Estimates of unresolved point sources contribution to WMAP 5

0.38 MB·English

by  Loris P. L. Colombo

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed20January2010 (MNLATEXstylefilev2.2)
Estimates of unresolved point sources contribution to WMAP 5
1 1
L.P.L. Colombo & E. Pierpaoli ,
1UniversityofSouthernCalifornia,LosAngeles,CA,90089-0484
0
1 Accepted???.Received???;inoriginalform???
0
2

n ABSTRACT
a
J We present an alternative estimate of the unresolved point source contribution to the

0 WMAPtemperaturepowerspectrumbasedoncurrentknowledgeofsourcesfromradiosur-
2 veysinthe1.4−90GHzrange.Weimplementastochasticextrapolationofradiopointsources
intheNRAO-VLASkySurvey(NVSS)catalog,fromtheoriginal1.4GHztothe∼100GHz

] frequencyrangerelevantforCMB experiments.With a bootstrapapproach,we generatean
O
ensembleofrealizationsthatprovidestheprobabilitydistributionforthefluxofeachNVSS
C sourceatthefinalfrequency.ThepredictedsourcecountsagreewithWMAPresultsforS >1
. Jy and the correspondingsky mapscorrelatewith WMAP observedmapsin Q-, V- and W-
h
bands, for sources with flux S > 0.2 Jy. The low–frequencyradio surveysfound a steeper
p
- frequencydependenceforsourcesjustbelowtheWMAPnominalthresholdthantheoneesti-
o matedbytheWMAPteam.Thisfeatureispresentinoursimulationsandtranslatesintoashift
r of0.3−0.4σ intheestimatedvalueofthetiltofthepowerspectrumofscalarperturbation,
t
s ns, aswell asωc. Thisapproachdemonstratesthe use ofexternalpointsourcesdatasetsfor
a
CMBdataanalysis.
[

1 Key words: cosmic microwave background- cosmology:observations- radio continuum:
v general-radiocontinuum:galaxies
9
5
6
3
1 INTRODUCTION 2009). Significant improvements will come from Planck and the
.
1 nextgenerationCMBmissions.
0
0 ModernCosmicMicrowaveBackground(CMB)observationspro- However,exploitationofthefullpotentialofCMBmeasure-
1 videapowerfultestofourunderstanding oftheUniverse.Within ments requires a deep understanding of instrumental systematics
: the generally accepted framework of a ΛCDM model, the 5- and a careful cleaning of foreground contaminants. On small an-
v
yearmeasurementsbytheWilkinsonMicrowaveAnisotropyProbe gular scales, extragalactic point sources are an important source
i
X (WMAP)constrainthefundamentalcosmologicalparameterswith ofcontamination.Brightsources,whichcanbedetectedwithhigh
arelativeaccuracyof1 10%(Dunkleyetal.2009;Komatsuetal. significanceinCMBmaps,aretypicallyaccountedforbymasking
r −
a 2009),whiletheupcomingobservationsbythePlancksatelliteare a small area around the source position during the estimation of
expected to improve these numbers by at least a factor 3 4 theCMBangularpower spectra,Cℓ.Ontheother hand,toafirst
∼ −
(e.g. ThePlanckCollaboration 2006; Colomboetal. 2009). One approximation,undetected,andthereforeunmasked,pointsources
of the main scientific goals of Planck, and other proposed future provideaPoissonnoisecontributiontothemeasuredCℓaswellas
CMBmissions(e.g.EPIC,Bocketal.2009),isunderstandingthe a non-Gaussian signature in the maps (e.g. Toffolattietal. 1998;
nature of Inflation. Detection of the B-mode of CMB polariza- Pierpaoli 2003; Pierpaoli&Perna 2004; Serra&Cooray 2008;
tionwouldprovidedirectevidenceofaprimordialbackgroundof Babich&Pierpaoli2008).Anincorrectdeterminationofthiscon-
Gravitational Waves arising from Inflation (Kamionkowskietal. tamination can lead to relevant biases on the estimated cosmo-
1997; Spergel&Zaldarriaga 1997). Even without such detection logical parameters, in particular on ns (Huffenbergeretal. 2006,
theCMBremainsthemostpowerfulprobeofInflationcurrentlyac- 2008),whoseaccuratemeasurementdependsonacarefulestimate
cessible.Ageneralpredictionofinflationarymodelsisthatthefrac- ofCℓoverthelargestrangeofscalesprobedbyanexperiment.In
tionalamplitudeofdensityfluctuationswouldbenearlyscaleinde- thisrespectthegreatestexpectationsarenowonPlanck,whichis
pendent, sothatthecorresponding power spectrumcouldbewell awholeskysurveywithasmallinstrumentalbeamandlownoise
approximatedbyanHarrison-ZeldovichformP(k) kns,where level.Ifforegroundsandsystematicsarewellundercontrol,Planck
∝
thespectralindexns 1.Theamount ofdeviationns 1con- will be able to confirm or falsify the WMAP findings about ns
≃ −
strainstheshapeoftheinflationarypotential,andcurrentWMAP beingsignificantlysmallerthanone.Asinflationpredictsthatthe
results already allow to rule out several models (Komatsuetal. levelofdepartureonnsfromoneisproportionaltotheamountof
(cid:13)c 0000RAS
2 L.P.L. Colomboet al.
gravitationalwavesexpected,thisalsohasimplicationsonthelevel ing.Here,however,werefrainfromusinganalternativelikelihood
ofB-modepolarizationsignalexpectedandisinturnrelevantfor approachtothetreatmentofpointsourcesintheparameterestima-
thedesignandplanningoffuturemissions.Itisthereforeappropri- tionandadopttheWMAPstrategytothataim.
atetotrytoapproachtheissueofpointsourcecontaminationfrom Theoutlineofthepaperisasfollow.InSection2webriefly
differentanglesandconceivedifferentapproachestodealwithit. reviewthecontributionofunresolvedsourcestoCMBtemperature
Inthispaper,weundertakethispathwithanapplicationtoWMAP spectra.Section3describesourextrapolationprocedureofNVSS
dataanalysiswhichcouldleadtonewapproachesforpointsources source to WMAP frequencies. In section 4 and 5 we discuss the
subtractionforPlanck. implicationofourapproachatthenumbercountsandmapslevel,
At frequencies around 1 GHz there is a wealth of infor- respectively,whileinsection6wedebatethecorrespondingimpact
∼
mationonextragalacticsources,fromnearlyfull-skysurveyswith ondeterminationofcosmologicalparameters.Finally,insection7
high angular resolution and flux sensitivity, like the Green Bank wedrawourconclusions.
TelescopeGB6survey(Gregoryetal.1996),theNRAO-VLASky
Survey(NVSSCondonetal.1998)ortheFaintImagesoftheRa-
dio Sky (FIRST Beckeretal. 2003). On the other hand, at fre-
quencies 100GHz,correspondingtotheoptimalobservational 2 UNRESOLVEDSOURCESCONTRIBUTIONTOCMB
windowf∼orCMBexperiments,thereareonlyfewdedicatedstud- SPECTRA
ies covering a handful of sources. Most information comes from
The angular resolutions of WMAP (& 13 arcmin) and Planck
sourcesdetectedinCMBexperimentsthemselves,typicallyrepre-
(& 5 arcmin) are significantly larger than the angular dimen-
sentingonlythebrightendofthepointsourcedistribution.Inaddi-
sionsoftypicalextragalacticradiosources,whichcanthenbeef-
tion,thislackofinformationlimitedthepossibilityofextrapolating
fectively considered as point–like objects in the resulting CMB
datafromlowfrequenciestotheCMB“goldspot”.
maps. Several works studied the secondary contribution to CMB
To estimate the residual point sources contribution, the
anisotropies due to point sources below the detection thresh-
WMAPteamassumedthatunresolvedsourcesfollowedthesame
old (e.g. Franceschinietal. 1989; Tegmark&Efstathiou 1996;
frequencyscalingasthebrightsourcesdetectedinthe5yearmaps.
Toffolattietal. 1998; Scott&White 1999). In this section we
Theoverallnormalizationoftheresidualpowerspectrumwasthen
brieflysummarizethemainresults.
determinedbyafittomapsatthreehighest frequencies. Thisap-
Unresolved sources contribution toCℓ can bedivided intoa
proachcanbeconsideredinternal,asittakesintoaccount(mostly)
Poissontermduetotherandomsourcespositioning,andacorrec-
onlytheWMAPdataitself,withoutrelyingonthelow-frequency
tionaccountingforclusteringinthesourcesdistribution.Wecon-
information form other available point sources surveys. As other
siderCMBobservationsatasinglefrequencyν0 andsupposethat
radio surveys provide alternative and further information on ra-
allsourcesabovealimitfluxScwillbedetectedandmaskedfrom
diosourcesproperties,incorporatingsuchresultsintheCMBdata
thefinalmap,andassumeasourcepopulationcharacterizedbythe
analysis pipeline could improve the estimation of the extragalac-
differentialcountsdN(>S)/dS.
ticsourcescontamination,orprovideanexternalcrosscheckofthe
The Poisson noise contribution to the measured CMB Cℓ is
WMAPresults.
thengivenby:
DifferentauthorsprovidedestimatesofsourcecountsatCMB
frequenciesonthebasisofthephysicalpropertiesofthedifferent src 2 Sc 2 dN
sourcepopulations(e.g.Toffolattietal.1998;deZottietal.2005). Cℓ =g (ν0)Z0 dSS (cid:12)(cid:12)dS(cid:12)(cid:12) (1)
However,multifrequencystudies,describingthescalingofsources (cid:12) (cid:12)
whereg(ν)convertsfromfluxdensitytothe(cid:12)rmod(cid:12)ynamictempera-
frommiddle frequencies, ν 20 GHz, to 100 GHz, are now
∼ ∼ ture:
starting to appear (Sadleretal. 2008) and more are expected in
the future. Together with existing studies describing sources’ be- c2h2 (ex 1)2 hν
g(ν)= − , x= . (2)
haviour fromlowtointermediatefrequencies, theseallow topre- 2k3T02x2 x2ex kT0
dictsourcescontaminationinCMBmapsstartingfromthelowfre-
The source correction does not depend on ℓ and, since approx-
quencysurveys.Suchpredictionscouldbeusedinfuturedataanal-
ysistomakespecificassessmentsonpointsourcescontamination imately Cℓ ∝ ℓ−2 at high multipoles up to ℓ ∼ 2000, the
sourcecorrectionbecomesrelevantonangularscalessmallerthan
inCMBmapsinrealspace,andaccountforthemwithtechniques
180/ℓ 180/1000 0.2deg.Thecorrectionduetoclusteringis
differentfromtheFourier–basedonescurrentlyused. ∼ ∼
givenby:
Inthisworkwefollowtheempiricalapproachoutlinedabove
tdcoaattaaaslosaengsdsaistthsaeimtuepnmarepcstlaootlenvecodofspsmooiounlrtocgesoicfluaurlcxepesasrcaaomnndetrtpieboruss.tiitWoionentsuosaetthterhaeWdiNoMVfASrePS- Cℓclust =g2(ν0)ωℓ(cid:18)Z0ScdSS (cid:12)(cid:12)ddNS(cid:12)(cid:12)(cid:19)2 ; (3)
(cid:12) (cid:12)
quency, and extrapolate thatinformation toCMB frequencies. At hereωℓ istheharmonictransformofthesou(cid:12)rcetw(cid:12)o-pointangular
difference with previous works, we extrapolate each individual correlationfunction.
source in the NVSS rather than generating source populations at Real CMB experiments typically have complicated noise
CMBfrequenciesbyrandomlysamplingatheoreticaldistribution, properties,duetoacombinationofeffectsincludinginstrumental
orbyextrapolatingonlythestatisticalpropertiesofthesourcepop- systematics,scanningstrategyandforegroundremoval.Asaconse-
ulation(e.g.Waldrametal.2007).Thisworkthereforeprovidesa quence,theprobabilityofdetectingapointsourcewithfluxS0will
crosscheck of WMAPestimatesof undetected point sources con- notbeuniformonthewholeskyandSc willnotbewelldefined.
tamination,andhenceofthedeterminationofnsandotherparam- Thisisnotirrelevant astheunresolved point sourcesspectrumin
eters.Asmentionedabove,sincethisapproachmaintainstheinfor- eq.1istypicallydominated bythestrongest amongst theresidual
mationontheactualspatialdistributionofsources,itcanbeuseful sources.Whilethesourcepopulationabove& 1JyintheWMAP
incleaningfuturePlanckmapsortoestimatetheimpactofcluster- 5yearcatalog(WMAP5,Wrightetal.2009)iswelldescribedby
(cid:13)c 0000RAS,MNRAS000,000–000
3
GHz). We will show this population to be dominated by sources
withflatspectrum(α> 0.5andS να),thereforecorrespond-
− ∝
ingtosourceswithfluxesfromafewhundredsofmJytoafewJy
at1.4GHz.
Inordertoperformtheextrapolation,weconsidered thefol-
lowingsetsofmeasurements:
TheNRAO-VLASkySurvey(NVSS):itcovers thefull sky
no•rth of 40deg declination with an angular resolution of 45′′
at 1.4 GH−z. It includes about 1.8 106 discrete sources with a
×
completenesslimitofS1.4 =2.5mJy.
TheGreenBankTelescopeSurveyat6cm(GB6)observedthe
•
skyinthedeclinationband0deg < δ < 75degwithanangular
resolution 3.5′ (however,theactualbeamisnotcircular).Itin-
∼
cludes75162sourcesaboveS4.8 =18mJy.
TheNVSSfollowupmeasurementsbyMasonetal.(2009).A
•
setof3165NVSSfaintsourceswithS1.4 < 100mJyandfalling
intheCosmicBackgroundImager(CBIReadheadetal.2004)field
wasobservedatν =31GHzusingtheGreenBankTelescopeand
theOwensValleyRadioObservatory.Theresultingsetofobserva-
tionsallowedtoderivethedistributionofspectralindexesα31.14for
sourceswithS1.4 60.1Jy.
Figure 1. Fraction of unresolved source contribution to WMAP spectra TheNinthCambridgeSurveyat15GHz(9C)followupobser-
attributed to sources with fluxes 6 S∗, assuming that all sources with vat•ionsbyWaldrametal.(2007).Thecatalogincludes121sources
S > 0.7Jyhavebeenremovedfromthemaps.WeusedthedeZottiet
selectedat15.2GHzwithacompletenesslimitofS15 = 25mJy
al.(2005)modelforthenumbercountsofsourceswithS<0.7Jy.Sources
andsimultaneousmeasurementsat4.85,15.2,22and43GHz.As
below0.1Jycontribute.10%ofthetotalunresolvedpower.
the9CsurveycoverstheskyareaobservedbytheVerySmallAr-
ray(VSAWatsonetal.2003),whichwasselectedtobedevoidof
anEuclideanscalingofthenumbercountsdN/dS S−2.5,de- brightsources,thesamplecontainsonlyahandfulofsourceswith
∝
tectedfaintersourcessuffersfromsignificantcompletenessissues. S >100mJy.
Dependingonthefrequencybandconsidered,& 90%ofdetected TheAT20GfollowupobservationsbySadleretal.(2008).The
•
sourceshaveaflux> 0.7Jy.Assumingthenanoptimisticdetec- workpresentstwosamplesofsourcesselectedat20GHzwithsi-
tionthresholdSc =0.7JyandthedeZottietal.(2005)modelfor multaneousmeasurementsat90GHz:afirstsetof59invertedspec-
dN/dS,wecanestimatethatsourceswithS . 0.1Jycontribute trasourceswithS20 >50mJy,andaflux-limitedsamplecompris-
. 10% of the total unresolved source power in WMAP data, as ing 70 sources with S20 > 150 mJy. We consider here only the
showninfigure1.Onthecontrary,forPlanck,whose90%detec- flux-limitedsample,whichconstitutesour mainreferencesample
tion confidence limit is expected to be Sc = 0.2 Jy at 100 GHz for the extrapolation of sources with S20 6 1 Jy from 20 to 94
(Vielvaetal.2003),sourceswithS 0.1Jywillplayadominant GHz.
roleindeterminationofCsrc.There∼fore,theaccuracyrequiredfor Atfrequenciesabove 23GHzandforfluxesabove 1Jy,
ℓ • ∼ ∼
theextrapolationsatagivenfluxleveldependsonthecharacteris- themostcomprehensivesurveyofpointsourcesisprovidedbythe
ticsofthetargetexperiment. WMAP5yearcatalog.WMAPobservedtheskyatfivebroadbands
K,Ka,Q,VandWwithcentralfrequenciesof 23,33,41,61,94
∼
GHzrespectively.Thecatalogincludesatotalof390sourceswhich
havebeendetectedwitha5σconfidencelevelinatleastoneofthe
3 EXTRAPOLATIONPROCEDURE
bands,andiscompleteforfluxesabove 1Jyinallbands.
∼
3.1 ReferenceCatalogs
Instead of extrapolating sources directly from NVSS to WMAP
3.2 Method
frequencies,weconsideranumberofintermediatesteps,depend-
ingontheinitialfluxofthesourceconsidered.Thisisparticularly Since the number of sources with high frequency information is
relevant at frequencies below 20 GHz, where a single power law smallerthanthenumber ofNVSSsources, reconstructingtheac-
is not an accurate description for many sources (e.g. Sadleretal. tualscalingofindividualsourcesisnotpossible.Ourassumptionis
2006). By introducing several intermediate steps, we account for thusthatfromthereferencecatalogsdiscussedabove,wecanbuild
thediversityofbehaviours observedinthedata.Thenumber and referencesampleswhicharerepresentativeofthebehaviourofthe
frequency range of the steps used here was determined by avail- wholesourcepopulationintherespectiverangeoffrequenciesand
able data. Inselecting thereference catalogs that wewill use for fluxes.Wethenperformasetofbootstrapsimulationsinwhichat
theextrapolation,ourgoalwastogetacoverageoffluxesrelevant eachextrapolationstep,asimulatedsourceisrandomlypairedtoa
for the estimation of unresolved sources contribution to WMAP referencesourceinthecatalogcoveringthenexthigherfrequency
data over the full range of frequencies considered. While this is range considered, and we use the measured spectral index of the
notanissueinthe GHzrange,coveragebecomessketchierand referencesourcetofurtherextrapolatethesimulatedsource.Some
∼
patchierathigherν.Accordingtothediscussioninsection2,the ofthecatalogsconsideredprovidemultifrequencyinformationfor
maincontaminationtoWMAPspectraisduetoundetectedsources theirsources.Whenthisisthecase,weusethesetofspectralin-
withfluxes 0.1 0.7 Jy in the CMB frequency range (40 90 dexesofthecatalogsourcetoextrapolatefluxesovertherangeof
∼ −
(cid:13)c 0000RAS,MNRAS000,000–000
4 L.P.L. Colomboet al.
frequenciesconsidered.ForeachsourceintheNVSScatalog,we
generate a set of 800 simulations. Whilein two different simula-
tionsagivensourcecanbeextrapolatedindifferentways,thesetof
simulationsprovidestheprobabilitydistributionforfluxatWMAP
frequencies.
Inbroadterms,wedefinetwomainfrequencyranges:1)the
interval from NVSS to the lowest WMAP band, 1.4 23 GHz,
−
and 2) the WMAP frequency range, 23 94 GHz. Sources with
−
S1.4 6 100 mJy are extrapolated directly to 23 GHz according
toMasonetal.(2009), whilefor thosewithS1.4 > 300 mJywe
introduceanumberofintermediatestepsbasedbasedontheGB6
andWaldrametal.(2007)measurements.Intheintermediateflux
range, we weight the two approaches in order to fit the 33 GHz
lowfluxcounts.From23GHzupwards,sourceswithS23 > 1Jy
arepropagatedbasedontheWMAP5catalog,whileforlowerflux
sources we base the extrapolation on the Sadleretal. (2008) and
Waldrametal.(2007)referencesamples.
Inthefollowing,wediscusseachstepindetail:
The1.4GHzto4.85GHzreferencesample.Kimball&Ivezic´
•
(2008)provideaunifiedcatalogofsourcesfromfourradiosurveys
NVSS,GB6, FIRST,WENSand fromtheoptical SDSSsurveys, Figure2.Theaverage1.4−4.85GHzspectralindexofGB6sourcesas
matching sources in the different catalogs. From the NVSS-GB6 afunctionof1.4GHzflux.BelowS1.4 ∼ 100mJyNVSSsourceswitha
pairings identified in that work, we construct a reference sample GB6counterpartarebiasedtohaveupturnspectra.
ofsourceswhichprovidesspectralindexesbetween1.4GHzand
4.85 GHz. Weconsider only pairsin which the GB6counterpart
isfoundwithin70′′ oftheNVSSsource, corresponding toanes- fallbelowtheGB6detectionthreshold.Waldrametal.(2009)find
timated completeness of 0.979 and an efficiency (fraction of de- that4.3%ofsourcesselectedat15.2GHzwith10mJy 6 S15 6
tectedpairscorrespondingtoactualphysicalmatches)of0.79(for 15mJy do not have an NVSS counterpart. GB6 sources are se-
furtherdetailsseeKimball&Ivezic´2008).Inaddition,weexclude lectedwithhigherthreshold(18mJy)andatlowerfrequenciesthan
sources flagged as extended in the GB6 survey and include only Waldrametal. (2009). Therefore, sources of type (a) would re-
unique pair of sources, i.e. sources in one catalog with multiple quire spectral indexes significantly more rising than those of the
matchesintheothercatalogareexcluded.Sincemultiplesidelobes rising population discussed in that work, an we expect the frac-
ofasinglesourcemayshowupasdifferentNVSScomponents,in tion of sources detected at 4.85 GHz but not in the NVSS to be
thiswaywemaybiasthepairstowardcompactsources.However significantly lower than Waldrametal. (2009) estimates. Regard-
NVSSresolutionissuch that lessthan 1% of sources isresolved ing point (b), we note that sources in the reference sample with
intomultiplecomponents(Bestetal.2005),comparedwith 7% 100 6 S1.4 6 200 mJy have an average spectral index α41..84 =
∼
ofGB6sourceswithmultipleNVSSmatchesinthereferencecat- 0.84 0.34.ThereforeweestimatethefractionofNVSSsources
− ±
alog. Wethenconclude that themajorityof multiplematcheswe withnoGB6counterpart tobe 5% at 100mJy and 0.5% at
∼ ∼
findarespuriouspairs. 150mJy.
The distribution of spectral indexes so obtained describes the From1.4GHzto23GHzforsourceswithS1.4 > 300mJy.
•
1.4 4.85GHzbehaviourofsourcesselectedattheGB6frequency, From the previous discussion, we conclude that the thinned ref-
−
andcanbeusedtoaccuratelypropagatesourcesfrom4.85GHzto erence sample defined above provide an accurate description of
the lower frequency. In order to use this distribution to extrapo- thefrequencyscalingfrom1.4to4.85GHzforthepopulation of
latefrom1.4GHztohigherfrequencies,someadjustmentsarere- sourceswithS1.4 &300mJy,whichareresponsibleforthebulkof
quired.TheNVSSsurveyhasafluxlimitof2.5mJy,whiletheGB6 sourcecontaminationatCMBfrequencies.Therefore,eachNVSS
includes sources above 18mJy, therefore low–flux NVSSsources sourcewithS1.4 > 300mJyisrandomlypairedtoaspectralin-
whichappearalsointheGB6surveywillbebiasedtohavearis- dex αi from the corresponding flux bin of the thinned reference
ingspectralindex. Figure2showstheaverageα41..84 asafunction sample.Wethenrandomlydrawaspectralindexα¯ fromaGaus-
ofS1.4.WhileforS1.4 &0.3Jythespectralindexisadecreasing sian with mean αi and width δαi. In this way we incorporate in
functionoftheNVSSflux,forS1.4 . 0.1Jyα41..84 showsasteep theextrapolation the measured uncertainty on αi, however, since
upturnduetocompletenessissues.Wethereforeexcludefromthe the distribution of the reference spectral indexes is not necessar-
referencesampleallsourceswithS1.4 60.1Jy.Wefurtherdivide ilyGaussian,theoveralldistributionoftheα¯bothwithinasingle
thereferencesamplein10logarithmicbinscoveringthefluxrange realizationandamongdifferentrealizationswillnotbeGaussian.
0.1 3.0 Jy, depending on the value of S1.4, with an additional ThefluxoftheNVSSsourceisextrapolatedto4.85GHzassuming
bini−ncludingallsourceswithS1.4 >3Jy.Intheendthereference S να¯.Thesourceisthenpropagatedto23GHzusingthespec-
∝
sampleincludes 31000sources,withcorrespondingspectralindex tralindexes by Waldrametal. (2007). Sincethefluxthreshold of
αianduncertaintyδαi. the15.2GHzsampleishigherthantheGB6limit,weneedtothin
Inprinciple,sincethereferencesampleisbasedonsourceswith thecatalogtoavoidthatsourceswithlowfluxesat4.85willbias
matchesinbothNVSSandGB6catalogs,wemaybemissing:a) thesampletowardflatterspectralindexes.Duetothelownumberof
sourceswithstronglyrisingspectra,whichareabsentinNVSSbut sources,wecannotstudythebehaviourofα(S)asdoneinthepre-
aredetectedbyGB6,orb)sourceswithverysteepspectra,which vioussection.Insteadwecomputetheaveragespectralindexα14.58
(cid:13)c 0000RAS,MNRAS000,000–000
5
using all the121 sources inthe sample. Using thisaverage α14.58, singlepower law withspectra index drawn randomly from the
we extrapolate the 15.2 GHz threshold to 4.85 GHz and remove referencesampleofSadleretal.(2008);
formthe4.8 15.2 GHz referencesample all sources withS4.8 – Sources with 100 < S23 < 400mJy are extrapolated
−
belowthisvalue.Theprocessisiterateduntilnomoresourcesare choosingrandomlybetweentheprevioustwomethods;theprob-
rejected.Weareleftwith76sourceswithS4.85 > 75mJy,which ability of selecting one method over the other is given by the
weassumeprovidearepresentativedescriptionofthebehaviourof relativenumberofsourcesintherelevantfluxrange.
sources between 4.85 and 23 GHz. Each source simulated at the
In addition, when at a given step wehave different extrapo-
previousstepispairedwitharandomsourceinthisnewreference
lation procedures depending of the flux of the source, we do not
sample,anditscorrespondingsetofspectralindex.Weusethisset
switchabruptlyfromoneregimetotheother,butweadoptalin-
topropagatethesourceto23GHz,goingthroughanintermediate
eartransitionbetweenthetwowithawidthwhichisthegreaterof
stepat15.2GHz.
100mJyand10%ofthethresholdflux.Moreover,wedonotcon-
From1.4GHzto23GHzforsourceswithS1.4 6 100mJy.
• siderpolarization.
Forlowfluxsources, S1.4 < 100mJy,weextrapolatedirectlyto
23GHzassumingS να,withαrandomlydrawnfromthedis-
∝
tributionofspectralindexesbyMasonetal.(2009).Inordertobe
consistentwiththetreatmentofsourceswithS1.4 > 100mJyde- 4 RESULTS
scribed above, we assumed that such distribution be valid in the
range 1.4 23 GHz, even if it is based on 31 GHz followup of 4.1 Fluxdistribution
−
NVSSsources.
Thisapproachprovidesboththeprobabilitythatasourcewithan
3up0•p0emrFrJvoyam.liSd1oit.u4yrGcliemHsizwttoiotfh2t13h0eG0MHmzaJsfyoonr<seotSua1rl..c4e(2s<0w03i9t0h)01smt0u0Jdmyy,aJaryned<abmoSav1ye.4tyh<eet wNeVllSaSsflthuexcSo1m.4plwemillehnatavreyadi9s4triGbHutzioflnuPx(SS914.4,|iS.e9.4P).(ISn9fi4g|Su1re.43),waes
plotthedistributionoftheinitial1.4GHzfluxesforsourceswith
beslightlyaffectedbyspuriouseffectsintheNVSS-GB6refer-
encesampledefinedabove.Therefore,inthisintermediateregime, final94GHzfluxesofS94 ∼ 0.01,∼ 0.10 and∼ 1.0Jy,which
werandomlychosebetweenthetwoapproachesdiscussedabove, providesanestimateofP(S1.4|S94).Theplotsrefertothefullset
and tune the selection function to reproduce low flux counts at of800simulations.Figure4showsinsteadP(S94|S1.4).Ageneral
expectationisthatforfluxlevelsrelevanttoCMBexperiments,the
higherfrequencies.Theprobabilityofextrapolatingsourcesusing
highfrequencypopulationofsourcesisdominatedbyobjectswith
thespectralindexdistributionofMasonetal.(2009)isgivenby:
flatspectralindexes.Werecoverthisbehaviourinthesimulations,
P(S1.4)= 1 1 tanh S1.4−S∗ . (4) asshowninfigures3and4.Inparticular,accordingtothediscus-
2(cid:20) − (cid:18) ∆ (cid:19)(cid:21) sionofsection2,theresidualcontaminationinWMAPmapswill
The values S∗ and ∆ are chosen by fitting the 33 GHz num- bedominatedbysourceswithS & 0.1Jyinthefrequencyrange
bercountsfromtheextrapolationproceduredescribedheretothe 60 90GHz,andweexpectthemajorityofthesesourcestohave
−
observed number counts by DASI and VSA in the 0.05 1 Jy S1.4 & 100mJy. Inaccuracy intheextrapolations ofsourceswith
range.Sourceswithlowerfluxesarenotasignificantconta−minant lowerS1.4willhaveaminimalimpactonestimatesoftheresidual
inWMAPmaps,asdiscussedabove.Notethatwedonotrequire contaminationinWMAPmaps.
theselectionfunctioninequation4tobecontinuousatthe100mJy
and300mJyboundaries.
From23GHzto94GHzforsourceswithS23 >1Jy.Sources 4.2 Numbercounts
•
with S23 > 1 Jy are extrapolated to higher frequencies accord-
For each simulation, we compute the corresponding differential
ing to the WMAP5 catalog itself. We divide the WMAP5 cata-
numbercountsdn(S)/dS andthenaverageoverthewholesetof
logintwosubcatalogscomprisingsourceswithS23 > 2Jyand
simulations. In figure 5 we compare the incomplete counts from
1Jy < S23 < 2Jy,respectivelycontaining69and166elements.
WMAP5 source catalog with the ensemble average differential
WhenasimulatedsourcehasS23 >1Jy,werandomlypairitwith
countsatthefrequenciesof33,41,61and94GHz,approximately
a spectral index from the subcatalog covering the corresponding
correspondingtothecentralfrequenciesofWMAPKa,Q,Vand
fluxrange,andextrapolatethesourcetohigherfrequencyusinga
Wbands.At33GHzwecompareourestimationalsowithresults
simplepowerlaw.
fromCBI,VSA,andDASIsurveys,whileat94GHzwealsoplot
From 23 GHz to 94 GHz for sources with S23 < 1 Jy.
• predictionfromearlierworks (deZottietal.2005;Waldrametal.
For fluxes below 1 Jy the WMAP catalog is not complete.
∼ 2007).
Waldrametal.(2007)suggestaproceduretoextrapolatefluxesinto
Comparison with lower flux data at 33 GHz, shows that the
theWMAPfrequencyrangebasedonthespectralbehaviourofthe
methods discussed here tends tounderestimate source counts be-
sourcearound∼ 20GHz.Forsourcesforwhichα4232 6 α2125 6 0 lowS 0.02 0.03Jy.Thesesourcesdonotprovidearelevant
theysuggestfittingaquadraticformtothelog(S) log(ν)relation, ∼ −
− contributiontothetotalresidualcontaminationtoWMAPspectra,
whilefortheremainingsourcestheyadoptalinearlog(S) log(ν),
using α4232. Alternatively, we consider the set of spectra−l indexes buuntcewritlalinptlyayinaomuorrmereethleovdasnitsrodloemfoinraPteladnmckodstaltyabaynatlhyesilso.wSinnucmebtheer
basedontheSadleretal.(2008)sample.Takingintoaccountboth
ofreferencesourcesabove4.85GHz,weexpectabetteragreement
setsofobservations,weconsiderthenthefollowingextrapolation
asthepoolofreferencesourcesincreases.Intherange0.5 1.0Jy,
strategyforsourceswithS23 <1Jy: −
simulationsshowevidentartifactsduetothetransitionbetweendif-
– SourceswithS23 <100mJyarerandomlypairedtosource ferentregimesoffluxextrapolation,i.e.usingspectralindexfrom
ofWaldrametal.(2007)andextrapolatedaccordingtothepro- theWMAP5catalogforS23 >1Jy.Theyareduetothelownum-
ceduredescribedthere; berofsourcesinthatfluxrangeinthebasecatalogs;asabove,we
– Sources with S23 > 400mJy are propagated assuming a expecttheseartifacttosignificantlydecreaseasmoredataoverthe
(cid:13)c 0000RAS,MNRAS000,000–000
6 L.P.L. Colomboet al.
Figure3.Distributionsoftheinitial1.4GHzfluxesforsourceswithafinal94GHzasindicatedintherespectivepanels.Histogramsrefertothefullsetof
800simulationsandhavebeennormalizedtohaveunitarea.
Figure4.Asinfigure3butshowingdistributionsofsimulated94GHzfluxesforsourceswithinitial1.4GHzfluxasindicatedinthedifferentpanels.
relevantfrequencyandfluxrangewillbeincludedinthereference deZottietal.2009,forarecentreview).Ifthisisactuallythecase,
catalogs. describingthesourcebehaviourinthe23 94GHzrangewitha
−
singlepowerlawmaynotbeentirelyaccurate.
WhileourextrapolationscorrectlyrecoverWMAP5resultsat
Alternatively,theremaybesomeunaccountedforsystematics
lowerfrequencies,countsat94GHzareoverestimatedby 50%.
∼ affectingWMAPdetectionefficiencyinWband.TheWMAPteam
Infigure6weplottheintegratedcountsN(>S =1Jy)fromour
requiredthatforasourcetobeincludedinthecatalogitneededto
simulationsandfromWMAP5data.Forreference, wealsoshow
ananalyticestimateofthecountsassumingdN/dS S−2.5 and bedetectedatmorethan5σ,inatleastoneband.Itsfluxintheother
S ν−0.09(Wrightetal.2009),normalizedtofitWM∝AP5counts bandswouldbeincludedif:a)itwasmeasuredatmorethan2σ,b)
∝ the fitted source width is within a factor of 2 of the actual beam
in K, Ka and Q bands. WMAP5 data show a steepening of the
(Wrightetal.2009).Sawangwit&Shanks(2009)pointedoutthat
countsinWbandcomparedtolowerfrequencies,whileourcounts
theWbandbeamresponsetodiscretesourcesappearstodependin
seemtobeinbetteragreement withalinearlog(S) log(ν)re-
− anon-linearfashiononsourceflux.Inparticular,theyfindthatthe
lation. As discussed in section 3.2, our extrapolations of sources
effectivebeamprofileissignificantlywiderforpointsourcesthan
with S23 > 1 Jy is mainly based on the family of spectral in-
fortheJupiterobservationsusedasabasisfortheWMAPteam’s
dexesprovidedbytheWMAP5team,whichcanbeapproximated
beamanalysis, which cancreateasignificant biasindetermining
byaGaussiandistributionwithmeanα = 0.09anddispersion
− if a source fitsrequirement (b). Since number counts from simu-
σα = 0.17. Therefore, as expected source counts in the simula- lationsdonotincludesystematicsduetoWMAPactualdetection
tionsareinagreement withtheanalyticestimateplottedinfigure
procedure,thiseffect,ifconfirmed,couldbeanalternativeexpla-
6; the slightly steeper behaviour seen in simulations is compati-
nationforthediscrepanciesbetweensimulatedandactualnumber
ble with the actual distribution of WMAP5 spectral indexes hav-
counts. Further data on source behaviour at high frequency, e.g.
ing a tail toward steeper α. This suggests that there issome ten-
fromPlanck,willhelpsolvethisissue.
sionbetweentheWMAP5W-bandcountsandtheirspectralindex
distribution.Apossible explanationcould beaprogressive steep- In addition, all WMAP sources with measured W band flux
ening of the spectral index of bright sources with increasing fre- have been detected also in at least one of the lower frequencies.
quency.Gonza´lez-Nuevoetal.(2008)noticedthatthespectralin- Aslong asaWMAPsourceisdetectedinatleast oneband, itis
dexofWMAPsourcesinthe41 64GHzrangeissteeper than maskedinthepowerspectrumanalysis(Noltaetal.2009),andwe
−
theirspectralindexinthe5 23GHzrange,andradiosourcemea- follow the same procedure in building sky mask for the analysis
−
surements in the 100 250 GHz range by the QUaD tele- of section 6. Therefore, any mismatch in the number of W band
∼ −
scope (Friedmanetal. 2009) and the South PoleTelescope (SPT, bright sources between simulations and real data will not impact
Vieiraetal.2009)provideadditionalevidenceinthisdirection(see ourestimatesoftheunresolvedsourcepoweraslongascountsat
(cid:13)c 0000RAS,MNRAS000,000–000
7
Figure5.ComparisonofthepredictedEuclideancountsfromtheensembleofNVSSsimulationsofthiswork(solidlines)withtheincompletecountsfrom
theWMAP5yearcatalog(squares),atthefrequenciesof33,41,61and94GHz.At33GHzwealsoplotdatafromtheCBI,VSAandDASIsurvey,whileat
94GHzwecomparewithpredictionsfromearlierworks.
thelowerfrequencyforthesimulationsareingoodagreementwith
200
WMAPfindings.
150
5 MAPS
Anadvantageofthisapproachconsistsinthatitnaturallyprovides
an estimates of bright sources position and accounts for cluster-
100
ingofunresolvedsources.However,agreementofnumbercounts WMAP5
estimatedfromsimulationswiththosecomputedbydatadoesnot Simulations
guarantee that the method correctly recovers the spatial distribu-
tionofsources.Inordertocheckwhetherthesimulationsmatchthe 50
structureobservedinWMAPmaps,foreachrealizationweproduce
source-only(i.e.withoutnoiseandCMB)mapsandcomparethem
Figure 6. Comparison ofintegrated counts in simulations (red/triangles)
with WMAP observed maps. Source maps also allow and inde- andinWMAP5yearcatalog(black/squares). Thedashedlineshowsthe
pendent determinationofunresolved sourcecontamination. Since expected behaviour assuming a scaling of differential counts dn/dS ∝
WMAPestimateofthiscontributionisbasedontheQ,VandW S−2.5andafrequencydependenceS ∝ ν−0.09.Thecurveamplitudeis
bandsdata,wegeneratemapforeachoftheabovefrequencies. fixedby afit toWMAP count in K,Ka andQ bands. Incomputing the
We place the sources on an HEALPIX1(Go´rskietal. 2005) numbercountsweexcludedtheskyareawithdeclination6−40degand
withintheWMAP5yrpointsourcecatalogmask.
1 http://healpix.jpl.nasa.gov/
(cid:13)c 0000RAS,MNRAS000,000–000
8 L.P.L. Colomboet al.
mapataresolutionNside =2048,correspondingtoapixelsizeof degreeofcorrelationbetweensimulationsandactualmapsislower
1.7′.ThisvalueisconsiderablygreaterthantheNVSSbeamand thanthatbetweensimulationsandmockdata.WhileforWbandthe
∼
angularresolution,sowecanconsiderthesourcesaspointswhen differenceiswithintheassociatedstandarddeviation,inQbandthe
placingthemonthemap,andconsiderablysmallerthanWMAPW discrepancyisoforder2σforbrightsourcesincreasingto 3σfor
beamof 13′,sothatanyartifactsduetopixelization,smoothing faintones.Foreground contaminationismorerelevant at∼41GHz
∼
etc.willbeminimal.TheactualWMAPbeamsarenotGaussian, than at 94 GHz and this may account for the significant loss of
theirexactprofilevaryingbetweeneachsetofDifferentialArrays correlationinQband.
(DAs). To reduce the number of maps we have to deal with, we
consider individual frequencies, ratherthanindividual DAs.Each
mapisthensmoothedwithaneffectivebeamgivenbytheaverage,
6 POWERSPECTRAANDEFFECTONPARAMETERS
in harmonic space, of the beam profilesof the DAs of the corre-
sponding frequency. The smoothed maps are then degraded to a TheWMAPteamestimatedtheamplitudeoftheunresolvedpoint
resolutionNside=512. source contribution to a cross spectrum CℓY1Y2 (Y1Y2 represents
Inordertocheckthatourpredictionsrecoverthepointsource apairofdifferentDAs,e.g.Q1Q2,Q1V1)assumingapower-law
structureseen inthemaps, wecompute thelinear correlationco- scalinginfrequency:
efficientbetweeneachindividualsimulationandthecorresponding
WMAP5yrmap: Cℓsrc =A0g(ν1)g(ν2)(cid:18)ννQ1(cid:19)α(cid:18)ννQ2(cid:19)α, (6)
r= h(Xi−hXi)σ(XMσiM−hMi)iS>Sc . (5) whereν1,ν2 arethefrequenciesoftheDAsconsidered,g(ν)was
defined in equation 2, and the spectral index α is assumed to be
Here X and M respectively refer to the simulated and WMAP either 0, asmost sources areexpected tohaveaflatspectrum, or
maps, after application of an harmonic space filter bℓ/(b2ℓCℓ + -0.09,corresponding tothemeanofofaGaussianfittothespec-
Cℓnoise), where bℓ isthe harmonic transform of the beam profile, tralindexesoftheactualWMAP5catalog(Wrightetal.2009).The
CℓisatheoryCMBpowerspectrumbasedonWMAP5bestfitpa- commonamplitudeA0 = 0.011 0.001µK2srisdeterminedby
rameters(Dunkleyetal.2009)andCℓnoiseisthenoisepowerspec- fittingthe spectral shape of equa±tion 6, to the cross spectra from
trum (Wrightetal. 2009). The means and variances of the maps, theQ,V,Wbands;theestimatedA0hasonlyasmalldependenceon
X and σX, and the correlationcoefficients are computed using thetwovaluesofαconsidered(Noltaetal.2009).Thefinalpower
h i
onlypixelswithinradiusequaltothenominalfrequencyFWHM, spectrum,instead,isalinearcombinationofthepossibleVV,VW,
of sources with flux above a threshold Sc. To assess the signifi- and WW cross spectra, weighted by the corresponding inverse
canceofthisresult,wecomparethevalueofrcomputedfromthe covariance matrix (Hinshawetal. 2003; Noltaetal. 2009). This
simulations-WMAP5correlations,withthatcomputedbycorrelat- method is internal, as it includes information only from WMAP
ing our simulations with a mock microwave sky which includes own measurements. On the contrary, the approach discussed in
contributionfromCMB,WMAP5-likenoise,Galacticforegrounds this paper, provides a partially external method, in the sense that
and a point sources population obtained according to the proce- it incorporates both information from WMAP data (to propagate
dure described in this work. In practice, we randomly chose one sourceswithS23 > 1Jy)andfromothersurveys,andcouldpro-
of the simulations asthe actual distributionof point sources, and videacrosscheckofWMAPresults.
addtothisaCMBrealization,ananisotropicGaussiannoiseterm Toestimatetheunresolvedpointsourcescontaminationfrom
and a foreground contribution. The pixel noise variance is given our simulations we need to define an individual mask for each
by n = n0/√Nobs, where Nobs is the effective number of ob- realization. We start by applying the same Galactic cut as for
servationsineachpixelandn0 = 2.197 mKfortheQband and the KQ853 mask, used in the WMAP5 power spectrum analysis
6.547 mK in W band (Limonetal. 2009). However, we do not (Noltaetal. 2009), and also mask the area outside of the NVSS
account for noisecorrelations between different pixels. Thefore- sky coverage. This base mask is the same for all realizations. In
ground contribution is based on the Maximum Entropy Method addition, theKQ85 maskremoves theskyareawithin0.6deg of
maps 2 (Goldetal.2009). Themaps are provided at a resolution sources intheWMAP5catalog. TheWMAP detection efficiency
Nside = 128andsmoothedwitha1degbeam,andarethushave for radio sources depends on a number of factors, including fre-
nopoweratscalesℓ&300.Thereforetheforegroundscontribution quency, position on the sky and the actual CMB and noise fluc-
tothevarianceandmeanofthemocksky,inparticulartothesmall tuationsaround thesource. Replicatingalltheseeffectsinallour
scalesrelevanttopointsources,willbesignificantlylowerthanthe simulationswouldbetootimeconsumingandinsteadweadopta
impactontheactualWMAPskymaps.Wegeneratedifferentsets simplifiedprocedure to“detect”and masksources. TheWMAP5
ofmockskies,correspondingtodifferentpointsources,CMBand catalogiscompleteforsources& 1Jyatallfrequencies, andthe
noise realizations. The mock sky then constitute a best case sce- numbercountsforthosesourcesfollowanEuclideandistribution.
nario, in which the point source distribution follows exactly the Therefore,ateachfrequencyweassumeapowerlawshapeforthe
model discussed inthiswork, and several systematiceffects, like number countsFνS−2.5,andfixthevalueof Fν byfittingtothe
foregroundsandcorrelatednoise,areabsentorplayareducedrole. observed WMAP5 differential number counts n¯ν(S) for S > 1
Tables1and2showresultsofthecomparisoninQandWbands, Jyattheappropriatefrequency.Wethenassigntoeachsimulated
respectively. source adetection probability P(S∗) = n¯ν(S∗)/(FνS∗−2.5). On
A general remark is that even in the best case scenario we average,ineachsimulationwedetect304 16sources,compared
expectonlyamoderatelevelofcorrelation:r.0.40andr.0.25 with321actualWMAP5sourcesinthesam±eskyarea,andwemask
forbrightsources,respectivelyinQandWband.Asexpected,the a 0.6deg radius around each detected source. Alternatively, we
2 http://lambda.gsfc.nasa.gov/product/map/dr3/mem mapsinfo.cfm 3 http://lambda.gsfc.nasa.gov/product/map/dr3/masks info.cfm
(cid:13)c 0000RAS,MNRAS000,000–000
9
Sc(Jy) WMAP5 mock mock mock mock mock mock
maps data1 data2 data3 data4 data5 data6
1.0 0.22±0.07 0.34±0.08 0.36±0.07 0.37±0.08 0.35±0.07 0.32±0.07 0.37±0.07
0.5 0.18±0.05 0.30±0.06 0.32±0.05 0.31±0.06 0.29±0.05 0.26±0.05 0.31±0.05
0.2 0.13±0.04 0.23±0.04 0.25±0.04 0.23±0.04 0.22±0.04 0.19±0.03 0.25±0.04
0.0 0.03±0.02 0.06±0.02 0.07±0.01 0.06±0.01 0.06±0.01 0.05±0.01 0.08±0.01
Table1.Linearcorrelationcoefficientrbetweensimulatedsourcemapsat41GHzandWMAP5yearQbandmaps,fordifferentfluxcutsSc.Onlypixels
within1FWHMradiusofsourceswithS > Scareincludedinthecomputationofr.Forcomparisons,columns3-8showthesamequantitycomputedby
correlatingoursetofsimulationswithmockWMAP-likeskiesforwhichthepointsourcecomponentfollowsthesamedistributionasoursimulations.
Sc(Jy) WMAP5 mock mock mock mock mock mock
maps data1 data2 data3 data4 data5 data6
1.0 0.14±0.08 0.21±0.09 0.24±0.08 0.20±0.08 0.18±0.07 0.16±0.07 0.18±0.07
0.5 0.14±0.06 0.18±0.07 0.21±0.06 0.16±0.06 0.15±0.05 0.14±0.05 0.15±0.05
0.2 0.12±0.04 0.13±0.04 0.16±0.04 0.11±0.03 0.11±0.03 0.11±0.03 0.11±0.03
0.0 0.014±0.003 0.011±0.003 0.014±0.003 0.010±0.002 0.010±0.002 0.010±0.003 0.011±0.002
Table2.Astable1butforWband.
consider theconservative approach of masking onlysources with forthesubsetof8sourceswithS20 >0.8Jy,α¯= 0.26 0.16,in
− ±
aQbandfluxS41 >1.0Jy. goodagreementwithestimatesforWMAPsourceswithS23 > 1
Jy.Oursimulationsreflectthisfinding.Asmoreinformationonthe
Weapplytheresultingmasktothecorrespondingsetofmaps
spectral behaviour of sources with K band flux in the 0.5 1.5
andcomputetheQQ,VV,VWandWWspectrausingaMASTER
−
Jy will become available, it will be possible to better establish if
approach (Hivonetal. 2002). The VV, VW and WW spectra are
thelowfluxsourcesofWMAPcatalogarebiasingtheestimatesof
thencombinedaccordingtoHinshawetal.(2003);noticethatwe
thespectralindexusedforestimatingtheunresolvedpointsources
onlyincludethediagonaltermsofthecovariancematrix.Thefinal
contribution.
estimate of the unresolved sources contribution is the average of
Inaddition,WMAPfixtheamplitudeoftheunresolvedsource
thespectrafromtheindividualrealizations.
contributionA0byasimultaneousfittotheensembleofcrossspec-
In figure 7 we compare our estimates with WMAP results.
tra obtained using Q, V and W bands, assuming a constant fre-
WhileourpredictionsareingoodagreementwithWMAP5yresti-
quency scalingacrossthecorresponding range offrequencies, 41
matesfortheQQspectra,theyshowasteeperfrequencybehaviour.
-94GHz.Therefore,althoughthefinalCMBpowerspectraarea
In particular, for the final combined power spectrum, we predict
combinationofonlyVandWmeasurements,theestimatedunre-
a residual point source contribution 50% lower than WMAP
∼ solvedsourcescontributiontosuchspectradependsalsoonQdata.
estimates for the source detection strategy discussed above, and
Inthiswork, instead,weestimatethepoint sourcecontamination
∼ 25% lowerif weonly masksources withS41 > 1Jy. Uncer- at agiven frequency directlyfrom the point sources maps at that
taintyonsimulationsresultsis 10%,comparabletotheerroron
∼ frequency,withoutincludinginformationfromotherbands.Inpar-
A0.Noticethatinfigure7wedonotplotourestimatesofresidual ticular,ourpredictionforthefinalcombinedVV+VW+WWspec-
sourcescontaminationatℓ>700fortheQband,astheuncertainty
trumdoesnotdependon41GHzdata.AsdiscussedinNoltaetal.
inthebeamasymmetriesforthatchannelintroducesignificantarti-
(2009), the WMAP5 estimated value of A0 using only V and W
factsatℓ&500(Hilletal.2009).SincewedonotusetheQband
datais 10% 40%(dependingonthechoiceofGalacticmask)
datatoestimatethepointsourcescontributiontothefinalspectra, ∼ −
lowerthanwhenusingQ,VandWdata,althoughtheuncertaintyin
theseartifactsdonotalterthefollowingdiscussion.
theformercaseis 3timestheuncertaintyinthelatter.Therefore,
∼
The differences between our prediction and WMAP5 esti- usinginformationfromQbandmayleadtoaslightoverestimateof
matesarisefromacombinationoffactors.Asdiscussedabove,the A0at60 90GHz.
−
referencecatalogweusetoextrapolatehighfluxsourcesfrom20 AspointedoutbyHuffenbergeretal.(2006,2008)thepoint
GHzto94GHzwasgivenbythesubsampleofWMAP5sources sourcescorrectionandthewayitistreatedinthelikelihoodfunc-
with with fluxes S23 > 1 Jy, corresponding to the approximate tion affects the determination of the cosmological parameters, in
completeness limit of WMAP 5yr catalog. WMAP sources with particular of the slope of the power spectrum of scalar pertur-
Sw2h3ile>fo1rsJoyurhcaevsewainthaSve23ra<ge1spJyecα¯tra=l ind0e.x09α¯ =0.6−6.0T.2h1e±spe0c.t3r0al, braamtioetnesr,enssti.mTaotiotne,stwteheuseeffCecOt,SMifOanMy,C4of(Loeuwriasp&prBoaricdhleo2n00p2a-)
indexusedbytheWMAPteaminorder−toder±ivetheamplitudeis + PICO5 (Fendt&Wandelt 2007) to run a set of Monte-Carlo
themeanofaGaussianfittothespectralindexdistributionofall Markov Chains (MCMC) replacing the source correction term
their detected sources, and is therefore affected by the shallower in WMAP5 likelihood code with our results. However, we did
spectralindexesofsourceswithS23 < 1Jy.Notethatwedidnot not change the shape of the likelihood function as suggested by
use the latter set of sources for the simulations, as for this range
of fluxes and frequencies we adopted the results of Sadleretal.
(2008).Theseresultssuggestthatsourceswithfluxesinthe0.2 1 4 http://cosmologist.info/cosmomc/
Jyrangehaveanaveragespectralindexα¯ = 0.43 0.31,wh−ile 5 http://cosmos.astro.uiuc.edu/pico/
− ±
(cid:13)c 0000RAS,MNRAS000,000–000
10 L.P.L. Colomboet al.
Figure7.EstimatesoftheunresolvedpointsourcescontributiontoCMBpowerspectraatdifferentWMAPfrequencieswhenapplyingthemaskingprocedure
describedinthetext(solidlines)orsimplyremovingallsourceswithQbandfluxS41>1Jy(dashedlines).ThedottedlinesshowaPoissonspectrumwith
normalizationCℓsrc ∝A0ν1αν2αwithA0 =0.011andα=−0.09.ThefinalpowerWMAPpowerspectrumisacombinationofVV,VWandWWcross
spectraanddoesnotdependonQQdata.ForreferencewealsoshowtheCMBtemperaturepowerspectrumforthebestfitWMAP5cosmologicalmodel.
Huffenbergeretal. (2008). We consider the six base ΛCDM pa- 800times,andtheensembleofsimulationsprovidestheprobabil-
rameters:physicalbaryon,ωb,andcolddarkmatter,ωc,densities; itydistributionforthefluxeachNVSSatthedesiredfrequencies.
opticaldepthtoreionization,τe;tilt,ns,andamplitude,Asofthe Wecomparedthestatisticalpropertiesofthesimulationswith
powerspectrumofscalardensityfluctuations;theanglesubtended thoseofthesourcesoftheWMAP5catalog,withothersurveysat
bythesoundhorizonatrecombination,θ.Weincludemarginaliza- 33GHzandwithpredictionsbypreviousauthorsat94GHz,find-
tionovertheamplitudeoftheSunayev-Zeldovicheffectandlens- inganagreementdownforfluxes& 0.1JyinKaband,whilewe
ing.Asexpected,sincechangingthesourcecorrectioneffectively underestimatethenumberofsourcesbelow 0.1Jy.TheWMAP
∼
alterstheshapeofthesecondandthirdacousticpeaksinthemea- catalog is not complete below S 1 Jy, so point sources with
sured spectrum, we find that τ and, to a lesser degree, θ and As S 0.1 Jy account for only a sm∼all fraction of the total power
∼
areunaffected.Forourdefaultsourceremovalprocedure,theshift duetounresolvedsources.Athigherfrequenciespredictionsarein
onns andωc canreachalmost 0.4σ, withanoticeable shiftalso generalagreementwithpreviousworks.
forthebaryondensity,ωb,whileifweadopt amoreconservative Anadvantageofthemethodisthatitmaintainstheinforma-
masking,theeffectcanbeupto0.3σ. tion about the sources position. In order to test this, we consid-
ered the sets of point-source-only (i.e. no CMB nor noise) maps
obtainedfromNVSSsimulationsat41and94GHz,andcorrelated
them with both the Q and W band WMAP5 sky maps and with
7 SUMMARYANDCONCLUSIONS
mockCMBskiesinwhichthepointsourcepopulationfollowsthe
We discussed a new method aimed at evaluating the residual samedistributionasourmodel. Wefoundthatalthoughtheover-
point source contribution to the WMAP5 power spectrum. We alllevelofcorrelationislow,Wbandcorrelationswithactualdata
implemented a stochastic approach to extrapolate the flux of ra- areingoodagreementwithcorrelationswiththemockdata.InQ
dio sources of the NRAO-VLA Sky Survey (NVSS) to WMAP band,instead,thedegreeofcorrelationbetweensimulationsandac-
frequencies. Using different point source catalogs with multifre- tualmapsissignificantlylowerthanwhatexpectedfromthemock
quencyinformation,wedividedthe1.4 90GHzinanumberof data.Inthisbandforegroundscontaminationissignificantlyhigher
frequencysteps.AteachstepanNVSS−sourceisrandomlypaired thanathigherfrequencyandmaycontributetodecreasingthelevel
toareferencesourceinthecatalogcoveringtherelevantfrequency ofcorrelationbetweensimulationsandrealmaps.
range, and extrapolated to theupper limitof that range using the Weusedthesimulatedmapstoestimatetheunresolvedsource
spectral index of the reference source. The procedure is iterated correction to WMAP5 measured spectra. As a general result, we
(cid:13)c 0000RAS,MNRAS000,000–000