlationprovidesspectraerrorbarssmaller upto50
errorbars,weperformasimilaranalysisasabovebutwithhigh- −
60%,dependingonthechosenkneefrequency.Wepointoutthat
pass frequencies f = 0.02,0.05and 0.1Hz and neglecting the
c this improvement could be crucial in constraining the B-mode
cross-correlatednoiseintheanalysisforthenoiseconfiguration
polarization.
ofcase2and f =0.1Hz.
k Averyrelevantissueistocomparetheseresultswithapos-
We comparethese results with the powerspectra errorbars
sible low-frequencydata filtering, commonly used for ground-
estimatedaccountingforcross-correlatednoisewithnodatafil-
based and balloon-borne experiments. We found that account-
tering. In Fig. 5 we show the fractional difference of the error
ingforcross-correlatednoisewithnofilteringisaviableoption,
barsbetweenthecaseswithcross-correlationandlow-frequency
whichiscertainlylesscrudethanacutoflow-frequencystreams,
filtering. It can be noticed that, by including the noise cross-
withtheadditionalpotentialadvantageofnotremovingcosmo-
correlationin the map-makingcode, we are recoveringsmaller
logicalinformationatthelargestscales.
BB power spectrum error bars compared to the case with data
filtering.Atverylowmultipoles,theimprovementisupto30% Acknowledgements. We acknowledge the use of the HEALPix package
and100%forthe f = 0.02Hzand f 0.05Hzcases,respec- (Górskietal.2005)andoftheFFTWlibrary(Frigo&Johnson2005).Wewish
c c ≥ tothankSilviaMasiandMarinaMigliaccioforusefulsuggestionsanddiscus-
tively.
sions,andLSPEcollaborationforprovidinguswiththeLSPE-SWIPEpointing
It is not the aim of this work to forecasta suitable filtering informations.
strategyforLSPE-SWIPEinrelationtotheinclusionofthenoise
cross-correlationintheanalysis.Thiswouldrequireamoreac-
curate specifications of many parameters, such as telescope el-
References
evation,azimuthscanvelocityandHWPvelocity.However,we
stress that the improvementsprovided by the extended ROMA Bersanelli, M., Mennella, A.,Morgante, G., etal. 2012, in Society ofPhoto-
code may result in a less dramatic filtering of low-frequency OpticalInstrumentationEngineers(SPIE)ConferenceSeries,Vol.8446,So-
data,thuspreservingmostofthecosmologicalinformation. cietyofPhoto-OpticalInstrumentationEngineers(SPIE)ConferenceSeries,
7
BICEP2/Keck and Planck Collaborations, Ade, P. A. R., Aghanim, N., et al.
2015,PhysicalReviewLetters,114,101301
4. Conclusions Buzzelli,A.2015,Master’sthesis,UniversitàdiRoma“TorVergata”,Italy
Buzzelli,A.,Cabella,P.,deGasperis,G.,&Vittorio,N.2016,JournalofPhysics
We presented a new version of the ROMA map-making code ConferenceSeries,689,012003
extendedtoapossiblecross-correlatednoisecomponentamong Cantalupo,C.M.,Borrill,J.D.,Jaffe,A.H.,Kisner,T.S.,&Stompor,R.2010,
ApJS,187,212
thedetectorsofaCMBexperiment.
deBernardis,P.,Aiola,S.,Amico,G.,etal.2012,inSocietyofPhoto-Optical
This effect must be properly taken into account in the data
Instrumentation Engineers(SPIE)ConferenceSeries,Vol.8452,Societyof
analysisofanyexperimentaimedatdetectionofB-modepolar- Photo-OpticalInstrumentationEngineers(SPIE)ConferenceSeries,3
ization,eitherfromspace,balloonsorground. deGasperis,G.,Balbi,A.,Cabella,P.,Natoli,P.,&Vittorio,N.2005,A&A,436,
1159
Amongthe nextgenerationofCMB experiments,we focus
Frigo,M.&Johnson,S.G.2005,ProceedingsoftheIEEE,93,216,specialissue
ontheforthcomingLSPEballoon-borneexperiment,devotedto
on“ProgramGeneration,Optimization,andPlatformAdaptation”
the accurate observation of CMB polarization at large angular Górski,K.M.,Hivon,E.,Banday,A.J.,etal.2005,ApJ,622,759
Articlenumber,page4of5
G.deGasperisetal.:OptimalCMBmap-makinginthepresenceofcross-correlatednoise(RN)
Gualtieri,R.,Battistelli,E.S.,Cruciani,A.,etal.2016,JournalofLowTemper-
aturePhysics[arXiv:1602.07744]
Hivon,E.,Górski,K.M.,Netterfield,C.B.,etal.2002,ApJ,567,2
Lyth,D.H.D.H.&Riotto,A.A.1999,Phys.Rep.,314,1
Masi,S.,Ade,P.A.R.,Bock,J.J.,etal.2006,A&A,458,687
Molinari,D.,Gruppuso,A.,Polenta,G.,etal.2014,MNRAS,440,957
Patanchon,G.,Ade,P.A.R.,Bock,J.J.,etal.2008,ApJ,681,708
PlanckCollaboration,Ade,P.A.R.,Aghanim,N.,etal.2015a,ArXive-prints
[arXiv:1502.01589]
Planck Collaboration, Aghanim, N.,Arnaud, M.,et al. 2015b, ArXive-prints
[arXiv:1507.02704]
Tristram,M.,Filliard,C.,Perdereau,O.,etal.2011,A&A,534,A88
Wright,E.L.1996,ArXivAstrophysicse-prints[astro-ph/9612006]
Articlenumber,page5of5
